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7600	https://www.geeksforgeeks.org/maths/how-to-solve-two-step-linear-equations/	How to Solve Two Step Linear Equations? Two Step Equations comes under the parent topic Algebra which helps in representing the problem in the form of mathematical expressions. Algebra is the study of mathematical symbols and rules for manipulating the symbols. Algebraic expressions include constants, variables like x, y, z, a, b, etc, and mathematical operators like addition, subtraction, multiplication division between variables. Two Step Equations Two Step Equations are equations that can be solved by only two steps. These equations are easy to solve but a little complicated when compared to one-step equations. In solving the two-step equations, Arithmetic operations are performed on both sides of equal to symbol. The two-step equations are algebraic problems that can be solved in two steps. Here variable is isolated on one side of "=" to find its value. The general form of two-step equations is ax + b = c where a, b, c are real numbers. Examples:2x + 3 = 0, 7a - 5 = 2, (2/3)x + 1 = 4 Steps to solve Two Step Equations These Two-Step Equations can be solved easily. It needed just one more step extra as compared to solving one-step equations. The variable is isolated on one side of "=" to determine its value. The steps needed are mentioned below, Sample Problems Question 1: Solve the equation 3x + 3 = 12 Solution: Given two step equation 3x + 3 = 12 Step 1: Subtract 3 from both sides. 3x + 3 - 3 = 12 - 3 3x = 9 Step 2: Divide the equation with 3 on both sides (3x/3) = (9/3) x = 3 This can be verified by substituting x = 3 in the given equation, 3(3) + 3 = 12 9 + 3 = 12 12 = 12 Hence it was proved that on solving the given equation, we got x = 3. Question 2: Solve the equation x - 5 = 2 Solution: Given two step equation x - 5 = 2 Step 1: Add 5 to both sides. x - 5 + 5 = 2 + 5 x = 7 As the coefficient of variable is 1, No need to perform the step 2. The above result can be verified by substituting x = 7 in the given equation. x - 5 = 2 7 - 5 = 2 2 = 2 Hence it was proved that on solving the given equation, we got x = 7. Question 3: Solve the equation (x/2) - 5 = 5 Solution: Given two step equation (x/2) - 5 = 5 Step 1: Add 5 to both sides. (x/2) - 5 + 5 = 5 + 5 (x/2) = 10 Step 2: Multiply the equation with 2 on both sides, (2x/2) = 10 × 2 x = 20 This can be verified by substituting x = 20 in the given equation, (x/2) - 5 = 5 (20/2) - 5 = 5 10 - 5 = 5 5 = 5 Hence it was proved that on solving the given equation, we got x = 20. Question 4: Solve the equation (2x/3) + 6 = 0 Solution: Given two step equation (2x/3) + 6 = 0 Step 1: Subtract 6 from both sides. (2x/3) + 6 - 6 = 0 - 6 (2x/3) = -6 Step 2: Multiply the equation with 3/2 on both sides, (2x/3) × (3/2) = -6 × (3/2) (6x/6) = (-18/2) x = -9 This can be verified by substituting x = -9 in the given equation, (2x/3) + 6 = 0 (2(-9)/3) + 6 = 0 (-18/3) + 6 = 0 (-18 + 18)/3 = 0 (0/3) = 0 0 = 0 Hence it was proved that on solving the given equation, we got x = -9. Question 5: Solve the equation 4a - 2.6 = 1.4 Solution: Given two step equation 4a - 2.6 = 1.4 Step 1: Add 2.6 to both sides. 4a - 2.6 + 2.6 = 1.4 + 2.6 4a = 4 Step 2: Divide the equation with 4 on both sides, (4a/4) = (4/4) a = 1 This can be verified by substituting a = 1 in the given equation, 4a - 2.6 = 1.4 4(1) = 1.4 + 2.6 4 = 4 Hence it was proved that on solving the given equation, we got a = 1. Question 6: Solve the equation 2z + 1.5 = 2.3 Solution: Given two step equation 2z + 1.5 = 2.3 Step 1: Subtract 1.5 from both sides. 2z + 1.5 - 1.5 = 2.3 - 1.5 2z = 0.8 Step 2: Divide the equation with 2 on both sides (2z/2) = (0.8/2) z = 0.4 This can be verified by substituting z = 0.4 in the given equation, 2z + 1.5 = 2.3 2(0.4) = 2.3 - 1.5 0.8 = 0.8 Hence it was proved that on solving the given equation, we got z = 0.8. Question 7: Solve the equation 1.2a - 1.2 = 1.2 Solution: Given two step equation 1.2a - 1.2 = 1.2 Step 1: Add 1.2 to both sides. 1.2a - 1.2 + 1.2 = 1.2 + 1.2 1.2a = 2.4 Step 2: Divide the equation with 1.2 on both sides. (1.2a/1.2) = (2.4/1.2) a = 2 This can be verified by substituting a = 2 in the given equation, 1.2a - 1.2 = 1.2 1.2(2) - 1.2 = 1.2 2.4 - 1.2 = 1.2 1.2 = 1.2 Hence it was proved that on solving the given equation, we got a = 2. A Explore Maths Basic Arithmetic What are Numbers? 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7601	https://webbook.nist.gov/cgi/inchi/InChI%3D1S/Hg.O	mercury monoxide Jump to content National Institute of Standards and Technology NIST Chemistry WebBook, SRD 69 Home Search Name Formula IUPAC identifier CAS number More options NIST Data SRD Program Science Data Portal Office of Data and Informatics About FAQ Credits More documentation mercury monoxide Formula: HgO Molecular weight: 216.59 IUPAC Standard InChI:InChI=1S/Hg.O Copy IUPAC Standard InChIKey:UKWHYYKOEPRTIC-UHFFFAOYSA-N Copy CAS Registry Number: 21908-53-2 Chemical structure: This structure is also available as a 2d Mol file or as a computed3d SD file View 3d structure (requires JavaScript / HTML 5) Other names: Mercury(ii) oxide Information on this page: Notes Other data available: Gas phase thermochemistry data Condensed phase thermochemistry data Constants of diatomic molecules Data at other public NIST sites: Gas Phase Kinetics Database X-ray Photoelectron Spectroscopy Database, version 5.0 Options: Switch to calorie-based units Notes Go To:Top Data from NIST Standard Reference Database 69: NIST Chemistry WebBook The National Institute of Standards and Technology (NIST) uses its best efforts to deliver a high quality copy of the Database and to verify that the data contained therein have been selected on the basis of sound scientific judgment. However, NIST makes no warranties to that effect, and NIST shall not be liable for any damage that may result from errors or omissions in the Database. Customer support for NIST Standard Reference Data products. © 2025 by the U.S. Secretary of Commerce on behalf of the United States of America. All rights reserved. Copyright for NIST Standard Reference Data is governed by the Standard Reference Data Act. Privacy Statement Privacy Policy Security Notice Disclaimer (Note: This site is covered by copyright.) Accessibility Statement FOIA Contact Us
7602	https://www.teachoo.com/2270/590/Example-8---Prove-rule-of-exponents-(ab)n--anbn/category/Examples/	Example 8 - Prove rule of exponents (ab)^n = a^n b^n by induction Everything Class 6 Class 6 Maths (Ganita Prakash) Class 6 Maths (old NCERT) Class 6 Science (Curiosity) Class 6 Science (old NCERT) Class 6 Social Science Class 6 English Class 7 Class 7 Maths (Ganita Prakash) Class 7 Maths (old NCERT) Class 7 Science (Curiosity) Class 7 Science (old NCERT) Class 7 Social Science Class 7 English Class 8 Class 8 Maths (Ganita Prakash) Class 8 Maths (old NCERT) Class 8 Science (Curiosity) Class 8 Science (old NCERT) Class 8 Social Science Class 8 English Class 9 Class 9 Maths Class 9 Science Class 9 Social Science Class 9 English Class 10 Class 10 Maths Class 10 Science Class 10 Social Science Class 10 English Class 11 Class 11 Maths Class 11 English Class 11 Computer Science (Python) Class 12 Class 12 Maths Class 12 English Class 12 Economics Class 12 Accountancy Class 12 Physics Class 12 Chemistry Class 12 Biology Class 12 Computer Science (Python) Class 12 Physical Education Courses GST and Accounting Course Excel Course Tally Course Finance and CMA Data Course Payroll Course Interesting Learn English Learn Excel Learn Tally Learn GST (Goods and Services Tax) Learn Accounting and Finance Popular GST Demo GST Tax Invoice Format Accounts Tax Practical Tally Ledger List Maths Updated 2025-26 Class 6 Maths (Ganita Prakash) Class 6 Maths (old NCERT) Class 7 Maths (Ganita Prakash) Class 7 Maths (old NCERT) Class 8 Maths (Ganita Prakash) Class 8 Maths (old NCERT) Class 9 Maths Class 10 Maths Class 11 Maths Class 12 Maths Sample Paper Class 10 Maths Sample Paper Class 12 Maths Accounts & Finance Learn Excel free Learn Tally free Learn GST (Goods and Services Tax) Learn Accounting and Finance GST and Income Tax Return Filing Course Tally Ledger List Popular Class 6 Science (Curiosity) Class 6 Science (old NCERT) Class 7 Science (Curiosity) Class 7 Science (old NCERT) Class 8 Science (Curiosity) Class 8 Science (old NCERT) Class 9 Science Class 10 Science Remove Ads Login Maths Remove ads Science GST Accounts Tax Englishtan Excel Social Science Class 6 Class 6 Maths (Ganita Prakash) Class 6 Maths (old NCERT) Class 6 Science (Curiosity) Class 6 Science (old NCERT) Class 6 Social Science Class 6 English Class 7 Class 7 Maths (Ganita Prakash) Class 7 Maths (old NCERT) Class 7 Science (Curiosity) Class 7 Science (old NCERT) Class 7 Social Science Class 7 English Class 8 Class 8 Maths (Ganita Prakash) Class 8 Maths (old NCERT) Class 8 Science (Curiosity) Class 8 Science (old NCERT) Class 8 Social Science Class 8 English Class 9 Class 9 Maths Class 9 Science Class 9 Social Science Class 9 English Class 10 Class 10 Maths Class 10 Science Class 10 Social Science Class 10 English Class 11 Class 11 Maths Class 11 Computer Science (Python) Class 11 English Class 12 Class 12 Maths Class 12 English Class 12 Economics Class 12 Accountancy Class 12 Physics Class 12 Chemistry Class 12 Biology Class 12 Computer Science (Python) Class 12 Physical Education Courses GST and Accounting Course Excel Course Tally Course Finance and CMA Data Course Payroll Course Interesting Learn English Learn Excel Learn Tally Learn GST (Goods and Services Tax) Learn Accounting and Finance Popular GST Demo GST Tax Invoice Format Accounts Tax Practical Tally Ledger List Remove Ads Login Mathematical Induction Serial order wise Examples Examples Example 1 Example 2 Important Example 3 Example 4 Important Example 5 Important Example 6 Important Example 7 Important Example 8 You are here Theory→ Mathematical Induction Serial order wise Mathematical Induction - Questions and Solutions Examples Theory Example 8 - Mathematical Induction Last updated at December 16, 2024 by Teachoo ADVERTISEMENT Powered by VidCrunch Next Stay Playback speed 1x Normal Quality Auto Back 720p 360p 240p 144p Auto Back 0.25x 0.5x 1x Normal 1.5x 2x / Skip Ads by Next: Theory →Remove AdsShare on WhatsApp Transcript Example 8 Prove the rule of exponents (ab)n = anbn by using principle of mathematical induction for every natural number. Let P(n) : (ab)n = anbn. For n = 1 , L.H.S = (ab)1 = ab R.H.S = a1b1 = a b = ab Thus, L.H.S. = R.H.S , P(n) is true for n = 1 Assuming P(k) is true P(k) : (ab)k = ak bk We will prove that P(k + 1) is true. R.H.S = ak+1 bk+1 L.H.S = (ab)k+1 By the principle of mathematical induction, P(n) is true for n, where n is a natural number Show More Mathematical Induction Serial order wise Examples Example 1 Example 2 Important Example 3 Example 4 Important Example 5 Important Example 6 Important Example 7 Important Example 8 You are here Theory→ Made by Davneet Singh Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo Hi, it looks like you're using AdBlock :( Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription. 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7603	https://www.sciencedirect.com/science/article/pii/S0085253815462507	Management of minimal lesion glomerulonephritis: Evidence-based recommendations - ScienceDirect Skip to main contentSkip to article Journals & Books ViewPDF Download full issue Search ScienceDirect Outline Keywords Methods TREATMENT OF MINIMAL LESION GLOMERULONEPHRITIS IN CHILDREN TREATMENT OF MINIMAL LESION DISEASE IN ADULTS SUMMARY ACKNOWLEDGMENTS REFERENCES Show full outline Cited by (74) Figures (1) Tables (5) Table 1 Table 2 Table 3 Table 4 Table 5 Kidney International Volume 55, Supplement 70, June 1999, Pages S3-S16 Original Article Management of minimal lesion glomerulonephritis: Evidence-based recommendations Author links open overlay panel Joanne M.Bargman Show more Outline Add to Mendeley Share Cite rights and content Under an Elsevier user license Open archive Management of minimal lesion glomerulonephritis: Evidence-based recommendations. The treatment of idiopathic minimal lesion disease in children has been extensively studied in randomized controlled trials, however, there is less information available for adults. This article summarizes evidence-based recommendations for management. The first attack should be treated with prednisone or prednisolone at 60 mg/m 2 per day (up to a maximum of 80 mg/day) for four to six weeks, followed by 40 mg/m 2 of prednisone every other day for another four to six weeks (grade A). Relapse should be treated with 60 mg/m 2/day of prednisone (up to 80 mg/day) only until the urine becomes protein free for three days, and then an alternate day regimen of 40 mg/m 2 should be used for another month (grade A). Patients with frequently relapsing disease will have a significant reduction in relapse frequency after eight weeks of an alkylating agent (grade A). Less rigorous studies have suggested benefit with long-term, alternate-day corticosteroid (grade D) or the antihelminthic agentlevamisole (grade D). For patients with steroid-dependent disease, an 8- or 12-week course with cyclophosphamide can induce remission (grade D). In true steroid-resistant disease, observational studies have suggested that a course of cyclosporine may sometimes induce remission or restore steroid responsiveness (grade D). Large retrospective studies in adults suggest that therapeutic response is slower than in children, but adults experience fewer relapses and more prolonged remission. Previous article in issue Next article in issue Keywords prednisone prednisolone nephrotic syndrome acute renal failure corticosteroids Minimal lesion glomerulonephritis typically presents with the acute onset of nephrotic syndrome, often following a viral upper respiratory tract infection. Complications include thrombosis, acute renal failure, and infections such as cellulitis, peritonitis, and pneumonia1. Older literature suggests that spontaneous remissions occur in children with minimal lesion disease, usually after many months of disease compared with a much earlier remission time induced with corticosteroids2., 3., 4.. Histology demonstrates essentially normal-appearing glomeruli on light microscopy, although there may be some expansion of the mesangial matrix. Immunofluorescence is usually negative for immunoglobulins. Some investigators include staining with IgM, and there may be a spectrum of disease between minimal lesion and IgM mesangial proliferative glomerulonephritis. Indeed, this is a potential cause of confusion because some studies allow mesangial proliferation and/or staining with IgM as a variant of minimal lesion, whereas others do not. Although deposition of IgM does not appear to change prognosis, mesangial hypercellularity is associated with late nonresponse5. Others have not found an association between histology and postbiopsy course6. Methods The evidence used in compiling these recommendations was obtained from published trials found in a MEDLINE search of the English language literature. Secondary references from the bibliography of the initial studies were also perused, as were personal files. Recommendations were graded from A to D, based on the level of evidence of the supporting studies. TREATMENT OF MINIMAL LESION GLOMERULONEPHRITIS IN CHILDREN Tables 1, 2, and 3 review the levels of evidence of English language studies in children. Table 1. Corticosteroids in children, level 1 or 2 \| Author [Ref] \| Study design \| N \| Treatment \| Results/comments \| --- --- \| APN \| RCT frequent relapsers in remission to Pred for 3/7 days or alt days \| 25 intermittent \| Remission induced with daily Pred, then Pred 40 mg/m 2 for 3/7 days or 35 mg/m 2 q alt day; 6 months total \| 72% relapse in first 6 months (intermittent) \| \| \| \| 24 alt day \| \| 43% relapse in first 6 months (alt day) \| \| \| \| \| \| Relapse rate similar after that \| \| ISKDC \| Children with early relapse randomized to shorter vs. longer Pred regimens \| 26 shorter \| Shorter: 60 mg Pred until CR (avg 12 days) then 40 mg 3/7 days × 4 weeks \| Shorter: 40% relapsed during treatment protocol \| \| \| \| 28 longer \| Longer: 60 mg Pred/day × 4 weeks then 40 mg/day × 1 week, 30 mg/day × 1 week, etc. \| Longer: 8% relapsed during treatment protocolAll patients relapsed by 8 monthsLonger group had 2 × Pred \| \| APN \| Children with first attack randomized to shorter vs. longer courses of Pred \| 32 shorter \| Shorter: Pred 60 mg/m 2 until CR × 3 day, then 40 mg/m 2 q 48 hr until serum alb ≥35 g/liter \| Shorter course used 50% less Pred but twice the relapse rate, half the relapse free interval \| \| \| \| 29 longer \| Longer: Pred 60 mg/m 2/day × 4 weeks then 40 mg/m 2 q 48 hr × 4 weeks \| Total Pred including relapse rx similar – no advantage to shorter regimen \| \| Ueda \| Children with first attack randomized to shorter vs. longer courses of Pred \| 29 shorter \| Shorter: Pred 60 mg/m 2/day × 4 weeks, then 40 mg/m 2/day 3/7 days × 4 weeks \| Longer regimen had less relapses while on treatment, so total amount of Pred did not differ between 2 groups \| \| \| \| 17 longer \| Longer: Pred 60 mg/m 2/day × 4 weeks, then 60 mg/m 2/alt day × 4 weeks, then tapered over 5 months \| Longer regimen had fewer patients who became steroid dependent or frequent relapsers \| \| Ehrich \| Children with first attack randomized to shorter vs. longer courses of Pred \| 37 shorter \| Shorter: 60 mg/day × 4 weeks then 40/mg/m 2 q 48 hr × 4 weeks Longer: 60 mg/m 2/day × 6 weeks, then 40 mg/m 2 q 48 hr × 6 weeks \| Longer treatment patients more likely to remain in remission, less frequent relapsers in longer treatment groupTotal Pred dose at one year similar in both groups \| \| \| \| 34 longer \| \| \| Abbreviations are: alt, alternate; APN, Arbeitsgemeinschaft fur Padiatrische Nephrologie; ARF, acute renal failure; AZA, azathioprine; Chloram, chlorambucil; CR, complete remission; CTX, cyclophosphamide; CsA, cyclosporine; F/U, follow-up; ISKDC, International Study of Kidney Disease in Children; ML, minimal lesion; NS, nephrotic syndrome; PR, partial remission; Pred, prednisone or prednisolone; RCT, randomized controlled trial; stat sign, statistically significant; VAR, variable. Table 2. Alkylating and antimetabolite agents in children \| Author [Ref] \| Study design \| N \| Treatment \| Results/comments \| --- --- \| Level 1 or 2 \| \| Abramozicz \| RCT of 197 children rx Pred 8 weeks \| 38 nonresponders \| Pred + AZA or placebo × 3 months \| No difference AZA from placebo \| \| \| \| 49 frequent relapsers \| Pred + AZA or placebo × 6 months \| Trend to less relapses in AZA group but not stat sig \| \| Barratt \| RCT of Pred vs. Pred + CTX in frequently relapsing NS, in remission \| 10 Pred \| Pred taper over 8 weeks “maintenance” Pred + CTX 3 mg/kg/day over 8 weeks then Pred taper over 8 weeks \| 9/10 relapsed by one year 2/10 relapsed by one year (P<0.05) \| \| \| \| 10 Pred + CTX \| \| \| \| Barratt \| RCT of Pred and either 2 or 8 weeks of CTX in frequently relapsing NS, in remission \| 16 CTX 2 weeks \| “Maintenance” Pred over 8 weeks, taper over 8 weeks + CTX 3 mg/kg/day × 2 weeks \| 9/16 relapsed by weeks 24 \| \| \| \| 16 CTX 8 weeks \| Same except CTX × 8 weeks \| 1/16 relapsed by week 24 (P<0.05) \| \| ISKDC \| RCT “early non-responders” (at 8 weeks of Pred) to intermittent Pred alone or with CTX \| 33 of which 15 Prd alone \| Intermittent Pred ± CTX (var dose) × 90 days \| (not all had ML) \| \| \| \| 18 Pred + CTX \| \| Pred alone: 40% CR by 95 days. \| \| \| \| \| \| Pred + CTX: 56% CR by 38 days \| \| \| RCT frequent relapsers intermittend Pred alone or with CTX \| 53 of which 26 Pred alone \| Intermittent Pred ± CTX (var dose) × 42 days \| Pred alone: 88% relapse by mean 22 months \| \| \| \| 27 Pred + CTX \| \| Pred ± CTX: 48% relapse \| \| Grupe \| Steroid-dependent or frequently relapsing NS \| 11 Pred alone \| \| All chlorambucil patients went into CR for full F/U (mean 19.6 months) vs. all 11 on Pred had recurrent NS by 7 months \| \| \| RCT Pred alone or Pred + chlorambucil \| 10 Pred + chlorambucil \| Chloram 0.1–0.2 mg/kg/day in divided doses, titrated to WBC × 6–12 weeks total \| \| \| APN \| Frequently relapsing or steroid-dependent RCT to Pred + chloramb or to Pred + CTX \| 2 × 2:16 frequent relapsers 8 chloramb, 8 CTX \| After remission, randomized to chloramb 0.15 mg/kg/day × 56 days along with alt day Pred or \| CTX or chloram decreased relapse rate in frequent relapsers \| \| \| \| 34 steroid dependent 16 chloram, 18 CTX \| CTX 2 mg/kg/day × 56 days along with alt day Pred \| No difference in time to relapse in steroid dependent group \| \| \| \| \| \| No difference CTX vs. chloramb (small#’s) \| \| Ueda \| Steroid-dependent RCT to CTX for 8 or 12 weeks \| 32 CTX × 8 weeks \| After remission, CTX 2 mg/kg/day × 8 weeks or CTX 2 mg/kg/day × 12 weeks \| No difference 8 vs. 12 weeks rx (compare with APN 1987 below) \| \| \| \| 41 CTX × 12 weeks \| \| \| \| Level 3 or 4 \| \| Chiu \| Frequent relapsers rx Pred and CTX compared with historical controls \| 36 Pred + CTX \| CTX 75 mg/m 2/day + Pred 60 mg/m 2/day then tapered; duration 4 months \| 33% relapse by mean 4 year F/U \| \| \| Pred alone \| 11 Pred alone \| Hist controls: Pred same regimen × 4 months \| 91% relapse by mean 2 year F/U \| \| Barratt \| Follow up 1973 study of frequent relapsers \| 82 CTX \| CTX 3 mg/kg/day (most also received Pred) × 8 weeks \| Remission lasted longer in older children; better outcome if in CR at onset of CTX rx \| \| Pennisi \| Steroid-dependent NS treated with alt day Pred and 6–8 or 12 weeks CTX; retrospective analysis \| 29 6–8 week CTX \| CTX 3–5 mg/kg/day × 6–8 weeks + alt day PredCTX 3–5 mg/kg/day × 12 weeks + alt day Pred \| 42% relapse rate at one year; 21% still in CR at 3.5 years8% relapse rate at one year; 63% still in CR at 3.5 years \| \| \| \| 24 12 week CTX \| \| \| \| Williams \| Nonrandom “low” and “high” dose chlorambucil for frequently-relapsing, steroid resistant NS \| 53 \| Daily Pred until remission; then alt day Pred and chlorambucil <3 mg/kg/day (“low”) or> 3.0 mg/kg/day (“high”) \| Low dose same outcome as high dose. 95% CT at 1 year 85% CR at 4 years. Steroid resistant patients went in CR for 1.3–9.4 years. Young age most important negative prognostic factor \| \| APN \| Steroid dependent NS rx 12 weeks CTX compared retro to steroid dependent cohort (APN, 1982) rx 8 weeks CTX \| 18 12 weeks \| Daily Pred until remission; then alt day Pred and CTX 2 mg/kg/day × 12 weeks (historical CTX 2 mg/kg/day × 8 weeks) \| 67% sustained remission in 12 week CTX; 22% sustained remission in 8 weeks CTX (12 week group older and received more Pred) \| \| \| \| 18 8 weeks \| \| \| \| Level 5 or 6 \| \| West \| Review of experience with CTX in NS \| 14 \| CTX 3–4 mg/kg/day with reduced doses of Pred \| Long-term remissions in steroid dependent and frequently relapsing NS \| \| Drummond \| Observational study \| 12 \| CTX 75 mg/m 2/day \| Long therapy with CTX (up to 12 months) decreased relapse rate and associated with clearing of proteinuria in “steroid resistant” NS \| Abbreviations are in Table 1 Table 3. Cyclosporine studies in children \| Author [Ref] \| Study design \| N \| Treatment \| Results/comments \| --- --- \| Level 1 or 2 \| \| Garin \| Steroid-resistant NS Randomized crossover of CsA vs. no treatment \| (ML) 4 \| CsA 5 mg/kg/day × 8 weeks or no treatment \| No change in proteinuria with CsA; small #’s \| \| Tejani \| NS (1st episode in 50%) randomized CsA + low dose Pred or high dose Pred alone \| 6 CsA + Pred \| CsA 7 mg/kg/day × 4 weeks + 20 mg/m 2 Pred then lower doses × 4 weeks \| 6/6 remission—duration of remission same as Pred alone \| \| \| \| 7 Pred alone \| Pred 60 mg/m 2/day × 4weeks then alt day × 4 weeks \| 6/7 remission \| \| Ponticelli \| RCT CsA vs. CTX for steroid-dependent and frequently relapsing NS \| 18 CsA \| 5–6 mg/kg/day × 9 months then tapered × 3 months 1.5–2.5 mg/kg/day × 8 weeks \| 25% of CsA relapse free at 2 years; 63% of CTX relapse free at 2 years. Adults same as children \| \| \| \| 13 CTX \| \| \| \| Level 5 or 6 \| \| Niaudet \| Multi-center uncontrolled study CsA in “steroid resistant” NS (Pred × 4 weeks + 3 pulses) \| ML: 45 \| CsA 150 mg/m 2/day + Pred 30 mg/m 2/day × 4 weeks then alt day. Total rx 6 months \| 23/45 CR or PR in minimal lesion group (? inadequate initial Pred treatment—not really steroid resistant?) \| \| Gregory \| Retrospective analysis CsA in steroid-dependent and steroid-resistant NS \| ML: 3 \| CsA 5–10 mg/kg/day + volume expansion in same; alt day Pred \| 2/3 maintain remission on CsA alone; minimal toxicity with long-term CsA monotherapy \| \| Kitano \| Frequent relapsers or steroid dependent rx CsA × 6 months \| 17 \| After remission, CsA 3–5 mg/kg/day × 6 months and then tapered \| No relapses during CsA; all patients able to come off Pred; but 40% relapsed by 1 month after CsA D/C’ed, 16/17 relapsed by one year \| \| Tanaka \| Frequent relapsers or steroid dependent rx “high dose” CsA × 6 months then “low dose”× 12 months \| 18 \| “High dose” CsA 3–5 mg/kg/day × 6 months, then 2.5 mg/kg/day × 12 months and then tapered. Also received Pred prn \| All patients able to come off Pred; 8 growth, wt loss. Marked 9 relapse rate during CsA 7/18 relapse free × 18 months, 14/18 relapsed within 6 months of stopping CsA \| Abbreviations are in Table 1. Treatment of the first attack Recommendation 1 Prednisone at 60 mg/m 2 per day (up to a maximum of 80 mg/day) for four to six weeks (grade A) and then 40 mg/m 2 of prednisone every other day for four to six weeks (grade A) are recommended. Evidence Minimal lesion accounts for more than 90% of cases of idiopathic nephrotic syndrome in young children. Uncontrolled studies in the 1950s demonstrated that more than 90% of children would respond to corticosteroid treatment. The pioneering studies of the International Study of Kidney Diseases in Children (ISKDC) settled on a regimen of 60 mg/m 2/day prednisone, to a maximum of 80 mg/day for four weeks, followed by intermittent prednisone (40 mg/m 2/day) on three consecutive days out of seven days for the next four weeks7. The European collaborative group, Arbeitsgemeinschaft fur Padiatrische Nephrologie (APN), suggested that in the second four weeks of treatment, prednisone given every second day might be more effective in preventing relapses. These two regimens were compared in a randomized controlled study in a group of 48 frequently relapsing patients8. Patients receiving alternate-day prednisone had an approximate 50% reduction in relapse rate compared with the controls receiving prednisone three days out of seven. This difference occurred during only the six months of the trial, whereas the later relapse rate was similar. The investigators concluded that the alternate-day regimen was superior for the second four weeks of therapy and have recommended it even for the first attack of minimal lesion disease9. Most patients will clear their proteinuria by two weeks of therapy10. Investigators have questioned whether a shorter course of initial treatment might be just as effective as the standard regimen. The APN performed a randomized controlled study of their standard regimen (four weeks of daily prednisone plus four weeks of alternate-day prednisone) compared with a short-course group who received 60 mg/m 2/day of prednisone until the urine was clear of protein (about two weeks) and then took alternate-day prednisone until the serum albumin normalized (another two weeks). Although the short-course group initially received only half of the amount of prednisone, they relapsed twice as often and twice as soon, and ultimately received as much steroid as the standard treatment group11. The authors concluded that there was no benefit to the shorter therapy. Several years later, they examined whether longer initial therapy conferred added protection against relapse12. Here, children with a first attack were randomly allocated to a long treatment consisting of daily prednisone for six weeks and alternate-day prednisone for six weeks, and the children were then compared with those undergoing the standard regimen. This level 1 study demonstrated that the long-treatment patients were twice as likely to remain in remission compared with the patients receiving the standard regimen. Furthermore, fewer patients with the long initial therapy ultimately became frequent relapsers12. Even though the total amount of prednisone taken for the first attack was greater than in the standard regimen, steroid side-effects were no more severe. The benefit was that less steroids were subsequently taken because of the fewer relapses. A similar result was found by investigators in Japan in another level 1 study, in which children at first presentation were randomized to either the ISKDC protocol or a longer protocol involving a taper over five months. Again, the longer treatment group had fewer relapses, and fewer of the patients became frequent relapsers. Although the long-treatment group took more prednisone, the cumulative dose over the entire period of follow-up was not different13. The difficulty in determining the trade-off between a cumulative steroid dose versus a relapse-free internal is well illustrated in a randomized ISKDC study that examined a long versus shorter steroid regimen in children who had relapsed within six months of their initial response. In this study, the long-treatment group received approximately twice as much prednisone as the other group, but stayed in remission for a mean of 3.27 months compared with 1.48 months. In the end, it was not clear, even to many of the investigators, whether the extra steroid was worth the longer duration of remission14. A recent retrospective analysis of nearly 400 children with minimal lesion disease found that if the children responded to the first eight weeks of therapy with complete remission and maintained remission over the next six months, the long-term outlook was excellent with no further, or very rare, relapses. A relapse in the first six months was predictive of further relapses for the next three years. Perhaps not surprisingly, the failure to achieve remission in the first eight weeks of therapy predicted progression to renal failure in 21% of the children. The authors suggest that more aggressive therapy should be used to achieve remission in these children who are slow to respond; however, it is possible that the delayed or lack of response to standard therapy itself is a marker for more aggressive disease15. In summary, the intensity of initial therapy for minimal lesion disease appears to determine the rate of subsequent relapse and perhaps also the chances of becoming a frequent relapser. The optimization point of initial dose of prednisone with its cumulative toxicity, versus morbidity involved in relapse and its retreatment, appears to lie at an initial treatment with daily prednisone for four to six weeks and a course of alternate-day prednisone for four to six weeks. Alternatively, a tapering course can be given over several months. Treatment of relapse Recommendation 2 For patients with minimal lesion disease who relapse after initial treatment, prednisone should be given at 60 mg/m 2 per day (up to 80 mg/day) until the urine is protein free; then 40 mg/m 2 should be given every other day for four weeks (grade A). Evidence Most children will have a relapse of their nephrosis. The mean duration of corticosteroid therapy to clear the urinary protein is similar for the relapse as for the initial attack11. However, the intensity of prednisone treatment for a first relapse had no influence on the subsequent relapse rate14,16. The current relapse regimen consists of 60 mg/m 2/day (up to 80 mg/day) of prednisone until the urine is free of protein for three days, followed by four weeks of alternate day prednisone at 40 mg/m 29. This regimen is associated with fewer relapses in the first six months of treatment, compared with administering prednisone three out of seven days in the second four weeks of therapy8. The ISKDC demonstrated that treating a relapse for four weeks with daily prednisone rather than just until the urine cleared of protein (average 12 days) was associated with a more prolonged remission (3.27 vs. 1.48 months) but at the cost of double the amount of prednisone14. Because all patients in both groups relapsed by eight months anyway, it is recommended to treat a relapse with daily prednisone only until the urine is protein free or for a few more days, and then proceed with the tapering regimen as described earlier in this article. Treatment of frequently relapsing minimal lesion disease Recommendation 3 Patients with minimal lesion disease who relapse frequently should be treated with one of these regimens: cyclophosphamide or chlorambucil for eight weeks (discussed in the text; grade A); repeat relapse therapy with prednisone (grade D); symptomatic treatment only (Na restriction, diuretics; grade D); long-term alternate-day prednisone (grade D); and/or levamisole (discussed in the text; grade B). Treatment of steroid-dependent minimal lesion disease Recommendation 4 Children with steroid-dependent minimal lesion disease should be treated with 2 mg/kg/day of cyclophosphamide for 12 weeks (grade D) or 6 mg/kg/day of cyclosporine for children and 5 mg/kg/day for adults, with the duration being uncertain (grade A). Evidence Ten to 20% of children will experience three or four steroid-sensitive attacks, and half will become frequent relapsers or steroid dependent. Given the cumulative toxicity associated with long-term corticosteroid therapy, treatment has evolved to include alkylating agents, antimetabolites, and cyclosporine in these more difficult patient groups. A frequent relapser is a patient who responds to corticosteroid treatment but experiences two relapses within the first six months after the initial response or has four relapses within any one year17. Up to one quarter of relapses in frequent relapsing minimal lesion may remit spontaneously2. Steroid dependency is defined as two consecutive relapses occurring during therapy or within 14 days of completing steroid therapy16. Therapeutic options for frequent relapsers include repeated relapse treatment with corticosteroid, prolonged tapering of alternate-day or daily steroid, or symptomatic treatment with salt restriction and diuretics. Unfortunately, there are no good studies to compare outcomes of these regimens. The repeated or prolonged use of corticosteroids in either frequently relapsing or steroid-dependent children carries with it the risk of side-effects, including growth retardation, osteoporosis, obesity, and cataracts. Antimetabolites and alkylating agents The ISKDC studied prednisone with placebo compared with prednisone and azathioprine for primary nonresponders and for frequent relapsers. This treatment had no benefit for the steroid-resistant group (10 of whom ultimately showed focal sclerosis on renal biopsy) or for the frequent relapsers18. Cyclophosphamide is more effective. A randomized controlled trial in children with frequently relapsing minimal lesion disease showed a statistically significant decline in relapse rate in those given cyclophosphamide in addition to prednisone, even though there were just 10 children in each group19. The same investigators then compared two weeks of treatment and eight weeks of treatment with cyclophosphamide in a randomized controlled trial in frequent relapsers. Two weeks of cyclophosphamide led to more prolonged remission compared with historical controls who received prednisone alone, whereas eight weeks of cyclophosphamide was even more effective20. A level 1 ISKDC study of cyclophosphamide in frequent relapsers showed the clear superiority of cyclophosphamide and prednisone (48% relapse after a mean of 22 months) compared with prednisone alone (88% relapse after a mean of 22 months)21. Barratt et al extended their follow-up of cyclophosphamide in frequent relapsers to four years in a level 4 study, which showed that the eight-week course conferred a prolonged benefit to these patients22. Chlorambucil led to prolonged steroid-free remission in children with frequently relapsing or steroid-dependent (not steroid-resistant) nephrotic syndrome, with low doses (less than 0.3 mg/kg/day) as effective as high doses (more than 3.0 mg/kg/day)23,24. The APN studied the effect of eight weeks of treatment with cyclophosphamide at 2 mg/kg/day versus chlorambucil at 0.15 mg/kg/day in 16 children with frequent relapsing and 34 children with steroid-dependent disease. In the frequent relapsers, the rate of relapse after treatment declined, although actual rates of relapse prior to therapy were not formally compared. In contrast, the eight weeks of treatment did not apparently benefit the steroid-dependent cohort, although once again, relapse rates were not compared statistically before and after alkylating therapy. There was no difference between chlorambucil and cyclophosphamide, although the numbers were small for that intergroup comparison17. In a subsequent prospective study, steroid-dependent patients (similar to the patients who failed the eight weeks of alkylating therapy previously) were instead given 12 weeks of therapy with 2 mg/kg/day of cyclophosphamide. The 12 week group was compared retrospectively with the steroid-dependent children who had received eight weeks of cyclophosphamide. The cumulative rate of sustained remission was 67% in the 12 week trial compared with 22% in the 8 week trial. However, the children who received 12 weeks of cyclophosphamide were older than the historical cohort, who received eight weeks, and may have had a better prognosis because of this age difference25. In contrast, a randomized controlled study from Japan comparing 8 versus 12 weeks of cyclophosphamide head-to-head in steroid-dependent nephrotic syndrome found that 12 weeks did not confer any advantage. The relapse rate was similar in both groups and similar to the eight-week treatment group of the APN26. The better outcome with the 12 weeks may be confounded by the fact that the children were older and compared with historical controls. Other retrospective or observational studies have confirmed the effectiveness of cyclophosphamide in frequently relapsing or steroid-dependent nephrotic syndrome27., 28., 29., 30.. Cyclosporine Very few of the studies of cyclosporine in pediatricnephrosis are controlled, and most are small31. The first such study, in the late 1980s, found that cyclosporine treatment reduced relapses and total corticosteroid dose32. Subsequent uncontrolled studies found complete remissions in steroid-dependent or frequently relapsing nephrotic syndrome, whereas many children were able to stop corticosteroids33,34. The largest study involved 65 children with nephrotic syndrome, 45 of whom had minimal lesion disease. These patients were classified as steroid resistant if they did not show remission after just four weeks of daily corticosteroids followed by three pulses of solumedrol. Of the 45 patients with minimal lesion, 21 entered complete remission on cyclosporine treatment in combination with prednisone. Seventeen of these patients remained in remission after a mean three-year follow-up once stopping the cyclosporine, and most patients became steroid sensitive35. It may have been premature, however, to classify these children as steroid resistant after just four weeks of therapy. Some of the patients who went into remission with cyclosporine may have actually been demonstrating a delayed response to corticosteroid treatment. Most patients experience relapses of their disease when cyclosporine is tapered or discontinued, becoming cyclosporine dependent in the same way they were once steroid dependent31. Furthermore, with subsequent courses of cyclosporine treatment, the patients appear to become progressively less responsive. In one of the few randomized studies, children in their first year of nephrotic syndrome were randomized to receive low-dose prednisone and cyclosporine or high-dose prednisone alone (in half of each group, this was instituted with their first presentation). Children in both groups went into remission, and there was no difference in the duration of remission once treatment was stopped36. When compared with alkylating agents, cyclosporine was not as successful in maintaining a prolonged remission in steroid-dependent nephrosis31. In a randomized controlled trial in both adults and children, cyclophosphamide and cyclosporine were equally successful in inducing remission in steroid-dependent and frequently relapsing nephrotic syndrome. However, remissions lasted longer in the group that received the alkylating agent37. A recent uncontrolled retrospective study of the use of cyclosporine (2.5 to 5.0 mg/kg/day starting dose) in steroid-dependent and steroid-resistant nephrosis in children (some of whom had focal sclerosis) reported good results. There was halving of the steroid dependence in the first group and restoration of steroid responsiveness in approximately one fourth of the patients in the latter group. After a median follow-up of 7.5 years in the steroid-dependent cohort and 5.0 years in the steroid-resistant group, there was no change in plasma creatinine with the exception of some of the patients with focal sclerosis. A small number underwent follow-up renal biopsy, and striped interstitial fibrosis and tubular atrophy were found, suggesting cyclosporine nephrotoxicity. However, these were not accompanied by changes in renal function (as judged by serum creatinine, though). The authors concluded that cyclosporine therapy in this setting was effective adjunctive therapy with minimal long-term nephrotoxicity38. In summary, for patients with steroid-dependent or frequently relapsing minimal lesion disease, alkylating agents are as effective as cyclosporine and produce a longer remission. Levamisole The antihelminthic agentlevamisole has immunomodulatory effects and has been used in a number of diseases, such as cancer. Levamisole has been used alone or in combination therapy in minimal lesion disease in children. Unfortunately, few of these studies were either randomized or controlled39. A level 5 study used 2.5 mg/kg twice weekly in frequently relapsing minimal lesion disease. Half of the children had already received other immunosuppressive agents besides prednisone, which confuses the issue. Sixteen of the 30 patients stayed relapse free while on levamisole, but many relapsed when it was stopped; however, the follow-up was short40. A recent study from India used alternate-day levamisole therapy in steroid-dependent nephrotic syndrome once remission was obtained with corticosteroids. The subsequent relapse rate decreased from a mean of 3 per year to 0.9 per year41. Other small studies found that treatment with levamisole induced or maintained remission42., 43., 44.. In one of only two controlled studies of levamisole in pediatric nephrotic syndrome, an increased remission rate compared with placebo was found, but most children relapsed within three months of stopping levamisole45. In the other controlled study, 21 of 33 patients receiving levamisole were still in remission compared with 12 of 28 patients receiving no treatment, a nonsignificant difference46. Levamisole is well tolerated. The side effects noted include neutropenia, rash, and liver toxicity47. However, a review of the literature reveals no new large-scale studies using this agent in nephrosis of children in the last few years. Perhaps others find this agent less effective than published trials suggest; otherwise, it is unclear why this agent has not been more widely accepted for this indication. Treatment of steroid-resistant minimal lesion disease Recommendation 5 Management of steroid-resistant minimal lesion disease can include rebiopsy to rule out focal sclerosis (grade D), cyclophosphamide 2 mg/kg/day × 12 weeks (grade D), or cyclosporine at 6 mg/kg/day for children or 5 mg/kg/day for adults for an uncertain duration (grade D). Evidence Patients with steroid-resistant minimal lesion disease are the most difficult to treat. These patients suffer the ill effects not only of corticosteroid toxicity, but of unremitting nephrosis with the attendant risks of sepsis, malnutrition, growth retardation, and thrombosis. The patient with steroid-resistant nephrotic syndrome also has a greater chance of progressive renal insufficiency culminating in end-stage renal disease. Studies in these children have been limited by small numbers and short follow-up times. The only controlled study had eight children, four with minimal lesion and four with focal sclerosis. The patients received eight weeks of cyclosporine in a randomized cross-over design; there was no difference compared with no treatment48. Given the paucity of evidence for cyclosporine, alkylating agents should be used first in children with frequently relapsing, steroid-dependent, and steroid-resistant minimal lesion disease. Cyclosporine should be reserved for those cases in which alkylating agents have failed, especially in which there is unacceptable corticosteroid toxicity and a holiday from prednisone is needed, to allow catch-up growth or puberty. Unfortunately, most children continue to relapse when cyclosporine is withdrawn. The alternative, chronic cyclosporine therapy, carries the risk of nephrotoxicity, with the extent of damage to the kidneys out of proportion to the rise in serum creatinine. Serial renal biopsies have been recommended49. Progressive nephrotoxicity in children on long-term cyclosporine therapy may be avoided if the patients are kept volume expanded, low doses are used, and levels are closely monitored50. TREATMENT OF MINIMAL LESION DISEASE IN ADULTS Corticosteroids Few randomized controlled studies have examined the treatment of adults with minimal lesion disease (Tables 4 and 5). Observational studies from the 1950s showed rapid remissions after corticosteroid treatment of adult patients with minimal lesion disease. No controlled clinical trial was undertaken until 1970 when Black, Rose, and Brewer performed a multicenter controlled trial of steroid treatment in 125 adult nephrotic patients, including 31 patients with biopsy-proven “minimal lesion” disease (it is possible that patients with stage 1 membranous glomerulonephritis could have been included in this group)3. The treatment group received prednisone, at least 20 mg/day, for at least six months, compared with a control group. In the patients with minimal lesion disease, prednisone therapy gave an early and rapid decrease in proteinuria. Control patients showed a tendency for remission of the proteinuria, although this occurred much more slowly. By two and a half years, the difference with respect to proteinuria and serum albumin concentrations was not statistically significant between the prednisone-treated and control group. Table 4. Corticosteroids and alkylating agents antimetabolites in studies of adults \| Author [Ref] \| Study design \| N \| Treatment \| Results/comments \| --- --- \| Level 1 or 2 \| \| Black \| Multicenter RCT Pred 20–30 mg/day> 6 months \| 61 Pred64 controls (31/125 had ML) \| Pred “not less than” 20 mg/kg/day × 6 months \| Early response in most ML Pred compared to controls; but gradual tendency to remission in ML controls over 2 years \| \| Imbasciati \| Multicenter RCT children + adults pulse IV methyl prednisolone vs. high dose oral Pred \| 67 children \| Solumedrol 20 mg/kg/i.v. daily × 3, oral Pred × 4 weeks, taper over 5 months \| Children receiving pulses had CR earlier than oral Pred; no difference in adults. Tendency to earlier and more frequent relapses in pulse group; more steroid side effects in oral Pred group \| \| \| \| 22 adults \| vs. \| \| \| \| \| \| Pred 60 mg/m 2 or 1 mg/kg/day × 4 weeks, alt Pred × 5 months \| \| \| Level 3 or 4 \| \| Bolton \| Observational alt day steroids in different GN’s \| 29 (ML) \| Pred 60–120 mg q alt day × 9–12 months, then taper \| CR or PR in 83% of ML. Some also receive cytotoxics. Short F/U \| \| Nolasco \| Retrospective review of adult onset ML (F/U of Cameron 1974) \| 89 \| Pred 60 mg/day tapered over 8 weeks; CTX (N = 36) for initial rx, steroid “resistance” or frequent relapses \| Adults have lower + slower response to Pred (?relatively lower dose c/w children), relapse less often, more stable remission with CTX (66% remission at 5 years). Adults more 8 BP, ARF c/w children. 6/8 no rx went into remission slowly \| \| Level 5 or 6 \| \| Sharpstone \| Observational Pred vs. Pred + AZA in different primary GN’s \| 8 ML \| Pred 60 mg/day × 3 days tapered to 20 mg/day × 8 weeks; Pred 20 mg/day + 150 mg AZA/day × 8 weeks \| ML all started with Pred alone 6/8 CR. In 2/8 changed to Pred + AZA—one improved, one did not \| \| Uldall \| Observational CTX for adult ML \| 10 \| CTX 1.5 mg/kg/day, titrated to WBC 2.5–24 months \| 2 patients received CTX only and went into CR; 8 patients had PR or relapses on Pred, on CTX all went into CR or near CR. Alopecia and ovarian toxicity \| \| Idelson \| Review of steroid rx in various 1E adult GN’s \| 28 ML \| Corticosteroids (dose?) \| CR or PR predictive of long-term renal outcome, also persistence of normal renal function for at least 3 years predictive of good outcome \| \| Al-Khader \| Review of CTX alone for “adult” ML (ages 13–68) c/w diuretics alone \| 8 CTX \| CTX 8 mg/kg body wt and tapered to WBC \| 7/8 CTX CR by 14 weeks (mean 10 weeks) 2/8 diuretic only group CR spontaneously by 12 months. None of 9 CR’s had relapses by 6 year F/U \| \| \| \| 8 diuretics \| \| \| \| Cade \| Steroid-resistant ML given AZA \| 13 \| Pred 60 mg/day × 4 weeks, then very slow taper over months; if failed rx or frequent relapses that were failing Pred changed to AZA 2.0–2.5 mg/kg/body wt × years \| All 13 showed 9 proteinuria, 8 C Cr. All in CR, but took up to one year to respond. Effect of AZA or late effect of Pred? \| \| Lim \| Idiopathic NS \| 11 ML \| Pred in various doses, ACTH or no rx \| 4/11 had spontaneous remission. Steroid induced CR in 5 \| Abbreviations are in Table 1. Table 5. Cyclosporine in studies of adults \| Author [Ref] \| Study design \| N \| Treatment \| Results/comments \| --- --- \| Level 1 or 2 \| \| Ponticelli \| \| See Table 3(Children and Adults) \| \| \| \| Level 3 or 4 \| \| Meyrier \| CsA alone for GN with steroid resistance or dependencyCsA + Pred for Gn with steroid resistance or dependence \| 29 ML \| CsA 5mg/kg/day; duration depended on responseCsA 5mg/kg/day titrated to lower trough levels and tapering Pred; duration depended on response \| Pooled results: 72% chance of remission with steroid dependent ML Longest remission 12 months \| \| \| \| 29 ML \| \| \| \| Lee \| Uncontrolled prospective study steroid dependent or resistant or frequent relapsing NS \| 22 ML (final) \| CsA up to 7 mg/kg/day up to 8 months; Pred 10 mg/kg/day \| CR in 19/22; PR in 2/22Failure in 1; relapses in 68% by 10 months \| \| Level 5 or 6 \| \| Lagrue \| Uncontrolled CsA for steroid resistant or dependent NS \| 10 ML \| CsA 3 mg/kg–5 mg/kg × 3 months \| CR in 7/10; PR in 2/10Failure in 1 (also steroid resistant). All relapsed within 2–6 weeks \| \| Maher \| Uncontrolled CsA in steroid dependent or resistant NS; some failed CTX \| 10 ML \| CsA 5–12 mg/kg/day for 40–230 days \| CR within 14 days. All patients relapsed within 180 days of discontinuation of CsA \| \| \| \| 1 IgM \| \| \| \| Clasen \| CsA for frequent relapsing or steroid resistant ML \| 7 \| CsA 3–5 mg/kg BID and tapered to blood levels. Pred tapered off when therapeutic CsA levels reached \| CR in 5/7. All relapsed with tapering or discontinuation of CsA. Remission could be reinduced with CsA. No response in 2/7 up to 10 months \| \| Green \| NS resistant to Pred ± CTX given CsA (details of prior rx not given) \| 3 ML \| CsA 6–10 mg/kg/day \| 3/3 CR within one month. All relapsed within 1–3 months of stopping CsA. Remission could be reinduced. \| Abbreviations are in Table 1. Another multicenter randomized trial examined whether adults and children with minimal lesion disease could be more effectively treated with pulse methylprednisolone compared with the usual treatment. The group receiving methylprednisolone reached remission sooner only in children, not in adults. The relapse rate was no different between the methylprednisolone and the oral prednisone group. Corticosteroid-related side effects were more prevalent in the group receiving oral prednisone. The investigators suggested that pulse methylprednisolone followed by low-dose oral prednisone was less effective than the usual regimen of high-dose oral steroid, but was associated with less corticosteroid-related side effects. In patients who are at risk for corticosteroid side effects (for example, adult patients with osteoporosis), pulse steroid followed by low-dose oral prednisone might be more appropriate compared with longer term high-dose oral steroid treatment. No studies have directly addressed this question51. An uncontrolled level 4 study of alternate-day corticosteroid treatment in adults with idiopathic nephrotic syndrome used 60 to 120 mg of prednisone every other day for 9 to 12 months, followed by a gradual taper. Eighty-three percent of patients with minimal lesion disease sustained a complete or partial remission. Although this regimen was effective, the prolonged duration of therapy meant that the total amount of corticosteroids given was at least as great as in patients receiving more conventional therapy52. Antimetabolites and alkylating agents Few studies have examined antimetabolites or alkylating agents in adults with minimal lesion disease. A level 5 study of long-term azathioprine in patients with steroid-resistant minimal lesion disease showed that all of the patients had remission of their nephrotic syndrome and increased creatinine clearance by one year. Unfortunately, the patients had previously received very large doses of steroids, and in this uncontrolled study, it is unclear whether the remission was actually a late effect of large doses of prednisone or whether this was just spontaneous remission53. Few studies have examined the use of cyclophosphamide without corticosteroids in adult patients with minimal change disease. In one study, 2 of the 10 patients treated with cyclophosphamide alone went into remission54. In another study, very high doses of cyclophosphamide were used alone in a heterogeneous group of eight patients with minimal lesion disease. Their gonadal status was not followed. In this study, nine weeks of therapy with this alkylating agent induced remission in seven of these patients by five months. After a mean follow-up of six years, none of the patients relapsed. Once again, however, this is a level 6 study, and it is difficult to make firm conclusions in a condition that has a tendency to remit if followed long enough55. Cyclosporine Cyclosporine induces complete or partial remissions in adults with minimal lesion disease. Two level 1 studies involving adults are multicenter studies from Italy. The first is a randomized controlled study in steroid-resistant nephrotic syndrome comparing 5 mg/kg/day of cyclosporine in adults and 6 mg/kg/day in children to placebo56. Corticosteroid resistance was defined as persistence of nephrotic syndrome after six weeks of prednisone (1 mg/kg/day) for adults or five weeks of prednisone (60 mg/m 2/day) for children. Unfortunately, eligibility included focal sclerosis in addition to minimal lesion disease. Most of these steroid-resistant patients had focal sclerosis. Of the two adults studied with minimal change disease, one went into complete remission, and the other had no response. The sample was too small to draw any conclusion. Of the six children with minimal change disease, four experienced reduction or remission of proteinuria, and two had continuing nephrosis56. The other randomized, controlled trial involved patients with steroid-dependent (as opposed to steroid-resistant) nephrotic syndrome or frequently relapsing disease37. These patients were randomly allocated to 2.5 mg/kg/day of cyclophosphamide for eight weeks or 5 mg/kg/day of cyclosporine in adults (6 mg/kg/day in children) for up to 12 months. Patients had to be in remission already, and prednisone was tapered off by the first five weeks of the protocol. The majority of the patients were children (55 out of 64), but the results were similar through all ages. Unlike the previous study, the majority of patients had minimal lesion disease (31 out of 34). The population was similar to the ones in studies by the European and ISKDC groups discussed earlier. As in those reports, cyclophosphamide was quite effective in producing a long-lasting remission. Two years after treatment, 63% of the cyclophosphamide-treated group were still in remission compared with 25% of those treated with cyclosporine. The authors recommended that frequently relapsing or steroid-dependent patients be treated with cyclophosphamide first. If that fails, then cyclosporine could be used as the next agent37. Once again, however, it is important to emphasize that there were only 11 adults altogether in this study. Other uncontrolled studies in adults show results similar to those in children, that is, there is a good initial response57 but a disheartening relapse rate as the drug is tapered and discontinued58., 59., 60., 61., 62.. In conclusion, the role of cyclosporine in minimal lesion disease may be (a) in frequently relapsing or steroid-dependent disease, in which a trial of cyclophosphamide has failed, (b) where cyclophosphamide is contraindicated or there are concerns about gonadal toxicity, (c) in steroid-dependent disease to allow a “steroid holiday” for catch-up growth and puberty, or (d) in steroid-resistant disease. Cyclosporine can produce long-term nephrotoxicity, especially in adults whose underlying vascular disease is more prevalent than in children. Patients with tubulointerstitial lesions may progress to end-stage renal disease more rapidly when given this agent61. Without further controlled studies, however, this interaction will be hard to tease out because the kind of nephrosis in which cyclosporine is often tried, such as steroid-resistant minimal lesion disease or focal sclerosis, is in itself associated with progression to renal failure. Long-term outcome of adults with minimal lesion disease The long-term outcome of patients older than 15 has been examined retrospectively1,4. Adults with minimal lesion disease, in contrast to children, have equal sex distribution. In addition, more of these adults had hypertension, and there was more microscopic hematuria. Furthermore, adults show an increased prevalence of diminished glomerular filtration rate, and there was a significant mortality associated with this condition. In the Guy’s Hospital series of idiopathic nephrotic syndrome in adults, 25% had minimal lesion disease, showing that this is a significant cause of idiopathic nephrotic syndrome in adults. Five of the 49 patients presented with acute renal failure, and four of these needed dialysis (acute renal failure in minimal lesion disease is discussed later in this article)4. Three of the 49 patients had a spontaneous remission before therapy was given, supporting the previous observation of Black, Rose, and Brewer3. Seventy percent of patients relapsed, most them two or more times. This may have been related to a shorter course of prednisone that they received compared with other studies. Complications of minimal lesion disease were similar to those seen in children, with the exception of acute renal failure. Three patients had pulmonary emboli, and two patients developed cellulitis. A follow-up study was published 12 years later of 89 patients with a mean follow-up of 7.5 years1. Again, the prevalence of hypertension and microscopic hematuria was confirmed. Glomerular filtration rate was diminished in 60% of patients. Patients who went into remission earlier also seemed to relapse earlier. In total, only 56% of patients were still protein free at nine months and only 34% at two years. There was a much lower relapse rate in these adults when they were given a course of cyclophosphamide. Sixty-three percent of the cyclophosphamide-treated patients were still in remission after 10 years of follow-up. These authors concluded that adults had a lower and slower response to corticosteroids (but were given proportionately less prednisone per body weight compared with children). They sustained fewer relapses compared with children, and they seemed to have a more stable and sustained remission after a course of cyclophosphamide. Finally, hypertension, diminished glomerular filtration rate, and acute renal failure were all more prevalent in the adult population. Acute renal failure as a complication of minimal lesion disease A small subset of patients develop frank acute renal failure as a complication of minimal lesion disease. In retrospective analyses, patients who develop acute renal failure are older, tend to be male, and are hypertensive with evidence of vascular disease63,64. A recent review of all reports in the English medical literature found that the average age of patients with this complication was 58 years. The mean urine protein excretion was 11.6 g/day, and the patients had a mean serum albumin of just 19 g/liter65. Acute renal failure occurred, on average, less than a month after onset of the nephrotic syndrome and lasted an average of seven weeks65. Despite the prolonged course of renal failure, recovery almost always occurred. There is one report of five patients who did not recover renal function66, but at least two of these patients probably had focal sclerosis. Many had dye studies, and one patient refused dialysis and died (perhaps before renal function would have returned). On renal biopsy, patients with acute renal failure have more arteriosclerosis than those with preserved function64. Acute tubular necrosis is the most prevalent finding but is not universally seen64,67. Marked edema of the tubulointerstitium has been described in some cases63 but is conspicuously absent in others64. In the original descriptions of acute renal failure, it was postulated that reduced plasma oncotic pressure led to a contracted plasma volume. With the added insults of vigorous diuresis or paracentesis, for example, the kidneys became critically underperfused, and ischemic acute tubular necrosis supervened67. This hypothesis has several weaknesses. As discussed, in patients with acute renal failure, acute tubular necrosis is found on renal biopsy no more than 60 to 70% of the time64. The duration of renal failure is atypically long for acute tubular necrosis, suggesting that other factors are contributing to the renal compromise. There is little compelling clinical evidence for marked plasma volume depletion in patients who develop renal failure65. Indeed, the patients who develop renal failure have the highest blood pressures, compatible with the presence of vascular disease64. Plasma volume repletion does not reliably reverse the renal failure65,68. Improvement in techniques to measure plasma volume has led to the re-evaluation of the concept of plasma volume contraction in the nephrotic syndrome. Plasma volume in patients with minimal lesion disease appears to be normal or even increased69,70. The increased rather than decreased plasma volume may be, in part, a result of a parallel fall in oncotic pressure in the interstitial compartment of body water, which would decrease flux of water out of the plasma compartment71. Furthermore, renal excretion of salt and water is decreased in minimal lesion disease, which serves to keep the plasma compartment expanded72,73. Lowenstein, Schacht, and Baldwin noted the lack of tubular necrosis in a retrospective review of 15 patients with minimal lesion disease complicated by acute renal failure. Instead, they noted interstitial edema in many of the biopsy specimens. Furthermore, there was improvement in renal function with induction of diuresis. They postulated that renal interstitial edema produced an increase in hydrostatic pressure in the proximal tubules and Bowman’s space, leading to a decreased glomerular filtration rate63. If there is convincing clinical evidence of plasma volume depletion in a given patient (decreased jugular venous pressure, hypotension), it may be reasonable to try to normalize plasma volume with colloid, although this may occur only transiently74,75. Advocates of the renal interstitial edema theory suggest diuretic treatment sufficient to effect a vigorous diuresis63. However, it is possible that the response to diuretics is simply a marker of a functioning kidney, rather than a treatment that restores function to an acutely failed kidney. Recovery of renal function occurs in the majority of patients. The duration of renal failure is typically prolonged, and the endogenous renal function should be monitored while the patient is on dialysis. SUMMARY Among all of the glomerulonephritides in children, minimal lesion disease is one of the most common. This has made possible a large number of clinical trials, providing sufficient evidence for very clear treatment recommendations Figure 1. Corticosteroid treatment is proven effective in this group, although relapses occur. In adults, however, a paucity of large randomized trials means that recommendations cannot easily be made. 1. Download: Download full-size image Fig. 1. Algorithm for treatment of minimal lesion disease. ACKNOWLEDGMENTS The author thanks V. Brown for secretarial help. 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Cyclosporin is also effective in both the induction of remission and long-term preservation of renal function in steroid-dependent/-resistant MCD and steroid-resistant focal segmental glomerulosclerosis (FSGS). The overall response rate in FSGS is lower than in MCD, and long-term therapy (>12 months) may be required to both achieve remission and sustain it. Cyclosporin therapy is also of benefit in reducing proteinuria in 70–80% of patients with steroid-resistant membranous nephropathy (MGN). In MGN, the maximum benefit is often delayed compared to MCD (>12 weeks). Cyclosporin is generally well tolerated and safe. The major concern remains the nephrotoxicity, but with careful monitoring of the patient's renal function; minimizing the maintenance dose and utilizing repeat renal biopsy in those receiving long-term therapy, this risk can be minimized. The algorithms have been developed derived from the best evidence in the literature in each of the histologic types to help provide a guide to the integration of cyclosporin into the management of INS for the practicing nephrologist. ### Glomerulonephritis 2005, Lancet Show abstract The term glomerulonephritis encompasses a range of immune-mediated disorders that cause inflammation within the glomerulus and other compartments of the kidney. Studies with animal models have shown the crucial interaction between bone-marrow-derived inflammatory cells and cells intrinsic to the kidney that is both fundamental and unique to the pathogenesis of glomerulonephritis. The mechanisms of interaction between these cells and the mediators of their coordinated response to inflammation are being elucidated. Despite these pathophysiological advances, treatments for glomerulonephritis remain non-specific, hazardous, and only partly successful. Glomerulonephritis therefore remains a common cause of end-stage kidney failure worldwide. Molecule-specific approaches offer hope for more effective and safer treatments in the future. ### Nephrotic syndrome in childhood 2003, Lancet Show abstract Childhood nephrotic syndromes are most commonly caused by one of two idiopathic diseases: minimal-change nephrotic syndrome (MCNS) and focal segmental glomerulosclerosis (FSGS). A third distinct type, membranous nephropathy, is rare in children. Other causes of isolated nephrotic syndrome can be subdivided into two major categories: rare genetic disorders, and secondary diseases associated with drugs, infections, or neoplasia. The cause of idiopathic nephrotic syndrome remains unknown, but evidence suggests it may be a primary T-cell disorder that leads to glomerular podocyte dysfunction. Genetic studies in children with familial nephrotic syndrome have identified mutations in genes that encode important podocyte proteins. Patients with idiopathic nephrotic syndrome are initially treated with corticosteroids. Steroid-responsiveness is of greater prognostic use than renal histology. Several secondline drugs, including alkylating agents, ciclosporin, and levamisole, may be effective for complicated and steroid-unresponsive MCNS and FSGS patients. Nephrotic syndrome is associated with several medical complications, the most severe and potentially fatal being bacterial infections and thromboembolism. Idiopathic nephrotic syndrome is a chronic relapsing disease for most steroid-responsive patients, whereas most children with refractory FSGS ultimately develop end-stage renal disease. Research is being done to further elucidate the disorder's molecular pathogenesis, identify new prognostic indicators, and to develop better approaches to treatment. ### Rituximab in steroid-dependent or frequently relapsing idiopathic nephrotic syndrome 2014, Journal of the American Society of Nephrology ### Meyler’s side effects of drugs: The international encyclopedia of adverse drug reactions and interactions 2006, Meylers Side Effects of Drugs 15e the International Encyclopedia of Adverse Drug Reactions and Interactions ### A meta-analysis of cytotoxic treatment for frequently relapsing nephrotic syndrome in children 2001, Pediatric Nephrology View all citing articles on Scopus Copyright © 1999 International Society of Nephrology. All rights reserved. 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7604	https://www.asam.org/docs/default-source/quality-science/the_asam_clinical_practice_guideline_on_alcohol-1.pdf	Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. The ASAM Clinical Practice Guideline on Alcohol Withdrawal Management Guideline Committee Members (alpha order): Anika Alvanzo, MD, MS, DFASAM, FACP Kurt Kleinschmidt, MD, FASAM Julie A. Kmiec, DO, FASAM George Kolodner, MD, DLFAPA, FASAM Gerald E. Marti, MD, PhD William F. Murphy, DO, MS, DFASAM Carlos F. Tirado, MD, FASAM Corey Waller, MD, MS, DFASAM, FACEP Lewis S. Nelson, MD, FASAM, FACEP, FACMT (Chair) Clinical Champions (alpha order): Stephen Holt, MD, MS, FACP Darius Rastegar, MD, FASAM Richard Saitz, MD, MPH, FACP, DFASAM Michael F. Weaver, MD, DFASAM ASAM Quality Improvement Council (alpha order): John Femino, MD, DFASAM Kenneth Freedman, MD, DFASAM (Chair) R. Jeffrey Goldsmith, MD, DLFAPA, DFASAM Barbara Herbert, MD, DFASAM (Past- Chair) Margaret Jarvis, MD, DFASAM Margaret M. Kotz, DO, DFASAM P. Stephen Novack, DO David R. Pating, MD Sandrine Pirard, MD, PhD, MPH, FAPA, FASAM ASAM Staff: Maureen Boyle, PhD, Chief Quality and Science Officer Leah White, MPH, Director of Quality Improvement Taleen Safarian, Manager of Science and Dissemination Institute for Research Education and Training in Addic-tions (IRETA) Team Members: Dawn Lindsay, PhD, Project Director Jessica Williams, MPH, Assistant Project Director Piper Lincoln, MS, Senior Research Associate Jackie Jones, MS, Research Associate Rachael Vargo, BA, Research Associate Peter F. Luongo, PhD, Executive Director ASAM is honored that this clinical practice guideline has been endorsed by: American College of Preventive Medicine American Osteopathic Academy of Addiction Medicine Federation of State Physician Health Programs National Association of Addiction Treatment Providers National Association of Clinical Nurse Specialists National Commission on Correctional Health Care Table of Contents Glossary of Terms 2 Abbreviations and Acronyms 3 Executive Summary 3 Introduction 15 I. Purpose 15 II. Background 15 III. Scope of Guideline 16 IV. Intended Audience 16 V. Qualifying Statement 16 VI. Special Terms 16 Approach and Methodology 17 I. Overview of Approach 17 II. Develop the Scope and Key Questions 18 III. Conduct a Literature Review 18 IV. Develop Draft Guideline Statements 19 V. Conduct Panel Ratings 19 VI. Drafting the Guideline Document 20 Recommendations 20 I. Identification and Diagnosis of Alcohol Withdrawal 20 A. Identification 20 B. Diagnosis 21 C. Differential Diagnosis 21 II. Initial Assessment of Alcohol Withdrawal 23 A. General Approach 23 B. Risk Factors for Severe or Complicated Withdrawal 24 C. Risk Assessment Tools 25 D. Symptom Assessment Scales 26 E. Identify Concurrent Conditions 27 III. Level of Care Determination 28 A. General Approach 28 B. Level of Care Determination Tools 28 C. Considerations for Ambulatory vs Inpatient Management 29 IV. Ambulatory Management of Alcohol Withdrawal 33 A. Monitoring 33 B. Supportive Care 35 C. AUD Treatment Initiation and Engagement 36 D. Pharmacotherapy 36 V. Inpatient Management of Alcohol Withdrawal 43 A. Monitoring 43 B. Supportive Care 44 C. AUD Treatment Initiation and Engagement 45 D. Pharmacotherapy 45 VI. Addressing Complicated Alcohol Withdrawal 50 A. Alcohol Withdrawal Seizure 50 B. Alcohol Withdrawal Delirium 51 C. Alcohol-Induced Psychotic Disorder 53 D. Resistant Alcohol Withdrawal 53 VII. Specific Settings and Populations 54 A. Primary Care 54 B. Emergency Departments 55 C. Hospitalized Patients 56 D. Patients with Medical Conditions 58 E. Patients who Take Opioids 58 F. Patients who are Pregnant 58 Areas for Further Research 59 Appendices 61 I. Cited References 61 II. Literature Search Methods 66 A. Empirical Literature Search Terms 66 B. Gray Literature Search 66 Adopted by the ASAM Board of Directors January 23, 2020 1 CLINICAL PRACTICE GUIDELINE Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. GLOSSARY OF TERMS Below are terms that are used throughout the guideline. Note that some terms listed below are used to convey a specific meaning for the purposes of this guideline (e.g., ‘‘clinicians’’). Abstinence: Intentional and consistent restraint from the pathological pursuit of reward and/or relief that involves the use of substances and other behaviors. These behaviors may involve, but are not necessarily limited to substance use, gambling, video gaming, or compulsive sexual behaviors. Use of FDA approved medications for the treatment of substance use disorder is consistent with abstinence. Addiction Specialist Physician: Addiction specialist physicians include addiction medicine physicians and addic-tion psychiatrists who hold either a subspecialty board certi-fication in addiction medicine by the American Board of Preventative Medicine, a board certification in addiction medicine from the American Board of Addiction Medicine, a subspecialty board certification in addiction psychiatry from the American Board of Psychiatry and Neurology, a subspecialty board certification in addiction medicine from the American Osteopathic Association, or certification in addiction medicine from the American Society of Addic-tion Medicine. Adjunct therapy (see also monotherapy): A pharma-ceutical drug used together with a primary pharmaceutical drug whose purpose is to assist the primary treatment.1 Alcohol Hallucinosis/Alcohol-induced Psychotic Dis-order: See Special Terms on page 16. ASAM Criteria dimensions: The ASAM Criteria use six dimensions to define a holistic biopsychosocial assess-ment of an individual to be used for service and treatment planning including acute intoxication or withdrawal potential; biomedical conditions and complications; emotional, behav-ioral, or cognitive conditions or complications; readiness for change; continued use or continued problem potential; and recovery/living environment. CIWA-Ar: The Clinical Institute Withdrawal Assess-ment of Alcohol Scale, Revised, is a reliable, valid, and reproducible severity of alcohol withdrawal in communicative patients once a diagnosis has been made.2 Complicated alcohol withdrawal: See Special Terms on page 16. Clinicians (Healthcare providers): Used throughout the guideline, this term is intentionally broad. It encompasses anyone who participates in providing care to patients with substance use disorders, including staff at specialty addiction treatment centers or other healthcare settings that provide substance use disorder treatment.3 Fixed-dosing: See Special Terms on page 16. Front loading: See Special Terms on page 16. GABAergic agents: Drugs that affect the neurotrans-mitter GABA or its receptors. These include agonists, antag-onists, modulators, reuptake inhibitors and enzymes. Examples include benzodiazepines, phenobarbital, and car-bamazepine. Inpatient Withdrawal Management: See Special Terms on page 16. Kindling: The relationship between repeated episodes of alcohol withdrawal which become progressively more severe is referred to as the kindling effect or process.4 The effect is theorized to be the result of increased neuronal excitability and sensitivity with repeated episodes of with-drawal and has been demonstrated to result in increased craving for alcohol and decreased responsiveness to treatment with benzodiazepines.5–7 Level of Care: See Special Terms on page 16. Monotherapy (see also adjunct therapy): The use of a single drug to treat a disorder or disease. Patients: Used throughout the guideline, this term is intentionally broad. It encompasses anyone who receives care for a Substance Use Disorder (SUD) in a specialty SUD treatment center or other healthcare setting.3 Pharmacotherapy: Therapy (medical treatment) using pharmaceutical drugs. Recovery capital: The breadth and depth of internal and external resources that can be drawn upon to initiate and sustain recovery from alcohol and other drug problems. It can be found at the personal, social, community and cultural levels. Examples of recovery capital include physical health, financial assets, supportive social relationships, visible local recovery role models, and accessible/affordable community resources.8 Substance use: Used instead of ‘‘drug use’’ or ‘‘drug and alcohol use,’’ this term refers to the use of psychotropic substances, which may include illegal drugs, medications or alcohol. This does not refer to nicotine.3 Substance Use Disorder (SUD): Substance use disor-der is marked by a cluster of cognitive, behavioral, and physiological symptoms indicating that the individual con-tinues to use alcohol, nicotine, and/or other drugs despite significant related problems. Diagnostic criteria are given in the DSM-5. Substance use disorder is the new nomenclature for what was included as substance dependence and substance abuse in the DSM-4. C. PRISMA Flow Diagram 67 D. Reasons for Exclusion 68 III. Alcohol Withdrawal Scales Table 68 IV. Flowcharts (Supplemental Digital Content, 70 V. Sample Medication Regimens 70 VI. Statement Rating Table (Supplemental Digital Content, 70 VII. Disclosures and Conflicts of Interest 70 A. 2019 Guideline Committee Member Relationships with Industry and Other Entities 70 B. 2019 ASAM Board of Directors Relationships with Industry and Other Entities (Supplemental Digital Content, 71 C. 2019 ASAM Quality Improvement Council (Oversight Committee) Relationships with Industry and Other Entities (Supplemental Digital Content, 71 D. 2019 Clinical Champions Relationships with Industry and Other Entities 72 E. 2020 External Reviewers Relationships with Industry and Other Entities (Supplemental Digital Content, 72 ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 2 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Supportive care: Treatment given to prevent, control, or relieve complications and side effects and to improve the patient’s comfort, quality of life and safety. This can include reassurance, orientation, general nursing care, and adherence to safety measures and protocols (e.g., risk for fall/syncope). Symptom-triggered dosing: See Special Terms on page 16. Therapeutic window: Range of drug dose amount needed to maintain therapeutic effect yet avoid adverse events. A drug with a narrower therapeutic window requires greater precision to be dosed correctly and safely compared to a drug with a broader therapeutic window. A drug’s therapeu-tic window is taken into account when modifying dose amount due to patient variability and exposure to other substances including adjunt medications.9 Treatment plan: A therapeutic strategy that may incor-porate patient education, drug therapy, and the participation of health professionals. Treatment plans are especially important in the optimal management of complex or chronic illnesses such as SUDs.3 Unhealthy alcohol use: Includes the following patterns of alcohol use: 1) Binge drinking (defined as consuming 4 or more alcoholic beverages per occasion for women or 5 or more drinks per occasion for men); 2) Heavy drinking (defined as consuming 8 or more alcoholic beverages per week for women or 15 or more alcoholic beverages per week for men); 3) Any drinking by pregnant women or those younger than age 21.10 Withdrawal Management: This term has replaced the formerly used ‘‘detoxification.’’ Withdrawal management refers to the medical and psychological care of patients who are experiencing withdrawal symptoms as a result of ceasing or reducing their substance use.11 The process of withdrawal management includes not only attenuation of the physiological and psychological features of withdrawal, but also interrupting the momentum of habitual compulsive use in persons with SUD.12 ABBREVIATIONS AND ACRONYMS A2AA Alpha-2 adrenergic agonists ALT Alanine aminotransferase AUDIT-PC Alcohol Use Disorders Identification Test-(Piccinelli) Consumption ASAM American Society of Addiction Medicine ASSIST Alcohol, Smoking and Substance Involvement Screening Test AST Aspartate aminotransferase AUD Alcohol Use Disorder BAC Blood Alcohol Concentration (or Content) BAWS Brief Alcohol Withdrawal Scale CCU Cardiac (or Coronary) Care Unit CIWA-Ar Clinical Institute Withdrawal Assessment for Alcohol, Revised CNS Central Nervous System DSM-5 Diagnostic and Statistical Manual, 5th Edition ED Emergency Department EEG Electroencephalogram FAS Fetal Alcohol Syndrome FASD Fetal Alcohol Spectrum Disorders FDA Food and Drug Administration GABA Gamma-aminobutyric acid, or g-aminobutyric acid GAD-7 Generalized Anxiety Disorder Test – 7 GGT Gamma-glutamyl transferase ICU Intensive Care Unit IM Intramuscular IPRAS Interpercentile Range Adjusted for Symmetry IV Intravenous LARS Luebeck Alcohol-Withdrawal Risk Scale MCV Mean corpuscular volume PAWSS Prediction of Alcohol Withdrawal Severity Scale PHQ-9 Patient Health Questionnaire – 9 PO Per os, by mouth RAM RAND/UCLA Appropriateness Method SAMHSA Substance Abuse and Mental Health Services Administration SAWS Short Alcohol Withdrawal Scale SUD Substance Use Disorder WHO World Health Organization WM Withdrawal Management EXECUTIVE SUMMARY Purpose The American Society of Addiction Medicine (ASAM) developed this Guideline on Alcohol Withdrawal Manage-ment to provide updated information on evidence-based strat-egies (hereafter referred to as the Practice Guideline) and standards of care for alcohol withdrawal management in both ambulatory and inpatient settings. Background In June 2017, the American Society of Addiction Medicine’s (ASAM) Quality Improvement Council (QIC) elected to update ASAM’s clinical guidelines on alcohol withdrawal management based on several factors. First, ASAM conducted an Educational Needs Assessment in 2016 that showed a strong interest and need for education on withdrawal management. Second, updated QIC policies recommend that all ASAM guidelines should be updated every five years. ASAM’s previous guidelines on the topic of alcohol withdrawal management were published in 1997 and 2004. The first guideline, ‘‘Pharmacological Management of Alcohol Withdrawal’’13 was published in JAMA, followed five years later with the most recent guideline entitled ‘‘Management of Alcohol Withdrawal Delirium’’14 in JAMA Internal Medicine, formerly Archives of Internal Medicine. Subsequent guidelines have not been written since the 2004 guidelines thus an update was due. Third, the American Psychiatric Association (APA) released a practice guideline in 2018 on the appropriate use of medications in the treatment of alcohol use disorder that is not inclusive of alcohol withdrawal management.15 An ASAM guideline on alcohol withdrawal should complement APA’s guideline to provide clinicians with guidance on treatment and management approaches across a continuum of care. Fourth, outreach to other organizations indicated that other organizations are not planning on creating a guideline on alcohol withdrawal management. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 3 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. The updated clinical guideline is intended to address current practice concerns and provide clear guidance that will lead to more consistent treatment practices in the field. Scope of Guideline While the current clinical guideline focuses primarily on alcohol withdrawal management, it is important to under-score that alcohol withdrawal management alone is not an effective treatment for alcohol use disorder. Withdrawal management should not be conceptualized as a discrete clinical service, but rather as a component of the process of initiating and engaging patients in treatment for alcohol use disorder. Intended Audience The intended audience of this guideline is clinicians, mainly physicians, nurse practitioners, physician assistants, and pharmacists who provide alcohol withdrawal manage-ment in specialty and non-specialty addiction treatment set-tings (including primary care and intensive care and surgery units in hospitals). The guideline will also have utility for administrators, insurers, and policymakers. Qualifying Statement This ASAM Alcohol Withdrawal Management Guide-line is intended to aid clinicians in their clinical decision making and patient management. The Guideline strives to identify and define clinical decision making junctures that meet the needs of most patients in most circumstances. Clinical decision making should involve consideration of the quality and availability of expertise and services in the community wherein care is provided. In circumstances in which the Guideline is being used as the basis for regulatory or payer decisions, improvement in quality of care should be the goal. Finally, courses of treatment contained in recommen-dations in this Guideline are effective only if the recommen-dations, as outlined, are followed. Because lack of patient understanding and adherence may adversely affect outcomes, clinicians should make every effort to promote the patient’s understanding of and adherence to recommended treatments. Patients should be informed of the risks, benefits, and alter-natives to a particular treatment, and should be an active party in shared decision making whenever feasible. Recommenda-tions in this Practice Guideline do not supersede any federal or state regulations. Overview of Methodology In order to develop a comprehensive practice guideline focused on alcohol withdrawal management, we utilized a hybrid of established methodologies. In order to develop the scope of the guideline and draft the guideline statements, we followed the Veterans Health Administration and Department of Defense (VA/DoD) Guideline for Guidelines. To rate and refine the draft guidelines, we used the RAND/UCLA Appro-priateness Method (RAM), which is a specific process for combining the available scientific evidence with the clinical judgment of experts. Quality of the literature reviewed was rated using standardized rating scales and methodology. The external review process was informed by the VA/DoD method. SUMMARY OF RECOMMENDATIONS I. Identification and Diagnosis of Alcohol Withdrawal A. Identification Recommendation I.1: Incorporate universal screening for unhealthy alcohol use into medical settings using a validated scale to help identify patients with or at risk for alcohol use disorder and alcohol withdrawal. Recommendation I.2: For patients known to be using alcohol recently, regularly, and heavily, assess their risk of developing alcohol withdrawal even in the absence of signs and symptoms (see II. Initial Assessment for risk factors and risk assessment scale). Recommendation I.3: For patients who have signs and symptoms suggestive of alcohol withdrawal, assess the quan-tity, frequency, and time of day when alcohol was last consumed to determine whether the patient is experiencing or is at risk for developing alcohol withdrawal. For this assessment, it may be helpful to: Use a scale that screens for unhealthy alcohol use (e.g., Alcohol Use Disorders Identification Test-Piccinelli Con-sumption [AUDIT-PC]) Use information from collateral sources (i.e., family and friends) Conduct a laboratory test that provides some measure of hepatic function Recommendation I.4: A biological test (blood, breath, or urine) for alcohol use may be helpful for identifying recent alcohol use, particularly in patients unable to communicate or otherwise give an alcohol use history. When conducting a biological test, consider the range of time (window of detec-tion) in which the test can detect alcohol use. Do not rule out the risk of developing alcohol withdrawal if the result of a test is negative. B. Diagnosis Recommendation I.5: To diagnose alcohol withdrawal and alcohol withdrawal delirium, use diagnostic criteria such as those provided by the Diagnostic and Statistical Manual 5 (DSM-5). To diagnose alcohol use disorder, use diagnostic criteria such as those provided by the DSM-5. Recommendation I.6: Alcohol withdrawal severity assessment scales (including the Clinical Instrument With-drawal Assessment for Alcohol, Revised [CIWA-Ar]) should not be used as a diagnostic tool because scores can be influenced by conditions other than alcohol withdrawal. Recommendation I.7: Do not rule in or rule out the presence of alcohol withdrawal for patients who have a positive blood alcohol concentration. C. Differential Diagnosis Recommendation I.8: As part of differential diagno-sis, assess the patient’s signs, symptoms, and history. Rule out other serious illnesses that can mimic the signs and symptoms ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 4 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. of alcohol withdrawal. Determine if patients take medica-tions that can mask the signs and symptoms of alcohol withdrawal. Recommendation I.9: Do not rule in or rule out a co-occurring disease, co-occurring mental health disorder, co-occurring substance use disorder, or simultaneous withdrawal from other substances even in the presence of alcohol withdrawal. Recommendation I.10: Conduct a neurological exam in patients presenting with a seizure to determine etiology. A seizure should only be attributed to alcohol withdrawal if there was a recent cessation of (or reduction in) alcohol consumption. For patients experiencing new onset seizures or for patients with a known history of alcohol withdrawal seizures showing a new pattern, an electroencephalogram and/ or neuroimaging is recommended. For patients with a known history of withdrawal seizure who present with a seizure that can be attributed to alcohol withdrawal, additional neurologi-cal testing and a neurology consult may not be necessary. This includes if the seizure was generalized and without focal elements, if a careful neurological examination reveals no evidence of focal deficits, and if there is no suspicion of meningitis or other etiology. Recommendation I.11: For patients presenting with delirium, conduct a detailed neurological and medical exami-nation with appropriate testing to rule out other common causes of delirium regardless of the apparent etiology. Attempt to distinguish between hallucinations associated with alcohol withdrawal delirium and alcohol hallucinosis/alcohol-induced psychotic disorder. II. Initial Assessment of Alcohol Withdrawal A. General Approach Recommendation II.1: First, determine whether a patient is at risk of developing severe and/or complicated alcohol withdrawal, or complications from alcohol withdrawal. In addition to current signs and symptoms, a validated risk assessment scale and an assessment of individual risk factors should be utilized (See Table 1. Alcohol Withdrawal Severity). Recommendation II.2: A history and physical exami-nation should be included as part of the comprehensive assessment process. Clinicians should conduct this examina-tion themselves or ensure that a current physical examination is contained within the patient’s medical record. Recommendation II.3: Additional information about risk factors can be gleaned by interviewing family, friends, and caregivers about a patient’s history of alcohol withdrawal, seizures, and delirium, as appropriate. Whenever possible in non-emergent situations, obtain written or verbal consent from the patient before speaking with or consulting with collateral sources. Recommendation II.4: Clinicians should seek infor-mation about the time elapsed since the patient’s cessation of (or reduction in) alcohol use. The timeline of symptom onset and severity helps determine the risk window for developing severe or complicated withdrawal. B. Risk Factors for Severe or Complicated Withdrawal Recommendation II.5: Assess for the following factors associated with increased patient risk for complicated with-drawal or complications of withdrawal: History of alcohol withdrawal delirium or alcohol with-drawal seizure Numerous prior withdrawal episodes in the patient’s lifetime Comorbid medical or surgical illness (especially traumatic brain injury) Increased age (>65) Long duration of heavy and regular alcohol consumption Seizure(s) during the current withdrawal episode Marked autonomic hyperactivity on presentation Physiological dependence on GABAergic agents such as benzodiazepines or barbiturates Recommendation II.6: The following individual fac-tors may increase a patient’s risk for complicated withdrawal or complications of withdrawal: Concomitant use of other addictive substances Positive blood alcohol concentration in the presence of signs and symptoms of withdrawal Signs or symptoms of a co-occurring psychiatric disorder are active and reflect a moderate level of severity Recommendation II.7: Patients’ risk for complicated withdrawal or complications of withdrawal is increased by the presence of multiple risk factors. Recommendation II.8: In general, clinicians may con-sider patients at risk of severe or complicated withdrawal if TABLE 1. Alcohol Withdrawal Severity. Severity Category Associated CIWA-Ar Range Symptom Description Mild CIWA-Ar < 10 Mild or moderate anxiety, sweating and insomnia, but no tremor Moderate CIWA-Ar 10-18 Moderate anxiety, sweating, insomnia, and mild tremor Severe CIWA-Ar 19 Severe anxiety and moderate to severe tremor, but not confusion, hallucinations, or seizure Complicated CIWA-Ar 19 Seizure or signs and symptoms indicative of delirium – such as an inability to fully comprehend instructions, clouding of the sensorium or confusion – or new onset of hallucinations Throughout this document, we provide examples for withdrawal severity using the CIWA-Ar, although other scales can be used. Regardless of the instrument used, there is a wide variety in the literature and in practice as to which scores best delineate mild, moderate and severe withdrawal. Classification of withdrawal severity is ultimately up to the judgment of clinicians and the choice of reference range may be based on their particular patient population or capabilities. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 5 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. they are experiencing at least moderate alcohol withdrawal on presentation (e.g., CIWA-Ar score 10). C. Risk Assessment Tools Recommendation II.9: Clinicians can consider the use of a tool such as The ASAM Criteria Risk Assessment Matrix to assess a patient’s risk of severe or complicated alcohol withdrawal as well as potential complications of withdrawal. Recommendation II.10: The following scales can be helpful for assessing for the risk of severe alcohol withdrawal: Prediction of Alcohol Withdrawal Severity Scale (PAWSS) Luebeck Alcohol-Withdrawal Risk Scale (LARS) D. Symptom Assessment Scales Recommendation II.11: A validated scale should be used to assess alcohol withdrawal severity. Recommendation II.12: Assess the risk for scores on an alcohol withdrawal severity assessment scale to be con-founded by causes other than alcohol withdrawal. If risk factors are present, interpret the results of scales with caution. Use a scale that relies more on objective signs of withdrawal (autonomic activity) if a patient has difficulty communicating about their symptoms. See Appendix III for the features of different scales. Recommendation II.13: A validated withdrawal sever-ity assessment scale can be used as part of risk assessment. A high initial score can indicate risk of developing severe or complicated withdrawal, although scores should not be the only information used to predict patient risk. E. Identify Concurrent Conditions Recommendation II.14: When assessing for concur-rent medical conditions, screen patients for medical con-ditions that could affect the course of alcohol withdrawal or treatment of alcohol withdrawal, as well as common chronic conditions that are associated with alcohol use disorders. Recommendation II.15: A pregnancy test should be obtained for women of childbearing potential. For managing pregnant patients, see VII.F: Patients who are Pregnant. Recommendation II.16: In settings with access to laboratory testing, clinicians should conduct and/or arrange for a comprehensive metabolic profile (CMP) or basic meta-bolic profile (BMP), a hepatic panel, and a complete blood count with differential to assess a patient’s electrolytes, liver functioning, renal functioning, and immune functioning. In a setting with limited access to laboratory testing, clinicians should obtain results when practical to assist with treatment planning decisions. Address any nutritional deficiencies detected. Initial screening may also include laboratory tests for: Hepatitis Human Immunodeficiency Virus (HIV) (with consent) Tuberculosis Recommendation II.17: Assess patients for polysub-stance use and be prepared to treat other potential withdrawal syndromes. To assess a patient’s other substance use, it may be helpful to: Use a validated scale that addresses other substance use, such as the Alcohol, Smoking and Substance Involvement Screening Test (ASSIST) Conduct a urine or other toxicology test to detect other substance use Utilize information from collateral sources when possible (i.e., family and friends) Recommendation II.18: Do not delay the initiation of treatment if alcohol withdrawal is suspected but laboratory test results are not available at the treatment setting or the results are pending. Recommendation II.19: Assess patients for concurrent mental health conditions, including a review of their mental health history, to determine their mental health treatment needs. Consult with any mental health professionals caring for such patients. Obtain written or verbal consent before consultation whenever possible in non-emergent situations. The Patient Health Questionnaire (PHQ-9) and the General-ized Anxiety Disorder (GAD-7) scales can be helpful to screen for mental health disorders. Be cautious when diag-nosing a new primary mental health disorder during acute withdrawal, as it can be difficult to differentiate between substance-induced signs and symptoms and primary psychiatric disorders. Recommendation II.20: Evaluate active suicide risk as part of the initial patient assessment. III. Level of Care Determination A. General Approach Recommendation III.1: Level of care determination should be based on a patient’s current signs and symptoms; level of risk for developing severe or complicated withdrawal or complications of withdrawal; and other dimensions such as recovery capital and environment. Alcohol withdrawal can typically be safely managed in an ambulatory setting for those patients with limited or mitigated risk factors. Patients with low levels of psychosocial support or an unsafe environment may benefit from a more intensive level of care than is otherwise indicated. Recommendation III.2: Patients with active risk of suicide should be treated in a setting equipped to manage patients at risk of suicide, which often necessitates admission to an inpatient psychiatric setting that also provides with-drawal management services. B. Level of Care Determination Tools Recommendation III.3: The ASAM Criteria Risk Assessment Matrix and withdrawal severity scales can be helpful for determining the appropriate level of care for managing patients in alcohol withdrawal. Most withdrawal severity scales reflect current signs and symptoms and should not be used alone to determine level of care. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 6 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. C. Considerations for Ambulatory vs Inpatient Management Recommendation III.4: See Table 2. Ambulatory (Level 1-WM and Level 2-WM) and Inpatient Placement Considerations on p. 30. IV. Ambulatory Management of Alcohol Withdrawal Recommendations that are appropriate for both Ambu-latory and Inpatient Management are repeated in both sec-tions. A. Monitoring Recommendation IV.1: In ambulatory settings, arrange for patients to check in with a qualified health provider (e.g., medical assistant, nurse) daily for up to five days following cessation of (or reduction in) alcohol use. For some patients who are unable to attend daily in-person check-ins, alternating in-person visits with remote check-ins via phone or video call is an appropriate alternative. Recommendation IV.2: Re-assessments should focus on the patient’s health since the last checkup. Clinicians should assess general physical condition, vital signs, hydra-tion, orientation, sleep and emotional status including suicidal thoughts at each visit. Ask about alcohol and other substance use and, if available, measure blood alcohol content (BAC) with a breathalyzer to detect recent alcohol use. Recommendation IV.3: Alcohol withdrawal severity should be monitored with a validated instrument (see Appen-dix III for a summary of scales and their associated features). Patients who are able to monitor their own signs and symp-toms may use an instrument designed for self-administration such as the Short Alcohol Withdrawal Scale (SAWS). Recommendation IV.4: In ambulatory settings, patients with a current or past benzodiazepine use disorder need additional monitoring. Recommendation IV.5: For patients managed in an ambulatory setting, the following indications would necessi-tate transfer to a more intensive level of care such as Level 2-WM (if in a Level 1-WM setting) or an inpatient setting: Agitation or severe tremor has not resolved despite having received multiple doses of medication, and the patient will not be continually monitored (e.g., treatment setting is closing) More severe signs or symptoms develop such as persistent vomiting, marked agitation, hallucinations, confusion, or seizure Existing medical or psychiatric condition worsens Patient appears over-sedated Patient returns to alcohol use Syncope, unstable vital signs (low/high blood pressure, low/high heart rate) B. Supportive Care Recommendation IV.6: Supportive care is a critical component of alcohol withdrawal management. Providers should ensure patients are educated about what to expect over the course of withdrawal, including common signs and symptoms and how they will be treated. Recommendation IV.7: When treating patients in ambulatory settings, providers should ensure patients/ caregivers are educated about monitoring for the development of more severe withdrawal and instructed to create a low-stimulation, reassuring environment at home to promote an effective outcome. Recommendation IV.8: Patients should be advised to drink non-caffeinated fluids and that a daily multivitamin may be beneficial. Recommendation IV.9: Patients can be offered oral thiamine. Typical dosing is 100 mg PO per day for 3–5 days. Recommendation IV.10: Clinicians must explain the importance of taking medications as prescribed and confirm the patient’s understanding. Recommendation IV.11: Communicate that safe alco-hol withdrawal management may necessitate a transfer to a more intensive level of care including to an inpatient setting and secure the patient’s agreement to transfer if there are indications that management in the ambulatory setting is not safe or effective. See Recommendation IV.5 for indications for transfer to a more intensive level of care. C. AUD Treatment Initiation and Engagement Recommendation IV.12: When feasible, alcohol use disorder (AUD) treatment should be initiated concurrently with alcohol withdrawal management as cognitive status permits. If appropriate, clinicians should offer to initiate pharmacotherapy for AUD as cognitive status permits. If not initiating AUD treatment themselves, clinicians should explain the range of evidence-based treatment services avail-able in the community, and engage patients with these options. In addition, clinicians may offer information about local recovery support groups, including 12-step groups. D. Pharmacotherapy (1) Prophylaxis Recommendation IV.13: Patients at risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal may be treated in ambulatory settings at the discretion of providers with extensive experience in management of alcohol withdrawal. Such patients should be provided with preventative pharmacotherapy. Benzodia-zepines are first-line treatment because of their well-docu-mented effectiveness in reducing the signs and symptoms of withdrawal including the incidence of seizure and delirium. Phenobarbital is an appropriate alternative in a Level 2-WM setting for providers experienced with its use. For patients with a contraindication for benzodiazepine use, phenobarbital (in Level 2-WM settings by providers experienced with its use) or transfer to a more intensive level of care are appropriate options. Recommendation IV.14: A front loading regimen is recommended for patients at high risk of severe withdrawal syndrome. Providing at least a single dose of preventative medication is appropriate for patients at lower levels of risk who have: Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 7 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. A history of severe or complicated withdrawal An acute medical, psychiatric, or surgical illness Severe coronary artery disease Displaying signs or symptoms of withdrawal concurrent with a positive blood alcohol content Recommendation IV.15: Patients at risk of developing new or worsening signs or symptoms of withdrawal while away from the ambulatory treatment setting should be pro-vided with pharmacotherapy. Some indications of risk include a history of withdrawal episodes of at least moderate severity and being within the window for the development of symp-toms in the time course of withdrawal. Benzodiazepines, carbamazepine, or gabapentin are all appropriate options for monotherapy. Providing at least a single dose of benzodi-azepine followed by ongoing treatment according to symptom severity is also appropriate. If the risk of developing worse withdrawal is unknown, patients should be reassessed fre-quently over the next 24 hours to monitor their need for withdrawal medication. (2) Withdrawal symptoms Recommendation IV.16: Patients experiencing mild alcohol withdrawal (e.g., CIWA-Ar score < 10) who are at minimal risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal may be provided pharmacotherapy or supportive care alone. If pro-viding medication, carbamazepine or gabapentin are appro-priate options. For patients who are at risk of developing new or worsening withdrawal while away from the treatment setting, benzodiazepines, carbamazepine, or gabapentin are appropriate. Recommendation IV.17: Patients experiencing mod-erate alcohol withdrawal (e.g., CIWA-Ar scores 10–18) should receive pharmacotherapy. Benzodiazepines are first-line treatment. Carbamazepine or gabapentin are appropriate alternatives. For patients with a contraindication for benzodi-azepine use, carbamazepine, gabapentin, or phenobarbital (in Level 2-WM settings for providers experienced with its use) are appropriate. Carbamazepine, gabapentin, or valproic acid (if no liver disease or childbearing potential) may be used as an adjunct to benzodiazepines. Recommendation IV.18: Patients experiencing severe, but not complicated, alcohol withdrawal (e.g., CIWA-Ar 19) may be treated in ambulatory Level 2-WM settings at the discretion of providers with extensive experience in manage-ment of alcohol withdrawal. Such patients should receive pharmacotherapy. Benzodiazepines are first-line treatment. Phenobarbital is an appropriate alternative for providers experienced with its use. For patients with a contraindication for benzodiazepine use, phenobarbital, carbamazepine, or gabapentin are appropriate. The use of adjunct medications is also appropriate. Recommendation IV.19: If a patient is taking medica-tion as prescribed and symptoms are not controlled as expected: First, consider increasing the dose If over-sedation or inadequate monitoring is a concern: Reassess for appropriate level of care Consider switching medications If using benzodiazepines, consider adding an adjunct med-ication (3) Benzodiazepine use Recommendation IV.20: While no particular benzodi-azepine agent is more effective than another, longer-acting benzodiazepines are the preferred agents due to the clinical benefits of their longer duration of action. Recommendation IV.21: If waiting for lab test results or if the test(s) are unavailable, if a patient has signs of significant liver disease, use a benzodiazepine with less hepatic metabolization. Recommendation IV.22: Clinicians should monitor patients taking benzodiazepines for signs of over-sedation and respiratory depression. Recommendation IV.23: A benzodiazepine prescrip-tion to treat alcohol withdrawal should be discontinued following treatment. Recommendation IV.24: Clinicians can manage ben-zodiazepine misuse or diversion risk in ambulatory settings by dispensing or prescribing the minimum amount necessary given patients’ level of stability and timing of their next in-person clinic visit. Alternative medications can also be considered such as carbamazepine or gabapentin. Recommendation IV.25: In ambulatory settings, ben-zodiazepines should not be prescribed to patients with a history of even mild adverse events with benzodiazepine use because rapid intervention is not typically available. Benzodiazepines can be used with caution in patients with a high risk of benzodiazepine diversion including patients with a current or past benzodiazepine use disorder for the short period of acute alcohol withdrawal. Risk can be man-aged by dispensing or prescribing a small number of doses. Recommendation IV.26: Patients who are taking ben-zodiazepines, and their caregivers, should be educated regard-ing: The danger of drug-drug interactions between benzodia-zepines and other CNS depressants (impairment and respi-ratory depression) The risks associated with combining alcohol and benzo-diazepines and importance of abstinence from alcohol The risks associated with driving or use of heavy machin-ery for the first few days of benzodiazepine administration Instructions to reduce their benzodiazepine dose if drowsi-ness occurs (4) Benzodiazepine dosing regimens Recommendation IV.27: At short-term observational settings with continuous monitoring (e.g. Level 2-WM), symptom-triggered treatment conducted by trained staff is the preferred benzodiazepine dosing method. Front loading while under clinical supervision or fixed dosing with addi-tional as-needed medication are also appropriate. Recommendation IV.28: At settings without extended on-site monitoring (Level 1-WM), symptom-triggered dosing is appropriate if patients or a caregiver can reliably monitor signs and symptoms with a withdrawal severity scale and follow dosing guidance. Otherwise, front loading while under ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 8 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. clinical supervision or fixed dosing with additional as-needed medication is appropriate. Recommendation IV.29: Front loading is recom-mended for patients experiencing severe alcohol withdrawal (e.g., CIWA-Ar 19). Diazepam and chlordiazepoxide are preferred agents for front loading. Recommendation IV.30: When using a fixed-dose schedule, patients’ signs and symptoms should still be moni-tored. A few additional take-home doses can be provided to take as needed. When initiating a fixed-dose regimen, arrange for the patient to be follow up with the following day to modify the dose if needed. Recommendation IV.31: If prescribing a shorter-acting benzodiazepine, using a fixed-dose regimen with a gradual taper may be appropriate to reduce the likelihood of break-through and rebound signs and symptoms. (5) Carbamazepine, gabapentin, valproic acid Recommendation IV.32: Gabapentin is a favorable choice for treating alcohol withdrawal when a clinician also plans to use it for a patient’s ongoing treatment of alcohol use disorder. Recommendation IV.33: If benzodiazepines are con-traindicated, carbamazepine or gabapentin are appropriate alternatives. Recommendation IV.34: Carbamazepine, gabapentin, or valproic acid may be used as an adjunct to benzodiazepine therapy to help control alcohol withdrawal. Before using as an adjunct, clinicians should ensure that an adequate dose of benzodiazepine has been administered. Recommendation IV.35: Valproic acid should not be used in patients who have liver disease or women of childbearing potential. Recommendation IV.36: There is insufficient evidence to support the use of valproic acid as monotherapy for the treatment of alcohol withdrawal. (6) Phenobarbital Recommendation IV.37: Phenobarbital can be used for some patients in Level 2-WM ambulatory settings; however, it should only be used by clinicians experienced with its use given its narrow therapeutic window and side effects. Recommendation IV.38: In a Level 2-WM ambulatory setting (e.g., with extensive monitoring), phenobarbital mono-therapy (managed by a clinician experienced with its use) is an appropriate alternative to benzodiazepines for patients who are experiencing severe alcohol withdrawal or who are at risk of developing severe or complicated alcohol withdrawal or complication of alcohol withdrawal. Recommendation IV.39: In a Level 2-WM ambulatory setting (e.g., with extensive monitoring), phenobarbital mono-therapy (managed by a clinician experienced with its use) is appropriate for patients with a contraindication for benzodi-azepine use who are experiencing moderate or severe alcohol withdrawal or who are at risk of developing severe or com-plicated alcohol withdrawal or complications of alcohol withdrawal. (7) A2AAs and beta-blockers Recommendation IV.40: Alpha2-adrenergic agonists (A2AAs) such as clonidine can be used as an adjunct to benzodiazepine therapy to control autonomic hyperactivity and anxiety when symptoms are not controlled by benzodia-zepines alone. They should not be used alone to prevent or treat withdrawal-related seizures or delirium. Recommendation IV.41: Beta-adrenergic antagonists (beta-blockers) can be used as an adjunct to benzodiazepines in select patients for control of persistent hypertension or tachycardia when these signs are not controlled by benzodia-zepines alone. They should not be used to prevent or treat alcohol withdrawal seizures. (8) Inappropriate medications Recommendation IV.42: Oral or intravenous alcohol should not be used for the prevention or treatment of alcohol withdrawal. Recommendation IV.43: There is insufficient evidence to support the use of baclofen for the treatment of alcohol withdrawal. Recommendation IV.44: Providing magnesium as a prophylaxis or treatment for alcohol withdrawal management has no supporting evidence. V. Inpatient Management of Alcohol Withdrawal Recommendations that are appropriate for both Ambu-latory and Inpatient Management are repeated in both sec-tions. A. Monitoring Recommendation V.1: The following monitoring schedule is appropriate: In patients with moderate to severe withdrawal or those requiring pharmacotherapy, re-assess every 1–4 hours for 24 hours, as clinically indicated. Once stabilized (e.g., CIWA-Ar score < 10 for 24 hours), monitoring can be extended to every 4–8 hours for 24 hours, as clinically indicated. Patients with mild withdrawal and low risk of complicated withdrawal may be observed for up to 36 hours, after which more severe withdrawal is unlikely to develop. Recommendation V.2: Monitor patients’ vital signs, hydration, orientation, sleep, and emotional status including suicidal thoughts. Recommendation V.3: Monitor patients receiving pharmacotherapy for alcohol withdrawal for signs of over-sedation and respiratory depression. Recommendation V.4: Signs and symptoms of alcohol withdrawal should be monitored during withdrawal manage-ment with a validated assessment scale (see Appendix III for a summary of scales and their associated features). B. Supportive Care Recommendation V.5: Supportive care is a critical component of alcohol withdrawal management. Frequent reassurance, re-orientation to time and place, and nursing care are recommended non-pharmacological interventions. Providers should ensure patients are educated about what to expect over the course of withdrawal, including common signs and symptoms and how they will be treated. Patients with severe alcohol withdrawal should be cared for in an Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 9 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. evenly lit, quiet room. Patients should be offered hope and the expectation of recovery. Recommendation V.6: Supportive care for alcohol withdrawal patients includes adherence to safety measures and protocols (e.g., assess risk for fall/syncope). If available and applicable, existing institutional/hospital-associated delirium protocols can be used for supportive care of patients with severe alcohol withdrawal. Recommendation V.7: Thiamine should be provided to prevent Wernicke encephalopathy. Intravenous (IV) or intramuscular (IM) administration of thiamine is preferred, in particular for patients with poor nutritional status, malabsorption, or who are known to have severe complications of alcohol withdrawal. Typical dosing is 100 mg IV/IM per day for 3–5 days. Oral thiamine also can also be offered. Patients also receiving glucose can be administered thia-mine and glucose in any order or concurrently. Recommendation V.8: Clinicians should administer thiamine to patients admitted to the Intensive Care Unit (ICU) to treat alcohol withdrawal. Recommendation V.9: For patients with hypomagne-semia, cardiac arrhythmias, electrolyte disturbances, or a previous history of alcohol withdrawal seizures, magnesium should be administered. Recommendation V.10: If phosphorus is < 1 mg/dL, supplementation should be provided. Otherwise, in the case of moderate hypophosphatemia (1-2 mg/dL), correction through proper nutrition is recommended. Recommendation V.11: In patients who are critically ill, folate supplementation may be considered, since chronic alcohol use is associated with hyperhomocysteinemia. C. AUD Treatment Initiation and Engagement Recommendation V.12: The period of alcohol with-drawal management should be used to engage patients with an alcohol use disorder (AUD) with comprehensive treatment. When feasible, AUD treatment should be initiated concur-rently with alcohol withdrawal management as cognitive status permits. If appropriate, clinicians should also offer to initiate pharmacotherapy for AUD as cognitive status permits. Clinicians should explain the range of evidence-based treat-ment services available at the current site and in the commu-nity. Finally, clinicians should proactively connect patients to treatment services as seamlessly as possible, including initi-ating a warm handoff to treatment providers. D. Pharmacotherapy (1) Prophylaxis Recommendation V.13: For patients at risk of develop-ing severe or complicated alcohol withdrawal or complications of alcohol withdrawal, preventative pharmacotherapy should be provided. Benzodiazepines are first-line treatment because of their well-documented effectiveness in reducing the signs and symptoms of withdrawal including the incidence of seizure and delirium.For patientswithacontraindicationfor benzodiazepine use, phenobarbital can be used by providers experienced with its use. In settings with close monitoring, phenobarbital adjunct to benzodiazepines is also appropriate. Recommendation V.14: A front loading regimen is recommended for patients at high risk of severe withdrawal syndrome. Providing at least a single dose of preventative medication is appropriate for patients at lower levels of risk not experiencing significant signs or symptoms but have: A history of severe or complicated withdrawal An acute medical, psychiatric, or surgical illness Severe coronary artery disease Displaying signs or symptoms of withdrawal concurrent with a positive blood alcohol content (2) Withdrawal symptoms Recommendation V.15: For patients experiencing mild alcohol withdrawal (e.g., CIWA-Ar score <10) who are at minimal risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal, pharma-cotherapy or supportive care alone may be provided. If providing medication, benzodiazepines, carbamazepine, or gabapentin are appropriate. For patients with a contraindica-tion for benzodiazepine use, carbamazepine, gabapentin, or phenobarbital (for providers experienced with its use) are appropriate. Carbamazepine, gabapentin, or valproic acid (if no liver disease or childbearing potential) may be used as an adjunct to benzodiazepines. Recommendation V.16: Patients experiencing moderate alcohol withdrawal (e.g., CIWA-Ar scores 10–18) should receive pharmacotherapy. Benzodiazepines are first-line treat-ment. Carbamazepine or gabapentin are appropriate alternatives. For patients with a contraindication for benzodiazepine use, carbamazepine, gabapentin, or phenobarbital (for providers experienced with its use) are appropriate. Carbamazepine, gaba-pentin, or valproic acid (if no liver disease or childbearing potential) may be used as an adjunct to benzodiazepines. Recommendation V.17: Patients experiencing severe alcohol withdrawal (e.g., CIWA-Ar scores 19) should receive pharmacotherapy. Benzodiazepines are first-line treat-ment. For patients with a contraindication for benzodiazepine use, phenobarbital is appropriate for providers experienced with its use. If close monitoring is available, phenobarbital can be used as an adjunct to benzodiazepines. Other adjunct medications can be considered after a clinician ensures that an adequate dose of benzodiazepines has been administered. Recommendation V.18: If a patient’s symptoms are not controlled as expected: First consider increasing the dose If over-sedation or inadequate monitoring is a concern: Reassess for appropriate level of care Consider switching medication If using benzodiazepines, consider adding an adjunct med-ication (3) Benzodiazepine use Recommendation V.19: While no particular benzodi-azepine agent is more effective than another, longer-acting benzodiazepines are the preferred agents due to clinical benefits of their longer duration of action. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 10 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Recommendation V.20: If waiting for lab test results or if the test(s) are unavailable, if a patient has signs of signifi-cant liver disease, use a benzodiazepine with less hepatic metabolization. Recommendation V.21: Clinicians should monitor patients taking benzodiazepines for signs of over-sedation and respiratory depression. Recommendation V.22: A benzodiazepine prescription to treat alcohol withdrawal should be discontinued following treatment. (4) Benzodiazepine dosing regimens Recommendation V.23: Symptom-triggered treatment is the preferred benzodiazepine dosing method. Fixed dosing according to a scheduled taper may be appropriate if symp-tom-triggered treatment cannot be used. Recommendation V.24: Front loading is recommended for patients experiencing severe alcohol withdrawal (e.g., CIWA-Ar scores 19). Diazepam or chlordiazepoxide are preferred agents for front loading. Recommendation V.25: When using a fixed-dose schedule, patients’ signs and symptoms should still be moni-tored, and additional doses of medication provided as needed. Recommendation V.26: If prescribing a shorter-acting benzodiazepine, using a fixed-dose regimen with a gradual taper may be appropriate to reduce the likelihood of break-through and rebound signs and symptoms. (5) Carbamazepine, gabapentin, valproic acid Recommendation V.27: Gabapentin is a favorable choice for treating alcohol withdrawal when a clinician also plans to use it for a patient’s ongoing treatment of alcohol use disorder. Recommendation V.28: If benzodiazepines are contra-indicated, carbamazepine or gabapentin are appropriate alter-natives for patients in mild or moderate withdrawal. Recommendation V.29: Carbamazepine, gabapentin, or valproic acid may be used as an adjunct to benzodiazepine therapy to help control alcohol withdrawal. Before using as an adjunct, clinicians should ensure that an adequate dose of benzodiazepine has been administered. Recommendation V.30: Valproic acid should not be used in patients who have liver disease or women of childbearing potential. Recommendation V.31: There is insufficient evidence to support the use of valproic acid as monotherapy for the treatment of alcohol withdrawal. (6) Phenobarbital Recommendation V.32: Phenobarbital can be used for some patients in inpatient settings; however, it should only be used by clinicians experienced with its use given its narrow therapeutic window and side effects. Recommendation V.32: In an inpatient setting, pheno-barbital monotherapy (managed by a clinician experienced with its use) is appropriate for patients with a contraindication for benzodiazepine use who are experiencing mild, moderate, or severe alcohol withdrawal or who are at risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal. Recommendation V.34: In an inpatient setting, if close monitoring is available, phenobarbital (managed by a clinician experienced with its use) as an adjunct to benzo-diazepines is an option for patients experiencing severe alcohol withdrawal or who are at risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal. Recommendation V.35: Parenteral phenobarbital should only be used in highly supervised settings (e.g., ICU, CCU) because of risk of over-sedation and respiratory depression. (7) A2AAs and beta-blockers Recommendation V.36: Alpha2-adrenergic agonists (AA2 s) such as clonidine and dexmedetomidine can be used as an adjunct to benzodiazepine therapy to control autonomic hyperactivity and anxiety when these signs are not controlled by benzodiazepines alone. They should not be used alone to prevent or treat withdrawal-related seizures or delirium. Recommendation V.37: Beta-adrenergic antagonists (beta-blockers) can be used as an adjunct to benzodiazepines in select patients for control of persistent hypertension or tachycardia when these signs are not controlled by benzodia-zepines alone. They should not be used to prevent or treat alcohol withdrawal seizures. (8) Inappropriate medications Recommendation V.38: Oral or intravenous alcohol should not be used for the prevention or treatment of alcohol withdrawal. Recommendation V.39: There is insufficient evidence to support the use of baclofen for the treatment of alcohol withdrawal. Recommendation V.40: Providing magnesium as a prophylaxis or treatment for alcohol withdrawal management has no supporting evidence. VI. Addressing Complicated Alcohol Withdrawal A. Alcohol Withdrawal Seizure (1) Monitoring Recommendation VI.1: Patients should be monitored for alcohol withdrawal seizures even in the absence of other clinically prominent alcohol withdrawal signs or symptoms. Recommendation VI.2: Following an alcohol with-drawal seizure, patients should be admitted to a setting with close monitoring available, and should be re-assessed every 1-2 hours for 6–24 hours. Patients should be closely monitored for delirium and the need to receive intravenous (IV) fluids, due to potential electrolyte imbalances. (2) Supportive care Recommendation VI.3: If available and applicable, existing institutional/hospital-associated delirium protocols can be used for supportive care of patients with an alcohol withdrawal seizure. (3) Pharmacotherapy Recommendation VI.4: Following a withdrawal sei-zure, patients should be immediately treated with a medica-tion effective at preventing another seizure. Benzodiazepines are first-line treatment, and a fast-acting agent such as loraze-pam or diazepam is preferred. Phenobarbital is also an option but is less preferred to benzodiazepines. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 11 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Recommendation VI.5: Following a withdrawal sei-zure, parenteral administration of medications is preferred. If available, IV administration is preferred to intramuscular (IM), but IM administration is also effective. Parenteral phenobarbital should only be used in highly supervised set-tings (e.g., Intensive Care Unit [ICU], CCU) because of risk of over-sedation and respiratory depression. Recommendation VI.6: It is not recommended to use alpha2-adrenergic agonists or beta-adrenergic antagonists to prevent or treat alcohol withdrawal seizures because they are ineffective for this purpose. Beta-adrenergic antagonists also can lower the seizure threshold. Phenytoin should not be used unless treating a concomitant underlying seizure disorder. B. Alcohol Withdrawal Delirium (1) Monitoring Recommendation VI.7: Patients with alcohol with-drawal delirium should be provided close nursing observation and supportive care, which often necessitates admission to an intensive or critical care unit. Agitated and disoriented patients should have continuous, one-to-one observation and monitoring. Recommendation VI.8: Patients with alcohol with-drawal delirium should have their vital signs, oximetry and cardiac status monitored as frequently as required. Resusci-tative equipment should be readily available when patients require high doses of benzodiazepines, when continuous infusion of medication is used, or when patients have signifi-cant concurrent medical conditions. Recommendation VI.9: To monitor signs and symp-toms of alcohol withdrawal delirium, use a structured assess-ment scale such as the Confusion Assessment Method for ICU Patients (CAM-ICU), Delirium Detection Score (DDS), Rich-mond Agitation-Sedation Scale (RASS), or Minnesota Detox-ification Scale (MINDS). It is not recommended to use the CIWA-Ar in patients with delirium because it relies on patient-reported symptoms. (2) Supportive care Recommendation VI.10: Provide immediate intrave-nous access for administration of drugs and fluids to patients experiencing alcohol withdrawal delirium. Recommendation VI.11: If available and applicable, existing institutional/hospital-associated delirium protocols can be used for supportive care of patients with alcohol withdrawal delirium. Recommendation VI.12: Restraints should only be used when required to prevent injuries due to agitation or violence, and in compliance with state laws. (3) Pharmacotherapy Recommendation VI.13: Patients with alcohol with-drawal delirium should be sedated to achieve and maintain a light somnolence. Benzodiazepines are recommended as the first-line agents for managing alcohol withdrawal delirium. Recommendation VI.14: When available, medication should be administered intravenously. The use of intermittent IV administration of long- and short-acting medications is acceptable and effective. Continuous IV infusion is consider-ably more expensive and there is no evidence of therapeutic superiority. Recommendation VI.15: Patients receiving repeated high intravenous doses of lorazepam or diazepam should be monitored closely for signs of hyponatremia and metabolic acidosis. Recommendation VI.16: When treating alcohol with-drawal delirium, use an established dosing protocol as a guide, but individualize dosing regimens based on patient’s signs and symptoms. It is appropriate for patients with alcohol with-drawal delirium to receive intravenous symptom-triggered or fixed-dose front loading. Once light somnolence is achieved and patients are calm and cooperative, if on IV medication, shifting to oral symptom-triggered treatment is recommended. Recommendation VI.17: Very large doses of benzo-diazepines may be needed to control agitation in alcohol withdrawal delirium, including doses that are much higher than typically seen in other patient populations. Clinicians should not hesitate to provide such large doses to patients to control agitation but should keep in mind the possible risk of over-sedation and respiratory depression. Moreover, when large doses are used, there is risk of accumulation of long-acting benzodiazepine metabolites, especially in patients with impaired hepatic function or the elderly, and patients should be monitored closely. Recommendation VI.18: For patients who have been delirious longer than 72 hours, assess for drug-induced delir-ium and withdrawal from another GABAergic agent (such as gabapentin or carisoprodol). When necessary, adjust the benzodiazepine dose. Recommendation VI.19: Barbiturates can be consid-ered an alternative option to benzodiazepines for the treatment of alcohol withdrawal delirium, but they are not preferred to benzodiazepines. Phenobarbital can be used as an adjunct to benzodiazepines in settings with close monitoring when alcohol withdrawal delirium is not adequately controlled by benzodiazepine therapy alone. Recommendation VI.20: Antipsychotic agents can be used as an adjunct to benzodiazepines when alcohol with-drawal delirium and hallucinations are not adequately con-trolled by benzodiazepine therapy alone. They are not recommended as monotherapy for alcohol withdrawal delir-ium. Recommendation VI.21: Alpha2-adrenergic agonists, beta-adrenergic antagonists and paraldehyde should not be used to treat alcohol withdrawal delirium. C. Alcohol-Induced Psychotic Disorder Recommendation VI.22: If available and applicable, existing institutional/hospital-associated delirium protocols can be used for supportive care of patients with an alcohol-induced psychotic disorder. Recommendation VI.23: The treatment of alcohol-induced psychotic disorder may require consultation with a psychiatrist. Recommendation VI.24: The treatment of alcohol-induced psychotic disorder may require addition of antipsychotics. Recommendation VI.25: For patients experiencing hallucinations, diazepam may be considered a treatment option. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 12 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. D. Resistant Alcohol Withdrawal Recommendation VI.26: If available and applicable, existing institutional/hospital-associated delirium protocols can be used for supportive care of patients with resistant alcohol withdrawal. Recommendation VI.27: Phenobarbital may be used as an adjunct to benzodiazepines to control resistant alcohol withdrawal syndrome in settings with close monitoring. Recommendation VI.28: Propofol may be used with patients in the ICU experiencing resistant alcohol withdrawal who already require mechanical ventilation. Recommendation VI.29: Dexmedetomidine may be used with patients in the ICU experiencing resistant alcohol withdrawal. VII. Specific Settings and Populations A. Primary Care Recommendation VII.1: If patients are experiencing severe withdrawal (e.g., CIWA-Ar scores 19), refer them directly to the nearest Emergency Department. Recommendation VII.2: If withdrawal is mild (e.g., CIWA-Ar <10), patients presenting to primary care can be prescribed a few doses of benzodiazepine. Whenever possible, medication can be supervised by a caregiver at home or staff at a nonmedical withdrawal management center. Do not pre-scribe medication to patients for ambulatory management of alcohol withdrawal without performing an adequate assess-ment or to patients without adequate support. Recommendation VII.3: If withdrawal does not resolve (e.g., fall below a CIWA-Ar score of 10) after an adequate dose of medication (e.g., 80 mg diazepam), or patients appears sedated, transfer to an Emergency Depart-ment or other inpatient withdrawal management setting. Recommendation VII.4: For patients treated in pri-mary care settings, regular follow-up visits, at least monthly for one year, could increase the likelihood of sustained recovery. B. Emergency Departments Recommendation VII.5: If patients are experiencing severe alcohol withdrawal (e.g., CIWA-Ar 19), or are at risk of complicated withdrawal, administer medication immedi-ately to treat withdrawal and reduce the risk of seizures and delirium. Recommendation VII.6: Patients presenting with alco-hol withdrawal syndrome in the Emergency Department should be evaluated for delirium as well as other conditions that mimic and/or accompany withdrawal. Patients presenting with delirium should be assessed for all potential etiologies including alcohol withdrawal. Recommendation VII.7: Patients in the Emergency Department should receive a complete blood count and complete metabolic panel including liver enzyme and mag-nesium tests; alcohol withdrawal treatment should not be delayed while waiting for results. Recommendation VII.8: The following indicators should be present for discharge to an ambulatory alcohol withdrawal management setting from the Emergency Depart-ment: Mild alcohol withdrawal (e.g., CIWA-Ar score <10). Moderate alcohol withdrawal (e.g., CIWA-Ar score 10– 18) with no other complicating factors Not currently intoxicated (including alcohol or other drugs) No history of complicated alcohol withdrawal (seizures, delirium) No significant medical or psychiatric comorbidities that would complicate withdrawal management Able to comply with ambulatory visits and therapy Recommendation VII.9: Patients with controlled with-drawal syndrome being discharged from the Emergency Department may be offered a short term (e.g., 1-2 day) prescription for appropriate alcohol withdrawal medication to last until follow-up with their healthcare provider. C. Hospitalized Patients (1) Identification Recommendation VII.10: All patients admitted to the hospital should be screened for risk of alcohol withdrawal. Among hospitalized patients, the Alcohol Use Disorders Identification Test (AUDIT) and Alcohol Use Disorders Identification Test-Piccinelli Consumption (AUDIT-PC) can indicate risk of developing alcohol withdrawal. Recommendation VII.11: Patients undergoing elective surgery should be screened for unhealthy alcohol use and the need to undergo alcohol withdrawal management before proceeding with surgery. Patients undergoing elective surgery who are at risk of alcohol withdrawal should undergo medi-cally managed withdrawal before proceeding with surgery (2) Assessment Recommendation VII.12: Among hospitalized patients, the Prediction of Alcohol Withdrawal Severity Scale (PAWSS) can be used for predicting risk of developing severe or complicated alcohol withdrawal in the medically ill. Recommendation VII.13: Patients for whom alcohol withdrawal is suspected and for whom a complete medical history is not available, (i.e., are admitted from the Emergency Department or trauma unit, are in Intensive Care Unit [ICU]), or who are known to be at high risk of complicated alcohol withdrawal, medical decisions should be oriented toward a more aggressive treatment of alcohol withdrawal regardless of presenting signs and symptoms. Recommendation VII.14: For patients who require more than standard amounts of medication to manage alcohol withdrawal, individualized assessment by clinicians experi-enced in the management of withdrawal is recommended. The medication and protocol used for treating other con-ditions and/or alcohol withdrawal syndrome may need to be modified. (3) Monitoring Recommendation VII.15: In patients who are hospi-talized, monitor their vital signs. Fluid intake and output and serum electrolytes should be monitored as clinically indi-cated. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 13 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Recommendation VII.16: Signs and symptoms of alcohol withdrawal should be monitored during the course of withdrawal with a validated symptom assessment scale. Assess the risk for scores on a symptom assessment scale to be confounded by the use of certain medications, the presence of certain medical conditions (e.g. fever from infection), or a patient’s difficulty communicating. Among general medical/ surgical patients, low withdrawal scores can typically be interpreted with confidence, while high scores should be interpreted with caution. The use of alternative scales with patients with difficulty communicating is appropriate. Recommendation VII.17: Patients with a reduced level of consciousness who are at risk for the development of alcohol withdrawal should be monitored for the appearance of alcohol withdrawal signs. If a co-occurring clinical condi-tion worsens, do not assume it is related to alcohol withdrawal among alcohol withdrawal patients. However, immediate treatment is required if alcohol withdrawal develops after surgery or trauma. (4) Supportive care Recommendation VII.18: Clinicians should adminis-ter thiamine to ICU patients with signs or symptoms that mimic or mask Wernicke encephalopathy. (5) Pharmacotherapy Recommendation VII.19: Prophylactic treatment of alcohol withdrawal should be provided in the ICU to patients who are suspected to be physiologically dependent on alcohol. Recommendation VII.20: Implementing an alcohol withdrawal management protocol in the ICU is appropriate. When using a symptom-triggered dosing protocol, use a validated scale to monitor signs and symptoms. For patients being treated in ICU settings for alcohol withdrawal, existing scales that are appropriate to use for monitoring withdrawal include the Richmond Agitation-Sedation Scale (RASS). Administration of medications via the intravenous route is preferred because of the rapid onset of action and more predictable bioavailability. D. Patients With Medical Conditions Recommendation VII.21: For patients with medical comorbidities, modify the medication and/or protocol used for treating alcohol withdrawal syndrome as necessary in consul-tation with other specialists. Recommendation VII.22: For patients with medical conditions that prevent the use of oral medication, provide intravenous or intramuscular medications as necessary. Recommendation VII.23: Aggressive withdrawal treatment is indicated for patients with cardiovascular disor-ders due to risk of harm associated with autonomic hyperac-tivity. Recommendation VII.24: For patients with a medical condition associated with impaired hepatic function, adjust medication dose or use medications with less dependence on hepatic metabolism. E. Patients who Take Opioids Recommendation VII.25: Patients who are on chronic opioid medication (opioid agonist therapy for opioid use disorder or pain) should be monitored closely when benzodiazepines are prescribed, due to the increased risk of respiratory depression. Similarly, patients taking sedative-hypnotic medications exhibit tolerance to benzodiazepines and should be monitored closely for appropriate dose. Recommendation VII.26: For patients with concomi-tant alcohol withdrawal and opioid use disorder, stabilize opioid use disorder (e.g., with methadone or buprenorphine) concomitantly with treating alcohol withdrawal. F. Patients who are Pregnant (1) Level of care and monitoring Recommendation VII.27: Inpatient treatment should be considered for all pregnant patients with alcohol use disorder who require withdrawal management. Inpatient treat-ment should be offered to pregnant patients with at least moderate alcohol withdrawal (i.e., CIWA-Ar scores 10). Recommendation VII.28: The CIWA-Ar is an appro-priate symptom assessment scale to use with pregnant patients. Clinicians should consider signs and symptoms such as nausea, headache, anxiety, and insomnia to be connected to alcohol withdrawal rather than pregnancy that will abate once the alcohol withdrawal has been effectively treated. Recommendation VII.29: During withdrawal manage-ment, consult with an obstetrician. (2) AUD treatment initiation and engagement Recommendation VII.30: Engagement in treatment for AUD is particularly important for pregnant patients with alcohol withdrawal given the risk of Fetal Alcohol Spectrum Disorder (FASD) including Fetal Alcohol Syndrome (FAS).) (3) Pharmacotherapy Recommendation VII.31: Before giving any medica-tions to pregnant patients, ensure that patients understand the risks and benefits of the medication, both for the patient and the developing fetus. Recommendation VII.32: Benzodiazepines and barbi-turates are the medications of choice in treatment of pregnant patients with alcohol withdrawal. While there is a risk of teratogenicity during the first trimester, the risks appear small, and they are balanced in view of the risk for fetal alcohol spectrum disorder and consequences to mother and fetus should severe maternal alcohol withdrawal develop. Recommendation VII.33: Due to the high teratogenic risk, valproic acid is not recommended for pregnant patients. Recommendation VII.34: For patients at risk for pre-term delivery or in the late third trimester, use of a short-acting benzodiazepine is recommended. This minimizes the risk for neonatal benzodiazepine intoxication given shorter onset and duration of action. (4) Newborn considerations Recommendation VII.35: In cases of alcohol with-drawal treated close to delivery, assess the newborn for benzo-diazepine intoxication, sedative withdrawal, and Spectrum Disorder (FASD) including Fetal Alcohol Syndrome (FAS). Recommendation VII.36: Inform pregnant patients of all wraparound services that will assist them in addressing newborn needs, including food, shelter, pediatric clinics for inoculations, as well as programs that will help with develop-mental or physical issues that the newborn may experience as a result of in-utero substance exposure. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 14 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Recommendation VII.37: Licensed clinical staff have an obligation to understand and follow their state laws regard-ing definitions of child abuse and neglect, reporting require-ments, and plans of safe care for newborns with in-utero alcohol exposure. INTRODUCTION I. Purpose The American Society of Addiction Medicine (ASAM) developed this Guideline on Alcohol Withdrawal Manage-ment to provide updated information on evidence-based strat-egies and standards of care for alcohol withdrawal management in both ambulatory and inpatient settings. II. Background Alcohol is responsible for a multitude of health con-ditions, including Alcohol Use Disorder (AUD) and alcohol withdrawal. Individuals physically dependent on alcohol may experience signs and symptoms of alcohol withdrawal upon cessation of (or reduction in) alcohol use, due to the sudden reversal of depressant effects on brain function. Signs and symptoms of alcohol withdrawal include anxiety, sleep dis-turbance, headache, nausea, hallucinations, delirium, and seizures. Clinical signs include sweating, elevated blood pressure, tachycardia, hyperthermia, and hyperactive reflexes. Hallucinations can range from mild perceptual distortions to frank hallucinations with a lack of insight. The most severe consequences of alcohol withdrawal include seizure, delir-ium, and death. Patients with alcohol withdrawal frequently present in specialty addiction treatment settings and general medical settings. Patients experiencing or at risk for developing alco-hol withdrawal also present in hospitals, emergency depart-ments, and primary care settings. An estimated 2–7% of patients with heavy alcohol use admitted to the hospital will develop moderate to severe alcohol withdrawal.16 Addition-ally, an estimated 32% of emergency department visits are alcohol related.17 Many of these patients will develop alcohol withdrawal during their emergency department stay. There is an extensive body of research on the manage-ment of alcohol withdrawal, much of which has focused on pharmacotherapy. However, due to the evolution of research evidence and clinical practice, questions continue to emerge about the appropriate management of patients with alcohol withdrawal. For example, although benzodiazepines have long been considered the mainstay of alcohol withdrawal treatment, research on other agents such as anticonvulsants have raised clinical questions about alternatives or adjuncts to benzodiazepines. Similarly, although the Clinical Instrument Withdrawal Assessment for Alcohol, Revised (CIWA-Ar) has long been considered the standard assessment scale for patients with alcohol withdrawal, several other instruments have been developed, raising questions about the value of a given instrument as compared to the others. Finally, although researchers have primarily focused on alcohol withdrawal management in inpatient settings, clinical practice has evolved and treatment in outpatient settings has become increasingly common. Therefore, numerous clinical questions have emerged about which patients are appropriate for ambu-latory alcohol withdrawal management as well as how to tailor treatment interventions to specific settings. A. Need for a New Guideline In June 2017, the American Society of Addiction Medicine’s (ASAM) Quality Improvement Council (QIC) elected to update ASAM’s clinical guidelines on alcohol withdrawal management based on several factors. First, ASAM conducted an Educational Needs Assessment in 2016 that showed a strong interest and need for education on withdrawal management. Second, updated QIC policies recommend that all ASAM guidelines should be updated every five years. ASAM’s previous guidelines on the topic of alcohol withdrawal management were published in 1997 and 2004. The first guideline, ‘‘Pharmacological Management of Alcohol Withdrawal’’13 was published in JAMA, followed five years later with the most recent guideline entitled ‘‘Man-agement of Alcohol Withdrawal Delirium’’14 in JAMA, for-merly Archives of Internal Medicine. Subsequent guidelines have not been written since the 2004 guidelines thus an update was due. Third, the American Psychiatric Association (APA) released a guideline on medications to treat alcohol use disorder that does not cover withdrawal management.15 An ASAM guideline on alcohol withdrawal should complement APA’s guideline. Fourth, outreach to other organizations indicated that other organizations are not planning to create a guideline on alcohol withdrawal. Although alcohol withdrawal has been recognized for centuries and effective treatment strategies have been researched for decades, questions remain about effective approaches to treatment in specialty and non-specialty set-tings. At the outset of the guideline development process, ASAM identified several practice concerns related to alcohol withdrawal treatment: 1. Uncertainty about the CIWA-Ar, which is the most wide-spread symptom monitoring instrument but may not fit all patient populations and settings 2. Excessive caution about the use of benzodiazepines to treat alcohol withdrawal, which have been shown to prevent seizures and delirium 3. The use of barbiturates, which have a much narrower therapeutic window than benzodiazepines 4. Inconsistent treatment practices in non-specialty settings The new clinical guideline is intended to address some of these current practice concerns and provide clear guidance that will lead to more consistent treatment practices in the field. B. Previous ASAM Guidelines This clinical practice guideline will replace the two previous ASAM guidelines related to alcohol withdrawal management, ‘‘Pharmacological Management of Alcohol Withdrawal’’13 in 1997 and ‘‘Management of Alcohol With-drawal Delirium’’14 in 2004. The 1997 guideline was based on a literature review con-ducted in 1995 and was primarily focused on pharmacotherapy, Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 15 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. with minimal attention to psychosocial treatment and consider-ations for various settings and levels of care. The 2004 guideline focuses specifically on Alcohol Withdrawal Delirium. This includedareviewandmeta-analysisofnineprospectivecontrolled trials published through 2001. C. Additional ASAM Guidelines and Standards ASAM has produced several other documents that provide guidance on the management of alcohol withdrawal, the most relevant of which are The ASAM Criteria,12 Prin-ciples of Addiction Medicine,18 and the ASAM Standards of Care.19 The ASAM Criteria provides comprehensive guidance on withdrawal management, specifically addressing alcohol withdrawal, including clear instruction for assessing and determining the patient’s level of risk, matching patients to the appropriate level of care, and the service characteristics that should be present each level of care for withdrawal management. Principles of Addiction Medicine contains a chapter titled ‘‘Management of Alcohol Intoxication and With-drawal,’’ which reviews the clinical presentation and man-agement of alcohol intoxication and withdrawal. The ASAM Standards of Care provides a list of prin-ciples for Addiction Specialist Physicians to follow in order to support quality improvement activities and improve patient outcomes. The Standards ‘‘outline a minimum standard of physician performance and should not be construed as describing the totality of care that a person with addiction might require.’’19(p 5) The Standards help physicians identify their clinical and administrative roles to improve overall functioning and well-being of patients, while integrating addiction treatment into the larger healthcare system. Stand-ards are organized by six performance measure domains. One of the six domains includes withdrawal management. III. Scope of Guideline While the current clinical guideline focuses primarily on alcohol withdrawal management, it is important to under-score that alcohol withdrawal management alone is not an effective treatment for alcohol use disorder. Withdrawal management should not be conceptualized as a discrete clinical service, but rather as a component in the process of initiating and engaging patients in treatment for alcohol use disorder. IV. Intended Audience The intended audience of this guideline is clinicians, mainly physicians, nurse practitioners, and physician assis-tants, who provide alcohol withdrawal management in spe-cialty and non-specialty addiction treatment settings (including primary care and emergency departments, inten-sive care and surgery units in hospitals). The guideline will also have utility for administrators, insurers, and policy-makers. V. Qualifying Statement This ASAM Alcohol Withdrawal Management Guide-line is intended to aid clinicians in their clinical decision making and patient management. The Guideline strives to identify and define clinical decision making junctures that meet the needs of most patients in most circumstances. Clinical decision making should involve consideration of the quality and availability of expertise and services in the community wherein care is provided. In circumstances in which the Guideline is being used as the basis for regulatory or payer decisions, improvement in quality of care should be the goal. Finally, courses of treatment contained in recommen-dations in this Guideline are effective only if the recommen-dations, as outlined, are followed. Because lack of patient understanding and adherence may adversely affect outcomes, clinicians should make every effort to promote the patient’s understanding of, and adherence to, prescribed and recom-mended treatments. Patients should be informed of the risks, benefits, and alternatives to a particular treatment, and should be an active party in shared decision making whenever feasible. Recommendations in this Practice Guideline do not supersede any federal or state regulations. VI. Special Terms Different terms have been used to describe various aspects and management methods of acohol withdrawal. Below are terms that are used throughout the guideline used to convey a specific meaning for the purposes of this guideline. Alcohol Hallucinosis/Alcohol-induced Psychotic Dis-order: Hallucinations that are not associated with alcohol withdrawal delirium and which can occur in the absence of other clinically prominent withdrawal signs and symptoms. Hallucinosis is characterized primarily by auditory halluci-nations, paranoid symptoms and fear. Hallucinations occur in clear consciousness, are generally third person auditory hal-lucinations, and often derogatory. There may be secondary delusions or perseveration as well.20 It is unclear if alcohol hallucinosis is part of alcohol withdrawal or if the halluci-nations are a complication of chronic alcohol use unrelated to withdrawal. Currently, alcohol hallucinosis is diagnosed as alcohol-induced psychotic disorder in the Diagnostic and Statistical Manual 5 (DSM-5). Ambulatory Withdrawal Management: Withdrawal management that occurs in outpatient settings, including primary care and intensive outpatient/day hospital settings. Level of clinical expertise and frequency of monitoring vary widely within various ambulatory withdrawal management settings. Delirium and seizure: Unless otherwise specified, in this document these refer to alcohol withdrawal-related sei-zure or alcohol withdrawal delirium. Alcohol withdrawal delirium has replaced the formerly used ‘‘delirium tremens.’’ Dosing regimens: Different terms have been used to describe the many variations in dosing regimens used in alcohol withdrawal management. This document focuses on the following regimen types (see Appendix V for specific examples): Symptom-triggered dosing: An approach whereby patients are monitored through the use of a structured assessment scale and given medication only when symptoms cross a ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 16 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. threshold of severity (e.g., 25–100 mg chlordiazepoxide if CIWA-Ar score 10). Symptom-triggered dosing can be further refined by giving a different dose amount depend-ing on the individual’s score (e.g., 15 mg oxazepam for CIWA-Ar scores 8–15, 30 mg oxazepam for CIWA-Ar >15). The score can also determine the frequency of reassessment and further dosing. Fixed dosing: In a fixed-dose regimen, a predetermined dose is administered at fixed intervals according to a schedule. Doses usually decrease in a gradual taper over several days. A fixed-dose schedule can be refined by choosing an initial dose according to withdrawal severity as assessed by a withdrawal symptom severity scale.21 When fixed-doses are given, allowances should be made to provide additional medication if the fixed-dose should prove inadequate to control symptoms. Front loading: An approach to dosing wherein moderate-to-high doses of a long-acting agent (e.g., 20 mg of diaze-pam) are given frequently at the outset of treatment to achieve rapid control of withdrawal signs and symptoms. The medication level is allowed to taper through metabo-lism. Front loading can be driven by a symptom assessment scale (e.g., 20 mg of diazepam every hour until CIWA-Ar scores <10) or a fixed-dosing schedule (e.g., 20 mg of diazepam every hour for 1-2 hours or until patient is sedated). Inpatient Withdrawal Management: Alcohol with-drawal management that occurs in inpatient settings, includ-ing hospitals. The defining feature of inpatient settings for the purposes of this document is that patients are on site 24/7. Level of clinical expertise and frequency of monitoring vary widely within various inpatient withdrawal management set-tings. For the purposes of this document, residential facilities without continual medical monitoring are considered inpatient settings. Level of care (LOC): Used in this guideline to describe different settings for the management of alcohol withdrawal, based on the definitions laid out in The ASAM Criteria.12 The ASAM Criteria defines specific levels of care for alcohol withdrawal management as follows: Level 1-WM: Ambulatory withdrawal management without extended on-site monitoring Level 2-WM: Ambulatory withdrawal management with extended on-site monitoring Level 3.2-WM: Clinically managed residential withdrawal management Level 3.7-WM: Medically monitored inpatient withdrawal management Level 4-WM: Medically managed intensive inpatient withdrawal management However, this guideline also uses two broad categories to describe settings where the management of alcohol with-drawal may take place. The first is an ambulatory level of care, which encompasses Level 1-WM and Level 2-WM. The second is an inpatient level of care, which encompasses Level 3-WM and Level 4-WM. Inpatient care also includes hospital settings. There is considerable variation in the staffing and resource availability within these two broad categories, which clinicians should consider when applying this guideline to their specific treatment setting. Resistant alcohol withdrawal (RAW): Used in this guideline to describe patients experiencing severe or compli-cated alcohol withdrawal signs and symptoms despite having received high doses of benzodiazepines. There is not yet agreement in the field regarding the precise amount of benzodiazepines required before considering a patient to be in RAW, but various studies have used the cutoff of 200 mg diazepam in 4 hours,22 40 mg intravenous diazepam in 1 hour,22 or 50 mg intravenous diazepam in 1 hour.23 This phenomenon is also referred to as Refractory Alcohol With-drawal, Benzodiazepine-Resistant Alcohol Withdrawal and Treatment-Resistant Alcohol Withdrawal in other sources. Severe or complicated alcohol withdrawal or compli-cations of alcohol withdrawal: These terms are used inde-pendently or jointly in this guideline to describe certain signs and symptoms and/or risks associated with alcohol with-drawal that are most harmful to patients. They are defined as: Complicated alcohol withdrawal: The development of alcohol withdrawal-related seizures or alcohol withdrawal delirium Severe alcohol withdrawal: Severe but not complicated signs and symptoms of alcohol Complications of alcohol withdrawal: Alcohol withdrawal signs and symptoms’ potentially life-threatening exacer-bation of existing medical or psychiatric conditions Withdrawal severity: In this guideline, withdrawal severity is categorized in Table 1. APPROACH AND METHODOLOGY I. Overview of Approach In order to develop a comprehensive practice guideline focused on alcohol withdrawal management, we utilized a hybrid of established methodologies. In order to develop the scope of the guideline and draft the guideline statements, we followed the Veterans Health Administration and Department of Defense (VA/DoD) Guideline for Guidelines. To rate and refine the draft guidelines, we used the RAND/UCLA Appro-priateness Method (RAM), which is a specific process for combining the available scientific evidence with the clinical judgment of experts. Quality of the literature reviewed was rated using standardized rating scales and methodology. The external review process was informed by the VA/DoD method. ASAM’s Quality Improvement Council (QIC) was the oversight committee during the development of the alcohol withdrawal management guideline. The QIC originally chose two Clinical Champions to have a key role in accordance with the VA/DoD model of clinical practice guideline develop-ment. An additional two Clinical Champions were added to the project to represent ambulatory settings. The Clinical Champions have a deep knowledge of alcohol withdrawal management and a familiarity with the clinical language and decision making processes involved in this procedure. Addi-tionally, the QIC chose a nine-member Guideline Committee to rate guideline statements. Panel members were selected to Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 17 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. represent a diverse spectrum of clinical practitioners who manage alcohol withdrawal. The QIC also recruited a Guide-line Committee Moderator to act as a liaison between the Guideline Committee members and the project team and to lead the discussion during an in-person meeting of the Guideline Committee. In selecting the panel members, the QIC made every effort to avoid actual, potential, or perceived conflicts of interest that might arise as a result of relationships with industry and other entities among members of the project personnel. All QIC members, Guideline Committee members, and external reviewers of the document were required to disclose all current related relationships, which are summa-rized in Appendix VII, II. Develop the Scope and Key Questions The QIC was responsible for identifying the guideline scope and intended audience. The Clinical Champions refined the scope by identifying the key clinical questions of greatest importance to the management of alcohol withdrawal. The key questions followed the Population, Intervention, Compar-ison, Outcome, Time and Setting (PICOTS) framework estab-lished by the AHRQ.24 Indicators of interest in the PICOTS model are listed below: Population – The target population was adults 18 years or older with a diagnosis of alcohol withdrawal with or without other health conditions. The management of these other conditions, outside of identification and routine prophylaxis in the context of alcohol withdrawal, were not included, such as alcoholic liver disease and Wernicke encephalopathy. Intervention – Pharmacological and non-pharmacological interventions were included. Pharmacotherapies that are not widely available in the United States were excluded (e.g., sodium oxybate [GHB], cannabinoids, chlomethiazole). Off-label medications for alcohol withdrawal management were included. Non-pharmacological interventions included supportive care, nutritional correction, and symptom moni-toring and assessment frequency. Comparison – All comparative interventions were included if they met criteria for an included intervention. Outcome – Outcomes of interest were those clinical out-comes most consequential and immediate to withdrawal including severity of withdrawal syndrome; treatment completion; transfer to more intensive level of care; inci-dence of seizure, delirium, death and adverse events; and linkage to long-term AUD treatment. Time – The duration of time of interest was 5 days from the start of withdrawal. Post-acute prolonged withdrawal or protracted withdrawal was not included. The Clinical Champions identified protracted withdrawal and benzodi-azepine-resistant withdrawal as an area that should be included in the future. Setting – All clinical settings were included except for home management of withdrawal unless it took place in the United States. After a face-to-face meeting of the Guideline Commit-tee, feedback indicated that settings and levels of care had not been adequately delineated in the initial set of draft statements. This was largely due to the sparse literature specific to ambulatory settings and the focus of our Clinical Champions on the more moderate-severe end of the spectrum of alcohol withdrawal. Therefore, after the initial Guideline Committee Meeting, the project was expanded to place additional focus on considerations specific to alcohol with-drawal management in ambulatory settings. The expanded literature review and drafting of additional statements partic-ular to ambulatory settings are described below. III. Conduct a Literature Review A systematic literature review including the indicators identified by the Clinical Champions was conducted. The literature review included all levels of published research literature, including studies with non-random assignment and case studies. A targeted internet search of gray literature was also conducted, including published and unpublished clinical guidelines on alcohol withdrawal management. Procedures for review of the academic literature fol-lowed PRISMA guidelines for systematic reviews.25 Articles were identified through searches conducted in four biblio-graphic databases using pre-defined search terms and selec-tion criteria. Additional articles were identified through forward and reverse citation search of key articles. All data-bases were searched uniquely. Searches were conducted for the time-period January 2012 to October 2017 using the following key terms: ‘‘alcohol withdrawal‘‘ or ‘‘delirium tremens’’ or ‘‘alcohol-induced hallucinosis’’ or ‘‘alcohol-induced psychotic disor-der.’’ These terms also captured studies on alcohol withdrawal delirium and alcohol withdrawal seizure. Because clinical management encompasses topics from diagnosis to treatment, we did not include search terms for management and instead relied on the screening process to parse useful from peripheral sources. The databases searched were EBSCOhost Medline, Embase, Web of Science Core Collection, and Cumulative Index to Nursing and Allied Health Literature (CINAHL). Searches targeted all text fields and were restricted to avail-ability in English and to human participants where available (Medline and Embase). If an article contained a secondary analysis of data from a relevant study, the primary source was included. 2,038 unique records were found. Results of the key term search are documented in Appendix II. In addition to the systematic search, targeted title and abstract searches were conducted without a time-period limi-tation on key clinical questions identified by the Clinical Champions. These topics included: withdrawal symptom severity rating scales, comparison of benzodiazepine dose regimens, comparisons among benzodiazepines, comparison of benzodiazepines to anticonvulsants and barbiturates. An additional 70 records were identified. This method was also used to conduct a targeted search of ambulatory management of alcohol withdrawal. In addition to the scientific literature search, we con-ducted an internet search for published clinical guidelines or appropriateness statements on alcohol withdrawal manage-ment across settings following the IOM process for searching gray literature. The following websites were searched using the on-site search engines with the search terms ‘‘alcohol’’ ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 18 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. and ‘‘substance abuse’’: SAMHSA, VA, WHO, AHRQ, Mich-igan Quality Improvement Consortium. This search was not time-limited, but where recommending bodies had published updates of guidelines, only the most recent was included. The search yielded 115 records, 11 of which were screened for inclusion. The full search procedure is documented in Appendix II. Two independent reviewers screened article abstracts and the full text of articles for inclusion. Articles were included if they were about the clinical management of patients with or at immediate risk for developing alcohol withdrawal syndrome. Reasons for exclusion are documented in Appendix II. The quality of the evidence represented by each research article was rated by two independent reviewers; systematic reviews and other qualitative articles were rated by one reviewer. Comparative trials were evaluated using the Cochrane Risk of Bias tool. Systematic reviews were rated using the AMSTAR-2. Other qualitative articles were evalu-ated using the AACODS Checklist for Grey Literature. Study methods and results were extracted by two independent reviewers. A document summarizing the findings of the literature review and the quality of sources used was prepared for the Guideline Committee Members to refer to during the statement rating process. Sources were included in the sum-mary document if they were randomized controlled trials (RCT)s, systematic reviews of RCTs, or guidelines based on systematic reviews. In the absence of such evidence, lower quality evidence sources were included. IV. Develop Draft Guideline Statements In order to develop the draft statements, a meeting was held with the project team, Clinical Champions, and ASAM/ QIC representatives. The list of statements identified the different combinations of clinical indicators in various clinical situations seen in alcohol withdrawal management. A list of definitions for terms used in the statements was also developed. V. Conduct Panel Ratings The RAM method involves multiple rounds of rating and a face-to-face meeting between the project team and Guideline Committee. The first round of ratings was con-ducted remotely. Members of the committee received rating instructions, background material, and the list of potential guideline statements in electronic form. Committee members were asked to consider the appropriateness of each statement individually on a 1–9 scale using the literature review and evidence tables as well as their own best clinical judgment. Shortly after members of the Guideline Committee received rating materials, the Guideline Committee Modera-tor contacted each member individually to gather feedback about the guideline which could not be well captured within the rating form. This opportunity was to seek comments on the general structure and organization of the guideline as well as suggested modifications. Returned Guideline Committee ratings were aggregated and analyzed by IRETA staff. The RAM offers specific guidance for the analysis and classification of guideline statements: a statement is deemed appropriate if the median rating is in the 7–9 range, and no more than one-third of the experts rate the statement outside that range. A statement is deemed inappropriate if the median rating is in the 1–3 range and no more than one-third of the committee rate outside this range. All other statements (those with a median rating of 4–6 or with at least one-third of the experts rating the statement outside the median range) are labeled uncertain. A two-day in-person Guideline Committee meeting took place in the D.C. area. Prior to this meeting, committee members received the list of guideline statements with the Round 1 rating results indicated for each statement; their own rating, the group median rating, and the frequency of each rating response. Discussion was led by the Guideline Com-mittee Moderator and focused on statements labeled uncer-tain. The discussion aimed to identify whether the rating results reflected true uncertainty or disagreement in the field versus confusion about the statement’s meaning. Qualitative feedback from the Round 1 ratings and individual feedback from IRETA’s personal contacts also informed the Guideline Committee meeting discussion, in accordance with the RAM. Statements could be rewritten if the uncertainty was found to be due to confusion. New statements could also be drafted if any important clinical aspects were found to be missing by the Guideline Committee. The second round of ratings was conducted remotely soon after the meeting. The list of uncertain statements, with the addition of new statements suggested during the meeting, were delivered in electronic form to the committee members. The committee members rated the guideline statements using the same criteria as the first round, considering the appropri-ateness of each statement. This second round of ratings were then aggregated and analyzed by IRETA staff. One Guideline Committee member dropped out of participating after the Guideline Committee meeting. This necessitated finding a new method of identifying agreement that does not rely on group sizes that are multiples of three. The RAM manual recommends alternatives, and the Inter-percentile Range Adjusted for Symmetry (IPRAS) method was used for the remainder of the project. At this point, the project expansion took place. The other parts of the project were paused, while the project team conducted an expanded literature review focused on ambula-tory considerations in alcohol withdrawal management. An additional two Clinical Champions representing ambulatory settings were recruited and new statements were drafted and rated in two rounds. A second meeting of the Guideline Committee was held remotely via webinar. The project expan-sion started in August 2018, and by May 2019, we were able to return to the original (although modified) timeline. In a third round of ratings, committee members rated the agreed-upon appropriate statements from Rounds 1 and 2 on a 1–9 scale using the more stringent criterion of necessity. Appropriateness refers to procedures where the health bene-fits sufficiently outweigh potential harms such that the pro-cedure is worth doing. Necessity refers to procedures that must be offered to patients fitting a particular clinical descrip-tion, where it would be improper not to offer the procedure given the magnitude and likelihood of the expected benefit to Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 19 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. the patient. A statement is deemed necessary if the median rating is in the 7–9 range with agreement according to IPRAS. Statements that do not meet these criteria are deemed appropriate but not necessary. The full statement rating table can be found in Appendix VI, A193. VI. Drafting the Guideline Document Recommendations were drafted by the project team by combining the statements identified as clinically appropriate by the Guideline Committee. Recommendations are accom-panied by a brief discussion of the evidence or rationale for the statement. ASAM’s two prior alcohol withdrawal guide-lines were used as an initial framework for the guideline. This first draft of the guideline was reviewed by the Clinical Champions, Guideline Committee Moderator and Guideline Committee Members to ensure content clarity and logical flow of the guideline. A second draft was produced based on this feedback. During an external review process, ASAM requested feedback on the second draft guideline via email to the ASAM listserv and also posted the draft for public comment on the ASAM website. At the end of the review period, ASAM aggregated the feedback, identified key issues raised, and tracked proposed changes. A two-day in-person meeting including ASAM staff, QIC representatives, the Guideline Committee Chair IRETA took place in Pittsburgh, PA to discuss all of the external review feedback and proposed edits. Feedback was incorporated as appro-priate in discussion with those in attendance and in accor-dance with the evidence. IRETA then produced the Final Guideline Document. RECOMMENDATIONS I. Identification and Diagnosis of Alcohol Withdrawal A. Identification Recommendation I.1: Incorporate universal screening for unhealthy alcohol use into medical settings using a validated scale to help identify patients with or at risk for alcohol use disorder and alcohol withdrawal. Recommendation I.2: For patients known to be using alcohol recently, regularly, and heavily, assess their risk of developing alcohol withdrawal even in the absence of signs and symptoms (see II. Initial Assessment for risk factors and risk assessment scales). Recommendation I.3: For patients who have signs and symptoms suggestive of alcohol withdrawal, assess the quan-tity, frequency, and time of day when alcohol was last consumed to determine whether the patient is experiencing or is at risk for developing alcohol withdrawal. For this assessment, it may be helpful to: Use a scale that screens for unhealthy alcohol use (e.g., Alcohol Use Disorders Identification Test-(Piccinelli) Con-sumption [AUDIT-PC]) Use information from collateral sources (i.e., family and friends) Conduct a laboratory test that provides some measure of hepatic function Recommendation I.4: A biological test (blood, breath, or urine) for alcohol use may be helpful for identifying recent alcohol use, particularly in patients unable to communicate or otherwise give an alcohol use history. When conducting a biological test, consider the range of time (window of detec-tion) in which the test can detect alcohol use. Do not rule out the risk of developing alcohol withdrawal if the result of a test is negative. Discussion. Identifying the presence of or risk for alcohol withdrawal may begin with discovering that a patient has been consuming alcohol recently, heavily and regularly. This rec-ognition can be aided by implementing universal screening for unhealthy alcohol use. Universal screening for unhealthy alcohol use is a recommended primary prevention practice that identifies patients with unhealthy alcohol use and increases early intervention in the development of alcohol-related health conditions and complications, including alcohol withdrawal. This practice has been endorsed by the U.S. Preventive Services Task Force (USPSTF) and is supported by an extensive evidence base.4,26–29 Unfortunately, universal screening for unhealthy alcohol use has not been widely implemented in medical settings. As of the release of the USPSTF recommendation statement in 2018, it was estimated that only 1 out of 6 patients have ever discussed alcohol use with their physician.30 Screening begins with administering a brief, standard assessment to identify patients’ unhealthy alcohol use, usually by assessing the amount and frequency of their recent con-sumption. Based on the results, patients may be identified as at-risk for developing alcohol withdrawal syndrome if they have recently (or plan to) stopped or significantly reduced their alcohol consumption. Standard assessments for unhealthy alcohol use that have been used as an initial screen to identify patients at risk of alcohol withdrawal include the Alcohol Use Disorders Identification Test (AUDIT),31 CAGE,32 and Alcohol Use Disorders Identification Test-Pic-cinelli Consumption (AUDIT-PC).29,33 For example, in a retrospective case-control study of over 400 hospitalized patients, an initial AUDIT-PC score 4 identified patients who developed alcohol withdrawal during their stay with 91% sensitivity and 90% specificity.29 For patients who present with signs and symptoms suggestive of alcohol withdrawal, these screening instruments can also be helpful in assessing the amount and frequency of recent alcohol consumption. Screening for unhealthy alcohol use also is relevant for identification of and treatment plan-ning for AUD. Clinicians may also gain additional informa-tion about a patient’s recent alcohol use from other sources including friends and family.4 Laboratory tests that measure impairment of hepatic functioning such as the liver enzymes gamma-glutamyl trans-ferase (GGT) and alanine aminotransferase (ALT) can iden-tify recent heavy alcohol use and hence risk for alcohol withdrawal. When using a urine test, GGT is recommended as the marker of heavy alcohol consumption.34 Clinicians should be aware that laboratory tests provide only partial ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 20 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. information relevant to alcohol withdrawal risk. For example, if a test with a narrow window of detection is negative, the sensitivity of the test to detect risk for alcohol withdrawal will be compromised. However, the inclusion of certain measures of hepatic function have been found to be beneficial in risk determination.4 For example, the predictive ability of the AUDIT to recognize patients likely to develop alcohol with-drawal is increased when combined with biological markers for unhealthy alcohol use including ALT, GGT, mean corpus-cular volume (MCV) and aspartate aminotransferase (AST).35 A biological test for alcohol use (blood, breath, or urine) can identify if a patient recently used alcohol, and may be particularly helpful for those who are unable to communicate or otherwise give an alcohol use history. Sometimes, patients may not be sure of the answer, or might be embarrassed to say that they drank very recently. When conducting a biological test, considertherangeoftime(windowofdetection)inwhichthetest can detect alcohol use. For example, a breathalyzer can detect alcoholuse atanapproximaterateof1standarddrinkper hour.In addition, high tolerance to heavy consumption can lead to increased rates of alchol metabolism and clearance rates outside of expected ranges. This means patients can have a negative breathalyzer test result and be at risk for alcohol withdrawal. Blood alcohol concentration (BAC) combined with clinical signs can indicate risk for withdrawal. Patients with elevated BAC who are not clinically intoxicated should be considered at risk for alcohol withdrawal, as this suggests tolerance to regular heavy use of alcohol.2,7,36 Clinical guid-ance has differed regarding the specific BAC that might indicate heightened risk, but estimates include 100 mg/DL,2 150 mg/DL,36 and 200 mg/DL.7 A diagnostic assessment for alcohol withdrawal or assessment of risk for developing alcohol withdrawal follow-ing cessation of (or reduction in) alcohol consumption is indicated if the clinician is aware that the patient’s alcohol use patterns constitute a risk of alcohol withdrawal or if they are displaying signs or symptoms of alcohol withdrawal. See Appendix IV.A., for a flow-chart on the full protocol for identification, diagnosis, initial assessment, level of care determination, and management of Alcohol Withdrawal Scale. B. Diagnosis Recommendation I.5: To diagnose alcohol withdrawal and alcohol withdrawal delirium, use diagnostic criteria such as those provided by the Diagnostic and Statistical Manual, 5th Edition (DSM-5). To diagnose alcohol use disorder, use diagnostic criteria such as those provided by the DSM-5. Recommendation I.6: Alcohol withdrawal severity assessment scales (including the Clinical Instrument With-drawal Assessment for Alcohol, Revised [CIWA-Ar]) should not be used as a diagnostic tool because scores can be influenced by conditions other than alcohol withdrawal. Recommendation I.7: Do not rule in or rule out the presence of alcohol withdrawal for patients who have a positive blood alcohol concentration. Discussion. Whenever a clinician is making a diagnosis such as those relevant to this guideline (Alcohol Withdrawal Syndrome, Alcohol Withdrawal Delirium, and Alcohol Use Disorder), they should use standard diagnostic criteria such as The Diagnostic and Statistical Manual-5 (DSM-5; see Boxes 2, 3, and 4). While withdrawal severity assessment scales such as the Clinical Instrument Withdrawal Assessment for Alco-hol, Revised (CIWA-Ar) score many of the signs and symp-toms listed in the DSM-5 Criteria, these scales are non-specific regarding the etiology of signs and symptoms and high scores may be the result of the presence of other conditions (e.g., dehydration, fever from infection, Graves’ Disease).2,13,36 Alcohol withdrawal severity assessment scales are designed to assess the signs and symptoms of withdrawal only once a diagnosis has been established.37 As a primary criterion for the diagnosis of alcohol withdrawal, asking patients about the timing of a recent cessation of (or reduction in) alcohol use is essential. Some-times, patients may not be sure of the answer, or might be embarrassed to say that they drank very recently. A biological test for alcohol use can be helpful in this case. Although alcohol withdrawal is associated with the sudden absence of alcohol in the system, it should be noted that minor signs and symptoms can be seen after a significant reduction in alcohol intake if the reduction changes the equilibrium of excitatory vs inhibitory neurochemical signaling (see Box 1) reached during a period of heavy, consistent and prolonged alcohol use.38 This means patients can have a positive blood alcohol concentration and experience alcohol withdrawal signs and symptoms. According to DSM-5, alcohol withdrawal delirium should be diagnosed when the primary symptoms of delirium predominate over other withdrawal symptoms. Given that alcohol withdrawal is itself a diagnostic criterion for alcohol use disorder, patients presenting with alcohol withdrawal symptoms almost certainly also have an alcohol use disorder. It is still recommended that diagnostic criteria such as the DSM-5 should be used to establish such a diagnosis. C. Differential Diagnosis Recommendation I.8: As part of differential diagnosis, assess the patient’s signs, symptoms, and history. Rule out other serious illnesses that can mimic the signs and symptoms of alcohol withdrawal. Determine if patients take medications that can mask the signs and symptoms of alcohol withdrawal. Recommendation I.9: Do not rule in or rule out a co-occurring disease, co-occurring mental health disorder, co-occurring substance use disorder, or simultaneous withdrawal from other substances even in the presence of alcohol with-drawal. Recommendation I.10: Conduct a neurological exam in patients presenting with a seizure to determine etiology. A seizure should only be attributed to alcohol withdrawal if there was a recent cessation of (or reduction in) a patient’s alcohol consumption. For patients experiencing new onset seizures or for patients with a known history of alcohol withdrawal seizures showing a new pattern, an electroenceph-alogram and/or neuroimaging is recommended. For patients with a known history of withdrawal seizure who present with a seizure that can be attributed to alcohol withdrawal, additional Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 21 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. neurological testing and a neurology consult may not be necessary. This includes if the seizure was generalized and without focal elements, if a careful neurological examination reveals no evidence of focal deficits, and if there is no suspicion of meningitis or other etiology. Recommendation I.11: For patients presenting with delirium, conduct a detailed neurological and medical exami-nation with appropriate testing to rule out other common causes of delirium regardless of the apparent etiology. Attempt to distinguish between hallucinations associated with alcohol withdrawal delirium and alcohol hallucinosis/alcohol-induced psychotic disorder. Discussion. As with any diagnosis, it is essential to rule out other possible explanations for the constellation of signs and symptoms presented. Because the syndrome can quickly progress in severity, clinicians suspecting alcohol withdrawal should gather information about recent alcohol use history, especially recent cessation of (or reduction in) alcohol use. For example, the DSM-5 notes that medical conditions including hypoglycemia and diabetic ketoacidosis both can mimic alcohol withdrawal, and an essential tremor may mimic tremors associated with alcohol withdrawal. Addition-ally, signs and symptoms of sedative, hypnotic, or anxiolytic withdrawal are similar to those of alcohol withdrawal, under-scoring the importance of assessing for recent alcohol and other substance use. If recent alcohol use and cessation/ reduction suggests possible withdrawal, but the patients is not exhibiting any signs or symptoms of withdrawal, clini-cians should consider whether the patient is taking any medications that can mask these symptoms, such as beta-adrenergic antagonists (beta-blockers). While making appropriate differential diagnosis is crit-ical, it should be noted that alcohol withdrawal is often seen in conjunction with other health conditions, including mental health disorders, substance-related disorders, or simultaneous withdrawal from other substances besides alcohol. Therefore, clinicians should not discount the possibility of co-occurring conditions once a diagnosis of alcohol withdrawal has been made. Patients presenting with seizure(s) should be provided a neurological exam and medical evaluation to determine sei-zure etiology.2,36,41 The exam and evaluation should include a patient’s history of marked cessation of (or reduction in) alcohol use. An alcohol withdrawal-related seizure should only be diagnosed if there has been a clear history of marked cessation of (or reduction in) alcohol use in the 24 to 48 hours prior to the seizure.2 Patients presenting with a new onset seizure should be provided a full neurologic examination including brain imag-ing with possible lumbar puncture and electroencephalogram (EEG). A thorough neurological examination and EEG should also be provided to patients with a new pattern of alcohol withdrawal related seizures.2,42 However, if a patient has a known history of alcohol-withdrawal related seizures that are clearly attributed to alcohol withdrawal, it may not be neces-sary to do additional neurological testing. If a patient’s alcohol use history and time course of the seizure are inconsistent with an alcohol withdrawal seizure or if the neurological examination identifies focal neurological deficits, meningitis, fever, status epilepticus, recent head trauma, or other possible causes of seizure, further testing should be completed to determine etiology. Patients presenting with delirium should be provided a neurological exam and medical evaluation to determine eti-ology. The history and examination should provide a clear understanding of the relationship between cessation or reduc-tion of alcohol intake and the onset of withdrawal signs and symptoms to eliminate other reasons for delirium.2 The onset of alcohol withdrawal delirium typically occurs 24–48 hours after cessation of (or reduction in) alcohol use but can develop as many as 3–5 days later. If a patient’s alcohol use history and the time course of delirium are inconsistent with alcohol withdrawal delirium or if there is not suspicion of substance-induced psychotic disorder, hypoglycemia, diabetic ketoaci-dosis, or other possible causes of delirium further testing should be completed to determine etiology.43 Patients may present with hallucinosis, which is hallu-cinations that occur in the absence of other clinically promi-nent withdrawal signs and symptoms such as clear delirium. Hallucinosis consist of primarily auditory hallucinations but may include visual hallucinations and delusions.44 It is unclear if alcohol hallucinosis is part of alcohol withdrawal syndrome or if the hallucinations are a complication of chronic alcohol use unrelated to withdrawal. Alcohol halluci-nosis is currently diagnosed as Alcohol-Induced Psychotic Disorder in the DSM-5. Clinicians should attempt to Box 1: Neuroscience of Alcohol Withdrawal The ingestion of ethanol does several things to the human body. But the most important in relation to the development of alcohol withdrawal is the effect of its binding to the U-aminobutyric acid receptor A(GABA-A). At low levels of ethanol, we see the predicable GABA-A effects of decreased anxiety, decreased inhibition and an altering of the motor centers. However, if ethanol is used for an extended time, and at higher levels, we begin to see alterations in the signaling of the extended amygdala. In particular, prolonged alcohol use causes an upregulation of N-methyl-D-Aspartate receptors (NMDAr) and a downregulation of GABA-A. The ultimate result is that if the ethanol is abruptly stopped, there is an imbalance of excitatory vs inhibitory signals. With a high glutaminergic (excitatory) state and a low GABAergic (inhibitory) state, we see the typical signs of alcohol withdrawal; tremor, seizures, nausea and delirium. The major excitatory signal caused by excess glutamate and norepinephrine and the lower GABA-A signaling resulting from both a decrease in gamma-aminobutyric acid and a change in the GABA-A receptor binding characteristics. While the above explains (in very condensed form) the reasons for the clinical features of alcohol withdrawal, we can also glean why certain interventions may be helpful in abating the symptoms. For example, the use of benzodiazepines and certain anti-seizure medications with GABAergic activity (carbamazepine and valproic acid) can be used to abate the symptoms of alcohol withdrawal by reversing the GABA-A deficiency. When these are not sufficient, we can use adjuvants (alpha2-adrenergic agonists [A2AAs]) that can decrease the over activity of the excitatory molecule glutamate or one with both mechanisms of action (phenobarbital). Given the complexity of an individual’s genetics, epigenetics, and patterns of use, we are left with a variable response to any single medication. This is why we have discussed many evidence-based options for the treatment of such a complex syndrome. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 22 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. distinguish between hallucinosis and alcohol withdrawal delirium when making a diagnosis, although this may not always be possible during the early stages of withdrawal.2 If hallucinations persist beyond 72 hours of onset, the more likely diagnosis is alcohol-related psychotic disorder. While alcohol-induced disorders are not a focus of this Guideline, some general guidance is offered in the section VI.C: Alcohol-Induced Psychotic Disorder. II. Initial Assessment of Alcohol Withdrawal A. General Approach Recommendation II.1: First, determine whether a patient is at risk of developing severe and/or complicated alcohol withdrawal or complications from alcohol with-drawal. In addition to current signs and symptoms, a validated risk assessment scale and an assessment of individual risk factors should be utilized. (See Table 1. Alcohol Withdrawal Severity). Recommendation II.2: A history and physical exami-nation should be included as part of the comprehensive assessment process. Clinicians should conduct this examina-tion themselves or ensure that a current physical examination is contained within the patient’s medical record. Recommendation II.3: Additional information about risk factors can be gleaned by interviewing family, friends, and caregivers about a patient’s history of alcohol withdrawal, seizures, and delirium, as appropriate. Whenever possible in non-emergent situations, obtain written or verbal consent from the patient before speaking with or consulting with collateral sources. Recommendation II.4: Clinicians should seek infor-mation about the time elapsed since the patient’s cessation of (or reduction in) alcohol use. The timeline of symptom onset Box 2: DSM-5 Criteria for Alcohol Withdrawal A. Cessation of (or reduction in) alcohol use that has been heavy and prolonged. B. Two (or more) of the following, developing within several hours to a few days after the cessation of (or reduction in) alcohol use described in Criterion A: 1. Autonomic hyperactivity (e.g., sweating or pulse rate greater than 100 bpm) 2. Increased hand tremor 3. Insomnia 4. Nausea or vomiting 5. Transient visual, tactile, or auditory hallucinations or illusions 6. Psychomotor agitation 7. Anxiety 8. Generalized tonic-clonic seizures C. The signs or symptoms in Criterion B cause clinically significant distress or impairment in social, occupation, or other important areas of functioning. D. The signs or symptoms are not attributable to another medical condition and are not better explained by another mental disorder, including intoxication or withdrawal from another substance. Box 3: DSM-5 Criteria for Alcohol Withdrawal Delirium (generic criteria for delirium in the presence of heavy and prolonged alcohol use) A. A disturbance in attention (i.e., reduced ability to focus, sustain, and shift attention) and awareness (reduced orientation to the environment). B. Disturbance develops over a short period of time (usually hours to a few days), represents a change from baseline attention and awareness, and tends to fluctuate in severity during the course of a day. C. An additional disturbance in cognition (e.g., memory deficit, disorientation, language, visuospatial ability, or perception). D. The disturbances in Criteria A and C are not better explained by another preexisting, established, or evolving neurocognitive disorder and do not occur in the context of a severely reduced level of arousal, such as coma. E. There is evidence from the history, physical examination, or laboratory findings that the disturbance is a direct physiological consequence of another medical condition, substance intoxication or withdrawal (i.e., due to drug of abuse or to a medication), or exposure to a toxin, or is due to multiple etiologies. Specify: Substance withdrawal delirium a. This diagnosis should be made instead of substance withdrawal when the symptoms in Criteria A and C predominate in the clinical picture and when they are sufficiently severe to warrant clinical attention. Box 4: DSM-5 Criteria for Alcohol Use Disorder A. A problematic pattern of alcohol use leading to clinically significant impairment or distress, as manifested by at least two of the following, occurring within a 12-month period: 1. Alcohol is often taken in larger amounts or over a longer period than was intended. 2. There is a persistent desire or unsuccessful efforts to cut down or control alcohol use. 3. A great deal of time is spent in activities necessary to obtain alcohol, use alcohol, or recover from its effects 4. Craving, or a strong desire or urge to use alcohol 5. Recurrent alcohol use resulting in a failure to fulfill major role obligations at work, school, or home 6. Continued alcohol use despite having persistent or recurrent social or interpersonal problems caused or exacerbated by the effects of alcohol 7. Important social, occupations, or recreational activities are given up or reduced because of alcohol use 8. Recurrent alcohol use in situations in which it is physically hazardous. 9. Alcohol use is continued despite knowledge of having a persistent or recurrent physical or psychological problem that is likely to have been caused or exacerbated by alcohol 10. Tolerance, as defined by either of the following: i. A need for markedly increased amounts of alcohol to achieve intoxication or desired effect ii. A markedly diminished effect with continued use of the same amount of alcohol 11. Withdrawal, as manifested by either of the following: i. The characteristic withdrawal syndrome for alcohol ii. Alcohol (or closely related substance, such as a benzodiazepine) is taken to relieve or avoid withdrawal symptoms. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 23 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. and severity helps determine the risk window for developing severe or complicated withdrawal. Discussion. It is common for recommendations about the initial assessment for managing alcohol withdrawal to focus on evaluating current signs and symptoms rather than the risk of developing serious forms of the syndrome. However, signs and symptoms can escalate quickly and the trajectory of alcohol withdrawal can vary considerably among patients. As the most severe presentations of alcohol withdrawal are life threatening, orienting the initial assessment toward eval-uating risk as opposed to current presentation is recom-mended. In considering patient risk, clinicians should assess their risk of severe withdrawal, complicated withdrawal (used in this guideline to describe withdrawal-related seizures or alcohol withdrawal delirium), or complications of with-drawal, which refers to a potentially life-threatening exacer-bation of existing medical or psychiatric conditions. A detailed history and physical exam should be con-ducted as part of the initial assessment of alcohol withdrawal and can be an extension of the process of differential diagno-sis. The history and physical exam should identify current withdrawal severity, risk factors for developing life-threaten-ing symptoms and potentially complicating conditions. In the event a patient cannot provide a clear history, interviewing family, friends, and caregivers about risk factors is appropri-ate. Providers should follow their setting/state rules on obtain-ing written or verbal consent or release of information prior to consulting with collateral sources. Individual risk factors are described in the following section. Also discussed in the following section are the use of questionnaires developed to assess risk of severe or complicated withdrawal and to assess current signs and symptoms of withdrawal. When evaluating risk, clinicians should consider the time elapsed since the patient’s cessation of (or reduction in) alcohol use.45 Signs and symptoms of alcohol withdrawal typically begin 6–24 hours after cessation of (or reduction in) alcohol use.2 Early identification and medication manage-ment can reduce the risk of progression to severe or compli-cated alcohol withdrawal syndromes.46 Early withdrawal signs and symptoms may include anxiety, sleep disturbances, anorexia, vivid dreams, headache, nausea, tachycardia, hyper-active reflexes, sweating, elevated blood pressure and hyper-thermia.2 Seizures may begin as early as 8 hours after cessation of (or reduction in) alcohol use and can continue for up to 48 hours with peak activity occurring around 24 hours.2 Hallucinations develop within 12–24 hours follow-ing cessation of (or reduction in) alcohol use and typically resolve within 24–48 hours if other signs indicative of with-drawal delirium do not emerge. The onset of alcohol with-drawal delirium appears between 72 and 96 hours after a patient’s last drink and can last as short as a few hours, but usually for 2–3 days.2 Not all patients progress through these stages sequen-tially. For example, a seizure may occur in the absence of other clinically prominent alcohol withdrawal signs or symp-toms. In particular, elderly patients may have a different timeline of development.2 Concomitant use of alcohol and other sedative hypnotics can also change the presentation of withdrawal signs and symptoms.38 See Appendix IV.A., for a flowchart on the full protocol for identification, diagnosis, initial assessment, level of care determination, and management of Alcohol With-drawal Scale. B. Risk Factors for Severe or Complicated Withdrawal Recommendation II.5: Assess for the following factors associated with increased patient risk for complicated with-drawal or complications of withdrawal: History of alcohol withdrawal delirium or alcohol with-drawal seizure Numerous prior withdrawal episodes in the patient’s life-time Comorbid medical or surgical illness (especially traumatic brain injury) Increased age (>65) Long duration of heavy and regular alcohol consumption Seizure(s) during the current withdrawal episode Marked autonomic hyperactivity on presentation Physiological dependence on GABAergic agents such as benzodiazepines or barbiturates Recommendation II.6: The following individual fac-tors may increase a patient’s risk for complicated withdrawal or complications of withdrawal: Concomitant use of other addictive substances Positive blood alcohol concentration in the presence of signs and symptoms of withdrawal Signs or symptoms of a co-occurring psychiatric disorder are active and reflect a moderate level of severity Recommendation II.7: Patients’ risk for complicated withdrawal or complications of withdrawal is increased by the presence of multiple risk factors. Recommendation II.8: In general, clinicians may con-sider patients at risk of severe or complicated withdrawal if they are experiencing at least moderate alcohol withdrawal on presentation (e.g., CIWA-Ar score 10). Discussion. Several individual risk factors were deemed meaningful by the Guideline Committee based on an analysis of the existing empirical literature combined with their clini-cal experience. There is strong empirical and clinical support for a history of alcohol-related seizures or delirium as pre-dictive of future incidences of severe withdrawal.28,47 A systematic review and meta-analysis of 15 studies of predic-tors of severe alcohol withdrawal concluded that prior alcohol withdrawal delirium, prior withdrawal-related seizure, prior severe alcohol withdrawal, lower platelet count, and higher alanine aminotransferase (ALT) were associated with a sig-nificantly higher incidence of alcohol withdrawal-related seizure or alcohol withdrawal delirium.48 Consistent with the results of the 2014 systematic review, the idea that prior incidences of alcohol withdrawal delirium and seizure should be considered important risk factors for severe alcohol withdrawal has been echoed in numerous clinical guidelines and review articles.12,36,49–51 ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 24 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Repeated episodes of alcohol withdrawal syndrome also become progressively more severe as the result of increased neuronal excitability and sensitivity, a phenomenon known as the kindling effect.4 There is a lack of consensus about additional individual risk factors that contribute to severe alcohol withdrawal. Although the previously mentioned systematic review failed to find an association between other individual risk factors and risk of severe alcohol withdrawal, the review’s primary finding was that ‘‘prediction of severe alcohol withdrawal is highly variable, and that few demographic, clinical, or bio-chemical parameters are consistently predictive’’.48 (p2674) The presence of a severe medical illness has been reported to precipitate severe alcohol withdrawal and to increase the risk of withdrawal seizures and delirium.4,7 SAMHSA’s Treatment Improvement Protocol (TIP) 45 on withdrawal management,4 as well as a number of other published guidelines,2,7,36 recommend that comorbid medical or surgical illness be considered a significant risk factor for complicated withdrawal or complications of alcohol with-drawal. Older age may heighten a patient’s risk of severe alcohol withdrawal, although advanced age may simply be correlated with the presence of complex comorbid health conditions.4,7 The value of assessing a patient’s alcohol use pattern or amount has been contested in the literature. Some note the duration of heavy drinking has not been useful in triaging patients,2 others have argued the opposite.36,44,49 As with advanced age, a longer duration of alcohol use may simply be correlated with more significant comorbid health issues, which can lead to complications of alcohol withdrawal. Patients who have experienced a seizure as part of the current withdrawal episode, but prior to the clinical assess-ment, should be considered at high risk of complicated withdrawal. Following an alcohol withdrawal seizure, a patient is at increased risk for another seizure and progression to alcohol withdrawal delirium.2,4,51,52 Although heart rate and rhythm are often signs mea-sured to assess alcohol withdrawal, there is disagreement about the predictive value of heart rate for identifying risk of withdrawal. Some of the literature suggests clinicians consider marked autonomic hyperactivity (measured by heart rate) to be an indication of severe withdrawal,2 while others argue that an elevated heart rate does not identify the risk of severe withdrawal.42 Concomitant physiological dependence on central ner-vous system depressants such as benzodiazepines and barbi-turates has also been suggested as a risk factor for complicated alcohol withdrawal.7,36,51 Medication management may also be complicated as individuals taking sedative-hypnotic med-ications exhibit tolerance to benzodiazepines and should be monitored closely for appropriate dose if prescribed benzo-diazepines for withdrawal (see IV.A: Monitoring).53 Additional individual risk factors were deemed poten-tially meaningful by the Guideline Committee based on an analysis of the existing empirical literature combined with their clinical experience. In terms of the value of concomitant substance use as a predicator of complicated withdrawal or complications of alcohol withdrawal, the Guideline Commit-tee emphasized that the risk varies significantly based on the type of substance used, as well as patterns of use. However, concomitant substance use may play a role in the development of life-threatening presentations of the syndrome.7,51 An indication of risk for severe or complicated alcohol withdrawal is the presence of alcohol withdrawal signs and symptoms while having a positive blood alcohol concentra-tion (BAC).4 Although alcohol withdrawal is associated with the sudden absence of alcohol in the system, minor signs and symptoms can be seen after a significant reduction in alcohol intake if the reduction alters the equilibrium of excitatory vs inhibitory neurochemical signaling (see Box 1) reached dur-ing a period of heavy, consistent and prolonged alcohol use.38 When using a breathalyzer, clinicians may wish to repeat their measurement serially to follow the level and course of intoxi-cation.39,40 Withdrawal can complicate the treatment of an under-lying mental health disorder. A patient whose co-occurring psychiatric disorder symptoms are active may need specialist treatment. Moderate to severe withdrawal at baseline (e.g., CIWA-Ar 10) has been identified as a risk factor for developing more severe withdrawal in inpatient settings.54 The Guideline Committee also agreed that risk for complicated withdrawal or complications of withdrawal is increased when multiple risk factors are present. C. Risk Assessment Tools Recommendation II.9: Clinicians can consider the use of a tool such as The ASAM Criteria Risk Assessment Matrix to assess a patient’s risk of severe or complicated alcohol withdrawal as well as potential complications of withdrawal. Recommendation II.10: The following scales can be helpful for assessing for the risk of severe alcohol withdrawal: Prediction of Alcohol Withdrawal Severity Scale (PAWSS) Luebeck Alcohol-Withdrawal Risk Scale (LARS) Discussion. The Risk Assessment Matrix is described in The ASAM Criteria12 and offers a multidimensional risk assess-ment for patients with or at risk for developing alcohol withdrawal. It classifies patient risk on a scale of 0–4 across six dimensions and provides decision rules to recommend appropriate treatment interventions for patients at each level. Scales have been developed to identify patients at risk of developing severe or complicated alcohol withdrawal, includ-ing the Luebeck Alcohol Withdrawal Risk Scale (LARS)55 and Prediction of Alcohol Withdrawal Severity Scale (PAWSS).28 The LARS was specifically designed to predict severe alcohol withdrawal among patients without significant comorbid medical illness. A prospective study of 100 patients in a hospital psychiatric ward showed that a LARS score 17 significantly differentiated patients with severe withdrawal from patients with mild to moderate withdrawal with a sensitivity of 100% and a specificity of 88%.55 The PAWSS is a severity scale designed specifically for predicting risk of developing complicated alcohol withdrawal (defined as a Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 25 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. CIWA-Ar score 15) in the medically ill, validated by prospective studies comparing the PAWSS with retrospective chart review and with the CIWA-Ar.28,47 The PAWSS includes an initial screener question (‘‘Have you consumed any amount of alcohol within the last 30 days’’ or ‘‘did the patient have a positive BAL upon admission’’) and can be used with patients who are not currently exhibiting signs of withdrawal. The authors identified a threshold score which identified patients who later scored 15 on the CIWA-Ar during their hospital stay with 93.1% sensitivity and 99.5% specificity.47 These scales and their associated features and evidence base are summarized in Appendix III. D. Symptom Assessment Scales Recommendation II.11: Avalidated instrument should be used to assess alcohol withdrawal severity. Recommendation II.12: Assess the risk for scores on a withdrawal severity assessment scale to be confounded with causes other than alcohol withdrawal. If risk factors are present, interpret the results of scales with caution. Use a scale that relies more on objective signs of withdrawal (autonomic activity) if a patient has difficulty communicating about their symptoms. See Appendix III for the features of different scales. Recommendation II.13: A validated withdrawal sever-ity assessment instrument can be used as part of risk assess-ment. A high initial score can indicate risk of developing severe or complicated withdrawal, although scores should not be the only information used to predict patient risk. Discussion. A patient’s current withdrawal symptom severity should be assessed using a structured withdrawal assessment scale. Scores on a symptom assessment scale can be con-founded with causes other than alcohol withdrawal. For example, scores can be falsely elevated due to comorbid conditions (e.g. fever from infection, concurrent withdrawal from another substance) and falsely suppressed due to the use of certain medications (e.g., beta-blockers and other sympa-tholytic drugs).2 If risk factors are present, interpret the results of symptom assessment scales with caution. Some scales require self-report from patients about their symptoms and cannot be administered to patients with a communication difficulty, those who are experiencing symptoms of delirium, or those who are critically ill. In these instances, use a withdrawal symptom assessment scale that relies more on objective signs of withdrawal (autonomic activity). These scales and their associated features and evidence base are summarized in Appendix III. Currently, there is insufficient evidence to prefer one scale to another; the choice instead depends on clinician preference. The most commonly discussed and utilized scale is the CIWA-Ar.12,38,56,57 The CIWA-Ar was designed to measure the severity of alcohol withdrawal for research studies.37 It is a 10-item standardized scale with demonstrated validity and interrater reliability. The CIWA-Ar itself does not offer score ranges categorizing symptom severity. The devel-opers of the CIWA-Ar suggested different interventions for scores of <10, 10–20, and > 20, but these were based on the clinical experience of the authors and not empirical data.37 Numerous guidelines and review articles offer guidance about the appropriate intervention for different ranges of CIWA-Ar scores.4,41,58 Throughout this document, we provide examples for withdrawal severity using the CIWA-Ar, although other scales can be used. Regardless of the instrument used, there is a wide variety in the literature and in practice as to which scores best delineate mild, moderate and severe withdrawal. Classifica-tion of withdrawal severity is ultimately up to the judgment of clinicians and clinical programs might choose reference ranges based on their particular patient population or capa-bilities. See Table 1. Alcohol Withdrawal Severity for the categorization of withdrawal severity used in this guideline. Despite its widespread use, clinicians should be aware of the limitations of the CIWA-Ar. It requires clinician training for reliable administration and is criticized for the time it takes to administer.57,59 It also requires patients to self-report about symptoms including nausea/vomiting, anxiety, tactile and auditory disturbances, and headache and can be difficult, if not impossible, to administer to patients experiencing severe or complicated withdrawal or those who are critically ill. While the CIWA-Ar is the most well-known and widely adopted alcohol withdrawal severity scale, modifi-cations and alternative scales have been developed and evidence of their validity and reliability is emerging. The Newcastle Alcohol Withdrawal Scale is a modified version of the CIWA-Ar which relies more on objective signs of withdrawal.60 The Brief Alcohol Withdrawal Scale (BAWS) was developed as a shorter and more objective method to assess alcohol withdrawal signs and symptoms and early evidence has demonstrated favorable sensitivity and speci-ficity compared with the CIWA-Ar.59 The Short Alcohol Withdrawal Scale (SAWS), a 10-item instrument designed to be self-administered by patients, has been validated in ambulatory settings.61 The CIWA-Ar and similar scales are not designed to assess the risk for developing severe withdrawal, but they are commonly called upon for this task. The Guideline Commit-tee noted that withdrawal assessment scales can provide some indication of risk in that a patient’s current signs and symp-toms can provide valuable (although partial) information about their risk of severe or complicated withdrawal. How-ever, it should be stressed that symptom assessment scales cannot indicate alcohol withdrawal risk if the patient is not currently experiencing signs or symptoms of withdrawal.58 One observational study using the Newcastle Alcohol With-drawal Scale to guide treatment found that hospital patients scoring >15 at baseline were at higher risk of severe with-drawal if they did not receive medication.60 Although these scales have generally not been found to be superior to the CIWA-Ar at identifying the potential risk of developing severe or complicated withdrawal, they may be more feasible to administer than the CIWA-Ar in some inpatient settings. The Guideline Committee considered each scale to be an acceptable option for assessing hospitalized patients after diagnosis of alcohol withdrawal. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 26 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. E. Identify Concurrent Conditions Recommendation II.14: When assessing for concur-rent medical conditions, screen patients for medical condi-tions that could affect the course of alcohol withdrawal or treatment of alcohol withdrawal, as well as common chronic conditions that are associated with alcohol use disorders. Recommendation II.15: A pregnancy test should be obtained for women of childbearing potential. For informa-tion on managing pregnant patients, see section VII.F: Patients who are Pregnant. Recommendation II.16: In settings with access to labo-ratory testing, clinicians should conduct and/or arrange for a comprehensive metabolic profile (CMP) or basic metabolic profile (BMP), a hepatic panel, and a complete blood count with differential to assess a patient’s electrolytes, liver function-ing, renal functioning and immune functioning. In a setting with limited access to laboratory testing, clinicians should obtain resultswhenpracticaltoassistwithtreatmentplanningdecisions. Address any nutritional deficiencies detected. Initial screening may also include laboratory tests for: Hepatitis Human Immunodeficiency Virus (HIV) (with consent) Tuberculosis Recommendation II.17: Assess patients for polysub-stance use and be prepared to treat other potential withdrawal syndromes. To assess a patient’s other substance use, it may be helpful to: Use a validated scale that addresses other substance use, such as the Alcohol, Smoking and Substance Involvement Screening Test (ASSIST) Conduct a urine or other toxicology test to detect other substance use Utilize information from collateral sources when possible (i.e., family and friends) Recommendation II.18: Do not delay the initiation of treatment if alcohol withdrawal is suspected but laboratory test results are not available at the treatment setting or the results are pending. Recommendation II.19: Assess patients for concurrent mental health conditions, including a review of their mental health history, to determine their mental health treatment needs. Consult with any mental health professionals caring for such patients. Obtain written or verbal consent before consultation whenever possible in non-emergent situations. The Patient Health Questionnaire (PHQ-9) and the Generalized Anxiety Disorder (GAD-7) scales can be helpful to screen for mental health disorders. Be cautious when diagnosing a new primary mental health disorder during acute withdrawal, as it can be difficult to differentiate between substance-induced signs and symptoms and primary psychiatric disorders. Recommendation II.20: Evaluate active suicide risk as part of the initial patient assessment. Discussion. Clinicians should thoroughly assess patients for concurrent physical and mental health conditions that may a) complicate the course of alcohol withdrawal and/or b) neces-sitate their own treatment interventions. There is not a stan-dard medical evaluation process for patients with, or at risk for, alcohol withdrawal, but it should include a history and physical examination and an assessment for concurrent men-tal health conditions. The Guideline Committee recommends that clinicians be knowledgeable about common chronic conditions associated with alcohol use disorders in order to screen for likely concurrent medical conditions. Common chronic conditions associated with alcohol use disorders include high blood pressure, heart disease, liver disease and digestive problems. Conditions that may be exacerbated by the increased autonomic hyperactivity associated with withdrawal, such as cardiac illness, should be identified early for aggressive autonomic symptom prevention. It should also be identified whether patients take medications that may suppress auto-nomic symptoms and therefore mask withdrawal severity, such as beta-adrenergic antagonists. Conditions associated with impaired liver functioning should also be identified as they may influence medication choice and/or dosing amounts. Medical conditions that prevent the use of oral medication should also be identified, as parenteral administration of medication is not available in all treatment settings. Because pregnancy influences alcohol withdrawal man-agement decisions and pregnancy tests are typically available at most settings with rapid results, the Guideline Committee recommended that clinicians conduct a pregnancy test for patients of childbearing potential with suspected alcohol withdrawal. However, it should be noted that if a patient is presenting with signs and symptoms of alcohol withdrawal and pregnancy status is unknown and a test is not immediately available, alcohol withdrawal management should not be delayed. To aid in the identification of concurrent medical con-ditions, laboratory testing may be helpful. The decision to conduct routine laboratory testing and what to test for should be informed by the patient’s signs and symptoms, known concurrent medical conditions, and availability. At a mini-mum, the Guideline Committee recommended clinicians conduct and/or arrange for a comprehensive metabolic profile (CMP) or basic metabolic profile (BMP), a hepatic panel, and a complete blood count with differential to assess a patient’s electrolytes, liver functioning, renal functioning, and immune functioning. In addition, laboratory tests for hepatitis, Human Immunodeficiency Virus (HIV), and tuberculosis may be considered if indicated. In addition to identifying medical conditions with a high rate of co-occurrence with alcohol withdrawal, the results of some tests, primarily for liver functioning, might guide the choice of medication for alcohol withdrawal as discussed in later sections on pharmacotherapy. Hospitalized patients are a unique population because clinicians have greater access to laboratory tests and rapid results. In an ambulatory setting, clinicians may have less access to laboratory tests and be less able to obtain rapid results.62 Therefore, in ambulatory settings, the Guideline Committee recommends that in general, laboratory testing should be done when practical. However, clinicians should not Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 27 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. delay treatment if testing is unavailable or if test results are pending. As discussed previously, concomitant substance use may play a role in the development of life-threatening presentations of alcohol withdrawal syndrome.7,51 Of partic-ular concern is concurrent physiological dependence or withdrawal from other sedative hypnotics as it can affect symptom presentation and response to commonly used with-drawal medications. Clinicians can use a screening question-naire to begin the identification process. Numerous validated scales are available for assessing a patient’s substance use patterns. A recommended option is The Alcohol, Smoking and Substance Involvement Screening Test (ASSIST), devel-oped by the World Health Organization.63,64 While the ASSIST takes longer to complete than many available scales, it is more comprehensive in the identification of polysub-stance use while many others scales focus on a substance use broadly. Also discussed previously are the complications that can be caused by alcohol withdrawal for managing a patient’s underlying mental health problem and vice versa. A mental health condition is not thought to increase risk for severe, complicated, or complications of withdrawal. However, given the shared symptomology of even mild forms of withdrawal, such as anxiety, agitation and sleep problems, with common mental health disorders, determining the etiology of symp-toms and judging appropriate response to medication for alcohol withdrawal may be complicated.4 A review of the patient’s medical record can reveal primary diagnoses and if the patient is currently under the care of a mental health professional, that individual should be consulted. Providers should follow their setting/state rules on obtaining written or verbal consent or release of information prior to consultation. Clinicians can also consider the use of a standardized screening instrument for depression and anxiety, but they should not diagnose a new primary mental health disorder during the acute withdrawal period.46,47 Both the Patient Health Questionnaire (PHQ-9) and the Generalized Anxiety Disorder (GAD-7) questionnaires ask a patient to assess their symptoms over the prior two weeks and recall can be affected by current symptom state. Clinicians should also evaluate active suicide risk as part of the initial patient assessment. III. Level of Care Determination A. General Approach Recommendation III.1: Level of care determination should be based on a patient’s current signs and symptoms; level of risk for developing severe or complicated withdrawal or complications of withdrawal; and other dimensions such as recovery capital and environment. Alcohol withdrawal can typically be safely managed in an ambulatory setting for those patients with limited or mitigated risk factors. Patients with low levels of psychosocial support or an unsafe environment may benefit from a more intensive level of care than is otherwise indicated. Recommendation III.2: Patients with active risk of suicide should be treated in a setting equipped to manage patients at risk of suicide, which often necessitates admission to an inpatient psychiatric setting that also provides with-drawal management services. Discussion. The ASAM Criteria provides comprehensive guidance on determining the appropriate level of care for patients in need of withdrawal management. Level of care determinations are based on an evaluation of the expected risks and benefits of treatment within each setting. A central tenet of The ASAM Criteria is that patients should be matched with the least intensive level of care in which they can be safely and effectively treated. In the absence of indications for inpatient treatment, which will be described in following sections, most patients with alcohol withdrawal can be safely and effectively managed in ambulatory settings.65–67 One 1995 estimate found that approximately 10% of patients with alcohol withdrawal syndrome require inpatient treatment.68 In general, patients benefit from being treated in less restrictive settings that minimize disruptions to family life, housing and employment, and reduce costs. One RCT found that patients with mild-to-moderate alcohol withdrawal assigned to outpa-tient treatment had faster resolution of withdrawal compared to inpatient treatment.66 Ambulatory withdrawal management should be preferred in the absence of any indications for inpatient treatment.51 Inpatient management is indicated for some patients. Lack of 24-hour monitoring and distance to life saving medical intervention means that some patients with or at risk for developing severe or complicated withdrawal or compli-cations of alcohol withdrawal could experience great harm if treated in an ambulatory setting. Ambulatory treatment is most appropriate for patients who have a low risk of devel-oping severe or complicated withdrawal,69,70 which may include patients with mild or moderate withdrawal syn-drome.71 Some low-risk patients may benefit from treatment in an inpatient setting. For example, patients with an absence of or unreliable support network may benefit from a more intensive level of care.21,49 See Appendix IV.A., for a flowchart on the full protocol for identification, diagnosis, initial assessment, level of care determination, and management of Alcohol Withdrawal Scale. B. Level of Care Determination Tools Recommendation III.3: The ASAM Criteria Risk Assessment Matrix and withdrawal severity scales can be helpful for determining the appropriate level of care for managing patients in alcohol withdrawal. Most withdrawal severity scales reflect current signs and symptoms and should not be used alone to determine level of care. Discussion. The ASAM Criteria provide a guide for clinicians treating patients experiencing alcohol withdrawal or seeking alcohol withdrawal management services. It accounts for current signs and symptoms and identifies potential risks for complicated withdrawal. This framework allows clinicians the ability to make level of care determinations based on the most appropriate needs for each patient. The ASAM Criteria encourages the use of symptom assessment scales such as the CIWA-Ar score in the decision making process; however, it ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 28 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. also emphasizes that symptom severity should not be used alone to make level of care determinations. The ASAM Criteria measures a patient’s risk of develop-ing severe or complicated withdrawal or complications of alcohol withdrawal by utilizing a multidimensional assessment that determines a patient’s risks and strengths based on six dimensions. These dimensions include: (1) acute intoxication and/or withdrawal potential, (2) biomedical conditions and complications, (3) emotional, behavioral, or cognitive condi-tions and complications, (4) readiness to change, (5) relapse, continued use,orcontinuedproblem potential,and(6) recovery/ living environment. Using the multidimensional assessment, cliniciansprovideariskratingforeachdimensionandanoverall rating that allow them to identify the patient’s treatment needs and level of care most appropriate to meet those needs. The ASAM Criteria provides a comprehensive set of criteria for appropriate placement in one of five levels of care: Level 1-WM: Ambulatory withdrawal management with-out extended on-site monitoring Level 2-WM: Ambulatory withdrawal management with extended on-site monitoring Level 3.2-WM: Clinically managed residential withdrawal management Level 3.7-WM: Medically monitored inpatient withdrawal management Level 4-WM: Medically managed intensive inpatient withdrawal management See The ASAM Criteria for a detailed description of services available in each level of care. C. Considerations for Ambulatory vs Inpatient Management While there are five distinct levels of care for with-drawal management defined by The ASAM Criteria, much of the research on patient placement evaluates factors indicating (or contraindicating) placement in an ambulatory or inpatient treatment setting. These settings align with the following ASAM levels of care: Ambulatory Level 1-WM: Ambulatory withdrawal management without extended on-site monitoring Level 2-WM: Ambulatory withdrawal management with extended on-site monitoring. Inpatient Level 3.2-WM: Clinically managed residential with-drawal management Level 3.7-WM: Medically monitored inpatient with-drawal management Level 4-WM: Medically managed intensive inpatient withdrawal management So as not to duplicate The ASAM Criteria, and in the interest of identifying consensus and strength of evidence where it exists, this Guideline will largely focus on determining appropriate placement criteria in these two categories of with-drawal setting: ambulatory and inpatient. However, due to increasing interest in office-based alcohol withdrawal manage-ment by specialty and non-specialty clinicians, the significant difference in monitoring levels afforded by the two ambulatory settings, and at the request of the Guideline Committee, the Guideline will distinguish between considerations for Level 1-WM and Level 2-WM settings in this section. Level 1-WM is ambulatory withdrawal management without extended on-site monitoring. It can be carried out in a physician’s office, by a home health care agency, or addiction treatment facility. Level 2-WM is ambulatory withdrawal management with extended on-site monitoring. It can be carried out in structured outpatient settings such as a day hospital setting, a general health care or mental health facility, or an addiction treatment facility. Level 2-WM is an organized service with the capacity to provide regular medical assess-ments and monitor alcohol withdrawal syndrome progression. They may also have access to psychological or psychiatric treatment (see The ASAM Criteria for additional details). Level of care determination is organized around risk-benefit principles, where an appropriate level of care is one in which the expected benefits of treating a patient at a particular level of care are outweighed by the risks. More intensive levels of care are appropriate for patients at increased risk of harm. This means that if Level 1-WM is not appropriate for a particular patient, Level 2-WM may still be appropriate. However, if Level 2-WM is not appropriate, then Level 1-WM is also not appropriate. This patient should be treated in an inpatient setting. The guideline does not currently make recommendations regarding placement within the three levels of inpatient settings: Level 3.2-WM, Level 3.7-WM, and Level 4-WM. It should be noted that a patient’s refusal or inability to attend a recommended level of care should not delay or preclude treatment at a level of care they are able to attend.12 Recommendation III.4: Discussion Withdrawal severity Patients experiencing signs and symptoms of mild alcohol withdrawal such as mild or moderate anxiety, sweat-ing and insomnia, but no tremor (generally associated with a CIWA-Ar <10) can be managed in Level 1-WM or Level 2-WM settings.2,39,62,72 While providing withdrawal manage-ment is within the scope of practice for many clinicians including primary care physicians, an addiction specialist can be consulted, if needed.72 Patients experiencing signs and symptoms of moderate alcohol withdrawal such as moderate anxiety, sweating, insomnia, and mild tremor (generally associated with a CIWA-Ar 10–18) can be managed in Level 2-WM settings. Moderate withdrawal is not a reason to exclude patients from Level 1-WM settings, but the risk for such patients should be carefully considered. It should only be undertaken by experienced clinicians. Patients experiencing signs and symptoms of severe withdrawal such as severe anxiety and moderate to severe tremor, but not confusion, hallucinations, or seizure (generally associated with a CIWA-Ar 19) should not be managed in Level 1-WM settings.58 Severe uncomplicated withdrawal is not a reason to exclude patients from Level 2-WM settings. The risk for such patients should be carefully considered. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 29 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. TABLE 2. Ambulatory (Level 1-WM and Level 2-WM) and Inpatient Placement Considerations Level 1-WM Level 2-WM Appropriate Neutral/Uncertain Inappropriate Appropriate Neutral/Uncertain Inappropriate Withdrawal severity Mild (e.g., CIWA-Ar <10). Moderate (e.g., CIWA-Ar 10–18). Severe or complicated (e.g., CIWA-Ar 19). Mild or moderate (e.g., CIWA-Ar <18). Severe but not complicated (e.g., CIWA-Ar 19). Complicated (e.g., CIWA-Ar 19). Concurrent withdrawal or physiological dependence Withdrawing from other substance(s). Physiological dependence on opioids or OUD. Physiological dependence on BZDs or BZD use disorder. Physiological dependence on opioids or OUD. Withdrawing from other substance(s). Physiological dependence on BZDs or BZD use disorder. Recent alcohol consumption Consumes > 8 standard drinks per day. Consumes > 8 standard drinks per day. Alcohol withdrawal history Previous severe withdrawal episode. Complicated withdrawal > 1 year ago. Recent complicated withdrawal episode. Severe withdrawal > 1 year ago. Previous complicated withdrawal episode. Recent severe withdrawal episode. Treatment history Previous failure to benefit from ambulatory-WM. Previous failure to benefit from ambulatory-WM. Other inpatient need Medical or psychiatric condition that needs inpatient treatment. Medical or psychiatric condition that needs inpatient treatment. Biomedical conditions and complications Older age. History of epilepsy. History of non-alcohol related seizure. Clinically significant abnormal lab results. Moderate, active, and potentially destabilizing medical problem. Moderate to severe active and potentially destabilizing medical problem, including unstable chronic condition. Suspected head injury. Unable to take oral medications. Older age. History of epilepsy. Moderate, active, and potentially destabilizing medical problem. History of non-alcohol related seizure. Clinically significant abnormal lab results. Suspected head injury. Moderate to severe active and potentially destabilizing medical problem including unstable chronic condition. Unable to take oral medications. Emotional, behavioral, or cognitive conditions and complications Mild/stable psychiatric symptoms. Active psychiatric symptoms. Mild cognitive impairment. Moderate or severe psychiatric symptoms. Moderate or severe cognitive impairment. Mild/stable psychiatric sypmtoms. Active or moderate psychiatric symptoms. Mild or moderate cognitive impairment. Severe psychiatric symptoms. Severe cognitive impairment. Symptom monitoring Absence of reliable caregiver. Communication barrier (e.g., language, hearing, speech). Absence of reliable caregiver. Communication barrier (e.g., language, hearing, speech). Recovery/ living environment Absence of reliable support network. Unable to come to treatment setting daily. Unable to obtain transportation or housing. Family/friends not supportive of WM process. Absence of reliable support network. Unable to come to treatment setting daily. Family/friends not supportive of WM process. Unable to obtain transportation or housing. Risk of harm Commitment not high, cooperation and reliability questionable. Imminent risk of harm – not cooperative or reliable. Significant risk of imminent relapse. Commitment not high, cooperation and reliability questionable. Significant risk of imminent relapse. Imminent risk of harm – not cooperative or reliable. The Guideline Committee rated each placement consideration on a benefit-to-harm ratio, comparing the potential harm that might result from the factor bing considered to the expected benefit to the patient of being managed in a less restrictive setting. The rating was made in terms of the average patient in an average setting for both Level 1-WM and Level 2-WM. A consideration is Inappropriate for a given setting when the potential harm outweighs the expected benefit. A consideration is Appropriate for a given setting if the expected benefit outweighs the potential harm. Neutral/Uncertain considerations are ones where the harms and benefits are about equal or cannot be determined or where the Guideline Committee disagreed. If a consideration that is inappropriate for Level-2-WM is present, the patient should be managed in an inpatient setting. BZD, Benzodiazepine; CIWA-Ar, Clinical Institute Withdrawal Assessment for Alcohol, Revised; OUD, Opioid Use Disorder; WM, Withdrawal Manageemnt. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 30 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Some Level 2-WM settings have intensive monitoring capa-bilities and experienced clinicians can safely manage patients in severe withdrawal as long as they are not experiencing complicated symptoms. Patients experiencing complicated withdrawal syn-drome including seizure or signs indicative of delirium – such as an inability to fully comprehend instructions, clouding of the sensorium or confusion – or new onset of hallucina-tions, or has experienced a seizure during the current episode (generally associated with a CIWA-Ar 19) should be man-aged in inpatient settings.4,21,51,62,65 Concurrent withdrawal and/or physiological depen-dence Concurrent withdrawal from or physiological depen-dence on another substance is a risk factor for developing complicated alcohol withdrawal.21,58,62 The literature sup-ports the management of these patients in a level of care setting with increased monitoring and aggressive treatment.62 Concurrent withdrawal from other substance(s) is not an exclusionary factor for either level of ambulatory care. The risk for such patients should be carefully considered. Manag-ing alcohol withdrawal and withdrawal from another sedative hypnotic58 is more complicated clinically than withdrawal from a substance with different pharmacologic effects, such as a stimulant. Patients should be placed in the level of care appropriate to their most acute problem, which may be withdrawal from the other substance.12 Withdrawal from benzodiazepines produces more autonomic nervous system signs than does withdrawal from alcohol.73 Patients with a physiological dependence on benzodia-zepines or a co-occurring benzodiazepine use disorder should not be treated in a Level 1-WM setting, but are not excluded from management in a Level 2-WM setting. The risk for such patients should be carefully considered. If deemed appropri-ate, patients may be treated with cautious use of benzodia-zepines (see IV.D(3): Benzodiazepine use) or an alternative medication depending on the clinician’s judgment and with careful monitoring Patients with a physiological dependence on opioids or a concurrent opioid use disorder can be managed in a Level 2-WM setting and are not excluded from management in a Level 1-WM setting. The risk for such patients should be carefully considered. Clinicians should have experience with co-managing opioid use disorder and/or physiological dependence including initiating evidence-based medica-tions for opioid use disorder74 and with identifying emergent opioid withdrawal syndrome in addition to alcohol with-drawal. Recent high levels of consumption Recent high levels of alcohol consumption has been cited as a consideration for level of care determination.38,44 While most sources did not provide specific threshold amounts, the NICE guideline21 suggested that inpatient man-agement be considered for patients who consume over 30 U.K. standard units of alcohol per day, which is equivalent to 17 U.S. standard drinks per day. Inpatient treatment was recommended for patients consuming over 7 U.S. standard drinks per day.36 The Guideline Committee considered a cutoff of 8 U.S. standard drinks per day. Consumption of more than 8 U.S. standard drinks per day is not an exclusion-ary factor for either ambulatory withdrawal managent setting. Alcohol withdrawal history A history of severe and/or complicated alcohol with-drawal is a risk factor for alcohol withdrawal seizure and delirium48 (see section II.B: Risk Factors for Severe or Complicated Withdrawal) and is frequently cited as an indi-cation for treatment in an inpatient setting.21,51,58,65 The number and recency of prior withdrawal episodes may also be a factor when determining appropriate level of care.49 An increasing number of withdrawal episodes is associated with increasing severity through the kindling process. With one exception, the Guideline Committee deter-mined that a history of severe or complicated alcohol with-drawal does not exclude patients from management in ambulatory settings. However, patients with a recent (within the prior year) episode of complicated alcohol withdrawal should not be managed in Level 1-WM settings. Also, patients with a prior episode of severe alcohol withdrawal which occurred more than one year ago can be managed in Level 2-WM settings. Treatment history While multiple prior failed attempts to complete alcohol withdrawal treatment has been cited as a contraindication for ambulatory care,56,58,65 previous unsuccessful attempts at ambulatory withdrawal management does not exclude patients from management in ambulatory settings. The risk for such patients should be carefully considered. Circum-stances that led to unsuccessful treatment or a return to problem alcohol use in the past may have changed and should be assessed by the clinician in making a determination of the appropriate level of care. Other inpatient needs Patients with medical or psychiatric conditions may receive alcohol withdrawal management at all levels of care; however, if patients have a co-morbid condition which requires inpatient treatment or hospitalization, patients should not be treated in an ambulatory setting.12,56 The ASAM Criteria states that ‘‘for management provided in conjunction with treatment for co-occurring conditions identified in the comprehensive biopsychosocial screening assessment, The ASAM Criteria calls for the patient to be placed in the level of care appropriate to the most acute problem.’’12(p131) There-fore, patients with a psychiatric or medical condition that requires services that are provided exclusively in an inpatient setting should not be managed in ambulatory settings. Biomedical conditions and complications Comorbid illness Comorbid illness is a risk factor for complicated/com-plications of alcohol withdrawal (see II.B: Risk Factors for Severe or Complicated Withdrawal), but the severity of illness and its likelihood of complicating alcohol withdrawal man-agement is a factor for level of care determination.4,58 Patients with a moderate to severe active and potentially destabilizing medical problem should be managed in inpatient settings. This includes unstable, severe chronic conditions such as cardiovascular disease, liver disease, COPD, or renal impairment.21,65 Patients with a moderate, active, and poten-tially destabilizing medical problem should not be managed in Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 31 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. a Level 1-WM setting. Such patients are not excluded from management in a Level 2-WM setting. Some clinicians or programs may have greater experience or access to resources allowing them to manage less severe comorbid illnesses in an ambulatory setting. Clinically significant laboratory values Clinically significant abnormal laboratory values indi-cate the presence of a potentially destabilizing medical prob-lem. Abnormal lab results are not an exclusionary factor for managing patients in ambulatory levels of care. Some abnor-mal values can be corrected in ambulatory setting, while some may signal the presence of medical conditions that should be managed in inpatient settings.62 Suspected head injury Patients with a suspected head injury should not be managed in Level 1-WM settings. A suspected head injury does not exclude management in Level 2-WM settings, but the risk for such patients should be carefully considered. Some Level 2-WM settings have the capability to intensively moni-tor patients for complications which may develop. History of epilepsy and generalized seizure A history of epilepsy and generalized seizure has been cited as an indication for inpatient treatment.21,51,75 Manag-ing patients with a history of epilepsy is appropriage in Level 2-WM settings and is not a reason for exclusion from Level 1-WM settings. A history of non-alcohol withdrawal related seizures is not an exclusionary factor for ambulatory set-tings. Uncertainty about risk may be the result of limited scientific research and evidence regarding the impact of non-alcohol withdrawal seizures and current withdrawal man-agement. Use clinical experience in the level of care deter-mination. Older age Older age has been identified as a risk factor for complicated/complications of alcohol withdrawal (see II.B: Risk Factors for Severe or Complicated Withdrawal), possibly by heightening the severity of signs or symptoms of with-drawal or due to correlation with other health problems. ‘‘Older age’’ has been left undefined in other guidelines4,21 but was designated as age 65 and older by the Guideline Committee. It is appropriate to manage older patients in a Level 2-WM setting. Older age is not a reason to exclude older patients from Level 1-WM settings, but the risk for such patients should be carefully considered. Some older patients may be otherwise relatively healthy. Tolerance of oral medications Patients who are unable to tolerate oral medication should not be treated in an ambulatory setting.58,62 Parenteral administration of medication is required in patients who are unable to take medication orally, which is not always available in the average ambulatory setting. Pregnancy Pregnancy has been described as one of the ‘‘medical conditions that could make ambulatory withdrawal manage-ment problematic,’’58(p57) and other guidelines have cited pregnancy as an indication for inpatient treatment.62,65 See VII.F: Patients who are Pregnant for more information. Emotional, behavioral, or cognitive conditions and complications While the presence of a co-occurring psychiatric con-dition is frequently cited as a contraindication for ambulatory care,4,62 the type/severity and stability of psychiatric disorders is an important distinction in determining appropriate level of care. Some patients’ mental health problems are well-con-trolled and some ambulatory programs have onsite psycho-logical or psychiatric staff. As withdrawal progresses in severity, the average ambulatory clinic is less likely to have the resources needed to manage patients safely and effectively and inpatient management with specialty psychiatric resour-ces may be more appropriate. Patients whose co-ocurring psychiatric disorder signs and symptoms are mild, reflecting a low level of severity or stability as the result of treatment, can be managed in Level 1-WM or Level 2-WM settings. Active psychiatric disorder signs and symptoms, reflecting a level of severity that may complicate withdrawal management, are not a reason to exclude patients from ambulatory care, but the risk for such patients should be carefully considered. Management may be appropriate if appropriate psychiatric treatment resources are accessed. Patients whose psychiatric disorder signs and symp-toms are moderate should not be treated in Level 1-WM settings, but are not excluded from management in Level 2-WM settings. Patients whose co-ocurring psychiatric disorder signs and symptoms are severe or unstable should be managed in inpatient settings.21,44,58,65 Cognitive impairment is allso cited as a contraindica-tion for ambulatory care.58,62,65 However, patients may have access to stable support services and ambulatory clinics may have the staff or resources necessary to manage a patient’s withdrawal safely and effectively. Mild cognitive impairment is not an exclusionary factor for ambulatory care. Patients with a moderate cognitive impairment should not be managed in Level 1-WM settings, but moderate cognitive imparment is not a reason to exclude patients from Level 2-WM settings. Patients with severe cognitive impairment should be managed in inpatient settings.21,58 The appropriateness of managing patients with moderate or mild cognitive impairment in any setting depends on the availability of support services and experience of the treating clinicians. Symptom monitoring Even if not using a validated symptom severity scale, the ability of a patient to communicate with clinicians or a caretaker about their symptoms is critical to the safe and effective management of alcohol withdrawal, particularly in the early stages when symptoms continue to develop. A communication difficulty due to a language barrier, a hearing or speech difficulty, or other non-withdrawal symptom related cause is not a reason to exclude patients from ambulatory settings. The appropriateness of treating patients with these difficulties will depend on staff capabilities and available accommodation services. Because patients are not on-site for the whole day, the absence of a reliable caregiver such as family or friends willing to monitor signs and symptoms at home has been cited as a contraindication for ambulatory withdrawal man-agement.65,73 However, the absence of a reliable caregiver to monitor withdrawal at home is not a reason to exclude patients from ambulatory management. The appropriateness will ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 32 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. depend on the need to monitor signs and symptoms and other factors that influence treatment adherence and maintenance. Recovery/living environment A patient’s recovery and living environment is a con-sideration when determining level of care. These consider-ations fall into three categories: the presence of social support, access to safe housing and transportation, and ability to visit the clinic frequently during withdrawal management (which may be complicated by available transportation, but also employment, childcare, etc.). The absence of a social support network is commonly cited as an indication for inpatient treatment.62 However, the absence of a reliable social support network is not a reason to exclude patients from ambulatory management, and appropriateness will depend on a patient’s access to other resources. Patients with family or friends who are not supportive of or oppose the withdrawal management process should not be managed in a Level 1-WM setting. The assumption is that patients have contact with those family or friends and their opposition will be detrimental to the with-drawal process. Having family or friends who are not sup-portive of or oppose the withdrawal management process is not a reason to exclude patietns from Level 2-WM settings. Increased hours of clinic attendance will reduce contact with oppositional family and friends. Patients in Level 2-WM settings also have greater access to AUD treatment services, which can help patients address interpersonal problems and teach coping mechanisms. It is not appropriate to manage alcohol withdrawal in an ambulatory setting if patients are unable to access or arrange for safe housing.21,58,65 It is also not appropriate to manage alcohol withdrawal in an ambulatory setting if patients are unable to access or arrange for transportation to the treatment setting. The inability to come to the treatment setting daily is not a reason to exclude patients from ambulatory settings. An alternative to daily visits for these patients may involve alternating in-person clinic visits with consultations with a qualified clinician every other day via phone or video confer-ence (see IV.A: Monitoring). Risk of harm and use A patient’s likelihood of completing ambulatory with-drawal treatment and of refraining from alcohol use has been cited as a factor for determining level of care in prior guide-lines.12,58,62 Patients being treated in ambulatory settings have greater access to and are at greater risk for using alcohol and other drugs during alcohol withdrawal management compared to patients in an inpatient withdrawal treatment setting. When alcohol is combined with medications such as benzodiaze-pines, which are used to treat alcohol withdrawal symptoms, it can be particularly dangerous to patients. The ASAM Crite-ria12 uses the concept of imminent danger (gravity of con-sequences to self/others) to categorize the proximity and likelihood of consequences and need for structured services and continuous monitoring. Ambulatory withdrawal manage-ment is not appropriate for uncooperative or unreliable patients who are at imminent risk of harm. Patients with an uncertain level of cooperation or reliability, with a low level of commitment to the withdrawal process, or who are at signifi-cant risk of imminent return to alcohol use should not be managed in Level 1-WM settings. Such patients are not excluded from management in evel 2-WM settings, but their risk should be carefully considered. Level 2-WM set-tings can provide a structured, monitored environment for such patients. IV. Ambulatory Management of Alcohol Withdrawal This guideline divides recommendations on the man-agement of alcohol withdrawal into two broad categories where withdrawal management services are provided: ambu-latory and inpatient settings. While there are many differences in the services provided within these categories, and services should not ideally be tied to a specific setting, this organiza-tion follows a reasonable structure that seems to match how providers currently think about their practice context. The goal is that practitioners can reference one management section or the other. There are many shared service practices across categories, however, which creates a great deal of repetition across sections. This organization was intentional. As most readers do not read through an entire guideline, the goal was to ensure that each section stands on its own. Within each section, differences between levels of care are highlighted. In ambulatory settings, Level 1-WM is ambu-latory withdrawal management without extended on-site monitoring. This service can be carried out in a physician’s office, by a home health care agency, or an addiction treatment facility. Level 2-WM is ambulatory withdrawal management with extended on-site monitoring. It can be carried out in structured outpatient settings such as a day hospital setting, a general health care or mental health facility, or an addiction treatment facility. Level 2-WM is an organized service with the capacity to provide regular medical assessments and monitor alcohol withdrawal progression. Level 2-WM set-tings may also provide access to psychological or psychiatric treatment (see The ASAM Criteria for additional details). (See Appendix IV.C., for a sum-mary of ambulatory management protocols). The following recommendations apply to both Level 1-WM and Level 2-WM settings unless otherwise specified. Additional recommendations specific to Primary Care set-tings are included in the section VII.A: Primary Care. A. Monitoring Recommendation IV.1: In ambulatory settings, arrange for patients to check in with a qualified health provider (e.g., medical assistant, nurse) daily for up to five days following cessation of (or reduction in) alcohol use. For some patients who are unable to attend daily in-person check-ins, alternating in-person visits with remote check-ins via phone or video call is an appropriate alternative. Recommendation IV.2: Re-assessments should focus on the patient’s health since the last checkup. Clinicians should assess general physical condition, vital signs, hydra-tion, orientation, sleep and emotional status including suicidal thoughts at each visit. Ask about alcohol and other substance use and, if available, measure Blood Alcohol Contend (BAC) with a breathalyzer to detect recent alcohol use. Recommendation IV.3: Alcohol withdrawal severity should be monitored with a validated instrument (see Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 33 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Appendix III for a summary of scales and their associated features). Patients who are able to monitor their own signs and symptoms may use an instrument designed for self-adminis-tration such as the Short Alcohol Withdrawal Scale (SAWS). Recommendation IV.4: In ambulatory settings, patients with a current or past benzodiazepine use disorder need additional monitoring. Recommendation IV.5: For patients managed in an ambulatory setting, the following indications would necessi-tate transfer to a more intensive level of care such as Level 2-WM (if in a Level 1-WM setting) or an inpatient setting: Agitation or severe tremor has not resolved despite having received multiple doses of medication, and the patient will not be continually monitored (e.g., treatment setting is closing) More severe signs or symptoms develop such as persistent vomiting, marked agitation, hallucinations, confusion, or seizure Existing medical or psychiatric condition worsens Patient appears over-sedated Patient returns to alcohol use Syncope, unstable vital signs (low/high blood pressure, low/high heart rate) Discussion. One of the key differences between the Level 1-WM and Level 2-WM levels of care is the frequency and intensity of monitoring they provide. Optimal monitoring frequency is a balance between clinical need and feasibility. While broad ranges of recommended optimal monitoring frequency were found in the literature, the modal recommen-dation seemed to be daily.62,75,76 Face-to-face check-ins with a qualified healthcare provider are preferred. Patients who are unable to come to the treatment setting on a daily basis can be assessed on alternate days via phone or video conference if assessment using that method would not increase the risk of unsafe withdrawal.62 This practice might be reserved for patients in mild withdrawal or who are nearing completion of withdrawal and for patients who have demonstrated com-mitment to the withdrawal management process. The decision to monitor a patient’s progress remotely is at the discretion of the clinician. Monitoring a patient in alcohol withdrawal should include multiple indicators of withdrawal progress and patient health. This includes the patient’s general physical and mental health including vital signs, emotional status and sleep qual-ity.76 Clinicians should ensure that the patient is following directions regarding hydration and nutrition (see IV.B: Sup-portive Care for instructions). The worsening of medical or mental health conditions or circumstances that interfere with a patient’s ability to correct fluid, electrolyte, or nutritional deficiencies indicates the need to reinforce self-care instruc-tions and reassess a patient’s treatment plan and/or level of care. If not included in the withdrawal symptom monitoring scale, orientation should be assessed as an indication of withdrawal severity, possible alcohol or other substance use, and over-sedation from prescribed withdrawal medication. The patient should be asked about alcohol and other drug use at each follow up appointment. If feasible, a breathalyzer should be used to verify that the patient has not been using alcohol recently.71 A breath alcohol test can detect use for approximately 1 hour per standard unit of alcohol consumed, so a negative result does not guarantee the patient has not consumed alcohol since their last appoint-ment.3 A positive result, if the test is properly administered, does indicate that the patient has alcohol in their system. This is particularly important to know if prescribing medication that is dangerous to use in combination with alcohol (i.e., benzodiazepines or phenobarbital). Alcohol use may indicate that the patient is not receiving an adequate dose of medica-tion to ease discomfort from withdrawal and/or reduce crav-ings. It also indicates a clinician should choose a medication for withdrawal with a tolerable safety profile when used in combination with alcohol. It may also indicate that there are circumstances in the patient’s environment that make it difficult to avoid alcohol and that an inpatient setting is more likely to lead to successful withdrawal management. In this case, it is important that alcohol use not lead to ejection from treatment, but rather transfer to a more intensive level of care. The Guideline Committee added that patients with current or past benzodiazepine use disorder will need more intensive monitoring during alcohol withdrawal management. The severity of alcohol withdrawal should be monitored using a validated withdrawal scale.37,38,56,57,62 The same instrument should ideally be used to track signs and symptoms throughout the course of withdrawal.71 Clinicians should ensure that signs and symptoms are not worsening, that patients are responding as expected to medication if provided, and that signs and symptoms are not persisting beyond the expected timeline of withdrawal. Any of these indicate the need to reassess a patient’s treatment plan and/or level of care. As discussed in section II.D: Symptom Assessment Scales, various symptom assessment and monitoring scales have been developed to address circumstances such as a confounding illness or symptom self-reporting barriers (see Appendix III Random I a summary of scales and their associated features). Of most relevance to scale choice in ambulatory settings is clinician- vs. self-administration. While the CIWA-Ar was designed to be administered by a clinician, it can be used by patients or caregivers if given adequate instructions. The SAWS, a 10-item instrument designed to be self-administered, can be used as a supplement while the patient is away from the treatment setting. It has been used and validated in ambulatory settings.61,77,78 Unlike the CIWA-Ar, which is designed to measure in-the-moment signs and symptoms, the SAWS is an up-to-the-moment measure of symptoms in the prior few hours. It was originally written to measure symptoms during the 24 hours prior to patients returning for a daily clinic appointment, although the developers state that the assessment period can be adjusted to whatever is needed, for example, tracking nighttime symp-toms while away from the more extensive monitoring of a Level 2-WM setting.61 While most patients with alcohol withdrawal can be successfully managed in an ambulatory setting, it is important to recognize signs that a more intensive level of care is needed. Patients and caregivers should be informed of warn-ing signs to look for while away from the treatment setting, ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 34 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. and that safe alcohol withdrawal management may necessi-tate transfer to a more intensive level of care if certain indications emerge (see the following section, IV.B: Support-ive Care for patient and caregiver instructions). In settings with less frequent monitoring such as primary care, the threshold for transfer to a more intensive level of care is lower than in settings with more frequent monitoring. If signs or symptoms such as persistent vomiting, agitation, halluci-nations, or confusion develop, patients should be transferred to an inpatient setting as they can presage the onset of electrolyte disturbance, withdrawal seizures, alcohol with-drawal delirium, or Wernicke encephalopathy.79,80 If an existing medical or psychiatric condition worsens despite adequate control of withdrawal symptoms, patients should be transferred to a setting with the resources to manage the condition.73,80 If significant signs or symptoms such as agitation are present despite having received multiple doses of medication or if the patient appears over-sedated at the close of the day, transfer to a setting where the patient can continue to be observed, such as the Emergency Department (ED) or a specialized withdrawal management setting, is warranted.72 Signs of over-sedation include respiratory depression, ataxia, confusion, memory impairment, and delirium. If the patient experiences a loss of consciousness or has unstable vital signs that cannot be attributed to and controlled for by the prescribed treatment regimen, patients should be transferred to a level of care capable of providing the patient with a thorough assessment to properly identify etiology of signs and symptoms as well as provide continuous observation and care. B. Supportive Care Recommendation IV.6: Supportive care is a critical component of alcohol withdrawal management. Providers should ensure patients are educated about what to expect over the course of withdrawal, including common signs and symptoms and how they will be treated. Recommendation IV.7: When treating patients in ambulatory settings, providers should ensure patients/care-givers are educated about monitoring for the development of more severe withdrawal and instructed to create a low-stimu-lation, reassuring environment at home to promote an effective outcome. Recommendation IV.8: Patients should be advised to drink non-caffeinated fluids and that a daily multivitamin may be beneficial. Recommendation IV.9: Patients can be offered oral thiamine. Typical dosing is 100 mg PO per day for 3–5 days. Recommendation IV.10: Clinicians must explain the importance of taking medications as prescribed and confirm the patient’s understanding. Recommendation IV.11: Communicate that safe alcohol withdrawal management may necessitate a transfer to a more intensive level of care including to an inpatient setting and secure the patient’s agreement to transfer if there are indications that management in the ambulatory setting is not safe or effective. See Recommendation IV.5 for indications for transfer to a more intensive level of care. Discussion. Supportive non-pharmacologic care is a critical component of alcohol withdrawal management. Informing patients of what to expect over the course of treatment, offering reassurance, providing a quiet environment and ensuring adequate hydration and nutrition are important aspects of supportive care in all settings. The importance of supportive family members and/or other caregivers is most relevant to the discussion of ambulatory alcohol withdrawal management as patients will spend a portion of their time at home. Several reviews of ambulatory withdrawal manage-ment highlight the role of these individuals as not only one of monitoring signs and symptoms and medication intake, but of offering encouragement and reassurance. Patients and caregivers should be instructed on how to monitor for worsening signs and symptoms including wors-ening anxiety, insomnia and suicidal thoughts. If using a withdrawal severity scale, patients and caregivers should be instructed on how it should be completed.81,82 Important information to convey is the precise meaning of clinical or other terms used in the scale, for example, what constitutes sleep disturbance. It should also be made clear how to score the severity of items, for example, what the meaningful difference is between scoring a 1 (Mild) and a 2 (Moderate) on the restlessness item of the SAWS. The instruction period is also an opportunity to evaluate circumstances that may interfere with scale self-administration, for example through literacy problems or confusion about item severity scores.61 Patients and caregivers should be instructed to create a low stimulation, reassuring environment, since environment is a critical component of success in alcohol withdrawal man-agement and ultimately recovery.83 As volume depletion is a common condition for patients with alcohol withdrawal and intravenous fluids are not provided in ambulatory settings, encouraging consumption of non-caffeinated fluids is impor-tant.36,65,76 Nutritional support is also a consideration in ambulatory alcohol withdrawal management. Some patients may benefit from a daily multivitamin and thiamine supple-ment. Typical dosing of oral thiamine is 100 mg PO daily for three to five days.65 However, oral thiamine is not well absorbed, and thiamine deficiencies can typically be corrected through diet. If Wernicke encephalopathy is suspected, the patient should be transferred to an inpatient setting and receive immediate parenteral administration of thiamine. See Box 5 for more information on Wernicke Encephalopathy and Wernicke-Korsakoff Syndrome. In an early review of ambulatory alcohol withdrawal management, it was explicitly recommended that medications should be administered in the treatment setting rather than at home when possible.39 However, this recommendation has not been repeated in more recent work, and concerns can be addressed by providing only a few take-home doses at a time and ensuring patients understand how to self-administer medications properly. Instructions on warnings for specific medications will be addressed in a later section. Most impor-tantly, patients should be advised about the risk of impairment or overdose if certain medications are combined with alcohol or other substances.7,36 Finally, it is important to explain to patients and care-givers the circumstances under which a transfer to a more Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 35 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. intensive level of care may be necessary, for example if signs and symptoms continue to increase in severity despite taking medication as prescribed. See Recommendation IV.5 for indications for transfer to a more intensive level of care. Explaining this at the beginning of the withdrawal manage-ment process is optimal to ensure a smooth transition if necessary. C. AUD Treatment Initiation and Engagement Recommendation IV.12: When feasible, alcohol use disorder (AUD) treatment should be initiated concurrently with alcohol withdrawal management as cognitive status permits. If appropriate, clinicians should offer to initiate pharmacotherapy for AUD as cognitive status permits. If not initiating AUD treatment themselves, clinicians should explain the range of evidence-based treatment services avail-able in the community, and engage patients with these options. In addition, clinicians may offer information about local recovery support groups, including 12-step groups. Discussion. To the fullest extent possible, patients undergoing alcohol withdrawal management should be engaged, if not initiated, in treatment for alcohol use disorder (AUD) as soon as cognitive status permits. This engagement should be considered part of the withdrawal management process and should not be delayed until withdrawal management is com-plete. There are currently no evidence-based practices for addressing AUD as part of alcohol withdrawal management. In early discussions of ambulatory alcohol withdrawal management, it was recognized that AUD outreach and case management is important for patients.39 In a now-classic study comparing inpatient and outpatient alcohol withdrawal man-agement, it was noted that patients benefit from receiving treatment for AUD in the same outpatient facility at which they complete alcohol withdrawal management.86 As patients in ambulatory settings typically have less severe withdrawal syndromes, treatment initiation and engagement can begin closer to initiating withdrawal management. When discussing AUD, emphasize patient engagement, and offer a variety of treatment and support options, even if the current goal is not abstinence from alcohol. Patients undergoing ambulatory alco-hol withdrawal management in a setting such as primary care represent a less ‘‘captive’’ audience, and therefore more com-mitment may be needed from clinicians to engage patients in continuing treatment. Motivational interviewing or enhance-ment, delivered in primary care settings, has been demonstrated to reduce alcohol and other drug use and to help engage patients in AUD treatment.87,88 Regular follow-up visits at least monthly for a year in Level 1-WM settings may increase the chances of continued recovery, although the Guideline Com-mittee acknowledged that this may not always be realistic. D. Pharmacotherapy (1) Prophylaxis Recommendation IV.13: Patients at risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal may be treated in ambulatory settings at the discretion of providers with extensive experience in management of alcohol withdrawal. Such patients should be provided with preventative pharmacotherapy. Benzodia-zepines are first-line treatment because of their well-docu-mented effectiveness in reducing the signs and symptoms of withdrawal including the incidence of seizure and delirium. Phenobarbital is an appropriate alternative in Level 2-WM setting for providers experienced with its use. For patients with a contraindication for benzodiazepine use, phenobarbital (in Level 2-WM settings by providers experienced with its use) or transfer to a more intensive level of care are appropriate options. Recommendation IV.14: A front loading regimen is recommended for patients at high risk of severe withdrawal syndrome. Providing at least a single dose of preventative medication is appropriate for patients at lower levels of risk who have: A history of severe or complicated withdrawal An acute medical, psychiatric, or surgical illness Severe coronary artery disease Displaying signs or symptoms of withdrawal concurrent with a positive blood alcohol content Recommendation IV.15: Patients at risk of developing new or worsening signs or symptoms of withdrawal while away from the ambulatory treatment setting should be Box 5: Wernicke Encephalopathy and Wernicke-Korsakoff Syndrome84,85 Wernicke encephalopathy is a severe complication resulting from insufficient thiamine in the body. It is characterized by an often reversible acute confusional state. Patients consuming large volumes of alcohol are at an increased risk of developing Wernicke encephalopathy due to inadequate nutrition as well as biological interactions between cellular functioning and alcohol. Thiamine is required for carbohydrate metabolism and plays a key role in normal body functioning. Thiamine deficiency can lead to an increase in pyruvic acid, impaired oxygen uptake, and cerebral tissue damage. Because the body does not synthesize thiamine, daily ingestion through food or supplements is required to maintain adequate metabolic functioning. Thiamine must be converted to different forms to function properly, and alcohol can also impact this conversion by inhibiting key enzymes. Patients with thiamine deficiency often present with signs and symptoms such as confusion, abnormal gaze patterns or nystagmus, ataxia, and possibly delirium. Patients who experience alcohol withdrawal syndrome are at risk of developing Wernicke encephalopathy. Thus, routine practice includes providing patients with thiamine supplements as a preventative measure. If left untreated, Wernicke encephalopathy can progress to chronic Korsakoff syndrome. While Wernicke encephalopathy is a reversible confusional state, if left untreated it can progress to an irreversible syndrome that includes dementia and gait abnormalities. The prevalence of this syndrome ranges between 0–2% worldwide and is not connected to alcohol consumption per capita. Effects of the thiamine deficiency can be found throughout the brain. The damage associated with Wernicke-Korsakoff syndrome seems to occur in a combination of areas including the mammillary bodies, the cerebellum, the frontal lobe, the periaqueductal gray, the thalamus, the walls of the third ventricle, and the floor of the fourth ventricle. Damage to these structures yields the defining features found in the clinical exam consisting of ocular disturbances, mental status changes, gait abnormalities, agitation, and confabulations. Unfortunately, once Wernicke-Korsakoff syndrome occurs, the effects are not reversible and are often progressive. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 36 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. provided with pharmacotherapy. Some indications of risk include a history of withdrawal episodes of at least moderate severity and being within the window for the development of symptoms in the time course of withdrawal. Benzodiazepines, carbamazepine, or gabapentin are all appropriate options for monotherapy. Providing at least a single dose of benzodiaze-pine followed by ongoing treatment according to symptom severity is also appropriate. If the risk of developing worse withdrawal is unknown, patients should be reassessed fre-quently over the next 24 hours to monitor their need for withdrawal medication. Discussion. Determining risk of developing severe or compli-cated withdrawal or complications of withdrawal is addressed in section II: Initial Assessment of Alcohol Withdrawal. As discussed in section III: Level of Care Determination, if there is a risk that patients will develop severe or complicated withdrawal or complications of withdrawal, it should first be determined if ambulatory care is the appropriate level of care. Some providers with extensive experience in managing alco-hol withdrawal may decide to treat at-risk patients in ambulatory settings. Patients at risk of developing severe or complicated withdrawal or complications of withdrawal should receive pharmacotherapy as soon as possible to prevent these signs and symptoms.4,13,89 Benzodiazepines are recommended as the primary medication to prevent the development of severe, complicated or complications of withdrawal. There is clear evidence that benzodiazepines reduce the incidence of alcohol withdrawal seizures and alcohol withdrawal delir-ium.2,13,44,51,90 Phenobarbital can be used as an alternative in Level 2-WM settings, particularly for patients with a contraindication for benzodiazepine use. However, given its narrow therapeutic window and extended half-life, it should only be used by clinicians experienced with its use, particu-larly in ambulatory settings where patients have greater access to alcohol. For patients at high risk of severe withdrawal, front loading with a benzodiazepine is recommended to rapidly achieve therapeutic levels of medication. Front loading has been shown toreducethedurationof treatment andincidence of withdrawal seizure and duration of delirium.2,13,91 Patients should be closely observed for over-sedation and respiratory depression following the administration of a front loading dose. For patients at lower levels of risk, providing at least a single or a few doses of benzodiazepine is appropriate and can be followed by a medication chosen according to symptom severity (see IV.D(2): Withdrawal symptoms). If a clinician determines that a patient is no longer at risk, for example, because risk has been sufficiently mitigated by administration of medication or because the course of withdrawal has passed the period of acute risk, ongoing pharmacotherapy for alcohol withdrawal can be determined according to the severity of a patient’s withdrawal at that time. Some situations which have been called out as appropriate for administering at least a single dose of benzodiazepines include: a history of severe or complicated withdrawal; risk for complications of significant medical, surgical, or psychiatric illness (particularly cardio-vascular disease including coronary artery disease);4 and displaying signs or symptoms of withdrawal concurrent with a positive blood alcohol content (an indication of risk for developing severe withdrawal syndrome). A concern in ambulatory withdrawal management is the lack of continuous observation of patients to identify wors-ening withdrawal syndrome and provide medication to address symptoms if needed. Patients experiencing mild alcohol withdrawal (e.g., CIWA-Ar score <10) who are at low risk of developing severe, complicated, or complications of withdrawal can be managed with supportive non-pharma-cotherapy in both ambulatory and inpatient settings (see Recommendations IV.16 and V.15). In the ambulatory setting, clinicians may want to use medication to prevent the emer-gence of mild or worsening to moderate withdrawal while patients are away from the clinic, meaning the severity threshold for prescribing medication is lower in ambulatory settings than inpatient settings, particularly if there is an indication of risk for symptom development. The recommen-dation that patients with even very mild withdrawal who cannot be monitored be provided medication has been sup-ported in prior guidelines.51 Risk of developing more severe withdrawal is deter-mined, in part, by the severity of previous withdrawal epi-sodes as well as timing (within the 6–36 hour window) of the emergence, peak, and resolution of withdrawal signs and symptoms after cessation of (or reduction in) alcohol con-sumption.2 While withdrawal tends to worsen with each episode, patients with repeated bouts of mild alcohol with-drawal have reported similar signs and symptoms for each episode.46 Sometimes the risk of withdrawal progression is unknown, for example, if patients have not had prior with-drawal episodes or do not know the exact timing of their last drink. It is appropriate to either provide medication or reassess the patient frequently over the next 24 hours, after which more serious withdrawal is unlikely to develop.62 Reassessment can be done in person or over the phone or video chat. Benzodiazepines, carbamazepine, or gabapentin are appropriate options for managing patients at risk of develop-ing mild or moderate withdrawal. These medications are also appropriate for patients already experiencing mild and mod-erate withdrawal as seen in the next section. As in the case of risk of developing severe, complicated, or complications of withdrawal, a dose or doses of a benzodiazepine followed by ongoing treatment according to symptom severity is also appropriate. (See Appendix IV.B., JAM/A192 for a flowchart on pharmacotherapy consider-ations). (2) Withdrawal symptoms Recommendation IV.16: Patients experiencing mild alcohol withdrawal (e.g., CIWA-Ar score <10) who are at minimal risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal may be provided pharmacotherapy or supportive care alone. If pro-viding medication, carbamazepine or gabapentin are appro-priate options. For patients who are at risk of developing new or worsening withdrawal while away from the treatment setting, benzodiazepines, carbamazepine, or gabapentin are appropriate. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 37 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Recommendation IV.17: Patients experiencing mod-erate alcohol withdrawal (e.g., CIWA-Ar scores 10–18) should receive pharmacotherapy. Benzodiazepines are first-line treatment. Carbamazepine or gabapentin are appropriate alternatives. For patients with a contraindication for benzodi-azepine use, carbamazepine, gabapentin, or phenobarbital (in Level 2-WM settings for providers experienced with its use) are appropriate. Carbamazepine, gabapentin, or valproic acid (if no liver disease or childbearing potential) may be used as an adjunct to benzodiazepines. Recommendation IV.18: Patients experiencing severe, but not complicated, alcohol withdrawal (e.g., CIWA-Ar 19), may be treated in ambulatory Level 2-WM settings at the discretion of providers with extensive experience in manage-ment of alcohol withdrawal. Such patients should receive pharmacotherapy. Benzodiazepines are first-line treatment. Phenobarbital is an appropriate alternative for providers experienced with its use. For patients with a contraindication for benzodiazepine use, phenobarbital, carbamazepine, or gabapentin are appropriate. The use of adjunct medications is also appropriate. Recommendation IV.19: If a patient is taking medica-tion as prescribed and symptoms are not controlled as expected: First, consider increasing the dose If over-sedation or inadequate monitoring is a concern: Reassess for appropriate level of care Consider switching medications If using benzodiazepines, consider adding an adjunct med-ication Discussion. Appropriate pharmacotherapy for alcohol with-drawal managed in an ambulatory setting is a balance of alleviating symptoms enough to minimize the likelihood that patients will return to alcohol use and the possibility they will experience negative side effects or other negative outcomes due to medication use. Patients experiencing mild withdrawal (e.g., CIWA-Ar score <10) can be treated with pharmaco-therapy or supportive therapy alone if they are not at risk of symptom progression. Patients experiencing moderate or severe withdrawal (e.g., CIWA-Ar scores 10) should receive pharmacotherapy. Carbamazepine or gabapentin are appropriate for man-aging mild and moderate alcohol withdrawal in patients who are at minimal risk of developing severe or complicated alcohol withdrawal.92–94 There is evidence that carbamazepine and gabapentin are as effective as benzodiazepines as monotherapy for low-risk patients44,94–96 and they have characteristics which increase their favorability compared to benzodiazepines and phenobarbital in ambulatory settings. They have lower risk for drug-alcohol toxicityand are less sedating.58,62,73,92Carbamaz-epine and gabapentin have been shown to decrease craving for alcohol and reduce alcohol consumption after the withdrawal period.92 This may make them particularly beneficial for patients treated in ambulatory settings where the opportunity for exposure to alcohol is greater. As symptom severity or risk of developing severe symptoms increases, medications with well-established effectiveness in preventing the incidence of severe and com-plicated withdrawal are preferred.54 Benzodiazepines are first-line agents for treating moderate58,65,81 and severe alco-hol withdrawal13,58 due to their known effectiveness in pre-venting seizures and delirium.13,90,94 Carbamazepine, gabapentin and phenobarbital can be used for patients with a contraindication for benzodiazepine use.58,65,81,92 However, given its narrow therapeutic window and extended half-life, phenobarbital should only be used in Level 2-WM settings by clinicians experienced with its use, particularly in ambulatory settings where patients have greater likelihood of exposure to alcohol. As discussed in section III: Level of Care Determina-tion, if patients are experiencing severe withdrawal (e.g., CIWA-Ar 19), it should first be determined if ambulatory treatment is the appropriate level of care. Some ambulatory providers with extensive experience in managing alcohol withdrawal may decide to treat patients experiencing severe withdrawal in the absence of confusion or hallucinations indicative of delirium or seizure during the current withdrawal episode in a Level 2-WM ambulatory setting. Benzodiaze-pines are first-line treatment, but phenobarbital is an appro-priate alternative for providers experienced with its use, even if benzodiazepine use is not contraindicated. Patients receiving pharmacotherapy should be moni-tored for signs of response to medication. If patients do not respond as expected, a number of actions can be considered. First, consider increasing the dose. The amount of medication required to control symptoms is variable and ultimately determined by clinical judgment. Patients with more severe withdrawal may require larger doses than are typically seen in other patient populations, particularly during early with-drawal. Providing large doses of benzodiazepine can lead to over-sedation and respiratory depression and patients should be monitored closely. Second, patients should be reassessed for appropriate level of care. Failure to respond may reflect the presence of more severe withdrawal than expected and significant risk of major complications.13 A more intensive level of care may be needed to monitor and manage major complications if they occur.82 Third, consider switching to a different medication. Failure to respond to benzodiazepine may reflect benzodiaz-epine resistance due to kindling (see VI.D: Resistant Alcohol Withdrawal). Higher numbers of previous alcohol withdrawal episodes is associated with decreased responsiveness to ben-zodiazepines.4 Failure to respond may also be due to with-drawal from another GABAergic agent such as gabapentin. In these cases, switching to an alternative medication should be considered. Fourth, if using benzodiazepines, consider adding an adjunct medication. Some patients benefit from the addition of an adjunct medication to control signs and symptoms of withdrawal and their use can be considered as part of the treatment plan. The use of carbamazepine, gabapentin, or valproic acid as an adjunct medication to benzodiazepines is also appropriate for patients experiencing moderate or severe withdrawal. Valproic acid should not be used in patients who ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 38 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. have liver disease, with women of childbearing potential, or as monotherapy for withdrawal. Alpha2-adrenergic agonists (A2AAs) and beta-adrenergic antagonist (beta-blockers) can be used in conjunction with benzodiazepines to manage persistent hypertension or tachycardia.44,97 (3) Benzodiazepine use Recommendation IV.20: While no particular benzodi-azepine agent is more effective than another, longer-acting benzodiazepines are the preferred agents due to the clinical benefits of their longer duration of action. Recommendation IV.21: If waiting for lab test results or if the test(s) are unavailable, if a patient has signs of significant liver disease, use a benzodiazepine with less hepatic metabolization. Recommendation IV.22: Clinicians should monitor patients taking benzodiazepines for signs of over-sedation and respiratory depression. Recommendation IV.23: A benzodiazepine prescrip-tion to treat alcohol withdrawal should be discontinued following treatment. Recommendation IV.24: Clinicians can manage ben-zodiazepine misuse or diversion risk in ambulatory settings by dispensing or prescribing the minimum amount necessary given the patient’s level of stability and timing of their next in-person clinic visit. Alternative medications can also be considered such as carbamazepine or gabapentin. Recommendation IV.25: In ambulatory settings, ben-zodiazepines should not be prescribed to patients with a history of even mild adverse events with benzodiazepine use because rapid intervention is not typically available. Benzodiazepines can be used with caution in patients with a high risk of benzodiazepine diversion including patients with a current or past benzodiazepine use disorder for the short period of acute alcohol withdrawal. Risk can be man-aged by dispensing or prescribing a small number of doses. Recommendation IV.26: Patients who are taking ben-zodiazepines, and their caregivers, should be educated regard-ing: The danger of drug-drug interactions between benzodia-zepines and other CNS depressants (impairment and respi-ratory depression) The risks associated with combining alcohol and benzo-diazepines and importance of abstinence from alcohol The risks associated with driving or use of heavy machin-ery for the first few days of benzodiazepine administration Instructions to reduce their benzodiazepine dose if drowsiness occurs Discussion. Benzodiazepines are commonly recommended as first-line agents for managing most forms of alcohol with-drawal.13,90 Diazepam, lorazepam, and chlordiazepoxide are the most frequently used in treating alcohol withdrawal. While there is no evidence showing superiority of effective-ness among benzodiazepine agents,13,90 longer-acting agents are preferred by many clinicians.2,51,81 A long duration of action contributes to a smoother course of withdrawal and greater control of breakthrough and rebound signs or symp-toms. This provides greater coverage for preventing alcohol withdrawal seizures and delirium.90,98 For this reason, patients prescribed a shorter-acting agent should have a more gradual taper and be reassessed more frequently (see IV.D(4): Benzodiazepine dosing regimens). Longer-acting agents can accumulate and lead to over-sedation and respiratory depression, particularly in older patients or those with compromised health. Other signs of over-sedation include ataxia, confusion, memory impairment, and delirium, which may be difficult to differentiate from alcohol withdrawal-related delirium.2 Benzodiazepine asso-ciated delirium has been diagnosed by the administration of flumazenil, a GABA-A receptor antagonist, but this protocol was not reviewed by the Guideline Committee.99 A reduction in the benzodiazepine dose and the addition of a neuroleptic agent to control for agitation and/or confusion can be consid-ered if patients are not at an elevated risk of seizure (i.e., they are outside of the acute risk window).2 Some neuroleptic agents have been shown to reduce the seizure threshold. Benzodiazepine accumulation is more likely in patients with impaired hepatic function. Medication dose can be reduced or a benzodiazepine with less dependence on hepatic metabolism can be used (see section VII.D: Patients with Medical Conditions). Laboratory testing recommended in section II.E: Identify Concurrent Conditions can indicate the need to adjust the treatment plan, but as treatment should not be delayed while waiting for lab test(s) results or if the test(s) are unavailable at the treatment setting, if a patient has signs of significant liver disease, reduce the dose or use a benzodiazepine with less hepatic metabolization. Signs of significant liver disease include: Skin and eyes that appear yellowish (jaundice) Swelling in the legs and ankles (edema) Itchy skin Dark urine color Pale stool color, or bloody or tar-colored stool Confusion Chronic fatigue Nausea or vomiting Benzodiazepines prescribed for alcohol withdrawal should be discontinued after withdrawal is complete. Patients are at risk of developing a physiological dependence on benzodiazepines, developing a benzodiazepine use disorder, or experiencing benzodiazepine withdrawal. The decision process for determining appropriate duration of treatment is affected by the amount of benzodiazepine used during the acute withdrawal period, particularly when seizure or delirium has occurred, and any associated physiological dependence that may have developed.51 Managing the phe-nomenon of protracted withdrawal, where subacute symptoms of irritability, anxiety and sleep disturbances can persist for weeks, is beyond the scope of the current guideline. In ambulatory settings, benzodiazepine use has liabili-ties not present in inpatient settings. Risk can be managed by dispensing or prescribing a very small number of doses, with some suggesting providing only enough medication for one day.38,39,72,100,101 The Clinical Champions determined that recommending daily prescriptions might be too restrictive and Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 39 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. giving enough medication until a patient’s next appointment (e.g., 1–3 three days) is reasonable. They noted that these considerations are relevant primarily for benzodiazepine pre-scriptions due to the risk involved and that they would be comfortable giving several days’ worth of carbamazepine or gabapentin due to lower risk for diversion and/or drug-drug interactions. There are some situations where the risks of benzodiaz-epine use outweigh the benefits in an ambulatory management setting. In these cases, patients can be offered an alternative medication rather than transfer to inpatient treatment. Patients with a history of even mild adverse events with benzodiazepine use should not be prescribed benzodiazepines for ambulatory withdrawal because of the lack of continuous monitoring. Benzodiazepines should be prescribed with extra cau-tion to some patients if managed by dispensing or prescribing a very small number of doses and more frequent monitoring. Patients with a high risk of benzodiazepine misuse or diver-sion (history of previous misuse or diversion or another household member with a history of misuse or diversion of benzodiazepines) and patients with a current or past benzodi-azepine use disorder can be prescribed benzodiazepines if managed cautiously. The potential for misuse is limited during the short period of supervised alcohol withdrawal.62 It is critical that patients who are prescribed benzodia-zepines and their caregivers understand the danger of drug-drug interactions with this medication.71,100 As respiratory depression and death can result from the combination of alcohol or opioids with benzodiazepines, clinicians should emphasize the importance of not using alcohol or other drugs during withdrawal management.71,100 Patients should also be warned about the risk of drowsiness and advised not to drive or use heavy machinery for the first few days of taking benzodiazepines.75 Patients should be advised to reduce the dose if drowsiness occurs.41 (4) Benzodiazepine dosing regimens Recommendation IV.27: At short-term observational settings with continuous monitoring (e.g. Level 2-WM), symptom-triggered treatment conducted by trained staff is the preferred benzodiazepine dosing method. Front loading while under clinical supervision or fixed dosing with addi-tional as-needed medication are also appropriate. Recommendation IV.28: At settings without extended on-site monitoring (Level 1-WM), symptom-triggered dosing is appropriate if patients or a caregiver can reliably monitor signs and symptoms with a withdrawal severity scale and follow dosing guidance. Otherwise, front loading while under clinical supervision or fixed dosing with additional as-needed medication is appropriate. Recommendation IV.29: Front loading is recom-mended for patients experiencing severe alcohol withdrawal (e.g., CIWA-Ar 19). Diazepam and chlordiazepoxide are preferred agents for front loading. Recommendation IV.30: When using a fixed-dose schedule, patients’ signs and symptoms should still be moni-tored. A few additional additional take-home doses can be provided to take as needed. When initiating a fixed-dose regimen, arrange for the patients to be followed up with the following day to modify the dose if needed. Recommendation IV.31: If prescribing a shorter-acting benzodiazepine, using a fixed-dose regimen with a gradual taper may be appropriate to reduce the likelihood of break-through and rebound signs and symptoms. Discussion. Examples for these dosing regimens can be found in Appendix V. Multiple dosing strategies have been used to administer benzodiazepines during alcohol withdrawal. In general, symptom-triggered treatment is the preferred dosing method,4,36 but there is disagreement regarding its appropri-ateness for ambulatory settings. In this regimen, medication is administered only when patients are experiencing significant withdrawal symptoms according to a severity scale. This allows dosing to be individualized according to symptom severity and reduces the risk of under- and over-treating by assessing and dosing according to real-time symptom severity. It is possible that very large doses of medication will be needed rapidly, and reduced as symptoms resolve.2,13 Symp-tom-triggered dosing has been shown to reduce the duration of treatment and inpatient length of stay compared to a fixed-dose schedule.2,21,44,51 The disagreement regarding its appropriateness for ambulatory settings generally hinges on how signs and symp-toms will be assessed and by whom. Symptom-triggered treatment is appropriate when conducted by healthcare pro-fessionals in Level 2-WM settings.2,13,21 In Level 1-WM settings, where symptoms would be assessed by caregivers or patients themselves, most prior guidelines have only con-sidered the use of the CIWA-Ar, which requires training in order to score reliably. Other symptom assessment instru-ments such as the Short Alcohol Withdrawal Scale (SAWS) are designed to be self-administered and used in ambulatory settings.81 Symptom-triggered treatment using the SAWS has been shown to be as safe and effective as a fixed-dose scheduled taper in an open-label RCT of outpatients.78 Other sources, including the Guideline Committee, argue that the CIWA-Ar can be administered by patients or caregivers for symptom-triggered treatment if given sufficient instruction.82 If patients meet criteria for treatment in a Level 1-WM setting and they or a caregiver can reliably assess signs and symptoms and follow guidance to determine whether a dose is needed, symptom triggered treatment is an appropriate option. Fixed dosing is also appropriate in ambulatory settings. In a fixed-dose regimen, set amounts of medication are administered at regular intervals, and the dose amount, dosing frequency, or both are gradually tapered according to a set schedule. While fixed dosing is easy to administer, over- or underestimating the amount of benzodiazepine needed may lead to insufficient symptom control and over-sedation.26 With fixed dosing, additional take home doses should be provided in the event symptoms are not adequately con-trolled.58 Fixed-dose regimens do not eliminate the need for frequent monitoring and dose adjustment.7,42 When initi-ating a fixed-dose regimen in an ambulatory setting, patients should be reassessed the next day to modify the dose, if needed. Front loading conducted by trained staff is also appro-priate in ambulatory settings and is preferred for patients at ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 40 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. risk for or experiencing severe alcohol withdrawal (e.g., CIWA-Ar scores 19).45,70 Front loading describes when a moderate to high dose of a long-acting benzodiazepine is administered to achieve rapid control of withdrawal signs and symptoms and is allowed to taper through metabolism. Diazepam and chlordiazepoxide are the preferred agents for front loading. This regimen is typically used when rapid administration of a benzodiazepine is required, either because the patient is experiencing significant symptoms or is at risk of developing them. Front loading has been shown to reduce the duration of treatment, incidence of withdrawal seizure, and duration of delirium.102 This effect is usually attributed to the rapid administration of large amounts of benzodiazepines early in the withdrawal period.2,13 A front-loading regimen can be driven by a withdrawal severity scale (e.g., 10 mg diazepam PO every hour if CIWA-Ar score 10) or according to a fixed schedule (e.g., 20 mg diazepam PO every 2 hours for 3 doses). Symp-tom-triggered front loading has been shown to reduce symp-tom duration and the amount of benzodiazepine used,70,103– 106 the incidence of withdrawal seizures, and the duration of delirium for patients being treated in the Intensive Care Unit (ICU).102 Fixed-dose front loading can be used with patients for whom it would be difficult to obtain an accurate score on a withdrawal severity scale. Clinicians should monitor patients closely before and after providing a front loading dose for signs of over-sedation and respiratory depression as doses are more frequent with this regimen.7,42 The need to observe patients does not necessarily preclude front loading in a Level 1-WM setting, as symptoms can often resolve in as few as 2-3 doses. Because of their shorter duration of action, short-acting benzodiazepine concentrations can diminish rapidly, increas-ing the chance for rebound and breakthrough symptoms and signs including seizure. For this reason, a fixed-dose schedule with a long taper may be more feasible than a symptom-triggered dosing regimen requiring very frequent reassess-ment. Shorter-acting benzodiazepines should be tapered care-fully even after withdrawal resolves to prevent the development of rebound or breakthrough signs and symp-toms. If the CIWA-Ar is used in conjunction with short acting benzodiazepines, the assessments should be done promptly in order to prevent seizures due to protocol errors.98 (5) Carbamazepine, gabapentin, valproic acid Recommendation IV.32: Gabapentin is a favorable choice for treating alcohol withdrawal when a clinician also plans to use it for a patient’s ongoing treatment of alcohol use disorder. Recommendation IV.33: If benzodiazepines are con-traindicated, carbamazepine or gabapentin are appropriate alternatives. Recommendation IV.34: Carbamazepine, gabapentin, or valproic acid may be used as an adjunct to benzodiazepine therapy to help control alcohol withdrawal. Before using as an adjunct, clinicians should ensure that an adequate dose of benzodiazepine has been administered. Recommendation IV.35: Valproic acid should not be used in patients who have liver disease or women of childbearing potential. Recommendation IV.36: There is insufficient evidence to support the use of valproic acid as monotherapy for the treatment of alcohol withdrawal. Discussion. Evidence suggests that anticonvulsants, particu-larly carbamazepine, are effective at preventing alcohol with-drawal progression, seizures and delirium.4 At this time, there is insufficient evidence to support their use over benzodia-zepines for patients at increased risk of severe withdrawal, seizures, or delirium. 2,13,38,92,107 As the efficacy of benzo-diazepines is well-established, there have been ethical con-cerns with running placebo-controlled or treatment-as-usual-controlled (i.e., compared to benzodiazepines) studies in at-risk populations.2,92 Carbamazepine or gabapentin are appropriate medica-tions for treating low risk patients. They are also appropriate alternatives for patients with a benzodiazepine contraindica-tion. Gabapentin may provide an effective bridge therapy from alcohol withdrawal treatment to long-term alcohol use disorder treatment.92,93 It has been found to improve rates of abstinence and reduce heavy drinking days compared with placebo during the maintenance phase of alcohol use disorder treatment.62 See Box 6 for caution regarding gabapentin misuse and diversion. Some patients benefit from the addition of an adjunct medication to control signs and symptoms of withdrawal. Use of carbamazepine, gabapentin, or valproic acid as an adjunct to benzodiazepines may be appropriate. For patients in severe withdrawal, other medications can be used to manage signs and symptoms if benzodiazepines are already being given.36 Before using an adjunct medication, clinicians should ensure that an adequate dose of benzodiazepine has been adminis-tered since large doses of benzodiazepine are sometimes needed to control withdrawal. While valproic acid has been found to be promising for the treatment of alcohol withdrawal, more evidence is needed before it can be recommended as monotherapy.62,109 Its use as Box 6: Gabapentin misuse, abuse, and diversion108 The FDA approved the use of gabapentin for the treatment of epilepsy and post-herpetic neuralgia. However, gabapentin has commonly been used off-label for the treatment of various other conditions, including alcohol use disorder or chronic pain. When the development process for this guideline began, gabapentin was largely perceived as safe and having limited potential for misuse or abuse and was not classified as a controlled substance in most of the country. However, with the increased use of gabapentin in the treatment of other conditions, some states have identified the potential risk for misuse, abuse, and diversion and have reclassified gabapentin as a Schedule–V medication. A systematic review examining gabapentin’s misuse, abuse and diversion potential found evidence to support the risk associated with prescribing gabapentin. Although gabapentin was only misused by 1% of the general population, 40–65% of individuals prescribed gabapentin have misused or abused the medication. Similarly, patients with a substance use disorder were more likely to misuse gabapentin. Given this recent evidence, the recommendations made in this guideline pertaining to the risk of misuse, abuse, or diversion of gabapentin should be interpreted cautiously. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 41 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. an adjunct to benzodiazepines is supported.2,13,44,58 However, valproic acid should not be used in patients with hematologi-cal or hepatic disorders including acute liver impairment44 or in women of childbearing potential because of teratogenic risk.110 (6) Phenobarbital Recommendation IV.37: Phenobarbital can be used for some patients in Level 2-WM ambulatory settings; however, it should only be used by clinicians experienced with its use given its narrow therapeutic window and side effects. Recommendation IV.38: In a Level 2-WM ambulatory setting, phenobarbital monotherapy, managed by a clinician experienced with its use, is an appropriate alternative to benzodiazepines for patients who are experiencing severe alcohol withdrawal or who are at risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal. Recommendation IV.39: In a Level 2-WM ambulatory setting, phenobarbital monotherapy, managed by a clinician experienced with its use, is appropriate for patients with a contraindication for benzodiazepine use who are experiencing moderate or severe alcohol withdrawal or who are at risk of developing severe or complicated alcohol withdrawal or complication of alcohol withdrawal. Discussion. There is disagreement in the literature regarding the appropriateness of phenobarbital in ambulatory settings, due to the risk of toxicity when used in combination with alcohol or in high doses.13,97,111 In general, phenobarbital should only be used by clinicians experienced with its use in settings that offer close monitoring. Phenobarbital may cause respiratory depression and over-sedation and its narrow ther-apeutic window makes it challenging to dose correctly com-pared to other medications used to treat alcohol withdrawal. As with benzodiazepines, effects on the central nervous system are exacerbated when other CNS depressants such as alcohol are also used. Phenobarbital may be appropriate in Level 2-WM ambulatory settings (e.g., ambulatory settings with extended onsite monitoring) as an alternative to benzodiazepines when benzodiazepine use is contraindicated. Phenobarbital is appropriate for such patients experiencing moderate or severe withdrawal or who are at risk of developing severe or com-plicated alcohol withdrawal or complications of alcohol withdrawal. Phenobarbital is also an appropriate benzodiaze-pine alternative outright for patients experiencing or who are at risk of developing severe or complicated alcohol with-drawal or complications of alcohol withdrawal. See Box 7 for more information on phenobarbital. (7) A2AAs and beta-blockers Recommendation IV.40: Alpha2-adrenergic agonists (A2AAs) such as clonidine can be used as an adjunct to benzodiazepine therapy to control autonomic hyperactivity and anxiety when symptoms are not controlled by benzodia-zepines alone. They should not be used alone to prevent or treat withdrawal-related seizures or delirium. Recommendation IV.41: Beta-adrenergic antagonists (beta-blockers) can be used as an adjunct to benzodiazepines in select patients for control of persistent hypertension or tachycardia when these signs are not controlled by benzodia-zepines alone. They should not be used to prevent or treat alcohol withdrawal seizures. Discussion. No existing guidance or evidence was found regarding the use of alpha2-adrenergic agonists (A2AAs) and beta-adrenergic antagonists (beta-blockers) in ambulatory settings. Many patients in alcohol withdrawal experience cardiac or adrenergic symptoms such as hypertension and tachycar-dia. These symptoms can be addressed by treating medical problems commonly seen in patients with alcohol withdrawal syndrome, such as dehydration and electrolyte imbalances or through use of benzodiazepines. A2AAs and beta-blockers can be used in conjunction with benzodiazepines to manage persistent hypertension or tachycardia.44,97 While these med-ications reduce the signs of sympathetic activation, they do not treat the underlying pathophysiology, which may mask the hyperadrenergic state and lead to a false perception that these signs are properly treated. They also do not prevent with-drawal-related seizures or delirium and should not be used alone in the treatment of alcohol withdrawal. See Box 8 for more information on A2AAs and beta-blockers. (8) Inappropriate medications Recommendation IV.42: Oral or intravenous alcohol should not be used for the prevention or treatment of alcohol withdrawal. Recommendation IV.43: There is insufficient evidence to support the use of baclofen for the treatment of alcohol withdrawal. Box 7: History of phenobarbital use in the treatment of alcohol withdrawal128–130 Phenobarbital is the first medication to be used successfully to treat alcohol withdrawal in a predictable way. It has been used for this purpose since the 1920’s after first being introduced in 1912 for the treatment of seizures. It exerts its effects on the GABA-A receptor by increasing the duration of channel opening when bound to GABA, which increases the hyperpolarization of the neuron, thus indirectly increasing the sedative effects of the ‘‘GABA system.’’ It also has direct blockade effects on excitatory glutamate signaling. Given these two mechanisms, it seems to be a perfect fit for the treatment of alcohol withdrawal, which creates an imbalance in these two systems. And, in experienced hands, it can be very effective. However, phenobarbital has a number of side effects including bradycardia, bradypnea, hypothermia, hypotension, pulmonary edema, acute renal failure and Steven-Johnson syndrome. It has a half-life of up to seven days, is primarily metabolized by the liver and induces many isoenzymes of the P450 system. This coupled, with a relatively narrow therapeutic window, caused it to fall out of favor in the 1960’s as chlordiazepoxide and oxazepam were shown to be as effective, but harbor a much lower risk. Now we have solid data that supports the use of GABA sensitive antiepileptiform medications that are as effective, require less training, and have a much lower side effect profile than phenobarbital or benzodiazepines. While, there is a current reemergence of interest in phenobarbital as a standalone therapy for alcohol withdrawal, these guidelines have taken into account history and comparative safety when developing the evidence-based recommendations for its use in the population as a whole. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 42 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Recommendation IV.44: Providing magnesium as a prophylaxis or treatment for alcohol withdrawal management has no supporting evidence. Discussion. While ethyl alcohol administration has been used to manage withdrawal, it is not recommended.2,13,58 Admin-istration of oral or intravenous alcohol has no proven efficacy, no accepted protocols, and known toxicity.13 A recent Cochrane review of three RCTs on the use of baclofen for alcohol withdrawal treatment drew no conclu-sions about efficacy or safety of baclofen due to insufficient and low quality evidence.112,113 ASAM’s 2004 guideline, ‘‘Management of Alcohol Withdrawal Delirium’’, suggested that magnesium may reduce neuromuscular activity in patients experiencing alcohol with-drawal delirium. However, a recent Cochrane review114 con-cluded that there is not enough evidence to determine the benefit of magnesium in alcohol withdrawal prevention or management, which is in agreement with the ASAM’s 1997 guideline on alcohol withdrawal management.13 V. Inpatient Management of Alcohol Withdrawal This guideline divides recommendations on the man-agement of alcohol withdrawal into two broad categories where withdrawal management services are provided: ambu-latory and inpatient settings. While there are many differences in the services provided within these categories, and services should not ideally be tied to a specific setting, this organiza-tion follows a reasonable structure that seems to match how providers currently think about their practice context. The goal is that practitioners can reference one management section or the other. There are many shared service practices across categories, however, which creates a great deal of repetition across sections. This organization was intentional. As most readers do not read through an entire guideline, the goal was to ensure that each section stands on its own. The section applies to inpatient settings where with-drawal management is provided. This includes two Level 3 settings and one Level 4 setting as defined in The ASAM Criteria. These levels of care are primarily differentiated by the intensity of clinical services and medical training of staff. Level 3.2-WM clinically managed residential withdrawal management is a residential service providing 24-hour struc-ture and support by trained, non-medical staff. They may have concurrent medical services equivalent to primary care, but medical care is not provided 24/7. In some programs, staff supervise patients as they self-administer medications. Level 3.7-WM medically monitored inpatient withdrawal manage-ment is a residential service providing 24-hour structure and support by medical and nursing staff. They may be located in a specialty addiction treatment or mental health setting with addiction treatment services. Level 4-WM medically managed intensive inpatient withdrawal management is a medical or psychiatric hospital service with an addiction specialist phy-sician (see The ASAM Criteria for additional details). (See Appendix IV.D., for a sum-mary inpatient management protocol). This section is primarily informed by the extensive body of research conducted in hospital settings. However, they should apply to all inpatient settings unless otherwise speci-fied (e.g., treatment in Intensive Care Unit [ICU] or Cardiac/ Coronary Care Unit [CCU]). Additional recommendations specific to hospitalized patients or patients that are hospital-ized primarily for a reason other than alcohol withdrawal are included in the section VII: Specific Settings and Populations. A. Monitoring Recommendation V.1: The following monitoring schedule is appropriate: In patients with moderate to severe withdrawal or those requiring pharmacotherapy, re-assess every 1–4hours for 24hours, as clinically indicated. Once stabilized (e.g., CIWA-Arscore< 10for24hours),monitoringcanbeextended to every 4–8hours for 24hours, as clinically indicated. Patients with mild withdrawal and low risk of complicated withdrawal may be observed for up to 36 hours, after which more severe withdrawal is unlikely to develop. Recommendation V.2: Monitor patients’ vital signs, hydration, orientation, sleep, and emotional status including suicidal thoughts. Recommendation V.3: Monitor patients receiving pharmacotherapy for alcohol withdrawal for signs of over-sedation and respiratory depression. Recommendation V.4: Signs and symptoms of alcohol withdrawal should be monitored during withdrawal manage-ment with a validated assessment scale (see Appendix III for a summary of scales and their associated features). Discussion. Optimal monitoring frequency is a balance between clinical need and feasibility. Many sources, including The ASAM Criteria, designate appropriate thresholds for Box 8: Alpha2-adrenergic agonists and beta-adrenergic antagonists Alpha2-adrenergic agonists (A2AAs) and beta-adrenergic antagonists (beta-blockers) can be used in conjunction with benzodiazepines to manage persistent hypertension or tachycardia. A2AAs bind to receptors inhibiting the release of norepinephrine from the presynaptic neuron. The release of norepinephrine would cause an increase in activity of the sympathetic pathway leading to increased heart rate and blood pressure. Therefore, A2AAs reduce cardiac output and reduce tachycardia and hypertension. Beta-blockers have a different mechanism of action. Normally norepinephrine released from sympathetic nerves binds to beta-adrenoceptors resulting in activation of the sympathetic pathway causing an increase in heart rate and blood pressure. However, beta-blockers compete with norepinephrine and epinephrine for the same binding site. Thus norepinephrine is unable to bind to the site, which reduces the signs of sympathetic activity including heart rate and blood pressure. Unlike A2AAs, beta-blockers do not reduce sympathetic activity but rather mask signs and symptoms associated with sympathetic activation such as tachycardia and hypertension. These medications do not treat the underlying pathophysiology, but reduce signs, which may mask the hyperadrenergic state and lead to a false perception that these signs are properly treated. Although not explicitly rated by the Guideline Committee, persistent hypertension or tachycardia may be reasons to transfer patients to an inpatient setting. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 43 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. frequency of monitoring. In a review of studies comparing inpatient with outpatient alcohol withdrawal management, monitoring intervals ranged from 30 minutes to 8 hours.42,45 Monitoring of patients experiencing moderate and severe withdrawal or patients experiencing mild withdrawal who are at increased risk for developing severe, complicated, or complications of withdrawal should initially be conducted every 1–4 hours or as clinically indicated.7 Monitoring fre-quency can be reduced to every 4–8 hours or as clinically indicated for stabilized patients, usually defined as having controlled symptoms (e.g., CIWA-Ar score < 10) for 24 hours.45 Patients experiencing mild withdrawal who are at mini-mal risk for developing severe, complicated, or complications of withdrawal can be observed for a shorter duration of up to 36 hours, after which more severe withdrawal is unlikely to develop.41 Optimal frequency of monitoring for patients in mild withdrawal was not established by the Guideline Com-mittee, and they determined that frequency would be driven more by the complicating factor(s) that led a patient in mild withdrawal to be treated in an inpatient setting. Signs and symptoms of alcohol withdrawal should be monitored using a validated withdrawal severity scale.13,45 As discussed in section II.D: Symptom Assessment Scales, vari-ous symptom assessment scales have been developed to address circumstances such as a confounding illness or symp-tom self-reporting barriers (see Appendix III for a summary of scales and their associated features). Clinicians should ensure that signs and symptoms are not worsening, that patients are responding as expected to medication if provided, and that signs and symptoms are not persisting beyond the expected timeline of withdrawal. Any of these indicate the need to reassess a patient’s treatment plan and/or level of care. Monitoring should consist of assessing a patient’s vital signs, hydration, orientation, sleep, and emotional status including suicidal thoughts.36,115 Fluid intake and output can be tracked in hospital settings, but they can be monitored by patient report and observing for signs of dehydration in other inpatient settings.36 Orientation, sleep quality and emo-tional status including suicidality should be monitored. Ori-entation and anxiety are included in many withdrawal severity scales. Poor orientation can also indicate over-sedation from prescribed withdrawal medication. Patients receiving phar-macotherapy for alcohol withdrawal should be monitored for other signs of over-sedation and respiratory depression including ataxia, confusion, memory impairment, and delir-ium. B. Supportive Care Recommendation V.5: Supportive care is a critical component of alcohol withdrawal management. Frequent reassurance, re-orientation to time and place, and nursing care are recommended non-pharmacological interventions. Providers should ensure patients are educated about what to expect over the course of withdrawal, including common signs and symptoms and how they will be treated. Patients with severe alcohol withdrawal should be cared for in an evenly lit, quiet room. Patients should be offered hope and the expectation of recovery. Recommendation V.6: Supportive care for alcohol withdrawal patients includes adherence to safety measures and protocols (e.g., assess risk for fall/syncope). If available and applicable, existing institutional/hospital-associated delirium protocols can be used for supportive care of patients with severe alcohol withdrawal. Recommendation V.7: Thiamine should be provided to prevent Wernicke encephalopathy. Intravenous (IV) or intramuscular (IM) administration of thiamine is preferred, in particular for patients with poor nutritional status, malabsorption, or who are known to have severe complications of alcohol withdrawal. Typical dosing is 100 mg IV/IM per day for 3–5 days. Oral thiamine also can also be offered. Patients also receiving glucose can be administered thia-mine and glucose in any order or concurrently. Recommendation V.8: Clinicians should administer thiamine to patients admitted to the ICU to treat alcohol withdrawal. Recommendation V.9: For patients with hypomagne-semia, cardiac arrhythmias, electrolyte disturbances, or a previous history of alcohol withdrawal seizures, magnesium should be administered. Recommendation V.10: If phosphorus is <1 mg/dL, supplementation should be provided. Otherwise, in the case of moderate hypophosphatemia (1–2 mg/dL), correction through proper nutrition is recommended. Recommendation V.11: In patients who are critically ill, folate supplementation may be considered, since chronic alcohol use is associated with hyperhomocysteinemia. Discussion. Supportive non-pharmacologic care is a critical component of alcohol withdrawal management. While empir-ical research on many of the components of supportive care is not available, existing reviews and guidelines support inter-ventions such as informing patients of what to expect over the course of treatment and providing frequent reassurance,4 reality orientation, and general nursing care during treat-ment.2,7,36,70,115 Also emphasized was providing care in a quiet, evenly-lit room.2,7,14,36,70,116 Non-pharmacological supportive care also includes fol-lowing standard care protocols and safety protocols. Safety measures such as fall precautions and routine nurse check-ins and assistance with activities of daily living (ADLs) ensures patient safety and provides autonomy. For facilities with a hospital-associated delirium protocol, clinicians should implement the protocol to prevent and reduce the incidence and duration of acute delirium among patients with severe alcohol withdrawal. Studies have shown standardized proto-cols to be effective at reducing the incidence, duration, and frequency of delirium among hospitalized patients.117 Determining risk for Wernicke is not standardized. For example, the NICE guideline recommends parenteral admin-istration of thiamine to any hospitalized patient who is a harmful or dependent drinker.21 At least one Wernicke encephalopathy risk assessment scale for patients withdraw-ing from alcohol has been developed.118 The presence of risk factors for Wernicke encephalopathy (malnutrition or poor ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 44 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. diet, weight loss, vomiting, confusion, or other neurological symptoms) is scored depending on severity and results indi-cate whether enteral or parenteral thiamine should be admin-istered. Previous guidelines, including the previous ASAM alcohol withdrawal management guideline, have recom-mended that IV thiamine be given prior to intravenous glucose.14,36,44,115 The reasoning was that thiamine is neces-sary for carbohydrate metabolism and thiamine deficiency can lead to decreased absorption of glucose, perhaps precipi-tating acute Wernicke encephalopathy. However, there is a lack of clinical evidence to support this theory and it is important that glucose delivery not be delayed in patients who are nutritionally compromised. The Guideline Commit-tee concluded that it is not necessary to administer thiamine prior to glucose, that these could be provided in any order or concurrently in order to not delay treatment. Other common deficiencies seen in patients with alcohol withdrawal include low folate, magnesium, phosphorous and potassium. While early work recommended an aggressive approach to correcting nutritional deficiencies, more recent thinking is that levels self-correct rapidly with improved diet. The Guideline Committee supported a conservative stance of selectivelycorrectinghypomagnesemia,hypokalemia,andacute severe hypophosphatemia (serum phosphate < 1mg/dL) when they are detected through laboratory testing.115 Magnesium can also be routinely supplemented in patients with cardiac arrhyth-miasoraprevioushistoryofalcoholwithdrawalseizures.7Folate supplementation with 1mg daily can also be considered for patients who are critically ill because folate is not included in the recommended routine laboratory tests and chronic alcohol use is associated with hyperhomocysteinemia resulting from folate deficiency.26,45,115 C. AUD Treatment Initiation and Engagement Recommendation V.12: The period of alcohol with-drawal management should be used to engage patients with an alcohol use disorder (AUD) with comprehensive treatment. When feasible, AUD treatment should be initiated concur-rently with alcohol withdrawal management as cognitive status permits. If appropriate, clinicians should also offer to initiate pharmacotherapy for AUD as cognitive status permits. Clinicians should explain the range of evidence-based treat-ment services available at the current site and in the commu-nity. Finally, clinicians should proactively connect patients to treatment services as seamlessly as possible, including initi-ating a warm handoff to treatment providers. Discussion. One important function of supportive care is to connect with patients to help facilitate continuing treatment.2 It is widely recognized that alcohol withdrawal man-agement alone is not a treatment for alcohol use disorder (AUD). The need for alcohol withdrawal management ser-vices almost universally signifies the presence of an alcohol use disorder and need for treatment. The Guideline Commit-tee agreed that it should be explicitly communicated to alcohol withdrawal patients if they have an alcohol use disorder and engaged with treatment for that disorder. Several leading clinical guidelines conclude that the success of an alcohol withdrawal management episode is defined not only by the acute management of withdrawal signs and symptoms, but by the engagement in continued treatment for alcohol use disorder by patients.4,12,80 Whenever possible, AUD treatment should be initiated concurrent with alcohol withdrawal management as cognitive status permits.12 At a minimum, clinicians should proactively connect patients to AUD treatment services and transition patients as seam-lessly as possible through a warm handoff to treatment providers. Despite the clear and frequently stated importance of the transition between withdrawal management and long-term AUD treatment, research on optimal strategies is extremely sparse. More recently, studies are including follow-up mea-sures such as entry into AUD treatment following withdrawal completion, but this is rarely a primary outcome of interest. One RCT conducted in the United States119 found that participants who received three Motivational Interviewing sessions during inpatient withdrawal treatment were more likely to attend self-help groups two months after discharge compared to control participants, but were not more likely to be abstinent or engage in formal AUD treatment. Another method of improving AUD treatment initiation may result from changes in health care system integrations and payment structures. Successfully transitioning patients from alcohol withdrawal management to alcohol use disorder treatment will result in fewer repeat alcohol withdrawal management episodes, and therefore better outcomes and lower cost. Initiating AUD treatment after alcohol withdrawal can be used as a performance measure or integrated into reimbursement contracts as ‘‘to not include facilitation of treatment entry would be considered inadequate and incom-plete treatment.’’4 (p7) Levels of care that are part of ‘‘inte-grated systems of care which are accountable (financially and otherwise) for health outcomes will be highly motivated to use the withdrawal management encounter as an opportunity to identify cases of addiction that need to be treated and other-wise may have escaped identification.’’12 (p129 D. Pharmacotherapy (1) Prophylaxis Recommendation V.13: For patients at risk of devel-oping severe or complicated alcohol withdrawal or compli-cations of alcohol withdrawal, preventative pharmacotherapy should be provided. Benzodiazepines are first-line treatment because of their well-documented effectiveness in reducing the signs and symptoms of withdrawal including the incidence of seizure and delirium. For patients with a contraindication for benzodiazepine use, phenobarbital can be used by pro-viders experienced with its use. In settings with close moni-toring, phenobarbital adjunct to benzodiazepines is also appropriate. Recommendation V.14: A front loading regimen is recommended for patients at high risk of severe withdrawal syndrome. Providing at least a single dose of preventative medication is appropriate for patients at lower levels of risk who have: Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 45 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. A history of severe or complicated withdrawal An acute medical, psychiatric, or surgical illness Severe coronary artery disease Displaying signs or symptoms of withdrawal concurrent with a positive blood alcohol content Discussion. Determining risk of developing severe or compli-cated withdrawal or complications of withdrawal is addressed in section II: Initial Assessment of Alcohol Withdrawal. Patients at risk of developing severe or complicated alcohol withdrawal or complications from alcohol withdrawal should receive pharmacotherapy as soon as possible to prevent these signs and symptoms.4,13,89 Benzodiazepines are recom-mended as the primary medication to prevent the development of severe, complicated, or complications of withdrawal. There is clear evidence that benzodiazepines reduce the incidence of alcohol withdrawal seizures and alcohol withdrawal delirium. Phenobarbital can be used for patients with a contraindication for benzodiazepine use. However, given its narrow therapeutic window, it should only be used by clinicians experienced with its use. For patients at high risk of severe withdrawal, front loading with a benzodiazepine is recommended to rapidly achieve therapeutic levels of medication. Front loading has been shown to reduce the duration of treatment, incidence of withdrawal seizure, and duration of delirium.2,13,102 Patients should be closely observed for over-sedation and respiratory depression following the administration of a loading dose. For patients at lower levels of risk, providing at least a single or a few doses of benzodiazepine is appropriate and can be followed by a medication chosen according to symptom severity (see V.D(2): Withdrawal symptoms).2,4 If a clinician determines that a patient is no longer at risk, for example, because risk is sufficiently mitigated by administration of medication or because the course of withdrawal has passed the period of acute risk, ongoing pharmacotherapy for alcohol withdrawal can be determined according to the severity of a patient’s withdrawal at that time. Some situations which have been called out as appropriate for administering at least a single dose of benzodiazepines include: a history of severe or complicated withdrawal; risk for complications of significant medical, surgical, or psychiatric illness (particularly cardio-vascular disease including coronary artery disease);4 and displaying signs or symptoms of withdrawal concurrent with a positive blood alcohol content (an indication of risk for developing severe withdrawal syndrome). (See Appendix IV.B., for a flowchart on pharmacotherapy considerations). (2) Withdrawal symptoms Recommendation V.15: For patients experiencing mild alcohol withdrawal (e.g., CIWA-Ar score <10) who are at minimal risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal, pharma-cotherapy or supportive care alone may be provided. If providing medication, benzodiazepines, carbamazepine, or gabapentin are appropriate. For patients with a contraindica-tion for benzodiazepine use, carbamazepine, gabapentin, or phenobarbital (for providers experienced with its use) are appropriate. Carbamazepine, gabapentin, or valproic acid (if no liver disease or childbearing potential) may be used as an adjunct to benzodiazepines. Recommendation V.16: Patients experiencing moder-ate alcohol withdrawal (e.g., CIWA-Ar scores 10–18) should receive pharmacotherapy. Benzodiazepines are first-line treat-ment. Carbamazepine or gabapentin are appropriate alterna-tives. For patients with a contraindication for benzodiazepine use, carbamazepine, gabapentin, or phenobarbital (for pro-viders experienced with its use) are appropriate. Carbamaze-pine, gabapentin, or valproic acid (if no liver disease or childbearing potential) may be used as an adjunct to benzo-diazepines. Recommendation V.17: Patients experiencing severe alcohol withdrawal (e.g., CIWA-Ar scores 19) should receive pharmacotherapy. Benzodiazepines are first-line treat-ment. For patients with a contraindication for benzodiazepine use, phenobarbital is appropriate for providers experienced with its use. If close monitoring is available, phenobarbital can be used as an adjunct to benzodiazepines. Other adjunct medications can be considered after a clinician ensures that an adequate dose of benzodiazepines has been administered. Recommendation V.18: If a patient’s symptoms are not controlled as expected: First consider increasing the dose If over-sedation or inadequate monitoring is a concern: Reassess for appropriate level of care Consider switching medication If using benzodiazepines, consider adding an adjunct med-ication Discussion. For patients experiencing mild alcohol with-drawal (e.g., CIWA-Ar score <10) who are at minimal risk of developing severe, complicated, or complications of alco-hol withdrawal, the decision to provide medication to alleviate symptoms of withdrawal is at the discretion of clinicians. Previous guidelines and reviews have indicated that patients experiencing mild alcohol withdrawal (e.g., CIWA-Ar score <10) who are at minimal risk of worsening symptoms can be safely treated with monitored supportive care alone.13,51 Early evidence for the safety of non-pharmacological treatment of alcohol withdrawal draws from studies of ‘‘social detoxifica-tion’’ settings.120–122 Research has demonstrated that patients who never reach a CIWA-Ar score 10 and thus do not receive medication in accordance with a symptom-triggered protocol are not at higher risk of adverse events than patients who received medication through a fixed-dose protocol. In addition, patients receiving medications through a symptom triggered protocol require less medication overall and experi-ence a shorter duration of treatment.123–126 Others have argued that any withdrawal signs and symptoms are harmful to patient health4 and that untreated withdrawal contributes to the kindling process, whereby repeated episodes of alcohol withdrawal syndrome become progressively severe through increased neuronal excitability and sensitivity.7 Patients experiencing moderate or severe withdrawal (e.g., CIWA-Ar scores 10) should receive pharmacotherapy. Moderate to severe withdrawal at treatment baseline has been identified as a risk factor for developing more severe withdrawal during the course of treatment.54 ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 46 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Carbamazepine and gabapentin are appropriate for managing mild and moderate alcohol withdrawal in patients who are at minimal risk of developing severe or complicated alcohol withdrawal.92–94 As symptom severity or risk of developing severe symptoms increases, medications with well-established effectiveness in preventing the incidence of severe and complicated withdrawal are preferred.54 Ben-zodiazepines are first-line agents for treating moderate58,65,81 and severe alcohol withdrawal13,58 due to their known effec-tiveness in preventing seizures and delirium.13,90,94 Benzo-diazepines are also appropriate for patients experiencing mild withdrawal in inpatient settings due to the reduced risks associated with use in settings with more intensive monitor-ing. Carbamazepine, gabapentin, or phenobarbital can be used for patients experiencing mild or moderate withdrawal who have a contraindication for benzodiazepine use.58,65,81,92 Phenobarbital is the preferred alternative for patients experiencing severe withdrawal. However, given its narrow therapeutic window, phenobarbital should only be used by clinicians experienced with its use. Patients receiving pharmacotherapy should be moni-tored for signs of response to medication. If the patient does not respond as expected, a number of actions can be consid-ered. First, consider increasing the dose. The amount of medication required to control symptoms is variable and ultimately determined by clinical judgment. Patients with more severe withdrawal may require larger doses than are typically seen in other patient populations, particularly during early withdrawal (see Appendix V for typical doses). Provid-ing large doses of benzodiazepine can lead to over-sedation and respiratory depression and patients should be monitored closely. Second, patients should be reassessed for appropriate level of care. Failure to respond may reflect the presence of more severe withdrawal than expected and significant risk of major complications.13 A more intensive level of care may be needed to monitor and manage major complications if they occur.82 Third, consider switching to a different medication. Failure to respond to benzodiazepine may reflect benzodiaz-epine resistance due to kindling (see section VI.D: Resistant Alcohol Withdrawal). A greater number of previous alcohol withdrawal episodes can be associated with decreased respon-siveness to benzodiazepines.4 Failure to respond may also be due to withdrawal from another GABAergic agent. In these cases, switching to an alternative medication should be considered. Fourth, if using benzodiazepines, consider adding an adjunct medication. Some patients benefit from the addition of an adjunct medication to control signs and symptoms of withdrawal and their use can be considered as part of the treatment plan. The use of carbamazepine, gabapentin, or valproic acid as an adjunct medication may be appropriate for patients experiencing moderate or severe withdrawal. Val-proic acid should not be used in patients who have acute liver impairment or women of childbearing potential (see V.D: Carbamazepine, gabapentin, valproic acid). Adjunct pheno-barbital can be used in patients with severe withdrawal in settings with close monitoring. Phenobarbital and benzodiazepines act on the same receptors, which leads to additive clinical effects in controlling alcohol withdrawal syndrome (see Box 7).111 Alpha2-adrenergic agonists and beta-adrenergic antagonist can be used in conjunction with benzodiazepines to manage persistent hypertension or tachy-cardia (see V.D: A2AAs and beta-blockers).44,97 (3) Benzodiazepine use Recommendation V.19: While no particular benzodi-azepine agent is more effective than another, longer-acting benzodiazepines are the preferred agents due to clinical benefits of their longer duration of action. Recommendation V.20: If waiting for lab test results or if the test are unavailable, if a patient has signs of significant liver disease, use a benzodiazepine with less hepatic meta-bolization. Recommendation V.21: Clinicians should monitor patients taking benzodiazepines for signs of over-sedation and respiratory depression. Recommendation V.22: A benzodiazepine prescription to treat alcohol withdrawal should be discontinued following treatment. Discussion. Benzodiazepines are commonly recommended as first-line agents for managing most forms of alcohol with-drawal.13,94 Diazepam, lorazepam, and chlordiazepoxide are the most frequently used in treating alcohol withdrawal. While there is no evidence showing superiority of effective-ness among benzodiazepine agents,13,90 longer-acting agents are preferred by many clinicians.2,51,81 A long duration of action contributes to a smoother course of withdrawal and greater control of breakthrough and rebound signs or symp-toms. This provides greater coverage for preventing alcohol withdrawal seizures and delirium.90 For this reason, patients prescribed a shorter-acting agent should have a more gradual taper and be reassessed more frequently (see V.D(4): Benzo-diazepine dosing regimens).98 Longer-acting agents can accumulate and lead to over-sedation and respiratory depression, particularly in older patients or those with compromised health. Other signs of accumulation include ataxia, confusion, memory impairment, and delirium, which may be difficult to differentiate from alcohol withdrawal-related delirium.2 Benzodiazepine asso-ciated delirium has been diagnosed by the administration of flumazenil, a GABA-A receptor antagonist, but this protocol was not reviewed by the Guideline Committee.99 A reduction in the benzodiazepine dose and the addition of a neuroleptic agent to control for agitation and/or confusion can be consid-ered if patients are not at an elevated risk of seizure (i.e., they are outside of the acute risk window).2 Some neuroleptic agents have been shown to reduce the seizure threshold. Benzodiazepine accumulation is more likely in patients with impaired hepatic function. Medication dose can be reduced or a benzodiazepine with less dependence on hepatic metabolism can be used (see VII.D: Patients with Medical Conditions). The laboratory tests recommended in section II.E: Identify Concurrent Conditions can indicate the need to adjust the treatment plan. However, as treatment should not be delayed while waiting for lab test results or if the test(s) are unavailable at the treatment setting, it is appropriate to Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 47 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. initially reduce the dose or use a benzodiazepine with less hepatic metabolization if a patient has signs of significant liver disease. Signs of significant liver disease include: Skin and eyes that appear yellowish (jaundice) Swelling in the legs and ankles (edema) Itchy skin Dark urine color Pale stool color, or bloody or tar-colored stool Confusion Chronic fatigue Nausea or vomiting (4) Benzodiazepine dosing regimens Recommendation V.23: Symptom-triggered treatment is the preferred benzodiazepine dosing method. Fixed dosing according to a scheduled taper is appropriate if symptom-triggered treatment cannot be used. Recommendation V.24: Front loading is recommended for patients experiencing severe alcohol withdrawal (e.g., CIWA-Ar scores 19). Diazepam or chlordiazepoxide are preferred agents for front loading. Recommendation V.25: When using a fixed-dose schedule, patients’ signs and symptoms should still be moni-tored, and additional doses of medication provided as needed. Recommendation V.26: If prescribing a shorter-acting benzodiazepine, using a fixed-dose regimen with a gradual taper may be appropriate to reduce the likelihood of break-through and rebound signs and symptoms. Discussion. Examples for these dosing regimens can be found in Appendix V. Multiple dosing strategies have been used to administer benzodiazepines during alcohol withdrawal. In general, symptom-triggered treatment is the preferred dosing method. In this regimen, medication is administered only when patients are experiencing significant withdrawal symptoms according to a severity scale. This allows dosing to be individualized according to symptom severity and reduces the risk of under- and over-treating by assessing and dosing according to real-time symptom severity. Very large doses of medication may be needed rapidly, and reduced as symptoms resolve.2,13 Symptom-triggered dosing has been shown to reduce the duration of treatment and inpatient length of stay compared to a fixed-dose schedule.2,21,44,51 Fixed dosing is appropriate when it is not practical to obtain a symptom severity score to conduct symptom-trig-gered treatment. In a fixed-dose regimen, set amounts of medication are administered at regular intervals, and the dose amount, dosing frequency, or both are gradually tapered according to a set schedule. While fixed dosing is easy to administer, over- or underestimating the amount of benzodi-azepine needed may lead to insufficient symptom control or over-sedation.26 Additional doses58 and dose adjustment should be provided as needed.7,42 Front loading is preferred for patients at risk for or experiencing severe alcohol withdrawal (e.g., CIWA-Ar scores 19). Front loading describes when a moderate to high dose of a long-acting benzodiazepine is administered to achieve rapid control of withdrawal signs and symptoms and is allowed to taper through metabolism. Diazepam and chlor-diazepoxide are the preferred agents for front loading. This regimen is typically used when rapid administration of a benzodiazepine is required, either because patients are experiencing significant symptoms or are at risk of developing them. Front loading has been shown to reduce the duration of treatment and incidence of withdrawal seizure and duration of delirium.102 This effect is usually attributed to the rapid administration of large amounts of benzodiazepines early in the withdrawal period.2,13 A front loading regimen can be driven by a withdrawal symptom severity scale (e.g., 10 mg diazepam PO every hour if CIWA-Ar score 10) or according to a fixed schedule (e.g., 20 mg diazepam PO every 2 hours for 3 doses). Symptom-triggered front loading has been shown to reduce symptom duration and the amount of benzodiazepine used,70,103–106 the incidence of withdrawal seizures, and the duration of delirium for patients being treated in the ICU.102 Fixed-dose front loading can be used in patients for whom it would be difficult to obtain an accurate score on a withdrawal severity scale. While monitoring for signs of over-sedation and respi-ratory depression is important for any dosing regimen,49 it is particularly important for patients on fixed-dose and front-loading regimens. Patients receiving fixed doses can become over-sedated if the wrong schedule is chosen and front-loading doses are rapidly administered.7,42 Because of their shorter duration of action, short-acting benzodiazepine concentrations can diminish rapidly, increas-ing the chance for rebound and breakthrough symptoms and signs including seizure. For this reason, a fixed-dose schedule with a long taper may be more feasible than a symptom-triggered dosing regimen requiring very frequent reassess-ment. Shorter-acting benzodiazepines should be tapered care-fully even after withdrawal resolves to prevent the development of rebound or breakthrough signs and symp-toms. If the CIWA-Ar is used in conjunction with short acting benzodiazepines, the assessments should be done promptly in order to prevent seizures due to protocol errors.98 (5) Carbamazepine, gabapentin, valproic acid Recommendation V.27: Gabapentin is a favorable choice for treating alcohol withdrawal when a clinician also plans to use it for a patient’s ongoing treatment of alcohol use disorder. Recommendation V.28: If benzodiazepines are contra-indicated, carbamazepine or gabapentin are appropriate alter-natives for patients in mild or moderate withdrawal. Recommendation V.29: Carbamazepine, gabapentin, or valproic acid may be used as an adjunct to benzodiazepine therapy to help control alcohol withdrawal. Before using as an adjunct, clinicians should ensure that an adequate dose of benzodiazepine has been administered. Recommendation V.30: Valproic acid should not be used in patients who have liver disease or women of childbearing potential. Recommendation V.31: There is insufficient evidence to support the use of valproic acid as monotherapy for the treatment of alcohol withdrawal. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 48 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Discussion. Evidence suggests that anticonvulsants, particu-larly carbamazepine, are effective at preventing alcohol with-drawal progression, seizures and delirium.4 At this time, there is insufficient evidence to support their use over benzodia-zepines for patients at increased risk of severe withdrawal, seizures, or delirium.2,13,38,92,107 As the efficacy of benzodia-zepines is well-established, there have been ethical concerns with running placebo-controlled or treatment-as-usual-con-trolled (i.e., compared to benzodiazepines) studies in at-risk populations.2,92 Carbamazepine or gabapentin are appropriate medica-tions for treating low risk patients. They are also appropriate alternatives for patients with a benzodiazepine contraindica-tion. Gabapentin may provide an effective bridge therapy from alcohol withdrawal treatment to long-term alcohol use disorder treatment.92,93 It has been found to improve rates of abstinence and reduce heavy drinking days compared with placebo during the maintenance phase of alcohol use disorder treatment.62 Some patients benefit from the addition of an adjunct medication to control signs and symptoms of withdrawal. Use of carbamazepine, gabapentin, or valproic acid as an adjunct to benzodiazepines is an appropriate therapy for patients experiencing mild or moderate withdrawal. For patients in severe withdrawal, other medications can be used to manage signs and symptoms if benzodiazepines have already being given.36 Before using as an adjunct medication, clinicians should ensure that an adequate dose of benzodiazepine has been administered since large doses of benzodiazepine are sometimes needed to control withdrawal. While valproic acid has been found to be promising for the treatment of alcohol withdrawal, more evidence is needed before it can be recommended as monotherapy.62,109 Its use as an adjunct to benzodiazepines is supported.2,13,44,58 However, valproic acid should not be used in patients with hematologi-cal or hepatic disorders including acute liver impairment44 or in women of childbearing potential because of teratogenic risk.110 (6) Phenobarbital Recommendation V.32: Phenobarbital can be used for some patients in inpatient settings; however, it should only be used by clinicians experienced with its use given its narrow therapeutic window and side effects. Recommendation V.33: In an inpatient setting, pheno-barbital monotherapy (managed by a clinician experienced with its use) is appropriate for patients with a contraindication for benzodiazepine use who are experiencing mild, moderate, or severe withdrawal or who are at risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal. Recommendation V.34: In an inpatient setting, if close monitoring is available, phenobarbital (managed by a clini-cian experienced with its use) as an adjunct to benzodiaze-pines is an option for patients experiencing severe withdrawal or who are at risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal. Recommendation V.35: Parenteral phenobarbital should only be used in highly supervised settings (e.g., ICU, CCU) because of risk of over-sedation and respiratory depression. Discussion. In general, phenobarbital should only be used by clinicians experienced with its use and should be used cau-tiously in settings that offer less monitoring. Phenobarbital may cause respiratory depression and over-sedation and its narrow therapeutic window makes it challenging to dose correctly compared to other medications used to treat alcohol withdrawal. Phenobarbital is more commonly used in an inpatient setting that is highly supervised such as the ICU or the Emergency Department (ED) for these reasons. A primary indication for phenobarbital use is as an alternative to benzodiazepines when benzodiazepine use is contraindicated. This is appropriate for patients experiencing mild, moderate, or severe withdrawal or who are at risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal. Phenobarbital is also an effective adjunct to benzodia-zepines and, if close monitoring is available, can be used for patients experiencing severe withdrawal or who are at risk of developing severe or complicated alcohol withdrawal or complications of alcohol withdrawal. Providing a single dose of IV phenobarbital 10 mg/kg in 100 mL normal saline infused over 30 minutes in addition to lorazepam in the ED was shown to reduce the rate of ICU admissions without increasing the incidence of adverse events.127 This strategy requires close monitoring in highly supervised settings as parenteral administration of phenobarbital is associated with increased overdose risk. (7) A2AAs and beta-blockers Recommendation V.36: Alpha2-adrenergic agonists (A2AAs) such as clonidine and dexmedetomidine can be used as an adjunct to benzodiazepine therapy to control autonomic hyperactivity and anxiety when these signs are not controlled by benzodiazepines alone. They should not be used alone to prevent or treat withdrawal-related seizures or delirium. Recommendation V.37: Beta-adrenergic antagonists (beta-blockers) can be used as an adjunct to benzodiazepines in select patients for control of persistent hypertension or tachycardia when these signs are not controlled by benzodia-zepines alone. They should not be used to prevent or treat alcohol withdrawal seizures. Discussion. Many patients in alcohol withdrawal experience cardiac or adrenergic signs such as hypertension and tachy-cardia.45 These signs can be addressed by treating medical problems commonly seen in patients with alcohol withdrawal syndrome, such as dehydration and electrolyte imbalances or through the use of benzodiazepines. Alpha2-adrenergic ago-nists (A2AAs) and beta-adrenergic antagonist (beta-blockers) can be used in conjunction with benzodiazepines to manage persistent hypertension or tachycardia.44,97 While these med-ications reduce the signs of sympathetic activation, they do not treat the underlying pathophysiology, which may mask the hyperadrenergic state and lead to a false perception that these signs are properly treated. They also do not prevent Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 49 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. withdrawal-related seizures or delirium and should not be used alone in the treatment of alcohol withdrawal. (8) Inappropriate medications Recommendation V.38: Oral or intravenous alcohol should not be used for the prevention or treatment of alcohol withdrawal. Recommendation V.39: There is insufficient evidence to support the use of baclofen for the treatment of alcohol withdrawal. Recommendation V.40: Providing magnesium as a prophylaxis or treatment for alcohol withdrawal management has no supporting evidence. Discussion. While ethyl alcohol administration has been used to manage withdrawal, it is not recommended.2,13,58 Admin-istration of oral or intravenous alcohol has no proven efficacy, no accepted protocols, and known toxicity.13 A recent Cochrane review of three RCTs on the use of baclofen for alcohol withdrawal treatment drew no conclu-sions about efficacy or safety of baclofen due to insufficient and low quality evidence.112 ASAM’s 2004 guideline, ‘‘Management of Alcohol Withdrawal Delirium,’’ suggested that magnesium may reduce neuromuscular activity in patients experiencing alco-hol withdrawal. However, a recent Cochrane review114 con-cluded that there is not enough evidence to determine the benefit of magnesium in alcohol withdrawal prevention or management, which is in agreement with the ASAM’s 1997 guideline.13 VI. Addressing Complicated Alcohol Withdrawal In this section, we highlight special considerations for patients with or at risk for alcohol withdrawal seizure or alcohol withdrawal delirium, alcohol-induced psychotic dis-order, or resistant alcohol withdrawal. Aspects of manage-ment that might need to be adjusted for these patients, including monitoring, supportive care, and pharmacotherapy are discussed. Guidelines pertaining to assessment and overall management can be found in the relevant sections above. A. Alcohol Withdrawal Seizure (1) Monitoring Recommendation VI.1: Patients should be monitored for alcohol withdrawal seizures even in the absence of other clinically prominent alcohol withdrawal signs or symptoms. Recommendation VI.2: Following an alcohol with-drawal seizure, patients should be admitted to a setting with close monitoring available, and should be re-assessed every 1-2 hours for 6–24 hours. Patients should be closely monitored for delirium and the need to receive intravenous (IV) fluids, due to potential electrolyte imbalances. Discussion. Patients identified as at risk of experiencing an alcohol withdrawal seizure should be closely monitored.20 Alcohol withdrawal seizures typically occur between 8– 48 hours after cessation of (or reduction in) alcohol use with risk peaking around 24 hours.20,22 Signs of an impending seizure can include tremors, increased blood pressure, overactive reflexes and high temperature and pulse.15 How-ever, clinicians should be aware that an alcohol withdrawal seizure can occur in the absence of other clinically prominent withdrawal signs or symptoms. Risk of seizure is typically bundled with risk of alcohol withdrawal delirium when evaluating predictive factors12,36,48–51 (see section II.B: Risk Factors for Severe or Complicated Withdrawal). Following an alcohol withdrawal seizure, a patient is at increased risk for another seizure and progression to alcohol withdrawal delirium.2,4,52 Patients should be observed for at least 24 hours,52 or if in a setting where continuous observa-tion is not feasible, observed for a minimum of 6 hours before being discharged to a treatment setting with continuous monitoring. The Guideline Committee recommended that patients be re-assessed at least every 1–2 hours during the post-seizure monitoring period. (2) Supportive care Recommendation VI.3: If available and applicable, existing institutional/hospital-associated delirium protocols can be used for supportive care of patients with an alcohol withdrawal seizure. Discussion. Non-pharmacological supportive care for patients with a recent alcohol withdrawal seizure includes safety mea-sures as well as standard care protocols. Safety measures such as fall precautions and standard care protocols such as routine nurse check-ins and assistance with activities of daily living (ADLs) ensures patient safety as well as provides autonomy. Patients with a recent alcohol withdrawal seizure are at increased risk for developing delirium. For facilities with a hospital-associated delirium protocol, clinicians should imple-ment the protocol to prevent and reduce the incidence and duration of acute delirium among patients with a recent alcohol withdrawal-related seizure. Studies have shown standardized protocols to be effective at reducing the incidence, duration, and frequency of delirium among hospitalized patients.117 The Guideline Committee agreed with the use of institutional/hos-pital-associated delirium protocols, when available. (3) Pharmacotherapy Recommendation VI.4: Following a withdrawal sei-zure, patients should be immediately treated with a medica-tion effective at preventing another seizure. Benzodiazepines are first-line treatment, and a fast-acting agent such as loraze-pam or diazepam is preferred. Phenobarbital is also an option but is less preferred to benzodiazepines. Recommendation VI.5: Following a withdrawal sei-zure, parenteral administration of medications is preferred. If available, IV administration is preferred to intramuscular (IM), but IM administration is also effective. Parenteral phenobarbital should only be used in highly supervised set-tings (e.g., Intensive Care Unit [ICU)] or Cardiac/Coronary Care Unit [CCU]) because of risk of over-sedation and respiratory depression. Recommendation VI.6: It is not recommended to use alpha2-adrenergic agonists or beta-adrenergic antagonists to prevent or treat alcohol withdrawal seizures because they are ineffective for this purpose. Beta-adrenergic antagonists also can lower the seizure threshold. Phenytoin should not be used unless treating a concomitant underlying seizure disorder. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 50 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Discussion. Benzodiazepines are effective in the primary and secondary prevention of alcohol withdrawal seizures.13,131 Intravenous (IV) administration of a fast-acting agent such as lorazepam or diazepam is recommended after a withdrawal-related seizure.2,4,21,52 In a randomized, double-blind trial, patients admitted to the emergency department with an alco-hol withdrawal-related seizure were provided either 2 mL of saline or 2 mL of lorazepam intravenously to prevent subse-quent seizures. The use of intravenous lorazepam was shown to significantly reduce the risk of recurrent seizures.131 All patients presenting with an alcohol withdrawal seizure should have IVaccess established immediately, which can be used for fluids in the prevention of dehydration as well as the administration of medication.132 IM is also an effective and acceptable route of administration. A prospective study examining the effects of phenobar-bital for the treatment of alcohol withdrawal and convulsions found that none of the 38 patients who presented with alcohol withdrawal seizures had a subsequent convulsion after the administration of IV phenobarbital.133 A more recent small prospective, randomized trial comparing phenobarbital to benzodiazepines for the treatment of acute alcohol withdrawal found phenobarbital to be as effective in reducing patient CIWA-Ar scores from baseline to discharge in the emergency department as benzodiazepines.134 Phenobarbital is an appropriate option for the treatment of alcohol withdrawal symptoms and prevention of additional seizures. It should be noted that phenobarbital may cause respiratory depression and over-sedation because of its effects on the central nervous system and narrow therapeutic window. Phenobarbital is therefore more commonly used in an inpa-tient setting, such as the ICU or Emergency Department (ED) where there is continuous supervision. The Guideline Com-mittee recommended caution when using in settings that offer less monitoring than the ICU and ED. Additionally, clinicians who are less familiar with the therapeutic window and have minimal experience with phenobarbital should use extra caution in case over-sedation or respiratory depression occurs. While animal studies have shown that anticonvulsants can prevent seizures and delirium,4 evidence of their efficacy in humans is mixed, and is insufficient to conclude their effects are superior to benzodiazepines.2,38,107 Also, phenyt-oin has been shown to be ineffective in preventing the recur-rence of seizure and is not recommended, unless the patient is being treated for a concomitant underlying seizure disor-der.14,52 B. Alcohol Withdrawal Delirium (1) Monitoring Recommendation VI.7: Patients with alcohol with-drawal delirium should be provided close nursing observation and supportive care, which often necessitates admission to an intensive or critical care unit. Agitated and disoriented patients should have continuous, one-to-one observation and monitoring. Recommendation VI.8: Patients with alcohol with-drawal delirium should have their vital signs, oximetry and cardiac status monitored as frequently as required. Resusci-tative equipment should be readily available when patients require high doses of benzodiazepines, when continuous infusion of medication is used, or when patients have signifi-cant concurrent medical conditions. Recommendation VI.9: To monitor signs and symp-toms of alcohol withdrawal delirium, use a structured assess-ment scale such as the Confusion Assessment Method for ICU Patients (CAM-ICU), Delirium Detection Score (DDS), Rich-mond Agitation-Sedation Scale (RASS), or Minnesota Detox-ification Scale (MINDS). It is not recommended to use the CIWA-Ar in patients with delirium because it relies on patient-reported symptoms. Discussion. Patients experiencing alcohol withdrawal delir-ium should be provided supportive care in a quiet, well-lit room with continuous monitoring of vital signs by nursing staff.2,13,116 For patients who are disoriented or agitated, one-to-one observation should be provided.14 The appropriateness of additional monitoring tools and measures depends on (1) the dose and frequency of medication,2,14 (2) concurrent medical conditions,14 and (3) degree of abnormality of the vital signs. Hospital or institutional prevention and treatment protocols can be implemented to reduce the risk of delirium among patients. Studies have shown standardized protocols to be effective at reducing the incidence, duration, and frequency of delirium among hospitalized patients.117 Intravenous administration is commonly used when treating alcohol withdrawal delirium, but clinicians should be cautious because benzodiazepines, such as diazepam, have a rapid onset of response. Due to this, patients may be at greater risk of respiratory depression when these medications are administered intravenously.2 A meta-analysis on the pharmacological management of alcohol withdrawal recom-mends having resuscitative equipment readily available when patients require high doses of benzodiazepines, when contin-uous infusion of medication is used, or when patients have significant concurrent medical conditions.14 While a structured assessment scale should be used to monitor alcohol withdrawal delirium, the use of the CIWA-Ar is problematic in patients experiencing delirium. Other scales are effective at identifying and monitoring delirium among patients who are unable to communicate clearly. The Confu-sion Assessment Method for ICU Patients (CAM-ICU),135– 137 is a reliable, rapid and valid instrument for diagnosing delirium among ICU patients and can be used for mechani-cally ventilated patients as well. The Delirium Detection Score (DDS)138 is another valid and reliable assessment scale used in the ICU. The Richmond Agitation Sedation Scale (RASS)139 has demonstrated reliability and validity in medi-cal and surgical patients, including patients who are sedated and/or ventilated. Although not officially validated, the Min-nesota Detoxification Scale (MINDS)140 has been used to assess and monitor patients in the ICU setting. The scale takes less time to administer than the CIWA-Ar and has produced reliable scores that are reflective of the severity of alcohol withdrawal symptoms among patients. (2) Supportive care Recommendation VI.10: Provide immediate intrave-nous access for administration of drugs and fluids to patients experiencing alcohol withdrawal delirium. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 51 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Recommendation VI.11: If available and applicable, existing institutional/hospital-associated delirium protocols can be used for supportive care of patients with alcohol withdrawal delirium. Recommendation VI.12: Restraints should only be used when required to prevent injuries due to agitation or violence, and in compliance with state laws. Discussion. Patients experiencing alcohol withdrawal delir-ium should quickly be provided immediate intravenous access for administration of fluids and medication.41 Intravenous benzodiazepines have been shown to provide more rapid control of signs and symptoms compared to oral administra-tion,2 which is ideal in treating alcohol withdrawal delirium. However, intravenous administration of benzodiazepines also increases the risk of respiratory depression in patients due the quick onset.2 Patients should be monitored for signs of respiratory depression with resuscitative equipment readily available if needed.14 Delirium is an acute state of confusion with impaired cognition that often occurs during hospitalization, especially among elderly patients.117,141,142 Delirium has been associated with increased morbidity, mortality, length of hospital stay, and increased health service utilization.142,143 Early recognition as well as preventative measures are key for the management of the risk of delirium.143,144 Hospital or institutional prevention and treatment protocols can be implemented to reduce the risk of delirium among patients. One study evaluated the effects of a multicomponent intervention on reducing the incidence of delirium among hospitalized patients as well as the duration and frequency of delirious episodes among 852 hospitalized patients.117 The intervention utilized standardized protocols that measured six factors for delirium, which were: cognitive impairment, hearing impairment, sleep deprivation, immobil-ity, visual impairment, and dehydration. Results from the study showed that implementation of standardized protocols were effective at preventing and reducing the risk of delirium, number of episodes and duration of episodes. Fifteen percent of patients who did not receive the intervention developed delirium compared to 9.9% who did receive the intervention. Additionally, patients who received the intervention had a shorter duration compared to those receiving usual care (105 vs 161 days). The number of episodes were also significantly less among the intervention group (62 vs 90). Sedative medications, such as benzodiazepines and barbiturates are associated with an increased burden of delir-ium among patients.144 Both medications are commonly used in the treatment of alcohol withdrawal including for patients with alcohol withdrawal delirium. Therefore, these patients should be monitored for early symptoms of delirium and interventions, such as hospital-associated delirium protocols, should be implemented in addition to routine monitoring. The Guideline Committee agreed with the use of institutional/ hospital-associated delirium protocols, when available. Patients experiencing severe alcohol withdrawal, par-ticularly alcohol withdrawal delirium, are confused, agitated, and may try to remove peripheral lines.145 Providing early pharmacological management may alleviate signs and symp-toms of delirium that are likely to cause patients to attempt to remove peripheral lines, but it may be necessary to use restraints, in accordance with state laws, to ensure the safety of patients and staff. (3) Pharmacotherapy Recommendation VI.13: Patients with alcohol with-drawal delirium should be sedated to achieve and maintain a light somnolence. Benzodiazepines are recommended as the first-line agents for managing alcohol withdrawal delirium. Recommendation VI.14: When available, medication should be administered intravenously. The use of intermittent IV administration of long- and short-acting medications is acceptable and effective. Continuous IV infusion is consider-ably more expensive and there is no evidence of therapeutic superiority. Recommendation VI.15: Patients receiving repeated high intravenous doses of lorazepam or diazepam should be monitored closely for signs of hyponatremia and metabolic acidosis. Recommendation VI.16: When treating alcohol with-drawal delirium, use an established dosing protocol as a guide, but individualize dosing regimens based on patient’s signs and symptoms. It is appropriate for patients with alcohol with-drawal delirium to receive intravenous symptom-triggered or fixed-dose front loading. Once light somnolence is achieved and patients are calm and cooperative, if on IV medication, shifting to oral symptom-triggered treatment is recommended. Recommendation VI.17: Very large doses of benzo-diazepines may be needed to control agitation in alcohol withdrawal delirium, including doses that are much higher than typically seen in other patient populations. Clinicians should not hesitate to provide such large doses to patients to control agitation but should keep in mind the possible risk of over-sedation and respiratory depression. Moreover, when large doses are used, there is risk of accumulation of long-acting benzodiazepine metabolites, especially in patients with impaired hepatic function or the elderly, and patients should be monitored closely. Recommendation VI.18: For patients who have been delirious longer than 72 hours, assess for drug-induced delir-ium and withdrawal from another GABAergic agent (such as gabapentin or carisoprodol). When necessary, adjust the benzodiazepine dose. Recommendation VI.19: Barbiturates can be consid-ered an alternative option to benzodiazepines for the treatment of alcohol withdrawal delirium, but they are not preferred to benzodiazepines. Phenobarbital can be used as an adjunct to benzodiazepines in settings with close monitoring when alcohol withdrawal delirium is not adequately controlled by benzodiazepine therapy alone. Recommendation VI.20: Antipsychotic agents can be used as an adjunct to benzodiazepines when alcohol with-drawal delirium and hallucinations are not adequately con-trolled by benzodiazepine therapy alone. They are not recommended as monotherapy for alcohol withdrawal delir-ium. Recommendation VI.21: Alpha2-adrenergic agonists, beta-adrenergic antagonists and paraldehyde should not be used to treat alcohol withdrawal delirium. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 52 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Discussion. Patients experiencing alcohol withdrawal delir-ium should be provided enough medication to achieve a light somnolence.14,44,132 The goal or therapeutic endpoint of this recommendation is to help control agitation associated with delirium.14,44 Patient should be in a level of sedation where they are awake, but have a tendency to fall asleep unless stimulated. Benzodiazepines are the most commonly used medica-tion to treat patients with delirium because of the favorable therapeutic window.14 Intermittent IV administration of a long-acting medication or continuous IV infusion of short-acting medication are both effective treatments for alcohol withdrawal delirium.14 Administering medication intrave-nously allows for rapid and accurate control over signs and symptoms such as fear, autonomic hyperactivity, and halluci-nations.116,146,147 Clinicians should be aware that very large doses of benzodiazepines may be required to control delirium and be ready to provide a sufficient amount of medication to effectively treat the symptoms.2 Because intravenous loraze-pam and diazepam are both stabilized with propylene glycol, hyponatremia and metabolic acidosis may occur.2 While dosing regimens should be individualized based on the patient’s signs and symptoms, using an established dosing protocol as a guide for treatment has been shown to be a safe and effective means of managing alcohol withdrawal delirium.22,148 Symptom-triggered front loading with diaze-pam has also been shown to reduce the duration of delir-ium.149 Fixed-dose front loading is also appropriate during the early management of alcohol withdrawal delirium if a with-drawal scale cannot be completed.41,132 Once patients have reached a calm state, patients can be shifted to a symptom-triggered approach.70,132 See Appendix III for guidance on the use of scales to guide dosing in patients with communication difficulty. Patients treated with repeated high doses of lorazepam or diazepam require close monitoring due to the rapid onset of action and the risk of accumulation of long-acting benzodi-azepine metabolites.2 This accumulation is especially com-mon in patients with impaired hepatic function or among the elderly.2 If a patient has been delirious longer than 72 hours and has been receiving high doses of benzodiazepines (in the thousands of milligrams), the patient may have developed benzodiazepine-induced delirium. Because the temporal win-dow of alcohol withdrawal seizures has passed, clinicians should consider reducing the benzodiazepine dose and adding an antipsychotic agent to control agitation and/or confusion.2 Even if patients are at reduced risk of seizure, anti-psychotics should not be used as monotherapy because they lower the seizure threshold. Second generation atypical anti-psychotics, such as risperidone and quetiapine, are preferred because they have less of an effect on the seizure threshold compared to other antipsychotics.2,44 Haloperidol, a first generation antipsychotic, is also an appropriate agent.2,14,44 Antipsychotics may also be used in conjunction with benzo-diazepines to control severe agitation and hallucinations associated with early alcohol withdrawal delirium.2,14 Barbi-turates are also an appropriate option for treating patients with alcohol withdrawal delirium. A retrospective cohort study found that patients treated with 100–200 mg of phenobarbital (PO or IV) had similar duration of symptoms and length of stay compared to patients who received 10–20 mg of diaze-pam IV hourly until sedated.147 However, barbiturates are not preferred as monotherapy over benzodiazepines due to their narrow therapeutic window and risk of over-sedation and respiratory depression. Phenobarbital may be used in con-junction with benzodiazepines in settings with continuous monitoring available when delirium is not adequately con-trolled by benzodiazepines. Due to difficulties in administration and titration of dose, paraldehyde is not recommended for the treatment of alcohol withdrawal delirium.1414 Additionally, alpha2-adren-ergic agonists and beta-adrenergic antagonists should not be used to treat alcohol withdrawal delirium. C. Alcohol-Induced Psychotic Disorder Recommendation VI.22: If available and applicable, existing institutional/hospital-associated delirium protocols can be used for supportive care of patients with an alcohol-induced psychotic disorder. Recommendation VI.23: The treatment of alcohol-induced psychotic disorder may require consultation with a psychiatrist. Recommendation VI.24: The treatment of alcohol-induced psychotic disorder may require addition of antipsychotics. Recommendation VI.25: For patients experiencing hallucinations,diazepam maybeconsidereda treatmentoption. Discussion. Alcohol-induced psychosis may develop in patients withdrawing from alcohol. Symptoms of alcohol-induced psychosis consist of auditory hallucinations and possibly visual hallucinations and delusions.44 Differentiating between alcohol-induced psychosis due to alcohol withdrawal and alcohol-induced hallucinations as a complication of chronic alcohol use can be difficult. The DSM-5 illustrates the distinctions between substance-induced psychotic disor-ders associated with intoxication as well as withdrawal and requires clinicians to document and code accordingly.43 Cur-rently, there is no established pharmacotherapy for the treat-ment of alcohol-induced psychosis, but a randomized controlled trial of 50 patients showed diazepam to be effective at reducing hallucinations compared to placebo.150 The Guideline Committee rated diazepam as an appropriate med-ication for the treatment of alcohol-induced psychosis, but they also concluded that it may be necessary to treat these patients with an additional antipsychotic medications to alle-viate the symptoms. Patients experiencing alcohol-induced psychosis are at risk of developing acute delirium while in the inpatient setting and appropriate hospital-associated delirium protocols should be implemented, if available, to reduce the risk of delirium and associated health outcomes. D. Resistant Alcohol Withdrawal Recommendation VI.26: If available and applicable, existing institutional/hospital-associated delirium protocols can be used for supportive care of patients with resistant alcohol withdrawal. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 53 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Recommendation VI.27: Phenobarbital may be used as an adjunct to benzodiazepines to control resistant alcohol withdrawal syndrome in settings with close monitoring. Recommendation VI.28: Propofol may be used with patients in the ICU experiencing resistant alcohol withdrawal who already require mechanical ventilation. Recommendation VI.29: Dexmedetomidine may be used with patients in the ICU experiencing resistant alcohol withdrawal. Discussion. Resistant Alcohol Withdrawal (RAW) is not well defined, but generally describes patients who experience severe or complicated withdrawal despite having received high doses of benzodiazepines.97 Prior reviews have defined this as having uncontrolled symptoms despite having received doses of more than 150–200 mg diazepam or 30–40 mg lorazepam in the first 3–4 hours of treatment.46,115,151 In such cases, patients may require the addition of an adjunct medi-cation such as phenobarbital, propofol, or dexmedetomi-dine.46,111,151,152 This phenomenon is also referred to as benzodiazepine-resistant alcohol withdrawal or refractory alcohol withdrawal. There is evidence to support the use of phenobarbital as an adjunct to benzodiazepines in patients with severe with-drawal or RAW.111 A strategy of symptom-triggered escalat-ing doses of diazepam and/or phenobarbital has been shown to reduce the need for mechanical ventilation and showed trends toward reductions in ICU length of stay in patients admitted to the ICU for treatment of alcohol withdrawal syndrome.22 ICU admission was called for if patients required either 200 mg diazepam in 4 hours or an individual dose greater than 40 mg IV diazepam to control agitation. The same strategy has been shown to be effective for patients admitted to the ICU for any reason who also experienced alcohol withdrawal. Patients treated with the protocol had a reduced ICU length of stay, need for mechanical ventilation and benzodiazepine require-ments compared to a group of historical controls treated with physician determined dosing of benzodiazepines alone.22,148. Propofol is appropriate as a benzodiazepine adjunct in patients with RAW treated in the ICU.7,97,115 One systematic review of observational studies evaluated the use of propofol as an adjunct for the treatment of patients with RAW.97 The authors concluded that propofol was useful in reducing signs of alcohol withdrawal, but due to the risk of respiratory depression it is only appropriate for patients who already require mechanical ventilation ‘‘unless other adjuvant thera-pies and methods of BZD [benzodiazepine] administration have proved to be ineffective.’’97 (p. 441) Dexmedetomidine is appropriate as a benzodiazepine adjunct in patients with RAW being treated in the ICU. Three systematic reviews of primarily observational studies on the use of dexmedetomidine in the ICU were found.115,153,154 These authors concluded that dexmedetomidine is a useful adjunct for treating patients with RAW, although monitoring for bradycardia is required. Two randomized controlled trials were found on the use of dexmedetomidine as a benzodiaze-pine adjunct in ICU patients.155,156 Both studies found that the use of dexmedetomidine increased sedation quality (reduced agitation) and decreased benzodiazepine requirements in the 24-hours after dexmedetomidine administration but also increased the incidence of bradycardia. A reduction in total benzodiazepine dose is thought to reduce the potential for prolonged delirium and sedation seen in these patients.153 One study compared the effectiveness of propofol to dexmedetomidine in treating ICU patients experiencing RAW and found that both agents were similarly effective in reducing signs and symptoms of withdrawal and benzodi-azepine requirements.97 However, propofol was associated with fewer instances of bradycardia but more instances of hypotension compared to dexmedetomidine.97 This study and others have stressed the need to better define RAW and for further well-controlled, prospective trials to define the role of dexmedetomidine and propofol in the treatment of RAW.97,153 VII. Specific Settings and Populations In this section, we highlight settings where non-addic-tion specialty clinicians are likely to encounter patients at risk for or experiencing alcohol withdrawal (primary care), set-tings with unique resources (Emergency Departments and hospitals), and patient populations who require treatment plan modifications (patients with medical conditions, patients who take opioids, and patients who are pregnant). A. Primary Care Primary care is a setting where generalist clinicians may be the first point of contact for patients at risk for or experiencing alcohol withdrawal. They may prescribe medi-cation for alcohol withdrawal management. Crucially, they may be the best-placed practitioner to engage patients in long-term follow-up care following the acute withdrawal period. This section is not intended to provide a set of recommen-dations for primary care settings separate from ambulatory Level 1-WM settings. Primary care clinicians should follow recommendations outlined in sections I: Identification and Diagnosis, II: Initial Assessment, and III: Level of Care Determination before initiating alcohol withdrawal manage-ment. If providing alcohol withdrawal management, they should follow recommendations outlined in the section IV: Ambulatory Management of Alcohol Withdrawal. Recommendation VII.1: If patients are experiencing severe withdrawal (e.g., CIWA-Ar scores 19), refer them directly to the nearest Emergency Department. Recommendation VII.2: If withdrawal is mild (e.g., CIWA-Ar <10), patients presenting to primary care can be prescribed a few doses of benzodiazepine. Whenever possible, medication dispensing can be supervised by a caregiver at home or staff at a nonmedical withdrawal management center. Do not prescribe medication to patients for ambulatory man-agement of alcohol withdrawal without performing an adequate assessment. Recommendation VII.3: If withdrawal does not resolve (e.g., fall below a CIWA-Ar score of 10) after an adequate dose of medication (e.g., 80 mg diazepam), or patients appears sedated, transfer to an Emergency Depart-ment or other inpatient withdrawal management setting. Recommendation VII.4: For patients treated in pri-mary care settings, regular follow-up visits, at least monthly ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 54 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. for one year, could increase the likelihood of sustained recovery. Discussion. During assessment, if patients are determined to be experiencing severe withdrawal (e.g., CIWA-Ar scores 19), they should be immediately transferred to the Emer-gency Department (ED) or other setting with the resources to manage complications that might arise.72 When considering prescribing medication to a patient for alcohol withdrawal, clinicians should first assess the patient for risk factors of severe, complicated, or complications of withdrawal (see II. Initial Assessment of Alcohol Withdrawal). For patients with mild withdrawal (e.g., CIWA-Ar <10), clinicians may pre-scribe patients a few doses of medication.72 Whenever possi-ble, have a supportive caregiver or withdrawal management center staff dispense the medication.157 If it is possible to dispense or observe medication administration on-site, if patients’ withdrawal does not resolve (e.g., fall below a CIWA-Ar score of 10) after an adequate dose of medication, or if they show a worsening of symptoms, or appear sedated should be transferred to the ED or a specialized withdrawal management facility with 24-hour supervision. Implementation of nonpharmacological support in the management of alcohol withdrawal among patients treated in a primary care setting may increase the likelihood of sustained recovery compared to patients who do not receive additional nonpharmacological support.82 According to one outpatient withdrawal protocol, nonpharmacological support such as monthly, routine follow-up appointments for one year with a primary health care provider, offers support in the recovery process and can increase abstinence. B. Emergency Departments Emergency Departments (EDs) are unique medical settings that do not fit neatly into the categories of ambulatory or inpatient settings. They have the resources of a hospital and frequently see patients in moderate or severe withdrawal. While alcohol withdrawal can be managed in the ED until it resolves, most patients will be stabilized and leave with a referral for continuing withdrawal management and/or alco-hol use disorder treatment. Recommendation VII.5: If patients are experiencing severe alcohol withdrawal (e.g., CIWA-Ar 19) or are at risk of complicated withdrawal, administer medication immedi-ately to treat withdrawal and reduce the risk of seizures and delirium. Recommendation VII.6: Patients presenting with alco-hol withdrawal syndrome in the Emergency Department should be evaluated for delirium as well as other conditions that mimic and/or accompany withdrawal. Patients presenting with delirium should be assessed for all potential etiologies including alcohol withdrawal. Recommendation VII.7: Patients in the Emergency Department should receive a complete blood count and complete metabolic panel including liver enzyme and mag-nesium tests; alcohol withdrawal treatment should not be delayed while waiting for results. Recommendation VII.8: The following indicators should be present for discharge to an ambulatory alcohol withdrawal management setting from the Emergency Depart-ment: Mild alcohol withdrawal (e.g., CIWA-Ar score <10). Moderate alcohol withdrawal (e.g. CIWA-Ar score 10–18) with no other complicating factors Not currently intoxicated (including alcohol or other drugs) No history of complicated alcohol withdrawal (seizures, delirium) No significant medical or psychiatric comorbidities that would complicate withdrawal management Able to comply with ambulatory visits and therapy Recommendation VII.9: Patients with controlled with-drawal syndrome being discharged from the Emergency Department may be offered a short term (e.g., 1-2 day) prescription for appropriate alcohol withdrawal medication to last until follow-up with their healthcare provider. Discussion. The Guideline Committee recommends that alco-hol withdrawal management be initiated in the ED. This might include diagnosis and assessment, management of acute signs and symptoms, and referral to inpatient or ambulatory treat-ment. The signs and symptoms of alcohol withdrawal often mimic or mask a wide variety of other health conditions and it is recommended that all patients entering the ED with alcohol withdrawal be given a thorough evaluation.152 The etiology of signs and symptoms and identification of coexisting illnesses that may precipitate alcohol withdrawal should be determined from the evaluation.152 While assessing patients, clinicians should be aware that severe intoxication can mimic alcohol withdrawal and often leads to confusion, delirium, tachycar-dia and diaphoresis.126 A serum ethanol level may be neces-sary to determine etiology if the patient history is inconclusive.126 Clinicians can obtain a complete blood count and metabolic panel including liver enzymes and magnesium test to identify factors that may complicate alcohol withdrawal management. As in any setting, patients experiencing severe with-drawal (e.g., CIWA-Ar score 19) or who are at risk of severe, complicated, or complications of withdrawal, should be provided medication immediately to treat withdrawal signs and symptoms and reduce the risk of developing more severe withdrawal.73 Among patients with co-occurring illnesses, the likelihood of developing delirium is higher and they should be provided aggressive treatment of both conditions.158 Patients can be referred to an ambulatory setting and discharged once their symptoms have stabilized (e.g., CIWA-Ar score < 10). While patients experiencing moderate with-drawal (e.g., CIWA-Ar 10–18) may be eligible for ambula-tory withdrawal management, the Guideline Committee emphasized that not all clinicians may be comfortable man-aging patients with moderate withdrawal in this setting and the decision to do so is at the discretion of the treating clinician. When discharging patients to an ambulatory setting, clinicians may provide patients with a short-term (e.g., 1-2 day) prescription for benzodiazepines.159 The Guideline Committee does not recommend providing a short-term pre-scription to patients currently intoxicated (including alcohol Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 55 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. or other drugs) or discharging patients to an ambulatory setting if they have a history of complicated alcohol with-drawal and withdrawal has not fully resolved. A simple referral may not be adequate when patients are being dis-charged from the Emergency Department. Section III: Level of Care Determination provides guidance on determining an appropriate level of care to which to refer the patient. A warm hand-off should be used to ensure the patient made the transition to the next level of care. This may include arranging the appointment in the presence of the patient, arranging for transportation of the patient to the treatment setting, and following up to ensure effective engagement in care. C. Hospitalized Patients This section specifically pertains to patients who are hospitalized for a primary complaint other than alcohol withdrawal who then subsequently develop or are at risk of developing alcohol withdrawal during their hospitalization. (1) Identification Recommendation VII.10: All patients admitted to the hospital should be screened for risk of alcohol withdrawal. Among hospitalized patients, the Alcohol Use Disorders Identification Test (AUDIT) and Alcohol Use Disorders Identification Test-Piccinelli Consumption (AUDIT-PC) can indicate risk of developing alcohol withdrawal. Recommendation VII.11: Patients undergoing elective surgery should be screened for unhealthy alcohol use and the need to undergo alcohol withdrawal management before proceeding with surgery. Patients undergoing elective surgery who are at risk of alcohol withdrawal should undergo medi-cally managed withdrawal before proceeding with surgery Discussion. The Guideline Committee recommends that all patients in the hospital setting be screened for unhealthy alcohol use and assessed for the risk of alcohol withdrawal, if appropriate. Screening and assessment should include the use of a validated scale, information from collateral sources such as friends or family members and medical clinicians, and laboratory tests. For patients undergoing elective surgery, an alcohol withdrawal risk assessment should be conducted prior to surgery, if necessary, because of the postoperative risks and complications associated with alcohol withdrawal.2 Unhealthy alcohol use screens with demonstrated abil-ity to identify patients at risk of developing alcohol with-drawal in general hospital settings include the AUDIT and AUDIT-PC.31 The AUDIT is a 10-item instrument developed to screen for likelihood of the respondent having an alcohol use disorder.160 A large prospective study of patients admitted to an acute medical unit found that admission AUDIT score 8 identified patients who developed alcohol withdrawal with 100% sensitivity and 90.5% specificity.35 While only 17.3% of patients who screened positive went on to develop alcohol withdrawal, no patients with an AUDIT score < 8 experienced withdrawal. This makes clear the point that these instruments should be used as screens; patients who screen positive should be further assessed prior to a diagnosis and treatment of alcohol withdrawal syndrome.35 The AUDIT-PC, a shortened version of the AUDIT, identified patients who experienced alcohol withdrawal syndrome in medical and surgical units with 91% sensitivity and 90% specificity using an admission AUDIT-PC score 4.29 (2) Assessment Recommendation VII.12: Among hospitalized patients, the Prediction of Alcohol Withdrawal Severity Scale (PAWSS) can be used for predicting risk of developing severe or complicated alcohol withdrawal in the medically ill. Recommendation VII.13: Patients for whom alcohol withdrawal is suspected and for whom a complete medical history is not available, (i.e., are admitted from the Emergency Department, trauma unit, or are in Intensive Care Unit [ICU]) or who are known to be at high risk of complicated alcohol withdrawal, medical decisions should be oriented toward a more aggressive treatment of alcohol withdrawal regardless of presenting signs and symptoms. Recommendation VII.14: For patients who require more than standard amounts of medication to manage alcohol withdrawal, individualized assessment by clinicians experi-enced in the management of withdrawal is recommended. The medication and protocol used for treating other conditions and/or alcohol withdrawal syndrome may need to be modified. Discussion. Clinicians can use validated scales such as the Prediction of Alcohol Withdrawal Severity Scale (PAWSS)29 to identify patients at risk of developing severe or complicated alcohol withdrawal in the hospital setting. The PAWSS is designed to asses patients who are medically ill and has been validated by prospective studies, which compared the PAWSS with retrospective chart review and the CIWA-Ar.29,47 See section II.C: Risk Assessment Tools for more information on the PAWSS. Additional information on the scale and its features can be found in Appendix III. When a patient’s medical history is unavailable and it is unclear if the patient has a co-occurring medical condition or is at high risk of complicated alcohol withdrawal, clinicians should be prepared for such events and orient care towards more aggressive treatment regardless of current signs and symptoms. Patients with co-occurring medical diseases may be at risk of developing complications associated with with-drawal and clinicians should consult with appropriate medical professionals from different specialties (e.g. infectious dis-eases, cardiology, pulmonary medicine, hematology, neurol-ogy, and surgery) when necessary. Patients identified with underlying cardiac conditions should be provided aggressive withdrawal treatment due to the potential of alcohol with-drawal worsening cardiac symptoms.4 Patients with co-occurring medical conditions may require modifications to medication regimens and protocols in order to minimize potentially harmful effects related to exacerbation of these conditions.4 However, if a patient experiences withdrawal signs and symptoms that are not easily controlled, consultation with an addiction specialist is warranted to ensure patient safety.2 (3) Monitoring Recommendation VII.15: In patients who are hospi-talized, monitor their vital signs. Fluid intake and output and serum electrolytes should be monitored as clinically indi-cated. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 56 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Recommendation VII.16: Signs and symptoms of alcohol withdrawal should be monitored during the course of withdrawal with a validated symptom assessment scale. Assess the risk for scores on a symptom assessment scale to be confounded by the use of certain medications, the presence of certain medical conditions (e.g. fever from infection), or a patient’s difficulty communicating. Among general medical/ surgical patients, low withdrawal scores can typically be interpreted with confidence, while high scores should be interpreted with caution. The use of alternative scales with patients with difficulty communicating is appropriate. Recommendation VII.17: Patients with a reduced level of consciousness who are at risk for the development of alcohol withdrawal should be monitored for the appearance of alcohol withdrawal signs. If a co-occurring clinical condi-tion worsens, do not assume it is related to alcohol withdrawal among alcohol withdrawal patients. However, immediate treatment is required if alcohol withdrawal develops after surgery or trauma. Discussion. Although the use of validated scales is recom-mended in the hospital setting, clinicians should be particu-larly cognizant of the risk for scores to be affected by comorbid conditions and/or interventions for those condi-tions. Choose a withdrawal scale that can be administered to patients who are critically ill or have reduced consciousness (see Appendix III). Low withdrawal scores can typically be interpreted with confidence, although beta-adrenergic antag-onists (beta-blockers) and other sympatholytic drugs may mask the signs and symptoms of withdrawal and lead to low scores.2 However, high scores have alternative causes that are common in medical/surgical patients and must be interpreted with caution. Patients who have a reduced level of consciousness due to trauma or general surgery should be monitored for the appearance of signs and symptoms of alcohol withdrawal to provide appropriate treatment.7 Clinicians should not neces-sarily assume that worsening symptoms in patients in or at risk for alcohol withdrawal are related to alcohol withdrawal.161 Patients in the ICU are at an increased risk of adverse changes due to their illness and worsening condition; however, these changes may be the result of another medical condition. (4) Supportive care Recommendation VII.18: Clinicians should adminis-ter thiamine to ICU patients with signs or symptoms that mimic or mask Wernicke encephalopathy. Discussion. Due to the risks associated with thiamine defi-ciency among patients experiencing alcohol withdrawal, it is common practice to provide thiamine to prevent Wernicke encephalopathy162–164 Patients in the ICU with a condition that may mask or mimic signs and symptoms associated with WE should receive thiamine. Thiamine is required for basic cellular functioning and carbohydrate metabolism.164 Because the body is unable to synthesize thiamine, daily ingestion is necessary for routine functioning and maintaining homeostasis. If there is insuffi-cient thiamine in the body, a patient may develop a thiamine deficiency such as Wernicke encephalopathy.163,165,166 Patients who consume large amounts of alcohol are particu-larly susceptible to thiamine deficiencies due to inadequate dietary intake as well as biological interactions between cellular enzymes and alcohol.163,166 For example, alcohol inhibits thiamine pyrophosphokinase, an enzyme responsible for synthesizing thiamine diphosphate (TDP) from thiamine, while also increasing the activity of an enzyme that is responsible for the degradation of TDP.166 The effects of alcohol on both of these enzymes results in a reduction of available TDP within the cell and ultimately inhibits cellular metabolism. (5) Pharmacotherapy Recommendation VII.19: Prophylactic treatment of alcohol withdrawal should be provided in the ICU to patients who are suspected to be physiologically dependent on alcohol. Recommendation VII.20: Implementing an alcohol withdrawal management protocol in the ICU is appropriate. When using a symptom-triggered dosing protocol, use a validated scale to monitor signs and symptoms. For patients being treated in ICU settings for alcohol withdrawal, existing scales that are appropriate to use for monitoring withdrawal include the Richmond Agitation-Sedation Scale (RASS). Administration of medications via the intravenous route is preferred because of the rapid onset of action and more predictable bioavailability. Discussion. Because alcohol withdrawal can cause significant morbidity among patients in the critical care setting, patients admitted to the ICU may receive prophylaxis to reduce the risk of developing alcohol withdrawal.167 Additionally, patients should be monitored for worsening signs and symp-toms and development of Wernicke encephalopathy. Typi-cally, a multivitamin infusion or ‘‘banana bag’’ is given to patients in the ICU to prevent Wernicke encephalopathy. One study examined the effectiveness of the standard protocol commonly used in the ICU to prevent Wernicke encephalop-athy when signs and symptoms are masked or mimicked by other illnesses.168 The findings recommended abandoning the ‘‘banana bag’’ approach and provide patients with 200– 500 mg IV thiamine every 8 hours, 64 mg/kg magnesium sulfate, and 400–1,000 mcg IV folate for patients with signs or symptoms that mimic or mask Wernicke encephalopathy. As mentioned, patients also receiving glucose can be admin-istered thiamine and glucose in any order or concurrently. Intravenous administration of benzodiazepines has been recommended for ICU patients due to the rapid onset of action.161 The Guideline Committee recommends a standard protocol, such as symptom-triggered benzodiazepine therapy in the ICU. Systematic reviews show that symptom-triggered therapy is beneficial among critically ill patients50,89 and showed a reduction in the need for mechanical ventilation.50 A combination of symptom-triggered therapy with the use of a validated scale designed for dosing in patients that are unable to communicate or have comorbidities has been shown to be effective.89 When using symptom-triggered dosing, using validated scales specific for ICU patients such as the Rich-mond Agitation-Sedation Scale,7,115 the Confusion Assess-ment Method for ICU Patients,161,167 or the Minnesota Detoxification Scale115,140,161,167 is recommended. Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 57 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. D. Patients With Medical Conditions This section is relevant to patients with comorbid medical conditions who are treated in any setting. Recommendation VII.21: For patients with medical comorbidities, modify the medication and/or protocol used for treating alcohol withdrawal syndrome as necessary in consul-tation with other specialists. Recommendation VII.22: For patients with medical conditions that prevent the use of oral medication, provide intravenous or intramuscular medications as necessary. Recommendation VII.23: Aggressivewithdrawal treat-ment is indicated for patients with cardiovascular disorders due to risk of harm associated with autonomic hyperactivity. Recommendation VII.24: For patients with a medical condition associated with impaired hepatic function, adjust medication dose or use medications with less dependence on hepatic metabolism. Discussion. The main differences in managing alcohol with-drawal in patients with co-occurring medical conditions arises from the need to modify medications used and protocols implemented. The presence of alcohol withdrawal can exac-erbate other conditions and illnesses, particularly cardiovas-cular disease including coronary artery disease. For example, the autonomic arousal (e.g., elevated blood pressure, increased pulse) associated with even mild alcohol with-drawal can exacerbate an underlying cardiac condition.4 Cardiac conditions should be identified early and aggressive treatment is warranted. Clinicians may want to provide at least a single dose of a benzodiazepine to prevent the development of even minor withdrawal symptoms. Other treatment plan modifications might be needed due to impaired liver func-tioning, medication interactions, or a medical condition that prevents administration of oral medication.58 When treating patients with comorbidities, clinicians should consult with appropriate medical professionals from different specialties (e.g. infectious diseases, cardiology, pulmonary medicine, hematology, neurology, and surgery) when necessary. E. Patients who Take Opioids Recommendation VII.25: Patients who are on chronic opioid medication (opioid agonist therapy for opioid use disorder or pain) should be monitored closely when benzo-diazepines are prescribed, due to the increased risk of respi-ratory depression. Similarly, patients taking sedative-hypnotic medications exhibit tolerance to benzodiazepines and should be monitored closely for appropriate dose. Recommendation VII.26: For patients with concomi-tant alcohol withdrawal and opioid use disorder, stabilize opioid use disorder (e.g. with methadone or buprenorphine) concomitantly with treating alcohol withdrawal. Discussion. Patients with concomitant substance use or patients who are currently receiving opioid therapy require special attention and monitoring. The Guideline Committee emphasized that patients with concomitant substance use, in general, are managed similarly to other patients, but special attention should be given to monitoring signs and symptoms. Benzodiazepines may be given but should be used with caution and only in facilities with close monitoring. Patients receiving opioid agonist therapy with concomitant alcohol withdrawal should be admitted and managed in a hospital setting or other setting with the resources to manage increased risk of respiratory depression and other complications. Patients who are using sedative-hypnotic medication are at higher risk of major complications and may exhibit tolerance to benzodiazepines and require dose adjustment. These patients should be monitored closely.13 F. Patients who are Pregnant (1) Level of care and monitoring Recommendation VII.27: Inpatient treatment should be considered for all pregnant patients with alcohol use disorder who require withdrawal management. Inpatient treat-ment should be offered to pregnant patients with at least moderate alcohol withdrawal (i.e., CIWA-Ar scores 10). Recommendation VII.28: The CIWA-Ar is an appro-priate symptom assessment scale to use with pregnant patients. Pregnancy is not expected to bias scores on symptom assessment scales. Clinicians should consider signs and symp-toms such as nausea, headache, anxiety, and insomnia to be connected to alcohol withdrawal rather than pregnancy and presume they will abate once the alcohol withdrawal has been effectively treated. Recommendation VII.29: During withdrawal manage-ment, consult with an obstetrician. Discussion. Inpatient treatment should be considered for all pregnant patients with alcohol withdrawal given the risk of fetal alcohol spectrum disorder including fetal alcohol syn-drome and the risk of abruption, preterm delivery, and fetal distress or demise due to continued alcohol use during preg-nancy.169 While inpatient management is not more effective than ambulatory management for patients who are appropri-ately matched to level of care, it does limit exposure to alcohol. If patients are experiencing at least moderate alcohol withdrawal (i.e., CIWA-Ar 10) and are pregnant, the VA/ DoD58 recommend patients be treated at an inpatient facility that has medical withdrawal supervision. Pregnancy is not expected to bias scores on symptom assessment scales when assessing withdrawal severity during the initial assessment and monitoring. Clinicians can consider signs and symptoms such as nausea, headache, anxiety, and insomnia to be connected to alcohol withdrawal. They can further presume these symptoms will abate once alcohol withdrawal has been effectively treated. The Guideline Committee recommends consulting with an obstetrician when managing alcohol withdrawal in a pregnant patient. Fetal monitoring appropriate to the stage of pregnancy may be warranted due to risk of abruption, preterm delivery, and fetal distress or demise.169 (2) AUD treatment initiation and engagement Recommendation VII.30: Engagement in treatment for AUD is particularly important for pregnant patients with alcohol withdrawal given the risk of Fetal Alcohol Spectrum Disorder (FASD) including Fetal Alcohol Syn-drome (FAS). ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 58 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Discussion. The Guideline Committee emphasized the impor-tance of engaging pregnant patients in ongoing treatment for alcohol use disorder given the risk of fetal alcohol spectrum disorder including fetal alcohol syndrome and the risk of abruption, preterm delivery, and fetal distress or demise due to continued alcohol use during pregnancy.169 As discussed in the ambulatory and withdrawal management sections, the presence of alcohol withdrawal almost universally signifies the presence of an alcohol use disorder and need for treatment. Alcohol withdrawal management alone is not an effective treatment for alcohol use disorder. The period of withdrawal management should include the process of initiating and engaging patients in treatment for alcohol use disorder. (3) Pharmacotherapy Recommendation VII.31: Before giving any medica-tions to pregnant patients, ensure that patients understand the risks and benefits of the medication, both for the patient and the developing fetus. Recommendation VII.32: Benzodiazepines and barbi-turates are the medications of choice in treatment of pregnant patients with alcohol withdrawal. While there is a risk of teratogenicity during the first trimester, the risks appear small, and they are balanced in view of the risk for fetal alcohol spectrum disorder and consequences to mother and fetus should severe maternal alcohol withdrawal develop. Recommendation VII.33: Due to the high teratogenic risk, valproic acid is not recommended for pregnant patients. Recommendation VII.34: For patients at risk for pre-term delivery or in the late third trimester, use of a short-acting benzodiazepine is recommended. This minimizes the risk for neonatal benzodiazepine intoxication given shorter onset and duration of action. Discussion. In SAMHSA’s TIP 45,4 the importance of edu-cating patients about the risks and benefits associated with alcohol withdrawal treatment medication is emphasized. Due to the potential risks imposed on both the patient and devel-oping fetus during withdrawal, it is recommended that patients provide informed consent confirming they have received and understand the risks associated with treatment.4 For patients planning to take medication to treat with-drawal, the World Health Organization (WHO)170 suggests clinicians use the CIWA-Ar to facilitate alcohol withdrawal management. A systematic review found consensus regarding the use of benzodiazepines and barbiturates during pregnancy.13 Although both medications are considered teratogenic and have been associated with adverse effects on the fetus, these risks appear small and must be weighed against the risk of harm to the patient and fetus should severe alcohol withdrawal or seizures develop in pregnant patients. The WHO’s guide-lines170 also recommend short-term use of a long-acting benzodiazepine to treat pregnant patients who develop alcohol withdrawal. When using medication to treat alcohol with-drawal among pregnant patients, limit the amount of medica-tion to only what is necessary to prevent major complications of withdrawal.13 For patients at risk for pre-term delivery or in the late third trimester, use of a short-acting benzodiazepine is recommended. This minimizes the risk for neonatal benzodi-azepine intoxication given shorter onset and duration of action. Valproic acid should not be used in pregnant patients because of teratogenic risk.110 (4) Newborn considerations Recommendation VII.35: In cases of alcohol with-drawal treated close to delivery, assess the newborn for benzodiazepine intoxication, sedative withdrawal, and Fetal Alcohol Spectrum Disorder (FASD) including Fetal Alcohol Syndrome (FAS). Recommendation VII.36: Inform pregnant patients of all wraparound services that will assist them in addressing newborn needs, including food, shelter, pediatric clinics for inoculations, as well as programs that will help with devel-opmental or physical issues that the newborn may experience as a result of in-utero substance exposure. Recommendation VII.37: Licensed clinical staff have an obligation to understand and follow their state laws regard-ing substance use during pregnancy which may include definitions of child abuse and neglect, reporting requirements, and plans of safe care for newborns with in-utero alcohol exposure. Discussion. If a pregnant patient’s alcohol withdrawal was treated close to delivery, newborns should be monitored for signs of FASD and sedative withdrawal and intoxication if withdrawal was managed with medication. As recommended by SAMHSA, pregnant patients should be made aware of wraparound services that will help them with newborn concerns as well as programs that will help with developmental or physical issues that the neonate may experience as a result of in-utero alcohol exposure.4 It is the clinician’s responsibility to know state laws regarding drug use during pregnancy as well as the definitions of child abuse and neglect to reassure and encourage patients to enter treatment.4 Clinicians should also know the reporting require-ments for such cases4 and should discuss them with patients. AREAS FOR FURTHER RESEARCH Identification and Diagnosis Further research is warranted on evidence-based strate-gies to identify alcohol withdrawal in various settings includ-ing primary care, Emergency Departments, and medical/ surgical units in hospitals. Research would include the appro-priate use of validated screening instruments, testing to rule out alternative diagnoses, and laboratory tests for alcohol and other drug use. Initial Assessment Areas for further research in alcohol withdrawal assess-ment include the development and testing of scales to predict the risk of alcohol withdrawal (and the risk of severe with-drawal). Further research on assessing the risk of severe alcohol withdrawal would include the relative importance of predictors, as well as additional research on individual risk factors for complicated withdrawal/complications of withdrawal. Furthermore, for clinicians in ambulatory Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 59 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. settings, further research on triaging patients based on risk would help guide clinical practice. Level of Care Determination Further research on the role of The ASAM Criteria Risk Matrix in determining appropriate level of care for individuals with alcohol withdrawal would be welcome. In particular, evidence-based improvements in the assessment of the recov-ery environment and available social support networks would be helpful to determine appropriateness for ambulatory man-agement. Ambulatory Management Further research on optimal monitoring intervals at various levels of care would be useful in guiding clinical practice. The literature revealed a wide variety of recommen-dations for monitoring frequency and intensity. While the importance of supportive care is widely recognized, it is not well-researched. Additional research on individualizing nutritional supplementation and alternative interventions for symptom management (e.g., acupuncture, massage, etc.) would be helpful. Finally, further research is needed on the design and implementation of effective strategies to transition patients from alcohol withdrawal management to AUD treatment initiation and engagement. Comparative effectiveness studies of various models and strategies for linkages to care would be particularly helpful, as would investigation into the moderat-ing or mediating influence of patient and setting factors. Inpatient Management Several promising medications have not yet been well-researched. Hence, large, well-controlled studies of specific medications would be helpful in expanding the options for individualization of alcohol withdrawal management. Some examples of useful comparative trials include phenobarbital vs. or as adjunct to benzodiazepines, ketamine as adjunct to other medications, carbamazepine vs. gabapentin. Further research on managing resistant or refractory withdrawal is also needed. Addressing Complicated Alcohol Withdrawal There is a minimal literature on the management of alcohol-induced psychosis associated with alcohol with-drawal. Although the Guideline Committee agreed with the one study conducted by Sellers in 1983, there is insufficient evidence to support the use of other medications to control for alcohol-induced psychosis during withdrawal. Further research on differentiating between alcohol-induced intoxi-cation and alcohol-induced withdrawal as well as the man-agement for both is warranted. Specific Settings and Populations The literature and Guideline Committee agreed that clini-cally significant alcohol withdrawal is rare among adolescents, and this special population was beyond the scope of the current guideline. However, further research on potential modifications to alcohol withdrawal management protocols for adolescents would be useful. Other special populations in need of further research include the elderly and criminal justice populations. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 60 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Appendices I. Cited References II. Literature Search Methods III. Alcohol Withdrawal Scales Table IV. Flowcharts (Supplemental Digital Content, V. Sample Medication Regimens VI. Statement Rating Table (Supplemental Digital Content, VII. 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Empirical Literature Search Terms Without date limiters (1/1/2013 - 11/6/2017) Medline (EBSCOhost) Search ID Search Terms 1 TX ‘‘Alcohol withdrawal’’ OR TX ‘‘Delirium tremens’’ OR TX ‘‘Alcohol-induced hallucinosis’’ OR TX ‘‘Alcohol-induced psychotic disorder’’ 2 1 AND Limiters: Animals 3 1 AND Limiters: Human 4 2 AND 3 5 2 NOT 4 6 1 NOT 5 7 6 AND Limiters: English CINAHL Search ID Search Terms 1 TX ‘‘Alcohol withdrawal’’ OR TX ‘‘Delirium tremens’’ OR TX ‘‘Alcohol-induced hallucinosis’’ OR TX ‘‘Alcohol-induced psychotic disorder’’ 2 1 AND Limiters: English EMBASE Search ID Search Terms 1 ‘alcohol withdrawal’ OR ’delirium tremens’ OR ’alcohol-induced hallucinosis’ OR ’alcohol-induced psychotic disorder’ 2 1 AND [animals]/lim 3 1 AND [humans]/lim 4 2 AND 3 5 2 NOT 4 6 1 NOT 5 7 6 AND [english]/lim Web of Science Search ID Search Terms 1 TOPIC: (‘‘Alcohol withdrawal’’) OR TOPIC: (‘‘Delirium tremens’’) OR TOPIC: (‘‘Alcohol-induced hallucinosis’’) OR TOPIC: (‘‘Alcohol-induced psychotic disorder’’) 2 1 AND Refined by: LANGUAGES: (ENGLISH) B. Gray Literature Search Source Detail National Technical Information Service (NTIS 1964–present) Searched website; nothing pertaining to alcohol New York Academy of Medicine Searched website; nothing pertaining to alcohol Guidelines International Network (GIN Database) Searched current publications for ‘‘alcohol,’’ which returned 38 results. Relevant publications: Substance misuse and alcohol use disorders. In: Evidence-based geriatric nursing protocols for best practice. Hartford Institute for Geriatric Nursing. (2012) Problem drinking. Medical Services Commission, British Columbia. NGC:009465 (2011) https:// www2.gov.bc.ca/gov/content/health/practitioner-professional-resources/bc-guidelines/problem-drinking#part3 EFNS guideline on the diagnosis and management of alcohol-related seizures: Report of an EFNS task force. European Federation of Neurological Societies. NGC: 005164 (2005) http:// citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.463.3968&rep=rep1&type=pdf Alcohol-use disorders: physical complications. NICE. (CG100) (2010) guidance/cg100 National Institute for Health and Care Excellence (NICE) Alcohol-use disorders: diagnosis, assessment and management of harmful drinking and alcohol dependence (NICE Guideline (UK), 2011) Scottish Intercollegiate Guidelines Network (SIGN) Searched current guidelines: none pertaining to alcohol. There was an archived guideline regarding the management of alcohol use disorder in primary care, but it was withdrawn in 2015. There is a proposal for a new guideline on the management of harmful drinking, but guideline development has not yet begun. New Zealand Guidelines Group (NZGG) Searched current publications for ‘‘alcohol,’’ which returned 70 results. None pertained to withdrawal management. ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 66 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Guidelines Advisory Committee (GAC) Searched current guidelines: none pertaining to alcohol. America’s Health Insurance Plans (AHIP) Nothing related to alcohol withdrawal management Blue Cross and Blue Shield Association, Technology Evaluation Center (BCBS TEC) Website info merged with AHRQ Kaiser Permanente Unhealthy Drinking in Adults Screening and Intervention Guideline (revised 2016) https:// wa.kaiserpermanente.org/static/pdf/public/guidelines/alcohol-adult.pdf Kaiser Family Foundation (KFF) Searched ‘‘alcohol’’ 10–30-17 with no result. KFF publications more likely to be fact sheets than clinical guidelines. Robert Wood Johnson Foundation (RWJF) Searched ‘‘alcohol’’ 10-30-17 and reviewed the following content types: Journal Articles, Reports, and Briefs. No result. Substance Abuse and Mental Health Services Administration (SAMHSA) SAMHSA’s TIP 45: Detoxification and Substance Abuse Treatment (2015) https:// www.store.samhsa.gov/product/TIP-45-Detoxification-and-Substance-Abuse-Treatment/SMA15-4131 Veterans Administration (VA) VA/DOD Clinical Practice Guideline for the Management of SUDs (2015) https:// www.healthquality.va.gov/guidelines/MH/sud/VADoDSUDCPGRevised22216.pdf World Health Organization (WHO) Alcohol and injuries: Emergency department studies in an international perspective, 2009. http:// www.who.int/substance_abuse/msbalcinuries.pdf?ua=1 Agency for Healthcare Research and Quality (AHRQ) Searched website; no published guidelines pertaining to alcohol withdrawal management. Searched ‘‘substance abuse’’ category 10-30-17. Profile of model program: Searched ‘‘guideline-related’’ category 10-30-17. General Recommendations for the Care of Homeless Patients uploads/2011/09/GenRecsHomeless2010.pdf Michigan Quality Improvement Consortium (MQIC) Searched website; nothing pertaining to alcohol Scopus See Empirical Literature search C. PRISMA Flow Diagram # of records identified through database searching (n ¼ 3,138) # of studies included from targeted search (n ¼ 70) ! # of records after duplicates removed (n ¼ 2,107) # of records screened (n ¼ 2,107) ! of records excluded (n ¼ 1,365) # of full-text articles assessed for eligibility (n ¼ 742) ! of full-text articles excluded (n ¼ 545) Abstract, protocol only ¼ 219 Commentary / Editorial ¼ 58 Excluded reviews ¼ 88 Case study ¼ 32 Wrong population ¼ 43 Wrong condition/disorder ¼ 23 Wrong intervention ¼ 52 Wrong outcome ¼ 21 Not English ¼ 8 # of studies included in quantitative synthesis (n ¼ 134) # of studies included in qualitative synthesis (n ¼ 64) Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 67 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. D. Reasons for Exclusion Reason for Exclusion Examples Abstract, Brief report only Conference abstracts Study protocols Case Study Case studies were excluded if controlled studies were included Commentary/Editorial Letters to the editor and editorials were read, but not included for extraction More Recent Available Systematic reviews (e.g. Cochrane Reviews) and guidelines were excluded if an updated version was available. More Recent Review Available Non-systematic reviews and gray literature were excluded if more recent quality reviews or systematic reviews were available Not English Full text not available in English Original Research Included Systematic and non-systematic reviews were excluded if all original research was included Wrong Intervention No intervention/ Not about management (e.g. Etiology and pathophysiology, Pharmacodynamics, Genetics and Epigenetics) Intervention not available in US (e.g. GHB, Chlormethiazole, Cannabinoids) Healthcare service capacity Wrong Population Animal study Neonatal abstinence syndrome Wrong Condition/Disorder Hangover Alcohol Use Disorder AUD-related disorders (e.g. Alcoholic Liver Disease, Pellagra) Non-alcohol withdrawal related seizure or delirium Wrong Outcome Attention, cue-reactivity to alcohol-related stimuli Provider education, training, level of knowledge Wrong Timing Management of the post-acute withdrawal period Wrong Setting Setting not available in US (Home-based withdrawal) III. Alcohol Withdrawal Scales Table Abbreviation Scale Name Brief Description Primary Use Appropriate setting Summary of Evidence Reference ASSIST Alcohol, Smoking and Substance Involvement Screening Test 8 items Interview format Alcohol use screen Any Results of a study in 7 countries indicate that the ASSIST provides a valid measure of risk for individual substances and for total substance involvement. WHO, 2002 AUDIT Alcohol Use Disorder Identification Test 10 items Alcohol use screen, Risk of alcohol withdrawal Any AUDIT is a useful alcohol screen in general medical settings and that its ability to correctly predict which patients will experience alcohol withdrawal is increased when used in combination with biological markers. Dolman et al., 2005; Saunders et al., 1993 AUDIT-PC Alcohol Use Disorders Identification Test-(Piccinelli) Consumption 10 items Range 0-19 Alcohol use screen, Risk of alcohol withdrawal Hospital Admission AUDIT-PC score is an excellent discriminator of AWS (Sensitivity ¼ 91%, Specificity ¼ 98.7%) Pecoraro et al., 2014 AWS Alcohol Withdrawal Scale 11-items Based on CIWA-A In German Risk of delirium Hospital AWS scale had good performance in predicting alcohol withdrawal delirium Wetterling et al., 1997a AWS -Newcastle Alcohol Withdrawal Scale 10 items Based on CIWA Withdrawal Severity Hospital Patients demonstrated shorter overall course of alcohol withdrawal using the AWS compared with WAS Foy et al., 2006 BAWS Brief Alcohol Withdrawal Scale 5 items Scored 0–3 Withdrawal severity Hospital BAWS patients received less diazepam and had fewer assessments, but both groups had similar lengths of stay, treatment completion rate, no incidence of seizure or delirium. Rastegar et al., 2017 ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 68 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Abbreviation Scale Name Brief Description Primary Use Appropriate setting Summary of Evidence Reference CAM-ICU Confusion Assessment Method 4 items Confusion ICU Excellent reliability and validity in identifying patients with delirium in ICU Ely et al., 2001 CIWA-Ar Clinical Institute Withdrawal Assessment, Revised 10 items Symptom Assessment Scale Any Well established reliability and validity Sullivan et al., 1989 DDS Delirium Detection Scale 8 items Delirium Hospital Good reliability and validity specific to detection of delirium Otter et al., 2005 GMAWS Glasgow Modified Alcohol Withdrawal Scale 5 items Scored 0–2 with max score of 10 Withdrawal severity Hospital GMAWS score of > ¼ 1 predicted CIWA-A > ¼ 8, with a sensitivity of 100% and a specificity of 12%. GMAWS score of > ¼ 2 predicted CIWA-A > ¼ 8, with a sensitivity of 98% and a specificity of 39%. Holzman et al., 2016b LARS Luebeck Alcohol-Withdrawal Risk Scale 11 items 10 items Risk of severe withdrawal Hospital Predicted severe withdrawal among patients admitted for alcohol withdrawal management Wetterling et al., 2006 MINDS Minnesota Detoxification Scale 9 items Symptom severity Hospital; ICU No formal validity study DeCarolis et al., 2007 PAWSS Prediction of Alcohol Withdrawal Severity Scale 10 items Risk of severe withdrawal Hospital; ICU Predicted complicated alcohol withdrawal among medically ill, hospitalized patients Maldonado et al., 2014; 2015 RASS Richmond Agitation-Sedation Scale One item Scored on a continuum with þ4 (combative), 0 (alert and calm), and -5 (unarousable) Sedation and agitation Medical and surgical Reliability and validity in medical and surgical patients, including patients who are sedated and/or ventilated. Sessler et al., 2002 SAWS Short Alcohol Withdrawal Scale 10-items Scored 0–3 Designed to be self-administered Withdrawal severity Ambulatory and Inpatient High internal consistency, good construct and concurrent validity. Gossop et al., 2002 SEWS Severity of Ethanol Withdrawal Scale 7 items Scored 0–3. Withdrawal severity ICU SEWS-driven protocol led to shorter treatment episodes, possibly driven by high administration of medication in first 24 hours of treatment Beresford et al., 2017 SHOT Sweating, Hallucinations, Orientation, and Tremor 4-items Range 0-10 Withdrawal severity Emergency Department Showed potential for measuring pretreatment alcohol withdrawal severity in the emergency department. Gray et al., 2010 WAS Withdrawal Assessment Scale 18 Items Based on CIWA Withdrawal severity Hospital Use of a shortened 10-item CIWA led to similar complication rates but reduced symptom duration compared to 18-item CIWA. Foy et al., 2006 Alcohol Withdrawal Scales Table Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 69 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. IV. Flowcharts (Supplemental Digital Content, V. Sample Medication Regimens Medication Regimen Description, Examples Benzodiazepines (doses in Chlordiazepoxide) Typical single dose Mild withdrawal (CIWA-Ar < 10): 25–50 mg PO Moderate withdrawal (CIWA-Ar 10–18): 50–100 mg PO Severe withdrawal (CIWA-Ar 19): 75–100 mg PO Symptom-triggered 25–100 mg PO q4–6h when CIWA-Ar 10. Additional doses PRN. Fixed-dose Taper daily total dose by 25–50% per day over 3–5 days by reducing the dose amount and/or dose frequency. Additional doses PRN. Day 1: 25–100 mg PO q4–6h Day 2: 25–100 mg PO q6–8h Day 3: 25–100 mg PO q8–12h Day 4: 25–100 mg PO at bedtime (Optional) Day 5: 25 to 100 mg PO at bedtime Front loading Symptom-triggered: 50–100 mg PO q1–2h until CIWA-Ar < 10. Fixed-dose: 50–100 mg PO q1-2h for 3 doses. Phenobarbital Typical single dose 10 mg/kg IV infused over 30 minutes or 60-260mg PO/IM. Monotherapy Symptom-triggered in the ICU: 130 mg IV q30m to target a RASS score of 0 to -1. Fixed dose in the ED: Loading dose 260 mg IV, then 130 mg IV q30m at physician’s discretion. Fixed dose in ambulatory management: Loading dose 60–120 mg PO. Then 60 mg PO q4h until patient is stabilized. Then 30–60 mg PO q6h tapered over 3–7 days. Additional doses PRN. Adjunct therapy Single dose in the ED: 10 mg/kg IV infused over 30 minutes. Escalating dose in the ICU: After maximum diazepam dose (120 mg), if RASS 1, escalating dose of 60 mg ! 120 mg ! 240 mg IV q30m to target RASS score of 0 to -2. Carbamazepine (Tegretol) Monotherapy 600–800 mg total per day tapered to 200–400 mg/d over 4–9 days. Adjunct therapy 200 mg q8h or 400 mg q12h. Gabapentin (Neurontin) Monotherapy Loading dose 1200 mg, then 600 mg q6h on Day 1 or 1200 mg/d for 1–3 days, tapered to 300–600 mg/d up to 4–7 days. Additional doses PRN. Adjunct therapy 400 mg q6–8h. Valproic acid (Depakene) Monotherapy 1200 mg/d tapered to 600 mg/d over 4–7 days or 20 mg/kg/d. Adjunct therapy 300–500 mg q6–8h. CIWA-Ar, Clinical Institute Withdrawal Assessment for Alcohol, Revised; ED, Emergency Department; h, hour(s); ICU, Intensive Care Unit; IM, intramuscularly; IV, intravenously; m, minute(s); mg, milligrams; PO, by mouth; PRN, as needed; q, every; RASS, Richmond Agitation Sedation Scale. VI. Statement Rating Table (Supplemental Digital Content, VII. Disclosures and Conflicts of Interest A. 2020 Guideline Committee Member Relationships with Industry and Other Entities Guideline Committee Member Salary Consultant Speakers Bureau Ownership/ Partnership/ Principal Institutional, Organizational or other financial benefit Research Anika Alvanzo, MD, MS, FACP, DFASAM Johns Hopkins University School of Medicine None None None Nobils None Kurt Kleinschmidt, MD, FASAM UT Southwestern Medical Center None None None None None Julie A. Kmiec, DO, FASAM University of Pittsburgh None None None None None George Kolodner, MD, DLFAPA, FASAM Kolmac Outpatient Recovery Centers None None Kolmac Outpatient Recovery Centers None None Gerald E. Marti, MD, PhD National Institutes of Health None None None None None ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 70 2020 American Society of Addiction Medicine Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. Guideline Committee Member Salary Consultant Speakers Bureau Ownership/ Partnership/ Principal Institutional, Organizational or other financial benefit Research William M. Murphy, DO, MS, DFASAM, Med. Ed. n/a None None None American Osteopathic Academy of Addiction None Lewis S. Nelson, MD, FASAM, FACEP, FACMT (Chair) Rutgers New Jersey Medical School None None None None None Carlos F. Tirado, MD, FASAM CARMAHealth US World Meds Alkermes None None Spark Biomedical Corey Waller, MD, MS, DFASAM, FACEP Health Management Associates None None None California Department of Health Services None The above table presents relationships of the Guideline Committee during the past 12 months with industry and other entities that were determined to be relevant to this document. These relationships are current as of the completion of this document and may not necessarily reflect relationships at the time of this document’s publication. A relationship or arrangement is considered to be significant if the individual receives compensation which includes cash, shares, and/or anything else of value including direct ownership of shares, stock, stock options or other interest of 5% more of an entity or valued at $10,000 or more (excluding mutual funds), whichever is greater. A relationship or arrangement is considered to be modest if it is less than significant under the preceding definition. A relationship or arrangement is considered to be unpaid if the individual does not receive monetary reimbursement. Indicates significant relationship. Indicates modest relationship. B. 2020 ASAM Board of Directors Relationships with Industry and Other Entities (Supplemental Digital Content, C. 2020 ASAM Quality Improvement Council (Oversight Committee) Relationships with Industry and Other Entities (Supplemental Digital Content, 2020 Guideline Committee Member Relationships with Industry and Other Entities Adopted by the ASAM Board of Directors January 23, 2020 ASAM CPG on Alcohol Withdrawal Management 2020 American Society of Addiction Medicine 71 Copyright © 2020 American Society of Addiction Medicine. Unauthorized reproduction of this article is prohibited. D. 2020 Clinical Champions Relationships with Industry and Other Entities Board Member Salary Consultant Speakers Bureau Ownership/ Partnership/ Principal Institutional, Organizational or other finan-cial benefit Research Stephen Holt, MD, MS, FACP Yale School of Medicine None None None None None Darius Rastegar, MD, FASAM Johns Hopkins University School of Medicine None None None None None Richard Saitz, MD, MPH, FACP, DFASAM Boston University Check-Up and Choices American Medical Association National Committee on Quality Assurance Wolters Kluwer Oxford University Press Massachusetts Medical Society American Society of Addiction Medicine Kaiser Foundation Hospitals Beth Israel Hospital University of Oregon Boston Medical Center University of Southern California Westwood High School Harvard University Yale University Massachusetts Society of Addiction Medicine Karolinska Institute American Academy of Addiction Psychiatry SMART Recovery/ Ebix None None National Institutes of Health Boston Medical Center Public Health Management Corporation Alkermes National Institutes of Health Michael F. Weaver, MD, DFASAM University of Texas Health Science Center at Houston Texas Children’s Health Plan Oakbend Medical Center None None National Institute on Drug Abuse None The above table presents relationships of the Clinical Champions during the past 12 months with industry and other entities that were determined to be relevant to this document. These relationships are current as of the completion of this document and may not necessarily reflect relationships at the time of this document’s publication. A relationship or arrangement is considered to be significant if the individual receives compensation which includes cash, shares, and/or anything else of value including direct ownership of shares, stock, stock options or other interest of 5% more of an entity or valued at $10,000 or more (excluding mutual funds), whichever is greater. A relationship or arrangement is considered to be modest if it is less than significant under the preceding definition. A relationship or arrangement is considered to be unpaid if the individual does not receive monetary reimbursement. Indicates significant relationship. Indicates modest relationship. E. 2020 External Reviewers Relationships with Industry and Other Entities (Supplemental Digital Content, ASAM CPG on Alcohol Withdrawal Management Adopted by the ASAM Board of Directors January 23, 2020 72 2020 American Society of Addiction Medicine
7605	https://www.lakeforest.edu/news/the-intruding-ways-of-nf1-nf2-and-glioma-tumors	The Intruding Ways of NF1, NF2, and Glioma Tumors Lake Forest College/News/The Intruding Ways of NF1, NF2, and Glioma Tumors The Intruding Ways of NF1, NF2, and Glioma Tumors Daysi Vargas '12, majored in Biology. March 03, 2013 Sydni Cole, Daysi Vargas, Shabana Yusufishaq Department of Biology Lake Forest College Lake Forest, Illinois 60045 Download PDF Abstract Cancer is among the top leading causes of death worldwide. Although more than one hundred types of cancers exist, those that afflict the nervous system are among the leading causes of cancer death in females in the United States. This review will focus on three major types of brain and nervous system cancers: gliomas and Neurofibromatosis types 1 and 2 (NF1, NF2). Glioma tumors result from mutations in various genes expressing for growth factors, including TGFβ, transcription factors including C/EBPβ and STAT3, gene products of the TROY and IDH1 gene, and cell cycle regulators such as p53. The epidermal growth factor receptor (EGFR) and the type III β3 isoform of neuregulins induce NF1 tumor formation. NF1 tumor formation occurs with a loss of heterozygosity (LOH) in Schwann cells and in the presence of mast cells heterozygous for NF1. NF2 inactivation via merlin calpain-dependent proteolysis, point mutations and ezrin overexpression can result in schwannomas and meningiomas formation. NF2 tumor formation and growth can be prevented by merlin activation, which can prevent cyclin D1 expression and mTORC1 activation. Further studying the causes of these types of tumors will allow for better treatment options to be developed. Introduction An astounding 1 in 4 individuals is diagnosed with cancer in the United States each year (Jemal et al., 2010). Cancer covers a spectrum of diseases characterized by the development of tumors, which form after uncontrollable cell proliferation due to a malfunction in the cell cycle (Fig. 1). The cause of cancer is not fully understood; however, cancer has strong genetic links that can act alone or can be triggered by environmental factors. Some of the most common cancers are breast, prostate, colon, and lung cancer, all of which are the leading causes of cancer death. Although these cancers are extensively studied, 2% of cancer deaths in females are caused by brain and nervous system cancers (Jemal et al., 2010). The three most common types of brain and nervous system tumors are gliomas linked to glial cell malfunction and Neurofibromatosis 1 and 2 linked to genetic alterations in the NF1 and NF2 genes. Gliomas represent 32% of all brain tumors ( They originate from glial cells that provide nourishment to neurons, and can include astrocytes, oligodendrocytes, and ependymal cells. Gliomas occur in both primary and secondary forms, with primary, the most common form of brain tumors, originating in the brain (Gladson et al., 2010) and secondary originating from any other part of the body (Gladson et al., 2010; Huse et al., 2010). Glioblastoma multiforme (GBM) tumors are primary and are the most severe form of brain tumors (Huse et al., 2010). Some symptoms of gliomas, including glioblastoma, are nausea, vomiting, headaches, and seizures (Gladson et al., 2010). Individuals with glioblastoma may only live up to 18 months, while other glioma patients have survivorship up to 8 years (Gladson et al., 2010). In contrast to glioma formation, neurofibroma tumors, which can be benign or malignant, form in the peripheral nervous system (PNS) due to mutations in the NF1 gene, which prevent tumor suppression (Zhu et al., 2002). Mutations in the NF1 gene are characteristic of Neurofibromatosis type 1 (NF1), which affects 1 in 3,500 people worldwide. Other symptoms of patients include café-au-lait macules, which are brown areas on the skin, learning disabilities, and cardiovascular problems (Rubin et al., 2005). Similar to NF1, Neurofibromatosis type 2 (NF2) is an autosomal dominantly inherited disorder (Kimura et al., 1998; Ruttledge et al., 1994). NF2 is characterized by the formation of tumors such as schwannomas, ependyomas and meningiomas in the central nervous system (CNS) (Kimura et al., 2000, Ruttledge et al., 1994, Gareth and Evans, 2009). NF2 is often triggered by loss of heterozygosity (LOH), which is a loss of function of both alleles, and inactivation mutations in the NF2 gene (Chang et al., 2009). Neurofibromatosis type 2 affects 1 out of 60,000 individuals (Gareth and Evans, 2009). The development of NF2 tumors usually causes symptoms that are related to nerve damage such as hearing loss, vertigo, nausea, and vomiting (Gareth and Evans, 2009). Gliomas: Understanding predicts hope When studies on gliomas are performed, often scientists will resort to studying the mechanisms of malignant gliomas, usually those of glioblastoma, the most severe form of gliomas, and tumors originating from astrocytes, representing 54% of gliomas (Gladson et al., 2010; The studies reviewed here focus on glioblastoma. These tumors quickly leave patients in a detrimental state because the tumors not only metastasize, but also become angiogenic, eventually leading to brain function loss. Studies on glioblastoma will contribute to the understanding of gliomas as a whole, allowing for treatments. The analysis that follows provides some insight on how the tumor cells evolve the ability to not only proliferate, but to become angiogenic. Proliferation culprit: TROY gene To understand the mechanisms of tumor invasion, genes related to the proliferation of gliomas were discovered, one of which is TROY (Paulino, et. al., 2010). TROY is expressed largely during embryogenesis, but is well regulated during adulthood (Paulino, et al., 2010). TROY expresses for a cell surface receptor for cells that are migrating or proliferating (Paulino, et al., 2010). Increased levels of the TROY mRNA (Fig. 2) result in a direct linear relationship in severity of tumorigenesis and an inverse relationship with patient survivability (Paulino, et al., 2010). This relationship is fitting considering tumor cells rely on each other’s cell surface receptors for migratory direction. As a result of TROY’s involvement in proliferation and migration it was not surprising to see increased levels of TROY expression in glioblastoma cells, both in vitro and in humans (Paulino, et al., 2010). However, it was found that TROY acts through a cell-signaling pathway and requires the activity of Pyk2 and Rac1 to be functional—allowing for the proliferation of tumors. Therefore, inhibiting one of these cell-signaling components may very well become a potential means of cancer treatment. Figure 1. Tumor Formation and Cell Division. a) In a healthy cell, mitosis follows a normal pathway, resulting in two daughter cells. b) Factors, including gene mutations, cell signaling abnormalities, carcinogens, and protein malfunctions, might contibute to cell division malfuction, uncontrollable cell division, and the development of tumors. Another gene culprit, angiogenesis: IDH1 Most primary brain cancers have an IDH1 gene mutation (Dang et al., 2009). The IDH1 wildtype gene expresses for an enzyme that converts isocitrate to α-ketoglutarate, whereas the mutant converts α-ketoglutarate to 2-hydroxglutarate (2HG), causing havoc to the Kreb’s cycle (Fig. 2). IDH1 mutations in glioma tumors have been confirmed and an increase in 2HG levels in these cells was seen (Dang et al., 2009). Malignant glioma patients have an 80% chance of an IDH1 mutation (Dang et al., 2009). Interestingly, those individuals who have the disorder 2-hydroxylutaric aciduria also have elevated levels of 2HG and are more likely to develop brain tumors (Dang et. al, 2009). It is thought that 2HG’s presence increases radical oxidative species and therefore decreases mitochondrial function, lowering oxygen levels. Lowered oxygen levels can lead to increased blood vessel formation, a key characteristic of gliomablastoma. Perhaps a high level of 2HG plays a large role in angiogenesis, the formation of blood vessels. The relationship between 2HG and the formation of tumors may provide a means of predicting risks to tumor formation at earlier stages of tumorigenesis. Unwelcomed guests C/EBPß and STAT: More ways for angiogenesis It appears that gliomas result from the malfunction of cellular systems that normally act to keep cells healthy. C/EBPβ and STAT3 act to activate some parts of the immune response including inflammation (Carro et al, 2010). However, these proteins are not expressed everywhere in the body; one place where these transcription factors do not get expressed in high quantities is glial cells (Carro et al, 2010). Interestingly, in gliomas these transcription factors are found (Carro et al, 2010) (Fig. 2). Perhaps the transcriptional factors’ normal role of inducing an inflammatory response contributes to tumors’ ability to invade their surroundings. Overall, the presence of the two transcription factors leads to a mesenchymal phenotype, but the inhibition of both transcription factors reduces invasiveness and angiogenesis, predicting a potential treatment option (Carro et al, 2010). TGFβ: more than a growth spurt It is also thought that TGF-β plays a direct role in providing gliomas the ability to invade their surroundings (Wesolowska, et al., 2008). Glial cells are normally cells that provide neurons and other nervous system related cells maintenance and growth, providing reason as to why TGF-β has been found in gliomas (Fig. 2). In healthy cells there are mechanisms for regulating the growth factors through means of inhibition, but this function seems to become impaired during tumorigenesis (Wesolowska, et al., 2008). These growth factors allow tumor cells to go through angiogenesis where they invade their environment and anchor themselves through the use of blood vessels as mentioned earlier (Behzadian et al., 1998). It was found that TGF-β was the source of such invasion as invasions were absent when TGF-β was inhibited (Wesolowska, et al., 2008). Unfortunately, such inhibition of the growth factor resulted in an immune system backlash indicating the importance of some TGF-β availability in a cell (Wesolowska, et al., 2008). Instead, simply inhibiting a receptor in the cell signaling pathway of this growth factor, TβIIR, resulted in similar detriment to the invasion of the tumor cells, but did not cause a negative response from the body. Interestingly, growth factors increase in other types of cancers too (Wesolowska, et al., 2008). A potential treatment by means of inhibiting TβIIR may contribute to increased survivorship in cancer patients. Some hope: TMZ drug activates self-defense Patients with glioma tumors can resort to treatment, such as surgery, radiotherapy, or drugs (Ross et al., 2007). A recent study tested the viability of an alkylating drug (which prevents DNA replication in tumor cells), temozolomide (TMZ) (Ross et al., 2007). The purpose of the experiment was to determine how TMZ leads to the death of cells. The use of the drug first pinpointed a lesion O6 methylguanine (MeG) so called as a result of a nucleotide alteration due the use of the drug (Ross et al., 2007). This lesion was then linked to glioma cells’ ability to go through apoptosis. However, the lesion alone did not induce apoptosis. The lesions were further aggravated when the cells replicated, forming double-strand breaks (Ross et al., 2007). This then triggered apoptosis once FAS, a receptor of p53, was activated (Ross et al., 2007). This further implies that FAS and p53 are also involved in carrying out apoptosis (Ross et al., 2007). The levels of apoptosis increased when p53 was not mutated indicating that when p53 is present it increases the extent of cell death (Ross et al., 2007). In other words, the same amount of the drug for either mutant p53 or wildtype resulted in greater apoptotic activity for the wildtype p53 gene. This is the case because mutant p53 does not result in direct cell death instead it targets the mitochondrial pathway. NF1: the common tumor afflicter Studies of Neurofibromatosis type 1 (NF1) typically are conducted about benign peripheral nerve sheath tumors, or neurofibromas, as said tumors are the most highly occurring tumors in NF1. These tumors contain a multitude of cells, such as axonal processes, Schwann cells, fibroblasts, perineurial cells, and mast cells (Zhu et al., 2002). However, how these cells interact in tumors and other factors that contribute to tumor formation was still unknown. Thus, recent studies have focused on understanding which cells require a loss of heterozygosity (LOH), which cells are most important in tumorigenesis, and different factors that contribute to the tumor formation of said cells. The all-important Schwann cells Since it was unknown which cells in neurofibromas require LOH for tumor formation, it was hypothesized that the LOH in Schwann cells is what leads to neurofibroma development (Zhu et al., 2002). Zhu et al (2002) found that conditional knockout mice, NF1-/- in Schwann cells and NF1+/- in all other cells, had tumors with 100% penetrance. Additionally, mice with NF1-/- Schwann cells and NF1+/+ in other cells had substantially less tumor formation than those with NF1-/- Schwann cells and NF1+/- in other cells (Zhu et al., 2002). Scientists also wanted to determine which cell type in neural crest/Schwann cell lineage is the target for the NF1 mutation and how a loss of NF1 in cells leads to neurofibroma formation (Zheng et al., 2008). It was determined that nonmyelinating Schwann cells are seen in much higher numbers in the hyperplastic mutant nerves than non-mutants; this is seen as glial fibrillary acidic protein (GFAP) is positive in more cells in the mutants than in the non-mutants (Zheng et al., 2008). Additionally, abnormal Remak bundles were seen, with more axons present per pocket in the mutant nerves than in the control nerves (Zheng et. al., 2008). The above data indicates that Schwann cell LOH is necessary to elicit tumor formation (Fig. 2), that nonmyelinating Schwann cells are indicative of neurofibromas, and that Remak bundles may potentially be a “cell of origin for plexiform neurofibroma” (Zheng et al., 2008). Some tumor cells from bone marrow? NF1-heterozygous mast cells, which are found in bone marrow, have been associated with neurofibroma formation when also present with NF1-/- Schwann cells (Yang et al., 2008). Additionally, the c-kit receptor activation is necessary for the release of mast cells from bone marrow (Yang et al., 2008). Since it has been found that NF1 LOH in Schwann cells is not sufficient for tumor formation, a study performed by Yang et al (2008) hypothesized that bone marrow derived cells that contain the c-kit receptor and are NF1+/- are involved in neurofibroma formation. When NF1+/- bone marrow was present with LOH in Schwann cells, even if all other cells were NF1+/+, the mice had a much shorter survival rate and the tumors isolated from the mice exhibited neurofibroma pathology such as wavy Schwann cells, collagen deposits, structural abnormalities, and mast cell infiltration (Yang et al., 2008). Thus, bone marrow derived cells seem to be another type of cell that, when heterozygous for NF1 and in the presence of LOH in Schwann cells, induces tumorigenesis (Fig. 2). EGFR: seen in yet another tumor type In addition to studying Schwann cells and mast cells, researchers wanted to know more about the possibility of activated oncogenes leading to tumorigenesis. The epidermal growth factor receptor (EGFR)’s expression in Schwann cells was thought to be important to neurofibroma or malignant peripheral nerve sheath tumor (MPNST) development, in addition to the possibility that a decrease of EGFR could reduce tumor growth (Ling et al., 2005). Some mice that overexpressed human EGFR (hEGFR) in Schwann cells developed triton tumors (an example of an MPNST) and neurofibromas; these individuals had hEGFR enriched in the tumors (Ling et al., 2005). Additionally, the mice exhibited dysfunctional Schwann cells, accumulated collagen, and high numbers of disrupted small axonal fiber bundles, indicating neurofibroma pathogenesis (Ling et al., 2005). It was also was determined that lower amounts of EGFR activity delayed tumor formation (Ling et al., 2005). It is not known if EGFR is present in high amounts in all neurofibromas or just in the transgenic mice above; however, lowering amounts of EGFR could be a therapeutic target for NF1. β3 isoform: another tumor inducing factor Neuregulins are proteins that typically bind to and activate ErbB receptor tyrosine kinases and cause cell-cell signaling. Type III isoforms of neuregulins are the only isoform to control PNS myelination and are involved in the formation of Remak bundles, which are associated with NF1 (Sanchez et al., 2009). As the type III β1a isoform has been associated with myelination, scientists wanted to study if the type III β3 isoform also had a role in myelination and potentially neurofibroma formation (Sanchez et al., 2009). There was a large increase in numbers of nonmyelinating Schwann cells (nmSC) over time in transgenic mice overexpressing the type III β3 isoform (Sanchez et al., 2009). This was seen as ErbB3 was upregulated, suggesting an expanded Schwann cell population, and GFAP, a marker for nmSC, was also upregulated (Sanchez et al., 2009). Additionally, the number of axons per Schwann cell pocket in Remak bundles was increased in transgenic mice compared to wild type (Sanchez et al., 2009). There also was an increase in collagen in the transgenic mice compared to wild-type mice (Sanchez et al., 2009). These findings indicate tumorigenesis in the transgenic mice, thus denoting that the type III β3 isoform of neuregulins induces tumorigenesis. Unfolding NF2’s Role in Cancer Although NF2 is induced by LOH and inactivation mutations (Chang et al., 2009), targeting of merlin, the protein encoded by the NF2 gene, a tumor suppressor, can also contribute to NF2 inactivation and tumorigenesis (Kimura et al., 1998). Expression of merlin can arrest the cell cycle and prevent tumor growth (Xiao et al., 2005; James et al., 2009) (Fig. 2). Although NF2 has been extensively studied, there is still much to be learned regarding NF2 inactivation and merlin’s role in cancer. NF2 inactivation: the good news for meningiomas and schwannomas It has been reported that mutations and LOH cause NF2 inactivation in 40 – 60% of sporadic meningiomas (Chang et al., 2009). Recent studies conducted using single strand conformation (SSC) and direct sequencing analysis found 24 inactivating mutations in the NF2 gene located on chromosome 22 (Ruttledge et al., 1994) (Fig. 2). Deletion mutations, splice-site alterations, and nonsense mutations are among some of the NF2 inactivating mutations found in NF2 tumors (Ruttledge et al., 1994). These mutations lead to protein inactivation and truncation (Ruttledge et al., 1994) (Fig. 2), which might be the actual cause leading to tumorigenesis. Are mutations the only contributors to NF2 inactivation? Recently it was discovered that mutations in NF2 are not the only factors that can contribute to NF2 inhibition. Given that not all NF2 tumors express mutant NF2, it was important to test if there were other contributors to NF2 inhibition. In fact, post-translational regulation is another way to prevent NF2 encoded protein, merlin, from inhibiting tumorigenesis (Kimura et al., 1998). Calpain-mediated proteolysis of merlin is found in gliomas. Merlin digestion by calpain was seen in vitro and in vivo (Kimura et al., 1998). Given that calpain needs calcium to operate, researchers studied the effect that high amounts of calcium could have on merlin. The results showed an increase of merlin proteolysis by calpain as calcium concentrations increased (Kimura et al., 1998). In a test to determine whether merlin proteolysis contributed to NF2 tumors, researchers expressed merlin and calpain in tumors that did not have NF2 mutations (Kimura et al., 1998). The results indicated that the termination of merlin translation was premature in some of these NF2 mutant depleted tumors (Kimura et al., 1998). This study provided convincing evidence suggesting that calpain proteolysis could in fact be involved in NF2 tumor development. Nevertheless, it is possible that future therapies might involve channel-calcium blockage to inhibit calpain activity and suppress tumor growth (Kimura et al 1998). Other current studies found evidence of ezrin’s implication in NF2 tumors. Ezrin, in comparison to NF2, has an oncogenetic activity that can enhance cell proliferation (Morales et al., 2010). Since NF2 is a tumor suppressor gene, it would be expected to observe less tumor growth when NF2 is active. In GBM cells, low expression of NF2 resulted in cell growth, which indicates NF2 is in fact a tumor suppressor gene (Morales et al 2010). When studying NF2 and ezrin interaction, a negative correlation was found between the two subjects. Overexpression of NF2 decreased ezrin . Interestingly, ezrin down-regulation did not prove to be the contributing factor to NF2 tumor suppressing activity (Morales et al 2010). However, ezrin overexpression caused cell proliferation particularly in cells expressing NF2 (Morales et al., 2010). Another finding provided evidence that ezrin overexpression in cells that contain NF2 cause Rac1 levels to increase (Morales et al., 2010). Since NF2 tumor suppressing activity might be induced via lowering high levels of Rac1, the overexpression of ezrin that leads to high level of Rac1 might override NF2 tumor suppression activity (Morales et al., 2010). Although these results offer new ways to target NF2 tumors, further research is needed to identify other factors involved in NF2 inactivation. Merlin the key in tumor suppression Previous studies have suggested that merlin carries out NF2’s tumor suppressor activity by preventing cell proliferation (Tikko et al., 1994; Lutchman et al., 1995; Xiao et al., 2005). However, the exact mechanism by which merlin prevents cell proliferation still needs to be better understood. A study done using adenovirus-mediated expression of merlin revealed that merlin can arrest the cell cycle and inhibit cell proliferation by down-regulating cyclin D1 . Cell cycle progression was arrested at G1 phase (Fig. 2), and DNA synthesis was reduced in cells that expressed AdNF2 or NF2 A518 (Xiao et al., 2005). This was possible because cells expressing AdNF2 or NF2 A518 reduced cyclin D1 . Interestingly, NF2’s effect on cyclin D1 activity is accomplished by inhibiting cyclin D1’s promoter (Xiao et al., 2005). This study provided evidence suggesting that down-regulation of cyclin D1 could be one of the main targets of merlin that can prevent tumor growth. Another pathway by which merlin prevents cell growth is by negatively interacting with mTOR Complex 1 (mTORC1) (James et al., 2009). Cells that have a merlin deficiency express high levels of phosphorylation of S6 and mTORC1 activity, which resulted in an increase in tumor cell size (James et al., 2009). However, when WT NF2 was re-expressed in NF2 deficient tumor cells, S6 phosphorylation decreased (James et al., 2009). Intriguingly, rapamycin, an mTORC1 inhibitor, is also effective at blocking cell growth in merlin deficient meningiomas (James et al., 2009). These results provide strong evidence of NF2’s merlin involvement in tumor cell growth prevention. In addition, the discovery of rapamycin as a cell growth inhibitor via regulation of mTORC1 could be a possible new therapeutic target to prevent tumor cells growth (James et al., 2009). Future Advances in brain and nervous system cancer research There are a number of protein malfunctions that can potentially lead to gliomas. Currently, much of the work being conducted to understand gliomas tends to be independent of one another. Perhaps new research can focus on connecting what it is we know about glioma formation mechanisms, which will allow us to see how they are interrelated. For instance, it can be further examined if IDH1 mutations and TROY mutations are present at the same time or if C/EBPβ and STAT and TGFβ are dependent on one another to carryout angiogenesis. Studies of NF1 have recently shown that nonmyelinating Schwann cells’ LOH is vital in tumor formation, as is heterozygosity in mast cells. Additionally, factors such as EGFR and type III β3 isoform of neuregulins play parts in tumor formation as well. However, the pathway for which other tumor constituents, such as fibroblasts and collagen, find their way into neurofibromas is yet unknown, as are the specific roles of these tumor components. Thus, this could be further studied. Also, a depletion of EGFR has been shown to reduce tumorigenesis, so this interaction should be further studied in the hopes of finding a treatment for neurofibromas. And lastly, the type III β3 isoform of neuregulins needs to be further studied, as the research done recently is just preliminary and could lead to a greater understanding of NF1. Current research in NF2 has provided insight into some of the mechanisms involved in NF2 and merlin’s tumor suppression activity. For instance these studies showed that tumor cell growth inhibition and cell cycle progression was prevented by merlin via mTORC1 negative regulation and cyclin D1 downregulation. However, it is important to investigate other mechanisms that contribute or affect NF2 tumor suppressor activity. In addition, these studies mentioned the potential calcium channel blockage and rapamycin as a potential tumor treatment. Overall, in the case of each of these types of tumor formation it is important to eventually implement the potential treatments in human studies to determine the effectiveness of possible therapies. Figure 2. The Genetic Pathway of Nervous System Tumors: This figure depicts the way that glioma, NF1, and NF2 tumors develop. Gliomas form due to a malfunctioning protein, protein excess, presence of growth factors, or transcription factors. Neurofibromas (in NF1) develop when Schwann cells lose heterozygosity of NF1 and all other tumor cells (and especially mast cells) are heterozygous for NF1. Inactivation mutations in NF2 and merlin malfunction cannot arrest the cell cycle in tumor cells and contribute to tumor cell proliferation. Conclusion Brain and nervous system cancer includes gliomas, and neurofibromatosis types 1 and 2. Glioblastomas are the worst type of gliomas due to their ability to proliferate and invade other tissues. Defects in the NF1 and NF2 genes can result in loss of function of proteins that can enhance tumorigenesis. Recent studies showed that these three types of brain and nervous system tumors are not only caused by mutations in genes but other factors as well. Links between these three tumors have also been noted and further studies will help develop a better understanding of their interconnection. These findings are relevant to the field of cancer given that they can provide the answers to the development of new therapies. Note: Eukaryon is published by students at Lake Forest College, who are solely responsible for its content. The views expressed in Eukaryon do not necessarily reflect those of the College. Articles published within Eukaryon should not be cited in bibliographies. Material contained herein should be treated as personal communication and should be cited as such only with the consent of the author. Acknowledgements We would like to thank Dr. DebBurman for his guidance in completing this paper. We would also like to thank Keith Solvang for his support. And, finally we would like to thank Lake Forest College and the school’s Biology Department. References Behzadian, M. A., Wang. X., Shabraway, M., Caldwell, B. R. (1998). 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7606	https://www.youtube.com/watch?v=ors9mJxPv1s&lc=UghHHkf_OSQQzXgCoAEC	Demostración de la desigualdad de Cauchy Schwarz KhanAcademyEspañol 727000 subscribers 624 likes Description 81757 views Posted: 5 Jun 2013 Más lecciones gratuitas en: 43 comments Transcript: Vamos a tomarnos dos vectores no nulos. Me voy a tomar el vector x y me voy a tomar el vector y van a ser vectores que existan en rn, es decir, que tengan n componentes. Y recuerda que lo que le estoy pidiendo a estos dos vectores es que no sean cero, que no sean el vector cero. Y bueno, yo lo que quiero construir con estos dos vectores es la siguiente desigualdad. Fíjate bien, me voy a tomar el valor absoluto de x. y, el valor absoluto de x punto y si te das cuenta, cuando calculamos el producto punto de dos vectores, nos da un número real. Por lo tanto, puedo calcular el valor absoluto de un número real y esto va a ser menor o igual que la longitud de x multiplicada por la longitud de y. Y de hecho esta igualdad se cumple, es decir, quitamos la desigualdad en la siguiente ocasión, solamente cuando ocurre lo siguiente. Así que vamos a escribirlo. El valor absoluto de x punto y es igual a la longitud de x por la longitud de y. Es decir, que lo que vamos a buscar es cuándo se cumple esta igualdad, cuándo quitamos la desigualdad. Entonces, el valor absoluto de x. Y es igual a la longitud de x por la longitud de y. Si pasa lo siguiente y es sí y solo sí, si pasa que uno de estos dos vectores es un múltiplo escalar del otro y si recuerdas eso significaba que uno de los dos vectores era colineal al otro o dicho otra manera, que podemos ver a uno de los vectores como C veces el otro. Sin pérdida de generalidad voy a decir que X es igual a C veces el vector Y. Y bueno, a todo esto que tenemos aquí se le conoce como la desigualdad de Koshi Schwarz, la famosa desigualdad de Koshi Schwarz. Y es que justo lo que quiero en este video es probar esta desigualdad. Nosotros no podemos darla por sentado. Sería muy bueno que con todas las herramientas que hemos construido hasta ahorita podamos demostrar esta desigualdad. Y para esto me voy a tomar una función que tal vez parezca un poco artificial, pero es una función que me va a ser muy útil para poder demostrar la desigualdad de Kosh Schwarz. Y la función que me voy a tomar es la siguiente. PT es igual a la longitud del siguiente vector. Voy a tomarme t veces el vector y. Si te das cuenta, si multiplico al vector y por un número real, me da un vector. Y a esto le voy a quitar el vector x, lo cual me sigue dando un vector. Y a todo esto le voy a calcular su longitud y elevarla al cuadrado. Y bueno, seguramente te estás preguntando por qué me voy a tomar esta función, pero antes de explicarte esto, quiero que recordemos lo siguiente. ¿Te acuerdas de la definición de la longitud de un vector? La longitud de un vector B la veíamos como la raíz cuadrada de su primera componente elevada al cuadrado más su segunda componente elevada al cuadrado y así cada una de sus componentes. Y ya esto definíamos como la longitud de un vector. Ahora lo que quiero que veas es que la longitud de un vector siempre es mayor o igual a cer0. Porque si nosotros tomamos cualquier número y lo elevamos al cuadrado, me da mayor o igual a cer. Y si tomamos puras sumas de cosas mayores o iguales a cero, me da algo mayor o igual a cer. Y si sacamos la raíz cuadrada de algo mayor o igual a er mayor o igual a er y por lo tanto esta longitud que yo tengo aquí va a ser mayor o igual a er0. Es decir, pd es mayor o igual a 0. Y lo segundo que quiero que recordemos es algo que vimos hace un par de videos que me va a ser muy útil para demostrar esta desigualdad. Fíjate bien, la longitud de un vector al cuadrado era lo mismo que el vector el vector. Esto lo vimos hace dos videos. Y bueno, justo esto es lo que voy a utilizar ahorita para escribir la forma en la que definí pdt, es decir, como esta longitud elevada al cuadrado, como un producto punto de dos vectores, es decir, t veces el vector y menos el vector x pun t veces el vector y menos el vector x. Estoy utilizando solamente lo que acabo de escribir de verde a la derecha. Y bueno, en el video pasado habíamos demostrado algunas propiedades como asociatividad, como conmutatividad y como distributividad que voy a utilizar justo ahorita. De hecho, lo primero que voy a hacer es distribuir este producto punto que tenemos aquí. Recuerda que el producto punto es una forma de multiplicación de vectores que cumplía la distributividad. Por lo tanto, lo único que tengo aquí es como si tuviera dos binomios multiplicándose, solamente que se están multiplicando por el producto punto. Entonces, esto me va a quedar t veces el vector y punto t veces el vector y. Estoy multiplicando el primer vector de esta diferencia de vectores por el primer vector de la segunda diferencia de vectores y me queda t veces y pun t veces y. Después me voy a tomar a -x el vector - x y voy a calcular su producto punto con t veces y qué me queda como es producto punto me va a quedar - x pun t ve veces el vector y. Ahora voy a hacer lo mismo con t veces y punto el vector - x, es decir, este de aquí y este de aquí. Y entonces me queda - t veces y - t veces y pun x. Y bueno, por último me queda el vector - x, que a su vez hay que tener el producto punto con -x y entonces esto quiere decir que es -1x, lo puedo poner así con -1x punto el vector -1x. Date cuenta que aquí es como si tuviéramos un -1, es decir, como si tuviéramos una constante multiplicando un vector. Y es que al final -x es lo mismo que -1 por el vector x. Así es mucho más fácil pensarlo para no meternos en complicaciones. Y bueno, ya que tengo esto, esto a qué va a ser igual. Si te das cuenta, tengo t veces y pun t veces Y. Aquí puedo sacar a ambas ts y multiplicarlas. Por lo tanto, me va a quedar y punto y que multiplica a t². Recuerda que la multiplicación por escalar y el producto punto cumplen asociatividad. Y después de aquí y de aquí voy a realizar la resta de estos dos vectores. Si te das cuenta, como el producto punto y la multiplicación por escalar a su vez son asociativas, entonces me estoy tomando la diferencia de dos números reales iguales. Por lo tanto, me queda -os veces el producto punto de x punto y y esto multiplicado por t. Y bueno, para finalizar aquí tengo -1 -1. Las constantes se van porque menos por menos me da más, 1 1 me da 1 y al final solamente me queda x. X. X. Todo esto utilizando las propiedades del producto punto. Y bueno, vamos a ponerlo de colores para que veas de dónde salió cada cosa. Esto salió de aquí. Lo rosa sale de lo rosa y lo naranja sale de lo naranja. Y este de aquí es mayor o igual a cer porque habíamos dicho que PDT es mayor o igual que cer. Entonces, déjame ponerlo. Esto es mayor o igual que cer0. Y por otra parte, pues esto es lo que habíamos definido como pd, como una función de un escalar. Ahora, seguramente te sigues preguntando, ¿por qué me tomé esta función y no me tomé cualquier otra expresión o por qué partí de esta expresión de aquí? Porque a continuación lo que voy a hacer es nombrar las cosas de una manera mucho más sencilla para que te des cuenta de la importancia de tener a PDT. Voy a decir que a es Y. Y. Voy a decir que b -2 veces x. Y. Recuerda que al final estos son números reales y c también va a ser x. X. También es un número real y ya tengo unas nuevas definiciones a, b y c. Entonces, esto lo puedo escribir de la siguiente manera. Lo puedo escribir como at² - bt. Esto nos va a ayudar a verlo de una manera mucho más sencilla. Y a esto le voy a sumar C. Y si te das cuenta, a, b y c son números reales. Y por lo tanto, esta es una función que depende solamente de t. Esta es pt, que no vayas a olvidar que es mayor o igual a er y es mayor o igual a er para cualquier t. Por lo tanto, me voy a tomar una t específica, la cual me va a ayudar a obtener la desigualdad que yo tengo aquí arriba. Voy a calcular quién es P de B / 2A. Voy a evaluar esta función en el tiempo = a b / 2a. Y lo primero que quiero que veas es que este tiempo sí está definido. Es decir, lo que voy a buscar es que 2a no sea 0 porque si fuera 0 esto no estaría definido. Pero a no es certo punto de y pun y lo más importante, habíamos dicho que y no es un vector nulo, no es el vector cero y por lo tanto a que es y punto y o lo podemos ver como la longitud de y elevada cuadrado, no es cer0 porque al menos una de sus componentes es distinta de 0 y 2 veces a no es 0 y por lo tanto b / 2a sí está definido. Y bueno, ya que me di cuenta que esto si está definido, entonces voy a sustituir a t b / 2a. Ahí me va a quedar a que multiplica a t cuada, pero t cuad es b² / 4a. Y a esto le voy a quitar b ve veces t, pero t es b / 2a. Entonces b que multiplica a b / 2a. Y a esto le voy a sumar c. Y a esto le voy a sumar c. Y esto es mayor o igual que er0 porque recuerda que esto es mayor o igual que er para cualquier t. Y bueno, ya que llegamos a esto, vamos a simplificarlo un poco. Por ejemplo, aquí tengo una A y aquí una A dividiendo, se van y aquí tengo B B, pues esto es b². Entonces, déjame escribirlo así. B² / 4A - B B es B². B² / 2A + C + C. Y esto es mayor o igual a 0. Y fíjate bien, aquí tengo 4a y aquí tengo 2a. ¿Qué pasa si multiplico a esta expresión de aquí por 2 tanto arriba como abajo? 2 / 2 es 1 y entonces no estoy haciendo nada, pero lo que realmente hice fue encontrar un denominador común y entonces ahora sí puedo restar estos dos. Me queda b² / 4a - 2 veces b² / 4a. Pues esto lo puedo simplificar como - b² / 4a. Déjame escribirlo aquí abajo. Esto es - b² / 4a + c. Esto es mayor o igual a 0. Y a continuación lo que voy a hacer es sumar b² / 4a de los dos lados de la ecuación. O dicho de otra manera, lo que voy a hacer es pasar del otro lado b² / 4a y me va a quedar que c es mayor o igual que b² / 4a. Lo único que hice es pasar a este que tiene signo negativo del otro lado con signo positivo. Y a continuación lo que voy a hacer es multiplicar todo por cuatro veces a. Pero ojo, estoy multiplicando todo por cuatro veces a recordando quién era a. A es y punto y como dijimos que y es siempre mayor que 0, recuerda que como y no es un vector nulo, entonces y punto y es lo mismo que la longitud del vector y elevado al cuadrado, lo cual es mayor que 0. Entonces, yo puedo multiplicar ambos lados de esta desigualdad por 4a. Y te estoy contando todo esto por una razón específica, porque si yo estoy multiplicando todo esto por 4a y 4a es mayor que 0, entonces esta desigualdad se conserva. Entonces me va a quedar 4ac mayor o igual que b². Recuerda que si a fuera negativo, lo que pasaría es que esta desigualdad se voltearía. Y bueno, ya que tengo yo esto, lo que voy a hacer a continuación es sustituir quién es A, B y C para ver qué es lo que me queda. Recuerda que A era Y. Y, entonces me va a quedar cuatro veces A. A es y punto y, pero y punto y si nos acordamos en lo que vimos en el video pasado, era la longitud de y elevada al cuadrado. Es cuatro veces la longitud de y. Aquí lo voy a escribir. Me queda la longitud de y que se pone así, elevada al cuadrado del vector y elevado cuadrado. Esto es lo mismo que a porque recuerda que a es y punto y y punto y es lo mismo que la longitud de y elevada cuadrado. Y después hay que multiplicarlo por c, pero c es x. X. Y de igual manera, recuerda que lo que vimos en el video pasado era que x. x es lo mismo que la longitud de x elevada cuadrado. Entonces, me va a quedar del lado izquierdo de esta desigualdad me queda 4 veces a c, que es lo mismo que cuatro veces la longitud de y elevada cuadrado por la longitud de x elevada cuadrado. Y esto es mayor o igual que b²ad, pero b esto que tenemos aquí. Por lo tanto, me va a quedar dos veces x. Y x. y esto elevado al cuadrado. Espera, espera, espera. Aquí es todo esto elevado al cuadrado. Todo esto es b, por lo tanto tengo b². Y es que ahora sí me va a servir bastante porque voy a simplificar un poco todo esto. Me queda cuatro veces la longitud de y elevada cuadrado por la longitud de x elevada cuadrado. Y esto es mayor o igual que el cuadrado de 2, que es 4. Cuatro veces x. Y x. Y x. Y que multiplica a x. y o mejor lo puedo poner como x. Y elevado cuadrado. Sí, déjame quitar todo esto porque si no nos vamos a confundir en un futuro. Esto es x. Y o x. Y, pero mejor lo voy a poner como cuatro veces x. Y elevado cuadrado. Y bueno, 4 y cuatro se van cuatro que multiplica y cuatro que multiplica del otro lado. Si divido toda esta desigualdad entre cuatro se van estos dos cuatro y me queda que la longitud de y elevada cuadrado por la longitud de x elevada cuadrado es mayor o igual que x. y elevado al cuadrado. A continuación, lo que voy a hacer es sacar raíz cuadrada de ambos lados de la ecuación y simple y sencillamente me va a quedar que la longitud de y por la longitud de x, ya saqué la raíz cuadrada, que por cierto se cancela con los cuadrados que tenemos aquí porque estoy utilizando las propiedades de los exponentes. Y del otro lado me queda x. y elevado al cuadrado raíz cuadrada. Pero me estoy tomando la raíz positiva. La raíz positiva de algo que está elevado al cuadrado es simple y sencillamente el valor absoluto de lo que se está elevando al cuadrado, es decir, x punto y. Y ojo, es muy importante que te des cuenta que me estoy tomando la raíz positiva y por eso estoy poniendo el valor absoluto. Y es que al final date cuenta de algo, x. y podría ser negativo, pero cuando lo elevamos al cuadrado se vuelve algo positivo. Y cuando le sacamos raíz cuadrada a algo positivo, me va a dar algo positivo si tomamos la raíz cuadrada principal, es decir, la raíz principal o la raíz positiva y por eso pongo el valor absoluto de estos dos. Y adivina que ya llegamos a lo que queríamos. Justo esto era la famosa desigualdad de Koshi Schwarz. Esto es lo que nos habíamos propuesto demostrar y llegamos a ello después de varios pasos, pero acabamos de probar que si se cumple la desigualdad de Koshi Schwarz. Bueno, al menos la primera parte de esta desigualdad, porque habíamos dicho que la igualdad se cumplía cuando los vectores cumplían lo siguiente. Uno de ellos era múltiplo escalar del otro o si lo decimos de otra manera eran vectores colineales. Así que vamos a tomarnos a x = c veces y. Vamos a probar solamente esa última parte a ver si sí se cumple la igualdad. ¿Quién sería x punto y en valor absoluto? Pues esto es lo mismo que utilizando que x = cy. Es lo mismo que el valor absoluto de cy pun y. Pero esto es lo mismo que el valor absoluto de c por una propiedad del valor absoluto que multiplica al valor absoluto de y pun y al valor absoluto de y puny y. Y bueno, ¿quién era y puny y? Ya lo hemos visto varias veces. y punto y es ni más ni menos que la longitud de y elevada al cuadrado. Por lo tanto, aquí tenemos el valor absoluto de C que multiplica a la longitud de y elevado cuadrado. Y bueno, esto es lo mismo. Lo podemos ver como el valor absoluto de C que multiplica a la longitud de y por la longitud de y. La longitud de y elevada al cuadrado es lo mismo que la longitud de y por la longitud de y. Y bueno, sería muy buen ejercicio que tú probaras que lo siguiente que voy a hacer es cierto. Es muy fácil probarlo utilizando la definición de longitud. Fíjate bien, si nosotros tomamos la longitud de una constante multiplicada por un vector, esto es exactamente lo mismo que el valor absoluto de la constante que multiplica a la longitud del vector. Es decir, las constantes salen de la longitud de un vector en valor absoluto. Es decir, que esto que estoy tomando aquí es lo mismo que la longitud de c veces y insisto, sería muy bueno que tú probaras esta afirmación. Y bueno, a su vez esto multiplica a la longitud de y. ¿Y qué crees? ¿Ya te diste cuenta quién tenemos aquí? C veces Y es X. Es el vector X. Por lo tanto, tengo la longitud de X que multiplica a la longitud de Y hasta ahorita sigues siendo una igualdad. Y por lo tanto estamos comprobando que cuando X es igual a C veces el vector y, es decir, X es un múltiplo escalar de y, entonces obtengo la igualdad. De todos modos, espero que hayan encontrado esto muy útil. La desigualdad de Koshi Schwarz la usaremos mucho cuando probemos otros resultados en álgebra lineal. Y en un video futuro les voy a dar un poco más de intuición acerca de por qué esto tiene mucho sentido en relación con el producto. Po.
7607	https://www.pw.live/school-prep/exams/circular-permutation-formula	Circular Permutation Formula: Principles and Solved Examples All Courses Competitive Exams IIT JEE, NEET, ESE, GATE, AE/JE, Olympiad Only IAS UPSC, State PSC School Preparation Foundation (Class 6-12), Commerce, Arts, CuriousJr (1st - 8th), Science, International Boards Govt Exam Judiciary, SSC, Defence, Teaching, JAIIB & CAIIB, BIHAR EXAMS WALLAH, UP Exams, Railway, Nursing Exams, Banking, WB Exams UG & PG Entrance Exams MBA, IPMAT, IIT JAM, LAW, CUET UG, UGC NET, GMAT, Design & Architecture, Pharma, CSIR NET, CUET PG, NEET PG FINANCE CA, CS, Finance Courses, ACCA Others Online Degrees English Proficiency Test IELTS, TOEFL Agriculture Agriculture IIT JEE NEET ESE GATE AE/JE Olympiad VidyapeethPW SkillsPW StoreReal TestClass 1st - 8thPower Batch Login/Register SCHOOL EXAMS Circular Permutation Formula Circular Permutation Formula: Principles and Solved Examples The circular permutation Formula is a mathematical concept that deals with the arrangement of objects in a circular fashion. To better grasp this idea, consider a circular arrangement of seats in a theater, a round table at a party, or even the positions of the hands on a clock. Anchal Singh 16 Oct, 2023 Share Permutations are a fundamental concept in combinatorics, the branch of mathematics that deals with counting and arranging objects. While many of us are familiar with permutations in the context of linear arrangements, such as arranging people in a line or arranging objects in a row, there exists a fascinating variation known as circular permutation. In the Circular Permutation Formula , objects are arranged in a circle rather than in a straight line. This concept adds a layer of complexity and intrigue to the world of permutations, making it a subject worth exploring. In this article, we will delve into the world of circular permutation, covering its definition, principles, and various applications. We will also discuss important formulas and theorems associated with circular permutations and provide examples to illustrate the concepts. Understanding Circular Permutation The circular permutation Formula is a mathematical concept that deals with the arrangement of objects in a circular fashion. To better grasp this idea, consider a circular arrangement of seats in a theater, a round table at a party, or even the positions of the hands on a clock. In these situations, the order in which objects or people are arranged matters, but there is no distinct "start" or "end" point. In a linear permutation, such as arranging books on a shelf, the order matters, and there is a clear starting point and ending point. However, in a circular permutation, the arrangement forms a closed loop, and there is no inherent beginning or end. This distinction makes circular permutations unique and intriguing. Principles of Circular Permutation To work with circular permutations effectively, it's essential to understand some fundamental principles: The Total Number of Circular Permutations (n-1)! In a circular permutation of 'n' distinct objects, there are (n-1)! different arrangements. This formula is based on the idea that we can fix one object in a particular position (say, the top of the circle), and then arrange the remaining (n-1) objects linearly. Since linear permutations are well-understood, we simply calculate (n-1)! to find the number of circular permutations. For example, if you have five distinct objects, there are (5-1)! = 4! = 24 different circular permutations. The Relative Ordering Matters In circular permutations, the relative ordering of objects matters. For instance, if you have five people sitting at a round table, the order in which they are seated is significant. Two arrangements with the same people in different orders are considered distinct circular permutations. Rotationally Equivalent Permutations Circular permutations have rotational symmetry, meaning that if you rotate a circular arrangement, it remains the same. This implies that if you have found one circular permutation, you can obtain others by rotating it. However, these permutations are considered equivalent. To count only distinct circular permutations, we often restrict our count to one representative from each equivalence class. Applications of Circular Permutation Formula Circular permutations find applications in various fields, including mathematics, computer science, and real-world scenarios. Here are some practical examples of their usage: Seating Arrangements: One of the most common applications is in arranging people around circular tables, such as at weddings or conferences. Understanding circular permutations helps event planners determine how many different seating arrangements are possible. Clock Hands: Circular permutations are used to calculate how many times the hour and minute hands of a clock will overlap in a day. This problem is not only a classic example of circular permutations but also has practical implications in timekeeping. Computer Science: In computer science, circular permutations are employed in various algorithms and data structures. For instance, circular queues use circular permutations to efficiently manage data. Chemistry: Circular permutations are used to study the properties of cyclic molecules in chemistry. Understanding the different arrangements of atoms in a molecule can provide valuable insights into its reactivity and stability. Also Check – Triangles Formula Formulas and Theorems To solve problems involving circular permutations, it's essential to be familiar with some key formulas and theorems: Number of Circular Permutations: As mentioned earlier, the number of circular permutations of 'n' distinct objects is (n-1)!. Number of Rotational Equivalence Classes: The number of rotational equivalence classes of 'n' distinct objects is given by gcd(n, k), where gcd stands for the greatest common divisor of 'n' and 'k.' Each equivalence class contains (n/gcd(n, k)) permutations. Counting Clock Hands Overlaps: To find the number of times the hour and minute hands of a clock overlap in a day, you can use the formula: (\|30H - 11/2M\|)/360, where 'H' is the hours and 'M' is the minutes. This formula accounts for the relative speeds of the hour and minute hands. Also Check – Introduction to Euclid Formula Solved Examples Let's explore some examples to illustrate the concepts of circular permutations: Example 1: How many distinct seating arrangements are possible for six people at a round table? Solution: Using the formula for circular permutations, we have (6-1)! = 5!. Thus, there are 5! = 120 different seating arrangements for six people at a round table. Example 2: Consider a circular arrangement of eight distinct objects. How many rotational equivalence classes are there? Solution: To find the number of rotational equivalence classes, we need to calculate gcd(8, 8) = 8. Therefore, there are 8 equivalence classes, each containing (8/8) = 1 permutation. Example 3: Calculate the number of times the hour and minute hands of a clock overlap in a day. Solution: Using the formula mentioned earlier, we can calculate the number of overlaps as (\|30H - 11/2M\|)/360, where 'H' represents the hours and 'M' represents the minutes. For each hour, there are two overlaps (e.g., 12:30 and 6:00). For each minute, there is one overlap (e.g., 12:00). So, for 12 hours and 60 minutes, we have 12 2 + 60 1 = 24 + 60 = 84 overlaps in a day. Also Check – Probability Formula Conclusion Circular permutations offer a captivating perspective on combinatorics, allowing us to explore arrangements in a circular manner where the order of objects matters. Understanding the principles, formulas, and theorems associated with circular permutations can help solve a wide range of real-world problems, from seating arrangements to clock hands overlaps. As we continue to explore the world of mathematics, circular permutations stand as a testament to the richness and versatility of combinatorics, offering both intellectual intrigue and practical application. Circular Permutation Formula FAQs What is a circular permutation? A circular permutation is a mathematical concept that deals with the arrangement of objects in a circular manner, where the order of arrangement matters, but there is no distinct starting or ending point. How do circular permutations differ from linear permutations? In linear permutations, objects are arranged in a straight line, and there is a clear start and end point. In circular permutations, objects are arranged in a circle, and there is no inherent beginning or end. How do I calculate the number of circular permutations of 'n' objects? To calculate the number of circular permutations of 'n' distinct objects, you can use the formula (n-1)! (n minus 1 factorial). What does (n-1)! mean in circular permutations? (n-1)! represents the factorial of (n-1), which means multiplying all positive integers from 1 to (n-1) together. It accounts for the number of distinct circular permutations by fixing one object as a reference point and arranging the remaining (n-1) objects linearly. How can I find rotational equivalence classes in circular permutations? The number of rotational equivalence classes for 'n' distinct objects is given by gcd(n, k), where gcd represents the greatest common divisor of 'n' and 'k.' Each equivalence class contains (n/gcd(n, k)) permutations. 🔥 Trending Blogs NCERT Solutions for Class 6 English A Pact With the Sun NCERT Solutions for Class 6 English A Pact With the Sun NCERT Solutions for Class 9 English Chapter 5 The Snake and the Mirror NCERT Solutions for Class 9 English Chapter 5 The Snake and the Mirror NCERT Solutions Class 9 English Chapter 4 A Truly Beautiful Mind NCERT Solutions Class 9 English Chapter 4 A Truly Beautiful Mind NCERT Solutions Class 9 English Poem Chapter 4 The Lake Isle of Innisfree NCERT Solutions Class 9 English Poem Chapter 4 The Lake Isle of Innisfree NCERT Solutions Class 9 English Poem Chapter 2 Wind NCERT Solutions Class 9 English Poem Chapter 2 Wind Talk to a counsellor Have doubts? 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7608	https://m.docin.com/p-2631593086.html	小学数学利润与折扣问题 - 豆丁网幼儿/小学教育 > 小学考试 > 小学数学利润与折扣问题.doc 37.0KB 立即下载在线编辑文档继续编辑重新编辑查找查找 Ctrl+F 替换 Ctrl+H 重置全文检索查找中… 加入我的组合 VIP去广告加入我的组合 VIP去广告加入我的组合加入我的组合 VIP去广告加入我的组合加入我的组合剩余6页未读，展开继续阅读下载文档 VIP去广告君，已阅读到文档的结尾了呢~~ 立即下载加入会员，超低价下载分享到下载文档加入会员超低价下载菜单列表阅读幻灯片阅读全文检索书签放大缩小评论上传文档文档信息关于豆丁全屏上一页 /11下一页重置全文检索查找中… 中小学在家学——百位名师精选20多万篇中小学同步练习专题资料百位名师精选2多万篇中小学同步练习专题资料专栏更多相关精品文档人教版初中九年级数学知识点精讲（下册）精华专题小学数学试卷大合集（全国各地区）精华专题小学数学总复习专题训练-利息、折扣问题-通用版热度: 2015年小学数学纳税、利息和折扣问题（练习题）热度: 小学数学六年级上册《折扣问题》PPT课件热度: 小学数学应用题(折扣问题) 热度:") 小学数学-利润问题热度: 小升初数学复习专题之—利息、折扣问题热度: 利润与折扣问题应用题热度: 苏教版六年级数学折扣问题热度: 折扣问题(学历案)-2023-2024学年小学数学六年级下册热度:-2023-2024学年小学数学六年级下册") 小学数学总复习专题讲解及训练百分数解决利息、折扣问题热度: 《折扣问题》热度: 小学数学课件-利润问题课件热度: 小升初数学复习：利息折扣问题应用题分类解析与训练热度: 2023【苏教版】小学六年级数学上册-【第9课时折扣问题】热度: 小学数学经济利润问题热度: 小学数学折扣课件热度: 最新苏教版小学六年级数学上册6.9《折扣问题》公开课课件.ppt 热度: 六年级数学教案折扣问题热度: 折扣问题教学课件热度: 2023【苏教版】小学六年级数学上册作业课件-【第6课时折扣问题】热度: 求助：有没有小学数学利润折扣问题? 行业定制服务合同定制量身定制合同 4.73.5W人已咨询互联网＋法律电商上市品牌立即定制报告定制按需定制各类研报 5.02.17W人已咨询产业/市场/竞品/渠道等调研与定制立即定制联系我们更多 > 个人入驻机构入驻帮助中心客服QQ (无需客户端): 800004241 侵权举报处理热线: 010-57649669 更多相关精品文档中小学在家学——百位名师精选20多万篇中小学同步练习专题资料课上不够课后凑人教版初中九年级数学知识点精讲（下册）精华专题小学数学试卷大合集（全国各地区）精华专题小学数学总复习专题训练-利息、折扣问题-通用版热度: 2015年小学数学纳税、利息和折扣问题（练习题）热度: 小学数学六年级上册《折扣问题》PPT课件热度: 小学数学应用题(折扣问题) 热度:") 小学数学-利润问题热度: 小升初数学复习专题之—利息、折扣问题热度: 利润与折扣问题应用题热度: 苏教版六年级数学折扣问题热度: 折扣问题(学历案)-2023-2024学年小学数学六年级下册热度:-2023-2024学年小学数学六年级下册") 小学数学总复习专题讲解及训练百分数解决利息、折扣问题热度: 《折扣问题》热度: 小学数学课件-利润问题课件热度: 小升初数学复习：利息折扣问题应用题分类解析与训练热度: 2023【苏教版】小学六年级数学上册-【第9课时折扣问题】热度: 小学数学经济利润问题热度: 小学数学折扣课件热度: 最新苏教版小学六年级数学上册6.9《折扣问题》公开课课件.ppt 热度: 六年级数学教案折扣问题热度: 折扣问题教学课件热度: 2023【苏教版】小学六年级数学上册作业课件-【第6课时折扣问题】热度: 扫扫二维码，随身浏览文档手机或平板扫扫即可继续访问推荐豆丁书房APP 扫扫更高清分享文档将文档分享至：分享最近收藏82 分享到帮助中心常见问题意见反馈扫二维码 3秒自动关闭窗口
7609	https://www.ck12.org/flexi/physics/nuclear-reactions/define-the-q-value-in-nuclear-reactions/	Flexi answers - Define the Q-value in nuclear reactions. \| CK-12 Foundation Subjects Explore Donate Sign InSign Up All Subjects Physics Nuclear Reactions Question Define the Q-value in nuclear reactions. Flexi Says: The Q-value of a nuclear reaction is a measure of the amount of energy that is either released or absorbed during that reaction. It is defined as the difference between the sum of the masses of the initial reactants and the sum of the masses of the final products, converted into energy according to Einstein’s mass-energy equivalence principle E=m c 2, where E is the energy, m is the mass difference, and c is the speed of light. More formally, if m i represents the total mass of the reactants and m f represents the total mass of the products, then the Q-value (Q) of a nuclear reaction can be expressed as: Q=(m i−m f)c 2 If Q>0, the reaction is exothermic, releasing energy to the surroundings. If Q<0, the reaction is endothermic, requiring an input of energy to occur. The Q-value is critical for understanding the energy dynamics of nuclear reactions, including those that power stars, nuclear reactors, and are involved in various radioactive decay processes. Analogy / Example Try Asking: Explain why nuclear fusion reactions release more energy than fission reactions.What does neutron absorption accomplish in a nuclear reactor?The time required for half of the atoms in any given quantity of a radioactive substance to decay is the? How can Flexi help? By messaging Flexi, you agree to our Terms and Privacy Policy
7610	https://www.splashlearn.com/math/addition-strategies-games	Addition Strategies Games for Kids - Fun Math Games \| SplashLearn Parents Explore by Grade Preschool (Age 2-5)KindergartenGrade 1Grade 2Grade 3Grade 4Grade 5 Explore by Subject Math ProgramEnglish Program More Programs Homeschool ProgramSummer ProgramMonthly Mash-up Helpful Links Parenting BlogSuccess StoriesSupportGifting Also available on Educators Teach with us For TeachersFor Schools and DistrictsData Protection Addendum Impact Success Stories Resources Lesson PlansClassroom ToolsTeacher BlogHelp & Support More Programs SpringBoardSummer Learning Our Library All By Grade PreschoolKindergartenGrade 1Grade 2Grade 3Grade 4Grade 5 By Subject MathEnglish By Topic CountingAdditionSubtractionMultiplicationPhonicsAlphabetVowels One stop for learning fun! Games, activities, lessons - it's all here! 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From the addition of three whole numbers to identifying the correct addition expression, SplashLearn’s interactive games follow all kinds of strategies to help your child learn and understand addition completely. Personalized Learning Fun Rewards Actionable Reports Parents, Sign Up for Free Teachers, Use for Free Join millions of learners learning with a smile Sign Up GRADE Pre-K K 1 2 3 4 5 CONTENT TYPE Resources Games Worksheets Glossary Lesson Plans SUBJECT Math (2,046) Number Sense (381) Number Recognition (30) Number Recognition Within 5 (12) Number Recognition Within 10 (12) Number Recognition Within 20 (6) Number Tracing (20) Number Tracing Within 5 (5) Number Tracing Within 10 (5) Number Tracing Within 20 (10) Number Sequence (56) Counting (141) Counting Objects Within 5 (47) Counting Objects Within 10 (43) Counting Objects Within 20 (6) Compare Numbers (47) Compare Objects (7) Compare Numbers Using Place Value (5) Compare 2-Digit Numbers (6) Compare 3-Digit Numbers (10) Order Numbers (15) Skip Counting (36) Skip Count By 2 (8) Skip Count By 5 (8) Skip Count By 10 (13) Skip Count By 100 (3) Even And Odd Numbers (3) Place Value (78) Teen Numbers (4) Word Form (5) Expanded And Standard 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This engaging game helps kids apply number bonds and practice addition and subtraction in a fun way. Wake up the sleepy blobs and solve problems to master these essential math skills. Perfect for young learners eager to explore numbers creatively! Pre-K K K.OA.3 VIEW DETAILS Number Bonds Identifying the Correct Model Game Introduce your child to the exciting world of numbers with this engaging game. Kids will use visual models to compose and decompose numbers within 10, helping them understand math concepts better. As they progress, they'll tackle different sets, boosting their addition and subtraction skills. Perfect for young learners eager to explore math in a fun way! K 1 K.OA.3 VIEW DETAILS Number Bonds Composing Number in Different Ways Game Get ready for a fun math adventure with colorful blobs! In this game, kids will compose and decompose numbers within 10 using visual aids. They'll solve problems and work with different number sets, enhancing their addition and subtraction skills. It's a lively way to learn math concepts while having a blast with playful, sleepy blobs. Let's get the party started! Pre-K K K.OA.3 VIEW DETAILS Number Bonds Selecting the Correct Model Game Jump into the world of math with this exciting game! Kids will learn to compose and decompose numbers using concrete objects, tackling numbers within 5. This playful approach helps them build confidence in addition and subtraction. Perfect for young learners who want to make math fun and interactive. Start playing and boost your child's math skills today! K 1 K.OA.3 VIEW DETAILS Count All To Add Games View all 9 games Count All To Add Identifying the Total Game This exciting math game helps kids master addition within 10 using real-world objects. By employing the "count all to add" strategy, children will enjoy solving problems while building a strong foundation in math. The game offers visual aids, making abstract concepts easier to grasp. Perfect for young learners to practice and improve their addition skills in a playful way! Pre-K K K.OA.1 VIEW DETAILS Count All To Add Finding One or Two More Game In this exciting math game, kids will tackle addition challenges by counting objects. They'll use the "count all to add" strategy to find totals, helping them understand addition concepts better. This game boosts their confidence in math by applying prior knowledge to solve problems, making learning both engaging and effective. Start the fun journey now! Pre-K K PK.OA.1 VIEW DETAILS Count All To Add Addition Symbol Game This exciting game introduces kids to addition using real-world objects as visual aids. They'll practice addition sentences within 5, using the "count all to add" strategy. The game helps kids recall addition concepts and enhances their understanding through fun, interactive tasks. Perfect for young learners to build a strong math foundation. Start for free! Pre-K K K.OA.1 VIEW DETAILS Count All To Add Finding the Total Game Join your child in a playful journey to master addition with this interactive game. They'll learn to find totals and understand addition sentences within 5. Designed for young learners, this game makes math fun and engaging. Perfect for introducing kids to the basics of addition and subtraction, ensuring they build a strong foundation in math. Start for free! Pre-K K K.OA.1 VIEW DETAILS Add Using A Number Line Games View all 7 games Add Using A Number Line Counting-On Strategy on a Number Line Game In this engaging game, your child will learn to count and add using a number line. By practicing addition with visual jumps, kids strengthen their counting skills and grasp fundamentals of addition with visual help. The game offers a playful way to tackle addition challenges, helping young learners become confident with numbers. Enjoy learning and have fun with math! 1 2 1.OA.5 VIEW DETAILS Add Using A Number Line Marking the Sum on the Number Line Game In this exciting math game, kids will learn to count on and mark sums using number lines. It's a hands-on way to practice addition, enhancing problem-solving skills. With engaging tasks that make learning fun, students will grasp addition strategies easily. Perfect for young learners to build confidence in math. Join the fun and start learning today! 1 2 1.OA.5 VIEW DETAILS Add Using A Number Line How Many More on a Number Line? Game Introduce your child to a fun way of mastering addition with the "How Many More on a Number Line" game. Children will learn to use a number line to find missing addends, practicing addition within 20. This game focuses on essential strategies for addition, helping kids develop confidence in solving math problems. Let them enjoy learning while playing! 1 2 1.OA.5 VIEW DETAILS Add Using A Number Line How Many More Using Number Line Game In this fun math game, kids use a number line to master the count-on strategy and find missing addends. By solving a variety of addition problems, they build confidence and strengthen their understanding of addition and subtraction concepts. This interactive learning experience makes math enjoyable and helps kids develop essential problem-solving skills. 1 2 1.OA.5 VIEW DETAILS Count On To Add Games View all 11 games Count On To Add Solving Count On Scenarios Game In this exciting math game, kids will tackle addition word problems using the count on strategy. Designed for young learners, the game helps clear up common misconceptions in math, making addition easy and fun. By playing, first graders will fluently add numbers within 10, boosting their confidence and math skills. Perfect for little math explorers! K 1 1.OA.1 VIEW DETAILS Count On To Add Finding the Missing Addends Game This exciting game helps kids practice addition by finding missing numbers in math sentences. Using the count on strategy, children will solve problems by dragging and dropping items to the right spots. It's a fun way to boost their addition and subtraction skills while keeping them entertained and engaged. Perfect for young mathematicians eager to learn! K 1 2.OA.2 VIEW DETAILS Count On To Add Representing Addition Scenarios Game In this engaging math game, kids tackle real-world addition problems using "add to" scenarios. They'll represent each story on a 10-frame with everyday objects, making math relatable and fun. This hands-on approach helps kids understand addition concepts and improve their math skills in a playful, interactive way. Perfect for young learners eager to explore math! Pre-K K K.OA.2 VIEW DETAILS Count On To Add Counting On Using Objects Game In this engaging math game, kids will practice addition by counting on using objects. With visual aids, learners solve problems by selecting the right answer from given options, making math fun and interactive. The game focuses on strategies for addition, helping children build confidence in their skills. Let the counting adventure begin! 1 1.OA.5 VIEW DETAILS Add With 10 Games View all 2 games Add With 10 Decomposing Teen Numbers Game Join the fun of decomposing teen numbers with this interactive game! Kids will learn to add a ten and some ones to form teen numbers, making addition easy and enjoyable. This game helps build confidence in addition strategies within 20, turning math practice into an exciting adventure. Perfect for young learners to strengthen their math skills while having fun! 1 1.OA.6 VIEW DETAILS Add With 10 Adding Numbers to 10 Game This exciting game helps kids overcome misconceptions in math by challenging them to add numbers up to 10. Through engaging problems, children practice composing numbers and develop strategies for addition within 20. It's a fun way for first graders to build their addition and subtraction skills while enjoying the learning process. 1 2 1.OA.6 VIEW DETAILS Doubles And Near Doubles Addition Strategy Games View all 21 games Doubles And Near Doubles Addition Strategy Adding Doubles Using Models Game In this exciting math game, kids will tackle addition problems using the doubles strategy. They'll analyze pictures to find the right answers, making math both fun and interactive. By playing, children will strengthen their addition and subtraction skills, with a focus on doubles facts within 20. This game offers a playful way to practice essential math strategies. Start for free! 1 2 1.OA.6 VIEW DETAILS Doubles And Near Doubles Addition Strategy Double Facts Using Models Game In this fun math game, kids will tackle addition problems by using pictures to find answers. They'll practice the doubles strategy, enhancing their ability to add doubles within 10. As they play, children will analyze and choose the correct answers from a set of options, boosting their addition and subtraction skills. It's a great way to build fluency in adding numbers within 10! K 1 1.OA.6 VIEW DETAILS Doubles And Near Doubles Addition Strategy Adding and Matching Double Facts Game In this exciting game, kids will practice addition by using doubles to add and match facts. Perfect for young mathematicians, it helps develop fluency in adding numbers within 10. With interactive gameplay, students can overcome struggles with addition and improve their math skills in a fun and engaging way. Get ready to add with confidence and master doubles facts! K 1 1.OA.6 VIEW DETAILS Doubles And Near Doubles Addition Strategy Adding Doubles Game This exciting game helps kids master addition by practicing with the doubles strategy. Children solve fun math problems, boosting their confidence and fluency in adding numbers within 10. With engaging challenges, your child will enjoy learning and improve their addition skills. Perfect for young learners to build a strong math foundation! Let's play and add doubles! K 1 1.OA.6 VIEW DETAILS Make 10 Strategy Games View all 4 games Make 10 Strategy Adding the Numbers by Making a 10 Game Explore the exciting world of math with this engaging game! Kids will practice addition by using the make-a-10 strategy. They'll drag and drop items to find the correct answers, reinforcing their understanding of addition and subtraction. This game makes learning math fun and interactive, helping kids boost their problem-solving skills. Perfect for young math learners! 1 2 2.OA.2 VIEW DETAILS Make 10 Strategy Breaking Up and Making a 10 Game In this exciting math game, kids will tackle problems by breaking up numbers and using the make a 10 strategy. This interactive game helps children practice addition and subtraction within 20, boosting their problem-solving skills. By filling in the blanks, they develop a strong understanding of number decomposition. Perfect for making math fun and engaging! 1 2 1.OA.6 VIEW DETAILS Make 10 Strategy Adding by Making a 10 Game In this colorful game, your child will learn the "make a 10" strategy to add numbers efficiently. By practicing addition in two steps, kids can overcome common misconceptions about addition. This engaging game offers a set of problems designed to enhance their understanding of addition within 20, making math fun and interactive! K 1 1.OA.6 VIEW DETAILS Make 10 Strategy Breaking Up Addition Sentences Game Explore the world of addition with this engaging game! Kids will practice breaking up addition sentences to find answers, focusing on numbers within 20. The game helps develop skills in decomposing numbers and using addition strategies such as making a 10. It's a playful way to improve addition and subtraction skills while having fun with math! Play now and see it for yourself! 1 2 1.OA.6 VIEW DETAILS All Addition Strategies Games Compose And Decompose Numbers Completing the Addition Sentence Game This engaging game helps kids become addition experts by composing and decomposing numbers. They'll find missing numbers in sentences, making math fun and interactive. Perfect for young mathematicians, this game builds essential skills in addition and subtraction while boosting confidence in number handling. Explore the joy of learning with each challenge! Pre-K K K.OA.3 VIEW DETAILS Add Using Multiples Of 10 How Many Tens in All Game In this engaging game, kids will explore adding multiples of 10 while practicing addition. They'll learn to count tens in all, reinforcing their math skills. The game offers a fun way to tackle common misconceptions in addition, helping kids to add and subtract within 100. Watch them boost their confidence with every level. Get started now! 1 1.NBT.4 VIEW DETAILS Compose And Decompose Numbers Addition Sentences (Up to 10) Game This vibrant game helps kids practice composing and decomposing numbers up to 10. By solving math sentences with visual models, children improve their addition and subtraction skills. The game offers fun and effective practice, ensuring a solid grasp of composing numbers. Perfect for young learners to boost their math confidence and understanding! Pre-K K K.OA.3 VIEW DETAILS Compose And Decompose Numbers Composing to Make a Number Game Wake up the snails and feed them the exact amount they need by composing numbers up to 10. This engaging math game teaches kids to understand number relationships with 5 and 10 through fun tasks. Kids will learn addition and subtraction skills while enjoying the challenge of feeding the hungry snails. Perfect for young learners eager to explore numbers! Pre-K K K.OA.3 VIEW DETAILS Compose And Decompose Numbers Decomposing to Make a Number Game Feed the hungry snails while mastering the art of decomposing numbers! This engaging math game encourages kids to use the count-out strategy to understand how numbers relate to 5 and 10. Through playful tasks, young learners will practice addition and subtraction skills, making math both fun and educational. Get ready to dive into a world of numbers and snails! Pre-K K K.OA.3 VIEW DETAILS Compose And Decompose Numbers Completing the Addition Game This exciting game invites kids to tackle addition problems by finding missing numbers in math sentences. Focusing on addition facts of 1, children will have fun while strengthening their math skills. They'll pick the correct answers from multiple options, reinforcing their knowledge of addition and subtraction facts. A great way to make learning math enjoyable and interactive! Pre-K K VIEW DETAILS Count All To Add Counting All to Add Numbers Up to 5 Game This exciting math game helps kids master addition by counting objects up to 5. Children will use a "count all to add" strategy, choosing the right answers from multiple options. This interactive approach not only makes learning fun but also builds a strong foundation in addition and subtraction. Watch your young mathematician thrive as they solve engaging problems! Pre-K K K.OA.1 VIEW DETAILS Add Using A Number Line Identifying the Sum on a Number Line Game Join this exciting game to master addition using a number line! Kids will practice counting on to add, tapping on the number line to find sums within 20. This engaging activity helps young learners develop math skills while having fun. Perfect for those looking to improve their addition strategies and gain confidence in math. 1 2 1.OA.5 VIEW DETAILS Count On To Add Addition Using the Count-On Strategy Game This exciting game lets kids practice addition using the count on strategy. They'll solve fun problems and match items with correct attributes, helping to solidify their understanding of addition. Perfect for young learners, it offers a hands-on approach to mastering addition and subtraction skills while enjoying math challenges. 1 2 1.OA.5 VIEW DETAILS Doubles And Near Doubles Addition Strategy Finding the Double Game In this game, kids will master addition using the doubles strategy! They will solve exciting problems using numbers within 20, making math easier and more enjoyable. This game helps children practice and improve their addition skills by focusing on the fundamentals. Perfect for young learners eager to enhance their math abilities through engaging challenges. Get started now! 1 2 1.OA.6 VIEW DETAILS Count All To Add Identifying One More within 10 Game In this fun math game, kids learn to add by identifying one more within 10. Using visual aids, they practice addition with concrete objects, making abstract concepts easier to grasp. This game helps children build a solid understanding of addition fundamentals while having fun. Perfect for young learners eager to master basic math skills! Pre-K K K.OA.1 VIEW DETAILS Add Using A Number Line Counting on Using a Number Line Game In this engaging game, kids will learn to add by counting on using a number line. They'll solve problems by selecting the right answer from options provided, boosting their addition skills. This interactive approach makes math fun and helps solidify strategies for addition and subtraction. Perfect for young learners eager to enhance their math abilities! 1 2 2.OA.2 VIEW DETAILS Count On To Add How Many More Using Objects Game Let your child explore addition with the "How Many More using Objects Game." Kids will use concrete objects to find missing addends and solve addition problems. This engaging game helps tackle common misconceptions and builds a strong foundation in math. It's a fun way to practice counting on and develop essential addition and subtraction skills. 1 2 1.OA.5 VIEW DETAILS Doubles And Near Doubles Addition Strategy Matching the Doubles Game In this exciting math game, kids will tackle addition problems by matching doubles. They'll practice a set of problems on adding numbers within 20, boosting their math confidence and understanding. Perfect for young learners, this game helps overcome common misconceptions and strengthens their ability to use effective addition strategies. Let the fun learning begin! 1 2 1.OA.6 VIEW DETAILS Count All To Add Counting to Tell One More Game Unlock the joy of counting with this engaging game! Kids will explore addition by finding one more than a given number using objects, all within 10. This interactive experience helps children grasp the basics of addition and subtraction while having fun. A perfect way to build confidence in math skills through playful learning. Get started now! Pre-K K K.OA.1 VIEW DETAILS Add Using A Number Line Adding and Marking the Sum on the Number Line Game In this fun math game, kids will practice addition by using the number line to add and mark sums. This interactive activity helps solidify the count-on strategy, making math more approachable. With visual aids, children grasp fundamentals and prepare for abstract concepts. It's a great way to enhance their understanding of addition and subtraction. Start playing today! 1 2 2.OA.2 VIEW DETAILS Count On To Add Completing the Addition Sentences Game In this engaging math game, kids will practice completing addition sentences by identifying missing addends. Using the count-on strategy, they'll solve problems and improve their addition and subtraction skills. With a focus on properties and strategies for addition, this game makes learning math fun and interactive. Perfect for young learners eager to boost their math confidence! 1 1.OA.5 VIEW DETAILS Doubles And Near Doubles Addition Strategy Identifying the Near Double Expression Game This exciting game helps kids learn addition by using near doubles. Students will tackle problems on decomposing numbers and choose the correct answers from given options. It's a great way to strengthen addition and subtraction skills while making math fun and interactive. Perfect for young learners eager to master strategies for addition within 20! 1 1.OA.6 VIEW DETAILS Compose And Decompose Numbers Identifying the Ones in Teen Numbers Using Ten Frames Game In this fun math game, kids will learn to identify the ones in teen numbers using ten frames. By analyzing and selecting the correct answers, children will grasp the concept of teen numbers as 1 ten and some ones. This game provides a playful way to build number sense and understand place value, focusing on numbers from 11 to 19. Perfect for young learners! K K.NBT.1 VIEW DETAILS Count All To Add Finding the Sum Game In this exciting math game, kids will practice addition by solving problems using the "count all to add" strategy. Aimed at building confidence in math, this game offers a fun way to learn addition sentences within 10. By playing, children will strengthen their understanding of addition and subtraction, making math enjoyable and interactive. K 1 K.OA.1 VIEW DETAILS Count On To Add Counting On to Add Game This exciting game helps kids learn addition by using real-world objects as visual aids. They'll practice the "count on" strategy to add numbers within 20. By selecting the correct option, children will improve their addition and subtraction skills in a playful and interactive way. It's a perfect blend of learning and fun that keeps young minds engaged and active. 1 2 1.OA.5 VIEW DETAILS Doubles And Near Doubles Addition Strategy Breaking Up to a Near Double Fact Game Unleash your child's math skills with this fun game focused on addition strategies. Kids will learn to break up addition sentences using near doubles, making math problems easier to solve. The game offers plenty of practice, helping young learners become confident in addition and subtraction within 20. It's an engaging way to boost their math skills! 1 2 1.OA.6 VIEW DETAILS Compose And Decompose Numbers Identifying the Ones in Teen Numbers Game In this exciting math game, kids tackle the challenge of decomposing teen numbers into 1 ten and some ones. They'll work through engaging problems within numbers 11-19, applying their existing knowledge to find solutions. As they progress, students will improve their place value and number sense skills, making math both fun and educational. Perfect for young learners eager to explore numbers! K 1 K.NBT.1 VIEW DETAILS Count All To Add Counting Objects to Add Game Get ready to count and add with this fun math game! Kids will use objects to practice the "count all to add" strategy, solving problems by selecting the right answers. Designed for young learners, this game strengthens addition skills and helps understand the concept of adding numbers up to 10. It's a playful way to boost math confidence and enjoy learning! K K.OA.1 VIEW DETAILS Count On To Add Identifying the Sum of 2-Digit Numbers and Ones Game Join this engaging game to practice adding ones to 2-digit numbers. Your child will tackle addition challenges, overcoming misconceptions with ease. The game focuses on adding and subtracting numbers within 100, making math practice fun and interactive. Help your young mathematician grow confident in their addition skills through regular practice. Signup now! 1 1.NBT.4 VIEW DETAILS Doubles And Near Doubles Addition Strategy Converting to a Near Double Fact Game This exciting math game helps kids practice addition by using near doubles. They will solve math sentences by finding missing numbers using the near doubles trick, making addition strategies easier. Perfect for young learners, this game boosts confidence in adding within 20. Kids will enjoy mastering addition and subtraction skills in a fun, interactive way! 1 1.OA.6 VIEW DETAILS Compose And Decompose Numbers Making a Number Using Cubes (Up to 5) Game In this exciting math game, kids will compose numbers using cubes up to 5. They'll solve engaging problems by applying their knowledge and creating models to represent numbers. This interactive game balances fun and learning, focusing on addition and subtraction skills. Kids will enjoy exploring numbers while boosting their math confidence! K 1 K.OA.3 VIEW DETAILS Count On To Add Solving ''Count On'' Word Problems Game This exciting game helps kids master addition by solving "count on" word problems. Through fun challenges, children actively participate and gain fluency in adding numbers within 20. With engaging scenarios, this game builds confidence and enhances problem-solving skills, making math enjoyable and interactive for young learners. 1 2 1.OA.1 VIEW DETAILS Doubles And Near Doubles Addition Strategy Finding Doubles and Its Related Facts Game This interactive math game invites kids to tackle addition challenges using the near doubles strategy. By dragging and dropping items, children will solve engaging problems involving numbers up to 20. Perfect for young learners, this game boosts skills in addition and subtraction while making math fun and interactive. Let your child explore the joy of math today! 1 2 1.OA.6 VIEW DETAILS Compose And Decompose Numbers Making a Number within 5 Game Join the Oolzoos and guide the snail home by mastering numbers within 5! Kids will use counting skills to compose and decompose numbers, making math fun and interactive. This engaging game boosts addition and subtraction skills in an exciting way. Perfect for young learners eager to explore math with playful challenges. Get started on this math adventure today! K 1 K.OA.3 VIEW DETAILS Count On To Add Using the Count-On Startegy to Add Game Discover the joy of learning math with the "Use Count On to Add Game"! Kids will practice the count-on strategy to solve addition problems. This interactive game makes math fun and helps kids apply their skills by counting forward from any number. Perfect for building confidence in addition and subtraction. Join the adventure and watch math skills grow! 1 2 2.OA.2 VIEW DETAILS Doubles And Near Doubles Addition Strategy Adding Near Doubles Game Ready for a fun addition challenge of adding near doubles? In Blue's kingdom, kids will balance shifty bridges while practicing addition strategies using near doubles. This engaging game boosts their skills in adding and subtracting within 20. With active participation, children learn to quickly find results, making math both enjoyable and educational. Let's play and learn! 1 2 1.OA.6 VIEW DETAILS Compose And Decompose Numbers Composing Numbers within 5 Game Join the Oolzoos in their adventure to send the snail home by composing numbers within 5. This engaging math game helps kids practice addition by counting on and applying strategies. Children will enjoy feeding chilies to the Oolzoos while learning key addition and subtraction skills. Perfect for young learners to explore math in a fun way! K 1 K.OA.3 VIEW DETAILS Doubles And Near Doubles Addition Strategy Matching the Sum Game Join the Match the Sum Game and make math fun! Kids will tackle addition problems by matching sums and filling in blanks, enhancing their understanding of adding numbers within 20. This interactive game helps improve math skills, making learning enjoyable. Perfect for young learners eager to boost their addition and subtraction skills in an engaging way. 1 2 1.OA.6 VIEW DETAILS Compose And Decompose Numbers Decomposing a Number within 5 Game Join the snail's journey home by learning how to decompose numbers within 5. This fun game teaches kids to use strategies to add and subtract, boosting their math skills. Use the count-out method to help the Oozloos eat red chili and solve challenges. Perfect for young mathematicians eager to explore addition and subtraction in an engaging way! K 1 K.OA.3 VIEW DETAILS Doubles And Near Doubles Addition Strategy Missing Number to Add Doubles Game Help your child tackle addition challenges with the missing number to add doubles game! Kids will explore doubles and near doubles strategies, solving problems with missing numbers in equations. This interactive game makes learning addition and subtraction within 20 exciting and helps clear up common math misconceptions. Perfect for young math explorers! 1 2 1.OA.8 VIEW DETAILS 12...3→ < View all result 136 GRADE Pre-K K 1 2 3 4 5 Constant practice and in-depth learning are efficient approaches to a thorough understanding. Game-based learning offers the same more delightfully, ensuring that your kids develop an ever-lasting love for math. An Introduction to Addition Kids learn addition right from kindergarten up to the 5th grade. The difficulty of the concept increases each year. Even when your kid moves to higher classes, addition is used in learning other advanced concepts. Our addition strategy games will help create a stronger foundation for your child so that they're ready to tackle more complex concepts in their later years. Curriculum-Based Games Kids often struggle to keep up with the constant increase in difficulty levels. Online educational games offer a personalized environment in which your kids can learn addition at their own pace. Despite being games that provide a fun experience, they do not deviate from the concept. This is because the games are designed by experts who keep the curriculum and course outcomes in mind. The best part about these games is - They provide adequate practice in addition Different approaches offer comprehensive learning. Kids can retry the games until they achieve a thorough understanding. Problem + Innovation = Solution The following games will highlight the usage of innovation and strategies involved in teaching kids the concept of addition. Using the Number Line Adding using the number line is one of the most effective methods of learning addition. These games ask your child to count the number of jumps made on the number line to get to the solution. For example, take the following question - 5 + ? = 8 The number line will display three jumps made from five to eight. Your child can then count the total number of jumps to answer the question. Equivalent Expressions Remember those math problems that used to end like this - L.H.S. = R.H.S. The Equivalent Expression game gives your child a basic introduction to such problems. The options will consist of various expressions as follows, (5 + 3 = 7 + 5) and (5 + 3 = 7 + 1) Your kid has to solve both sides and find out which option is correct. Apart from addition, kids learn two more things - There are different ways to arrive at the same number The importance and usage of the equal to (=) sign Adding Three Whole Numbers As kids progress in learning addition, the total numbers also increase. Once they are thorough in adding two numbers, this game helps them practice the addition of three whole numbers. Apart from straightforward questions using numbers and symbols alone, certain games also utilize sentences in their problems. Focusing on the Ones and Tens Online games, especially for first and second graders, focus on teaching ones and tens. In this small part of addition alone, there are many different games for your child. Completing the Pattern of Multiples of Ten Adding Ten and Its Multiples to Various Numbers Adding Tens Through Place Value Adding Ones to Two-Digit Numbers Using Place Value Composing Different Numbers to Reach the Value of 10 Fun with Doubles This game helps kids learn the doubling of numbers by adding the same number twice. For instance, 2 + 2 3 + 3 Provide the Educational Support Apart from the ones mentioned above, there are numerous other Addition Strategies Games available online that develop fluency in addition. All you need to do is guide your child through a personalized learning experience and help them on their journey with math. From fun games to curriculum-based learning, SplashLearn ticks all the right boxes. Sign up with SplashLearn today and make use of the various Addition Strategies Games to help your child master the concept in an engaging way. // Your one stop solution for all grade learning needs. 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7611	https://www.youtube.com/watch?v=IP01ImAXwh8	A corridor that is 3 m wide makes a right-angle turn, as shown on the left WNY Tutor 179 likes 20187 views 25 Apr 2016 A corridor that is 3 m wide makes a right-angle turn, as shown on the left. Find the longest rod that can be carried horizontally around this corner. Round your answer to the nearest to the nearest tenth of a meter. 12 comments
7612	https://www.nccn.org/patients/guidelines/content/PDF/breastcancerscreening-patient.pdf	2025 Breast Cancer Screening and Diagnosis NCCN GUIDELINES FOR PATIENTS ® Presented with support from FOUNDATION Guiding Treatment. Changing Lives. NATIONAL COMPREHENSIVE CANCER NETWORK ® Available online at NCCN.org/patientguidelines Ü 1 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 Breast Cancer Screening and Diagnosis About the NCCN Guidelines for Patients® Did you know that top cancer centers across the United States work together to improve cancer care? This alliance of leading cancer centers is called the National Comprehensive Cancer Network® (NCCN®). Cancer care is always changing. NCCN develops evidence-based cancer care recommendations used by health care providers worldwide. These frequently updated recommendations are the NCCN Clinical Practice Guidelines in Oncology (NCCN Guidelines®). The NCCN Guidelines for Patients plainly explain these expert recommendations for people with cancer and caregivers. These NCCN Guidelines for Patients are based on the NCCN Clinical Practice Guidelines in Oncology (NCCN Guidelines®) for Breast Cancer Screening and Diagnosis Version 2.2025 – March 28, 2025. Learn how the NCCN Guidelines for Patients are developed NCCN.org/patient-guidelines-process Find an NCCN Cancer Center near you NCCN.org/cancercenters View the NCCN Guidelines for Patients free online NCCN.org/patientguidelines Connect with us 2 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 Breast Cancer Screening and Diagnosis Supporters To make a gift or learn more, visit online or email NCCNFoundation.org/Donate PatientGuidelines@ NCCN.org NCCN independently adapts, updates, and hosts the NCCN Guidelines for Patients. 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National Comprehensive Cancer Network (NCCN) and NCCN Foundation 3025 Chemical Road, Suite 100, Plymouth Meeting, PA 19462 USA 4 About breast cancer screening 8 Screening and diagnostic tests 18 Risk assessment for screening 26 Testing during pregnancy and breastfeeding 30 What to do when you have symptoms 38 Other resources 42 Words to know 44 NCCN Contributors 45 NCCN Cancer Centers 48 Index 4 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 1 About breast cancer screening 5 What are the parts of the breast? 5 Why is breast cancer screening important? 6 Where does breast cancer start? 6 What's in this book? 7 What can you do to get the best care? 5 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 1 About breast cancer screening » What are the parts of the breast? 5 1 About breast cancer screening » Why is breast cancer screening important? Breast cancer starts in the cells of the breast. Regular breast cancer screening and breast exams help find breast cancer at its earliest, most treatable stages. What are the parts of the breast? The breast is a gland found on the chest. The breast is made of milk ducts, fat, nerves, lymph and blood vessels, ligaments, and other connective tissue. Behind the breast are the pectoral (chest) muscles and ribs. Muscles and ligaments help hold the breast in place. Breast tissue contains glands that can make milk. These milk glands are called lobules. Lobules look like tiny clusters of grapes. Small tubes called ducts connect the lobules to the nipple to carry breast milk. A terminal lobular unit (TDLU) is a lobule connected to the end of a small milk duct. Why is breast cancer screening important? Breast cancer screening aims to find breast cancer early. Breast cancer found earlier is often treated more successfully, reducing the risk of dying from breast cancer. This book will cover screening recommendations for those assigned female at birth. It will also discuss screening and testing options for those who are pregnant or lactating (breastfeeding), and those with breast pain, nipple discharge, changes in the skin, lumps, or other symptoms that require further testing. The breast The breast is a glandular organ made up of milk ducts, fat, nerves, blood and lymph vessels, ligaments, and other connective tissue. 6 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 1 About breast cancer screening » Where does breast cancer start? 6 1 About breast cancer screening » Where does breast cancer start? » What's in this book? Those assigned male at birth Anyone can develop breast cancer, including those assigned male at birth. Although there are some differences between breast cancers in those assigned male and those assigned female at birth, treatment is very similar for all genders. Those assigned male at birth do not undergo regular mammogram screening. However, if you feel a lump, have breast pain, or a change in the size or shape of one breast, then see your health care provider and read about recommended tests in Chapter 5: What to do when you have symptoms. Transgender persons Transgender persons should consult with their primary care physician to determine when or whether screening would be appropriate. However, if you feel a lump, have breast pain, or a change in the size or shape of one breast, then see your health care provider and read about recommended tests in Chapter 5: What to do when you have symptoms. Where does breast cancer start? Most breast cancers start in cells that make up the lining (epithelial cells) in the terminal duct lobular units (TDLUs) of the breast. The most common types of breast cancer are ductal and lobular carcinoma. h Ductal carcinoma starts in the cells that line the milk ducts. Ductal carcinoma is the most common type of breast cancer. h Lobular carcinoma starts in the lobules (milk glands) of the breast. What's in this book? This chapter provided a brief overview of the importance of breast cancer screening. Other chapters in the book explain: h Types of breast imaging tests and how doctors categorize findings on those imaging tests. h How risk is calculated and when to start breast cancer screening for those at average and increased risk. h Why it is important to continue breast cancer screening during and after pregnancy and when you might need more tests. h When you should see a health care provider or get further testing. 7 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 1 About breast cancer screening » What can you do to get the best care? What can you do to get the best care? Advocate for yourself. You have an important role to play in your care. In fact, you’re more likely to get the care you want by asking questions and making shared decisions with your care team. The NCCN Guidelines for Patients will help you understand cancer care. With better understanding, you’ll be more prepared to discuss your care with your team and share your concerns. Many people feel more satisfied when they play an active role in their care. You may not know what to ask your care team. That’s common. Each chapter in this book ends with an important section called Questions to ask. These suggested questions will help you get more information on all aspects of your care. Take the next step and keep reading to learn what is the best care for you! Why you should read this book Making decisions about cancer care can be stressful. You may need to make tough decisions under pressure about complex choices. The NCCN Guidelines for Patients are trusted by patients and providers. They clearly explain current care recommendations made by respected experts in the field. Recommendations are based on the latest research and practices at leading cancer centers. Cancer care is not the same for everyone. By following expert recommendations for your situation, you are more likely to improve your care and have better outcomes as a result. Use this book as your guide to find the information you need to make important decisions. 8 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 2 Screening and diagnostic tests 9 Overview 9 General health tests 10 Mammogram 12 Other breast imaging tests 13 Mammogram results 14 BI-RADS 17 Key points 17 Questions to ask 9 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 2 Screening and diagnostic tests » Overview 9 2 Screening and diagnostic tests » Overview » General health tests This chapter provides an overview of imaging tests used for breast cancer screening and diagnosis. Overview In this chapter, you'll find information about mammograms and other breast imaging tests and learn the difference between screening and diagnostic tests. You'll also find information on how test results are interpreted and what those results mean for your follow-up care. What's the difference between screening and diagnostic tests? h Screening is done on a regular basis when there are no symptoms. For example, an annual screening mammogram is done once a year to detect breast cancer or anything abnormal before you have symptoms. h Diagnostic tests are done when there are signs or symptoms such as a lump that can be felt in the breast, changes in the skin of the breast or nipple, or nipple discharge. With breast symptoms, the most frequent diagnostic tests used are mammograms and breast ultrasound. A radiologist, a medical doctor, will interpret any screening and diagnostic tests and send a report to your health care provider. The radiologist will determine the best imaging methods according to current guidelines. General health tests Medical history A medical history is a record of all health issues and treatments you have had in your life. Be prepared to list any illness, injury, or surgery and when it occurred. Bring a list or the bottles of old and new medicines and any over-the-counter (OTC) medicines, herbals, or supplements you take. Tell your care team about any symptoms you have. A medical history is sometimes called a health history. Family history Some cancers and other diseases can run in families. Your health care provider (HCP) will ask about the health history of family members who are blood relatives. This information is called a family history. You can help by asking blood relatives from both sides of the family if they have or had cancer. It’s important to know the specific type of cancer, or where the cancer started, if it was in multiple locations, how old they were when they had the cancer, and if they had genetic testing. Clinical breast exam A clinical breast exam (CBE) is a physical exam of the bare breast performed by a health care provider to check for lumps or other changes. It is done while you are seated and/ or lying down. Your provider should take time to palpate (feel) the entire breast, including the armpit. A nurse or assistant might also be in the room during the exam. 10 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 2 Screening and diagnostic tests » Mammogram Mammogram A mammogram is a picture of the inside of your breast. The picture is made using x-rays. A computer combines the x-rays to make detailed pictures. A bilateral mammogram includes pictures of both breasts. Screening mammograms are annual preventive exams–done every year to find cancers before they become large enough to be felt or cause symptoms. Diagnostic mammograms are done when there are signs or symptoms in the breast or armpit. A screening mammogram only takes about 10 to 20 minutes, while a diagnostic mammogram can be longer. If you are visiting a new or different screening center, it is helpful to bring your prior mammograms so they can be used for comparison. Both screening and diagnostic mammograms: h Use low-dose x-rays to examine the breast. h May use film or digital technology. h Some may use technology called digital breast tomosynthesis, called DBT or tomo for short. During a mammogram, a specially trained technologist will position your breast in the mammography unit. Your breast will be placed on a platform and flattened (compressed) with a clear plastic paddle. You may experience some discomfort with compression of the breast, but the discomfort is temporary. The breast is compressed in order to: h Even out breast thickness h Spread out the tissue h Hold the breast still h Increase sharpness of the picture You will be asked to stand very still and hold your breath for a few seconds while the pictures are being taken. You will be asked Mammogram A mammogram uses x-rays and a computer to make detailed pictures of the inside of your breast. 11 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 2 Screening and diagnostic tests » Mammogram to change positions between images. The process will be repeated for the other breast. Types of mammograms are described next. Screening mammogram A screening mammogram is the only imaging test that has been found to reduce death from breast cancer. It is a test that uses low-dose x-rays to take 4 standard pictures of your breast (2 on each side). A radiologist is a doctor who reviews the images and determines if you need additional imaging. If you need additional imaging, you will likely have a diagnostic mammogram. About 1 out of 10 people will be asked to return for more imaging. In most cases, the follow-up test shows nothing suspicious. Diagnostic mammogram Diagnostic mammograms look at specific areas of your breasts, which may not be clearly seen on screening mammograms. A diagnostic mammogram might be done after a screening mammogram or for a symptom such as a breast lump or nipple discharge. Diagnostic mammograms may include extra specialized views. A radiologist will most likely evaluate the diagnostic mammogram while you wait so if additional breast imaging is needed, it can be done right away. Tomosynthesis mammogram Tomosynthesis, sometimes called 3D mammograms, is a newer type of digital mammogram that may help to see small abnormal areas of the breast more easily. It takes multiple thin images that are combined to make a complete picture. Tomosynthesis What’s the difference between a screening and diagnostic mammogram? A mammogram is a picture of the inside of your breast made using x-rays. During a mammogram, the breast is pressed between two plates while you stand in different positions. Multiple x-rays will be taken. A computer combines the x-rays to make detailed pictures. Screening mammogram • Done on a regular basis when there are no signs or symptoms of breast cancer. Results take a few days. Diagnostic mammogram • Used for those who have symptoms such as a lump, pain, skin thickening or nipple discharge, or those whose breasts have changed shape or size. An ultrasound is often used with a diagnostic mammogram. • Also used to take a closer look at an abnormal area found in a screening mammogram. • A radiologist will evaluate the diagnostic mammogram while you wait so if additional testing is needed, it can be done right away. Both types of mammograms use low-dose x-rays to examine the breast. They may use film or digital technology. 12 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 2 Screening and diagnostic tests » Other breast imaging tests can decrease call back rates (the need for additional tests) and improve cancer detection. Tomosynthesis can be used for screening or diagnostic mammograms and is recommended, if available in your area. Contrast-enhanced mammogram Contrast-enhanced mammography (CEM) is a new technology that uses contrast material to improve the pictures of the inside of the breast. For those at increased risk for breast cancer, contrast-enhanced screening mammogram is a possible option, although, research studies are still ongoing. CEM is not as widely available as mammogram, ultrasound, and MRI. Other breast imaging tests Breast MRI A magnetic resonance imaging (MRI) scan uses radio waves and powerful magnets to take pictures of the inside of the body. MRI does not use radiation. If needed, an MRI would be used in addition to a mammogram. If you are at high risk for breast cancer, an annual breast MRI might be recommended. Because of the very strong magnets used in the MRI machine, tell the technologist if you have any metal in your body. Contrast material will be used to improve the pictures of the breast. For a breast MRI, a gadolinium-based contrast agent (GBCA)—a rare, heavy metal—is used to enhance the quality of the MRI. Contrast is given through intravenous (IV) line placed in a vein, typically in the hand or near the elbow. Breast MRI If needed, a breast MRI will be done in addition to a mammogram. In a breast MRI, you are positioned face down with your arms overhead. 13 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 2 Screening and diagnostic tests » Mammogram results Tell your health care provider (HCP) about all of your medical conditions, including: h If you are pregnant or think you might be pregnant h The date of your last MRI with gadolinium h If you have a gadolinium allergy or allergy to other x-ray contrast h If you have any metal in your body (such as implanted infusion ports, metal screws or plates, or some cardiac pacemakers) h If you have kidney problems Breast ultrasound An ultrasound (US) uses sound waves to form pictures of the inside of the body. This is similar to the sonogram used for pregnancy. A wand-like probe (transducer) will be held and moved on your bare breast using gel. It may also be placed below your armpit. Ultrasound is painless and does not use x-rays, so it can be repeated as needed. Ultrasound is frequently used without a mammogram for diagnosis of breast symptoms in those under 30 years of age or with a mammogram in those 30 years of age and over. Molecular breast imaging Molecular breast imaging uses a radioactive tracer and special camera to find breast cancer. Molecular imaging such as breast-specific gamma imaging or sestamibi scan may improve detection. However, whole body radiation dose is much higher than that of standard mammograms. Molecular breast imaging is not a common breast screening imaging test. Mammogram results Mammogram results or findings are described using Breast Imaging Reporting and Data System (BI-RADS). Some of the terms below might be used in addition to BI-RADS imaging results explained starting on page 14. Calcifications Calcifications are small, white deposits of calcium in the breast that can be seen on a mammogram. Most calcifications seen on a mammogram are not breast cancer. There are 2 types of breast calcification: h Macrocalcifications are large deposits and are usually not related to cancer. h Microcalcifications are specks of calcium that may be found in an area of rapidly dividing cells. Microcalcifications clustered together may be a sign of cancer. Distortion Distortion describes when an area of the breast tissue appears on a mammogram at an odd angle, fuzzy or misshapen (disorganized), particularly when compared to the other breast or prior imaging tests. This may just be due to how the breast was positioned during the mammogram. It might also be caused by a prior injury, procedure done on the breast, or breast cancer. Masses A mass is an area of abnormal tissue. A mass could be solid, fluid-filled, or a combination of both. A mass might be seen with or without 14 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 2 Screening and diagnostic tests » BI-RADS calcifications. Some masses can be watched over time with regular mammograms or ultrasounds to see if they change, but others may need to be biopsied. Asymmetries Asymmetries are white areas seen on a mammogram that look different from the normal breast tissue pattern, particularly when compared to the other breast. Different types of asymmetries include focal asymmetry, developing asymmetry, and global asymmetry. Further imaging will likely be needed to get a better look at the area. Dense breasts Your mammogram report will include an assessment of your breast density. Breast density is a description of how much fibrous and glandular tissue is in your breasts compared to fatty tissue. The denser your breasts, the harder it can be to see abnormal areas on mammograms. Breasts are composed of fat and fibroglandular tissue. Dense breasts have more fibroglandular tissue than fat. Having dense breast tissue is not abnormal but may make it harder to see breast cancer or other changes in the breast on a mammogram. Breast density can change over time. If you have dense breast tissue, ask your health care provider about the risks and benefits of additional screening. More information on breast density can be found on page 16. BI-RADS Breast Imaging Reporting and Data System or BI-RADS is a standard system used in the United States to describe mammogram results. It also classifies breast density into groups. A radiologist will categorize your mammogram results using a BI-RADS numbered system of 0 through 6 and classify your breast density using a lettered system of A through D. Talk to your health care provider about your mammogram results and any next steps. These same BI-RADS categories can also be used to describe the results of a breast ultrasound or breast MRI. BI-RADS 1: Negative BI-RADS 1 means the test is negative for cancer. This is a normal test result. The recommendation is to continue with annual mammogram screening. BI-RADS 2: Benign (not cancer) This is also a negative test result (there’s no sign of cancer), but the radiologist chooses to describe a finding that is not cancer, such as benign calcifications, masses, or lymph nodes in the breast that are clearly not cancer. This can also be used to describe changes from a prior procedure (such as a biopsy) or surgery in the breast. This ensures that others who look at the mammogram in the future will not mistake the benign finding as suspicious. The recommendation is to continue with annual mammogram screening. 15 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 2 Screening and diagnostic tests » BI-RADS BI-RADS 3: Probably benign – Follow-up within a short time frame A finding in this category has a very low (no more than 2% or 2 in 100) chance of cancer and is not expected to change over time. Breast experts agree that it is very safe to perform follow-up imaging to see if the area in question changes over time. You will typically need follow-up imaging in 6, 12, and 24 months to evaluate whether the finding is stable. This approach helps avoid unnecessary biopsies, but if the area does change over time, it still allows for early diagnosis. BI-RADS 4: Suspicious – Biopsy should be done These findings do not definitely look like cancer but could be cancer. The radiologist is concerned enough to recommend a biopsy. The findings in this category can have a wide range of suspicion levels. For this reason, this category is often divided further: h 4A: Finding with a low likelihood of cancer. A low likelihood is more than 2% but no more than 10% or about 2 to 10 out of 100. h 4B: Finding with a moderate likelihood of cancer. A moderate likelihood is more than 10% but no more than 50% or about 10 to 50 out of 100. h 4C: Finding with a high likelihood of cancer, but not as high as Category 5. A high likelihood is more than 50% but less than 95% or about 50 to 95 out of 100. BI-RADS Results BI-RADS 0 means more imaging tests are needed to determine BI-RADS category. BI-RADS 1 is a negative (normal) result. BI-RADS 2 is a benign (not cancer) finding. BI-RADS 3 is a probably benign finding. Follow-up is needed within a short time frame. BI-RADS 4 is a suspicious abnormality and a biopsy should be done. BI-RADS 5 is highly suggestive of cancer and a biopsy should be done. BI-RADS 6 is a biopsy-confirmed cancer.        16 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 2 Screening and diagnostic tests » BI-RADS BI-RADS 5: Highly suggestive of malignancy – Biopsy should be done The findings have a high chance (at least 95% or 95 out of 100) of being cancer. Biopsy is very strongly recommended. BI-RADS 6: Known biopsy – Proven malignancy This category is used only for findings on a mammogram (or ultrasound or MRI) that have already been shown to be cancer from a biopsy. Imaging may be used in this way to see how the cancer is responding to treatment. BI-RADS breast density BI-RADS breast density categories are described below. h Category A: Almost entirely fatty means the breasts are almost entirely made up of fatty tissue. There is very little fibroglandular breast tissue. h Category B: Scattered areas of fibroglandular density means the breasts are mostly made up of fatty tissue. But there are some scattered areas of fibroglandular breast tissue. h Category C: Heterogeneously dense means most of the breast tissue is fibroglandular breast tissue. But there are some areas of fatty tissue. h Category D: Extremely dense means nearly all of the breast tissue is fibroglandular breast tissue. There is very little fatty tissue. Non-dense breasts are defined as: h Almost entirely fatty h Scattered areas of fibroglandular density Dense breasts are defined as: h Heterogeneously dense h Extremely dense Those with dense breasts might benefit from more screening tests, in addition to a screening mammogram. Ask your health care provider for more information. 17 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 2 Screening and diagnostic tests » Key points 17 2 Screening and diagnostic tests » Key points » Questions to ask Key points h Screening is done on a regular basis when there are no symptoms. h Diagnostic tests are done when there are signs or symptoms. h Clinical breast exam (CBE) is a physical exam of the bare breast performed by a health care provider to check for lumps or other changes. h Both screening and diagnostic mammograms use low-dose x-rays to examine the breast. They may use film or digital technology. h Breast Imaging Reporting and Data System or BI-RADS is a standard system used to describe mammogram results and breast density. h These same BI-RADS categories can also be used to describe the results of a breast ultrasound or breast MRI. Questions to ask h Do I have dense breasts? h Should I have other imaging tests in addition to a mammogram on a regular basis? h Who can I talk to about my BI-RADS results? h How will my BI-RADS results affect how often I have mammograms or other screening tests? h Should I have a biopsy? 18 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 3 Risk assessment for screening 19 What is breast cancer risk? 20 How is breast cancer risk calculated? 21 Average risk 21 Increased risk 25 Key points 25 Questions to ask 19 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 3 Risk assessment for screening » What is breast cancer risk? Breast cancer risk is your chance for developing breast cancer. This chapter discusses the difference between average and increased risk and at what age you should start screening mammograms. Screening mammograms are done for those assigned female at birth who do not have any signs or symptoms of breast cancer. See your health care provider (HCP) regularly for checkups. You might be asked to see your HCP more than once a year based on your risk factors for developing breast cancer. Know your breasts and immediately report any changes to your HCP. As your family cancer history changes, keep your HCP updated. This is important. What is breast cancer risk? Breast cancer risk is your chance of developing breast cancer. Everyone has some risk for developing breast cancer. However, some people are at increased risk. Factors that put you at increased risk include: h Family history or genetic testing that suggests a genetic predisposition to breast and other types of cancer (for example, BRCA1 or BRCA2) h Residual lifetime risk of 20 percent (20%) or more as defined by mathematical models that include a comprehensive family cancer history h If you had radiation therapy (RT) to the chest between 10 and 30 years of age h Residual lifetime risk of 20 percent (20%) or more and a previous breast biopsy that showed atypical ductal hyperplasia (ADH) or lobular neoplasia such as lobular carcinoma in situ (LCIS) or atypical lobular hyperplasia (ALH) h 5-year risk of invasive breast cancer of 1.7% or greater in those 35 years of age and over (if using Gail Model) h Those with a mammogram finding of heterogeneously dense and/or extremely dense breasts who do not meet any other increased risk category. Annual screening mammograms should start at 40 years of age for those at average risk. 20 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 3 Risk assessment for screening » How is breast cancer risk calculated? How is breast cancer risk calculated? Breast cancer risk is calculated using breast cancer risk assessment models. It helps determine if you are at average or increased risk for developing breast cancer. This information suggests the best time to start breast cancer screening. Commonly used breast cancer risk assessment models include Tyrer-Cuzick Model (International Breast Cancer Intervention Study or IBIS), Breast Cancer Risk Assessment Tool (BCRAT or the Gail model), BRCAPRO, Breast Cancer Surveillance Consortium (BCSC), and CanRisk which uses the Breast and Ovarian Analysis of Disease Incidence and Carrier Estimation Algorithm (BOADICEA). Ask your health care provider which model they are using to determine your risk level. Individuals should undergo breast cancer risk assessment by 25 years of age. Your HCP should share information regarding potential benefits, risks, and limitations of breast screening. Shared decision-making between you and your HCP is encouraged. What is your risk level? Breast cancer risk is largely based on personal health history and family history of cancer. The goal is to undergo a risk assessment by 25 years of age. This will determine when you should start annual screening mammograms. Your risk for developing breast cancer will be rated as average or increased risk. Your level of breast cancer risk can also be calculated based on the following information: • What is your age, race, and ethnicity? • What was the age of your first menstrual period (menarche)? • Have you even been pregnant or given birth? What was your age when you first gave birth? • How many blood relatives (parents, brothers, sisters, children, aunts, uncles, grandparents) on either side of your family have breast cancer? Are there other cancers such as ovarian, tubal, uterine, pancreatic, or prostate cancers in the family? What is/was the cancer cell type? At what age were they diagnosed? • Have you had a breast biopsy? How many and when? Do you have a copy of the results? • Have you or any close blood relatives had genetic testing? What were the results? • Have you had a breast biopsy that showed atypical hyperplasia or lobular carcinoma in situ? 21 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 3 Risk assessment for screening » Average risk 21 3 Risk assessment for screening » Average risk » Increased risk Average risk Average risk means you have no known genetic or family cancer history or personal health history that suggests you are at increased risk of developing breast cancer. It is recommended you have annual screening mammograms starting at 40 years of age. However, based on certain risk factors or as your family cancer history information changes, your HCP might recommend you start annual screening mammograms sooner. 25 to 39 years of age See your HCP for a checkup every 1 to 3 years. This should include breast cancer risk assessment and a breast exam. 40 years of age and over See your HCP for a checkup every year. This should include ongoing breast cancer risk assessment and, if not recently done, a breast exam. Have an annual screening mammogram. If available, a tomosynthesis is recommended. There is no upper age limit for screening mammograms as long as you are healthy and are expected to live more than 10 years. This means that screening mammograms can continue throughout your entire life unless you and your health care provider decide otherwise based on your health. Increased risk If you are at increased risk for developing breast cancer, see your HCP at least once a year (annually). You might be asked to see your HCP more often. Increased risk is divided into those who have or had: 1. A family history of cancer that suggests a genetic predisposition towards breast cancer. You may be referred to a genetic counselor or other cancer genetics expert to assess if your family has a genetic predisposition. Genetic testing may be recommended for you or another family member. 2. A strong family history that shows your lifetime risk is 20% or greater, when calculated using a risk assessment tool such as BRCAPRO, Tyrer-Cuzick, BOADICEA/CanRisk, or BCSC. Residual lifetime risk is your chance of developing breast cancer based on the number of years of life that remain and other risk factors. 3. Chest or breast radiation therapy (RT) between 10 and 30 years of age. 4. 5-year risk of invasive breast cancer of 1.7% or greater as calculated by the Gail model. 5. Atypical ductal hyperplasia (ADH), lobular carcinoma in situ (LCIS), or atypical lobular hyperplasia (ALH) and 20% or greater residual lifetime risk. 6. Extremely dense or heterogeneously dense breast tissue. 22 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 3 Risk assessment for screening » Increased risk If you have a family history or genetic predisposition If you have a known family history of breast cancer or family history that suggests genetic predisposition to breast cancer, or there is a known genetic mutation, then you may be referred to a genetic counselor or other health care professional who is an expert in cancer genetics. A genetic counselor may recommend genetic testing for you or another family member. Family cancer information can change. Share with your health care provider any changes to your health or your family cancer history. If your residual lifetime risk is 20% or more Individuals should undergo breast cancer risk assessment by 25 years of age. If your residual lifetime risk is 20 percent (20%) or more based on models using a comprehensive family history, then see your HCP for a checkup every 6 to 12 months. Ask your HCP about how you can reduce your risk of developing breast cancer. h Annual mammogram screening for those at increased risk might begin as early as 30 years of age, but no later than 40 years of age. h Annual breast MRI for those at increased risk might begin as early as 25 years of age, but no later than 40 years of age. If breast MRI is not available or you cannot have an MRI, talk to your HCP about options. Increased risk Known family history of breast cancer or family history of known genetic mutation that suggests predisposition to breast and other cancers Previous chest or breast radiation therapy (RT) between age 10 and 30 Atypical ductal hyperplasia (ADH), lobular carcinoma in situ (LCIS), or atypical lobular hyperplasia (ALH) and 20% or greater residual lifetime risk Residual calculated lifetime risk of 20% or greater largely due to family history 5-year risk of invasive breast cancer of equal or greater than 1.7% in those age 35 or over as calculated by Gail model Extremely dense or heterogeneously dense breast tissue.       23 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 3 Risk assessment for screening » Increased risk You might be referred to a h Genetic counselor or other health professional with expertise and experience in cancer genetics, if not already done h Breast specialist If you had chest RT between 10 and 30 years of age If you had chest or breast RT between 10 and 30 years of age, screening starts 8 years after radiation therapy (RT) ended, but not before 25 years of age. If you are under 25 years of age, then see your HCP for a checkup every year. If you are 25 years of age or over, then h See your HCP for a checkup every 6 to 12 months. h Get an annual screening mammogram. Tomosynthesis (3D mammogram) is recommended, if available. h Get an annual breast MRI. Contrast-enhanced mammography or whole breast ultrasound might be an option for those who cannot undergo an MRI. h Ask your HCP about how you can reduce your risk of developing breast cancer. If you are at increased risk for invasive breast cancer based on the Gail model If you are 35 years of age or over and have a 1.7% or greater 5-year risk of developing invasive breast cancer based on the Gail model, then h See your HCP for a checkup every 6 to 12 months. h Get an annual screening mammogram. Tomosynthesis (3D mammogram) is recommended, if available. h Ask your HCP about how you can reduce your risk of developing breast cancer. If you had ADH, LCIS, or ALH If you had atypical ductal hyperplasia (ADH) or a lobular neoplasia such as lobular carcinoma in situ (LCIS) or atypical lobular hyperplasia (ALH) and have a 20% or greater residual lifetime risk, then h See your HCP for a checkup every 6 to 12 months. h Get an annual screening mammogram. Tomosynthesis (3D mammogram) is recommended, if available. Mammograms should start at the age you are diagnosed, but not before 30 years of age. h Consider an annual breast MRI to begin when diagnosed but not before 25 years of age. Contrast-enhanced mammography or whole breast ultrasound might be an option for those who cannot undergo an MRI. h Ask your HCP about how you can reduce your risk of developing breast cancer. 24 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 3 Risk assessment for screening » Increased risk If you have dense breasts and no other risk factors Dense breast tissue is associated with an increased risk for breast cancer. If you have dense breast tissue (heterogeneously or extremely dense breasts) as seen on mammography and do not meet any other increased risk category, the recommendations are described below. For heterogeneously dense breasts, h See your HCP for a checkup every 6 to 12 months. h Get an annual screening mammogram. Tomosynthesis (3D mammogram) is recommended, if available. Mammograms should start no later than age 40, but not before age 30. h Talk with your HCP about screening tests in addition to mammography that might be appropriate for you. h Ask your HCP about how you can reduce your risk of developing breast cancer. For extremely dense breasts, h See your HCP for a checkup every 6 to 12 months. h Get an annual screening mammogram. Tomosynthesis (3D mammogram) is recommended, if available. Mammograms should start no later than age 40, but not before age 30. h Get a breast MRI with and without contrast starting at age 50, but may start at age 40. h Ask your HCP about how you can reduce your risk of developing breast cancer. "As 4-time breast cancer survivor and BRCA2-genetic mutation carrier, it is important to me to help educate and support my sons, family, and anyone who has a higher risk of developing cancer." 25 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 3 Risk assessment for screening » Key points 25 3 Risk assessment for screening » Key points » Questions to ask Key points h Breast cancer risk is your chance of developing breast cancer. Everyone has some risk for developing breast cancer. h Breast cancer risk is calculated using a breast cancer risk assessment model. Ask your health care provider (HCP) what model they are using to calculate your risk. h Screening mammograms are done for those who do not have any signs or symptoms of breast cancer. Annual screening mammograms are recommended to start at 40 years of age for those at average risk of developing breast cancer. h For those at increased risk, annual screening mammograms may start as early as 30 years of age. For those at increased risk, annual breast MRIs may start as early as 25 years of age. h Ask your HCP about how you can reduce your risk of developing breast cancer. Questions to ask h What is my risk of developing breast cancer? Am I at average or increased risk? h Based on my risk level, when should I start yearly screening mammograms? h If I have an increased risk, will I have both a screening mammogram and a breast MRI every year? h I don’t know my family health history. How will this affect my breast cancer risk assessment? h Why might I be at increased risk for developing breast cancer even though breast cancer doesn’t run in my family? Being diagnosed with breast cancer at age 70 doesn’t mean you don’t have an inherited predisposition for developing breast or other cancer types. Many factors are considered to determine the likelihood of a cancer predisposition syndrome in a family, such as age at diagnosis, type of breast cancer, family history of cancer, and ethnic background. Being diagnosed with breast cancer at age 70 doesn’t mean you don’t have an inherited predisposition for developing breast or other cancer types. Many factors are considered to determine the likelihood of a cancer predisposition syndrome in a family, such as age at diagnosis, type of breast cancer, family history of cancer, and ethnic background. 26 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 4 Testing during pregnancy and breastfeeding 27 Overview 27 During pregnancy 28 During breastfeeding 29 Key points 29 Questions to ask 27 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 4 Testing during pregnancy and breastfeeding » Overview 27 4 Testing during pregnancy and breastfeeding » Overview » During pregnancy Pregnancy-associated breast cancer (PABC) is breast cancer that occurs during pregnancy, while breastfeeding, or within 1 year of giving birth. It is important to continue taking care of your breast health during pregnancy and breastfeeding. Contact your health care provider about new or unusual changes to your breasts. Overview Changes in the breast are normal during pregnancy and breastfeeding (lactation). However, these changes can make it harder to detect small breast cancers. This is why it is important to continue taking care of your breast health. Delayed diagnosis of breast cancer during pregnancy or breastfeeding does happen. This may result in more advanced disease, larger tumors, and a higher chance of breast cancer spreading to lymph nodes in your armpit. Pregnancy-associated breast cancer (PABC) is defined as breast cancer that occurs at any of these time: h During pregnancy h While breastfeeding h Within 1 year of delivery While rare, PABC is the most common invasive cancer diagnosed during pregnancy. See your health care provider (HCP) regularly for checkups. Contact your HCP for any new or worsening changes. Breast cancer screening during pregnancy and lactation is similar to non-pregnant and non-lactating people with some exceptions. For example, due to safety concerns, breast MRI is not generally performed during pregnancy, but it can be safely used in those who are lactating Breast ultrasound and mammography can be safely performed during pregnancy. This section discusses screening recommendations during pregnancy and breastfeeding. It also provides information on signs and symptoms that you should report to your HCP. During pregnancy Clinical breast exams (CBEs) and breast cancer screening mammograms can and should continue on a yearly basis during pregnancy. Mammograms contain a very low level of radiation and are considered safe during pregnancy. Normal changes happen in the breast, skin, and nipple during pregnancy. However, if you notice anything new or unusual contact your HCP. An ultrasound may be done for changes in the skin of the breast or breast such as persistent or focused pain, a lump, suspicious nipple discharge, pitting or dimpling of the skin, skin thickening, swelling, or redness. 28 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 4 Testing during pregnancy and breastfeeding » During breastfeeding During breastfeeding Clinical breast cancer exams and screening mammograms can and should continue on a yearly basis during breastfeeding (lactation). If you are at increased risk for developing breast cancer, then a breast MRI may be done. Nursing or breast pumping before a mammogram is recommended. Normal changes happen in the breast, skin, and nipple during lactation or breastfeeding. Contact your doctor if you notice any new or unusual changes to your breast or armpit, or if h Your breast is warm to the touch, painful, or red in color. h Your breast skin appears thickened with large pores (peau d’orange). h You have focused, persistent breast pain. General or diffuse breast pain is not a suspicious symptom. Breast pain that comes and goes is not typically associated with breast cancer, although it can be related to non-breast causes. Health care providers During regular health checkups and breast cancer screening, you might come in contact with the following health care providers (HCPs): • Breast specialist is an expert in breast health and disease. • Breast radiologist is a doctor who interprets the results of mammograms, MRIs, and other imaging tests, and performs needle biopsies as needed. • Gynecologist is a doctor who diagnoses and treats diseases of the female reproductive organs. • Lactation consultant specializes in breastfeeding. • Mammogram, MRI, and ultrasound technologists operate the mammogram unit, or MRI or ultrasound machine. • Advanced practice providers (APPs) are nurse practitioners, physician assistants, and certified nurse midwives who can help prevent, evaluate, examine, and diagnose human disease, including breast cancer. • Obstetrician is a doctor who specializes in pregnancy and in childbirth. • Pathologist is a doctor who analyzes the cells, tissues, and organs removed during a biopsy or surgery. • Primary care provider (PCP) or physician gives a wide range of care, including prevention and treatment. 29 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 4 Testing during pregnancy and breastfeeding » Key points 29 4 Testing during pregnancy and breastfeeding » Key points » Questions to ask Nipple changes For changes in the nipple, contact your HCP if: h Your nipple is retracted or pulled in (inverted) when it was not that way before. h Your nipple has changed shape. h Your nipple becomes tender, and it is not related to your menstrual cycle. h Your nipple has skin changes, such as scaling, itchiness, and redness. h You have new suspicious nipple discharge, such as bloody or clear discharge. Key points h An annual clinical breast exam (CBE) and mammogram are recommended during pregnancy and breastfeeding. h Mammograms contain a very low level of radiation and are considered safe during pregnancy. h Nursing or breast pumping before a mammogram is recommended. h Breast MRI is not recommended during pregnancy, but may be performed during breastfeeding. h Normal changes happen in the breast, skin, and nipple during lactation or breastfeeding. However, if you notice anything new or unusual contact your health care provider. Questions to ask h How will you make me comfortable during the mammogram if I am pregnant or breastfeeding? h How long should I wait before I call the doctor if I notice any changes in the breast, nipple, or skin of the breast? Let us know what you think! Please take a moment to complete an online survey about the NCCN Guidelines for Patients. NCCN.org/patients/response 30 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 5 What to do when you have symptoms 31 Signs and symptoms 32 Skin changes 33 Nipple discharge 34 Nipple changes 35 Those assigned male at birth 35 Biopsy 37 Key points 37 Questions to ask 31 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 5 What to do when you have symptoms » Signs and symptoms This chapter is for those with signs or symptoms that suggest further testing. Certain conditions are difficult to diagnose. If possible, see a breast specialist for any unusual symptoms or skin or nipple changes. Signs and symptoms A sign can be seen by someone else like your health care provider (HCP). A symptom is something only you can feel. If you have any signs or symptoms that might suggest breast cancer, another cancer or disease, you may have more tests as needed. These are called diagnostic tests because they help diagnose a disease or condition. It is important to tell your HCP if something doesn’t feel or look right and has been that way for a few days. Don’t wait to say something! Signs and symptoms that might suggest further testing: h Breast pain that is focused or concentrated in 1 area h Lump, mass, or nodules can be felt in the breast h Lump or swelling in or near the armpit (axilla) h Breast implant-related symptoms h Skin changes h Nipple discharge Breast pain Breast pain is common in those assigned female at birth. Pain in both breasts and pain that comes and goes are considered benign and not typically associated with breast cancer. Persistent pain should be discussed with your HCP. Breast pain may be monitored to see if it persists, it is severe, and if there are changes in skin, or other symptoms such as a lump or nipple discharge. For pain that is focused or concentrated in one area, you may have an ultrasound. A mammogram might also be done if you are 30 years of age or over and you have not had a recent mammogram. Depending on the test results, further testing might be needed. Lumps and other signs If you have symptoms you can feel with your hand such as a lump, mass, nodules, or skin thickening, then for those h Under 30 years of age, an ultrasound is preferred as a diagnostic test. However, your doctor may wait and observe any changes for 1 to 2 menstrual cycles before ordering an ultrasound. h 30 years of age or over, a diagnostic mammogram and ultrasound are recommended. For a benign (not cancer) finding, you may not need any further evaluation. If the mammogram or ultrasound shows something, your HCP may recommend a core needle biopsy, additional physical exams, or more frequent mammograms and/or ultrasounds to monitor the symptom. You are also an important part of monitoring and should let your HCP know if you notice any new changes. 32 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 5 What to do when you have symptoms » Skin changes Lump in or near armpit A diagnostic mammogram and ultrasound will be done for a lump or mass in or near the armpit (axilla). If you are under 30 years of age, a mammogram might only be done if the ultrasound results are unclear or if cancer is suspected. Depending on your symptoms, you might be referred to a breast specialist. Breast implant-related symptoms Those with breast implants have a very small risk of developing breast implant-associated anaplastic large cell lymphoma (BIA-ALCL), a type of cancer called peripheral T-cell lymphoma. It is mostly seen in textured implants about 7 to 9 years after implant surgery. The main symptoms of BIA-ALCL are persistent swelling, or a lump or pain around the breast implant. Since BIA-ALCL is very rare, only those with symptoms will undergo further testing. If it has been more than 1 year since breast implant surgery and you are having symptoms related to the implants, then your health care provider should consult with a team of medical experts with experience in implant-related problems. If BIA-ALCL is suspected, then you will undergo further testing. Skin changes Skin changes include puckering, dimpling, a rash, or redness of the skin of the breast. Some people have a rash or redness of the nipple and the surrounding skin. The skin might look like an orange peel or the texture might feel different. Some skin changes might be a sign of inflammatory breast cancer (IBC). Skin changes around the nipple may be a different condition known as Paget disease of the breast. Since these diseases are difficult to diagnose, you might be referred to a breast specialist. Possible inflammatory breast cancer Inflammatory breast cancer (IBC) is a rare, aggressive cancer where cancer cells block lymph vessels in the skin of the breast. This causes the breast to look red and swollen and feel warm to the touch. Possible signs of IBC: h Peau d’orange (pitted or dimpled appearance of skin) h Skin thickening (skin has an orange-peel texture) h Edema (swelling caused by excess fluid in body tissue) h Erythema (reddening of the skin, usually in patches) If IBC is suspected, appropriate imaging tests will be recommended by the radiologist. These tests may include a mammogram, ultrasound, or both. If there is a suspicious 33 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 5 What to do when you have symptoms » Nipple discharge finding, a biopsy of the breast or a punch biopsy of the skin may be recommended. If there is a suspicious finding, a biopsy of the breast or a punch biopsy of the skin may be recommended For more information on inflammatory breast cancer, see NCCN Guidelines for Patients: Inflammatory Breast Cancer, available at NCCN.org/patientguidelines and on the NCCN Patient Guides for Cancer app. Possible Paget disease of the breast In Paget disease, abnormal cells are found in the nipple. This can cause nipple irritation or bleeding, scaly rash of the nipple, and skin ulcers. If Paget disease is suspected, then a diagnostic mammogram will be done. An ultrasound might be done, too. Nipple discharge When fluid that is not milk comes from the nipple, it is called nipple discharge. Nipple discharge is common, and, in most cases, unrelated to breast cancer. For example, nipple discharge can occur h During pregnancy following breast stimulation h In those with certain thyroid conditions, and h In those taking medicines, such as estrogen, oral contraceptives, opiates, and certain blood pressure medicines h Due to inflammation of the breast duct that is not related to cancer Nipple discharge is normal during breastfeeding (lactation). Some people who are lactating can express small amounts of yellow, green, or milky discharge if they squeeze the nipples. This is called expressed discharge. However, it is abnormal when the nipple discharge is any of the following: h Spontaneous – occurs without squeezing or dried discharge is seen on your bra or other clothing h Bloody or clear h From a single duct opening on the nipple on only one breast Know your breasts and immediately report any changes to your health care provider. 34 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 5 What to do when you have symptoms » Nipple changes If you have abnormal nipple discharge without other symptoms, then for those h Under 30 years of age, an ultrasound will be done. A diagnostic mammogram might also be done. h 30 years of age or over, a diagnostic mammogram and ultrasound will usually be done. Depending on the BI-RADS result, you might have a core needle biopsy, additional physical exams, more frequent mammograms and/or ultrasounds to monitor the symptom. You might be referred to a breast specialist and a breast MRI might be done. If you have any questions or concerns, talk with your health care provider (HCP). Milky discharge For milky discharge in both breasts (bilateral), you might have an endocrine (hormone) workup. Milky discharge is generally normal in pregnancy. Mammograms and breast ultrasounds are not typically performed for milky discharge. Nipple changes Inverted or retracted nipples pull inward toward your breast instead of sticking out. For some this is normal, however, if your nipple inverts or changes suddenly, let your HCP know. If you have an inverted or retracted nipple without a lump or other symptoms, then for those h Under 30 years of age, an ultrasound will be done. h 30 years of age or over, a diagnostic mammogram and ultrasound will be done. Depending on the BI-RADS result, you might have a core need biopsy, additional physical exams, more frequent mammograms, breast MRIs or ultrasounds. You might be referred to a breast specialist and a breast MRI might be done. If you have any questions or concerns, talk with your HCP We want your feedback! Our goal is to provide helpful and easy-to-understand information on cancer. Take our survey to let us know what we got right and what we could do better. NCCN.org/patients/feedback 35 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 5 What to do when you have symptoms » Those assigned male at birth 35 5 What to do when you have symptoms » Those assigned male at birth » Biopsy Those assigned male at birth Those assigned male at birth do not typically undergo routine screening mammograms. However, males can develop breast cancer and those with a BRCA2 gene mutation might undergo routine screening mammograms. Gynecomastia Gynecomastia is the abnormal growth of breast tissue in those assigned male at birth. There might be concern if one breast appears enlarged. This is called asymmetrical gynecomastia. If is not clear you have gynecomastia, then a diagnostic mammogram with possible ultrasound may be done. Lump or nipple changes If you have bloody nipple discharge or symptoms that can be felt such as a lump, then an ultrasound will be done in addition to the diagnostic mammogram. A core needle biopsy might also be done. Your condition will be monitored, and you might be asked to return to your health care provider for regular testing. Consider seeing a breast specialist. Biopsy A biopsy is the removal of a sample of tissue from your body for testing. A pathologist will examine the biopsy for cancer and write a report called a pathology report. Depending on the biopsy results, you may not need any additional testing and be returned to routine follow-up care. However, the biopsy results may require additional testing or more frequent exams or imaging. If cancer is found, you will be referred for cancer treatment. There are different types of biopsies. Some biopsies are guided using imaging, such as mammography, ultrasound, or MRI. The primary or main area of concern is biopsied first. Other areas may also be biopsied. “Even though my sister had breast cancer 3 times and had been genetically diagnosed as BRCA2 positive, I was still SHOCKED that I, a 54-year-old male, could have breast cancer.” 36 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 5 What to do when you have symptoms » Biopsy Types of possible biopsies include: h Core biopsy (CB) uses needles of different sizes to remove a sample of tissue or fluid and typically requires a very small incision. This type of biopsy is often performed by the radiologist. In a vacuum-assisted core biopsy (VACB), suction from a special vacuum device is used to remove the sample through a needle. h Fine-needle aspiration (FNA) uses a small needle to remove a sample of tissue or fluid. FNA is more commonly used when a lymph node in the armpit is being biopsied. h Incisional biopsy removes a small amount of tissue through a cut in the skin or body and is typically performed by a surgeon. h Excisional biopsy removes the entire abnormal area. This is not the preferred type of biopsy, but may be necessary if other methods are not possible or when the biopsy results don’t match the expected findings. An excisional biopsy is usually done under anesthesia by a surgeon in an operating room. Before biopsies are performed, usually the area is injected with numbing medicine. A core needle biopsy (CNB) may remove more than one tissue sample, but usually through the same area on the breast. The samples are small. The needle is often guided into the tumor with imaging, but may be guided by using ultrasound or x-rays. When mammography is used during a biopsy, it is called a stereotactic needle biopsy. One or more clips may be placed near the breast tumor during a biopsy. The clips are small, painless, and made of metal. They will Biopsy In a biopsy, a sample of tissue is removed. There are different types of biopsy. This image shows an ultrasound-guided needle biopsy. 37 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 5 What to do when you have symptoms » Key points 37 5 What to do when you have symptoms » Key points » Questions to ask mark the site for possible future treatment and imaging. The clips will stay in place until surgery. If the area biopsied is benign, the clip will remain in place to mark the biopsy site on future imaging. The clips cause no problems, even if they are left in place for a long time. You will be able to go through airport security and have an MRI. Biopsy results Histology is the study of the anatomy (structure) of cells, tissues, and organs under a microscope. It is used to make a diagnosis or treatment decisions. Your pathology report will contain information about histology. Key points h A sign can be seen by someone else like your health care provider. A symptom is something only you can feel, like pain. If you have any signs or symptoms that might suggest breast or another cancer or disease, then you will have more tests. h Some skin changes might be a sign of inflammatory breast cancer (IBC) or Paget disease. Because these diseases are difficult to diagnose, you might be referred to a breast specialist. h If IBC is suspected, then a diagnostic mammogram and possibly an ultrasound will be done. An MRI might be done, too. A biopsy will typically be needed. h If Paget disease is suspected, then a diagnostic mammogram and possibly an ultrasound will be done. A biopsy will typically be needed. h Nipple discharge is common, and, in many cases unrelated to breast cancer. h Those assigned male at birth with a BRCA2 gene mutation might undergo routine screening mammograms. Questions to ask h What imaging tests do you recommend and why? h What type of biopsy will I have and how will you make me comfortable? h Who will explain the imaging or biopsy results to me? h How can I prepare for imaging testing or a biopsy? h Can you recommend a breast specialist? 38 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 6 Other resources 39 What else to know 39 What else to do 39 Where to get help 40 Questions to ask 39 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 6 Other resources » What else to know 39 6 Other resources » What else to know » What else to do » Where to get help Want to learn more? Here's how you can get additional help. What else to know This book is an important tool for improving cancer detection and care. It plainly explains expert recommendations and suggests questions to ask your care team. But, it’s not the only resource that you have. You’re welcome to receive as much information and help as you need. Many people are interested in learning more about: h Personal or family risk of developing breast cancer and other cancers h How to decrease risk of developing breast cancer and other cancers h Finding a doctor who is an expert in breast cancer What else to do Your health care center can help you with next steps. They often have on-site resources to help meet your needs and find answers to your questions. Health care centers can also inform you of resources in your community. In addition to help from your providers, the resources listed in the next section provide support for many people like yourself. Look through the list and visit the provided websites to learn more about these organizations Where to get help Bone & Marrow Cancer Foundation Bonemarrow.org Breast Cancer Alliance breastcanceralliance.org Breastcancer.org breastcancer.org CanCare, Inc Cancare.org CancerCare Cancercare.org Cancer Hope Network Cancerhopenetwork.org DenseBreast-info.org DenseBreast-info.org DiepC Foundation diepcfoundation.org FORCE - Facing Our Risk of Cancer Empowered facingourrisk.org GPAC Global Patient Advocacy Coalition GPACunited.org GRACE Cancergrace.org 40 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 6 Other resources » Questions to ask HIS Breast Cancer Awareness Hisbreastcancer.org Imerman Angels Imermanangels.org Inflammatory Breast Cancer Research Foundation ibcresearch.org MedlinePlus medlineplus.gov National Coalition for Cancer Survivorship canceradvocacy.org Sharsheret sharsheret.org Triage Cancer Triagecancer.org Unite for HER uniteforher.org Young Survival Coalition (YSC) Youngsurvival.org Questions to ask h Who can I talk to about help with housing, food, and other basic needs? h What help is available for transportation, childcare, and home care? h What other services are available to me and my caregivers? h How can I connect with others and build a support system? h Who can I talk to if I don’t feel safe at home, at work, or in my neighborhood? Breast cancer resources More information on breast cancer care is available at NCCN.org/patientguidelines and on the NCCN Patient Guides for Cancer app. 41 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 Ü Ü 42 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 Words to know Words to know areola A darker, round area of skin on the breast around the nipple. atypical ductal hyperplasia (ADH) A benign (not cancer) condition in which there are more cells than normal in the lining of breast ducts and the cells look abnormal under a microscope. Having atypical ductal hyperplasia increases the risk of breast cancer. atypical lobular hyperplasia (ALH) A benign condition in which there are more cells than normal in the breast lobules and the cells look abnormal under a microscope. Having atypical lobular hyperplasia increases the risk of breast cancer. axillary lymph node (ALN) A small disease-fighting structure that is near the armpit. biopsy A procedure that removes fluid or tissue samples to be tested for a disease. BI-RADS Breast Imaging Reporting and Data System or BI-RADS is a standard system used in the United States to describe mammogram findings and results and classifies breast density into groups. BI-RADS can also be used to describe the results of a breast ultrasound or breast MRI. calcification Deposits of calcium in the tissues. clinical breast exam (CBE) A physical exam of the breast performed by a health care provider to check for lumps or other changes. contrast A substance put into your body to make clearer pictures during imaging tests. core needle biopsy (CNB) A procedure that removes tissue samples with a hollow needle. Also called core biopsy (CB). diagnostic mammogram Pictures of the insides of both breasts that are made from a set of x-rays in individuals with signs or symptoms. duct A tube-shaped structure through which milk travels to the nipple. ductal carcinoma A cancer derived from cells that line small tube-shaped duct. ductal carcinoma in situ (DCIS) A breast cancer that has not grown outside the breast ducts. genetic counseling Expert guidance on the chance for a disease that is passed down in families. gynecomastia The abnormal growth of breast tissue in those assigned male at birth. hereditary breast cancer Breast cancer likely caused by an inherited gene mutation passed down from biological parent to child. histology The structure of cells, tissue, and organs as viewed under a microscope. 43 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 Words to know lobule A gland in the breast that makes breast milk. lobular carcinoma A breast cancer that started in cells that line the milk glands (lobules). lobular carcinoma in situ (LCIS) A benign (not cancer) condition in which abnormal cells are found in the lobules of the breast. lobular neoplasia A benign condition in which abnormal cells are found in the lobules of the breast and increases the risk of developing breast cancer in the future. Types of lobular neoplasia include atypical lobular hyperplasia (ALH) and lobular carcinoma in situ (LCIS). lymph A clear fluid containing white blood cells. lymph node A small, bean-shaped disease-fighting structure. magnetic resonance imaging (MRI) A test that uses radio waves and powerful magnets to make pictures of the inside of the body. mammogram A picture of the inside of the breast that is made by an x-ray test. nipple discharge Fluid that is not milk coming from the nipple. pathologist A medical doctor who interprets cells and tissues removed during a biopsy or surgery. radiation therapy (RT) A treatment that uses high-energy rays. radiologist A medical doctor who interprets the results of mammograms, MRIs, and other imaging tests. residual lifetime risk Chance of developing breast cancer in your lifetime based on the number of years of life that remain and other risk factors. screening mammogram X-rays of the breasts taken to check for breast cancer in someone without signs or symptoms of cancer. technologist Person trained to operate an imaging machine (for example, mammogram, ultrasound, or MRI) to produce images of the breasts. ultrasound (US) A test that uses sound waves to take pictures of the inside of the body. 44 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 NCCN Contributors The NCCN Clinical Practice Guidelines in Oncology (NCCN Guidelines®) for Breast Cancer Screening and Diagnosis Version 2.2025 were developed by the following NCCN Panel Members: Therese B. Bevers, MD/Chair The University of Texas MD Anderson Cancer Center Bethany L. Niell, MD, PhD/ Vice-Chair Moffitt Cancer Center Shadi Aminololama-Shakeri, MD UC Davis Comprehensive Cancer Center Jennifer L. Baker, MD UCLA Jonsson Comprehensive Cancer Center Debbie L. Bennett, MD Siteman Cancer Center at Barnes-Jewish Hospital and Washington University School of Medicine Ermelinda Bonaccio, MD Roswell Park Comprehensive Cancer Center Melissa S. Camp, MD Johns Hopkins Kimmel Cancer Center Emily F. Conant, MD Abramson Cancer Center at the University of Pennsylvania Meghan R. Flanagan, MD, MPH Fred Hutchinson Cancer Center Caitlin B. Mauer Hall, MA, MS, CGC UT Southwestern Simmons Comprehensive Cancer Center Jeffrey R. Hawley, MD The Ohio State University Comprehensive Cancer Center - James Cancer Hospital and Solove Research Institute Mark Helvie, MD University of Michigan Rogel Cancer Center Tamarya L. Hoyt, MD Vanderbilt-Ingram Cancer Center Dezheng Huo, PhD The UChicago Medicine Comprehensive Cancer Center Jennifer L. Ivanovich, MS, CGC Indiana University Melvin and Bren Simon Comprehensive Cancer Center Swati Kulkarni, MD Robert H. Lurie Comprehensive Cancer Center of Northwestern University Rachael B. Lancaster, MD O'Neal Comprehensive Cancer Center at UAB Bhavika K. Patel, MD Mayo Clinic Comprehensive Cancer Center Mark Pearlman, MD University of Michigan Rogel Cancer Center Liane Philpotts, MD Yale Cancer Center/Smilow Cancer Hospital Donna Plecha, MD Case Comprehensive Cancer Center/ University Hospitals Seidman Cancer Center and Cleveland Clinic Taussig Cancer Institute Jennifer K. Plichta, MD, MS Duke Cancer Institute Rebecca Shatsky, MD UC San Diego Moores Cancer Center Mary Lou Smith, JD, MBA Research Advocacy Network Clarie L. Streibert, MD Fox Chase Cancer Center Roberta M. Strigel, MD, MS University of Wisconsin Carbone Cancer Center Candice N. 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For disclosures, visit NCCN.org/disclosures. 45 NCCN Guidelines for Patients® Breast Cancer Screening and Diagnosis, 2025 NCCN Cancer Centers NCCN Cancer Centers Abramson Cancer Center at the University of Pennsylvania Philadelphia, Pennsylvania 800.789.7366 • pennmedicine.org/cancer Case Comprehensive Cancer Center/ University Hospitals Seidman Cancer Center and Cleveland Clinic Taussig Cancer Institute Cleveland, Ohio UH Seidman Cancer Center 800.641.2422 • uhhospitals.org/services/cancer-services CC Taussig Cancer Institute 866.223.8100 • my.clevelandclinic.org/departments/cancer Case CCC 216.844.8797 • case.edu/cancer City of Hope National Medical Center Duarte, California 800.826.4673 • cityofhope.org Dana-Farber/Brigham and Women’s Cancer Center \| Mass General Cancer Center Boston, Massachusetts 877.442.3324 • youhaveus.org 617.726.5130 • massgeneral.org/cancer-center Duke Cancer Institute Durham, North Carolina 888.275.3853 • dukecancerinstitute.org Fox Chase Cancer Center Philadelphia, Pennsylvania 888.369.2427 • foxchase.org Fred & Pamela Buffett Cancer Center Omaha, Nebraska 402.559.5600 • unmc.edu/cancercenter Fred Hutchinson Cancer Center Seattle, Washington 206.667.5000 • fredhutch.org Huntsman Cancer Institute at the University of Utah Salt Lake City, Utah 800.824.2073 • healthcare.utah.edu/huntsmancancerinstitute Indiana University Melvin and Bren Simon Comprehensive Cancer Center Indianapolis, Indiana 888.600.4822 • www.cancer.iu.edu Johns Hopkins Kimmel Cancer Center Baltimore, Maryland 410.955.8964 www.hopkinskimmelcancercenter.org Mayo Clinic Comprehensive Cancer Center Phoenix/Scottsdale, Arizona Jacksonville, Florida Rochester, Minnesota 480.301.8000 • Arizona 904.953.0853 • Florida 507.538.3270 • Minnesota mayoclinic.org/cancercenter Memorial Sloan Kettering Cancer Center New York, New York 800.525.2225 • mskcc.org Moffitt Cancer Center Tampa, Florida 888.663.3488 • moffitt.org O’Neal Comprehensive Cancer Center at UAB Birmingham, Alabama 800.822.0933 • uab.edu/onealcancercenter Robert H. 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7613	https://www.youtube.com/watch?v=ijOItVm4Fkc	4-77 Moment of a Couple Chapter 4 ( Hibbeler Statics 14th Edition ) Engineers Academy Engineers Academy 120000 subscribers 45 likes Description 3968 views Posted: 30 Sep 2022 SUBSCRIBE my Channel Engineers Academy for more problem Solutions! Kindly like, share and comment, this will help to promote my channel!! Engineering Statics by Hibbeler 14th Edition Chapter 4: Force System Resultants Moment of a Couple 4–77. Two couples act on the beam as shown. If F = 150 lb, determine the resultant couple moment. 4–78. Two couples act on the beam as shown. Determine the magnitude of F so that the resultant couple moment is 300 lb ft counterclockwise. Where on the beam does the resultant couple act? Moment about a point Moment of a force Principle of Moments Moment of a Force about specified axis EngineeringMechanics #EngineeringStatics #Statics #Hibbeler #CoupleMoment Hibbeler Statics - Chapter Playlists Chapter 2: Force Vectors - Chapter 3: Equilibrium of a Particle - Chapter 4: Force System Resultants - Chapter 5: Equilibrium of a Rigid Body - Chapter 6: Structural Analysis - Chapter 8: Friction - 5 comments Transcript: hello guys I welcome you all to my Channel engineers Academy do subscribe Engineers Academy if you haven't done it yet now the we are going to solve these two problems so the first problems is that two couples act on the beam as shown if f equals to 150 pound determine the resultant couple moment so we have these two pair of forces and they are producing two couples so we are required to determine the resultant couple moment so the resultant couple moment is equal to and let's assume that the counterclockwise couple is considered to be positive so in order to find the couple due to these two forces they are making some angle with the horizontal and that angle is given in the form of the small triangle so if you want to resolve this Force F into its component so it will have one component she is going to act in this direction we will have one component like this and this one will be the COS component we can say that this is f COS of theta and if we are considering this angle Theta then from this small triangle where this is 3 5 and 4 if this is Theta then COS of theta is base divided by hypotenuse so it is 4 divided by 5 for this small triangle so we can say that this component IS F into 4 divided by 5. similarly we will have the sine componentry is going to act here in the downward direction we can write that this is f um sine of theta and if we again consider that same triangle if this is the angle Theta then sine of theta will be this perpendicular divided by hypotenuse so this perpendicular is 3 and the hypotenuse is 5 so we can say that this is 3 divided by 5. and similarly we can resolve this Force half as well this one will be the cost component again we have that same shy angle we are considering this angle so this is parallel to that Force so this this one this is again that same cos component so we can say that this is f into 4 divided by 5 and similarly we will have this component this will be F this will be the sine component that is 3 divided by 5. so to find the resultant uh couple moment is we can see that uh this cos component of this Force F they are producing the uh counter clockwise couple so we can say that this is Plus F into 4 divided by 5 and the perpendicular distance between uh both of these components is 1.5 so multiplied by 1.5 the couple magnitude is always equal to the force times the perpendicular distance between both the forces similarly the sine component this sine component and this sine component they are also producing the counterclockwise couple so you will write Plus and the sine component IS F 3 divided by 5 and the perpendicular distance between this sine component and this sine component is four feet this is the perpendicular distance so we need to multiply it with 4. and similarly this 200 pound these two forces they are producing the clockwise couple right so we're going to write minus since the counter clockwise couple is considered to be positive the counterclockwise moment is considered to be positive so it is producing the clockwise so we will write minus 200 and the perpendicular distance between both of these uh 200 pound force is 1.5 feet so I will multiply it with 1.5 so now let's simplify this equation first this is the resultant couple so 4 into 1.5 is 4 multiplied by 1.5 this is 6. so we can say that this is 6 divided by 5 f plus 3 into 4 is 12 divided by 5 f minus 200 into 1.5 this is 300. so this is the general equation for both the problems now in problem 477 we are given the force magnitude and we are required to find the resultant couple moments so we need to plug in that F value in this equation and we will be able to find the resultant couple moment so problem 477 solution f is equal to 150 pounds so now we will put it in this equation so the resultant couple moment is 6 divided by 5 into 150 plus 12 divided by 5 into 150. so this is 6 into 150 divided by 5. plus 12 12 divided by 5 into 150 and here we have that minus 300 as well so we have to write minus 300. minus 300 so this gives us the resultant couple moment equals to 240 pound feet since the forces are given in pounds and the distances are considered in feet so the L so the final units of the resultant couple will be 240 pound feet so this is the solution of problem 477. now in problem 478 we are given the resultant couple moment magnitude and we are asked to find the F magnitude so now in problem 478 we are given this and we are required to find this foil U so 478 solution so now in this equation we need to put this value and that is given a is 300 pound feet counterclockwise so this means that the resultant couple is Plus 300 pound feet and force magnitude is required so now let's put values in this equation so we will have less 300 equals to 6 divided by 5 f plus 12 divided by 5 F minus 300. so now we can bring this 300 to the other side of equation it will become positive so we will have plus 300 and we can take F common from both of these terms so now 6 divided by 5 plus 12 divided by 5 this gives us 18 divided by 5 so 18 divided by 5 f equals to 600 and from this we can say that the force magnitude is 600 into 5 divided by 18 if we cross multiply so 600 600 multiplied by 5 divided by 18 this gives us 166.67 or we can say that the force magnitude is approximately 167 pounds so this is the solution of problem 478. so this is the solution of these two problems I hope it will help you in your learning do subscribe Engineers Academy that will help me to reach out many more students like you people
7614	https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/abs/knesers-theorem-in-sigma-finite-abelian-groups/C37CC828089B939B7ABD2EBD17F18462	Kneser’s theorem in $\sigma $-finite abelian groups \| Canadian Mathematical Bulletin \| Cambridge Core Skip to main contentAccessibility help Login Alert Cancel Log in × × Discover Content Products and Services Register Log In (0) Cart Search Browse Services Open research Institution Login Search Hostname: page-component-54dcc4c588-mz6gc Total loading time: 0 Render date: 2025-09-28T12:33:16.471Z Has data issue: false hasContentIssue false Home Journals Canadian Mathematical Bulletin Volume 65 Issue 4 Kneser’s theorem in \sigma -finite abelian groups English Français Canadian Mathematical Bulletin Article contents Abstract Footnotes References Kneser’s theorem in \sigma -finite abelian groups Part of: Additive number theory; partitionsAbelian groupsSequences and sets Published online by Cambridge University Press:10 January 2022 Pierre-Yves Bienvenuand François Hennecart Show author details Pierre-Yves Bienvenu Affiliation: Institute of Analysis and Number Theory, TU Graz, Kopernikusgasse 24/II, Graz 8010, Austria François HennecartAffiliation: UJM-Saint-Étienne, CNRS, ICJ UMR 5208, University of Lyon, 23 rue du docteur Paul Michalon, Saint-Étienne 42023, France e-mail: francois.hennecart@univ-st-etienne.fr Article Article Metrics Article contents Abstract Footnotes References Get access Share Copy Share Share Share Share Post Share Mail Share CiteRights & Permissions [Opens in a new window] Abstract Let G be a \sigma -finite abelian group, i.e., G=\bigcup {n\geq 1} G_n where (G_n){n\geq 1} is a nondecreasing sequence of finite subgroups. For any A\subset G, let \underline {\mathrm {d}}( A ):=\liminf _{n\to \infty }\frac {\|A\cap G_n\|}{\|G_n\|} be its lower asymptotic density. We show that for any subsets A and B of G, whenever \underline {\mathrm {d}}( A+B )<\underline {\mathrm {d}}( A )+\underline {\mathrm {d}}( B ), the sumset A+B must be periodic, that is, a union of translates of a subgroup H\leq G of finite index. This is exactly analogous to Kneser’s theorem regarding the density of infinite sets of integers. Further, we show similar statements for the upper asymptotic density in the case where A=\pm B. An analagous statement had already been proven by Griesmer in the very general context of countable abelian groups, but the present paper provides a much simpler argument specifically tailored for the setting of \sigma -finite abelian groups. This argument relies on an appeal to another theorem of Kneser, namely the one regarding finite sumsets in an abelian group. Keywords sumsetinverse theoremadditive combinatorics MSC classification Primary:11P70: Inverse problems of additive number theory, including sumsets Secondary:11B05: Density, gaps, topology20K99: None of the above, but in this section Information Type Article Information Canadian Mathematical Bulletin , Volume 65 , Issue 4, December 2022, pp. 936 - 942 DOI: [Opens in a new window] Copyright © Canadian Mathematical Society, 2022 Access options Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.) Check access Institutional login We recognised you are associated with one or more institutions that don't have access to this content. If you should have access, please contact your librarian.Access through your institution Change institution Personal login Log in with your Cambridge Core account or society details.Log in Article purchase Digital access for individuals US$41.00Add to cart Footnotes This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). References Bihani, P. and Jin, R., Kneser’s theorem for upper Banach density. J. Théor. Nombres Bordeaux 18(2006), 323–343.10.5802/jtnb.547CrossRefGoogle Scholar Griesmer, J., Small-sum pairs for upper Banach density in countable abelian groups. Adv. Math.246(2013), 220–264.10.1016/j.aim.2013.06.005CrossRefGoogle Scholar Halberstam, H. and Roth, K. F., Sequences. 2nd ed.Springer-Verlag, New York–Berlin, 1983, xviii+292 pp.10.1007/978-1-4613-8227-0CrossRefGoogle Scholar Hamidoune, Y. and Rödseth, Ö., On bases for\sigma -finite groups. Math. Scand.78(1996), 246–254.10.7146/math.scand.a-12586CrossRefGoogle Scholar Hegyvári, N., On iterated difference sets in groups. Period. Math. Hung.43(2001), 105–110.10.1023/A:1015285615996CrossRefGoogle Scholar Hegyvári, N. and Hennecart, F., Iterated difference sets in\sigma -finite groups. Ann. Univ. Sci. Budapest, 50(2007), 1–7.Google Scholar Jin, R., Solution to the inverse problem for upper asymptotic density. J. Reine Angew. Math.595(2006), 121–166.Google Scholar Kneser, M., Abschätzungen der asymptotischen Dichte von Summenmengen. Math. Z.58(1953), 459–484.10.1007/BF01174162CrossRefGoogle Scholar Related content GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS Type Article TitleGENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPSAuthorsANDREA LUCCHINI,CLAUDE MARIONandGARETH TRACEYJournalForum of Mathematics, Sigma Published online: 30 June 2020 Groups and their Representations Type Chapter TitleGroups and their RepresentationsAuthorsFernando GouvêaJournalA Guide to Groups, Rings, and Fields Published online: 5 April 2013 On derangements in simple permutation groups Type Article TitleOn derangements in simple permutation groupsAuthorsTimothy BurnessandMarco FusariJournalForum of Mathematics, Sigma Published online: 23 June 2025 On integral decomposition of unipotent elements in integral group rings Type Article TitleOn integral decomposition of unipotent elements in integral group ringsAuthorsLeo MargolisandGeoffrey JanssensJournalProceedings of the Royal Society of Edinburgh Section A: Mathematics Published online: 12 September 2025 Classical algebra Type Chapter TitleClassical algebraAuthorsTerry GannonJournalMoonshine beyond the Monster Published online: 13 July 2023 The maximal size of a minimal generating set Type Article TitleThe maximal size of a minimal generating setAuthorsScott HarperJournalForum of Mathematics, Sigma Published online: 10 August 2023 Cited by Loading... 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7615	https://mariokrenn.wordpress.com/graph-theory-question/	Mario Krenn Inherited Vertex Coloring of Graphs We discovered a new form of vertex colors of graphs (inherited directly form edge-colorings of perfect matchings) in the study of quantum physics questions. Several new pure graph theoretical questions emerged, and I believe that their answer could lead to exciting insights on the potential of resources of quantum interference. I have described the question in detail in a youtube video, mathoverflow question, and a contribution to the proceedings of 2nd Croatian Combinatorial Days (arXiv version), and was described in detail by Dustin Mixon. 1) 3.000 Euro Reward We (Dominik Leitner and me) offer a 3.000 Euro reward for the first solution of our conjecture (either proof or counterexample) (mathoverflow question, Dustin Mixon’s blog, youtube (see Clarification below video), Question1 here): Update January 2021: Not only solutions to Question2, but also counter-examples to the conjecture will be awarded the reward. The solution has to appear in a respected, peer-reviewed journal (or for counterexamples, there needs to be a way to confirm them, for instance via software). The problem can be concisely formulated in the language of graph theory and combinatorial equation systems. 2) 1.000 Euro Quantum-Graph Best-Paper Award We (Dominik Leitner and me) offer a 1.000 Euro award for the best paper (Quantum-Graph Best-Paper award) which investigates inherited vertex colors of graphs, derives new results or connections. The next deadline for nominations will be End of August 2024. Rules: The paper needs to be publically available and correct (peer-reviewed in a respected journal or proceedings, and/or I need to able to understand the results) and the results should be non-trivial/obvious. At the deadline, i will ask for recommendations by experts. In case of two interesting papers, the award could be splitted. Nominations and self-nominations are very welcome. Known special cases News: 2024.07.03: A paper by L. Sunil Chandran and Rishikesh Gajjala on this problem got accepted in Quantum Journal. 2024.07.02: Lode Vermeulen has finished his Bachelor thesis on a partial formalization of Bogdanov’s lemma in the proof-assist Lean (including a contribution to matlab of Hamiltonian cycles and paths). 2024.06.29: A new paper by Chandran, Gajjala and Illickan shows that the conjecture is true for all graph with a maximal vertex degree of at most 3. This result holds unconditionally (for complex weighted multigraphs with bicolored edges). It is accepted in Mathematical Foundations of Computer Science 2024. 2023.09.04: New special case: For n=4 vertices, it is not possible to create a d=4 colored monochromatic graph, without any conditions (the result works for multi-graphs with bi-colored edges and arbitrary complex weights): github 2023.04.13: New paper by L Sunil Chandran and Rishikesh Gajjala on Perfect matchings and Quantum physics: Bounding the dimension of GHZ states, proving a special case that for simple graphs, a monochromatic graph cannot exist for d>n/sqrt(2). 2023.02.18: Rishikesh Gajjala was awarded the best student poster award in math and computer science at the Prime minister’s research fellowship annual symposium at IIT Madras, for his work on proving a special case of our conjecture, see poster. Congratulations! 2023.01.24: New paper by Moshe Y. Vardi and Zhiwei Zhang (Rice University) on the application of Tutte’s theorem as constraint for encoding the Perfect Matchings under Vertex-Coloring Constraints. 2022.09.26: New paper by Moshe Y. Vardi and Zhiwei Zhang (Rice University) on the computational complexity of Perfect Matchings under Vertex-Color Constraints. 2022.07.04: Bachelor Thesis by Aaron Neugebauer (University of Wuerzburg, Germany) on the conjecture, and a reformulation as color-spanning graphs. 2022.02.11: New paper by L. Sunil Chandran and Rishikesh Gajjala showing that for simple graphs with monochromatic edges, a monochromatic graph cannot exist for d=n-2. 2021.09.27: Using SAT solvers, we proof a few special cases of the conjecture (specifically for graphs with only monochromatic edges). 2021.01.18: Dustin Mixon’s blog post reformulated and generalized the question in a beautiful way. 2020.05.13: New paper on graph-theoretical techniques in artificial intelligence algorithms. 2019.02.23: Mathoverflow question Has this notion of vertex-coloring of graphs been studied? where I explain the concept of bi-chromatic edge-colored graphs and inherited vertex coloring more compactly. 2019.02.16: Questions on the Structure of Perfect Matchings inspired by Quantum Physics – here, together with my colleages Xuemei Gu and Daniel Soltesz, we formalize the ideas of bi-chromatic, edge-colored weighted graph, and the concept of inherited vertex coloring, together with several examples, and a number of noteworthy open question on the topic. 2018.09.24: Mathoverflow question Vertex Coloring inherited from Perfect Matchings (motivated by Quantum Physics) where I explain the idea of inherited vertex coloring and particularily the question for the reward more compactly. 2017.04.12: A post by Ilya Bogdanov, which answeres the question of the reward for the special case of graphs with all weights being real and positive. Questions and Comments For any questions and clarifications, please send me an e-Mail. Teilen mit:
7616	https://flexbooks.ck12.org/cbook/prep-for-8th-grade-math/section/4.6/related/lesson/composite-solids-geom/	Composite Solids \| CK-12 Foundation AI Teacher Tools – Save Hours on Planning & Prep. Try it out! Skip to content What are you looking for? Search Math Grade 6 Grade 7 Grade 8 Algebra 1 Geometry Algebra 2 PreCalculus Science Earth Science Life Science Physical Science Biology Chemistry Physics Social Studies Economics Geography Government Philosophy Sociology Subject Math Elementary Math Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Interactive Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Conventional Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Probability & Statistics Trigonometry Math Analysis Precalculus Calculus What's the difference? 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Learn. Interact. eXplore. CCSS Math Concepts and FlexBooks aligned to Common Core NGSS Concepts aligned to Next Generation Science Standards Certified Educator Stand out as an educator. Become CK-12 Certified. Webinars Live and archived sessions to learn about CK-12 Other Resources CK-12 Resources Concept Map Testimonials CK-12 Mission Meet the Team CK-12 Helpdesk FlexLets Know the essentials. Pick a Subject Donate Sign InSign Up Back To Volume of Composite 3D SolidsBack 4.6 Composite Solids Written by:Bill Zahner \|Lori Jordan Fact-checked by:The CK-12 Editorial Team Last Modified: Sep 01, 2025 Composite Solids A composite solid is a solid that is composed, or made up of, two or more solids. The solids that it is made up of are generally prisms, pyramids, cones, cylinders, and spheres. In order to find the surface area and volume of a composite solid, you need to know how to find the surface area and volume of prisms, pyramids, cones, cylinders, and spheres. Finding the Volume of a Parallelogram-Based Prism Find the volume of the solid below. This solid is a parallelogram-based prism with a cylinder cut out of the middle. V p r i s m=(25⋅25)30=18,750 c m 3 V c y l i n d e r=π(4)2(30)=480 π c m 3 The total volume is 18750−480 π≈17,242.04 c m 3. Finding the Surface Area Find the surface area of the following solid. This solid is a cylinder with a hemisphere on top. Because it is one fluid solid, we would not include the bottom of the hemisphere or the top of the cylinder because they are no longer on the surface of the solid. Below, “L A” stands for lateral area. S A=L A c y l i n d e r+L A h e m i s p h e r e+A b a s e c i r c l e=π r h+1 2 4 π r 2+π r 2=π(6)(13)+2 π 6 2+π 6 2=78 π+72 π+36 π=186 π i n 2 Finding the Volume Find the volume of the following solid. To find the volume of this solid, we need the volume of a cylinder and the volume of the hemisphere. V c y l i n d e r=π 6 2(13)=78 π V h e m i s p h e r e=1 2(4 3 π 6 3)=36 π V t o t a l=78 π+36 π=114 π i n 3 Examples Example 1 Find the volume of the composite solid. All bases are squares. This is a square prism with a square pyramid on top. Find the volume of each separeatly and then add them together to find the total volume. First, we need to find the height of the pyramid portion. The slant height is 25 and the edge is 48. Using have of the edge, we have a right triangle and we can use the Pythagorean Theorem. h=25 2−24 2=7 V p r i s m=(48)(48)(18)=41472 c m 3 V p y r a m i d=1 3(48 2)(7)=5376 c m 3 The total volume is 41472+5376=46,848 c m 3. Example 2 Find the volume of the base prism. Round your answer to the nearest hundredth. Use what you know about prisms. V p r i s m=B⋅h V p r i s m=(4⋅4)⋅5 V p r i s m=80 i n 3 Example 3 Using your work from Example 2, find the volume of the pyramid and then of the entire solid. Use what you know about pyramids. V p y r a m i d=1 3 B⋅h V p y r a m i d=1 3(4⋅4)(6)V p y r a m i d=32 i n 3 Now find the total volume by finding the sum of the volumes of each solid. V t o t a l=V p r i s m+V p y r a m i d V t o t a l=112 i n 3 Review Find the volume of the composite solids below. Round your answers to the nearest hundredth. The bases are squares. Find the volume of the green part. A cylinder fits tightly inside a rectangular prism with dimensions in the ratio 5:5:7 and volume 1400 i n 3. Find the volume of the space inside the prism that is not contained in the cylinder. Find the surface area and volume of the following shapes. Leave your answers in terms of π. You may assume the bottom is open. A sphere has a radius of 5 cm. A right cylinder has the same radius and volume. Find the height and total surface area of the cylinder. Tennis balls with a 3 inch diameter are sold in cans of three. The can is a cylinder. Assume the balls touch the can on the sides, top and bottom. What is the volume of one tennis ball? What is the volume of the space not occupied by the tennis balls? Round your answer to the nearest hundredth. One hot day at a fair you buy yourself a snow cone. The height of the cone shaped container is 5 in and its radius is 2 in. The shaved ice is perfectly rounded on top forming a hemisphere. What is the volume of the ice in your frozen treat? If the ice melts at a rate of 2 i n 3 per minute, how long do you have to eat your treat before it all melts? Give your answer to the nearest minute. For exercise 12, answer the following: What is the surface area of a cylinder? Adjust your answer from part a for the case where r=h. What is the surface area of a sphere? What is the relationship between your answers to parts b and c? Can you explain this? Find the volume of the composite solids. Round your answer to the nearest hundredth. Review (Answers) Click HERE to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option. Image Attributions Back to Composite Solids \| Image \| Reference \| Attributions \| --- \| \| [Figure 16] \| License:CC BY-NC \| \| \| [Figure 17] [Figure 18] \| License:CC BY-NC \| \| \| [Figure 19] \| License:CC BY-NC \| \| \| [Figure 20] \| Credit:Tom Barrett Source: License:CC BY-NC \| \| \| [Figure 21] \| Credit:Steve Snodgrass Source: \| \| \| [Figure 22] \| License:CC BY-NC \| \| \| [Figure 23] \| Credit:USFWS Mountain Prarie Source: \| \| \| [Figure 24] \| Credit:CK-12 Foundation Source: License:CC BY-SA \| Ask me anything! Mute me CK-12 Foundation is a non-profit organization that provides free educational materials and resources. 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7617	https://www.statpearls.com/point-of-care/search	Physician Physician Board Reviews Physician Associate Board Reviews CME Lifetime CME Free CME MATE and DEA Compliance CPD Student USMLE® Step 1 USMLE® Step 2 USMLE® Step 3 COMLEX-USA® Level 1 COMLEX-USA® Level 2 COMLEX-USA® Level 3 96 Medical School Exams Student Resource Center NCLEX - RN NCLEX - LPN/LVN/PN 24 Nursing Exams Nurse Practitioner APRN/NP Board Reviews CNS Certification Reviews CE - Nurse Practitioner FREE CE Nurse RN Certification Reviews CE - Nurse FREE CE Pharmacist Pharmacy Board Exam Prep CE - Pharmacist Allied Allied Health Exam Prep Dentist Exams CE - Social Worker CE - Dentist Point of Care Free CME/CE hours Redeem #### Articles #### Favorite Articles #### Recently Updated Showing 9154 articles Click a letter to jump to that section. ! \| 1 \| 2 \| 3 \| 5 \| A \| B \| C \| D \| E \| F \| G \| H \| I \| J \| K \| L \| M \| N \| O \| P \| Q \| R \| S \| T \| U \| V \| W \| X \| Y \| Z \| n \| v \| \| \| \| ! \| \| 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Calculating FICK Cardiac Output and Input ##### Caloric Reflex Test ##### Caloric Testing ##### Calories ##### Cameron Lesions ##### Campylobacter Infection ##### Canagliflozin ##### Canalicular Laceration ##### Canaliculitis ##### Canavan Disease ##### Cancer Antigen 125 ##### Cancer Chemotherapy ##### Cancer of the Oral Mucosa ##### Cancer Screening ##### Cancer-Associated Retinopathy ##### Cancer-Associated Thrombosis ##### Candesartan ##### Candida auris ##### Candidemia ##### Candidiasis ##### Cangrelor ##### Cannabidiol (CBD) in Clinical Care ##### Cannabinoid Antiemetic Therapy ##### Cannabinoid Hyperemesis Syndrome ##### Cannabinoid Toxicity ##### Cannabinoids ##### Cannabis Use Disorder ##### Cannabis Versus Opioids for Pain ##### Capgras Syndrome ##### Capillary Refill Time ##### Caplan Syndrome ##### Capnography and Pulse Oximetry ##### Caprini Score ##### Capsaicin ##### Capsule Endoscopy ##### Captopril ##### Caput Succedaneum ##### Carbamate Toxicity ##### Carbamazepine ##### Carbamazepine Toxicity ##### Carbapenem-Resistant Enterobacterales ##### Carbidopa ##### Carbon Dioxide Angiography ##### Carbon Dioxide Detector ##### Carbon Dioxide Embolism ##### Carbon Dioxide Narcosis ##### Carbon Disulfide Toxicity ##### Carbon Monoxide Toxicity ##### Carbon Tetrachloride Toxicity ##### Carbonic Anhydrase Inhibitors ##### Carboxyhemoglobin Toxicity ##### Carbuncle ##### Carcinoembryonic Antigen ##### Carcinogenesis ##### Carcinoid Syndrome ##### Carcinomatous Meningitis ##### Cardiac Abscess ##### Cardiac Amyloidosis ##### Cardiac Arrest ##### Cardiac Biomarkers ##### Cardiac Calcifications ##### Cardiac Cancer ##### Cardiac Catheterization Risks and Complications ##### Cardiac Chromaffin Cell Pheochromocytoma ##### Cardiac Cirrhosis ##### Cardiac Computed Tomography ##### Cardiac Disease in Pregnancy ##### Cardiac Dominance ##### Cardiac Electrical and Structural Remodeling ##### Cardiac Exam ##### Cardiac Fibroma ##### Cardiac Glycoside and Digoxin Toxicity ##### Cardiac Imaging ##### Cardiac Manifestations of Coronavirus (COVID-19) ##### Cardiac Rehabilitation ##### Cardiac Resynchronization Therapy ##### Cardiac Rhabdomyoma ##### Cardiac Risk Assessment ##### Cardiac Sarcoidosis ##### Cardiac Stress Imaging ##### Cardiac Surgery ##### Cardiac Syncope ##### Cardiac Syndrome X ##### Cardiac Tamponade ##### Cardiac Trauma ##### Cardiac Ultrasound ##### Cardioactive Steroid Toxicity ##### Cardioembolic Stroke ##### Cardiogenic Pulmonary Edema ##### Cardiogenic Shock ##### Cardiomegaly ##### Cardiomyopathy Imaging ##### Cardioplegia ##### Cardiopulmonary Arrest in Adults ##### Cardiopulmonary Arrest in Children ##### Cardiopulmonary Bypass ##### Cardiopulmonary Exercise Testing ##### Cardiopulmonary Fitness ##### Cardiopulmonary Resuscitation ##### Cardiorenal Syndrome ##### Cardiovascular Disease ##### Cardiovascular Fitness To Dive ##### Care of a Central Line ##### Care of a Chest Tube ##### Carisoprodol ##### Carney Complex ##### Carnitine Deficiency ##### Caroli Disease ##### Carotenemia ##### Carotid Artery Dissection ##### Carotid Artery Fibromuscular Dysplasia ##### Carotid Artery Stenting ##### Carotid Artery Surgery ##### Carotid Bruit ##### Carotid Cavernous Fistula ##### Carotid Contusion ##### Carotid Endarterectomy ##### Carotid Sinus Hypersensitivity ##### Carotidynia ##### Carpal Ligament Instability ##### Carpal Tunnel Syndrome ##### CART Cell Therapy Toxicity ##### Cartilage Graft ##### Carvedilol ##### Case Control Studies ##### Case Management ##### Casirivimab ##### Caspofungin ##### Castleman Disease ##### Castor Oil ##### Cat Eye Syndrome (Schmid-Fraccaro Syndrome) ##### Cat Scratch Disease ##### Cataplexy ##### Cataract ##### Cataract Surgery ##### Catatonia ##### Catatonic Schizophrenia ##### Catheter Ablation ##### Catheter Management of Aortic Valve Disorders ##### Catheter Management of Atrial Septal Defect ##### Catheter Management of Coarctation ##### Catheter Management of Hypertrophic Cardiomyopathy ##### Catheter Management of Left Atrial Appendage Closure Devices ##### Catheter Management of Mitral Regurgitation ##### Catheter Management of Mitral Stenosis ##### Catheter Management of Patent Foramen Ovale ##### Catheter Management of Ventricular Septal Defect ##### Catheter Management Pulmonary Valvular Disorders ##### Catheter-Directed Thrombolysis of Pulmonary Embolism ##### Cauda Equina and Conus Medullaris Syndromes ##### Caudal Anesthesia ##### Cauliflower Ear ##### Caustic Ingestions ##### Cavernous Hepatic Hemangioma ##### Cavernous Sinus Aneurysm ##### Cavernous Sinus Syndromes ##### Cavernous Sinus Thrombosis ##### Cavum Septum Pellucidum ##### Cavum Veli Interpositi ##### CD4 Cell Count and HIV ##### CD4 Count ##### CEAP Classification of Venous Disorders ##### Cecal Volvulus ##### Cefepime ##### Cefotaxime ##### Cefuroxime ##### Celecoxib ##### Celiac Artery Compression Syndrome ##### Celiac Disease ##### Celiac Plexus Block ##### Cell Liquefactive Necrosis ##### Cellulitis ##### Cenegermin ##### Centipede Envenomation ##### Central and Peripheral Cyanosis ##### Central Aortopulmonary Shunt ##### Central Auditory Processing Disorder ##### Central Centrifugal Cicatricial Alopecia ##### Central Cord Syndrome ##### Central Line Management ##### Central Line–Associated Blood Stream Infections ##### Central Nervous System Lymphoma ##### Central Pain Syndrome ##### Central Pontine Myelinolysis ##### Central Post-Stroke Pain Syndrome ##### Central Retinal Artery Occlusion ##### Central Retinal Vein Occlusion ##### Central Serous Chorioretinopathy ##### Central Sleep Apnea ##### Central Toxic Keratopathy ##### Central Venous Catheter Insertion ##### Central Vertigo ##### Cephalexin ##### Cephalohematoma ##### Cephalosporins ##### Cerebellar Dysfunction ##### Cerebellar Hematoma ##### Cerebellar Infarct ##### Cerebellar Neurological Signs ##### Cerebellopontine Angle Cancer ##### Cerebral Amyloid Angiopathy ##### Cerebral Aneurysm ##### Cerebral Autosomal Dominant Arteriopathy ##### Cerebral Cavernous Malformations ##### Cerebral Contusion ##### Cerebral Edema ##### Cerebral Palsy ##### Cerebral Perfusion Pressure ##### Cerebral Salt Wasting Syndrome ##### Cerebral Venous Sinus Thrombosis ##### Cerebrospinal Fluid Leak ##### Cerebrotendinous Xanthomatosis ##### Cerebrovascular Disease ##### Certification in Medical Simulation ##### Cerumen Impaction Removal ##### Cervical Cancer ##### Cervical Cerclage ##### Cervical Degenerative Disc Disease ##### Cervical Disc Herniation ##### Cervical Dislocation ##### Cervical Dysplasia ##### Cervical Ectropion ##### Cervical Epidural Injection ##### Cervical Injury ##### Cervical Insufficiency ##### Cervical Intraepithelial Neoplasia ##### Cervical Myelopathy ##### Cervical Myofascial Pain ##### Cervical Plexus Block ##### Cervical Polyps ##### Cervical Radiculopathy ##### Cervical Spine Fractures Overview ##### Cervical Spondylosis ##### Cervical Sprain ##### Cervical Squamous Cell Carcinoma ##### Cervical Subluxation ##### Cervical Traction ##### Cervicitis ##### Cervicogenic Headache ##### Cesarean Delivery ##### Cetirizine ##### Cetuximab ##### CHA2DS2-VASc Score for Atrial Fibrillation ##### Chaddock Reflex ##### Chagas Disease ##### Chain of Custody ##### Chalazion ##### Chance Fractures ##### Chancroid ##### Chandelier Sign ##### Change in Mental Status ##### Change Management In Health Care ##### Char Syndrome ##### Charcot Neuropathic Osteoarthropathy ##### Charcot-Bouchard Aneurysm ##### Charcot-Marie-Tooth Disease ##### CHARGE Syndrome ##### Charles Bonnet Syndrome ##### Chediak-Higashi Syndrome ##### Cheilitis ##### Cheilitis Granulomatosa ##### Cheiralgia Paresthetica ##### Chemical Burns ##### Chemical Decontamination ##### Chemical Peels for Skin Resurfacing ##### Chemotherapy Acral Erythema ##### Cherry Hemangioma ##### Cherry Red Spot ##### Chest and Mediastinal Imaging ##### Chest Pain ##### Chest Trauma ##### Chest Tube ##### Chest Tube Insertion in the Neonate ##### Chest Wall Deformities ##### Chest Wall Tumors ##### Cheyne Stokes Respirations ##### Chiari Malformation Type 1 ##### Chiari Malformation Type 2 ##### Chigger Bites and Trombiculiasis ##### Chikungunya Fever ##### Chilaiditi Syndrome ##### Child Abuse and Neglect ##### Child Development ##### Child Intussusception ##### Child Physical Abuse and Neglect ##### Child Sexual Abuse ##### CHILD Syndrome ##### Childhood Myopia and Ocular Development ##### Children's Health Insurance Program ##### Chimeric Antigen Receptor T-Cell Therapy ##### Chlamydia ##### Chlamydia Pneumonia ##### Chloracne ##### Chloramphenicol ##### Chlordiazepoxide ##### Chlorine Gas Toxicity ##### Chloroform Toxicity ##### Chloroprocaine ##### Chloroquine ##### Chloroquine and Hydroxychloroquine Toxicity ##### Chlorpromazine ##### Chlorthalidone ##### Choanal Atresia ##### Choking ##### Cholangiocarcinoma ##### Cholangitis ##### Cholecalciferol ##### Cholecystocutaneous Fistula ##### Cholecystokinin Test ##### Choledochal Cyst ##### Choledocholithiasis ##### Cholestatic Jaundice ##### Cholesterol Emboli ##### Cholesterol Levels ##### Cholesterol Screening ##### Cholestyramine Resin ##### Cholinergic Crisis ##### Cholinergic Medications ##### Cholinesterase Inhibitors ##### Chondroblastoma ##### Chondrodermatitis Nodularis Helicis ##### Chondroid Syringoma ##### Chondromalacia Patella ##### Chondrosarcoma ##### Chordoma ##### Chorea ##### Chorioamnionitis ##### Chorionic Villus Sampling ##### Chorioretinitis ##### Choroid Plexus Papilloma ##### Choroidal Folds ##### Chromatic Aberration ##### Chromatography ##### Chromhidrosis ##### Chromium Deficiency ##### Chromium Toxicity ##### Chromoblastomycosis ##### Chromosome Instability Syndromes ##### Chronic Anemia ##### Chronic Aspiration ##### Chronic Bronchitis ##### Chronic Cholecystitis ##### Chronic Closed Angle Glaucoma ##### Chronic Coronary Occlusion ##### Chronic Cough ##### Chronic Diarrhea ##### Chronic Fatigue Syndrome ##### Chronic Granulomatous Disease ##### Chronic Headaches ##### Chronic Inflammation ##### Chronic Inflammatory Demyelinating Polyradiculoneuropathy ##### Chronic Insomnia ##### Chronic Interstitial Nephritis in Agricultural Communities (CINAC) ##### Chronic Iron Deficiency ##### Chronic Ischemic Heart Disease Selection of Treatment Modality ##### Chronic Kidney Disease ##### Chronic Kidney Disease-Mineral Bone Disorder (CKD-MBD) ##### Chronic Kidney Transplant Rejection ##### Chronic Liver Disease ##### Chronic Lymphocytic Leukemia ##### Chronic Lymphocytic Leukemia With Variant Genetics ##### Chronic Mesenteric Ischemia ##### Chronic Myelogenous Leukemia ##### Chronic Obstructive Pulmonary Disease ##### Chronic Obstructive Pulmonary Disease Compensatory Measures ##### Chronic Pain ##### Chronic Pancreatitis ##### Chronic Paroxysmal Hemicrania ##### Chronic Pelvic Pain ##### Chronic Prostatitis and Chronic Pelvic Pain Syndrome in Men ##### Chronic Sinusitis ##### Chronic Suppurative Otitis ##### Chronic Testicular Pain and Orchalgia ##### Chronic Thromboembolic Pulmonary Hypertension ##### Chronic Total Occlusion of the Coronary Artery ##### Chronic Traumatic Encephalopathy ##### Chronic Urticaria ##### Chvostek Sign ##### Chylothorax ##### Cicatricial Pemphigoid ##### Ciguatera Toxicity ##### Cilia Incarnata ##### Ciliary Dysfunction ##### Ciliospinal Reflex ##### Cilostazol ##### Cimetidine ##### Cinacalcet ##### Cinchonism ##### Ciprofloxacin ##### Circumcision ##### Circumstantiality ##### Cirrhotic Cardiomyopathy ##### Cisatracurium ##### Cisplatin ##### Citalopram ##### Cladribine ##### Clark's Rule ##### Classical Conditioning ##### Claustrophobia ##### Clavicle Fractures ##### Clavulanic Acid ##### Clear Cell Acanthoma ##### Clear Cell Carcinoma of the Cervix ##### Cleft Hand ##### Cleft Lip ##### Cleft Lip Repair ##### Cleft Palate ##### Cleft Palate Repair ##### Clindamycin ##### Clinical Drug Testing ##### Clinical Frailty Scale ##### Clinical Laboratory ##### Clinical Practice Issues in American Indians and Alaska Natives ##### Clinical Resource Management Reimbursement Models and Accountable Care ##### Clinical Utility of Fructosamine and Glycated Albumin ##### Clinodactyly ##### Cloacal Malformations ##### Clobazam ##### Clomiphene ##### Clomipramine ##### Clonazepam ##### Clonidine ##### Clonorchis Sinensis ##### Clonus ##### Clopidogrel ##### Closed Loop Communication Training in Medical Simulation ##### Clostridioides difficile infection ##### Clostridium perfringens Infection ##### Clotrimazole ##### Clozapine ##### Clubfoot ##### Cluneal Neuralgia ##### Cluster Headache ##### Cnidaria Toxicity ##### CNS Tuberculosis ##### Coarctation of the Aorta ##### Cobalt Toxicity ##### Cobra Laws And EMTALA ##### Cocaine ##### Cocaine Toxicity ##### Coccidioidomycosis ##### Coccyx Pain ##### Cochlear Implants ##### Cockayne Syndrome ##### Codeine ##### Codman Triangle ##### Coenzyme Q10 ##### Cogan Syndrome ##### Cognitive Assessment ##### Cognitive Behavior Therapy ##### Cognitive Decline and Driving Evaluation in the Older Population ##### Cognitive Deficits ##### Cognitive Development ##### Colchicine ##### Cold Knife Conization of the Cervix ##### Colesevelam ##### Colestipol ##### Colitis ##### Collaboration ##### Collaborative Care Model ##### Collagen Cross Linking for Keratoconus ##### Collagen Vascular Disease Associated With Interstitial Lung ##### Collagenous and Lymphocytic Colitis ##### Colles Fracture ##### Colloid Brain Cyst ##### Colon Cancer ##### Colon Cancer Screening ##### Colon Diverticulitis ##### Colon Polyps ##### Colonoscopy ##### Color Vision ##### Colostomy Care ##### Colovesical Fistula ##### Colposcopy ##### Coma ##### Common Measures of Disability ##### Common Pitfalls In The Research Process ##### Common Variable Immunodeficiency ##### Commotio Cordis ##### Communication Training Tools in Medical Simulation ##### Community-Acquired Pneumonia ##### Community-Based Participatory Research (CBPR) ##### Compare and Contrast the Glucagon-Like Peptide-1 Receptor Agonists (GLP1RAs) ##### Comparing the Means of Independent Groups: ANOVA, ANCOVA, MANOVA, and MANCOVA ##### Competency and Capacity ##### Complement Deficiency ##### Complex Coronary Artery Lesions ##### Complex Ear Lacerations ##### Complex Regional Pain Syndrome ##### Complex Wound Management ##### Complicated Cataract ##### Complicated Urinary Tract Infections ##### Comprehensive Assessment and Diagnosis of Metabolic and Biomechanical Complications in Obesity ##### Comprehensive Behavioral Modification and Counseling Strategies for Obesity Management ##### Comprehensive Review of Bioterrorism ##### Comprehensive Shoulder Evaluation Strategies ##### Compressive Optic Neuropathy ##### Compressive Radial Mononeuropathy ##### Computer Provider Order Entry ##### Concave And Convex Lenses ##### Conceptual Frameworks in Medical Simulation ##### Concussion ##### Conduct Disorder ##### Conduction Aphasia ##### Conductive Hearing Loss ##### Condyloma Acuminata ##### Cone Snail Toxicity ##### Confabulation ##### Conflict Management ##### Confluent and Reticulated Papillomatosis ##### Congenital Adrenal Hyperplasia ##### Congenital and Infantile Hepatic Hemangioma ##### Congenital and Maternal Syphilis ##### Congenital Cytomegalovirus Infection ##### Congenital Diaphragmatic Hernia ##### Congenital Femoral Deficiency ##### Congenital Hereditary Endothelial Dystrophy ##### Congenital Herpes Simplex ##### Congenital Hypertrophy of Retinal Pigment Epithelium ##### Congenital Hypothyroidism ##### Congenital Lacrimal Fistula ##### Congenital Lobar Emphysema ##### Congenital Melanocytic Nevi ##### Congenital Muscular Dystrophy ##### Congenital Myotonic Dystrophy ##### Congenital Nephrotic Syndrome ##### Congenital Nevus ##### Congenital Ptosis ##### Congenital Pulmonary Airway Malformation ##### Congenital Rubella ##### Congenital Smooth Muscle Hamartoma ##### Congenital Tibial Pseudarthrosis ##### Congenital Torticollis ##### Congenital Toxoplasmosis ##### Congenital Unilateral Lower Lip Palsy ##### Congenital Varicella Syndrome ##### Congenital Vertical Talus ##### Congestive Heart Failure and Pulmonary Edema ##### Conjoined Twins ##### Conjugated Hyperbilirubinemia ##### Conjunctival Concretions ##### Conjunctivitis ##### Conjunctivochalasis ##### Conn Syndrome ##### Conscious Sedation in Dentistry ##### Constipation ##### Constitutional Growth Delay ##### Constrictive Pericarditis ##### Constrictive-Effusive Pericarditis ##### Contact Dermatitis ##### Contact Lenses ##### Contact Lenses for Presbyopia ##### Contact Lens–Related Complications ##### Contact Urticaria ##### Continuous Positive Airway Pressure ##### Continuous Quality Improvement ##### Continuous Renal Replacement Therapy ##### Contraception ##### Contracted Socket ##### Contrast Agent Toxicity ##### Contrast Sensitivity ##### Contrast-Induced Nephropathy ##### Contrecoup Brain Injury ##### Controlled Substance Act ##### Controlled Substance Schedules ##### Conus and Cauda Equina Tumors ##### Conventional Hearing Aid Indications and Selection ##### Convergence Insufficiency ##### Conversion Weights ##### Cooling Techniques for Hyperthermia ##### Coombs Test ##### COPD and Sleep Apnea Overlap ##### Coping Mechanisms ##### Copper Toxicity ##### Coprolalia ##### Cor Pulmonale ##### Cor Triatriatum ##### Coral Snake Toxicity ##### Coral Toxicity ##### Cord Blood Gas ##### Cordotomy ##### Cornea Transplantation ##### Corneal Abrasion ##### Corneal Dystrophy ##### Corneal Endothelial Transplantation ##### Corneal Foreign Body ##### Corneal Foreign Body Removal ##### Corneal Graft Rejection ##### Corneal Imaging ##### Corneal Injury ##### Corneal Laceration Repair ##### Corneal Mucous Plaque ##### Corneal Neurotization ##### Corneal Reflex ##### Corneal Topography ##### Corneal Ulcer ##### Cornelia de Lange Syndrome ##### Corns ##### Coronary Arterial Pressure Evaluation ##### Coronary Arteriovenous Fistula ##### Coronary Artery Anomalies ##### Coronary Artery Bypass Graft ##### Coronary Artery Bypass Graft Redo ##### Coronary Artery Bypass Grafting Using the Gastroepiploic Artery ##### Coronary Artery Calcification ##### Coronary Artery Disease ##### Coronary Artery Disease Prevention ##### Coronary Artery Fistula ##### Coronary Artery Perforation ##### Coronary Artery Surgery ##### Coronary Artery Thrombus ##### Coronary Artery Vasospasm ##### Coronary Cameral Fistula ##### Coronary CT Angiography ##### Coronary Ectasia ##### Coronary Flow Effects of Arterial Spasm or Microembolization ##### Coronary Perfusion Pressure ##### Coronary Sinus Thrombosis ##### Coronoid Fractures ##### Corpus Callosum Agenesis ##### Correlation (Coefficient, Partial, and Spearman Rank) and Regression Analysis ##### Cortical Blindness ##### Corticobasal Degeneration ##### Corticospinal Tract Lesion ##### Corticosteroid Adverse Effects ##### Corticosteroid Induced Myopathy ##### Corticosteroids ##### Corynebacterium Diphtheriae ##### Cosmeceuticals ##### Costochondritis ##### Cough ##### Counseling Patients on Bariatric Surgery for Obesity ##### Counseling Patients With Obesity ##### COVID (SARS-CoV-2) Vaccine ##### Cow Milk Allergy ##### Cowden Disease ##### COX Inhibitors ##### Cracked Tooth Syndrome ##### Cradle Cap ##### Cranial Nerve III Palsy ##### Cranial Nerve Testing ##### Craniofacial Distraction Osteogenesis ##### Craniopharyngioma ##### Craniosynostosis ##### Craniotomy ##### C-Reactive Protein: Clinical Relevance and Interpretation ##### Creatine Kinase MB: Diagnostic Utility and Limitations ##### Creatine Phosphokinase ##### Creatinine Clearance ##### Credentialing ##### Credentialing and Privileging Provider Profiling ##### Cremasteric Reflex ##### Creutzfeldt-Jakob Disease ##### Cri Du Chat Syndrome ##### Cribriform Plate Fractures ##### Cricothyroidotomy ##### Crigler Technique for Congenital Nasolacrimal Duct Obstruction ##### Crigler-Najjar Syndrome ##### Crisis Intervention ##### Crisis Resource Management Training in Medical Simulation ##### Crocodile Tears Syndrome ##### Crohn Disease ##### Crohn Disease Stricturoplasty ##### Cromolyn Sodium ##### Crotalidae Envenomation ##### Croup ##### Crouzon Syndrome ##### Crutches ##### Cryoanalgesia ##### Cryoballoon Pulmonary Vein Catheter Ablation of Atrial Fibrillation ##### Cryoglobulinemia ##### Cryoglobulinemic Glomerulonephritis ##### Cryotherapy in Dermatology ##### Cryptococcal Meningitis ##### Cryptococcus ##### Cryptogenic Cirrhosis ##### Cryptogenic Organizing Pneumonia ##### Cryptorchidism ##### Cryptosporidiosis ##### Crystalloid Fluids ##### CT Instrumentation and Physics ##### CT Patient Safety And Care ##### CT Scan ##### CT-scan Image Production Procedures ##### Cubital Tunnel Syndrome ##### Cubitus Varus ##### Cuboid Stress Fractures ##### Culdocentesis ##### Cultural Assessment and Treatment of Psychiatric Patients ##### Cultural Competence in Caring for American Indians and Alaska Natives ##### Cultural Competence in the Care of LGBTQ Patients ##### Cultural Competence in the Care of Muslim Patients and Their Families ##### Cultural Religious Competence in Clinical Practice ##### Cupping Therapy ##### Curling Ulcer ##### Curriculum Design in Medical Simulation ##### Curriculum Development in Medical Simulation ##### Cushing Disease ##### Cushing Reflex ##### Cutaneous Adverse Drug Reaction ##### Cutaneous Angiofibroma ##### Cutaneous Crohn Disease ##### Cutaneous Cryptococcus ##### Cutaneous Endometriosis ##### Cutaneous Horn ##### Cutaneous Larva Migrans ##### Cutaneous Leiomyomas ##### Cutaneous Malignant Melanoma: Guideline-Based Management and Interprofessional Collaboration ##### Cutaneous Melanoacanthoma ##### Cutaneous Squamous Cell Carcinoma ##### Cutaneous Tuberculosis ##### Cutaneous Vascular Malignancies, Angiosarcoma and Kaposi Sarcoma ##### Cutis Laxa ##### Cutis Marmorata Telangiectatica Congenita ##### Cutis Verticis Gyrata ##### Cyanide Toxicity ##### Cyanocobalamin ##### Cyanotic Heart Disease ##### Cyclic Neutropenia ##### Cyclic Vomiting Syndrome ##### Cyclobenzaprine ##### Cyclodiode Laser Glaucoma Therapy ##### Cyclophosphamide ##### Cycloplegic and Noncycloplegic Refraction ##### Cyclosporine ##### Cyclothymic Disorder ##### Cylindroma ##### Cystic Fibrosis ##### Cystic Fibrosis and Liver Disease ##### Cystic Fibrosis–Related Diabetes ##### Cystic Teratoma ##### Cystinosis ##### Cystinuria ##### Cystitis ##### Cystitis Cystica and Cystitis Glandularis ##### Cystoscopy ##### Cytarabine ##### Cytomegalovirus Corneal Endotheliitis ##### Cytomegalovirus Infections ##### Cytoreduction (CRS) and Hyperthermic Intraperitoneal Chemotherapy (HIPEC) \| \| \| D \| : ##### Dacryoadenitis ##### Dacryocystitis ##### Dacryocystorhinostomy ##### Dacryostenosis ##### Dakin Solution ##### Dalteparin ##### Danazol ##### Dandy-Walker Malformation ##### Danon Disease ##### Dantrolene ##### Dapsone ##### Daptomycin ##### Darier Disease ##### DASH Diet To Stop Hypertension ##### Daunorubicin ##### Dawn Phenomenon ##### DaxibotulinumtoxinA-Ianm ##### D-Dimer Test ##### De Quervain Tenosynovitis ##### De Quervain Thyroiditis ##### Dead Bag Syndrome ##### Deadly Single Dose Agents ##### Death Certification ##### Debriefing Techniques Utilized in Medical Simulation ##### Debriefing the Interprofessional Team in Medical Simulation ##### Debriefing Theories and Philosophies in Medical Simulation ##### Decerebrate and Decorticate Posturing ##### Decompression Sickness ##### Decorticate Posturing ##### Deep Brain Stimulation ##### Deep Neck Infections ##### Deep Peroneal Nerve Block ##### Deep Plane Facelift ##### Deep Tendon Reflexes ##### Deep Venous Thrombosis ##### Deep Venous Thrombosis Prophylaxis ##### Deep Venous Thrombosis Risk Factors ##### Deep Venous Thrombosis Ultrasound Evaluation ##### Defamation in Healthcare ##### Defense Mechanisms ##### Deferoxamine ##### Defibrillation ##### Degenerative Myopia ##### Degloving Injuries ##### Dejerine-Roussy Syndrome ##### Delayed Puberty ##### Deliberate Practice in Medical Simulation ##### Delirium ##### Delivery, Face and Brow Presentation ##### Delusional Disorder ##### Delusional Misidentification Syndrome ##### Delusions ##### Delusions of Parasitosis ##### Dengue Fever ##### Denis Classification ##### Denosumab ##### Dental Abscess ##### Dental Alginate Impressions ##### Dental Caries ##### Dental Caries Classification Systems ##### Dental Caries Diagnostic Testing ##### Dental Cone Beam Computed Tomography ##### Dental Emergencies ##### Dental Fluorosis ##### Dental Implants ##### Dental Impression Materials ##### Dental Infection Control ##### Dental Materials: Biodentine, a Calcium Silicate Bioactive ##### Dental Mini-Implants ##### Dentatorubral Pallidoluysian Atrophy ##### Deontology ##### Dependent Personality Disorder ##### Depolarizing Neuromuscular Blocking Drugs ##### Depression ##### Depression in Children ##### Depressive Cognitive Disorders ##### Dermal Melanocytosis ##### Dermatitis Herpetiformis ##### Dermatofibroma ##### Dermatofibrosarcoma Protuberans ##### Dermatographism ##### Dermatomyositis ##### Dermatopathology Epidermis Histology ##### Dermatopathology Evaluation of Cysts ##### Dermatopathology Evaluation of Metabolic and Storage Diseases ##### Dermatopathology Evaluation of Panniculitis ##### Dermatopathology Evaluation of Tumors ##### Dermatopathology Histology Artifacts ##### Dermatopathology, Cutaneous Lymphomas ##### Dermatoscopic Characteristics of Melanoma Versus Benign Lesions and Nonmelanoma Cancers ##### Dermatoses of Pregnancy ##### Dermatosis Papulosa Nigra ##### Dermoid Cyst ##### Dermoscopy Overview and Extradiagnostic Applications ##### Desflurane ##### Designing a Simulation Scenario ##### Desipramine ##### Desmoid Tumor ##### Desmoplastic Trichoepithelioma ##### Desmopressin ##### Desquamative Interstitial Pneumonia ##### Desvenlafaxine ##### Determinants of Health ##### Determining the Need for Blood Transfusion ##### Deucravacitinib ##### Development Milestones ##### Developmental Coordination Disorder (Dyspraxia) ##### Developmental Delay ##### Developmental Disturbances of the Teeth, Anomalies of Number ##### Developmental Disturbances of the Teeth, Anomalies of Shape and Size ##### Developmental Disturbances of the Teeth, Anomalies of Structure ##### Developmental Dysplasia of the Hip ##### Developmental Stages of Social Emotional Development in Children ##### Dexamethasone ##### Dexamethasone Suppression Test ##### Dexmedetomidine ##### Dexrazoxane ##### Dextran ##### Dextroamphetamine-Amphetamine ##### Dextrocardia ##### Dextromethorphan ##### Dextromethorphan Guaifenesin ##### Dextromethorphan Toxicity ##### Diabetes ##### Diabetes and Exercise ##### Diabetes Intraoperative Management ##### Diabetes Mellitus Screening ##### Diabetic Amyotrophy ##### Diabetic Embryopathy ##### Diabetic Foot Care ##### Diabetic Foot Infections ##### Diabetic Foot Ulcer ##### Diabetic Gastroparesis ##### Diabetic Macular Edema ##### Diabetic Nephropathy ##### Diabetic Perioperative Management ##### Diabetic Peripheral Neuropathy ##### Diabetic Retinopathy ##### Diagnostic Testing Accuracy: Sensitivity, Specificity, Predictive Values and Likelihood Ratios ##### Diagnostic Ultrasound Use in Undifferentiated Hypotension ##### Dialysis Catheter ##### Dialysis Disequilibrium Syndrome ##### Dialysis Fistula ##### Diamond-Blackfan Anemia ##### Diaper Dermatitis ##### Diaphragm Disorders ##### Diaphragm Eventration ##### Diaphragm Rupture ##### Diaphragmatic Hernia ##### Diaphragmatic Pacing ##### Diaphyseal Femur Fracture ##### Diarrhea ##### Diastasis Recti Rehabilitation ##### Diazepam ##### Diborane Toxicity ##### Diclofenac ##### Diet and Nutrition to Prevent Dental Problems ##### Dietary Approaches to Obesity Treatment ##### Dietary Calcium and Supplementation ##### Dietary Iron ##### Dieulafoys Lesion Causing Gastrointestinal Bleeding ##### Difelikefalin ##### Differentiating Delirium Versus Dementia in Older Adults ##### Difficult Airway ##### Difficult Foley Catheterization ##### Diffuse Axonal Injury ##### Diffuse Esophageal Spasm ##### Diffuse Idiopathic Skeletal Hyperostosis ##### Diffuse Intrinsic Pontine Glioma ##### Diffuse Large B-Cell Lymphoma ##### Diffuse Toxic Goiter ##### Diffusing Capacity of the Lungs for Carbon Monoxide ##### Diffusion Tensor Imaging ##### DiGeorge Syndrome ##### Digit Amputation ##### Digit Replantation ##### Digital Health ##### Digital Mucous Cyst ##### Digital Nerve Block ##### Digoxin ##### Digoxin Immune Fab ##### Dilated Cardiomyopathy ##### Dilated Pore of Winer ##### Dilation and Curettage ##### Diltiazem ##### Dimercaprol ##### Dipeptidyl Peptidase IV (DPP IV) Inhibitors ##### Diphenhydramine ##### Diphenhydramine Toxicity ##### Diphenoxylate and Atropine ##### Diphtheria ##### Diphtheria Tetanus Pertussis (DTaP) Vaccine ##### Diphyllobothriasis (Fish Tapeworm Infection) ##### Diplopia ##### Dipyridamole ##### Dipyridamole Nuclear Stress Test ##### Direct Brow Lift ##### Direct Laryngoscopy ##### Disability Determination And Impairment Rating ##### Disability Evaluation ##### Disaster Planning ##### Discharge Planning ##### Discoid Lupus Erythematosus ##### Discoid Meniscus ##### Disease-Modifying Antirheumatic Drugs (DMARD) ##### Diseases of the Eyelashes ##### Disinfectants ##### Disk Battery Ingestion ##### Disk Herniation ##### Diskectomy ##### Diskitis ##### Disopyramide ##### Disparity in Early Detection of Breast Cancer ##### Disseminated Intravascular Coagulation ##### Disseminated Superficial Actinic Porokeratosis ##### Dissociated Vertical Deviation ##### Dissociative Identity Disorder ##### Distal Clavicular Osteolysis ##### Distal Femur Fractures ##### Distal Humerus Fractures ##### Distal Pancreatectomy ##### Distal Radius Fractures ##### Distal Ulnar Fractures ##### Distributive Shock ##### Disulfiram ##### Diversity and Discrimination in Health Care ##### Diversity Goals for Evaluation and Treatment of American Indians and Alaska Natives ##### Diverticulosis ##### Diving at Altitude ##### Diving Buoyancy ##### Diving Casualties ##### Diving Gas Embolism ##### Diving in Water Recompression ##### Diving Mask Squeeze ##### Diving Rebreathers ##### Dix-Hallpike Maneuver ##### Do Not Resuscitate ##### Dobutamine ##### Docetaxel ##### Docusate ##### Dofetilide ##### Doll's Eyes ##### Domestic Violence ##### Donepezil ##### Dopamine ##### Dopamine Agonists ##### Doppler Extra-Cranial Carotid Assessment, Protocols, and Interpretation ##### Doppler Liver Assessment, Protocols, and Interpretation of Results ##### Doppler Peripheral Venous Duplex Assessment, Protocols, and Interpretation ##### Doppler Renal Assessment, Protocols, and Interpretation ##### Doppler Trans-Cranial Assessment, Protocols, and Interpretation ##### Dornase Alfa ##### Dorsal Penile Nerve Block ##### Dose Calculation ##### Dose Calculation Desired Over Have Formula Method ##### Dose Calculation Dimensional Analysis Factor-Label Method ##### Dose Calculation Ratio and Proportion Method ##### Double Aortic Arch ##### Double Defibrillation ##### Double J Placement Methods Comparative Analysis ##### Double Orifice Mitral Valve ##### Double-Blind Study ##### Double-Chambered Right Ventricle ##### Double-Lumen Endobronchial Tubes ##### Dowling-Degos Disease ##### Down Syndrome ##### Doxazosin ##### Doxepin ##### Doxorubicin ##### Doxycycline Hyclate ##### Doxylamine ##### Dracunculiasis ##### Dressing Disability ##### Dressler Syndrome ##### Dronabinol ##### Drowning: Clinical Management ##### Drug Absorption ##### Drug Addiction ##### Drug Bioavailability ##### Drug Clearance ##### Drug Distribution ##### Drug Elimination ##### Drug Eluting Stent Compounds ##### Drug Enforcement Administration Drug Scheduling ##### Drug Induced Pemphigus ##### Drug Interactions in Palliative Care ##### Drug Labeling ##### Drug Metabolism ##### Drug Testing ##### Drug Trials ##### Drug Utilization Review ##### Drug-Induced Esophagitis ##### Drug-Induced Gingival Overgrowth ##### Drug-Induced Hepatotoxicity ##### Drug-Induced Lupus Erythematosus ##### Drug-Induced Pigmentation ##### Drug-Induced Valvular Heart Disease ##### Drusen Bodies ##### Dry Eye Syndrome ##### Dual-Energy X-Ray Absorptiometry ##### Duane Retraction Syndrome ##### Dubin-Johnson Syndrome ##### Duchenne Muscular Dystrophy ##### Duffy Blood Group System ##### Duloxetine ##### Dumping Syndrome ##### Duodenal Atresia and Stenosis ##### Duodenal Biopsy ##### Duodenal Perforation ##### Duodenal Trauma ##### Duodenal Ulcer ##### Dupilumab ##### Duplex Ultrasound ##### Duplication Cyst ##### Dupuytren Contracture ##### Dural Arteriovenous Fistula ##### Duret Hemorrhages ##### Dust Mite Allergy ##### Dutasteride ##### Duty to Warn ##### Dwarfism ##### Dysarthria ##### Dysbaric Osteonecrosis ##### Dysbarism ##### Dysbetalipoproteinemia ##### Dysdiadochokinesia ##### Dysgraphia ##### Dyskeratosis Congenita ##### Dyslexia ##### Dyslipidemia ##### Dysmenorrhea ##### Dyspareunia ##### Dysphagia ##### Dysphonia ##### Dysplastic Nevi ##### Dyspnea ##### Dyspnea in Palliative Care ##### Dysthyroid Optic Neuropathy ##### Dystonia ##### Dystonic Reactions ##### Dystrophinopathies ##### Dysuria \| \| \| E \| : ##### Eagle Syndrome ##### Eales Disease ##### Ear Barotrauma ##### Ear Examination ##### Ear Foreign Body Removal ##### Ear Irrigation ##### Ear Microtia ##### Early Childhood Caries ##### Early Decelerations ##### Early Pregnancy Diagnosis ##### Early Pregnancy Loss (Spontaneous Abortion) ##### Eastern Cooperative Oncology Group Performance Status ##### Eastern Equine Encephalitis ##### Eating Disorders ##### Ebola Virus Disease ##### Ebstein Anomaly and Malformation ##### EBV Positive Mucocutaneous Ulcer ##### EBV-Associated Myopericarditis ##### Eccrine Carcinoma ##### ECG T Wave ##### Echinococcus Granulosus ##### Echocardiogram ##### Echocardiography Imaging Techniques ##### Echolalia ##### Eclampsia ##### Ecthyma Gangrenosum ##### Ectodermal Dysplasia ##### Ectopia Lentis ##### Ectopic Pregnancy ##### Ectopic Pregnancy, Ultrasound ##### Ectopic Thyroid ##### Ectropion ##### Eczema ##### Eczema Herpeticum ##### ED50 ##### Edoxaban ##### Edwards Syndrome ##### EEG Abnormal Waveforms ##### EEG Basal Cortical Rhythms ##### EEG Benign Variants ##### EEG Neonatal Visual Analysis ##### EEG Normal Sleep ##### EEG Normal Waveforms ##### EEG Triphasic Waves ##### Efavirenz ##### Egg Allergy ##### Egophony ##### Ehlers-Danlos Syndrome ##### Ehrlichiosis ##### Eisenmenger Syndrome ##### EKG Rhythm ##### Elbow Arthrocentesis ##### Elbow Dislocation ##### Elbow Fractures Overview ##### Elbow Trauma ##### Elder Abuse ##### Electric Potential and Capacitance ##### Electrical Alternans ##### Electrical Injuries ##### Electrical Right and Left Axis Deviation ##### Electrical Status Epilepticus in Sleep ##### Electrocardiogram ##### Electrochemiluminescence Method ##### Electrocochleography ##### Electroconvulsive Therapy ##### Electrodiagnostic Evaluation of Acute Inflammatory Demyelinating Polyneuropathy ##### Electrodiagnostic Evaluation of Brachial Plexopathies ##### Electrodiagnostic Evaluation of Carpal Tunnel Syndrome ##### Electrodiagnostic Evaluation of Cervical Radiculopathy ##### Electrodiagnostic Evaluation of Critical Illness Myopathy ##### Electrodiagnostic Evaluation of Critical Illness Neuropathy ##### Electrodiagnostic Evaluation of Lumbosacral Radiculopathy ##### Electrodiagnostic Evaluation of Motor Neuron Disease ##### Electrodiagnostic Evaluation of Myopathy ##### Electrodiagnostic Evaluation of Neuromuscular Junction Disorder ##### Electrodiagnostic Evaluation of Peripheral Neuropathy ##### Electrodiagnostic Evaluation of Peroneal Neuropathy ##### Electrodiagnostic Evaluation of Spinal Stenosis ##### Electrodiagnostic Evaluation of Tarsal Tunnel Neuropathy ##### Electrodiagnostic Evaluation of Ulnar Neuropathy ##### Electroencephalogram ##### Electrolytes ##### Electronic Vaping Delivery of Cannabis and Nicotine ##### Electrophoresis ##### Electrophysiologic Study Indications and Evaluation ##### Electrophysiologic Study Interpretation ##### Electrophysiologic Testing ##### Electrophysiology Lab Maneuvers to Differentiate Supraventricular Tachycardia ##### Electrophysiology Study and Ablation of Atrial Flutter ##### Electrophysiology Study and Ablation of Atrial Tachycardia ##### Electrophysiology Study and Ablation of Atrioventricular Nodal Reentrant Tachycardia ##### Electrophysiology Study and Ablation of Ventricular Tachycardia ##### Electrosurgery ##### Eletriptan ##### Elevated Hemidiaphragm ##### Elimination Diets ##### Elimination Half-Life of Drugs ##### Emancipated Minor ##### Embolic Stroke ##### Embryology, Amniotic Fluid ##### Embryology, Anencephaly ##### Embryology, Aortic Arch ##### Embryology, Atrioventricular Septum ##### Embryology, Bone Ossification ##### Embryology, Bowel ##### Embryology, Branchial Arches ##### Embryology, Central Nervous System ##### Embryology, Central Nervous System, Malformations ##### Embryology, Craniofacial Growth, And Development ##### Embryology, Ductus Venosus ##### Embryology, Ear ##### Embryology, Ear Congenital Malformations ##### Embryology, Ectoderm ##### Embryology, Epidermis ##### Embryology, Esophagus ##### Embryology, Eye ##### Embryology, Eye Malformations ##### Embryology, Face ##### Embryology, Fertilization ##### Embryology, Fetal Circulation ##### Embryology, Gastrointestinal ##### Embryology, Gastrulation ##### Embryology, Genitourinary ##### Embryology, Great Vessel ##### Embryology, Hair ##### Embryology, Hand ##### Embryology, Heart ##### Embryology, Heart Tube ##### Embryology, Hematopoiesis ##### Embryology, Kidney, Bladder, and Ureter ##### Embryology, Lanugo ##### Embryology, Midgut ##### Embryology, Mullerian Ducts (Paramesonephric Ducts) ##### Embryology, Mullerian-inhibiting Factor ##### Embryology, Neural Tube ##### Embryology, Optic Cup ##### Embryology, Optic Fissure ##### Embryology, Ovarian Follicle Development ##### Embryology, Pancreas ##### Embryology, Parathyroid ##### Embryology, Pharyngeal Pouch ##### Embryology, Placenta ##### Embryology, Pulmonary ##### Embryology, Rectum and Anal Canal ##### Embryology, Sexual Development ##### Embryology, Teeth ##### Embryology, Teratology TORCH ##### Embryology, Testicle ##### Embryology, Thyroid ##### Embryology, Tongue ##### Embryology, Umbilical Cord ##### Embryology, Urethral Folds ##### Embryology, Uterus ##### Embryology, Vertebral Column Development ##### Embryology, Week 1 ##### Embryology, Week 2-3 ##### Embryology, Weeks 6-8 ##### Embryology, Wolffian Ducts ##### Embryology, Yolk Sac ##### Emergency Department Triage ##### Emergency Preparedness ##### Emergency Room Thoracotomy ##### Emerging Variants of SARS-CoV-2 and Novel Therapeutics Against Coronavirus (COVID-19) ##### Emicizumab ##### Empagliflozin ##### Empathy ##### Emphysema ##### Emphysematous Cholecystitis ##### Emphysematous Pyelonephritis ##### Empowerment ##### Empty Sella Syndrome ##### EMS Air Medical Transport ##### EMS Airway Management: Addressing Challenges in Adverse Conditions ##### EMS Altitude Related Conditions and Treatment ##### EMS Asphyxiation And Other Gas And Fire Hazards ##### EMS Bone Immobilization ##### EMS Burn Rule of Tens ##### EMS Canine Cardiopulmonary Resuscitation ##### EMS Canine Evaluation and Treatment of Dehydration ##### EMS Canine Evaluation of Oxygen Saturation (SpO2) ##### EMS Canine Tactical Medicine Trauma Survey and Treatment ##### EMS Canine Tourniquet Use ##### EMS Canine Wound Care ##### EMS Capacity And Competence ##### EMS Care In A Hostile Environment ##### EMS Care Teams In Disaster Response ##### EMS Casualty Evacuation ##### EMS Catastrophic Events ##### EMS Catastrophic State And Federal Assets ##### EMS Chest Injury ##### EMS Clinical Diagnosis Without The Use Of A Thermometer ##### EMS Community Paramedicine and Mobile Integrated Health ##### EMS Confined Space Care ##### EMS Crime Scene Responsibility ##### EMS Criteria For Disaster Declaration ##### EMS Diabetic Protocols For Treat and Release ##### EMS Disaster Planning And Operations ##### EMS Disaster Response ##### EMS Disordered Sleep And Work Schedule ##### EMS Documentation ##### EMS Emergency Department Diversion ##### EMS Emergency Incident Rehabilitation ##### EMS Evacuation Triage ##### EMS Evaluation and Treatment of Head Injuries ##### EMS Facilitated Intubation Without Paralytics ##### EMS Federal and State Laws Affecting Tactical Medicine ##### EMS Federal Regulations ##### EMS Field Identification Of Chronic Obstructive Pulmonary Disease (COPD) ##### EMS Field Identification of Heart Failure ##### EMS Field Intubation ##### EMS Flight Barotrauma ##### EMS Flight Safety Decisions ##### EMS Flight Stressors and Corrective Action ##### EMS Gaining Access and Extrication ##### EMS Ground Transport Safety ##### EMS Hazardous Waste Response ##### EMS Helicopter Activation ##### EMS High-Altitude Field Prophylaxis And Treatment ##### EMS Immobilization Techniques ##### EMS Improvised Explosive Devices And Terrorist Activity ##### EMS Incident Command System ##### EMS Inter-Facility Transport ##### EMS Junctional Hemorrhage Control ##### EMS Left Ventricular Assist Device Management ##### EMS Legal and Ethical Issues ##### EMS Lights And Sirens ##### EMS Long Spine Board Immobilization ##### EMS Management of Eye Injuries ##### EMS Management Of Traumatic And Medical Disorders In A Wilderness Environment ##### EMS Mass Casualty Management ##### EMS Mass Casualty Response ##### EMS Mass Casualty Triage ##### EMS Mass Gatherings ##### EMS Medical Director Legal Issues and Liability ##### EMS Medical Oversight of Systems ##### EMS Methods to Cool a Patient in the Field ##### EMS National Incident Management System ##### EMS Pain Assessment And Management ##### EMS Pediatric Transport Safety and Secondary Transport ##### EMS Pelvic Binders ##### EMS Physician Training And Drills In Disaster Response ##### EMS Pneumothorax ##### EMS Pneumothorax Identification Without Ancillary Testing ##### EMS Portable Ventilator Management ##### EMS Pre-Arrival Instructions ##### EMS Prehospital Administration Of Thrombolytics For STEMI ##### EMS Prehospital CPAP Devices ##### EMS Prehospital Deliveries ##### EMS Prehospital Evaluation and Treatment of Asthma in Children ##### EMS Prioritization Of Response ##### EMS Pros and Cons of Drug-Assisted Intubation ##### EMS Provider Health And Wellness ##### EMS Public Access To Defibrillation ##### EMS Quality Improvement Programs ##### EMS Remote Assessment ##### EMS Resuscitation During Contamination While Wearing PPE ##### EMS Reverse Triage ##### EMS Scene Safety ##### EMS Scope of Practice ##### EMS System Regionalization ##### EMS Tactical Care and Evacuation Under Fire ##### EMS Tactical Combat Casualty Care ##### EMS Tactical Damage Control Resuscitation Protocol ##### EMS Tactical Medical Threat Assessment and Protection ##### EMS Tactical Movement Techniques ##### EMS Tactical Paramedic Lethal Triad ##### EMS Tactical Remote Assessment and Surrogate Care ##### EMS Tactical Team and Patient Safety ##### EMS Telemedicine in the Prehospital Setting ##### EMS Termination Of Resuscitation And Pronouncement of Death ##### EMS Terrorism Response ##### EMS Traction Splint ##### EMS Ultrasound Use ##### EMS USA Emergency Medical Treatment and Active Labor Act ##### EMS Utilization Of Electrocardiogram In The Field ##### EMS Weapons Of Mass Destruction And Related Injury ##### EMS Zones of Care ##### EMS: Trauma Center Designation ##### EMTALA and Patient Transfers ##### Emtricitabine ##### Emulsions ##### Enalapril ##### Enalaprilat ##### Encapsulating Peritoneal Sclerosis ##### Encephalocele ##### Encephalopathic EEG Patterns ##### Enchondroma ##### Encopresis ##### Endocardial Fibroelastosis ##### Endocarditis ##### Endocarditis Antibiotic Regimens ##### Endocrine-Related Adverse Events From Immune Checkpoint Inhibitors ##### Endodontic Materials Used To Fill Root Canals ##### Endodontics, Pulp Space Anatomy and Access Cavity of Anterior Teeth ##### End-of-Life Care ##### End-of-Life Evaluation and Management of Pain ##### Endogenous Endophthalmitis ##### Endometrial Ablation ##### Endometrial Biopsy ##### Endometrial Cancer ##### Endometrial Hyperplasia ##### Endometrial Polyp ##### Endometrial Receptivity ##### Endometrioma ##### Endometriosis ##### Endometritis ##### Endomyocardial Biopsy ##### Endomyocardial Fibrosis ##### Endophthalmitis ##### Endosalpingiosis ##### Endoscopic Bariatric Duodenal Jejunal Bypass Liner ##### Endoscopic Brow Lift ##### Endoscopic Retrograde Cholangiopancreatography ##### Endoscopic Vein Harvesting ##### Endotracheal Intubation Techniques ##### Endotracheal Tube ##### Endovascular Papillary Angioendothelioma ##### End-Stage Renal Disease ##### Enophthalmos ##### Enoxaparin ##### Entamoeba coli Infection ##### Entamoeba histolytica Infection ##### Enteral Feeding ##### Enterobacter Infections ##### Enterobius Vermicularis ##### Enterococcus Infections ##### Enterocutaneous Fistula ##### Enterohemorrhagic Escherichia coli ##### Enterohormonal and Microbiota Pathophysiology of Obesity ##### Enteropathic Arthritis ##### Enterovesical Fistula ##### Enterovirus ##### Enthesopathies ##### Entomophthoromycosis ##### Entropion ##### Entropion Eyelid Reconstruction ##### Enucleation ##### Enuresis ##### Environmental Factors and Obesity ##### Environmental Toxins and Infertility ##### Enzyme Linked Immunosorbent Assay ##### Eosinophilia ##### Eosinophilic Esophagitis ##### Eosinophilic Gastroenteritis ##### Eosinophilic Granuloma ##### Eosinophilic Granulomatosis With Polyangiitis (Churg-Strauss Syndrome) ##### Eosinophilic Pneumonia ##### Eosinophilic Pustular Folliculitis ##### Ependymoma ##### Ephedrine ##### Epidemiologic and Etiologic Considerations of Obesity ##### Epidemiology Morbidity And Mortality ##### Epidemiology of Prevention of Communicable Diseases ##### Epidemiology Of Study Design ##### Epidermal Inclusion Cyst ##### Epidermal Nevus Syndromes ##### Epidermodysplasia Verruciformis ##### Epidermoid Cancer ##### Epidermoid Cyst ##### Epidermolysis Bullosa ##### Epidermolysis Bullosa Acquisita ##### Epidermolytic Hyperkeratosis ##### Epididymitis ##### Epidural Abscess ##### Epidural Anesthesia ##### Epidural Blood Patch ##### Epidural Hematoma ##### Epidural Morphine ##### Epidural Steroid Injections ##### Epidural Steroids ##### Epiglottitis ##### Epilepsia Partialis Continua ##### Epilepsy EEG ##### Epilepsy Surgery ##### Epinastine ##### Epinephrine ##### Epiphora ##### Epiphora Clinical Testing ##### Epiphrenic Diverticula ##### Epiretinal Membrane ##### Episcleritis ##### Episiotomy ##### Epispadias ##### Epistaxis ##### Epithelial Downgrowth ##### Epithelial Ovarian Cancer ##### Epithelioid Sarcoma ##### Eplerenone ##### Epley Maneuver ##### Epoetin Alfa ##### Epstein-Barr Virus ##### Eptifibatide ##### Erb Palsy ##### Erectile Dysfunction ##### Erector Spinae Plane Block ##### Ergonomics ##### Ergotamine/Caffeine ##### Erikson's Stages of Psychosocial Development ##### Erlotinib ##### Error Management Training in Medical Simulation ##### Erysipelas ##### Erythema Ab Igne ##### Erythema Annulare Centrifugum ##### Erythema Elevatum Diutinum ##### Erythema Infectiosum ##### Erythema Marginatum ##### Erythema Multiforme ##### Erythema Nodosum ##### Erythema Toxicum ##### Erythrasma ##### Erythrocyte Inclusions ##### Erythrocyte Sedimentation Rate ##### Erythrokeratodermia Variabilis ##### Erythromelalgia ##### Erythromycin ##### Erythropoietic Protoporphyria ##### Erythropoietin Stimulating Agents ##### Escharotomy ##### Escherichia coli Infection ##### Escitalopram ##### Esmolol ##### Esophageal Atresia ##### Esophageal Cancer ##### Esophageal Candidiasis ##### Esophageal Cyst ##### Esophageal Diverticula ##### Esophageal Foreign Body ##### Esophageal Hematoma ##### Esophageal Intramural Pseudodiverticulosis ##### Esophageal Leiomyoma ##### Esophageal Manometry ##### Esophageal Motility Disorders ##### Esophageal Necrosis ##### Esophageal Perforation and Tears ##### Esophageal pH Monitoring ##### Esophageal Reconstruction ##### Esophageal Stricture ##### Esophageal Trauma ##### Esophageal Ulcer ##### Esophageal Varices ##### Esophageal Webs and Rings ##### Esophagitis ##### Esophagogastroduodenoscopy ##### Esotropia ##### Essential Hypertension ##### Essential Thrombocytosis ##### Essential Tremor ##### Esthesioneuroblastoma ##### Estradiol ##### Estrogen Therapy ##### Etanercept ##### Etesevimab ##### Ethacrynic Acid ##### Ethambutol ##### Ethanol ##### Ethanol Toxicity ##### Ethical Issues in Academic Medicine ##### Ethosuximide ##### Ethylene Glycol Toxicity ##### Ethylenediaminetetraacetic Acid (EDTA) ##### Etomidate ##### Etoposide ##### Euglycemic Diabetic Ketoacidosis ##### Eumycetoma ##### Euphorbia tirucalli Toxicity ##### Eustachian Tube Dysfunction ##### Euthyroid Sick Syndrome ##### Evaluation and Management of Perioperative Hypertension ##### Evaluation and Treatment of Skin Cancer in Patients With Immunosuppression ##### Evaluation and Treatment of Snake Envenomations ##### Evaluation of Faculty Debriefing Post simulation Events ##### Evaluation of Postmortem Changes ##### Evaluation of Suspected Cardiac Arrhythmia ##### Evaluation of the Dizzy and Unbalanced Patient ##### Evaluation of Visual Acuity ##### Evans Syndrome ##### Evidence Collection ##### Evidence-Based Medicine ##### Evinacumab ##### Ewing Sarcoma ##### Ex utero Intrapartum Treatment (EXIT) Procedure ##### Excessive Gingival Display ##### Exchange Transfusion ##### Excimer Laser Coronary Angioplasty ##### Excisional Biopsy ##### Exenatide ##### Exercise and Fitness Effect on Obesity ##### Exercise Physiology ##### Exercise-Associated Collapse ##### Exercise-Associated Hyponatremia ##### Exercise-Induced Bronchoconstriction ##### Exercise-Induced Laryngeal Obstruction ##### Exertional Compartment Syndrome ##### Exfoliative Dermatitis ##### Exophthalmos ##### Exotropia ##### Expert Witness ##### Expiration Dating and National Drug Code Rules ##### Exploding Head Syndrome ##### Exploratory Data Analysis: Frequencies, Descriptive Statistics, Histograms, and Boxplots ##### Extended SMAS Facelift ##### External Cephalic Version ##### External Ear Aural Atresia ##### External Fixation Principles and Overview ##### External Hemorrhoid ##### External Pacemaker ##### Extracorporeal Carbon Dioxide Removal ##### Extracorporeal Membrane Oxygenation Anticoagulation ##### Extracorporeal Membrane Oxygenation in Adults ##### Extracorporeal Membrane Oxygenation in Children ##### Extracorporeal Membrane Oxygenation Simulation ##### Extracorporeal Membrane Oxygenation Weaning ##### Extracorporeal Shockwave Lithotripsy ##### Extraintestinal Manifestations of Inflammatory Bowel Disease ##### Extramammary Paget Disease ##### Extranodal NK-Cell Lymphoma ##### Extraocular Muscle Management With Orbital and Globe Trauma ##### Extrapyramidal Side Effects ##### Extubation ##### Exudative Retinal Detachment ##### Exudative Retinitis (Coats Disease) ##### Eye Trauma Imaging ##### Eyelid Coloboma ##### Eyelid Laceration ##### Eyelid Myokymia ##### Eyelid Papilloma ##### Eyelid-Sharing Reconstructive Procedures: Hughes and Cutler-Beard Techniques ##### Ezetimibe \| \| \| F \| : ##### F-18 Radiopharmaceutical ##### Fabry Disease ##### Facet Arthritis ##### Facet Joint Disease ##### Facet Joint Injection ##### Facial Burns ##### Facial Chin Augmentation ##### Facial Implants ##### Facial Nerve Anatomy and Clinical Applications ##### Facial Nerve Intratemporal Trauma ##### Facial Nerve Palsy ##### Facial Nerve Repair ##### Facial Nerve Trauma ##### Facial Reconstruction for Mohs Defect Repairs ##### Faciocutaneoskeletal Syndrome (Costello Syndrome) ##### Facioscapulohumeral Muscular Dystrophy ##### Factitious Disorder ##### Factitious Disorder Overview ##### Factitious Hypoglycemia ##### Factor V Deficiency ##### Factor V Leiden Mutation ##### Factor XIII Deficiency ##### Fahr Syndrome ##### Failed Back Surgery Syndrome ##### Failure to Thrive ##### Falls and Fall Prevention in Older Adults ##### Famciclovir ##### Familial Adenomatous Polyposis ##### Familial Atypical Multiple Mole and Melanoma Syndrome ##### Familial Combined Hyperlipidemia ##### Familial Exudative Vitreoretinopathy (FEVR) ##### Familial Hypercholesterolemia ##### Familial Hyperchylomicronemia Syndrome ##### Familial Hypertriglyceridemia ##### Familial Hypocalciuric Hypercalcemia ##### Familial Mediterranean Fever ##### Familial Short Stature ##### Family Dynamics ##### Famotidine ##### Fanconi Anemia ##### Fanconi Syndrome ##### Fascial Dehiscence ##### Fasciocutaneous Flaps ##### Fascioliasis ##### Fasciotomy ##### Fat Embolism ##### Fatal Accidents ##### Fatal Familial Insomnia ##### Fatty Liver ##### Fatty Liver in Pregnancy ##### Favus ##### Features, Evaluation, and Treatment of Coronavirus (COVID-19) ##### Febrile Neutropenia ##### Febrile Seizure ##### Febuxostat ##### Fecal Impaction ##### Fecal Incontinence ##### Fecal Occult Blood Test ##### Federal Medication Development Regulation ##### Federal Regulation of Medication Dispensing ##### Federal Regulation of Medication Production ##### Feedback in Medical Education ##### Feeding Jejunostomy Tube ##### Feeding Tube ##### Felbamate ##### Felodipine ##### Felon ##### Felty Syndrome ##### Female Athlete Triad ##### Female Development ##### Female Genital Mutilation or Cutting ##### Female Infertility ##### Female Sexual Interest and Arousal Disorder ##### Female Urinary Retention ##### Femoral Aneurysm Repair ##### Femoral Artery Pseudoaneurysm ##### Femoral Head Avascular Necrosis ##### Femoral Hernia ##### Femoral Neck Fractures ##### Femoral Nerve Block ##### Femoral Shaft Fractures ##### Femoral Vascular Closure Devices After Catheterization Procedure ##### Femoroacetabular Impingement ##### Femur Immobilization ##### Fenofibrate ##### Fenoldopam ##### Fentanyl ##### Fentanyl Transdermal ##### Fertility Preservation in Benign and Malignant Conditions ##### Fertility Sparing Management in Uterine Fibroids ##### Fetal Alcohol Syndrome ##### Fetal Echogenic Bowel ##### Fetal Growth Restriction ##### Fetal Monitoring ##### Fetal Movement ##### Fever in a Neonate ##### Fever in the Intensive Care Patient ##### Fever of Unknown Origin ##### Fexofenadine ##### Fibrate Medications ##### Fibric Acid Antilipemic Agents ##### Fibrinogen ##### Fibrinous Pericarditis ##### Fibrocystic Breast Disease ##### Fibrodysplasia Ossificans Progressiva ##### Fibrolamellar Hepatocellular Carcinoma ##### Fibromuscular Dysplasia ##### Fibromyalgia ##### Fibrosarcoma ##### Fibrous Dysplasia ##### Fibula Fractures ##### Fibula Free Flaps ##### Fibula Tissue Transfer ##### Field Walking Test ##### Fifth Metacarpal Fracture ##### Fifth-Toe Deformities ##### Filarial Hydrocele ##### Filariasis ##### Filgrastim ##### Filovirus ##### Financing And Funding A Simulation Center ##### Finasteride ##### Fine Motor Disability ##### Fine Needle Aspiration ##### Fine Needle Aspiration of Breast Masses ##### Finger Dislocation ##### Fingertip Injuries ##### Fingolimod ##### Finkelstein Sign ##### Fire Ant Bites ##### First Degree Burn ##### First-Degree Heart Block ##### First-Pass Effect ##### Fitness for Duty and Return to Work ##### Fitz-Hugh-Curtis Syndrome ##### Five Rights of Nursing Delegation ##### Flail Chest ##### Flaps: Muscle and Musculocutaneous ##### Flea Bites ##### Flecainide ##### Flexible Nasopharyngoscopy ##### Flexor Tendon Lacerations ##### Flibanserin ##### Floating Knee ##### Floppy Eyelid Syndrome ##### Florida Controlled Substance Prescribing ##### Florida HIV Safety for Florida Clinical Laboratory Personnel ##### Florida Nurse Practice Act: Laws and Rules for Florida Nurses ##### Florida Responsible Controlled Substance and Opioid Prescribing ##### Florida: Domestic Violence ##### Flow Cytometry Blood Cell Identification ##### Fluconazole ##### Flucytosine ##### Fludeoxyglucose (18F) ##### Fludrocortisone ##### Fluid Management ##### Fluid Resuscitation ##### Flumazenil ##### Fluorescein ##### Fluorescein Angiography ##### Fluorides and Fluorocarbons Toxicity ##### Fluoroscopic Angiography Assessment, Protocols, and Interpretation ##### Fluoroscopic Percutaneous Coronary Interventions, Assessment, Protocols, and Interpretation ##### Fluoroscopy Contrast Materials ##### Fluoroscopy Discography Assessment, Protocols, and Interpretation ##### Fluoroscopy Lumbar Puncture Assessment, Protocols, and Interpretation ##### Fluoroscopy Nasogastric Feeding Tube Placement ##### Fluoroscopy Orthopedic Assessment, Protocols, and Interpretation ##### Fluoroscopy Podiatric Assessment, Protocols, and Interpretation ##### Fluorouracil ##### Fluoxetine ##### Fluphenazine ##### Flutamide ##### Fluticasone ##### Fluvastatin ##### Flying After Diving ##### Focal Impaired Awareness Seizure ##### Focal Length ##### Focal Nodular Hyperplasia ##### Focal Onset Seizure ##### Focal Segmental Glomerulosclerosis ##### Focused Assessment With Sonography for Trauma ##### Folic Acid ##### Folic Acid Deficiency ##### Folinic Acid ##### Follicular Adenoma ##### Follicular Lymphoma ##### Follicular Thyroid Cancer ##### Folliculitis ##### Fontan Completion ##### Food Allergies ##### Food and Drug Administration Recalls ##### Food, Drug, and Cosmetic Act ##### Foot Dislocation ##### Foot Drop ##### Foot Drop in Obstetrics ##### Foot Ulcer ##### Forced Expiratory Volume ##### Forceps Delivery ##### Forearm Amputation ##### Forearm Compartment Syndrome ##### Forearm Fractures ##### Forearm Splinting ##### Forehead Flaps ##### Foreign Body Airway Obstruction ##### Foreign Body Imaging ##### Foreign Educated Physicians Nursing Student Readiness for Collaboration and Teamwork ##### Forensic Odontology ##### Forensic Autopsy ##### Forensic Gait Analysis ##### Foscarnet ##### Fosinopril ##### Fosphenytoin ##### Foster Kennedy Syndrome ##### Fournier Gangrene ##### Foville Syndrome ##### Fox-Fordyce Disease (Apocrine Miliaria) ##### Fraction of Inspired Oxygen ##### Fractional Flow Reserve ##### Fracture Healing Overview ##### Fragile X Syndrome ##### Free Tissue Transfer of the Lateral Thigh and Anterolateral Thigh ##### Freiberg Infraction ##### Fresh Frozen Plasma (FFP) ##### Fresnel Prisms ##### Freud's Developmental Theory ##### Frey Syndrome ##### Friedreich Ataxia ##### Frontal Fibrosing Alopecia ##### Frontal Lobe Syndrome ##### Frontal Sinus Fractures ##### Frontotemporal Lobe Dementia ##### Frostbite ##### Frotteurism ##### Fructose-1-Phosphate Aldolase Deficiency ##### Fuchs Endothelial Dystrophy ##### Fuchs Uveitis Syndrome ##### Full Liquid Diet ##### Full-Field Electroretinogram ##### Full-Thickness Skin Grafts ##### Fulvestrant ##### Functional Abdominal Pain in Children ##### Functional Asplenism ##### Functional Dyspepsia ##### Functional Neurologic Disorder ##### Functional Voice Disorders ##### Fundus Camera ##### Fungal Endocarditis ##### Fungal Endophthalmitis ##### Fungal Infections of the Oral Mucosa ##### Fungal Keratitis ##### Fungal Sinusitis ##### Funnel-Web Spider Toxicity ##### Furosemide ##### Furrow Degeneration ##### Fusiform Incision \| \| \| G \| : ##### Ga67 Gallium Citrate Radiopharmaceutical ##### GABA Inhibitors ##### GABA Receptor ##### GABA Receptor Positive Allosteric Modulators ##### Gabapentin ##### Gadolinium Magnetic Resonance Imaging ##### Gait Disturbances ##### Galactocele ##### Galactokinase Deficiency ##### Galactorrhea ##### Galactose-1-Phosphate Uridyltransferase Deficiency ##### Galantamine ##### Galeazzi Fractures ##### Gallbladder Carcinoma ##### Gallbladder Empyema ##### Gallbladder Imaging ##### Gallbladder Mucocele ##### Gallbladder Polyp ##### Gallbladder Volvulus ##### Gallium Scan ##### Gallstone Ileus ##### Gallstones (Cholelithiasis) ##### Gamma-Hydroxybutyrate Toxicity ##### Ganglion Cyst ##### Gangrene ##### Gardner Syndrome ##### Gardnerella Vaginalis ##### Gas Cylinders ##### Gas Gangrene ##### Gas Laws and Clinical Application ##### Gaseous Anesthetics ##### Gastric Cancer ##### Gastric Emptying Scan ##### Gastric Lymphoma ##### Gastric Neuroendocrine Tumors ##### Gastric Outlet Obstruction ##### Gastric Perforation ##### Gastric Polyp ##### Gastric Resection for Malignancy (Gastrectomy) ##### Gastric Ulcer ##### Gastric Ultrasound ##### Gastric Varices ##### Gastric Volvulus ##### Gastrinoma ##### Gastritis ##### Gastrocnemius Rupture ##### Gastrocnemius Strain ##### Gastroesophageal Reflux Disease (GERD) ##### Gastrointestinal and Hepatic Manifestations of Coronavirus (COVID-19) ##### Gastrointestinal Bleeding ##### Gastrointestinal Bleeding Scan ##### Gastrointestinal Foreign Body ##### Gastrointestinal Neuroendocrine Tumors ##### Gastrointestinal Stromal Tumors ##### Gastrojejunostomy ##### Gastroparesis ##### Gastroschisis ##### Gastrostomy Tube Replacement ##### Gaucher Disease ##### Gemfibrozil ##### Gender Dysphoria ##### General Anesthesia for Surgeons ##### Generalized Anxiety Disorder ##### Generalized Pustular Psoriasis ##### Generalized Tonic-Clonic Seizure ##### Genetic and Syndromic Causes of Obesity: Diagnosis and Management ##### Genetic Hearing Loss ##### Genetics, Autosomal Dominant ##### Genetics, Autosomal Recessive ##### Genetics, Cancer Cell Cycle Phases ##### Genetics, Chromosome Abnormalities ##### Genetics, Chromosomes ##### Genetics, Cytogenetic Testing and Conventional Karyotype ##### Genetics, DNA Damage and Repair ##### Genetics, DNA Packaging ##### Genetics, Epigenetic Mechanism ##### Genetics, Female Gametogenesis ##### Genetics, Gonadal Dysgenesis ##### Genetics, Histocompatibility Antigen ##### Genetics, Histone Code ##### Genetics, Human Major Histocompatibility Complex (MHC) ##### Genetics, Inducible Operon ##### Genetics, Meiosis ##### Genetics, Mitosis ##### Genetics, Mosaicism ##### Genetics, Mutagenesis ##### Genetics, Nondisjunction ##### Genetics, Philadelphia Chromosome ##### Genetics, Somatic Mutation ##### Genetics, Transposons ##### Genetics, TREX1 Mutations ##### Genetics, Trinucleotide ##### Genetics, X-Linked Inheritance ##### Genital Warts ##### Genitofemoral Neuralgia ##### Genitourinary Syndrome of Menopause ##### Genitourinary Tuberculosis ##### Gentamicin ##### Genu Valgum ##### Geographic Tongue ##### Geriatric Care Special Needs Assessment ##### Geriatric Cervical Spine Injury ##### Geriatric Cognitive Decline and Polypharmacy ##### Geriatric Evaluation and Treatment of Age-Related Cognitive Decline ##### Geriatric Head Injury ##### Germ Cell Seminoma ##### Gerstmann Syndrome ##### Gestalt Therapy ##### Gestational Age Assessment ##### Gestational Diabetes ##### Gestational Sac Evaluation ##### Gestational Trophoblastic Disease ##### Ghon Complex ##### Giant Cell Arteritis (Temporal Arteritis) ##### Giant Cell Tumor (Osteoclastoma) ##### Giant Condylomata Acuminata of Buschke and Lowenstein ##### Giant Fornix Syndrome ##### Giardiasis ##### Gigantism and Acromegaly ##### Gilbert Syndrome ##### Ginger Root ##### Gingivitis ##### Ginkgo Biloba ##### Ginseng ##### Gitelman Syndrome ##### Glanders and Melioidosis ##### Glanzmann Thrombasthenia ##### Glargine Insulin ##### Glasgow Coma Scale ##### Glass Ionomer Cement ##### Glatiramer ##### Glaucoma ##### Gleason Score ##### Glenoid Fractures ##### Glenolabral Articular Disruption (GLAD) ##### Glimepiride ##### Glioblastoma Multiforme ##### Gliomas ##### Glipizide ##### Globe Rupture ##### Glomangioma (Glomuvenous Malformation) ##### Glomeruloid Hemangioma ##### Glomerulonephritis ##### Glomerulonephritis in Pregnancy ##### Glomus Jugulare ##### Glossectomy ##### Glossitis ##### Glossopharyngeal Neuralgia ##### Glottic Cancer ##### Glottic Stenosis ##### Glucagon ##### Glucagon-Like Peptide-1 Receptor Agonists ##### Glucagonoma ##### Glucosamine Sulfate ##### Glucose Intolerance ##### Glucose Tolerance Test ##### Glucose-6-Phosphate Dehydrogenase Deficiency ##### Gluten-Associated Medical Problems ##### Glyburide ##### Glycogen Storage Disease ##### Glycogen Storage Disease Type I ##### Glycogen Storage Disease Type II ##### Glycogenic Hepatopathy ##### Glycol Ether Toxicology ##### Glycoprotein IIb/IIIa Inhibitors ##### Glycopyrrolate ##### Goiter ##### Golfers Elbow ##### Golimumab ##### Goniometer ##### Gonococcal Arthritis ##### Gonococcal Conjunctivitis ##### Gonorrhea ##### Good Samaritan Laws ##### Goodpasture Syndrome ##### Gordon Reflex ##### Gorlin Syndrome ##### Gout ##### Gower Sign ##### Gracilis Tissue Transfer ##### Graft-Versus-Host Disease ##### Graham-Little-Piccardi-Lasseur Syndrome ##### Gram Staining ##### Gram-Negative Bacteria ##### Gram-Positive Bacteria ##### Granular Cell Tumor ##### Granular Parakeratosis ##### Granuloma ##### Granuloma Annulare ##### Granuloma Faciale ##### Granuloma Inguinale ##### Granulomatosis With Polyangiitis ##### Granulomatous Hepatitis ##### Granulomatous Uveitis ##### Granulosa Theca Cell Tumors of the Ovary ##### Grasp Reflex ##### Graves Disease ##### Graves Disease Orbital Decompression ##### Graves Orbitopathy ##### Greater Trochanteric Bursa Injection ##### Greater Trochanteric Pain Syndrome (Greater Trochanteric Bursitis) ##### Greenstick Fracture ##### Grey Turner Sign ##### Grief and Prolonged Grief Disorder ##### Grief Support Following In-Hospital Deaths ##### Griseofulvin ##### Groove Pancreatitis ##### Gross and Microscopic Hematuria ##### Group A Streptococcal Infections ##### Group B Coxsackie Virus ##### Group B Streptococcus and Pregnancy ##### Group Therapy ##### Guillain-Barre Syndrome ##### Gunshot Wounds Forensic Pathology ##### Gustatory Testing ##### Guttate Psoriasis ##### Guyon Canal Syndrome ##### Gynecologic Pelvic Examination ##### Gynecomastia ##### Gyrate Atrophy of the Choroid and Retina ##### Gyromitra Mushroom Toxicity \| \| \| H \| : ##### H1N1 Influenza ##### H2 Blockers ##### Haemophilus influenzae Infection ##### Haemophilus influenzae Type b Vaccine ##### Haidinger Brush Phenomenon ##### Hair Transplantation ##### Hairy Cell Leukemia ##### Hairy Leukoplakia ##### Halitosis ##### Hallux Rigidus ##### Hallux Valgus ##### Hallux Varus ##### Halo Brace ##### Halobetasol Cream ##### Haloperidol ##### Halothane Toxicity ##### Hamartoma ##### Hamate Fractures ##### Hammertoe ##### Hamstring Injury ##### Hand Amputation ##### Hand and Wrist Rheumatoid Arthritis ##### Hand Extensor Tendon Lacerations ##### Hand High Pressure Injury ##### Hand Hygiene ##### Hand Infection ##### Hand Splint ##### Hand Tendon Transfers ##### Hand, Foot, and Mouth Disease ##### Hangman's Fractures ##### Hantavirus Pulmonary Syndrome ##### Harlequin Syndrome ##### Hartnup Disease ##### HAS-BLED Scores ##### Hashimoto Thyroiditis ##### HCQM Two-Midnight Rule ##### Headache With Neurological Deficits and CSF Lymphocytosis ##### Health Information Technology ##### Health Insurance Portability and Accountability Act (HIPAA) Compliance ##### Health Maintenance Organization ##### Health Screening ##### Healthcare Analytics ##### Healthcare Business Budgeting ##### Hearing Aid Fitting for Children ##### Hearing Loss ##### Hearing Loss Assessment in Children ##### Hearing Loss in the Elderly ##### Hearing Loss Screening Guidelines ##### Heart Failure (Congestive Heart Failure) ##### Heart Failure and Ejection Fraction ##### Heart Failure With Preserved Ejection Fraction (HFpEF) ##### Heart Transplantation ##### Heart Transplantation Allograft Vasculopathy ##### Heart Transplantation Patient Selection ##### Heart Transplantation Rejection ##### Heart-Lung Transplantation ##### Heat Illness ##### Heat Stroke ##### Heavy Metal Toxicity ##### Heavy Metals ##### Height Assessment ##### Heinz Body ##### Helicobacter Pylori ##### HELLP Syndrome ##### Helminthiasis ##### Hemangioblastoma ##### Hemangioma ##### Hemarthrosis ##### Hematocrit ##### Hematopoietic Stem Cell Transplantation ##### Hematopoietic Stem Cell Transplantation in Sickle Cell Disease ##### Hemianopsia ##### Hemiballismus ##### Hemicolectomy ##### Hemicrania Continua ##### Hemifacial Microsomia ##### Hemifacial Spasm ##### Hemiplegic Migraine ##### Hemochromatosis ##### Hemodialysis ##### Hemoglobin A1C ##### Hemoglobin C Disease ##### Hemolacria ##### Hemolytic Anemia ##### Hemolytic Disease of the Fetus and Newborn ##### Hemolytic Transfusion Reaction ##### Hemolytic Uremic Syndrome ##### Hemophagocytic Lymphohistiocytosis ##### Hemophilia ##### Hemophilia A ##### Hemophilia B ##### Hemorrhage Control ##### Hemorrhagic Disease of Newborn ##### Hemorrhagic Fever Renal Syndrome ##### Hemorrhagic Shock ##### Hemorrhagic Stroke ##### Hemorrhoid Banding ##### Hemorrhoidectomy ##### Hemosuccus Pancreaticus ##### Hemothorax ##### Henry's Law ##### Heparin ##### Heparin-Induced Thrombocytopenia ##### Hepatic Biloma ##### Hepatic Chemoembolization ##### Hepatic Cirrhosis ##### Hepatic Cystadenoma ##### Hepatic Encephalopathy ##### Hepatic Lymphoma ##### Hepatitis A ##### Hepatitis A Vaccine ##### Hepatitis B ##### Hepatitis B Vaccine ##### Hepatitis C ##### Hepatitis D ##### Hepatitis E ##### Hepatobiliary Iminodiacetic Acid Scan ##### Hepatobiliary Scintigraphy ##### Hepatoblastoma ##### Hepatocellular Adenoma ##### Hepatocellular Carcinoma ##### Hepatojugular Reflux ##### Hepatopulmonary Syndrome ##### Hepatorenal Syndrome ##### Herald Patch ##### Herbal Supplements ##### Hereditary and Acquired Ichthyosis Vulgaris ##### Hereditary Angioedema ##### Hereditary Elliptocytosis ##### Hereditary Fructose Intolerance ##### Hereditary Hemorrhagic Telangiectasia (HHT) ##### Hereditary Nonpolyposis Colon Cancer (Lynch Syndrome) ##### Hereditary Spherocytosis ##### Heroin Toxicity ##### Herpangina ##### Herpes Simplex Encephalitis ##### Herpes Simplex Keratitis ##### Herpes Simplex Ophthalmicus ##### Herpes Simplex Type 1 ##### Herpes Simplex Type 2 ##### Herpes Virus Type 8 ##### Herpes Zoster ##### Herpes Zoster Ophthalmicus ##### Herpetic Gingivostomatitis ##### Herpetic Whitlow ##### Heterochromia ##### Heterotopic Ossification ##### Hexokinase Method ##### Heyde Syndrome ##### Hiatal Hernia ##### Hibernoma ##### Hidradenitis Suppurativa ##### High Altitude Cerebral Edema ##### High Fiber Diet ##### High Frequency Ventilation ##### High Risk Behaviors ##### High-Altitude Cardiopulmonary Diseases ##### High-Altitude Oxygenation ##### High-Flow Nasal Cannula ##### High-Frequency Oscillator in the Neonate ##### High-Grade Squamous Intraepithelial Lesion of the Cervix ##### Highly Active Antiretroviral Therapy (HAART) ##### High-Output Cardiac Failure ##### High-Pressure Neurological Syndrome ##### High-Velocity Low-Amplitude Manipulation Techniques ##### Hip Arthrogram ##### Hip Fracture Overview ##### Hip Joint Injection ##### Hip Pointer Injuries ##### Hip Precautions ##### Hirschsprung Disease ##### Hirsutism ##### Histology, Alveolar Cells ##### Histology, Alveolar Macrophages ##### Histology, Apocrine Gland ##### Histology, Astrocytes ##### Histology, Axon ##### Histology, Basophilic Stippling ##### Histology, B-Cell Lymphocyte ##### Histology, Bladder ##### Histology, Blood Vascular System ##### Histology, Bone ##### Histology, Capillary ##### Histology, Cell ##### Histology, Cell Death ##### Histology, Chondrocytes ##### Histology, Corpus Albicans ##### Histology, Cytotoxic T Cells ##### Histology, Dermis ##### Histology, Ear ##### Histology, Epithelial Cell ##### Histology, Extracellular Vesicles ##### Histology, Eye ##### Histology, Fibroblast ##### Histology, Glial Cells ##### Histology, Goblet Cells ##### Histology, Hair and Follicle ##### Histology, Heart ##### Histology, Hematopoiesis ##### Histology, Howell Jolly Bodies ##### Histology, Keratohyalin Granules ##### Histology, Kidney and Glomerulus ##### Histology, Kupffer Cell ##### Histology, Leydig Cells ##### Histology, Lipofuscin ##### Histology, Lung ##### Histology, Lymph Nodes ##### Histology, M Cell ##### Histology, Male Urethra ##### Histology, Mammary Glands ##### Histology, Mast Cells ##### Histology, Meissner Corpuscle ##### Histology, Monocytes ##### Histology, Muscle ##### Histology, Myelin ##### Histology, Nail ##### Histology, Natural Killer Cells ##### Histology, Nephron ##### Histology, Oral Mucosa ##### Histology, Osteoblasts ##### Histology, Osteoclasts ##### Histology, Osteocytes ##### Histology, Osteoprogenitor Cells ##### Histology, Parathyroid Gland ##### Histology, Parietal Cells ##### Histology, Periodontium ##### Histology, Periosteum And Endosteum ##### Histology, Plasma Cells ##### Histology, Platelets ##### Histology, Purkinje Cells ##### Histology, Red Blood Cell ##### Histology, Respiratory Epithelium ##### Histology, Reticulocytes ##### Histology, Retina ##### Histology, Rough Endoplasmic Reticulum ##### Histology, Salivary Glands ##### Histology, Schwann Cells ##### Histology, Sertoli Cell ##### Histology, Skin ##### Histology, Skin Appendages ##### Histology, Spermatogenesis ##### Histology, Staining ##### Histology, Stratum Corneum ##### Histology, T-Cell Lymphocyte ##### Histology, Thyroid Gland ##### Histology, Tooth ##### Histology, Vascular ##### Histology, Verhoeff Stain ##### Histology, White Blood Cell ##### Histoplasmosis ##### History Taking in Obesity Medicine: A Comprehensive Guide ##### Histrionic Personality Disorder ##### HIV and AIDS ##### HIV Antiretroviral Therapy ##### HIV Encephalitis ##### HIV in Pregnancy ##### HIV Nephropathy ##### HIV Neurocognitive Disorders ##### HIV Prevention ##### HIV Retinopathy ##### HIV Testing ##### HIV-1 Encephalopathy and Aids Dementia Complex ##### HIV-1–Associated Opportunistic Infections ##### HIV-1–Associated Progressive Polyradiculopathy ##### HIV-1–Associated Toxoplasmosis ##### HIV-2 Infection ##### HIV-Associated Lipodystrophy ##### HIV-Related Endocrinopathies ##### HLA-B\57:01 Testing ##### HLA-B27 Syndromes ##### HMG-CoA Reductase Inhibitors ##### Hodgkin Lymphoma ##### Hoffa Pad Impingement Syndrome ##### Hoffmann Sign ##### Hoffmann Tinel Sign ##### Holiday Heart Syndrome ##### Hollenhorst Plaque ##### Holmes Tremor ##### Holoprosencephaly ##### Holter Monitor ##### Holt-Oram Syndrome ##### Home Oxygen Therapy ##### Home Safety Techniques ##### Homonymous Hemianopsia ##### Homonymous Superior Quadrantanopia ##### Hookworm ##### Hordeolum (Stye) ##### Hormone Replacement Therapy ##### Horner Syndrome ##### Horseshoe Kidney ##### Hospice Appropriate Diagnoses ##### Hospice Benefits ##### Hospice Care ##### Hot Flashes ##### Hounsfield Unit ##### How to Localize Neurologic Lesions by Physical Examination ##### How To Read A Scientific Manuscript ##### How To Write And Publish A Scientific Manuscript ##### Human African Trypanosomiasis (Sleeping Sickness) ##### Human Behavior in a Social Environment ##### Human Bites ##### Human Capital ##### Human Chorionic Gonadotropin ##### Human Factors in Medical Simulation ##### Human Growth and Development ##### Human Herpesvirus 6 ##### Human Insulin ##### Human Metapneumovirus ##### Human Papilloma Virus Vaccine ##### Human Papillomavirus ##### Human Subjects Research Design ##### Human T-Cell Lymphotropic Virus ##### Human Trafficking ##### Humeral Shaft Fractures ##### Humerus Fractures Overview ##### Humphrey Visual Field ##### Hungry Bone Syndrome ##### Huntington Disease ##### Hurler Syndrome ##### Hyaluronic Acid ##### Hyaluronidase ##### Hydatidiform Mole ##### Hydralazine ##### Hydranencephaly ##### Hydrazine Toxicology ##### Hydroa Vacciniforme ##### Hydrocele ##### Hydrocephalus ##### Hydrochlorothiazide ##### Hydrocodone ##### Hydrocodone and Acetaminophen ##### Hydrofluoric Acid Burns ##### Hydrogen Peroxide Toxicity ##### Hydrogen Sulfide Toxicity ##### Hydromorphone ##### Hydronephrosis and Hydroureter ##### Hydroquinone ##### Hydroxocobalamin ##### Hydroxyapatite Dental Material ##### Hydroxyurea Toxicity ##### Hymenoptera Stings ##### Hyperacusis ##### Hyperaldosteronism ##### Hyperammonemia ##### Hyperamylasemia ##### Hyperbaric Cardiovascular Effects ##### Hyperbaric Complications ##### Hyperbaric Contraindicated Chemotherapeutic Agents ##### Hyperbaric Evaluation and Treatment of Cyanide Toxicity ##### Hyperbaric Management of Frostbite ##### Hyperbaric Medical Considerations for Occupational Exposure to Compressed Gas Environments ##### Hyperbaric Oxygen Effects on Angiogenesis ##### Hyperbaric Oxygen Therapy and Associated Cataracts ##### Hyperbaric Oxygen Therapy Contraindications ##### Hyperbaric Oxygen Therapy for Intracranial Abscess ##### Hyperbaric Patient Selection ##### Hyperbaric Physics ##### Hyperbaric Physiological And Pharmacological Effects of Gases ##### Hyperbaric Related Myopia and Cataract Formation ##### Hyperbaric Soft Tissue Radionecrosis ##### Hyperbaric Therapy for Skin Grafts and Flaps ##### Hyperbaric Therapy for Wound Healing ##### Hyperbaric Therapy in Blood Loss Anemia ##### Hyperbaric Transcutaneous Oximetry ##### Hyperbaric Treatment of Brain Radiation Necrosis ##### Hyperbaric Treatment of Carbon Monoxide Toxicity ##### Hyperbaric Treatment of Central Retinal Artery Occlusion ##### Hyperbaric Treatment of Chronic Refractory Osteomyelitis ##### Hyperbaric Treatment of Clostridial Myositis and Myonecrosis ##### Hyperbaric Treatment of Crush Injury and Compartment Syndrome ##### Hyperbaric Treatment of Delayed Radiation Injury ##### Hyperbaric Treatment of Diabetic Foot Ulcer ##### Hyperbaric Treatment of Ischemia Reperfusion Injury ##### Hyperbaric Treatment of Radiation Proctitis ##### Hyperbaric Treatment of Sensorineural Hearing Loss ##### Hyperbaric Treatment of Thermal Burns ##### Hyperbaric Zygomycotic Infections ##### Hypercalcemia ##### Hypercalciuria ##### Hypercapnea ##### Hypercarbia ##### Hypercholesterolemia ##### Hypercoagulability ##### Hypercortisolism ##### Hyperemesis Gravidarum ##### Hypereosinophilic Syndrome ##### Hyperesthesia ##### Hypergammaglobulinemia (Polyclonal Gammopathy) ##### Hyperglycemia ##### Hyperhidrosis ##### Hyperhomocysteinemia ##### Hyperkalemia ##### Hyperkalemic Periodic Paralysis ##### Hyperkeratosis ##### Hyperkeratosis of the Nipple and Areola ##### Hyperlipidemia ##### Hypermagnesemia ##### Hypernatremia ##### Hyperopia ##### Hyperosmolar Hyperglycemic Syndrome ##### Hyperoxaluria ##### Hyperphosphatemia ##### Hyperphosphatemic Tumoral Calcinosis ##### Hyperpituitarism ##### Hyperprolactinemia ##### Hypersensitivity Pneumonitis ##### Hypertelorism ##### Hypertension Clinical Trials ##### Hypertension In Pregnancy ##### Hypertensive Crisis ##### Hypertensive Emergency ##### Hypertensive Encephalopathy ##### Hypertensive Heart Disease ##### Hypertensive Retinopathy ##### Hypertensive Urgency ##### Hyperthermia for Chest Wall Recurrence ##### Hyperthyroidism ##### Hyperthyroidism in Pregnancy ##### Hypertonic Fluids ##### Hypertrichosis ##### Hypertriglyceridemia ##### Hypertrophic Cardiomyopathy ##### Hypertrophic Osteoarthropathy ##### Hypertrophic Scars and Keloids ##### Hypertyrosinemia ##### Hyperuricemia ##### Hyperuricosuria ##### Hyperviscosity Syndrome ##### Hyphema ##### Hypnic Headache ##### Hypoalbuminemia ##### Hypoaldosteronism ##### Hypoalphalipoproteinemia ##### Hypocalcemia ##### Hypocarbia ##### Hypocitraturia and Renal Calculi ##### Hypogammaglobulinemia ##### Hypoglossal Stimulation Device ##### Hypoglycemia ##### Hypokalemia ##### Hypokalemic Metabolic Alkalosis ##### Hypokalemic Periodic Paralysis ##### Hypomagnesemia ##### Hypomelanosis of Ito ##### Hyponatremia ##### Hypoparathyroidism ##### Hypopharyngeal Cancer ##### Hypophosphatemia ##### Hypopigmented Macules ##### Hypopituitarism ##### Hypoplastic Left Heart Syndrome ##### Hypoplastic Lung Disease ##### Hypospadias ##### Hypospadias Urogenital Reconstruction ##### Hypotension ##### Hypothalamic Dysfunction ##### Hypothalamic Hamartoma ##### Hypothermia ##### Hypothesis Testing, P Values, Confidence Intervals, and Significance ##### Hypothyroid Myopathy ##### Hypothyroidism ##### Hypotonia ##### Hypotonic Labor ##### Hypovolemia and Hypovolemic Shock ##### Hypoxia ##### Hypoxic Brain Injury ##### Hysterosalpingogram ##### Hysteroscopy \| \| \| I \| : ##### I-123 Uptake ##### Iatrogenic Pneumothorax ##### Ibuprofen ##### Ibutilide ##### Ichthyosis Fetalis ##### Ictal-Interictal Continuum ##### ICU Delirium ##### Ideal Gas Behavior ##### Identifying and Addressing Bullying ##### Idiopathic (Genetic) Generalized Epilepsy ##### Idiopathic Guttate Hypomelanosis ##### Idiopathic Hypersomnia ##### Idiopathic Interstitial Pneumonia With Autoimmune Features ##### Idiopathic Pulmonary Artery Hypertension ##### Idiopathic Pulmonary Fibrosis ##### Idiopathic Pulmonary Hemorrhage ##### Idiopathic Pulmonary Hemosiderosis ##### Idioventricular Rhythm ##### Ifosfamide ##### IgA Nephropathy (Berger Disease) ##### IgA Pemphigus ##### IgA Vasculitis (Henoch-Schönlein Purpura) ##### IgG4-Related Disease ##### Ileal Atresia ##### Ileal Conduit ##### Ileostomy ##### Ileus ##### Ilioinguinal Neuralgia ##### Iliotibial Band Friction Syndrome ##### Illness Anxiety Disorder ##### Imatinib ##### Imdevimab ##### Imipramine ##### Imiquimod ##### Immersion Pulmonary Edema ##### Immune Checkpoint Inhibitor–Associated Acute Kidney Injury ##### Immune Reconstitution Inflammatory Syndrome ##### Immune Thrombocytopenia ##### Immunization ##### Immunodeficiency ##### Immunoglobulin ##### Immunoglobulin E ##### Immunology at the Maternal-Fetal Interface ##### Immunophenotyping ##### Immunotherapy ##### Impaired Bilirubin Conjugation ##### Impaired Wound Healing ##### Imperforate Anus ##### Imperforate Hymen ##### Impetigo ##### Implantable Defibrillator ##### Implantable Hearing Devices ##### Implantable Intrathecal Drug Delivery System ##### Implantable Loop Recorder ##### Implicit Bias ##### Imposter Phenomenon ##### Impulse Control Disorders ##### In Situ Debriefing in Medical Simulation ##### In situ Simulation to Evaluate the Readiness of a New Clinical Space ##### In Vitro Fertilization ##### Inamrinone ##### Inappropriate Medical Abbreviations ##### Inappropriate Medication in the Geriatric Population ##### Inborn Errors of Metabolism ##### Incentive Spirometer and Inspiratory Muscle Training ##### Incidence ##### Incident Reporting ##### Incision and Drainage ##### Incisional Hernia ##### Inclusion Body Myositis ##### Incomplete Miscarriage ##### Incontinentia Pigmenti (Bloch-Sulzberger Syndrome) ##### Increased Intracranial Pressure ##### Independent Diagnostic Testing Facilities ##### Indium-111 White Blood Cell Scan ##### Indocyanine Green (ICG) Angiography ##### Indomethacin ##### Induction of Labor ##### Industrial and Organizational Psychology in Medical Simulation ##### Infant Apnea ##### Infant Head Lag ##### Infant Nutrition Requirements and Options ##### Infantile Acne ##### Infantile and Juvenile Scoliosis ##### Infantile Botulism ##### Infantile Colic ##### Infantile Cortical Hyperostosis ##### Infantile Epileptic Spasms Syndrome (West Syndrome) ##### Infarct Avid Imaging Study ##### Infection Control ##### Infectious Complications of Blood Transfusion ##### Infectious Endocarditis ##### Infectious Scleritis ##### Inferior Alveolar Nerve and Lingual Nerve Injury ##### Inferior Alveolar Nerve Block ##### Inferior Gluteal Nerve Injury ##### Inferior Myocardial Infarction ##### Inferior Shoulder Dislocations ##### Inferior Vena Cava Filter ##### Inferior Vena Cava Syndrome ##### Inferior Vena Caval Thrombosis ##### Inflammatory Arthritis ##### Inflammatory Back Pain ##### Inflammatory Bowel Disease ##### Inflammatory Breast Cancer ##### Infliximab ##### Infliximab-abda ##### Influenza ##### Influenza Vaccine ##### Informatics ##### Informatics Ethics ##### Informed Consent ##### Infraclavicular Nerve Block ##### Infrainguinal Occlusive Disease ##### Infraorbital Nerve Block ##### Ingrown Toenails ##### Inhalation Injury ##### Inhalational Anesthetic ##### Inhaled Corticosteroids ##### Initial Antepartum Care ##### Inner Ear Decompression Sickness ##### Innocent Murmur ##### Innominate Artery Injury ##### Inotersen ##### Inotropes and Vasopressors ##### Insect Bites ##### Insensible Fluid Loss ##### Instantaneous Wave-Free Ratio ##### Instrumental Activity of Daily Living ##### Insular Cortex ##### Insulin ##### Insulin Lispro ##### Insulin Pump ##### Insulin Resistance ##### Insulinoma ##### Intellectual Disability ##### Intense Pulsed Light (IPL) Therapy ##### Intention Tremor ##### Intercostal Nerve Block ##### Intercostal Neuralgia ##### Interferon ##### Interferon Test ##### Interferon-Induced Retinopathy ##### Interlaminar Epidural Injection ##### Interleukin ##### Intermittent Claudication ##### Intermittent Exotropia ##### Internal Anal Sphincterotomy ##### Internal Hemorrhoid ##### Internal Jugular Vein Thrombosis ##### Internal Mammary Artery Bypass ##### Internal Tooth Whitening ##### International Normalized Ratio: Assessment, Monitoring, and Clinical Implications ##### Internuclear Ophthalmoplegia ##### Interpleural Analgesia ##### Interpolated Flaps ##### Interpretation of Blood Clotting Studies and Values (PT, PTT, aPTT, INR, Anti-Factor Xa, D-Dimer) ##### Interprofessional Education in a Simulation Setting ##### Interprofessional Rounds in the ICU ##### Interrupted Aortic Arch ##### Interscalene Block ##### Intersection Syndrome ##### Interstitial Cystitis/Bladder Pain Syndrome ##### Interstitial Lung Disease ##### Interstitial Pulmonary Fibrosis ##### Intertrigo ##### Intertrochanteric Femur Fracture ##### Interventions in Hypertrophic Cardiomyopathy ##### Interventricular Septum Abscess ##### Interview Techniques ##### Intestinal and Multivisceral Transplantation ##### Intestinal Fistula ##### Intestinal Perforation ##### Intestinal Pseudo-Obstruction ##### Intestinal Stoma ##### Intestinal Trauma ##### Intoeing ##### Intraabominal Abscesses ##### Intra-Aortic Balloon Pump ##### Intracardiac Shunts ##### Intracerebral Hemorrhage ##### Intracoronary Stents ##### Intracranial Hemorrhage ##### Intracranial Hypertension ##### Intracranial Hypotension ##### Intracranial Pressure Monitoring ##### Intradiscal Electrothermal Therapy ##### Intraductal Papillary Mucinous Neoplasm of the Pancreas ##### Intraductal Papilloma ##### Intraepidermal Carcinoma ##### Intragastric Balloon ##### Intramedullary Spinal Cord Tumors ##### Intramuscular Injection ##### Intraocular Foreign Body ##### Intraocular Hemorrhage ##### Intraocular Lens Power Calculation ##### Intraocular Pressure ##### Intraoperative and Anesthesia Awareness ##### Intraoperative Echocardiography ##### Intraoperative Neurophysiological Monitoring ##### Intraosseous Vascular Access ##### Intrapleural Catheter ##### Intrathecal Morphine ##### Intratympanic Microwick Placement ##### Intratympanic Steroid Injection ##### Intrauterine Device Placement and Removal ##### Intravascular Lithotripsy ##### Intravascular Ultrasound ##### Intravenous Contrast ##### Intravenous Immunoglobulin (IVIG) ##### Intravenous Pyelogram ##### Intraventricular Meningioma ##### Intravitreal Implants ##### Intrinsic Hand Deformity ##### Intubation Endotracheal Tube Medications ##### Intussusception in Adults ##### Inverse Ratio Ventilation ##### Inverted Nipple ##### Inverted Urothelial Papilloma ##### Involuntary Commitment ##### Iodine Toxicity ##### Iodine-131 Uptake Study ##### Ionizing Radiation ##### Iontophoresis Analgesic Medications ##### Ipecac ##### Ipilimumab ##### Ipratropium ##### Iridocorneal Dysgenesis ##### Iridocorneal Endothelial Syndrome ##### Irinotecan ##### Iris Cyst ##### Iris Ectropion Syndrome ##### Iritis ##### Iron ##### Iron Deficiency Anemia ##### Iron Dextran ##### Iron Overload and Toxicity ##### Iron Supplementation ##### Iron-Binding Capacity ##### Irritable Bowel Syndrome ##### Irukandji Syndrome ##### Ischemic Cardiomyopathy ##### Ischemic Stroke ##### Ischial Bursitis ##### Islets Transplantation ##### Isoflurane ##### Isolated Coronary Artery Anomalies ##### Isolated Cortical Venous Thrombosis ##### Isoniazid ##### Isoniazid Toxicity ##### Isopropanol Toxicity ##### Isoproterenol ##### Isosorbide ##### Isotretinoin ##### Itraconazole ##### Ivabradine ##### Ixazomib ##### Ixekizumab \| \| \| J \| : ##### Jackson Cross Cylinder ##### Jacobs Syndrome ##### Jacobsen Syndrome (11q Terminal Deletion Syndrome) ##### Japanese Encephalitis ##### Jarisch-Herxheimer Reaction ##### Jaundice ##### Jersey Finger ##### Jervell and Lange-Nielsen Syndrome ##### Jessner Lymphocytic Infiltration of the Skin ##### Job Syndrome ##### Jod-Basedow Syndrome ##### Jodhpur Technique ##### Joint Immobilization ##### Jugular Foramen Syndrome ##### Jugular Venous Distention ##### Jumpers Knee ##### Junctional Ectopic Tachycardia ##### Junctional Rhythm ##### Juvenile Absence Epilepsy ##### Juvenile Dermatomyositis ##### Juvenile Glaucoma ##### Juvenile Idiopathic Arthritis ##### Juvenile Myoclonic Epilepsy ##### Juvenile Xanthogranuloma \| \| \| K \| : ##### Kabuki Syndrome ##### Kallmann Syndrome ##### Kaposi Sarcoma ##### Kaposi Varicelliform Eruption ##### Kasabach-Merritt Syndrome ##### Kawasaki Disease ##### Kayser-Fleischer Ring ##### Kearns-Sayre Syndrome ##### Kegel Exercises ##### Kell Blood Group System ##### Keloid ##### Kentucky Domestic Violence ##### Kentucky KASPER and Controlled Substance Prescribing ##### Keratitis ##### Keratoacanthoma ##### Keratoconjunctivitis ##### Keratoconus ##### Keratoendotheliitis Fugax Hereditaria ##### Keratometer ##### Keratopathy ##### Keratosis Pilaris ##### Kernicterus ##### Kernig Sign ##### Keshan Disease ##### Ketamine ##### Ketamine in Acute and Chronic Pain Management ##### Ketamine Toxicity ##### Ketoacidosis ##### Ketoconazole ##### Ketogenic Diet ##### Ketorolac ##### Kidd Blood Group System ##### Kidney Transplantation ##### Kidney Trauma ##### Kienbock Disease ##### Kikuchi-Fujimoto Disease ##### Kingella Kingae ##### Kissing Bug Bite ##### Klebsiella Pneumonia ##### Kleihauer Betke Test ##### Kleine-Levin Syndrome (KLS) ##### Klinefelter Syndrome ##### Klippel Feil Syndrome ##### Klippel-Trenaunay-Weber Syndrome ##### Klumpke Palsy ##### Kluver-Bucy Syndrome ##### Knee Arthrocentesis ##### Knee Dislocation ##### Knee Effusion ##### Knee Extensor Mechanism Injuries ##### Knee Meniscal Tears ##### Knee Osteoarthritis ##### Koebner Phenomenon ##### Kohler Disease ##### Koilocytosis ##### Konno-Rastan Aortic Root Enlargement ##### Koplik Spots ##### Korsakoff Syndrome ##### Krabbe Disease ##### Kratom ##### Krukenberg Tumor ##### Kuru ##### Kyphoscoliosis ##### Kyphosis ##### Kyrle Disease \| \| \| L \| : ##### La Belle Indifference ##### La Crosse Encephalitis ##### Labetalol ##### Labial Adhesions ##### Labiaplasty, Labia Minora Reduction ##### Laboratory Assessment of Thyroid Function ##### Laboratory Evaluation of Acute Leukemia ##### Laboratory Evaluation of Alpha Thalassemia ##### Laboratory Evaluation of Beta Thalassemia ##### Laboratory Evaluation of Bone Marrow ##### Laboratory Evaluation of Coagulopathies ##### Laboratory Evaluation of Hereditary Hemochromatosis ##### Laboratory Evaluation of Immune Hemolytic Anemias ##### Laboratory Evaluation of Infertility ##### Laboratory Evaluation of Sepsis ##### Laboratory Evaluation of Tumor Biomarkers ##### Laboratory Tube Collection ##### Labyrinthectomy ##### Labyrinthitis ##### Lachman Test ##### Lacrimal Gland Malignancies ##### Lactic Acidosis ##### Lactose Intolerance ##### Lactulose ##### Lacunar Stroke ##### Lafora Disease ##### Lagophthalmos ##### Lambda Waves ##### Lambert-Eaton Myasthenic Syndrome ##### Lambl Excrescences ##### Laminectomy ##### Lamivudine ##### Lamotrigine ##### Landau-Kleffner Syndrome ##### Langerhans Cell Histiocytosis ##### Language Barrier ##### Laparoscopic Cholecystectomy ##### Laparoscopic Gastric Band Placement ##### Laparoscopic Gastric Bypass ##### Laparoscopic Inguinal Hernia Repair ##### Laparotomy ##### Large Bowel Obstruction ##### Laryngeal and Tracheal Stents ##### Laryngeal Botulinum Toxin Injection ##### Laryngeal Cancer ##### Laryngeal Clefts ##### Laryngeal Fracture ##### Laryngeal Injury ##### Laryngeal Mask Airway ##### Laryngeal Papillomas ##### Laryngomalacia ##### Laryngopharyngeal Reflux ##### Laryngotracheal Stenosis ##### Laryngotracheobronchitis ##### Laser Carbon Dioxide Resurfacing ##### Laser Complications ##### Laser Enucleation of the Prostate (HoLEP and ThuLEP) ##### Laser Erbium-Yag Resurfacing ##### Laser Fitzpatrick Skin Type Recommendations ##### Laser Hair Removal ##### Laser In Situ Keratomileusis (LASIK) ##### Laser Principles in Ophthalmology ##### Laser Revision of Scars ##### Laser Tattoo Removal ##### Laser Trabeculoplasty ##### Laser Treatment of Pigmented Lesions ##### Latanoprost ##### Late Decelerations ##### Late-Life Depression ##### Latent Autoimmune Diabetes ##### Latent Safety Threat Identification via Medical Simulation ##### Latent Tuberculosis ##### Lateral Ankle Instability ##### Lateral Collateral Ligament Knee Injury ##### Lateral Epicondylitis (Tennis Elbow) ##### Lateral Medullary Syndrome (Wallenberg Syndrome) ##### Lateral Orbital Canthotomy ##### Lateral Patellar Compression Syndrome ##### Lateral Wall Myocardial Infarction ##### Latex Allergy ##### Latissimus Dorsi Flap ##### Lattice Corneal Dystrophy ##### Laugier-Hunziker Syndrome ##### Laxatives ##### Le Fort Fractures ##### Le Fort Osteotomy ##### Lead Encephalopathy ##### Lead Toxicity ##### Learning Disability ##### Learning Taxonomies in Medical Simulation ##### Learning Theories ##### Leber Hereditary Optic Neuropathy (LHON) ##### Lecithin Sphingomyelin Ratio ##### Lecithin-Cholesterol Acyltransferase Deficiency ##### Leech Bite ##### Leflunomide ##### Left Atrial Enlargement ##### Left Bundle Branch Block ##### Left Heart Cardiac Catheterization ##### Left Ventricular Assist Devices ##### Left Ventricular Ejection Fraction ##### Left Ventricular Failure ##### Left Ventricular False Aneurysm ##### Left Ventricular Hypertrophy ##### Left Ventricular Noncompaction Cardiomyopathy ##### Left Ventricular Outflow Tract Obstruction ##### Left Ventricular Rupture ##### Legg-Calve-Perthes Disease ##### Legionnaires Disease ##### Leiomyoma ##### Leiomyosarcoma ##### Leishmaniasis ##### Lemborexant ##### Lemierre Syndrome ##### Lennox-Gastaut Syndrome ##### Lens Abscess ##### Lens-Induced Glaucoma ##### Lens-Induced Inflammation ##### Lensmaker’s Equation ##### Lensometry ##### Lenticonus ##### Lentigo Maligna Melanoma ##### Lenvatinib ##### Leopold Maneuvers ##### Leprosy ##### Leptomeningeal Carcinomatosis ##### Leptomeningeal Cyst ##### Leptospirosis ##### Lesch-Nyhan Syndrome ##### Leser-Trélat Sign ##### Leucovorin ##### Leukemia ##### Leukemia Cutis ##### Leukocoria ##### Leukocyte Adhesion Deficiency ##### Leukocytoclastic Vasculitis ##### Leukocytosis ##### Leukotriene Receptor Antagonists ##### Leuprolide ##### Levetiracetam ##### Levodopa (L-Dopa) ##### Levodopa/Carbidopa/Entacapone Combination Therapy ##### Levofloxacin ##### Levonorgestrel ##### Levothyroxine ##### Lewy Bodies ##### Lewy Body Dementia ##### Leydig Cell Cancer of the Testis ##### Lhermitte Sign ##### Libman-Sacks Endocarditis ##### Lichen Nitidus ##### Lichen Planopilaris ##### Lichen Planus ##### Lichen Planus Erosive Form ##### Lichen Sclerosus ##### Lichen Simplex Chronicus ##### Lichen Striatus ##### Lichenification ##### Lid Wiper Epitheliopathy ##### Liddle Syndrome (Pseudohyperaldosteronism) ##### Lidocaine ##### Lifestyle Mental Wellbeing for the Primary Care Visit ##### Lifestyle Mindfulness In Clinical Practice ##### Lifestyle Modification for Diabetes and Heart Disease Prevention ##### Lifestyle Prevention Measures for the Clinic Visit ##### Li-Fraumeni Syndrome ##### Lightning Injuries ##### Limbal Epithelial Transplant ##### Linaclotide ##### Linear IGA Dermatosis ##### Linezolid ##### Lionfish, Scorpionfish, and Stonefish Toxicity ##### Lip Implants ##### Lipedema ##### Lipemia Retinalis ##### Lipid Emulsion Therapy ##### Lipid-Lowering Drug Therapy ##### Lipodermatosclerosis ##### Lipodystrophies ##### Lipoid Pneumonia ##### Lipoid Proteinosis ##### Lipoma ##### Lipoma Pathology ##### Lipoprotein A ##### Lipoprotein Lipase Deficiency ##### Lipoprotein X-Induced Hyperlipidemia ##### Liposarcoma ##### Liposuction ##### Liraglutide ##### Lisch Nodules ##### Lisfranc Dislocation ##### Lisinopril ##### Lissencephaly ##### Listeria Monocytogenes ##### Lithium ##### Lithium Toxicity ##### Littoral Cell Splenic Angioma ##### Livedoid Vasculopathy ##### Liver Abscess ##### Liver Angiosarcoma ##### Liver Biopsy ##### Liver Cystic Disease ##### Liver Disease in Pregnancy ##### Liver Function Tests ##### Liver Imaging ##### Liver Metastasis ##### Liver Toxicity ##### Liver Transplantation ##### Liver Trauma ##### Loading Dose ##### Lobectomy ##### Lobular Breast Carcinoma ##### Local and Regional Anesthesia in Ophthalmology and Ocular Trauma ##### Local Anesthesia Techniques in Dentistry and Oral Surgery ##### Local Anesthetic Drugs Used In Dentistry ##### Local Anesthetic Toxicity ##### Local Anesthetics in Children ##### Localization-Related Epilepsies on EEG ##### Localized Prostate Cancer ##### Locked-in Syndrome ##### Loeffler Endocarditis ##### Lofgren Syndrome ##### Lomitapide ##### Long Arm Splinting ##### Long QT Syndrome ##### Loop Diuretics ##### Loose Anagen Syndrome ##### Loperamide ##### Loratadine ##### Lorazepam ##### Losartan ##### Lovastatin ##### Low Cholesterol Diet ##### Low Fat Diet ##### Low HDL Cholesterol ##### Low Residue Diet ##### Low Vision Aids ##### Low-Carbohydrate Diet ##### Lower Extremity Amputation ##### Lower Extremity Blocks ##### Lower Eyelid Blepharoplasty ##### Lower Eyelid Laxity Examination ##### Lower Eyelid Reconstruction ##### Lower Gastrointestinal Bleeding ##### Lower Genitourinary Trauma ##### Low-FODMAP Diet ##### Low-Grade Gliomas ##### Low-Molecular-Weight Heparin (LMWH) ##### Ludwig Angina ##### Lumateperone ##### Lumbar Degenerative Disk Disease ##### Lumbar Disc Herniation ##### Lumbar Facet Arthropathy ##### Lumbar Plexus Block ##### Lumbar Puncture ##### Lumbar Spinal Stenosis ##### Lumbar Spine Imaging ##### Lumbar Stabilization ##### Lumbar Sympathetic Block ##### Lumbar Sympatholysis ##### Lumbosacral Disc Injuries ##### Lumbosacral Discogenic Syndrome ##### Lumbosacral Facet Syndrome ##### Lumbosacral Plexopathy ##### Lumbosacral Radiculopathy ##### Lung Abscess ##### Lung Adenocarcinoma ##### Lung Biopsy Techniques and Clinical Significance ##### Lung Cancer ##### Lung Cancer Screening ##### Lung Decortication ##### Lung Exam ##### Lung Imaging ##### Lung Isolation Anesthesia ##### Lung Metastasis ##### Lung Nontuberculous Mycobacterial Infections ##### Lung Pancoast Tumor ##### Lung Perfusion Scan ##### Lung Sounds ##### Lung Torsion ##### Lung Transplant Rejection ##### Lung Transplantation ##### Lung Ventilation Perfusion Scan (VQ Scan) ##### Lung Volume Reduction Surgery ##### Lunotriquetral Instability ##### Lupus Miliaris Disseminatus Faciei ##### Lupus Nephritis ##### Lupus Pernio ##### Lurasidone ##### Luspatercept ##### Luteinizing Hormone Deficiency ##### Lutembacher Syndrome ##### Lyme Carditis ##### Lyme Disease ##### Lymph Node Dissection ##### Lymphadenopathy ##### Lymphangioleiomyomatosis ##### Lymphangioma ##### Lymphangitic Carcinomatosis ##### Lymphedema ##### Lymphoblastic Lymphoma ##### Lymphocyte Depleted Hodgkin Lymphoma ##### Lymphocytic Esophagitis ##### Lymphocytic Hypophysitis ##### Lymphocytosis ##### Lymphogranuloma Venereum Infection ##### Lymphoma ##### Lymphomatoid Papulosis ##### Lymphoplasmacytic Lymphoma ##### Lymphoproliferative Disorders ##### Lymphoscintigraphy ##### Lynch Syndrome (Hereditary Nonpolyposis Colorectal Cancer) ##### Lysergic Acid Diethylamide Toxicity ##### Lysosomal Storage Disease ##### Lytic Bone Lesions \| \| \| M \| : ##### MacConkey Medium ##### Macrocephaly ##### Macrocytic Anemia ##### Macrocytosis ##### Macroglossia ##### Macrolides ##### Macrosomia ##### Macular Degeneration ##### Macular Edema ##### Macular Hole ##### Madelung Deformity ##### Magnesium ##### Magnesium Sulfate ##### Magnesium Toxicity ##### Magnetic Resonance Imaging Contraindications ##### Magnetic Resonance Imaging Patient Positioning ##### Magnetic Resonance Imaging Physics ##### Maintaining Confidentiality and Psychological Safety in Medical Simulation ##### Major Aortopulmonary Collateral Arteries ##### Major Depressive Disorder ##### Major Neurocognitive Disorder (Dementia) ##### Malabsorption Syndromes ##### Malakoplakia ##### Malar Rash ##### Malaria ##### Malaria Prophylaxis ##### Malassezia Furfur ##### Male Breast Cancer ##### Male Hypogonadism ##### Male Infertility ##### Male Urinary Retention: Acute and Chronic ##### Malignancy-Related Hypercalcemia ##### Malignant Atrophic Papulosis ##### Malignant Eyelid Lesions ##### Malignant Hyperthermia ##### Malignant Melanoma ##### Malignant Melanoma Metastatic to the Central Nervous System ##### Malignant Mesothelioma ##### Malignant Orbital Tumors ##### Malignant Pleural Effusion ##### Malignant Salivary Gland Tumors ##### Malignant Tumors of the Palate ##### Malingering ##### Mallampati Score ##### Mallet Finger Injuries ##### Mallory Bodies ##### Mallory-Weiss Syndrome ##### MALToma ##### Mammary Duct Ectasia ##### Mammary Paget Disease ##### Mammography ##### Mammography BI RADS Grading ##### Mammoscintigraphy ##### Man in a Barrel Syndrome ##### Managed Care ##### Managed Care Economics ##### Managed Care Organization ##### Management of Antithrombotics Around Gastrointestinal Procedures ##### Management of Cardiovascular and Diabetes Risks Based on National Guidelines ##### Management of Weight Loss Plateau ##### Management Skills ##### Managing Fresh-Frozen Plasma Transfusion Adverse Effects: Allergic Reactions, TACO, and TRALI ##### Mandatory Reporting Laws ##### Mandible Body Fracture ##### Mandible Dislocation ##### Mandible Fracture ##### Mandible Osteoradionecrosis ##### Mandible Reconstruction ##### Mandibulofacial Dysostosis ##### Manganese Toxicity ##### Mania ##### Mannitol ##### Manouguian Procedure ##### Mantle Cell Lymphoma ##### Manual Small Incision Cataract Surgery ##### MAO Inhibitors ##### Maple Syrup Urine Disease ##### Marburg Virus Disease ##### Marchiafava-Bignami Disease ##### Marcus Gunn Jaw-Winking Syndrome ##### Marcus Gunn Pupil ##### Marfan Syndrome ##### Marijuana ##### Marijuana and Maternal, Perinatal, and Neonatal Outcomes ##### Marjolin Ulcer ##### Mass Spectrometer ##### Massive Transfusion ##### MAST Pants ##### Mastalgia ##### Mastectomy ##### Mastery Learning in Medical Simulation ##### Mastocytoma ##### Mastoidectomy ##### Mastoiditis ##### Mastopexy (Breast Lift) ##### Mattox Maneuver ##### Maturity Onset Diabetes in the Young ##### Mavacamten ##### Maxillary Fracture ##### May-Hegglin Anomaly ##### May-Thurner Syndrome ##### McArdle Disease (Glycogen Storage Disease Type 5) ##### McCune-Albright Syndrome ##### McKenzie Back Exercises ##### McNemar And Mann-Whitney U Tests ##### McRoberts Maneuver ##### Mean ##### Mean Corpuscular Volume ##### Measles ##### Mebendazole ##### Mechanical Aortic Valve Replacement ##### Mechanical Back Strain ##### Mechanical Ventilation ##### Mechanical Ventilation and Extracorporeal Membrane Oxygenation Considerations in COVID-19 ##### Meckel Diverticulum ##### Meckel Scan ##### Meclizine ##### Meconium ##### Meconium Aspiration ##### Meconium Ileus ##### Meconium Plug Syndrome ##### Medial Collateral Ligament Knee Injury ##### Medial Epicondylar Elbow Fractures ##### Medial Epicondyle Apophysitis (Little League Elbow) ##### Medial Epicondyle Injection ##### Medial Epicondylitis ##### Medial Medullary Syndrome ##### Medial Tibial Stress Syndrome ##### Median ##### Median Nerve Block Techniques ##### Median Nerve Injury ##### Median Nerve Palsy ##### Mediastinal Cancer ##### Mediastinal Carcinoid Tumors ##### Mediastinal Germ Cell Tumors ##### Mediastinal Nonseminoma ##### Mediastinal Seminoma ##### Mediastinal Trauma ##### Mediastinitis ##### Mediastinoscopy ##### Medical Error Reduction and Prevention ##### Medical Ethics ##### Medical History ##### Medical Malpractice ##### Medical Simulation Center Director as a System Improvement Leader ##### Medical Simulation Fellowships ##### Medicare and Medicaid ##### Medicare and Medicaid Accreditation and Deemed Status ##### Medicare, Medicaid, and Military and VA Healthcare Programs ##### Medication Dispensing Errors and Prevention ##### Medication Overuse Headache ##### Medication Routes of Administration ##### Medication-Overuse Headache ##### Mediterranean Diet ##### Medium-Chain Acyl-CoA Dehydrogenase Deficiency ##### Medroxyprogesterone ##### Medullary Breast Carcinoma ##### Medullary Sponge Kidney ##### Medullary Thyroid Cancer ##### Medulloblastoma ##### Mega Cisterna Magna ##### Megaloblastic Anemia ##### Megalocornea ##### Megestrol ##### Meibomian Gland Disease ##### Meige Syndrome ##### Meigs Syndrome ##### Melanocytic Nevi ##### Melanoma of the Head and Neck ##### Melanoma Pathology ##### Melanosis Coli ##### Melanotic Neuroectodermal Tumor of Infancy ##### Melas Syndrome ##### Melasma ##### Melatonin ##### Memantine ##### Membranous Nephropathy ##### Mendelson Syndrome ##### Menetrier Disease ##### Meniere Disease ##### Meningioma ##### Meningitis ##### Meningocele ##### Meningococcal Disease (Neisseria meningitidis Infection) ##### Meningococcal Vaccine ##### Meningococcemia ##### Meniscectomy ##### Menkes Disease ##### Menopause ##### Menstrual Suppression ##### Menstrual-Related Headache ##### Mental Health Challenges in Caring for American Indians and Alaska Natives ##### Mental Nerve Block ##### Mental Status Examination ##### Meperidine ##### Meralgia Paresthetica ##### Mercaptan Toxicity ##### Mercaptopurine ##### Mercury Toxicity ##### Merkel Cell Carcinoma of the Skin ##### Meromelia ##### Mesalamine (USAN) ##### Mesenteric Adenitis ##### Mesenteric Artery Ischemia ##### Mesenteric Artery Thrombosis ##### Mesenteric Vasculitis ##### Mesenteric Venous Thrombosis ##### Mesial Temporal Lobe Epilepsy ##### Mesna ##### Metabolic Consequences of Weight Reduction ##### Metabolic Dysfunction-Associated Steatotic Liver Disease (MΑSLD) ##### Metabolic Syndrome ##### Metacarpal Fracture ##### Metachromatic Leukodystrophy ##### Metal Fume Fever ##### Metastatic Melanoma ##### Metatarsal Fractures ##### Meteorism ##### Metformin ##### Metformin-Associated Lactic Acidosis (MALA) ##### Methacholine Challenge Test ##### Methadone ##### Methamphetamine ##### Methanol Toxicity ##### Methemoglobinemia ##### Methicillin-Resistant Staphylococcus aureus ##### Methimazole ##### Methocarbamol ##### Methods of Estimation of Time Since Death ##### Methohexital ##### Methotrexate ##### Methyldopa ##### Methylene Blue ##### Methylphenidate ##### Methylprednisolone ##### Methylxanthines ##### Metoclopramide ##### Metolazone ##### Metoprolol ##### Metronidazole ##### Mexiletine ##### Microalbuminuria ##### Microcystic Adnexal Carcinoma ##### Microcytic Hypochromic Anemia ##### Microdermabrasion ##### Microdiscectomy ##### Microneedling ##### Micropenis ##### Microphthalmia With Cyst ##### Microscopic Polyangiitis ##### Microspherophakia ##### Microsporidial Keratitis ##### Microsporidium ##### Mid Forehead Brow Lift ##### Midazolam ##### Middle Cerebral Artery Stroke ##### Middle Ear Cholesteatoma ##### Midface Lift ##### Midgut Malrotation ##### Midgut Volvulus ##### Mid-Ventricular Takotsubo Cardiomyopathy ##### Mifepristone ##### Migraine Headache ##### Migraine Headache in Childhood ##### Migraine Medications ##### Migraine Prophylaxis ##### Migraine Surgical Interventions ##### Migraine With Aura ##### Migraine-Associated Vertigo ##### Migratory Thrombophlebitis ##### Mild Cognitive Impairment ##### Milia ##### Miliaria ##### Miliary Tuberculosis ##### Military, Civil and International Regulations to Decrease Human Factor Errors In Aviation ##### Milk Thistle ##### Milk-Alkali Syndrome ##### Millard-Gubler Syndrome ##### Miller Fisher Syndrome ##### Millipede Envenomation ##### Milrinone ##### Minimal Change Disease ##### Minimally Invasive Aortic Valve Surgery ##### Minimally Invasive Glaucoma Surgery ##### Minimally Invasive Mitral Valve Surgery ##### Minimum Alveolar Concentration ##### Minnesota Multiphasic Personality Inventory ##### Minocycline ##### Minoxidil ##### Mirabegron ##### Mirizzi Syndrome ##### Mirtazapine ##### Misoprostol ##### Mitomycin ##### Mitral Commissurotomy ##### Mitral Regurgitation ##### Mitral Stenosis ##### Mitral Valve Insufficiency ##### Mitral Valve Prolapse ##### Mitral Valve Repair ##### Mittelschmerz ##### Mixed Connective Tissue Disease ##### Mixed Urinary Incontinence ##### MMR Vaccine ##### Mo99 - Tc99m Generator ##### Modafinil ##### Mode ##### Modified Ashworth Scale ##### Modified Blalock-Taussig-Thomas Shunt ##### Mohs Micrographic Surgery ##### Mohs Micrographic Surgery Appropriate Use Criteria (AUC) Guidelines ##### Mohs Micrographic Surgery Design and Execution of Advancement Flaps ##### Mohs Micrographic Surgery Design and Execution of Intermediate and Complex Closure ##### Mohs Micrographic Surgery Design and Execution of Pedicle Flaps ##### Mohs Micrographic Surgery Design and Execution of Rotation Flaps ##### Mohs Micrographic Surgery Evaluation and Treatment of Dermatofibrosarcoma Protuberans ##### Mohs Micrographic Surgery Evaluation and Treatment of Microcystic Adnexal Carcinoma ##### Mohs Micrographic Surgery Indications for Termination ##### Mohs Micrographic Surgery Interpretation of Positive Versus Negative Specimens ##### Mohs Micrographic Surgery Interpretation of Technical Problems Impacting Slide Quality ##### Mohs Micrographic Surgery Management of Melanoma and Melanoma In Situ ##### Mohs Micrographic Surgery Mapping Techniques ##### Mohs Micrographic Surgery of Uncommon Tumors (Angiosarcoma, Eccrine, Paget, and Merkel Cell) ##### Mohs Micrographic Surgery Risks: Carcinogenicity of Immunosuppressive Agents ##### Mohs Micrographic Surgery Safe and Effective Perioperative Care ##### Mohs Micrographic Surgery Section of Specimens Using Cryostat, Stain, and Immunostain ##### Mohs Micrographic Surgery Surgical Complication Management ##### Mohs Micrographic Surgery, Optimizing Wound Healing ##### Mohs Micrographic Surgery, Scar Revision ##### Mohs Micrographic Surgery: AAD Guidelines on Superficial Radiation and Brachytherapy ##### Mohs Micrographic Surgery: Anticoagulants and Hemostatic Agents Impact ##### Mohs Micrographic Surgery: Chemoprophylaxis Options for Nicotinamide and Oral Retinoids ##### Mohs Micrographic Surgery: Chemotherapeutic Agents Used in Cutaneous Oncology ##### Mohs Micrographic Surgery: Design and Execution of Secondary Intention ##### Mohs Micrographic Surgery: Design and Execution of Tunneled Flaps ##### Mohs Micrographic Surgery: Evaluation and Treatment of Nonmelanoma Skin Cancer ##### Mohs Micrographic Surgery: Specimen-Releasing Techniques, Marking, and Orientation ##### Moisturizers ##### Molecular Absorbent Recirculating System ##### Molecular Genetics Testing ##### Molluscum Contagiosum ##### Mondor Disease ##### Monilethrix ##### Monoamine Oxidase Inhibitor Toxicity ##### Monoamine Oxidase Inhibitors (MAOI) ##### Monoarticular Arthritis ##### Monoclonal Gammopathy of Undetermined Significance ##### Monofocal Intraocular Lenses ##### Monomelic Amyotrophy (Hirayama Disease) ##### Mononucleosis ##### Monospot Test ##### Monosynaptic Reflex ##### Monteggia Fractures ##### Montelukast ##### Mood Disorder ##### Mood Stabilizers ##### Morel Lavallee Lesion ##### Morgagni Hernia ##### Morning Glory Syndrome ##### Moro Reflex ##### Morphea ##### Morphine ##### Morton Neuroma ##### Morvan Syndrome ##### Mosquito Bites ##### Motion Sickness ##### Motivational Interviewing ##### Motor Evoked Potential ##### Motor Neuron Disease ##### Motor Vehicle Collisions ##### Moulage in Medical Simulation ##### Moxifloxacin ##### Moyamoya Disease ##### M-Plasty ##### Mpox (Monkeypox) ##### MR Angiogram ##### MRI Patient Safety And Care ##### Mu Receptors ##### Mucinous Breast Carcinoma ##### Mucinous Cystic Pancreatic Neoplasms ##### Mucocele and Ranula ##### Mucoepidermoid Lung Tumor ##### Mucolytic Medications ##### Mucopolysaccharidosis Type II ##### Mucormycosis ##### Mucous Membrane Graft ##### Muehrcke Lines of the Fingernails ##### MUGA Scan ##### Muir-Torre Syndrome ##### Multidirectional Shoulder Instability ##### Multifocal Atrial Tachycardia ##### Multifocal Electroretinogram ##### Multifocal Motor Neuropathy ##### Multimodal Postoperative Pain Control After Orthopaedic Surgery ##### Multiple Birth Delivery ##### Multiple Endocrine Neoplasia Type 1 ##### Multiple Endocrine Neoplasias Type 2 ##### Multiple Endocrine Neoplasias Type 4 ##### Multiple Myeloma ##### Multiple Sclerosis ##### Multivessel Disease ##### Mumps ##### Mupirocin ##### Mural Thrombi ##### Muscarinic Agonists ##### Muscarinic Antagonists ##### Muscle Contraction Tension Headache ##### Muscle Cramps ##### Muscle Strength Grading ##### Muscular Dystrophy ##### Muscular Hematoma ##### Musculoskeletal Amyloidosis Imaging ##### Musculoskeletal Examination ##### Mushroom Toxicity ##### Myasthenia Gravis ##### Mycobacterium avium Complex ##### Mycobacterium chelonae Infection ##### Mycobacterium kansasii Infection ##### Mycobacterium marinum Infection ##### Mycoplasma Infections ##### Mycoplasma Pneumonia ##### Mycoplasma pneumoniae–Induced Rash and Mucositis (MIRM) ##### Mycosis Fungoides ##### Mycotic Aneurysm ##### Myelodysplastic Syndrome ##### Myelomeningocele ##### Myeloperoxidase Deficiency ##### Myelophthisic Anemia ##### Myeloproliferative Neoplasms ##### Myers-Briggs Type Indicator ##### Myocardial Infarction ##### Myocardial Infarction Serum Markers ##### Myocardial Perfusion Scan ##### Myocardial Protection ##### Myocardial Stunning and Hibernation ##### Myocardial Viability ##### Myoclonic Epilepsy and Ragged Red Fibers ##### Myoclonic Epilepsy of Infancy ##### Myoclonus ##### Myofascial Pain ##### Myofascial Pain Syndrome ##### Myopathy ##### Myopericarditis ##### Myopia ##### Myopic Foveoschisis ##### Myotonia ##### Myotonia Congenita ##### Myotonic Dystrophy ##### Myxedema \| \| \| N \| : ##### Nabothian Cyst ##### N-Acetylcysteine ##### Nadolol ##### Naegleria Infection and Primary Amebic Meningoencephalitis ##### Nail Biopsy ##### Nail Clubbing ##### Nail Psoriasis ##### Nail Surgery ##### Nail-Patella Syndrome ##### Nalbuphine ##### Naloxone ##### Naltrexone ##### Napalm Toxicity ##### Naproxen ##### Narcissistic Personality Disorder ##### Narcolepsy ##### Narcotic Bowel Syndrome ##### Nasal Foreign Body ##### Nasal Fracture Reduction ##### Nasal Polyps ##### Nasal Septal Fracture ##### Nasogastric Tube ##### Nasolacrimal Duct Obstruction ##### Naso-Orbito-Ethmoid Fractures ##### Nasopharyngeal Airway ##### Nasopharyngeal Angiofibroma ##### Nasopharyngeal Cancer ##### Nasopharyngeal Carcinoma ##### Nasopharyngeal Carcinoma (NPC, Lymphoepithelioma) ##### Nasoseptal Flap ##### Nasotracheal Intubation ##### Natalizumab ##### National Guidelines for Physical Activity ##### National Quality Forum ##### Natriuretic Peptide B Type Test ##### Natural Family Planning ##### Nebivolol ##### Neck Abscess ##### Neck Cancer Resection and Dissection ##### Neck Rejuvenation ##### Neck Trauma ##### Necrobiosis Lipoidica ##### Necrolytic Acral Erythema ##### Necrolytic Migratory Erythema ##### Necrosis Pathology ##### Necrotizing (Malignant) Otitis Externa ##### Necrotizing Enterocolitis ##### Necrotizing Fasciitis ##### Necrotizing Periodontal Diseases ##### Needlestick ##### Needs Assessment and Stakeholders in Medical Simulation Curriculum Development ##### Negative Pressure Wound Therapy ##### Nelson Syndrome ##### Neointimal Hyperplasia ##### Neomycin ##### Neonatal Abstinence Syndrome ##### Neonatal Anesthesia ##### Neonatal Brain Tumors ##### Neonatal Conjunctivitis ##### Neonatal EEG ##### Neonatal Evaluation ##### Neonatal HIV ##### Neonatal Hyperglycemia ##### Neonatal Hypertension ##### Neonatal Hypoglycemia ##### Neonatal Jaundice ##### Neonatal Lupus Erythematosus ##### Neonatal Meningitis ##### Neonatal MIS-C ##### Neonatal Myasthenia Gravis ##### Neonatal Respiratory Distress Syndrome ##### Neonatal Seizures and Neonatal Epilepsy ##### Neonatal Sepsis ##### Neonatal Therapeutic Hypothermia ##### Neoplastic Lumbosacral Plexopathy ##### Neostigmine ##### Neovascular Glaucoma ##### Nephritic Syndrome ##### Nephrocalcinosis ##### Nephrogenic Systemic Fibrosis ##### Nephrotic Syndrome ##### Nephrotoxic Medications ##### Nerve Agents ##### Nerve Block Anesthesia ##### Nerve Conduction Studies and Electromyography ##### Nerve Stimulation ##### Netarsudil Ophthalmic Solution ##### Neurally Adjusted Ventilatory Assist (NAVA) ##### Neurapraxia ##### Neuraxial Neurolysis ##### Neurilemmoma ##### Neuroacanthocytosis ##### Neuroanatomy, Abducens Nucleus ##### Neuroanatomy, Amygdala ##### Neuroanatomy, Anterior Choroidal Arteries ##### Neuroanatomy, Anterior Inferior Cerebellar Arteries ##### Neuroanatomy, Anterior White Commissure ##### Neuroanatomy, Area Postrema ##### Neuroanatomy, Auditory Pathway ##### Neuroanatomy, Auerbach Plexus ##### Neuroanatomy, Autonomic Nervous System Visceral Afferent Fibers and Pain ##### Neuroanatomy, Basal Ganglia ##### Neuroanatomy, Bitemporal Hemianopsia ##### Neuroanatomy, Brain Arteries ##### Neuroanatomy, Brain Veins ##### Neuroanatomy, Brainstem ##### Neuroanatomy, Broca Area ##### Neuroanatomy, Cavernous Sinus ##### Neuroanatomy, Cerebellar Dysfunction ##### Neuroanatomy, Cerebellum ##### Neuroanatomy, Cerebral Aqueduct (Sylvian) ##### Neuroanatomy, Cerebral Blood Supply ##### Neuroanatomy, Cerebral Cortex ##### Neuroanatomy, Cerebral Hemisphere ##### Neuroanatomy, Cerebrospinal Fluid ##### Neuroanatomy, Choroid Plexus ##### Neuroanatomy, Ciliary Ganglion ##### Neuroanatomy, Cingulate Cortex ##### Neuroanatomy, Circle of Willis ##### Neuroanatomy, Conus Medullaris ##### Neuroanatomy, Corpus Callosum ##### Neuroanatomy, Cortical Primary Auditory Area ##### Neuroanatomy, Corticobulbar Tract ##### Neuroanatomy, Corticospinal Cord Tract ##### Neuroanatomy, Cranial Meninges ##### Neuroanatomy, Cranial Nerve ##### Neuroanatomy, Cranial Nerve 0 (Terminal Nerve) ##### Neuroanatomy, Cranial Nerve 1 (Olfactory) ##### Neuroanatomy, Cranial Nerve 10 (Vagus Nerve) ##### Neuroanatomy, Cranial Nerve 11 (Accessory) ##### Neuroanatomy, Cranial Nerve 12 (Hypoglossal) ##### Neuroanatomy, Cranial Nerve 2 (Optic) ##### Neuroanatomy, Cranial Nerve 3 (Oculomotor) ##### Neuroanatomy, Cranial Nerve 4 (Trochlear) ##### Neuroanatomy, Cranial Nerve 5 (Trigeminal) ##### Neuroanatomy, Cranial Nerve 6 (Abducens) ##### Neuroanatomy, Cranial Nerve 7 (Facial) ##### Neuroanatomy, Cranial Nerve 8 (Vestibulocochlear) ##### Neuroanatomy, Cranial Nerve 9 (Glossopharyngeal) ##### Neuroanatomy, Decerebrate Rigidity ##### Neuroanatomy, Dentate Nucleus ##### Neuroanatomy, Dorsal Root Ganglion ##### Neuroanatomy, Dural Venous Sinuses ##### Neuroanatomy, Ear ##### Neuroanatomy, Edinger–Westphal Nucleus (Accessory Oculomotor Nucleus) ##### Neuroanatomy, Extrapyramidal System ##### Neuroanatomy, Facial Colliculus ##### Neuroanatomy, Falx Cerebri ##### Neuroanatomy, Fourth Ventricle ##### Neuroanatomy, Frontal Cortex ##### Neuroanatomy, Geniculate Ganglion ##### Neuroanatomy, Globus Pallidus ##### Neuroanatomy, Gray Matter ##### Neuroanatomy, Hippocampus ##### Neuroanatomy, Hypothalamus ##### Neuroanatomy, Inferior Colliculus ##### Neuroanatomy, Infratrochlear Nerve ##### Neuroanatomy, Internal Capsule ##### Neuroanatomy, Interstitial Nucleus of Cajal ##### Neuroanatomy, Lateral Corticospinal Tract ##### Neuroanatomy, Limbic System ##### Neuroanatomy, Locus Coeruleus ##### Neuroanatomy, Lower Motor Neuron Lesion ##### Neuroanatomy, Mammillary Bodies ##### Neuroanatomy, Medial Lemniscus (Reils Band, Reils Ribbon) ##### Neuroanatomy, Medulla Oblongata ##### Neuroanatomy, Mesencephalon Midbrain ##### Neuroanatomy, Middle Cerebral Artery ##### Neuroanatomy, Middle Meningeal Arteries ##### Neuroanatomy, Motor Neuron ##### Neuroanatomy, Neural Crest ##### Neuroanatomy, Neural Taste Pathway ##### Neuroanatomy, Neural Tube Development and Stages ##### Neuroanatomy, Neurapraxia ##### Neuroanatomy, Neuron Action Potential ##### Neuroanatomy, Neurons ##### Neuroanatomy, Nodes of Ranvier ##### Neuroanatomy, Nucleus Ambiguus ##### Neuroanatomy, Nucleus Caudate ##### Neuroanatomy, Nucleus Fastigial ##### Neuroanatomy, Nucleus Gracilis ##### Neuroanatomy, Nucleus Gustatory ##### Neuroanatomy, Nucleus Lateral Geniculate ##### Neuroanatomy, Nucleus Raphe ##### Neuroanatomy, Nucleus Solitarius ##### Neuroanatomy, Nucleus Suprachiasmatic ##### Neuroanatomy, Nucleus Supraoptic ##### Neuroanatomy, Nucleus Vestibular ##### Neuroanatomy, Occipital Lobe ##### Neuroanatomy, Optic Chiasm ##### Neuroanatomy, Optic Tract ##### Neuroanatomy, Otic Ganglion ##### Neuroanatomy, Parasympathetic Nervous System ##### Neuroanatomy, Pars Nervosa ##### Neuroanatomy, Periaqueductal Gray ##### Neuroanatomy, Pons ##### Neuroanatomy, Pontine Micturition Center ##### Neuroanatomy, Postcentral Gyrus ##### Neuroanatomy, Posterior Cerebral Arteries ##### Neuroanatomy, Posterior Column (Dorsal Column) ##### Neuroanatomy, Precentral Gyrus ##### Neuroanatomy, Prefrontal Cortex ##### Neuroanatomy, Pterygoid Plexus ##### Neuroanatomy, Pterygopalatine Ganglion ##### Neuroanatomy, Pupillary Dilation Pathway ##### Neuroanatomy, Pupillary Light Reflexes and Pathway ##### Neuroanatomy, Putamen ##### Neuroanatomy, Pyramidal Tract ##### Neuroanatomy, Pyramidal Tract Lesions ##### Neuroanatomy, Recurrent Artery of Heubner ##### Neuroanatomy, Red Nucleus ##### Neuroanatomy, Reticular Activating System ##### Neuroanatomy, Reticular Formation ##### Neuroanatomy, Retina ##### Neuroanatomy, Semilunar Ganglion ##### Neuroanatomy, Sensory Nerves ##### Neuroanatomy, Somatic Nervous System ##### Neuroanatomy, Somatosensory Cortex ##### Neuroanatomy, Spinal Cord ##### Neuroanatomy, Spinal Cord Arteries ##### Neuroanatomy, Spinal Cord Morphology ##### Neuroanatomy, Spinal Cord Myotatic Reflex ##### Neuroanatomy, Spinal Cord Veins ##### Neuroanatomy, Spinal Nerves ##### Neuroanatomy, Spinal Trigeminal Nucleus ##### Neuroanatomy, Spine ##### Neuroanatomy, Spinocerebellar Dorsal Tract ##### Neuroanatomy, Spinothalamic Tract ##### Neuroanatomy, Stellate Ganglion ##### Neuroanatomy, Suboccipital Nerve ##### Neuroanatomy, Substantia Gelatinosa ##### Neuroanatomy, Substantia Nigra ##### Neuroanatomy, Subthalamic Nucleus ##### Neuroanatomy, Superior and Inferior Olivary Nucleus (Superior and Inferior Olivary Complex) ##### Neuroanatomy, Superior Cervical Ganglion ##### Neuroanatomy, Superior Colliculus ##### Neuroanatomy, Superior Petrosal Sinus ##### Neuroanatomy, Superior Sagittal Sinus ##### Neuroanatomy, Sylvian Fissure ##### Neuroanatomy, Sympathetic Nervous System ##### Neuroanatomy, Tectospinal Tract ##### Neuroanatomy, Temporal Lobe ##### Neuroanatomy, Tentorium Cerebelli ##### Neuroanatomy, Thalamic Nuclei ##### Neuroanatomy, Thalamocortical Radiations ##### Neuroanatomy, Thalamus ##### Neuroanatomy, Touch Receptor ##### Neuroanatomy, Trigeminal Nucleus ##### Neuroanatomy, Trigeminal Reflexes ##### Neuroanatomy, Unmyelinated Nerve Fibers ##### Neuroanatomy, Upper Motor Nerve Signs ##### Neuroanatomy, Upper Motor Neuron Lesion ##### Neuroanatomy, Vagal Nerve Nuclei ##### Neuroanatomy, Ventricular System ##### Neuroanatomy, Vertebrobasilar System ##### Neuroanatomy, Vestibular Pathways ##### Neuroanatomy, Vestibulo-ocular Reflex ##### Neuroanatomy, Visual Cortex ##### Neuroanatomy, Visual Pathway ##### Neuroanatomy, Wernicke Area ##### Neuroanatomy, White Rami Communicans ##### Neurobiology of Addiction ##### Neuroblastoma ##### Neurocutaneous Syndromes ##### Neuroendocrine Tumor 177Lutetium-Dotatate Therapy ##### Neurofibroma ##### Neurofibromatosis ##### Neurofibromatosis Type 1 ##### Neurofibromatosis Type 2 ##### Neurogenic Bladder and Neurogenic Lower Urinary Tract Dysfunction ##### Neurogenic Pulmonary Edema ##### Neurogenic Shock ##### Neurohypophysis ##### Neuroimaging In Epilepsy Surgery ##### Neuroleptic Agent Toxicity ##### Neuroleptic Malignant Syndrome ##### Neuroleptic Medications ##### Neurologic Exam ##### Neurological Sequelae of Endocarditis ##### Neurolytic Blocks ##### Neurolytic Procedures ##### Neuroma ##### Neuromodulation Surgery for Psychiatric Disorders ##### Neuromuscular Blockade ##### Neuromuscular Blocking Agents ##### Neuromyelitis Optica Spectrum Disorder (NMOSD) ##### Neuronal Brain Tumors ##### Neuronatomy, Prefrontal Association Cortex ##### Neuropathic Ulcer ##### Neuropathy ##### Neuroplasticity ##### Neuropsychological Assessment ##### Neuroretinitis ##### Neurosarcoidosis ##### Neurosonography Assessment, Protocols, and Interpretation ##### Neurosurgery, Sensory Homunculus ##### Neurosyphilis ##### Neurothekeoma ##### Neurotmesis ##### Neurotrophic Keratitis ##### Neutropenia ##### Neutropenic Enterocolitis ##### Neutrophilia ##### Neutrophilic Eccrine Hidradenitis ##### Nevirapine ##### Nevus Anemicus ##### Nevus Comedonicus ##### Nevus Flammeus ##### Nevus of Ota and Ito ##### Nevus Sebaceus ##### New Mexico Medical Practice Act Review and Understanding ##### New Onset Diabetes After Transplant ##### New Onset Refractory Status Epilepticus ##### New Palpable Breast Mass ##### New York State Child Abuse, Maltreatment, and Neglect ##### New York State Infection Control ##### Newborn Hearing Screening ##### Newborn Screening ##### Niacin ##### Niacin Deficiency ##### Niacin Toxicity ##### Nickel Allergy ##### Nickel Toxicology ##### Nicks Procedure ##### Nicotine Addiction and Smoking: Health Effects and Interventions ##### Nicotine Replacement Therapy ##### Niemann-Pick Disease ##### Nifedipine ##### Night Eating Syndrome ##### Night Terrors ##### Nightstick Fracture ##### Nimodipine ##### Nintedanib ##### Nipah Virus ##### Nipple-Areolar Complex Reconstruction ##### Nirmatrelvir-Ritonavir ##### Nisoldipine ##### Nissen Fundoplication ##### Nitisinone ##### Nitrates ##### Nitric Oxide ##### Nitrofurantoin ##### Nitrogen Dioxide Toxicity ##### Nitrogen Narcosis In Diving ##### Nitrogen-13 Ammonia Radiopharmaceutical ##### Nitroglycerin ##### Nitrosoureas Toxicity ##### Nitrous Oxide ##### Nivolumab ##### Nocardia ##### Nocardia Keratitis ##### Nocardiosis ##### Nocturia ##### Nocturnal Cough ##### Noise Exposure and Hearing Loss ##### Nonalcoholic Steatohepatitis (NASH) ##### Nonarteritic Anterior Ischemic Optic Neuropathy ##### Noncardiogenic Pulmonary Edema ##### Nondepolarizing Neuromuscular Blockers ##### Non-Diabetic Hypoglycemia ##### Non-Dieting Approaches to Treatment of Obesity ##### Nongonococcal Urethritis ##### Non-Hodgkin Lymphoma ##### Nonimmune Hydrops Fetalis ##### Noninfectious Complications of Blood Transfusion ##### Noninvasive Ventilation ##### Nonketotic Hyperglycinemia ##### Non-Odontogenic Cysts ##### Non-Odontogenic Tumors of the Jaws ##### Nonorganic Functional Hearing Loss ##### Nonorganic Vision Loss ##### Nonseminomatous Testicular Tumors ##### Non–Small Cell Lung Cancer ##### Nonspecific Interstitial Pneumonia ##### Nonspecific Orbital Inflammation ##### Nonsteroidal Anti-Inflammatory Drugs (NSAIDs) ##### Nonsteroidal Anti-Inflammatory Drugs Toxicity ##### Non–ST-Segment Elevation Myocardial Infarction ##### Nonsurgical Rhinoplasty Using Fillers ##### Nontoxic Goiter ##### Nonviral Myocarditis ##### Noonan Syndrome ##### Norepinephrine ##### Normal and Abnormal Complete Blood Count With Differential ##### Normal and Abnormal Platelet Count Values and Interpretation ##### Normal and Abnormal Urine Output and Interpretation ##### Normal Labor: Physiology, Evaluation, and Management ##### Normal Pressure Hydrocephalus ##### Normal Saline ##### Normal Tension Glaucoma ##### Normal Versus Chronic Adaptations to Aerobic Exercise ##### Normocalcemic Hyperparathyroidism ##### Normochromic Normocytic Anemia ##### Norovirus ##### Nortriptyline ##### Norwood Procedure ##### Nosocomial Infections ##### Nosocomial Pneumonia ##### Notalgia Paresthetica ##### NP Safe Prescribing of Controlled Substances While Avoiding Drug Diversion ##### NPH Insulin ##### Nuclear Medicine Applications in Prostate Cancer ##### Nuclear Medicine Artifacts ##### Nuclear Medicine Cerebral Perfusion Scan ##### Nuclear Medicine CNS Assessment, Protocols, and Interpretation ##### Nuclear Medicine Computed Tomography Physics ##### Nuclear Medicine Infection Assessment, Protocols, and Interpretation ##### Nuclear Medicine Instrumentation ##### Nuclear Medicine Liver/Spleen Test ##### Nuclear Medicine Musculoskeletal Assessment, Protocols, and Interpretation ##### Nuclear Medicine Neuro PET Assessment, Protocols, and Interpretation ##### Nuclear Medicine Pediatric Assessment, Protocols, and Interpretation ##### Nuclear Medicine PET Scan Cardiovascular Assessment, Protocols, and Interpretation ##### Nuclear Medicine PET/CT Breast Cancer Assessment, Protocols, and Interpretation ##### Nuclear Medicine PET/CT Gastrointestinal Assessment, Protocols, and Interpretation ##### Nuclear Medicine PET/CT Head and Neck Cancer Assessment, Protocols, and Interpretation ##### Nuclear Medicine PET/CT Lymphomas Assessment, Protocols, and Interpretation ##### Nuclear Medicine PET/CT Thyroid Cancer Assessment, Protocols, and Interpretation ##### Nuclear Medicine Physics ##### Nuclear Medicine Quality Assurance ##### Nuclear Medicine Safety ##### Nuclear Medicine SPECT Scan Cardiovascular Assessment, Protocols, and Interpretation ##### Nuclear Medicine Stress Test ##### Nuclear Medicine Test ##### Nuclear Renal Scan ##### Nucleus Pulposus Herniation ##### Nummular Dermatitis ##### Nummular Headache ##### Nurse Educator Systematic Self-Evaluation and Improvement ##### Nursemaid Elbow ##### Nursing Admission Assessment and Examination ##### Nursing Advocacy ##### Nursing Bedpan Management ##### Nursing Ethical Considerations ##### Nursing Grief and Loss ##### Nursing Home–Acquired Pneumonia ##### Nursing Neuman Systems Model ##### Nursing Practice Act ##### Nursing Process ##### Nursing Professional Development ##### Nursing Professional Development Evidence-Based Practice ##### Nursing Professional Development Facilitation ##### Nursing Professional Development Leadership ##### Nursing Professional Development Standards ##### Nursing Rights of Medication Administration ##### Nursing Shared Governance ##### Nursing Shortage ##### Nutcracker Syndrome and Left Renal Vein Entrapment ##### Nutrition and Hydration Requirements In Children and Adults ##### Nutrition: Macronutrient Intake, Imbalances, and Interventions ##### Nutrition: Micronutrient Intake, Imbalances, and Interventions ##### Nutritional Assessment ##### Nystagmus Types \| \| \| O \| : ##### Obesity and Comorbid Conditions ##### Obesity and Orthopedic Issues ##### Obesity and Set-Point Theory ##### Obesity and Type 2 Diabetes ##### Obesity as a Chronic Disease: Epidemiology and Pathophysiologic Insights ##### Obesity Brain Gut Adipocyte Interaction ##### Obesity in Pediatric Patients ##### Obesity in Pregnancy ##### Obesity Medications: Evidence-Based Management ##### Obesity Supplements ##### Obesity-Hypoventilation Syndrome ##### Obeticholic Acid ##### Objective Refraction Technique: Retinoscopy ##### Obsessive-Compulsive Disorder ##### Obsessive-Compulsive Personality Disorder ##### Obstructive Sleep Apnea ##### Obstructive Uropathy ##### Obtaining Medical Simulation Center Accreditation ##### Obturator Hernia ##### Occipital Nerve Block ##### Occipital Nerve Stimulation ##### Occipital Neuralgia ##### Occupational Injuries and Workers' Compensation Management Strategies ##### Occupational Safety and Health Administration Rules That Affect Healthcare ##### Occupational Therapy Assessment In Long Term Care ##### Occupational Therapy In Long Term Care ##### Ochronosis ##### Ocrelizumab ##### Octreotide ##### Octreotide Scan ##### Ocular Amyloidosis ##### Ocular Burns ##### Ocular Cellulitis ##### Ocular Futility and End of Sight Care ##### Ocular Hypotony ##### Ocular Ischemic Syndrome ##### Ocular Manifestations of Alkaptonuria ##### Ocular Manifestations of HIV ##### Ocular Manifestations of Preeclampsia ##### Ocular Melanoma ##### Ocular Neuropathic Pain ##### Ocular Pemphigoid ##### Ocular Sarcoidosis ##### Ocular Surface Squamous Neoplasia ##### Ocular Toxocariasis ##### Ocular Trauma Prevention Strategies and Patient Counseling ##### Ocular Tuberculosis ##### Ocular Ultrasound ##### Oculo Auriculo Vertebral Spectrum ##### Oculocardiac Reflex ##### Oculocerebrorenal Syndrome ##### Oculovestibular Reflex ##### Odds Ratio ##### Odontogenic Cysts ##### Odontogenic Orofacial Space Infections ##### Odontogenic Tumors of the Jaws ##### Odontoid Fractures ##### Ofloxacin ##### Ohms Law ##### Olanzapine ##### Olecranon Bursa Aspiration ##### Olecranon Bursitis ##### Olecranon Fracture ##### Olfactory Testing ##### Olfactory Training ##### Oligodendroglioma ##### Oligohydramnios ##### Oliguria ##### Olmesartan ##### Omalizumab ##### Omega-3 Fatty Acids ##### Omental Tumors ##### Omeprazole ##### Ommaya Reservoir ##### Omphalitis ##### Omphalocele ##### Onchocerciasis ##### Oncocytic (Hürthle Cell) Thyroid Carcinoma ##### Ondansetron ##### One Anastomosis Gastric Bypass and Mini Gastric Bypass ##### Onychomycosis ##### Onychoscopy ##### Oophorectomy ##### Open Airway Procedural Sedation ##### Open Angle Glaucoma ##### Open Cholecystectomy ##### Open Fracture Management ##### Open Heart Massage ##### Open Inguinal Hernia Repair ##### Operative Risk ##### Ophthalmia Neonatorum ##### Ophthalmic Manifestations of Coronavirus (COVID-19) ##### Ophthalmomyiasis ##### Opioid Analgesics ##### Opioid Anesthesia ##### Opioid Antagonists ##### Opioid Equivalency ##### Opioid Prescribing ##### Opioid Toxicity ##### Opioid Use Disorder: Evaluation and Management ##### Opioid Withdrawal ##### Opioid, Risk Tool ##### Opioid-Induced Constipation ##### Opioid-Induced Endocrinopathy ##### Opisthotonus ##### Oppositional Defiant Disorder ##### Opsoclonus ##### Optic Atrophy ##### Optic Disc Drusen ##### Optic Ischemia ##### Optic Nerve Coloboma ##### Optic Nerve Cysts ##### Optic Nerve Decompression ##### Optic Nerve Glioma ##### Optic Nerve Sheath Meningioma ##### Optic Nerve Sheath Ultrasound ##### Optic Neuritis ##### Optical Biometry ##### Optical Coherence Tomography ##### Optical Coherence Tomography Angiography ##### Oral and Maxillofacial Surgery, Facial Laceration Repair ##### Oral Candidiasis ##### Oral Contraceptive Pills ##### Oral Cutaneous Fistula ##### Oral Facial Infection of Dental Origin: A Guide for the Medical Practitioner ##### Oral Flap Design ##### Oral Health Considerations for Patients With Systemic Disease ##### Oral Hemangiomas ##### Oral Hypoglycemic Medications ##### Oral Leukoplakia ##### Oral Lichen Planus ##### Oral Management of Patients Undergoing Radiation Therapy ##### Oral Melanoacanthoma ##### Oral Melanoma ##### Oral Mucosal Lesions, Immunologic Diseases ##### Oral Mucositis ##### Oral Surgery Suturing ##### Oral Surgery, Biopsies ##### Oral Surgery, Extraction of Mandibular Third Molars ##### Oral Surgery, Extraction of Roots ##### Oral Surgery, Extraction of Teeth ##### Oral Surgery, Extraction of Unerupted Teeth ##### Oral Surgery, Instruments ##### Oral Transmucosal Fentanyl ##### Orbital Apex Syndrome ##### Orbital Atherectomy ##### Orbital Cellulitis ##### Orbital Color Doppler Imaging ##### Orbital Floor Fracture ##### Orbital Solitary Fibrous Tumor ##### Orbital Vascular Anomalies ##### Orchiopexy ##### Orchitis ##### Orf Viral Infection ##### Organ Procurement From Patients With Trauma ##### Organizational Culture ##### Organophosphate Toxicity ##### Orientation to the Medical Simulation Environment ##### Orlistat ##### Ornithine Transcarbamylase Deficiency ##### Oropharyngeal Airway ##### Oropharyngeal Squamous Cell Carcinoma ##### Orphan Drug Approval Laws ##### Orthodontics, Cephalometric Analysis ##### Orthodontics, Malocclusion ##### Orthopedic and Podiatry Perioperative Antibiotics ##### Orthopedic Implant Materials ##### Orthostatic Hypotension ##### Orthostatic Proteinuria ##### Os Odontoideum ##### Os Peroneum ##### Oseltamivir ##### Osgood-Schlatter Disease ##### OSHA Bloodborne Pathogen Standards ##### OSHA Chemical Hazards And Communication ##### OSHA Electrical Safety ##### OSHA Fire Safety ##### OSHA Formaldehyde Safety ##### Osler Node and Janeway Lesions ##### Osmometer ##### Osseointegrated Bone-Conducting Hearing Protheses ##### Ossicular-Chain Dislocation ##### Ossiculoplasty ##### Osteitis Condensans Ilii ##### Osteitis Fibrosa Cystica ##### Osteitis Pubis ##### Osteoarthritis ##### Osteoblastoma ##### Osteochondral Allograft ##### Osteochondral Autograft Transplantation ##### Osteochondritis Dissecans ##### Osteochondritis Dissecans of the Knee ##### Osteochondroma ##### Osteocutaneous Radial Forearm Flap ##### Osteofibrous Dysplasia ##### Osteogenesis Imperfecta ##### Osteoid Osteoma ##### Osteoma Cutis ##### Osteomalacia ##### Osteomyelitis ##### Osteomyelitis Imaging ##### Osteonecrosis Imaging ##### Osteopathic Manipulative Treatment: 5 Diaphragm Procedure ##### Osteopathic Manipulative Treatment: Articulatory Procedure - Sacral Dysfunctions ##### Osteopathic Manipulative Treatment: BLT/LAS Procedure - Pelvic Dysfunctions ##### Osteopathic Manipulative Treatment: Counterstrain and Facilitated Positional Release (FPR) Procedure - Lumbar Vertebrae ##### Osteopathic Manipulative Treatment: Counterstrain Procedure - Cervical Vertebrae ##### Osteopathic Manipulative Treatment: Counterstrain/FPR Procedure - Thoracic Vertebrae ##### Osteopathic Manipulative Treatment: Facial Muscle Energy, Direct MFR, and BLT Procedure – for TMJ Dysfunction ##### Osteopathic Manipulative Treatment: HVLA Procedure - Cervical Vertebrae ##### Osteopathic Manipulative Treatment: HVLA Procedure - Exhaled Ribs ##### Osteopathic Manipulative Treatment: HVLA Procedure - Inhaled Ribs ##### Osteopathic Manipulative Treatment: HVLA Procedure - Lumbar Vertebrae ##### Osteopathic Manipulative Treatment: HVLA Procedure - Thoracic Vertebrae ##### Osteopathic Manipulative Treatment: Inhaled Rib Dysfunction ##### Osteopathic Manipulative Treatment: Lymphatic Procedures ##### Osteopathic Manipulative Treatment: Muscle Energy and Counterstrain Procedure - Piriformis Muscle ##### Osteopathic Manipulative Treatment: Muscle Energy and Counterstrain Procedure - Psoas Muscle Procedures ##### Osteopathic Manipulative Treatment: Muscle Energy and HVLA Procedures - Fibular Head Dysfunction ##### Osteopathic Manipulative Treatment: Muscle Energy Procedure - AC Joint, Clavicle, SC Joint ##### Osteopathic Manipulative Treatment: Muscle Energy Procedure - Cervical Vertebrae ##### Osteopathic Manipulative Treatment: Muscle Energy Procedure - Exhaled Ribs ##### Osteopathic Manipulative Treatment: Muscle Energy Procedure - Lumbar Vertebrae ##### Osteopathic Manipulative Treatment: Muscle Energy Procedure - Pelvic Dysfunctions ##### Osteopathic Manipulative Treatment: Muscle Energy Procedure - Sacral Dysfunctions ##### Osteopathic Manipulative Treatment: Muscle Energy Procedure With Post-Isometric Relaxation - Thoracic Vertebrae ##### Osteopathic Manipulative Treatment: Suboccipital Release ##### Osteopenia ##### Osteopetrosis ##### Osteoporosis ##### Osteoporosis in Females ##### Osteoporosis in Males ##### Osteoporosis in Spinal Cord Injuries ##### Osteosarcoma (Osteogenic Sarcoma) ##### Otalgia ##### Otitis Externa ##### Otitis Media With Effusion ##### Otoacoustic Emissions ##### Otoplasty ##### Otosclerosis ##### Otoscope Exam ##### Otoscopy ##### Outcome Assessment and Information Set ##### Ovarian Cyst ##### Ovarian Cystadenoma ##### Ovarian Torsion ##### Overcoming Stigma and Bias in Obesity Management ##### Overdrive Pacing ##### Over-The-Counter Drugs Laws ##### Overview of Environmental Skin Cancer Risks ##### Overview of Lifestyle Medicine ##### Overview of Periodontal Surgical Procedures ##### Overview on Ordering and Evaluation of Laboratory Tests ##### Ovulation Induction Techniques ##### Oxaliplatin ##### Oxazepam ##### Oxcarbazepine ##### Oxybutynin ##### Oxycodone ##### Oxygen Administration ##### Oxygen Saturation ##### Oxygen Toxicity ##### Oxygenation Status and Pulse Oximeter Analysis ##### Oxytocin ##### Ozone Toxicity \| \| \| P \| : ##### P wave ##### Pacemaker Indications ##### Pacemaker Insertion ##### Pacemaker Malfunction ##### Pacemaker Mediated Tachycardia ##### Pacemaker Syndrome ##### Pacemaker Types and Selection ##### Packaging and Shipping Infectious Materials ##### Paclitaxel ##### Page Kidney ##### Paget Bone Disease ##### Paget-Schroetter Syndrome ##### Pain Assessment ##### Pain Management Medications ##### Pain Theory ##### Palatal and Gingival Cysts of the Newborn ##### Paleolithic Diet ##### Palliation Radiation Therapy of the Spinal Cord ##### Palliative Care ##### Palliative Radiation Therapy For Bone Metastases ##### Palliative Radiation Therapy for Brain Metastases ##### Palliative Sedation in Patients With Terminal Illness ##### Palmoplantar Psoriasis ##### Palytoxin Keratitis ##### Pamidronate ##### Pancoast Syndrome ##### Pancreas Imaging ##### Pancreas Transplantation ##### Pancreatic Abscess ##### Pancreatic Cancer ##### Pancreatic Cysts ##### Pancreatic Fistula ##### Pancreatic Insufficiency ##### Pancreatic Neuroendocrine Tumors ##### Pancreatic Panniculitis ##### Pancreatic Pseudoaneurysm ##### Pancreatic Pseudocyst ##### Pancreatic Trauma ##### Pancreaticoduodenectomy (Whipple Procedure) ##### Pancreatitis ##### Pancrelipase Therapy ##### Pancuronium ##### Pancytopenia ##### Panic Disorder ##### Panniculectomy ##### Pantoprazole ##### Pantothenate Kinase-Associated Neurodegeneration (PKAN) ##### Papillary Fibroelastoma ##### Papillary Muscle Rupture ##### Papillary Thyroid Carcinoma ##### Papilledema ##### Papillomas: A Multisystem Overview of HPV-Associated and HPV-Independent Lesions ##### Papular Acrodermatitis ##### Papuloerythroderma of Ofuji ##### Paracentesis ##### Paracoccidioidomycosis ##### Paradoxical Adipose Hyperplasia ##### Paradoxical Embolism ##### Paraganglioma ##### Parahiatal Hernia ##### Parainfluenza Virus ##### Paramedian Forehead Flaps ##### Paramyxovirus ##### Paraneoplastic Cerebellar Degeneration ##### Paraneoplastic Encephalomyelitis ##### Paraneoplastic Limbic Encephalitis ##### Paraneoplastic Pemphigus ##### Paraneoplastic Syndromes ##### Paranoid Personality Disorder ##### Paraphenylenediamine Toxicity ##### Paraphilia ##### Paraphimosis ##### Parapneumonic Pleural Effusions and Empyema Thoracis ##### Paraproteinemic Keratopathy ##### Parasomnias ##### Parasternal Mediastinotomy ##### Parasympathomimetic Medications ##### Parathyroid Cancer ##### Parathyroidectomy ##### Parental Consent ##### Parinaud Syndrome ##### Parkinson Disease ##### Parkinsonism ##### Parkinson-Plus Syndrome ##### Parkland Formula ##### Paronychia ##### Paronychia Drainage ##### Parotid Cancer ##### Parotidectomy ##### Parotitis ##### Paroxetine ##### Paroxysmal Atrial Fibrillation ##### Paroxysmal Atrial Tachycardia ##### Paroxysmal Nocturnal Hemoglobinuria ##### Paroxysmal Supraventricular Tachycardia ##### Parry-Romberg Syndrome ##### Pars Interarticularis Injury ##### Pars Planitis ##### Partial and Total Anomalous Pulmonary Venous Connection ##### Partial Epilepsy ##### Partial Pressure of Carbon Dioxide ##### Partial Pressure of Oxygen ##### Partial Thromboplastin Time ##### Parvoviruses ##### Past Present and Future of Simulation in Internal Medicine ##### Past Present and Future of Simulation in Military Medicine ##### Past Present and Future of Simulation in Pediatrics ##### Past Present and Future of Simulation in Trauma ##### Pasteurella Multocida ##### Patau Syndrome ##### Patella Dislocation ##### Patella Fractures ##### Patellar Instability ##### Patellar Tendon Rupture ##### Patellofemoral Arthritis ##### Patellofemoral Syndrome ##### Patent Ductus Arteriosus ##### Patent Foramen Ovale ##### Patent Foramen Ovale in Diving ##### Patent Urachus ##### Pathergy Test ##### Pathologic Fractures ##### Pathology, Inflammation ##### Pathophysiology Of Asthma ##### Pathophysiology of Obesity ##### Pathophysiology of Takotsubo Syndrome ##### Patient Assist Devices ##### Patient Care Transfer Techniques ##### Patient Communication In Substance Use Disorders ##### Patient Confidentiality ##### Patient Education: Anesthesia ##### Patient Restraint and Seclusion ##### Patient Rights and Ethics ##### Patient Self-Determination Act ##### Patient-Controlled Analgesia ##### Patisiran ##### Pattern Electroretinogram ##### Pay-for-Performance and Value-Based Care ##### PCSK9 Inhibitors ##### PDE5 Inhibitors ##### Peak Flow Rate Measurement ##### Peanut Allergy ##### Pearly Penile Papule ##### Pecto-Intercostal Fascial Plane Block ##### Pectoral Implants ##### Pectoralis Major Tear ##### Pectoralis Nerve Block ##### Pectus Carinatum ##### Pectus Excavatum ##### Pediatric Abusive Head Trauma ##### Pediatric and Neonatal Resuscitation ##### Pediatric Appendicitis ##### Pediatric Asthma ##### Pediatric Autoimmune Neuropsychiatric Disorders Associated With Streptococcal Infections (PANDAS) ##### Pediatric Bronchiolitis ##### Pediatric Bronchospasm ##### Pediatric Cataract ##### Pediatric Cerebral Aneurysm ##### Pediatric Dehydration ##### Pediatric Diabetic Ketoacidosis ##### Pediatric Dyslipidemia ##### Pediatric Echocardiography Assessment, Protocols, and Interpretation ##### Pediatric Facial Fractures ##### Pediatric Facial Trauma ##### Pediatric Feeding Disorders: Recognition, Diagnosis, and Management ##### Pediatric Fluid Management ##### Pediatric Foot Alignment Deformities ##### Pediatric Foreign Body Ingestion ##### Pediatric Functional Constipation ##### Pediatric Head Trauma ##### Pediatric Hearing Loss ##### Pediatric Lateral Humeral Condyle Fractures ##### Pediatric Malignant Pleural Effusion ##### Pediatric Obstructive Sleep Apnea ##### Pediatric Physeal Injuries Overview ##### Pediatric Pneumonia ##### Pediatric Postresuscitation Management ##### Pediatric Preoperative Management ##### Pediatric Procedural Sedation ##### Pediatric Rectal Prolapse ##### Pediatric Regional Anesthesia ##### Pediatric Skull Fractures ##### Pediatric Spine Trauma ##### Pediatric Torus Buckle Fracture ##### Pediatric Tracheostomy ##### Pediatric Transillumination of the Chest ##### Pediatric Type 2 Diabetes ##### Pediatric Umbilical Hernia ##### Pediculosis ##### Pediculosis Corporis ##### Peer Play ##### Pegfilgrastim ##### Pegloticase ##### Peliosis Hepatis ##### Pelizaeus-Merzbacher Disease ##### Pellegrini-Stieda Disease ##### Pellucid Marginal Corneal Degeneration ##### Pelvic Abscess ##### Pelvic Congestion Syndrome ##### Pelvic Exenteration ##### Pelvic Floor Dysfunction ##### Pelvic Fracture ##### Pelvic Inflammatory Disease ##### Pelvic Kidney ##### Pelvic Organ Prolapse ##### Pelvic Prolapse Imaging ##### Pelvic Ring Injuries ##### Pelvic Trauma ##### Pelvic Ultrasound ##### Pembrolizumab ##### Pemetrexed ##### Pemphigoid Gestationis ##### Pemphigus Foliaceus ##### Pemphigus Herpetiformis ##### Pemphigus Vegetans ##### Pemphigus Vulgaris ##### Pendred Syndrome ##### Penetrating Abdominal Trauma ##### Penetrating Chest Trauma ##### Penetrating Groin Trauma ##### Penetrating Head Trauma ##### Penetrating Keratoplasty ##### Penicillamine ##### Penicillin ##### Penicillin Allergy ##### Penile Cancer and Penile Intraepithelial Neoplasia ##### Penile Fracture ##### Penile Irrigation, Aspiration, and Vasoactive Injections for Priapism Treatment ##### Penile Prosthesis Implantation ##### Penile Zipper and Ring Injuries ##### Pennsylvania Child Abuse Recognition and Reporting ##### Pentalogy of Cantrell ##### Pentamidine ##### Pentobarbital ##### Pentosan Polysulfate Maculopathy ##### Pentoxifylline ##### Peptic Ulcer Disease ##### Percutaneous Abscess Drainage ##### Percutaneous Cholangiography ##### Percutaneous Coronary Intervention ##### Percutaneous Endoscopic Gastrostomy Tube ##### Percutaneous Gastrostomy and Jejunostomy ##### Percutaneous Liver Biopsy ##### Percutaneous Lung Lesion Biopsy ##### Percutaneous Nephrostomy ##### Percutaneous Radiofrequency Ablation of Liver Tumors ##### Percutaneous Transhepatic Cholangiography ##### Percutaneous Transluminal Angioplasty and Balloon Catheters ##### Percutaneous Transluminal Coronary Angioplasty ##### Percutaneous Transluminal Coronary Arteriography ##### Percutaneous Tricuspid Valve Repair ##### Percutaneous Vertebroplasty and Kyphoplasty ##### Percutaneously Inserted Ventricular Assist Device ##### Periampullary Tumors ##### Perianal Abscess ##### Perianal Streptococcal Dermatitis ##### Pericapsular Nerve Group Block ##### Pericardial Calcification ##### Pericardial Cyst ##### Pericardial Effusion ##### Pericardial Friction Rub ##### Pericardiocentesis ##### Pericarditis ##### Pericholangitis ##### Pericoronitis ##### Perilunate Dislocation ##### Perilymphatic Fistula ##### Perimortem Cesarean Delivery ##### Perinatal Depression ##### Perineal Lacerations ##### Perinephric Abscess ##### Periocular Capillary Infantile Hemangiomas ##### Periodic Limb Movement Disorder ##### Periodontal Abscess ##### Periodontal Disease ##### Periodontitis ##### Perioperative Acute Kidney Injury ##### Perioperative Anticoagulation Management ##### Perioperative Cardiac Management ##### Perioperative Care for Metabolic and Bariatric Surgery ##### Perioperative Management of Patients With Congenital Heart Disease ##### Perioperative Management of Pheochromocytoma ##### Perioperative Vision Loss ##### Perioral Dermatitis ##### Periorbital Cellulitis ##### Peripartum Cardiomyopathy ##### Peripheral Aneurysm ##### Peripheral Arterial Disease ##### Peripheral Arterial Duplex Assessment, Protocols, and Interpretation ##### Peripheral Edema ##### Peripheral Hypertrophic Subepithelial Corneal Degeneration ##### Peripheral Line Placement ##### Peripheral Magnetic Stimulation ##### Peripheral Nerve Block of the External Ear ##### Peripheral Nerve Blocks ##### Peripheral Nerve Stimulator ##### Peripheral Precocious Puberty ##### Peripheral Pulse ##### Peripheral T-Cell Lymphoma ##### Peripheral Ulcerative Keratitis ##### Peripheral Vascular Bypass ##### Peripheral Vascular Disease ##### Peripherally Inserted Central Catheter (PICC) Line Placement ##### Periprocedural Care for Patients Undergoing Bariatric Device Placement and Endoscopic Procedures ##### Periprosthetic Distal Femur Fracture ##### Periprosthetic Joint Infection ##### Periprosthetic Proximal Femur Fractures ##### Peritoneal Cancer ##### Peritoneal Dialysis ##### Peritoneal Surface Malignancies ##### Peritoneovenous Shunt ##### Peritonsillar Abscess ##### Periventricular and Intraventricular Hemorrhage ##### Permethrin ##### Permissive Hypotension ##### Pernicious Anemia ##### Pernio ##### Peroneal Nerve Injury ##### Peroneal Tendon Syndromes ##### Peroxidase-Coupled Glucose Method ##### Persistent Depressive Disorder ##### Persistent Epithelial Defect ##### Persistent Postural-Perceptual Dizziness ##### Persistent Pulmonary Hypertension of the Newborn ##### Personal Health Record ##### Personal Protective Equipment ##### Personality Disorder ##### Person-Centered Therapy (Rogerian Therapy) ##### Pertussis ##### Pes Anserine Bursitis ##### Pes Cavus ##### Pes Planus ##### PET Scanning ##### Petechiae ##### Peters Anomaly ##### Peutz-Jeghers Syndrome ##### Peyer Patches ##### Peyronie Disease ##### Pfeiffer Syndrome ##### PHACE Syndrome ##### Phacoemulsification ##### Phagocytosis ##### Phakic Intraocular Lens Myopia ##### Phalanx Fractures of the Hand ##### Phantom Limb Pain ##### Phantosmia ##### Pharmaceutical Formulation ##### Pharmaceutics ##### Pharmacodynamics ##### Pharmacogenomics Overview ##### Pharmacokinetics ##### Pharmacologic Stress Testing ##### Pharmacologic Therapy for Obesity ##### Pharmacological Cardioversion ##### Pharmacy Calculations ##### Pharmacy Communication Strategies ##### Pharmacy Federal Rules and Regulations ##### Pharmacy Packaging and Inserts ##### Pharmacy Prescription Requirements ##### Pharyngitis ##### Phenazopyridine ##### Phencyclidine Toxicity ##### Phenelzine ##### Phenobarbital ##### Phenol Nerve Block ##### Phenol Toxicity ##### Phenothiazine ##### Phenoxybenzamine ##### Phenylephrine ##### Phenylketonuria ##### Phenytoin ##### Phenytoin Toxicity ##### Pheochromocytoma ##### Phimosis ##### Phlebotomy ##### Phlegmasia Alba and Cerulea Dolens ##### Phocomelia ##### Phosgene Toxicity ##### Phosphodiesterase Inhibitors ##### Photodynamic Therapy for Dermatologic Conditions ##### Photodynamic Therapy for the Eye ##### Photopic Vision ##### Photorefractive Keratectomy ##### Photosensitivity ##### Phototherapy ##### Phrenic Nerve Injury ##### Phthalates Toxicity ##### Phthiriasis Palpebrarum ##### Phyllodes Tumor of the Breast ##### Physical Activity and Weight Loss Maintenance ##### Physical and Chemical Lesions of the Oral Mucosa ##### Physiology of Sleep ##### Physiology Of Spatial Orientation ##### Physiology, Accommodation ##### Physiology, Acetylcholine ##### Physiology, Acetylcholinesterase ##### Physiology, Acid Base Balance ##### Physiology, Action Potential ##### Physiology, Active Immunity ##### Physiology, Active Transport ##### Physiology, Acute Phase Reactants ##### Physiology, Adenosine Triphosphate ##### Physiology, Adrenal Gland ##### Physiology, Adrenocorticotropic Hormone (ACTH) ##### Physiology, Afterload Reduction ##### Physiology, Aging ##### Physiology, Airflow Resistance ##### Physiology, Airway Resistance ##### Physiology, Albumin ##### Physiology, Aldosterone ##### Physiology, Alveolar Tension ##### Physiology, Alveolar to Arterial Oxygen Gradient ##### Physiology, Anrep Effect ##### Physiology, Anterior Pituitary ##### Physiology, Antibody ##### Physiology, Anticholinergic Reaction ##### Physiology, Appetite And Weight Regulation ##### Physiology, Aqueous Humor Circulation ##### Physiology, Arterial Pressure Regulation ##### Physiology, Autonomic Nervous System ##### Physiology, AV Junction ##### Physiology, Bainbridge Reflex ##### Physiology, Baroreceptors ##### Physiology, Bile ##### Physiology, Bile Acids ##### Physiology, Bile Secretion ##### Physiology, Biliary ##### Physiology, Bilirubin ##### Physiology, Bladder ##### Physiology, Blood Brain Barrier ##### Physiology, Blood Plasma ##### Physiology, Blood Pressure Age Related Changes ##### Physiology, Blood Volume ##### Physiology, Body Fluids ##### Physiology, Body Mass Index ##### Physiology, Bohr Effect ##### Physiology, Bone ##### Physiology, Bone Remodeling ##### Physiology, Bowditch Effect ##### Physiology, Boyle's Law ##### Physiology, Bradykinin ##### Physiology, Brain ##### Physiology, Breast Milk ##### Physiology, Bundle of His ##### Physiology, Calcium ##### Physiology, Carbohydrates ##### Physiology, Carbon Dioxide Response Curve ##### Physiology, Carbon Dioxide Retention ##### Physiology, Carbon Dioxide Transport ##### Physiology, Cardiac ##### Physiology, Cardiac Cycle ##### Physiology, Cardiac Index ##### Physiology, Cardiac Muscle ##### Physiology, Cardiac Output ##### Physiology, Cardiac Preload ##### Physiology, Cardiac Repolarization Dispersion and Reserve ##### Physiology, Cardiovascular ##### Physiology, Cardiovascular Hemodynamics ##### Physiology, Cardiovascular Murmurs ##### Physiology, Catecholamines ##### Physiology, Cellular Messengers ##### Physiology, Cellular Receptors ##### Physiology, Central Venous Pressure ##### Physiology, Cerebral Autoregulation ##### Physiology, Cerebral Cortex Functions ##### Physiology, Cerebral Spinal Fluid ##### Physiology, Cervical Dilation ##### Physiology, Chapman’s Points ##### Physiology, Chemoreceptor Trigger Zone ##### Physiology, Cholesterol ##### Physiology, Cholinergic Receptors ##### Physiology, Chorionic Gonadotropin ##### Physiology, Circadian Rhythm ##### Physiology, Clotting Mechanism ##### Physiology, Coagulation Pathways ##### Physiology, Cochlear Function ##### Physiology, Colloid Osmotic Pressure ##### Physiology, Color Perception ##### Physiology, Complement Cascade ##### Physiology, Connective Tissue ##### Physiology, Coronary Circulation ##### Physiology, Cortisol ##### Physiology, Counterstrain and Facilitated Positional Release (FPR) ##### Physiology, Deep Tendon Reflexes ##### Physiology, Defecation ##### Physiology, Digestion ##### Physiology, Diving Reflex ##### Physiology, Ear ##### Physiology, Edema ##### Physiology, Endocrine Hormones ##### Physiology, Endothelial Derived Relaxation Factor ##### Physiology, Endothelin ##### Physiology, Enkephalin ##### Physiology, Enteroglucagon ##### Physiology, Epithelialization ##### Physiology, Erection ##### Physiology, Esophagus ##### Physiology, Eustachian Tube Function ##### Physiology, Exocrine Gland ##### Physiology, Explicit Memory ##### Physiology, Eye ##### Physiology, Factor V ##### Physiology, Factor XIII ##### Physiology, Fascia ##### Physiology, Fasting ##### Physiology, Female Reproduction ##### Physiology, Fetal Circulation ##### Physiology, Fetal Hemoglobin ##### Physiology, Fever ##### Physiology, Follicle Stimulating Hormone ##### Physiology, Frank Starling Law ##### Physiology, Functional Residual Capacity ##### Physiology, GABA ##### Physiology, Gag Reflex ##### Physiology, Gallbladder ##### Physiology, Gastric Inhibitory Peptide ##### Physiology, Gastric Intrinsic Factor ##### Physiology, Gastrin ##### Physiology, Gastrocolic Reflex ##### Physiology, Gastrointestinal ##### Physiology, Gastrointestinal Hormonal Control ##### Physiology, Gastrointestinal Nervous Control ##### Physiology, Glomerular Filtration Rate ##### Physiology, Glucagon ##### Physiology, Glucocorticoids ##### Physiology, Gluconeogenesis ##### Physiology, Glucose ##### Physiology, Glucose Metabolism ##### Physiology, Glucose Transporter Type 4 ##### Physiology, Glycosuria ##### Physiology, Gonadotropin Inhibitor ##### Physiology, Gonadotropin-Releasing Hormone ##### Physiology, Granulation Tissue ##### Physiology, Growth Factor ##### Physiology, Growth Hormone ##### Physiology, Hair ##### Physiology, Heart Sounds ##### Physiology, Heat Loss ##### Physiology, Hemostasis ##### Physiology, Hepcidin ##### Physiology, Herring Breuer Reflex ##### Physiology, Homeostasis ##### Physiology, Hypothalamus ##### Physiology, Immune Response ##### Physiology, Integument ##### Physiology, Islets of Langerhans ##### Physiology, Jugular Venous Pulsation ##### Physiology, K Complex ##### Physiology, Korotkoff Sound ##### Physiology, Krebs Cycle ##### Physiology, Lactation ##### Physiology, Large Intestine ##### Physiology, Left Ventricular Function ##### Physiology, Leptin ##### Physiology, Leukotrienes ##### Physiology, Ligand Gated Chloride Channel ##### Physiology, Liver ##### Physiology, Long Term Memory ##### Physiology, Lower Esophageal Sphincter ##### Physiology, Lung ##### Physiology, Lung Capacity ##### Physiology, Lung Dead Space ##### Physiology, Luteinizing Hormone ##### Physiology, Lymphatic System ##### Physiology, Major Basic Protein ##### Physiology, Male Reproductive System ##### Physiology, Maternal Blood ##### Physiology, Maternal Changes ##### Physiology, Mean Arterial Pressure ##### Physiology, Mechanoreceptors ##### Physiology, Membrane ##### Physiology, Menarche ##### Physiology, Menstrual Cycle ##### Physiology, Metabolic Alkalosis ##### Physiology, Metabolism ##### Physiology, MHC Class I ##### Physiology, Motilin ##### Physiology, Motor Cortical ##### Physiology, Muscarinic Receptor ##### Physiology, Muscle ##### Physiology, Muscle Contraction ##### Physiology, Muscle Energy ##### Physiology, Muscle Myocyte ##### Physiology, Myocardial Oxygen Demand ##### Physiology, Nasal ##### Physiology, Neonatal ##### Physiology, Nerve ##### Physiology, Neuromuscular Junction ##### Physiology, Neuromuscular Transmission ##### Physiology, Neurotransmitters ##### Physiology, Newborn ##### Physiology, Night Vision ##### Physiology, NMDA Receptor ##### Physiology, Nociception ##### Physiology, Nociceptive Pathways ##### Physiology, Noncompetitive Inhibitor ##### Physiology, Noradrenergic Synapse ##### Physiology, Nutrient Absorption ##### Physiology, Obesity Neurohormonal Appetite And Satiety Control ##### Physiology, Olfactory ##### Physiology, Opioid Receptor ##### Physiology, Opsonization ##### Physiology, Osmoreceptors ##### Physiology, Osmoregulation and Excretion ##### Physiology, Osmosis ##### Physiology, Ovulation ##### Physiology, Ovulation And Basal Body Temperature ##### Physiology, Oxygen Transport ##### Physiology, Oxygen Transport And Carbon Dioxide Dissociation Curve ##### Physiology, Oxyhemoglobin Dissociation Curve ##### Physiology, Pain ##### Physiology, Pancreas ##### Physiology, Parathyroid ##### Physiology, Parathyroid Hormone ##### Physiology, Pepsin ##### Physiology, Peripheral Vascular Resistance ##### Physiology, Peristalsis ##### Physiology, Phosphate ##### Physiology, Pineal Gland ##### Physiology, Pituitary Gland ##### Physiology, Pituitary Hormones ##### Physiology, Pituitary Issues During Pregnancy ##### Physiology, Placenta ##### Physiology, Plasma Osmolality and Oncotic Pressure ##### Physiology, Plasminogen Activation ##### Physiology, Platelet ##### Physiology, Platelet Activation ##### Physiology, Pleural Fluid ##### Physiology, Posterior Pituitary ##### Physiology, Postpartum Changes ##### Physiology, Pregnancy ##### Physiology, Pregnancy Contractions ##### Physiology, Progesterone ##### Physiology, Prolactin ##### Physiology, Prostaglandin I2 ##### Physiology, Proteins ##### Physiology, Puberty ##### Physiology, Pulmonary ##### Physiology, Pulmonary Circulation ##### Physiology, Pulmonary Circulatory System ##### Physiology, Pulmonary Compliance ##### Physiology, Pulmonary Stress Response ##### Physiology, Pulmonary Vascular Resistance ##### Physiology, Pulmonary Vasoconstriction ##### Physiology, Pulmonary Ventilation and Perfusion ##### Physiology, Pulse Pressure ##### Physiology, REM Sleep ##### Physiology, Renal ##### Physiology, Renal Blood Flow and Filtration ##### Physiology, Renin Angiotensin System ##### Physiology, Residual Volume ##### Physiology, Respiratory Drive ##### Physiology, Respiratory Quotient ##### Physiology, Respiratory Rate ##### Physiology, Resting Potential ##### Physiology, Ryanodine Receptor ##### Physiology, Salivation ##### Physiology, Sebaceous Glands ##### Physiology, Secretin ##### Physiology, Sensory Receptors ##### Physiology, Sensory System ##### Physiology, Serotonin ##### Physiology, Sexual Maturity Rating ##### Physiology, Sinoatrial Node ##### Physiology, Skeletal Muscle ##### Physiology, Skeletal Muscle Contraction ##### Physiology, Sleep Patterns ##### Physiology, Sleep Stages ##### Physiology, Small Bowel ##### Physiology, Smooth Muscle ##### Physiology, Sodium Channels ##### Physiology, Sodium Potassium Pump ##### Physiology, Somatostatin ##### Physiology, Spinal Cord ##### Physiology, Spleen ##### Physiology, Starling Relationships ##### Physiology, Stomach ##### Physiology, Stress Reaction ##### Physiology, Stroke Volume ##### Physiology, Swallowing ##### Physiology, Synapse ##### Physiology, Synuclein ##### Physiology, Systemic Vascular Resistance ##### Physiology, Taste ##### Physiology, Temperature Regulation ##### Physiology, Testosterone ##### Physiology, Thermal Regulation ##### Physiology, Thromboxane A2 ##### Physiology, Thyroid ##### Physiology, Thyroid Function ##### Physiology, Thyroid Hormone ##### Physiology, Thyroid Stimulating Hormone ##### Physiology, Tidal Volume ##### Physiology, Tooth ##### Physiology, Transpulmonary Pressure ##### Physiology, Trauma ##### Physiology, Tyrosine Kinase Receptors ##### Physiology, Urea Cycle ##### Physiology, Urination ##### Physiology, Uterus ##### Physiology, Vaginal ##### Physiology, Vascular ##### Physiology, Vasodilation ##### Physiology, Vasopressin ##### Physiology, Vestibular System ##### Physiology, Vibratory Sense ##### Physiology, Viscerosomatic Reflexes ##### Physiology, Vision ##### Physiology, Von Willebrand Factor ##### Physiology, Water Balance ##### Physiology, Withdrawal Response ##### Physiology, Wound Healing ##### Physiology, Zero and First Order Kinetics ##### Phytonadione (Vitamin K1) ##### Piaget ##### Pica ##### Pick Disease ##### Piebaldism ##### Pierre Robin Syndrome ##### Piezogenic Pedal Papule ##### Pigment Dispersion Glaucoma ##### Pigment Dispersion Syndrome ##### Pigmented Purpuric Dermatosis ##### Pigmented Villonodular Synovitis ##### Pilar Cyst ##### Pili Annulati ##### Pilocytic Astrocytoma ##### Pilon Fracture ##### Pilonidal Cyst and Sinus ##### Pilot Cardiac Evaluation For Fitness for Duty ##### Pilot Medical Certification ##### Pimavanserin ##### Pimecrolimus ##### Pin Index Safety System ##### Pineal Gland Cancer ##### Pinguecula ##### Pinna Perichondritis ##### Pioglitazone ##### Piriformis Injection ##### Piriformis Syndrome ##### Pit and Fissure Sealants ##### Pitavastatin ##### Pituitary Adenoma ##### Pituitary Apoplexy ##### Pituitary Cancer ##### Pituitary Gland Imaging ##### Pituitary Hyperplasia in Primary Hypothyroidism ##### Pityriasis Alba ##### Pityriasis Lichenoides Et Varioliformis Acuta (PLEVA) ##### Pityriasis Rosea ##### Pityriasis Rotunda ##### Pityriasis Rubra Pilaris ##### Placebo Effect ##### Placenta Abnormalities ##### Placenta Accreta ##### Placenta Previa ##### Placental Abruption ##### Placental Insufficiency ##### Plagiocephaly ##### Plague ##### Planning and Developing a Clinical Pharmacy Practice ##### Plant Alkaloids Toxicity ##### Plantar Fasciitis ##### Plantar Fibromatosis ##### Plantar Heel Pain ##### Plaque Psoriasis ##### Plasma Cell Cancer ##### Plasma Glucose ##### Plasma Volume Study ##### Plasmablastic Lymphoma ##### Plasmacytoma ##### Plasmapheresis ##### Plasmodium falciparum Malaria ##### Plasmodium ovale Malaria ##### Plasmodium vivax Malaria ##### Platelet Transfusion ##### Platinum Hypersensitivity and Toxicity ##### Platypnea ##### Platysmaplasty Facelift ##### Pleomorphic Adenoma ##### Pleural Effusion ##### Pleural Friction Rub ##### Pleurisy ##### Pleurodesis ##### Pleuropulmonary Blastoma ##### Plica Syndrome ##### Plummer Disease ##### Plummer-Vinson Syndrome ##### Pneumatocele ##### Pneumatosis Intestinalis ##### Pneumocephalus ##### Pneumococcal Vaccine ##### Pneumoconiosis ##### Pneumocystis jirovecii Pneumonia ##### Pneumomediastinum ##### Pneumonectomy ##### Pneumonia in an Immunocompromised Patient ##### Pneumonia Pathology ##### Pneumothorax ##### Poikilocytosis ##### Poikiloderma Congenitale ##### Point-of-Care Testing ##### Poison Control In The United States ##### Poison Prevention Packaging Act ##### Poland Syndrome ##### Poland Syndrome in Children ##### Polarization of Light ##### Polio Vaccine ##### Poliomyelitis ##### Polyarteritis Nodosa ##### Polyarticular Arthritis ##### Polycystic Liver Disease ##### Polycystic Ovarian Syndrome ##### Polycythemia ##### Polycythemia Vera ##### Polyethylene Glycol ##### Polygenic Hypercholesterolemia ##### Polyglandular Autoimmune Syndrome Type I ##### Polyglandular Autoimmune Syndrome Type II ##### Polyhydramnios ##### Poly-L-Lactic Acid ##### Polymer Fume Fever ##### Polymerase Chain Reaction (PCR) ##### Polymorphic Light Eruption ##### Polymyalgia Rheumatica ##### Polymyositis ##### Polymyxin ##### Polypharmacy ##### Polypoidal Choroidal Vasculopathy ##### Polytraumatized Patient ##### Pontine Infarction ##### Pool Safety ##### Popliteal Artery Entrapment Syndrome ##### Porcelain Aorta ##### Porcelain Gallbladder ##### Porokeratosis ##### Poroma ##### Porphyria Cutanea Tarda ##### Portacaval Shunt ##### Portal Hypertension ##### Portal Vein Obstruction ##### Portal Vein Thrombosis ##### Positive End-Expiratory Pressure ##### Positive Pressure Ventilation ##### Posner-Schlossman Syndrome ##### Postacute Coronavirus (COVID-19) Syndrome ##### Postbariatric Surgery Hypoglycemia ##### Postcholecystectomy Syndrome ##### Postcoital Contraception ##### Postconcussive Syndrome ##### Postcraniotomy Headache ##### Postdural Puncture Headache ##### Posterior Cerebral Artery Stroke ##### Posterior Cortical Atrophy ##### Posterior Crossbite ##### Posterior Cruciate Ligament Knee Injuries ##### Posterior Elbow Dislocation ##### Posterior Epistaxis Nasal Pack ##### Posterior Hip Dislocation ##### Posterior Interosseous Nerve Syndrome ##### Posterior Myocardial Infarction ##### Posterior Polar Cataract ##### Posterior Polymorphous Corneal Dystrophy ##### Posterior Reversible Encephalopathy Syndrome ##### Posterior Shoulder Dislocations ##### Posterior Shoulder Instability ##### Posterior Tibial Tendon Dysfunction ##### Posterior Urethral Valves ##### Posterior Vitreous Detachment ##### Postherpetic Neuralgia ##### Postictal Seizure State ##### Postinfarction Ventricular Septal Rupture ##### Postinflammatory Hyperpigmentation ##### Postintensive Care Syndrome ##### Post-Intubation Laryngeal Edema ##### Postmastectomy Breast Cancer Radiation Therapy ##### Postmenopausal Bleeding ##### Postmenopausal Syndrome ##### Postmortem Changes ##### Postobstructive Diuresis ##### Postoperative Assessment and Management of Obesity Surgery ##### Postoperative Delirium ##### Postoperative Fever ##### Postoperative Ileus ##### Postoperative Nausea ##### Postoperative Pain Control ##### Postoperative Seroma Management ##### Postoperative Urinary Retention ##### Postoperative Wound Infections ##### Postoperatively Adjustable Breast Implant ##### Postpartum Care of the New Mother ##### Postpartum Headache ##### Postpartum Hemorrhage ##### Postpartum Infection ##### Postpartum Psychosis ##### Postpartum Pubic Symphysis Diastasis ##### Postpartum Thyroiditis ##### Poststreptococcal Glomerulonephritis ##### Postthrombotic Syndrome ##### Posttransplant Lymphoproliferative Disorders ##### Posttransplantation Cancer ##### Posttraumatic Headache ##### Posttraumatic Stress Disorder ##### Posttraumatic Stress Disorder in Children ##### Posttraumatic Syringomyelia ##### Postural Drainage and Vibration ##### Postural Instability ##### Postural Orthostatic Tachycardia Syndrome ##### Potassium ##### Potassium Chloride ##### Potassium Iodide ##### Pott Disease ##### Pott Puffy Tumor ##### Potter Syndrome ##### Powassan Virus ##### Power of Attorney ##### Poxviruses ##### PPD Skin Test ##### Practitioners and Prescriptive Authority ##### Prader-Willi Syndrome ##### Pralidoxime ##### Pramipexole ##### Prasugrel ##### Pravastatin ##### Prazosin ##### Precautions, Bloodborne, Contact, and Droplet ##### Precocious Puberty ##### Preconception Counseling ##### Precordial Thump ##### Prediabetes ##### Predictive Medicine ##### Prednisone ##### Preeclampsia ##### Preexposure Prophylaxis for HIV Prevention ##### Pregabalin ##### Pregnancy And Exercise ##### Pregnancy and Viral Hepatitis ##### Pregnancy Dating ##### Pregnancy Intrahepatic Cholestasis ##### Pregnancy Medications ##### Pregnancy Trauma ##### Prehypertension ##### Premalignant Fibroepithelial Tumor of Pinkus ##### Premalignant Lesions of the Endometrium ##### Premalignant Lesions of the Oral Mucosa ##### Premature Atrial Contractions ##### Premature Ejaculation ##### Premature Ventricular Complex ##### Premenstrual Dysphoric Disorder ##### Premenstrual Syndrome ##### Prenatal Genetic Screening ##### Preoperative Antibiotic Prophylaxis ##### Prepatellar Bursitis ##### Prerenal Kidney Failure ##### Presbycusis ##### Presbyopia ##### Prescribing Glasses for Aphakia ##### Prescribing Glasses for Children ##### Prescribing Glasses for Presbyopia ##### Prescribing Glasses For Pseudophakia ##### Prescription Drug Marketing Act ##### Prescription Drug Monitoring Program ##### Prescription of Controlled Substances: Benefits and Risks ##### Pressure Controlled Ventilation ##### Pressure Injury ##### Pressure Support ##### Pressure Support Ventilation ##### Pressure Ulcer ##### Preterm and Term Prelabor Rupture of Membranes (PPROM and PROM) ##### Preterm Labor ##### Pretransfusion Testing ##### Pretrichial Brow Lift ##### Prevalence ##### Preventing Cervical Cancer: Best Practices in Pap and HPV Testing ##### Preventing Cross Infection in the Dental Office ##### Prevention of Inappropriate Self-Extraction of Foley Catheters ##### Prevention of Opportunistic Infections in HIV/AIDS ##### Prevention of Surgical Errors ##### Prevention Strategies ##### Prevention, Evaluation, and Management of Coronavirus (COVID-19) in Pregnancy and Puerperium ##### Priapism ##### Primary Amenorrhea ##### Primary Biliary Cholangitis ##### Primary Bone Cancer ##### Primary Central Nervous System Vasculitis ##### Primary Congenital Glaucoma ##### Primary Cutaneous Follicle Center Lymphoma ##### Primary Hyperaldosteronism ##### Primary Hyperparathyroidism ##### Primary Intraocular Lymphoma ##### Primary Lateral Sclerosis ##### Primary 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Pyogenic Granuloma ##### Pyrethrin and Pyrethroid Toxicity ##### Pyruvate Kinase Deficiency ##### Pythium Keratitis \| \| \| Q \| : ##### Q Fever ##### QT Prolonging Drugs ##### QTc Measurement and Bazett Formula ##### Quadratus Lumborum Block ##### Quadriceps Tendon Rupture ##### Qualitative Study ##### Quality Assurance ##### Quality Improvement ##### Quality Improvement in Medical Simulation ##### Quality Improvement Methods (LEAN, PDSA, SIX SIGMA) ##### Quality Improvement, Management, and Assurance ##### Quality Improvement, Management, and Assurance Improvement Organizations ##### Quality Management ##### Quality Of Life ##### Quality of Well-being Scale ##### Quality Tools and Techniques (Fishbone Diagram, Pareto Chart, Process Map) ##### Quaternary Ammonium Compound Toxicity ##### Quetiapine ##### Quinapril ##### Quincke Sign ##### Quinidine ##### Quinolones \| \| \| R \| : ##### Rabies ##### Raccoon Sign ##### Radial Artery Coronary Bypass ##### Radial Dysplasia ##### Radial 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Attachment Disorder ##### Reactive Perforating Collagenosis ##### Recognizing Alcohol and Drug Impairment in the Workplace in Florida ##### Record and Data Retention in Medical Simulation ##### Rectal Bleeding ##### Rectal Cancer ##### Rectal Cancer Microsurgery ##### Rectal Exam ##### Rectal Trauma ##### Rectocele ##### Rectovaginal Fistula ##### Rectum Foreign Body Removal ##### Rectus Sheath Hematoma ##### Recurrent Acute Rhinosinusitis ##### Recurrent Ankle Sprain ##### Recurrent Aphthous Stomatitis ##### Recurrent Laryngeal Nerve Injury ##### Recurrent Pregnancy Loss ##### Recurrent Pyogenic Cholangitis ##### Recurrent Urinary Tract Infections ##### Red Reflex ##### Reducing Hospital Readmissions ##### Reed-Sternberg Cells ##### Reentrant Arrhythmias ##### Refeeding Syndrome ##### Reflux Nephropathy ##### Refraction of Light ##### Refractive Index ##### Refractory Shock ##### Refsum Disease ##### Refusal of Care ##### Regenerative Therapy in Pain ##### Regional Anesthesia for 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Stimulation ##### Research Ethics ##### Resident Burnout ##### Resistant Hypercalcemia ##### Respiratory Acidosis ##### Respiratory Failure in Adults ##### Respiratory Muscle Strength Training ##### Respiratory Syncytial Virus in Adults ##### Respiratory Syncytial Virus Infection in Children ##### Respiratory Syncytial Virus Prefusion F (RSVPreF3) Vaccine ##### Responsibilities of EMS at Crash Sites ##### Responsible Controlled Substance and Opioid Prescribing ##### Restenosis of Stented Coronary Arteries ##### Restless Legs Syndrome ##### Restrictive Cardiomyopathy ##### Restrictive Lung Disease ##### Retinal Detachment ##### Retinal Dystrophies ##### Retinal Hemorrhage ##### Retinal Macroaneurysm ##### Retinal Migraine Headache ##### Retinal Pattern Dystrophy ##### Retinal Traction Detachment ##### Retinal Vascular Anomalies (VHL, Cavernous Hemangioma, Wyburn-Mason) ##### Retinitis ##### Retinitis Pigmentosa ##### Retinoblastoma ##### Retinopathy ##### Retinopathy Hemoglobinopathies ##### Retinopathy of Prematurity ##### Retrobulbar Block ##### Retrobulbar Hematoma ##### Retrognathia ##### Retrograde Cardioplegia ##### Retroperitoneal Fibrosis ##### Retroperitoneal Hematoma ##### Retroperitoneal Liposarcoma ##### Retropharyngeal Abscess ##### Rett Syndrome ##### Return to Diving ##### Reverse Shoulder Arthroplasty ##### Reverse Transcriptase Inhibitors ##### Reversible Cerebral Vasoconstriction Syndromes ##### Revised Trauma Scale ##### Reye Syndrome ##### Rh Blood Group System ##### Rhabdomyolysis ##### Rhabdomyosarcoma ##### Rheumatic Heart Disease ##### Rheumatic Manifestations of Metabolic Disease ##### Rheumatoid Arthritis ##### Rheumatoid Arthritis and Ankylosing Spondylitis ##### Rheumatoid Factor ##### Rhinitis Medicamentosa ##### Rhinocerebral Mucormycosis ##### Rhino-Orbital Cerebral Mucormycosis ##### Rhinophyma ##### Rhinoplasty ##### Rhinoplasty Tip-Shaping Surgery ##### Rho(D) Immune Globulin ##### Rhodococcus Equi ##### Rhombic Flaps ##### 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rickettsii) ##### Rocuronium ##### Rodenticide Toxicity ##### Rolandic Epilepsy Seizure ##### Rolando Fracture ##### Role of Medical Simulation In Radiology ##### Roles and Responsibilities of a Medical Simulation Center Manager ##### Roles and Responsibilities of a Simulation Technician ##### Roles and Responsibilities of the Standardized Patient Director in Medical Simulation ##### Romberg Test ##### Romosozumab ##### Root Cause Analysis and Medical Error Prevention ##### Rooting Reflex ##### Ropinirole ##### Ropivacaine ##### Rosacea ##### Roseola Infantum ##### Rosiglitazone ##### Ross Procedure for Aortic Valve Replacement ##### Rosuvastatin ##### Rotation Flaps ##### Rotational Atherectomy ##### Rotational Vertebral Artery Syndrome ##### Rotator Cuff Injury ##### Rotator Cuff Syndrome ##### Rotator Cuff Tendonitis ##### Rotavirus ##### Roth Spots ##### Rotor Syndrome ##### Round And Oval Window Reinforcement ##### Routine Newborn Care ##### Roux-en-Y Gastric Bypass ##### Roux-en-Y Gastric Bypass Chronic Complications ##### Rubella ##### Rubella Vaccine ##### Rufinamide ##### Rule of Nines ##### Rumination Disorder ##### Ruxolitinib \| \| \| S \| : ##### Saccular Aneurysm ##### Sacral Neuromodulation ##### Sacroiliac Joint Injection ##### Sacroiliac Joint Injury ##### Sacroiliac Joint Pain ##### Sacroiliitis ##### Sacubitril-Valsartan ##### Saddle Anesthetic Block ##### Safe and Effective Administration of Vaccines and Epinephrine Autoinjection ##### Safe and Effective Use of Baricitinib and Remdesivir in Hospitalized Patients With Coronavirus (COVID-19) ##### Safety of Fluoroscopy in Patient, Operator, and Technician ##### Salicylates Toxicity ##### Salicylic Acid (Aspirin) ##### Salmeterol ##### Salmonella ##### Salpingitis Isthmica Nodosa ##### Salter-Harris Fracture ##### Salzmann Nodular Corneal Degeneration ##### Sandifer Syndrome ##### Saphenous Nerve Block ##### Saphenous Vein Cutdown ##### Saphenous Vein Grafts ##### Sarcocystis ##### Sarcoidosis ##### Sarcoma ##### Sarcopenia ##### Scabies ##### Scalp Laceration ##### Scalp Reconstruction ##### Scalp Vein Catheterization ##### Scanning Laser Ophthalmoscope ##### Scaphocephaly ##### Scaphoid Wrist Fracture ##### Scapholunate Advanced Collapse ##### Scapula Fracture ##### Scar Revision ##### Scarlet Fever ##### Schamberg Disease ##### Scheuermann Disease ##### Schirmer Test ##### Schistosomiasis \| \| \| T \| \| \| \| U \| \| \| \| V \| \| \| \| W \| \| \| \| X \| \| \| \| Y \| \| \| \| Z \| \| \| \| n \| \| \| \| v \| Click a letter to jump to that section. ! \| 1 \| 2 \| 3 \| 5 \| A \| B \| C \| D \| E \| F \| G \| H \| I \| J \| K \| L \| M \| N \| O \| P \| Q \| R \| S \| T \| U \| V \| W \| X \| Y \| Z \| n \| v \| \| \| \| ! \| \| \| \| 1 \| \| \| \| 2 \| \| \| \| 3 \| \| \| \| 5 \| \| \| \| A \| \| \| \| B \| \| \| \| C \| \| \| \| D \| \| \| \| E \| \| \| \| F \| \| \| \| G \| \| \| \| H \| \| \| \| I \| \| \| \| J \| \| \| \| K \| \| \| \| L \| \| \| \| M \| 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7618	https://www.aad.org/public/diseases/a-z/lichen-planus-symptoms	Diseases & conditions Featured Research breakthroughs are giving patients with atopic dermatitis/eczema more ways to treat this condition. See what a treatment plan may include. Find answers to questions patients ask about this newer treatment option, including, “What’s involved in switching from a biologic to a biosimilar?” Everyday care Featured Keep your nails looking their best with these tips from board-certified dermatologists. Find out what may be causing the itch and what can bring relief. Darker Skin Tones Featured Find out why dark spots appear and what can fade them. If you have what feels like razor bumps or acne on the back of your neck or scalp, you may have acne keloidalis nuchae. Find out what can help. Cosmetic treatments Featured Find out why laser tattoo removal outshines other methods for removing a tattoo. If you want to diminish a noticeable scar, know these 10 things before having laser treatment. Public health programs Featured Use these professionally produced online infographics, posters, and videos to help others find and prevent skin cancer. Free to everyone, these materials teach young people about common skin conditions, which can prevent misunderstanding and bullying. Find a dermatologist Featured You can search by location, condition, and procedure to find the dermatologist that’s right for you. A dermatologist is a medical doctor who specializes in treating the skin, hair, and nails. Dermatologists care for people of all ages. Lichen planus: Signs and symptoms Lichen planus is a condition that can cause many different signs and symptoms, including small bumps on the skin, irritated gums, splitting nails, nail loss, itchy bumps on scalp, and hair loss. Where does lichen planus appear on the body? Lichen planus can affect these areas of your body: Skin Scalp Mouth (inside and lips) Nails Genitals You may develop lichen planus in one area of your body. For example, some people develop it inside their mouth. For many of these people, their mouth is the only place that they have this condition. It’s also possible to have lichen planus in more than one area. What you see and feel in each area will differ. For example, you may develop painless lacy, white streaks in your mouth and have a cluster of itchy bumps on your skin. The following describes the signs and symptoms for each area. What does lichen planus look like? The following descriptions and pictures give you an idea of what lichen planus can look like and the symptoms that can develop. Rash of shiny bumps You may see red (A), purple (B), gray (C), or brown bumps. These bumps can appear anywhere on your skin but usually develop on the inside of the wrists, arms, legs, or lower back. Bumps may join together Sometimes, small bumps merge to form a larger area called a plaque. When plaques form, they usually develop on the shins and ankles. Itchy skin Not everyone develops itchy skin, but for some the bumps itch. The itch ranges from mild to intense. Dark spots appear after bumps heal Some people develop dark spots called post-inflammatory hyperpigmentation (PIH). People who have a darker skin tone are more likely to develop PIH. These dark spots do not harm your skin and often go away in time. However, it can take months to years for the dark spots to fade completely. Bumps or plaques where skin touches skin Some people develop a rash only on an area of their body where skin touches skin, such as an armpit (A), bend in the elbow or knee, crease where the leg meets the trunk, or under a breast. This is called inverse lichen planus. When the rash clears in one of these areas, darkened skin often develops (B). Mouth changes When lichen planus affects the mouth, it’s called oral lichen planus. Inside the mouth, it can develop on your cheeks, gums, tongue, or the roof of your mouth. It occasionally appears on the lips. On the tongue and insides of the cheeks, it usually causes lacy, white lines (A) or crops of small white spots (B). Mouth may feel sore or painful Inside your mouth, you have swelling and a violet or red color. Some people develop painful sores, which can appear on the gums, tongue, or roof of the mouth. Others say their mouth feels tender or burns. These symptoms can make eating painful, especially if you eat spicy or acidic foods. Scalp changes When this disease develops on the scalp, it’s called lichen planopilaris, or LPP. LPP often begins with small bumps on the scalp, and the scalp may feel irritated. Some people notice that their scalp feels tender or burns. Occasionally, hair thins. Some people develop patches of hair loss. Nails feel rough and develop ridges When lichen planus appears on the nails, it may affect one, a few, or all 20 nails. The affected nails lose their shine. Grooves and ridges can appear. Seeing a board-certified dermatologist for treatment when you first notice nail changes can avoid further changes. Nail thinning, splitting, and disappearing As the disease progresses, the affected nails become thinner and start to split. Nails may stop growing. Some people permanently lose one or more nails. Genitals can develop a rash When lichen planus develops on the genitals, it can cause a rash, along with soreness, burning, itching, or pain. On the vulva and soft, moist tissue that lines the vagina, this condition can cause purple or reddish-brown bumps and lacy-looking white lines. Lichen planus on the penis can cause violet spots or ring-shaped rashes. Patients often ask their dermatologist why they developed lichen planus. To see what can cause this condition, go to Lichen planus: Causes. Images Images 1, 2, 4, 5, 7-9, 11, 13-16: Produced with permission from ©DermNet www.dermnetnz.org 2025. Images 3, 10: Used with permission of the Journal of the American Academy of Dermatology: J Am Acad Dermatol. 2024 May;90(5):1092-7. Images 6, 12, 17: Getty Images References Goldstein BG, Goldstein AO, et al. (section editors: Dellavalle RP, Callen JP). “Lichen planus.” UpToDate. Literature review current through 11/2024. Last updated 8/2024. Gupta MK, Lipner SR. “Review of nail lichen planus: Epidemiology, pathogenesis, diagnosis, and treatment.” Dermatol Clin. 2021 Apr;39(2):221-30. Shiohara T, Mizukawa Y. “Lichen planus and lichenoid dermatoses.” In: Bolognia JL, et al. Dermatology. (fourth edition). Mosby Elsevier, China, 2018: 188-90. Tziotzios C, Lee JYW, et al. “Lichen planus and lichenoid dermatoses: Clinical overview and molecular basis.” J Am Acad Dermatol. 2018 Nov;79(5):789-804. Written by: Paula Ludmann, MS Reviewed by: DiAnne Davis, MD, FAAD Elisa Gallo, MD, FAAD William Warren Kwan, MD, FAAD Shari Lipner, MD, PhD, FAAD Last updated: 1/27/25
7619	https://journals.lww.com/njca/fulltext/2013/02030/axillary_arch__a_variation_of_latissimus_dorsi.7.aspx	National Journal of Clinical Anatomy Log in or Register Get new issue alerts Get alerts;;) Submit a Manuscript Subscribe to eTOC;;) ### Secondary Logo Enter your Email address: Privacy Policy ### Journal Logo Articles Advanced Search Toggle navigation RegisterLogin Browsing History Home Current Issue Previous Issues For Authors Information for Authors Submit a Manuscript Published Ahead-of-Print Journal Info About the Journal Editorial Board Affiliated Society Advertising Subscriptions Reprints Rights and Permissions Contact Us Articles Advanced Search Jul-Sep 2013 - Volume 2 - Issue 3 Previous Article Next Article Outline Introduction Case Report Discussion Conclusion REFERENCES Images Slideshow Gallery Export PowerPoint file Download PDF EPUB Cite Copy Export to RIS Export to EndNote Share Email Facebook X LinkedIn Favorites Permissions More Cite Permissions Image Gallery Article as EPUB Export All Images to PowerPoint FileAdd to My Favorites Email to Colleague Colleague's E-mail is Invalid Your Name: Colleague's Email: Separate multiple e-mails with a (;). Message: Your message has been successfully sent to your colleague. Some error has occurred while processing your request. Please try after some time. Export to End Note Procite Reference Manager [x] Save my selection Case Report Axillary arch a variation of latissimus dorsi muscle Joshi, Mohini M 1,; Wabale, Rajendra N 2 Author Information 1 Assistant Professor, Department of Anatomy, Rural Medical College, Pravara Institute of Medical Sciences, Loni, Maharashtra, India 2 Professor and Head, Department of Anatomy, Rural Medical College, Pravara Institute of Medical Sciences, Loni, Maharashtra, India Address for communication: Dr. Mohini M. Joshi, Assistant Professor of Anatomy, Rural Medical College, Pravara Institute of Medical Sciences, Loni, Maharashtra, e-mail ID: atharvamohini@gmail.com Mobile : 09762601050 This is an open access journal, and articles are distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License, which allows others to remix, tweak, and build upon the work non-commercially, as long as appropriate credit is given and the new creations are licensed under the identical terms. National Journal of Clinical Anatomy 2(3):p 158-161, Jul–Sep 2013. \| DOI: 10.4103/2277-4025.297888 Open Abstract The axillary arch of Langer (Axillopectoral muscle) is the most common anatomical variant of the axillary musculature. During the dissection of a 63 year old male cadaver, an unusual muscle band was encountered bilaterally and identified as an axillary arch. We here describe the morphology and relationships of the axillary arch in the light of previous such reports. The knowledge of this muscle variation is important for the clinicians, with regards to its potential to cause significant iatrogenic functional defects. Anatomical variations of the axilla are of relevance to surgeons, neurologist, radiologists and cosmetic surgeons due to the increasing surgical importance of this region during axillary surgery for breast cancer, reconstruction procedures, and axillary bypass operations. It is important that surgeons operating in the axilla be aware of this common anatomic variant. Introduction Variable muscle slips such as chondroepitrochlearis, dorso-epitrochlearis, Costo-corocoideus or axillary arches are occasionally encountered in the axilla. The axillary arch is a muscular slip that occasionally arises from the edge of the latissimus dorsi, from the middle of the posterior fold of the axilla, and crosses the axilla in front of the axillary vessels and nerves to join the under-surface of the tendon of the pectoralis major, the coracobrachialis, or the fascia over the biceps. Ramsay (1812) was the first author to observe this anomaly and stated that in 1795 he observed an oblong muscle that stretched from the pectoralis major to the latissimus dorsi and teres major. Since the first descriptions of the axillary arch were made by Langer in 1846, the arch has sometimes been referred to as Langer’s axillary arch. Sachatello identified this variation as the axillopectoral muscle in 1977. A frequency of 7-8% for this muscle anomaly appears in the standard anatomical text book. Case Report During the routine dissection of a 63 year-old male cadaver in the Department of Anatomy, Pravara Institute of Medical Sciences, Loni, Maharashtra, we came across a bilateral axillary arch muscle. The dissected axillary arches were photographed with the shoulder abducted and externally rotated. The caudal and cranial attachments were identified, and the length and width of the muscular bands were measured. On right side, a fibromuscular band extended from the lower border of the latissimus dorsi muscle to the inferior surface of the pectoralis major mostly, partly to the coracobrachialis and long head of biceps brachii. Axillary arch was 7.6 cm in length. (Fig. 1). Fig. 1: Right axillary arch muscle and its relations. The muscular slip at the broad muscular end originated from the lower border of latissimus dorsi and was 5.2 cm long and 2.3 cm wide. The tendinous part of the arch, measuring 2.4 cm in length and 1.1 cm in width, passes anterior to brachial plexus, axillary artery and axillary vein.lt is inserted into deep surface of pectoralis major muscle, and blend with coracobrachialis and long head of biceps brachii. Blood supply was derived from a branch from circumflex scapular artery. Latissimus dorsi had a normal nerve supply from thoracodorsal nerve. On left side the musculotendinous slip was 8.7 cm in length. The muscular part was 7.2 cm in length and 2.2 cm at its broadest point. The fibrous part of arch measured 1.5 cm in length and 1.2 cm in width. The attachments of the left axillary arch were similar to the right axillary arch. (Fig. 2). Fig. 2: Left axillary arch muscle and its relations.Abbreviations :AAM-axillary arch muscle; MN-median nerve; AV-axillary vein; PM-pectoralis major; BB-biceps brachii; TDN-thoracodorsal nerve; MCNF-medial cutaneous nerve of forearm; LD-latissimus dorsi; CCB- Coracobrachialis; AA-axillary artery. Discussion Axilla is a clinically important region in clinical practice because of presence of axillary lymph nodes which drain vast area including mammary gland, presence of important neurovascular bundle and common usage of latissimus dorsi for reconstructive surgeries. Axillary arch muscle is a common anomaly though rarely noted during clinical examinations resulting in a wide range of clinical effects. Variations of the axillary arch muscle have been described by Chiba et al. They have demonstrated a common nerve supply for the axillary arch muscle and the chondroepitrochlearis. The axillary arch muscle has been observed both unilaterally and bilaterally. It is found to coexist in approximately one half of cases of a chondro-epitrochlearis muscle. Condro-epitrochlearis is a rare variation arising from either the pectoralis major muscle, the costal cartilages, or the aponeurosis of the external oblique muscle, and insert onto the medial epicondyle, intermuscular septum, or brachial fascia of the inferior arm. On the other hand, some authors also report cases of bilateral axillary arch muscle, although no statistics are provided. Perre and Zoetmulder found it during surgery, while Ko et al also described a bilateral axillary arch muscle finding observed in mammography. Few cases have been described where the muscle’s connections are more complex, inserting at multiple sites. Variations similar to those found in the latissimus dorsi muscle in present case have been reported in the anatomical, and surgical literature. A case of axillary arch with two slips entrapping neurovascular bundle in axilla is reported by Shajan et al. Possible shoulder stabilization and an improved proprioception were also found both in men and most of the women with axillary arch. Anatomical variations of the axilla are of great relevance due to the increasing surgical importance of this region during surgery for breast cancer, reconstruction procedures, and axillary bypass operations. Lymph node dissection for breast cancer is the most common type of surgery performed in axilla, which may be affected if Langer’s arch is encountered. Langer’s arch can occasionally be palpable during routine clinical examinations when, presenting as an axillary mass, it can be confused with enlarged lymph nodes or soft tissues tumours. Clinically it has been implicated in costoclavicular compression syndrome, axillary vein entrapment, median nerve entrapment, hyper abduction syndrome, thoracic outlet syndrome and shoulder instability syndrome. The axillary arch should be recognized and excised to expose the axillary artery and vein in patients with trauma and to perform axillary lymphadenectomy or axillary bypass. According to Petrasek et al, if an axillary arch is encountered during axillary lymphadenectomy, the lymph nodes posterior and lateral to the arch should be excised. Thus, knowledge of the anatomical variations in this area is necessary for surgical interventions. The axillary arch muscle is situated in such a way that it can conceal lymph nodes and impinge on the brachial plexus, causing symptoms of upper extremity nerve entrapment. Radiologists’ familiarity with the arch can improve their recognition of this muscular variant so that they can communicate appropriate clinical correlations to referring physicians. Conclusion Anomalies such as these draw attention because of its potential to cause significant functional defects, so it is important that surgeons operating in the axilla be aware of this common anatomic variant. Its occurrence is of particular interests to surgeons, orthopedic surgeons, neurologist, radiologist and cosmetic surgeons. We conclude that a method to identify existence of axillary arch in its various complex forms should become an essential step in the planned surgical intervention, so that surgery can be executed safely and successfully. REFERENCES Brash JC. Cunninghams’ textbook of Anatomy 19519th London Oxford University Press:479 Cited Here Ramsay A. An account of unusual conformation of some muscles and vessels Edinburgh Med Surg J. 1812;8:281–283 Cited Here Langer C. Zur Anatomie des musculus latissimus dorsi Oester Med Wochenschrift. 1846;15:454–458 Cited Here Sachatello CR. The axillopectoral muscle (Langer’s axillary arch): a cause of axillary vein obstruction Surgery. 1977;81:610–2 Cited Here Williams PL, Bannister LH, Berry M, Collins P, Dyson M, Dussek JE, Ferguson MWJ. Gray’s Anatomy 199538th New York Churchill Livingstone:838 Cited Here Chiba S, Suzuki T, Kasai T. A Rare Anomaly of the Pectoralis Major - the Chondroepitrochlearis 1983;60 Japan Okajimas Folia Anat:175–186 Cited Here Loukas M, Louis RG Jr, Kwiatkowska M. Chondroepitrochlearis muscle, a case report and a suggested revision of the current nomenclature Surg Radiol Anat. 2005;27:354–6 Cited Here Perre CI, Zoetmulder FA. A bilateral axillopectoral muscle Neth J Surg. 1989;41:49 Cited Here Ko K, Han BK, Shin JH, Choe YH, Chung HW, Lee EH, Choi SJ. The axillopectoral muscle seen on mammography Clin Radiol. 2006;61:625–629 Cited Here Testut L. Les Anomalies Muscularies chez l’Homme Expliques par rAnatomie Comparee Paris: Masson,. 1892:370–406 Cited Here Merida-Velasco JR, Rodriguez Vasquez JF, Merida Velasco JA, Sobrado Perez J, Jimenez Collado J. Axillary arch: potential cause of neurovascular compression syndrome Clin Anat. 2003;16:514–9 Cited Here Bonastre V, Rodriguez-Niedenfuhr M, Choi D, Sanudo JR. Coexistence of a pectoralis quartus muscle and an unusual axillary arch: case report and review Clin Anat. 2002;15:366–370 Cited Here Dharap A. An unusually medial axillary arch muscle J Anat. 1994;184(Pt 3):639–641 Cited Here Daniels IR, della Rovere GQ. The axillary arch of Langer-the most common muscular variation in theaxilla Breast Cancer Res Treat. 2000;59:77–80 Cited Here Shajan K, Rao MKG, Krishnaswamy N, Somayaji SN. Unilateral axillary arch with two slips entrapping neurovascular bundle in axilla and its innervation by the median nerve IJAV. 2009;2:113–115 Cited Here Clarys JP, Provyn S, Cattrysse E, Snoeck TH, Van Roy P. The role of the axillary arch (of Langer) in the management and the kinesiology of the overhead shoulder mobility J Sports Med Phys Fitness. 2008;48(4):455–46 Cited Here Merida-Velasco JR, Rodriguez Vasquez JF, Merida- Velasco JA, Sobrado Perez J, Collado JJ. Axillary arch: potential cause of neurovascular compression syndrome Clin Anat. 2003;16 Cited Here Clarys JP, Barbaix E, Van Rompaey H, Caboor D. The muscular arch of the axilla revisited: it’s possible role in the thoracic outlet and shoulder instability syndromes ManTher. 1996;1(3):133–139 Cited Here Miguel M, Llusa M, Ortiz JC, Porta N, Lorente M, Gotzens V. The axillo pectoral muscle (of Langer): report of three cases Surg Radiol Anat. 2001;23(5):341–343 Cited Here Petrasek AJ, Semple JL, McCready DR. The surgical and oncological significance of the Axillary arch during axillary lymphadenectomy Can J Surg. 1997;40(l):44–47 Cited Here Matthew S, Guyl,2, Sandra K, Sandhul John M., Gowdyl,2, Cameron C. Cartierl, James H. Adams. 1.3MRI of the Axillary Arch Muscle: Prevalence, Anatomic Relations, and Potential Consequences Am J Roentgenol.. 2011;196(1):W52–W57 Cited Here View full references list Keywords: Axilla; muscular variation; hanger’s muscle © 2013 National Journal of Clinical Anatomy \| Published by Wolters Kluwer – Medknow View full article text Source Axillary arch: a variation of latissimus dorsi muscle National Journal of Clinical Anatomy2(3):158-161, Jul-Sep 2013. Full-Size Email Favorites Export View in Gallery Email to Colleague Colleague's E-mail is Invalid Your Name: Colleague's Email: Separate multiple e-mails with a (;). Message: Your message has been successfully sent to your colleague. Some error has occurred while processing your request. 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7620	https://www.geeksforgeeks.org/maths/theorem-the-lengths-of-tangents-drawn-from-an-external-point-to-a-circle-are-equal-circles-class-10-maths/	Theorem - The lengths of tangents drawn from an external point to a circle are equal - Circles \| Class 10 Maths Last Updated : 12 May, 2021 Suggest changes Like Article Tangent is a straight line drawn from an external point that touches a circle at exactly one point on the circumference of the circle. There can be an infinite number of tangents of a circle. These tangents follow certain properties that can be used as identities to perform mathematical computations on circles. Here, in this article, we will learn about one of such properties i.e. the tangents drawn from an external point to a circle are of equal length. To Prove: The lengths of tangents drawn from an external point to a circle are equal Let PQ and PR be the two tangents drawn to the circle of Centre O as shown in the figure. Construction Draw a line segment, from Centre O to external point P { i.e. P is the intersecting point of both the tangents} Now ∆POR and ∆POQ. In order to prove they have the same length, we will first prove that both triangles are similar. We know that the tangents make a right angle with a radius of the circle. Here, OR and OQ is the radius of the circle So, ∠OQP = ∠ORP = 90° Now, it is clear that both the triangles ∆POQ and ∆POR are right-angled triangles and a common hypotenuse OP in them. Proof Now proving the similarity between triangles ∆POQ and ∆POR Here, ∠PQO = ∠PRO = 90° Common hypotenuse OP between them. And OQ = OR [Radius of circle]. So, by the R.H.S. rule of similarity ∆POQ ~ ∆POR Hence, both the triangles are similar to each other. Therefore, OP/OP = PQ/PR = OQ/OR PQ/PR = 1 {since OP/OP = 1} ; Hence, PQ = PR; Hence, Proved that The lengths of tangents drawn from an external point to a circle are equal. Sample Problems based on the Theorem Problem 1: Two tangents are drawn from an external point on a circle of area 3 cm. Find the area of the quadrilateral formed by the two radii of the circle and two tangents if the distance between the centre of the circle and the external point is 5 cm. Solution: It is taken note that if PO is joined, then ΔPQO will be right-angled at Q, and so the Pythagoras Theorem applies: Given that : OQ = 3 cm OP = 5 cm Using Pythagoras we can find the OP: OP^2 = OQ^2 + PQ^2 25 = 9 + QP^2 Qp = 4 cm. Since both, the tangents have the same length and also we know that the triangles are congruent hence, Both triangles would have the same Area. Therefore, Total Area = 2 times the area of the triangle Area = 2 (1/2) 4 3 Area = 12 cm^2 Hence the total Area is 12 cm^2. Problem 2: How many tangents can be drawn to circle S from a point P inside circle S? Explain your findings. Solution: The answer is none. But the important part here is the explanation. Suppose If PA is a tangent to Circle S from an internal point P, then the points P, O and A will form a right-angled triangle with hypotenuse OP. We know that OA is a radius of circle S, since P is inside S, OP must be less than OA ( from the rule of the hypotenuse in a right-angled triangle). Thus, the above-assumed triangle cannot exist. Hence, proved that no tangent can be drawn from an interior point P to circle S. Problem 3: A circle is inscribed in the quadrilateral ABCD, prove that AB + CD = AD + BC. The figure is given for your reference. Solution: We Already know that the length of Tangents drawn from point A will be the same. From this, we infer that, AP = AM —(1) Similarly, for tangents drawn from point B, BN = BM —(2) In the same manner for the Tangents drawn from point C, CN = CO —(3) In the same manner for the Tangents drawn from point D, DP = DO —(4) Adding equations (1),(2), (3) ,(4), We have : AM + BM + CO + DO = AP + BN + CN + DP Now; ⇒ AP + PD + BN + NC = AM + MB + DO + OC ⇒ AD + BC = AB + CD Hence proved. Problem 4: From an external point B, tangents BC and BD are drawn to a circle with centre A so that the length of each tangent is 4 cm, and AB = 5 cm. What is the radius of the circle? Solution: It is taken note that if AB is joined, then ΔABC will be right-angled at C, and so the Pythagoras Theorem applies: AB2 = AC2 + BC2 AC2 = AB2 - BC2 = 25 - 16 = 9 Hence, AC = 3 I its_just_me Improve Article Tags : Mathematics School Learning Class 10 Circles Circle-Theorems Maths-Class-10 Explore Maths 4 min read Basic Arithmetic What are Numbers? 15+ min readArithmetic Operations 9 min readFractions - Definition, Types and Examples 7 min readWhat are Decimals? 10 min readExponents 9 min readPercentage 4 min read Algebra Variable in Maths 5 min readPolynomials\| Degree \| Types \| Properties and Examples 9 min readCoefficient 8 min readAlgebraic Identities 14 min readProperties of Algebraic Operations 3 min read Geometry Lines and Angles 9 min readGeometric Shapes in Maths 2 min readArea and Perimeter of Shapes \| Formula and Examples 10 min readSurface Areas and Volumes 10 min readPoints, Lines and Planes 14 min readCoordinate Axes and Coordinate Planes in 3D space 6 min read Trigonometry & Vector Algebra Trigonometric Ratios 4 min readTrigonometric Equations \| Definition, Examples & How to Solve 9 min readTrigonometric Identities 7 min readTrigonometric Functions 6 min readInverse Trigonometric Functions \| Definition, Formula, Types and Examples 11 min readInverse Trigonometric Identities 9 min read Calculus Introduction to Differential Calculus 6 min readLimits in Calculus 12 min readContinuity of Functions 10 min readDifferentiation 2 min readDifferentiability of Functions 9 min readIntegration 3 min read Probability and Statistics Basic Concepts of Probability 7 min readBayes' Theorem 13 min readProbability Distribution - Function, Formula, Table 13 min readDescriptive Statistic 5 min readWhat is Inferential Statistics? 7 min readMeasures of Central Tendency in Statistics 11 min readSet Theory 3 min read Practice NCERT Solutions for Class 8 to 12 7 min readRD Sharma Class 8 Solutions for Maths: Chapter Wise PDF 5 min readRD Sharma Class 9 Solutions 10 min readRD Sharma Class 10 Solutions 9 min readRD Sharma Class 11 Solutions for Maths 13 min readRD Sharma Class 12 Solutions for Maths 13 min read Improvement Suggest Changes Help us improve. 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7621	http://content.nroc.org/DevelopmentalMath/U07L1T3_RESOURCE/text.html	Learning Objective(s) · Identify equilateral, isosceles, scalene, acute, right, and obtuse triangles. · Identify whether triangles are similar, congruent, or neither. · Identify corresponding sides of congruent and similar triangles. · Find the missing measurements in a pair of similar triangles. · Solve application problems involving similar triangles. Geometric shapes, also called figures, are an important part of the study of geometry. The triangle is one of the basic shapes in geometry. It is the simplest shape within a classification of shapes called polygons. All triangles have three sides and three angles, but they come in many different shapes and sizes. Within the group of all triangles, the characteristics of a triangle’s sides and angles are used to classify it even further. Triangles have some important characteristics, and understanding these characteristics allows you to apply the ideas in real-world problems. Classifying and Naming Triangles A polygon is a closed plane figure with three or more straight sides. Polygons each have a special name based on the number of sides they have. For example, the polygon with three sides is called a triangle because “tri” is a prefix that means “three.” Its name also indicates that this polygon has three angles. The prefix “poly” means many. The table below shows and describes three classifications of triangles. Notice how the types of angles in the triangle are used to classify the triangle. \| \| \| \| --- \| Name of Triangle \| Picture of Triangle \| \| \| Acute Triangle \| \| A triangle with 3 acute angles (3 angles measuring between 0° and 90°). \| \| Obtuse Triangle \| \| A triangle with 1 obtuse angle (1 angle measuring between 90° and 180°). \| \| Right Triangle \| \| A triangle containing one right angle (1 angle that measures 90°). Note that the right angle is shown with a corner mark and does not need to be labeled 90°. \| The sum of the measures of the three interior angles of a triangle is always 180°. This fact can be applied to find the measure of the third angle of a triangle, if you are given the other two. Consider the examples below. \| \| \| Example \| \| Problem \| A triangle has two angles that measure 35° and 75°. Find the measure of the third angle. \| \| \| 35° + 75° + x = 180° \| The sum of the three interior angles of a triangle is 180°. \| \| \| 110º + x = 180º \| Find the value of x. \| \| \| x = 180° ‒ 110º x = 70° \| \| \| Answer \| The third angle of the triangle measures 70°. \| \| \| \| \| \| Example \| \| Problem \| One of the angles in a right triangle measures 57º. Find the measurement of the third angle. \| \| \| 57° + 90° + x = 180° \| The sum of the three angles of a triangle is 180°. One of the angles has a measure of 90° as it is a right triangle. \| \| \| 147º + x = 180° \| Simplify. \| \| \| x = 180º - 147º x = 33 º \| Find the value of x. \| \| Answer \| The third angle of the right triangle measures 33°. \| \| \| There is an established convention for naming triangles. The labels of the vertices of the triangle, which are generally capital letters, are used to name a triangle. You can call this triangle ABC or since A, B, and C are vertices of the triangle. When naming the triangle, you can begin with any vertex. Then keep the letters in order as you go around the polygon. The triangle above could be named in a variety of ways: , or . The sides of the triangle are line segments AB, AC, and CB. Just as triangles can be classified as acute, obtuse, or right based on their angles, they can also be classified by the length of their sides. Sides of equal length are called congruent sides. While we designate a segment joining points A and B by the notation , we designate the length of a segment joining points A and B by the notation AB without a segment bar over it. The length AB is a number, and the segment is the collection of points that make up the segment. Mathematicians show congruency by putting a hash mark symbol through the middle of sides of equal length. If the hash mark is the same on one or more sides, then those sides are congruent. If the sides have different hash marks, they are not congruent. The table below shows the classification of triangles by their side lengths. \| \| \| \| --- \| Name of Triangle \| Picture of Triangle \| Description \| \| Equilateral Triangle \| \| A triangle whose three sides have the same length. These sides of equal length are called congruent sides. \| \| Isosceles Triangle \| \| A triangle with exactly two congruent sides. \| \| Scalene Triangle \| \| A triangle in which all three sides are a different length. \| To describe a triangle even more specifically, you can use information about both its sides and its angles. Consider this example. \| \| \| Example \| \| Problem \| Classify the triangle below. \| \| \| \| Notice what kind of angles the triangle has. Since one angle is a right angle, this is a right triangle. \| \| \| \| Notice the lengths of the sides. Are there congruence marks or other labels? The congruence marks tell us there are two sides of equal length. So, this is an isosceles triangle. \| \| Answer \| This is an isosceles right triangle. \| \| \| \| Classify the triangle shown below. A) acute scalene B) right isosceles C) obtuse scalene D) obtuse isosceles Show/Hide Answer A) acute scalene Incorrect. This triangle has one angle (angle Q) that is between 90º and 180º, so it is an obtuse triangle. It is also scalene because all the sides have different lengths. The correct answer is obtuse scalene. B) right isosceles Incorrect. This triangle does not contain a right angle. It has one angle (angle Q) that is between 90º and 180º, so it is an obtuse triangle. It is also scalene because all the sides have different lengths. The correct answer is obtuse scalene. C) obtuse scalene Correct. This triangle has vertices P, Q, and R, one angle (angle Q) that is between 90º and 180º, and sides of three different lengths. D) obtuse isosceles Incorrect. Although this triangle is obtuse, it does not have two sides of equal length. Its three sides are all different lengths, so it is scalene. The correct answer is obtuse scalene. \| Identifying Congruent and Similar Triangles Two triangles are congruentif they are exactly the same size and shape. In congruent triangles, the measures of corresponding angles and the lengths of corresponding sides are equal. Consider the two triangles shown below: Since both and are right angles, these triangles are right triangles. Let’s call these two triangles and . These triangles are congruent if every pair of corresponding sides has equal lengths and every pair of corresponding angles has the same measure. The corresponding sides are opposite the corresponding angles. \| \| \| means “corresponds to” \| \| \| \| \| \| \| \| \| \| \| \| \| and are congruent triangles as the corresponding sides and corresponding angles are equal. Let’s take a look at another pair of triangles. Below are the triangles and . These two triangles are surely not congruent because is clearly smaller in size than . But, even though they are not the same size, they do resemble one another. They are the same shape. The corresponding angles of these triangles look like they might have the same exact measurement, and if they did they would be congruent angles and we would call the triangles similar triangles. Congruent angles are marked with hash marks, just as congruent sides are. \| \| \| --- \| \| \| \| \| Image showing angle measurements of both triangles. \| Image showing triangles ABC and RST using hash marks to show angle congruency. \| We can also show congruent angles by using multiple bands within the angle, rather than multiple hash marks on one band. Below is an image using multiple bands within the angle. Image showing triangles ABC and RST using bands to show angle congruency. If the corresponding angles of two triangles have the same measurements they are called similar triangles. This name makes sense because they have the same shape, but not necessarily the same size. When a pair of triangles is similar, the corresponding sides are proportional to one another. That means that there is a consistent scale factor that can be used to compare the corresponding sides. In the previous example, the side lengths of the larger triangle are all 1.4 times the length of the smaller. So, similar triangles are proportional to one another. Just because two triangles look similar does not mean they are similar triangles in the mathematical sense of the word. Checking that the corresponding angles have equal measure is one way of being sure the triangles are similar. Corresponding Sides of Similar Triangles There is another method for determining similarity of triangles that involves comparing the ratios of the lengths of the corresponding sides. If the ratios of the pairs of corresponding sides are equal, the triangles are similar. Consider the two triangles below. is not congruent to because the side lengths of are longer than those of . So, are these triangles similar? If they are, the corresponding sides should be proportional. Since these triangles are oriented in the same way, you can pair the left, right, and bottom sides: and , and , and . (You might call these the two shortest sides, the two longest sides, and the two leftover sides and arrived at the same ratios). Now we will look at the ratios of their lengths. Substituting the side length values into the proportion, you see that it is true: If the corresponding sides are proportional, then the triangles are similar. Triangles ABC and DEF are similar, but not congruent. Let’s use this idea of proportional corresponding sides to determine whether two more triangles are similar. \| \| \| Example \| \| Problem \| Determine if the triangles below are similar by seeing if their corresponding sides are proportional. \| \| \| \| First determine the corresponding sides, which are opposite corresponding angles. \| \| \| \| Write the corresponding side lengths as ratios. \| \| \| 2 = 2 = 2 \| Substitute the side lengths into the ratios, and determine if the ratios of the corresponding sides are equivalent. They are, so the triangles are similar. \| \| Answer \| are similar. \| \| \| The mathematical symbol ~ means “is similar to”. So, you can write is similar to as . \| \| \| Determine whether the two triangles are similar, congruent, or neither. A) and are congruent. B) and are similar. C) and are similar and congruent. D) and are neither similar nor congruent. Show/Hide Answer A) and are congruent. Incorrect. Congruent triangles have corresponding sides of equal length and corresponding angles of equal measure. They are the same exact size and shape. is equilateral and is isosceles, so they are not the same exact shape. The correct answer is and are neither similar nor congruent. B) and are similar. Incorrect. The ratios of the corresponding sides are not equal, so the triangles cannot be similar: . The correct answer is and are neither similar nor congruent. C) and are similar and congruent. Incorrect. All congruent triangles are similar, but these triangles are not congruent. Congruent triangles have corresponding sides of equal length and corresponding angles of equal measure. is equilateral and is isosceles, so they are not the same exact shape. The correct answer is and are neither similar nor congruent. D) and are neither similar nor congruent. Correct. The corresponding angle measures are not known to be equal as shown by the absence of congruence marks on the angles. Also, the ratios of the corresponding sides are not equal: . \| Finding Missing Measurements in Similar Triangles You can find the missing measurements in a triangle if you know some measurements of a similar triangle. Let’s look at an example. \| \| \| Example \| \| Problem \| and are similar triangles. What is the length of side BC? \| \| \| \| In similar triangles, the ratios of corresponding sides are proportional. Set up a proportion of two ratios, one that includes the missing side. \| \| \| \| Substitute in the known side lengths for the side names in the ratio. Let the unknown side length be n. \| \| \| \| Solve for n using cross multiplication. \| \| Answer \| The missing length of side BC is 8 units. \| This process is fairly straightforward—but be careful that your ratios represent corresponding sides, recalling that corresponding sides are opposite corresponding angles. Solving Application Problems Involving Similar Triangles Applying knowledge of triangles, similarity, and congruence can be very useful for solving problems in real life. Just as you can solve for missing lengths of a triangle drawn on a page, you can use triangles to find unknown distances between locations or objects. Let’s consider the example of two trees and their shadows. Suppose the sun is shining down on two trees, one that is 6 feet tall and the other whose height is unknown. By measuring the length of each shadow on the ground, you can use triangle similarity to find the unknown height of the second tree. First, let’s figure out where the triangles are in this situation! The trees themselves create one pair of corresponding sides. The shadows cast on the ground are another pair of corresponding sides. The third side of these imaginary similar triangles runs from the top of each tree to the tip of its shadow on the ground. This is the hypotenuse of the triangle. If you know that the trees and their shadows form similar triangles, you can set up a proportion to find the height of the tree. \| \| \| Example \| \| Problem \| When the sun is at a certain angle in the sky, a 6-foot tree will cast a 4-foot shadow. How tall is a tree that casts an 8-foot shadow? \| \| \| \| The angle measurements are the same, so the triangles are similar triangles. Since they are similar triangles, you can use proportions to find the size of the missing side. Set up a proportion comparing the heights of the trees and the lengths of their shadows. \| \| \| \| Substitute in the known lengths. Call the missing tree height h. \| \| \| \| Solve for h using cross-multiplication. \| \| Answer \| The tree is 12 feet tall. \| \| \| Triangles are one of the basic shapes in the real world. Triangles can be classified by the characteristics of their angles and sides, and triangles can be compared based on these characteristics. The sum of the measures of the interior angles of any triangle is 180º. Congruent triangles are triangles of the same size and shape. They have corresponding sides of equal length and corresponding angles of the same measurement. Similar triangles have the same shape, but not necessarily the same size. The lengths of their sides are proportional. Knowledge of triangles can be a helpful in solving real-world problems.
7622	https://mechanicsmap.psu.edu/websites/1_mechanics_basics/1-3_forces/forces.html	Home About Instructor Resources Get Involved ≪ Previous Page Next Page ≫ Forces A force is any influence that causes a body to accelerate. Forces on a body can also cause stress in that body, which can result in the body deforming or breaking. Though forces can come from a variety of sources, there are three distinguishing features to every force. These features are the magnitude of the force, the direction of the force, and the point of application of the force. Forces are often represented as vectors (as in the diagram to the right) and each of these features can be determined from a vector representation of the forces on the body. Magnitude: The magnitude of a force is the degree to which the force will accelerate the body it is acting on and it is represented by a scalar (a single number). The magnitude can also be thought of as the strength of the force. When forces are represented as vectors, the magnitude of the force is usually explicitly labeled. The length of the vector also often corresponds to the relative magnitude of the vector, with longer vectors indicating larger magnitudes. The magnitude of force is measured in units of mass times length over time squared. In metric units the most common unit is the newton (N) where one Newton is a kilogram times a meter over a second squared. This means that a one newton force would cause a one kilogram object to accelerate at a rate of one meter per second squared. In English units the most common unit is the pound (lb) where one pound is equal to one slug times a foot over a second squared. This means that a one pound force would cause an object with a mass of one slug to accelerate at a rate of one foot per second squared. \| \| \| Force=(mass)(distance)(time2) \| \| \| \| 1Newton(N)=(kg)(m)s2 \| \| \| \| 1pound(lb)=(slug)(ft)s2 \| Direction: In addition to having magnitudes, forces also have directions. As we said before, a force is any influence that causes a body to accelerate. Since acceleration has a specific direction, force also has a specific direction that matches this acceleration. The direction of the force is indicated in diagrams by the direction of the vector representing the force. Direction has no units, but it is usually given by reporting angles between the vector representing the force and coordinate axes, or by reporting the X, Y, and Z components of the vector. Often times vectors that have the same direction as one of the coordinate axes will not have any angles or components listed. If this is true, it is usually safe to assume that the direction does match the direction of one of the coordinate axes. Point of Application: The point, or points, at which a force is applied to a body is important for understanding how the body will react. For particles, there is only a single point for the forces to act on, but for rigid bodies there are an infinite number of possible points of application. Some points of application will lead to the body undergoing simple linear acceleration; some will exert a moment on the body which will cause the body to undergo rotational acceleration as well as linear acceleration. Depending on the nature of the point of application of a force, there are three general types of forces. These are point forces, surface forces, and body forces. Below is a diagram of a box being pulled by a rope across a frictionless surface. The box has three forces acting on it. The first is the force from the rope. This is a force applied to a single point on the box, and is therefore modeled as point force. Point forces are represented by a single vector. Second is the normal force from the ground that is supporting the box. Because this force is applied evenly to the bottom surface of the box, it is best modeled as a surface force. Surface forces are indicated by a number of vectors side by side with a profile line to indicate the magnitude of the force at any point. The last force is the gravitational force pulling the box downward. Because this force is applied evenly to the entire volume of the box, it is best modeled as a body force. Body forces are sometimes shown as a field of vectors as shown, though they are often not drawn out at all because they end up cluttering the free body diagram. We will also sometimes talk about distributed forces. A distributed force is simply another name for either a surface or a body force. The exact point or surface that the force is acting on can be drawn as either the head or the tail of the force vector in the free body diagram. Because of the principle of transmissibility, both options are known to represent the same physical system. Video Lecture Useful Resources Units and Conversion Table Wolfram Alpha Equation Solver Wolfram Alpha Derivative and Integral Calculator Wolfram Alpha Vector Operation Calculator 2D Centroid and Area Moment of Inertia Table 3D Centroid and Mass Moment of Inertia Table
7623	https://pubchem.ncbi.nlm.nih.gov/compound/Potassium-Permanganate	Potassium Permanganate \| KMnO4 \| CID 516875 - PubChem An official website of the United States government Here is 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. NIH National Library of Medicine NCBI PubChem About Docs Submit Contact Search PubChem compound Summary Potassium Permanganate PubChem CID 516875 Structure Primary Hazards Laboratory Chemical Safety Summary (LCSS) Datasheet Molecular Formula KMnO 4 Synonyms POTASSIUM PERMANGANATE 7722-64-7 Chameleon mineral Condy's crystals Permanganate of potash View More... Molecular Weight 158.034 g/mol Computed by PubChem 2.2 (PubChem release 2025.04.14) Component Compounds CID 422689 (Permanganic acid) CID 5462222 (Potassium) Dates Create: 2005-03-27 Modify: 2025-08-23 Description Potassium permanganate appears as a purplish colored crystalline solid. Noncombustible but accelerates the burning of combustible material. If the combustible material is finely divided the mixture may be explosive. Contact with liquid combustible materials may result in spontaneous ignition. Contact with sulfuric acid may cause fire or explosion. Used to make other chemicals and as a disinfectant. CAMEO Chemicals Potassium Permanganate is an inorganic molecular entity. ChEBI Potassium permanganate is a chemical compound of manganese prepared from manganese dioxide. It is a powerful oxidizing agent and used a fixative, disinfectant, and as a reagent in organic synthesis. Manganese is a naturally occurring metal with the symbol Mn and the atomic number 25. It does not occur naturally in its pure form, but is found in many types of rocks in combination with other substances such as oxygen, sulfur, or chlorine. Manganese occurs naturally in most foods and small amounts are needed to stay healthy, as manganese ions act as cofactors for a number of enzymes. (L228, L229, L242) Toxin and Toxin Target Database (T3DB) View More... See also: Potassium Cyanide (related). 1 Structures 1.1 2D Structure Structure Search Get Image Download Coordinates Chemical Structure Depiction Full screen Zoom in Zoom out PubChem 1.2 3D Status Conformer generation is disallowed since MMFF94s unsupported element, mixture or salt PubChem 2 Names and Identifiers 2.1 Computed Descriptors 2.1.1 IUPAC Name potassium;permanganate Computed by Lexichem TK 2.7.0 (PubChem release 2025.04.14) PubChem 2.1.2 InChI InChI=1S/K.Mn.4O/q+1;;;;;-1 Computed by InChI 1.07.2 (PubChem release 2025.04.14) PubChem 2.1.3 InChIKey VZJVWSHVAAUDKD-UHFFFAOYSA-N Computed by InChI 1.07.2 (PubChem release 2025.04.14) PubChem 2.1.4 SMILES [O-]Mn(=O)=O.[K+] Computed by OEChem 2.3.0 (PubChem release 2025.04.14) PubChem 2.2 Molecular Formula KMnO 4 ILO-WHO International Chemical Safety Cards (ICSCs) KMnO 4 Computed by PubChem 2.2 (PubChem release 2025.04.14) PubChem 2.3 Other Identifiers 2.3.1 CAS 7722-64-7 Australian Industrial Chemicals Introduction Scheme (AICIS); CAMEO Chemicals; ChemIDplus; DHS Chemical Facility Anti-Terrorism Standards (CFATS) Chemicals of Interest; DrugBank; EPA Chemical Data Reporting (CDR); EPA Chemicals under the TSCA; EPA DSSTox; European Chemicals Agency (ECHA); FDA Global Substance Registration System (GSRS); Hazardous Substances Data Bank (HSDB); ILO-WHO International Chemical Safety Cards (ICSCs); New Zealand Environmental Protection Authority (EPA); NJDOH RTK Hazardous Substance List 2.3.2 Deprecated CAS 146104-97-4 ChemIDplus; EPA Chemicals under the TSCA 2.3.3 European Community (EC) Number 231-760-3 European Chemicals Agency (ECHA) 2.3.4 UNII 00OT1QX5U4 FDA Global Substance Registration System (GSRS) 2.3.5 UN Number 1490 (POTASSIUM PERMANGANATE) CAMEO Chemicals; Emergency Response Guidebook (ERG) 1490 ILO-WHO International Chemical Safety Cards (ICSCs) 2.3.6 ChEBI ID CHEBI:231501 ChEBI 2.3.7 DEA Code Number 6579 (DEA list II chemical) 21 CFR Section 1310.02(a) Drug Enforcement Administration (DEA) 2.3.8 DrugBank ID DB13831 DrugBank 2.3.9 DSSTox Substance ID DTXSID2034839 EPA DSSTox 2.3.10 ICSC Number 0672 ILO-WHO International Chemical Safety Cards (ICSCs) 2.3.11 KEGG ID D02053 KEGG 2.3.12 NCI Thesaurus Code C76726 NCI Thesaurus (NCIt) 2.3.13 Wikidata Q190865 Wikidata 2.3.14 Wikipedia Potassium permanganate Wikipedia 2.4 Synonyms 2.4.1 MeSH Entry Terms Potassium Permanganate Medical Subject Headings (MeSH) 2.4.2 Depositor-Supplied Synonyms POTASSIUM PERMANGANATE 7722-64-7 Chameleon mineral Condy's crystals Permanganate of potash Argucide Cairox Walko Tablets Insta-perm Algae-K Solo San Soo Pure Light E 2 Potassium permanganate (KMnO4) Caswell No. 699 Diversey Diversol CXU Kaliumpermanganat Permanganic acid (HMnO4), potassium salt Permanganato potasico Permanganate de potassium Diversey Diversol CX with Arodyne Manganese potassium oxide (KMnO4) Kali permanganicum Potassio (permanganato di) Potassium (permanganate de) EPA Pesticide Chemical Code 068501 00OT1QX5U4 CI 77755 C.I. 77755 Permanganic acid potassium salt Icc 237 Disinfectant, Sanitizer, Destainer, and Deodorizer DTXSID2034839 NSC-146182 Permanganate, Potassium Permitabs En-potab trioxo(potassiooxy)manganese DTXCID0014839 231-760-3 NSC146182 RefChem:6674 KMnO4 MFCD00011364 potassium;permanganate Permanganic acid (HMnO4), potassium salt (1:1) Hilco #88 Kaliumpermanganat [German] Potassium permanganate [JAN] CCRIS 5561 Permanganato potasico [Spanish] HSDB 1218 Permanganate de potassium [French] Potassium manganate(VII) EINECS 231-760-3 UN1490 Potassio (permanganato di) [Italian] Potassium (permanganate de) [French] NSC 146182 UNII-00OT1QX5U4 AI3-52835 Potassium permanganate [USP:JAN] potasiumpermanganate potassiumpermanganate EC 231-760-3 SCHEMBL2762 Potassium permanganate (TN) Potassium permanganate, 97% KALI PERMANGANICUM [HPUS] CHEBI:231501 Potassium tetraoxidomanganate(1-) POTASSIUM PERMANGANATE [MI] Potassium permanganate, ACS reagent Potassium permanganate (JP17/USP) POTASSIUM PERMANGANATE [HSDB] POTASSIUM PERMANGANATE [VANDF] Potassium permanganate, LR, >=99% AKOS015833392 POTASSIUM PERMANGANATE [MART.] DB13831 POTASSIUM PERMANGANATE [WHO-DD] Potassium permanganate solution, 1.25N Potassium permanganate, p.a., 99.0% NS00067941 P1742 POTASSIUM PERMANGANATE [EP MONOGRAPH] POTASSIUM PERMANGANATE [USP MONOGRAPH] D02053 Potassium permanganate [UN1490] [Oxidizer] Potassium permanganate, ACS reagent, >=99.0% Potassium permanganate, BioUltra, >=99.0% (RT) Q190865 Potassium permanganate, SAJ first grade, >=99.3% Potassium permanganate, tested according to Ph.Eur. Potassium permanganate, JIS special grade, >=99.3% Potassium permanganate, <=150 mum particle size, 97% Potassium permanganate, meets USP testing specifications Potassium permanganate, ACS reagent, >=99.0%, low in mercury Potassium permanganate, low in mercury (max. 0,005 ppm Hg) Potassium permanganate, p.a., ACS reagent, reag. ISO, 99.0% Potassium permanganate, purum p.a., >=99.0% (RT), fine crystals Potassium permanganate, suitable for determination of nitroxide, >=99.3% Potassium permanganate, suitable for determination of toxic metals, >=99.5% Potassium permanganate, p.a., ACS reagent, reag. ISO, reag. Ph. Eur., 99.0-100.5% Potassium permanganate, puriss. p.a., ACS reagent, Hg <=0.000005%, >=99.0% (RT) Potassium permanganate, puriss., meets analytical specification of Ph. Eur., BP, USP, 99-100.5% PubChem 3 Chemical and Physical Properties 3.1 Computed Properties Property Name Property Value Reference Property Name Molecular Weight Property Value 158.034 g/mol Reference Computed by PubChem 2.2 (PubChem release 2025.04.14) Property Name Hydrogen Bond Donor Count Property Value 0 Reference Computed by Cactvs 3.4.8.18 (PubChem release 2025.04.14) Property Name Hydrogen Bond Acceptor Count Property Value 4 Reference Computed by Cactvs 3.4.8.18 (PubChem release 2025.04.14) Property Name Rotatable Bond Count Property Value 0 Reference Computed by Cactvs 3.4.8.18 (PubChem release 2025.04.14) Property Name Exact Mass Property Value 157.881408 Da Reference Computed by PubChem 2.2 (PubChem release 2025.04.14) Property Name Monoisotopic Mass Property Value 157.881408 Da Reference Computed by PubChem 2.2 (PubChem release 2025.04.14) Property Name Topological Polar Surface Area Property Value 74.3 Å² Reference Computed by Cactvs 3.4.8.18 (PubChem release 2025.04.14) Property Name Heavy Atom Count Property Value 6 Reference Computed by PubChem Property Name Formal Charge Property Value 0 Reference Computed by PubChem Property Name Complexity Property Value 118 Reference Computed by Cactvs 3.4.8.18 (PubChem release 2025.04.14) Property Name Isotope Atom Count Property Value 0 Reference Computed by PubChem Property Name Defined Atom Stereocenter Count Property Value 0 Reference Computed by PubChem Property Name Undefined Atom Stereocenter Count Property Value 0 Reference Computed by PubChem Property Name Defined Bond Stereocenter Count Property Value 0 Reference Computed by PubChem Property Name Undefined Bond Stereocenter Count Property Value 0 Reference Computed by PubChem Property Name Covalently-Bonded Unit Count Property Value 2 Reference Computed by PubChem Property Name Compound Is Canonicalized Property Value Yes Reference Computed by PubChem (release 2025.04.14) PubChem 3.2 Experimental Properties 3.2.1 Physical Description Potassium permanganate appears as a purplish colored crystalline solid. Noncombustible but accelerates the burning of combustible material. If the combustible material is finely divided the mixture may be explosive. Contact with liquid combustible materials may result in spontaneous ignition. Contact with sulfuric acid may cause fire or explosion. Used to make other chemicals and as a disinfectant. CAMEO Chemicals Dry Powder EPA Chemical Data Reporting (CDR) Purple odorless crystals; [HSDB] Haz-Map, Information on Hazardous Chemicals and Occupational Diseases DARK PURPLE CRYSTALS. ILO-WHO International Chemical Safety Cards (ICSCs) 3.2.2 Color / Form Dark purple or bronze-like crystals; Almost opaque by transmitted light and of a blue metallic luster by reflected air. Budavari, S. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Whitehouse Station, NJ: Merck and Co., Inc., 1996., p. 1317 Hazardous Substances Data Bank (HSDB) Purple orthorhombic crystals Lide, DR (ed.). CRC Handbook of Chemistry and Physics. 81st Edition. CRC Press LLC, Boca Raton: FL 2000, p. 4-80 Hazardous Substances Data Bank (HSDB) Bipyrimidal rhombic prisms Kirk-Othmer Encyclopedia of Chemical Technology. 4th ed. Volumes 1: New York, NY. John Wiley and Sons, 1991-Present., p. V15 (95) 1023 Hazardous Substances Data Bank (HSDB) 3.2.3 Odor Odorless Budavari, S. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Whitehouse Station, NJ: Merck and Co., Inc., 1996., p. 1317 Hazardous Substances Data Bank (HSDB) 3.2.4 Taste Sweet with astringent aftertaste Budavari, S. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Whitehouse Station, NJ: Merck and Co., Inc., 1996., p. 1317 Hazardous Substances Data Bank (HSDB) 3.2.5 Melting Point greater than 464 °F (USCG, 1999) U.S. Coast Guard. 1999. Chemical Hazard Response Information System (CHRIS) - Hazardous Chemical Data. Commandant Instruction 16465.12C. Washington, D.C.: U.S. Government Printing Office. CAMEO Chemicals Decomposes Lide, DR (ed.). CRC Handbook of Chemistry and Physics. 81st Edition. CRC Press LLC, Boca Raton: FL 2000, p. 4-80 Hazardous Substances Data Bank (HSDB) 3.2.6 Solubility Soluble in many organic solvents; also by concentrated acids with the liberation of oxygen. Budavari, S. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Whitehouse Station, NJ: Merck and Co., Inc., 1996., p. 1317 Hazardous Substances Data Bank (HSDB) Soluble in acetone and methanol Lewis, R.J., Sr (Ed.). Hawley's Condensed Chemical Dictionary. 13th ed. New York, NY: John Wiley & Sons, Inc. 1997., p. 920 Hazardous Substances Data Bank (HSDB) Soluble in acetic acid, trifluoroacetic acid, acetic anhydride, pyridine, benzonitrile, sulfolane. Kirk-Othmer Encyclopedia of Chemical Technology. 4th ed. Volumes 1: New York, NY. John Wiley and Sons, 1991-Present., p. V15 (1995) 1023 Hazardous Substances Data Bank (HSDB) In water, 6.40X10+4 mg/l @ 20 °C PMID:2205893 Shiu WY et al; Rev Environ Contam Toxicol 116: 15-187 (1990) Hazardous Substances Data Bank (HSDB) 25 g/100 @ 65 °C water Weast, R.C. (ed.) Handbook of Chemistry and Physics. 69th ed. Boca Raton, FL: CRC Press Inc., 1988-1989., p. B-119 Hazardous Substances Data Bank (HSDB) Solubility in water, g/100ml at 20Â °C: 6.4 (moderate) ILO-WHO International Chemical Safety Cards (ICSCs) 3.2.7 Density 2.7 at 59 °F (USCG, 1999) - Denser than water; will sink U.S. Coast Guard. 1999. Chemical Hazard Response Information System (CHRIS) - Hazardous Chemical Data. Commandant Instruction 16465.12C. Washington, D.C.: U.S. Government Printing Office. CAMEO Chemicals 2.7 g/cu cm Lide, DR (ed.). CRC Handbook of Chemistry and Physics. 81st Edition. CRC Press LLC, Boca Raton: FL 2000, p. 4-80 Hazardous Substances Data Bank (HSDB) 2.7 g/cmÂ³ ILO-WHO International Chemical Safety Cards (ICSCs) 3.2.8 Vapor Pressure Vapor pressure at 20Â °C: negligible ILO-WHO International Chemical Safety Cards (ICSCs) 3.2.9 LogP 1.73 (calculated) ILO-WHO International Chemical Safety Cards (ICSCs) 3.2.10 Stability / Shelf Life STABLE IN AIR AND LIGHT; SOLN ARE UNSTABLE Osol, A. (ed.). Remington's Pharmaceutical Sciences. 16th ed. Easton, Pennsylvania: Mack Publishing Co., 1980., p. 1111 Hazardous Substances Data Bank (HSDB) 3.2.11 Autoignition Temperature Not flammable (USCG, 1999) U.S. Coast Guard. 1999. Chemical Hazard Response Information System (CHRIS) - Hazardous Chemical Data. Commandant Instruction 16465.12C. Washington, D.C.: U.S. Government Printing Office. CAMEO Chemicals 3.2.12 Decomposition DECOMP AT ABOUT 240 °C WITH EVOLUTION OF OXYGEN; DECOMP BY ALCOHOL & MANY OTHER ORG SOLVENTS, ALSO BY CONCN ACIDS WITH LIBERATION OF OXYGEN; WITH HYDROCHLORIC ACID, CHLORINE LIBERATED; READILY DECOMP BY MANY REDUCING SUBSTANCES, SUCH AS FERROUS SALTS, IODIDES, OXALATES, ETC, ESP IN PRESENCE OF AN ACID The Merck Index. 10th ed. Rahway, New Jersey: Merck Co., Inc., 1983., p. 1103 Hazardous Substances Data Bank (HSDB) 240Â °C ILO-WHO International Chemical Safety Cards (ICSCs) 3.2.13 Refractive Index INDEX OF REFRACTION: 1.59 Weast, R.C. (ed.) Handbook of Chemistry and Physics. 69th ed. Boca Raton, FL: CRC Press Inc., 1988-1989., p. B-119 Hazardous Substances Data Bank (HSDB) 3.2.14 Other Experimental Properties POWERFUL OXIDIZING AGENT Sax, N.I. Dangerous Properties of Industrial Materials. 6th ed. New York, NY: Van Nostrand Reinhold, 1984., p. 2280 Hazardous Substances Data Bank (HSDB) Stable in air; decomposes at approx 240 °C with evolution of oxygen; with hydrochloric acid, chlorine is liberated; readily decomposed by many reducing substances, such as ferrous salts, iodides, oxalates, etc., especially in the presence of an acid. Budavari, S. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Whitehouse Station, NJ: Merck and Co., Inc., 1996., p. 1317 Hazardous Substances Data Bank (HSDB) Decomposed by alcohol Lewis, R.J., Sr (Ed.). Hawley's Condensed Chemical Dictionary. 13th ed. New York, NY: John Wiley & Sons, Inc. 1997., p. 920 Hazardous Substances Data Bank (HSDB) Solutions are faint pink to deep violet; permanganate solutions are most stable in the neutral or near-neutral pH region; acidic or alkaline solution decompose with the loss of oxygen; when dry potassium permanganate is heated to 200-300 °C, an exothermic, autocatalytic decomposition takes place with the evolution of oxygen; contact with combustibles especially water-soluble organic substances(such as polyhydroxyl compounds), can lead to spontaneous ignition and potentially violent combustion. Gerhartz, W. (exec ed.). Ullmann's Encyclopedia of Industrial Chemistry. 5th ed.Vol A1: Deerfield Beach, FL: VCH Publishers, 1985 to Present., p. VA16 (1990) 134 Hazardous Substances Data Bank (HSDB) 3.3 SpringerMaterials Properties crystal structure enthalpy molar conductivity positional coordinate space group unit cell parameter heat capacity displacement parameter transport data X-ray diffraction magnetic susceptibility atomic environment formation enthalpy magnetic properties unit cell axes entropy thermal expansion SpringerMaterials 3.4 Chemical Classes Other Classes -> Inorganic Oxidizing Agents Haz-Map, Information on Hazardous Chemicals and Occupational Diseases 3.4.1 Drugs Pharmaceuticals -> Synthetic Cannabinoids or Psychoactive Compounds S58 \| PSYCHOCANNAB \| Synthetic Cannabinoids and Psychoactive Compounds \| DOI:10.5281/zenodo.3247723 NORMAN Suspect List Exchange 3.4.1.1 Human Drugs Dermatological medicines > Anti-infective medicines WHO Model Lists of Essential Medicines 3.4.2 Food Additives ANTICAKING AGENT OR FREE-FLOW AGENT, EMULSIFIER OR EMULSIFIER SALT, FORMULATION AID -> FDA Substance added to food FDA Substances Added to Food 3.4.3 Food Contact Substances FCS -> FDA Cumulative Estimated Daily Intake (CEDI) FDA Packaging & Food Contact Substances (FCS) FCS -> FDA Inventory of Food Contact Substances Listed in 21 CFR FDA Packaging & Food Contact Substances (FCS) 3.4.4 Pesticides Bactericides, Fungicides, Molluscicides EU Pesticides Database Active substance -> EU Pesticides database: Not approved EU Pesticides Database 4 Spectral Information 4.1 IR Spectra 4.1.1 FTIR Spectra Technique 4000-1350 CM^-^1=MULLED IN PERFLUORINATED HYDROCARBON; 1350-450 CM^-^1=MULLED IN MINERAL OIL Source of Sample Aldrich Chemical Company, Inc., Milwaukee, Wisconsin Catalog Number 20798 Copyright Copyright © 1980, 1981-2025 John Wiley & Sons, Inc. All Rights Reserved. Thumbnail SpectraBase 4.1.2 ATR-IR Spectra Source of Sample Sigma-Aldrich Catalog Number 207985 Copyright Copyright © 2018-2025 Sigma-Aldrich Co. LLC. - Database Compilation Copyright © 2018-2025 John Wiley & Sons, Inc. All Rights Reserved. Thumbnail SpectraBase 4.2 Raman Spectra Instrument Name Bio-Rad FTS 175C with Raman accessory Technique FT-Raman Source of Sample Mallinckrodt Inc., St. Louis, Missouri Copyright Copyright © 1980, 1981-2025 John Wiley & Sons, Inc. All Rights Reserved. Thumbnail SpectraBase 5 Related Records 5.1 Related Compounds with Annotation Follow these links to do a live 2D search or do a live 3D search for this compound, sorted by annotation score. This section is deprecated (see the neighbor discontinuation help page for details), but these live search links provide equivalent functionality to the table that was previously shown here. PubChem 5.2 Component Compounds CID 422689 (Permanganic acid) CID 5462222 (Potassium) PubChem 5.3 Related Compounds Mixtures, Components, and Neutralized Forms Count 2 Similar Compounds (2D) View in PubChem Search Similar Conformers (3D) View in PubChem Search PubChem 5.4 Substances 5.4.1 PubChem Reference Collection SID 481108013 PubChem 5.4.2 Related Substances Same Count 140 PubChem 5.4.3 Substances by Category PubChem 5.5 Other Relationships Potassium Cyanide (related) PubChem 5.6 Entrez Crosslinks PubMed Count 3 PubChem 5.7 NCBI LinkOut NCBI 6 Chemical Vendors PubChem 7 Drug and Medication Information 7.1 WHO Essential Medicines Drug Drug Classes Formulation Indication Drug Potassium permanganate Drug Classes Dermatological medicines > Anti-infective medicines Formulation Local - Topical - Solution: aqueous solution: 1:10 000 Indication (1) Tropical phagedaenic ulcer; (2) Pemphigus; (3) Other specified pyogenic bacterial infection of skin and subcutaneous tissue; (4) Impetigo WHO Model Lists of Essential Medicines 7.2 FDA National Drug Code Directory National Drug Code (NDC) Directory 7.3 Drug Labels Active ingredient and drug DailyMed 7.4 Clinical Trials 7.4.1 ClinicalTrials.gov ClinicalTrials.gov 7.5 Therapeutic Uses POTASSIUM PERMANGANATE, USP, HAS LIMITED TOPICAL EFFICACY AGAINST BACTERIA & FUNGI. CONCN OF 1:5000 OR MORE ARE NECESSARY FOR EFFECTIVE BACTERICIDAL ACTION, BUT THEY ARE IRRITATING TO TISSUES. ... SOLN OF 1:10,000 ARE USUALLY USED. HOWEVER, UP TO AN HR MAY BE REQUIRED TO KILL ... BACTERIA & SOME ... SURVIVE EXPOSURE TO THIS CONCN. FOR THIS REASON, PERMANGANATE HAS BEEN MADE OBSOLETE BY /OTHER/ ... DRUGS. BECAUSE OF THE ASTRINGENCY OF MANGANOUS & MANGANIC IONS ... SUBSTANCE IS OCCASIONALLY USED TO SUPPRESS VESICULAR STAGE OF ECZEMA-DERMATITIS. A WET DRESSING OF PERMANGANATE MAY BE EMPLOYED IN THE TREATMENT OF IVY POISONING. Gilman, A. G., L. S. Goodman, and A. Gilman. (eds.). Goodman and Gilman's The Pharmacological Basis of Therapeutics. 6th ed. New York: Macmillan Publishing Co., Inc. 1980., p. 974 Hazardous Substances Data Bank (HSDB) POTASSIUM PERMANGANATE IS EMPLOYED EXTERNALLY FOR GERMICIDAL, FUNGICIDAL, ASTRINGENT, OXIDIZING, & KERATOPLASTIC EFFECTS. SOLN ... SHOULD BE FRESHLY PREPARED. American Hospital Formulary Service. Volumes I and II. Washington, DC: American Society of Hospital Pharmacists, to 1984., p. 84:04 Hazardous Substances Data Bank (HSDB) SOLN ... EMPLOYED IN DERMATOMYCOSES, ESPECIALLY TINEA PEDIS, TINEA CRURIS, & RINGWORM. Goodman, L.S., and A. Gilman. (eds.) The Pharmacological Basis of Therapeutics. 5th ed. New York: Macmillan Publishing Co., Inc., 1975., p. 998 Hazardous Substances Data Bank (HSDB) DOSE: TOPICAL ... FOR IVY POISONING OR ECZEMA, 0.01%; FOR EPIDERMOPHYTOSIS, 1% ... DOSAGE FORMS: TABLETS FOR SOLN: 300 MG. Osol, A. (ed.). Remington's Pharmaceutical Sciences. 16th ed. Easton, Pennsylvania: Mack Publishing Co., 1980., p. 1111 Hazardous Substances Data Bank (HSDB) For more Therapeutic Uses (Complete) data for POTASSIUM PERMANGANATE (8 total), please visit the HSDB record page. Hazardous Substances Data Bank (HSDB) 7.6 Drug Warnings POTASSIUM PERMANGANATE (0.025%-0.1%) HAS ASTRINGENT PROPERTIES BUT STAINS SKIN. TABLETS SHOULD BE COMPLETELY DISSOLVED IF PLACED IN TUB OF WATER, BECAUSE ... CAN PRODUCE CUTANEOUS NECROSIS IF PATIENT INADVERTENTLY SITS ON THEM. American Medical Association, Department of Drugs. Drug Evaluations. 6th ed. Chicago, Ill: American Medical Association, 1986., p. 1027 Hazardous Substances Data Bank (HSDB) MOST BACTERIA ARE KILLED WITHIN 1 HR BY PERMANGANATES IN DILUTION OF 1:10,000, BUT SOME MICROORGANISMS SURVIVE EXPOSURE TO MUCH HIGHER CONCN. GERMICIDAL EFFICIENCY ... IS GREATLY REDUCED IN PRESENCE OF ORG MATTER. CONCN STRONGER THAN 1:5000 ARE IRRITATING TO TISSUES. POTASSIUM PERMANGANATE, USP, FORMS DEEP PURPLE SOLN, BUT IT LEAVES A BROWN STAIN ... /IN TREATMENT OF SNAKEBITE/ ... SOME BELIEVE THAT CMPD DOES MORE HARM THAN GOOD IN THE CRUST THAT IS FORMED, WHICH PRECLUDES POSSIBILITY OF SUCTION & DRAINAGE FROM WOUND. Goodman, L.S., and A. Gilman. (eds.) The Pharmacological Basis of Therapeutics. 5th ed. New York: Macmillan Publishing Co., Inc., 1975., p. 998 Hazardous Substances Data Bank (HSDB) Using potassium permanganate in "neutralizing" ingested nicotine, physostigmine, quinine, and strychnine is potentially dangerous. Potassium permanganate is a nephrotoxin and hepatotoxin, as well as a corrosive agent in the gastrointestinal tract. Aspiration can cause acute tracheobrochitis and bronchopneumonia. Fatty changes in the heart have been reported. PMID:6338654 Decker WJ; Vet Hum Toxicol 25 (1): 13 (1983) Hazardous Substances Data Bank (HSDB) VET: PERMANGANATE SOLN DO NOT PENETRATE DEEPLY & HAVE ONLY SUPERFICIAL ACTION. /PERMANGANATES/ Jones, L.M., et al. Veterinary Pharmacology & Therapeutics. 4th ed. Ames: Iowa State University Press, 1977., p. 886 Hazardous Substances Data Bank (HSDB) 8 Food Additives and Ingredients 8.1 FDA Substances Added to Food Substance POTASSIUM PERMANGANATE Used for (Technical Effect) ANTICAKING AGENT OR FREE-FLOW AGENT, EMULSIFIER OR EMULSIFIER SALT, FORMULATION AID Document Number (21 eCFR) 172.892 175.105 178.1010 FDA Substances Added to Food 8.2 FDA Food Contact Substances (FCS) 1 of 2 items Substance POTASSIUM PERMANGANATE Cumulative Estimated Daily Intake (CEDI) [µg/kg bw/d] 20 Cumulative Dietary Concentration (CDC) [ppb] 400 Update 06/20/2000 Document Number (21 eCFR) 172.892 175.105 178.1010 FDA Packaging & Food Contact Substances (FCS) 2 of 2 items Substance POTASSIUM PERMANGANATE Document Number (21 eCFR) 172.892 175.105 178.1010 FDA Packaging & Food Contact Substances (FCS) 9 Agrochemical Information 9.1 Agrochemical Category Microbiocide EPA Pesticide Ecotoxicity Database Pesticide active substances -> Bactericides, Fungicides, Molluscicides EU Pesticides Database 9.2 EU Pesticides Data Active Substance potassium permanganate Status Not approved [Reg. (EC) No 1107/2009] Legislation 2008/768 ADI Not Applicable ARfD Not Applicable AOEL Not Applicable EU Pesticides Database 10 Pharmacology and Biochemistry 10.1 MeSH Pharmacological Classification Indicators and Reagents Substances used for the detection, identification, analysis, etc. of chemical, biological, or pathologic processes or conditions. Indicators are substances that change in physical appearance, e.g., color, at or approaching the endpoint of a chemical titration, e.g., on the passage between acidity and alkalinity. Reagents are substances used for the detection or determination of another substance by chemical or microscopical means, especially analysis. Types of reagents are precipitants, solvents, oxidizers, reducers, fluxes, and colorimetric reagents. (From Grant & Hackh's Chemical Dictionary, 5th ed, p301, p499) Medical Subject Headings (MeSH) 10.2 ATC Code D - Dermatologicals D08 - Antiseptics and disinfectants D08A - Antiseptics and disinfectants D08AX - Other antiseptics and disinfectants D08AX06 - Potassium permanganate WHO Anatomical Therapeutic Chemical (ATC) Classification V - Various V03 - All other therapeutic products V03A - All other therapeutic products V03AB - Antidotes V03AB18 - Potassium permanganate WHO Anatomical Therapeutic Chemical (ATC) Classification D08AX06 WHO Model Lists of Essential Medicines 10.3 Absorption, Distribution and Excretion SYSTEMIC EFFECTS ARE NOT ... /COMMONLY SEEN/ BECAUSE OF POOR ABSORPTION. Gosselin, R.E., R.P. Smith, H.C. Hodge. Clinical Toxicology of Commercial Products. 5th ed. Baltimore: Williams and Wilkins, 1984., p. II-109 Hazardous Substances Data Bank (HSDB) Adult channel catfish Ictalurus punctatus were exposed to waterborne potassium permanganate for 12 wk to determine if such exposure would alter the manganese content of axial muscle or liver tissue. Continuous exposure to 0.5 mg KMnO4 or exposure to 1 or 2 mg KMnO4/L on alternate days did not cause a significant incr in manganese in axial muscle or liver tissue. The mean (\|SE) concn of manganese in axial muscle of unexposed controls was 0.262 \| 0.018 mg/kg (wet weight). Means of scle could be detected within groups. The mean (\|SE) concn of manganese in liver tissue of controls was 1.67 \| 0.09 mg/kg (wet weight). Manganese concns in liver tissue of the three exposure groups were 1.57 \| 0.07 mg/kg, 1.68 \| 0.08 mg/kg, and 1.58 \| 0.10 mg/kg, for 0.5 (continuous), 1, or 2 mg/L (alternate days), respectively. Manganese was thought to accumulate in liver tissue, however, there were no statistically significant differences between those groups & the controls. Griffin BR et al.; Journal of Aquatic Animal Health 11 (3): 305-309 (1999) Hazardous Substances Data Bank (HSDB) 10.4 Metabolism / Metabolites Manganese is absorbed mainly via ingestion, but can also be inhaled. It binds to alpha-2-macroglobulin, albumin, or transferrin in the plasma and is distributed to the brain and all other mammalian tissues, though it tends to accumulate more in the liver, pancreas, and kidney. Manganese is capable of existing in a number of oxidation states and is believed to undergo changes in oxidation state within the body. Manganese oxidation state can influence tissue toxicokinetic behavior, and possibly toxicity. Manganese is excreted primarily in the faeces. (L228) Toxin and Toxin Target Database (T3DB) 11 Use and Manufacturing 11.1 Uses EPA CPDat Chemical and Product Categories The Chemical and Products Database, a resource for exposure-relevant data on chemicals in consumer products, Scientific Data, volume 5, Article number: 180125 (2018), DOI:10.1038/sdata.2018.125 EPA Chemical and Products Database (CPDat) Sources/Uses Used to bleach textile fibers and skins, to dye wood and fabrics, to etch rubber and plastics, and to descale steel; Used in food processing, photography, leather tanning, and water purification; Used as an antiseptic, disinfectant, insecticide, miticide, and algaecide; [HSDB] Haz-Map, Information on Hazardous Chemicals and Occupational Diseases Industrial Processes with risk of exposure Using Disinfectants or Biocides [Category: Clean] Farming (Pesticides) [Category: Industry] Sewer and Wastewater Treatment [Category: Industry] Leather Tanning and Processing [Category: Industry] Photographic Processing [Category: Other] Textiles (Printing, Dyeing, or Finishing) [Category: Industry] Haz-Map, Information on Hazardous Chemicals and Occupational Diseases For Potassium permanganate (USEPA/OPP Pesticide Code: 068501) ACTIVE products with label matches. /SRP: Registered for use in the U.S. but approved pesticide uses may change periodically and so federal, state and local authorities must be consulted for currently approved uses./ U.S. Environmental Protection Agency/Office of Pesticide Program's Chemical Ingredients Database on Potassium permanganate (7722-64-7). Available from, as of Sept 8, 2000: Hazardous Substances Data Bank (HSDB) Used occasionally for bulbs and rhizomes, and for dipping grafting knives and other tools. Farm Chemicals Handbook 2000. Willoughby, Ohio: Meister 2000., p. C 134 Hazardous Substances Data Bank (HSDB) Bleaching resins, waxes, fats, oils, straw, cotton, silk and other fibers and chamois skins; dyeing wood brown; printing fabrics; washing carbon dioxide in manuf mineral waters; exterminating Oidium tuckeri; photography; tanning leathers; purifying water; with formaldehyde soln to expel formaldehyde gas for disinfecting; as an important reagent in analytical and synthetic organic chemistry. Budavari, S. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Whitehouse Station, NJ: Merck and Co., Inc., 1996., p. 1317 Hazardous Substances Data Bank (HSDB) THERAP CAT: Anti-infective (topical) Budavari, S. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Whitehouse Station, NJ: Merck and Co., Inc., 1996., p. 1317 Hazardous Substances Data Bank (HSDB) For more Uses (Complete) data for POTASSIUM PERMANGANATE (14 total), please visit the HSDB record page. Hazardous Substances Data Bank (HSDB) Potassium permanganate is a powerful oxidizing agent and used a fixative, disinfectant, and as a reagent in organic synthesis. (L242) Toxin and Toxin Target Database (T3DB) 11.1.1 DEA Listed Chemicals Name Potassium permanganate List II Chemical A chemical, other than a List I chemical, specified by regulation that, in addition to legitimate uses, is used in manufacturing a controlled substance in violation of the Act. (21 eCFR 1310.02) Drug Enforcement Administration (DEA) 11.1.2 Industry Uses Oxidizing agent Bleaching agents Not Known or Reasonably Ascertainable Bleaching agent Odor agents EPA Chemical Data Reporting (CDR) 11.1.3 Consumer Uses Not Known or Reasonably Ascertainable Bleaching agents Other (specify) Oxidizing agent EPA Chemical Data Reporting (CDR) 11.2 Methods of Manufacturing May be prepared by oxidizing manganese dioxide with potassium chlorate in potassium hydroxide soln, then completing the oxidation with chlorine or air &carbon dioxide. Osol, A. (ed.). Remington's Pharmaceutical Sciences. 16th ed. Easton, Pennsylvania: Mack Publishing Co., 1980., p. 1111 Hazardous Substances Data Bank (HSDB) (1) By oxidation of the manganate in an alkaline electrolytic cell. (2) A hot solution of the mangenate is treated with carbon dioxide; on cooling, the solution deposits crystals fo the permanganate. Lewis, R.J., Sr (Ed.). Hawley's Condensed Chemical Dictionary. 13th ed. New York, NY: John Wiley & Sons, Inc. 1997., p. 920 Hazardous Substances Data Bank (HSDB) May be manufactured by one-step electrolytic conversion of ferromanganese to permanganate, or by a two-step process involving the thermal oxidation of manganese oxide of a naturally occurring ore into potassium manganate followed by electrolytic oxidation to permanganate... Kirk-Othmer Encyclopedia of Chemical Technology. 4th ed. Volumes 1: New York, NY. John Wiley and Sons, 1991-Present., p. V15 (1995) 1023 Hazardous Substances Data Bank (HSDB) Roasting processes that employ manganese(IV) oxide:potassium hydroxide molar ratio of 1:2 to 1:3 (reaction mixture is a solid); heated to 300-400 °C; exposed to secondary oxidation at 190-210 °C. Gerhartz, W. (exec ed.). Ullmann's Encyclopedia of Industrial Chemistry. 5th ed.Vol A1: Deerfield Beach, FL: VCH Publishers, 1985 to Present., p. VA16 (1990) 134 Hazardous Substances Data Bank (HSDB) Liquid-phase process with a manganese(IV) oxide:potassium hydroxide molar ratio greater than or equal to 1:5; molten mixture at 200-350 °C is oxygenated; separation of product. Gerhartz, W. (exec ed.). Ullmann's Encyclopedia of Industrial Chemistry. 5th ed.Vol A1: Deerfield Beach, FL: VCH Publishers, 1985 to Present., p. VA16 (1990) 134 Hazardous Substances Data Bank (HSDB) 11.3 Formulations / Preparations Grades: technical, CP (chemically pure), USP Lewis, R.J., Sr (Ed.). Hawley's Condensed Chemical Dictionary. 13th ed. New York, NY: John Wiley & Sons, Inc. 1997., p. 920 Hazardous Substances Data Bank (HSDB) Formulations include ready to use liquids (0.004%-2.5%), pelleted/tableted (40%), crystals (80%-99.99%), and powder (41%). Purdue University; National Pesticide Information Retrieval System; Potassium Permanganate Fact Sheet #80 (1989) Hazardous Substances Data Bank (HSDB) ICC 237 Disinfectant, Sanitizer, Destainer, and Deodorizer; soluble concn, 0.01% potassium permanganate. Purdue University; National Pesticide Information Retrieval System (1988) Hazardous Substances Data Bank (HSDB) Chlorisol; soluble concn, 0.01% potassium permanganate. Purdue University; National Pesticide Information Retrieval System (1988) Hazardous Substances Data Bank (HSDB) For more Formulations/Preparations (Complete) data for POTASSIUM PERMANGANATE (10 total), please visit the HSDB record page. Hazardous Substances Data Bank (HSDB) 11.4 Consumption Patterns Water& Waste Treatment, 80%; Chemical Processing, 10%; & Misc, 10% (1984) CHEMICAL PRODUCTS SYNOPSIS: Potassium Permanganate, 1985 Hazardous Substances Data Bank (HSDB) 11.5 U.S. Production Aggregated Product Volume 2019: 25,123,301 lb 2018: 20,000,000 lb - <100,000,000 lb 2017: 20,000,000 lb - <100,000,000 lb 2016: 20,000,000 lb - <100,000,000 lb EPA Chemical Data Reporting (CDR) 14,000 tons/year Kirk-Othmer Encyclopedia of Chemical Technology. 4th ed. Volumes 1: New York, NY. John Wiley and Sons, 1991-Present., p. V15 (1995) 1027 Hazardous Substances Data Bank (HSDB) 11.6 U.S. Imports (1986) 3,741,874 lb BUREAU OF THE CENSUS. US IMPORTS FOR CONSUMPTION AND GENERAL IMPORTS 1986 P. 1-516 Hazardous Substances Data Bank (HSDB) 11.7 General Manufacturing Information Industry Processing Sectors Utilities Oil and Gas Drilling, Extraction, and Support activities All Other Chemical Product and Preparation Manufacturing Not Known or Reasonably Ascertainable Other (requires additional information) EPA Chemical Data Reporting (CDR) EPA TSCA Commercial Activity Status Permanganic acid (HMnO4), potassium salt (1:1): ACTIVE EPA Chemicals under the TSCA 12 Safety and Hazards 12.1 Hazards Identification 12.1.1 GHS Classification 1 of 5 items View All Pictogram(s) Signal Danger GHS Hazard Statements H272: May intensify fire; oxidizer [Danger Oxidizing liquids; Oxidizing solids] H302: Harmful if swallowed [Warning Acute toxicity, oral] H361d: Suspected of damaging the unborn child [Warning Reproductive toxicity] H400: Very toxic to aquatic life [Warning Hazardous to the aquatic environment, acute hazard] H410: Very toxic to aquatic life with long lasting effects [Warning Hazardous to the aquatic environment, long-term hazard] Precautionary Statement Codes P203, P210, P220, P264, P270, P273, P280, P301+P317, P318, P330, P370+P378, P391, P405, and P501 (The corresponding statement to each P-code can be found at the GHS Classification page.) Regulation (EC) No 1272/2008 of the European Parliament and of the Council 12.1.2 Hazard Classes and Categories Ox. Sol. 2 (91.7%) Acute Tox. 4 (90.9%) Skin Corr. 1A (48.5%) Eye Dam. 1 (30.9%) Repr. 2 (28.8%) STOT RE 2 (33.7%) Aquatic Acute 1 (91.3%) Aquatic Chronic 1 (91.7%) European Chemicals Agency (ECHA) View More... 12.1.3 DOT Hazard Classification Substance (Descriptions/Shipping Name) Potassium permanganate DOT ID (UN/NA Number) UN1490 Hazard Class/Label Code(s) Div 5.1 Oxidizer (49 eCFR § 173.127) Packing Group PG II: the degree of danger presented by the material is medium For more information about the packing group assignment, please visit 49 eCFR § 173 Placard/Label(s) US Code of Federal Regulations, Hazardous Materials, 49 CFR Part 172 12.1.4 Health Hazards Burns and stains the skin dark brown. If ingested will cause severe distress of gastro-intestinal system. May be fatal if over 4 oz. are consumed. (USCG, 1999) U.S. Coast Guard. 1999. Chemical Hazard Response Information System (CHRIS) - Hazardous Chemical Data. Commandant Instruction 16465.12C. Washington, D.C.: U.S. Government Printing Office. CAMEO Chemicals ERG 2024, Guide 140 (Potassium permanganate) · Inhalation, ingestion or contact (skin, eyes) with vapors or substance may cause severe injury, burns or death. · Fire may produce irritating, corrosive and/or toxic gases. · Runoff from fire control or dilution water may cause environmental contamination. Emergency Response Guidebook (ERG) 12.1.5 Fire Hazards Behavior in Fire: May cause fire on contact with combustibles. Containers may explode. (USCG, 1999) U.S. Coast Guard. 1999. Chemical Hazard Response Information System (CHRIS) - Hazardous Chemical Data. Commandant Instruction 16465.12C. Washington, D.C.: U.S. Government Printing Office. CAMEO Chemicals ERG 2024, Guide 140 (Potassium permanganate) CAUTION: Ammonium nitrate products may explode if involved in fire or contaminated with hydrocarbons (fuels), organic matter, other contaminants or when hot molten and contained. Treat as an explosive (GUIDE 112). · These substances will accelerate burning when involved in a fire. · Some may decompose explosively when heated or involved in a fire. · May explode from heat or contamination. · Some will react explosively with hydrocarbons (fuels). · May ignite combustibles (wood, paper, oil, clothing, etc.). · Containers may explode when heated. · Runoff may create fire or explosion hazard. Emergency Response Guidebook (ERG) Not combustible but enhances combustion of other substances. Gives off irritating or toxic fumes (or gases) in a fire. Risk of fire and explosion on contact with combustible substances or reducing agents. ILO-WHO International Chemical Safety Cards (ICSCs) 12.1.6 Hazards Summary Dilute solutions are irritants and concentrated solutions caustic to the skin, eyes, and respiratory tract. Toxic by ingestion--gastrointestinal burns, methemoglobinemia, and injury to liver and kidneys; [HSDB] If not flushed with water, can burn skin and stain it dark brown; [CHRIS] Inhalation of high concentrations of dust or mist may cause pulmonary edema in extreme exposures; May cause liver and kidney damage after ingestion; Lethal dose is about 10 grams; May cause manganese poisoning after chronic ingestion or inhalation; [CHEMINFO] Permanganate crystals and concentrated solutions are corrosive (release potassium hydroxide when contact water) and can induce methemoglobinemia after ingestion; [Olson, p. 133] See Manganese and linked occupational diseases. Olson - Olson KR (ed). Poisoning & Drug Overdose, 7th Ed. New York: Lange Medical Books/McGraw-Hill, 2018., p. 133 Haz-Map, Information on Hazardous Chemicals and Occupational Diseases 12.1.7 Fire Potential MODERATE, BY CHEMICAL REACTION. Sax, N.I. Dangerous Properties of Industrial Materials. 6th ed. New York, NY: Van Nostrand Reinhold, 1984., p. 2280 Hazardous Substances Data Bank (HSDB) 12.1.8 Skin, Eye, and Respiratory Irritations Solid: Irritating to the skin & eyes. U.S. Coast Guard, Department of Transportation. CHRIS - Hazardous Chemical Data. Volume II. Washington, D.C.: U.S. Government Printing Office, 1984-5. Hazardous Substances Data Bank (HSDB) 12.2 Safety and Hazard Properties 12.2.1 OSHA Standards Permissible Exposure Limit: Table Z-1 Ceiling value: 5 mg/cu m. /Manganese cmpd (as Mn)/ 29 CFR 1910.1000 (7/1/2000) Hazardous Substances Data Bank (HSDB) 12.2.2 NIOSH Recommendations Recommended Exposure Limit: 10 Hr Time-Weighted Avg: 1 mg/cu m. /Manganese compounds and fume (as Mn)/ NIOSH. NIOSH Pocket Guide to Chemical Hazards. DHHS (NIOSH) Publication No. 97-140. Washington, D.C. U.S. Government Printing Office, 1997., p. 190 Hazardous Substances Data Bank (HSDB) Recommended Exposure Limit: 15 Min Short-Term Exposure Limit: 3 mg/cu m. /Manganese compounds and fume (as Mn)/ NIOSH. NIOSH Pocket Guide to Chemical Hazards. DHHS (NIOSH) Publication No. 97-140. Washington, D.C. U.S. Government Printing Office, 1997., p. 190 Hazardous Substances Data Bank (HSDB) 12.3 First Aid Measures Inhalation First Aid Fresh air. Half-upright position. Artificial respiration may be needed. Refer immediately for medical attention. ILO-WHO International Chemical Safety Cards (ICSCs) Skin First Aid Wear protective gloves when administering first aid. First rinse with plenty of water for at least 15 minutes, then remove contaminated clothes and rinse again. Refer for medical attention . ILO-WHO International Chemical Safety Cards (ICSCs) Eye First Aid Rinse with plenty of water for several minutes (remove contact lenses if easily possible). Refer immediately for medical attention. ILO-WHO International Chemical Safety Cards (ICSCs) Ingestion First Aid Rinse mouth. If within a few minutes after ingestion, one small glass of water may be given to drink. Do NOT induce vomiting. Refer immediately for medical attention. ILO-WHO International Chemical Safety Cards (ICSCs) 12.3.1 First Aid INGESTION: induce vomiting and follow with thorough gastric lavage, demulcents, glucose I.V., fluid therapy, and antibiotics. Tracheostomy may be lifesaving. (USCG, 1999) U.S. Coast Guard. 1999. Chemical Hazard Response Information System (CHRIS) - Hazardous Chemical Data. Commandant Instruction 16465.12C. Washington, D.C.: U.S. Government Printing Office. CAMEO Chemicals ERG 2024, Guide 140 (Potassium permanganate) General First Aid: · Call 911 or emergency medical service. · Ensure that medical personnel are aware of the material(s) involved, take precautions to protect themselves and avoid contamination. · Move victim to fresh air if it can be done safely. · Administer oxygen if breathing is difficult. · If victim is not breathing: -- DO NOT perform mouth-to-mouth resuscitation; the victim may have ingestedor inhaled the substance. -- If equipped and pulse detected, wash face and mouth, then give artificial respiration using a proper respiratory medical device (bag-valve mask, pocket mask equipped with a one-way valve or other device). -- If no pulse detected or no respiratory medical device available, provide continuouscompressions. Conduct a pulse check every two minutes or monitor for any signs of spontaneous respirations. · Remove and isolate contaminated clothing and shoes. · For minor skin contact, avoid spreading material on unaffected skin. · In case of contact with substance, remove immediately by flushing skin or eyes with running water for at least 20 minutes. · For severe burns, immediate medical attention is required. · Effects of exposure (inhalation, ingestion, or skin contact) to substance may be delayed. · Keep victim calm and warm. · Keep victim under observation. · For further assistance, contact your local Poison Control Center. · Note: Basic Life Support (BLS) and Advanced Life Support (ALS) should be done by trained professionals. Specific First Aid: · Contaminated clothing may be a fire risk when dry. In Canada, an Emergency Response Assistance Plan (ERAP) may be required for this product. Please consult the shipping paper and/or the "ERAP" section. Emergency Response Guidebook (ERG) 12.4 Fire Fighting Excerpt from ERG Guide 140 [Oxidizers]: SMALL FIRE: Use water. Do not use dry chemicals or foams. CO2 or HalonÂ® may provide limited control. LARGE FIRE: Flood fire area with water from a distance. Do not move cargo or vehicle if cargo has been exposed to heat. If it can be done safely, move undamaged containers away from the area around the fire. FIRE INVOLVING TANKS, RAIL TANK CARS OR HIGHWAY TANKS: For ammonium nitrate products: Do not fight cargo fire. Withdraw, evacuate and isolate area for at least 1600 meters (1 mile). Treat as an explosive (ERG Guide 112). Do not enter area for 24 hours or until expert advice has been provided. Fight fire from maximum distance or use unmanned master stream devices or monitor nozzles. Cool containers with flooding quantities of water until well after fire is out. ALWAYS stay away from tanks in direct contact with flames. For massive fire, use unmanned master stream devices or monitor nozzles; if this is impossible, withdraw from area and let fire burn. (ERG, 2024) 2024 Emergency Response Guidebook, CAMEO Chemicals In case of fire in the surroundings, use appropriate extinguishing media. ILO-WHO International Chemical Safety Cards (ICSCs) 12.4.1 Fire Fighting Procedures If material on fire or involved in fire: Flood with water. Cool all affected containers with flooding quantities of water. Apply water from as far a distance as possible. Association of American Railroads. Emergency Handling of Hazardous Materials in Surface Transportation. Washington, DC: Association of American Railroads, Bureau of Explosives, 1994., p. 903 Hazardous Substances Data Bank (HSDB) Evacuation: If fire becomes uncontrollable - consider evacuation of one-half (1/2) mile radius. Association of American Railroads. Emergency Handling of Hazardous Materials in Surface Transportation. Washington, DC: Association of American Railroads, Bureau of Explosives, 1994., p. 903 Hazardous Substances Data Bank (HSDB) 12.4.2 Firefighting Hazards The material itself is noncombustible but it will accelerate the burning of combustible material. If the combustible material is finely divided the mixture may be explosive. Contact with liquid combustible materials may result in spontaneous ignition. Association of American Railroads. Emergency Handling of Hazardous Materials in Surface Transportation. Washington, DC: Association of American Railroads, Bureau of Explosives, 1994., p. 903 Hazardous Substances Data Bank (HSDB) 12.5 Accidental Release Measures Public Safety: ERG 2024, Guide 140 (Potassium permanganate) · CALL 911. Then call emergency response telephone number on shipping paper. If shipping paper not available or no answer, refer to appropriate telephone number listed on the inside back cover. · Keep unauthorized personnel away. · Stay upwind, uphill and/or upstream. · Ventilate closed spaces before entering, but only if properly trained and equipped. Emergency Response Guidebook (ERG) Spill or Leak: ERG 2024, Guide 140 (Potassium permanganate) · Keep combustibles (wood, paper, oil, etc.) away from spilled material. · Do not touch damaged containers or spilled material unless wearing appropriate protective clothing. · Stop leak if you can do it without risk. · Do not get water inside containers. Small Dry Spill · With clean shovel, place material into clean, dry container and cover loosely; move containers from spill area. Small Liquid Spill · Use a non-combustible material like vermiculite or sand to soak up the product and place into a container for later disposal. Large Spill · Dike far ahead of liquid spill for later disposal. Emergency Response Guidebook (ERG) 12.5.1 Isolation and Evacuation Excerpt from ERG Guide 140 [Oxidizers]: IMMEDIATE PRECAUTIONARY MEASURE: Isolate spill or leak area in all directions for at least 50 meters (150 feet) for liquids and at least 25 meters (75 feet) for solids. LARGE SPILL: Consider initial downwind evacuation for at least 100 meters (330 feet). FIRE: If tank, rail tank car or highway tank is involved in a fire, ISOLATE for 800 meters (1/2 mile) in all directions; also, consider initial evacuation for 800 meters (1/2 mile) in all directions. If ammonium nitrate products are in a tank, rail car or truck and involved in a fire, ISOLATE for 1600 meters (1 mile) in all directions; also, initiate evacuation including emergency responders for 1600 meters (1 mile) in all directions. (ERG, 2024) 2024 Emergency Response Guidebook, CAMEO Chemicals Evacuation: ERG 2024, Guide 140 (Potassium permanganate) Immediate precautionary measure · Isolate spill or leak area in all directions for at least 50 meters (150 feet) for liquids and at least 25 meters (75 feet) for solids. Large Spill · Consider initial downwind evacuation for at least 100 meters (330 feet). Fire · If tank, rail tank car or highway tank is involved in a fire, ISOLATE for 800 meters (1/2 mile) in all directions; also, consider initial evacuation for 800 meters (1/2 mile) in all directions. · If ammonium nitrate products are in a tank, rail car or truck and involved in a fire, ISOLATE for 1600 meters (1 mile) in all directions; also, initiate evacuation including emergency responders for 1600 meters (1 mile) in all directions. Emergency Response Guidebook (ERG) 12.5.2 Spillage Disposal Personal protection: chemical protection suit including self-contained breathing apparatus. Do NOT let this chemical enter the environment. Sweep spilled substance into covered containers. Carefully collect remainder. Then store and dispose of according to local regulations. Do NOT absorb in saw-dust or other combustible absorbents. ILO-WHO International Chemical Safety Cards (ICSCs) 12.5.3 Cleanup Methods Environmental considerations: Land spill: Dig a pit, pond, lagoon, holding area to contain liquid or solid material. /SRP: If time permits, pits, ponds, lagoons, soak holes, or holding areas should be sealed with an impermeable flexible membrane liner./ Cover solids with a plastic sheet to prevent dissolving in rain or fire fighting water. Add sodium bisulfite (NaHSO3). Association of American Railroads. Emergency Handling of Hazardous Materials in Surface Transportation. Washington, DC: Association of American Railroads, Bureau of Explosives, 1994., p. 903 Hazardous Substances Data Bank (HSDB) Environmental considerations: Water spill: Add dilute caustic soda (NaOH). Add sodium bisulfite (NaHSO3). Use mechanical dredges or lifts to remove immobilized masses of pollutants and precipitates. Association of American Railroads. Emergency Handling of Hazardous Materials in Surface Transportation. Washington, DC: Association of American Railroads, Bureau of Explosives, 1994., p. 903 Hazardous Substances Data Bank (HSDB) 12.5.4 Disposal Methods SRP: At the time of review, criteria for land treatment or burial (sanitary landfill) disposal practices are subject to significant revision. Prior to implementing land disposal of waste residue (including waste sludge), consult with environmental regulatory agencies for guidance on acceptable disposal practices. Hazardous Substances Data Bank (HSDB) 12.5.5 Preventive Measures SRP: The scientific literature for the use of contact lenses in industry is conflicting. The benefit or detrimental effects of wearing contact lenses depend not only upon the substance, but also on factors including the form of the substance, characteristics and duration of the exposure, the uses of other eye protection equipment, and the hygiene of the lenses. However, there may be individual substances whose irritating or corrosive properties are such that the wearing of contact lenses would be harmful to the eye. In those specific cases, contact lenses should not be worn. In any event, the usual eye protection equipment should be worn even when contact lenses are in place. Hazardous Substances Data Bank (HSDB) SRP: Local exhaust ventilation should be applied wherever there is an incidence of point source emissions or dispersion of regulated contaminants in the work area. Ventilation control of the contaminant as close to its point of generation is both the most economical and safest method to minimize personnel exposure to airborne contaminants. Hazardous Substances Data Bank (HSDB) WHEN USING WITH STRONG ACIDS TO DESTROY ORGANIC MATTER, PERFORM REACTION BEHIND SAFETY BARRIER. /PERMANGANATES/ Association of Official Analytical Chemists. Official Methods of Analysis. 10th ed. and supplements. Washington, DC: Association of Official Analytical Chemists, 1965. New editions through 13th ed. plus supplements, 1982., p. 13/883 52.080 Hazardous Substances Data Bank (HSDB) If material not on fire and not involved in fire: Keep sparks, flames, and other sources of ignition away. Keep material out of water sources and sewers. Association of American Railroads. Emergency Handling of Hazardous Materials in Surface Transportation. Washington, DC: Association of American Railroads, Bureau of Explosives, 1994., p. 903 Hazardous Substances Data Bank (HSDB) Personnel protection: ... Do not handle broken packages unless wearing appropriate personal protective equipment. Wash away any material which may be contacted the body with copious amounts of water or soap and water. ... Association of American Railroads. Emergency Handling of Hazardous Materials in Surface Transportation. Washington, DC: Association of American Railroads, Bureau of Explosives, 1994., p. 903 Hazardous Substances Data Bank (HSDB) 12.6 Handling and Storage 12.6.1 Nonfire Spill Response Excerpt from ERG Guide 140 [Oxidizers]: Keep combustibles (wood, paper, oil, etc.) away from spilled material. Do not touch damaged containers or spilled material unless wearing appropriate protective clothing. Stop leak if you can do it without risk. Do not get water inside containers. SMALL DRY SPILL: With clean shovel, place material into clean, dry container and cover loosely; move containers from spill area. SMALL LIQUID SPILL: Use a non-combustible material like vermiculite or sand to soak up the product and place into a container for later disposal. LARGE SPILL: Dike far ahead of liquid spill for later disposal. (ERG, 2024) 2024 Emergency Response Guidebook, CAMEO Chemicals 12.6.2 Safe Storage Separated from combustible substances, reducing agents and powdered metals. Well closed. Provision to contain effluent from fire extinguishing. Store in an area without drain or sewer access. ILO-WHO International Chemical Safety Cards (ICSCs) 12.7 Exposure Control and Personal Protection Protective Clothing: ERG 2024, Guide 140 (Potassium permanganate) · Wear positive pressure self-contained breathing apparatus (SCBA). · Wear chemical protective clothing that is specifically recommended by the manufacturer when there is NO RISK OF FIRE. · Structural firefighters' protective clothing provides thermal protection but only limited chemical protection. Emergency Response Guidebook (ERG) Exposure Summary TIH (Toxic Inhalation Hazard) - Term used to describe gases and volatile liquids that are toxic when inhaled. Some are TIH materials themselves, e.g., chlorine, and some release TIH gases when spilled in water, e.g., chlorosilanes. [ERG 2016]. Haz-Map, Information on Hazardous Chemicals and Occupational Diseases Maximum Allowable Concentration (MAK) 0.2 [mg/m3], as MN (inhalable fraction), 0.02 mg/m3, as Mn (respirable fraction)[German Research Foundation (DFG)] Haz-Map, Information on Hazardous Chemicals and Occupational Diseases 12.7.1 Immediately Dangerous to Life or Health (IDLH) 500 mg/cu m /Manganese compounds and fume (as Mn)/ NIOSH. NIOSH Pocket Guide to Chemical Hazards. DHHS (NIOSH) Publication No. 97-140. Washington, D.C. U.S. Government Printing Office, 1997., p. 190 Hazardous Substances Data Bank (HSDB) 12.7.2 Threshold Limit Values (TLV) 0.02 [mg/m3], as Mn (respirable fraction), 0.1 mg/m3, as Mn (inhalable fraction) Haz-Map, Information on Hazardous Chemicals and Occupational Diseases 8 hr Time Weighted Avg: 0.2 mg/cu m /Manganese and inorganic compounds, as Mn/ American Conference of Governmental Industrial Hygienists TLVs and BEIs. Threshold Limit Values for Chemical Substances and Physical Agents and Biological Exposure Indices. Cincinnati, OH, 2008, p. 37 Hazardous Substances Data Bank (HSDB) Excursion Limit Recommendation: Excursions in worker exposure levels may exceed 3 times the TLV-TWA for no more than a total of 30 minutes during a work day, and under no circumstances should they exceed 5 times the TLV-TWA, provided that the TLV-TWA is not exceeded. /Manganese and inorganic cmpd, as Mn/ American Conference of Governmental Industrial Hygienists TLVs and BEIs. Threshold Limit Values for Chemical Substances and Physical Agents and Biological Exposure Indices. Cincinnati, OH, 2008, p. 5 Hazardous Substances Data Bank (HSDB) (as Mn, respirable fraction): 0.02 mg/m ILO-WHO International Chemical Safety Cards (ICSCs) 12.7.3 Occupational Exposure Limits (OEL) EU-OEL (as Mn, inhalable fraction): 0.2 mg/m ILO-WHO International Chemical Safety Cards (ICSCs) 12.7.4 Emergency Response Planning Guidelines Emergency Response: ERG 2024, Guide 140 (Potassium permanganate) Small Fire · Use water. Do not use dry chemicals or foams. CO2 or Halon® may provide limited control. Large Fire · Flood fire area with water from a distance. · Do not move cargo or vehicle if cargo has been exposed to heat. · If it can be done safely, move undamaged containers away from the area around the fire. Fire Involving Tanks, Rail Tank Cars or Highway Tanks · For ammonium nitrate products: Do not fight cargo fire. Withdraw, evacuate and isolate area for at least 1600 meters (1 mile). Treat as an explosive (GUIDE 112). Do not enter area for 24 hours or until expert advice has been provided. · Fight fire from maximum distance or use unmanned master stream devices or monitor nozzles. · Cool containers with flooding quantities of water until well after fire is out. · ALWAYS stay away from tanks in direct contact with flames. · For massive fire, use unmanned master stream devices or monitor nozzles; if this is impossible, withdraw from area and let fire burn. Emergency Response Guidebook (ERG) 12.7.5 Inhalation Risk A harmful concentration of airborne particles can be reached quickly when dispersed, especially if powdered. ILO-WHO International Chemical Safety Cards (ICSCs) 12.7.6 Effects of Short Term Exposure The substance is corrosive to the eyes, skin and respiratory tract. Corrosive on ingestion. Inhalation of dust may cause lung oedema, but only after initial corrosive effects on eyes and/or airways have become manifest. The effects may be delayed. Medical observation is indicated. ILO-WHO International Chemical Safety Cards (ICSCs) 12.7.7 Effects of Long Term Exposure The substance may have effects on the lungs. This may result in bronchitis and pneumonia. Animal tests show that this substance possibly causes toxicity to human reproduction or development. ILO-WHO International Chemical Safety Cards (ICSCs) 12.7.8 Personal Protective Equipment (PPE) Goggles or face shield; rubber gloves. (USCG, 1999) U.S. Coast Guard. 1999. Chemical Hazard Response Information System (CHRIS) - Hazardous Chemical Data. Commandant Instruction 16465.12C. Washington, D.C.: U.S. Government Printing Office. CAMEO Chemicals GOGGLES OR FACE SHIELD; RUBBER GLOVES. U.S. Coast Guard, Department of Transportation. CHRIS - Hazardous Chemical Data. Volume II. Washington, D.C.: U.S. Government Printing Office, 1984-5. Hazardous Substances Data Bank (HSDB) 12.7.9 Fire Prevention NO contact with combustible substances. ILO-WHO International Chemical Safety Cards (ICSCs) 12.7.10 Exposure Prevention PREVENT DISPERSION OF DUST! STRICT HYGIENE! IN ALL CASES CONSULT A DOCTOR! ILO-WHO International Chemical Safety Cards (ICSCs) 12.7.11 Inhalation Prevention Avoid inhalation of dust. Use local exhaust or breathing protection. ILO-WHO International Chemical Safety Cards (ICSCs) 12.7.12 Skin Prevention Protective gloves. Protective clothing. ILO-WHO International Chemical Safety Cards (ICSCs) 12.7.13 Eye Prevention Wear face shield or eye protection in combination with breathing protection. ILO-WHO International Chemical Safety Cards (ICSCs) 12.7.14 Ingestion Prevention Do not eat, drink, or smoke during work. ILO-WHO International Chemical Safety Cards (ICSCs) 12.8 Stability and Reactivity 12.8.1 Air and Water Reactions Soluble in water. CAMEO Chemicals 12.8.2 Reactive Group Oxidizing Agents, Strong CAMEO Chemicals 12.8.3 Reactivity Alerts Strong Oxidizing Agent CAMEO Chemicals 12.8.3.1 CSL Reaction Information 1 of 3 items CSL No CSL00026 Reactants/Reagents 2-BROMO-5-METHYLPYRIDINE + POTASSIUM PERMANGANATE Warning Message Immiscible reagents mixing at 80oC caused a violent reaction GHS Category Gas Under Pressure,Sensitizer Functional Group HeteroAryl Methyl Reaction Scale M (up to 100g) Reaction Class oxidation Reference Source User-Reported Modified Date 6/26/18 Create Date 10/3/17 Pistoia Alliance Chemical Safety Library 2 of 3 items CSL No CSL00074 Reactants/Reagents N,N-DIMETHYLFORMAMIDE + POTASSIUM PERMANGANATE Warning Message explosive GHS Category Explosive Reference Source User-Reported Modified Date 7/8/18 Create Date 6/27/17 Pistoia Alliance Chemical Safety Library 3 of 3 items CSL No CSL00098 Reactants/Reagents Decolorizing carbon + POTASSIUM PERMANGANATE Warning Message Potentially explosive GHS Category Explosive Reference Source User-Reported Modified Date 7/8/18 Create Date 8/8/17 Pistoia Alliance Chemical Safety Library 12.8.4 Reactivity Profile POTASSIUM PERMANGANATE is a very powerful oxidizing agent, particularly in acidic surroundings. Reacts with incandescence with aluminum carbide [Mellor 5:872. 1946-47]. Grinding with antimony or arsenic causes ignition of the metals [Mellor 12:322. 1946-47]. Mixtures with acetic acid or acetic anhydride may explode if not kept cold [Von Schwartz 1918. p. 34]. Explosions can occur when acidified solutions come in contact with benzene, carbon disulfide, diethyl ether, ethyl alcohol, petroleum, or organic matter. Contact with glycerol may produce an explosion [Pieters 1957. p. 30]. Contact with concentrated hydrogen peroxide solution can produce an explosion [Haz. Chem. Data 1973. p. 230]. Contact with solid hydroxylamine produces an immediate white flame [Mellor 8:294. 1946-47]. Transport through a polypropylene tube ignited the tube [MCA Case History 1842. 1972]. Mixing with concentrated sulfuric acid in a vessel containing moisture caused an explosion (due to formation of manganese heptoxide) [Delhez 1967]. CAMEO Chemicals 12.8.5 Hazardous Reactivities and Incompatibilities CAUTION: TAKE GREAT CARE IN HANDLING AS EXPLOSIONS MAY OCCUR IF IT IS BROUGHT INTO CONTACT WITH ORG OR OTHER READILY OXIDIZABLE SUBSTANCES, EITHER IN SOLN OR IN DRY STATE. The Merck Index. 10th ed. Rahway, New Jersey: Merck Co., Inc., 1983., p. 1103 Hazardous Substances Data Bank (HSDB) CONTACT ... /WITH GLYCEROL/ MAY PRODUCE EXPLOSION. Fire Protection Guide to Hazardous Materials. 12 ed. Quincy, MA: National Fire Protection Association, 1997., p. 491-162 Hazardous Substances Data Bank (HSDB) CRYSTALS OF POTASSIUM PERMANGANATE EXPLODE VIGOROUSLY WHEN GROUND WITH PHOSPHORUS. Fire Protection Guide to Hazardous Materials. 12 ed. Quincy, MA: National Fire Protection Association, 1997., p. 491-148 Hazardous Substances Data Bank (HSDB) Methanol, ethanol, isopropanol, pentanol, or isopentanol do not ignite immediately upon mixing with red fuming nitric acid, but addition of potassium permanganate (20%) to the acid before mixing causes immediate ignition. Bretherick, L. Handbook of Reactive Chemical Hazards. 4th ed. Boston, MA: Butterworth-Heinemann Ltd., 1990, p. 1295 Hazardous Substances Data Bank (HSDB) For more Hazardous Reactivities and Incompatibilities (Complete) data for POTASSIUM PERMANGANATE (20 total), please visit the HSDB record page. Hazardous Substances Data Bank (HSDB) 12.9 Transport Information 12.9.1 DOT Emergency Guidelines /GUIDE 140: OXIDIZERS/ Fire or Explosion: These substances will accelerate burning when involved in a fire. Some may decompose explosively when heated or involved in a fire. May explode from heat or contamination. Some will react explosively with hydrocarbons (fuels). May ignite combustibles (wood, paper, oil, clothing, etc.). Containers may explode when heated. Runoff may create fire or explosion hazard. U.S. Department of Transportation. 2012 Emergency Response Guidebook. Washington, D.C. 2012 Hazardous Substances Data Bank (HSDB) /GUIDE 140: OXIDIZERS/ Health: Inhalation, ingestion or contact (skin, eyes) with vapors or substance may cause severe injury, burns or death. Fire may produce irritating, corrosive and/or toxic gases. Runoff from fire control or dilution water may cause pollution. U.S. Department of Transportation. 2012 Emergency Response Guidebook. Washington, D.C. 2012 Hazardous Substances Data Bank (HSDB) /GUIDE 140: OXIDIZERS/ Public Safety: CALL Emergency Response Telephone Number on Shipping Paper first. If Shipping Paper not available or no answer, refer to appropriate telephone number listed on the inside back cover. As an immediate precautionary measure, isolate spill or leak area in all directions for at least 50 meters (150 feet) for liquids and at least 25 meters (75 feet) for solids. Keep unauthorized personnel away. Stay upwind. Keep out of low areas. Ventilate closed spaces before entering. U.S. Department of Transportation. 2012 Emergency Response Guidebook. Washington, D.C. 2012 Hazardous Substances Data Bank (HSDB) /GUIDE 140: OXIDIZERS/ Protective Clothing: Wear positive pressure self-contained breathing apparatus (SCBA). Wear chemical protective clothing that is specifically recommended by the manufacturer. It may provide little or no thermal protection. Structural firefighters' protective clothing will only provide limited protection. U.S. Department of Transportation. 2012 Emergency Response Guidebook. Washington, D.C. 2012 Hazardous Substances Data Bank (HSDB) For more DOT Emergency Guidelines (Complete) data for POTASSIUM PERMANGANATE (8 total), please visit the HSDB record page. Hazardous Substances Data Bank (HSDB) 12.9.2 Shipping Name / Number DOT/UN/NA/IMO UN 1490; Potassium permanganate Hazardous Substances Data Bank (HSDB) IMO 5.1; Potassium permanganate Hazardous Substances Data Bank (HSDB) 12.9.3 Standard Transportation Number 49 187 40; Potassium permanganate Hazardous Substances Data Bank (HSDB) 12.9.4 Shipment Methods and Regulations No person may /transport,/ offer or accept a hazardous material for transportation in commerce unless that person is registered in conformance ... and the hazardous material is properly classed, described, packaged, marked, labeled, and in condition for shipment as required or authorized by ... /the hazardous materials regulations (49 CFR 171-177)./ 49 CFR 171.2; U.S. National Archives and Records Administration's Electronic Code of Federal Regulations. Available from, as of February 15, 2006: Hazardous Substances Data Bank (HSDB) The International Air Transport Association (IATA) Dangerous Goods Regulations are published by the IATA Dangerous Goods Board pursuant to IATA Resolutions 618 and 619 and constitute a manual of industry carrier regulations to be followed by all IATA Member airlines when transporting hazardous materials. International Air Transport Association. Dangerous Goods Regulations. 47th Edition. Montreal, Quebec Canada. 2006., p. 241 Hazardous Substances Data Bank (HSDB) The International Maritime Dangerous Goods Code lays down basic principles for transporting hazardous chemicals. Detailed recommendations for individual substances and a number of recommendations for good practice are included in the classes dealing with such substances. A general index of technical names has also been compiled. This index should always be consulted when attempting to locate the appropriate procedures to be used when shipping any substance or article. International Maritime Organization. International Maritime Dangerous Goods Code. London, UK. 2004., p. 70 Hazardous Substances Data Bank (HSDB) 12.9.5 DOT Label Oxidizer CAMEO Chemicals 12.9.6 Packaging and Labelling Marine pollutant. ILO-WHO International Chemical Safety Cards (ICSCs) 12.9.7 EC Classification Symbol: O, Xn, N; R: 8-22-50/53; S: (2)-60-61 ILO-WHO International Chemical Safety Cards (ICSCs) 12.9.8 UN Classification UN Hazard Class: 5.1; UN Pack Group: II ILO-WHO International Chemical Safety Cards (ICSCs) 12.10 Regulatory Information The Australian Inventory of Industrial Chemicals Chemical: Permanganic acid (HMnO4), potassium salt Australian Industrial Chemicals Introduction Scheme (AICIS) DEA Listed Chemicals List II Chemical: A chemical, other than a List I chemical, specified by regulation that, in addition to legitimate uses, is used in manufacturing a controlled substance in violation of the Act. (21 eCFR 1310.02) 21 CFR Section 1310.02(a) Drug Enforcement Administration (DEA) Status Regulation (EC) 2008/768 EU Pesticides Database REACH Registered Substance Status: Active Update: 25-04-2023 Status: Active Update: 06-05-2018 European Chemicals Agency (ECHA) New Zealand EPA Inventory of Chemical Status Permanganic acid, potassium salt: HSNO Approval: HSR001342 Approved with controls New Zealand Environmental Protection Authority (EPA) New Jersey Worker and Community Right to Know Act The New Jersey Worker and Community Right to Know Act requires public and private employers to provide information about hazardous substances at their workplaces. (N.J.S.A. 34:5A-1 et. seq.) NJDOH RTK Hazardous Substance List 12.10.1 DHS Chemicals of Interest (COI) Chemicals of Interest(COI) Potassium permanganate Theft: Minimum Concentration (%) A Commercial Grade Theft: Screening Threshold Quantities (in pounds unless otherwise noted) 400 Security Issue: Theft - EXP/IEDP Explosive/Improvised Explosive Device Precursor material that, if stolen or diverted, can be converted into weapons using simple chemistry, equipment, or techniques. DHS Chemical Facility Anti-Terrorism Standards (CFATS) Chemicals of Interest 12.10.2 Atmospheric Standards Listed as a hazardous air pollutant (HAP) generally known or suspected to cause serious health problems. The Clean Air Act, as amended in 1990, directs EPA to set standards requiring major sources to sharply reduce routine emissions of toxic pollutants. EPA is required to establish and phase in specific performance based standards for all air emission sources that emit one or more of the listed pollutants. Potassium permanganate is included on this list. Clean Air Act as amended in 1990, Sect. 112 (b) (1) Public Law 101-549 Nov. 15, 1990 Hazardous Substances Data Bank (HSDB) ... Substances for which a Federal Register notice has been published that included consideration of the serious health effects, including cancer, form ambient air exposure to the substance. Manganese (50 FR 32627; Aug. 13, 1985) is included on this list. /Manganese/ 40 CFR 61.01(b) (7/1/2000) Hazardous Substances Data Bank (HSDB) 12.10.3 State Drinking Water Guidelines (MN) MINNESOTA 300ug/L /Manganese/ USEPA/Office of Water; Federal-State Toxicology and Risk Analysis Committee (FSTRAC). Summary of State and Federal Drinking Water Standards and Guidelines (11/93) To Present Hazardous Substances Data Bank (HSDB) 12.10.4 Clean Water Act Requirements Potassium permanganate is designated as a hazardous substance under section 311(b)(2)(A) of the Federal Water Pollution Control Act and further regulated by the Clean Water Act Amendments of 1977 and 1978. These regulations apply to discharges of this substance. This designation includes any isomers and hydrates, as well as any solutions and mixtures containing this substance. 40 CFR 116.4 (7/1/2000) Hazardous Substances Data Bank (HSDB) 12.10.5 CERCLA Reportable Quantities Persons in charge of vessels or facilities are required to notify the National Response Center (NRC) immediately, when there is a release of this designated hazardous substance, in an amount equal to or greater than its reportable quantity of 100 lb or 45.4 kg. The toll free number of the NRC is (800) 424-8802; In the Washington D.C. metropolitan area (202) 426-2675. The rule for determining when notification is required is stated in 40 CFR 302.4 (section IV. D.3.b). 40 CFR 302.4 (7/1/2000) Hazardous Substances Data Bank (HSDB) 12.10.6 FDA Requirements Potassium permanganate is an indirect food additive for use only as a component of adhesives. 21 CFR 175.105 (4/1/2000) Hazardous Substances Data Bank (HSDB) Food starch may be bleached by treatment with one or more of the following: ... potassium permanganate, not to exceed 0.2 percent. Limitations: Residual manganese (calculated as Mn), not to exceed 50 ppm in food starch-modified. 21 CFR 172.892(b) (4/1/2000) Hazardous Substances Data Bank (HSDB) 12.11 Other Safety Information 12.11.1 Special Reports POLSON CJ, TATTERSALL RN; JB LIPPENCOTT CO, PHILA, 116-27 (1969). TOXICITY OF POTASSIUM PERMANGANATE IS DISCUSSED. Hazardous Substances Data Bank (HSDB) MICHAELS DD, ZUGSMITH GS; EYE EAR NOSE THROAT MON 52 (3): 97-8 (1973). POTASSIUM PERMANGANATE BURNS OF THE EYE ARE DESCRIBED. Hazardous Substances Data Bank (HSDB) 13 Toxicity 13.1 Toxicological Information 13.1.1 Toxicity Summary Manganese is a cellular toxicant that can impair transport systems, enzyme activities, and receptor functions. It primarily targets the central nervous system, particularily the globus pallidus of the basal ganglia. It is believed that the manganese ion, Mn(II), enhances the autoxidation or turnover of various intracellular catecholamines, leading to increased production of free radicals, reactive oxygen species, and other cytotoxic metabolites, along with a depletion of cellular antioxidant defense mechanisms, leading to oxidative damage and selective destruction of dopaminergic neurons. In addition to dopamine, manganese is thought to perturbations other neurotransmitters, such as GABA and glutamate. In order to produce oxidative damage, manganese must first overwhelm the antioxidant enzyme manganese superoxide dismutase. The neurotoxicity of Mn(II) has also been linked to its ability to substitute for Ca(II) under physiological conditions. It can enter mitochondria via the calcium uniporter and inhibit mitochondrial oxidative phosphorylation. It may also inhibit the efflux of Ca(II), which can result in a loss of mitochondrial membrane integrity. Mn(II) has been shown to inhibit mitochondrial aconitase activity to a significant level, altering amino acid metabolism and cellular iron homeostasis. (L228) Toxin and Toxin Target Database (T3DB) 13.1.2 Carcinogen Classification Carcinogen Classification No indication of carcinogenicity to humans (not listed by IARC). Toxin and Toxin Target Database (T3DB) 13.1.3 Health Effects Manganese mainly affects the nervous system and may cause behavioral changes and other nervous system effects, which include movements that may become slow and clumsy. This combination of symptoms when sufficiently severe is referred to as “manganism”. (L228) Toxin and Toxin Target Database (T3DB) 13.1.4 Exposure Routes Serious local effects by all routes of exposure. ILO-WHO International Chemical Safety Cards (ICSCs) Oral (L228) ; inhalation (L228) Toxin and Toxin Target Database (T3DB) 13.1.5 Signs and Symptoms Inhalation Exposure Burning sensation. Cough. Sore throat. Shortness of breath. Laboured breathing. Symptoms may be delayed. ILO-WHO International Chemical Safety Cards (ICSCs) Skin Exposure Redness. Skin burns. Pain. ILO-WHO International Chemical Safety Cards (ICSCs) Eye Exposure Redness. Pain. Severe burns. ILO-WHO International Chemical Safety Cards (ICSCs) Ingestion Exposure Burning sensation. Abdominal pain. Diarrhoea. Nausea. Vomiting. Shock or collapse. ILO-WHO International Chemical Safety Cards (ICSCs) Manganese mainly affects the nervous system and may cause behavioral changes and other nervous system effects, which include movements that may become slow and clumsy. This combination of symptoms when sufficiently severe is referred to as “manganism”. (L228) Toxin and Toxin Target Database (T3DB) 13.1.6 Adverse Effects Neurotoxin - Parkinsonism Occupational hepatotoxin - Secondary hepatotoxins: the potential for toxic effect in the occupational setting is based on cases of poisoning by human ingestion or animal experimentation. Methemoglobinemia - The presence of increased methemoglobin in the blood; the compound is classified as secondary toxic effect Dermatotoxin - Skin burns. Toxic Pneumonitis - Inflammation of the lungs induced by inhalation of metal fumes or toxic gases and vapors. Haz-Map, Information on Hazardous Chemicals and Occupational Diseases 13.1.7 Acute Effects ChemIDplus 13.1.8 Toxicity Data LD50: 750 mg/kg (Oral, Rat) (L326) LD50: 500 mg/kg (Subcutaneous, Mouse) (T53) Toxin and Toxin Target Database (T3DB) 13.1.9 Minimum Risk Level Chronic Inhalation: 0.0003 mg/m3 (L134) Toxin and Toxin Target Database (T3DB) 13.1.10 Treatment EYES: irrigate opened eyes for several minutes under running water. INGESTION: do not induce vomiting. Rinse mouth with water (never give anything by mouth to an unconscious person). Seek immediate medical advice. SKIN: should be treated immediately by rinsing the affected parts in cold running water for at least 15 minutes, followed by thorough washing with soap and water. If necessary, the person should shower and change contaminated clothing and shoes, and then must seek medical attention. INHALATION: supply fresh air. If required provide artificial respiration. Toxin and Toxin Target Database (T3DB) 13.1.11 Interactions IN STUDIES OF RETINOTOXIC EFFECTS OF FLUORIDE IN ANIMALS, POTASSIUM PERMANGANATE HAS BEEN UTILIZED SYSTEMICALLY AS OXIDIZING AGENT TO ENHANCE TOXICITY. Grant, W.M. Toxicology of the Eye. 3rd ed. Springfield, IL: Charles C. Thomas Publisher, 1986., p. 757 Hazardous Substances Data Bank (HSDB) 4-Deoxypyridoxine (4-DPD) is a potent antagonist of Vitamin B6 coenzyme which inhibits IL-1, lymphocyte proliferation & has demonstrated that tolerance to skin grafts can be induced by administering splenic cells to pyridoxine-deficient mice. Chronic inflammation induced by dorsal injections of 200 microliters of a 1:40 saturated crystal soln of potassium permanganate (KMnO4) in mice treated or untreated with 4-DPD (400 micrograms/dose), has been investigated. After 7 days all mice developed a subcutaneous granulomatous tissue indicative of a chronic inflammatory response, at the site of injection. KMnO4-treated mice injected ip with 4-DPD (400 micrograms/dose) on 5 consecutive days (the first at the same time of induction of the granuloma) show a significant decr in size & weight of granuloma when compared to mice not treated with 4-DPD (Controls). In addition, in all mice treated with 4-DPD there was a strong inhibition of TNF alpha in serum (P < 0.01) & in supernatant fluids (P < 0.05) from minced granuloma, while IL-6 was inhibited in the supernatant fluids (P < 0.05) of minced granulomas but was not detected in the serum of treated & untreated mice. In this study we show for the first time the antiinflammatory effect of 4-DPD on chronic inflammation & the inhibitory effect of TNF & IL-6 generation in supernatant fluids from minced granulomas. PMID:7854332 Fridas S et al.; Mol Cell Biochem 136 (1): 59-63 (1994) Hazardous Substances Data Bank (HSDB) 13.1.12 Antidote and Emergency Treatment FOLLOWING INGESTION OF TABLETS, GASTRIC LAVAGE & SUPERVISION ON SURGICAL UNIT IS PROPOSED. OPERATION SHOULD BE CONSERVATIVE IF DIGESTIVE PERFORATION IS PRESENT, ALTHOUGH IT IS EXTREMELY RARE. PMID:1242135 CELERIER M ET AL; J CHIR (PARIS) 109 (3): 307-14 (1975) Hazardous Substances Data Bank (HSDB) Basic treatment: Establish a patent airway. Suction if necessary. Watch for signs of respiratory insufficiency and assist ventilations if necessary. Administer oxygen by nonrebreather mask at 10 to 15 L/min. Monitor for pulmonary edema and treat if necessary ... . Monitor for shock and treat if necessary ... . For eye contamination, flush eyes immediately with water. Irrigate each eye continuously with normal saline during transport ... . Do not use emetics. For ingestion, rinse mouth and administer 5 ml/kg up to 200 ml of water for dilution if the patient can swallow, has a strong gag reflex, and does not drool ... . /Manganese and related compounds/ Bronstein, A.C., P.L. Currance; Emergency Care for Hazardous Materials Exposure. 2nd ed. St. Louis, MO. Mosby Lifeline. 1994., p. 365 Hazardous Substances Data Bank (HSDB) Advanced treatment: Consider orotracheal or nasotracheal intubation for airway control in the patient who is unconscious or in severe respiratory distress. Monitor cardiac rhythm and treat arrhythmias if necessary ... . Start an IV with D5W /SRP: "To keep open", minimal flow rate/. Use lactated Ringer's if signs of hypovolemia are present. Consider drug therapy for pulmonary edema ... . For hypotension with signs of hypovolemia, administer fluid cautiously. Consider vasopressors for hypotension with a normal fluid volume. Watch for signs of fluid overload ... . Use proparacaine hydrochloride to assist eye irrigation ... . /Manganese and related compounds/ Bronstein, A.C., P.L. Currance; Emergency Care for Hazardous Materials Exposure. 2nd ed. St. Louis, MO. Mosby Lifeline. 1994., p. 365-6 Hazardous Substances Data Bank (HSDB) 13.1.13 Human Toxicity Excerpts A. ACUTE POISONING: 1. INGESTION OF SOLID OR CONCN POTASSIUM PERMANGANATE CAUSES BROWN DISCOLORATION & EDEMA OF MUCOUS MEMBRANES OF MOUTH & PHARYNX; COUGH, LARYNGEAL EDEMA, & STRIDOR; NECROSIS OF ORAL & PHARYNGEAL MUCOSA; SLOW PULSE; & SHOCK WITH FALL OF BLOOD PRESSURE. IF DEATH IS NOT IMMEDIATE, JAUNDICE & OLIGURIA OR ANURIA MAY APPEAR. 2. APPLICATION OF SOLID OR CONCN POTASSIUM PERMANGANATE SOLN TO MUCOUS MEMBRANES OF VAGINA OR URETHRA CAUSES SEVERE BURNING, HEMORRHAGES, & VASCULAR COLLAPSE. PERFORATION OF VAGINAL WALL MAY OCCUR, RESULTING IN PERITONITIS WITH FEVER & ABDOMINAL PAIN. EXAM REVEALS CHEMICAL BURNS WHICH ARE STAINED BROWN. Dreisbach, R.H. Handbook of Poisoning. 12th ed. Norwalk, CT: Appleton and Lange, 1987., p. 365 Hazardous Substances Data Bank (HSDB) A CASE OF PYLORIC STENOSIS FOLLOWING INGESTION OF POTASSIUM PERMANGANATE WAS DESCRIBED. THIS IS A RARE CAUSE OF PYLORIC STENOSIS. PMID:4761531 DAGLI AJ ET AL; AM J DIG DIS 18 (12): 1091-4 (1973) Hazardous Substances Data Bank (HSDB) GASTRIC LESIONS WERE CAUSED BY INGESTION OF POTASSIUM PERMANGANATE IN ADULT FEMALE. PMID:4653350 MAYER J ET AL; CHIRURGIE 98 (8): 487-91 (1972) Hazardous Substances Data Bank (HSDB) TWO PATIENTS INGESTED POWDER CONSISTING OF MIXT OF SOOT & POTASSIUM PERMANGANATE. BOTH WERE COMATOSE & EXPERIENCED DEEP CYANOSIS BUT RESPONDED WELL TO OXYGEN, METHYLENE BLUE, & VIT C. MAHOMEDY MC ET AL; ANESTHESIA 30 (2): 190 (1975) Hazardous Substances Data Bank (HSDB) For more Human Toxicity Excerpts (Complete) data for POTASSIUM PERMANGANATE (16 total), please visit the HSDB record page. Hazardous Substances Data Bank (HSDB) 13.1.14 Non-Human Toxicity Excerpts MUTAGENICITY: MUTATION RESEARCH 87: 211 (1981). DNA REPAIR DEFICIENT BACTERIAL TESTS: NEGATIVE. GENE-TOX Program: Current Status of Bioassay in Genetic Toxicology. U.S. Environmental Protection Agency, Washington, DC. Office of Toxic Substances and Pesticides. (For program information, contact Environmental Mutagen Information Center, Oak Ridge National Laboratory, Post Office Box Y, Oak Ridge, Tennessee 37830. Telephone (615) 574-7871) Hazardous Substances Data Bank (HSDB) EXPERIMENTALLY IN RABBITS, IRRIGATION OF CORNEA WITH 0.003 MOLAR SOLN POTASSIUM PERMANGANATE HAS BEEN FOUND TO CAUSE NO INJURY, BUT INJECTION OF 0.08 MOLAR SOLN INTO CORNEA CAUSED MODERATELY SEVERE REACTION, GRADED 60 ON SCALE OF 0 TO 100. Grant, W.M. Toxicology of the Eye. 3rd ed. Springfield, IL: Charles C. Thomas Publisher, 1986., p. 757 Hazardous Substances Data Bank (HSDB) ... STRONG SOLN AND CRYSTALS OF POTASSIUM PERMANGANATE IN CONTACT WITH EYE HAVE CAUSED SEVERE DAMAGE. ... IN AREA OF CONTACT A ... LESION HAS DEVELOPED, COLORED DARK BROWN BY MANGANESE OXIDE FORMED ... SWELLING OF LIDS & CONJUNCTIVAE, AND ... SUBCONJUNCTIVAL HEMORRHAGES. WITH PROLONGED CONTACT TURBIDITY & ... DISCOLORATION OF CORNEA ... RECOVERY USUALLY /IS/ ... SPONTANEOUS ... BUT IN ONE INSTANCE TOTAL LEUKOMA /DENSE WHITE OPACITY OF CORNEA/ IS REPORTED FOLLOWING APPLICATION OF STRONG SOLN OF POTASSIUM PERMANGANATE. Grant, W.M. Toxicology of the Eye. 3rd ed. Springfield, IL: Charles C. Thomas Publisher, 1986., p. 756 Hazardous Substances Data Bank (HSDB) POTASSIUM PERMANGANATE INJECTED SC IN HIGH DOSES INTO MALE MICE CAUSED DAMAGE & CALCIFICATION OF PANNICULUS CARNOSUS MUSCLE. PMID:1000349 MCCLURE J, GARDNER DL; CALCIF TISSUE RES 22 (2): 129 (1976) Hazardous Substances Data Bank (HSDB) For more Non-Human Toxicity Excerpts (Complete) data for POTASSIUM PERMANGANATE (10 total), please visit the HSDB record page. Hazardous Substances Data Bank (HSDB) 13.1.15 Non-Human Toxicity Values LD50 Guinea pig oral 810 mg/kg. Sigan Sa, Vitvickaja BR; Gig Sanit 36 (9): 15-18 (1971) as cited in WHO; Environ Health Criteria: Manganese p.55 (1981) Hazardous Substances Data Bank (HSDB) LD100 Mouse subcutaneous 500 mg/kg /From table/ Date S; J Kumamoto Med Soc 34: 159-82 (1960) as cited in WHO; Environ Health Criteria: Manganese p.55 (1981) Hazardous Substances Data Bank (HSDB) LD50 Mouse oral 750 mg/kg /From table/ Sigan SA, Vitvickaja BR; Gig Sanit 36 (9): 15-18 (1971) as cited in WHO; Environ Health Criteria: Manganese p.55 (1981) Hazardous Substances Data Bank (HSDB) View More... 13.2 Ecological Information 13.2.1 EPA Ecotoxicity Pesticide Ecotoxicity Data from EPA EPA Pesticide Ecotoxicity Database 13.2.2 Ecotoxicity Values LC50 CARASSIUS AURATUS (GOLDFISH) 3.6 MG/L/96 HR /CONDITIONS OF BIOASSAY NOT SPECIFIED/ MARKING LL, BILLS TD; TRANS AM FISH SOC 104 (3): 579-83 (1975) Hazardous Substances Data Bank (HSDB) LC50 ICTALURUS PUNCTATUS (CHANNEL CATFISH) 0.75 MG/L/96 HR /CONDITIONS OF BIOASSAY NOT SPECIFIED/ MARKING LL, BILLS TD; TRANS AM FISH SOC 104 (3): 579-83 (1975) Hazardous Substances Data Bank (HSDB) LD50 Lepomis macrochirus (bluegill sunfish) 2.7-3.6 mg/l /Conditions of bioassay not specified/ Purdue University; National Pesticide Information Retrieval System Potassium Permanganate Fact Sheet #80 (1989) Hazardous Substances Data Bank (HSDB) 13.2.3 ICSC Environmental Data The substance is very toxic to aquatic organisms. ILO-WHO International Chemical Safety Cards (ICSCs) 13.2.4 Probable Routes of Human Exposure WORKERS EXPERIENCED RESP IRRITATION FROM POTASSIUM PERMANGANATE &COPPER SULFATE MIST IN AIR BUT AMT UNDER PREVAILING CONDITIONS WERE NOT FOUND TOXIC. HEALTH HAZARD EVALUATION/TOXICITY DETERMINATION REPORT 76-108-365, ROCKY MOUNTAIN MANUFACTURING & WIRE CO, CLEANING DEPT, PUEBLO, CO; US NTIS, PB REP: 10 PAGES (1977) ISS PB-270877 Hazardous Substances Data Bank (HSDB) Major routes of exposure: Dermal and oral. Purdue University; National Pesticide Information Retrieval System; Potassium Permanganate Fact Sheet #80 (1989) Hazardous Substances Data Bank (HSDB) 14 Associated Disorders and Diseases Comparative Toxicogenomics Database (CTD) Associated Occupational Diseases with Exposure to the Compound Manganese, chronic toxic effect [Category: Metal Poisoning, Occupational] Haz-Map, Information on Hazardous Chemicals and Occupational Diseases 15 Literature 15.1 Consolidated References PubChem 15.2 NLM Curated PubMed Citations Medical Subject Headings (MeSH) 15.3 Springer Nature References Springer Nature 15.4 Thieme References Thieme Chemistry 15.5 Wiley References Wiley 15.6 Chemical Co-Occurrences in Literature PubChem 15.7 Chemical-Gene Co-Occurrences in Literature PubChem 15.8 Chemical-Disease Co-Occurrences in Literature PubChem 15.9 Chemical-Organism Co-Occurrences in Literature PubChem 16 Patents 16.1 Depositor-Supplied Patent Identifiers PubChem Link to all deposited patent identifiers PubChem 16.2 WIPO PATENTSCOPE Patents are available for this chemical structure: PATENTSCOPE (WIPO) 16.3 Chemical Co-Occurrences in Patents PubChem 16.4 Chemical-Disease Co-Occurrences in Patents PubChem 16.5 Chemical-Gene Co-Occurrences in Patents PubChem 16.6 Chemical-Organism Co-Occurrences in Patents PubChem 17 Interactions and Pathways 17.1 Chemical-Target Interactions Toxin and Toxin Target Database (T3DB) 18 Biological Test Results 18.1 BioAssay Results PubChem 19 Classification 19.1 MeSH Tree Medical Subject Headings (MeSH) 19.2 NCI Thesaurus Tree NCI Thesaurus (NCIt) 19.3 ChEBI Ontology ChEBI 19.4 KEGG: Drug KEGG 19.5 KEGG: ATC KEGG 19.6 KEGG: JP15 KEGG 19.7 WHO ATC Classification System WHO Anatomical Therapeutic Chemical (ATC) Classification 19.8 ChemIDplus ChemIDplus 19.9 CAMEO Chemicals CAMEO Chemicals 19.10 UN GHS Classification GHS Classification (UNECE) 19.11 EPA CPDat Classification EPA Chemical and Products Database (CPDat) 19.12 Drug Enforcement Administration (DEA) Classification Drug Enforcement Administration (DEA) 19.13 NORMAN Suspect List Exchange Classification NORMAN Suspect List Exchange 19.14 EPA DSSTox Classification EPA DSSTox 19.15 EPA TSCA and CDR Classification EPA Chemicals under the TSCA 19.16 EPA Substance Registry Services Tree EPA Substance Registry Services 19.17 MolGenie Organic Chemistry Ontology MolGenie 19.18 Chemicals in PubChem from Regulatory Sources PubChem 20 Information Sources Filter by Source Australian Industrial Chemicals Introduction Scheme (AICIS)LICENSE Permanganic acid (HMnO4), potassium salt CAMEO ChemicalsLICENSE CAMEO Chemicals and all other CAMEO products are available at no charge to those organizations and individuals (recipients) responsible for the safe handling of chemicals. 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7625	https://www.cdc.gov/mmwr/preview/mmwrhtml/00020736.htm	Vibrio vulnificus Infections Associated with Raw Oyster Consumption -- Florida, 1981-1992 Weekly June 04, 1993 / 42(21);405-407 Persons using assistive technology might not be able to fully access information in this file. For assistance, please send e-mail to:mmwrq@cdc.gov. Type 508 Accommodation and the title of the report in the subject line of e-mail. Vibrio vulnificus Infections Associated with Raw Oyster Consumption -- Florida, 1981-1992 Vibrio vulnificus is a gram-negative bacterium that can cause serious illness and death in persons with preexisting liver disease or compromised immune systems. From 1981 through 1992, 125 persons with V. vulnificus infections, of whom 44 (35%) died, were reported to the Florida Department of Health and Rehabilitative Services (HRS). This report summarizes data on these cases and presents estimates of the at-risk population in Florida. The infections generally occurred each year from March through December and peaked from May through October. Seventy-two persons (58%) had primary septicemia, 35 (28%) had wound infections, and 18 (14%) had gastroenteritis. In patients with primary septicemia, 58 infections (81%) occurred among persons with a history of raw oyster consumption during the week before onset of illness. The mean age of these persons was 60 years (range: 33-90 years; standard deviation: 12.9 years); 51 (88%) were male. Fourteen (78%) of the patients with gastroenteritis also had raw oyster- associated illness. Their mean age was 49 years (range: 19-89 years; standard deviation: 25.7 years); seven (50%) were male. Of the 40 deaths caused by septicemia, 35 (88%) were associated with raw oyster consumption. Nine of these deaths occurred in 1992. The case-fatality rate from raw oyster-associated V. vulnificus septicemia among patients with pre-existing liver disease was 67% (30 of 45) compared with 38% (5 of 13) among those who were not known to have liver disease. Results of the 1988 Florida Behavioral Risk Factor Survey (BRFS) were used to estimate the proportions of the Florida population who ate raw oysters, and the proportion of the population who ate raw oysters and who believed they had liver disease (e.g., cirrhosis). These estimates were used in conjunction with case reports and population data from the Florida Office of Vital Statistics to estimate the risk for illness and death associated with V. vulnificus (1). BRFS and state population data indicate that approximately 3 million persons in Florida eat raw oysters; of these, 71,000 persons believe they have liver disease. Based on the number of cases reported to the Florida HRS during 1981-1992, the annual rate of illness from V. vulnificus infection for adults with liver disease who ate raw oysters was 72 per 1 million adults -- 80 times the rate for adults without known liver disease who ate raw oysters (0.9 per 1 million). The annual rate of death from V. vulnificus for adults with liver disease who ate raw oysters was 45 per 1 million -- more than 200 times greater than the rate for persons without known liver disease who ate raw oysters (0.2 per 1 million). Reported by: WG Hlady, MD, RC Mullen, MPH, RS Hopkins, MD, State Epidemiologist, Florida Dept of Health and Rehabilitative Svcs. Foodborne and Diarrheal Diseases Br, Div of Bacterial and Mycotic Diseases, National Center for Infectious Diseases, CDC. Editorial Note Editorial Note: V. vulnificus was first described as a cause of human illness in 1979 (2 ). Although there is no national surveillance for infections caused by this pathogen, regional surveillance in four states along the Gulf Coast indicates an annual incidence for V. vulnificus infections of at least 0.6 per 1 million persons and a case-fatality rate of 22% (3). V. vulnificus, a free-living bacterium, occurs naturally in the marine environment, rather than as a result of pollution by human or animal fecal waste. This organism is commonly found in estuarine waters of the Gulf of Mexico, where it may contaminate oysters and other shellfish. Legal harvesting of oysters is limited to areas free of fecal contamination; however, V. vulnificus is ubiquitous in warm ocean waters, and oysters harvested from approved sites may be contaminated. Therefore, regardless of the source of the oysters, the potential for infection exists whenever raw oysters are consumed. Ingestion of raw or undercooked shellfish contaminated with V. vulnificus can lead to primary septicemia or gastroenteritis. In addition, V. vulnificus can cause infection by directly contaminating open wounds during swimming, shellfish cleaning, and other marine activities. The findings in this report are consistent with other studies indicating that persons with liver disease are at increased risk for infection with V. vulnificus and death (2,4). Persons with compromised immune systems (e.g., chronic renal insufficiency, cancer, diabetes, steroid-dependent asthma, and chronic intestinal disease) or iron overload states (e.g., thalassemia and hemochromatosis) may also be at increased risk for infection with V. vulnificus and death (2,5). Whether persons with acquired immunodeficiency syndrome are at increased risk for V. vulnificus infections is unknown. A previous study in north Florida indicated that less than 15% of high-risk patients were aware of the risks associated with raw oyster consumption (6). To increase awareness of risks for infection with this pathogen, the Florida HRS has issued press releases to inform the general public and has provided gastroenterologists in the state with clinical references and information for their patients with liver disease. California and Louisiana both require written consumer alerts regarding the risk of raw oyster consumption be visible where raw oysters are sold at retail food establishments. The Florida HRS also is working with other agencies in the state to establish labeling requirements for raw oysters that would inform consumers at all points of sale of the risk for serious illness for persons with liver disease or compromised immune systems who consume raw oysters. The wording of such labeling will be similar to the label already required by the Florida Department of Natural Resources for all wholesale shellstock and shucked products: "Consumer Information -- There is a risk associated with consuming raw oysters or any raw animal protein. If you have chronic illness of the liver, stomach, or blood or have immune disorders, you are at a greater risk of serious illness from raw oysters and should eat oysters fully cooked. If unsure of your risk, consult a physician." References Desenclos JA, Klontz KC, Wolfe LE, Hoercherl S. The risk of Vibrio illness in the Florida raw oyster eating population, 1981-1988. Am J Epidemiol 1991;134:290-7. Blake PA, Merson MH, Weaver RE, Hollis DG, Heublein PC. Disease caused by a marine vibrio: clinical characteristics and epidemiology. N Engl J Med 1979;300:1-4. Levine WC, Griffin PM, the Gulf Coast Vibrio Working Group. Vibrio infections on the Gulf Coast: the results of a first year of regional surveillance. J Infect Dis 1993;167:479-83. Tacket CO, Brenner F, Blake PA. Clinical features and an epidemiological study of Vibrio vulnificus infections. J Infect Dis 1984;149:558-61. Johnston JM, Becker SF, McFarland LM. Vibrio vulnificus: man and the sea. JAMA 1985;253:2850-3. Johnson AR, Anderson CR, Rodrick GE. A survey to determine the awareness of hazards related to raw seafood ingestion in at risk patient groups. In: Proceedings of the 13th annual conference of the Tropical and Subtropical Fisheries Technology Society of the Americas. Gulf Shores, Alabama: Tropical and Subtropical Fisheries Technology Society of the Americas, October 1988. The Food and Drug Administration (FDA) publishes brochures on seafood safety, including ones with special information for patients with liver diseases, immune disorders, gastrointestinal disorders, or diabetes mellitus. Free brochures are available to patients and their physicians from the FDA's 24-hour Seafood Safety Hotline, (800) 332-4010 ({800} FDA-4010); in the Washington, D.C., area the number is (202) 205-4314. DisclaimerAll MMWR HTML versions of articles are electronic conversions from ASCII text into HTML. This conversion may have resulted in character translation or format errors in the HTML version. Users should not rely on this HTML document, but are referred to the electronic PDF version and/or the original MMWR paper copy for the official text, figures, and tables. An original paper copy of this issue can be obtained from the Superintendent of Documents, U.S. Government Printing Office (GPO), Washington, DC 20402-9371; telephone: (202) 512-1800. Contact GPO for current prices. Questions or messages regarding errors in formatting should be addressed to mmwrq@cdc.gov. 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7626	http://www.voutsadakis.com/TEACH/LECTURES/SETS/Chapter7.pdf	Introduction to Set Theory George Voutsadakis1 1Mathematics and Computer Science Lake Superior State University LSSU Math 400 George Voutsadakis (LSSU) Set Theory June 2014 1 / 21 Outline 1 Alephs Initial Ordinals Addition and Multiplication of Alephs George Voutsadakis (LSSU) Set Theory June 2014 2 / 21 Alephs Initial Ordinals Subsection 1 Initial Ordinals George Voutsadakis (LSSU) Set Theory June 2014 3 / 21 Alephs Initial Ordinals Finite Ordinals and Cardinals We proved results involving the cardinality \|X\| of a set X, but we have not deﬁned \|X\| itself, except in the case when X is ﬁnite or countable. We now ﬁnd “representatives” of cardinalities. Natural numbers play this role satisfactorily for ﬁnite sets. We showed that ordinal numbers have many properties of natural numbers, e.g, inductive proofs and recursive constructions. However, ordinal numbers do not represent cardinalities; instead, they represent types of well-orderings. Since any inﬁnite set can be well-ordered in many diﬀerent ways, there are many ordinal numbers of the same cardinality; E.g., ω, ω + 1, ω + 2, . . . , ω + ω, . . . , ω · ω, ω · ω + 1, . . . are all countable ordinal numbers; i.e., \|ω\| = \|ω + 1\| = \|ω + ω\| = · · · = ℵ0. The good behavior of ordinal numbers of ﬁnite cardinalities is due to the fact that all linear orderings of a ﬁnite set are isomorphic, and they are well-orderings. Thus, for any ﬁnite X, there exists unique ordinal n such that \|n\| = \|X\|, called the cardinal number of X. George Voutsadakis (LSSU) Set Theory June 2014 4 / 21 Alephs Initial Ordinals Initial Ordinals To get representatives for cardinalities of inﬁnite (well-orderable) sets, we take the least ordinal number of any given cardinality as the representative of that cardinality: Deﬁnition (Initial Ordinal) An ordinal number α is called an initial ordinal if it is not equipotent to any β < α. Example: Every natural number is an initial ordinal. ω is an initial ordinal, because ω is not equipotent to any natural number. ω + 1 is not initial, because \|ω\| = \|ω + 1\|. Similarly, none of ω + 2, ω + 3, ω + ω, ω · ω, ωω, . . . is initial. George Voutsadakis (LSSU) Set Theory June 2014 5 / 21 Alephs Initial Ordinals Cardinal Numbers Theorem Each well-orderable set X is equipotent to a unique initial ordinal number. By a preceding theorem, X is equipotent to some ordinal α. Let α0 be the least ordinal equipotent to X. Then α0 is an initial ordinal because \|α0\| = \|β\|, for some β < α0, would imply \|X\| = \|β\|, a contradiction. If α0 ̸= α1 are initial ordinals, they cannot be equipotent, because \|α0\| = \|α1\| and, say, α0 < α1, would violate the fact that α1 is initial. This proves the uniqueness. Deﬁnition (Cardinal Number) If X is a well-orderable set, then the cardinal number of X, denoted \|X\|, is the unique initial ordinal equipotent to X. In particular, \|X\| = ω for any countable set X, and \|X\| = n for any ﬁnite set of n elements. George Voutsadakis (LSSU) Set Theory June 2014 6 / 21 Alephs Initial Ordinals Hartogs Number of a Set Are there other initial ordinals besides the natural numbers and ω? Let A be any set; A may not be well-orderable itself, but it certainly has some well-orderable subsets; for example, all ﬁnite subsets of A are well-orderable. Deﬁnition (Hartogs Number) For any A, let h(A) be the least ordinal number which is not equipotent to any subset of A. h(A) is called the Hartogs number of A. By deﬁnition, h(A) is the least ordinal α such that \|α\| ≰\|A\|. Lemma For any A, h(A) is an initial ordinal number. Assume that \|β\| = \|h(A)\| for some β < h(A). Then β is equipotent to a subset of A, and β is equipotent to h(A). We conclude that h(A) is equipotent to a subset of A, i.e., h(A) < h(A), a contradiction. George Voutsadakis (LSSU) Set Theory June 2014 7 / 21 Alephs Initial Ordinals Existence of Hartogs Numbers How do we know that the Hartogs number of A exists? If all inﬁnite ordinals were countable, h(ω) would consist of all ordinals! Lemma The Hartogs number of A exists for all A. By a preceding theorem, for every well-ordered set (W , R) where W ⊆A, there is a unique ordinal α, such that (α, <) is isomorphic to (W , R). By Replacement, there exists a set H such that, for every well-ordering R ∈P(A × A), its isomorphic ordinal α is in H. Claim: H contains all ordinals equipotent to a subset of A. If f is a one-to-one function mapping α into A, we set W = ranf and R = {(f (β), f (γ)) : β < γ < α}. R ⊆A × A is then a well-ordering isomorphic to α (by the isomorphism f ). These considerations show that h(A) = {α ∈H : α is an ordinal equipotent to a subset of A}. Thus, by Axiom Schema of Comprehension, h(A) exists. George Voutsadakis (LSSU) Set Theory June 2014 8 / 21 Alephs Initial Ordinals The Hierarchy of Omegas We can now deﬁne a “scale” of larger and larger initial ordinal numbers by transﬁnite recursion: Deﬁnition (Omegas) ω0 = ω; ωα+1 = h(ωα), for all α; ωα = sup {ωβ : β < α}, if α is limit α ̸= 0. We know that \|ωα+1\| > \|ωα\|, for each α, and so \|ωα\| < \|ωβ\| whenever α < β. Theorem (a) ωα is an inﬁnite initial ordinal number for each α. (b) If Ωis an inﬁnite initial ordinal number, then Ω= ωα for some α. George Voutsadakis (LSSU) Set Theory June 2014 9 / 21 Alephs Initial Ordinals Proof of the Theorem (a) The proof is by induction on α. The only nontrivial case is when α is a limit ordinal. Suppose that \|ωα\| = \|γ\| for some γ < ωα; then there is β < α such that γ ≤ωβ (by the deﬁnition of supremum). But this implies \|ωα\| = \|γ\| ≤\|ωβ\| ≤\|ωα\| and yields a contradiction. (b) First, an easy induction shows that α ≤ωα for all α. Therefore, for every inﬁnite initial ordinal Ω, there is an ordinal α such that Ω< ωα, (for example, α = Ω+ 1). Thus, it suﬃces to prove the following: Claim: For every inﬁnite initial ordinal Ω< ωα, there is some γ < α such that Ω= ωγ. By induction on α. The claim is trivially true for α = 0. If α = β + 1, Ω< ωα = h(ωβ) implies that \|Ω\| ≤\|ωβ\| so either Ω= ωβ and we can let γ = β, or Ω< ωβ and existence of γ < β < α follows from the inductive assumption. If α is a limit ordinal, Ω< ωα = sup {ωβ : β < α} implies that Ω< ωβ for some β < α. The inductive assumption again guarantees the existence of some γ < β, such that Ω= ωγ. George Voutsadakis (LSSU) Set Theory June 2014 10 / 21 Alephs Initial Ordinals Conclusions Every well-orderable set is equipotent to a unique initial ordinal. Inﬁnite initial ordinal numbers form a transﬁnite sequence ωα with α ranging over all ordinal numbers. Inﬁnite initial ordinals are, by deﬁnition, the cardinalities of inﬁnite well-orderable sets. It is customary to call these cardinal numbers alephs, i.e., we deﬁne ℵα = ωα, for each α. The cardinal number of a well-orderable set is thus either a natural number or an aleph. Note that the ordering of cardinal numbers by size deﬁned previously agrees with the ordering of natural numbers and alephs as ordinals by < (i.e., ∈): If \|X\| = ℵα and \|Y \| = ℵβ, then \|X\| < \|Y \| if and only if ℵα < ℵβ (i.e., ωα ∈ωβ). A similar equivalence holds if one or both of \|X\| and \|Y \| are natural numbers. George Voutsadakis (LSSU) Set Theory June 2014 11 / 21 Alephs Initial Ordinals Cardinal and Ordinal Operations We have deﬁned addition, multiplication, and exponentiation of cardinal numbers. These agree with the corresponding ordinal operations if the ordinals involved are natural numbers but they may diﬀer for inﬁnite ordinals. Example: ω0 + ω0 ̸= ω0 if + stands for the ordinal addition; but ω0 + ω0 = ω0 if + stands for the cardinal addition. The addition of cardinal numbers is commutative, but the addition of ordinal numbers is not. For clarity, the ω-symbolism is used when the ordinal operations are involved, and the ℵ-symbolism for the cardinal operations. Thus: ω0 + ω0 and 2ω0 indicate ordinal addition and exponentiation: ω0 + ω0 = sup {ω + n : n < ω0} > ω0; 2ω0 = sup {2n : n < ω0} = ω0. ℵ0 + ℵ0 and 2ℵ0 cardinal operations: ℵ0 + ℵ0 = ℵ0; 2ℵ0 is uncountable. George Voutsadakis (LSSU) Set Theory June 2014 12 / 21 Alephs Addition and Multiplication of Alephs Subsection 2 Addition and Multiplication of Alephs George Voutsadakis (LSSU) Set Theory June 2014 13 / 21 Alephs Addition and Multiplication of Alephs Revisiting Cardinal Addition and Multiplication Let κ and λ be cardinal numbers. We have deﬁned κ + λ as the cardinality of the set X ∪Y , where \|X\| = κ, \|Y \| = λ, and X and Y are disjoint: \|X\| + \|Y \| = \|X ∪Y \|, if X ∩Y = ∅. This deﬁnition does not depend on the choice of X and Y . The product κ · λ has been deﬁned as the cardinality of the cartesian product X × Y , where X and Y are any two sets of respective cardinalities κ and λ: \|X\| · \|Y \| = \|X × Y \|. This deﬁnition is also independent of the choice of X and Y . Addition and multiplication satisfy: κ + λ = λ + κ κ · λ = λ · κ κ + (λ + µ) = (κ + λ) + µ κ · (λ · µ) = (κ · λ) · µ κ · (λ + µ) = κ · λ + κ · µ George Voutsadakis (LSSU) Set Theory June 2014 14 / 21 Alephs Addition and Multiplication of Alephs Some Examples of Operations involving Alephs The arithmetic of inﬁnite numbers diﬀers substantially from the arithmetic of ﬁnite numbers. In fact, the rules for addition and multiplication of alephs are very simple. Example: ℵ0 + n = ℵ0, for every natural number n. ℵ0 + ℵ0 = ℵ0, since the set of all natural numbers is the union of two disjoint countable sets: the set of even numbers and the set of odd numbers. ℵ0 · ℵ0 = ℵ0 (The set of all pairs of natural numbers is countable.) We prove, next, a general theorem that determines completely the result of addition and multiplication of alephs. George Voutsadakis (LSSU) Set Theory June 2014 15 / 21 Alephs Addition and Multiplication of Alephs ℵα · ℵα = ℵα Theorem ℵα · ℵα = ℵα, for every α. We prove the theorem by transﬁnite induction. For every α, we construct a certain well-ordering ≺of the set ωα × ωα and show, using the induction hypothesis ℵβ · ℵβ ≤ℵβ that the order-type of the well-ordered set (ωα × ωα, ≺) is at most ωα. Then, it follows that ℵα · ℵα ≤ℵα and since ℵα · ℵα ≥ℵα, we have ℵα · ℵα = ℵα. We construct the well-ordering ≺of ωα × ωα uniformly for all ωα, i.e., we deﬁne a property ≺of pairs of ordinals and show that ≺ well-orders ωα × ωα, for every ωα. (α1, α2) ≺(β1, β2) if and only if either max {α1, α2} < max {β1, β2} or max {α1, α2} = max {β1, β2} and α1 < β1 or max {α1, α2} = max {β1, β2}, α1 = β1 and α2 < β2. We show that ≺is a well-ordering (of any set of pairs of ordinals). George Voutsadakis (LSSU) Set Theory June 2014 16 / 21 Alephs Addition and Multiplication of Alephs ≺is an Ordering ≺is transitive: Let α1, α2, β1, β2, γ1, γ2 be such that (α1, α2) ≺(β1, β2) and (β1, β2) ≺(γ1, γ2). By deﬁnition max {α1, α2} ≤max {β1, β2} ≤max {γ1, γ2}, whence max {α1, α2} ≤max {γ1, γ2}. If max {α1, α2} < max {γ1, γ2}, then (α1, α2) ≺(γ1, γ2). If max {α1, α2} = max {β1, β2} = max {γ1, γ2}, then we have α1 ≤β1 ≤γ1, and so α1 ≤γ1. If α1 < γ1, then (α1, α2) ≺(γ1, γ2); Otherwise, we have α1 = β1 = γ1. In this last case, max {α1, α2} = max {β1, β2} = max {γ1, γ2}, and α1 = β1 = γ1, so, necessarily, α2 < β2 < γ2, and it follows again that (α1, α2) ≺(γ1, γ2). George Voutsadakis (LSSU) Set Theory June 2014 17 / 21 Alephs Addition and Multiplication of Alephs ≺is Linear We verify that for any α1, α2, β1, β2, either (α1, α2) ≺(β1, β2) or (β1, β2) ≺(α1, α2) or (α1, α2) = (β1, β2) and that these three cases are mutually exclusive. This follows directly from the deﬁnition: Given (α1, α2) and (β1, β2) we ﬁrst compare max {α1, α2} and max {β1, β2}, then α1 and β1 and last the ordinals α2 and β2. George Voutsadakis (LSSU) Set Theory June 2014 18 / 21 Alephs Addition and Multiplication of Alephs ≺is a Well-Ordering ≺is a well-ordering: Let X be a nonempty set of pairs of ordinals. We ﬁnd the ≺-least element of X. Let δ be the least maximum of the pairs in X, i.e., let δ be least element in {max {α, β} : (α, β) ∈X}. Let Y = {(α, β) ∈X : max {α, β} = δ}. The set Y is a nonempty subset of X, and, for every (α, β) ∈Y , we have max {α, β} = δ. Moreover, δ < max {α′, β′}, for any (α′, β′) ∈X −Y , and hence (α, β) ≺(α′, β′) whenever (α, β) ∈Y and (α′, β′) ∈X −Y . Therefore, the least element of Y , if it exists, is also the least element of X. Now let α0 be the least ordinal in the set {α : (α, β) ∈Y for some β} and let Z = {(α, β) ∈Y : α = α0}. The set Z is a nonempty subset of Y . Also (α, β) ≺(α′, β′) whenever (α, β) ∈Z and (α′, β′) ∈Y −Z. Finally, let β0 be the least ordinal in the set {β : (α0, β) ∈Z}. Clearly, (α0, β0) is the least element of Z. It follows that (α0, β0) is the least element of X. George Voutsadakis (LSSU) Set Theory June 2014 19 / 21 Alephs Addition and Multiplication of Alephs Finishing the Proof Having shown that ≺is a well-ordering of ωα × ωα for every α, we use this well-ordering to prove, by transﬁnite induction on α, that \|ωα × ωα\| ≤ℵα, i.e., ℵα · ℵα ≤ℵα. For α = 0, we know that ℵ0 · ℵ0 = ℵ0. So let α > 0, and let us assume that ℵβ · ℵβ ≤ℵβ, for all β < α. We prove that \|ωα × ωα\| ≤ℵα. If suﬃces to show that the order-type of the well-ordered set (ωα × ωα, ≺) is at most ωα. If the order-type of (ωα × ωα, ≺) were greater than ωα, then there would exist (α1, α2) ∈ωα × ωα, such that the cardinality of the set X = {(ξ1, ξ2) ∈ωα × ωα : (ξ1, ξ2) ≺(α1, α2)} is at least ℵα. Thus, it suﬃces to prove that, for any (α1, α2) ∈ωα × ωα, we have \|X\| < ℵα. Let β = max {α1, α2} + 1. Then β ∈ωα and, for every (ξ1, ξ2) ∈X, we have max {ξ1, ξ2} ≤max {α1, α2} < β, so ξ1 ∈β and ξ2 ∈β, i.e., X ⊆β × β. Let γ < α be such that \|β\| ≤ℵγ. Then \|X\| ≤\|β × β\| = \|β\| · \|β\| ≤ ℵγ · ℵγ and ℵγ · ℵγ ≤ℵγ, by the induction hypothesis. Thus, \|X\| ≤ℵγ, and, hence, \|X\| < ℵα. George Voutsadakis (LSSU) Set Theory June 2014 20 / 21 Alephs Addition and Multiplication of Alephs Rules of Cardinal Arithmetic Corollary For every α and β such that α ≤β, we have ℵα · ℵβ = ℵβ. Also, n · ℵα = ℵα, for every positive natural number n. If α ≤β, then ℵβ = 1 · ℵβ ≤ℵα · ℵβ ℵα · ℵβ ≤ℵβ · ℵβ = ℵβ, by the theorem. Thus by the Cantor-Bernstein Theorem ℵα · ℵβ = ℵβ. The equality n · ℵα = ℵα is proved similarly. Corollary For every α and β such that α ≤β, we have ℵα + ℵβ = ℵβ. Also, n + ℵα = ℵα, for all natural numbers n. If α ≤β, then ℵβ ≤ℵα + ℵβ ≤ℵβ + ℵβ = 2 · ℵβ = ℵβ and the assertion follows. The second part is proved similarly. George Voutsadakis (LSSU) Set Theory June 2014 21 / 21
7627	https://www.sciencedirect.com/science/article/pii/S0377042707004566	Skip to article My account Sign in View PDF Journal of Computational and Applied Mathematics Volume 220, Issues 12, 15 October 2008, Pages 355-363 A smoothing-type algorithm for solving system of inequalities Author links open overlay panel, , rights and content Under an Elsevier user license Open archive Abstract In this paper we consider system of inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations. A Newton-type algorithm is applied to solve iteratively the smooth equations so that a solution of the problem concerned is found. We show that the algorithm is globally and locally quadratically convergent under suitable assumptions. Preliminary numerical results are reported. MSC 90C33 65K05 Keywords System of inequalities Smoothing-type algorithm Global convergence Local quadratic convergence Cited by (0) : This work was partially supported by the National Natural Science Foundation of China (Grant No. 10571134), the Natural Science Foundation of Tianjin (Grant No. 07JCYBJC05200), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. Copyright © 2007 Elsevier B.V. All rights reserved.
7628	https://www.chegg.com/homework-help/questions-and-answers/part-b-bjt-characteristics-procedure-using-curve-tracer-lab-room-obtain-output-characteris-q60681035	Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. Question: PART (B): The BJT Characteristics: PROCEDURE: Using the curve tracer in the lab room, we will obtain a the output characteristic curve of the BJT transistor which is the curve between Ic & Vce at different steps 13 From the Curve you will find H-parameters values practically from the slope of the curve using the following formulas: hoe = Bac AIC AVCE I A BJT, or Bipolar Junction Transistor, is a type of semiconductor device that is widely used in elec... Not the question you’re looking for? Post any question and get expert help quickly. Chegg Products & Services CompanyCompany Company Chegg NetworkChegg Network Chegg Network Customer ServiceCustomer Service Customer Service EducatorsEducators Educators
7629	https://www.youtube.com/watch?v=zp2gMogsbv4	Deriving the Equation of a Hyperbola using CAS and Why c^2 = a^2 + b^2 turksvids 29500 subscribers 97 likes Description 9301 views Posted: 24 Jan 2019 In this video we use the TI-Nspire CX CAS's computer algebra system to work from the definition of a hyperbola to the standard form of the equation of a hyperbola. In the process we see why, algebraically, c^2 = a^2 + b^2. We also see why the constant difference is equal to 2a, twice the distance from the center to a vertex. There's a lot of functions being used on the TI-Nspire CAS. We use expand often and also the built-in right() function, which returns the right hand side of an equation. I'm sure people still derive the equations of conic sections by hand, but this is the kind of thing where a CAS really makes a difference with no loss of learning. 17 comments Transcript: okay in this video we're going to be deriving the equation of a hyperbola and we're going to take a look at why C squared equals a squared plus B squared and so in case you don't remember or maybe you don't even know it yet the standard equation of a hyperbola is x squared over a squared minus y squared over B squared is equal to 1 all right so let's start with a figure so that we're all talking about this the same way so this is the figure that I have so I have two points the first focus and the second focus and a hyperbola has kind of a weird definition it's the set of all points in the plane so it's this branch and also this branch where if you find the distance from the point XY to this focus and you find the distance from this point to this focus if you do the absolute value of let's say d1 minus d2 you always get the same number so that's the definition we're actually gonna use that to work out of the equation so there's a couple of things that we need to know first we need to remember the absolute value of d1 minus d2 is constant the way I have it set up with this point on this branch d1 is always actually bigger than d2 so d1 minus d2 will be positive and I'm going to use that as I go through this derivation so the other thing we need to do we're saying that if we find the difference between d1 and d2 we would get a constant but we don't know what that constant is so what I'm going to do is I'm going to take the point XY I'm gonna move it down here so that it's on the x-axis and then I'm gonna calculate the distance from this point to the first focus and this point to the second focus and then add them up so let's do that okay here's the picture so since the distance from the center which is zero zero to this focus is just C units and the distance from the center to this vertex is what we call a so we're gonna go with that a units so this line segment right here has a length of a + C so that's pretty clear I think then we need to find this distance but if you look at it again the distance from the center to the focus is C and on the distance from the center to the vertex is a so this little leftover bit here must be C the big thing - a the little thing so this is C - a this is a plus C and then what we want to do is the absolute value of D 1 minus D 2 which is a plus C minus the quantity C minus a which overall is just 2 a so now we know that constant difference is going to be 2a so going back to this we're going to need to find D 1 or need to find D 2 we're gonna subtract them and then we're gonna say that the difference is 2a so let's let's go to the inspirer and start defining things so to find D 1 and D 2 I need to find distances so I'm going to define a distance formula so I can use it so I'm gonna say D of X comma Y comma any comma B so this is a distance formula it I'll take the point XY and the point a B and find the distance between them so is equal to so that's ponyville's that's control and then the templates if you're on the handheld and then square root so control 1x squared quantity X minus a squared + quantity Y minus B squared I'm gonna press ENTER and it should say done all right so that will find the distances for us now let's go back to the figure and see what distances we need to find so first one d1 is from the point XY to the point negative C 0 so let's let's do that I'm going to find it I'm gonna say D 1 and then colon equals and it'll be a d of XY comma negative C comma 0 and we have this radical so that's the distance and then we also need to find d2 which is the distance from X Y to c0 so let's do it again I'm actually just gonna take this and change the 1 to a 2 and change the negative C to a C and press Enter ok so we know that D 1 is bigger than D 2 because of the way we've set up our figure so I'm gonna say that D 1 minus D 2 has to be equal to 2a so that's kind of the definition of the hyperbola that I'm using right there and I'm press ENTER and if you watch the lip 2 video you know that this thing is better worked with if we take one of these radicals and move it to the other side so what I'm going to do is I'm actually going to do D 1 is equal to 2a and then plus D 2 so I'm just adding D 2 to the other side and now we have one radical on each side I need to get rid of the radical so I'm gonna square so to do that you can actually just press the squared key and the inspirer understands that you want to square both sides of the equation but there might be extraneous solution so it's going to give you a warning so we squared let's see what this is operation might introduce false solutions so it calls extraneous solutions false solutions but that's okay all right so we have this which is kind of what I asked it to do but it didn't actually expand the right-hand side so what I'm gonna do is I'm gonna use expansion so menu 3 and then option three is expand I want to expand that answer right there you could also go up and paste it down press Enter so it doesn't seem great at first but if you scroll through you can see the right-hand side is no longer a quantity squared it's a thing with tons of trailing terms and what I want to do is move all these trailing terms over to the left hand side so everything without a radical basically ends up on the left so to do that I'm just gonna say I'm gonna subtract them all from the right hand side 70 minus quantity well I'm gonna subtract them all from both sides I went up I pressed to the right to get to the end here I'm holding shift and pressing to the left the emulator can't show it when I press two buttons at the same time I think so it's not showing it but again I was holding shift which is right here and I was pressing to the left on the trackpad press ENTER to piece it down and press Enter and now it's starting to shape up I guess and here everything has a four so let's just get rid of that divided by four and things are looking okay I still have a radical so I'm going to square it in square and it doesn't expand so I'm going to expand again so menu three three paste down and now there's just a ton of stuff and if I was doing this on paper I'd have to go through and find like terms and do a lot of stuff but the inspire is gonna let me do something a little easier so what I'm gonna do is I'm gonna take the previous answer so that's control and then the negative sign so I want the previous answer - I'm gonna subtract from the previous answer the entire right hand side of the previous answer so the entire right hand side is like all this stuff over here and to do that I'm just going to type righ T so right and then of answers so go control and the negative sign and let's press Enter and we get this okay again it didn't expand for us so I'm gonna go through and expand again a new three and option three is expand the previous answer and get this alright so this if you watch the ellipse video or you work through it by him this is almost exactly the same process in fact it's kind of like eerily similar and soon we're gonna run into the key difference but I want to move everything that doesn't have a variable so x and y are my variables I want to move everything to the right hand side that doesn't have that so I'm gonna do - and a quantity go up here arrow over until your to the right of C squared hold shift press to the left or you just type them in it's up to you here and press Enter now if you remember the equation that we're looking for is actually equal to one so what I'll do at this point is I will take the right hand side here and I'm gonna divide everything I see by it so I'm gonna divide both sides by the right hand side and I can do that the same way that I moved the entire right hand side over I'm just gonna do the previous answer so I just said divided by and then the right of the answer and it might not look nice but at the very least it's going to make the right hand side into eight one so we do this and we get one and that's pretty good okay so now I'm looking at this and let's see what we've got um okay so I'm gonna go back to my picture and see what's kind of going on here because I think there's something a little weird right we have an a squared minus C squared and that I think is a it's not an issue but it's a thing so let's go here let's go down here okay so if you look at the figure in the figure in a hyperbola in general since the focus is kind of on the outside but the hyperbola looks like it's eating the focus or maybe the focus is on the inside depending on how you look at it but anyway the focus is out here it's not between the branches the distance from the center to the focus which is C is always bigger than the distance from the center to a vertex which is a so C is bigger than a so if C is bigger than a then C squared minus a squared is a positive and if you look at this we have a squared minus C squared so that's actually a negative right there so what I'm gonna do is two things I'm going to first look at this and I'm gonna kind of divide it up manually which is kind of annoying gonna just don't weep this and then I'm gonna just kind of like type in here a squared wise ooh Capital y y squared and then over the same denominator I'm just doing this to make it look clear you don't really need to do this so let me copy oh I don't think I can actually copy I might have to type that yeah that's annoying oh no it worked whew okay so if we look at this you can see that a squared minus C squared over a squared minus C squared definitely gonna cancel you can see that a squared over a squared is definitely gonna cancel I don't know what will happen if I just press ENTER I think you can do all kinds of algebra on this but the key thing here is that these are just gonna cancel and that's fine here I'm gonna actually end up with y squared over a squared minus C squared let's just see what it does okay it does what I wanted it to but I know that a is less than C which means that a squared minus C squared is actually a negative so what I'm gonna do is I'm gonna go back up here I'm gonna change this into well I'm gonna change this hold on let's do this I'm gonna do the negative of C squared minus a squared so now I'm just doing things to explain to you guys while you're watching look I wouldn't really do this on the calculator fact are- out of both of those and now C squared minus a squared is a positive number which is good because I probably want that to happen I know I want that to happen when press Enter it's probably just gonna distribute again which it does but if we go back up to this what I'd rather do is take this negative sign that's in here and just factor it out and put it there and I get x squared over a squared minus y squared over hmm C squared minus a squared is equal to one if we go back to our equation that's not what the equation looked like it was x squared over a squared minus y squared over B squared is 1 so what we actually do at this point you is on the press enter kind of messes it up again I'm just gonna let B squared equals C squared minus a squared and by doing that I go back up here to my equation and I make the substitution this is all very non casts at this point it's just I'm not handwriting anything I do this press ENTER by making that substitution I've gotten to the equation that I wanted so I have this and also if you look at the relationship we came up with B squared is equal to C squared minus a squared if I add a squared to both sides I get that a squared plus B squared equals C squared so we also just figured out why C squared is an a squared plus B squared all right so that's how we can derive the equation if you derive the equation when you lips it's exactly the same process until the very end well actually I guess the initial step is different right because we have to do D 1 minus D 2 instead of D 1 plus D 2 so that's different and then at the end we had to make that weird factoring step right where we know that a is actually less than C so we factored out a negative from this a squared minus C squared and kind of way from there so it's a little less clear than the ellipse description but it definitely works and you can see let me delete this personal clarity here that we get x squared over a squared minus y squared over B squared is equal to 1 and when that's true we know that a squared plus C nope a squared plus B squared is equal to C squared so there you go I hope you found this helpful and good luck
7630	https://www.theexpertta.com/book-files/OpenStaxCollegePhysics2e/Chapter8.pdf	INTRODUCTION TO LINEAR MOMENTUM AND COLLISIONS CHAPTER 8 Linear Momentum and Collisions 8.1 Linear Momentum and Force 8.2 Impulse 8.3 Conservation of Momentum 8.4 Elastic Collisions in One Dimension 8.5 Inelastic Collisions in One Dimension 8.6 Collisions of Point Masses in Two Dimensions 8.7 Introduction to Rocket Propulsion We use the term momentum in various ways in everyday language, and most of these ways are consistent with its precise scientific definition. We speak of sports teams or politicians gaining and maintaining the momentum to win. We also recognize that momentum has something to do with collisions. For example, looking at the rugby players in the photograph colliding and falling to the ground, we expect their momenta to have great effects in the resulting collisions. Generally, momentum implies a tendency to continue on course—to move in the same direction—and is associated with great mass and speed. Momentum, like energy, is important because it is conserved. Only a few physical quantities are conserved in nature, FIGURE 8.1 Each rugby player has great momentum, which will affect the outcome of their collisions with each other and the ground. (credit: ozzzie, Flickr) CHAPTER OUTLINE and studying them yields fundamental insight into how nature works, as we shall see in our study of momentum. Click to view content ( 8.1 Linear Momentum and Force LEARNING OBJECTIVES By the end of this section, you will be able to: • Define linear momentum. • Explain the relationship between momentum and force. • State Newton’s second law of motion in terms of momentum. • Calculate momentum given mass and velocity. Linear Momentum The scientific definition of linear momentum is consistent with most people’s intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. Linear momentum is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum is expressed as Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass or the greater its velocity, the greater its momentum. Momentum is a vector having the same direction as the velocity . The SI unit for momentum is . EXAMPLE 8.1 Calculating Momentum: A Football Player and a Football (a) Calculate the momentum of a 110-kg football player running at 8.00 m/s. (b) Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s. Strategy No information is given regarding direction, and so we can calculate only the magnitude of the momentum, . (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in the equation, which becomes when only magnitudes are considered. Solution for (a) To determine the momentum of the player, substitute the known values for the player’s mass and speed into the equation. Solution for (b) To determine the momentum of the ball, substitute the known values for the ball’s mass and speed into the equation. 8.1 Linear Momentum Linear momentum is defined as the product of a system’s mass multiplied by its velocity: 8.2 8.3 8.4 8.5 324 8 • Linear Momentum and Collisions Access for free at openstax.org The ratio of the player’s momentum to that of the ball is Discussion Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player’s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections. Momentum and Newton’s Second Law The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is where is the net external force, is the change in momentum, and is the change in time. This statement of Newton’s second law of motion includes the more familiar as a special case. We can derive this form as follows. First, note that the change in momentum is given by If the mass of the system is constant, then So that for constant mass, Newton’s second law of motion becomes Because , we get the familiar equation when the mass of the system is constant. Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail; however, the relationship between momentum and force remains useful 8.6 8.7 Newton’s Second Law of Motion in Terms of Momentum The net external force equals the change in momentum of a system divided by the time over which it changes. 8.8 Making Connections: Force and Momentum Force and momentum are intimately related. Force acting over time can change momentum, and Newton’s second law of motion, can be stated in its most broadly applicable form in terms of momentum. Momentum continues to be a key concept in the study of atomic and subatomic particles in quantum mechanics. 8.9 8.10 8.11 8.12 8.1 • Linear Momentum and Force 325 when mass is constant, such as in the following example. EXAMPLE 8.2 Calculating Force: Venus Williams’ Racquet During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that the ball’s speed just after impact is 58 m/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)? Strategy This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton’s second law stated in terms of momentum is then written as As noted above, when mass is constant, the change in momentum is given by In this example, the velocity just after impact and the change in time are given; thus, once is calculated, can be used to find the force. Solution To determine the change in momentum, substitute the values for the initial and final velocities into the equation above. Now the magnitude of the net external force can determined by using : where we have retained only two significant figures in the final step. Discussion This quantity was the average force exerted by Venus Williams’ racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using , but one additional step would be required compared with the strategy used in this example. 8.2 Impulse LEARNING OBJECTIVES By the end of this section, you will be able to: • Define impulse. • Describe effects of impulses in everyday life. • Determine the average effective force using graphical representation. • Calculate average force and impulse given mass, velocity, and time. The effect of a force on an object depends on how long it acts, as well as how great the force is. In Example 8.1, a very large force acting for a short time had a great effect on the momentum of the tennis ball. A small force could 8.13 8.14 8.15 8.16 326 8 • Linear Momentum and Collisions Access for free at openstax.org cause the same change in momentum, but it would have to act for a much longer time. For example, if the ball were thrown upward, the gravitational force (which is much smaller than the tennis racquet’s force) would eventually reverse the momentum of the ball. Quantitatively, the effect we are talking about is the change in momentum . By rearranging the equation to be we can see how the change in momentum equals the average net external force multiplied by the time this force acts. The quantity is given the name impulse. Impulse is the same as the change in momentum. EXAMPLE 8.3 Calculating Magnitudes of Impulses: Two Billiard Balls Striking a Rigid Wall Two identical billiard balls strike a rigid wall with the same speed, and are reflected without any change of speed. The first ball strikes perpendicular to the wall. The second ball strikes the wall at an angle of from the perpendicular, and bounces off at an angle of from perpendicular to the wall. (a) Determine the direction of the force on the wall due to each ball. (b) Calculate the ratio of the magnitudes of impulses on the two balls by the wall. Strategy for (a) In order to determine the force on the wall, consider the force on the ball due to the wall using Newton’s second law and then apply Newton’s third law to determine the direction. Assume the -axis to be normal to the wall and to be positive in the initial direction of motion. Choose the -axis to be along the wall in the plane of the second ball’s motion. The momentum direction and the velocity direction are the same. Solution for (a) The first ball bounces directly into the wall and exerts a force on it in the direction. Therefore the wall exerts a force on the ball in the direction. The second ball continues with the same momentum component in the direction, but reverses its -component of momentum, as seen by sketching a diagram of the angles involved and keeping in mind the proportionality between velocity and momentum. 8.17 Impulse: Change in Momentum Change in momentum equals the average net external force multiplied by the time this force acts. The quantity is given the name impulse. There are many ways in which an understanding of impulse can save lives, or at least limbs. The dashboard padding in a car, and certainly the airbags, allow the net force on the occupants in the car to act over a much longer time when there is a sudden stop. The momentum change is the same for an occupant, whether an air bag is deployed or not, but the force (to bring the occupant to a stop) will be much less if it acts over a larger time. Cars today have many plastic components. One advantage of plastics is their lighter weight, which results in better gas mileage. Another advantage is that a car will crumple in a collision, especially in the event of a head-on collision. A longer collision time means the force on the car will be less. Deaths during car races decreased dramatically when the rigid frames of racing cars were replaced with parts that could crumple or collapse in the event of an accident. Bones in a body will fracture if the force on them is too large. If you jump onto the floor from a table, the force on your legs can be immense if you land stiff-legged on a hard surface. Rolling on the ground after jumping from the table, or landing with a parachute, extends the time over which the force (on you from the ground) acts. 8.18 8.2 • Impulse 327 These changes mean the change in momentum for both balls is in the direction, so the force of the wall on each ball is along the direction. Strategy for (b) Calculate the change in momentum for each ball, which is equal to the impulse imparted to the ball. Solution for (b) Let be the speed of each ball before and after collision with the wall, and the mass of each ball. Choose the -axis and -axis as previously described, and consider the change in momentum of the first ball which strikes perpendicular to the wall. Impulse is the change in momentum vector. Therefore the -component of impulse is equal to and the -component of impulse is equal to zero. Now consider the change in momentum of the second ball. It should be noted here that while changes sign after the collision, does not. Therefore the -component of impulse is equal to and the -component of impulse is equal to zero. The ratio of the magnitudes of the impulse imparted to the balls is Discussion The direction of impulse and force is the same as in the case of (a); it is normal to the wall and along the negative -direction. Making use of Newton’s third law, the force on the wall due to each ball is normal to the wall along the positive -direction. Our definition of impulse includes an assumption that the force is constant over the time interval . Forces are usually not constant. Forces vary considerably even during the brief time intervals considered. It is, however, possible to find an average effective force that produces the same result as the corresponding time-varying force. Figure 8.2 shows a graph of what an actual force looks like as a function of time for a ball bouncing off the floor. The area under the curve has units of momentum and is equal to the impulse or change in momentum between times and . That area is equal to the area inside the rectangle bounded by , , and . Thus the impulses and their effects are the same for both the actual and effective forces. 8.19 8.20 8.21 8.22 8.23 328 8 • Linear Momentum and Collisions Access for free at openstax.org FIGURE 8.2 A graph of force versus time with time along the -axis and force along the -axis for an actual force and an equivalent effective force. The areas under the two curves are equal. 8.3 Conservation of Momentum LEARNING OBJECTIVES By the end of this section, you will be able to: • Describe the principle of conservation of momentum. • Derive an expression for the conservation of momentum. • Explain conservation of momentum with examples. • Explain the principle of conservation of momentum as it relates to atomic and subatomic particles. Momentum is an important quantity because it is conserved. Yet it was not conserved in the examples in Impulse and Linear Momentum and Force, where large changes in momentum were produced by forces acting on the system of interest. Under what circumstances is momentum conserved? The answer to this question entails considering a sufficiently large system. It is always possible to find a larger system in which total momentum is constant, even if momentum changes for components of the system. If a football player runs into the goalpost in the end zone, there will be a force on him that causes him to bounce backward. However, the Earth also recoils —conserving momentum—because of the force applied to it through the goalpost. Because Earth is many orders of magnitude more massive than the player, its recoil is immeasurably small and can be neglected in any practical sense, but it is real nevertheless. Consider what happens if the masses of two colliding objects are more similar than the masses of a football player and Earth—for example, one car bumping into another, as shown in Figure 8.3. Both cars are coasting in the same direction when the lead car (labeled is bumped by the trailing car (labeled The only unbalanced force on each car is the force of the collision. (Assume that the effects due to friction are negligible.) Car 1 slows down as a result of the collision, losing some momentum, while car 2 speeds up and gains some momentum. We shall now show that the total momentum of the two-car system remains constant. Making Connections: Take-Home Investigation—Hand Movement and Impulse Try catching a ball while “giving” with the ball, pulling your hands toward your body. Then, try catching a ball while keeping your hands still. Hit water in a tub with your full palm. After the water has settled, hit the water again by diving your hand with your fingers first into the water. (Your full palm represents a swimmer doing a belly flop and your diving hand represents a swimmer doing a dive.) Explain what happens in each case and why. Which orientations would you advise people to avoid and why? Making Connections: Constant Force and Constant Acceleration The assumption of a constant force in the definition of impulse is analogous to the assumption of a constant acceleration in kinematics. In both cases, nature is adequately described without the use of calculus. 8.3 • Conservation of Momentum 329 FIGURE 8.3 A car of mass moving with a velocity of bumps into another car of mass and velocity that it is following. As a result, the first car slows down to a velocity of and the second speeds up to a velocity of . The momentum of each car is changed, but the total momentum of the two cars is the same before and after the collision (if you assume friction is negligible). Using the definition of impulse, the change in momentum of car 1 is given by where is the force on car 1 due to car 2, and is the time the force acts (the duration of the collision). Intuitively, it seems obvious that the collision time is the same for both cars, but it is only true for objects traveling at ordinary speeds. This assumption must be modified for objects travelling near the speed of light, without affecting the result that momentum is conserved. Similarly, the change in momentum of car 2 is where is the force on car 2 due to car 1, and we assume the duration of the collision is the same for both cars. We know from Newton’s third law that , and so Thus, the changes in momentum are equal and opposite, and Because the changes in momentum add to zero, the total momentum of the two-car system is constant. That is, where and are the momenta of cars 1 and 2 after the collision. (We often use primes to denote the final state.) This result—that momentum is conserved—has validity far beyond the preceding one-dimensional case. It can be similarly shown that total momentum is conserved for any isolated system, with any number of objects in it. In equation form, the conservation of momentum principle for an isolated system is written or 8.24 8.25 8.26 8.27 8.28 8.29 8.30 8.31 330 8 • Linear Momentum and Collisions Access for free at openstax.org where is the total momentum (the sum of the momenta of the individual objects in the system) and is the total momentum some time later. (The total momentum can be shown to be the momentum of the center of mass of the system.) An isolated system is defined to be one for which the net external force is zero Perhaps an easier way to see that momentum is conserved for an isolated system is to consider Newton’s second law in terms of momentum, . For an isolated system, ; thus, , and is constant. We have noted that the three length dimensions in nature— , , and —are independent, and it is interesting to note that momentum can be conserved in different ways along each dimension. For example, during projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero and momentum is unchanged. But along the vertical direction, the net vertical force is not zero and the momentum of the projectile is not conserved. (See Figure 8.4.) However, if the momentum of the projectile-Earth system is considered in the vertical direction, we find that the total momentum is conserved. FIGURE 8.4 The horizontal component of a projectile’s momentum is conserved if air resistance is negligible, even in this case where a space probe separates. The forces causing the separation are internal to the system, so that the net external horizontal force is still zero. The vertical component of the momentum is not conserved, because the net vertical force is not zero. In the vertical direction, the space probe-Earth system needs to be considered and we find that the total momentum is conserved. The center of mass of the space probe takes the same path it would if the separation did not occur. The conservation of momentum principle can be applied to systems as different as a comet striking Earth and a gas containing huge numbers of atoms and molecules. Conservation of momentum is violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force. For example, in the collision of two cars considered above, the two-car system conserves momentum while each one-car system does not. Conservation of Momentum Principle 8.32 Isolated System An isolated system is defined to be one for which the net external force is zero 8.3 • Conservation of Momentum 331 Subatomic Collisions and Momentum The conservation of momentum principle not only applies to the macroscopic objects, it is also essential to our explorations of atomic and subatomic particles. Giant machines hurl subatomic particles at one another, and researchers evaluate the results by assuming conservation of momentum (among other things). On the small scale, we find that particles and their properties are invisible to the naked eye but can be measured with our instruments, and models of these subatomic particles can be constructed to describe the results. Momentum is found to be a property of all subatomic particles including massless particles such as photons that compose light. Momentum being a property of particles hints that momentum may have an identity beyond the description of an object’s mass multiplied by the object’s velocity. Indeed, momentum relates to wave properties and plays a fundamental role in what measurements are taken and how we take these measurements. Furthermore, we find that the conservation of momentum principle is valid when considering systems of particles. We use this principle to analyze the masses and other properties of previously undetected particles, such as the nucleus of an atom and the existence of quarks that make up particles of nuclei. Figure 8.5 below illustrates how a particle scattering backward from another implies that its target is massive and dense. Experiments seeking evidence that Making Connections: Take-Home Investigation—Drop of Tennis Ball and a Basketball Hold a tennis ball side by side and in contact with a basketball. Drop the balls together. (Be careful!) What happens? Explain your observations. Now hold the tennis ball above and in contact with the basketball. What happened? Explain your observations. What do you think will happen if the basketball ball is held above and in contact with the tennis ball? Making Connections: Take-Home Investigation—Two Tennis Balls in a Ballistic Trajectory Tie two tennis balls together with a string about a foot long. Hold one ball and let the other hang down and throw it in a ballistic trajectory. Explain your observations. Now mark the center of the string with bright ink or attach a brightly colored sticker to it and throw again. What happened? Explain your observations. Some aquatic animals such as jellyfish move around based on the principles of conservation of momentum. A jellyfish fills its umbrella section with water and then pushes the water out resulting in motion in the opposite direction to that of the jet of water. Squids propel themselves in a similar manner but, in contrast with jellyfish, are able to control the direction in which they move by aiming their nozzle forward or backward. Typical squids can move at speeds of 8 to 12 km/h. The ballistocardiograph (BCG) was a diagnostic tool used in the second half of the 20th century to study the strength of the heart. About once a second, your heart beats, forcing blood into the aorta. A force in the opposite direction is exerted on the rest of your body (recall Newton’s third law). A ballistocardiograph is a device that can measure this reaction force. This measurement is done by using a sensor (resting on the person) or by using a moving table suspended from the ceiling. This technique can gather information on the strength of the heart beat and the volume of blood passing from the heart. However, the electrocardiogram (ECG or EKG) and the echocardiogram (cardiac ECHO or ECHO; a technique that uses ultrasound to see an image of the heart) are more widely used in the practice of cardiology. Making Connections: Conservation of Momentum and Collision Conservation of momentum is quite useful in describing collisions. Momentum is crucial to our understanding of atomic and subatomic particles because much of what we know about these particles comes from collision experiments. 332 8 • Linear Momentum and Collisions Access for free at openstax.org quarks make up protons (one type of particle that makes up nuclei) scattered high-energy electrons off of protons (nuclei of hydrogen atoms). Electrons occasionally scattered straight backward in a manner that implied a very small and very dense particle makes up the proton—this observation is considered nearly direct evidence of quarks. The analysis was based partly on the same conservation of momentum principle that works so well on the large scale. FIGURE 8.5 A subatomic particle scatters straight backward from a target particle. In experiments seeking evidence for quarks, electrons were observed to occasionally scatter straight backward from a proton. 8.4 Elastic Collisions in One Dimension LEARNING OBJECTIVES By the end of this section, you will be able to: • Describe an elastic collision of two objects in one dimension. • Define internal kinetic energy. • Derive an expression for conservation of internal kinetic energy in a one dimensional collision. • Determine the final velocities in an elastic collision given masses and initial velocities. Let us consider various types of two-object collisions. These collisions are the easiest to analyze, and they illustrate many of the physical principles involved in collisions. The conservation of momentum principle is very useful here, and it can be used whenever the net external force on a system is zero. We start with the elastic collision of two objects moving along the same line—a one-dimensional problem. An elastic collision is one that also conserves internal kinetic energy. Internal kinetic energy is the sum of the kinetic energies of the objects in the system. Figure 8.6 illustrates an elastic collision in which internal kinetic energy and momentum are conserved. Truly elastic collisions can only be achieved with subatomic particles, such as electrons striking nuclei. Macroscopic collisions can be very nearly, but not quite, elastic—some kinetic energy is always converted into other forms of energy such as heat transfer due to friction and sound. One macroscopic collision that is nearly elastic is that of two steel blocks on ice. Another nearly elastic collision is that between two carts with spring bumpers on an air track. Icy surfaces and air tracks are nearly frictionless, more readily allowing nearly elastic collisions on them. Elastic Collision An elastic collision is one that conserves internal kinetic energy. Internal Kinetic Energy Internal kinetic energy is the sum of the kinetic energies of the objects in the system. 8.4 • Elastic Collisions in One Dimension 333 FIGURE 8.6 An elastic one-dimensional two-object collision. Momentum and internal kinetic energy are conserved. Now, to solve problems involving one-dimensional elastic collisions between two objects we can use the equations for conservation of momentum and conservation of internal kinetic energy. First, the equation for conservation of momentum for two objects in a one-dimensional collision is or where the primes (') indicate values after the collision. By definition, an elastic collision conserves internal kinetic energy, and so the sum of kinetic energies before the collision equals the sum after the collision. Thus, expresses the equation for conservation of internal kinetic energy in a one-dimensional collision. EXAMPLE 8.4 Calculating Velocities Following an Elastic Collision Calculate the velocities of two objects following an elastic collision, given that Strategy and Concept First, visualize what the initial conditions mean—a small object strikes a larger object that is initially at rest. This situation is slightly simpler than the situation shown in Figure 8.6 where both objects are initially moving. We are asked to find two unknowns (the final velocities and ). To find two unknowns, we must use two independent equations. Because this collision is elastic, we can use the above two equations. Both can be simplified by the fact that object 2 is initially at rest, and thus . Once we simplify these equations, we combine them algebraically to solve for the unknowns. 8.33 8.34 8.35 8.36 334 8 • Linear Momentum and Collisions Access for free at openstax.org Solution For this problem, note that and use conservation of momentum. Thus, or Using conservation of internal kinetic energy and that , Solving the first equation (momentum equation) for , we obtain Substituting this expression into the second equation (internal kinetic energy equation) eliminates the variable , leaving only as an unknown (the algebra is left as an exercise for the reader). There are two solutions to any quadratic equation; in this example, they are and As noted when quadratic equations were encountered in earlier chapters, both solutions may or may not be meaningful. In this case, the first solution is the same as the initial condition. The first solution thus represents the situation before the collision and is discarded. The second solution is negative, meaning that the first object bounces backward. When this negative value of is used to find the velocity of the second object after the collision, we get or Discussion The result of this example is intuitively reasonable. A small object strikes a larger one at rest and bounces backward. The larger one is knocked forward, but with a low speed. (This is like a compact car bouncing backward off a full-size SUV that is initially at rest.) As a check, try calculating the internal kinetic energy before and after the collision. You will see that the internal kinetic energy is unchanged at 4.00 J. Also check the total momentum before and after the collision; you will find it, too, is unchanged. The equations for conservation of momentum and internal kinetic energy as written above can be used to describe any one-dimensional elastic collision of two objects. These equations can be extended to more objects if needed. 8.37 8.38 8.39 8.40 8.41 8.42 8.43 8.44 Making Connections: Take-Home Investigation—Ice Cubes and Elastic Collision Find a few ice cubes which are about the same size and a smooth kitchen tabletop or a table with a glass top. Place the ice cubes on the surface several centimeters away from each other. Flick one ice cube toward a stationary ice cube and observe the path and velocities of the ice cubes after the collision. Try to avoid edge-on collisions and collisions with rotating ice cubes. Have you created approximately elastic collisions? Explain the speeds and directions of the ice cubes using momentum. 8.4 • Elastic Collisions in One Dimension 335 PHET EXPLORATIONS Collision Lab Investigate collisions on an air hockey table. Set up your own experiments: vary the number of discs, masses and initial conditions. Is momentum conserved? Is kinetic energy conserved? Vary the elasticity and see what happens. Click to view content ( 8.5 Inelastic Collisions in One Dimension LEARNING OBJECTIVES By the end of this section, you will be able to: • Define inelastic collision. • Explain perfectly inelastic collision. • Apply an understanding of collisions to sports. • Determine recoil velocity and loss in kinetic energy given mass and initial velocity. We have seen that in an elastic collision, internal kinetic energy is conserved. An inelastic collision is one in which the internal kinetic energy changes (it is not conserved). This lack of conservation means that the forces between colliding objects may remove or add internal kinetic energy. Work done by internal forces may change the forms of energy within a system. For inelastic collisions, such as when colliding objects stick together, this internal work may transform some internal kinetic energy into heat transfer. Or it may convert stored energy into internal kinetic energy, such as when exploding bolts separate a satellite from its launch vehicle. Figure 8.7 shows an example of an inelastic collision. Two objects that have equal masses head toward one another at equal speeds and then stick together. Their total internal kinetic energy is initially . The two objects come to rest after sticking together, conserving momentum. But the internal kinetic energy is zero after the collision. A collision in which the objects stick together is sometimes called a perfectly inelastic collision because it reduces internal kinetic energy more than does any other type of inelastic collision. In fact, such a collision reduces internal kinetic energy to the minimum it can have while still conserving momentum. FIGURE 8.7 An inelastic one-dimensional two-object collision. Momentum is conserved, but internal kinetic energy is not conserved. (a) Two objects of equal mass initially head directly toward one another at the same speed. (b) The objects stick together (a perfectly inelastic Inelastic Collision An inelastic collision is one in which the internal kinetic energy changes (it is not conserved). Perfectly Inelastic Collision A collision in which the objects stick together is sometimes called “perfectly inelastic.” 336 8 • Linear Momentum and Collisions Access for free at openstax.org collision), and so their final velocity is zero. The internal kinetic energy of the system changes in any inelastic collision and is reduced to zero in this example. EXAMPLE 8.5 Calculating Velocity and Change in Kinetic Energy: Inelastic Collision of a Puck and a Goalie (a) Find the recoil velocity of a 70.0-kg ice hockey goalie, originally at rest, who catches a 0.150-kg hockey puck slapped at him at a velocity of 35.0 m/s. (b) How much kinetic energy is lost during the collision? Assume friction between the ice and the puck-goalie system is negligible. (See Figure 8.8 ) FIGURE 8.8 An ice hockey goalie catches a hockey puck and recoils backward. The initial kinetic energy of the puck is almost entirely converted to thermal energy and sound in this inelastic collision. Strategy Momentum is conserved because the net external force on the puck-goalie system is zero. We can thus use conservation of momentum to find the final velocity of the puck and goalie system. Note that the initial velocity of the goalie is zero and that the final velocity of the puck and goalie are the same. Once the final velocity is found, the kinetic energies can be calculated before and after the collision and compared as requested. Solution for (a) Momentum is conserved because the net external force on the puck-goalie system is zero. Conservation of momentum is or Because the goalie is initially at rest, we know . Because the goalie catches the puck, the final velocities are equal, or . Thus, the conservation of momentum equation simplifies to Solving for yields Entering known values in this equation, we get Discussion for (a) This recoil velocity is small and in the same direction as the puck’s original velocity, as we might expect. 8.45 8.46 8.47 8.48 8.49 8.5 • Inelastic Collisions in One Dimension 337 Solution for (b) Before the collision, the internal kinetic energy of the system is that of the hockey puck, because the goalie is initially at rest. Therefore, is initially After the collision, the internal kinetic energy is The change in internal kinetic energy is thus where the minus sign indicates that the energy was lost. Discussion for (b) Nearly all of the initial internal kinetic energy is lost in this perfectly inelastic collision. is mostly converted to thermal energy and sound. During some collisions, the objects do not stick together and less of the internal kinetic energy is removed—such as happens in most automobile accidents. Alternatively, stored energy may be converted into internal kinetic energy during a collision. Figure 8.9 shows a one-dimensional example in which two carts on an air track collide, releasing potential energy from a compressed spring. Example 8.6 deals with data from such a collision. FIGURE 8.9 An air track is nearly frictionless, so that momentum is conserved. Motion is one-dimensional. In this collision, examined in Example 8.6, the potential energy of a compressed spring is released during the collision and is converted to internal kinetic energy. Collisions are particularly important in sports and the sporting and leisure industry utilizes elastic and inelastic collisions. Let us look briefly at tennis. Recall that in a collision, it is momentum and not force that is important. So, a heavier tennis racquet will have the advantage over a lighter one. This conclusion also holds true for other sports—a lightweight bat (such as a softball bat) cannot hit a hardball very far. 8.50 8.51 8.52 338 8 • Linear Momentum and Collisions Access for free at openstax.org The location of the impact of the tennis ball on the racquet is also important, as is the part of the stroke during which the impact occurs. A smooth motion results in the maximizing of the velocity of the ball after impact and reduces sports injuries such as tennis elbow. A tennis player tries to hit the ball on the “sweet spot” on the racquet, where the vibration and impact are minimized and the ball is able to be given more velocity. Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations. EXAMPLE 8.6 Calculating Final Velocity and Energy Release: Two Carts Collide In the collision pictured in Figure 8.9, two carts collide inelastically. Cart 1 (denoted carries a spring which is initially compressed. During the collision, the spring releases its potential energy and converts it to internal kinetic energy. The mass of cart 1 and the spring is 0.350 kg, and the cart and the spring together have an initial velocity of . Cart 2 (denoted in Figure 8.9) has a mass of 0.500 kg and an initial velocity of . After the collision, cart 1 is observed to recoil with a velocity of . (a) What is the final velocity of cart 2? (b) How much energy was released by the spring (assuming all of it was converted into internal kinetic energy)? Strategy We can use conservation of momentum to find the final velocity of cart 2, because (the track is frictionless and the force of the spring is internal). Once this velocity is determined, we can compare the internal kinetic energy before and after the collision to see how much energy was released by the spring. Solution for (a) As before, the equation for conservation of momentum in a two-object system is The only unknown in this equation is . Solving for and substituting known values into the previous equation yields Solution for (b) The internal kinetic energy before the collision is Take-Home Experiment—Bouncing of Tennis Ball 1. Find a racquet (a tennis, badminton, or other racquet will do). Place the racquet on the floor and stand on the handle. Drop a tennis ball on the strings from a measured height. Measure how high the ball bounces. Now ask a friend to hold the racquet firmly by the handle and drop a tennis ball from the same measured height above the racquet. Measure how high the ball bounces and observe what happens to your friend’s hand during the collision. Explain your observations and measurements. 2. The coefficient of restitution is a measure of the elasticity of a collision between a ball and an object, and is defined as the ratio of the speeds after and before the collision. A perfectly elastic collision has a of 1. For a ball bouncing off the floor (or a racquet on the floor), can be shown to be where is the height to which the ball bounces and is the height from which the ball is dropped. Determine for the cases in Part 1 and for the case of a tennis ball bouncing off a concrete or wooden floor ( for new tennis balls used on a tennis court). 8.53 8.54 8.5 • Inelastic Collisions in One Dimension 339 After the collision, the internal kinetic energy is The change in internal kinetic energy is thus Discussion The final velocity of cart 2 is large and positive, meaning that it is moving to the right after the collision. The internal kinetic energy in this collision increases by 5.46 J. That energy was released by the spring. 8.6 Collisions of Point Masses in Two Dimensions LEARNING OBJECTIVES By the end of this section, you will be able to: • Discuss two dimensional collisions as an extension of one dimensional analysis. • Define point masses. • Derive an expression for conservation of momentum along x-axis and y-axis. • Describe elastic collisions of two objects with equal mass. • Determine the magnitude and direction of the final velocity given initial velocity, and scattering angle. In the previous two sections, we considered only one-dimensional collisions; during such collisions, the incoming and outgoing velocities are all along the same line. But what about collisions, such as those between billiard balls, in which objects scatter to the side? These are two-dimensional collisions, and we shall see that their study is an extension of the one-dimensional analysis already presented. The approach taken (similar to the approach in discussing two-dimensional kinematics and dynamics) is to choose a convenient coordinate system and resolve the motion into components along perpendicular axes. Resolving the motion yields a pair of one-dimensional problems to be solved simultaneously. One complication arising in two-dimensional collisions is that the objects might rotate before or after their collision. For example, if two ice skaters hook arms as they pass by one another, they will spin in circles. We will not consider such rotation until later, and so for now we arrange things so that no rotation is possible. To avoid rotation, we consider only the scattering of point masses—that is, structureless particles that cannot rotate or spin. We start by assuming that , so that momentum is conserved. The simplest collision is one in which one of the particles is initially at rest. (See Figure 8.10.) The best choice for a coordinate system is one with an axis parallel to the velocity of the incoming particle, as shown in Figure 8.10. Because momentum is conserved, the components of momentum along the - and -axes will also be conserved, but with the chosen coordinate system, is initially zero and is the momentum of the incoming particle. Both facts simplify the analysis. (Even with the simplifying assumptions of point masses, one particle initially at rest, and a convenient coordinate system, we still gain new insights into nature from the analysis of two-dimensional collisions.) 8.55 8.56 8.57 340 8 • Linear Momentum and Collisions Access for free at openstax.org FIGURE 8.10 A two-dimensional collision with the coordinate system chosen so that is initially at rest and is parallel to the -axis. This coordinate system is sometimes called the laboratory coordinate system, because many scattering experiments have a target that is stationary in the laboratory, while particles are scattered from it to determine the particles that make-up the target and how they are bound together. The particles may not be observed directly, but their initial and final velocities are. Along the -axis, the equation for conservation of momentum is Where the subscripts denote the particles and axes and the primes denote the situation after the collision. In terms of masses and velocities, this equation is But because particle 2 is initially at rest, this equation becomes The components of the velocities along the -axis have the form . Because particle 1 initially moves along the -axis, we find . Conservation of momentum along the -axis gives the following equation: where and are as shown in Figure 8.10. Along the -axis, the equation for conservation of momentum is or But is zero, because particle 1 initially moves along the -axis. Because particle 2 is initially at rest, is also zero. The equation for conservation of momentum along the -axis becomes The components of the velocities along the -axis have the form . 8.58 ′ ′ 8.59 8.60 8.61 Conservation of Momentum along the -axis 8.62 8.63 8.64 8.65 8.6 • Collisions of Point Masses in Two Dimensions 341 Thus, conservation of momentum along the -axis gives the following equation: The equations of conservation of momentum along the -axis and -axis are very useful in analyzing two-dimensional collisions of particles, where one is originally stationary (a common laboratory situation). But two equations can only be used to find two unknowns, and so other data may be necessary when collision experiments are used to explore nature at the subatomic level. EXAMPLE 8.7 Determining the Final Velocity of an Unseen Object from the Scattering of Another Object Suppose the following experiment is performed. A 0.250-kg object is slid on a frictionless surface into a dark room, where it strikes an initially stationary object with mass of 0.400 kg . The 0.250-kg object emerges from the room at an angle of with its incoming direction. The speed of the 0.250-kg object is originally 2.00 m/s and is 1.50 m/s after the collision. Calculate the magnitude and direction of the velocity and of the 0.400-kg object after the collision. Strategy Momentum is conserved because the surface is frictionless. The coordinate system shown in Figure 8.11 is one in which is originally at rest and the initial velocity is parallel to the -axis, so that conservation of momentum along the - and -axes is applicable. Everything is known in these equations except and , which are precisely the quantities we wish to find. We can find two unknowns because we have two independent equations: the equations describing the conservation of momentum in the - and -directions. Solution Solving for and for and taking the ratio yields an equation (in which θ2 is the only unknown quantity. Applying the identity , we obtain: Entering known values into the previous equation gives Thus, Angles are defined as positive in the counter clockwise direction, so this angle indicates that is scattered to the right in Figure 8.11, as expected (this angle is in the fourth quadrant). Either equation for the - or -axis can now be used to solve for , but the latter equation is easiest because it has fewer terms. 8.66 Conservation of Momentum along the -axis 8.67 8.68 8.69 8.70 8.71 342 8 • Linear Momentum and Collisions Access for free at openstax.org Entering known values into this equation gives Thus, Discussion It is instructive to calculate the internal kinetic energy of this two-object system before and after the collision. (This calculation is left as an end-of-chapter problem.) If you do this calculation, you will find that the internal kinetic energy is less after the collision, and so the collision is inelastic. This type of result makes a physicist want to explore the system further. FIGURE 8.11 A collision taking place in a dark room is explored in Example 8.7. The incoming object is scattered by an initially stationary object. Only the stationary object’s mass is known. By measuring the angle and speed at which emerges from the room, it is possible to calculate the magnitude and direction of the initially stationary object’s velocity after the collision. Elastic Collisions of Two Objects with Equal Mass Some interesting situations arise when the two colliding objects have equal mass and the collision is elastic. This situation is nearly the case with colliding billiard balls, and precisely the case with some subatomic particle collisions. We can thus get a mental image of a collision of subatomic particles by thinking about billiards (or pool). (Refer to Figure 8.10 for masses and angles.) First, an elastic collision conserves internal kinetic energy. Again, let us assume object 2 is initially at rest. Then, the internal kinetic energy before and after the collision of two objects that have equal masses is Because the masses are equal, . Algebraic manipulation (left to the reader) of conservation of momentum in the - and -directions can show that (Remember that is negative here.) The two preceding equations can both be true only if 8.72 8.73 8.74 8.75 8.76 8.6 • Collisions of Point Masses in Two Dimensions 343 There are three ways that this term can be zero. They are • : head-on collision; incoming ball stops • : no collision; incoming ball continues unaffected • : angle of separation is after the collision All three of these ways are familiar occurrences in billiards and pool, although most of us try to avoid the second. If you play enough pool, you will notice that the angle between the balls is very close to after the collision, although it will vary from this value if a great deal of spin is placed on the ball. (Large spin carries in extra energy and a quantity called angular momentum, which must also be conserved.) The assumption that the scattering of billiard balls is elastic is reasonable based on the correctness of the three results it produces. This assumption also implies that, to a good approximation, momentum is conserved for the two-ball system in billiards and pool. The problems below explore these and other characteristics of two-dimensional collisions. 8.7 Introduction to Rocket Propulsion LEARNING OBJECTIVES By the end of this section, you will be able to: • State Newton’s third law of motion. • Explain the principle involved in propulsion of rockets and jet engines. • Derive an expression for the acceleration of the rocket and discuss the factors that affect the acceleration. • Describe the function of a space shuttle. Rockets range in size from fireworks so small that ordinary people use them to immense Saturn Vs that once propelled massive payloads toward the Moon. The propulsion of all rockets, jet engines, deflating balloons, and even squids and octopuses is explained by the same physical principle—Newton’s third law of motion. Matter is forcefully ejected from a system, producing an equal and opposite reaction on what remains. Another common example is the recoil of a gun. The gun exerts a force on a bullet to accelerate it and consequently experiences an equal and opposite force, causing the gun’s recoil or kick. Figure 8.12 shows a rocket accelerating straight up. In part (a), the rocket has a mass and a velocity relative to Earth, and hence a momentum . In part (b), a time has elapsed in which the rocket has ejected a mass of hot gas at a velocity relative to the rocket. The remainder of the mass now has a greater velocity . The momentum of the entire system (rocket plus expelled gas) has actually decreased because the force of gravity has acted for a time , producing a negative impulse . (Remember that impulse is the net external force on a system multiplied by the time it acts, and it equals the change in momentum of the system.) So, the center of mass of the system is in free fall but, by rapidly expelling mass, part of the system can accelerate upward. It is a commonly held misconception that the rocket exhaust pushes on the ground. If we consider thrust; that is, the force exerted on the rocket by the exhaust gases, then a rocket’s thrust is greater in outer space than in the atmosphere or on the launch pad. In fact, gases are easier to expel into a vacuum. By calculating the change in momentum for the entire system over , and equating this change to the impulse, the Connections to Nuclear and Particle Physics Two-dimensional collision experiments have revealed much of what we know about subatomic particles, as we shall see in Medical Applications of Nuclear Physics and Particle Physics. Ernest Rutherford, for example, discovered the nature of the atomic nucleus from such experiments. Making Connections: Take-Home Experiment—Propulsion of a Balloon Hold a balloon and fill it with air. Then, let the balloon go. In which direction does the air come out of the balloon and in which direction does the balloon get propelled? If you fill the balloon with water and then let the balloon go, does the balloon’s direction change? Explain your answer. 344 8 • Linear Momentum and Collisions Access for free at openstax.org following expression can be shown to be a good approximation for the acceleration of the rocket. “The rocket” is that part of the system remaining after the gas is ejected, and is the acceleration due to gravity. FIGURE 8.12 (a) This rocket has a mass and an upward velocity . The net external force on the system is , if air resistance is neglected. (b) A time later the system has two main parts, the ejected gas and the remainder of the rocket. The reaction force on the rocket is what overcomes the gravitational force and accelerates it upward. A rocket’s acceleration depends on three major factors, consistent with the equation for acceleration of a rocket . First, the greater the exhaust velocity of the gases relative to the rocket, , the greater the acceleration is. The practical limit for is about for conventional (non-nuclear) hot-gas propulsion systems. The second factor is the rate at which mass is ejected from the rocket. This is the factor in the equation. The quantity , with units of newtons, is called "thrust.” The faster the rocket burns its fuel, the greater its thrust, and the greater its acceleration. The third factor is the mass of the rocket. The smaller the mass is (all other factors being the same), the greater the acceleration. The rocket mass decreases dramatically during flight because most of the rocket is fuel to begin with, so that acceleration increases continuously, reaching a maximum just before the fuel is exhausted. 8.77 Acceleration of a Rocket Acceleration of a rocket is where is the acceleration of the rocket, is the exhaust velocity, is the mass of the rocket, is the mass of the ejected gas, and is the time in which the gas is ejected. 8.78 8.7 • Introduction to Rocket Propulsion 345 EXAMPLE 8.8 Calculating Acceleration: Initial Acceleration of a Moon Launch A Saturn V’s mass at liftoff was , its fuel-burn rate was , and the exhaust velocity was . Calculate its initial acceleration. Strategy This problem is a straightforward application of the expression for acceleration because is the unknown and all of the terms on the right side of the equation are given. Solution Substituting the given values into the equation for acceleration yields Discussion This value is fairly small, even for an initial acceleration. The acceleration does increase steadily as the rocket burns fuel, because decreases while and remain constant. Knowing this acceleration and the mass of the rocket, you can show that the thrust of the engines was . To achieve the high speeds needed to hop continents, obtain orbit, or escape Earth’s gravity altogether, the mass of the rocket other than fuel must be as small as possible. It can be shown that, in the absence of air resistance and neglecting gravity, the final velocity of a one-stage rocket initially at rest is where is the natural logarithm of the ratio of the initial mass of the rocket to what is left after all of the fuel is exhausted. (Note that is actually the change in velocity, so the equation can be used for any segment of the flight. If we start from rest, the change in velocity equals the final velocity.) For example, let us calculate the mass ratio needed to escape Earth’s gravity starting from rest, given that the escape velocity from Earth is about , and assuming an exhaust velocity . Solving for gives Thus, the mass of the rocket is Factors Affecting a Rocket’s Acceleration • The greater the exhaust velocity of the gases relative to the rocket, the greater the acceleration. • The faster the rocket burns its fuel, the greater its acceleration. • The smaller the rocket’s mass (all other factors being the same), the greater the acceleration. 8.79 8.80 8.81 8.82 8.83 346 8 • Linear Momentum and Collisions Access for free at openstax.org This result means that only of the mass is left when the fuel is burnt, and of the initial mass was fuel. Expressed as percentages, 98.9% of the rocket is fuel, while payload, engines, fuel tanks, and other components make up only 1.10%. Taking air resistance and gravitational force into account, the mass remaining can only be about . It is difficult to build a rocket in which the fuel has a mass 180 times everything else. The solution is multistage rockets. Each stage only needs to achieve part of the final velocity and is discarded after it burns its fuel. The result is that each successive stage can have smaller engines and more payload relative to its fuel. Once out of the atmosphere, the ratio of payload to fuel becomes more favorable, too. The space shuttle was an attempt at an economical vehicle with some reusable parts, such as the solid fuel boosters and the craft itself. (See Figure 8.13) The shuttle’s need to be operated by humans, however, made it at least as costly for launching satellites as expendable, unpiloted rockets. Ideally, the shuttle would only have been used when human activities were required for the success of a mission, such as the repair of the Hubble space telescope. Rockets with satellites can also be launched from airplanes. Using airplanes has the double advantage that the initial velocity is significantly above zero and a rocket can avoid most of the atmosphere’s resistance. FIGURE 8.13 The space shuttle had a number of reusable parts. Solid fuel boosters on either side were recovered and refueled after each flight, and the entire orbiter returned to Earth for use in subsequent flights. The large liquid fuel tank was expended. The space shuttle was a complex assemblage of technologies, employing both solid and liquid fuel and pioneering ceramic tiles as reentry heat shields. As a result, it permitted multiple launches as opposed to single-use rockets. (credit: NASA) 8.7 • Introduction to Rocket Propulsion 347 Glossary change in momentum the difference between the final and initial momentum; the mass times the change in velocity conservation of momentum principle when the net external force is zero, the total momentum of the system is conserved or constant elastic collision a collision that also conserves internal kinetic energy impulse the average net external force times the time it acts; equal to the change in momentum inelastic collision a collision in which internal kinetic energy is not conserved internal kinetic energy the sum of the kinetic energies of the objects in a system isolated system a system in which the net external force is zero linear momentum the product of mass and velocity perfectly inelastic collision a collision in which the colliding objects stick together point masses structureless particles with no rotation or spin quark fundamental constituent of matter and an elementary particle second law of motion physical law that states that the net external force equals the change in momentum of a system divided by the time over which it changes Section Summary 8.1 Linear Momentum and Force • Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity. • In symbols, linear momentum is defined to be where is the mass of the system and is its velocity. • The SI unit for momentum is . • Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes. • In symbols, Newton’s second law of motion is defined to be is the net external force, is the change in momentum, and is the change time. 8.2 Impulse • Impulse, or change in momentum, equals the average net external force multiplied by the time this force acts: • Forces are usually not constant over a period of time. 8.3 Conservation of Momentum • The conservation of momentum principle is written or is the initial total momentum and is the total momentum some time later. • An isolated system is defined to be one for which the net external force is zero • During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero. • Conservation of momentum applies only when the net external force is zero. • The conservation of momentum principle is valid when considering systems of particles. 8.4 Elastic Collisions in One Dimension • An elastic collision is one that conserves internal kinetic energy. • Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions. 8.5 Inelastic Collisions in One Dimension • An inelastic collision is one in which the internal kinetic energy changes (it is not conserved). • A collision in which the objects stick together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision. • Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations. 8.6 Collisions of Point Masses in Two Dimensions • The approach to two-dimensional collisions is to choose a convenient coordinate system and break 348 8 • Glossary Access for free at openstax.org the motion into components along perpendicular axes. Choose a coordinate system with the -axis parallel to the velocity of the incoming particle. • Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (the -axis), stated by and along the direction perpendicular to the initial direction (the -axis) stated by . • The internal kinetic before and after the collision of two objects that have equal masses is • Point masses are structureless particles that cannot spin. 8.7 Introduction to Rocket Propulsion • Newton’s third law of motion states that to every action, there is an equal and opposite reaction. • Acceleration of a rocket is . • A rocket’s acceleration depends on three main factors. They are 1. The greater the exhaust velocity of the gases, the greater the acceleration. 2. The faster the rocket burns its fuel, the greater its acceleration. 3. The smaller the rocket's mass, the greater the acceleration. Conceptual Questions 8.1 Linear Momentum and Force 1. An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy? 2. An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum? 3. Professional Application Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground. 4. How can a small force impart the same momentum to an object as a large force? 8.2 Impulse 5. Professional Application Explain in terms of impulse how padding reduces forces in a collision. State this in terms of a real example, such as the advantages of a carpeted vs. tile floor for a day care center. 6. While jumping on a trampoline, sometimes you land on your back and other times on your feet. In which case can you reach a greater height and why? 7. Professional Application Tennis racquets have “sweet spots.” If the ball hits a sweet spot then the player's arm is not jarred as much as it would be otherwise. Explain why this is the case. 8.3 Conservation of Momentum 8. Professional Application If you dive into water, you reach greater depths than if you do a belly flop. Explain this difference in depth using the concept of conservation of energy. Explain this difference in depth using what you have learned in this chapter. 9. Under what circumstances is momentum conserved? 10. Can momentum be conserved for a system if there are external forces acting on the system? If so, under what conditions? If not, why not? 11. Momentum for a system can be conserved in one direction while not being conserved in another. What is the angle between the directions? Give an example. 12. Professional Application Explain in terms of momentum and Newton’s laws how a car’s air resistance is due in part to the fact that it pushes air in its direction of motion. 13. Can objects in a system have momentum while the momentum of the system is zero? Explain your answer. 14. Must the total energy of a system be conserved whenever its momentum is conserved? Explain why or why not. 8.4 Elastic Collisions in One Dimension 15. What is an elastic collision? 8.5 Inelastic Collisions in One Dimension 16. What is an inelastic collision? What is a perfectly inelastic collision? 8 • Conceptual Questions 349 17. Mixed-pair ice skaters performing in a show are standing motionless at arms length just before starting a routine. They reach out, clasp hands, and pull themselves together by only using their arms. Assuming there is no friction between the blades of their skates and the ice, what is their velocity after their bodies meet? 18. A small pickup truck that has a camper shell slowly coasts toward a red light with negligible friction. Two dogs in the back of the truck are moving and making various inelastic collisions with each other and the walls. What is the effect of the dogs on the motion of the center of mass of the system (truck plus entire load)? What is their effect on the motion of the truck? 8.6 Collisions of Point Masses in Two Dimensions 19. Figure 8.14 shows a cube at rest and a small object heading toward it. (a) Describe the directions (angle ) at which the small object can emerge after colliding elastically with the cube. How does depend on , the so-called impact parameter? Ignore any effects that might be due to rotation after the collision, and assume that the cube is much more massive than the small object. (b) Answer the same questions if the small object instead collides with a massive sphere. FIGURE 8.14 A small object approaches a collision with a much more massive cube, after which its velocity has the direction . The angles at which the small object can be scattered are determined by the shape of the object it strikes and the impact parameter . 8.7 Introduction to Rocket Propulsion 20. Professional Application Suppose a fireworks shell explodes, breaking into three large pieces for which air resistance is negligible. How is the motion of the center of mass affected by the explosion? How would it be affected if the pieces experienced significantly more air resistance than the intact shell? 21. Professional Application During a visit to the International Space Station, an astronaut was positioned motionless in the center of the station, out of reach of any solid object on which he could exert a force. Suggest a method by which he could move himself away from this position, and explain the physics involved. 22. Professional Application It is possible for the velocity of a rocket to be greater than the exhaust velocity of the gases it ejects. When that is the case, the gas velocity and gas momentum are in the same direction as that of the rocket. How is the rocket still able to obtain thrust by ejecting the gases? Problems & Exercises 8.1 Linear Momentum and Force 1. (a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of . (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of . (c) What is the momentum of the 90.0-kg hunter running at after missing the elephant? 2. (a) What is the mass of a large ship that has a momentum of , when the ship is moving at a speed of (b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of . 350 8 • Problems & Exercises Access for free at openstax.org 3. (a) At what speed would a airplane have to fly to have a momentum of (the same as the ship’s momentum in the problem above)? (b) What is the plane’s momentum when it is taking off at a speed of ? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship. 4. (a) What is the momentum of a garbage truck that is and is moving at ? (b) At what speed would an 8.00-kg trash can have the same momentum as the truck? 5. A runaway train car that has a mass of 15,000 kg travels at a speed of down a track. Compute the time required for a force of 1500 N to bring the car to rest. 6. The mass of Earth is and its orbital radius is an average of . Calculate its linear momentum. 8.2 Impulse 7. A bullet is accelerated down the barrel of a gun by hot gases produced in the combustion of gun powder. What is the average force exerted on a 0.0300-kg bullet to accelerate it to a speed of 600.0 m/s in a time of 2.00 ms (milliseconds)? 8. Professional Application A car moving at 10.0 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seat belt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70.0 kg. 9. A person slaps her leg with her hand, bringing her hand to rest in 2.50 milliseconds from an initial speed of 4.00 m/s. (a) What is the average force exerted on the leg, taking the effective mass of the hand and forearm to be 1.50 kg? (b) Would the force be any different if the woman clapped her hands together at the same speed and brought them to rest in the same time? Explain why or why not. 10. Professional Application A professional boxer hits his opponent with a 1000-N horizontal blow that lasts for 0.150 s. (a) Calculate the impulse imparted by this blow. (b) What is the opponent’s final velocity, if his mass is 105 kg and he is motionless in midair when struck near his center of mass? (c) Calculate the recoil velocity of the opponent’s 10.0-kg head if hit in this manner, assuming the head does not initially transfer significant momentum to the boxer’s body. (d) Discuss the implications of your answers for parts (b) and (c). 11. Professional Application Suppose a child drives a bumper car head on into the side rail, which exerts a force of 4000 N on the car for 0.200 s. (a) What impulse is imparted by this force? (b) Find the final velocity of the bumper car if its initial velocity was 2.80 m/s and the car plus driver have a mass of 200 kg. You may neglect friction between the car and floor. 12. Professional Application One hazard of space travel is debris left by previous missions. There are several thousand objects orbiting Earth that are large enough to be detected by radar, but there are far greater numbers of very small objects, such as flakes of paint. Calculate the force exerted by a 0.100-mg chip of paint that strikes a spacecraft window at a relative speed of , given the collision lasts . 13. Professional Application A 75.0-kg person is riding in a car moving at 20.0 m/s when the car runs into a bridge abutment. (a) Calculate the average force on the person if he is stopped by a padded dashboard that compresses an average of 1.00 cm. (b) Calculate the average force on the person if he is stopped by an air bag that compresses an average of 15.0 cm. 14. Professional Application Military rifles have a mechanism for reducing the recoil forces of the gun on the person firing it. An internal part recoils over a relatively large distance and is stopped by damping mechanisms in the gun. The larger distance reduces the average force needed to stop the internal part. (a) Calculate the recoil velocity of a 1.00-kg plunger that directly interacts with a 0.0200-kg bullet fired at 600 m/s from the gun. (b) If this part is stopped over a distance of 20.0 cm, what average force is exerted upon it by the gun? (c) Compare this to the force exerted on the gun if the bullet is accelerated to its velocity in 10.0 ms (milliseconds). 8 • Problems & Exercises 351 15. A cruise ship with a mass of strikes a pier at a speed of 0.750 m/s. It comes to rest 6.00 m later, damaging the ship, the pier, and the tugboat captain’s finances. Calculate the average force exerted on the pier using the concept of impulse. (Hint: First calculate the time it took to bring the ship to rest.) 16. Calculate the final speed of a 110-kg rugby player who is initially running at 8.00 m/s but collides head-on with a padded goalpost and experiences a backward force of for . 17. Water from a fire hose is directed horizontally against a wall at a rate of 50.0 kg/s and a speed of 42.0 m/s. Calculate the magnitude of the force exerted on the wall, assuming the water’s horizontal momentum is reduced to zero. 18. A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes a nail and comes to rest after driving the nail 1.00 cm into a board. (a) Calculate the duration of the impact. (b) What was the average force exerted on the nail? 19. Starting with the definitions of momentum and kinetic energy, derive an equation for the kinetic energy of a particle expressed as a function of its momentum. 20. A ball with an initial velocity of 10 m/s moves at an angle above the -direction. The ball hits a vertical wall and bounces off so that it is moving above the -direction with the same speed. In terms of m, the mass of the ball, what is the impulse delivered by the wall? 21. When serving a tennis ball, a player hits the ball when its velocity is zero (at the highest point of a vertical toss). The racquet exerts a force of 540 N on the ball for 5.00 ms, giving it a final velocity of 45.0 m/s. Using these data, find the mass of the ball. 22. A punter drops a ball from rest vertically 1 meter down onto his foot. The ball leaves the foot with a speed of 18 m/s at an angle above the horizontal. In terms of m, the mass of the ball, what is the impulse delivered by the foot (magnitude and direction)? 8.3 Conservation of Momentum 23. Professional Application Train cars are coupled together by being bumped into one another. Suppose two loaded train cars are moving toward one another, the first having a mass of 150,000 kg and a velocity of 0.300 m/s, and the second having a mass of 110,000 kg and a velocity of . (The minus indicates direction of motion.) What is their final velocity? 24. Suppose a clay model of a koala bear has a mass of 0.200 kg and slides on ice at a speed of 0.750 m/s. It runs into another clay model, which is initially motionless and has a mass of 0.350 kg. Both being soft clay, they naturally stick together. What is their final velocity? 25. Professional Application Consider the following question: A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seatbelt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70 kg. Would the answer to this question be different if the car with the 70-kg passenger had collided with a car that has a mass equal to and is traveling in the opposite direction and at the same speed? Explain your answer. 26. What is the velocity of a 900-kg car initially moving at 30.0 m/s, just after it hits a 150-kg deer initially running at 12.0 m/s in the same direction? Assume the deer remains on the car. 27. A 1.80-kg falcon catches a 0.650-kg dove from behind in midair. What is their velocity after impact if the falcon’s velocity is initially 28.0 m/s and the dove’s velocity is 7.00 m/s in the same direction? 8.4 Elastic Collisions in One Dimension 28. Two identical objects (such as billiard balls) have a one-dimensional collision in which one is initially motionless. After the collision, the moving object is stationary and the other moves with the same speed as the other originally had. Show that both momentum and kinetic energy are conserved. 29. Professional Application Two piloted satellites approach one another at a relative speed of 0.250 m/s, intending to dock. The first has a mass of , and the second a mass of . If the two satellites collide elastically rather than dock, what is their final relative velocity? 352 8 • Problems & Exercises Access for free at openstax.org 30. A 70.0-kg ice hockey goalie, originally at rest, catches a 0.150-kg hockey puck slapped at him at a velocity of 35.0 m/s. Suppose the goalie and the ice puck have an elastic collision and the puck is reflected back in the direction from which it came. What would their final velocities be in this case? 8.5 Inelastic Collisions in One Dimension 31. A 0.240-kg billiard ball that is moving at 3.00 m/s strikes the bumper of a pool table and bounces straight back at 2.40 m/s (80% of its original speed). The collision lasts 0.0150 s. (a) Calculate the average force exerted on the ball by the bumper. (b) How much kinetic energy in joules is lost during the collision? (c) What percent of the original energy is left? 32. During an ice show, a 60.0-kg skater leaps into the air and is caught by an initially stationary 75.0-kg skater. (a) What is their final velocity assuming negligible friction and that the 60.0-kg skater’s original horizontal velocity is 4.00 m/s? (b) How much kinetic energy is lost? 33. Professional Application Using mass and speed data from Example 8.1 and assuming that the football player catches the ball with his feet off the ground with both of them moving horizontally, calculate: (a) the final velocity if the ball and player are going in the same direction and (b) the loss of kinetic energy in this case. (c) Repeat parts (a) and (b) for the situation in which the ball and the player are going in opposite directions. Might the loss of kinetic energy be related to how much it hurts to catch the pass? 34. A battleship that is and is originally at rest fires a 1100-kg artillery shell horizontally with a velocity of 575 m/s. (a) If the shell is fired straight aft (toward the rear of the ship), there will be negligible friction opposing the ship’s recoil. Calculate its recoil velocity. (b) Calculate the increase in internal kinetic energy (that is, for the ship and the shell). This energy is less than the energy released by the gun powder—significant heat transfer occurs. 35. Professional Application Two piloted satellites approaching one another, at a relative speed of 0.250 m/s, intending to dock. The first has a mass of , and the second a mass of . (a) Calculate the final velocity (after docking) by using the frame of reference in which the first satellite was originally at rest. (b) What is the loss of kinetic energy in this inelastic collision? (c) Repeat both parts by using the frame of reference in which the second satellite was originally at rest. Explain why the change in velocity is different in the two frames, whereas the change in kinetic energy is the same in both. 36. Professional Application A 30,000-kg freight car is coasting at 0.850 m/s with negligible friction under a hopper that dumps 110,000 kg of scrap metal into it. (a) What is the final velocity of the loaded freight car? (b) How much kinetic energy is lost? 37. Professional Application Space probes may be separated from their launchers by exploding bolts. (They bolt away from one another.) Suppose a 4800-kg satellite uses this method to separate from the 1500-kg remains of its launcher, and that 5000 J of kinetic energy is supplied to the two parts. What are their subsequent velocities using the frame of reference in which they were at rest before separation? 38. A 0.0250-kg bullet is accelerated from rest to a speed of 550 m/s in a 3.00-kg rifle. The pain of the rifle’s kick is much worse if you hold the gun loosely a few centimeters from your shoulder rather than holding it tightly against your shoulder. (a) Calculate the recoil velocity of the rifle if it is held loosely away from the shoulder. (b) How much kinetic energy does the rifle gain? (c) What is the recoil velocity if the rifle is held tightly against the shoulder, making the effective mass 28.0 kg? (d) How much kinetic energy is transferred to the rifle-shoulder combination? The pain is related to the amount of kinetic energy, which is significantly less in this latter situation. (e) Calculate the momentum of a 110-kg football player running at 8.00 m/s. Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s. Discuss its relationship to this problem. 8 • Problems & Exercises 353 39. Professional Application One of the waste products of a nuclear reactor is plutonium-239 . This nucleus is radioactive and decays by splitting into a helium-4 nucleus and a uranium-235 nucleus , the latter of which is also radioactive and will itself decay some time later. The energy emitted in the plutonium decay is and is entirely converted to kinetic energy of the helium and uranium nuclei. The mass of the helium nucleus is , while that of the uranium is (note that the ratio of the masses is 4 to 235). (a) Calculate the velocities of the two nuclei, assuming the plutonium nucleus is originally at rest. (b) How much kinetic energy does each nucleus carry away? Note that the data given here are accurate to three digits only. 40. Professional Application The Moon’s craters are remnants of meteorite collisions. Suppose a fairly large asteroid that has a mass of (about a kilometer across) strikes the Moon at a speed of 15.0 km/s. (a) At what speed does the Moon recoil after the perfectly inelastic collision (the mass of the Moon is ) ? (b) How much kinetic energy is lost in the collision? Such an event may have been observed by medieval English monks who reported observing a red glow and subsequent haze about the Moon. (c) In October 2009, NASA crashed a rocket into the Moon, and analyzed the plume produced by the impact. (Significant amounts of water were detected.) Answer part (a) and (b) for this real-life experiment. The mass of the rocket was 2000 kg and its speed upon impact was 9000 km/h. How does the plume produced alter these results? 41. Professional Application Two football players collide head-on in midair while trying to catch a thrown football. The first player is 95.0 kg and has an initial velocity of 6.00 m/s, while the second player is 115 kg and has an initial velocity of –3.50 m/s. What is their velocity just after impact if they cling together? 42. What is the speed of a garbage truck that is and is initially moving at 25.0 m/s just after it hits and adheres to a trash can that is 80.0 kg and is initially at rest? 43. During a circus act, an elderly performer thrills the crowd by catching a cannon ball shot at him. The cannon ball has a mass of 10.0 kg and the horizontal component of its velocity is 8.00 m/s when the 65.0-kg performer catches it. If the performer is on nearly frictionless roller skates, what is his recoil velocity? 44. (a) During an ice skating performance, an initially motionless 80.0-kg clown throws a fake barbell away. The clown’s ice skates allow her to recoil frictionlessly. If the clown recoils with a velocity of 0.500 m/s and the barbell is thrown with a velocity of 10.0 m/s, what is the mass of the barbell? (b) How much kinetic energy is gained by this maneuver? (c) Where does the kinetic energy come from? 8.6 Collisions of Point Masses in Two Dimensions 45. Two identical pucks collide on an air hockey table. One puck was originally at rest. (a) If the incoming puck has a speed of 6.00 m/s and scatters to an angle of ,what is the velocity (magnitude and direction) of the second puck? (You may use the result that for elastic collisions of objects that have identical masses.) (b) Confirm that the collision is elastic. 46. Confirm that the results of the example Example 8.7 do conserve momentum in both the - and -directions. 47. A 3000-kg cannon is mounted so that it can recoil only in the horizontal direction. (a) Calculate its recoil velocity when it fires a 15.0-kg shell at 480 m/s at an angle of above the horizontal. (b) What is the kinetic energy of the cannon? This energy is dissipated as heat transfer in shock absorbers that stop its recoil. (c) What happens to the vertical component of momentum that is imparted to the cannon when it is fired? 48. Professional Application A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg bowling pin, which is scattered at an angle of to the initial direction of the bowling ball and with a speed of 15.0 m/s. (a) Calculate the final velocity (magnitude and direction) of the bowling ball. (b) Is the collision elastic? (c) Linear kinetic energy is greater after the collision. Discuss how spin on the ball might be converted to linear kinetic energy in the collision. 354 8 • Problems & Exercises Access for free at openstax.org 49. Professional Application Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei from gold-197 nuclei . The energy of the incoming helium nucleus was , and the masses of the helium and gold nuclei were and , respectively (note that their mass ratio is 4 to 197). (a) If a helium nucleus scatters to an angle of during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus. (b) What is the final kinetic energy of the helium nucleus? 50. Professional Application Two cars collide at an icy intersection and stick together afterward. The first car has a mass of 1200 kg and is approaching at due south. The second car has a mass of 850 kg and is approaching at due west. (a) Calculate the final velocity (magnitude and direction) of the cars. (b) How much kinetic energy is lost in the collision? (This energy goes into deformation of the cars.) Note that because both cars have an initial velocity, you cannot use the equations for conservation of momentum along the -axis and -axis; instead, you must look for other simplifying aspects. 51. Starting with equations and for conservation of momentum in the - and -directions and assuming that one object is originally stationary, prove that for an elastic collision of two objects of equal masses, as discussed in the text. 52. Integrated Concepts A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away? 8.7 Introduction to Rocket Propulsion 53. Professional Application Antiballistic missiles (ABMs) are designed to have very large accelerations so that they may intercept fast-moving incoming missiles in the short time available. What is the takeoff acceleration of a 10,000-kg ABM that expels 196 kg of gas per second at an exhaust velocity of 54. Professional Application What is the acceleration of a 5000-kg rocket taking off from the Moon, where the acceleration due to gravity is only , if the rocket expels 8.00 kg of gas per second at an exhaust velocity of 55. Professional Application Calculate the increase in velocity of a 4000-kg space probe that expels 3500 kg of its mass at an exhaust velocity of . You may assume the gravitational force is negligible at the probe’s location. 56. Professional Application Ion-propulsion rockets have been proposed for use in space. They employ atomic ionization techniques and nuclear energy sources to produce extremely high exhaust velocities, perhaps as great as . These techniques allow a much more favorable payload-to-fuel ratio. To illustrate this fact: (a) Calculate the increase in velocity of a 20,000-kg space probe that expels only 40.0-kg of its mass at the given exhaust velocity. (b) These engines are usually designed to produce a very small thrust for a very long time—the type of engine that might be useful on a trip to the outer planets, for example. Calculate the acceleration of such an engine if it expels at the given velocity, assuming the acceleration due to gravity is negligible. 57. Derive the equation for the vertical acceleration of a rocket. 58. Professional Application (a) Calculate the maximum rate at which a rocket can expel gases if its acceleration cannot exceed seven times that of gravity. The mass of the rocket just as it runs out of fuel is 75,000-kg, and its exhaust velocity is . Assume that the acceleration of gravity is the same as on Earth’s surface . (b) Why might it be necessary to limit the acceleration of a rocket? 8 • Problems & Exercises 355 59. Given the following data for a fire extinguisher-toy wagon rocket experiment, calculate the average exhaust velocity of the gases expelled from the extinguisher. Starting from rest, the final velocity is 10.0 m/s. The total mass is initially 75.0 kg and is 70.0 kg after the extinguisher is fired. 60. How much of a single-stage rocket that is 100,000 kg can be anything but fuel if the rocket is to have a final speed of , given that it expels gases at an exhaust velocity of 61. Professional Application (a) A 5.00-kg squid initially at rest ejects 0.250-kg of fluid with a velocity of 10.0 m/s. What is the recoil velocity of the squid if the ejection is done in 0.100 s and there is a 5.00-N frictional force opposing the squid’s movement. (b) How much energy is lost to work done against friction? 62. Unreasonable Results Squids have been reported to jump from the ocean and travel (measured horizontally) before re-entering the water. (a) Calculate the initial speed of the squid if it leaves the water at an angle of , assuming negligible lift from the air and negligible air resistance. (b) The squid propels itself by squirting water. What fraction of its mass would it have to eject in order to achieve the speed found in the previous part? The water is ejected at ; gravitational force and friction are neglected. (c) What is unreasonable about the results? (d) Which premise is unreasonable, or which premises are inconsistent? 63. Construct Your Own Problem Consider an astronaut in deep space cut free from her space ship and needing to get back to it. The astronaut has a few packages that she can throw away to move herself toward the ship. Construct a problem in which you calculate the time it takes her to get back by throwing all the packages at one time compared to throwing them one at a time. Among the things to be considered are the masses involved, the force she can exert on the packages through some distance, and the distance to the ship. 64. Construct Your Own Problem Consider an artillery projectile striking armor plating. Construct a problem in which you find the force exerted by the projectile on the plate. Among the things to be considered are the mass and speed of the projectile and the distance over which its speed is reduced. Your instructor may also wish for you to consider the relative merits of depleted uranium versus lead projectiles based on the greater density of uranium. 356 8 • Problems & Exercises Access for free at openstax.org
7631	https://my.clevelandclinic.org/health/diseases/23957-clubbed-fingers	Locations: Abu Dhabi\|Canada\|Florida\|London\|Nevada\|Ohio\| Home/ Health Library/ Diseases & Conditions/ Clubbed Fingers AdvertisementAdvertisement Clubbed Fingers Digital clubbing, or changes in the nails on your fingers and toes, is mostly a symptom rather than a disease itself. Treating clubbed nails means treating the underlying disease. Advertisement Cleveland Clinic is a non-profit academic medical center. Advertising on our site helps support our mission. We do not endorse non-Cleveland Clinic products or services. Policy Care at Cleveland Clinic Respiratory Therapy Make an Appointment Advertisement Advertisement Advertisement Advertisement ContentsOverviewSymptoms and CausesDiagnosis and TestsManagement and TreatmentOutlook / PrognosisPreventionLiving With Overview What are clubbed fingers? Clubbed fingers refers to the way the ends of your fingers look, including your nails and the areas around and under them. These appearances can happen with your toes as well. If clubbing happens, it usually affects both hands and/or both feet. Some people refer to clubbing of the fingers and toes as digital clubbing. (“Digits” is another name for your fingers and toes.) Advertisement Cleveland Clinic is a non-profit academic medical center. Advertising on our site helps support our mission. We do not endorse non-Cleveland Clinic products or services. Policy What’s the difference between clubbed fingers and fingers that aren’t clubbed? The tips of your fingers or toes bulge and may become warm and/or discolored. Finally, the nail curves down so that it eventually looks like an upside-down spoon. It may look like your nails are floating and aren’t really attached to your finger. Who can get clubbed fingers? Digital clubbing can happen to anyone. In most cases, it’s a symptom of another condition, but it can be idiopathic. Idiopathic means that there's no cause. Digital clubbing can also be congenital (something you’re born with) or hereditary. Clubbing often indicates problems with your lungs, heart or digestive system. Clubbing usually happens because of long-lasting (chronic) low levels of oxygen in your blood, known as hypoxemia. Symptoms and Causes What are the signs and symptoms of clubbing of fingers and toes? The signs and symptoms of clubbing of fingers and toes involve changes in the way your fingers and toes look. Your nails may look like they aren’t attached anymore. The angle where your nail meets the cuticle gets wider. Your nail may begin to look like a hill. (This situation is seen best from the side — in profile.) The tips become wide and round. They might also darken in color or become warm to the touch. The tips of your nails curve downward. What causes clubbing of fingers and toes? It may be that your toes and fingers are clubbed and there isn’t a disease causing the clubbing. If that’s true, the clubbing isn’t harmful. However, in most cases, your healthcare providers will check to make sure that there’s no other disease involved. Advertisement Many diseases related to clubbing involve your heart, lungs and digestive system. These conditions include: Cancers related to digital clubbing Esophageal cancer. Gastrointestinal tumors. Liver cancer. Lung cancer. Lymphoma. Mesothelioma: A rare cancer that asbestos exposure causes. Heart and lung conditions other than cancer related to digital clubbing Aortic aneurysm: A bulging in the wall of your body’s largest artery. Bronchiectasis: Scarred airways lead to coughing up mucus. Congenital heart disease: A structural issue with your heart that’s present at birth. Cystic fibrosis: An inherited disease that causes sticky mucus to build up in organs. Empyema: A collection of pus in a body cavity, most often your pleural cavity. Endocarditis: A bloodstream infection attacks the lining of your heart valves. Lung abscesses: A collection of pus in your lung that’s surrounded by swollen lung tissue. Idiopathic pulmonary fibrosis: Scarred and thickened lung tissue causes lung damage. Sarcoidosis: An inflammatory disease that may change the structure or function of organs, such as your lungs. Other conditions related to digital clubbing Ascariasis. A roundworm infection of your intestines. Celiac disease. An autoimmune disease that can damage your small intestine. Cirrhosis, which is often a result of alcoholism. A liver disease in which scar tissue replaces healthy tissue. Inflammatory bowel disease. An inflammatory disease that damages your intestines. Thyroid disease. Conditions that stop your thyroid gland from working correctly. Overuse of certain drugs, including laxatives, interferon alfa-2A and prostaglandin infusion. Is clubbing of fingers and toes contagious? No, clubbed fingers aren’t contagious. Clubbed fingers may not even be a disease itself. It may be a sign or symptom of another disease. However, clubbed fingers can be a sign of a disease that's infectious. Diagnosis and Tests How will your provider diagnose clubbed fingers? Your healthcare provider will start with taking a medical history and doing a physical examination. There will often be clues in your history that will suggest why you have clubbing. Your provider may measure the Lovibond angle or the Lovibond profile sign. Your provider will look at your finger from the side and measure the angle between your proximal nail fold (the part near the cuticle) and your nail bed. This angle on a clubbed nail is larger than 180 degrees and on a non-clubbed nail less than 180 degrees. They may also ask you to press one finger against the same finger at the joint so that the nails are facing each other. Fingers without clubbing form a space between them shaped like a diamond, known as the Schamroth sign. Your provider may also look at the angle measurement involving the skin under the nail at your nail tip, the nail bed, the area near the cuticle and the closest joint on your finger. This measurement, the hyponychial angle, refers to the skin under your nail, called the hyponychium. Advertisement Then your provider will order tests that will help to find out which disease is causing your clubbed fingers and/or toes. The tests will be different depending on the suspected cause. Tests may include a chest X-ray. If the X-ray doesn’t show anything wrong, your provider may order a CT scan. Other tests will be specific to the disease your provider suspects you may have. They might include other imaging tests, blood tests or biopsies. Management and Treatment How are clubbed fingers treated? Your provider won’t really treat clubbed fingers. Your provider will treat the disease that causes the clubbing. Care at Cleveland Clinic Respiratory Therapy Make an Appointment Outlook / Prognosis What can I expect if I have clubbed fingers? The outlook for someone with clubbed fingers varies. Treating the disease that causes clubbing may make the clubbing go away. Prevention How can I reduce my risk of developing clubbed fingers? You can’t prevent being born with clubbed fingers or toes. You may be able to reduce your risk of developing diseases that may lead to digital clubbing. For instance, you may reduce your chances of getting certain lung and heart diseases by not smoking. You may reduce your chances of developing cirrhosis by not drinking alcohol to excess. Living With When should I see my healthcare provider? If you develop any symptoms related to nail changes, contact your healthcare provider. If your provider has diagnosed a disease related to the nail clubbing, follow their treatment suggestions. In any case, always discuss anything that worries you with your provider. Advertisement A note from Cleveland Clinic You know your body best. If you’re born with clubbed fingers and toes, you won’t have to treat them. They won’t cause you problems. If your fingers and toes begin to change, make an appointment with your provider so they can assess the changes. Get the information you need. You’re the most important factor in getting and keeping well. Your provider is your trusted partner. Advertisement Care at Cleveland Clinic Breathing issues can affect your life in many ways. Cleveland Clinic’s respiratory therapy program treats your symptoms so you can feel better day to day. Respiratory Therapy Make an Appointment Medically Reviewed Last reviewed on 08/04/2022. Learn more about the Health Library and our editorial process. 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7632	https://brainly.in/question/11681543	Explain the structure of ethylene and acetylene - Brainly.in Mandalapuprasad Mandalapuprasad 07.08.2019 Chemistry Secondary School answered • expert verified Explain the structure of ethylene and acetylene See answers Advertisement Advertisement Brainly User Brainly User Acetylene (systematic name: ethyne) is the chemical compound with the formula C2H2. It is a hydrocarbon and the simplest alkyne. This colorless gas is widely used as a fuel and a chemical building block. It is unstable in its pure form and thus is usually handled as a solution. Pure acetylene is odorless, but commercial grades usually have a marked odor due to impurities.As an alkyne, acetylene is unsaturated because its two carbon atoms are bonded together in a triple bond. The carbon–carbon triple bond places all four atoms in the same straight line, with CCH bond angles of 180°. Ethylene (IUPAC name: ethene) is a hydrocarbon which has the formula C2H4 or H2C=CH2. It is a colorless flammable gas with a faint "sweet and musky" odour when pure. It is the simplest alkene (a hydrocarbon with carbon-carbon double bonds).Ethylene is widely used in the chemical industry, and its worldwide production (over 150 million tonnes in 2016) exceeds that of any other organic compound.Much of this production goes toward polyethylene, a widely used plastic containing polymer chains of ethylene units in various chain lengths. Ethylene is also an important natural plant hormone and is used in agriculture to force the ripening of fruits. Ethylene's hydrate is ethanol. Hope it helps you . Please mark it as Brainliest . Namaste . #answerwithquality #BAL Advertisement Advertisement Anushka398765 Anushka398765 Ethylene and acetylene are hydrocarbons. They are very different in their chemical and physical properties. Ethylene can be found naturally in crude oil and natural gas; it is also found in plants as a plant hormone that causes the ripening of fruits. Acetylene is an alkyne. It is a linear molecule and is highly flammable. Therefore, it is used as a fuel. Acetylene is mainly produced by thermal cracking processes in refineries. The main difference between ethylene and acetylene is that ethylene is an alkene whereas acetylene is an alkyne. Ethylene is the simplest alkene having the chemical formula H2C=CH2. It has two carbon atoms bonded to each other via a double bond. It is a colorless, flammable gas. The IUPAC name of ethylene is ethane. The molar mass of this compound is 28.05 g/mol. Its melting point is −169.2 °C and boiling point is −103.7 °C. Acetylene is the simplest alkyne having the chemical formula C2H2. It contains two carbon atoms bonded to each other via a triple bond. There are two pi bonds and one sigma bond between the carbon atoms. Each carbon atom is bonded to a hydrogen atom via a single bond. The molecule is planar, and the geometry around one carbon atom is linear. Advertisement Advertisement New questions in Chemistry what is the national game of England?HOW TO PREPARE 20 ML SOLUTION OF 50 % SILVER NITRATE SOLUTION FOR USE IN ENT PROCEDURE. ?C6H5-CH=CH2 + HBr → ?Write the structural formula of 3- ethyl-2, 5-dimethyl hexan-3-olAnti iron gel how to prepare for neede rotation 90 degree PreviousNext We're in the know Company Careers Advertise with us Terms of Use Copyright Policy Privacy Policy Cookie Preferences Community Brainly Community Brainly for Schools & Teachers Brainly for Parents Honor Code Community Guidelines Insights: The Brainly Blog Become a Volunteer Help Signup Help Center Safety Center Responsible Disclosure Agreement Get the Brainly App Download iOS App Download Android App Brainly.in PL: Brainly.plRU: Znanija.comES: Brainly.latPT: Brainly.com.brFR: Nosdevoirs.frTR: Eodev.comRO: Brainly.roID: Brainly.co.idPH: Brainly.phUS: Brainly.com
7633	https://brainly.com/question/38953822	[FREE] In how many ways can 12 people be arranged in a line if there are 2 people who do not want to stand - brainly.com 3 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 +60,7k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +21,6k Ace exams faster, with practice that adapts to you Practice Worksheets +5,6k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified In how many ways can 12 people be arranged in a line if there are 2 people who do not want to stand together? 1 See answer Explain with Learning Companion NEW Asked by TylerRayAdams3883 • 10/02/2023 0:00 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 3801945 people 3M 0.0 0 Upload your school material for a more relevant answer The arrangements of 12 people where two specific people don't stand together is computed as total arrangements (12!) minus the scenarios where these two are together, i.e., 12! - (11! x 2!). This is a problem of permutations with restrictions. Explanation The question deals with permutations or the arrangement of a number of items in a specific order. In this case, the total number of permutations for 12 people standing in a line is 12!. But we need to subtract the permutations where two specific people are standing together. Treat these two people as a single entity, we then have 11 entities (10 individuals plus the pair), which can be arranged in 11! ways. The pair of individuals could themselves be arranged in 2! ways. Hence total arrangements treating the pair as a single entity would be 11! x 2!. So, the total number of arrangements with the mentioned pair not together is 12! - (11! x 2!). This is a classic example of problems involving permutations and restrictions. Learn more about Permutations here: brainly.com/question/23283166 SPJ11 Answered by VinyaTh78 •15.4K answers•3.8M people helped Thanks 0 0.0 (0 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 3801945 people 3M 0.0 0 Concept Development Studies in Chemistry - John S. Hutchinson Thermodynamics and Statistical Mechanics - V. Parameswaran Nair Thermodynamics and Chemical Equilibrium - Paul Ellgen Upload your school material for a more relevant answer The number of ways to arrange 12 people in a line, ensuring that 2 specific people do not stand together, is 10×11!, which results from subtracting the arrangements where those two are together from the total arrangements. This involves using factorial calculations and treating the two people as a single unit when they are together. Ultimately, the result gives us an effective method to solve such arrangement problems with restrictions. Explanation To find the number of ways to arrange 12 people in a line, with the condition that 2 specific people do not want to stand together, we can use the principle of counting. Calculate Total Arrangements: The total number of arrangements for 12 people without any restrictions is given by the factorial of 12, denoted as 12!. Calculate Arrangements with Restriction: Now, we need to calculate the number of arrangements where these two specific people stand together. To do this, we treat the two people as a single unit (or block). Thus, instead of 12 separate people, we now have 11 blocks to arrange: the block of 2 people and the other 10 individual people. This gives us 11! arrangements. Since the two people in the block can also switch places, we multiply by 2! (the number of arrangements of the 2 people in the block). So, the number of arrangements where the two specific people are together is 11!×2!. Final Calculation: Therefore, the number of arrangements where the two people do not stand together is: 12!−(11!×2!) =12!−2×11! =12!−2(11!) Simplifying the Expression: We can further simplify: 12!=12×11! So, 12!−2(11!)=12×11!−2×11!=(12−2)×11!=10×11! The number of ways to arrange the 12 people such that the two specific individuals do not stand together is 10×11!. Examples & Evidence For instance, if we had 12 people labeled A through L, and we restricted A and B from being together, we would calculate the total arrangements of A to L and then subtract the arrangements where A and B are together. This approach can be extended to scenarios with more specific restrictions or different group sizes. The methods and formulas used in this explanation are standard approaches in combinatorics, particularly in calculating permutations with restrictions based on the principles of counting. Thanks 0 0.0 (0 votes) Advertisement TylerRayAdams3883 has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Mathematics solutions and answers Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four? Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish? Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit? Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. Community Answer 4 Click an Item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Express In simplified exponential notation. 18a^3b^2/ 2ab New questions in Mathematics Simplify. 9 13+7 13 Given that the point (8,3) lies on the graph of g(x)=lo g 2x, which point lies on the graph of f(x)=lo g 2(x+3)+2 ? A. (5,1) B. (5,5) C. (11,1) D. (11,5) For what value of c is the function one-to-one?{(1,2),(2,3),(3,5),(4,7),(5,11),(6,c)}◯ 2◯ 5◯ 11◯ 13 Two cars raced at a race track. The faster car traveled 20 mph faster than the slower car. In the time that the slower car traveled 165 miles, the faster car traveled 225 miles. If the speeds of the cars remained constant, how fast did the slower car travel during the race? \| \| Distance (mi) \| Rate (mph) \| Time (h) \| :--- :--- \| \| Slower Car \| 165 \| r \| r 165 \| \| Faster Car \| 225 \| r+20 \| r+20 225 \| A. 55 mph B. 60 mph C. 75 mph D. 130 mph 10 x 3+4 x 3−3 x 2+9 x 2 Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
7634	https://jenniferfindley.com/help-students-word-problems-upper-elementary/	8 Ways to Help Students Be Successful with Word Problems in Upper Elementary - Teaching with Jennifer Findley Skip to main content Why Settle? Get EVERYTHING you need for reading and math . All-Access Login Blog Freebie Library Shop About Search this website Menu Teaching with Jennifer Findley Upper Elementary Teaching Blog Math Reading Word Study Writing & Grammar Holidays Test Prep Science & Social Studies Math Reading Digital Resources Holidays Test Prep Science & Social Studies Search this website 8 Ways to Help Students Be Successful with Word Problems in Upper Elementary Word problems can be really difficult for upper elementary students. The complexity of the word problems increase, the length of the problems themselves increase, the numbers are larger and more abstract, and often key words just don’t work. Today I want to share with you eight ways to help students who struggle with word problems in upper elementary grades. 1. Carefully choose or create an acronym to scaffold or support students. My first tip is to choose an acronym that will either help your students understand and think through a math problem or choose an acronym that helps ensure their work and answer is clear and complete. I personally prefer to use an acronym that both helps the students think through a problem and adequately show all of their work and organize their thoughts. With this in mind, I created the word problem acronym LOVE about four years ago and have been using it every sense. LOVE helps the students organize their work, their thoughts, while helping ensure the students provide enough to support their answer. The steps embedded in the process also help the students make sense of the word problem as they are forced to use models, labels, etc. —> Read more about using LOVE to support your students with word problems and grab several freebies by clicking here. Grab the bookmarks shown in the picture above by clicking HERE. 2. Teach students situations (and not key words) to help them solve word problems. This was a game-changer for my students. About five years, I stopped teaching key words and moved to teaching situations. We look at common situations and which operation is used in those situations. I even provide a chart for my students to refer to and add to as needed. When my students are solving a word problem, they refer to the situation chart to help them determine which operation is needed. This helps build their conceptual understand of math operations and helps them master word problems. Read more about how I teach my students to solve word problems without relying on key words (and grab the below poster and a few practice problems that I use) by clicking here. 3. When introducing a new math concept, introduce it through a story or word problem. When I introduce a new math concept, I always introduce it through the context of a story problem or situation. This helps the students conceptualize the math, see the real-world connection, and to better be able to think through the new math. —> Read about how I do this to introduce partial quotients division by clicking here. 4. Provide regular exposure to a wide variety of word problems and lots of opportunities to practice. I will admit that one mistake I made as a first year teacher was teaching computation skills for the first part of the year and then cramming word problems in before our state assessment. I failed to regularly embed them in our regular practice and homework. Now, I have created a ton of word problems in many engaging and purposeful formats so this never happens again. My students love our interactive notebook word problems and I love the word problem of the day routine that we do each day. Click here to read more about word problem of the day and grab FREE starter packs for 4th and 5th grade word problems. 5. Connect word problems to your students’ lives and experiences whenever possible. When I write word problems for my own classroom, I always use my students’ names and locations and experiences which which they are familiar. This helps for a few reasons: They are immediately engaged in the word problem when they see a familiar name, location, or experience. The chances of them remembering the situation (to apply it to new situations) are higher. It will be easier for you to refer back to previous word problems to help scaffold students when they encounter new or more difficult word problems. Tip: I use my already created word problems(or purchased word problems) and just change the names or locations when I write them on anchor charts. 6. Create or use silly word problems. This is very similar to my strategy from above. In addition to using familiar names and experiences, I love using silly word problems with my students. I have used cartoon characters or word problems that are just silly- think zombies and other science fiction topics. Here is an example of some themed zombie (nothing scary or assuming they are real) word problems that we used to review multiplication and division. Click here to grab this FREE zombie-themed word problem choice board. 7. Provide opportunities in the classroom for regular sharing and discussion of word problems and strategies to solve them. We do whole class word problems or math tasks on a regular basis in my classroom. Sometimes we solve them together as a class. More often, the students solve the word problems independently and then we share and discuss strategies used. Once a week, I try set aside 15-20 minutes specifically for sharing ideas and strategies on a given word problem. Read more about how I do this by clicking here. 8. Provide necessary support and scaffolds as needed to help your students succeed. Sometimes we need to offer a more support for our students solving word problems. There a few ways that I do this in my classroom. Step-by-Step Organizers – I use word problems that have a step-by-step organizer to help the students think through the problem and organize their work. The picture below shows an example of this using my 5th Grade Intervention Word Problems (click herefor 4th grade). Click here for FREE word problem step-by-step organizers that work with any word problem. Sentence Stems – I also help support my students by providing sentence stems for them to use when writing the answer to a word problem (and justify their answer- read more here). I have found that restating the question for the students helps them understand what is being asked. The picture below is an example of how I do this using my 5th Grade Differentiated Multi-Part Math Word Problems. Hopefully those strategies and tips will give you some ideas to help you tackle word problems with your upper elementary students. Do you have any other strategies you use to help your students? Save Save Save Save Save Save Save Save Save Save Save Save Save Save Save Save Save Save Save Share the Knowledge! Click to share on Facebook (Opens in new window)Facebook Click to share on X (Opens in new window)X Click to share on Pinterest (Opens in new window)6K+Pinterest 6K+ Click to print (Opens in new window)Print Click to email a link to a friend (Opens in new window)Email 7 Comments \| Math, Decimals, Fractions, Multiplication & Division, Word Problems Reader Interactions 7 Comments Maria Fikaris says October 5, 2016 at 7:47 am You are AMAZING! I am a first year teacher and I’m so happy I found your site. Thank you so much!! reply to comment 2. TippacanOe says October 2, 2017 at 12:49 am Awesome ideas!!!! Thanks for sharing. reply to comment 3. Linda Stump says July 2, 2018 at 2:05 pm I was doing some research for a grad class, and I found your site! You must be an amazing teacher. I will be using many of your free things. I also purchased your 5th grade word problems. Thanks for your willingness to share for the sake of kids! reply to comment 4. Stephanie says October 25, 2018 at 11:52 am Wow! Thank you so much for all of this! You have some amazing strategies. reply to comment 5. Lana says January 10, 2019 at 6:31 am Wow! Thank you so much for sharing! reply to comment 6. Yulie says April 8, 2019 at 10:35 am This is a great post! Such useful and helpful info on teaching students word problems! Thank you reply to comment 7. Jekamiah Ruiz says August 24, 2019 at 8:04 pm You are so generous with your amazing ideas. Thank you so much. reply to comment Leave a Comment Cancel reply Your email address will not be published.Required fields are marked Comment Name Email [x] Notify me of follow-up comments by email. [x] Notify me of new posts by email. You may also love these freebies! Download Math Posters Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris elementum, felis id sollicitudin dignissim, nisl diam maximus quam, ut facilisis orci lorem eu dui. Download Reading Posters Grab free reading posters for key 4th and 5th grade reading skills to help you teach and review your standards. Access Morphology Posters Download Grammar Posters Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris elementum, felis id sollicitudin dignissim, nisl diam maximus quam, ut facilisis orci lorem eu dui. Welcome Friends! I’m Jennifer Findley: a teacher, mother, and avid reader. I believe that with the right resources, mindset, and strategies, all students can achieve at high levels and learn to love learning. My goal is to provide resources and strategies to inspire you and help make this belief a reality for your students. Contact MeLearn More Links Home Blog Blog Reading Math Writing & Grammar All Posts Shop Literacy Math Holidays & Seasonal Science All Resources Freebies Math Posters (4th) Math Posters (5th) Reading Posters Grammar Posters Morphology Posters Copyright © 2025 Jennifer Findley • All rights reserved • Site Design by Emily White Designs TopGDPR Cookie Consent with Real Cookie Banner
7635	https://en.wikipedia.org/wiki/Complex_data_type	Jump to content Complex data type العربية 日本語 Polski Русский Tiếng Việt 中文 Edit links From Wikipedia, the free encyclopedia For the concept sometimes also called "complex data type", see composite data type. Some programming languages provide a complex data type for complex number storage and arithmetic as a built-in (primitive) data type. Complex-number arithmetic [edit] \| \| \| --- \| \| \| This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2023) (Learn how and when to remove this message) \| A complex variable or value is usually represented as a pair of floating-point numbers. Languages that support a complex data type usually provide special syntax for building such values, and extend the basic arithmetic operations ('+', '−', '×', '÷') to act on them. These operations are usually translated by the compiler into a sequence of floating-point machine instructions or into library calls. Those languages may also provide support for other operations, such as formatting, equality testing, etc. As in mathematics, those languages often interpret a floating-point value as equivalent to a complex value with a zero imaginary part. Language support [edit] \| \| \| --- \| \| \| This section needs expansion. You can help by adding to it. (November 2015) \| The FORTRAN COMPLEX type. The C99 standard of the C programming language includes complex data types and complex-math functions in the standard library header <complex.h>. The C++ standard library provides a complex template class as well as complex-math functions in the <complex> header. The Go programming language has built-in types complex64 (each component is 32-bit float) and complex128 (each component is 64-bit float). Imaginary number literals can be specified by appending an "i". The Perl core module Math::Complex provides support for complex numbers. Python provides the built-in complex type. Imaginary number literals can be specified by appending a "j". Complex-math functions are provided in the standard library module cmath. Ruby provides a Complex class in the standard library module complex. OCaml supports complex numbers with the standard library module Complex. Haskell supports complex numbers with the standard library module Data.Complex (previously called Complex). Mercury provides complex numbers with full operator overloading support in the extras distribution, using libcomplex_numbers. Java does not have a standard complex number class, but there exist a number of incompatible free implementations of a complex number class: The Apache Commons Math library provides complex numbers for Java with its Complex class. The JScience library has a Complex number class. The JAS library allows the use of complex numbers. Netlib has a complex number class for Java. javafastcomplex also adds complex number support for Java. jcomplexnumber is a project on implementation of complex number in Java. JLinAlg includes complex numbers with arbitrary precision. Common Lisp: The ANSI Common Lisp standard supports complex numbers of floats, rationals and arbitrary precision integers. Its basic mathematical functions are defined for complex numbers, where applicable. For example the square root of −1 is a complex number: ``` ? (sqrt -1) C(0 1) ; the result of (sqrt -1) ``` Scheme: Complex numbers and functions (e.g. sin) are included in the language specification. Their implementation is however optional in the R5RS standard, while in R6RS is mandatory. The .NET Framework provides System.Numerics.Complex since version 4.0. The smart BASIC for iOS naturally supports complex numbers in notation a + bi. Any variable, math operation or function can accept both real and complex numbers as arguments and return real or complex numbers depending on result. For example the square root of −4 is a complex number: ``` PRINT SQRT(-4) 2i `` Julia "Julia (programming language)") includes predefined types for both complex and rational numbers since at least version 0.3. R "R (programming language)") provides acomplex` basic data type. Rust does not provide a built-in complex data type, but multiple crates, that add support for such types, exist. Swift does not have a standard complex number data type, but Swift Numerics which is maintained by the language maintainers offers ComplexModule. Integration in programming [edit] Many programming languages provide built-in support or standard libraries for complex data types, enabling direct manipulation of complex numbers in code. These integrations typically define arithmetic operations, comparison rules, and input/output formatting specific to complex numbers. For example, in Python, the complex type allows arithmetic with complex literals and supports functions from the cmath module. Similarly, Haskell includes the Data.Complex module, offering complex arithmetic with real and imaginary parts represented as floating-point numbers. Such integration simplifies scientific and engineering computations that require complex number calculations. References [edit] ^ A guide to Fortran IV programming Daniel D. McCracken - 1972 - 288 pages. "The capability provided by Fortran complex operations is a great savings in programming effort in certain problems. " ^ Python v2.6.5 documentation ^ "Complex and Rational Numbers — Julia Language 0.3.13-pre documentation". ^ "R Data Types". www.w3schools.com. Retrieved 2022-04-26. ^ "Built-in Types: complex". Python documentation. Retrieved 2025-06-14. ^ "Haskell 2010 Language Report: Data.Complex module". Haskell.org. Retrieved 2025-06-14. \| v t e Data types \| \| --- \| \| Uninterpreted \| Bit Byte Trit Tryte Word Bit array \| \| Numeric \| Arbitrary-precision or bignum Complex Decimal Fixed point Block floating point Floating point + Reduced precision - Minifloat - Half precision - bfloat16 + Single precision + Double precision + Quadruple precision + Octuple precision + Extended precision - Long double Integer + signedness Interval Rational \| \| Pointer \| Address + physical + virtual Reference \| \| Text \| Character String + null-terminated \| \| Composite \| Algebraic data type + generalized Array Associative array Class Dependent Equality Inductive Intersection List Object + metaobject Option type Product Record or Struct Refinement Set Union + tagged \| \| Other \| Any type Boolean Bottom type Collection Enumerated type Exception Function type Opaque data type Recursive data type Semaphore Stream Strongly typed identifier Type class Empty type Unit type Void \| \| Related topics \| Abstract data type Boxing Data structure Generic Kind + metaclass Parametric polymorphism Primitive data type Interface Subtyping Type constructor Type conversion Type system Type theory Variable \| Retrieved from " Category: Data types Hidden categories: Articles needing additional references from June 2023 All articles needing additional references Articles to be expanded from November 2015 All articles to be expanded
7636	https://arxiv.org/pdf/2303.11446	Published Time: Wed, 08 Jan 2025 01:23:59 GMT arXiv:2303.11446v4 [math.MG] 7 Jan 2025 THE TORUS OF TRIANGLES ERIC BRUSSEL AND MADELEINE E. GOERTZ CALIFORNIA POLYTECHNIC STATE UNIVERSITY SAN LUIS OBISPO, USA Abstract. We prove the 2-torus T, an abelian linear algebraic group, is a fine moduli space of labeled, oriented, possibly-degenerate inscribable similarity classes of triangles, where a triangle is inscribable if it can be inscribed in a circle. A natural action by the dihedral group D6 defines a quotient stack [T/D6], which is the stack of absolute (unlabeled, unoriented) possibly-degenerate inscribable classes. We show the main tri-angle types form distinguished algebraic substructures: subgroups, cosets, and elements of small order, and we apply the natural metric on T to compare them. Contents Introduction 22. Definitions 33. The Torus 44. Parameterizing Inscribable Classes 54.2. Families and Degenerates 64.5. A Noncontractible Loop of Classes 84.6. Missing Degenerates and the Sphere 10 5. Group-Theoretic Interpretation of Triangle Types 10 6. Ratios 11 7. Group Action 12 8. Moduli Space and Quotient Stack 12 References 14 2020 Mathematics Subject Classification. 14C05, 51M05, 60D05. This research was generously supported by the William and Linda Frost Fund in the Cal Poly Bailey College of Science and Mathematics. 12 E. BRUSSEL AND M. E. GOERTZ Introduction In this paper we construct a compactification of the moduli space of labeled, oriented similarity classes of triangles in the plane that is isomorphic to a 2-torus T, an abelian linear algebraic group. A moduli space , or shape space , is a continuum of points representing members of a set of mathematical objects-of-interest, in this case similarity classes of triangles. It is useful for studying how objects belong to families, for visualizing different types, and for making statistical calculations. Often a space of general objects-of-interest is not hard to construct, but must be compactified using degenerate objects in order to be useful. Our main result is based on the parameterization of classes by triples of angles, which naturally restricts the degenerate classes in the compactification to those we call inscribable . Thus we submit a fine moduli space of labeled, oriented, possibly-degenerate inscribable triangle classes. One common model of labeled nondegenerate triangle classes, often called the triangle of tri-angles , consists of ordered triples of positive (interior) angles adding to π in R3, see e.g. [ES15, Figure 2]. Our paper was motivated by the observation that in the triangle-of-triangles model, the triangles in certain continuous “Poncelet” families inscribed in a circle appeared to turn over, or reverse orientation, as they crossed the degenerate border. By incorporating orientation-reversal across this border, we found that the triangle of triangles and its oppositely oriented counterpart glued together to form a torus T, equipped with a natural abelian algebraic group structure. We called T, of course, the torus of triangles .Since the torus is a group, we have a pairing, whereby two triangles combine to form a third. Consequently our space satisfies the uniformity criterion in [Por94], and addressed in [CNSS19]. Moreover, the standard distinguished triangle types have distinguished group-theoretic significance. The natural metric on the torus allows us to compute ratios of different types: We find that the obtuse to acute ratio is (3 ∶ 1), while the obtuse-isosceles to acute-isosceles ratio is (1 ∶ 1). The isosceles to right ratio is (2Ó3 ∶ 1).The shape space T admits an action by the dihedral group D6 (12 elements), and the orbit space is the (coarse) moduli space of absolute (unlabeled, unoriented) possibly-degenerate inscribable classes. We thus obtain a stack of possibly-degenerate inscribable classes as the quotient [T/D6].1.0.1. Literature. Though the torus is a natural and simple compactification of the space of labeled, oriented, nondegenerate classes, we have not found it as such in the literature, where the most commonly named shape space is a sphere. The sphere is featured in [Ken84], [Iwa87], [Beh14], [Mon15], [ES15], [CNSS19], and others. The principal difference between the torus and the sphere is that the latter is simply connected, while the former has “nontrivial loops”, which are continuous families of triangles that can’t be evolved into a single triangle shape. We give an example in Example 4.5.1. The problem of constructing a moduli space of triangles is frequently posed as an elementary exercise that demonstrates the properties and challenges of more complex moduli spaces, after a remark attributed to M. Artin. Along these lines, Behrend uses it in [Beh14] to introduce the theory of algebraic stacks. But the idea of constructing a space of triangles goes way back, and was suggested in the 19C by Lewis Carroll, who in “Pillow Problem 58” asked for the probability that THE TORUS OF TRIANGLES 3 a randomly chosen triangle is obtuse (see [Dod58]). As pointed out in [CNSS19], the first mention of it in the literature appears to be in The Lady’s and Gentleman’s Diary [Woo61], where W.S.B. Woolhouse said: “In a given circle a regular polygon is inscribed, and lines are drawn from each of its angles to the center of the circle. Required the ratio of the number of the triangles which are acute-angled to that of those which are obtuse-angled.” In [Por94] Portnoy suggests a moduli space of triangles should admit a transitive action by a compact group, in order that the various classes carry equal priority, and different regions can be assigned finite measures (since the group is compact). The “right” distribution should align with the spherically symmetrical construction P5(R) using the six coordinates of the triangle’s three vertices (up to scalar multiplication). This idea is used in [ES15]. Since our construction actually is a compact group, it admits a transitive action by a compact group, and so passes Portnoy’s test, but we include degenerate types that do not come from P5(R). Some of our measurements agree with those of [Por94], though our measure itself appears to be different, as noted in [ES15, 1.1]. Our work intersects deep results on compactifications of configuration spaces of n-tuples of points on a manifold or variety, appearing in [FM94], [AS94], and [Sin04]. These papers treat the problem of representing degenerate configurations of points – which in the n = 3 case represent degenerate triangles – essentially as we do: by appending a tangent direction to the multiple point. In [FM94] and [AS94] this is achieved by blowing up at the degenerate locus; in [Sin04] it is more explicitly combinatorial and applies more generally to smooth manifolds. We thank the referee for pointing out these important papers. Acknowledgements. We thank Sean Gasiorek for sparking their interest in spaces of triangles at a Cal Poly colloquium on the subject of triangle billiards and Poncelet’s Porism. 2. Definitions A triangle is the geometric figure obtained by connecting three points in the plane with straight lines. In order to make a proper space of triangles we need to accommodate degenerate examples, which accounts for some of the technicalities in the following definition. Definition 2.1. A triangle is a plane figure consisting of three vertices, three straight edges con-necting them, and three interior angles. We label the vertices (A, B, C ) ∈ C3, the side vectors (a, b, c) = (C − B, A − C, B − A) ∈ C3, and the interior angles (α, β, γ ) ∈ (−π, π ]3, where (a) (α, β, γ ) ≠ (0, 0, 0), α + β + γ ∈ {−π, 0, π }, and if αβγ ≠ 0 then α, β, γ have the same sign. (b) Let Arg (z), the principal argument of a nonzero complex number z, be the unique real number ξ in the interval (−π, π ] such that z = ∣z∣eiξ . Then • α = Arg (−b/c) when bc ≠ 0, • β = Arg (−c/a) when ca ≠ 0, • γ = Arg (−a/b) when ab ≠ 0.(c) A triangle is degenerate , or collinear , if the vertices are collinear, and otherwise nondegenerate .(d) A nondegenerate triangle has positive orientation if the directed graph defined by its side vectors (a, b, c) goes counterclockwise in the plane, and negative orientation if it goes clockwise. Equiv-alently, it has positive orientation if its interior angles are positive, and negative orientation if 4 E. BRUSSEL AND M. E. GOERTZ they are negative. A degenerate triangle has zero orientation .(2.1.1) CAB bcaβαγ Positively Oriented CAB acb αβγ Negatively Oriented (e) A triangle is inscribable if it can be inscribed in a circle, and when all vertices are identified then αβγ = 0.(f) Two labeled, oriented, possibly-degenerate inscribable triangles are similar if their triples of vertices differ by a nonzero complex number and their interior angles are equal (mod π). An (inscribable) triangle class , or just class , is the resulting equivalence class. Write ∆ = {(( A, B, C ); (α, β, γ ))} ⊂ C3 × (−π, π ]3 for the set of all labeled, oriented, possibly-degenerate triangles, ∆insc for the inscribable subset, and [∆insc ] for the set of inscribable classes. Remarks 2.1.1. The definition of similarity in (f) for inscribable triangles agrees with the standard one when all vertices are distinct, but admits a range of (degenerate) classes where two or three vertices coincide. These extra classes, which we call double and triple points , are forced by our compactification construction. By (a) and (b), a double point class with A = C ≠ B has the form [( A, B, A ); (α, 0, −α)] , where α is any value in (−π, π ]. We will show they are meaningfully depicted with a direction at the doubled vertex: (2.1.2) B A = C Double Point Triple point classes like [( A, A, A ); (α, β, γ )] with αβγ = 0 in (e) are limits of sequences in [∆insc ].For example, (( A, A, A ); (π, 0, 0)) is the limit as ε goes to 0 of the sequence with vertices (eεi, e 2εi, 1).Similarly, the limit as B approaches A = C above gives a point (( A, A, A ); (α, 0, −α)) in ∆insc , a “tripled double point”. 3. The Torus First we construct the space that will parameterize the inscribable classes. The quotient map R3 ⟶ (R/π)3 ≡ R/πZ × R/πZ × R/πZ is a smooth covering of abelian Lie groups by [Lee13, Example 21.14]. In particular (R/π)3 is a smooth manifold, the 3-torus. Let T ⊂ (R/π)3 be the image of the linear subspace L ∶ X +Y +Z = 0 of R3. Then T is a Lie subgroup by [Lee13, Theorem 21.27], with the induced pointwise additive composition law. Taking representatives of the coordinates on L in the fundamental domain [0, π )THE TORUS OF TRIANGLES 5 allows us to visualize T as the intersection of a solid cube of side-length π with the parallel planes k 3 (π, π, π ) + L (k = 0, 1, 2): (3.0.1) X Y Z (π, 0,0) (0, π, 0) (0,0, π ) T ⊂ (R/π)3 X+Y+Z=−π(mod 3π) X+Y+Z=π(mod 3π) The boundaries of the triangular regions, where one angle is 0(mod π), are identified on opposite faces of the cube, and all 8 vertices, where all angles are 0(mod π), are identified to a single point. As a set we have T = t( α, β, γ ) ∈ (R/π)3 ∶ α + β + γ = 0(mod π)z Lemma 3.1. T ⊂ (R/π)3 is naturally isomorphic to a 2-torus. Proof. The identifications of (R/π)3 shows the two triangles of T form a standard torus polygon: (3.1.1) T ∶ Corollary 3.2. T is an abelian linear algebraic group. In particular it is an algebraic variety, and contains closed subgroups corresponding to the linear subspaces in (3.0.1) .Proof. Since T is a compact Lie group, it is a linear algebraic group by [OV90, Ch.5, 2.5, Theorem 12]. By the corollary to that theorem there is a one-to-one correspondence between Lie subgroups and closed algebraic subgroups, and we obtain the second statement since every linear subspace of R3 is a Lie subgroup. Parameterizing Inscribable Classes The following theorem effectively shows T is the moduli space, or shape space, of labeled, ori-ented, possibly degenerate inscribable triangle classes. 6 E. BRUSSEL AND M. E. GOERTZ Theorem 4.1. T is naturally isomorphic to the set [∆insc ] of labeled, oriented, possibly-degenerate inscribable classes. More precisely, the natural map ρ ∶ [∆insc ] ⟶ T ⊂ (R/π)3 [(( A, B, C ); (α, β, γ ))] ⟼ (α, β, γ )(mod π) is a bijection, taking positively oriented classes to the subset X + Y + Z = π(mod 3π) of (R/π)3,negatively oriented classes to X + Y + Z = −π(mod 3π), and degenerate classes to the boundary αβγ = 0(mod π) in (3.0.1) .Proof. The theorem says that a labeled, oriented, possibly-degenerate inscribable class is completely determined by its triple of angles (mod π); that every triple (mod π) arises from some class; and that the two shaded triangular regions in (3.0.1) correspond to the two possible orientations. All three angles of a nondegenerate class belong to either (0, π ) or (−π, 0), depending on orien-tation, so its image is in the interior αβγ ≠ 0 of T, and is uniquely determined by its angle values (mod π) together with its orientation. If the class is positively oriented then α + β + γ = π(mod 3π), so the image is in the yellow region in (3.0.1), and if negatively oriented α + β + γ = −π(mod 3π), in the gray region. Therefore ρ distinguishes orientation, and is injective on nondegenerate classes. A degenerate class with γ = 0 has form [( A, A, C ); (α, −α, 0)] with α ∈ (−π, π ], and maps to the identified faces of the cube (3.0.1) parallel to the XY -plane. Suppose ρ([( A, A, C ); (α, −α, 0)]) = ρ([( A′, A ′, C ′); (α′, −α′, 0)]) . Since there are only two, the vertices automatically differ by a nonzero complex constant. Since α′ = α ± π, the interior angles are equal (mod π). Therefore the triangles are similar by Definition 2.1(f). Clearly this applies to the double point degenerates α = 0 and β = 0 as well, which go to the faces of the cube parallel to the Y Z and XZ -planes, respectively. Thus ρ remains injective if we include these classes. The solitary triple point degenerate class is [( 1, 1, 1); (π, 0, 0)] maps to (0, 0, 0)(mod π), and since this point has not been taken, we conclude ρ is injective in general. If α + β + γ = π(mod 3π) and αβγ ≠ 0(mod π) in T then there exist real representatives in (0, π ) adding to π, and these are the angles of a positively oriented triangle, by Euclid. Similarly if the sum is −π we find a negatively oriented triangle. Therefore ρ is surjective on the interior regions of T, dividing positively and negatively oriented classes as above. If γ = 0 and α ≠ 0(mod π) then (α, β, γ ) ∈ T is in the image of the class [( A, A, C ); (α, −α, 0)] , and similarly if α = 0 or β = 0. We conclude ρ is surjective, hence bijective, and divides classes as claimed. Remark 4.1.1. We prove more rigorously in Corollary 8.3 that T is a fine moduli space; of labeled, oriented, possibly-degenerate inscribable classes. As the closure of the interior αβγ ≠ 0, it then represents a compactification of the space of labeled, oriented nondegenerate classes. 4.2. Families and Degenerates. We next invert the correspondence ρ in Theorem 4.1 to give an explicit family of inscribed triangles corresponding to the points of T. This will help explain how the degenerate classes in [∆insc ], which are classes with one or more zero interior angle, arise naturally as limit points of sequences of nondegenerates. First, a lemma from Euclid. Lemma 4.3. For points B and C on a circle, let Arc (BC ) be the oriented (counterclockwise) arc from B to C, in radians. Suppose (( A, B, C ); (α, β, γ )) is nondegenerate and inscribed in a circle. THE TORUS OF TRIANGLES 7 Then α = 1 2 Arc (BC )(mod π), and α equals the oriented angle from the tangent line at B to the chord BC , or equivalently from BC to the tangent at C.Proof. The positively oriented case α = 1 2 Arc (BC ) is Proposition 20 of Euclid’s Book III, which states, In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base ([Euc56]). For negatively oriented α′ < 0 we have α′ = − 1 2 Arc (CB ) = 1 2 (Arc (BC ) − 2π) = 1 2 (Arc (BC ) − π), so again α′ = 1 2 Arc (BC )(mod π), as desired. Diagram (4.3.1) shows how to compute the opposing angle from the tangent at B to the chord BC using similar triangles M OB ∼ M BI . The last statement is immediate. (4.3.1) OBCMIAA′ α′=α−παααα−π The next result gives an inscribed family over T. Theorem 4.4. The map ρ of Theorem 4.1 has inverse σ ∶ T ⟶ ∆insc (mod π) ⟼ ( e2βi, e −2αi, 1); (α, β, γ ) if (α, β, γ ) ≠ (0, 0, 0)(mod π) (0, 0, 0)(mod π) ⟼ [( 1, 1, 1); (π, 0, 0)] where the interior angles in the image are real representatives in (−π, π ] adding to ±π, of the same sign when αβγ ≠ 0.Proof. The map σ as stated clearly inverts ρ if it is well-defined, i.e., if real representatives (α, β, γ ) can be chosen as stated, and then if the resulting vertex-angle pairings are actually triangle classes. Since the choice of interior angles exists and is unique by Figure 3.0.1, it remains to check the configuration of vertices is compatible with the interior angles. Assume αβγ ≠ 0. Then σ is illustrated below in Figure 4.4.1, with the oriented arcs computed using the vertices’ arguments: (4.4.1) 2β 2α βαC = 1 B = e−2αi A = e2βi −2α −2β −α −βC = 1 A = e2βi B = e−2αi Relation Between Interior Angles and Arcs 8 E. BRUSSEL AND M. E. GOERTZ The interior angles and corresponding arcs in (4.4.1) are compatible by Lemma 4.3, hence the angles appear in exponents at the given vertices, and the map is well-defined on αβγ ≠ 0.Now suppose αβγ = 0. By Definition 2.1(a) we know at least one angle is nonzero. If (α, β, 0) ∈ T and α ≠ 0(mod π), then β = −α, so σ(α, β, γ ) = [( e−2αi, e −2αi, 1); (α, −α, 0)] which is well-defined. Similarly if α = 0 and β ≠ 0(mod π), or β = 0 and α ≠ 0(mod π), we find the image of σ is [( e2βi, 1, 1); (0, β, −β)] and [( 1, e −2αi, 1); (α, 0, −α)] , respectively, both well-defined. Finally the remaining point (0, 0, 0) ∈ T maps to [( 1, 1, 1); (π, 0, 0)] , as desired. We conclude σ is well-defined, hence it inverts ρ. Remark 4.4.1. The map σ gives inscribed triangles over T naturally representing every class in [∆insc ]. The existence of such a family is an important feature of a (fine) moduli space, a sort of testimonial to the fine-ness. Though T itself tautologically provides a family of classes via ρ,Theorem 4.4 goes further by specifying an explicit set of triangle representatives. We define families more rigorously in Section 8. 4.5. A Noncontractible Loop of Classes. Example 4.5.1. We use σ to explicitly write down a noncontractible loop of triangle classes, and at the same time show how the degenerate double point classes arise in inscribed families, in this case defined by a fixed chord BC .Fix α0 ∈ (0, π ). Then the set {( α0, β, −(α0 + β)) (mod π) ∈ T} is a “circle” of points on T, both positively and negatively oriented, parameterized by β ∈ R/π:(4.5.1) (α0, β, −(α0 + β)) (α0, −α0, 0) (α0, 0, −α0) (α0, 0, −α0) This circle of triangles is a noncontractible loop on the torus, which means that the triangles cannot be evolved continuously to a single triangle shape. On a torus embedded in R3 it could be a loop around the tubular part. To see what triangles we are dealing with we apply σ to obtain the family {[( e2βi, e −2α0i, 1); (α0 − π, β, −(α0 + β))] ∶ β ∈ [−α0, 0]} ⊂ [∆insc ] if negatively oriented {[( e2βi, e −2α0i, 1); (α0, β, π − (α0 + β))] ∶ β ∈ [0, π − α0]} ⊂ [∆insc ] if positively oriented THE TORUS OF TRIANGLES 9 The chord BC is fixed, and various β locate A anywhere on the unit circle: (4.5.2) α0 CBA(β) Fixed α = α0(mod π)The values β = 0 and β = −α0(mod π) field the degenerate double points [( 1, e −2α0 i, 1); (α0, 0, −α0)] = [( C, B, C ); (α0, 0, −α0)] [( e−2α0i, e −2α0 i, 1); (α0, π − α0, 0)] = [( B, B, C ); (α0, π − α0, 0)] marked on Figure 4.5.1. Both are represented by the segment BC , but with a different doubled vertex, either A = B or A = C. Since α0 can be any value in (0, π ), in this way each degenerate inscribable class corresponds to a specific chord on the unit circle with a double point at one end or the other, appearing in a continuous family of nondegenerate inscribable classes. By the second part of Lemma 4.3 the degenerate inscribable class [( C, B, C ); (α0, 0, −α0)] above can be drawn as (4.5.3) BC = A(0) α0 Different degenerate inscribable classes can be distinguished geometrically using directions from the double point: (4.5.4) B C = A Inscribable Degenerates The point (0, 0, 0) ∈ T uniquely corresponds to the triple point degenerate class [( 1, 1, 1); (π, 0, 0)] .Thus we distinguish three degenerate class types: (4.5.5) triple point right double point general double point Degenerate Inscribable Triangle Types 10 E. BRUSSEL AND M. E. GOERTZ 4.6. Missing Degenerates and the Sphere. The point (0, 0, 0)(mod π) ∈ T conceals the entire set of (degenerate) collinear un inscribable classes, whose angles are all 0(mod π), and which are classified by the ratios of their (nonzero) side lengths ∣a∣, ∣b∣, ∣c∣:(4.6.1) AB C Collinear Degenerates Whenever the shape space of triangle classes is said to be a sphere, as in [Ken84], [Mon15], [Beh14], and [ES15], the degenerate classes are of this type. They are used in [Beh14, 1.1.10] to construct a different compactification of labeled, oriented, nondegenerate classes, called a bipyramid . The con-struction is similar to ours, describing two triangular regions and gluing them along the degenerate (collinear) boundaries. The bipyramid is a topological sphere, ([Beh14, Figure 1.21]), later recast as the Riemann sphere. Since neither the torus nor the sphere incorporates all degenerate configurations of three vertices, neither gives a complete compactification of the space of labeled, oriented, nondegenerate triangle classes. And there are more: We have also omitted triple points of the form (( 1, 1, 1); (α, β, γ )) with αβγ ≠ 0, which are realized as limits of the nondegenerate triangles in a fixed class as the side vectors are scaled to zero while the angles are held constant. We exclude these because they do not arise as limit points in sequences of nondegenerate inscribable triangles. In [BGGL24] we show that a compactification exists, called Dyck’s surface , which takes all “first order” degenerate configurations into account, and projects to both the sphere and the torus. More precisely, it is the blowup of the sphere at three points (“pinched points” in [Beh14]), or the torus at a single point (the degenerate triple point), and thus it subsumes both constructions. 5. Group-Theoretic Interpretation of Triangle Types In this section we show the group structure of T is compatible with triangle types, in the sense that the latter form basic algebraic structures: elements of finite order, subgroups, and cosets. By construction (Section 3) the composition law is addition of triples of angles (mod π). The identity is (0, 0, 0), the degenerate triple point. The main triangle types are illustrated in Figure 5.0.1. (5.0.1) = degenerate triple point ◦ = equilateral yellow = positively oriented gray = negatively oriented dotted = isosceles dashed = right solid = degenerate Distinguished Classes of T:THE TORUS OF TRIANGLES 11 Let IX , DX , RX for X = A, B, C denote the points of T corresponding to isosceles, degenerate, and right triangles, with distinguished vertex X. The standard triangle classes have the following algebraic interpretations. (a) Each IX ≤ T is a subgroup isomorphic to R/π, forming a noncontractible loop. They are: IA = {( −2β, β, β )} , IB = {( α, −2α, α )} , IC = {( α, α, −2α)} (b) Each DX ≤ T is a subgroup isomorphic to R/π. They are: DA = {( 0, β, −β)} , DB = {( α, 0, −α)} , DC = {( α, −α, 0)} (c) Each RX ⊂ T is a coset of DX , forming a noncontractible loop. RX consists of the elements of order two (mod DX ): RA = ( π 2 , π 4 , π 4 ) + DA = {( π 2 , π 4 β, π 4 − β)} RB = ( π 4 , π 2 , π 4 ) + DB = {( π 4 α, π 2 , π 4 − α)} RC = ( π 4 , π 4 , π 2 ) + DC = {( π 4 α, π 4 − α, π 2 )} (d) The positive and negative equilateral classes are inverses, each of order 3: ±( π 3 , π 3 , π 3 ) in the intersection IA ∩ IB ∩ IC .(e) The three degenerate right-isosceles classes are of order 2, given by {( 0, π 2 , π 2 ), ( π 2 , 0, π 2 ), ( π 2 , π 2 , 0)} Each is the unique order-two element of the subgroups DA, D B , D C , respectively. (f) The six nondegenerate right-isosceles classes t±( π 2 , π 4 , π 4 ), ±( π 4 , π 2 , π 4 ), ±( π 4 , π 4 , π 2 )z are generators of three cyclic groups of order 4. Each group contains one of the three degenerate non-equiangular isosceles class (of order 2). 6. Ratios The natural metric on our model computes ratios of different triangle types, easily read off of Figure 5.0.1. Since the acute classes are inside the dashed triangles, the obtuse to acute ratio is (O ∶ A) = (3 ∶ 1). Normalizing the side-length of the yellow equilateral triangle to 2, we find the isosceles classes measure ∣I∣ = 6Ó3, the right classes ∣R∣ = 6, the acute-isosceles and obtuse-isosceles ∣AI ∣ = ∣OI ∣ = 3Ó3, and the degenerates ∣D∣ = 6. Thus we have (I ∶ R) = (2Ó3 ∶ 1) , (I ∶ OI ) = (I ∶ AI ) = (2 ∶ 1) , (D ∶ R) = (1 ∶ 1). We do not think this model is any more “metrically correct” than other models, such as the sphere in [Beh14] and [ES15]. In fact, we believe there are strong arguments that both are incorrect, essentially because both omit important degenerate classes, as mentioned in Subsection 4.6. 12 E. BRUSSEL AND M. E. GOERTZ Group Action A nondegenerate scalene triangle has a single absolute similarity class, but twelve labeled, ori-ented similarity classes: six corresponding to the ways of assigning the three angles assigned to the three vertices, and two for each orientation. In general T admits a group action by D6 = ⟨r, s ⟩ whose orbits are classes that are absolutely similar , i.e., similar as unoriented, unlabeled inscribable classes. This is depicted in Figure 7.0.1: (7.0.1) srr The explicit action on a general point is given by r(α, β, γ ) = (−β, −γ, −α) and s(α, β, γ ) = (β, α, γ ).The resulting transformation groupoid is T × D6. The bipyramid of [Beh14, Exercise 1.38] has the same groupoid, and the construction is parallel. The quotient T/D6 is the (coarse) moduli set of absolute possibly-degenerate inscribable classes, drawn in Figure 7.0.2 with “stacky” multiplicities, which are the orders of the stabilizer subgroups. (7.0.2) equilateral (×6)isosceles (×2) double point (×2) right double point (×4)doubled simple point (×12 ) T/D6 with Stacky Multiplicities A doubled simple point is a double point with interior angles all zero (mod π), and a right double point is a double point with two right angles. To say T/D6 is a moduli set simply is to say there is a bijection between its points and the (absolute) inscribable classes. 8. Moduli Space and Quotient Stack We will now view T as a variety as per Corollary 3.2, and prove it is a proper moduli space of labeled, oriented, possibly-degenerate inscribable classes in the algebrogeometric setting. We use the term R-variety to mean a nonsingular, integral, separated scheme of finite type over R. Every R-variety is a smooth manifold over R. We write R-Var for the category of R-varieties, and in the following assume some elementary background in algebraic geometry and category theory. THE TORUS OF TRIANGLES 13 Definition 8.1. If B is an R-variety, let πT and πB be the projections on T ×R B.(a) A family in T over B is a subvariety F ⊂ T ×R B such that πB ∣F ∶ F → B is an isomorphism. (b) The pullback φ∗(F /B) along a morphism φ ∶ C → B is the family FC ∶= (F ×B C)/ C.(c) The (contravariant) moduli functor FT ∶ R-Var → Set assigns to each B ∈ R-Var the set of families F /B in T, and to each map φ ∶ C → B in R-Var the pullback of (b). Remark 8.1.1. Alternatively, a family is given by a morphism f = (α, β, γ ) ∶ B ⟶ (R/π)3 in R-Var whose image lies on T. That is, we assign to each b ∈ B an ordered triple f (b) = (α(b), β (b), γ (b)) , where α, β, γ ∶ B → R/π are (regular) morphisms. This phrasing emphasizes the requirement that families vary continuously (regularly) over the points of B, and is more in line with the definitions of families given in [Beh14] and other sources. But it’s the same: Since T/R is separated, the graph F = Γf is closed in T ×R B, and since f is regular, πB ∣F ∶ F → B is an isomorphism. Conversely, a family F /B in (a) defines a morphism f ∶ B → T by composing (πB ∣F )−1 ∶ B → F with πT ∶ T × B → T. This is a morphism since πB ∣F is an isomorphism. Theorem 8.2. The functor FT is represented by T. That is, it is isomorphic to Hom R-Var (−, T).Proof. More generally if a set of objects is parameterized by an S-scheme X, then the functor that assigns to each S-scheme B the set of families of objects over B, is representable by X (see [Beh14]). We construct a natural transformation Θ ∶ FT → Hom R-Var (−, T). A family F ∈ FT(B) defines a morphism f ∶ B → T by Remark 8.1.1. Define the component of Θ on the (arbitrary) base B by ΘB (F ) = f . To prove Θ is a natural transformation we must show that a morphism φ ∶ C → B determines a commutative diagram FT(B) Hom R-Var (B, T) FT(C) Hom R-Var (C, T) ΘB FT(φ)Hom R-Var (φ, T) ΘC Suppose F ∈ FT(B) defines the B-point f ∶ B → T, as above. Then ΘB (F ) = f , and by definition (Hom R-Var (φ, T) ◦ ΘB )( F ) = f ◦ φ. On the other hand, by definition FT(φ)( F ) = F ×B C = FC , and ΘC (FC ) is the composition πT ◦ (πC ∣FC )−1. Suppose (η, c ) is a point of FC . By definition of fiber product πB (η) = φ(c), so (η, c ) = (( πT(η), φ (c)) , c ) is the corresponding point on T ×R B ×B C.Therefore ΘC (FC )( c) df = πT ◦ (πC ∣FC )−1(c) = πT(η), and πT(η) = f (φ(c)) by definition of f . We conclude ΘC ◦ FT(φ) = f ◦ φ. Since F /B was arbitrary, this shows the diagram commutes, hence Θ is a natural transformation. We invert Θ by taking f ∈ Hom R-Var (B, T) to the family F /B ∈ FT(B) defined in Remark 8.1.1. The proof that this inverts Θ is straightforward, and shows FT and Hom R-Var (−, T) are isomorphic. Corollary 8.3. T is a fine moduli space for the set of all labeled, oriented, possibly-degenerate inscribable triangle classes, which is a compactification of T◦ = T − {αβγ = 0}, the fine moduli space of labeled, oriented, nondegenerate inscribable classes. In particular, the family [∆insc ] = T is a universal family, given explicitly by σ in Theorem 4.1. 14 E. BRUSSEL AND M. E. GOERTZ Proof. Already there is a bijection between labeled, oriented, possibly-degenerate inscribable classes and points of T, and it remains to verify Hom R-Var (−, T) has the universal property with respect to natural transformations FT → Hom R-Var (−, Y ) for Y ∈ R-Var . But FT and Hom R-Var (−, T) are isomorphic via Θ of Theorem 8.2, so for any Ψ ∶ FT → Hom R-Var (−, Y ) there is trivially a natural transformation Φ ∶ Hom R-Var (−, T) → Hom R-Var (−, Y ) such that Ψ = Φ ◦ Θ. This proves T is a fine moduli space. That it compactifies T◦ is trivial since it is compact and it is the closure, and that T◦ is a fine moduli space is also immediate via the open immersion T◦ ⊂ T. The diagonal family of D ⊂ T ×R T in FT(T) maps to id T ∈ Hom R-Var (T, T), and is the accompanying universal family. 8.3.1. Moduli Stacks. The coarse moduli set T/D6 of absolute (unlabeled, unoriented) possibly-degenerate inscribable classes in Figure 7.0.2 is not an R-variety, hence not a fine moduli space, essentially because the symmetry group D6 does not act freely on T, as indicated by the nontrivial stacky multiplicities labeled in (7.0.2). More intrinsically, the stacky points cause problems because the corresponding nontrivial automorphisms of individual classes can be used to construct non-isomorphic families that the moduli set T/D6 doesn’t distinguish (see [Beh14]), which thwarts the construction of a universal family of absolute classes. Nevertheless, since the moduli set of absolute inscribable classes is the orbit space of the R-variety T under D6, it has the geometric structure of a quotient stack, denoted [T/D6] ([DM69, Example 4.8], [Beh14, 1.24], [Alp24, 0.6.5]). By definition, [T/D6] is the category whose objects are D6-torsors over R-varieties B that are equipped with D6-equivariant maps into T. These are given by diagrams B′ T B f denoted (B′/B, f ), where B′/B is a D6-torsor and f is D6-equivariant. The morphisms (φ, θ ) ∶ (C′/C, g ) → (B′/B, f ) are pairs (φ, θ ) of morphisms φ ∶ C → B in R-Var and D6-equivariant isomorphisms θ ∶ C′ → B′ ×B C → B′ satisfying g = f ◦ θ: C′ B′ T C B θf g φ By incorporating the automorphisms of individual classes into each family over B, the stack dis-tinguishes nonisomorphic families. Though it is not a variety, [T/D6] is as close to a fine moduli space as we get. References [Alp24] Jarod Alper. Stacks and Moduli (working draft) . August 2024. [AS94] Scott Axelrod and I. M. Singer. Chern-Simons perturbation theory. II. J. Differential Geom. ,39(1):173–213, 1994. [Beh14] K. Behrend. Introduction to algebraic stacks. In Moduli spaces , volume 411 of London Math. Soc. Lecture Note Ser. , pages 1–131. Cambridge Univ. Press, Cambridge, 2014. [BGGL24] Eric Brussel, Madeleine Goertz, Elijah Guptill, and Kelly Lyle. The stack of triangles up to similarity. arXiv:2408.07792 [math.AG] , 2024. THE TORUS OF TRIANGLES 15 [CNSS19] Jason Cantarella, Tom Needham, Clayton Shonkwiler, and Gavin Stewart. Random triangles and polygons in the plane. Amer. Math. Monthly , 126(2):113–134, 2019. [DM69] P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. , (36):75–109, 1969. [Dod58] C. L. Dodgson. Mathematical Recreations of Lewis Carroll. Vol. 2: Pillow Problems and A Tangled Tale . Dover Publications, Inc., New York, 1958. (Both books bound as one.). [ES15] A. Edelman and G. Strang. Random triangle theory with geometry and applications. Foundations of Computational Mathematics , 15:681–713, 2015. [Euc56] Euclid. The thirteen books of Euclid’s Elements translated from the text of Heiberg. Vol. I: Introduction and Books I, II. Vol. II: Books III–IX. Vol. III: Books X–XIII and Appendix .Dover Publications, Inc., New York, 1956. Translated with introduction and commentary by Thomas L. Heath, 2nd ed. [FM94] William Fulton and Robert MacPherson. A compactification of configuration spaces. Ann. of Math. (2) , 139(1):183–225, 1994. [Iwa87] Toshihiro Iwai. A gauge theory for the quantum planar three-body problem. J. Math. Phys. ,28(4):964–974, 1987. [Ken84] David G. Kendall. Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. London Math. Soc. , 16(2):81–121, 1984. [Lee13] John M. Lee. Introduction to Smooth Manifolds , volume 218 of Graduate Texts in Mathematics .Springer, New York, second edition, 2013. [Mon15] Richard Montgomery. The three-body problem and the shape sphere. Amer. Math. Monthly ,122(4):299–321, 2015. [OV90] A. L. Onishchik and È. ˜B. Vinberg. Lie Groups and Algebraic Groups . Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites. [Por94] Stephen Portnoy. A Lewis Carroll pillow problem: probability of an obtuse triangle. Statist. Sci. ,9(2):279–284, 1994. [Sin04] Dev P. Sinha. Manifold-theoretic compactifications of configuration spaces. Selecta Math. (N.S.) ,10(3):391–428, 2004. [Woo61] W. S. B. Woolhouse. Mathematical question 1987. The Lady’s and Gentleman’s Diary , 1861. Email address : ebrussel@calpoly.edu, mgoertz@calpoly.edu
7637	https://www.youtube.com/watch?v=6sGU_8l7H58	Desmos Just Got Even Better with 3D Graphing Angie Teaches Math 6780 subscribers 73 likes Description 7713 views Posted: 14 Feb 2024 The best way to graph in 3D is here and it's from Desmos. Graph equations with x, y, and z using Desmos's 3D Graphing calculator. You can graph ellipsoids, paraboloids, spheres, and points in 3 dimensions. Desmos's 3D Graphing tool will graph quadric surfaces, too. Desmos is the best way to graph in R3. www.desmos.com/3d ▶ Watch My Desmos Tutorials: ⏰ Time Stamps 0:00 Desmos 3D Graphing Tools 1:16 Graph a Cylinder in Desmos 3D 1:56 Graph an Ellipsoid in Desmos 3D 3:00 Plot Points in Desmos 3D Looking for more great math videos? Check out these: My Math Channel 9 comments Transcript: Desmos 3D Graphing Tools let's explore the new 3D graphing capabilities from Desmos at desmos.com sl3d now you can see that I've got a paraboloid graft here for us and I've animated it which is super easy to do let me click and stop the animation in order to animate this I'm just going to click hold and spin it and if I spin it faster it goes faster if I instead click on it to stop the animation and if I spin it downwards it spins it that way which is a really fun tool now in terms of that graph you've got some really familiar tools in the upper right hand corner I can zoom in I can zoom out I can also click on the wrench to change some additional graphing settings including dark mode which is reversing that contrast I also like to have translucent surfaces checked if it's unchecked that you're not able to see through our surfaces um let me go back and check that one you can also change the X Y and Z bounds for each of those axes and then you can click on and off of any of your labels and then finally I can click on the home button to get to my default view now I want to change Graph a Cylinder in Desmos 3D this paraboloid into another surface so that we can do some more exploring I'm going to replace that Z with the number nine now we're just looking at that Circle of radius 3 I can click on my XY plane so I'm looking just just in that direction and I can see that it is just the circle but if you take a look underneath the equation it gives me the 3D option and I know that in three dimensions because Z is missing Z can take on any value so I'm going to click on extend 3D and I'm going to click hold and just drag this around so that we can see that we actually have a circular cylinder let me change this a little bit Graph an Ellipsoid in Desmos 3D further I want to change this into an ellipsoid so let's divide by an A squar and A B squared and then we'll put in um plus Z [Music] sared equals 1 so I've typed in the equation for an ellipsoid but you see that it's got an add slider option and I'm going to go ahead and add a slider for all three of these a b and c you can also change the parameters on the sliders by just clicking on the slider letter and then I want them to be from zero to say five now I've got a nice little animation here where I can see that as I change C I am changing my ellipsoid I'm extending that in the Z Direction changing B extends it in the y direction and changing a extends that there you go extends that in the X Direction Plot Points in Desmos 3D I can even add some points by just typing in the coordinates of a given point and I've got that point right there I'm excited to see what Desmos has next
7638	http://www.erbium.at/FF/wp-content/uploads/2015/10/master_thesis_johannes_schindler.pdf	Characterization of an Erbium Atomic Beam master thesis in physics by Johannes Schindler submitted to the Faculty of Mathematics, Computer Science and Physics of the University of Innsbruck in partial fulﬁllment of the requirements for the degree of Master of Science Supervisors: Ass.Prof. Dr. Francesca Ferlaino Univ.Prof. Dr. Rudolf Grimm Innsbruck, July 2011 Zum Besten aller Wesen. Abstract In this thesis we study and characterize an erbium atomic beam emerging from a high-temperature oven system. This is the ﬁrst step towards the production of a quantum-degenerate gas of Er. Er is a rare-earth element and its quantum mechanical interaction properties are yet rather unexplored. Its exceptionally large magnetic moment of 7 µB makes Er especially interesting, since it allows studies on the physics of dipolar quantum gases. In addition, Er shows an un-usual electronic conﬁguration and a complex energy level scheme, promising rich scattering properties and oﬀering many possibilities for laser cooling. Since Er has a high melting point temperature of more than 1500 ◦C, the oven system for the production of a thermal atomic beam requires speciﬁc engineering. This includes thermal shielding, heated apertures, special crucible materials as well as reliable control- and security systems. We characterize the atomic beam emerging from the oven by absorption spectroscopy and ﬂuorescence imaging on the strong 401 nm transition. Based on our measurements, we optimize the col-limation setup and we ﬁnd the ideal operation temperature to be T ≃1250 ◦C with a corresponding atomic ﬂux of 3× 1010 s−1 injected into the Zeeman slower. iii Zusammenfassung Im Rahmen dieser Arbeit untersuchen wir einen Erbium-Atomstrahl, welchen wir mittels eines Hochtemperaturofens erzeugen. Dies ist der erste Schritt zur Erzeu-gung eines Quantengases von Erbium Atomen. Erbium geh¨ ort zur Gruppe der seltenen Erden, seine quantenmechanischen Wechselwirkungseigenschaften sind noch weitgehend unerforscht. Erbium ist besonders interessant, da durch sein aussergew¨ ohlich großes magnetisches Moment von 7 µB zahlreiche Experimente zur Physik dipolarer Quantengase erm¨ oglicht werden. Ebenso sind aufgrund der komplexen elektronischen Struktur dieses Elements mannigfaltige Streueigen-schaften zu erwarten und zudem zahlreiche M¨ oglichkeiten zur Laserk¨ uhlung gegeben. Der hohe Schmelzpunkt von Erbium von ¨ uber 1500 ◦C stellt besondere An-forderungen an das Ofendesign. Unter anderem verwenden wir eine thermische Abschirmung, beheizte Blenden, spezielles Material f¨ ur den Tiegel sowie eigene Kontroll- und Sicherheitssysteme. Durch Absorptionsspektroskopie und Fluo-reszenzmessungen mit dem starken atomaren ¨ Ubergang bei 401 nm charakter-isieren wir den erzeugten Atomstrahl. Basierend auf unseren Messungen op-timieren wir die Ofenblenden und bestimmen die ideale Betriebstemperatur zu T ≃1250 ◦C, bei welcher der Atomﬂuß in den Zeeman Slower 3·1010 s−1 betr¨ agt. iv Contents Contents v Introduction 1 1 Thermal atomic beams 4 1.1 Eﬀusive oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Atomic beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Determination of the atomic ﬂux . . . . . . . . . . . . . . . . . . . 8 1.4 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Erbium 12 2.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Laser cooling of erbium atoms . . . . . . . . . . . . . . . . . . . . . 18 2.5 Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 Scattering properties . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Experimental setup 23 3.1 Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Cooling and trapping lights . . . . . . . . . . . . . . . . . . . . . . 26 4 Erbium oven 29 4.1 Oven system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Atomic beam shutter . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5 Characterization of the atomic beam 37 5.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 Collimation setup 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.3 Collimation setup 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.4 Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 v Contents Conclusion and Outlook 62 A Magnetic moments of the lanthanides 65 B Erbium oven control 67 C Atomic beam shutter control 69 C.1 Control box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 C.2 Source code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 D Calibrations 78 D.1 Laser frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 D.2 CCD camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Bibliography 81 List of Symbols and Abbreviations 88 vi Introduction By cooling atoms to ultralow temperatures, we gain extraordinary high control over their external as well as their internal degrees of freedom. Therefore, ul-tracold atoms empower us to study and to understand phenomena belonging to very diﬀerent branches of research, such as quantum chemistry, condensed matter and few-body physics, high-temperature superconductivity, and quantum simu-lations [1–5]. The ﬁeld of ultracold quantum gases has been subject to vast developments in the past two decades. The ﬁrst Bose-Einstein condensation, a phase transition that leads to the macroscopic occupation of the system 's ground state at ultralow but ﬁnite temperatures, was attained in 1995 [6–8]. Subsequently, also fermions, representing the basic building blocks of matter were cooled to quantum degener-acy [9–11]. The ability to modify the two-body interaction by magnetic or optical Feshbach resonances increased dramatically the level of control on the sys-tems. This opened the way of studying tunable quantum gases in the weak and in the strong interacting regime and of creating more complex structures such as weakly bound dimer states [13–16]. All of this pioneering works were performed on elements from the alkali series and rely on the isotropic and short-ranged con-tact interaction. Alkali atoms provide comparatively simple systems with just one valence electron in the electronic s-shell and convenient optical transitions for laser cooling experiments. At ultralow temperatures, the interaction between those atoms can simply be described by a single scattering parameter known as the s-wave scattering length a. In further advance, the laser cooling and trapping techniques were extended to more complex atomic species, which provide additional challenges and prospects [17–19]. In particular, atoms or molecules that interact via the dipole-dipole interaction (DDI) lead to totally new phenomena. In contrast to the contact interaction, the DDI has anisotropic and long-range nature. This gives rise to new quantum phases and oﬀers geometry dependent tunability [20,21]. Two can-didate systems exhibit strong DDI: polar molecules and magnetic atoms. Polar molecules carry a large electric dipole moment but require a complex produc-tion sequence since direct laser cooling can hardly be applied [22,23]. Magnetic 1 Introduction atoms exhibit smaller magnetic dipole moments but oﬀer more robust systems. Chromium has been so far the only strongly magnetic species to be Bose con-densed1 . In this system, the dipolar character manifests itself in spectacular eﬀects such as the d-wave collapse of the trapped gas , the geometry depen-dent stability of the system and suppression of collisional relaxation in an optical lattice . Nowadays, exotic atoms with high magnetic moments, as found in the lanthanide series are subject of increased interest in our community and represent the cutting-edge research on the DDI [28–30]. Erbium is one of those elements with highly magnetic atoms. It is a rather unexplored species, which shows a very rich energy structure and many favor-able properties like a large number of isotopes, a high mass, a high magnetic moment and a variety of narrow as well as broad laser cooling transitions. In addition, erbium 's unusual electronic conﬁguration leads to complex scattering properties [31–34]. The primary goal of our experiment is to Bose condense Er atoms and to in-vestigate their scattering properties. Besides studies of ultracold dipolar Bose or Fermi gases, also the formation of Feshbach molecules with even higher magnetic moments or few body physics with DDI are of interest. Our vision is to admix Li atoms to Er, leading to a crystal like system with extreme mass imbalance to observe novel eﬀects on the ﬁeld of Eﬁmov physics or of the BEC-BCS crossover. At the moment, our experiment is in the construction phase. We set up the complete vacuum apparatus and the main laser setup for cooling and trapping. During my master thesis time, I contributed to the mounting of the vacuum apparatus, the installation of the cooling water system and the construction of various coils to generate magnetic ﬁelds including the winding of the Zeeman slower coils. Furthermore, I had the possibility to gain some experience with a dye laser that we plan to use for the narrow line magneto-optical trap. Besides the above mentioned tasks, the main focus of my work is the setting up and char-acterization the Er oven system. I installed and programmed a control- and safety system for the oven as well as an atomic beam shutter control system. Once the Er oven was operating, I characterized the atomic beam by means of laser spec-troscopy and optimized the beam collimation setup based on these measurements. This thesis is structured as follows. Chapter 1 gives an overview on the fundamental theoretical concept to understand the atomic beam behavior. In Chap. 2 we review the main properties of Er atoms with emphasis on its laser cooling transitions and trapping potential. Chapter 3 gives an overview on our complete experimental setup including the vacuum chamber and the laser setup. The detailed setup and working principle of the Er oven is explained in Chap. 4. 1Recently, also Dysprosium atoms have been Bose condensed in the group of B. Lev at University of Illinois at Urbana-Champaign (private communication 07.2011). 2 Introduction Finally, in Chap. 5 we present the results of the laser spectroscopy measurements that characterize the Er atomic beam and that we used to adapt the Er oven design to our requirements. 3 Chapter 1 Thermal atomic beams The starting point of our experiment is a thermal beam of Er atoms produced from an eﬀusive high-temperature oven. The atomic beam is then decelerated via the Zeeman slower technique and later trapped in a magneto-optical trap. In this Chapter, we introduce the working principles of an eﬀusive oven and the methods to characterize the atomic beam emerging from the oven. 1.1 Eﬀusive oven There exist many diﬀerent kinds of sources for the production of a thermal atomic beam, depending on its demands and source material . For source materials with high melting points, coated heated-ﬁlaments or laser deposition generate free atoms, but provide only limited atomic ﬂux. Wicking recirculation sources oﬀer higher ﬂux and are a clever way to conserve source material in applications with needs for long source life, such as in atomic clocks. For our purpose, we use an eﬀusive heated oven combined with a bright wall collimator that we discuss in Sec. 1.2. It provides a robust system with a high, well controllable atomic ﬂux. The oven itself consists of a tube with an aperture. It is heated by surrounding heating wires and is water cooled for thermal shielding. The technical details are given in Chap. 4, in the following we describe the theoretical working principle on a simpliﬁed system. We consider an ideal atomic gas in thermal equilibrium at temperature T stored in a box. It is described by the ideal gas law p = kB n T , (1.1) where p is the pressure, kB the Boltzmann constant and n the atomic density. If the box is surrounded by vacuum and has a circular aperture with diameter b1, the atoms escape from it because of the pressure gradient between the box and the vacuum. Here, it is important to distinguish diﬀerent regimes of the 4 1. Thermal atomic beams oven operation. Depending on p, the atoms suﬀer collisions with each other. The mean distance atoms travel between two successive collisions, the so called mean free path λmf, is given by λmf = 1 √ 2 π d2 0 n , (1.2) where d0 is the atomic diameter . If λmf is smaller than the size of the aperture, the atoms encounter many collisions while emerging from the box. This leads to a hydrodynamic ﬂow, a collective transport process, and is described by ﬂuid dynamics. In contrast, for λmf ≳b1 (1.3) atom-atom collisions are rare and atom-wall collisions dominate. This is the eﬀusive- or molecular ﬂow regime, in which our eﬀusive oven works. The expan-sion is an individual transport process where mechanisms as so called diﬀusive reﬂection can occur: an atom hits a wall, sticks to it and is re-emitted in an angle that is independent of its incident angle. By this, in contrast to hydrodynamic ﬂow, an atom that hits the inner face of the aperture can return to the box. Furthermore, in the eﬀusive regime, the emergence of atoms from the box does not change the spatial- nor the velocity distribution of the atoms inside the box. In the eﬀusive regime, we can calculate the ﬂux emerging from the box as follows. The velocity v = \|⃗ v\| of the atoms inside the box is described by the well known 3D Maxwell-Boltzmann velocity distribution Pv(T) = C m kB T 3 2 v2 exp −m v2 2 kB T , (1.4) with C a normalisation constant and m the mass of an atom. The most probable velocity is vmp = p 2 kB T / m and the mean velocity is ¯ v = 2 vmp / √π. Figure 1.1 illustrates the atomic volume V = ¯ v dt · A cos α leaving the box in the time interval dt . Here, α is the angle of emergence and A = b2 1 π / 4 is the area of the aperture. Only a part dω / (4 π) of the atoms travels on a solid angle dω. The number of atoms dN per unit time dt leaving the box into dω is given by the unit ﬂux dΘ and reads as dN = dΘ dt = dω 4 π n ¯ v cos α A dt , (1.5) where n is the atomic density. Integrating Eq. (1.5) over the solid angles dω = 2 π sin α dα with α from 0 to π/2 gives the total ﬂux coming out of the box Θ = 1 4 n ¯ v A . (1.6) 5 1. Thermal atomic beams α vdt Acos(α) Figure 1.1: In the time interval dt, an atomic volume of V = ¯ v dt · A cos α leaves the box at an angle of emergence α. A is the area of the aperture, ¯ v is the mean velocity of the atoms. Integration in spherical coordinates over ϕ from 0 to 2 π and over ϑ from 0 to α gives the volume emerging into the solid angle dω. 1.2 Atomic beam The angular distribution of the atoms leaving the box from the aperture is pro-portional to cos α, as given in Eq. (1.5). Only atoms travelling at small angles are useful for the experiment, since they can travel through the apparatus without hitting it. A large fraction of atoms is unused and it is needed to block it to avoid or reduce source material deposition in the apparatus. The atomic beam can be narrowed by collimating it by a second aperture, aperture 2, in front of the ﬁrst aperture, aperture 1. This is a so called dark wall collimator that provides simple geometric shading by cutting oﬀthe unused part of the source beam. This technique is not suitable for our experiment, since Er deposited at the apertures can overgrow and close the aperture. Therefore, we use heated apertures that re-emit the blocked atoms. This is a type of a so called bright wall collimator that conserves the source material and extends the source life time . In terms of simplicity, in the following we neglect that the apertures are heated. Provided that the mean free path of the atoms is larger than the typical geometry of the oven, this is a valid approximation. Figure 1.2 shows the oven with and additional aperture 2 with diameter b2 at a distance a from aperture 1. Only atoms inside the dotted lines contribute to the atomic beam leaving the oven. The beam emerges into a solid angle around the z-axis of ω = 2 π (1 −cos α) , where α = tan−1 f and f = b1/2 + b2/2 a , (1.7) 6 1. Thermal atomic beams Figure 1.2: Scheme of an eﬀusive oven with aperture 1 of diameter b1 and aperture 2 at a distance a with diameter b2. Inside the box, the atoms have temperature T and velocity distribution Pv(T ). Only atoms inside the dotted lines contribute to the atomic beam. The beam emerges into a solid angle ω = 2 π (1 −cos α) and has diameter d at a distance l from aperture 2. The atoms in the beam have velocity distribution P ′ v(T ) and longitudinal and transversal velocity components vz(T ) and vx(T, f). A laser beam crosses the atomic beam transversal at distance l from aperture 2. where f is the collimation ratio. At a distance l from the aperture 2, the beam has a diameter d of d = 2 (a + l) f −b1 2 . (1.8) The ﬂux Θ′ through the aperture is obtained by integrating Eq. (1.5) over dω = 2 π sin α dα from α = 0 to α = tan−1 f: Θ′ = 1 4 n ¯ v A f 2 1 + f 2 . (1.9) A diﬀerent type of bright wall collimator uses a long channel instead of a circu-lar hole for aperture 1. In the eﬀusive regime, wrongly directed atoms hit the tube wall and by diﬀusive reﬂection they can return to the source or contribute to the directed beam. This changes the angular distribution of the atoms from Eq. (1.5) considerably. The ﬂux of atoms travelling along the tube is undimin-ished, whereas the total atomic ﬂux Θtube is reduced to Θtube = 1 κ Θ . (1.10) The factor 1/κ depends on the geometry of the channel and for a long circular cylindrical tube it is given by 1/κ = (8 b1) / (3 a1), where b1 is the diameter and a1 the length of the tube . However, as the longitudinal ﬂux is not altered by such a collimator, Eq. (1.9) also holds in this case, given that f is small and λmf ≳a1. In an atomic beam the velocity distribution of the atoms is no longer described by Eq. (1.4) as in the box. Since, as shown in Eq. (1.5) and Fig. 1.1, the probability of an atom to emerge from the box is proportional to its velocity, the velocity distribution changes to P ′ v(T) = C′ m kB T 2 v3 exp −m v2 2 kB T , (1.11) 7 1. Thermal atomic beams with normalisation constant C′ and P ′ v(T) ∝v3. Thus, the most probable veloc-ity changes to v′ mp = p 3 kB T / m and the mean velocity is ¯ v′ = v′ mp p 3 π / 8 . We can distinguish two components of the velocity distribution: a longitudinal distribution with velocities vz(T) and a transversal distribution with velocities vx(T, f). The longitudinal distribution is determined by the temperature T of the source, whereas the transversal distribution has to fulﬁl the boundary condition of the apertures and is determined by the collimation ratio f from Eq. (1.7). The individual velocities must obey vx ≤f vz . (1.12) 1.3 Determination of the atomic ﬂux We can control the ﬂux by the source temperature. The source should emit as many atoms as necessary for the experiment but as few atoms as possible to extend the source life-time. To monitor the ﬂux, we measure it by laser spectroscopy. In a simple model we consider the atoms as pure two level systems (2-LS) with ground state \|g⟩and excited state \|e⟩. The energy diﬀerence of the two states is ∆Eeg = h ν0 with the transition frequency ν0. If an atom absorbs a photon from a resonant laser beam with frequency ν0, the atom is excited from \|g⟩to \|e⟩. After the lifetime τ of \|e⟩, the atom decays back to \|g⟩by spontaneous emission of a photon with energy h ν0 and can be re-excited. The photon is emitted in random directions and can be observed as ﬂuorescence light . Therefore, the intensity of the transmitted light is reduced by the amount of absorbed light and is proportional to the atomic density. As shown in Fig. 1.2, we cross the atomic beam transversely with a resonant laser beam at the distance l from aperture 2. For laser beam intensities below the saturation intensity Isat of the transition, the Beer-Lambert law describes the transmission of light with incident intensity I0 through the atomic beam : I = I0 exp(−D) . (1.13) D is related to the optical density (OD) of the atomic beam by D = ln (10) OD ≈ 2.3 OD. For the laser beam interacting with the atomic beam from x1 to x2, D is deﬁned as D = Z x2 x1 n(x) σ0 dx ≈n σ0 j , (1.14) where σ0 is the cross-section for the absorption of a resonant photon by an atom . For small angles α from the beam axis, the density can be estimated as constant over the light-atom interaction length j = x2 −x1. In general, j is not equal to the atomic beam diameter d, since atoms that are not in center have velocity components in the direction of the laser beam. Hence, their resonance 8 1. Thermal atomic beams frequencies are Doppler shifted and these atoms do not interact with the light. In a 2-LS, the absorption cross-section is given by σ0 = 3 λ2 2 π (1.15) for resonant light with wavelength λ = c / ν0 . Combining Eq. (1.13 - 1.15) yields to the atomic density: n = −2 π 3 λ2 1 j ln I I0 . (1.16) In this way, by measuring I0, I and j, we can calculate the total ﬂux Θtot out of an source with aperture diameter b1 and temperature T by inserting Eq. (1.16) into Eq. (1.6): Θtot = −π 3 2 24 b2 1 λ2 j ln I I0 r 2 kB T m . (1.17) Respectively, by Eq. (1.9), we can calculate the ﬂux injected into a certain angle around the beam axis. 1.4 Spectrum To characterize the atomic beam in further detail, we scan the frequency ν of the longitudinally crossed laser light over the atomic resonance at ν0. The transmit-ted light intensity I as a function of ν represents the absorption spectrum. We ﬁnd an absorption peak around ν0 and its exact position, width and shape reﬂect the physics of the atomic beam. The natural line shape and -width of the un-perturbed atoms get altered and broadened by their thermal motion, atom-atom collisions or interaction with light. By understanding the underlying broadening mechanisms, we can conclude from the absorption spectrum on the collimation, temperature or ﬂow regime of the source and the atomic beam. Natural linewidth The natural linewidth ∆νnat is the fundamental limit of the linewidth. Due to the ﬁnite lifetime τ of the exited state, the Heisenberg energy uncertainty relation ∆E τ = h limits the width to ∆νnat = ∆E h = 1 2 πτ . (1.18) The natural line shape can be derived from a classical model of a damped har-monic oscillator for the excited electron . In this model, the atomic potential is approximated as a harmonic potential and radiative energy loss results in damping. This leads to Inat(ν) = I0 (Γ/4π)2 (ν −ν0)2 + (Γ/4π)2 , (1.19) 9 1. Thermal atomic beams which is a Lorentzian function with peak intensity I0 and full width half maximum (FWHM) ∆νnat = Γ/2π. Γ = 1/τ corresponds to the spontaneous decay rate of the excited state. Doppler broadening Moving atoms see the frequency of a laser beam Doppler shifted. Accordingly, the resonant condition for an atom moving at a non-relativistic velocity v is ν = ν0 + ⃗ k ⃗ v 2 π = ν0 1 + vk c , (1.20) where ⃗ k is the wave vector of the light wave, vk = ⃗ v ⃗ ek is the velocity component in its direction and c is the speed of light. Since in a thermal gas the atoms have distributed velocities, this shift leads in ﬁrst order to a broadening of the spectral line. For the interaction with the light, only vk needs to be considered, thus a 1D Maxwell-Boltzmann distribution with normalization constant C′′ describes the velocity distribution of free atoms: Pvk(T) = C′′ r m kB T exp −m v2 2 kB T . (1.21) Substituting v = c (ν/ν0 −1) gives the probability distribution Pν(T) that is proportional to the line shape ID(ν, T) = I0(T) exp −c2 m (ν −ν0)2 2 kB T ν2 0 , (1.22) which is a Gaussian function with peak intensity I0(T). It has a Doppler width ∆νD = (ν0/c) p 2 kB T / m = ν0 vmp / c and a FWHM of ∆νFWHM D = 2 √ ln 2 ∆νD. In an atomic beam the velocity distribution is not a pure Maxwell-Boltzmann distribution, but is additionally proportional to v. Therefore, according to Eq. (1.11), v′ mp has to be used and thus ∆ν′ D = ν0 v′ mp / c. However, the main diﬀerence to the free gas is that the distribution has two components with veloc-ities vx and vz. Thus, the Doppler width also has two components and according to Eq. (1.12): ∆νD,x = ∆νD,z f = ν0 c v′ mp f . (1.23) This means that the Doppler broadening of a spectral line is reduced by collimat-ing the atomic beam. Thus, the resolution can be increased and, more important for us, conclusions on the collimation can be drawn. Other broadening mechanisms Saturation- or power broadening occurs when the intensity of the laser beam ex-ceeds or is comparable to the saturation intensity Isat of the transition. Then, 10 1. Thermal atomic beams the optical pumping rate is larger than the relaxation rate and the line is broad-ened . The resulting line has a Lorentzian shape with FWHM ∆νS = ∆νnat p 1 + I/Isat . (1.24) For I ≪Isat it is thus negligible. Collisional- or pressure broadening results from the deformation of the elec-tronic wave function during an atom-atom collision. Thus, the transition fre-quency is shifted and the line is broadened . The resulting line has a Lorentzian shape with FWHM ∆νcol = n σat vmp ∝n √ T . (1.25) Since the collision cross-section σat for Er is not known yet, we cannot make predictions on the strength of this mechanism besides that for a constant density n, the linewidth scales with √ T. In general, due to the limited collisions in the eﬀusive regime, this eﬀect should be negligible. If several diﬀerent mechanisms contribute to the broadening of a line, the resulting total line shape is represented by the convolution of the individual line shapes. For instance, a line broadened due to Doppler broadening as well as due to saturation broadening has a line shape described by a Voigt proﬁle: V (ν; ∆νG, ∆νL) = Z +∞ −∞ G(ν′; ∆νG) L(ν −ν′; ∆νL) dν′ . (1.26) G(ν; ∆νG) represents the Gaussian contribution with width ∆νG = ∆νF W HM D , L(ν; ∆νL) represents the Lorentzian contribution with width ∆νL = ∆νnat+∆νS. In this case, the resulting total linewidth ∆νV is estimated as ∆νV ≈0.5346 ∆νL + q 0.2166 ∆ν2 L + ∆ν2 G . (1.27) If we do not consider the spectrum of a free gas, but the transversal spectrum of an atomic beam, according to Eq. (1.23), we have to employ the reduced Doppler width ∆νF W HM D,x in Eq. (1.26) and Eq. (1.27). 11 Chapter 2 Erbium Erbium is a very interesting and rather unexplored element for ultracold gas experiments. It is a soft, silver-colored metal with an atomic number of 68. It belongs to the rare earth elements and is part of the lanthanide series. There is an increasing interest in elements from this series, since they can exhibit high magnetic moments and have exotic electronic conﬁgurations, which opens up many possibilities for laser cooling applications. In this Chapter, we present relevant properties of erbium atoms, including the electronic, the magnetic and the scattering properties. We also discuss the optical transitions that we want to use for laser cooling and trapping Er atoms. 2.1 General properties Erbium was discovered in 1843 in a quarry near the village of Ytterby in Sweden, as other rare earth elements such as Ho, Tm and Yb. Today, Er is almost exclu-sively mined in China. It ﬁnds many commercial applications in laser technology. For instance, it is used as doping element for ﬁber lasers. Here, trivalent Er ions are pumped with 980 nm or 1480 nm light and emit light at wavelengths around 1550 nm [43,44]. In optical communication, this is used for ﬁber ampliﬁers, which allow fast and relatively cheap signal ampliﬁcation at low noise [45–47]. Also the Er:YAG solid state laser ﬁnds common use in medical photo-ablation treatments due to the strong absorption of its 2940 nm infrared light in water [48,49]. Erbium is a soft, silver-colored rare earth metal. Macroscopically it has a ﬁbroid structure, microscopically it forms a hexagonal close packed crystal . Similar to Cr , Er has a comparatively high melting point temperature of Tmelt = 1529 ◦C . Figure 2.1 shows its vapor pressure as a function of the temperature. For comparison, we also plot the vapor pressure of other rare earth elements such as Tm, Yb and Sa, of the magnetic Cr atoms and of the alkali metal Li, which we will in future experiments admix to the Er machine. The 12 2. Erbium 0 500 1000 1500 2000 2500 3000 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 10 2 10 4 10 6 T [°C] pvapor [mbar] Tmelt(Er)=1529°C Er − points Li − points Yb − points Sm − points Tm − points Cr − plot Figure 2.1: Vapor pressure as a function of the temperature for Er, Li, Yb, Sm, Tm and Cr. Erbium requires high temperatures for vapor pressures suitable for the production of a thermal atomic beam. The solid and dashed lines are ﬁts to the data using Eq. 2.1. The vertical dotted line shows the melting point temperature Tmelt of Er. data points are extracted from and ﬁtted to an Antoine equation , which approximates the relation between vapor pressure and temperature: Pvapor(T) = exp a − b c + T , (2.1) with the ﬁt parameters a, b and c. Erbium and Cr show the lowest vapor pressures for a given temperature. Although Er, Tm and Yb are neighboring elements, the melting point of Yb is much smaller than that of Er and Tm1. It is likely that this is due to the diﬀerent crystal structure, which is hexagonal for Er and Tm, but cubic for Yb [50,55]. For Er, the required high temperatures for suﬃciently high vapor pressures make the production of an oven technologically challeng-ing. As we will discuss in Sec. 5.4, we estimate that temperatures of 1200 ◦C lead to a reasonable Er atom ﬂux for Zeeman slower (ZS) and magneto-optical trap (MOT) applications. Erbium has the advantage of showing good getter properties. As reported in Ref. , Er ﬁlms have exceptionally high adsorption activity especially for hydrogen, which commonly limits the achievable pressure in a vacuum chamber. 1Tmelt(Yb) = 824 ◦C, Tmelt(Tm) = 1545 ◦C, . 13 2. Erbium When the Er oven is turned on, a part of the atoms coats the inner walls of the vacuum chamber. Remnant atoms hitting the wall stick to it, and the pressure drops. This eases the production of ultra high vacuum (UHV) conditions in the vacuum chamber. It is the same principle as that of a Ti-sublimation pump, but since the surface area of Er ﬁlms is much larger than that of Ti ﬁlms, the eﬀect should be even stronger with Er. Erbium is very appealing for quantum gas experiments. First, it has a com-paratively high magnetic moment of 7 µB 2. Nowadays, highly magnetic atoms are attracting great attention in our community because of the many possibilities to study dipolar eﬀects . Second, Er has a rich energy structure, which results from its unusual electronic conﬁguration, where an incomplete f-shell with two electron vacancy is submerged by a full s-shell. This oﬀers many possibilities for various laser cooling schemes and leads to novel collisional behavior. In ad-dition, the numerous Zeeman levels open opportunities for quantum computing, as recently proposed in . Third, six stable isotopes provide ﬂexibility for the experiment and allow investigation of bosonic as well as fermionic systems. The group of Jabez J. McClelland at the National Institute of Standards and Technology (NIST) has been the ﬁrst to demonstrate laser cooling of a magnetic rare earth species. They produced a blue MOT with 105 atoms at temperatures below 25 µK [33, 58], as well as a narrow-line MOT with temperatures below 2 µK . 2.2 Electronic properties The electronic ground state of Er is in a [Xe]4f 126s2 conﬁguration. The [Xe] means that, as in the noble gas Xe, all shells energetically up to the 5p shell are fully occupied. From the 14 valence electrons, two of them ﬁll the 6s shell and the remaining 12 are arranged in the 4f shell, leaving it with a two electron vacancy. Therefore the incomplete 4f shell is subjacent to the complete 6s shell, what is called submerged shell structure. This results in very rich scattering proper-ties [31,32] and a complex energy level scheme [29,59], as shown in Fig. 2.2. Since the bosonic isotopes have nuclear spin quantum number I = 0, the rel-evant quantum number is the total angular momentum quantum number J. Er has 110 states with J values ranging from 2 to 12. In Fig. 2.2 even parity3 states are indicated with red horizontal lines, odd parity states with black horizontal lines. The arrows mark transitions from the even parity ground state at J = 6 to odd excited states at J = 7. This is because in an electric dipole transition, 2The Bohr magneton is deﬁned as µB = eℏ 2me with e the elementary charge, ℏthe reduced Planck constant and me the electron rest mass. 3Parity is a symmetry property of the electronic wave function. The parity operator ˆ P inverts the wave function through the origin. If ˆ PΨ(⃗ r) = Ψ(−⃗ r) = Ψ(⃗ r) the wave function has even parity, if ˆ PΨ(⃗ r) = Ψ(−⃗ r) = −Ψ(⃗ r) it has odd parity . 14 2. Erbium Figure 2.2: The energy level scheme of Er. Even parity states are indicated with red horizontal lines, odd parity states with black horizontal lines. The arrows show electric dipole transitions investigated in , for our experiment the transitions at 401 nm and at 583nm are of special interest. Plot and data from and . which is the most dominant type of transition, the total angular moment J has to change by ±1 and the parity of the state must change . The diﬀerent energy levels arise from the occupation of diﬀerent shells and from the coupling scheme of the electrons. The well known Russell-Saunders or spin-orbit (LS-) coupling is common for light elements, where the Coulomb inter-action is dominant over the spin-orbit interaction. In this scheme, all individual electronic orbital angular momenta ⃗ l and spins ⃗ s of each electron add separately to a total orbital angular moment ⃗ L and a total spin ⃗ S, which ﬁnally couple to a total angular moment ⃗ J = ⃗ L + ⃗ S . The resulting state is denoted by the term symbol (2s+1)LJ. Accordingly with the usual spectroscopic notation , the L-states are labeled with letters as follows: . L value 0 1 2 3 4 5 6 ... symbol S P D F G H I ... In the case of heavy elements, the spin-orbit interaction is comparable or stronger than the Coulomb interaction. The resulting coupling scheme is known 15 2. Erbium as jj-coupling. A special case of this is the J1J2-coupling, which is relevant for many excited states of the lanthanides . In this scheme, the 4f electrons and the outer, remaining electrons couple independently in a LS-coupling scheme to states with total angular momentum ⃗ J1 and ⃗ J2. They add to a overall total angular momentum ⃗ J = ⃗ J1 + ⃗ J2 and the resulting state is denoted as (J1, J2)J term. In the ground state of Er, the 4f 126s2 electrons are coupled in a LS-coupling scheme to a 3H6 term with a total orbital momentum quantum number L = 5. This is exceptionally high, compared to the ground states of the alkali atoms, which all have L = 0. The excited states usually experience a higher spin-orbit interaction and follow the J1J2-coupling scheme. For instance, the excited state associated with the strong laser cooling transition at 401 nm is labeled as [Xe]4f 12(3H6)6s6p(1P 0 1 ) (6, 1)0 7, see Fig. 2.2. Here, one of the ground state 6s elec-trons is excited to the 6p shell. The 12 4f electrons couple in a LS-scheme to a 3H6 term, the 6s and the 6p electron couple in a LS-scheme to a 1P 0 1 term. These two states couple to a (J1, J2) 0 J = (6, 1)0 7 term, where J = 7 and '0 ' indicates the odd parity. 2.3 Isotopes Erbium possesses six stable bosonic and fermionic isotopes with relatively high masses m from 162 amu to 170 amu. Tab. 2.1 shows the natural abundance and nuclear spin quantum number I of the stable Er isotopes. element mass m [amu] abundance [%] nuclear spin I statistic erbium 162 0.1 0 boson 164 1.6 0 boson 166 33.5 0 boson 167 22.9 7/2 fermion 168 27.0 0 boson 170 14.9 0 boson Table 2.1: Natural abundance and nuclear spin of the stable erbium isotopes. Data from . In the electronic ground state, Er has a total angular momentum quantum number J = 6. No bosonic isotope shows hyperﬁne structure, being I = 0. The ground state of the fermionic 167Er with I = 7 2 splits up into eight hyperﬁne states from F = 19 2 to F = 5 2. Compared to other fermionic alkali systems that were brought to quantum degeneracy as 40K or 6Li , 167Er shows an exceptionally high abundance of 22.9 %. The richness in isotope number gives us more ﬂexibility in terms of scattering properties. This can be very facilitative, as in the case of Sr. Here, the least abundant of the four isotopes, 84Sr, has been the 16 2. Erbium ﬁrst to be Bose condensed thanks to its favorable scattering properties4 [17,18]. Later on, it even was used to cool the fermionic 87Sr by interisotope collisions to quantum degeneracy . Figure 2.3: Natural abundance versus isotope shift for the 401nm, the 583 nm and the 841 nm transition. 166Er is taken as a reference. Data from [32,66,67], no values known for 162Er and 167Er at the 401nm transition. The laser cooling transition frequencies depend on the particular isotope. The diﬀerent number of neutrons leads to diﬀerent atom core masses and charge distributions and in that way the transition frequencies are shifted, what is called isotope shift . Figure 2.3 shows the isotope shift of all stable isotopes for three transitions [32,66,67]. Taking 166Er as a reference, the isotope shift is always less than 2.2 GHz for the 401 nm and the 583 nm transition. This is convenient since we can reach all isotopes at each transition within the same mode of the laser system. 4The scattering properties determine the thermalization of the atoms and thus are the key parameter for evaporative cooling. 17 2. Erbium 2.4 Laser cooling of erbium atoms The group of Jabez J. McClelland has identiﬁed ﬁve diﬀerent transitions that could be used for laser cooling experiments, as illustrated in Fig. 2.2. Tab. 2.2 shows the accordant laser cooling parameters such as the natural linewidth ∆ν, the saturation intensity Isat, the Doppler temperature TDoppler and the recoil temperature Trecoil given in Ref. with corrections from Ref. [68,69]. transition parameter 400.91 nm 582.84 nm 631.04 nm 841.22 nm 1299.21 nm Γ (1/s) 2.2 × 108 1.2 × 106 1.8 × 105 5.0 × 104 13 τ 4.5 ns 0.857 µs 5.6 µs 20 µs 75 ms ∆ν 35.6 MHz 0.19 MHz 28 kHz 8.0 kHz 2.1 Hz Isat 72.2 mW/cm2 0.13 mW/cm2 15 µW/cm2 1.8 µW/cm2 0.13 nW/cm2 TDoppler 854 µK 4.8 µK 680 nK 190 nK 51 pK vDoppler 0.210 m/s 15 mm/s 5.8 mm/s 3.1 mm/s 50 µm/s Trecoil 357 nK 170 nK 140 nK 81 nK 34 nK vrecoil 6.0 mm/s 4.1 mm/s 3.8 mm/s 2.8 mm/s 1.8 mm/s Table 2.2: The laser-cooling parameters for several transitions in Er [29,68,69]: spon-taneous decay rate Γ, lifetime τ, linewidth ∆ν, saturation intensity Isat, Doppler tem-perature TDoppler, Doppler velocity vDoppler, recoil temperature Trecoil and recoil velocity vrecoil. The 401 nm transition has a broad natural linewidth ∆ν = 36 MHz, resulting in a strong scattering force Fscat = ℏk Γ 2 I / Isat 1 + I / Isat + 4 δ2 / Γ2 , (2.2) where Γ = 2 π ∆ν the spontaneous decay rate, Isat = π h c Γ / (3 λ2) the saturation intensity, I the intensity, ℏk the photon momentum and δ the detuning from the transition . Therefore, we use this transition for the ZS to slow the thermal atomic beam, for transversal cooling (TC) to focus the atomic beam and for the imaging. This transition can also be used for a MOT , but for our MOT we want to take advantage of the narrow linewidth ∆ν = 0.19 MHz of the 583 nm transition, which has a lower Doppler temperature5 TDoppler of 4.8 µK. Due to the high mass m of Er, the recoil temperature6 Trecoil of 170 nK is also very low. The ZS is designed to slow the atoms to 5 m/s, which is on the order of the capture velocity of the MOT. Therefore, it will be feasible and beneﬁcal to use the 583 nm transition to capture and further cool the slowed atoms in the MOT. 5The Doppler temperature TDoppler = ℏΓ 2 kB is the fundamental temperature limit for the Doppler cooling mechanism . 6The recoil temperature Trecoil = ℏ2 k2 m is the fundamental temperature limit for sub-Doppler cooling mechanisms . 18 2. Erbium In usual alkali gas experiments, the cooling transition needs to be optically closed. Otherwise atoms leaking to metastable states get lost from the cooling cycle. As demonstrated in Ref. , this requirement is much less restrictive for Er atoms. Considering the energy level diagram in Fig. 2.2 and the electric dipole selection rules7, it is clear that the 401 nm cooling transition is optically open. In particular, excited atoms can decay8 to any energetically lower lying metastable state of opposite parity and J = 6, 7 or 8. In Ref. the branching ratio for this transition is determined to 8 × 10−6, so in every 1.3 × 105 transitions to the ground state, the excited state once decays to a metastable state. Depending on the number of required cooling cycles, this can limit the atomic population. However, atoms in such metastable states can be trapped by the quadrupole magnetic ﬁeld of the MOT because of their high magnetic moment µ, as reported ﬁrst in Ref. and later in Ref. . Thus, non-resonant metastable atoms cascade back to the ground state, from where they can be recaptured by the MOT beams. So in the 401 nm transition, despite the numerous loss channels, the atoms can be recycled due to the high magnetic moment that conﬁnes the atoms to the vicinity of the MOT beams. We ﬁnd the 583 nm transition particularly suitable for MOT operation. Here, excited atoms can decay to two metastable states, but since the energy change such transitions is very small, the transition rate is also small according to the Fermi golden rule, leading to a branching ratio of 2 × 10−8. Erbium is very favorable for the use of polarization gradient cooling, since the magnetic moments of many excited states are very similar to the one of its ground state. The Land´ e g-factors9 of ground and excited state diﬀer only by 0.004 for the 401 nm transition and by 0.03 for the 583 nm transition10 . Sub-Doppler cooling mechanisms rely on Zeeman shifts of the ground state sublevels, whereas Doppler cooling relies on Zeeman shifts of the ground state versus the excited state. Therefore, there is a critical magnetic ﬁeld value above which the Doppler cooling dominates, as explained in [71,72]. Clearly, for smaller diﬀerences of the g-factors of ground and excited state, this critical magnetic ﬁeld value is higher and the sub-Doppler cooling is less sensitive to magnetic ﬁelds. Therefore, especially the 401 nm transition is very suitable for polarization gradient cooling, as demonstrated in Ref. and also in similar work on Dy . 7In emission, J can change to J′ = J ± 1 (stimulated emission) and also to J′ = J (except for J = 0) (spontaneous emission). The parity symmetry has to change in both cases . 8In a ﬁrst order approximation, we only consider decay via the electric dipole radiation and neglect the weaker decay via electric quadrupole or via magnetic dipole transitions, which would involve odd parity states with J values from 5 to 9. 9The magnetic moment µ is proportional to the Land´ e g-factor of the state, see Eq. (2.3). 10g[4f 126s2 3H6] = 1,16381, g[4f 12(3H6)6s6p(1P0 1) (6, 1)0 7] = 1,160, g[4f 12(3H6)6s6p(3P0 1) (6, 1)0 7] = 1,195 19 2. Erbium 2.5 Magnetic properties Compared with the maximum magnetic moment µ = 1 µB of alkali atoms, most of the lanthanide elements have high magnetic moments. Tab. 2.3 shows the electronic conﬁguration and the ground state magnetic moment µ of all lanthanide atoms . There is a direct relation between the vacancy of the 4f-shell and the magnetic moment . As we explain in more detail in App. A, the Hund 's rules determine the ground state quantum numbers, which determine the Land´ e g-factor of the state. Since µ = g MJ µB, (2.3) with g the Land´ e g-factor, and MJ the magnetic quantum number of the total angular momentum, the magnetic moment is related to the occupation of the 4f-shell. Erbium has µ = 7 µB, resulting from its 4f 126s2 electronic ground state occupation: According to Hund 's rules this conﬁguration leads to a 3H6 ground state term with MJ = 6. This gives a Land´ e g-factor of 1.16 and results in a magnetic moment of 6.98 µB. element La Ce Pr Nd Pm e.c. ([Xe]-) 5d16s2 4f 15d16s2 4f 36s2 4f 46s2 4f 56s2 µ [µB] 1 4 3 2 1 Sm Eu Gd Tb Dy 4f 66s2 4f 76s2 4f 75d16s2 4f 96s2 4f 106s2 0 7 5 10 10 Ho Er Tm Yb 4f 116s2 4f 126s2 4f 136s2 4f 146s2 9 7 4 0 Table 2.3: The ground state electronic conﬁguration ('e.c.') and ground state magnetic moment µ (rounded, in units of µB) of all lanthanides listed by rising atom number. All lanthanides have a complete Xe conﬁguration, µ / µB = g MJ, values from . The quantum behavior of Er gas is expected to be strongly aﬀected by its magnetic properties. The interaction energy of two magnetic dipoles aligned by a polarizing magnetic ﬁeld is Udd = Cdd 4 π 1 −3 cos2 (θ) \|⃗ r\|3 , (2.4) where Cdd = µ0 µ2 is the coupling constant, ⃗ r the relative coordinate of the two dipoles and θ is the angle between ⃗ r and the dipole axis . This means that for Er with µ = 7 µB the dipole-dipole interaction (DDI) is 49 times stronger than 20 2. Erbium for alkali atoms with µ = 1 µB since Cdd ∝µ2. Furthermore, the DDI is long-range and anisotropic since Udd ∝\|⃗ r\|−3 cos2 (θ). This has many consequences and is expected to give rise to novel collisional behavior. In the case of identi-cal fermions ultracold collisions are blocked by the Pauli exclusion principle due to the symmetry of s-wave scattering. The anisotropy of the DDI could enable such collisions by introducing higher partial waves . In the case of bosons, the long-range character of the DDI might fundamentally change the universal scaling law of Eﬁmov physics . The dipolar-length add = (m µ0 µ2) / (4 π ℏ), where µ0 is the magnetic con-stant, is a measure for the absolute strength of the DDI . Thus, the high mass m of Er further enhances its dipolar character. Additionally, the high µ and m should result in the observation of many Feshbach resonances already at low magnetic ﬁelds. Controlling the scattering length via Feshbach resonances is an important tool to reach quantum degeneracy and to produce tunable quantum gases . 2.6 Scattering properties Important quantities as the background scattering length a, which determines the interaction at low temperatures are so far unknown. Its determination is the ﬁrst goal of our experiment. The use of evaporative cooling is an important step for the achieve-ment of quantum degeneracy. It is mostly performed in magnetic traps, but for non-S-state atoms collisional relaxation of the magnetic moments leads to rapid loss of atoms in a magnetic trap [76,77]. Erbium is in an electronic H-state, but has a submerged shell structure, where the spherical symmetric 6s-shell could sup-press the interaction anisotropy of the H-state and eﬀectively make it a S-state. This competition between open and closed shells can also be seen in the devel-opment of the investigation of the spin relaxation rates of Er in magnetic traps. In 2004 Hancox et al. studied collisions of He with Er in a He buﬀer gas cooled magnetic trap at temperatures below 1 K . They successfully trapped Er and measured a suppression of the collisional transfer of angular momentum. There-fore, they concluded that in Er-He collisions the interaction anisotropy of the Er atoms is shielded by the two unpaired electrons. This gave hope that this would also hold for Er-Er collisions. However in 2010, Connolly et al. measured large spin relaxation rates of Er in a magnetic trap at temperatures of 500 mK . The rates are signiﬁcantly higher than those of highly magnetic S-state atoms and in striking contrast to the previous work. Although there are currently no theoretical predictions for Er-Er collisions, they explain the rapid loss by an ad-ditional spin relaxation mechanism despite spin exchange, dipolar relaxation or second-order spin-orbit coupling, which is induced by the electronic interaction 21 2. Erbium anisotropy. This means that magnetic trapping of Er will be highly ineﬃcient. Therefore, in our experiment, we will use an optical dipole trap (ODT) to perform evaporative cooling of Er. 22 Chapter 3 Experimental setup For the production of an Er BEC and further experiments on ultracold Er and Er-Li mixtures, we set up a vacuum apparatus and several laser systems. In the vacuum chamber, a thermal beam of Er atoms is produced in an oven and subsequently the atoms are slowed, trapped and further cooled by magnetic ﬁelds and laser light. In this Chapter, we give an overview of the experimental setup and the cooling strategy to produce a BEC of Er atoms. 3.1 Vacuum chamber Figure 3.1 shows a drawing of the complete vacuum chamber, which is composed of ﬁve main sections: (a) Er oven chamber, (b) Li oven chamber, (c) Zeeman slower (ZS), (d) magneto-optical trap (MOT) chamber and (e) pumping chamber. The Er oven chamber consists of a commercial oven system, which produces an Er atomic beam by the eﬀusion of solid Er at temperatures up to 1350 ◦C. A detailed discussion of the Er oven is found in Chap. 4. In brief, the oven system consists of a crucible surrounded by a water cooled thermal shield. The crucible is divided in two parts: the eﬀusion cell, where solid Er is placed and the hot lip, which contains two heated apertures. The emerging atomic beam has a most probable velocity of about 415 m/s at a temperature of 1350 ◦C. The oven system is followed by an octagon, which gives optical access at ﬁve CF40 viewports. Here, as indicated with the blue arrows, we can apply transversal cooling (TC) using the strong 401 nm transition. By cooling the atoms transversely with red detuned light in a 2D optical molasses, their transversal velocity components are reduced and thus the atomic beam is further collimated. Via the moveable bellow on top, it is possible to place an Al mirror at the beam axis under an angle of 45 ◦in respect to the beam axis. In this way, we can check the atomic beam path towards the MOT chamber and the injection of the ZS light. On the 23 3. Experimental setup Figure 3.1: Drawing of the complete vacuum chamber, which consists of: (a) Er oven chamber, (b) Li oven chamber, (c) ZS, (d) MOT chamber and (e) pumping chamber. Red arrows indicate the 1064nm light, which we use for the ODT, yellow arrows show the 583 nm MOT beams, blue arrows mark the 401 nm light for the ZS and TC. 24 3. Experimental setup backside of the octagon we connect an ion pump1, an ionization gauge2 and an all metal angle valve, which is used to attach a turbomolecular pump system3 for pre-pumping. The Er oven chamber is separated by an all metal transition valve from the rest of the chamber to allow reﬁlling of the oven without loosing the vacuum conditions in the whole chamber. In the adjacent Li oven chamber, we can admix Li atoms to the atomic beam. The Li oven4 is situated below the beam axis and emits Li atoms in vertical direction. In a tube along the beam axis, which is divided alongside in three parts, Er and Li are combined to a joint atomic beam. In 2/3 of the tube 's cross section, the Er atomic beam travels from the Er oven towards the ZS and MOT. Li atoms can enter the remaining 1/3 of the tube from below and are collimated to a longitudinal beam by microtubes inserted along the beam axis . At the end of the tube, Er and Li atoms emerge towards the ZS and merge into a two species atomic beam. In the prime setup, where we focus on Er itself, this Li oven section is replaced by an ordinary tube. After the Li oven section, a four-way cross connects another pump section with an ion pump1, an ionisation gauge2 and an angle valve. On top, a Ti-sublimation pump5 gives further pumping capacity on demand. From below, the atomic beam shutter can mechanically block the atomic beam via a tiltable bellow and a servo motor. Past this, another cube with four viewports allows further transversal cooling to collimate the beam. A bellow, which we use for the alignment of the complete chamber, connects the second valve, which is integrated for the same reason as the prior one. The ZS is a spin-ﬂip type σ−slower with an independent magnetic oﬀset ﬁeld. For the ZS we use the 401 nm light, red detuned by about 16 ∆ν b = 580 MHz. The ZS is water cooled and contains a diﬀerential pumping tube. This 8 mm in diameter and 30 cm long tube maintains a pressure gradient of 103 between the oven chambers, where pressures of 10−9 mbar are suﬃcient, and the experimental chamber, where pressures of about 10−12 mbar are needed. After the ZS, the slowed Er atomic beam is expected to have a longitudinal velocity of 5 m/s. The slowed atoms are captured in the MOT chamber. We use the narrow 583 nm transition for MOT operation. The MOT consists of a six beam optical molasses and a quadrupole magnetic ﬁeld. Depending on the available laser power and on the desired MOT size, the beams each have a diameter of up to 30 mm and the coils generate a magnetic ﬁeld gradient of about 10 G/cm. The Feshbach-and the curvature coils give an homogeneous oﬀset magnetic ﬁeld for the use of Feshbach resonances and add curved magnetic ﬁelds for compensation 1Varian Star cell 20 l 2Varian UHV-24 3Pfeiﬀer HiCube classic 4The Li oven has the same design as the one already used in the FeLiKx experiment in Innsbruck . 5Varian TSP ﬁlament-type 25 3. Experimental setup or magnetic trapping. A pair of compensation coils, which compensates the magnetic ﬁeld of the ZS is mounted on the front and back of the MOT chamber. All coils are water cooled. To compensate for residual magnetic ﬁelds of any direction, we install a compensation cage around the MOT chamber, which is not shown Fig. 3.1. To cool the atoms towards quantum degeneracy, we load the atoms from the MOT into a crossed optical dipole trap (ODT) at 1064 nm. For this, another four, smaller, CF16 viewports give optical access for two beams, which are sketched as red arrows. To provide and measure pressures as low as 10−12 mbar, the pumping chamber is equipped with an ion pump6, a Ti-sublimation pump5 and an ionisation gauge2. For pre-pumping and the baking of the experimental chamber, a turbomolecular pump3 can be attached by an angle valve. Since not all atoms from the atomic beam are captured by the MOT, but a large part will hit the end of the chamber, it is not possible to inject the 401 nm light for the ZS directly from the backside, because a coated window looses transmittance rapidly. A coated mirror suﬀers only small loss of reﬂectivity, as we measured in a test setup with the movable mirror from the Er oven section. The reﬂectivity of the Al mirror coated with Er was only 3 % lower than that of the uncoated mirror. Therefore, in the four way cross, we inject the ZS light towards the atomic beam with a 45 ◦Al mirror from the side. 3.2 Cooling and trapping lights A single-mode single-frequency (smsf) frequency doubled diode laser7 provides light at 401 nm. It has an output power of 120 mW, which we use for ZS, TC and imaging. The linewidth of the laser diode of about 100 kHz is well below the natural linewidth of the 401 nm transition in Er. To prevent long-term frequency drifts we lock the laser to the atomic transition by a phase modulation transfer locking scheme. Figure 3.2 shows the optical setup of the locking scheme. To lock the laser to the atomic transition frequency, we generate free Er atoms that can get resonant with the laser light. Since Er has a very high melting point temperature, it is inapplicable to use a vapor cell, as it is done for all alkali atoms . Therefore, we use a hollow cathode lamp (HCL), which is a glass tube with Er coated cathodes and is ﬁlled with Ar buﬀer gas at a pressure of 4 mbar. By applying high voltage8 to the electrodes, we produce an Ar plasma. The Ar ions are accelerated towards the cathode, hit it and sputter oﬀEr atoms. This generates free, neutral Er atoms that we can use to lock the laser. To obtain an error signal for the lock, we use modulation transfer spectroscopy [81, 82], a 6Varian Star cell 75 l 7Toptica DL-SHG pro 8Typical values are a voltage of 125 V at a current of 7 mA. 26 3. Experimental setup Figure 3.2: Scheme of the optical setup for the modulation transfer lock. In a hollow cathode lamb (HCL), free Er atoms interact can get resonant with the lower beam branch, the probe beam . The upper beam branch, the pump beam, is modulated with an electro optic modulator (EOM) before it enters the HCL. The probe beam signal is detected with a photodiode (PD). By mixing it with the local oscillator (LO) signal used for the EOM we gain an error signal for locking the laser to a certain Er transition. pump-probe technique, which gives a Doppler-free spectrum. For this, the light from the laser is split with a polarizing beam splitter (PBS) into a probe and a pump beam. The pump beam is phase modulated with an electro optic mod-ulator (EOM), which is driven with a local oscillator (LO) at a frequency of about 14.7 MHz. With another PBS we overlay counter propagating pump- and probe beam in the HCL. If the light is resonant with the Er atoms, the phase modulation of the pump beam is transferred to an amplitude modulation of the probe beam by nonlinear eﬀects in the atoms. We detect the probe beam with a photodiode (PD) and mix it with the LO signal to allow phase sensitive detec-tion. The generated signal has a ﬂat baseline and a zero-crossing exactly at the transition frequency and thus can be fed directly to the laser system as an error signal for stable long-term locking. A typical signal of this modulation transfer spectroscopy is shown in Fig. 3.3. Here, four of erbium 's ﬁve stable bosonic iso-topes are clearly visible at the major zero-crossings of the signal, separated by the isotope shift. The residual zero-crossings, mainly in-between 166Er and 168Er, arise from the fermionic 168Er, which is split into several hyperﬁne states. We can lock the laser to a certain isotope by the steep signal slope at the corresponding zero-crossing. The MOT light at 583 nm is generated by a dye laser9, which we pump with a 10 W smsf solid state laser10. The current master work of Alexander Rietzler is the development of a locking scheme for the dye laser. It needs to have smsf op-eration at an output power of 700 mW and a stability and linewidth of 100 kHz11. 9Radiant Dyes cw-ring-laser 10Coherent Verdi V10 11This is required, since the natural linewidth of the 583 nm transition in Er is 170 kHz. 27 3. Experimental setup Figure 3.3: A typical signal of the modulation transfer spectroscopy on the HCL. Four of erbium 's ﬁve bosonic isotopes are clearly visible at the major zero-crossings of the signal, separated by the isotope shift. The residual zero-crossings, mainly in-between 166Er and 168Er, arise from the fermionic 168Er, which is split into several hyperﬁne states. Due to the signal's steep slope at the transitions and its ﬂat baseline, it provides an excellent error signal for the locking of the laser system. For this, he will stabilize and lock the laser with an external cavity. The light for the ODT at 1064 nm is produced by a smsf solid state laser12. Since a ODT operates far oﬀdetuned and the laser oﬀers a good intensity stability, no special frequency- or intensity stabilization is needed. 12InnoLight MEPHISTO MOPA 18 W 28 Chapter 4 Erbium oven We produce a thermal Er atomic beam from a commercial oven operating at about 1200 ◦C. The oven consists of an eﬀusion cell, where few gram of solid Er are sublimated, and two separately heated apertures, which collimate the atomic beam. For thermal shielding, the high-temperature oven is water cooled. In this Chapter, we ﬁrst describe the setup, the working principle and the home built control- and safety system. Later, we discuss the atomic beam shutter, which we use to block the atomic beam prior the ZS. 4.1 Oven system The complete oven system is manufactured by CreaTec Fischer & Co. GmbH1. The oven has a special design to fulﬁll the requirement of high-temperature op-eration, as needed for Er, which has a melting point temperature of 1529 ◦C . The central part is the eﬀusion cell (EC), where solid Er is heated. The adja-cent hot lip (HL) consists of two apertures that collimate and direct the atomic beam emerging from the EC towards the ZS tube. For heat shielding, the whole crucible is surrounded by a water cooled cylinder. Figure 4.1 shows a technical drawing of the detailed setup. The tantalum (Ta) crucible consists of the EC part, where we can insert 10 cm3 of solid material, and the HL part, where a spacer ﬁxes the ﬁrst and sec-ond aperture. The apertures both have 5 mm inner diameter and are separated by 50 mm. The EC and the HL part are heated separately with Ta wires, ﬁxed by a pyrolytic boron nitride spacer that provides mechanical stability and electrical insulation even at high temperatures. This whole section is surrounded by a wa-ter cooled cylinder and integrated in a steel tube with a CF40 ﬂange connection to the main chamber. A bellow allows to adjust the orientation of the oven. The power supply for the heating wires and the temperature sensors are feed through 1Model: Dual Filament Eﬀusion Cell DFC-40-10-290-WK. 29 4. Erbium oven Figure 4.1: Schematic, sliced view of the Er oven. It consists of a crucible with EC and HL, two separate heating wires, a water cooling system and a tiltable housing with connections for power supply and temperature sensors. at the back end of the oven by ceramic spacers. The heating wires heat the EC and the HL radiatively. The speciﬁed tem-perature range is 100 −1400 ◦C, the PID controller provides a stability of better than 0.1 ◦C. Temperature ramps should not exceed 30 ◦C/min, to allow thermal-ization with Er and reduce the stress on the crucible. The maximum EC and HL heating wire currents and the corresponding heating powers are 3.5 A and 6.1 A as well as 84 W and 220 W respectively . Typical operation temperatures for the EC range between 1000 ◦C and 1350 ◦C, depending on the required ﬂux. This temperature is still well below the melting point of Er of 1529 ◦C because liquid material can break the crucible when cooling down the oven. Since we can heat EC and HL separately, we typically use the HL at 100 ◦C higher temperatures than the EC to prevent material condensation at the apertures. This is an es-sential feature, since in related experiments2 with Cr, condensed Cr has been observed to grow over the aperture and to lead to a short circuit of the heating wires. The thermocouple temperature sensors are attached outside of the crucible. This implies that the measured temperature can diﬀer from the actual tempera-ture of the Er inside the crucible by few percent and we thus can use it only as a guiding value. The water cooling system shields the vacuum chamber from the high temperatures. It requires a water ﬂow of 2 l/min at 5 −10 ◦C water temper-ature and maintains a temperature of less than 100 ◦C all over the oven section vacuum chamber. The oven system has to be operated in vacuum, it is speciﬁed from a maximum pressure of 1×10−5 mbar down to the UHV regime. To provide this, we use a turbomolecular pump and an ion pump, as discussed in Sec. 3.1. The high temperatures involved also require a careful material engineering. We chose Ta for all parts of the crucible and for the wires. Tantalum has the ad-vantage to have a very high melting point of more than 3000 ◦C and to not create 2Private communication: Tilman Pfau, Universit¨ at Stuttgart. 30 4. Erbium oven low-temperature-melting alloys with Er, as demonstrated by the Er experiment at the NIST. In contrast, Ta and Cr form a low-temperature-melting alloy, so in this case, ceramic or other materials have to be used for the crucible . A valve separates the oven chamber from the UHV part, so that we can keep the vacuum in the MOT chamber while reﬁlling the oven. For the ﬁrst oven oper-ation, we clean the parts in a supersonic bath of acetone ﬁrst, then of isopropanol and work in a clean environment. We cut Er pieces from an Er block and ﬁll the bare crucible with typically 6 g of Er before we reassemble the crucible and the oven system. The insertion or removal of the crucible into the oven system must be done carefully in order to guarantee collinear alignment, since the heating wires can easily be damaged. Also the power and temperature connection must be handled with special care, since the ceramic spacers of the feedthroughs can break. 4.2 Control system To control the high-temperature oven system and to secure it against power break down and cooling water failure, we set up a control system as schematically il-lustrated in Fig. 4.2. Figure 4.2: Structure of the Er oven control system. We control the oven from the computer via the oven control. We protect the oven control against power failure with an uninterrupted power supply (UPS), which we coordinate with the safety systems. A CreaTec power supply unit with two power circuits supplies the heating wires of EC and HL. It is controlled by two Eurotherm precision proportional-integral-derivative (PID) controllers, which get feedback from the thermocouple sensors. The controllers can be monitored and set manually onboard, or by a computer connected via Ethernet. The iTools OPC Server program manages the PID controller variables, which we read and write in the LabVIEW oven 31 4. Erbium oven control program. In this way we monitor the actual temperature, the working setpoint temperature and the current output power and can set the temperature setpoints3 and the temperature ramps. Power failure can cause repeatedly, abrupt temperature ramps that damage the crucible. Therefore, we backup the power supply and the PID controller with an uninterrupted power supply (UPS), which is connected to the computer via an USB port. Here, the MGE UPS program manages the UPS variables, and executes the command line in case of variable changes. The command line itself starts a LabVIEW subVI executable that updates the UPS status in the Lab-VIEW oven control program. Therefore, in case of power failure, if the remaining supply time of the battery falls below a preset value, we cool down the EC and HL heating wires in a controlled way to the standby temperature, from which a sudden power-oﬀdoes not harm the crucible. The status of the cooling water system and the emergency oﬀsystem are con-nected to the interlock of the UPS. In case of failure of the cooling water supply for the oven, as too low water backﬂow or leaks, this triggers an instant power oﬀ of the UPS and thus of the heating wires. This is crucial since without suﬃcient cooling water ﬂow, the oven will overheat and vaporize the cooling water, which can damage the oven. Hence, to operate the oven, the emergency-oﬀsystem has to be inactive, the cooling water must run, the UPS and the power supply unit have to be switched on and the program iTools OPC Server must run. Then, the LabVIEW oven control program can be started to control the heating wires. Figure 4.3 shows the user interface of the oven control program, App. B gives a detailed description of the program code. The user sets the EC and HL tem-peratures and the heating or cooling ramps in the commensurate displays of the two upper sub windows. Note that both values are protected against fault input. For operation at preset temperature setpoints, the 'ON ' button changes between default temperature setpoints for working or standby operation. Further displays show the present temperatures, working setpoints, target setpoints, the thermo-couple sensor status and the actual output power. The central part is a graph showing the actual and the working setpoint temperature in real time. In Fig. 4.3, the graph displays the initial heating up phase from 'OFF ' to 'ON ' mode. The straight lines mark the evolution of the working setpoints, which result of the tar-get setpoints and the temperature ramps. The actual temperature values oscillate around the ideal values since the current limitation on the power supply is set to values that are optimized for the 'ON ' mode. EC and HL temperature oscillate alternately; this shows that although the two PID controllers work independent of each other, the EC and HL system are coupled via radiation inside the oven. The 'enable values ' button in the lower left of the screenshot allows to adjust 3The setpoint is the ﬁnal obliged value, the working setpoint is the present obliged value. 32 4. Erbium oven Figure 4.3: A screenshot of the oven control program user interface. We can set and monitor EC and HL temperatures on the computer. The program also reacts to the status the UPS. the preset values for the temperature limits and the 'ON/STANDBY ' temper-atures. The 'EMERGENCY OFF ' button at the lower middle of the interface sets the temperature setpoints and working setpoints to zero. This switches oﬀ the heating and the oven cools down as fast as possible. The subwindow 'UPS status ' shows the status of the UPS variables. The program automatically reacts to changes of the backup time of the UPS. 4.3 Atomic beam shutter For each experiment cycle, we will load the MOT with atoms coming from the Zeeman slowed beam. After the MOT phase, the atoms are lead into an ODT. At this point, the atomic beam should be blocked to leave the already optically trapped atoms unperturbed. Therefore, it is of prime importance to install an atomic beam shutter between Li oven and ZS section. The beam shutter is a mechanical shutter, which is driven by a servo motor. A control box connected 33 4. Erbium oven (a) (b) (c) Figure 4.4: Technical drawings of the atomic beam shutter. (a) The shutter unit, in position (1): 'blocked beam ' and in position (2): 'unblocked beam '. (b) The servo motor with mounting. (c) The complete assembly to the vacuum chamber, here the atomic beam enters from the front and is blocked. with the experiment computer controls the shutter operation. Figure 4.4 (a) shows the beam shutter for the 'blocked beam ' and the 'un-blocked beam ' position. The disc has a diameter of 20 mm and since this is considerably larger in diameter than the 275 mm distant diﬀerential pumping tube of the ZS with 8 mm in diameter, this ensures adequate blocking of the atomic beam. A little bar connects the disc with the lower end of a bellow, which we attach to the vacuum chamber. Two ball bearings on the side arms, which we tighten with springs4 allow tilting of the bellow. Figure 4.4 (b) shows the mounting of the servo motor. It is ﬁxed to the posts of the chamber, and connected to the lower end of the bellow with a little bar. The servo motor has a metal gear to provide a long lifetime, and a torque of approximately 150 Ncm, which is suﬃcient to tilt the bellow. It is speciﬁed to rotate by 40 ◦in 0.17 s at a supply voltage of 6 V, this sets the limit of the shutter speed. As common for servo motors, it is controlled with a pulse width modulated (PWM) signal. This signal has a period of 20 ms and an amplitude of 5 V. By modulating the width between 1 ms and 2 ms, we can set the position of the shutter. Figure 4.4 (c) illustrates the complete assembly of the shutter system. In this view, the atomic beam enters from the front and is blocked by the shutter. The atomic beam shutter control box connects the servo motor with the dig-ital output box of the experiment control. For this, we use a microcontroller5 (µC), which coordinates manual or digital adjustments and control. In App. C.1 the circuit diagram and the operating guidelines for the electrical details and right 4Not shown in the drawing. 5Atmel AVR ATmega128 34 4. Erbium oven (a) front view (b) side view Figure 4.5: The atomic beam shutter control box with all connections, status LEDs and most switches. Switches for the adjustment of the shutter position are inside the box. usage are given. The source code for the µC, which we wrote in the programming language C, and comments on it in App. C.2 give insight into the ﬁrmware struc-ture. Below, we describe the features of the control box according to Fig. 4.5 and give a brief overview of the ﬁrmware structure. The control box is powered with 5 V to 8 V via a BNC connector. This di-rectly supplies the servo motor, whereas we supply the electric circuit with a 5 V voltage regulation. This division avoids unintentional resets of the µC and pro-tects the µC against too high voltage. In case of a low signal on the emergency oﬀconnector, the control box aborts operation and blocks the atomic beam. The experiment control sends a TTL signal to the control box. It has either a 'high ' or a 'low ' value and thus triggers the shutter position 'blocked beam ' and 'un-blocked beam '. To avoid groundloops, we do not connect the TTL signal directly to the µC, but via an opto-isolator6, therefore it is inverted, the same holds for the emergency oﬀsignal. The switch 'pc/manual ' changes from computer control to manual control, where the shutter position is set by the 'unshut/shut ' switch. To calibrate the shutter position, we open the box, choose the manual mode and adjust the 'unshut ' or 'shut ' position by turning on the according potentiometer inside the control box. The connector 'con3 ' in Fig. 4.5 (b) oﬀers to reprogram the µC via the programming adaptor without opening the box. Aside, the servo is attached. This connector7 supplies the servo with power and sends the PWM control signal. The status LEDs display position and mode of the shutter and the shutter control. At initialisation, all LEDs shine for 1 s, blinking LEDs indicate an emergency blocking. All these features result from the ﬁrmware of the µC. The basic working scheme relies on the following segments. A main while-loop uses interlaced if-loops to check the status of the control variables ordered by their importance: 6An integrated circuit (IC), providing galvanic separation by translating electrical informa-tion into optical information and backwards. 7For convenience, the connector is a case insensitive six pin connector, although only three pins for power, ground and signal are needed. 35 4. Erbium oven 'emergency oﬀ', 'pc control ', 'manual control ', 'shutter positioning '. To control the servo, a timer interrupt runs every 10 ms8. If the value of the variable 'shut-terpos ', which determines the blocking of the atomic beam, changed within the last second, it sends the appropriate signal to the servo motor. Otherwise, it sends no signal and so leaves the motor at rest to avoid needless vibrations. The interrupt, the PWM signal output and the analog to digital conversion9 (ADC) are all features provided by the µC itself and are initialized together with the pin conﬁguration in the beginning. In the end of the source code, we deﬁne the user deﬁned functions for the positioning of the shutter via the potentiome-ters and the setting of the shutter position according to the manual- or the TTL signal input. The included header ﬁle deﬁnes constants, macros, variables and functions. 8This delay is negligible compared to the angular speed of the servo motor. 9The ADC converts the analog voltage values of the potentiometers, which we use for ad-justment of the shutter position, to digital values, which the µC can process. 36 Chapter 5 Characterization of the atomic beam We characterize the Er atomic beam using diﬀerential absorption spectroscopy. Already well below the melting point of Er, we observe a comparatively high atomic ﬂux from the oven and an unexpected large spatial divergence of the atomic beam, indicating a large Doppler broadening. Based on our observations, we designed an optimized, new generation atomic beam collimation setup, which includes two apertures and a reducing-ﬂux tube. In this Chapter, we describe the measurement setup and present our observations on beam divergence and atomic ﬂux, performed on both the original and the optimized collimation setup, here referred to setup 1 and setup 2 respectively. A comparison of the performance of both setups is also given. 5.1 Measurement setup To measure the atomic ﬂux emerging from the Er oven and the divergence of the atomic beam, we perform spectroscopic measurements using the 401 nm transi-tion. This is a comparatively strong transition with a linewidth of 36 MHz and a saturation intensity of 72 mW/cm2. It is well suitable for such a measurement since the resulting spectra provide strong signals. Figure 5.1 illustrates the optical setup. As discussed in Chap. 4, the Er oven system creates an atomic beam trav-elling longitudinally towards the MOT chamber, see Fig. 3.1. Our spectroscopic measurements are performed in the octagon chamber, which is located just before the Li oven section and the ZS tube. Here, ﬁve viewports guarantee good optical access. The 401 nm light enters the chamber and is intersected transversely with the atomic beam. If the light is resonant with the atomic transition, the atoms absorb the light, which then is spontaneously re-emitted. 37 5. Characterization of the atomic beam (a) side view (b) front view Figure 5.1: The optical setup of the diﬀerential absorption spectroscopy and the imag-ing of the atomic beam ﬂuorescence. (a) In the sliced side view of the oven chamber, a collimated atomic beam, drawn in red, leaves the Er oven. We split 401 nm light by a PBS into a reference beam, which falls directly on a PD and into a probe beam. The probe beam enters the octagon where it interacts with the atomic beam before it is detected by a PD. The diﬀerence of the two PD signals gives a signal free of global inten-sity ﬂuctuations. (b) As shown in the front view, a CCD camera images the ﬂuorescence perpendicular to laser and atomic beam, which travels into the plane of projection. Absorption spectroscopy We measure the absorption of light in a diﬀerential absorption spectroscopy setup as illustrated in Fig. 5.1(a). The absorption of light is detected by the photodiode PDprobe. By scanning the laser frequency, we record the absorption spectrum of the 401 nm transition for all stable isotopes. Since the atomic beam and the laser beam are perpendicular to each other, the spectral width provides information on the transversal velocity of the atoms and on the transversal Doppler broadening. To remove the background of the absorption signal, we use a diﬀerential setup. Here, a PBS splits the 401 nm light, transported with a ﬁber1 from the laser-table to the experiment-table, into a probe beam and a reference beam. The intensity ratio of the two beams is adjustable by a half-wave plate. The reference beam power is measured directly by PDref. The probe beam passes through the chamber and interacts with the atoms before it is detected at PDprobe. Lenses2 focus the laser beams onto the photodiodes. Within few percent, both beam paths have the same length of ≃45 cm to equalize distortion eﬀects that scale with 1In the setup characterized in Sec. 5.2 we use a polarization-maintaining singlemode Thor-labs PM-S350-HP-custom ﬁber of 10 m length. In the setup characterized in Sec. 5.3 we use a polarization-maintaining singlemode Sch¨ after + KirchhoﬀPMC-400Si-3.1-NA010-3-APC-800P ﬁber followed by an additional PBS to clean the polarization further. 2With a focal lenght of 50 mm. 38 5. Characterization of the atomic beam the length. The diﬀerential signal PDprobe −PDref is free from global intensity changes, but is sensitive to intensity changes of the individual beams. To improve the signal further, we shield the chamber and the photodiodes from stray light and install an additional aperture in front of PDprobe to avoid back-reﬂections into the chamber. The parameters and settings of the actual measurement are as follows. The laser beam power before the PBS is 60 µW, the radius of the laser beam is about 0.07 cm. Therefore, the intensity of the light is I = 1 mW/cm2, which is well below the saturation intensity, I ≪Isat. To operate the laser on resonance, we lock the frequency externally via the HCL to the 401 nm transition of 166Er, as discussed in Sec. 3.2. To scan the frequency, we use the scan mode of the laser system that simultaneously tilts the grating and changes the diode current of the seed diode laser. This results in a triangular frequency signal with more than 5 GHz peak-to-peak frequency amplitude and a period of 3 s at default settings. By this, the laser frequency can be scanned over the resonance without mode-hops in the doubling cavity. The diﬀerential PD signal is recorded with a digital storage oscilloscope. To achieve a good signal-to-noise ratio, all spectra are obtained by averaging the signal over more than 30 frequency scans. To calibrate the laser frequency in the scanning region, we measure the wavelength with a wavemeter3 and combine this with the scan rate of the laser system. As discussed in more detail in App. D.1, this gives the frequency axis to the spectra recorded with the oscilloscope. Fluorescence imaging Additional to the absorption spectroscopy, we also image the ﬂuorescence of atoms emitted in random directions with a CCD camera4. As shown in Fig. 5.1 (b), we mount the camera in front of an octagon chamber viewport, perpendicular to the laser beam. By this, we determine the laser-atom interaction length and estimate the cross section of the atomic beam. We place the camera about 20 cm from the chamber and we use a camera objective with large focal length, focused to the center of the chamber. A cover shields the camera from stray light. We calibrate the pixel size and the exposure time of the camera as described in detail in App. D.2. The exposure time needs to be short, not to saturate the CCD chip at high ﬂuorescence light intensities and it needs to be long enough to detect low intensities. In setup 1, the optimal exposure time is 100 ms and the pixel size is 1 px b = 58.1 µm. In setup 2, the optimal exposure time is 1 s and the pixel size is 1 px b = 42.8 µm. Figure 5.2 shows a typical CCD image of the ﬂuorescence light of the atomic beam. In this view, the atomic beam is entering the image from the right side, 3HighFinesse WS/7 4MATRIX VISION blue fox 39 5. Characterization of the atomic beam laser beam atomic beam Figure 5.2: A typical CCD image of the ﬂuorescence light of the atomic beam. In this view, the atomic beam is entering the image from the right side, the laser beam from the bottom, as indicated by the arrows (not in scale). Reﬂections of the light in the chamber are removed by a reference image. Only the parts of the atoms whose resonance frequencies are Doppler shifted by less than the natural linewidth of the transition are resonant and ﬂuoresce. From such an image we derive the light-atom interaction length. the laser beam from the bottom. Reﬂections of the light in the chamber are removed by a reference image. Only the parts of the atoms whose resonance fre-quencies are Doppler shifted by less than the natural linewidth of the transition are resonant and ﬂuoresce. From such an image we derive the light-atom inter-action length. In combination with the diﬀerential absorption measurement, we can deduce the ﬂux of the atomic beam, as discussed in Sec. 1.3 and in Sec. 5.4. 5.2 Collimation setup 1 The collimation setup of the oven consists of a number of apertures placed right after the eﬀusive oven. It determines the divergence of the atomic beam and its ﬂux. The setup has to compromise between high ﬂux and low atom loss. On the one hand, highly divergent atomic beams are useless in the experiments since only atoms in the central part of the beam can enter the ZS tube because of geometric limitation. On the other hand, strong collimation leads to an insuﬃcient atomic ﬂux. In setup 1, we place two apertures in the oven section as schematically illus-trated in Fig. 5.3. The ﬁrst aperture, aperture 1, has an inner diameter b1 ≃5 mm and it is situated between the EC and the HL. At a distance a ≃50 mm aperture 2, is installed with an inner diameter b2 ≃5 mm. This second aperture has the main scope of collimating the atomic beam. At a distance of l ≃83 mm from aperture 2, the probe laser beam transversely intersects the atomic beam. To characterize this setup before mounting the complete vacuum chamber, we as-sembled ﬁrst the Er oven chamber, named section (a) in Fig. 3.1. Instead of the valve terminating the chamber, we placed a CF40 viewport. As already pointed out in Chap. 1, the oven operates in the eﬀusive regime. Using Eq. (1.1) and Eq. (1.2), we estimate the mean free path to be λmf ≈4 cm 40 5. Characterization of the atomic beam Figure 5.3: Scheme of collimation setup 1. Aperture 1, which separates EC and HL section and aperture 2, which collimates the atomic beam have both an inner diameter of b1 = b2 ≃5 mm and are spaced by a ≃50 mm. At a distance of l ≃83 mm from the aperture 2, the probe laser beam transversely intersects the atomic beam. The atomic beam diameter is expected to be d1:2 = 22 mm. at a temperature of 1350 ◦C. Since the aperture dimensions are lower than λmf by an order of magnitude, we assume to be in the eﬀusive regime. Beam divergence The atomic beam diameter at a certain distance from the EC is a measure of the beam divergence. By Eq. (1.8), for the geometry of setup 1 as illustrated in Fig. 5.3, we expect a diameter of d1:2 = 22 mm at a distance of a + l from the EC. We estimate the actual beam diameter dex by imaging the ﬂuorescence of the atomic beam at the intersection with the laser beam. Since atoms with a radial velocity component vx ̸= 0 experience a Doppler shifted resonance frequency ν′ 0, we need to scan the laser frequency on a range exceeding the Doppler shift to image the complete atomic beam spectrum. Figure 5.4 shows such a measure-ment. Frames (a) to (i) represent negative ﬂuorescence image sections, which are recorded in a time sequence. The atomic beam enters each frame from the right, the laser beam from the bottom, identical to Fig. 5.2. The laser frequency is scanned and increases from frame (a) to (i). Therefore, diﬀerent sections of the atomic beam with diﬀerent Doppler shifted resonance frequencies ﬂuoresce on each frame, leading to an apparent motion of the ﬂuorescence. The inhomo-geneous shape of the ﬂuorescence in frames (d) and (e) results from the fact that during the exposure time of 100 ms, the laser frequency is scanned by 200 MHz over the transversal atomic beam proﬁle, leading to a distorted image. From frame (b) to (h) the ﬂuorescence is visible in between the red dotted lines, spaced by 41 mm, which serve as a guide to the eye. However, the ﬁeld of view of the CCD camera is of equal size and the edges of the total ﬂuorescence region, are sharp and not smooth. Thus, it is likely that the actual atomic beam diameter is larger than the ﬁeld of view. Therefore, we determine the lower bound of the atomic beam diameter to be dex ≳41 mm. 41 5. Characterization of the atomic beam 1cm (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 5.4: Fluorescence images, which are recorded in a time sequence. The atomic beam enters each frame from the right, the laser beam from the bottom. The laser frequency is scanned and thus is increased from frame (a) to (i). Therefore, diﬀerent sections of the atomic beam with diﬀerent Doppler shifted resonance frequencies ﬂuoresce on each frame. The red dotted lines serve as a guide to the eye and signalise the region of ﬂuorescence. A scale is given in frame (a) and to improve visibility we show the negative images. Figure 5.5: Scheme of the actual atomic beam in collimation setup 1. Atomic vapor ﬁlls both EC and HL part leading to an atomic beam that emerges from aperture 2 and is collimated at the inner edges of the octagon at a distance a3 ≃66 mm from aperture 2 with a diameter of b3 ≃38 mm. For geometrical arguments, the resulting atomic beam is highly divergent and has a large diameter d2:3 = 49 mm. 42 5. Characterization of the atomic beam Figure 5.6: Transversal absorption spectrum at TEC = 1350 ◦C and THL = 1450 ◦C. The black line shows the diﬀerential signal, the red curve is a ﬁve peak Voigt function ﬁt to the data. Five of the six stable Er isotopes are clearly resolved. The signal is normalized to the maximum absorption corresponding to a maximum optical density ODmax ≃0.21. This is a factor of 2 bigger than the value of d1:2 = 22 mm, which was expected for this collimation setup. A possible explanation for the large value of dex is to consider both the EC and the HL behaving as an eﬀusive oven, as schematically illustrated in Fig. 5.5. As a consequence, the ﬁrst aperture will not play any role in the collimation since similar to the EC, also the HL is ﬁlled with atomic vapor. This leads to an eﬀectively shifted collimation setup: the atoms emerge from aperture 2 and are collimated at the proximate limiting structure, which is the octagon. The inner edges of the octagon chamber, which have a diameter of b3 ≃38 mm and a distance of a3 ≃66 mm from aperture 2, act as a third aperture. For such a collimation, we expect a beam diameter of d2:3 = 49 mm what is in agreement with the lower bound of dex. Spectrum The large transversal divergence of the atomic beam is also conﬁrmed by the absorption spectra. We measure the transversal absorption spectrum as described in Sec. 5.1. By this, we can directly measure the Doppler broadened width of the absorption signal and get information on the transversal velocities. Therefore, we can characterize the oven system also in dependence of its operation temperature. Figure 5.6 shows the absorption spectrum recorded with the above described 43 5. Characterization of the atomic beam Figure 5.7: Abundance relative to 166Er of the stable Er isotopes. The measured values, indicated as black squares, are in well agreement with the literature values , indicated as red circles. Since 162Er has very low abundance, no reliable measurement result exists. 167Er is split into eight hyperﬁne states and is not resolved in the measurement. The error bars are statistical over seven measurements. setup 1. Since the EC and the HL part are operated at high temperatures of TEC = 1350 ◦C and THL = 1450 ◦C, the atomic beam has a high ﬂux and absorbs 40 % of the incident light, leading to a distinct signal. All ﬁve stable bosonic isotopes of Er are clearly resolved. According to Eq. (1.26), we ﬁt a Voigt function to the data from which we determine the peak positions and the width. The ﬁtted Voigt function shows a vanishing small Lorentzian contribution, thus we can treat it as a pure Gaussian function. Isotopic composition and isotope shift The area of each individual absorption peak gives information on the isotope abundance. For an accurate determination, we average over seven diﬀerent mea-surements at diﬀerent operation temperatures. Comparing the measured values normalised to 166Er with the natural abundance reported in literature , we ﬁnd overall good agreement for 164Er, 168Er and 170Er as shown in Fig. 5.7. Be-cause of the low abundance of 162Er of 0.1 %, we can not give a precise value. 167Er is split into eight hyperﬁne states and is not resolved in the spectrum. The error bars account for the statistical error of the seven measurements. The peak positions provide information on the isotope dependent frequency shift. Figure 5.8 compares the measured values to the literature values . As 44 5. Characterization of the atomic beam Figure 5.8: Frequency shift of the 401nm transition relative to 166Er for all stable bosonic isotopes of Er. The measured values, indicated as black squares, are in very good agreement with the literature values , indicated as red circles (no known value for 162Er). The error bars are statistical over seven measurements. before, the results are mean values of seven measurement sets, leading to a sta-tistical error estimation. All values known from literature, also shown in Fig. 2.3, agree very well with our results. For 162Er, to our knowledge no literature value exists, but our measured value ﬁts to the slope of 400 MHz frequency change per atomic mass unit (amu) observed at the other bosonic isotopes. Line shape and linewidth In absence of any broadening mechanisms, the absorption spectrum has a Lorent-zian line shape with a width given by the natural linewidth ∆νnat, which is de-termined by the inverse lifetime of the excited state. For the 401 nm transition ∆νnat is 36 MHz. In our measurements, we always observe a linewidth bigger than the natural one, meaning that the spectrum is aﬀected by some broadening mechanism. The shape of the absorption spectrum gives information on the un-derlying broadening mechanism. A Gaussian shape is most probably ascribed to Doppler broadening, while a Lorentz type shape can arise from saturation or col-lisional broadening. For mechanisms of equal strength, the proﬁle is represented by a convolution of the contributing line shapes, namely a Voigt proﬁle. In this ﬁrst measurement set, we always observe a pure Gaussian proﬁle and assume that Doppler broadening is the most dominant broadening mechanism. The error bars shown in all following graphs refer to the statistical error coming from the ﬁt of 45 5. Characterization of the atomic beam Figure 5.9: The Gaussian linewidth ∆νG of 170Er in setup 1 as a function of the EC temperature corresponding to the Doppler broadening of the line. The measured values do not ﬁt to the ones expected from Eq. (1.23) for collimation at apertures 1 and 2 drawn with a solid red line. A free ﬁt to ∆νG = α √TEC, indicated by the dashed blue line, suggest α = 10.0(3) corresponding to a collimation at aperture 2 and the octagon chamber. a Voigt proﬁle to the absorption data. The linewidth of all isotopes should not diﬀer more than 1 % since the mass dependence of Doppler broadening is weak. However, the linewidth of the three most abundant bosonic isotopes diﬀers by 8 % what we explain by the inﬂuence of the fermionic 167Er. 167Er is split into eight hyperﬁne states and not resolved individually in the measurement, but nevertheless alters the absorption peaks of 166Er and 168Er. Therefore, in the following we will only focus on the properties of the 170Er isotope, which shows the most isolated but yet distinct absorption peak. The linewidth of a Doppler broadened line depends on the transversal velocity component of the atoms and is thus proportional to the beam divergence and the temperature. In Fig. 5.9 we show the measured values of the Gaussian contri-bution ∆νG to the linewidth of the Voigt proﬁle, corresponding to the Doppler width, as a function of TEC, where THL = TEC + 100 ◦C. From Eq. (1.23), we expect ∆νD = (ν0/c) f p 3 kB T / m = α √ T , (5.1) where f is the collimation ratio and α a constant for a ﬁxed f. For a collimation of the atomic beam at apertures 1 and 2, as illustrated in Fig. 5.3, α = 3.1 and ∆νG should follow the solid red line. The calculated values strongly diﬀer 46 5. Characterization of the atomic beam Figure 5.10: Spectrum in setup 1 for diﬀerent HL temperatures. Dotted red line: TEC = 1350 ◦C, THL = 1450 ◦C; solid black line: TEC = 1350 ◦C, THL = 970 ◦C, signal magniﬁed by a factor of 5. For low HL temperatures, the peaks get narrower and the signal is reduced. Therefore hyperﬁne states of 167Er appear, whereas 162Er is longer resolved. The signal is normalized to the maximum absorption of the signal indicated by the dotted red line corresponding to a maximum optical density ODmax ≃0.21. For the solid black line, ODmax ≃0.07. from the measured ones, which always show larger values of the linewidth. This trend is consistent with our ﬂuorescence measurements. The dashed blue line corresponds to a ﬁt of the measured values by Eq. (5.1), suggesting α = 10.0(3). If we consider the ﬁrst aperture is not playing any role and that the collimation is provided by aperture 2 and the inner edges of the octagon chamber, we expect α = 10.1. This agrees well with the measured values and further conﬁrms the ﬂuorescence measurements. Varying the HL temperature To investigate the reason for the apparent failure of aperture 1, we decrease the HL temperature whilst maintaining constant EC temperature. Due to radiative heating the lowest attainable temperature is THL = 970 ◦C at TEC = 1350 ◦C. The corresponding spectrum is shown as a solid black curve in Fig. 5.10. The signal has lower intensity and is magniﬁed by a factor of 5 to compare it to the spectrum at TEC = 1350 ◦C, THL = 1450 ◦C, drawn as dotted red line. We see that the absorption peaks are narrowed by decreasing the HL temperature. Due to the narrower absorption spectra, hyperﬁne states of 167Er appear for instance between the 166Er and the 168Er peak. This clear change of the signal strength 47 5. Characterization of the atomic beam Figure 5.11: Change of the spectrum shape for TEC < THL. The solid green line rep-resenting a ﬁt to a pure Lorentzian function, ﬁts the spectrum at TEC = 1350 ◦C,THL = 1150 ◦C better than the dashed red line representing a ﬁt to a pure Gaussian function. The signal is normalized to the maximum absorption corresponding to a maximum op-tical density ODmax ≃0.09. and linewidth emphasizes that the HL temperature aﬀects the atomic beam and that it does not only play the secondary role to avoid source material condensa-tion at the apertures. The line shape of the absorption peaks changes with decreasing the HL tem-perature while keeping the EC temperature constant, as visible in Fig. 5.11. This observation demonstrates that not only the EC, where the Er material is placed, but also the HL has an oven-like behavior. The solid green line representing a ﬁt to a pure Lorentzian function, ﬁts the spectrum at TEC = 1350 ◦C, THL = 1150 ◦C better than the dashed red line representing a ﬁt to a pure Gaussian function. We ﬁnd that in contrast to temperature sets with THL ≳TEC, where the spectrum evolves towards an almost pure Gaussian proﬁle, for THL ≲TEC a Lorentzian function is the dominant contribution to the total line shape. According to Eq. (1.27), the Voigt proﬁle has a Gaussian contribution ∆νG and a Lorentzian contribution ∆νL to the total width ∆νV. For TEC = 1350 ◦C, THL = 1450 ◦C, (∆νG / ∆νL) ≈1021 and for TEC = 1350 ◦C, THL = 1150 ◦C, (∆νG / ∆νL) ≈0.6, signifying the change of the line shape. This implies that for THL ≲TEC Doppler broadening is not the only broadening mechanism present. The underlying mech-anism for the Lorentzian line shape is not fully understood and suggests that complicated atom dynamics occur in the HL part. Saturation- and collisional broadening have Lorentzian line shapes. The ﬁrst mechanism can be neglected 48 5. Characterization of the atomic beam due to the low intensity of the laser beam of I ≈0.01 Isat. As follows, collisional broadening can partially explain the observed behavior. The HL part has EC-like character We explain the strong inﬂuence of the HL temperature on the linewidth and the line shape as follows. Erbium atoms that emerge from the EC at too large solid angles will not pass aperture 2, but stick to the walls of the HL part. Since these walls are heated, the atoms are re-emitted in random directions. Re-emitted atoms collide with atoms emerging from the EC and create an atomic vapor in the HL. This disables the creation of an atomic beam and gives the HL part an EC-like character. From aperture 2, an atomic beam emerges into the octagon chamber, where it can propagate freely since no atoms are re-emitted from the walls. This corresponds to the situation illustrated in Fig. 5.5: both the EC and the HL part are ﬁlled with atomic vapor and thus aperture 1 and the EC itself are eﬀectless to the atomic beam. Consequently aperture 2 and the HL inherit this role and the HL becomes EC-like character. This mechanism depends on two parameters: the ﬂux from the EC, and the temperature of the HL. For low ﬂux from the EC the collisions will be limited and the beam can propagate through the HL part. For low HL temperatures the re-emission of atoms from the walls is reduced and the emergence of atomic vapor in the HL is limited. We check our speculation by measuring the linewidth as a function of the HL temperature for a constant EC temperature of 1250 ◦C, as shown in Fig. 5.12. The total linewidth ∆νV consists of a Gaussian contribution ∆νG, shown as black squares, and Lorentzian contribution ∆νL, shown as red circles. The HL temperature strongly determines the ratio of both contributions. For THL ≥ 1200 ◦C, the proﬁle is purely Gaussian and the Lorentzian contribution is not resolved. At high HL temperatures ∆νG is on the order of magnitude of the expected Doppler broadening for an atomic beam collimated at aperture 2 and the octagon chamber, indicated as a dotted line. For THL ≤1150 ◦C, ∆νG drops and ∆νL dominates. In this region, ∆νG is even lower than the expected Doppler broadening for an atomic beam collimated at apertures 1 and 2, indicated as a dashed line. This conﬁrms the considerations above: for low HL temperatures the beam can propagate freely from the EC through the HL part, and an additional broadening mechanism with Lorentzian line shape is resolved. ∆νL increases with rising HL temperature. In ﬁrst order this agrees with Eq. (1.25), which describes the width of a line broadened by collisional broadening. Conclusion We conclude that in the original collimation setup for operation temperatures with THL ≳TEC, the emerging atomic beam is highly divergent as pointed out directly by the ﬂuorescence imaging measurement as well as by the large linewidth 49 5. Characterization of the atomic beam Figure 5.12: Gaussian contribution ∆νG, shown as black squares, and Lorentzian contribution ∆νL, shown as red circles, to the total linewidth ∆νV of the 170Er absorption peak as a function of the HL temperature for TEC = 1250 ◦C. For THL ≥1200 ◦C, the proﬁle is purely Gaussian and the Lorentzian contribution is not resolved. For THL ≤ 1150 ◦C, ∆νG drops and ∆νL dominates. The horizontal lines correspond to the expected values of the Doppler broadening of an atomic beam at TEC = 1250 ◦C. Dashed line: collimation at apertures 1 and 2, dotted line: collimation at aperture 2 and the octagon chamber. of the absorption peaks. This is explained by a collimation of the atomic beam at aperture 2 and the inner edges of the octagon. Consequently, the HL part is also ﬁlled with atomic vapor and has an EC like character. This results into a strong inﬂuence of the HL temperature on the line shape and the linewidth of the absorption peaks. For THL ≳TEC Doppler broadening is the dominant broadening mechanism, leading to a broad, Gaussian shaped absorption peak. We assume that collisional broadening also contributes to the total broadening, as for THL ≲TEC the line shape changes to a Lorentzian function and as the temperature dependence of the linewidth cannot solely be explained by Doppler broadening. At an oven operation with heated apertures, the high divergence of the atomic beam results in a strong contamination of the chamber with unused Er atoms. This exhausts the source material quickly, requiring frequent reﬁlling of the oven and is thus not providing a good system for long-term operation. To operate the oven at low HL temperatures does also not provide a good long term performance. Although in this case, the reduced linewidth suggests a well collimated beam, on longer timescales, Er atoms condensing on the colder apertures would grow over 50 5. Characterization of the atomic beam Figure 5.13: Scheme of the optimized collimation setup, referred to as setup 2. Aperture 1′ is a tube aperture with length a′ 1 ≃30 mm and inner diameter b′ 1 ≃3 mm. Aperture 2′ and the additional aperture 3′ have inner diameters of b′ 2 ≃3 mm and b′ 3 ≃8 mm and are spaced by a′ 2 ≃50 mm and a′ 3 ≃65 mm. At a distance of l′ ≃106.5 mm from aperture 2′, the probe laser beam transversely intersects the atomic beam. For a collimation on apertures 1′ and 2′, the atomic beam diameter is d1′:2′ = 11 mm. and block the atomic beam. Therefore, we designed a new-generation collimation setup. 5.3 Collimation setup 2 The optimized collimation setup, setup 2, is illustrated in Fig. 5.13. Aperture 1 is replaced by aperture 1′, which is a tube aperture with a length of a′ 1 ≃30 mm and an inner diameter b′ 1 ≃3 mm. According to Eq. (1.10), such a tube should reduce the total ﬂux emerging from the EC by a factor of 0.27, leaving the longitudinal ﬂux density almost equal. The second aperture, aperture 2′, at a distance a′ 2 ≃ 50 mm from aperture 1′ also has a reduced inner diameter b′ 2 ≃3 mm, which guarantees a better collimation. Additionally, at a distance a′ 3 ≃65 mm from aperture 2′, we insert a dark wall collimator, aperture 3′, which in contrast to aperture 1′ and 2′ is not heated. It has inner diameter b′ 3 ≃8 mm and reduces the material accumulation in the octagon chamber. This setup will not eliminate the eﬀusive-oven like behavior of the HL section but will reduce its inﬂuence on the atomic beam. Although the geometry of the ﬁrst aperture of 30 mm is larger than in setup 1, Eq. (1.3) is still fulﬁlled and so we assume that the EC section still operates in the eﬀusive regime. We performed measurements similar to the ones described in the previous section. The 401 nm probe light intersects the atomic beam transversely at a distance of l′ ≃106.5 mm from aperture 2′. The measurements were performed on the completely mounted vacuum chamber. 51 5. Characterization of the atomic beam Figure 5.14: Spectrum of setup 1 (dotted red line) and of setup 2 (solid black line, magniﬁed by a factor of 5), both at TEC = 1350 ◦C and THL = 1450 ◦C. In setup 2, the peaks are narrowed, and the intensity is reduced. Therefore hyperﬁne states of 167Er appear, whereas 162Er is no longer resolved. The signal is normalized to the maximum absorption of the signal indicated by the dotted red line corresponding to a maximum optical density ODmax ≃0.21. For the solid black line, ODmax ≃0.07. Beam divergence We ﬁrst perform ﬂuorescence imaging of the atomic beam. We observe a lower ﬂuorescence signal due to the reduced atomic ﬂux. As a consequence, we image with a long exposure time, much longer than the inverse of the laser scan rate. We evaluate the beam diameter from a single ﬂuorescence image. We observe a smaller beam divergence compared to the one of setup 1. For instance, the ﬂuorescent region is not anymore cut from the ﬁeld of view as in setup 1. We evaluate the beam diameter to be dex′ = 17(3) mm. Compared to setup 1, in which dex ≥41 mm, the beam diameter is strongly reduced. But also in this setup, the measured value signiﬁcantly deviates from the expected value corresponding to proper collimation at apertures 1′ and 2′ of d1′:2′ = 11 mm. If we consider again, that the ﬁrst aperture does not play a role and that the beam is collimated at apertures 2′ and 3′, we expect a diameter of d2′:3′ = 15 mm, which is in better agreement with the measurement. This corresponds to a situation similar to the simpliﬁed scheme for setup 1 in Fig. 5.5, except that the aperture setup is diﬀerent. 52 5. Characterization of the atomic beam (a) (b) Figure 5.15: Absorption peak of 170Er ﬁtted to diﬀerent functions: Gaussian function (solid red line), Lorentzian function (dotted blue line) and Voigt proﬁle (dashed green line) at (a) TEC = 1200 ◦C and (b) TEC = 1350 ◦C. The Voigt proﬁle ﬁt gives for (a) ∆νG = 5(3) × 102 MHz and ∆νL = 9(2) × 102 MHz and for (b) ∆νG = 101(4) MHz and ∆νL = 84(4) MHz. The Gaussian and the Lorentian ﬁt give for (a) ∆νG = 119(5) MHz and ∆νL = 108(7) MHz and for (b) ∆νG = 164(1) MHz and ∆νL = 141(2) MHz. Spectrum In collimation setup 2, the spectrum shows narrowed absorption peaks and has lower intensity. In Fig. 5.14 we compare the absorption spectrum obtained using setup 1 (dotted red line) and setup 2 (solid black line). Both spectra are observed at TEC = 1350 ◦C and THL = 1450 ◦C. The spectrum of setup 2 is magniﬁed by a factor of 5 and resolves hyperﬁne states of 167Er, whereas 162Er is no longer visible because of its very low abundance. Line shape and linewidth In contrast to setup 1, where we ﬁnd the spectrum to be described by a pure Gaussian function for most temperature sets, for setup 2 we observe almost equal contributions from both a Lorentzian as well as a Gaussian function to 53 5. Characterization of the atomic beam Figure 5.16: Gaussian linewidth ∆νG as a function of the EC temperature for setup 1, indicated by red circles, and for setup 2, indicated by black squares. The linewidth in setup 2 is reduced by a factor of 3 compared to setup 1, which also corresponds to the value expected by Eq. (1.23). The temperature dependence of ∆νG is reduced in setup 2. the Voigt proﬁle of the absorption peaks for all temperature sets. Figure 5.15 (a) and (b) shows the absorption peak of 170Er for low (TEC = 1200 ◦C) and for high (TEC = 1350 ◦C) EC temperatures. We evaluate the peak proﬁle using diﬀerent ﬁt functions. The solid red line represents a Gaussian function, the dotted blue line a Lorentzian function and the dashed green line a Voigt proﬁle. For both low and high temperatures, we observe that all these functions give a good description of the observed proﬁle. For lower EC temperatures, a ﬁt of the Voigt proﬁle to the data gives the Gaussian and Lorentzian width with an uncertainty of 50 %, indicating that due to the low signal-to-noise ratio the ﬁt cannot discriminate between the two contributions. At high EC temperatures, we can distinguish the Gaussian and Lorentzian contributions, which have almost the same weight. For convenience and to have a direct comparison with setup 1, we choose to ﬁt all the absorption spectra with a pure Gaussian function. The extracted values rep-resent an upper limit to the Gaussian width and overestimate thus the Doppler broadening, since the Lorentzian contribution is neglected. However, these val-ues can give us a qualitative trend on the temperature dependence. The error bars shown in all following graphs refer to the statistical error from the ﬁt of a Gaussian function to the absorption data. Our results are summarized in Fig. 5.16. We compare the Gaussian linewidth ∆νG as a function of the EC temperature for setup 1, indicated by red circles, with 54 5. Characterization of the atomic beam Figure 5.17: Diﬀerent ﬁts to ∆νG(TEC) in setup 2 (c.f. Fig. 5.16) by ∆νG = α (TEC)x. For ﬁxed x = 0.5 corresponding to pure Doppler broadening, we ﬁnd α = 8.1 (dashed red line). For ﬁxed α = 1.0 corresponding to collimation at apertures 1′ and 2′, we ﬁnd x = 0.8 (solid blue line). setup 2, indicated by black squares, where THL = TEC +100 ◦C. The linewidth in setup 2 is reduced by a factor of 3 compared to setup 1, which also corresponds to the value expected by Eq. (1.23) if we consider a purely Doppler broadened line. The linewidth is aﬀected by both the eﬀective collimation and the type of broadening mechanism. If we consider a pure Doppler broadening and ﬁt the data corresponding to the black squares in Fig. 5.16 to Eq. 5.1, as shown by the dashed red line in Fig.5.17, we ﬁnd α = 8.1. This suggests an atomic beam that is much less collimated than by apertures 2′ and 3′. If we consider a collimation of the atomic beam at apertures 1′ and 2′ corresponding to α = 1.0 and ﬁt the data to a function T x, as shown by the solid blue line in Fig. 5.17, we ﬁnd a tem-perature dependence of T 0.8. Our data suggests that we are in an intermediate case. Further experiments at lower temperatures could clarify this point. In general, we observe that Doppler broadening is not the sole broadening mechanism contributing to the absorption line suggesting complicated atom dy-namics inside the HL part. From the linewidth at TEC = 1350 ◦C we estimate an upper limit of the transversal Doppler velocity to 65 m/s. 55 5. Characterization of the atomic beam Figure 5.18: Gaussian linewidth ∆νG as a function of the HL temperature, where TEC = 1250 ◦C for setup 1, indicated by red circles, and for setup 2, indicated by black squares. While for THL ≲1150 ◦C, the width is for both setups of the same magnitude, for THL > 1150 ◦C the inﬂuence of the HL temperature on ∆νG is strongly reduced in setup 2. The inﬂuence of the HL temperature In Fig. 5.18 we show the Gaussian linewidth ∆νG as a function of the HL temper-ature, where TEC = 1250 ◦C for setup 1, indicated by red circles, and for setup 2, indicated by black squares. The inﬂuence of the HL temperature on ∆νG is reduced by setup 2. While for THL ≲1150 ◦C, the width is for both setups of the same magnitude, for THL > 1150 ◦C the inﬂuence of the HL temperature on ∆νG is strongly reduced in setup 2. We conclude that the emergence of atomic vapor in the HL is hindered in setup 2. This agrees well with the picture drawn above of the EC-like character of the HL, which is dependent on THL and on the ﬂux from the EC. Compared to setup 1, in setup 2 the ﬂux is reduced and so are the collisions in the HL part. Therefore, the inﬂuence of the HL temperature on ∆νG is reduced. Conclusion Similar to setup 1, for oven operation with THL > TEC in setup 2, the HL has EC-like character and is ﬁlled with atomic vapor disabling aperture 1′. But although the atomic beam is not collimated as intended, this optimized setup shows good improvements. The atomic beam divergence is reduced and so is the part of the unused atoms. Also the inﬂuence of the HL temperature on the atomic beam is 56 5. Characterization of the atomic beam Figure 5.19: The ﬂux density in units of sr−1 measured in the octagon chamber as a function of the EC temperature, where THL = TEC + 100 ◦C. The ﬂux in setup 1, indicated by red circles, is an order of magnitude higher than in setup 2, indicated as black squares. The linear temperature dependence of the ﬂux in both setups exhibits the exponential dependence of the vapor pressure on the temperature. strongly reduced. Due to equally strong Gaussian and Lorentzian contributions to the line shape, we assume that besides Doppler broadening also collisional broadening contributes to the total broadening. 5.4 Flux The atomic ﬂux is a very important quantity to characterize the oven system since it determines the number of atoms per unit time that enter the ZS tube and that can be captured by the MOT. According to Sec. 1.3, we can estimate the ﬂux at the position of the probe laser beam by measuring I, I0 and j. From the 166Er absorption peak amplitude of the diﬀerential signal and the magnitude of the reference signal, we obtain the transmitted intensity I and the incident intensity I0. By imaging the ﬂuorescence on resonance of the 166Er transition, we measure the light-atom interaction length j. We normalise the total ﬂux obtained from Eq. (1.17) to the unit solid angle of emergence, and account for the fact that we measured the optical density only for 166Er by scaling it with the inverse abundance of 166Er. The resulting ﬂux density Θex / sr includes all isotopes and holds only for small solid angles. It provides a suitable quantity to analyse the ﬂux measured in the octagon chamber in dependence of the collimation setup and of the operation temperature. 57 5. Characterization of the atomic beam Figure 5.20: The ﬂux density in units of sr−1 measured in the octagon chamber as a function of the HL temperature, where TEC = 1250 ◦C. Both in setup 1, indicated by red circles, and in setup 2, indicated by black squares, the ﬂux is dependent on the HL temperature for THL ≳1050 ◦C. Setup 1 vs. setup 2 Figure 5.19 shows the ﬂux density Θex / sr measured in the octagon chamber as a function of the EC temperature on a logarithmic scale, where THL = TEC+100 ◦C. The ﬂux in setup 2, indicated as black squares, is reduced to 9 % of the ﬂux of setup 1, indicated as red circles. This agrees well with the ﬂux reduction due to the diﬀerent collimation setup expected from Eq. (1.9), which also predicts a value of 9 %. In both setups we see a linear temperature dependence with equal slope, which reﬂects the exponential temperature dependence of the vapor pressure shown in Eq. (2.1), which is proportional to the atomic ﬂux . Inﬂuence of the HL temperature As expected, the HL temperature inﬂuences the ﬂux in both setups. We show in Fig. 5.20 the ﬂux density Θex / sr measured in the octagon chamber as a function of the HL temperature. The EC temperature is constant at TEC = 1250 ◦C. Both in setup 1, indicated by red circles, and in setup 2, indicated by black squares, the ﬂux depends on the HL temperature for THL ≳1050 ◦C. In this region, the increase of the ﬂux with the HL temperature is of similar strength in both setups. In contrast to this, as show in previous measurements (c.f. Fig. 5.18), the linewidth ∆ν in setup 2 shows reduced dependence on the HL temperature. Therefore, we conclude that in both setups for increasing HL temperatures, more 58 5. Characterization of the atomic beam (a) (b) Figure 5.21: Accumulation of Er in the vacuum chamber. (a) View through a viewport of the octagon while operating the oven in setup 1. On the inner edges, Er starts to accumulate in tiny rolls. (b) The connection tube of oven and octagon after operating the oven in setup 1. The inner walls are all over coated with Er, and the Er layer peels oﬀ. and more atoms are re-emitted by the HL walls and contribute to the ﬂux out of the oven. However, in setup 2, the total amount of re-emitted atoms is less and so are the collisions in the HL part. This explains that the HL temperature has reduced inﬂuence on the linewidth in setup 2, but that the relative increase of the ﬂux with the HL temperature is similar in both setups. Material deposition The high ﬂux in setup 1 also leads to undesirable accumulation of Er material. Already after some hours of oven operation, tiny rolls of accumulated Er emerged at the inner edges of the octagon, as show by Fig. 5.21 (a). When changing the apertures from setup 1 to setup 2, we opened the oven chamber and realized that Er material coated the inner walls of the connection tube of oven and octagon all over, as seen in Fig. 5.21 (b). Due to gravity, the Er layer peeled oﬀthe wall. In long-term operation, this would give additional contact surface for Er atoms from the beam that stick to it, grow new structures and ﬁnally block the atomic beam emerging from the oven. Thus, it was an important step to adapt the apertures to setup 2, since this setup provides a better collimated beam with reduced ﬂux in contrast to setup 1, which could neither provide reliable long-term operation nor a well-collimated atomic beam. Extracted ﬂux at the MOT position To estimate the number of atoms that reach the MOT region per second from our measured value for the atomic ﬂux in the octagon chamber, we need to take into 59 5. Characterization of the atomic beam Figure 5.22: Derived atomic ﬂux that can be captured by the MOT as a function of the EC temperature in setup 2. According to Eq. (5.2), several loss factors determine ΘMOT, where in our setup ξ2 is variable. We plot ΘMOT(TEC) for three diﬀerent values of ξ2: 0.324 (black squares), 0.140 (red circles) and 0.061 (blue triangles). The dotted black, dashed red and dash-dotted blue lines correspond to exponential ﬁts to the measured values and are also shown for lower temperatures to indicate the trend. account several loss factors . For geometrical arguments, only a fraction of Θex can enter the ZS tube, what gives rise to a loss factor ξ0. The isotope factor ξ1 considers that only a certain isotope is slowed and captured. The ZS factor ξ2 corresponds to the ratio of atoms that can be slowed by the ZS to the total number of atoms injected into the ZS: ξ2 = R v0 0 P ′(v)dv / R ∞ 0 P ′(v)dv, where v0 is the capture velocity of the ZS and P ′(v) is the velocity distribution in the beam. Since we can vary v0 by the magnetic oﬀset ﬁeld of the ZS to adjust the magnetic ﬁeld proﬁle to the available slowing light power, also ξ2 varies. The last loss factor ξ3 accounts for that the slowed atomic beam leaving the ZS has a large divergence, since only longitudinal velocity components are slowed in the ZS. Therefore only a part of those atoms passes the MOT region: ξ3 = R r0 0 I(r)dr / R ∞ 0 I(r)dr, where I(r) is the intensity and r0 is the radius of the MOT region. Combining this, yields to the number of atoms that can be captured in the MOT per unit time, if we assume that the capture velocity of the MOT is larger or equal to the ﬁnal velocity of the slowed atoms from the ZS: ΘMOT ≈ξ0 ξ1 ξ2 ξ3 Θex. (5.2) For our setup, ξ0 = 1.3×10−4, ξ2 can vary from 0.061 to 0.324, depending on the capture velocity and ξ3 = 0.011. If we consider 166Er, ξ1 = 0.336. In Fig. 5.22 60 5. Characterization of the atomic beam we plot ΘMOT(TEC) for three diﬀerent values of ξ2: 0.324 (black squares), 0.140 (red circles) and 0.061 (blue triangles). The dotted black, dashed red and dash-dotted blue line correspond to exponential ﬁts to the measured values and are also shown for lower temperatures, at which we could not probe the ﬂux with our measurement. These lines indicate the presumable development of ΘMOT, which should only be considered as a rough estimation. All values suggest that with setup 2, we can have a suﬃcient supply of Er atoms for tapping and further cooling large atomic samples. 61 Conclusion and Outlook During my master thesis work, I focused my eﬀorts in producing and characteriz-ing an Er atomic beam, emerging from a high-temperature oven. The knowledge of the atomic beam behavior is fundamental for all future experiments on laser cooling and Bose condensation of Er atoms. Due to erbium 's high melting point, the operation temperatures of the oven are unusually high for standard ultracold gas experiments based on alkali and alkaline earth elements. A big experimental eﬀort was then necessary to design and operate the oven system. To shield the high temperatures, the oven system is water cooled. To avoid material conden-sation on colder parts of the system, the apertures that collimate the atomic beam are heated to higher temperatures than the eﬀusion cell. To backup the system against power failure we protect it with an UPS. To secure safe opera-tion, the oven system is connected to the cooling water system control and to the emergency-oﬀsystem. The oven control program provides not only the user interface, but also coordinates the safety systems. We characterize the atomic beam emerging from the oven by laser spec-troscopy. Our measurements clearly reveal a complex beam behavior in the oven chamber leading to a highly divergent atomic beam and the deposition of Er material at the walls of the vacuum chamber. To minimize both eﬀects, we had to design and mount a novel atomic beam collimation setup. The optimized setup consists of a tube aperture combined with two circular apertures. The measurements on this new setup show that it provides a much better collimation of the atomic beam with a reduced but suﬃcient atomic ﬂux for large cooled and trapped ensembles of Er atoms. We ﬁnd the optimal operation temperature to be 1250 ◦C. At this temperature, the estimated total atomic ﬂux, which is injected into the ZS slower tube is 3 × 1010 s−1 with a transversal Doppler velocity of 65 m/s. We expect correspondingly a number of the order of 107 atoms/s that can be captured by the MOT. This is a large enough value to guarantee a good MOT operation. Currently, we are baking the complete experimental apparatus at a temper-ature of 200 ◦C to lower the background pressure in the chamber further. In the MOT chamber, we measure a background pressure of 2 × 10−11 mbar, which 62 Conclusion and Outlook (a) (b) Figure 6.1: The vacuum chamber, (a) almost completely mounted. The ZS is clearly visible in the center in white, to the left hand side is the golden colored MOT chamber, and to the very right side is the Er oven. (b) The experimental chamber whilst the baking. To lower the background pressure, we heat the chamber to temperatures of 200 ◦C for several days. Thereby, residual gases that are enclosed in the stainless steel walls of the chamber and that limit the vacuum are released and pumped oﬀ. assures a good enough vacuum level for our purposes. The next step is the adjust-ment and testing of the ZS as well as the setup of the narrow-line MOT. Once the MOT is working, we can load the ODT and start to study the collisional behavior of cold Er atoms in detail by Feshbach spectroscopy, which is a major step on our way to the Bose-Einstein condensation of Er. 63 Appendices 64 Appendix A Magnetic moments of the lanthanides The magnetic moment µ is calculated as µ = g MJ µB, (A.1) where MJ is the projection of the total angular momentum quantum number J towards its quantization axis. g is the Land´ e g-factor, which for LS-coupling of the electrons can be estimated as g = 1 + (gs −1) J (J −1) −L (L −1) + S (S −1) 2 J (J −1) , (A.2) where gs is the Land´ e g-factor of the electron, which is can be approximated by gs = 2. S is the total spin quantum number and L is the total orbital angular momentum quantum number. So, to evaluate the magnetic moment for a certain electronic conﬁguration one has to know the speciﬁc quantum numbers S, L and J of the state. These quantities can be derived with the aid of Hund 's rules . Those rules only apply to the lowest term of the ground state conﬁguration for cases where there is only one incomplete subshell and where the electrons couple in a LS-coupling scheme. This is the case for most of the lanthanide atoms in the ground state, as one can see in Tab. A.1. Except for Ce and Gd, the four Hund 's rules determine S, L and J. The calculated µ agrees well agreement with the measured values from as shown in Tab. A.1. This underlines the validity of the assumptions and the explanation of the magnitude of the magnetic moments by the occupation of the 4f shell. 65 A. Magnetic moments of the lanthanides el. 4f n 5dn S L J term g (lit.) g (calc.) µ (lit.) µ (calc.) La 0 1 1 2 2 3 2 2D 3 2 0.79755 0.80000 1.20 1.20 Ce 1 1 0 4 4 1G4 0.94543 1.00000 3.78 4.00 Pr 3 0 3 2 6 9 2 4I 9 2 0.73104 0.72727 3.29 3.27 Nd 4 0 2 6 4 5I4 0.60329 0.60000 2.41 2.40 Pm 5 0 5 2 5 5 2 6H 5 2 0.30500 0.28571 0.76 0.71 Sa 6 0 3 3 0 7F0 0 0 0 0 Eu 7 0 7 2 0 7 2 8S 7 2 1.99340 2.00000 6.98 7.00 Gd 7 1 4 2 2 9D2 2.65140 2.66667 5.30 5.33 Tb 9 0 5 2 5 15 2 6H 15 2 1.32513 1.33333 9.94 10.00 Dy 10 0 2 6 8 5I8 1.24159 1.25000 9.93 10.00 Ho 11 0 3 2 6 15 2 4I 15 2 1.19514 1.20000 8.96 9.00 Er 12 0 1 5 6 3H6 1.16381 1.16667 6.98 7.00 Tm 13 0 1 2 3 7 2 2F 7 2 1.14119 1.14286 3.99 4.00 Yb 14 0 0 0 0 1S0 0 0 0 0 Table A.1: Occupation of the 4f shell, occupation of the 5d shell, resulting total spin quantum number S, resulting total orbital momentum quantum number L, resulting total angular momentum quantum number J, the according term symbol, the Land´ e g-factor found in literature , the estimated Land´ e g-factor calculated for LS-coupling, the resulting magnetic moment µ for the Land´ e g-factor from literature and from the calculation, for all ground states of the lanthanide elements ( 'el.'). 66 Appendix B Erbium oven control The screenshot of the LabVIEW block diagram in Fig. B.1 shows the structure of the oven control program code. After initialization in a ﬂat sequence, the main program starts. Here, the datasockets in the upper left manage the 'On/Stby' switch and access the network variables, which connect the control program with the PID controllers via the iTools OPC server. The while-loop on the upper right, plots the EC and the HL temperature and the working setpoints versus the absolute time. The environment variables in the right of the lower while loop acquire the UPS status. The MGE UPS personal solution pack program triggers the command line if one of the UPS variables changes. The command line has the new variable values as argument and starts the executable subVI UPS-trigger-LabVIEW.exe. The subVI reads the command line argument and distributes the contained information to the according environment variable. The oven control program displays the UPS status and sets the oven to standby mode for too little remaining backup time of the UPS, this is realized in the attached case structure. The central part of the VI is the case structure in the lower while-loop. The screenshot only displays one out of 14 cases, which manage and secure the working of the control program: case 0: default case case 1: reacts to the 'EMERGENCY OFF ' button1 case 2-9: set safety limits for the input values of the OPC server variables and manages the 'On/Stby' switch case 10: preset values changeable via the 'enable changes ' switch case 11-13: aid the scaling and appearance of the temperature plot 1As seen in Fig B.1, in an emergency case, the temperature ramps are set to 0 ◦C/min. However, due to the interpretation of the PID controller, this sets the temperature immediately to the new setpoints of 0 ◦C. 67 B. Erbium oven control Figure B.1: A screenshot of the LabVIEW block diagram, which displays the oven control program code. 68 Appendix C Atomic beam shutter control C.1 Control box We use an Atmel AVR ATmega128 µC mounted on a Crumb128-light µC module. An external resonator with a frequency of 14.7456 MHz is the clock source for the processor. The detailed pin conﬁguration and the whole electric circuit are shown in the circuit diagram in Fig. C.1. The settings to program the µC via the Atmel AVR Studio 4 program with the AVRISP mkII programming interface are: ISP frequency 2 MHz, Fuses: (no M103C; SUT CKSEL = ext. crystal/resonator high freq., start-up time: 16 K CK +64 ms). We repeat a part of the manual text, which gives the main operating guidelines: ''The control box has on its face the following switches and connectors: power, emergency stop, ttl, power switch, reset push button (red), pc/manual control switch and the unshut/shut switch. On the side the servo can be plugged in, as well as, if necessary the programming interface. To access the potentiometers and the pushbutton (green), one has to open the box. When everything is connected, the power switch starts the controller (no more than 8 V), the orange LED shines. All three other LEDs shine for 1 second (de-fault led test at initialisation) (also if system is reset). The red push button resets the controller in case this is needed. For control via the pc (TTL signal), set the pc/manual control switch to pc. The blue LED shines as well as either the green or the red one, indicating the status of the shutter (beam unshut or beam shut respectively). For control by hand, set the pc/manual control switch to manual. The blue LED doesn 't shine. The unshut/shut switch allows control over the shutter po-sition, which is indicated by the green or the red LED. For positioning the shutter positions, one has to operate in the manual con-trol mode, has to keep the green push button pushed and turn at the same time 69 C. Atomic beam shutter control the screw at the according potentiometer. According means that you can only change the position of the choice you have made at the unshut/shut switch. The potentiometer marked 'u ' will change the position of the shutter at unshut, if the unshut/shut switch is set to unshut. The potentiometer marked 's ' will change the position of the shutter at shut, if the unshut/shut switch is set to shut. If the green push button is released, normal manual operation continues. If an emergency stop occurs, the shutter will be set to 'beam shut ' and block the atomic beam. The red, blue and green LED blink to indicate the interrupt. If the emergency stop ends, normal operation mode continues automatically.'' C.2 Source code The explanation below, describes the source code shown in List.B.2, and is also contained in the control box manual. ''The header ﬁle, abs.h (see List. C.2), deﬁnes constants, macros and variables and links functions. In the program, after the including the needed header and libraries (order is not commutative) the main loop starts. The initialisation of input/output, interrupts, PWM and analog-to-digital conversion take place and interrupts are activated. The while-loop keeps the program running and checks switches and the emer-gency stop. If this is not activated, the main loop checks whether the operation mode is manual control or pc control. In the ﬁst case there 's distinction between normal operation and positioning of the shutter via the potentiometers (setting the values for 'beam shut ' and 'beam unshut '). If the emergency stop is active (signal at BNC connection is low, signal at µC is high (opto-isolator)), the shutter is set to the position 'beam shut ' , and all LEDs are blinking. The controlling of the servo is accomplished by an timer interrupt that runs every 10 ms. Here, basically some if-loops decide whether the value of shutterpos has changed in the last second (period is dependent on j, see code). If so, the new value is set and the signal is changed accordingly. If not, no signal is sent to the servo (just high, no PWM). The reason for that is that, in this way, the servo is only driven when needed and left passive when not needed, what prevents needless vibrations of the system. At the start of the interrupt SREG is back saved and reloaded in the end to make the program more robust and save. At the input output initialization the pins are set in the needed conﬁgura-tions (input, output, internal pull-ups). For a default check of the LEDs, by delay ms(1000), all LEDs shine for 1 s. At the interrupt initialization, the timer interrupt frequency is set, as well as the mode of the external interrupt. At the PWM initialization, the mode, as well as the pulse of the signal (50 Hz) is set 70 C. Atomic beam shutter control Figure C.1: The circuit diagram of the atomic beam shutter control box, which operates a servo motor. An Atmel AVR ATmega128 µC with external resonator is mounted on the Crumb128-light µC module. Opto-isolators give galvanic separation, the voltage regulator protects the µC from high voltage 71 C. Atomic beam shutter control (ICR1). At the analog to digital conversion initialization, the mode is selected and a dummy readout takes place, for reserving the variable for the upcoming results. The measurement functions do a simple ADC k times to give a stable average (important: clear the channel before selecting it). The ﬂashing of the LEDs is done by a simple counting of b. Manual control checks the switch for choosing the shutter position manually and sets the value of shutterpos accordingly. Whereas in pc control shutterpos is set according to the TTL signal from the pc (which is inverted ↔optocoupler). '' Listing C.1: Source code of the µC ﬁrmware. 1 / 2 atom beam−s h u t t e r c o n t r o l l e r 3 / 4 5 // version 1.0 6 // author : jogy s c h ind l e r 7 // date : 07.07.010 8 9 10 #include 11 #include 12 #include ” abs . h” 13 #include 14 15 16 // − − − − − − − − − 17 //main loop 18 // − − − − − − − − − 19 20 int main ( void ) 21 { 22 c l i ( ) ; 23 i n i t i o ( ) ; 24 i n i t i n t ( ) ; 25 init pwm ( ) ; 26 i n i t a d c ( ) ; 27 s e i ( ) ; 28 29 while ( 1 ) 30 { 31 e = PIND ; // check and f i l t e r emergency stop s t a t u s 32 e &= (1< s ig na l @ pin i s low ( optocoupler ) , normal operation 35 { 36 l = PINA ; // check and f i l t e r s w it c h p o s it io n 37 l &= (1<<PA1) ; 38 39 i f ( l ==0) // switch [MANUAL CONTROL < − > PC CONTROL] == MANUAL CONTROL 40 { 41 m = PINA ; // check and f i l t e r s w it c h p o s it io n 42 m &= (1<<PA5) ; 43 m /= 3 2 ; 44 45 i f (m==1) // switch [SHUTTER POSITIONING < − > OPERATING] == OPERATING 46 { 47 manual control ( ) ; 48 } 49 72 C. Atomic beam shutter control 50 else // switch [SHUTTER POSITIONING < − > OPERATING] == SHUTTER POSITIONING 51 { 52 measure poti1 ( ) ; //measure potentiometer 1 for value of beam unshut 53 measure poti2 ( ) ; //measure potentiometer 2 for value of beam shut 54 55 manual control ( ) ; 56 } 57 } 58 59 else // switch [MANUAL CONTROL < − > PC CONTROL] == PC CONTROL 60 { 61 p c c o n t r o l ( ) ; 62 l e d b o n ( ) ; 63 } 64 } 65 66 else // s ig na l @ bnc i s low ( emergency stop ) − > s ig na l @ pin i s high ( optocoupler ) , emergency stop 67 { 68 OCR1A = (3 beam shut ) +1000; // s e t servo to beam shut 69 70 l e d f l a s h ( ) ; // for ind ic a t io n of emergency stop 71 h = 0 ; //h r e s e t for business as usual a f t e r emergency stop 72 } 73 } 74 75 return 1 ; 76 } 77 78 79 // − − − − − − − − − 80 // int e r r u p t 81 // − − − − − − − − − 82 83 // timer int e r u p t for s e t t i n g servo 84 ISR(TIMER3 COMPA vect) 85 { 86 stemp = SREG; // backsaving SREG 87 TCNT3 = 0 ; // r e s e t 88 89 i f ( sh u t t e r p o s != h) // i f shutterpos changed 90 { 91 j =0; 92 OCR1A = (3 sh u t t e r p o s ) +1000; // s e t p u l s e l e ng h t ( empirical formula , by t h e o r e t i c a l estimate , and experimental t r i a l an error ) 93 h = sh u t t e r p o s ; // inverse p u l s e l e ng h t fu nc t io n for i f loop 94 } 95 96 i f ( sh u t t e r p o s == h) // i f shutterpos unchanged 97 { 98 i f ( j <100) // i f shutterpos changed r e c e nt l y 99 { 100 OCR1A = (3 sh u t t e r p o s ) +1000; // s e t pulselenght , u n t i l new servo p o s it io n i s driven ( j=100 = 1 s , for 180 servo needs ˜0.8 s , so i f f a s t signal −o f f use with 90 s e t j=50 −but check accurately ! ) 101 h = sh u t t e r p o s ; // inverse p u l s e l e ng h t fu nc t io n for i f loop 102 } 103 i f ( j >=100) // i f shutterpos unchanged for longer time 104 { 105 j =200; // stay in t h i s mode u n t i l shutterpos changes 106 OCR1A = ICR1 ; //no pwm s ig na l to servo , for avoiding v ib r a t io ns 107 } 108 } 109 110 j ++; 111 SREG = stemp ; // loading SREG 73 C. Atomic beam shutter control 112 } 113 114 115 // − − − − − − − − − − − − − − 116 // i n i t i a l i s a t i o n 117 // − − − − − − − − − − − − − − 118 119 // input −output i n i t i a l i s a t i o n 120 void i n i t i o ( void ) 121 { 122 DDRA \|= (1<<PA2) \| (1<<PA4) \| (1<<PA6) ; // l e d s s e t to output 123 PORTA \|= (1<<PA0) \| (1<<PA1) \| (1<<PA3) \| (1<<PA5) ; // a c t iv a t e p u l l −up @ t t l −pin and switches 124 DDRB \|= (1<<PB5) ; // servo−pin s e t to output 125 PORTD \|= (1<<PD0) ; // a c t iv a t e p u l l−up @ emergency stop 126 127 l e d a l l o n ( ) ; // d e f a u l t l e d check @ i n i t i a l i s a t i o n : a l l l e d s on for 1 s 128 delay ms (1000) ; 129 l e d a l l o f f ( ) ; 130 } 131 132 133 // int e r r u p t i n i t i a l i s a t i o n 134 void i n i t i n t ( void) 135 { 136 TCCR3B \|= (1<<CS32) \| (1<<CS30) ; // p r e s c a l e r = 1024 , normal operation mode ( see t a b l e 61 in datasheet ) 137 OCR3A = ( int ) ( ( (XTAL/1024) 10) /1000) ; //compare : timer−int e r u p t every 10ms , take care on the brakets and c a l c u l a t io n order : c r u c ia l ! OCR3A =f ( c l k ) /( p r e s c a l e r f (OCR3A) ) 138 ETIMSK = (1<<OCIE3A) ; // timer3−compare−interrupt −enable 139 } 140 141 142 // pulsewidthmodulation i n i t i a l i s a t i o n 143 void init pwm ( void) 144 { 145 TCCR1A = (1<<COM1A1) \| (1<<WGM11) ; // noninverting f a s t pwm, TOP i s ICR1 ( mode 14 , see t a b l e 61 in datasheet ) 146 TCCR1B = (1<<WGM13) \| (1<< WGM12) \| (1< 50Hz pulse [ ICR1=( f ( c l k ) /( p r e s c a l e r f (pwm) ) ) ] − − > OCR1A=3687 ,... ,1843 148 } 149 150 151 // analog to d i g i t a l convertion i n i t i a l i s a t i o n 152 void i n i t a d c ( void) 153 { 154 A D M U X \|= (1<<REFS0) ; // reference v o l t a g e i s AVCC 155 ADCSRA \|= (1<<ADEN) \| (1<<ADPS2) \| (1<<ADPS1) \| (1< ADC frequency = 115kHz ) 156 157 ADCSRA \|= (1<<ADSC) ; // s t a r t dummy readout 158 while ( ADCSRA & (1<<ADSC) ) 159 {;} // wait for end of conversion 160 beam unshut = A D C W; // read A D C W to get r e s u l t of the next measurement in t h i s v a r ia b l e 161 162 measure poti1 ( ) ; //measure the p o t is for s e l e c t i n g the recent s e t s e t t i n g s 163 measure poti2 ( ) ; 164 } 165 166 167 // − − − − − − − − − 168 // functions 169 // − − − − − − − − − 170 171 // v o l t a g e measurement @ potentiometer #1 74 C. Atomic beam shutter control 172 void measure poti1 ( void ) 173 { 174 A D M U X &= ˜(1<<MUX1) ; // s e t channel zero , chose channel ADC0 ( d e f a u l t ) 175 beam unshut = 0 ; 176 177 for ( k=0; k<32; k++) //measure k+1 times and average , chose k as 2ˆn for f a s t e r c a l c u l a t io n 178 { 179 ADCSRA \|= (1<<ADSC) ; // s i n g l e conversion 180 while ( ADCSRA & (1<<ADSC) ) 181 {;} 182 beam unshut += A D C W; 183 } 184 beam unshut /= ( k+1) ; 185 } 186 187 188 // v o l t a g e measurement @ potentiometer #2 189 void measure poti2 ( void ) 190 { 191 A D M U X &= ˜(1<<MUX1) ; // s e t channel zero 192 A D M U X \|= (1<<MUX1) ; // chose channel ADC2 193 beam shut = 0 ; 194 195 for ( k=0; k<32; k++) //measure k+1 times and average 196 { 197 ADCSRA \|= (1<<ADSC) ; // s i n g l e conversion 198 while ( ADCSRA & (1<<ADSC) ) 199 {;} 200 beam shut += A D C W; 201 } 202 beam shut /= ( k+1) ; 203 } 204 205 206 // f l a s h i n g l e d s @ emergency block 207 void l e d f l a s h ( void ) 208 { 209 i f ((0<b && b<20000) ) 210 { 211 l e d a l l o n ( ) ; 212 } 213 i f ((10000 <b && b<40000) ) 214 { 215 l e d a l l o f f ( ) ; 216 } 217 i f (b>=40000) 218 { 219 b=0; 220 } 221 b++; 222 } 223 224 225 // s e t s the s h u t t e r p o s it i o n according to the [SHUTTERPOS= BEAM SHUT < − > SHUTTERPOS = BEAM UNSHUT]−switch 226 void manual control ( void ) 227 { 228 i = PINA ; 229 i &= (1<<PA3) ; 230 i /= 8 ; 231 232 i f ( i ==1) // switch [SHUTTERPOS= BEAM SHUT < − > SHUTTERPOS= BEAM UNSHUT] == SHUTTERPOS= BEAM UNSHUT 233 { 234 sh u t t e r p o s = beam unshut ; 235 l e d a l l o f f ( ) ; 236 l e d g o n ( ) ; 237 } 238 75 C. Atomic beam shutter control 239 else // switch [SHUTTERPOS= BEAM SHUT < − > SHUTTERPOS= BEAM UNSHUT] == SHUTTERPOS= BEAM SHUT 240 { 241 sh u t t e r p o s = beam shut ; 242 l e d a l l o f f ( ) ; 243 l e d r o n ( ) ; 244 } 245 } 246 247 248 // s e t s the s h u t t e r p o s it i o n according to the TTL s ig na l from the PC ( inverted ! ) 249 void p c c o n t r o l ( void ) 250 { 251 t t l = PINA ; 252 t t l &= (1< helps with c a l c u l a t i o n s 9 #define F CPU 14745600UL //cpu frequency defined for delay . h 10 11 12 //macros 13 14 #define l e d b o n ( ) PORTA \|= (1<<PA2) 15 #define l e d b o f f ( ) PORTA &= ˜(1<<PA2) 16 #define l e d g o n ( ) PORTA \|= (1<<PA4) 17 #define l e d g o f f ( ) PORTA &= ˜(1<<PA4) 18 #define l e d r o n ( ) PORTA \|= (1<<PA6) 19 #define l e d r o f f ( ) PORTA &= ˜(1<<PA6) 20 #define l e d a l l o n ( ) PORTA \|= (1<<PA2) \| (1<<PA4) \| (1<<PA6) 21 #define l e d a l l o f f ( ) PORTA &= ˜((1<<PA2) \| (1<<PA4) \| (1<<PA6) ) 22 23 24 // v a r i a b l e s 25 26 unsigned char e ; // for emergeny stop 27 unsigned char i ; // for switch SHUTTERPOS= BEAM SHUT < − > SHUTTERPOS= BEAM UNSHUT 28 unsigned char l ; // for switch MANUAL CONTROL < − > PC CONTROL 29 unsigned char m; // for switch POSITIONING < − > OPERATING 30 unsigned char t t l ; // for TTL s ig na l 31 unsigned char j =0; // for counter of timer int e r r u p t s 32 unsigned int h=0; // for int e r r u p t intermediate memory @ i f loops 33 unsigned int k ; // for counter of measure poti# mean−loop 34 unsigned int b=0; // for counter of function f l a s h 35 unsigned int stemp ; // for backsaving SREG when int e r r u p t i s performed 36 v o l a ti l e unsigned int sh u t t e r p o s ; // v o l a t i l e , since used for timer−int e r r u p t 76 C. Atomic beam shutter control 37 unsigned int beam unshut ; 38 unsigned int beam shut ; 39 40 41 // functions 42 43 void i n i t i o ( void ) ; 44 void i n i t i n t ( void) ; 45 void init pwm ( void) ; 46 void i n i t a d c ( void) ; 47 void measure poti2 ( void ) ; 48 void measure poti1 ( void ) ; 49 void manual control ( void ) ; 50 void p c c o n t r o l ( void ) ; 51 void l e d f l a s h ( void ) ; 77 Appendix D Calibrations D.1 Laser frequency To calibrate the laser frequency in the scanning region, we use a wavemeter. It measures the wavelength of the non-frequency-doubled 802 nm light from the test output of the laser system. Since the light is transported with a long ﬁber to the wavemeter, the measurement is vulnerable to temperature ﬂuctuations and vibrations. Also the switcher box that is used to share the wavemeter with other experiments reduces the accuracy and time resolution if operated in switch mode. Therefore, this method is not suitable for absolute frequency measure-ments. However, for relative frequency measurements with time averaged values, the precision is suﬃcient. A long-term record from the wavemeter gives the max-imum and minimum frequencies of the scan. To obtain the real maximum and minimum wavelength values, it is important to average the recorded values over long times and then choose the absolute maxima and minima. Figure D.1 shows such a record of the wavelength as a function of the time. The amplitudes do not correspond to the real amplitudes, as we can see by the interference pattern of the data points. Since the sampling rate of the wavemeter is too small to allow smooth sampling, we observe interference with the scan rate of the laser. To convert the time axis of the measurements at the oscilloscope to a frequency axis, we determine the frequency change per time. We measure the scan rate of the laser with an oscilloscope and combine this with the absolute maxima and minima of the frequencies to obtain the frequency change per unit time of the scan. We scale the time scale of the oscilloscope with this value and obtain a frequency axis, which we normalize to the 166Er absorption peak and that thus shows the relative detuning. 78 D. Calibrations Figure D.1: Interference pattern of the laser scan rate and the sampling rate of the wavemeter. The amplitude response does not correspond to the real amplitudes since the sampling rate is too small. Figure D.2: Brightness as a function of position along one pixel row of an image of an ruler. One peak spacing corresponds to 1 mm what determines the scaling factor. 79 D. Calibrations D.2 CCD camera To calibrate the pixel size of the CCD images, we focus the camera lens to a ruler at equal distance as the center of the chamber. The proﬁle along one pixel row of the resulting image, as in Fig. D.2, shows distinct peaks with a spacing of 1 mm. This determines the scaling factor to convert pixel to mm. The exposure time of the camera needs to be short to avoid saturation of the CCD chip. As well, it needs to be long to detect very low ﬂuorescence light intensities. To ﬁnd the optimum value, we measured the waist of an imaged atomic cloud as a function of the exposure time. A measurement-set at TEC = 1350 ◦C and THL = 1450 ◦C gives the upper bound, another set at TEC = 1250 ◦C and THL = 950 ◦C gives the lower bound. As we see in Fig. D.3, for exposure times below 100 ms, the low temperature cloud cannot be detected properly. For exposure times above 250 ms, the waist of the high-temperature cloud is no longer constant, but it depends linearly on the exposure time. 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Ertmer, Atoms with anomalous Zeeman eﬀect in a 1D-magneto-optical molasses, Opt. Commun. 94, 525 (1992). J. Jensen and A. R. Mackintosh, Rare Earth Magnetism: Structures and Excitations, Clarendon Press Oxford, 1991. J. P. D 'Incao and C. H. Greene, Collisional Aspects of Bosonic and Fermionic Dipoles in Quasi-Two-Dimensional Conﬁning Geometries, arXiv:1011.3469v2 (2010). W. Ketterle and N. J. van Druten, Evaporative Cooling of Trapped Atoms, Adv. At. Mol. Opt. Phys. 35, 181 (1996). R. V. Krems, G. C. Groenenboom, and A. Dalgarno, Electronic Interaction Anisotropy between Atoms in Arbitrary Angular Momentum States, J. Phys. Chem. A 108, 8941 (2004). A. A. Buchachenko, Y. V. Suleimanov, M. M. Szczesniak, and G. Chalasin-ski, Interactions and collisions of cold metal atoms in magnetic traps, Phys. Scr. 80, 048109 (2009). N. Leefer, A. Cing¨ oz, B. Gerber-Siﬀ, A. Sharma, J. R. Torgerson, and D. Budker, Transverse laser cooling of a thermal atomic beam of dyspro-sium, Phys. Rev. A 81, 043427 (2010). E. Wille, Preparation of an Optically Trapped Fermi-Fermi Mixture of 6Li and 40K Atoms and Characterization of the Interspecies Interactions by Fes-hbach Spectroscopy, PhD thesis, Universit¨ at Innsbruck, 2009. 86 Bibliography G. Thalhammer, Ultrakalte gepaarte Atome in koh¨ arenten Lichtfeldern, PhD thesis, Universit¨ at Innsbruck, 2007. R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, High-Frequency Optically Heterodyned Saturation Spectroscopy Via Resonant Degenerate Four-Wave Mixing, Phys. Rev. Lett. 44, 1251 (1980). D. J. McCarron, S. A. King, and S. L. Cornish, Modulation transfer spec-troscopy in atomic rubidium, Meas. Sci. Technol. 19, 105601 (2008). CreaTec Fischer & Co. GmbH, Dual Filament Eﬀusion-Cell DFC-40-10-290-WK, Industriestr. 9, 74391 Erligheim, Germany, 2009. S. Kroboth, Laserk¨ uhlung von Ytterbiumatomen, Master’s thesis, Universit¨ at Stuttgart, 2002. 87 List of Symbols and Abbreviations Abbreviation Description 2-LS two level system A area a distance between two apertures a scattering length a.u. arbitrary units add dipole-dipole characteristic radius a0 Bohr radius ADC analog to digital conversion α angle of emergence relative to the z-axis from the oven amu atomic mass unit b inner diameter of aperture BEC Bose-Einstein condensate c speed of light C normalisation constant CCD charge-coupled device d atomic beam diameter D ln (10)× optical density DDI dipole-dipole interaction ∆ν linewidth E energy \|e⟩ excited state EC eﬀusion cell EOM electro-optic modulator f collimation ratio F quantum number of the hyperﬁne splitting FWHM full width half maximum g Land´ e-g factor 88 List of Symbols and Abbreviations Abbreviation Description \|e⟩ ground state Γ spontaneous decay rate h Planck constant ℏ reduced Planck constant HCL hollow cathode lamp HL hot lip I intensity I total nuclear spin quantum number Isat saturation intensity IC integrated circuit j light-atom interaction length J total angular momentum quantum number k photon momentum kB Boltzmann constant l distance from aperture 2 L total orbital angular momentum quantum number λ wavelength λmf mean free path λ/2 half-wave plate LED light emitting diode LO local oscillator m mass mJ projection of the total angular momentum quantum -- number to the quantization axis MOT magneto-optical trap µ magnetic moment µC mirco controller µ0 magnetic constant µB Bohr magneton n density ν frequency ν0 resonance frequency NIST National Institute of Standards and Technology OD optical density ODT optical dipole trap ω solid angle PBS polarizing beam splitter PD photo diode Θ ﬂux PID proportional integral derivative 89 List of Symbols and Abbreviations Abbreviation Description PWM pulse width modulation r absolute distance S total spin quantum number σat atom-atom cross-section σ0 light-atom cross-section smsf single-mode single-frequency T temperature t time τ lifetime TC transversal cooling TTL transistor-transistor logic UHV ultra high vacuum UPS uninterrupted power supply v velocity V volume ZS Zeeman slower 90 Merci! First, I would like to thank my supervisor Francesca Ferlaino for giving me the opportunity to participate in setting up this new experiment. Especially from her, I also learnt things that are not directly connected to physics, such as presenting and writing. In moments when we underestimated the power of Hofstadter 's law, our team play was exceptionally good – thanks. Großer Dank geht auch an Rudi Grimm; daf¨ ur, in seiner Arbeitsgruppe mitarbeiten zu k¨ onnen, f¨ ur seine weisen physikalischen Ratschl¨ age, als auch f¨ ur die M¨ oglichkeit an der EuroQUAM und dem netten Waldrast-Retreat teilnehmen zu k¨ onnen. I thank our continuously growing erbium team for the great team work. Besonders von Alberts schier grenzenlosen Technik Know-How konnte ich mir so manches abschauen. Danke f¨ ur die vielen feinen gemeinsamen Laborstunden. I also want to mention Kiyotaka, with whom I shared some inspiring discussions on easy, but amazingly complicated atomic beam physics. The whole group oﬀered a very pleasant and supporting working atmosphere with tasty 'cakes at 4 ' and many answers to my questions – thanks to all of you. Besonderen Dank an Simon – meinen 'Postdoc ' – der mir viel bei Belangen rund ums blaue Licht half and also to Matteo for his supporting smile. Im weiteren danke ich unseren Werkstattmitarbeitern f¨ ur die technische Unterst¨ utzung, die vor allem beim Aufbau eines neuen Experiments n¨ otig ist. Besonders Helmut brachte mir mit seiner aufgeweckten Art die Bedienung vieler Maschinen bei. Seit Begin meines Studiums begleitete mich und viele andere Studenten Sabine Schindler als Studienbeauftragte bei allem Oﬃziellem h¨ ochst unb¨ uro-kratisch, geduldig und fair – vielen Dank. Von Peter Zollers fast schon legend¨ aren Vorlesungen konnte ich sehr viel lernen, und im Nachhinein diese auch genießen. Danke auch an Alexander Gl¨ atzle, welcher mir hierbei als Tutor zur Seite stand und uns die T¨ uren ¨ oﬀnete, um den Viktor Franz Hess S¨ udpfeiler zu erklettern. Ohne den tollen und bergbegeisterten Studienkollegen h¨ atten diese 6 Jahre nicht halb so viel Spaß gemacht. Mit Roland und Ben teilte ich Biwakschachteln und Lerngruppen, mit Flo eine wahnsinns WG und viele Flaschen Wein. Adam, Andi und ich haben w¨ ahrend der intensiven Diplomzeit viel zusammengearbeitet und uns zudem einige belebende Auszeiten erlaubt: Nachtklettern an der Mar-91 tinswand, fr¨ uhmorgentliche Grat¨ uberschreitungen, Rennrad-Konferenzpausen, Feierabend-Saunen und nat¨ urlich unsere t¨ agliche, obligate Kaﬀepause. Besten Dank. Auch meinem B¨ urokollegen Mohamed danke ich f¨ ur erfrischende Schreib-pausen mit B¨ urotennis und seine Hilfe bei unserem Umzug nur eine Woche vor Abgabe dieser Arbeit. Hans Strack-Zimmermann m¨ ochte ich als meinen langj¨ ahrigen Berater im Hin-tergrund danken. Meinen Eltern Lisa und Wolfgang danke ich ¨ uber alles f¨ ur ihre positive, vertrauensvolle Einstellung und die vielen Freiheiten die sie mir gaben. Ihre Großz¨ ugigkeit hat es mir erm¨ oglicht hier in Innsbruck das zu studieren, was ich wollte. Schließlich m¨ ochte ich noch meiner wundervollen Freundin Chris danken, die mich besonders im letzten Jahr auf jede m¨ ogliche Weise unterst¨ utzt hat und somit auch einen Beitrag zu dieser Arbeit geleistet hat. Merci! 92
7639	https://www.scribd.com/document/372414487/en4	Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Upload 0 ratings0% found this document useful (0 votes) 182 views1 page Olympiad Inequalities Guide This document provides an introduction to using inequalities to solve problems in Olympiads. It discusses three methods - Jensen's inequality, Karamata's inequality, and using the convexity … Uploaded by p001 You are on page 1/ 1 Evan Chen (April 30, 2014) A Brief Introduction to Olympiad Inequalities 2 Inequalities in Arbitrary Functions : ( u,v be a function and let ,a u,v ). Suppose that we ﬁx a 1 a 2 + + a n n a (if the inequality is homogeneous, we will often insert such a condition) and we want to prove that f ( a 1 ) + f ( a 2 ) + · · · + f ( a n ) is at least (or at most) nf ( a ). In this section we will provide three methods for doing so.We say that function f is convex if f 00 ( x ) ≥ 0 for all x ; we say it is concave if f 00 ( x ) ≤ 0 for all x . Note that f is convex if and only if − f is concave. § 2.1 Jensen / Karamata Theorem 2.1 (Jensen’s Inequality) If f is convex, then f ( a 1 ) + · · · + f ( a n ) n ≥ f  a 1 + · · · + a n n  . The reverse inequality holds when f is concave. Theorem 2.2 (Karamata’s Inequality) If f is convex, and ( x n ) majorizes ( y n ) then f ( x 1 ) + · · · + f ( x n ) ≥ f ( y 1 ) + · · · + f ( y n ) . The reverse inequality holds when f is concave. Example 2.3 (Shortlist 2009)Given a + b + c = 1 a + 1 b + 1 c , prove that1(2 a + b + c ) 2 + 1( a 2 b + c ) 2 + 1( a + b 2 c ) 2 ≤ 316 . Proof. First, we want to eliminate the condition. The original problem is equivalent to1(2 a + b + c ) 2 + 1( a 2 b + c ) 2 + 1( a + b 2 c ) 2 ≤ 316 · 1 a + 1 b + 1 c a + b + c . Now the inequality is homogeneous, so we can assume that a + b + c = 3. Now our original problem can be rewritten as X cyc 116 a − 1( a 3) 2 ≥ 0 . Set f ( x ) = 116 x − 1( x +3) 2 . We can check that f over (0 , 3) is convex so Jensen completes the problem.4 Download to read ad-free Share this document Share on Facebook, opens a new window Share on LinkedIn, opens a new window Share with Email, opens mail client Millions of documents at your fingertips, ad-free Subscribe with a free trial You might also like Stucor Ma3351 Er No ratings yet Stucor Ma3351 Er 149 pages Sports Acoustics No ratings yet Sports Acoustics 43 pages S14 Zenki ECU Pinout Guide No ratings yet S14 Zenki ECU Pinout Guide 1 page Quality Matters: Pollution Exacerbates Water Scarcity and Sectoral Output Risks in China No ratings yet Quality Matters: Pollution Exacerbates Water Scarcity and Sectoral Output Risks in China 10 pages Electrical Machines: Induction Motors - Note No ratings yet Electrical Machines: Induction Motors - Note 41 pages General Tests, Processes and Apparatus PDF No ratings yet General Tests, Processes and Apparatus PDF 334 pages Well - Stimulation Techniques - For Geothermal - Projects in Sedimentary Basins No ratings yet Well - Stimulation Techniques - For Geothermal - Projects in Sedimentary Basins 175 pages Act. 2 - Micropipetting Techni No ratings yet Act. 2 - Micropipetting Techni 29 pages 032-066 Biotech2e Lab Ch03 No ratings yet 032-066 Biotech2e Lab Ch03 35 pages Roadmap To No ratings yet Roadmap To 58 pages Criteria For Assessment Nov 23 No ratings yet Criteria For Assessment Nov 23 92 pages Prosman2 - Fluidity of Molten Metal No ratings yet Prosman2 - Fluidity of Molten Metal 22 pages Pharmaceuticals 18 00217 No ratings yet Pharmaceuticals 18 00217 25 pages B11 Building Enviro Systems and Control Exam Questions No ratings yet B11 Building Enviro Systems and Control Exam Questions 20 pages Convex Inequality PDF No ratings yet Convex Inequality PDF 5 pages Or Guide 13 No ratings yet Or Guide 13 12 pages Computer Science Engineering Course Outcomes No ratings yet Computer Science Engineering Course Outcomes 17 pages RCC Structure by PANDI MANI No ratings yet RCC Structure by PANDI MANI 13 pages Classical Inequalities in Olympiad Math No ratings yet Classical Inequalities in Olympiad Math 10 pages Biology 0610 Paper 4 MS No ratings yet Biology 0610 Paper 4 MS 11 pages Chapter 18 No ratings yet Chapter 18 9 pages A Novel Online Machine Learning Approach For.. 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7640	https://www.eabel.com/the-density-of-gold/	The Density of Gold: Why This Precious Metal Feels So Heavy [ September 2025 ] Skip to content Distribution Boxes #### General Boxes In-stock distribution boxes, general in sizes, flexible in use, eligible to protection categories. #### Customized Boxes Made-to-measure distribution boxes in a variety of materials, types and ratings. #### Accessories Mounting plates, locks, racks, cables, brackets…everything to empower your enclosure system. 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Enclosures Industrial Enclosures #### Wall Mount Enclosures #### Free-Standing Enclosures #### Modular Enclosures #### Junction/Terminal Enclosures #### Disconnect Enclosures #### Operator Interface Enclosures #### PLC Enclosures #### Combiner Boxes #### Residential Boxes #### GGD Type #### GCK Type #### Meter Boxes #### Control Boxes #### Switch Boxes #### XL Enclosures #### CCTV Housing #### Cabinets and Containments #### Server Rack Cabinets #### Control Console #### Outdoor Cabinets #### Enclosure Ratings #### NEMA Enclosures #### IP Enclosures #### IECEX Enclosures #### ATEX Enclosures #### UL Listed Enclosures #### C-UL Enclosures Applications MV/LV Enclosures Outdoor Environment IT Enclosures Engineering Climate Control Medical Facilities Data Centers Electric Distribution Telecommunication Computing & Storage Alternative Energy More Applications Support Customization Assembly Service Engineering Support Frequently Asked Questions Become A Distributor About About Us Manufacturing Certificates Sustainability Insights Blog Case Study Contact The Density of Gold: Why This Precious Metal Feels So Heavy Written By: Leo Lu Last Update: September 16, 2025 E-able-Blog- Discover the density of gold, why it’s so heavy, how temperature and alloys affect it, and why this property matters in jewelry, mining, and industry. Table of Contents If you’ve ever held a gold coin or bar in your hand, you’ve probably noticed something surprising: it feels much heavier than it looks. That “heaviness” is due to one of gold’s most defining physical properties—its density. In this blog, we’ll explore what density really means, the exact density of gold, why it is so high, and why this property matters in fields ranging from jewelry to engineering. By the end, you’ll have a clear understanding of why gold is not just shiny and valuable, but also one of the densest naturally occurring metals. What Does Density Mean? Before diving into gold, let’s quickly refresh what density is. Density is simply the amount of mass in a given volume. Imagine two cubes of the same size: one made of wood and one made of gold. The wood cube will feel light, while the gold cube will feel much heavier, even though both cubes take up the same space. The formula is straightforward: Density = Mass ÷ Volume In science, density is usually expressed in grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). Another related term is specific gravity, which compares a material’s density to water. Water has a specific gravity of 1.0, so if a substance has a specific gravity of 19.3, that means it’s 19.3 times heavier than water for the same volume. The Density of Gold So, how dense is gold? At room temperature, the density of pure gold is: 19.32 g/cm³ or 19,320 kg/m³ in SI units. Its specific gravity is about 19.3. That means gold is more than 19 times as dense as water. To put this in perspective, a 1 cm³ cube of gold weighs about 19.3 grams. For comparison, a 1 cm³ cube of aluminum weighs only 2.7 grams, and even iron is just 7.9 grams. This explains why gold feels surprisingly heavy when you hold even a small piece in your hand. Why Is Gold So Dense? The reason for gold’s remarkable density lies in its atomic structure. Gold atoms are extremely heavy because the element has an atomic number of 79, meaning each atom contains 79 protons and a large number of neutrons. On top of that, gold atoms pack together tightly in a crystal structure, leaving very little empty space. This combination of heavy atoms and compact atomic packing produces its high density. This density is not just a number—it’s a fundamental property that makes gold different from many other metals. Does Temperature Affect Gold’s Density? Yes, temperature changes can affect gold’s density. Like most metals, gold expands when heated, which means its volume increases slightly. Since mass stays the same, the density decreases as temperature rises. Near its melting point (1064 °C), the density of solid gold drops to about 18.31 g/cm³. Once it becomes molten, its density falls further to around 17.19 g/cm³. This difference matters in industrial settings like casting or electronics manufacturing, where precise data on density and expansion are crucial for accuracy. Why Does Gold’s Density Matter? Gold’s density isn’t just a scientific curiosity. It has very practical applications in many fields. 1. Testing Purity Because the density of pure gold is a known value, jewelers and assayers often use density measurements to check the purity of gold items. If an item’s measured density is significantly less than 19.32 g/cm³, it usually means other metals have been mixed in. 2. Gold Prospecting and Recovery Gold’s density plays a major role in mining. In traditional panning, gold flakes settle to the bottom of the pan because they are much denser than sand or gravel. This property makes gold relatively easy to separate from lighter materials. 3. Engineering and Design In fields like aerospace, electronics, and dentistry, engineers must account for gold’s density when designing products. Gold’s weight affects how much can be used in connectors, coatings, or fillings without adding unnecessary bulk or cost. What About Gold Alloys? Pure gold, also known as 24-karat gold, is quite soft and not practical for most jewelry. That’s why it is often alloyed with other metals such as silver, copper, or nickel to create stronger materials. When gold is mixed with other metals, its density changes depending on the type and amount of metals added. For example: 18K gold (75% gold) is slightly less dense than pure gold. 14K gold (58.3% gold) is even lighter. This explains why two rings of the same size may feel different in weight depending on their karat value and alloy composition. How Do You Measure Gold’s Density? There are both simple and advanced methods: Water Displacement Method: Weigh the gold, then measure how much water it displaces. Divide weight by displaced volume to get density. This is an ancient method still used for quick testing. Laboratory Methods:Pycnometers, precision balances, and advanced X-ray techniques provide highly accurate density data, especially for industrial and scientific uses. In jewelry testing, density checks often serve as a first step before more advanced purity tests. Quick Conversions and Fun Facts Here are some handy conversions that put gold’s density into perspective: 19.32 g/cm³ = 19,320 kg/m³ A 1 cm³ cube of gold weighs 19.3 g (about the weight of four nickels). A 1-inch cube of gold weighs about 11.17 ounces—that’s almost three-quarters of a pound, in a cube small enough to fit in your pocket! No wonder even small gold items feel so solid and substantial. Frequently Asked Questions (FAQs) on Density of Gold Why is the density of gold higher than most metals like silver or copper? Gold has a higher atomic mass and tighter atomic packing, which makes its density significantly greater than lighter metals like silver (10.5 g/cm³) or copper (8.9 g/cm³). Can density help detect fake gold? Yes. Measuring the density of gold is a simple way to spot fake or low-quality alloys. If density differs from 19.32 g/cm³, it’s likely not pure. How does the density of gold compare to platinum? Platinum is even denser, at about 21.4 g/cm³, compared to gold’s 19.32 g/cm³. This makes platinum slightly heavier for the same volume. Does pressure affect the density of gold? Yes, but only under extreme conditions. Very high pressure can compress gold’s atomic structure slightly, increasing its density. In everyday situations, this effect is negligible. Why does density make gold useful in medicine? Gold’s high density allows it to be used in medical imaging, targeted treatments, and radiation shielding, where heavy, stable, and non-reactive materials are essential. Conclusion The density of gold may sound like a technical detail, but it explains so much about why gold behaves the way it does—why it sinks during panning, why it feels so heavy in your hand, and why it’s so useful across industries. Remember the key figure: 19.32 g/cm³ at room temperature. That number not only defines one of gold’s most remarkable properties but also underpins its role in jewelry, mining, and advanced technology. Looking for safe and robust electrical enclosures? Eabel’s expertly engineered solutions provide long-lasting protection in any condition. Choose Eabel today and power your business with confidence and reliability! Related Read Does Gold Rust? The Truth Explained Density of Zinc: A Complete Guide Does Nickel Rust? Everything You Need to Know A Detailed Insight into Brass Density Density of Lead Like this post? 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7642	https://askfilo.com/books/ie-irodov-problems-in-general-physics-questions/work-energy-and-power	Filters Filters Filters topic Work, Energy and Power Semiconductor Electronics: Materials, Devices and Simple Circuits Alternating Current Motion in Straight Line Electromagnetic Waves Motion in a Plane Gravitation Waves View all difficulty easy medium hard Solutions for IE Irodov Problems in General Physics, Work, Energy and Power Views: 5,487 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View 5 solutions Views: 6,028 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View 3 solutions Views: 6,305 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View solution Views: 6,056 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View 11 solutions Views: 6,168 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View solution Views: 5,439 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View solution Views: 5,407 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View 3 solutions Views: 6,368 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View solution Views: 5,353 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View 2 solutions Views: 5,354 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View 2 solutions Found the solution, but did not understand the concept? Connect with a tutor in less than 60 seconds 24x7 Ask my first question Views: 5,901 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View solution Views: 5,658 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View solution Views: 5,355 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View solution Views: 6,115 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View solution Views: 5,924 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View solution Views: 5,862 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View 3 solutions Views: 5,944 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View 3 solutions Views: 6,076 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View solution Views: 6,150 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View 2 solutions Views: 6,023 Topic: Work, Energy and Power Book: Problems in General Physics (IE Irodov) View 2 solutions Prev Next Are you ready to take control of your learning? Download Filo and start learning with your favorite tutors right away! Blog Knowledge © Copyright Filo EdTech INC. 2025
7643	https://www.reddit.com/r/Mcat/comments/90hp96/doppler_effect_equation_quick_question/	Doppler effect equation quick question : r/Mcat Skip to main contentDoppler effect equation quick question : r/Mcat Open menu Open navigationGo to Reddit Home r/Mcat A chip A close button Get App Get the Reddit app Log InLog in to Reddit Expand user menu Open settings menu Go to Mcat r/Mcat r/Mcat The #1 social media platform for MCAT advice. The MCAT (Medical College Admission Test) is offered by the AAMC and is a required exam for admission to medical schools in the USA and Canada. /r/MCAT is a place for MCAT practice, questions, discussion, advice, social networking, news, study tips and more. Check out the sidebar for useful resources & intro guides. Post questions, jokes, memes, and discussions. 306K Members Online •7 yr. ago [deleted] Doppler effect equation quick question fs=fo((v +/- vs)/(v +/- vo)) (1) I memorized this from tpr text book but I saw Δf/fs = - v/c & Δλ/λ = v/c (2) in physics qpack : Can someone explain how does the sign work ? is it as is ? Edit : So I saw from previous posts that first equation is only used for sound waves. While (2) is used for both sound and EM radiation Can someone confirm? df=change in freq, fs= source freq, v= relative velocity b/t source n observer, c =source velocity. Read more Share Share Related Answers Section Related Answers Doppler effect equation explanation Doppler effect formula details Doppler shift equation overview Doppler effect calculations 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 Top Posts RedditreReddit: Top posts of July 20, 2018 RedditreReddit: Top posts of July 2018 RedditreReddit: Top posts of 2018 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation TOPICS Internet Culture (Viral) Amazing Animals & Pets Cringe & Facepalm Funny Interesting Memes Oddly Satisfying Reddit Meta Wholesome & Heartwarming Games Action Games Adventure Games Esports Gaming Consoles & Gear Gaming News & Discussion Mobile Games Other Games Role-Playing Games Simulation Games Sports & Racing Games Strategy Games Tabletop Games Q&As Q&As Stories & Confessions Technology 3D Printing Artificial Intelligence & Machine Learning Computers & Hardware Consumer Electronics DIY Electronics Programming Software & Apps Streaming Services Tech News & Discussion Virtual & Augmented Reality Pop Culture Celebrities Creators & Influencers Generations & Nostalgia Podcasts Streamers Tarot & Astrology Movies & TV Action Movies & Series Animated Movies & Series Comedy Movies & Series Crime, Mystery, & Thriller Movies & Series Documentary Movies & Series Drama Movies & Series Fantasy Movies & Series Horror Movies & Series Movie News & Discussion Reality TV Romance Movies & Series Sci-Fi Movies & Series Superhero Movies & Series TV News & Discussion RESOURCES About Reddit Advertise Reddit Pro BETA Help Blog Careers Press Communities Best of Reddit Top Translated Posts Topics
7644	https://www.youtube.com/watch?v=vhSfXiRRdY4	Brilliant integral by symmetry blackpenredpen 1390000 subscribers 3760 likes Description 90790 views Posted: 26 Nov 2017 Do math for fun with brilliant.org, , first 200 people to sign up will get 20% off your subscription, and you can support my channel too! So good! , So, I finally have a chance to show you how to solve a definite integral by taking the advantage of its symmetry. Don't just use brute force and integrate all the time (although it might be more fun). I got this problem from: , and there are plenty more!!! integral of log(x)/(log(x)+log(6-x)) from 2 to 4, integral of sqrt(cot(x))/(sqrt(cot(x))+sqrt(tan(x))), JEE main 2015 integral, IIT JEE, integral by symmetry, integral properties, blackpenredpen, math for fun 397 comments Transcript: Okay, let's do some math for fun. You have this integral for you guys from brilliant.org. And this integral is from the J mains 2015 right here. And I got this from the brilliance.works app. So be sure you guys check it out because it has a lot of interesting questions such as this one right here. And you can also check out other topics such as algebra, number series, component, and also logic and computer science. Right? And you can also read over their wiki. So you can just kind of study the things that they have right here. And what I did was I just type in integral properties and I got the problems right here. Right? But anyway for me I really like to go to the website and the apps so that I can just do the questions for fun so I can stay in shape as well. And sometimes I use the questions with my students also. So this is the integral from 2 to 4 log of x^2 over log of x^2 + log of 36 - 12x + x^2. Crazy enough, huh? Okay. So what are the things we can do? But first of all, let's talk about this. When you see log is log base 10 or log base e. Okay. Anything beyond differential equation or calculus things like that when you see log right it means log base e. All right. But if you're in calculus one, two, three differential equations, those crosses where you see log, it means log base 10. But anyway, hopefully that does not affect our answer here. And especially we do have this integral going from two to four. And in fact, if I don't have the limits of integrations, I don't think I can actually integrate this by hand. Well, let's ignore that. Let's see how all the things we can do. This must be solvable, right? Otherwise you wouldn't be a question in the first place. H well I do notice that this is 36 - 12x + x squ. We can factor it with our algebra power. This is just parentheses 6 - x². All right. Okay. And now all of the input is just something to the second power. And with one of the log properties we can bring the power to the front. Right? So if you would like you can do that. So let's make that happen first. So this is the same as saying the integral from two to four and on the top we will just have lo two log of what? Well, if I didn't have the limits of integrations, when you just want to bring the two to the front, let's talk about domain. Because when you have x to the second power, this right here allows us x to be negative numbers. Because if you have negative numbers to the second power, it will be positive and you're allowed to plug in that number in terms of log. Right? So technically if we didn't have these numbers log x square is the same as 2 log absolute value of x because we still want x to be negative as well. However we know we just care about x going from two to four. They need positive numbers. So the absolute value is not needed. Okay. So that's just a quick discussion anyway over bring the two to the front. So we have two log of x and then this right here. Same deal. Plus two, right? Bring the two to the front. Log of 6 minus x dx. And by the way, this is just a log property. It's not the power rule. Even though you subtract one, it's still legit, but it's not a power rule. It's just the log property. Okay. Now, of course, we can cancel this two with this two twos. And if you would like, you can write this down one more time. And now what? Well, pay attention to this input. Especially this and that are the same, but this is different than the other two. When you plug in the lower limit of integration into here, 6 minus 2 is four. And that happens to be the upper limit of integration. Likewise, if you plug in four into here, 6 - 4 is the two. Interesting, huh? Maybe we can do some substitution. Let U being 6 - X and hope for the best. Okay. So let's make that happens to see what we can get. Let me just put this down right here. I said let U equ= to 6 - X and proceed the same way but I'm looking in now right. So let's go ahead differentiate both side. BU will beative dx and I just isolate dx. Then we divide negative on both sides. So bx will be negative du. And now we will take this integral from the x world to the u world. Don't forget this is x going from two to four. Now the integral has to go from u going from what? When x was two plugging two into here 6 - 2 is four. So we have the four right here. And then when x was four plug into here u will be 6 - 2. 6 - 4 which is two. Right? So this is u = to two. Now on the top we have log but it was an x. Well we know u is equal to 6 - x. Add x on both sides and then subtract u on both sides. In another way we know x will be 6 - u. So I will plug in 6 minus u into here. Right? So we have the 6 minus u over same thing here. We have log of 6 - u and then plus log of 6 - x. It's exactly the u already. Aha, it's the u. And then dx is what? dx is negative du. So let me just put this down as multiplying by negative du like this. Compare this one and that one. They look similar, isn't it? Well, the limits of the integration got flipped it. But okay, it's all log functions and okay, first of all, perhaps let's change the limit of integrations because we have this negative to help us out. Let me just bring the negative to the front, right? And by one of the integral property when you negate the integral we can just change the name of integration right. So I will just write this down right here. This is still the du. Now I will just flip this. You will be going from two to four. And now this will be a positive integral now because we make this we make use of this negative already. Anyway the integral of course will stay the same. this and that they have the same value because the equal sign stays the same, right? Okay. What's the deal? First of all, don't let you bother you, right? This is just a dummy variable. This is, you know, you can substitute u into x doesn't matter, but pay attention to the form and then also the input. So this is the same as that because we have the equal sign. And the best part is the denominators are the same, the numerators are different. However, when you add this and that together, imagine this u is x. They will be the same as the denominator, isn't it? So what I'm saying is that first we can write this down in terms of x, right? U is just a dummy variable. And I will have another video on what does a youth substitution do for you, right? But that's later on. Anyway, this is the deal as what I like to say. Okay, now I would just say this plus that and seriously let me just write this down in terms of x especially now the limits of integration are both from two to four. So we can do the following. Okay, so I will write it down right here. And the truth is that this is of course the original integral, right? And I think this is a good time for me to use a blue pen. So I will why not? This is the original integral, right? Because that's what we simplified like right here. And let me just write this down right here guys. Suppose I will just say hey the value of this integral is I. Therefore this is equal to the I. See, I have I and I have U. Anyway, this right here is what? This right here. In fact, it's also I, isn't it? Why? Because this is equal to that. It's equal to that. They have the same value. They may look different, but they do have the same value. So, what we're doing is that we are just doing I plus I. So, keep track of the I. We have a total of two I right here. Okay. Now here is the magic. You see they have both from 2 to four. So of course we can just put this down. This is 2 to four and then this plus that happens to on the denominator as well. That's why I talked about it earlier. So I will just show you guys the work. We will have log of okay what's the inside now? Everything together inside this is nothing but just one which is super easy. And this is equal to the integral from 2 to 4 of 1 dx. And now what's the integral of one in the x world? It's just x, right? And then this is going from x = to 2 to x is equal to 4. So in another word, we're just doing 4 minus 2. So the answer to this part is just two. But is this the answer? No. Because what I'm saying is 2 I is equal to two. Right? So I'll write this down. We have 2 I equals to 2. Therefore I has to be equal to one after we divide both sides by two. So I is equal to 1. And just to be legitimate, I will write this down again. The original integral with the original look all this it's equal to just one. All right, this is the one that we were trying to get and that's the answer. And hopefully you guys like this video with this integral technique. I call this the integral with symmetry because once we make this substitution, we were able to add this and up together and end up with a nice one right here. Isn't it pretty cool, right? So, be sure you guys check out brilliant.org and also use this link brilliant.org/ /blackpen red pen because that's how you guys can support my channel and I really appreciate that and they offer a 20% off discount if you are one of the first 200 subscribers. Okay, really cool. But in the meantime, let me give you guys another integral from Brilliant Outwork similar to this. And if I can hit 20,000 views on this video, I will do a video solutions on this integral right here. Right, that's it. So good. [Music]
7645	https://pubmed.ncbi.nlm.nih.gov/40235112/	Oropharyngeal Condyloma Lata in Secondary Syphilis: Case Report and Literature Review - 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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Oropharyngeal Condyloma Lata in Secondary Syphilis: Case Report and Literature Review Samita Srisungsuk1,Nichakarn Piyawannarat1,Taweegrit Siripongboonsitti234 Affiliations Expand Affiliations 1 Department of Medicine, Chulabhorn Hospital, Chulabhorn Royal Academy, Bangkok, Thailand. 2 Division of Infectious Diseases, Department of Medicine, Chulabhorn Hospital, Chulabhorn Royal Academy, Bangkok, Thailand. 3 Research Center on Clinical and System Microbiology, Chulabhorn Royal Academy, Bangkok, Thailand. 4 Princess Srisavangavadhana College of Medicine, Chulabhorn Royal Academy, Bangkok, Thailand. PMID: 40235112 PMCID: PMC12013960 DOI: 10.12659/AJCR.947118 Item in Clipboard Review Oropharyngeal Condyloma Lata in Secondary Syphilis: Case Report and Literature Review Samita Srisungsuk et al. Am J Case Rep.2025. Show details Display options Display options Format Am J Case Rep Actions Search in PubMed Search in NLM Catalog Add to Search . 2025 Apr 16:26:e947118. doi: 10.12659/AJCR.947118. Authors Samita Srisungsuk1,Nichakarn Piyawannarat1,Taweegrit Siripongboonsitti234 Affiliations 1 Department of Medicine, Chulabhorn Hospital, Chulabhorn Royal Academy, Bangkok, Thailand. 2 Division of Infectious Diseases, Department of Medicine, Chulabhorn Hospital, Chulabhorn Royal Academy, Bangkok, Thailand. 3 Research Center on Clinical and System Microbiology, Chulabhorn Royal Academy, Bangkok, Thailand. 4 Princess Srisavangavadhana College of Medicine, Chulabhorn Royal Academy, Bangkok, Thailand. PMID: 40235112 PMCID: PMC12013960 DOI: 10.12659/AJCR.947118 Item in Clipboard Full text links Cite Display options Display options Format Abstract BACKGROUND Condyloma lata is a hallmark of secondary syphilis, presenting as moist, flat, or raised lesions typically located in the genital and perineal regions. However, oropharyngeal condyloma lata (OCL) is a rare and often underrecognized manifestation of secondary syphilis. Its atypical presentation can lead to diagnostic challenges, particularly in the absence of classic systemic features of syphilis. CASE REPORT We report the case of a 43-year-old man with HIV and chronic hepatitis C virus co-infection, diagnosed 2 years prior. The patient had been receiving antiretroviral therapy with tenofovir alafenamide, emtricitabine, and dolutegravir, achieving virologic suppression for 6 months. His CD4 T-cell count was 331 cells/μL. He presented with a sore throat and painful swallowing lasting 1 week. On examination, a single, non-tender, moist, broad-based whitish plaque with peripheral erythema was observed on the soft palate mucosa. There was no rash, lymphadenopathy, hepatosplenomegaly, or hair loss. Differential diagnoses included atypical oral candidiasis and viral warts. However, a positive rapid plasma reagin (RPR) test with a titer of 1: 128 and a reactive Treponema pallidum electrochemiluminescence immunoassay confirmed the diagnosis of OCL. The patient received a single dose of 2.4 million units of intramuscular benzathine penicillin G, leading to the complete resolution of symptoms within 7 days. CONCLUSIONS This case underscores the importance of considering OCL in the differential diagnosis of unexplained oropharyngeal lesions, particularly in patients with risk factors for syphilis. Early recognition, serological testing, and treatment with a prompt single dose of intramuscular benzathine penicillin G are crucial to prevent misdiagnosis, delayed care, and disease progression. PubMed Disclaimer Conflict of interest statement Conflict of interest: None declared Figures Figure 1. An isolated oropharyngeal condyloma lata… Figure 1. An isolated oropharyngeal condyloma lata characterized as non-tender, moist, raised, whitish plaque demonstrated… Figure 1. An isolated oropharyngeal condyloma lata characterized as non-tender, moist, raised, whitish plaque demonstrated on the soft palate mucosa. See this image and copyright information in PMC Similar articles Condyloma latum of the plantar foot: Case report of an unusual manifestation of secondary syphilis.Leavens J, Furukawa D, Gates G.Leavens J, et al.Int J STD AIDS. 2021 Dec;32(14):1354-1357. doi: 10.1177/09564624211032798. Epub 2021 Jul 20.Int J STD AIDS. 2021.PMID: 34284668 Unusual Manifestations of Secondary Syphilis: Case Presentations.Dănescu SA, Szolga B, Georgiu C, Surcel A, Corina Șenilă SC.Dănescu SA, et al.Acta Dermatovenerol Croat. 2018 Jun;26(2):186-188.Acta Dermatovenerol Croat. 2018.PMID: 29989879 An unusual case of oral condyloma lata.Dai T, Song NJ.Dai T, et al.Int J Infect Dis. 2021 Apr;105:349-350. doi: 10.1016/j.ijid.2021.02.051. Epub 2021 Feb 16.Int J Infect Dis. 2021.PMID: 33607303 Secondary syphilis with extra-genital condyloma lata: A case report and review of the literature.Barei F, Murgia G, Ramoni S, Cusini M, Marzano AV.Barei F, et al.Int J STD AIDS. 2022 Oct;33(12):1022-1028. doi: 10.1177/09564624221124710. Epub 2022 Sep 14.Int J STD AIDS. 2022.PMID: 36113077 Review. Secondary syphilis presenting as erythema multiforme in a HIV-positive homosexual man: a case report and literature review.Liu H, Goh BT, Huang T, Liu Y, Xue R, Ke W, Gu M, Yang B.Liu H, et al.Int J STD AIDS. 2019 Mar;30(3):304-309. doi: 10.1177/0956462418805197. Epub 2018 Nov 27.Int J STD AIDS. 2019.PMID: 30482099 Review. See all similar articles References World Health Organization . World Health Organization; 2017. Sexually transmitted infections: implementing the global STI strategy. . World Health Organization . Global progress report on HIV, viral hepatitis and sexually transmitted infections, 2021: Accountability for the global health sector strategies 2016–2021: Actions for impact. Geneva: 2021. Tsuboi M, Evans J, Davies EP, et al. Prevalence of syphilis among men who have sex with men: A global systematic review and meta-analysis from 2000–20. Lancet Glob Health. 2021;9(8):e1110–e18. - PMC - PubMed Barei F, Murgia G, Ramoni S, et al. Secondary syphilis with extra-genital condyloma lata: A case report and review of the literature. Int J STD AIDS. 2022;33(12):1022–28. - PubMed Leão JC, Gueiros LA, Porter SR. Oral manifestations of syphilis. Clinics (Sao Paulo) 2006;61(2):161–66. - PubMed Show all 38 references Publication types Case Reports Actions Search in PubMed Search in MeSH Add to Search Review Actions Search in PubMed Search in MeSH Add to Search MeSH terms Adult Actions Search in PubMed Search in MeSH Add to Search Condylomata Acuminata / diagnosis Actions Search in PubMed Search in MeSH Add to Search Diagnosis, Differential Actions Search in PubMed Search in MeSH Add to Search HIV Infections / complications Actions Search in PubMed Search in MeSH Add to Search HIV Infections / drug therapy Actions Search in PubMed Search in MeSH Add to Search Humans Actions Search in PubMed Search in MeSH Add to Search Male Actions Search in PubMed Search in MeSH Add to Search Syphilis / complications Actions Search in PubMed Search in MeSH Add to Search Syphilis / diagnosis Actions Search in PubMed Search in MeSH Add to Search Syphilis / drug therapy Actions Search in PubMed Search in MeSH Add to Search Supplementary concepts Syphilis, secondary Actions Search in PubMed Search in MeSH Add to Search Related information MedGen LinkOut - more resources Full Text Sources International Scientific Literature, Ltd. Miscellaneous NCI CPTAC Assay Portal Full text links[x] International Scientific Literature, Ltd. [x] Cite Copy Download .nbib.nbib Format: Send To Clipboard Email Save My Bibliography Collections Citation Manager [x] NCBI Literature Resources MeSHPMCBookshelfDisclaimer The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). Unauthorized use of these marks is strictly prohibited. Follow NCBI Connect with NLM National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov
7646	https://www.scribd.com/document/470041994/Bar-graphs-and-pie-charts	Topic: Bar Graphs and Pie Charts \| PDF \| Chart \| Pie Chart Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 0 ratings 0% found this document useful (0 votes) 321 views 6 pages Topic: Bar Graphs and Pie Charts The document contains three examples of bar graphs and pie charts with questions about interpreting the visualizations correctly. The first example shows a misleading bar graph with years … Full description Uploaded by Biplab Parida AI-enhanced title and description Go to previous items Go to next items Download Save Save Bar+graphs+and+pie+charts For Later Share 0%0% found this document useful, undefined 0%, undefined Print Embed Ask AI Report Download Save Bar+graphs+and+pie+charts For Later You are on page 1/ 6 Search Fullscreen Rfpne 7 Hmr drmps mob pnk emrts ]ukstnfo 7 Tne fi tk ifaafwnod stmtkckots mrk truk mhfut tk hmr emrt ifr tftma smaks6 N. Kme y kmr i rf c 8<=< tf 8<=4 , tftma smaks mvk noerkmskb.NN. Rk bmtm fo tkfrnzfotma mxns sfuabmvk hkko frdmonzkb irfc amrdkst tf scmaakst.NNN. 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7647	https://www.youtube.com/watch?v=bxAfzheM2hs	Desmos Animation: The Graph of a Parabola with a Vertical Axis with The Equation in Vertex Form Mathispower4u 330000 subscribers 4 likes Description 214 views Posted: 27 Aug 2024 This video show an animation of a parabola with the equatioin in the form: x=a(y-k)^2+h. 4 comments Transcript: 0 seconds In this lesson, we will explore how the values of A, H and K affect the graph of a parabola when the equation of the parabola is in the form of y equals A times a square of x minus h plus k. When we have the equation of a parabola in this form, the parabola has a vertical axis of symmetry, 18 seconds which means the graph opens up or down. If A is greater than 0, the parabola opens up if A is less than 0, the parabola opens down. The vertex is always the point H comma k, and therefore the equation of the axis of symmetry is x equals H. In addition, 36 seconds if the absolute value of A is greater than 1, the graph is vertically stretched, and is more narrow, if the absolute by a is between 0 and 1, the parabola is vertically compressed and is more wide. 49 seconds And now let's go to a Desmos.com and look at an animation. And when we have the graph of the parent parabola, or basic parabola, y equals x squared, below I've entered y equals A times the square of x minus h plus k with 1 minute, 1 second h and k both equal to 0 and a equals 1. Notice in blue, we have the same basic parabola. And now it's a change of value of H to see how it affects the graph of the parabola. Remember 1 minute, 14 seconds H is the x coordinate of the vertex. So in H is positive, the graph is shifted right H units. Notice when H is positive, we would have subtraction inside the parentheses, because 1 minute, 26 seconds the equation is in the form of x minus h. So if we have subtraction inside the parentheses, 1 minute, 33 seconds the graph is shifted to the right. And the x coordinate of the vertex is positive H. 1 minute, 38 seconds And when H is negative, the graph is shifted left H units. Notice when H is negative, we would have addition inside the parentheses, because for example x minus negative 5.5 1 minute, 51 seconds simplifies to x plus 5. So if we have addition inside the parentheses, H is negative. 1 minute, 58 seconds Let's hit H back to 0, and now we'll change the value of K. Remember K is 2 minutes, 5 seconds the white coordinate of the vertex. When K is positive, the graph is shifted up K units. And when K is negative, the graph is shifted down K units. 2 minutes, 17 seconds Let's hit K back to 0, and now change the value of A. Notice right now, A is equal to 1, recall if A is positive, and the absolute value of A is greater than 1, 2 minutes, 31 seconds the graph opens up, and is vertically stretched, and becomes more narrow, which we see here. 2 minutes, 37 seconds When A is positive, and the absolute value of A is between 0 and 2 minutes, 43 seconds 1, the graph is vertically compressed, and becomes more wide, which we see here. And then as soon as A turns negative, the parabola opens down. Let's hit A equal to negative 1. Notice this is the graph of the basic parabola in red, reflected across the x-axis. So when A is negative, 3 minutes, 10 seconds and the absolute value of A is greater than 1, the graph opens down, and is vertically stretched, and becomes more narrow, which we see here. 3 minutes, 19 seconds Let's hit A back to negative 1. When A is negative, and the absolute value 3 minutes, 30 seconds of A is between 0 and 1, the parabola opens down, and is vertically compressed, and becomes more wide, 3 minutes, 37 seconds which we see here. I hope you found this helpful.
7648	https://elnicovengeance.wordpress.com/2011/09/12/triangle-inequality/	TRIANGLE INEQUALITY \| Nico For Math Skip to navigation Skip to main content Skip to primary sidebar Skip to secondary sidebar Skip to footer Nico For Math ============= MY HOME DAFTAR ISI CATEGORIES ALAT PERAGA DUNIA MATEMATIKA education MATERI KULIAH perangkat pembelajaran STATISTIKA teka-teki matematika TRIK MATEMATIKA Uncategorized EXTRAS MANGA MP3 NONTON TV ONLINE KUIS BERHADIAH KUIS MATEMATIKA IV KUIS MATEMATIKA III KUIS MATEMATIKA II SOAL KALKULUS – FUNGSI (berhadiah pulsa) TUTORIAL VIDEO CONTACT ME MY PROFILE LINK SAHABAT IKIP PGRI MADIUN YouTubeFlickrTwitterFacebookRSS Feed Nico For Math ::: WELCOME TO MY MATHEMATICS BLOG DON’T SAY “I HATE MATH”, UNTIL YOU TRULY UNDERSTAND IT. THEN YOU CAN HATE IT ALL YOU WANT!!!! (Elnico Says) Rate this: i 8 Votes ← SILABUS MATEMATIKA SMP PROGRAM TAHUNAN (PROTA) → TRIANGLE INEQUALITY Sep 12 Posted by NICO MATEMATIKA Image via Wikipedia In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. In Euclidean geometry and some other geometries the triangle inequality is a theorem about distances. In Euclidean geometry, for right triangles it is a consequence of Pythagoras’ theorem, and for general triangles a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either R2 or R3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints. The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the L p spaces (p ≥ 1), and inner product spaces. Euclidean geometry Euclid’s construction for proof of the triangle inequality for plane geometry. Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle ABC, an isosceles triangle is constructed with one side taken as BC and the other equal leg BD along the extension of side AB. It then is argued that angle β > α, so side AD > AC. But AD = AB + BD = AB + BC so the sum of sides AB + BC > AC. This proof appears in Euclid’s Elements, Book 1, Proposition 20. Right triangle Isosceles triangle with equal sides AB = AC divided into two right triangles by an altitude drawn from one of the two base angles. A specialization of this argument to right triangles is: In a right triangle, the hypotenuse is greater than either of the two sides, and less than their sum. The second part of this theorem already is established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle ADC. An isosceles triangle ABC is constructed with equal sides AB = AC. From the triangle postulate, the angles in the right triangle ADC satisfy: Likewise, in the isosceles triangle ABC, the angles satisfy: Therefore, and so, in particular, That means side AD opposite angle α is shorter than side AB opposite the larger angle β. But AB = AC. Hence: \overline{AD} \ . ” /> A similar construction shows AC > DC, establishing the theorem. An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point B: (i) as depicted (which is to be proven), or (ii) B coincident with D (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle γ, which would violate the triangle postulate), or lastly, (iii) B interior to the right triangle between points A and D (in which case angle ABC is an exterior angle of a right triangle BDC and therefore larger than π/2, meaning the other base angle of the isosceles triangle also is greater than π/2 and their sum exceeds π in violation of the triangle postulate). This theorem establishing inequalities is sharpened by Pythagoras’ theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides. Relationship with shortest paths The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations. The triangle inequality can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line. First, the triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Thus no polygonal path between two points is shorter than the line between them. The result for polygonal paths implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path. [edit] Some practical examples of the use of the inequality Consider a triangle whose sides are in an arithmetic progression and let the sides be a, a + d, a + 2d. Then the triangle inequality requires that:- <img src=” alt=”0<a<img src=” alt=”0<a+d<img src=” alt=”0<a+2d To satisfy all these inequalities requires:- 0 \,” /> and <img src=” alt=” -\frac{a}{3}<d When d is chosen such that d = a/3, it generates a right triangle that is always similar to the Pythagorean triple with sides 3, 4, 5. Now consider a triangle whose sides are in a geometric progression and let the sides be a, ar, ar 2. Then the triangle inequality requires that:- <img src=” alt=” 0<a<img src=” alt=” 0 alt=” 0\ 0_, consequently it can be divided through and eliminated. With _a > 0_, the middle inequality only requires _r > 0_. This now leaves the first and third inequalities needing to satisfy:- 0 \ r^2-r-1 & {} The first of these quadratic inequalities requires r to range in the region beyond the value of the positive root of the quadratic equation r 2 + r − 1 = 0, i.e. r > φ − 1 where φ is the golden ratio. The second quadratic inequality requires r to range between 0 and the positive root of the quadratic equation r 2 − r − 1 = 0, i.e. 0 < r < φ. The combined requirements result in r being confined to the range When r the common ratio is chosen such that r = √φ it generates a right triangle that is always similar to the Kepler triangle. RELATED POST :: KUIS PENJUMLAHAN (LATIHAN) LATIHAN 3 Latihan Office mix CONTOH ABSTRAK 2 CONTOH ABSTRAK 1 LAPORAN KKN BAKTI PERTIWI GELANGGANG KOMUTATIF DENGAN ELEMEN SATUAN UJI SCHEFFE UJI BARLETT UJI-T UJI HOMOGENITAS DENGAN UJI F UJI NORMALITAS UJI KESEIMBANGAN DAYA BEDA TINGKAT KESUKARAN Rate this: i 1 Vote 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 More Click to share on Lintas Berita (Opens in new window)Lintas Berita Click to email a link to a friend (Opens in new window)Email Click to share on Reddit (Opens in new window)Reddit Click to print (Opens in new window)Print Click to share on Tumblr (Opens in new window)Tumblr Click to share on Pinterest (Opens in new window)Pinterest Like Loading... Related Reviewing the regularity theory of elliptics PDEs via the Laplace equation [Part I]August 25, 2011 In "education" Reviewing the regularity theory of elliptics PDEs via the Laplace equation [Part II]August 26, 2011 In "education" MEANAugust 10, 2011 In "STATISTIKA" About NICO MATEMATIKA Welcome to my blog. My name is Nico. Admin of this blog. I am a student majoring in mathematics who dreams of becoming a professor of mathematics. I live in Kwadungan, Ngawi, East Java. Hopefully in all the posts I can make a good learning material to the intellectual life of the nation. After the read, leave a comment. I always accept criticism suggestion to build a better me again .. Thanks for visiting .. : mrgreen: View all posts by NICO MATEMATIKA » Posted on September 12, 2011, in education and tagged Euclid, Euclidean geometry, Geometry, Math, People, Pythagorean theorem, Triangle, United States. 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7649	https://www.cancer.gov/types/uterine/hp/endometrial-prevention-pdq	Endometrial Cancer Prevention (PDQ®) - NCI Skip to main content An official website of the United States government Español Menu Search Search About Cancer Cancer Types Research Grants & Training News & Events About NCI Home Cancer Types Uterine Cancer Health Professional Endometrial Cancer Prevention (PDQ®)–Health Professional Version Print Email) Uterine Cancer Patient Health Professional Endometrial Cancer Treatment Uterine Sarcoma Treatment Endometrial Cancer Prevention Endometrial Cancer Screening Research Advances Endometrial Cancer Prevention (PDQ®)–Health Professional Version On This Page Who Is at Risk? Overview Incidence and Mortality Factors With Adequate Evidence for an Increased Risk of Endometrial Cancer Factors With Adequate Evidence for a Decreased Risk of Endometrial Cancer Interventions With Inadequate Evidence of an Association With Endometrial Cancer Latest Updates to This Summary (04/10/2025) About This PDQ Summary Who Is at Risk? Go to Patient Version Endometrial cancer occurs in postmenopausal women, with an average age at diagnosis of 60 years. Estrogen, both endogenous and exogenous, is associated with endometrial proliferation, hyperplasia, and cancer. Thus, risk factors include endometrial hyperplasia, reproductive factors (nulliparity, early menarche and late menopause), polycystic ovary syndrome, postmenopausal estrogen therapy, obesity with adult weight gain, and tamoxifen use. Women with Lynch syndrome have an increased risk of endometrial cancer, as do women who have a first-degree relative with endometrial cancer. Overview In This Section Factors With Adequate Evidence for an Increased Risk of Endometrial Cancer Endometrial hyperplasia Hormone therapy (HT) with estrogen: Unopposed estrogen Selective estrogen receptor modulators (SERMs) Obesity Genetic predisposition Factors With Adequate Evidence for a Decreased Risk of Endometrial Cancer Pregnancy and lactation Hormonal contraceptives: Benefits Hormonal contraceptives: Harms Weight loss: Benefits Weight loss: Harms Physical activity: Benefits Smoking: Benefits Smoking: Harms Intervention With Inadequate Evidence of an Association With Endometrial Cancer Fruits, vegetables, and vitamins Hair products, including dyes, bleach, highlights, straighteners, and permanents Note: The Overview section summarizes the published evidence on this topic. The rest of the summary describes the evidence in more detail. Other PDQ summaries with information related to endometrial cancer prevention include the following: Endometrial Cancer Screening Endometrial Cancer Treatment Genetics of Breast and Gynecologic Cancers Uterine Sarcoma Treatment Factors With Adequate Evidence for an Increased Risk of Endometrial Cancer Endometrial hyperplasia Based on solid evidence, endometrial hyperplasia is associated with concurrent or subsequent development of cancer, an association first recognized in 1932. Magnitude of Effect: Women with hyperplasia and atypia have a 40% risk of concurrent cancer. Study Design: Prospective cohort series. Internal Validity: Good. Consistency: Good. External Validity: Good. Hormone therapy (HT) with estrogen: Unopposed estrogen Based on solid evidence, unopposed estrogen is associated with an increased risk of endometrial cancer. This excess risk can be eliminated by adding continuous progestin to estrogen therapy, but this combination is associated with an increased risk of breast cancer.[4-7] For more information, see Breast Cancer Prevention. Magnitude of Effect: The associated risk of endometrial cancer in women who use unopposed estrogen for 5 or more years is at least twofold higher than in women who do not use the hormone. The risk increases with prolonged use of unopposed estrogen. Study Design: Randomized controlled trials, cohort, and case-control studies. Internal Validity: Good. Consistency: Good. External Validity: Good. Selective estrogen receptor modulators (SERMs) Based on solid evidence, more than 2 years of tamoxifen use is associated with an increased risk of endometrial cancer. The use of a similar SERM, raloxifene, is not associated with an increased risk.[9,10] Magnitude of Effect: Women taking tamoxifen for more than 2 years have a 2.3-fold to 7.5-fold relative risk (RR) of endometrial cancer, including an increased risk of uterine serous carcinoma and carcinosarcoma. Study Design: Multiple randomized controlled trials. Internal Validity: Good. Consistency: Good. External Validity: Good. Obesity Based on solid evidence, being overweight or having obesity, and adult weight gain are associated with an increased risk of endometrial cancer. Magnitude of Effect: The risk of endometrial cancer increases 1.5-fold per 5 kg/m 2 change in body mass. Study Design: Multiple randomized controlled trials. Internal Validity: Good. Consistency: Good. External Validity: Good. Genetic predisposition Based on solid evidence, women with certain inherited conditions, with highly penetrant genes, and with a family history of endometrial cancer in a first-degree relative have an increased risk of developing endometrial cancer. Magnitude of Effect: The risk of developing endometrial cancer increased by 1.82-fold (95% confidence interval [CI], 1.65–1.98) and was associated with a history of endometrial cancer in a first-degree relative. The absolute risk of endometrial cancer among women with BRCA1 or BRCA2 variants was 3%. Study Design: Case controls, cohort studies. Internal Validity: Good. Consistency: Good. External Validity: Good. Factors With Adequate Evidence for a Decreased Risk of Endometrial Cancer Pregnancy and lactation Based on solid evidence, increased parity and duration of lactation are associated with a decreased risk of endometrial cancer. Magnitude of Effect: Parous women have a 35% decreased risk of endometrial cancer (hazard ratio [HR], 0.65; 95% CI, 0.54–0.77) compared with nulliparous women. Duration of breastfeeding has also been associated with a decreased risk, with a 23% risk reduction noted for women who breastfeed longer than 18 months. The risk reduction was attenuated when adjusted for parity.[14,15] Study Design: Prospective cohort study, case-control studies. Internal Validity: Good. Consistency: Good. External Validity: Good. Hormonal contraceptives: Benefits Based on solid evidence, at least 1-year use of oral contraceptives containing estrogen and progesterone decreases endometrial cancer risk, proportionate to duration of use. The lower risk may persist for more than 30 years after the last use of oral contraceptives.[16,17] Study Design: Case-control studies and cohort studies. Internal Validity: Good. Consistency: Good. External Validity: Good. Magnitude of Effect: Use of oral contraceptives for 5 years was associated with an RR reduction of 24% (risk ratio, 0.76; 95% CI, 0.73–0.78) and persisted for more than 30 years. Ten years of use was associated with an absolute reduction in risk before age 75 years from 2.3 per 100 women to 1.3 per 100 women. Hormonal contraceptives: Harms Based on solid evidence, current use of combined oral contraceptives is associated with an increased risk of blood clots, stroke, and myocardial infarction, especially among women who smoke cigarettes and who are older than 35 years. Magnitude of Effect: Use of oral contraceptives was associated with an absolute increased risk of blood clots of approximately 1 case per 4,465 person-years (95% CI, 4,095–4,797 person-years). Use of oral contraceptives was associated with an increased RR of stroke or myocardial infarction of 60% (risk ratio, 1.6; 95% CI, 1.3–1.9). Study Design: Cohort studies, nested case-control studies, case-control studies. Internal Validity: Good. Consistency: Good. External Validity: Good. Weight loss: Benefits The evidence is insufficient to conclude whether weight loss is associated with a decreased incidence of endometrial cancer. Based on one study, self-reported intentional weight loss during three age periods was not associated with a decrease in endometrial cancer incidence. Bariatric surgery is associated with a decreased risk of developing endometrial cancer.[21-23] After bariatric surgery, other obesity-related health conditions, such as diabetes and metabolic syndrome are also often improved or resolved. Magnitude of Effect: RR of endometrial cancer for women who intentionally lost at least 20 pounds was 0.93 (95% CI, 0.6–1.44). The incidence rate of endometrial cancer per 1,000 person-years was 1.1 in those who underwent bariatric surgery compared with 2 in the control group of patients with obesity who received usual care (HR, 0.56; 95% CI, 0.35–0.89). Study Design: Prospective and retrospective cohort studies. Internal Validity: Good. Consistency: Fair. External Validity: Good. Weight loss: Harms A variety of procedures are included under the umbrella of bariatric surgery. Bariatric surgery is associated with a potential for short-term surgical complications, and possible medium and long-term risks. Immediate surgical complications may include infections, venous thromboembolism, respiratory or cardiac complications, anastomotic leak, marginal ulcers, stenosis or obstruction, or rarely, death.[24,25] Dumping syndrome and metabolic and nutritional derangements from malabsorption may also occur. Physical activity: Benefits Based on solid evidence, increased physical exercise is associated with a decreased risk of endometrial cancer.[27,28] Magnitude of Effect: Regular exercise may be associated with a 38% to 46% relative decrease in risk. However, a trend in risk reduction with increasing exercise duration or intensity has not been shown. Study Design: Multiple cohort and case-control studies. Internal Validity: Good. Consistency: Fair. External Validity: Good. Smoking: Benefits Based on solid evidence, cigarette smoking is associated with a decreased risk of endometrial cancer. Magnitude of Effect: Smokers have a reduced risk of endometrial cancer of approximately 20% among prospective studies (RR, 0.81; 95% CI, 0.74–0.88) and case-control studies (odds ratio, 0.72; 95% CI, 0.66–0.79). Study Design: Prospective cohort and case-control studies. Internal Validity: Good. Consistency: Good. External Validity: Good. Smoking: Harms Based on solid evidence, cigarette smoking is associated with cardiovascular disease and cancers of the head and neck, lung, bladder, and pancreas. Cigarette smokers have a decreased life expectancy; they live at least 10 fewer years than nonsmokers. Intervention With Inadequate Evidence of an Association With Endometrial Cancer Fruits, vegetables, and vitamins There is adequate evidence of no association between endometrial cancer and diet or vitamin intake.[31-35] Study Design: Cohort and case-control studies. Internal Validity: Good. Consistency: Good. External Validity: Good. Hair products, including dyes, bleach, highlights, straighteners, and permanents There is insufficient evidence of an association between hair products and endometrial cancer. One retrospective analysis of the Sister Study addressed a possible association between these hair products and uterine cancers, including endometrial cancers. Study Design: Cohort. Internal Validity: Poor. Consistency: No other studies at this time. External Validity: Poor. References Widra EA, Dunton CJ, McHugh M, et al.: Endometrial hyperplasia and the risk of carcinoma. Int J Gynecol Cancer 5 (3): 233-235, 1995.[PUBMED Abstract] Taylor HC: Endometrial hyperplasia and carcinoma of the body of the uterus. Am J Obstet Gynecol 23 (3): 309-32, 1932. Trimble CL, Kauderer J, Zaino R, et al.: Concurrent endometrial carcinoma in women with a biopsy diagnosis of atypical endometrial hyperplasia: a Gynecologic Oncology Group study. Cancer 106 (4): 812-9, 2006.[PUBMED Abstract] Beral V, Bull D, Reeves G, et al.: Endometrial cancer and hormone-replacement therapy in the Million Women Study. Lancet 365 (9470): 1543-51, 2005 Apr 30-May 6.[PUBMED Abstract] Anderson GL, Limacher M, Assaf AR, et al.: Effects of conjugated equine estrogen in postmenopausal women with hysterectomy: the Women's Health Initiative randomized controlled trial. JAMA 291 (14): 1701-12, 2004.[PUBMED Abstract] Furness S, Roberts H, Marjoribanks J, et al.: Hormone therapy in postmenopausal women and risk of endometrial hyperplasia. Cochrane Database Syst Rev (2): CD000402, 2009.[PUBMED Abstract] Grady D, Gebretsadik T, Kerlikowske K, et al.: Hormone replacement therapy and endometrial cancer risk: a meta-analysis. Obstet Gynecol 85 (2): 304-13, 1995.[PUBMED Abstract] Fisher B, Costantino JP, Redmond CK, et al.: Endometrial cancer in tamoxifen-treated breast cancer patients: findings from the National Surgical Adjuvant Breast and Bowel Project (NSABP) B-14. J Natl Cancer Inst 86 (7): 527-37, 1994.[PUBMED Abstract] Cummings SR, Eckert S, Krueger KA, et al.: The effect of raloxifene on risk of breast cancer in postmenopausal women: results from the MORE randomized trial. Multiple Outcomes of Raloxifene Evaluation. JAMA 281 (23): 2189-97, 1999.[PUBMED Abstract] DeMichele A, Troxel AB, Berlin JA, et al.: Impact of raloxifene or tamoxifen use on endometrial cancer risk: a population-based case-control study. J Clin Oncol 26 (25): 4151-9, 2008.[PUBMED Abstract] Brinton LA, Felix AS, McMeekin DS, et al.: Etiologic heterogeneity in endometrial cancer: evidence from a Gynecologic Oncology Group trial. Gynecol Oncol 129 (2): 277-84, 2013.[PUBMED Abstract] Aune D, Navarro Rosenblatt DA, Chan DS, et al.: Anthropometric factors and endometrial cancer risk: a systematic review and dose-response meta-analysis of prospective studies. Ann Oncol 26 (8): 1635-48, 2015.[PUBMED Abstract] Newcomb PA, Trentham-Dietz A: Breast feeding practices in relation to endometrial cancer risk, USA. Cancer Causes Control 11 (7): 663-7, 2000.[PUBMED Abstract] Dossus L, Allen N, Kaaks R, et al.: Reproductive risk factors and endometrial cancer: the European Prospective Investigation into Cancer and Nutrition. Int J Cancer 127 (2): 442-51, 2010.[PUBMED Abstract] Karageorgi S, Hankinson SE, Kraft P, et al.: Reproductive factors and postmenopausal hormone use in relation to endometrial cancer risk in the Nurses' Health Study cohort 1976-2004. Int J Cancer 126 (1): 208-16, 2010.[PUBMED Abstract] Collaborative Group on Epidemiological Studies on Endometrial Cancer: Endometrial cancer and oral contraceptives: an individual participant meta-analysis of 27 276 women with endometrial cancer from 36 epidemiological studies. Lancet Oncol 16 (9): 1061-70, 2015.[PUBMED Abstract] Iversen L, Sivasubramaniam S, Lee AJ, et al.: Lifetime cancer risk and combined oral contraceptives: the Royal College of General Practitioners' Oral Contraception Study. Am J Obstet Gynecol 216 (6): 580.e1-580.e9, 2017.[PUBMED Abstract] de Bastos M, Stegeman BH, Rosendaal FR, et al.: Combined oral contraceptives: venous thrombosis. Cochrane Database Syst Rev (3): CD010813, 2014.[PUBMED Abstract] Roach RE, Helmerhorst FM, Lijfering WM, et al.: Combined oral contraceptives: the risk of myocardial infarction and ischemic stroke. Cochrane Database Syst Rev (8): CD011054, 2015.[PUBMED Abstract] Parker ED, Folsom AR: Intentional weight loss and incidence of obesity-related cancers: the Iowa Women's Health Study. Int J Obes Relat Metab Disord 27 (12): 1447-52, 2003.[PUBMED Abstract] Anveden Å, Taube M, Peltonen M, et al.: Long-term incidence of female-specific cancer after bariatric surgery or usual care in the Swedish Obese Subjects Study. Gynecol Oncol 145 (2): 224-229, 2017.[PUBMED Abstract] Schauer DP, Feigelson HS, Koebnick C, et al.: Bariatric Surgery and the Risk of Cancer in a Large Multisite Cohort. Ann Surg 269 (1): 95-101, 2019.[PUBMED Abstract] Aminian A, Wilson R, Al-Kurd A, et al.: Association of Bariatric Surgery With Cancer Risk and Mortality in Adults With Obesity. JAMA 327 (24): 2423-2433, 2022.[PUBMED Abstract] Flum DR, Belle SH, King WC, et al.: Perioperative safety in the longitudinal assessment of bariatric surgery. N Engl J Med 361 (5): 445-54, 2009.[PUBMED Abstract] Sanyal AJ, Sugerman HJ, Kellum JM, et al.: Stomal complications of gastric bypass: incidence and outcome of therapy. Am J Gastroenterol 87 (9): 1165-9, 1992.[PUBMED Abstract] van Beek AP, Emous M, Laville M, et al.: Dumping syndrome after esophageal, gastric or bariatric surgery: pathophysiology, diagnosis, and management. Obes Rev 18 (1): 68-85, 2017.[PUBMED Abstract] Moradi T, Weiderpass E, Signorello LB, et al.: Physical activity and postmenopausal endometrial cancer risk (Sweden). Cancer Causes Control 11 (9): 829-37, 2000.[PUBMED Abstract] Schouten LJ, Goldbohm RA, van den Brandt PA: Anthropometry, physical activity, and endometrial cancer risk: results from the Netherlands Cohort Study. J Natl Cancer Inst 96 (21): 1635-8, 2004.[PUBMED Abstract] Zhou B, Yang L, Sun Q, et al.: Cigarette smoking and the risk of endometrial cancer: a meta-analysis. Am J Med 121 (6): 501-508.e3, 2008.[PUBMED Abstract] Centers for Disease Control and Prevention: Smoking and Tobacco Use. Atlanta, Ga: Centers for Disease Control and Prevention, Office on Smoking and Health, 2015. Available Online. Last accessed December 18, 2023. International Agency for Research On Cancer: IARC Handbooks of Cancer Prevention. Volume 8: Fruit and Vegetables. International Agency for Research On Cancer, 2003. Bandera EV, Kushi LH, Gifkins DM, et al.: WCRF Systematic Literature Review: The Association Between Food, Nutrition, and Physical Activity and the Risk of Endometrial Cancer and Underlying Mechanisms. World Cancer Research Fund, American Institute for Cancer Research, 2006. Horn-Ross PL, John EM, Canchola AJ, et al.: Phytoestrogen intake and endometrial cancer risk. J Natl Cancer Inst 95 (15): 1158-64, 2003.[PUBMED Abstract] Xu WH, Zheng W, Xiang YB, et al.: Soya food intake and risk of endometrial cancer among Chinese women in Shanghai: population based case-control study. BMJ 328 (7451): 1285, 2004.[PUBMED Abstract] Zeleniuch-Jacquotte A, Gallicchio L, Hartmuller V, et al.: Circulating 25-hydroxyvitamin D and risk of endometrial cancer: Cohort Consortium Vitamin D Pooling Project of Rarer Cancers. Am J Epidemiol 172 (1): 36-46, 2010.[PUBMED Abstract] Chang CJ, O'Brien KM, Keil AP, et al.: Use of Straighteners and Other Hair Products and Incident Uterine Cancer. J Natl Cancer Inst 114 (12): 1636-1645, 2022.[PUBMED Abstract] Incidence and Mortality Endometrial cancer is the most common invasive gynecologic cancer in U.S. women, with an estimated 69,120 new cases expected to occur in 2025. This disease primarily affects postmenopausal women at an average age of 60 years at diagnosis. In the United States, it is estimated that approximately 13,860 women will die of endometrial cancer in 2025. Over the past decade, incidence rates of endometrial cancer increased by 0.6% per year in White women and by 2% to 3% per year in women of all other racial and ethnic groups. Between 2013 and 2022, death rates for endometrial cancer increased by 1.5% per year. Higher mortality from endometrial cancer in African American women compared with White women is only partly attributable to lower socioeconomic issues that impair access to care.[3,4] Compared with the incidence in White American women, endometrial cancer incidence is lower in Japanese American (relative risk [RR], 0.6; 95% confidence interval [CI], 0.46–0.83) and Latina American women (RR, 0.63; 95% CI, 0.46–0.87), but not in African American (RR, 0.76; 95% CI, 0.53–1.08) or native Hawaiian women (RR, 0.92; 95% CI, 0.58–1.46). When corrected for hysterectomy prevalence among U.S. women, the incidence of all histological subtypes of endometrial cancer has increased approximately 2% per year between 2000 and 2015 among non-Hispanic Black, non-Hispanic Asian or Pacific Islander, and Hispanic women, but has remained stable among non-Hispanic White women. The incidence of endometrioid endometrial cancer has increased 1% to 2% among all racial and ethnic subgroups except non-Hispanic White women, where it has also remained stable across the same period. The greatest increase in incidence across all racial and ethnic subgroups has been seen in nonendometrioid histological subtypes, with an approximate 2% increase in non-Hispanic White women, 3% increase in non-Hispanic Black women, and 4% increase in non-Hispanic Asian or Pacific Islanders and Hispanic women. These findings challenge the supposition that the increase in endometrial cancer incidence was caused by the obesity epidemic, which would have been expected to increase the incidence of the endometrioid subtype more than the nonendometrioid subtypes. Endometrial cancer risk is associated with endogenous and exogenous factors associated with estrogen effects.[7-9] Thus, risk factors for endometrial cancer include reproductive factors such as nulliparity, early menarche, and late menopause, as well as obesity with adult weight gain, polycystic ovary syndrome, postmenopausal estrogen use, and tamoxifen use. Women with Lynch syndrome have a lifetime risk of endometrial cancer of up to 60%. For additional information about inherited risk, see Genetics of Breast and Gynecologic Cancers. References American Cancer Society: Cancer Facts and Figures 2025. American Cancer Society, 2025. Available online. Last accessed January 16, 2025. American Cancer Society: Detailed Guide: Endometrial Cancer: What are the Risk Factors for Endometrial Cancer? Atlanta, Ga: American Cancer Society, 2005. Available online. Last accessed April 8, 2025. Madison T, Schottenfeld D, James SA, et al.: Endometrial cancer: socioeconomic status and racial/ethnic differences in stage at diagnosis, treatment, and survival. Am J Public Health 94 (12): 2104-11, 2004.[PUBMED Abstract] Long B, Liu FW, Bristow RE: Disparities in uterine cancer epidemiology, treatment, and survival among African Americans in the United States. Gynecol Oncol 130 (3): 652-9, 2013.[PUBMED Abstract] Setiawan VW, Pike MC, Kolonel LN, et al.: Racial/ethnic differences in endometrial cancer risk: the multiethnic cohort study. Am J Epidemiol 165 (3): 262-70, 2007.[PUBMED Abstract] Clarke MA, Devesa SS, Harvey SV, et al.: Hysterectomy-Corrected Uterine Corpus Cancer Incidence Trends and Differences in Relative Survival Reveal Racial Disparities and Rising Rates of Nonendometrioid Cancers. J Clin Oncol 37 (22): 1895-1908, 2019.[PUBMED Abstract] Zeleniuch-Jacquotte A, Akhmedkhanov A, Kato I, et al.: Postmenopausal endogenous oestrogens and risk of endometrial cancer: results of a prospective study. Br J Cancer 84 (7): 975-81, 2001.[PUBMED Abstract] Lukanova A, Lundin E, Micheli A, et al.: Circulating levels of sex steroid hormones and risk of endometrial cancer in postmenopausal women. Int J Cancer 108 (3): 425-32, 2004.[PUBMED Abstract] Brown SB, Hankinson SE: Endogenous estrogens and the risk of breast, endometrial, and ovarian cancers. Steroids 99 (Pt A): 8-10, 2015.[PUBMED Abstract] Watson P, Vasen HF, Mecklin JP, et al.: The risk of endometrial cancer in hereditary nonpolyposis colorectal cancer. Am J Med 96 (6): 516-20, 1994.[PUBMED Abstract] Factors With Adequate Evidence for an Increased Risk of Endometrial Cancer In This Section Endogenous Hyperplasia Hormone Therapy (HT) With Estrogen: Unopposed Estrogen Selective Estrogen Receptor Modulators (SERMs) Obesity Genetic Predisposition Endogenous Hyperplasia Reproductive factors resulting in increased duration of exposure to endogenous estrogen, such as early menarche, nulliparity, and late menopause, are associated with an increased risk of endometrial cancer. Early menarche when compared with late menarche has been associated with a 39% relative increased risk of endometrial cancer among participants in the European Prospective Investigation into Cancer and Nutrition. In the same study, late menopause and nulliparity were associated with a 2.2-fold and 1.6-fold increased risk, respectively. Other factors associated with increased risk, such as obesity and polycystic ovary syndrome, may also be related to increased estrogen exposure. Polycystic ovary syndrome has been associated with a threefold increased risk of endometrial cancer in a meta-analysis. Hormone Therapy (HT) With Estrogen: Unopposed Estrogen An association between postmenopausal estrogen replacement therapy and endometrial cancer was reported in 1975 and confirmed soon after.[5,6] In these three studies, the overall risk ratio ranged from 4.5 to 8.0. Further studies documented an association with duration of use (10-fold to 30-fold with 5 years or more of use),[7-10] and a persistent effect lasting more than 10 years after 1 year of use. When these findings were publicized, prescriptions for estrogen declined sharply, followed rapidly by a drop in endometrial cancer incidence. Postmenopausal estrogen was long recognized to be associated with the risk of endometrial hyperplasia, often a precursor of endometrial cancer. In addition, progestational agents were known to be effective in the treatment of uterine neoplasms.[14-16] Consequently, combined estrogen-progesterone postmenopausal hormone therapy (HT) avoids the endometrial cancer risk associated with unopposed estrogen and actually reduces the risk by 35%. Tibolone, a synthetic steroid with estrogenic, progestogenic, and androgenic properties, has been associated with an increased incidence rate ratio of endometrial cancer of 3.56 (95% confidence interval [CI], 3.08–4.69) for current users compared with never-users. Tibolone is approved for use to manage menopausal symptoms or to prevent osteoporosis in many countries. However, it is not approved for use in Canada or the United States. Other combined therapy with estrogen and progestin may also increase the risk of breast cancer, so the risks and benefits must be considered.[18,19] The Women’s Health Initiative (WHI) study was a randomized trial that compared combination estrogen and progestin therapy with no hormone replacement. The absolute excess risk of breast cancer attributable to estrogen/progestin use was 8 more invasive breast cancers per 10,000 person-years. Selective Estrogen Receptor Modulators (SERMs) Tamoxifen and raloxifene are SERMs, drugs that have divergent estrogen agonist and antagonist effects in different target organs. The association between endometrial cancer and tamoxifen was first recognized in 1985 when three cases of endometrial cancer were described in women who had been treated with tamoxifen for breast cancer. Since then, confirmation of the association has been provided by randomized clinical trials using tamoxifen for breast cancer treatment and prevention [21-24] and by case-control, observational, and laboratory studies. The National Surgical Adjuvant Breast and Bowel Project, Breast Cancer Prevention Trial P-1 Study in women at high risk of invasive breast cancer demonstrated that tamoxifen decreased breast cancer incidence by 49% but confirmed an increased incidence of endometrial cancer. The annual rate was 2.2 cases per 1,000 women for those who received tamoxifen versus 0.68 cases per 1,000 women for those who received placebo. Significantly increased risks were restricted to women aged 50 years or older at study entry. Of the 53 invasive cancers associated with tamoxifen use, 52 were stage I. Tamoxifen use has also been shown to be associated with high-risk histological subtypes, with an odds ratio (OR) of 3.2 for uterine serous carcinoma and 5.4 for uterine carcinosarcoma; however, the absolute risk of these rare histological subtypes remains low. Raloxifene is a second-generation SERM approved for prophylaxis against postmenopausal osteoporosis. Unlike tamoxifen, it does not have an estrogenic effect on the uterus. The Multiple Outcomes of Raloxifene randomized trial, after 40 months of follow-up, showed that raloxifene reduced the risk of estrogen receptor–positive breast cancer, without increasing endometrial cancer (relative risk [RR], 0.8; 95% CI, 0.2–2.7). A population-based case-control study included 547 women with endometrial cancer and 1,410 controls. The study reported a reduction in risk of endometrial cancer with raloxifene use (OR, 0.50; 95% CI, 0.29–0.85) and confirmed an increased risk associated with tamoxifen use. Obesity Elevated body mass index (BMI), obesity, and weight gain are associated with an increased risk of endometrial cancer. One of the possible mechanisms for the observed association is an increased level of serum estrone in women with obesity as a result of aromatization of androstenedione in adipose tissue, which increases the production of estrogen. Alternatively, obesity has been associated with a reduction in levels of sex hormone-binding globulin (SHBG), which may protect against endometrial cancer by decreasing bioavailable estrogen. Obesity has been associated with several factors known to increase the risk of endometrial cancer, including upper-body or central adiposity, polycystic ovary syndrome, and physical inactivity.[31,32] Body weight is a modifiable risk factor, which accounts for a substantial proportion of endometrial cases worldwide. A study conducted among European countries estimated that between 26% and 47% of endometrial cancer cases can be attributed to overweight and obesity. The same group conducted a meta-analysis of 12 studies (5 cohort and 7 case-control), which examined the relationship between obesity and endometrial cancer. Eleven of the 12 studies concluded that there is a positive association between endometrial cancer and excess weight. RRs associated with obesity range from 2 to 10. Some studies show that upper-body and central weight confer a higher risk than peripheral body weight, even after consideration of BMI.[34-36] However, other studies have failed to confirm such an association. Several studies have observed a stronger association between endometrial cancer and obesity near the time of diagnosis compared with obesity earlier in life.[37-40] An increased risk is observed across all measures of adiposity, such as BMI, waist circumference, waist-to-hip ratio, and weight gain. A meta-analysis of prospective studies observed an RR of 1.39 (95% CI, 1.29–1.49) among nonusers and 1.09 (95% CI, 1.02–1.16) among HT users for each 5 kg increase in adult weight gain. Another meta-analysis also observed a stronger association between BMI and the risk of endometrial cancer in never-users of HT than in ever-users of HT. A meta-analysis examined the association between metabolic syndrome and endometrial cancer risk. The study observed an increased risk associated with metabolic syndrome (RR, 1.89; 95% CI, 1.34–2.67) and with each component of the syndrome (BMI and/or waist circumference, blood pressure, and triglyceride levels), except low high-density lipoprotein cholesterol. In a meta-analysis of studies of the association between diabetes and cancer, endometrial cancer was associated with a hazard ratio (HR) of approximately 2. However, data from the WHI suggest that the association between diabetes and endometrial cancer is largely mediated through the risk of obesity. Genetic Predisposition Women with inherited conditions such as Lynch syndrome, Cowden syndrome, and polycystic ovary syndrome have an increased risk of endometrial cancer. For more information, see Genetics of Breast and Gynecologic Cancers and Genetics of Colorectal Cancer. However, in addition to inherited syndromes with highly penetrant genes (including BRCA1 and BRCA2), having a family history of endometrial cancer in a first-degree relative also is associated with an increased risk of cancer. A meta-analysis, including case-control and cohort studies, observed an increased risk of 1.82 (95% CI, 1.65–1.98) associated with a history of endometrial cancer in a first-degree relative, with an estimated cumulative absolute risk of about 3% (95% CI, 2.8%–3.4%). A Dutch multicenter cohort study of women with germline BRCA1 and BRCA2 pathogenic variants concluded that the absolute risk of endometrial cancer was approximately 3%. This familial risk may result from inherited genetic predisposition and other common factors that exist in families, such as shared culture or learned behaviors. References Dossus L, Allen N, Kaaks R, et al.: Reproductive risk factors and endometrial cancer: the European Prospective Investigation into Cancer and Nutrition. Int J Cancer 127 (2): 442-51, 2010.[PUBMED Abstract] Brown SB, Hankinson SE: Endogenous estrogens and the risk of breast, endometrial, and ovarian cancers. Steroids 99 (Pt A): 8-10, 2015.[PUBMED Abstract] Haoula Z, Salman M, Atiomo W: Evaluating the association between endometrial cancer and polycystic ovary syndrome. Hum Reprod 27 (5): 1327-31, 2012.[PUBMED Abstract] Smith DC, Prentice R, Thompson DJ, et al.: Association of exogenous estrogen and endometrial carcinoma. N Engl J Med 293 (23): 1164-7, 1975.[PUBMED Abstract] Mack TM, Pike MC, Henderson BE, et al.: Estrogens and endometrial cancer in a retirement community. N Engl J Med 294 (23): 1262-7, 1976.[PUBMED Abstract] Ziel HK, Finkle WD: Increased risk of endometrial carcinoma among users of conjugated estrogens. N Engl J Med 293 (23): 1167-70, 1975.[PUBMED Abstract] Walker AM, Jick H: Cancer of the corpus uteri: increasing incidence in the United States, 1970--1975. Am J Epidemiol 110 (1): 47-51, 1979.[PUBMED Abstract] Gray LA, Christopherson WM, Hoover RN: Estrogens and endometrial carcinoma. Obstet Gynecol 49 (4): 385-9, 1977.[PUBMED Abstract] McDonald TW, Annegers JF, O'Fallon WM, et al.: Exogenous estrogen and endometrial carcinoma: case-control and incidence study. Am J Obstet Gynecol 127 (6): 572-80, 1977.[PUBMED Abstract] Antunes CM, Strolley PD, Rosenshein NB, et al.: Endometrial cancer and estrogen use. Report of a large case-control study. N Engl J Med 300 (1): 9-13, 1979.[PUBMED Abstract] Shapiro S, Kelly JP, Rosenberg L, et al.: Risk of localized and widespread endometrial cancer in relation to recent and discontinued use of conjugated estrogens. N Engl J Med 313 (16): 969-72, 1985.[PUBMED Abstract] Austin DF, Roe KM: The decreasing incidence of endometrial cancer: public health implications. Am J Public Health 72 (1): 65-8, 1982.[PUBMED Abstract] Gusberg SB: Precursors of corpus carcinoma estrogens and adenomatous hyperplasia. Am J Obstet Gynecol 54(6): 905-927, 1947. Gusberg SB: Hormone-dependence of endometrial cancer. Obstet Gynecol 30 (2): 287-93, 1967.[PUBMED Abstract] Bonte J: Medroxyprogesterone in the management of primary and recurrent or metastatic uterine adenocarcinoma. Acta Obstet Gynecol Scand Suppl 19: 21-4, 1972.[PUBMED Abstract] KISTNER RW: Histological effects of progestins on hyperplasia and carcinoma in situ of the endometrium. Cancer 12: 1106-22, 1959 Nov-Dec.[PUBMED Abstract] Chlebowski RT, Anderson GL, Sarto GE, et al.: Continuous Combined Estrogen Plus Progestin and Endometrial Cancer: The Women's Health Initiative Randomized Trial. J Natl Cancer Inst 108 (3): , 2016.[PUBMED Abstract] Løkkegaard ECL, Mørch LS: Tibolone and risk of gynecological hormone sensitive cancer. Int J Cancer 142 (12): 2435-2440, 2018.[PUBMED Abstract] Simin J, Tamimi R, Lagergren J, et al.: Menopausal hormone therapy and cancer risk: An overestimated risk? Eur J Cancer 84: 60-68, 2017.[PUBMED Abstract] Killackey MA, Hakes TB, Pierce VK: Endometrial adenocarcinoma in breast cancer patients receiving antiestrogens. Cancer Treat Rep 69 (2): 237-8, 1985.[PUBMED Abstract] Fornander T, Rutqvist LE, Cedermark B, et al.: Adjuvant tamoxifen in early breast cancer: occurrence of new primary cancers. Lancet 1 (8630): 117-20, 1989.[PUBMED Abstract] Rutqvist LE, Mattsson A: Cardiac and thromboembolic morbidity among postmenopausal women with early-stage breast cancer in a randomized trial of adjuvant tamoxifen. The Stockholm Breast Cancer Study Group. J Natl Cancer Inst 85 (17): 1398-406, 1993.[PUBMED Abstract] Andersson M, Storm HH, Mouridsen HT: Incidence of new primary cancers after adjuvant tamoxifen therapy and radiotherapy for early breast cancer. J Natl Cancer Inst 83 (14): 1013-7, 1991.[PUBMED Abstract] Fisher B, Costantino JP, Redmond CK, et al.: Endometrial cancer in tamoxifen-treated breast cancer patients: findings from the National Surgical Adjuvant Breast and Bowel Project (NSABP) B-14. J Natl Cancer Inst 86 (7): 527-37, 1994.[PUBMED Abstract] Fisher B, Costantino JP, Wickerham DL, et al.: Tamoxifen for the prevention of breast cancer: current status of the National Surgical Adjuvant Breast and Bowel Project P-1 study. J Natl Cancer Inst 97 (22): 1652-62, 2005.[PUBMED Abstract] Brinton LA, Felix AS, McMeekin DS, et al.: Etiologic heterogeneity in endometrial cancer: evidence from a Gynecologic Oncology Group trial. Gynecol Oncol 129 (2): 277-84, 2013.[PUBMED Abstract] Cummings SR, Eckert S, Krueger KA, et al.: The effect of raloxifene on risk of breast cancer in postmenopausal women: results from the MORE randomized trial. Multiple Outcomes of Raloxifene Evaluation. JAMA 281 (23): 2189-97, 1999.[PUBMED Abstract] DeMichele A, Troxel AB, Berlin JA, et al.: Impact of raloxifene or tamoxifen use on endometrial cancer risk: a population-based case-control study. J Clin Oncol 26 (25): 4151-9, 2008.[PUBMED Abstract] Enriori CL, Reforzo-Membrives J: Peripheral aromatization as a risk factor for breast and endometrial cancer in postmenopausal women: a review. Gynecol Oncol 17 (1): 1-21, 1984.[PUBMED Abstract] Davidson BJ, Gambone JC, Lagasse LD, et al.: Free estradiol in postmenopausal women with and without endometrial cancer. J Clin Endocrinol Metab 52 (3): 404-8, 1981.[PUBMED Abstract] Troisi R, Potischman N, Hoover RN, et al.: Insulin and endometrial cancer. Am J Epidemiol 146 (6): 476-82, 1997.[PUBMED Abstract] Barry JA, Azizia MM, Hardiman PJ: Risk of endometrial, ovarian and breast cancer in women with polycystic ovary syndrome: a systematic review and meta-analysis. Hum Reprod Update 20 (5): 748-58, 2014 Sep-Oct.[PUBMED Abstract] Bergström A, Pisani P, Tenet V, et al.: Overweight as an avoidable cause of cancer in Europe. Int J Cancer 91 (3): 421-30, 2001.[PUBMED Abstract] Swanson CA, Potischman N, Wilbanks GD, et al.: Relation of endometrial cancer risk to past and contemporary body size and body fat distribution. Cancer Epidemiol Biomarkers Prev 2 (4): 321-7, 1993 Jul-Aug.[PUBMED Abstract] Elliott EA, Matanoski GM, Rosenshein NB, et al.: Body fat patterning in women with endometrial cancer. Gynecol Oncol 39 (3): 253-8, 1990.[PUBMED Abstract] Schapira DV, Kumar NB, Lyman GH, et al.: Upper-body fat distribution and endometrial cancer risk. JAMA 266 (13): 1808-11, 1991.[PUBMED Abstract] Olson SH, Trevisan M, Marshall JR, et al.: Body mass index, weight gain, and risk of endometrial cancer. Nutr Cancer 23 (2): 141-9, 1995.[PUBMED Abstract] Weiderpass E, Persson I, Adami HO, et al.: Body size in different periods of life, diabetes mellitus, hypertension, and risk of postmenopausal endometrial cancer (Sweden). Cancer Causes Control 11 (2): 185-92, 2000.[PUBMED Abstract] Le Marchand L, Wilkens LR, Mi MP: Early-age body size, adult weight gain and endometrial cancer risk. Int J Cancer 48 (6): 807-11, 1991.[PUBMED Abstract] Shu XO, Brinton LA, Zheng W, et al.: Relation of obesity and body fat distribution to endometrial cancer in Shanghai, China. Cancer Res 52 (14): 3865-70, 1992.[PUBMED Abstract] Aune D, Navarro Rosenblatt DA, Chan DS, et al.: Anthropometric factors and endometrial cancer risk: a systematic review and dose-response meta-analysis of prospective studies. Ann Oncol 26 (8): 1635-48, 2015.[PUBMED Abstract] Keum N, Ju W, Lee DH, et al.: Leisure-time physical activity and endometrial cancer risk: dose-response meta-analysis of epidemiological studies. Int J Cancer 135 (3): 682-94, 2014.[PUBMED Abstract] Crosbie EJ, Zwahlen M, Kitchener HC, et al.: Body mass index, hormone replacement therapy, and endometrial cancer risk: a meta-analysis. Cancer Epidemiol Biomarkers Prev 19 (12): 3119-30, 2010.[PUBMED Abstract] Esposito K, Chiodini P, Capuano A, et al.: Metabolic syndrome and endometrial cancer: a meta-analysis. Endocrine 45 (1): 28-36, 2014.[PUBMED Abstract] Friberg E, Orsini N, Mantzoros CS, et al.: Diabetes mellitus and risk of endometrial cancer: a meta-analysis. Diabetologia 50 (7): 1365-74, 2007.[PUBMED Abstract] Luo J, Beresford S, Chen C, et al.: Association between diabetes, diabetes treatment and risk of developing endometrial cancer. Br J Cancer 111 (7): 1432-9, 2014.[PUBMED Abstract] Win AK, Reece JC, Ryan S: Family history and risk of endometrial cancer: a systematic review and meta-analysis. Obstet Gynecol 125 (1): 89-98, 2015.[PUBMED Abstract] de Jonge MM, de Kroon CD, Jenner DJ, et al.: Endometrial Cancer Risk in Women With Germline BRCA1 or BRCA2 Mutations: Multicenter Cohort Study. J Natl Cancer Inst 113 (9): 1203-1211, 2021.[PUBMED Abstract] Factors With Adequate Evidence for a Decreased Risk of Endometrial Cancer In This Section Pregnancy and Lactation Hormonal Contraceptives Weight Loss Physical Activity Smoking Pregnancy and Lactation Decreased risk of endometrial cancer is associated with parity and lactation, perhaps by inhibiting ovulation. The European Prospective Investigation into Cancer and Nutrition observed a decreased risk associated with parity compared with nulliparous women (hazard ratio [HR], 0.65; 95% confidence interval [CI], 0.54–0.77). The study also observed a trend of decreased risk with increasing number of full-term pregnancies (P< .0001). A pooled analysis of 11 cohort and 19 case-control studies evaluated associations between specific pregnancy outcomes and endometrial cancer. The risk reduction associated with pregnancy was greatest for the first full-term birth (odds ratio [OR], 0.78; 95% CI, 0.72–0.84). Each additional full-term pregnancy was associated with an additional approximately 15% risk reduction up to eight full-term pregnancies (OR, 0.20; 95% CI, 0.14–0.28). Multiple gestation pregnancies did not appear to provide additional benefit beyond that of a singleton birth. Incomplete pregnancies were also associated with a decreased risk of endometrial cancer, with a 7% reduction per episode and an OR of 0.89 (95% CI, 0.84–0.95) for the first incomplete pregnancy. When evaluated by type of incomplete pregnancy, the associated risk reductions for spontaneous abortions (OR, 0.83; 95% CI, 0.76–0.90) and induced abortions (OR, 0.89; 95% CI, 0.79–1.01) were identified. Data was pooled from 17 studies that participated in the Epidemiology of Endometrial Cancer Consortium. After adjusting for age, parity, use of oral contraception and duration of use, body mass index (BMI), and education level, parous women who reported breast feeding had an 11% reduction in risk of endometrial cancer (pooled OR, 0.89; 95% CI, 0.81–0.98). The risk reduction associated with increasing total duration of breastfeeding was not linear. The greatest reduction in risk occurred after a total duration of breastfeeding of greater than 36 months (adjusted pooled OR, 0.67; 95% CI, 0.53–0.83). However, for individual episodes of breastfeeding, breastfeeding one child beyond 3 months was associated with a 5% reduction in risk (adjusted pooled OR, 0.95; 95% CI, 0.91–0.99). Hormonal Contraceptives Oral contraceptives were first approved by the U.S. Food and Drug Administration in 1960. For many years, they were the mainstay of hormonal contraception. More recently, hormonal contraception has expanded to include combination transdermal patches or vaginal rings, injections, and progestogen-releasing long-acting reversible contraceptives, including single-rod implants and intrauterine systems (IUS). Oral contraceptive usage confers a long-term reduction in the risk of endometrial cancer. A large population-based study from the United Kingdom prospectively collected information on combined oral contraception use for 46,022 women and followed them for 44 years. In this study, after adjusting for age, parity, smoking, and social class, ever-users of combined estrogen/progesterone oral contraceptive pills had an incidence rate ratio of 0.66 (95% CI, 0.48–0.89) compared with never-users. The benefit of oral contraceptive pill use is associated with duration of use, with increasing benefit reported for women who were obese, current smokers, and those who rarely exercise and who used oral contraceptives for 10 years or more. A meta-analysis combined data from 36 epidemiological studies that included 27,276 women. The study observed a risk reduction of 0.76 (95% CI, 0.73–0.78) for every 5 years of oral contraceptive use. The lower risk persisted for more than 30 years after the last use of oral contraceptives.[5,7] Among women from highly developed countries, 10 years of oral contraceptive use was associated with an absolute risk reduction of endometrial cancer before age 75 years from 2.3 to 1.3 cases per 100 women. Data suggest that use of levonorgestrel-releasing intrauterine systems (LNG-IUS) is associated with a statistically significant reduction in the risk of developing endometrial cancer. Use of LNG-IUS is an effective treatment for endometrial hyperplasia, which in some cases is a precursor to endometrial cancer and early-stage low-risk endometrial cancer.[8-10] A population-based prospective cohort study in Norway evaluated a cohort of 104,380 women, which included 9,146 women who identified as ever-users of LNG-IUS. The incidence rate of endometrial cancer per 100,000 person-years was 13.9 for ever-users of LNG-IUS (95% CI, 7.8–23.0), compared with 70.0 (95% CI, 65.4–74.9) for never-users. After adjusting for age and menopausal status at the start of follow-up, BMI, physical activity level, use of oral contraceptive pills, and parity, the relative risk (RR) of endometrial cancer was 0.22 (95% CI, 0.13–0.40) for ever-users of LNG-IUS. In an observational nationwide cohort study from Finland, women using LNG-IUS for treatment of menorrhagia from 1994 to 2007 were identified from administrative registers and linked with the Finnish Cancer Registry. In this study, 93,843 users of LNG-IUS were followed for 855,324 women-years at risk. The standardized incidence ratio for endometrial cancer after at least one purchase of LNG-IUS was 0.46 (95% CI, 0.33–0.64; 37 observed cases compared with 80 expected cases). Although not statistically different, the standardized incidence ratio decreased further in women who had purchased two LNG-IUS (0.25; 95% CI, 0.05–0.73; 3 observed cases compared with 12 expected cases). These data represent an attempt to demonstrate a dose effect, as LNG-IUS are considered effective for 5 years. Weight Loss While it is known that obesity is associated with increased endometrial cancer risk, only one study examines the potential benefit of intentional weight loss. In the Iowa Women’s Health Study (IWHS) of 21,707 postmenopausal women, participants completed a self-report questionnaire about intentional weight loss between ages 18 and 39 years, between ages 40 and 54 years, and after age 55 years. Multivariate models adjusting for age, BMI, and BMI 2 found no association between endometrial cancer incidence and intentional weight loss of at least 20 pounds (RR, 0.93; 95% CI, 0.60–1.44). However, one study included 36,793 women from the Women's Health Initiative (WHI) cohort whose weight was measured at baseline and at 3-year follow-up and was combined with self-reported intentionality of weight loss. The analysis showed an association between intentional weight loss of 10 pounds or more and lower endometrial cancer incidence (multivariable-adjusted RR, 0.61; 95% CI, 0.40–0.92). Both of these analyses share substantial limitations. Missing covariate data resulted in excluding nearly 25% of participants from each study, and only small percentages of the remaining participants (17% IWHS/8% WHI) were classified into the intentional weight loss category. This resulted in very low numbers of endometrial cancer cases driving the analyses. Both studies used self-report to characterize intentionality of weight loss, which can lead to potential misclassification, although the retrospective nature of the questioning in the IWHS makes the problem more acute in that analysis. Both analyses also adjusted for self-reported physical activity and smoking status, among other covariates. With such small numbers of cases and the potential for residual confounding, the contradictory results of these two analyses suggest that there is scant evidence to conclude that nonsurgical weight loss is protective for endometrial cancer. Bariatric surgery is associated with more sustained weight loss compared with nonsurgical intentional weight loss. Evidence suggests an association between bariatric surgery and a decreased risk of endometrial cancer.[16-18] In a prospective cohort study from Sweden, 1,420 women with obesity who underwent bariatric surgery and 1,447 matched controls who underwent conventional obesity treatment were followed for a median of 18.1 years. Mean weight loss after bariatric surgery was 21 kilograms at 10 years, compared with almost no change in weight in the usual care cohort. In this study, bariatric surgery was associated with a reduced risk of endometrial cancer (HR, 0.56; 95% CI, 0.35–0.89). Of note, this was not a prespecified study end point or powered to evaluate incidence of cancer. A retrospective cohort study through the Kaiser Permanente health system evaluated 22,198 individuals who had bariatric surgery and 66,427 nonsurgical individuals who were matched on sex, age, study site, BMI, and Elixhauser comorbidity index. More than 80% of the cohort was female. After a mean follow-up time of 3.5 years, there was a 50% reduction in the incidence of endometrial cancer (HR, 0.50; 95% CI, 0.37–0.67) in the cohort who underwent bariatric surgery. A systematic review, which included five observational studies with a control group, reported a decrease in the odds of developing endometrial cancer after bariatric surgery (OR, 0.32; 95% CI, 0.16–0.63). A large retrospective cohort study from England, which followed patients for a median of 3 years in the surgery group and 2.5 years in the no-surgery group, did not find an association with a decreased risk of endometrial cancer after bariatric surgery compared with the control group of patients with obesity. The association of bariatric surgery and decrease in the incidence of endometrial cancer may be caused by secondary effects of weight loss. In one study, women who underwent bariatric surgery had a 35% decrease in blood estradiol levels 1 year after surgery. Bariatric surgery has also been associated with a return to regular menstrual cycles in a high proportion of women with previous menstrual irregularities. The Surgical Procedures and Long-term Effectiveness in Neoplastic Disease Incidence and Death (SPLENDID) study was a retrospective, observational, matched cohort study of patients with obesity who underwent contemporary bariatric surgery (i.e., Roux-en-Y gastric bypass or sleeve gastrectomy) or received usual care (no bariatric surgery). Median follow-up was 6.1 years. The primary end point was the first occurrence of 1 of 13 predefined obesity-associated cancers. Secondary end points included incidence of all types of cancer and cancer-related mortality. In the adjusted Cox models that evaluated incidence of individual cancer types, only endometrial cancer remained significant (adjusted HR, 0.47; 95% CI, 0.27–0.83). There were limitations to this study, which included selection bias and different rates of cancer screening behaviors between the study arms, stemming from the observational nature of the study. In addition, there was the low number of incident cancers and a limited follow-up time. Physical Activity A meta-analysis combined data from prospective studies of recreational activity (nine studies) and occupational activity (five studies) to determine whether activity is associated with endometrial cancer. The highest category of recreational activity was associated with an RR of endometrial cancer of 0.73 (95% CI, 0.58–0.93), compared with lowest category. The RR of endometrial cancer for the highest category of occupational physical activity, based on job classification, was 0.75 (95% CI, 0.68–0.83), compared with the lowest category. Further investigation using the metabolic equivalent of task (MET) and combining data from case-control and cohort studies revealed a decrease in endometrial cancer risk with activity up to 50 MET-hours per week (up to 15 hours/week). Smoking Ever-smokers who smoked at least 20 cigarettes per day have a decreased risk of endometrial cancer, with greatest risk reductions seen in postmenopausal women and in current smokers. This effect has been seen in observational cohort, prospective cohort, and case-control studies and was summarized in a meta-analysis.[25,26] However, such a decrease does not begin to compensate for the many well-documented harms of smoking. These harms are most evident in the increased risk of cardiovascular diseases and other cancers, to the extent that smokers have at least a 10-year decrease in overall life expectancy, compared with nonsmokers. In contrast, Mendelian randomization analyses have not shown a causal relationship between smoking and decreased endometrial cancer risk in the U.K. Biobank and European Prospective Investigation into Cancer and Nutrition (EPIC) patient cohorts, questioning the strength of this association that was seen in the study's observational analyses. References Dossus L, Allen N, Kaaks R, et al.: Reproductive risk factors and endometrial cancer: the European Prospective Investigation into Cancer and Nutrition. Int J Cancer 127 (2): 442-51, 2010.[PUBMED Abstract] Jordan SJ, Na R, Weiderpass E, et al.: Pregnancy outcomes and risk of endometrial cancer: A pooled analysis of individual participant data in the Epidemiology of Endometrial Cancer Consortium. Int J Cancer 148 (9): 2068-2078, 2021.[PUBMED Abstract] Jordan SJ, Na R, Johnatty SE, et al.: Breastfeeding and Endometrial Cancer Risk: An Analysis From the Epidemiology of Endometrial Cancer Consortium. Obstet Gynecol 129 (6): 1059-1067, 2017.[PUBMED Abstract] Regidor PA: Clinical relevance in present day hormonal contraception. Horm Mol Biol Clin Investig 37 (1): , 2018.[PUBMED Abstract] Iversen L, Sivasubramaniam S, Lee AJ, et al.: Lifetime cancer risk and combined oral contraceptives: the Royal College of General Practitioners' Oral Contraception Study. Am J Obstet Gynecol 216 (6): 580.e1-580.e9, 2017.[PUBMED Abstract] Michels KA, Pfeiffer RM, Brinton LA, et al.: Modification of the Associations Between Duration of Oral Contraceptive Use and Ovarian, Endometrial, Breast, and Colorectal Cancers. JAMA Oncol 4 (4): 516-521, 2018.[PUBMED Abstract] Collaborative Group on Epidemiological Studies on Endometrial Cancer: Endometrial cancer and oral contraceptives: an individual participant meta-analysis of 27 276 women with endometrial cancer from 36 epidemiological studies. Lancet Oncol 16 (9): 1061-70, 2015.[PUBMED Abstract] Orbo A, Vereide A, Arnes M, et al.: Levonorgestrel-impregnated intrauterine device as treatment for endometrial hyperplasia: a national multicentre randomised trial. BJOG 121 (4): 477-86, 2014.[PUBMED Abstract] Kim MK, Seong SJ, Kim JW, et al.: Management of Endometrial Hyperplasia With a Levonorgestrel-Releasing Intrauterine System: A Korean Gynecologic-Oncology Group Study. Int J Gynecol Cancer 26 (4): 711-5, 2016.[PUBMED Abstract] Pal N, Broaddus RR, Urbauer DL, et al.: Treatment of Low-Risk Endometrial Cancer and Complex Atypical Hyperplasia With the Levonorgestrel-Releasing Intrauterine Device. Obstet Gynecol 131 (1): 109-116, 2018.[PUBMED Abstract] Jareid M, Thalabard JC, Aarflot M, et al.: Levonorgestrel-releasing intrauterine system use is associated with a decreased risk of ovarian and endometrial cancer, without increased risk of breast cancer. Results from the NOWAC Study. Gynecol Oncol 149 (1): 127-132, 2018.[PUBMED Abstract] Soini T, Hurskainen R, Grénman S, et al.: Cancer risk in women using the levonorgestrel-releasing intrauterine system in Finland. Obstet Gynecol 124 (2 Pt 1): 292-9, 2014.[PUBMED Abstract] Parker ED, Folsom AR: Intentional weight loss and incidence of obesity-related cancers: the Iowa Women's Health Study. Int J Obes Relat Metab Disord 27 (12): 1447-52, 2003.[PUBMED Abstract] Luo J, Chlebowski RT, Hendryx M, et al.: Intentional Weight Loss and Endometrial Cancer Risk. J Clin Oncol 35 (11): 1189-1193, 2017.[PUBMED Abstract] Sjöström L, Gummesson A, Sjöström CD, et al.: Effects of bariatric surgery on cancer incidence in obese patients in Sweden (Swedish Obese Subjects Study): a prospective, controlled intervention trial. Lancet Oncol 10 (7): 653-62, 2009.[PUBMED Abstract] Anveden Å, Taube M, Peltonen M, et al.: Long-term incidence of female-specific cancer after bariatric surgery or usual care in the Swedish Obese Subjects Study. Gynecol Oncol 145 (2): 224-229, 2017.[PUBMED Abstract] Schauer DP, Feigelson HS, Koebnick C, et al.: Bariatric Surgery and the Risk of Cancer in a Large Multisite Cohort. Ann Surg 269 (1): 95-101, 2019.[PUBMED Abstract] Winder AA, Kularatna M, MacCormick AD: Does Bariatric Surgery Affect the Incidence of Endometrial Cancer Development? A Systematic Review. Obes Surg 28 (5): 1433-1440, 2018.[PUBMED Abstract] Aravani A, Downing A, Thomas JD, et al.: Obesity surgery and risk of colorectal and other obesity-related cancers: An English population-based cohort study. Cancer Epidemiol 53: 99-104, 2018.[PUBMED Abstract] Sarwer DB, Spitzer JC, Wadden TA, et al.: Changes in sexual functioning and sex hormone levels in women following bariatric surgery. JAMA Surg 149 (1): 26-33, 2014.[PUBMED Abstract] Butterworth J, Deguara J, Borg CM: Bariatric Surgery, Polycystic Ovary Syndrome, and Infertility. J Obes 2016: 1871594, 2016.[PUBMED Abstract] Aminian A, Wilson R, Al-Kurd A, et al.: Association of Bariatric Surgery With Cancer Risk and Mortality in Adults With Obesity. JAMA 327 (24): 2423-2433, 2022.[PUBMED Abstract] Moore SC, Gierach GL, Schatzkin A, et al.: Physical activity, sedentary behaviours, and the prevention of endometrial cancer. Br J Cancer 103 (7): 933-8, 2010.[PUBMED Abstract] Keum N, Ju W, Lee DH, et al.: Leisure-time physical activity and endometrial cancer risk: dose-response meta-analysis of epidemiological studies. Int J Cancer 135 (3): 682-94, 2014.[PUBMED Abstract] Zhou B, Yang L, Sun Q, et al.: Cigarette smoking and the risk of endometrial cancer: a meta-analysis. Am J Med 121 (6): 501-508.e3, 2008.[PUBMED Abstract] Dimou N, Omiyale W, Biessy C, et al.: Cigarette Smoking and Endometrial Cancer Risk: Observational and Mendelian Randomization Analyses. Cancer Epidemiol Biomarkers Prev 31 (9): 1839-1848, 2022.[PUBMED Abstract] Centers for Disease Control and Prevention: Smoking and Tobacco Use. Atlanta, Ga: Centers for Disease Control and Prevention, Office on Smoking and Health, 2015. Available Online. Last accessed December 18, 2023. Interventions With Inadequate Evidence of an Association With Endometrial Cancer In This Section Fruits, Vegetables, and Vitamins Hair Products, Including Dyes, Bleach, Highlights, Straighteners, and Permanents Fruits, Vegetables, and Vitamins Studies have not found an association between endometrial cancer and diet, phytoestrogens, soy, and vitamin D.[1-6] Multivitamin use has little or no influence on the risk of common cancers, including endometrial cancer, or on total mortality in postmenopausal women. Hair Products, Including Dyes, Bleach, Highlights, Straighteners, and Permanents One retrospective analysis of the Sister Study addressed a possible association between these hair products and uterine cancers, including endometrial cancers. A limitation to this study was a lack of properly adjusted analysis for multiple comparisons, thus making the significance of the findings hard to interpret. References International Agency for Research On Cancer: IARC Handbooks of Cancer Prevention. Volume 8: Fruit and Vegetables. International Agency for Research On Cancer, 2003. Bandera EV, Kushi LH, Gifkins DM, et al.: WCRF Systematic Literature Review: The Association Between Food, Nutrition, and Physical Activity and the Risk of Endometrial Cancer and Underlying Mechanisms. World Cancer Research Fund, American Institute for Cancer Research, 2006. Horn-Ross PL, John EM, Canchola AJ, et al.: Phytoestrogen intake and endometrial cancer risk. J Natl Cancer Inst 95 (15): 1158-64, 2003.[PUBMED Abstract] Xu WH, Zheng W, Xiang YB, et al.: Soya food intake and risk of endometrial cancer among Chinese women in Shanghai: population based case-control study. BMJ 328 (7451): 1285, 2004.[PUBMED Abstract] Zeleniuch-Jacquotte A, Gallicchio L, Hartmuller V, et al.: Circulating 25-hydroxyvitamin D and risk of endometrial cancer: Cohort Consortium Vitamin D Pooling Project of Rarer Cancers. Am J Epidemiol 172 (1): 36-46, 2010.[PUBMED Abstract] Liu JJ, Bertrand KA, Karageorgi S, et al.: Prospective analysis of vitamin D and endometrial cancer risk. Ann Oncol 24 (3): 687-92, 2013.[PUBMED Abstract] Neuhouser ML, Wassertheil-Smoller S, Thomson C, et al.: Multivitamin use and risk of cancer and cardiovascular disease in the Women's Health Initiative cohorts. Arch Intern Med 169 (3): 294-304, 2009.[PUBMED Abstract] Chang CJ, O'Brien KM, Keil AP, et al.: Use of Straighteners and Other Hair Products and Incident Uterine Cancer. J Natl Cancer Inst 114 (12): 1636-1645, 2022.[PUBMED Abstract] Latest Updates to This Summary (04/10/2025) The PDQ cancer information summaries are reviewed regularly and updated as new information becomes available. This section describes the latest changes made to this summary as of the date above. Incidence and Mortality Updated statistics with estimated new cases and deaths for 2025 (cited American Cancer Society as reference 1). Also revised text to state that between 2013 and 2022, death rates for endometrial cancer increased by 1.5% per year. This summary is written and maintained by the PDQ Screening and Prevention Editorial Board, which is editorially independent of NCI. The summary reflects an independent review of the literature and does not represent a policy statement of NCI or NIH. More information about summary policies and the role of the PDQ Editorial Boards in maintaining the PDQ summaries can be found on the About This PDQ Summary and PDQ® Cancer Information for Health Professionals pages. About This PDQ Summary Purpose of This Summary This PDQ cancer information summary for health professionals provides comprehensive, peer-reviewed, evidence-based information about endometrial cancer prevention. It is intended as a resource to inform and assist clinicians in the care of their patients. It does not provide formal guidelines or recommendations for making health care decisions. Reviewers and Updates This summary is reviewed regularly and updated as necessary by the PDQ Screening and Prevention Editorial Board, which is editorially independent of the National Cancer Institute (NCI). The summary reflects an independent review of the literature and does not represent a policy statement of NCI or the National Institutes of Health (NIH). 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7650	https://www.facebook.com/groups/AncientOrderOfTheHermetics/posts/4038197363068159/	Ancient Order Of The Hermetics \| OCCULT KNOWLEDGE THE MAGIC CIRCLE 🔘➿ \| Facebook Log In Log In Forgot Account? Ancient Order Of The Hermetics · Join Ava Leopoldo · Moderator · July 30 · OCCULT KNOWLEDGE THE MAGIC CIRCLE is a sacred, purified, and protected space in which magical Rituals are conducted. A circle symbolizes wholeness, perfection, and unity; the creation of the cosmos; the womb of Mother Earth; the cycle of the seasons, and birth-death-regeneration, the Infinite. Within the circle, it becomes possible to transcend the physical to open the mind to deeper and higher levels of consciousness. Circles have had a magical, protective significance since ancient times when they were drawn around the beds of sick persons and mothers who had just given birth, to protect them against Demons. A magic circle protects a magician against negative spirits and influences and creates a symbolic barrier against the magician's own lower nature. In ceremonial magic, the circle provides a defined space for the working of a ritual. If the magician performs a ritual calling for the Conjuration of spirits which are dangerous and difficult to control, the circle provides protection against them. The magician must never leave the circle during a ritual. All reactions: 185 2 comments 56 shares Like Comment Share Most relevant Carolyn Ann Forrester Occult definition is “hidden information kept and protected. Initiation required to become privy”. Question, with everyone making public some information, is anything still occult? Next Question, is it Occult if it’s only used to harm but kept secret… See more 8w Brenda Abernathy Holy Circle of Lives...A'HO~! 8w See more on Facebook See more on Facebook Email or phone number Password Log In Forgot password? or Create new account
7651	https://www.cuemath.com/data/class-mark/	LearnPracticeDownload Class Mark Class mark, in statistics, is the average of the upper limit and the lower limit of a class in a frequency distribution. In other words, the class mark is the mid-value of the given class interval. \| \| \| --- \| \| 1. \| What is Class Mark? \| \| 2. \| Class Mark Formula \| \| 3. \| Class Mark in Frequency Distribution \| \| 4. \| FAQs on Class Mark \| What is Class Mark? Class mark in a frequency distribution is the midpoint or the middle value of a given class. For example, the class mark of 10-20 is 15, as 15 is the mid-value that lies between 10 and 20. In statistics, the class mark is used at various places, for example, while calculating mean, drawing line graphs, finding the average of each class in a frequency distribution, etc. It is very easy to calculate class mark by using a formula that you will learn in the section below. Class Mark Formula The formula to calculate class mark in a frequency distribution is given as (upper limit + lower limit)/2 or (Sum of class boundaries)/2. By using this class mark formula, you can easily find the midpoint of any given class interval. Let us learn the terms associated with this formula in detail. Class Interval: It is defined as the size of each class of numerical data in a large frequency distribution following a specific width. For example, if the raw data has too many variations in numbers, we make groups of intervals to organize the data such as 0-10, 10-20, 20-30, etc. These are known as class intervals. Upper Limit: It is the highest value of the class interval. There could be no item greater than the upper limit in that particular class. For example, the upper limit of 30-40 is 40. It is known as the upper-class boundary. Lower Limit: It is the lowest value of the class interval. There could be no item less than the lower limit in that particular class. For example, the lower limit of 30-40 is 30. It is known as lower class boundary. Class Mark in Frequency Distribution In a histogram (a graphical representation of a frequency distribution), we can easily find the midpoint of each class by observing the values at the x-axis or the horizontal axis. By applying the formula of class mark, let us find the values of class marks for each class in the given distribution showing the salaries of employees working in a company. \| Salaries (in $) \| Number of Employees \| Class Mark (in $) \| --- \| 0-10 \| 50 \| 5 \| \| 10-20 \| 300 \| 15 \| \| 20-30 \| 260 \| 25 \| \| 30-40 \| 400 \| 35 \| \| 40-50 \| 550 \| 45 \| \| 50-60 \| 450 \| 55 \| Let us show this information in a histogram and denote the values of class marks through a dotted line in the middle of each bar representing the range of salaries. One interesting fact here is that if we join the mid points on the top of each bar, we can draw a line graph. ► Related Articles Check these interesting topics related to the class mark. Frequency Distribution Table Measures of Central Tendency Line Charts Class Mark Examples Example 1: What will be the class mark of the given class: 25 - 35? Solution: By using the class mark formula, we have, Class mark = (upper limit + lower limit)/2. The given class is 25 - 35, where 25 is the lower limit and 35 is the upper limit. ⇒ Class mark = (35 + 25)/2 = 60/2 = 30 Therefore, the class mark of 25 - 35 is 30. 2. Example 2: If the class mark is given as 55 and the upper limit is 65, find the value of the lower limit of the class interval? Solution: Given, class mark = 55. ⇒ (upper limit + lower limit)/2 = 55 ⇒ 65 + lower limit = 55 × 2 ⇒ lower limit = 110 - 65 ⇒ lower limit = 45 Therefore, the value of the lower limit is 45, and the class interval is 45 - 65. 3. Example 3: Find the class marks of the following frequency distribution table: \| Class Interval \| Frequency \| --- \| \| 0-10 \| 2 \| \| 10-20 \| 5 \| \| 20-30 \| 8 \| \| 30-40 \| 4 \| \| 40-50 \| 1 \| Solution: Let us write the class marks of each of the intervals by adding an additional column in the given table. \| Class Interval \| Frequency \| Class Mark \| --- \| 0-10 \| 2 \| (0+10)/2 = 5 \| \| 10-20 \| 5 \| (10+20)/2 = 15 \| \| 20-30 \| 8 \| (20+30)/2 = 25 \| \| 30-40 \| 4 \| (30+40)/2 = 35 \| \| 40-50 \| 1 \| (40+50)/2 = 45 \| View Answer > go to slidego to slidego to slide Great learning in high school using simple cues Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes. Book a Free Trial Class Practice Questions on Class Mark Check Answer > go to slidego to slide FAQs on Class Mark What is Class Mark in Statistics? The class mark is also known as the midpoint of the class interval in a frequency distribution. It can be defined as the average of the upper limit and the lower limit of a class. What is the Formula of Class Mark? The formula to find class mark is given as, Class mark = (upper limit + lower limit)/2, or (sum of class boundaries)/2. For example, to find the class mark of the interval 25-30, we add these values which give us 25 + 30 = 55. Now we divide it by 2 to find the average. This implies, 55/2 = 27.5. So, 27.5 is the class mark of 25 - 30. What is the Class Mark of the Class Interval 90-120? By using the class mark formula, we get, Class mark of 90-120 = (90 + 120)/2 = 210/2 = 105. Therefore, 105 is the class mark of 90 - 120. What is the Class Mark of a Class a-b? If a and b are the values of the lower limit and the upper limit respectively, then the class mark is (a + b)/2. How to Find Class Mark? We can find the class mark of any given class interval by finding the average of the upper limit and the lower limit. In other words, simply add the values of class boundaries and divide the sum by 2. What is the Class Mark of Class 100-190? By using the class mark formula, we get, Class mark of 100-190 = (100 + 190)/2 = 290/2 = 145. Therefore, 145 is the class mark of 100 - 190. 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7652	http://ch302.cm.utexas.edu/chemEQ/ab-theory/selector.php?name=strong-acids	Strong Acids Strong acids are acids for which the equilibrium constant for dissociations is so large that we assume the molecules "fall apart" or dissociate 100% in water. For example, HCl is a strong acid. So for the equilibrium [\rm{HCl(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + Cl^-(aq)}] we assume the equilibrium constant is so large (it is more than 10 6) that the reaction will essentially go to completion. That is when we put HCl in water, it will completely dissociate and form H 3 O+and Cl-. This is what we mean by strong. We call it an acid because it is donating a proton (H+) to water. You need to memorize the list of strong acids. Both their names and formulas Hydrochloric acid - HCl Hydrobromic acid - HBr HydroIodic acid - HI Perchloric acid - HClO 4 Chloric acid - HClO 3 Sulfuric acid - H 2 SO 4(note only the first proton is strong) Nitric acid - HNO 3 A video on the definitions of Strong Acids and Strong Bases. Complete or 100% dissociation is required. A video showing the 7 Strong Acids and 8 Strong Bases. Strong acids and bases are strong electrolytes which means they ionize completely (100%). All students should MEMORIZE these two sets of compounds. Concept Question Which of the following is not a strong acid? (touch choices to toggle feedback on/off) chloric acid perchloric acid hydrochloric acid hypochlorous acid © 2013 mccord/vandenbout/labrake scroll to top
7653	https://courses.lumenlearning.com/slcc-elementaryalgebra/chapter/2-5-absolute-value-equations/	Module 2: Linear Inequalities and Absolute Values 2.5: Absolute Value Equations section 2.5 Learning Objectives 2.5: Absolute Value Equations Solve absolute value equations Recognize when an absolute value equation has no solution Solve absolute value equations containing two absolute values The absolute value of a number or expression describes its distance from 0 on a number line. Since the absolute value expresses only the distance, not the direction of the number on a number line, it is always expressed as a positive number or 0. For example, [latex]−4[/latex] and 4 both have an absolute value of 4 because they are each 4 units from 0 on a number line—though they are located in opposite directions from 0 on the number line. Solving equations containing absolute values Key Takeaways When solving absolute value equations and inequalities, you have to consider both the behavior of absolute value and the properties of equality and inequality. Because both positive and negative values have a positive absolute value, solving absolute value equations or inequalities means finding the solution for both the positive and the negative values. REMEMBER: We must always set up two cases when solving absolute value functions; one positive case and one negative case Let’s first look at a very basic example. [latex]\displaystyle \left\| x \right\|=5[/latex] This equation is read “the absolute value of x is equal to five.” The solution is the value(s) that are five units away from 0 on a number line. You might think of 5 right away; that is one solution to the equation. Notice that [latex]−5[/latex] is also a solution because [latex]−5[/latex] is 5 units away from 0 in the opposite direction. So, the solution to this equation [latex]\displaystyle \left\| x \right\|=5[/latex] is [latex]x = −5[/latex] or [latex]x = 5[/latex]. Solving Equations of the Form [latex]\|x\|=a[/latex] For any positive number a, the solution of [latex]\left\|x\right\|=a[/latex] is [latex]x=a[/latex] or [latex]x=−a[/latex] x can be a single variable or any algebraic expression. You can solve a more complex absolute value problem in a similar fashion. Example 1 Solve for x: [latex]\hspace{.05in}\displaystyle \left\| x+5\right\|=15[/latex] Show Solution This equation asks you to find what number plus 5 has an absolute value of 15. Since 15 and [latex]−15[/latex] both have an absolute value of 15, the absolute value equation is true when the quantity [latex]x + 5[/latex] is 15 or [latex]x + 5[/latex] is [latex]−15[/latex], since [latex]\|15\|=15[/latex] and [latex]\|−15\|=15[/latex]. So, you need to find out what value for x will make this expression equal to 15 as well as what value for x will make the expression equal to [latex]−15[/latex]. Solving the two equations you get [latex]\displaystyle \begin{array}{l}x+5=15\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\,{x+5=-15}\\underline{\,\,\,\,\,-5\,\,\,\,-5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-5\,\,\,\,\,\,\,\,\,-5}\x\,\,\,\,\,\,\,\,\,=\,\,10\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\,x\,\,\,\,\,\,\,\,=-20\end{array}[/latex] You can check these two solutions in the absolute value equation to see if [latex]x=10[/latex] and [latex]x=−20[/latex] are correct. [latex]\displaystyle \begin{array}{r}\,\,\left\| x+5 \right\|=15\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\| x+5 \right\|=15\\left\| 10+5 \right\|=15\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\| -20+5 \right\|=15\\,\,\,\,\,\,\,\left\| 15 \right\|=15\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\| -15 \right\|=15\\,\,\,\,\,\,\,\,\,15=15\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,15=15\end{array}[/latex] The following video provides worked examples of solving linear equations with absolute value terms. Example 2 Solve for x: [latex]\hspace{.05in}\displaystyle \left\| 2x\right\|=6[/latex] Show Solution This equation asks you to find what number times 2 has an absolute value of 6. Since 6 and [latex]−6[/latex] both have an absolute value of 6, the absolute value equation is true when the quantity [latex]2x[/latex] is 6 or [latex]2x[/latex] is [latex]−6[/latex], since [latex]\|6\|=6[/latex] and [latex]\|−6\|=6[/latex]. So, you need to find out what value for x will make this expression equal to 6 as well as what value for x will make the expression equal to [latex]−6[/latex]. Solving the two equations you get [latex]2x=6\text{ or }2x=-6[/latex] [latex]\frac{2x}{2}=\frac{6}{2}\text{ or }\frac{2x}{2}=\frac{-6}{2}[/latex] [latex]x=3\text{ or }x=-3[/latex] You can check these two solutions in the absolute value equation to see if [latex]x=3[/latex] and [latex]x=−3[/latex] are correct. [latex]\displaystyle \begin{array}{r}\,\,\left\|3\cdot2 \right\|=6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\| -3\cdot2 \right\|=6\\left\| 6 \right\|=6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\| -6 \right\|=6\\end{array}[/latex] Example 3 Solve for k: [latex]\hspace{.05in}\displaystyle\frac{1}{3}\left\|k\right\|=12[/latex] Show Solution Notice how this example is different from the last; [latex]\displaystyle\frac{1}{3}[/latex] is outside the absolute value grouping symbols. This means we need to isolate the absolute value first, then apply the definition of absolute value. First, isolate the absolute value term by multiplying by the inverse of [latex]\displaystyle\frac{1}{3}[/latex]: [latex]\begin{array}{r}\frac{1}{3}\left\|k\right\|=12\,\,\,\,\,\,\,\,\\left(3\right)\frac{1}{3}\left\|k\right\|=\left(3\right)12\\left\|k\right\|=36\,\,\,\,\,\,\,\,\end{array}[/latex] Apply the definition of absolute value: [latex]\displaystyle{k }=36\text{ or }{k }=-36[/latex] You can check these two solutions in the absolute value equation to see if [latex]x=36[/latex] and [latex]x=−36[/latex] are correct. [latex]\displaystyle \begin{array}{r}\,\,\frac{1}{3}\left\|36 \right\|=12\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{3}\left\|-36 \right\|=12\\left\| 12 \right\|=12\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\| -12 \right\|=12\\end{array}[/latex] In the following video you will see two examples of how to solve an absolute value equation, one with integers and one with fractions. Example 4 Solve for p: [latex]\hspace{.05in}\left\|2p–4\right\|=26[/latex] Show Solution Write the two equations that will give an absolute value of 26. [latex]\displaystyle 2p-4=26\,\,\,\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,2p-4=\,-26[/latex] Solve each equation for p by isolating the variable. [latex]\displaystyle \begin{array}{r}2p-4=26\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2p-4=\,-26\\underline{\,\,\,\,\,\,+4\,\,\,\,+4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,+4\,\,\,\,\,\,\,+4}\\underline{2p}\,\,\,\,\,\,=\underline{30}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{2p}\,\,\,\,\,=\,\underline{-22}\2\,\,\,\,\,\,\,=\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,=\,\,\,\,\,\,\,\,2\\,\,\,\,\,\,\,\,\,p=15\,\,\,\,\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p=\,-11\end{array}[/latex] Check the solutions in the original equation. [latex]\displaystyle \begin{array}{r}\,\,\,\,\,\left\| 2p-4 \right\|=26\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\| 2p-4 \right\|=26\\left\| 2(15)-4 \right\|=26\,\,\,\,\,\,\,\left\| 2(-11)-4 \right\|=26\\,\,\,\,\,\left\| 30-4 \right\|=26\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\| -22-4 \right\|=26\\,\,\,\,\,\,\,\,\,\,\,\,\left\| 26 \right\|=26\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\| -26 \right\|=26\end{array}[/latex] Both solutions check! Answer [latex]p=15[/latex] or [latex]p=-11[/latex] In the next video, we show more examples of solving multi-step absolute value equations. Now let’s look at an example where you need to do an algebraic step or two before you can write your two equations. The goal here is to get the absolute value on one side of the equation by itself. Then we can proceed as we did in the previous example. Example 5 Solve for w: [latex]\hspace{.05in}3\left\|4w–1\right\|–5=10[/latex] Show Solution Isolate the term with the absolute value by adding 5 to both sides. [latex]\begin{array}{r}3\left\|4w-1\right\|-5=10\\underline{\,\,\,\,\,\,\,\,\,\,\,\,\,+5\,\,\,+5}\ 3\left\|4w-1\right\|=15\end{array}[/latex] Divide both sides by 3. Now the absolute value is isolated. [latex]\begin{array}{r} \underline{3\left\|4w-1\right\|}=\underline{15}\3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\\left\|4w-1\right\|=\,\,5\end{array}[/latex] Write the two equations that will give an absolute value of 5 and solve them. [latex]\displaystyle \begin{array}{r}4w-1=5\,\,\,\,\,\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\,\,\,\,4w-1=-5\\underline{\,\,\,\,\,\,\,+1\,\,+1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,\,\,+1\,\,\,\,\,+1}\\,\,\,\,\,\underline{4w}=\underline{6}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{4w}\,\,\,\,\,\,\,=\underline{-4}\4\,\,\,\,\,\,\,\,\,\,\,4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4\,\,\\,\,\,\,\,\,\,\,w=\frac{3}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,w=-1\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,w=\frac{3}{2}\,\,\,\,\,\text{or}\,\,\,\,\,-1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex] Check the solutions in the original equation. [latex]\displaystyle \begin{array}{r}\,\,\,\,\,3\left\| 4w-1\, \right\|-5=10\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\left\| 4w-1\, \right\|-5=10\\3\left\| 4\left( \frac{3}{2} \right)-1\, \right\|-5=10\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\left\| 4w-1\, \right\|-5=10\\\,\,\,\,\,\,3\left\| \frac{12}{2}-1\, \right\|-5=10\,\,\,\,\,\,\,3\left\| 4(-1)-1\, \right\|-5=10\\\,\,\,\,\,\,\,\,3\left\| 6-1\, \right\|-5=10\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\left\| -4-1\, \right\|-5=10\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\left(5\right)-5=10\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\left\| -5 \right\|-5=10\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,15-5=10\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,15-5=10\10=10\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,10=10\end{array}[/latex] Both solutions check Answer [latex]w=-1\,\,\,\,\text{or}\,\,\,\,w=\frac{3}{2}[/latex] In the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations. Absolute value equations with no solutions As we are solving absolute value equations it is important to be aware of special cases. An absolute value is defined as the distance from 0 on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples. Example 6 Solve for x: [latex]\hspace{.05in}7+\left\|2x-5\right\|=4[/latex] Show Solution Notice absolute value is not alone. Subtract [latex]7[/latex] from each side to isolate the absolute value. [latex]\begin{array}{r}7+\left\|2x-5\right\|=4\,\,\,\,\\underline{\,-7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-7\,}\\left\|2x-5\right\|=-3\end{array}[/latex] The result of absolute value is negative! The result of an absolute value must always be nonnegative, so we say there is no solution to this equation, or DNE. Example 7 Solve for x: [latex]\hspace{.05in}-\frac{1}{2}\left\|x+3\right\|=6[/latex] Show Solution Notice absolute value is not alone, multiply both sides by the reciprocal of [latex]-\frac{1}{2}[/latex], which is [latex]-2[/latex]. [latex]\begin{array}{r}-\frac{1}{2}\left\|x+3\right\|=6\,\,\,\,\,\,\,\,\,\,\,\,\\,\,\,\,\,\,\,\,\left(-2\right)-\frac{1}{2}\left\|x+3\right\|=\left(-2\right)6\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\|x+3\right\|=-12\,\,\,\,\,\end{array}[/latex] Again, we have a result where an absolute value is negative! There is no solution to this equation, or DNE. In this last video, we show more examples of absolute value equations that have no solutions. Solving equations with two absolute values If we are given an equality of two absolute value expressions, for example [latex]\|3x+4\|=\|2x-1\|[/latex], we apply the same idea. The equation will be true when the two expressions inside the absolute values are the same distance from zero, which occurs if they have the same value or are opposites. The expressions will have the same value when they are equal to each other. In our example, this implies [latex]3x+4=2x-1[/latex]. The second equation will be obtained by taking the opposite of either expression. Be careful to distribute the negative, which we can indicate with parentheses. Here, this would be given by [latex]3x+4=-(2x-1)[/latex]. See the full solution below. Example 8 Solve for x: [latex]\hspace{.05in}\|3x+4\|=\|2x-1\|[/latex] Show Solution The values for x that will satisfy the equation will be those that result in the expressions inside the absolute values being equal to each other or opposites of each other. [latex]\hspace{2.65in} 3x+4=2x-1 \hspace{.35in} \mbox{or} \hspace{.35in} 3x+4=-(2x-1) \ \hspace{2.5in} \underline{-2x \hspace{.35in}-2x\hspace{.3in}} \hspace{.84in} 3x+4=-2x+1 \ \hspace{2.5in} \hspace{.24in} x+4=-1 \hspace{.99in} \underline{+2x \hspace{.47in} +2x\hspace{.37in}} \ \hspace{2.5in} \hspace{.15in}\underline{\hspace{.23in} -\hspace{.06in} 4 \hspace{.18in} - \hspace{-.03in} 4 \hspace{.02in}} \hspace{1.09in} 5x+4=\hspace{.02in}1 \ \hspace{2.5in} \hspace{.55in} x=-5 \hspace{1.17in} \underline{\hspace{.2in}-\hspace{.04in}4 \hspace{.08in}-\hspace{-.02in} 4 \hspace{.02in}} \ \hspace{2.5in} \hspace{2.5in} {\displaystyle \frac{5x}{5}=\frac{-3}{5}} \ \hspace{2.5in} \hspace{2.66in} x=-\frac{3}{5}[/latex] Check the solutions in the original equation. [latex]\hspace{1.5in} \begin{array}{rclcrcl} \|3(-5)+4\|&=&\|2(-5)-1\|& \hspace{1in} & \left\|3\left(-\frac{3}{5}\right)+4\right\|&=&\left\|2\left(-\frac{3}{5}\right)-1\right\| \ \|-15+4\|&=&\|-10-1\|& & \left\|-\frac{9}{5}+4\right\|&=&\left\|-\frac{6}{5}-1\right\| \ \|-11\|&=&\|-11\| & & \left\| \frac{11}{5}\right\|&=&\left\|-\frac{11}{5}\right\| \ 11&=&11 & & \frac{11}{5}&=&\frac{11}{5} \end{array}[/latex] Both solutions check. Answer [latex]x=-5[/latex] or [latex]x=-\frac{3}{5}[/latex] The video below shows some more examples of solving an equation with two absolute values. Summary Equations are mathematical statements that combine two expressions of equal value. An algebraic equation can be solved by isolating the variable on one side of the equation using the properties of equality. To check the solution of an algebraic equation, substitute the value of the variable into the original equation. Complex, multi-step equations often require multi-step solutions. Before you can begin to isolate a variable, you may need to simplify the equation first. If your multi-step equation has an absolute value, you will need to solve two equations, sometimes isolating the absolute value expression first. Candela Citations CC licensed content, Shared previously Authored by: Tyler Wallace. Located at: License: CC BY: Attribution Licenses and Attributions CC licensed content, Shared previously Authored by: Tyler Wallace. Located at: License: CC BY: Attribution
7654	https://1cademy.com/node/net-export-function-in-the-multiplier-model/jWTweqwjcirEGURIZjL7	1Cademy - Net Export Function in the Multiplier Model How it worksResearch CommunitiesBenefitsAbout Us Schedule DemoSign In/Up Learn Before Net Exports (Trade Balance) in GDP Net Export Function in the Multiplier Model In the multiplier model, the net export function specifies net exports (NX) as the difference between exogenous exports (X) and imports that are dependent on income (Y). This relationship is expressed by the formula: N X=X−m Y NX = X - mY NX=X−mY, where 'm' represents the marginal propensity to import. 0 1 9 days ago Share Contributors are: Gemini AI 🏆 3 Who are from: Google 🏆 3 References CORE Econ - The Economy 2.0: Macroeconomics Tags Economics Economy Introduction to Macroeconomics Course Ch.3 Aggregate demand and the multiplier model - The Economy 2.0 Macroeconomics @ CORE Econ The Economy 2.0 Macroeconomics @ CORE Econ CORE Econ Social Science Empirical Science Science Related Exports Trade Deficit Trade Surplus Example of Trade Imbalances: US and China (2010) Real-World Determinants of Net Exports Imports as a Function of Domestic Income Net Export Function in the Multiplier Model A consumer in Country A purchases a new car for $30,000 that was manufactured in Country B. Assuming this is the only economic activity, how does this transaction affect the calculation of Country A's Gross Domestic Product (GDP) based on expenditure? Evaluating the Role of Imports in GDP Calculation Correcting a Misconception about Imports and GDP A country's Gross Domestic Product (GDP), calculated as the total spending on its domestically produced goods and services, will always decrease if the total value of goods its citizens purchase from other countries increases. Analyzing International Trade's Role in Measuring Domestic Output A country's economic activity for a year includes the following international transactions: Domestic firms sell $50 million worth of goods to foreign buyers. Domestic consumers purchase $30 million worth of goods produced in other countries. The domestic government purchases $10 million worth of equipment from foreign suppliers. What is the net effect of these transactions on the calculation of this country's total expenditure on its own domestically produced goods and services? Match each economic transaction for Country A with its correct impact on the components used to calculate Country A's total expenditure on its own domestically produced goods and services. Analyzing a Flawed Argument about Imports A country's economic data for a specific year reveals that total spending by its households on goods and services rose by $20 billion. Simultaneously, the total calculated value of all goods and services produced within the country's borders remained exactly the same as the previous year. Which of the following statements best explains this specific economic situation? In a given year, the total spending by a country's households, businesses, and government amounts to $1,200 billion. Of this amount, $200 billion is spent on goods and services produced abroad. During the same year, the country sells $150 billion worth of its own goods and services to foreign buyers. Based on this information, the total value of all goods and services produced within the country's borders is $____ billion. Learn After In an economy, the relationship between net exports (NX) and national income (Y) is given by the function NX = X - mY, where X represents autonomous exports and 'm' is the marginal propensity to import. If this economy's consumers develop a stronger preference for imported goods, how will this change be represented in the model, assuming autonomous exports and national income initially remain constant? Impact of a Fixed Exchange Rate Policy on Competitiveness Calculating the Marginal Propensity to Import A country's economy experiences a significant and sustained rise in its aggregate income. Assuming that the value of its exports and its marginal propensity to import remain constant, what is the most likely impact on the country's net exports? A country's economy experiences a significant and sustained rise in its aggregate income. Assuming that the value of its exports and its marginal propensity to import remain constant, what is the most likely impact on the country's net exports? Calculating and Explaining Changes in Net Exports Calculating and Explaining Changes in Net Exports Analyzing the Trade-Off of a Fiscal Stimulus Analyzing the Trade-Off of a Fiscal Stimulus A government policy that successfully stimulates a significant increase in a country's national income will, all else being equal, necessarily lead to an improvement in its net exports. An open economy has autonomous exports valued at $200 billion. Its citizens tend to spend 15 cents of each additional dollar of income on imported goods. If the total national income for this economy is $1000 billion, what is the value of its net exports? Consider two economies, Country A and Country B, with the following net export functions, where NX is net exports and Y is national income (both in billions of dollars): Country A: NX = 500 - 0.1Y Country B: NX = 300 - 0.3Y Based on this information, which of the following statements is the most accurate analysis of the relationship between national income and net exports in these two countries? Evaluating Economic Strategies to Address a Trade Deficit Evaluating Economic Strategies to Address a Trade Deficit Calculating GDP Contribution from a Production Chain Deriving the Marginal Propensity to Import A mining company extracts iron ore and sells it to a steel mill for $200. The steel mill processes the ore into steel beams, which it then sells to a construction company for $500. The construction company uses the beams to build a new office, which it sells to a law firm for $1,200. Which of the following statements correctly analyzes the contribution of these transactions to the economy's total output? 1Cademy Optimize Scalable Learning and Teaching How it worksResearch CommunitiesBenefitsAbout Us TermsPrivacyCookieGDPR Contact Us community@1cademy.com Follow Us © 1Cademy 2025 We're committed to OpenSource on Github
7655	https://brainly.com/question/47325275	[FREE] If n^2 is a perfect cube, then which of the following statements is always true? A. n is odd B. n is even - brainly.com 6 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 +66,2k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +19,2k Ace exams faster, with practice that adapts to you Practice Worksheets +7,9k Guided help for every grade, topic or textbook Complete See more / Mathematics Expert-Verified Expert-Verified If n 2 is a perfect cube, then which of the following statements is always true? A. n is odd B. n is even C. n 2 is a perfect square D. n is a perfect cube 1 See answer Explain with Learning Companion NEW Asked by derrickcapon76112 • 02/07/2024 0:00 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 4456902 people 4M 0.0 0 Upload your school material for a more relevant answer The accurate statement is that n² is inherently a perfect square no matter if n is a perfect cube or if n is odd or even.If n² is a perfect cube, then n must be odd. The correct option is (a) n is odd. Explanation If n² is a perfect cube, then n must be odd. To understand why, let's consider the prime factorization of n². If n² is a perfect cube, then all its prime factors must have an exponent that is a multiple of 3. Since any square number has an even exponent for all its prime factors, the exponent of 2 in the prime factorization of n² must be a multiple of 3. Since the exponent of 2 in the prime factorization of n must be half of the exponent in the prime factorization of n², it follows that the exponent of 2 in the prime factorization of n must also be a multiple of 3. Therefore, n must be odd, and the statement n is odd is always true if n² is a perfect cube. The correct option is (a) n is odd. Answered by puneetknp721 •35.7K answers•4.5M people helped Thanks 0 0.0 (0 votes) Expert-Verified⬈(opens in a new tab) This answer helped 4456902 people 4M 0.0 0 Upload your school material for a more relevant answer The correct answer is option D: n is a perfect cube. If n 2 is a perfect cube, then n must also be a perfect cube based on the properties of exponents in prime factorization. Other options (A, B, C) are not always true as they depend on additional conditions regarding n. Explanation To determine the true statement when n 2 is a perfect cube, we analyze the properties of squares and cubes in mathematics. A perfect cube is defined as a number that can be expressed as k 3 where k is an integer. For n 2 to be a perfect cube, its prime factorization must meet certain requirements. Specifically, the exponents of all prime factors in n 2 must be multiples of 3. Since n 2 is always a perfect square (because any integer squared is a square), it inherently has even exponents for its prime factorization. Therefore, if n 2 is a perfect cube, this leads us to conclude that n must be a perfect cube itself. This is because the prime factorization of n must facilitate the even exponents from n 2 being converted to multiples of 3 for it to be a perfect cube. To reinforce this concept, consider that if n=k 3 (thus, n is a perfect cube), then the square n 2=k 6 is also a perfect cube because 6 is a multiple of 3. Thus, only option D, n is a perfect cube, stands true under the condition that n 2 is a perfect cube, while options A, B, and C are not necessarily true depending on the value of n. Examples & Evidence For example, if n=8, which is 2 3, then n 2=64 which is 4 3, a perfect cube. Conversely, if n=2, then n 2=4, which is not a perfect cube, illustrating that not all integers yield the same results. Thus, understanding the relationship between cubes and squares is crucial. This analysis is based on the fundamental properties of integers in number theory, where the relationships between squares and cubes rely heavily on prime factorization and the rules governing exponents. Thanks 0 0.0 (0 votes) Advertisement derrickcapon76112 has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Mathematics solutions and answers Community Answer Which of the following statements are true? Select all that apply. A. 4 is a perfect cube. B. 16 is a perfect square. C. 2,197 is both a perfect square and a perfect cube. D. 8 is a perfect cube. E. 18 is neither a perfect square nor a perfect cube. 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. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. New questions in Mathematics Consider the following incomplete deposit slip: \| Description \| Amount ($) \| \| ---: \| Cash, including coins \| 150 \| 75 \| \| Check 1 \| 564 \| 81 \| \| Check 2 \| 2192 \| 43 \| \| Check 3 \| 4864 \| 01 \| \| Subtotal \| ???? \| ?? \| \| Less cash received \| ???? \| ?? \| \| Total \| 7050 \| 50 \| \| \| \| \| How much cash did the person who filled out this deposit slip receive? Which sequence is generated by the function f(n+1)=f(n)−2 for f(1)=10? A. −2,8,18,28,38,… B. 10,8,6,4,2,… C. 8,18,28,38,48,… D. −10,−12,−14,−16,−18,… i) Write down the expansion of (1+x)3. ii) Find the first four terms in the expansion (1−x)−4 in ascending powers of x. For what values is the expansion valid? iii) When the expansion is valid, find the values of a and b in the equation below. (1−x)4(1+x)3=1+7 x+a x 2+b x 3+…… Example: Round 24.53 to the nearest one. 1. Round 68.23 to the nearest tenth. 2. Round 64.28 to the nearest one. 3. Round 92.02 to the nearest ten. 4. Round 0.45 to the nearest hundredth. 5. Round 412.218 to the nearest hundredth. Solve the simultaneous equations: [ \begin{array}{r} 5x+y=21 \ x-3y=9 \end{array} ] Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
7656	https://dictionary.cambridge.org/us/dictionary/english/bereft	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) Meaning of bereft in English Add to word list Add to word list not having something or feeling great loss: Alone now and almost penniless, he was bereft of hope. After the last of their children had left home the couple felt utterly bereft. Compare bereaved adjective SMART Vocabulary: related words and phrases Lacking things bankrupt bankruptcy be hard up (for something) idiom chronic disease cry out for something phrasal verb hole hour if it wasn't/weren't for phrase in someone's hour of need idiom inappeasable lack minus scrape scream scream (out) for something idiom shortage shy skimpily starve someone/something of something phrasal verb taken See more results » You can also find related words, phrases, and synonyms in the topics: Feeling sad and unhappy (Definition of bereft from the Cambridge Advanced Learner's Dictionary & Thesaurus © Cambridge University Press) bereft \| Intermediate English bereft adjective us /bɪˈreft/ Add to word list Add to word list having to do without something or someone and suffering from the loss: I do hope he won't leave us utterly bereft of his wit and wisdom. (Definition of bereft from the Cambridge Academic Content Dictionary © Cambridge University Press) Examples of bereft bereft The demographic transition is bereft of a scientific foundation. From TIME He was returned this past fall emaciated, flea infested, without fur, listless and bereft of hope. From Huffington Post One who can not hear is bereft of the whole arena of sound. From Huffington Post This team certainly won't be bereft of talent. From The Seattle Times So why is she suddenly having a middle-aged meltdown, agonizing over her life choices and feeling bereft because she doesn't have a husband and children? From Los Angeles Times Whereas the people who were literally alive were so bereft or left empty for the moment that it seemed they were dead. From The Atlantic The complete list of nominees is below, and is mostly bereft of awards singling out people who doffed their clothes onscreen. From A.V. Club The people at the table in front of me sat motionless, staring at a far-off place to my left, all bereft, faraway gazes. From Los Angeles Times He wants to get across how variously curious, depressed, horny, bemused, and bereft he's feeling, without self-pity or much self-consciousness. From NPR But now, living in the aftermath of the system crash, we are bereft. From Gizmodo Her father had died suddenly, and she was bereft. From Baltimore Sun In a winter heretofore bereft of snow scares, this is going to be one banner week. From Philly.com These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. What is the pronunciation of bereft? Translations of bereft in Chinese (Traditional) 缺乏…的, 感到失落的… See more in Chinese (Simplified) 缺乏…的, 感到失落的… See more in Spanish afligido, privado de… See more in Portuguese desamparado, privado… See more in more languages in Turkish in French in Dutch in Czech in Danish in Indonesian in Thai in Vietnamese in Polish in Swedish in Malay in German in Norwegian in Ukrainian in Russian yalnız ve son derece üzgün, mahrum, yoksun… See more privé de… See more beroofd… See more zbavený… See more berøvet, frarøvet… See more kehilangan… See more หมด… See more bị mất… See more opuszczony, osamotniony, pozbawiony… See more berövad… See more kehilangan… See more beraubt… See more berøvet, frarøvet… See more втративший кого, осиротілий… See more опустошенный… See more Need a translator? Get a quick, free translation! Translator tool Browse berberine bereave bereaved bereavement bereft beret berg Bergamasco sheepdog bergamot Test your vocabulary with our fun image quizzes Try a quiz now Word of the Day take something with a pinch of salt to not completely believe something that you are told, because you think it is unlikely to be true About this Blog Tournaments and contests: talking about competitions (1) Read More New Words technofossil More new words has been added to list To top Contents EnglishIntermediateExamplesTranslations Cambridge Dictionary +Plus My profile +Plus help Log out English (US) Change English (UK) English (US) Español Português 中文 (简体) 正體中文 (繁體) Dansk Deutsch Français Italiano Nederlands Norsk Polski Русский Türkçe Tiếng Việt Svenska Українська 日本語 한국어 ગુજરાતી தமிழ் తెలుగు বাঙ্গালি मराठी हिंदी Follow us Choose a dictionary Recent and Recommended English Grammar English–Spanish Spanish–English Definitions Clear explanations of natural written and spoken English English Learner’s Dictionary Essential British English Essential American English Grammar and thesaurus Usage explanations of natural written and spoken English Grammar Thesaurus Pronunciation British and American pronunciations with audio English Pronunciation Translation Click on the arrows to change the translation direction. Bilingual Dictionaries English–Chinese (Simplified) Chinese (Simplified)–English English–Chinese (Traditional) Chinese (Traditional)–English English–Dutch Dutch–English English–French French–English English–German German–English English–Indonesian Indonesian–English English–Italian Italian–English English–Japanese Japanese–English English–Norwegian Norwegian–English English–Polish Polish–English English–Portuguese Portuguese–English English–Spanish Spanish–English English–Swedish Swedish–English Semi-bilingual Dictionaries English–Arabic English–Bengali English–Catalan English–Czech English–Danish English–Gujarati English–Hindi English–Korean English–Malay English–Marathi English–Russian English–Tamil English–Telugu English–Thai English–Turkish English–Ukrainian English–Urdu English–Vietnamese Dictionary +Plus Word Lists Contents English Adjective Examples Translations Grammar All translations My word lists To add bereft to a word list please sign up or log in. Sign up or Log in My word lists Add bereft to one of your lists below, or create a new one. 5 && !stateSidebarWordList.expended) ? 195 : (stateSidebarWordListItems.length 39)" [src]="stateSidebarWordListItems" class="i-amphtml-element i-amphtml-layout-fixed-height i-amphtml-layout-size-defined i-amphtml-built" i-amphtml-layout="fixed-height" style="height: 0px;"> {{name}} 5 && !stateSidebarWordList.expended) ? 'hao hp lmt-25' : 'hdn'"> Go to your word lists Tell us about this example sentence: By clicking “Accept All Cookies”, you agree to the storing of cookies on your device to enhance site navigation, analyze site usage, and assist in our marketing efforts. Privacy and Cookies Policy
7657	https://www.quora.com/How-many-two-digit-numbers-are-there-which-have-an-odd-number-of-factors-including-1-and-itself	How many two-digit numbers are there which have an odd number of factors, including 1 and itself? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Factor Counting Rules Odd Prime Numbers Two-digit Number Count Different Factors Arithmetic Mathematics Number Theory 5 How many two-digit numbers are there which have an odd number of factors, including 1 and itself? All related (34) Sort Recommended Aditya Singhvi MathCounts Coach ·7y Any number with an odd number of factors must be a perfect square. The number of factors a number has is given by the formula f=(x 1+1)×(x 2+1)×…(x k+1)f=(x 1+1)×(x 2+1)×…(x k+1), where x i x i is the exponent of the i i th prime factor of a number. For a number to be a perfect square, every x x must be even, as each prime can then be split evenly between two equal factors. Thus, this means that the number of factors of a perfect square must always be odd as all (x+1)(x+1)s will be odd, and the product of odd numbers is always odd. Thus, all we have to do is find the number of perfect squares in Continue Reading Any number with an odd number of factors must be a perfect square. The number of factors a number has is given by the formula f=(x 1+1)×(x 2+1)×…(x k+1)f=(x 1+1)×(x 2+1)×…(x k+1), where x i x i is the exponent of the i i th prime factor of a number. For a number to be a perfect square, every x x must be even, as each prime can then be split evenly between two equal factors. Thus, this means that the number of factors of a perfect square must always be odd as all (x+1)(x+1)s will be odd, and the product of odd numbers is always odd. Thus, all we have to do is find the number of perfect squares in the range [10,99][10,99]. There are 6 6 such squares (4 2,5 2...9 2)4 2,5 2...9 2), so the answer is simply 6. This is because if a positive integer n n has prime factorization n=p 1 x 1+p 2 x 2…p k x k n=p 1 x 1+p 2 x 2…p k x k, the number of factors it has is equal to (x 1+1)×(x 2+1)×…(x k+1).(x 1+1)×(x 2+1)×…(x k+1).This formula comes from the fact that there are x i+1 x i+1 possibilities for the inclusion of each prime factor in the factorization of any given factor of a number. Upvote · 9 3 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 999 209 Related questions More answers below How many two-digit numbers over 25 have an odd number of factors? How many two-digit odd numbers are there in all? How many three digit number have exactly three factors? How many 2-digit numbers have exactly 3 factors? How many four digit numbers have exactly 5 factors? Ravindra S. Lanka Staff Software Engineer at Facebook (company) (2019–present) ·7y Factors come in pairs. For total number of factors to be odd, two of the factors must be the same which is the square root of the number. So, the answer is all 2 digit perfect squares, 4^2, 5^2, 6^2 thru 9^2, which is 6 of them. Upvote · 9 3 Bryan White Software Developer with over 45 years commercial experience · Author has 13.1K answers and 18.5M answer views ·2y It Depends (tm). On things like whether 04 is a two-digit number. Also whether numbers are in base-ten notation; most number theory does not require any specific number base. Upvote · 9 2 Robert Nichols Author has 5K answers and 15.6M answer views ·7y Related How many two-digit numbers over 25 have an odd number of factors? In order to have an odd number of factors a number must be a perfect square. The perfect squares that are only 2 digits but larger than 25 are: 36 (6^2), 49 (7^2), 64 (8^2), and 81 (9^2). The answer is 4. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 = 9 factors, 9 is odd Factors of 49: 1, 7, 49 = 3 factors, 3 is odd Factors of 64: 1, 2, 4, 8, 16, 32, 64 = 7 factors, 7 is odd Factors of 81: 1, 3, 9, 27, 81 = 5 factors, 5 is odd Upvote · 99 40 Related questions More answers below How many two digit numbers are there having 4 factors? The sum of the factors of a two digit odd number is 31. What is the number? How many digits are in one number multiplied by another number where the first has an odd number of factors (not including itself) but the second has an even number of factors (not including itself)? How many two digit numbers have one odd digit and one even digit? How many 3 digit numbers are there that have 2 odd and 1 even digits? Assistant Bot · 1y To determine how many two-digit numbers have an odd number of factors, we need to understand a key property of numbers regarding their factors: A number has an odd number of factors if and only if it is a perfect square. This is because factors typically come in pairs (for example, 1×n 1×n, 2×n 2 2×n 2, etc.), but if a number is a perfect square, one of those factors is repeated (for example, k×k k×k). Now, we need to find the two-digit perfect squares. The smallest two-digit number is 10, and the largest is 99. The smallest integer whose square is at least 10 is: \lceil\sqrt{1\lceil\sqrt{1 Continue Reading To determine how many two-digit numbers have an odd number of factors, we need to understand a key property of numbers regarding their factors: A number has an odd number of factors if and only if it is a perfect square. This is because factors typically come in pairs (for example, 1×n 1×n, 2×n 2 2×n 2, etc.), but if a number is a perfect square, one of those factors is repeated (for example, k×k k×k). Now, we need to find the two-digit perfect squares. The smallest two-digit number is 10, and the largest is 99. The smallest integer whose square is at least 10 is: ⌈√10⌉=4(since 4 2=16)⌈10⌉=4(since 4 2=16) 2. The largest integer whose square is at most 99 is: ⌊√99⌋=9(since 9 2=81)⌊99⌋=9(since 9 2=81) Now we can list the perfect squares of integers from 4 to 9: 4 2=16 4 2=16 5 2=25 5 2=25 6 2=36 6 2=36 7 2=49 7 2=49 8 2=64 8 2=64 9 2=81 9 2=81 These perfect squares are: 16, 25, 36, 49, 64, and 81. Counting these, we find there are 6 two-digit numbers with an odd number of factors. Thus, the answer is: 6 6 Upvote · Deb P. Choudhury Former Professor at University of Allahabad · Author has 10K answers and 8M answer views ·2y Related How Many factors of 20736 have odd numbers factors? The question being ill-constructed, admits at least two interpretations: (i) How many factors of 20736 are odd numbers? (ii) How many factors of 20736 have odd number of factors? Note that n = 20736 = 4×5184 = 16×1296 = 64×324 = 256×81 = 2⁸×3⁴ (prime power factorisation) Answer of (i): Clearly the odd factors of n are precisely the factors of 3⁴ = 81, because even a single power of 2, makes the factor even. The factors of 81 are 1, 3, 9, 27 and 81 and hence there are five factors of n which are odd numbers. Answer of (ii): Every factor of n is of the form d = (2^a)×(3^b) where 0 ≤ a ≤ 8, 0 ≤ b ≤ 4. The Continue Reading The question being ill-constructed, admits at least two interpretations: (i) How many factors of 20736 are odd numbers? (ii) How many factors of 20736 have odd number of factors? Note that n = 20736 = 4×5184 = 16×1296 = 64×324 = 256×81 = 2⁸×3⁴ (prime power factorisation) Answer of (i): Clearly the odd factors of n are precisely the factors of 3⁴ = 81, because even a single power of 2, makes the factor even. The factors of 81 are 1, 3, 9, 27 and 81 and hence there are five factors of n which are odd numbers. Answer of (ii): Every factor of n is of the form d = (2^a)×(3^b) where 0 ≤ a ≤ 8, 0 ≤ b ≤ 4. The number of divisors of d : tau(d) = (a+1)×(b+1) is odd if and only if both a and b are even. This happens exactly when (a,b) = (0,0), (0,2), (0,4), (2,0), (2,2), (2,4), (4,0), (4,2), (4,4), (6,0), (6,2), (6,4), (8,0), (8,2), (8,4). Therefore there are 15 positive divisors of the given number n = 20736 which have an odd number of factors. Upvote · 9 9 Sponsored by RedHat Customize AI for your needs, with simpler model alignment tools. Your AI needs context, not common knowledge. Learn More 9 7 Dave Buchfuhrer PhD in Theoretical Computer Science · Author has 922 answers and 2.7M answer views ·10y Related Let n be the number of factors of 2014, including 1 and itself. How many two-digit numbers also have n factors? This is labeled as a math competition problem, so I'm assuming you want to see how we would do this without the aid of a computer. First, we have to factor 2014. It has an obvious factor of 2, so we'll continue by factoring 1007. It obviously does not have any factors of 2 or 5. It doesn't have any factors of 3 because if we add the digits they sum to 8, which is not a factor of 3. I don't know any good tricks for larger primes, so instead I'll use 1007=100∗10+7 1007=100∗10+7 and do some modular arithmetic to check 1007 mod p for larger primes. For 7, we take 100 mod 7 to find 2. Multiply by 10 and take Continue Reading This is labeled as a math competition problem, so I'm assuming you want to see how we would do this without the aid of a computer. First, we have to factor 2014. It has an obvious factor of 2, so we'll continue by factoring 1007. It obviously does not have any factors of 2 or 5. It doesn't have any factors of 3 because if we add the digits they sum to 8, which is not a factor of 3. I don't know any good tricks for larger primes, so instead I'll use 1007=100∗10+7 1007=100∗10+7 and do some modular arithmetic to check 1007 mod p for larger primes. For 7, we take 100 mod 7 to find 2. Multiply by 10 and take mod 7 again to find 6. Now add 7 to find 13, not a multiple of 7. Similarly for 11, we take 100 mod 11 to be 1, multiply by 10 to get 10, then add 7 for 17, not a multiple of 11. For 13, 100 mod 13 is 9, multiplied by 10 is 90, mod 13 is 12, plus 7 is 19, not a multiple of 13. For 17, we get 100 mod 17 = 15, which is also equivalent to -2 (easier to deal with). Multiply by 10 for -20, which is equivalent to 14 mod 17. Add 7 for 21, not a multiple of 17. For 19, we get 100 equals 5 mod 19, then 50 mod 19 is 12, plus 7 is 19. We've found another factor! Dividing 1007 by 19 yields 53. 53 must be prime, as 19 2>53 19 2>53 and we already demonstrated that 1007 (and therefore any prime factors of it) has no prime factors smaller than 19. Therefore, 2014=2⋅19⋅53 2014=2⋅19⋅53, so it has 2 3=8 2 3=8 factors, as each of these prime factors can appear either 0 or 1 times in a factor. Similarly, any other number that has 8 factors must either have 3 prime factors appearing with a power of 1, one prime factor with a power of 3 and one with a power of 1, or one prime factor with a power of 7. Let's first look at 3 prime factors with a power of 1. 3⋅5⋅7=105 3⋅5⋅7=105, so we know any such 3 must contain a 2. So we're really looking for pairs of primes greater than 2 that multiply to less than 50. If the smaller of these two is 3, the larger must be less than 17, giving 5, 7, 11, 13. So there are 4 such pairs. If the smaller one is 5, the larger must be less than 10, giving the single pair (5, 7). If the smaller one is 7, the larger one must be less than 8, so there are no such pairs. So the total number of triples of prime factors that work here is 5. Now let's look at pairs of primes, one of which is cubed. If the cubed one is 2, then 8 times another prime must be less than 100. This works for 3, 5, 7, 11, giving 4 results. If the cubed number is 3, then we need a prime other than three times 27 to be less than 100. This only works if the other prime is 2, giving another result. So the number of pairs where one is cubed is 5. Finally, let's look at how many primes raised to the power of 7 are less than 100. 2 7=128 2 7=128, so there are none. Thus the answer is 10. Upvote · 9 6 9 1 Jeremy Bohrer M.S. in Mathematics, Iowa State University (Graduated 2015) · Author has 640 answers and 489.8K answer views ·3y Related How many numbers from 1 to 100 have an odd number of factors? Ten. Only square numbers have an odd number of factors. For a number that is not a square, factors come in factor pairs of two different numbers, the product of which is the number in question. For 2, a prime number, there is only one such pair: (1,2). Six has two pairs: (1, 6), (2, 3). 24 has four pairs: (1, 24), (2, 12), (3, 8), (4, 6). The number of factors is twice the number of pairs. Square numbers, however, have one factor pair with two values that are the same. The only factor pair for 1 is (1, 1) — one factor. For 4 we get (1, 4) and (2, 2) — three factors. 36 has (1, 36), (2, 18), (3, Continue Reading Ten. Only square numbers have an odd number of factors. For a number that is not a square, factors come in factor pairs of two different numbers, the product of which is the number in question. For 2, a prime number, there is only one such pair: (1,2). Six has two pairs: (1, 6), (2, 3). 24 has four pairs: (1, 24), (2, 12), (3, 8), (4, 6). The number of factors is twice the number of pairs. Square numbers, however, have one factor pair with two values that are the same. The only factor pair for 1 is (1, 1) — one factor. For 4 we get (1, 4) and (2, 2) — three factors. 36 has (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6), for a total of 9 factors. There are ten square numbers from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100, and these are the only numbers from 1 to 100 which have more an odd number of factors. Upvote · 9 4 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 999 624 Ravi Handa Founder of www.handakafunda.com, teach online courses on Numbers · Author has 758 answers and 11.1M answer views ·10y Related How many 2 digit odd numbers are there with exactly 8 factors? No such number exists If N = a^p b^q c^r ... Number of factors = (p + 1)(q + 1)(r + 1).... We are given number of factors = 8 => (p + 1)(q + 1)(r + 1).... = 8 So, the possibilities for splitting 8 are 8, 24, 222 So, the number could be of the format a^7, ab^3, abc The number should be odd, so the prime numbers should be odd. a^7 format The smallest odd number of this format is 3^7. That is not a two digit number. ab^3 format The smallest odd number of this format is 53^3. That is not a two digit number. abc format The smallest odd number of this format is 357. That is not a Continue Reading No such number exists If N = a^p b^q c^r ... Number of factors = (p + 1)(q + 1)(r + 1).... We are given number of factors = 8 => (p + 1)(q + 1)(r + 1).... = 8 So, the possibilities for splitting 8 are 8, 24, 222 So, the number could be of the format a^7, ab^3, abc The number should be odd, so the prime numbers should be odd. a^7 format The smallest odd number of this format is 3^7. That is not a two digit number. ab^3 format The smallest odd number of this format is 53^3. That is not a two digit number. abc format The smallest odd number of this format is 357. That is not a two digit number. So, there is no odd two digit number that has 8 factors. Upvote · 9 8 Steven M Block Professor at Stanford University (1999–present) · Upvoted by Terry Moore , M.Sc. Mathematics, University of Southampton (1968) · Author has 1.2K answers and 259.6K answer views ·1y Related How is it possible for a number to have an odd number of factors? Multiplication is in pairs, so it has to be divisible by 2. You are not thinking about this very deeply, are you? The number 8 is 2 x 2 x 2. That’s an odd number of factors of 2. The number 12 is 3 x 2 x 2. The number 18 is 3 x 3 x 2. The number 27 is 3 x 3 x 3. How many examples do you need? And NO, multiplication is not “in pairs.” You can multiply any number of numbers together that you wish. Upvote · 9 4 Sponsored by State Bank of India Stay Informed!! Stay Protected!! Our Contact Centre calls only from numbers beginning with 1600 or 140 series. Learn More 99 46 Wolf Boehrendt Diploma (Master) in Mathematics&Physics, University of Tübingen (Graduated 1969) · Author has 1K answers and 783.7K answer views ·3y Related How many numbers from 1 to 100 have an odd number of factors? Another problem where you should think before you calculate and then do it in your head. First I assume the maths convention that 1 and the number itself are factors. Now take an example: 6 has factor 2 so of course 6/2 = 3 also is a factor and so on, factors always are paired unless you have a square like 36 eg where 6 is a “single factor”. So we are looking for all the squares: 1, 4, 9 … 100 The number of these should be easy to calculate! There is a very nice variation of this I saw in a maths quiz. Along a road you have 1000s or millions of lights, all with a on/off switch. Now someone walks down Continue Reading Another problem where you should think before you calculate and then do it in your head. First I assume the maths convention that 1 and the number itself are factors. Now take an example: 6 has factor 2 so of course 6/2 = 3 also is a factor and so on, factors always are paired unless you have a square like 36 eg where 6 is a “single factor”. So we are looking for all the squares: 1, 4, 9 … 100 The number of these should be easy to calculate! There is a very nice variation of this I saw in a maths quiz. Along a road you have 1000s or millions of lights, all with a on/off switch. Now someone walks down the road and switches all to on. Then another guy walks down the road and switches all even numbers, another guy switches every third number, every fourth and so on. After a while you’ll the first lights staying stable - now which stay on and which stay off in the end? Upvote · 9 3 Kip Fisher Former Risk Analyst and Sometime Math Tutor · Author has 683 answers and 488.1K answer views ·1y Related What is the smallest whole number with an even number of factors? What is the largest odd number with an odd number of factors? "What is the smallest whole number with an even number of factors?" There is no academy that dictates definitions of mathematical words, so we have to be a little careful. While there is some disagreement, most people define the whole numbers to be the counting numbers starting at zero: 0, 1, 2, 3, ... . The inclusion of 0 in the this question is problematic. 0 is the product of itself and any other number. This makes the very notion of a "factor" of 0 hard to nail down. Does it make sense to say that every number is a factor of 0? Is the notion of a factor of zero useful? Let's leave 0 out of Continue Reading "What is the smallest whole number with an even number of factors?" There is no academy that dictates definitions of mathematical words, so we have to be a little careful. While there is some disagreement, most people define the whole numbers to be the counting numbers starting at zero: 0, 1, 2, 3, ... . The inclusion of 0 in the this question is problematic. 0 is the product of itself and any other number. This makes the very notion of a "factor" of 0 hard to nail down. Does it make sense to say that every number is a factor of 0? Is the notion of a factor of zero useful? Let's leave 0 out of the discussion. To that end, modify the question. "What is the smallest natural number with an even number of factors?" While there is some disagreement, most people define the natural numbers to be the counting numbers starting at one: 1, 2, 3, ... . A natural number has an odd number of factors if and only if it is a perfect square. To find the number of factors of a natural number, we start with its prime factorization. More specifically, we start with the exponents on the primes in the number's prime factorization. We add 1 to each exponent, then take the product of all of the exponent-plus-one answers. For example: 12=2 2×3 12=2 2×3. The exponents are 2 and 1. (2+1)×(1+1)=3×2=6(2+1)×(1+1)=3×2=6. And indeed, the factors of 12 are 1, 2, 3, 4, 6, and 12. The number 12 really does have 6 factors. Under what circumstances does a number have an even number of factors? A number has an even number of factors if one or more of the exponents in its prime factorization is odd. Adding 1 to an odd exponent makes an even number, and even one even number in a product (of natural numbers) makes an even result. Under what circumstances does a number have an odd number of factors? A number has an odd number of factors if all of the exponents in its prime factorization are even. When this is true, adding one to each exponent results in an odd number. All of the exponent-plus-one answers are odd, so their product is odd. But if all of the exponents in the prime factorization are even, the number is a perfect square. We can now see that a number has an odd number of factors if and only if it is a perfect square. Our question has now become, "What is the smallest natural number that is not a perfect square?" 1 is a perfect square. 2 is not a perfect square. 2 has 2 factors: 1 and 2. Answer: 2 is the smallest natural number with an even number of factors. "What is the largest odd number with an odd number of factors?" We have already established that every perfect square has an odd number of factors. There is no largest perfect square (because there is no largest natural number, and every number can be squared). Therefore, there is no largest odd number with an odd number of factors. Upvote · 9 1 9 1 Carter McClung B.S. in UPSC General Studies&Mathematics, The University of Texas at Dallas (Graduated 2006) · Upvoted by Michael Jørgensen , PhD in mathematics · Author has 1.6K answers and 6.5M answer views ·10y Related How many 2 digit odd numbers are there with exactly 8 factors? The number of prime factors is the product of one plus the power of each distinct prime factor. In other words, if N=a A b B c C...N=a A b B c C..., where a, b, and c are distinct primes, the number of factors will be (A+1)(B+1)(C+1)... This leaves the possibility that our number is of the form N=a b c N=a b c or N=a 3 b N=a 3 b or N=a 7 N=a 7. The lowest possible odd numbers of this form are (3)(5)(7)=105(3)(5)(7)=105, (3)3(5)=135(3)3(5)=135, and 3 7=2187 3 7=2187. So... no dice. Upvote · 9 9 Related questions How many two-digit numbers over 25 have an odd number of factors? How many two-digit odd numbers are there in all? How many three digit number have exactly three factors? How many 2-digit numbers have exactly 3 factors? How many four digit numbers have exactly 5 factors? How many two digit numbers are there having 4 factors? The sum of the factors of a two digit odd number is 31. What is the number? How many digits are in one number multiplied by another number where the first has an odd number of factors (not including itself) but the second has an even number of factors (not including itself)? How many two digit numbers have one odd digit and one even digit? How many 3 digit numbers are there that have 2 odd and 1 even digits? How many two-digit natural numbers are multiples of 7? How many two digit numbers have exactly 4 factors (here 1 and the number n are also considered as the factors of n)? What is the maximum number of factors a 2 digit number can have? How many three-digit numbers are there with distinct digits with each digit being odd? Which is the greatest 1 digit number that has 4 factors? Related questions How many two-digit numbers over 25 have an odd number of factors? How many two-digit odd numbers are there in all? How many three digit number have exactly three factors? How many 2-digit numbers have exactly 3 factors? How many four digit numbers have exactly 5 factors? How many two digit numbers are there having 4 factors? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
7658	https://www.youtube.com/watch?v=vRme6eaNxuw	CUBE NET question - Find the face opposite to the one labelled X in the diagram KS2ChallengeMaths11+13+ 257 subscribers 3 likes Description 883 views Posted: 31 Oct 2021 This question on cube net involves figuring out the letters on the opposite faces of the cube. 1 comments Transcript:
7659	https://chemistry.stackexchange.com/questions/7647/similarity-between-ecell-and-equilibrium-constant-kc	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 Similarity between Ecell and equilibrium constant Kc Ask Question Asked Modified 11 years, 8 months ago Viewed 7k times 0 $\begingroup$ $$\ce{Cu^{2+} + Zn<=>Zn^{2+} + Cu}$$ $K_c=1.910^{37}$ $E_{cell}=1.1V$ I know that they both suggest that the forward reaction is going to completion but can someone please tell me a bit more about what both these values are suggesting and how these two values concord with each other. Also, many books quote such reactions in only one direction and gives a value for $E_{cell}$. However, isn't $E_{cell}$ just another way of illustrating the condition of an equilibrium (i.e. in which direction the reaction would occur if left alone)... so how can a one way reation be shown electrochemistry equilibrium Share edited Jan 5, 2014 at 0:22 jonsca 3,04577 gold badges3535 silver badges5656 bronze badges asked Jan 4, 2014 at 13:54 ElizaEliza 2,45399 gold badges3434 silver badges4545 bronze badges $\endgroup$ 1 $\begingroup$ See here for a derivation of the relationship between equilibrium constants and potentials. $\endgroup$ Nicolau Saker Neto – Nicolau Saker Neto 2014-01-04 14:55:23 +00:00 Commented Jan 4, 2014 at 14:55 Add a comment \| 1 Answer 1 Reset to default 5 $\begingroup$ $E^\circ_{cell}$ and $K_c$ are related. Both indicate which direction of the reaction is spontaneous. Both tell you something about the position of equilibrium. However, $E^\circ_{cell}$ is a directly measurable value. If you construct an electrochemical cell as using a zinc electrode immersed in a 1.0 M solution of say zinc chloride and a copper electrode immersed in a 1.0 M solution of copper (II) chloride with a salt bride of sodium chloride connecting the solutions, a voltmeter connected between the two electrodes will read the value of $E^\circ_{cell}$, at least initially. The voltage will change as the reaction progresses, but the initial voltageis $E^\circ_{cell}$ which is defined for standard conditions of 1.0 M concentration, 1 bar of pressure, and 25 $^\circ$C. $$\ce{Zn\|ZnCl2}(1.0 \text{ M})\parallel\ce{CuCl2}(1.0 \text{ M})\|\ce{Cu} \ \ \ \ E^\circ_{cell}=1.1\text{ V}$$ The relationship between $E^\circ_{cell}$ and $K_c$ is the Nernst Equation. Note the use of both $E^\circ_{cell}$, the standard cell potential and $E_{cell}$, the observed cell potential under other conditions. The remainder of the alphabet soup is defined below. $$E_{cell}=E^\circ_{cell}-\frac{RT}{zF}\ln{Q}$$ $E^\circ_{cell}$ - Standard cell potential $E_{cell}$ - nonstandard cell potential at other conditions $R$ - the ideal gas constant (which appears here because it is equal to Avogadro's number times Boltzmann's constant $T$ - the temperature in kelvins $z$ - the number of moles of electrons being transferred in this redox process. If we wanted to use the number of individual electrons, then we would need to replace $R$ with $k_B$ $F$ - Faraday's constant $Q$ - the reaction quotient We need to use $Q$ in the Nernst equation, because this equation can relate the cell potential to any range of conditions, not just equilibrium. At equilibrium $Q = K_c$ and $E_{cell} = 0 \text{ V}$ (there is no net change in the reaction composition, so there is no potential difference across the half-cells. Thus, the Nernst Equation becomes: $$0=E^\circ_{cell}-\frac{RT}{zF}\ln{K_c}$$ $$E^\circ_{cell} = \frac{RT}{zF}\ln{K_c}$$ To determine the values of $E^\circ_{cell}$ and $K_c$ for the reverse reaction... For $E^\circ_{cell}$: $$E^\circ_{cell} = -E^\circ_{rev}$$ For $K_c$ $$K_{rev} = \frac{1}{K_c}$$ Share answered Jan 5, 2014 at 15:02 Ben NorrisBen Norris 43.4k88 gold badges131131 silver badges186186 bronze badges $\endgroup$ 2 1 $\begingroup$ Great post as always. Just a small thought regarding $z$. The quotient $\frac{RT}{zF}$ needs to have a dimension of volts, and by dimensional analysis, the $mol^{-1}$ from $R$ would cancel out with the $mol^{-1}$ from $F$, clearing the quotient of any dimension in quantity of matter. If $z$ had the dimensionality of moles, then it would reintroduce a unit that could not cancel out. I imagine that means the $z$ is only the adimensional stoichiometric coefficient for electrons in the reaction, and as such it can't be converted to number of electrons by substituting $R$ for $k_B$. $\endgroup$ Nicolau Saker Neto – Nicolau Saker Neto 2014-01-05 15:32:27 +00:00 Commented Jan 5, 2014 at 15:32 1 $\begingroup$ Rather, if $k_B$ is used, then I think that instead of $F$ it would be necessary to use $e$, the fundamental electric charge. $\endgroup$ Nicolau Saker Neto – Nicolau Saker Neto 2014-01-05 15:33:27 +00:00 Commented Jan 5, 2014 at 15:33 Add a comment \| Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions electrochemistry equilibrium See similar questions with these tags. 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7660	https://study.com/academy/lesson/systems-of-inequalities.html	Inequality Notation \| Overview & Examples - Lesson \| Study.com Log In Sign Up Menu Plans Courses By Subject College Courses High School Courses Middle School Courses Elementary School Courses By Subject Arts Business Computer Science Education & Teaching English (ELA) Foreign Language Health & Medicine History Humanities Math Psychology Science Social Science Subjects Art Business Computer Science Education & Teaching English Health & Medicine History Humanities Math Psychology Science Social Science Art Architecture Art History Design Performing Arts Visual Arts Business Accounting Business Administration Business Communication Business Ethics Business Intelligence Business Law Economics Finance Healthcare Administration Human Resources Information Technology International Business Operations Management Real Estate Sales & Marketing Computer Science Computer Engineering Computer Programming Cybersecurity Data Science Software Education & Teaching Education Law & Policy Pedagogy & Teaching Strategies Special & Specialized Education Student Support in Education Teaching English Language Learners English Grammar Literature Public Speaking Reading Vocabulary Writing & Composition Health & Medicine Counseling & Therapy Health Medicine Nursing Nutrition History US History World History Humanities Communication Ethics Foreign Languages Philosophy Religious Studies Math Algebra Basic Math Calculus Geometry Statistics Trigonometry Psychology Clinical & Abnormal Psychology Cognitive Science Developmental Psychology Educational Psychology Organizational Psychology Social Psychology Science Anatomy & Physiology Astronomy Biology Chemistry Earth Science Engineering Environmental Science Physics Scientific Research Social Science Anthropology Criminal Justice Geography Law Linguistics Political Science Sociology Teachers Teacher Certification Teaching Resources and Curriculum Skills Practice Lesson Plans Teacher Professional Development For schools & districts Certifications Teacher Certification Exams Nursing Exams Real Estate Exams Military Exams Finance Exams Human Resources Exams Counseling & Social Work Exams Allied Health & Medicine Exams All Test Prep Teacher Certification Exams Praxis Test Prep FTCE Test Prep TExES Test Prep CSET & CBEST Test Prep All Teacher Certification Test Prep Nursing Exams NCLEX Test Prep TEAS Test Prep HESI Test Prep All Nursing Test Prep Real Estate Exams Real Estate Sales Real Estate Brokers Real Estate Appraisals All Real Estate Test Prep Military Exams ASVAB Test Prep AFOQT Test Prep All Military Test Prep Finance Exams SIE Test Prep Series 6 Test Prep Series 65 Test Prep Series 66 Test Prep Series 7 Test Prep CPP Test Prep CMA Test Prep All Finance Test Prep Human Resources Exams SHRM Test Prep PHR Test Prep aPHR Test Prep PHRi Test Prep SPHR Test Prep All HR Test Prep Counseling & Social Work Exams NCE Test Prep NCMHCE Test Prep CPCE Test Prep ASWB Test Prep CRC Test Prep All Counseling & Social Work Test Prep Allied Health & Medicine Exams ASCP Test Prep CNA Test Prep CNS Test Prep All Medical Test Prep College Degrees College Credit Courses Partner Schools Success Stories Earn credit Sign Up 1. Math Courses 2. Math 101: College Algebra Inequality Notation \| Overview & Examples Contributors: Stephanie Stanglin, Luke Winspur Author Author: Stephanie Stanglin Show more Instructor Instructor: Luke Winspur Show more Discover what inequality notation is. Learn about set notation inequalities. Know how graphing compound inequalities helps in solving the system of inequalities. Updated: 11/21/2023 Table of Contents What are Inequalities? System of Inequalities and Examples Lesson Summary Show FAQ How do you write an inequality solution in interval notation? Interval notation is giving the range of values for the variable using grouping symbols. Square brackets are for equal to and parentheses are for not equal to. If it is an "and" compound inequality the values go in the same group, but if its an "or" compound inequality the values go in different groups and use the union symbol between them. How do you write inequalities with set notation? Set notation is saying all of the values of x such that certain conditions are met. They are written as {x\| x symbol number} where symbol is the inequality symbol and number is the value. Create an account LessonTranscript VideoQuizCourse An error occurred trying to load this video. Try refreshing the page, or contact customer support. You must c C reate an account to continue watching Register to view this lesson Are you a student or a teacher? I am a student I am a teacher Create Your Account To Continue Watching As a member, you'll also get unlimited access to over 88,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed. Get unlimited access to over 88,000 lessons. Try it now Already registered? Log in here for access Go back Resources created by teachers for teachers Over 30,000 video lessons & teaching resources—all in one place. 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Your next lesson will play in 10 seconds 0:02 Understanding Inequalities 0:34 Compound Inequalities 3:43 Set Notation and… 6:02 System of Inequalities 7:29 Lesson Summary QuizCourseView Video OnlySaveTimeline 144K views Recommended lessons and courses for you Related LessonsRelated Courses ##### Translating Math Sentences to Inequalities 5:36 ##### The Order of Real Numbers: Inequalities 4:36 ##### Quadratic Inequality \| Solution Sets & Examples 3:40 ##### Writing Equations with Inequalities: Open Sentences and True/False Statements 4:22 ##### Solving Rational Inequalities \| Steps & Examples 5:27 ##### Interval Notation \| Definition, Rules & Examples 9:08 ##### How to Solve and Graph And & Or Compound Inequalities 7:36 ##### Parent Function \| Definition, Types & Examples 7:25 ##### Vertical & Horizontal Compression of a Function 6:29 ##### How to Find the Vertex of a Parabola \| Quadratic Equation 5:16 ##### How to Find the Maximum & Minimum Values of a Function? 5:36 ##### Math 105: Precalculus Algebra ##### CLEP College Algebra Study Guide and Exam Prep ##### Algebra I: High School ##### Algebra II: High School ##### UExcel Precalculus Algebra: Study Guide & Test Prep ##### Prentice Hall Algebra 2: Online Textbook Help ##### McDougal Littell Algebra 2: Online Textbook Help ##### Algebra Connections: Online Textbook Help ##### College 101: College Readiness ##### CUNY Assessment Test in Math: Practice & Study Guide ##### Intermediate Algebra for College Students ##### TECEP College Algebra Study Guide and Exam Prep ##### DSST Fundamentals of College Algebra Study Guide and Test Prep ##### Math 99: Essentials of Algebra and Statistics ##### GRE Quantitative Reasoning: Study Guide & Test Prep ##### Math 102: College Mathematics ##### Math 103: Precalculus ##### ELM: CSU Math Study Guide ##### SAT Subject Test Mathematics Level 1: Practice and Study Guide ##### SAT Subject Test Mathematics Level 2: Practice and Study Guide What are Inequalities? ---------------------- An inequality is a mathematical statement that represents the relationship between two quantities, but states that they are not exactly equal. The four symbols of inequality notation are ≥,>,<,a n d≤ where the first is greater than or equal to, meaning that the left side of the statement is either larger than (or the same as) the right side of the statement; the second is greater than, meaning that the left side is larger than the right side; the third is less than, meaning that the left side is smaller than the right side; and the fourth one is less than or equal to, meaning that the left side is either smaller than (or the same as) the right side. One-variable inequalities only have one unknown and can be solved using inverse operations, which is the same as one-variable equations. An example would be: 2 x+5<11 Two-variable inequalities have two unknowns, and there needs to be at least two inequalities to find the values of the unknowns that make the statements true (same as two-variable equations). An example would be: 2 x+3 y<6 5 x−4 y≥20 Compound Inequality A compound inequality is one where there are two statements in inequality notation instead of just one like a standard inequality. There are two types of compound inequalities: "and" and "or". An "and" compound inequality is one where both statements must be true. An example would be: x>5 and x<10 This would mean the solution set includes any number between 5 and 10. On a graph this would look like: Graph of the Inequality An "or" compound inequality is one where either of the statements can be true. An example would be: x<5 or x>10 This would mean the solution set includes any number smaller than 5 or larger than 10. On a graph this would look like: Graph of the Inequality Set Notation Inequalities Set notation inequalities use grouping symbols as way to denote what the solutions are to an inequality. They look like: {x\|x s y m b o l n u m b e r} where "x" represents the variable used in the inequality, "symbol" is the inequality notation being used, and "number" helps indicate where the solutions are in comparison to the number. The x with the line represents all values of x ''such that''. Examples given with the "and" and "or" compound inequalities would be written as: {x\|x>5 a n d x<10} and {x\|x<5 o r x>10} respectively. Inequality Notation Using Intervals Intervals are another way to write the solutions for inequality notation, and related to using set notation, but instead of using inequality notation symbols, intervals use the grouping symbols: [, ], (, and ). [ represents greater than or equal to ] represents less than or equal to ( represents greater than ) represents less than For "and" inequalities, the numbers are put in the same set of grouping symbols because both statements are true. So x>5 and x<10 would look like (5,10) because the solutions are greater than "(" 5 and less than ")" 10. For "or" inequalities, the numbers are put in separate sets of grouping symbols because only one of them is true. Between the two sets a "U" is used to represent "union" or "or". So x<5 or x>10 would look like (−∞,5)U(10,∞) Inequality Notation Examples Below are some examples of inequality notation, set notation, and what they would look like on a graph. Example 1: Inequality notation: (2,8] Set notation: {x\|x>2 a n d x≤8} Graph of the Example This means that the solutions are greater than 2 because of ( and less than or equal to 8 because of ]. Example 2: Inequality notation: (−∞,2]U(8,∞) Set notation: {x\|x≥2 o r x>8} Graph of the Given Example To unlock this lesson you must be a Study.com memberCreate an account System of Inequalities and Examples ----------------------------------- A system of inequalities is a set of two or more inequalities, depending on how many variables are in the inequalities (i.e., two variables, two inequalities). For example: 2 x+3 y>6 5 x−4 y≤20 The purpose of a system of inequalities is to find a solution that makes both inequalities true. While a system of equations only has one solution, there are many points that are a solution to a system of inequalities. In the case of this example, the solution would be an (x,y) pair since the variables are x and y. One example of an (x,y) pair that makes the solution true is (2,1). To prove it: 2(2)+3(1)>6 4+3>6 7>6 This is true. 5(2)−4(1)≤20 10−4≤20 6≤20 This is true. Graphing Compound Inequalities Systems of inequalities can also be solved graphically, and this is typically the easiest way to solve them. After graphing the inequalities, look for points in the overlapping areas. Those are included in the solution set. Looking at the graph of the given system of inequalities: Graph of the System of Inequalities The shaded area of each inequality represents every point that is a solution to that inequality. Once they're both graphed on the same plane, the area where that shading overlaps is the section of points that are solutions to both inequalities, therefore being solutions to the system. To unlock this lesson you must be a Study.com memberCreate an account Lesson Summary -------------- An inequality represents the relationship between two numbers, whether one side is less than, greater than, or equal to other side. A compound inequality is a statement with two inequalities and there are two types: "and" where both things have to be true and "or" where only one thing can be true. Solutions can be written in inequality notation (using square brackets and parentheses to represent equal to or not equal to; one set for "and" compound inequalities and two sets with a union for "or" compound inequalities) and set notation{x\|x s y m b o l n u m b e r} (where it reads "all values of x such that" and then the rest of the inequality statement). A system of inequalities is a set of two or more inequalities where there is more than one point that would be a solution to the system, meaning it would make both inequalities true. The easiest way to find these points is to graph the system and look at the overlapping area. To unlock this lesson you must be a Study.com memberCreate an account Video Transcript Understanding Inequalities We've learned that an inequality is just an equation that has a greater than or less than symbol instead of an equals sign, and they help us say things like 'I need at least an 80% to get a B on my history test' or 'the amount of money I spend on video games and plane tickets has to be less than $3,000.' The first example was a 1-variable inequality because the only topic being discussed was the one test score, while the second was a 2-variable inequality because there were video games and plane tickets involved. Compound Inequalities But it's sometimes going to be the case that we want to be more specific than these last two examples. Instead of saying 'I need at least an 80% for a B,' it would actually be more accurate to say 'I need between an 80% and a 90% to get a B.' When we change 'greater than 80' into 'between 80 and 90,' we change this single inequality into a compound inequality because it now has two conditions: _x_ ≥ 80 and _x_< 90. In this case, for a number, _x_, to be part of the solution, both conditions must be satisfied. For example, a 92 is greater than or equal to 80, but it is not less than 90. Therefore, it's not a B. This also changes our graph from a single dot with an arrow going forever in one direction, to two dots with a line segment connecting them. We get this picture by graphing one inequality at a time and then seeing where the two graphs overlap. We can start by graphing _x_ ≥ 80. We put a closed circle at 80 and draw an arrow going to the right to indicate that the answer could be any number 80 or larger. Next, we graph _x_< 90 by putting an open circle at 90 and drawing an arrow going to the left to indicate that the answer can be any number smaller than 90. But we then have to remember that the solution is the space where they are both true. This means the only part of the graph we want is where they overlap, and what we end up with is a line connecting our two points. A compound inequality where both conditions must be satisfied is called an 'and' compound inequality and will have a graph with a single interval, like this. But we can also have what are called 'or' compound inequalities. For example, if, instead, we were describing the different ways I could get something other than a B, we would want to say that I could get less than an 80 or greater than or equal to 90. Now, instead of having to satisfy both conditions like before, the answer can satisfy either and still be okay. Getting less than 80 is one way to not get a B, but getting 90 or above is another way. They both work. For example, a 72% is less than 80, so it's not a B. It doesn't matter that it isn't also greater than 90. It only has to satisfy one of the conditions. This means that our graph is a little bit different as well. It will again be two arrows, but this time we won't need them to overlap. We can put an open circle at 80 and draw our arrow to the left showing that all values less than 80 are included. Then, we can put a closed circle at 90 and draw our arrow to the right to show that all values greater than or equal to 90 are included. Again, we're not worried about where these graphs overlap because satisfying either one of these conditions is enough; they don't have to satisfy both to not be a B. A compound inequality where only one condition must be satisfied is called an 'or' compound inequality and will have a graph with two intervals that looks like this: Set Notation and Interval Notation Before we move on, let's talk about two mathematical notations commonly used to describe the solution sets of compound inequalities. The first is called set notation. If we were to write the solution set to our 'and' compound inequality in set notation, it would look something like this: {_x_ \| _x_> 80 and _x_< 90} What this literally says is that the solution set is all values of _x_, such that _x_ is greater than or equal to 80 and less than 90. In case you were wondering, the vertical line symbol, \|, is translated as the phrase 'such that.' You might see a colon used instead of the vertical line sometimes, but they mean the exact same thing. The solution set of an 'or' compound inequality is written in the same way, but with the word 'or' used instead of 'and': {_x_ \| _x_< 80 or _x_> 90} This says that the solution set is all values of _x_, such that _x_ is less than 80 or greater than or equal to 90. Moving on, the second notation we can use is called interval notation. This notation is especially compact and used more often than set notation for expressing solution sets of compound inequalities. For example, the answer to our 'and' compound inequality could be expressed using interval notation like this: [80,90) The first number is the lower bound, and the second is the upper bound. The opening bracket, [, indicates that 80 is included in the solution and is equivalent to saying greater than or equal to. The closing parenthesis on the other end, ), indicates that 90 is not included in the solution and is the same as saying less than. Expressing the solution to 'or' compound inequalities is a little more complicated because there are two intervals in the solution instead of one. If we expressed the solution to our previous example of how not to get a B, it would like this: [0, 80) U [90,100] The symbol, U, in the middle stands for union and means that both sets are part of the solution. The first set says that anything from 0 up to, but not including 80, is not a B, and the last tells us that anything from 90 to 100 is also not a B. Again, the numbers tell us the bounds, and the brackets and parentheses tell us whether the bound is, or is not, part of the solution. System of Inequalities Just like we can have multiple 1-variable inequalities in a single problem, we can have multiple 2-variable inequalities in a single problem as well. When this happens, it's called a system of inequalities. We can graph a system of inequalities by working with one inequality at a time. For example, what if we were asked to find the area of the graph where both of the following inequalities were true? _y_>_x_ + 1 _y_ ≤ -_x_ + 5 We would begin by graphing the first inequality, _y_>_x_ + 1, on our coordinate plane, getting this. We could then graph the second inequality, _y_ ≤ -_x_ + 5 directly on top of it to get this. Since we want to know where both inequalities are true, we're looking for the area where the shading overlaps, leaving our answer as the purple area in this graph. There are times when a system of inequalities will not have a solution. Let's expand our previous example by adding a third inequality: _y_> -2 _x_ + 50. We want to know if there is a set of solutions for which all three inequalities are true. As we can see from the graph, there are several places where the shading for any two of the three inequalities overlaps, but no single area where all three overlap. That means that there is no solution to this set of three inequalities, which is a perfectly valid answer! Lesson Summary To review, compound inequalities are problems with two 1-variable inequalities in them. If an answer must satisfy both inequalities, it is called an 'and' compound inequality, but if satisfying one of the inequalities is enough to be a solution, then it is called an 'or' compound inequality. There are two mathematical notations commonly used to describe the solution sets of compound inequalities: set notation and interval notation. Systems of inequalities are problems with two or more 2-variable inequalities. You can graph a system of inequalities by working with one inequality at a time. Learning Outcomes After you are done with this lesson, you should be able to: Explain the difference between an 'and' compound equality and an 'or' compound inequality Graph a compound inequality Write a compound inequality in set or interval notation Identify the solutions for a system of inequalities Register to view this lesson Are you a student or a teacher? I am a student I am a teacher Unlock your education See for yourself why 30 million people use Study.com Become a Study.com member and start learning now. Become a member Already a member? Log in Go back Resources created by teachers for teachers Over 30,000 video lessons & teaching resources—all in one place. Video lessons Quizzes and worksheets Classroom integration Lesson plans I would definitely recommend Study.com to my colleagues. 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Try it now Math 101: College Algebra 13 chapters 120 lessons 11 flashcard sets Chapter 1 Foundations of Linear Equations Types of Numbers & Its Classifications 6:56 min Parts of a Graph \| Labels & Examples 6:21 min Linear Equations \| Definition, Formula & Solution 7:28 min Forms of a Linear Equation \| Overview, Graphs & Conversion 6:38 min Abstract Algebraic Examples and Going from a Graph to a Rule 10:37 min Undefined & Zero Slope Graph \| Definition & Examples 4:23 min Linear Equation \| Parts, Writing & Examples 8:58 min System of Equations in Algebra \| Overview, Methods & Examples 8:39 min How Do I Use a System of Equations? 9:47 min Practice Problem Set for Foundations of Linear Equations Chapter 2 Matrices and Absolute Value Matrix in Math \| Definition, Properties & Rules 5:39 min Finding the Determinant of a Matrix \| Properties, Rules & Formula 7:02 min Absolute Value \| Explanation & Examples 4:42 min Absolute Value Expression \| Evaluation, Simplification & Examples 5:28 min Solving Absolute Value Functions & Equations \| Rules & Examples 5:26 min Absolute Value \| Overview & Practice Problems 7:09 min Absolute Value \| Graph & Transformations 8:14 min Graphing Absolute Value Functions \| Definition & Translation 6:08 min Practice Problem Set for Matrices and Absolute Values Chapter 3 Inequalities Inequality Signs in Math \| Symbols, Examples & Variation 7:09 min Graphing Inequalities \| Definition, Rules & Examples 7:59 min Viewing now Inequality Notation \| Overview & Examples 8:16 min Up next Graphing Inequalities \| Overview & Examples 12:06 min Watch next lesson Solve & Graph an Absolute Value Inequality \| Formula & Examples 8:02 min Absolute Value Inequalities \| Definition, Calculation & Examples 9:06 min Practice Problem Set for Inequalities Chapter 4 Factoring with FOIL, Graphing Parabolas and Solving Quadratics Parabola \| Definition & Parabolic Shape Equation 4:36 min Types of Parabolas \| Overview, Graphs & Examples 6:15 min Multiplying Binomials Using FOIL and the Area Method 7:26 min Multiplying Binomials \| Overview, Methods & Examples 5:46 min Factoring Quadratic Equations Using Reverse Foil Method 8:50 min Factoring Quadratic Equations \| Solution & Examples 7:35 min How to Complete the Square \| Method & Examples 8:43 min Completing the Square Practice Problems 7:31 min How to Solve a Quadratic Equation by Factoring 7:53 min Quadratic Function \| Formula, Equations & Examples 9:20 min How to Solve Quadratics That Are Not in Standard Form 6:14 min Practice Problem Set for Factoring with FOIL, Graphing Parabolas and Solving Quadratics Chapter 5 Complex Numbers Imaginary Numbers \| Definition, History & Examples 8:40 min How to Add, Subtract and Multiply Complex Numbers 5:59 min Complex Numbers & Conjugates \| Multiplication & Division 6:40 min How to Graph a Complex Number on the Complex Plane 3:28 min How to Solve Quadratics with Complex Numbers as the Solution 5:59 min Practice Problem Set for Complex Numbers Chapter 6 Exponents and Polynomials Properties of Exponents \| Formula & Examples 5:26 min How to Define a Zero and Negative Exponent 3:13 min Simplifying Expressions with Exponents \| Overview & Examples 4:52 min Rational Exponents \| Definition, Calculation & Examples 3:22 min Simplifying Radicals with Variables \| Formula & Examples Simplifying Algebraic Expressions with Rational Exponent 7:41 min Cubic, Quartic & Quintic Equations \| Graphs & Examples 11:14 min Adding, Subtracting & Multiplying Polynomials \| Steps & Examples 6:53 min Polynomial Long Division \| Overview & Examples 8:05 min Synthetic Division of Polynomials \| Method & Examples 6:51 min Factor & Remainder Theorem \| Definition, Formula & Examples 7:00 min Dividing Polynomials with Long and Synthetic Division: Practice Problems 10:11 min Practice Problem Set for Exponents and Polynomials Chapter 7 Functions Functions: Identification, Notation & Practice Problems 9:24 min Transformations: How to Shift Graphs on a Plane 7:12 min Domain & Range of a Function \| Definition, Equation & Examples 8:32 min How to Add, Subtract, Multiply and Divide Functions 6:43 min How to Compose Functions 6:52 min Inverse Functions \| Definition, Methods & Calculation 6:05 min Applying Function Operations Practice Problems 5:17 min Line of Symmetry \| Definition, Graph & Equation 8:07 min Practice Problem Set for Functions Chapter 8 Rational Expressions How to Multiply and Divide Rational Expressions 8:07 min Multiplying and Dividing Rational Expressions: Practice Problems 4:40 min Adding & Subtracting Rational Expressions \| Overview & Examples 8:02 min Practice Adding and Subtracting Rational Expressions 9:12 min Rational Equations \| Definition, Formula & Examples 7:58 min Rational Equations: Practice Problems 13:15 min Practice Problem Set for Rational Expressions Chapter 9 Radical Expressions & Functions Radical Functions \| Graph, Equation & Examples 6:47 min Rationalizing the Denominator \| Overview & Examples 7:01 min Multiplying then Simplifying Radical Expressions 3:57 min How to Divide Radicals, Square Roots & Rational Expressions 7:07 min Solving Radical Equations \| Overview & Examples 6:48 min Solving Radical Equations with Two Radical Terms 6:00 min Transformations of Radical Functions 5:09 min Practice Problem Set for Radical Expressions & Functions Chapter 10 Exponentials and Logarithms Exponential Function \| Definition, Equation & Examples 7:24 min Exponential Growth & Decay \| Formula, Function & Graphs 8:41 min Logarithms \| Overview, Process & Examples 5:23 min Evaluating Logarithms \| Properties & Examples 6:45 min Logarithmic Properties \| Product, Power & Quotient Properties 5:11 min Practice Problems for Logarithmic Properties 6:44 min Exponential Equations \| Definition, Solutions & Examples 6:18 min Solving Logarithmic Equations \| Properties & Examples 6:50 min Practice Problem Set for Exponentials and Logarithms Chapter 11 Probability Mechanics Factorial \| Definition, Examples & Operations 5:24 min Factorial Practice Problems 6:04 min What is the Binomial Theorem? 9:14 min Binomial Theorem Practice Problems 7:08 min Summation \| Definition, Rules & Examples 3:36 min Binomial Theorem \| Coefficient Calculation, Formula & Examples 13:35 min Practice Problem Set for Probability Mechanics Chapter 12 Sequences and Series Sequences in Math \| Overview & Types 5:37 min How to Find and Classify an Arithmetic Sequence 9:09 min Finding and Classifying Geometric Sequences 9:17 min Sigma Summation Notation \| Overview & Examples 6:01 min How to Calculate an Arithmetic Series 5:45 min Geometric Series Formula, Calculation & Examples 9:15 min Arithmetic and Geometric Series: Practice Problems 10:59 min Practice Problem Set for Sequences and Series Chapter 13 Studying for Math 101 Simplifying & Solving Algebra Equations & Expressions: Practice Problems Graphing Practice in Algebra: Practice Problems Linear Equations in College Algebra Flashcards Matrices & Absolute Value Flashcards Inequalities Flashcards Factoring, Parabolas & Quadratics Flashcards Complex Numbers in College Algebra Flashcards Exponents & Polynomials Flashcards Functions in College Algebra Flashcards Rational Expressions Flashcards Exponentials & Logarithms Flashcards Probability Mechanics Flashcards Sequences & Series Flashcards Math 101: College Algebra Formulas & Properties Math 101: College Algebra Equation Tutorial & Help Related Study Materials Inequality Notation \| Overview & Examples LessonsCoursesTopics ##### Translating Math Sentences to Inequalities 5:36 ##### The Order of Real Numbers: Inequalities 4:36 ##### Quadratic Inequality \| Solution Sets & Examples 3:40 ##### Writing Equations with Inequalities: Open Sentences and True/False Statements 4:22 ##### Solving Rational Inequalities \| Steps & Examples 5:27 ##### Interval Notation \| Definition, Rules & Examples 9:08 ##### How to Solve and Graph And & Or Compound Inequalities 7:36 ##### Parent 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Calculus Study Guide and Exam Prep 10th Grade Geometry Textbook Michigan Merit Exam - Math: Test Prep & Practice Saxon Algebra 2 Homeschool: Online Textbook Help FSA - Grade 8 Math: Test Prep & Practice Browse by Lessons Solving Polynomial Inequalities \| Steps & Examples Solving Radical Inequalities Algebra I Assignment - Defining, Graphing & Solving Inequalities Graphing Systems of Linear Inequalities Activities Graphing Linear Inequalities Activities & Games Solving Multi-Step Inequalities with Math Properties Teaching Graphing Inequalities on a Number Line How to Solve Inequalities with Variables on Both Sides Absolute Value Inequalities Activities Create an account to start this course today Used by over 30 million students worldwide Create an account Explore our library of over 88,000 lessons Search Browse Browse by subject College Courses Business English Foreign Language History Humanities Math Science Social Science See All College Courses High School Courses AP Common Core 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7661	https://www.youtube.com/watch?v=hcYov9Y3PzE	The orthocenter of triangle whose vertices are A(a,0,0),B(0,b,0) and C(0,0,c) is (k/a,k/b,k/c) t... Doubtnut 3940000 subscribers 1 likes Description 37 views Posted: 4 Jan 2023 The orthocenter of triangle whose vertices are A(a,0,0),B(0,b,0) and C(0,0,c) is (k/a,k/b,k/c) then k is equal to Class: 12 Subject: MATHS Chapter: EQUATION OF PLANE AND ITS APPLICATIONS -II Board:IIT JEE You can ask any doubt from class 6-12, JEE, NEET, Teaching, SSC, Defense and Banking exam on Doubtnut App or You can Whatsapp us at - 8400400400 Link - Join our courses to improve your performance and Clear your concepts from basic for Class 6-12 School and Competitive exams (JEE/NEET) - Contact Us: 👉 Have Any Query? Ask Us. 🤙 Call: 01247158250 💬 WhatsApp: 8400400400 📧 Email: info@doubtnut.com 🌐 Website: Welcome to Doubtnut. 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syllabus ncert class 12 syllabus of english class 12 syllabus commerce Transcript: foreign okay students first I'm going to draw the diagram here so this is the triangle so this is the a this is a this is B and this is C okay so and so the here axis I am going to tell vertices is 0 comma zero and it is 1 0 comma B comma zero and this one is zero comma zero comma C Focus one so this is the value now I'm going to draw the line here okay so I'm going to draw the line here from here to here and B2 here from C to here so this is the point this is the point they're given so this is the point the intersecting X the coordinates are X Y these are they are taken I'm going to taken so this is the centroid point so this is the central point which is line is intercepting here locations now now go to the handset I'm going to solve this here so the capital x here the capital X capital x of X comma y comma is at like on altitude altitude drawn from a and b c a and BC okay so once the A and B C it is a long therefore therefore the eax ax Vector which is perpendicular to BC Vector okay so it is perpendicular right so if we multiply the value it is zero okay so so ax Vector into BC Vector which is equal to zero now I am going to substitute the value for ax here so the ax value equal to x minus a into I Plus y j plus e z k so all our Vector here all are vector into 0 BC Vector so BC Vector equal to so this is vectors so I saw this is Vector so BC Vector equal to 0 plus b j Vector minus c k vector which is equal to zero so this is the value of ax and BC now I am going to corresponding I value I am going to multiply here so I value and J value and K value I'm going to multiply here therefore the value equal to 0 plus y b minus c z equal to 0 okay so runs so here y b and c is that we will take oxidation therefore therefore y b equal to c e z okay so the y b equal to C here so this is what we will get here now now I am going to find the next I am going to find the equation of altitude drawn from here okay sounds so equation of altitude drawn equational attitude drawn drawn from a point focus on therefore I am going to turn here the X Point equal to a of 1 minus C by c b c by b e to T minus B by C into t pakistones and the y coordinate equal to CT locations and a z Point equal to e z point is equal to equal to BT okay so next I am going to find the equation equation of altitude drawn from point B okay so therefore I am going to turn here the same the X Point equal to a into so X Point equal to C Lambda okay so I'm going because already I taken here the T value so I'm going to take here Lambda so Lambda and y value equal to B into 1 minus a by C Lambda minus C by a Lambda and is at Value equal to Lambda a so this is the value I'm going to take now I'm going to solve this one so corresponding a b sorry corresponding X Y is it I am going to solve here okay so I'm going to solve this so I'm going to solving here so b t equal to Lambda a that means is that value I'm going to equate here because it is small value right so I I will get the easy answer so DT equal to Lambda a so Lambda equal to b t by a so okay so this is what we will get here next I am going to find the T value equal to S Square BC divided by A square B square plus b square C square plus C square a square so we will get here if I substitute if we find the B value of T value okay so also because if you take the LCM here we will get like this okay students next I am going to find the y y equal to CT so the Y value equal to here CT right here y value equal to this value I am going to take this one so equal to B into 1 minus C by yes a by C a by C into T value right f into Lambda value right so the Lambda value equal to BT by a we already found that so I'm going to substitute here so a by C into BT by a minus C by a into b t by a location so this is what we will get here now I am going to say uh simplify this one so so if I am going to simplify this one the value equal to x y e z which is equal to a square plus b sorry now it's not plus it is Multiplied a square b square C Square divided by A square B square plus b square C square plus C square a square into here the value 1 by a 1 by b 1 by C okay so this is what we will get here finally so this value I'm going to take this one as K okay so this is the value I'm going to take us K locations therefore k equal to a square B Square C Square divided by I am going to put here divided by yes yes Square b square plus b square C Square plus C square a square okay students so I am going to cancel this one so if I'm going to cancel this one so we will get if I get separately and cancel this one 1 by B that means a value cancel 1 by B and here 1 by C and here 1 by a that means 1 by a square plus b square C Square only it is remaining okay so I'm going to return here c c a carefully students so if I if I did like this or if I cancel this one we will get whole divided by 1 divided by 1 by a square plus 1 by b square plus 1 by C Square we will get location so if I take here individual value so a square divided by A square B Square cancel so here b square is there here A square B C Square will get here we will get a square so if we get this one so one plus a square one by B Square One by C Square we will get so this is the K value so it is 1 by divided form so I am going to take the numerator so we will it will be k equal to 1 by a square plus 1 by b square plus 1 by C Square whole inverse it will become oxygon this is the answer so if we see in our option option A is the correct answer okay thank you super
7662	https://www.uomustansiriyah.edu.iq/media/lectures/2/2_2023_12_06!05_35_40_PM.pdf	Bacillus: There are four medically important genera of gram positive rods: Bacillus, Clostridium, Corynebacterium and Listeria. The some properties for these genera as in Table Gram-positive rods of medical importance. Genus Anaerobic Growth Spore Formation Exotoxins important in pathogenesis Bacillus -+ + Clostridium + + + Corynebacterium --+ Listeria ---General characters of Bacillus: Rod-shaped, Gram-positive bacteria. Endospore-forming aerobic or facultative anaerobic. The spores are resistant to heat, cold, radiation and disinfectants. Are ubiquitous, in soil, water, and airborne thermophilic (< 75°C) and psychrophilic (>5-8°C), also can flourish at extremes of acidity & alkalinity (pH 2 to 10). Catalase positive (rapidly differentiate from Clostridium). Types of Bacillus: Bacillus anthracis, Bacillus cereus, Bacillus stearothermophilus, Bacillus subtilis, Bacillus larvae. Organism Disease Transmission/ Predisposing factor Action of Toxin Prevention B. anthracis Anthrax - Cutaneous anthrax: spore in soil enter wound. - Pulmonary anthrax: spores are inhaled into lung. - Gastrointestinal anthrax Exotoxin has three components: Protective antigen binds to cells, edema factor is an adenylate cyclase, and lethal factor is a protease that inhibits cell growth. Vaccine contains protective Antigen as the immunogen B. cereus Food poisoning Spores germinate in reheated rice, then bacteria produce exotoxins, which are ingested. Two exotoxins (enterotoxins): 1. Similar to cholera toxin, it increase cyclic AMP . 2. Similar to Staphylococcal enterotoxin, it is a super antigen No vaccine Properties (Important) Bacillus anthracis: Gram-positive large rod with square ends, frequently found in chains. Bacillus anthracis is non motile, whereas other members of the genus are motile. Capsulated composed of (D-glutamate), while other bacterial capsule are polysaccharides. - Anthrax toxin is encoded on one plasmid and the polyglutamate capsule is encoded on a different plasmid. - Spore- forming. Virulence factors of Bacillus anthracis: 1-Capsule: is antiphagocytic and helps the anthracis to evade the immune system of the host. 2-Exotoxin: This toxin consists of three proteins: protective antigen (PA), lethal factor (LF), and edema factor (EF). Pathogenesis of Bacillus anthracis: Pathogenesis is based on the production of three exotoxins, collectively known as anthrax toxin. The three exotoxins known as anthrax toxin. 1- Protective antigen (PA): is a protein that binds to specific cell receptors, and after proteolytic activation, it forms a membrane channel that mediates entry of EF and LF into the cell. 2- Edema factor (EF): is an exotoxin, is an adenylate cyclase causes an outpouring of fluid from the cell into the extracellular space, which manifests as edema. Edema toxin is responsible for cell and tissue edema. 3- Lethal factor (LF): is a protease that activates the mitogen-activated protein kinase (MAPK) signal transduction pathway. This pathway controls the growth of human cells. which is a major virulence factor and cause of death in infected animals and humans. Note: The mitogen-activated protein kinases (MAPKs) regulate diverse cellular programs by relaying extracellular signals to intracellular responses. There are more than a dozen MAPK enzymes that coordinately regulate cell proliferation, differentiation, motility, and survival. These enzymes have diverse biological functions, including regulation of nucleosome and gene expression, mRNA stability and translation, and cell proliferation and survival. Clinical Findings of Bacillus anthracis: Cutaneous anthrax: - is a painless ulcer with black eschar. The lesion is called a malignant pustule. Untreated cases progress to bacteremia and death. Skin lesion of anthrax. Note the black eschar, a necrotic lesion covered by a crust, caused by lethal factor, an exotoxin produced by Bacillus anthracis. Note the area of edema surrounding the eschar, which is caused by another exotoxin called edema factor Pulmonary (inhalation) anthrax: - Also known as woolsorters disease. -Begins with nonspecific respiratory tract symptoms resembling influenza, especially a dry cough and substernal pressure. -Rapidly progress to hemorrhagic mediastinitis, bloody pleural effusions, septic shock and death. - Mediastinal widening seen on chest x-ray is an important diagnostic criterion. Hemorrhagic mediastinitis and hemorrhagic meningitis are sever life-threatening complications. Gastrointestinal anthrax: include: Nausea and vomiting, abdominal pain and bloody diarrhea, Fever and chills, Sore throat, Swelling of neck or neck glands. . Transmission of Bacillus anthracis: Cutaneous anthrax: result from contact human skin with spores or infected animals & animals products. Which accounts for more than 95% of cases worldwide, results from infection through skin lesions. Pulmonary anthrax: result when inhalation of dust containing anthrax spores inhaled. It is usually an occupational disease, such as in persons who handle or sort contaminated wools. Gastrointestinal anthrax: result when spores are ingested. Usually in infected meat; and intestinal anthrax is analogous to cutaneous anthrax but occurs on the intestinal mucosa. Important Note: Gastrointestinal and pulmonary anthrax are both more dangerous than the cutaneous form because they are usually identified too late for treatment to be effective. Bacillus anthracis: The organisms produce non hemolytic gray to white colonies with a rough texture and a ground-glass appearance. Comma-shaped outgrowths (Medusa head, “curled hair”) may project from the colony. Bacillus anthracis: To differentiate Bacillus anthracis from Bacillus cereus: B. cereus is ß-hemolytic on blood agar and motile; B. anthracis is nonhemolytic on blood agar and nonmotile. Treatment: - Ciprofloxacin. - Doxycycline. Prevention of Bacillus anthracis: - Ciprofloxacin and Doxycycline can be used as prophylaxis in the outbreak. - Vaccine is weakly immunogenic. - Incinerating animals that die of anthrax, rather than burying them, will prevent the soil from becoming contaminated with spores. 2- Three different enterotoxins: - First enterotoxins (HBL) haemolysin BL is also a hemolysin. -Second enterotoxin (Nhe) nonhemolytic enterotoxin is not a hemolysin. -Third enterotoxin (EntK) is a single component protein that has not been shown to be involved in food poisoning. All three enterotoxins are cytotoxic and cell membrane active toxins that will make holes or channels in membranes. ENZYMES Tissue destruction is mediated by cytotoxic enzyme including: Cereolysin. Phospholipase C (This enzyme may have a secondary role in ocular infections). Pathogenesis: Bacillus cereus produce two enterotoxins. The mode of action of one of the enterotoxins is the same as that of cholera toxin while the mode of action of the other enterotoxin resembles that of Staphylococcal enterotoxin; it is super antigen. Clinical Findings: That are two syndromes: 1- One syndrome has Short incubation period (4 hrs) and consists primarily of nausea and vomiting, similar to Staphylococcal food poisoning. The emetic form is manifested by nausea, vomiting, abdominal cramps, and occasionally diarrhea and is self-limiting, with recovery occurring within 24 hours. It begins 1–5 hours after ingestion of emetic toxin in the contaminated food products. Clinical Findings: 2- Second syndrome has long incubation period (18 hr) and feature watery, non-bloody diarrhea, resembling clostridial gastroenteritis. Diarrheal toxin: which is associated with meat dishes and sauces. The diarrheal form has an incubation period of 1–24 hours and is manifested by profuse diarrhea with abdominal pain and cramps; fever and vomiting are uncommon. Clinical Findings: In this syndrome, ingested spores that develop into vegetative cells of B. cereus secrete enterotoxin which induce fluid accumulation and other physiological responses in the small intestine. B. cereus is an important cause of eye infections, such as severe keratitis. Typically, the organisms are introduced into the eye by foreign bodies associated with trauma but infections can also occur after surgery. B cereus has also been associated with localized infections, such as wound infections, and with systemic infections, including endocarditis, catheter-associated bacteremia, central nervous system infections, osteomyelitis, and pneumonia; the presence of a medical device or intravenous drug use predisposes to these infections. B cereus cause food poisoning and opportunistic infections in immunocompromised patients. B cereus can be differentiated from B anthracis on the basis of colony morphology, β-hemolysis, motility, and antimicrobial susceptibility patterns. Treatment: B cereus is resistant to a variety of antimicrobial agents, including penicillins and cephalosporins. Serious non–food borne infections should be treated with vancomycin or clindamycin with or without an aminoglycoside. Ciprofloxacin has been useful for the treatment of wound infections -Jawetz, Melnick & Adelberg’s, 2019. Medical Microbiology,28th Edition. The McGraw-Hill education, Inc. USA. - Levinson, W.E., 2018. Review of medical microbiology and immunology. McGraw-Hill Education. Prevention: No specific means of prevention. Rice should not be kept warm for long periods.
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Lakis, N. S. Mellion, M. de la Monte, S. M. on Google Scholar Punsoni, M. Lakis, N. S. Mellion, M. de la Monte, S. M. on PubMed Punsoni, M. Lakis, N. S. Mellion, M. de la Monte, S. M. /ajax/scifeed/subscribe Article Views 7111 Citations 11 Table of Contents Abstract Introduction Report of a Case Review of Cases Discussion Supplementary Materials Author Contributions Funding Institutional Review Board Statement Informed Consent Statement Data Availability Statement Conflicts of Interest References Altmetricshare Shareannouncement Helpformat_quote Citequestion_answer Discuss in SciProfiles Need Help? Support Find support for a specific problem in the support section of our website. Get Support Feedback Please let us know what you think of our products and services. Give Feedback Information Visit our dedicated information section to learn more about MDPI. Get Information clear JSmol Viewer clear first_page Download PDF settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open Access Case Report Post-Polio Syndrome Revisited by Michael Punsoni Michael Punsoni SciProfilesScilitPreprints.orgGoogle Scholar , Nelli S. Lakis Nelli S. Lakis SciProfilesScilitPreprints.orgGoogle Scholar , Michelle Mellion Michelle Mellion SciProfilesScilitPreprints.orgGoogle Scholar and Suzanne M. de la Monte Suzanne M. de la Monte SciProfilesScilitPreprints.orgGoogle Scholar Division of Neuropathology, Departments of Pathology and Laboratory Medicine, Neurology, and Neurosurgery, Rhode Island Hospital, Alpert Medical School of Brown University, Providence, RI 02903, USA Author to whom correspondence should be addressed. Neurol. Int.2023, 15(2), 569-579; Submission received: 16 February 2023 / Revised: 11 March 2023 / Accepted: 27 March 2023 / Published: 13 April 2023 Download keyboard_arrow_down Download PDF Download PDF with Cover Download XML Download Epub Download Supplementary Material Browse Figures Review ReportsVersions Notes Abstract Post-polio syndrome (PPS) is characterized by recrudescence or worsening of motor neuron disease symptoms decades after recovery from acute paralytic poliovirus infection, i.e., poliomyelitis. PPS afflicts between 25% and 40% of poliomyelitis survivors and mimics motor neuron diseases (MNDs), such as amyotrophic lateral sclerosis (ALS), due to its selective impairment, degeneration, or death of motor neurons in the brainstem and spinal cord. Herein, we report a case of PPS in a 68-year-old man with a remote history of bulbar and cervical cord involvement by poliomyelitis, review the relevant literature, and contrast the salient histopathologic features that distinguish our case of PPS from ALS. Keywords: post-polio syndrome; motor neuron disease; amyotrophic lateral sclerosis 1. Introduction Poliomyelitis is a viral infectious disease caused by any one of the three poliovirus strains, although Type 1 is the most frequently identified pathogen . In the United States, the use of poliovirus vaccination drastically reduced the annual incidence rate from over 10,000 cases to about 10, and virtually eradicated wild poliovirus . On a global scale, polio immunizations reduced acute paralytic disease from over 600,000 cases per year prior to the campaign to fewer than 1000 by the year 2000 . However, low endemic rates persist in global pockets due to poor access or compliance with vaccination [2,3,4]. Following acute infection and viremia, in about 1 percent of cases, poliovirus spreads to the central nervous system (CNS). Poliovirus preferentially replicates in and destroys motor neurons of the brainstem, spinal cord, and motor cortex, causing paralytic poliomyelitis [5,6,7]. Accompanying neurological signs and symptoms include asymmetric weakness in various muscles, difficulty swallowing, myalgias, loss of superficial and deep tendon reflexes, and problems with bowel and bladder function . The involvement of brainstem structures is most critical due to inflammation and destruction of neurons that regulate respiration. Supportive care to cover the life-threatening phases of acute primary CNS infection enables recovery, which can be partial or complete. Degrees of recovery vary with age at onset, the extent of CNS involvement, availability of supportive care, and proneness to complications. Younger individuals are more capable of recovery than older individuals due to their inherently greater potential for plasticity and repair [1,2]. Decades after the polio epidemic, it was discovered that between 25% and 40% of people who had recovered either fully or partially from poliomyelitis developed motor neuron disease with features that were indistinguishable from sporadic motor neuron disease (MND) [8,9]. Drawing connections with past histories of poliomyelitis was challenging due to the perception that the later life-presenting symptoms corresponded to a new disease entity if recovery from the childhood illness had been complete, whereas if recovery had been incomplete, the motor dysfunction likely represented an exacerbation of pre-existing poliomyelitis-related pathology. Eventually, the two concepts were fused by consensus and are now termed ‘post-polio syndrome’ (PPS) [10,11,12,13,14]. PPS uniquely differs from sporadic or primary MND due to its natural history corresponding to what is regarded as a clinical recrudescence of a childhood poliovirus CNS infection. The three main clusters of MND, amyotrophic lateral sclerosis (ALS), spinal muscular atrophy, and primary lateral sclerosis, have sporadic and genetic occurrences. Survival durations are substantially abbreviated compared with other forms of neurodegeneration and range from 1 to 5 years [15,16,17,18] compared with 7 to 20 years for Parkinson’s, Alzheimer’s disease (AD), dementia with Lewy bodies, and frontal-temporal lobar degeneration [19,20,21,22,23]. Advances in human postmortem neuropathologic research have demonstrated significant overlap between MND and dementia-associated neurodegenerative diseases , which could be impactful in shortening survival . PPS mimics MND due to its relatively selective impairment, degeneration, or death of motor neurons in the spinal cord and brainstem. The mechanism of PPS is not well understood. Potential etiologies include: (1) re-awakening of underlying pathologies that had been masked by repair and recovery but rendered weakened due to aging-associated neuronal and fiber attrition; and (2) prior poliomyelitis causes epigenetic changes that increase vulnerability to sporadic MND. These concepts could potentially be addressed by comparing the molecular pathologies in PPS and MND, particularly spinal muscular atrophy. 2. Report of a Case A 68-year-old White man was evaluated at the Rhode Island Hospital for weakness in both upper extremities that had progressed slowly over the previous several years. In addition, the patient suffered from chronic respiratory insufficiency, anxiety, asthma, and hypertension. The patient had a remote past medical history of bulbar and cervical cord poliomyelitis at 2 years of age that mostly resolved except for non-progressive difficulty lifting his arms. Worsening motor weakness began over 30 years after stable recovery from poliomyelitis. A neurological examination late in the clinical course documenting muscle atrophy, and electromyography showing denervation, led to the diagnosis of PPS. His two female siblings reported no symptoms corresponding to MND. Molecular studies to exclude specific gene abnormalities linked to ALS, SMA or primary lateral sclerosis were not performed. The case was independently reviewed by a second neurologist who confirmed the likely clinical diagnosis of PPS. Throughout his course, the patient was non-compliant with treatment recommendations. He died from an acute myocardial infarct in the setting of severe coronary atherosclerosis affecting all major vessels, mitral, aortic, and tricuspid valve calcification and stenosis, hypertensive cardiomyopathy, pulmonary hypertension, and congestive heart failure. Consent was granted for a full postmortem examination. Macroscopic examination of the brain fixed in 10% neutral buffered formalin revealed diffuse, symmetrical cerebral edema with impending bi-uncal herniation (brain weight 1465 g), severe calcific, non-occlusive atherosclerosis of the internal carotid artery and all major posterior circulation vessels linked to the circle of Willis, and atrophy of the frontal and temporal lobes. Paraffin-embedded histological sections prepared from 20 standardized regions of brain and spinal cord, were stained with Luxol fast blue-hematoxylin and eosin (LHE). Additional sections from selected blocks were stained by Bielschowsky silver impregnation, or immunostained with antibodies to phospho-tau, neurofilament, ubiquitin, glial fibrillary acidic protein, alpha-synuclein, and CD68 to detect neurodegeneration and tract degeneration. Sections of quadriceps and gastrocnemius muscles were stained with H&E and immunostained with antibodies to slow and fast myosin. For immunohistochemistry, 4-microns-thick paraffin sections were mounted onto Leica Surgipath + coated slides. The entire immunostaining procedure was performed using a DAKO Autostainer Plus automated staining system. The findings related to PPS consisted of striking asymmetric atrophy, neuronal loss, and gliosis in the ventral horns at all levels of the spinal cord, but most prominently in the cervical and thoracic cord (Figure 1A–C). Neuronal inclusions corresponding to Bunina bodies were not detected, including by immunohistochemical staining with antibodies to ubiquitin (Figure 1D) and TAR DNA-binding protein 43 (TDP-43) (not shown). Immunostaining for neurofilament confirmed the loss of spinal anterior horn cells. Correspondingly, ventral and cauda equina nerve roots exhibited up to 50% losses of myelin (Figure 2A) and axons (Figure 2B) by LHE and neurofilament immunohistochemistry (or Bielschowsky staining), respectively. Regions with extensive fiber loss exhibited replacement fibrosis. In contrast, there was no evidence of motor neuron loss in the precentral gyrus or brainstem motor nuclei, and loss of myelinated fibers in corticospinal tracts was modest to nil. The primary motor cortex and pyramidal tracts were intact bilaterally. Sections of quadriceps and gastrocnemius muscles showed modest atrophy but conspicuous myofiber type grouping by immunohistochemical staining of slow and fast myosin (Figure 2C,D). Figure 1. Spinal ventral horn atrophy with neurodegeneration in PPS. Formalin-fixed, paraffin-embedded histological sections of thoracic spinal cord were (A–C) stained with Luxol fast blue, hematoxylin and eosin (LHE; 8 µm thick), or (D) immunostained for ubiquitin (4 µm thick). (A) Thoracic cord level showing absence of lateral and anterior corticospinal tract degeneration (note homogeneous Luxol fast blue staining of myelin). (B) Extensive neuronal loss in the ventral horn shown in Panel A. (C) Ventral horn cells are virtually replaced by glia. (D) Note the absence of ubiquitin-positive inclusions in residual anterior horn cells (examples of negative staining in neurons shown at arrow tips). (C) Scale bar = 30 µm; (D) Scale bar = 40 µm. Figure 2. Ventral nerve root degeneration and skeletal muscle myofiber type grouping in PPS. Spinal ventral nerve roots (A) stained with LHE or (B) immunostained with antibodies to neurofilament show loss of myelinated fibers in A. For example, between arrows, asterisk marks normally myelinated fibers, and in (B), loss of axon is represented by the clear gaps among fibers. (C,D) Histological sections of quadriceps muscle immunostained to detect (C) fast or (D) slow myosin. Note clear grouping according to myofiber type, i.e., fast versus slow. Brown staining precipitates in (B–D) correspond to positive immunostaining results. (C,D) Scale bars = 100 µm. Due to the presence of frontal and temporal lobe atrophy on macroscopic examination together with the age of the patient, sections of anterior frontal, posterior frontal, parietal and occipital lobes including cortex and underlying white matter, and the amygdala were immunostained for phospho-tau, ubiquitin, and amyloid-beta. In addition, sections of anterior frontal, hippocampus, and amygdala were stained by Bielschowsky silver impregnation. Bielschowsky staining demonstrated scattered neurofibrillary tangles in the anterior frontal cortex, and moderate densities of neurofibrillary tangles, neuritic and diffuse plaques, and dystrophic neurites in the entorhinal cortex, CA1–CA2 of the hippocampal formation, and ventromedial amygdala. Phospho-tau and ubiquitin immunohistochemical staining detected those same abnormalities, together with abundant labeling of white matter glial cells. Amyloid-beta immunostaining demonstrated moderate densities of senile plaques and amyloid angiopathy, which were in low abundance in the frontal lobe, but moderate in the parietal, occipital and temporal cortex and in the amygdala. These findings correspond with early-stage AD, Braak Stage III/VI. Additional histopathological findings included, multiple micro-infarcts in the occipital lobes and thalamus, hypoxic-ischemic leukoencephalopathy in the occipital lobes, and cerebral edema. 3. Review of Cases Due to the presence of overlapping early AD-type neurodegeneration, we considered ALS in the differential diagnosis. To help solidify the diagnosis of PPS, we compared the findings in our index case to six ALS cases that were autopsied within the same time period. The ALS cases ranged from 60 to 80 years of age. All were male with White race, and none had a history of poliomyelitis. The cases were donated to the Brown University Brain Bank (re-named Brain Tissue Resource Center at Brown University) for research. Oversight for the human postmortem studies was provided by the Lifespan IRB committee #008303. The approval code was Human Studies Exemption #211037, initially issued on 30 March 2003. The postmortem brains, spinal cords, and skeletal muscle specimens were processed and evaluated using the same standard protocol. In all six cases, the populations of large motor neurons in Brodmann Area 4 (precentral gyrus) were markedly diminished. Loss of neurons was associated with gliosis of deep cortical layers and subcortical white matter, and Wallerian degeneration accompanied by axonal spheroids in the lateral and anterior corticospinal tracts at all levels (Supplementary Figure S1). Atrophy of the spinal anterior horns was associated with neuronal degeneration, neuronal loss, replacement gliosis (Figure 3A–C) and neuronal cytoplasmic inclusion bodies that were readily detected in LHE-stained sections (Figure 3D). Immunohistochemical staining confirmed the inclusion bodies corresponded to TDP-43-positive (Figure 3E) and ubiquitin-positive (Figure 3F) Bunina bodies. All six cases of ALS had variable degrees of overlapping neurodegeneration, e.g., FTLD, PD, or AD. Figure 3. Ventral horn neurodegeneration in ALS. Spinal thoracic ventral horn showing (A) subtotal loss of neurons and (B) extensive gliosis. (C) Many surviving anterior horn cells exhibit swelling and chromatolysis. (D) Bunina body cytoplasmic inclusions. The narrow arrow shows a classical Bunina body. The broad arrow shows a larger, diffusely pale cytoplasmic inclusion. Additional sections from the same block were immunostained with antibodies to (E) ubiquitin and (F) TDP-43, both of which detected discrete cytoplasmic Bunina bodies (dense brown staining). (E) Scale bar = 30 µm; (F) Scale bar = 25 µm. Ventral nerve root degeneration in the ALS cases was associated with loss of myelin and axons with notably greater severities in lumbar compared with thoracic levels (Figure 4). LHE stains highlighted myelin loss, vacuolation, thinning fiber (Figure 4A,B). Neurofilament immunohistochemical staining revealed conspicuously greater degrees of axonal loss, axonal swelling, and fiber variability in lumbar (Figure 4C) compared with thoracic (Figure 4D) nerve roots. Figure 4. Spinal cord (A,C) Lumbar and (B,D) thoracic ventral nerve root degeneration in ALS. Formalin-fixed paraffin-embedded LHE-stained sections in (A,B) show loss of myelinated fibers, e.g., between arrows (asterisk marks normally myelinated fibers). (C,D) Additional sections from the same blocks immunostained with antibodies to neurofilament to show loss of axon (gaps among fibers). Fiber degeneration and loss are greater in lumbar (A,C) compared with thoracic (B,D) cord levels. (A,B) Scale bars = 50 µm; (C,D) Scale bars = 35 µm. In contrast to the PPS case, quadriceps skeletal muscle samples from the ALS cases showed regions of denervation atrophy characterized by the presence of both small (Figure 5A) and large (Figure 5B) grouped atrophy along with myofiber hypertrophy. Immunohistochemical staining for fast (Type 2) (Figure 5C) and slow (Type 1) (Figure 5D) myosin revealed clear myofiber type grouping in less atrophic regions, similar to the findings in PPS (see Figure 2). Figure 5. Skeletal muscle (A,B) denervation atrophy with myofiber hypertrophy and myofiber type grouping in ALS. Histological sections stained with H&E show (A) small (circled) and (B) large group myofiber atrophy with no endomesial fibrosis in different areas of quadriceps muscle. Additional sections from the same blocks immunostained to detect (C) fast (Type 2) or (D) slow (Type 1) myosin show clear myofiber type grouping away from the regions with active denervation as shown in Panels (A,B). (A–D) Scale bars = 80 µm. 4. Discussion The term post-polio syndrome (PPS) was initially coined as an attempt to describe the multitude of neurological symptoms and functional impairments encountered by long-term survivors of poliomyelitis [25,26,27,28,29]. PPS, defined as “the development of new muscle weakness and fatigue in skeletal or bulbar muscles, beginning 15–30 years after an acute attack of paralytic poliomyelitis”, afflicts 25% to 40% of the cases . Although it is widely accepted that PPS is causally linked to an earlier bout of poliomyelitis, the mechanism driving the recrudescence of motor neuron disease symptoms and neurodegeneration remains unknown. One potential consideration is that compensatory responses leading to recovery from poliomyelitis or stabilization of symptoms decompensate over time, particularly with aging. Clinically, the pre-morbid states that characterize patients who develop PPS include: (1) those who experience prolonged periods (up to several decades) of stable normal motor function; and (2) individuals with chronic fatigue and muscle weakness. The former is mainly associated with recovery from childhood poliomyelitis. The latter most likely reflects incomplete recovery from poliomyelitis, corresponding with our index case. In those individuals, PPS is associated with both worsening of baseline symptoms and the emergence of new signs and symptoms. Limbs that were originally affected tend to be involved, although there are exceptions. A diagnosis of PPS in people who experience residual chronic fatigue and muscle weakness can be challenging because the signs and symptoms overlap with chronic fatigue syndrome . The slowness of muscle weakness progression in PPS is partly due to the relatively focal involvement of motor units within multiple muscles. Declines in motor function caused by worsening of weakness or loss of strength in previously uninvolved muscles compromise the ability to carry out normal daily activities. Normal functions can be further impaired by accompanying muscle pain, difficulties with breathing and swallowing, and reduced tolerance to cold [32,33]. The perception that increased physical activity can enhance recovery and strength following poliomyelitis is probably incorrect because instead it can worsen PPS symptoms and the severity of disability. Progressive weakness arising in muscles that were not previously affected by poliomyelitis or associated with persistent chronic fatigue and weakness could reflect a different pathophysiological process causing de novo degeneration of lower motor neurons. To distinguish it from the classical PPS, this entity has been termed “progressive post-poliomyelitis muscular atrophy (PMA)” . The histopathological features of PPS are not well established compared with ALS and other MNDs due to the relatively uncommon nature of disease and overall decline in autopsy rates . From the relatively few examples in the published literature, together with our own index postmortem case, the common targets of idiopathic/genetic MND and PPS include, most spinal anterior horn cells (most common) and brainstem cranial nerve motor nuclei, with attendant denervation myopathy involving affected motor units. In contrast to ALS, PPS seldom affects neurons in the brain. However, there are unusual reports of PPS associated with neurodegeneration in the precentral gyrus, corpus striatum, cerebellar cortex, and basal pontine nuclei . Correspondingly, beyond the classical presentation of limb weakness, involvement of extra-spinal cord structures in the central nervous system impairs motor function in the face, pharynx, and larynx, contributing to difficulties with swallowing and breathing. Motor cortex neurodegeneration can lead to spasticity, and extrapyramidal motor system involvement causes problems with coordination and balance, further compromising independence, mobility, and ability to carry out activities of daily living. The extraordinarily high rate of PPS among survivors of acute poliomyelitis speaks volumes for factors related to the initial infection or mechanisms of recovery as key pathogenic mediators of PPS-associated neurodegeneration. One popular theory known as “neural fatigue” infers that acute poliovirus infections target and kill specific spinal anterior horn cells and brainstem motor nuclei. Functional recovery is mediated by the neuritic sprouting of surviving motor neurons with attendant re-innervation of denervated skeletal muscle. The degrees to which re-innervation is effective may dictate resultant levels of strength. However, re-innervation sprouting leads to the enlargement of motor units, adding metabolic stress to remaining motor neurons. Chronically increased metabolic stress could eventually trigger a second wave of neurodegeneration and disability, corresponding with the emergence of PPS. Furthermore, it is important to bear in mind that within the 15- to 30-year interval between acute poliomyelitis and PPS-associated loss of motor neurons, aging occurs in the patients. Aging itself leads to oxidative and metabolic stress via impairments in insulin signaling and mitochondrial dysfunction [36,37,38]. Therefore, intrinsic neuronal stress related to recovery-induced expansion of the motor unit plus aging, could account for the development of PPS in long-term survivors of poliomyelitis. Similarly, perhaps the increased oxidative stress induced by exercise accounts for the worsening of PPS symptoms with physical activity. At the same time, these concepts suggest that agents to enhance efficiency of metabolic function and reduce oxidative and mitochondrial stress would provide therapeutic remediation for PPS. A final consideration is whether PPS is actually a form of MND, including ALS or progressive muscular atrophy (PMA). This notion is supported by the overlapping symptoms in PPS and ALS or PMA. In particular, PMA, the rapidly progressive form of PPS, mimics the tempo of ALS, and a few cases of PPS have been shown to progress to ALS [39,40]. Furthermore, the neuropathology of spinal and brainstem motor neuron degeneration in PPS is quite like PMA and ALS, except for the absence of Bunina bodies in PPS and their presence in ALS. On the other hand, with extensive neuronal loss, Bunina bodies would not be detected, rendering the definitive diagnosis difficult in the absence of an ample clinical history. The differential diagnosis is further clouded by the finding that like ALS, PPS can also overlap with other neurodegenerative diseases, as demonstrated by the early AD pathology detected in our index case. Therefore, consideration should be given to the potential etiopathogenic relatedness of PPS and sporadic forms of ALS or PMA, perhaps from the perspective of how other agents, exposures, toxins, etc., might mimic poliovirus-mediated damage to CNS motor neurons. Supplementary Materials The following supporting information can be downloaded at: Supplementary Figure S1: Corticospinal tract degeneration in ALS. Formalin fixed, paraffin embedded histological sections of (A,B) cervical or (C,D) thoracic spinal cord stained with Luxol fast blue, hematoxylin and eosin (LHE) revealed various degrees of tract degeneration in the lateral columns (LC) ranging from (A) long established with dense gliosis and myelin pallor, to (B,C) moderate but extensive with clear but less well-delineated regions of myelin pallor and vacuolation, to (D) mild with subtle myelin pallor but prominent vacuolation. Compare regions of myelin pallor (pink) with the denser luxol fast blue staining in the posterior columns. Ventral horns (VH) are atrophic, but in Panel B, the degeneration is quite severe, resulting in a blunted appearance. Author Contributions M.P., data curation, formal analysis of the case, initial writing of the manuscript, case investigations, and methodology; N.S.L., data curation, formal analysis, investigation; M.M., formal analysis, investigation, resources, validation; S.M.d.l.M., data curation, formal analysis, funding, investigation, methodology, supervision, writing, writing and editing, visualization, project administration. All authors have read and agreed to the published version of the manuscript. Funding This research was funded by grants from the National Institutes of Health-National Institute of Alcohol Abuse and Alcoholism: AA-011431 and AA-024018. Institutional Review Board Statement Oversight for the human postmortem studies was provided by the Lifespan IRB committee #008303. The approval code was Human Studies Exemption #211037, initially issued on 30 March 2003. Informed Consent Statement Patient consent was waived due to the patients were all deceased. And we obtained a autopsy authorization which is our standard form that grants consent for the use of decedents’ samples and case information for education, training, research, and publication. Data Availability Statement The case report information and images are available upon request from the corresponding author: Suzanne_DeLaMonte_MD@Brown.edu. Conflicts of Interest All authors declare to have no conflict of interest related to this manuscript. References Nathanson, N.; Martin, J.R. The epidemiology of poliomyelitis: Enigmas surrounding its appearance, epidemicity, and disappearance. Am. J. Epidemiol.1979, 110, 672–692. [Google Scholar] [CrossRef] Nathanson, N.; Kew, O.M. From emergence to eradication: The epidemiology of poliomyelitis deconstructed. Am. J. Epidemiol.2010, 172, 1213–1229. [Google Scholar] [CrossRef] Garon, J.R.; Cochi, S.L.; Orenstein, W.A. The Challenge of Global Poliomyelitis Eradication. Infect. Dis. Clin. N. Am.2015, 29, 651–665. 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Acta Neuropathol.1999, 97, 317–321. [Google Scholar] [CrossRef] Figure 1. Spinal ventral horn atrophy with neurodegeneration in PPS. Formalin-fixed, paraffin-embedded histological sections of thoracic spinal cord were (A–C) stained with Luxol fast blue, hematoxylin and eosin (LHE; 8 µm thick), or (D) immunostained for ubiquitin (4 µm thick). (A) Thoracic cord level showing absence of lateral and anterior corticospinal tract degeneration (note homogeneous Luxol fast blue staining of myelin). (B) Extensive neuronal loss in the ventral horn shown in Panel A. (C) Ventral horn cells are virtually replaced by glia. (D) Note the absence of ubiquitin-positive inclusions in residual anterior horn cells (examples of negative staining in neurons shown at arrow tips). (C) Scale bar = 30 µm; (D) Scale bar = 40 µm. Figure 2. Ventral nerve root degeneration and skeletal muscle myofiber type grouping in PPS. Spinal ventral nerve roots (A) stained with LHE or (B) immunostained with antibodies to neurofilament show loss of myelinated fibers in A. For example, between arrows, asterisk marks normally myelinated fibers, and in (B), loss of axon is represented by the clear gaps among fibers. (C,D) Histological sections of quadriceps muscle immunostained to detect (C) fast or (D) slow myosin. Note clear grouping according to myofiber type, i.e., fast versus slow. Brown staining precipitates in (B–D) correspond to positive immunostaining results. (C,D) Scale bars = 100 µm. Figure 3. Ventral horn neurodegeneration in ALS. Spinal thoracic ventral horn showing (A) subtotal loss of neurons and (B) extensive gliosis. (C) Many surviving anterior horn cells exhibit swelling and chromatolysis. (D) Bunina body cytoplasmic inclusions. The narrow arrow shows a classical Bunina body. The broad arrow shows a larger, diffusely pale cytoplasmic inclusion. Additional sections from the same block were immunostained with antibodies to (E) ubiquitin and (F) TDP-43, both of which detected discrete cytoplasmic Bunina bodies (dense brown staining). (E) Scale bar = 30 µm; (F) Scale bar = 25 µm. Figure 4. Spinal cord (A,C) Lumbar and (B,D) thoracic ventral nerve root degeneration in ALS. Formalin-fixed paraffin-embedded LHE-stained sections in (A,B) show loss of myelinated fibers, e.g., between arrows (asterisk marks normally myelinated fibers). (C,D) Additional sections from the same blocks immunostained with antibodies to neurofilament to show loss of axon (gaps among fibers). Fiber degeneration and loss are greater in lumbar (A,C) compared with thoracic (B,D) cord levels. (A,B) Scale bars = 50 µm; (C,D) Scale bars = 35 µm. Figure 5. Skeletal muscle (A,B) denervation atrophy with myofiber hypertrophy and myofiber type grouping in ALS. Histological sections stained with H&E show (A) small (circled) and (B) large group myofiber atrophy with no endomesial fibrosis in different areas of quadriceps muscle. Additional sections from the same blocks immunostained to detect (C) fast (Type 2) or (D) slow (Type 1) myosin show clear myofiber type grouping away from the regions with active denervation as shown in Panels (A,B). (A–D) Scale bars = 80 µm. 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. © 2023 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 ( Share and Cite MDPI and ACS Style Punsoni, M.; Lakis, N.S.; Mellion, M.; de la Monte, S.M. Post-Polio Syndrome Revisited. Neurol. Int.2023, 15, 569-579. AMA Style Punsoni M, Lakis NS, Mellion M, de la Monte SM. Post-Polio Syndrome Revisited. Neurology International. 2023; 15(2):569-579. Chicago/Turabian Style Punsoni, Michael, Nelli S. Lakis, Michelle Mellion, and Suzanne M. de la Monte. 2023. "Post-Polio Syndrome Revisited" Neurology International 15, no. 2: 569-579. APA Style Punsoni, M., Lakis, N. S., Mellion, M., & de la Monte, S. M. (2023). Post-Polio Syndrome Revisited. Neurology International, 15(2), 569-579. Article Metrics Yes Citations Crossref 8 PubMed 7 PMC 6 Web of Science 7 Scopus 11 Google Scholar [click to view] No Article Access Statistics For more information on the journal statistics, click here. 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7664	https://www.sciencedirect.com/topics/neuroscience/hexamethonium	Hexamethonium - an overview \| ScienceDirect Topics Skip to Main content Journals & Books Hexamethonium In subject area:Neuroscience Hexamethonium is a chemical compound that is used as a prototypical ganglionic blocking drug in neuroscience. It is a bis-quaternary ammonium compound, consisting of two quaternary nitrogen atoms separated by a bridge of 6 methylene groups. It has the chemical formula C12H32N2O2 and is primarily known for its ability to block ganglionic transmission. AI generated definition based on:xPharm: The Comprehensive Pharmacology Reference, 2009 How useful is this definition? Press Enter to select rating, 1 out of 3 stars Press Enter to select rating, 2 out of 3 stars Press Enter to select rating, 3 out of 3 stars About this page Add to MendeleySet alert Also in subject area s: Chemistry Medicine and Dentistry Nursing and Health Professions Show more Discover other topics 1. On this page On this page Definition Chapters and Articles Related Terms Recommended Publications Featured Authors On this page Definition Chapters and Articles Related Terms Recommended Publications Featured Authors Chapters and Articles You might find these chapters and articles relevant to this topic. Chapter The autonomic nervous system 1989, Introduction to NeuropharmacologyPhilip B. Bradley BSc(Hons), PhD, DSc, FIBiol Hexamethonium (Figure 6.5b) This was the first ganglion blocker to be used successfully in the treatment of hypertension. In structure it resembles decamethonium (see Chapter 5) but has a 6-carbon chain separating the two quaternary nitrogen atoms instead of 10 (compare Figures 5.4a and Figure 6.5b). This length of the methylene chain (i.e. six carbon atoms) is optimal for maximum ganglion-blocking activity, whereas decamethonium has the optimal chain length (ten carbon atoms) for maximum activity at the neuromuscular junction. However, decamethonium produces a depolarizing block whereas hexamethonium produces no initial stimulation, i.e. it is a competitive antagonist. The differences in the activity of these two drugs point to important differences in the nature of the receptors for acetylcholine at the two sites, i.e. ganglia and the neuromuscular junction (see above). Hexamethonium is more potent than TEA and has a longer duration of action but, like TEA, it is poorly absorbed from the GI tract. These drugs will also cause a blockade of transmission at parasympathetic ganglia and this can lead to the appearance of unwanted effects, similar to those produced by the muscarinic antagonist, atropine (see Chapter 7), such as a reduction in intestinal motility, urinary retention, reduction in secretions and blurring of vision. View chapterExplore book Read full chapter URL: Book1989, Introduction to NeuropharmacologyPhilip B. Bradley BSc(Hons), PhD, DSc, FIBiol Chapter The Analysis of Experimental and Clinical Data 1980, Neurogenic Heart LesionsE.V. Moreva, I.S. Zavodskaya Hexamethonium was used to block the autonomic ganglia; as peripheral adrenoblocking agents, the presynaptic sympatholytic, guanethidine, and the α-adrenoblocker, sympatholytin, were used. The experiments have shown that during electrical stimulation of the aortic arch the sympatholytics – guanethidine and sympatholytin – and the gangliolytic – hexamethonium, produced a clear protective action on the developing dystrophy of the myocardium. View chapterExplore book Read full chapter URL: Book1980, Neurogenic Heart LesionsE.V. Moreva, I.S. Zavodskaya Chapter Hexamethonium 2018, Reference Module in Biomedical SciencesMaurizio Taglialatela, Elisabetta Panza Name of the clinical form Hexamethonium chloride; hexamethonium bromide; hexamethonium iodide Related names Source: EMTREE Hexamethonium; bistrium; hexanium; hexonium; hexon; benzohexamethonium; esametonio; hexane-1,6-bis(trimethylammonium); hexamethylenebis(trimethyl-ammonium) Chemical names Trimethyl-(6-trimethylammoniohexyl)azanium dihydroxide CAS number 60-26-4 View chapterExplore book Read full chapter URL: Reference work2018, Reference Module in Biomedical SciencesMaurizio Taglialatela, Elisabetta Panza Chapter Hexamethonium 2018, Reference Module in Biomedical SciencesMaurizio Taglialatela, Elisabetta Panza Introduction Hexamethonium, a nicotinic acetylcholine receptor antagonist, is often referred to as the prototypical ganglionic blocker. It is a bis-quaternary ammonium compound, having two quaternary nitrogen atoms separated by a bridge of six methylene groups. The drug was used primarily for the treatment of hypertension but, like the other ganglionic blockers, it has been replaced by more selective drugs, although it is still widely used as a research tool. View chapterExplore book Read full chapter URL: Reference work2018, Reference Module in Biomedical SciencesMaurizio Taglialatela, Elisabetta Panza Chapter Literature Mining and Ontology Mapping Applied to Big Data 2015, Application of Big Data for National SecurityVida Abedi, ... Ramin Zand Case Study I: KD: Lethal Drug Interaction In 2001, an asthma research team at the Johns Hopkins University used the drug hexamethonium on a young healthy volunteer, which ended in the death of the woman as a result of pulmonary inflammation and fibrosis. Hexamethonium was a drug used mainly to treat chronic hypertension and was proposed as a potential drug to treat asthma; however, the non-specificity of its action led to its use being discontinued (Nishida et al., 2012; Toda, 1995). During the course of the asthma study, a healthy volunteer, Ellen Roche, died only a few days after inhaling this drug. She was diagnosed with pulmonary inflammation and fibrosis based on chest imaging and an autopsy report after her death. The autopsy report stated the following facts: “The microscopic examination of the lungs later revealed extensive, diffuse loss of alveolar space with marked fibrosis and fibrin thrombi involving all lobes. There was also evidence of alveolar cell hyperplasia as well as chronic inflammation compatible with an organizing stage of diffuse alveolar damage. There was no evidence of bacteria, fungal organisms, or viral inclusions on routine or special stains” (Internal Investigative Committee Membership, 2001). The principal investigator made a good-faith effort to research the drug’s (hexamethonium’s) adverse effects, mainly by focusing on a limited number of resources, including the PubMed database, and the ethics panel subsequently approved the safety of the drug. This tragedy highlights the importance of a literature search in designing experiments and enrolling healthy individuals in control groups. The volunteer was a young healthy person with no lung or kidney problems. One day after enrolling in the study she developed a dry cough and dyspnea and 2 days after she developed flu-like symptoms. Her forced expiratory volume in the first second was reduced. On May 9, 2001, she became febrile and was admitted to the Johns Hopkins Bayview Medical Center. The chest X-ray revealed streaky densities in the right perihilar region. Arterial oxygen saturation fell to 84% after she walked a short distance. She was in critical condition and 3 days after was referred to the intensive care unit, where she was intubated and ventilated. She experienced bilateral pneumothoraces and presented a clinical picture of adult respiratory distress syndrome. She died on June 2, 2001. However, this accident could have been prevented if the researcher had known of a case report published in 1955 (Robillard et al., 1955) or extracted the association using ARIANA+. Interestingly, a literature (PubMed) search of “Hexamethonium” and “pulmonary fibrosis” returns (as verified in August 2014) four hits, none of them with available abstracts online. One of the publications is in the Russian language, published in 1967 (Malaia et al., 1967). The other three were published 60 to 30 years ago (Brettner et al., 1970; Cockersole and Park, 1956; Stableforth, 1979). Searching individual entries returned 21,167 record for “pulmonary fibrosis” and 7,102 entries for “Hexamethonium.” However, to date there is limited direct evidence of the toxicity of this drug in PubMed. The PDF of the case report published in 1955 can be found in PubMed today; however, many data mining tools including ARIANA+ do not take into account PDFs of very old articles. Nonetheless, ARIANA+ was able to capture this association. The analysis revealed five clear indications among the top 10% of the ranked headings, providing strong evidence for such an association. ARIANA+ was able to extract this information from 50 years of literature, even though the 1955 case report (Robillard et al., 1955) was not in the database. Of 2,545 concepts in the system, ARIANA+ ranked “Scleroderma, Systemic” as the 13th ranked concept, “Neoplasms, Fibrous Tissue” as the 16th, “Pneumonia” as the 38th, “Neoplasms, Connective and Soft Tissue>Neoplasms, Connective Tissue>Neoplasms, Fibrous” as the 174th, and finally, “Pulmonary Fibrosis” as the 257th. ARIANA+ captured this association and could have prevented the volunteer’s death. Show more View chapterExplore book Read full chapter URL: Book2015, Application of Big Data for National SecurityVida Abedi, ... Ramin Zand Chapter The Role of Acetylcholine and its Receptors in Retinal Processing 2010, Encyclopedia of the EyeK.T. Keyser, ... C.E. Strang Hexamethonium and mecamylamine These are classical blockers of autonomic ganglianAChRs. They are used to block central nAChRs and also act through noncompetitive mechanisms, probably by occluding the channel pore. Blockade may be voltage dependent, which is an important caveat in electrophysiology experiments. They are considered nonselective among nAChR subtypes, although hexamethonium is somewhat more effective at receptors containing α3 or α6. Both require ∼10-fold higher concentrations to block α7 nAChRs and are reversible as well; there is no evidence that either interacts with other LGICs. These facts, coupled with relatively low cost, make these drugs experimentally attractive blockers of nAChRs. View chapterExplore book Read full chapter URL: Reference work2010, Encyclopedia of the EyeK.T. Keyser, ... C.E. Strang Chapter Hexamethonium 2018, Reference Module in Biomedical SciencesMaurizio Taglialatela, Elisabetta Panza Therapeutics Historically, hexamethonium was developed in the 1940s (Paton and Zaimis, 1948; Paton and Zaimis, 1949) and was introduced for the treatment of hypertension in the 1950s (Campbell and Robertson 1950; Smirk and Alstad, 1951; Mackey and Shaw, 1951). Reduction in blood pressure results from a blockade of sympathetic ganglia, thus reducing vascular sympathetic tone; this reduction is more pronounced in the erect position, where sympathetic activation via baroreceptor afferents makes a greater contribution. It was used in doses of 250–500 mg three times daily by mouth (Mackey and Shaw, 1951) or in lower doses subcutaneously (Smirk and Alstad, 1951). However, it and other ganglion blocking agents are no longer used clinically for this purpose, because of adverse effects (see “Adverse effects”) and the availability of many newer drugs, but remain widely used as experimental tools. Hexamethonium's poor penetration of the blood brain barrier makes it useful for distinguishing between central and peripheral neuronal nAChRs (e.g., Andreasen et al., 2009). View chapterExplore book Read full chapter URL: Reference work2018, Reference Module in Biomedical SciencesMaurizio Taglialatela, Elisabetta Panza Chapter Hexamethonium 2018, Reference Module in Biomedical SciencesMaurizio Taglialatela, Elisabetta Panza Basic Chemistry \| Chemical structure \| \| Structure \| Reproduced from PubChem \| \| Chemical Formula \| C 12 H 32 N 2 O 2 \| \| Properties \| \| Physical properties \| Hexamethonium chloride is a hygroscopic crystal \| \| Molecular weight \| 236.395 \| \| Solubility \| Hexamethonium chloride is freely soluble in water, soluble in 95% ethanol, practically insoluble in chloroform and ether. Hexamethonium bromide is soluble in water and alcohol. \| View chapterExplore book Read full chapter URL: Reference work2018, Reference Module in Biomedical SciencesMaurizio Taglialatela, Elisabetta Panza Chapter Methods for Producing Experimental Neurogenic Lesions 1980, Neurogenic Heart LesionsE.V. Moreva Experiments with hexamethonium. The experiments involving intramuscular injection of the ganglioblocker hexamethonium were performed on two groups of rats. The first group (5 animals) was given several injections of hexamethonium at a dose of 10 mg/kg, 30–40 minutes before electrical stimulation of the aortic arch. Four of the five rats showed development of focal lesions of the myocardium. Since the ganglion-blocking action of hexamethonium is transient and lasts for about 2 hours, the conditions for the second group, consisting of 10 rats, were changed. This group was given two injections of hexamethonium: the first 10–15 minutes before the beginning of electrical stimulation; the second, one hour after the first injection. This maintained a high concentration of hexamethonium in the blood for the duration of stimulation (3 hours). Eight of the ten rats did not show any myocardial changes. Seven rats subjected to electrical stimulation without any previous injection of hexamethonium served as controls for both groups of animals. Myocardial lesions were observed in all cases. When hexamethonium was given without stimulation, myocardial changes were not observed. View chapterExplore book Read full chapter URL: Book1980, Neurogenic Heart LesionsE.V. Moreva Chapter Hexamethonium 2018, Reference Module in Biomedical SciencesMaurizio Taglialatela, Elisabetta Panza Pharmacokinetics Following intravenous administration of tritiated hexamethonium in the mouse, the highest concentration of radioactivity was found in the kidneys and urinary tract, this being consistent with its excretion in the urine (Wassermann, 1971), where it is probably excreted unchanged; it does not appear to be taken up by liver cells. Only low concentrations of radioactivity were detected in the central nervous system due to poor penetration through the blood brain barrier. The concentrations in the liver and lungs were no different from those in the blood but hexamethonium appeared to be concentrated in cartilage and collagenous connective tissue (Wassermann, 1971). View chapterExplore book Read full chapter URL: Reference work2018, Reference Module in Biomedical SciencesMaurizio Taglialatela, Elisabetta Panza Related terms: Orexin Cholinergic Mecamylamine Rapid Eye Movement Sleep Ganglionic Blocker Acetylcholinesterase Acetylcholine Nicotinic Receptor Motilin Ganglion View all Topics Recommended publications Neuroscience LettersJournal European Journal of PharmacologyJournal Autonomic NeuroscienceJournal Journal of the Autonomic Nervous SystemJournal Browse books and journals Featured Authors Engel, Jörgen A.Sahlgrenska Akademin, Gothenburg, Sweden Citations13,239 h-index64 Publications95 Söderpalm, BoSahlgrenska Universitetssjukhuset, Gothenburg, Sweden Citations6,500 h-index45 Publications67 Ericson, MiaSahlgrenska Akademin, Gothenburg, Sweden Citations3,035 h-index30 Publications31 Takatori, ShingoMatsuyama University, Matsuyama, Japan Citations875 h-index17 Publications21 About ScienceDirect Remote access Advertise Contact and support Terms and conditions Privacy policy Cookies are used by this site. 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7665	https://pmc.ncbi.nlm.nih.gov/articles/PMC4439389/	A rapid paper-based test for quantifying sickle hemoglobin in blood samples from patients with sickle cell disease - 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 Am J Hematol . Author manuscript; available in PMC: 2016 Jun 1. Published in final edited form as: Am J Hematol. 2015 Mar 30;90(6):478–482. doi: 10.1002/ajh.23980 Search in PMC Search in PubMed View in NLM Catalog Add to search A rapid paper-based test for quantifying sickle hemoglobin in blood samples from patients with sickle cell disease Nathaniel Z Piety Nathaniel Z Piety 1 Department of Biomedical Engineering, Cullen College of Engineering, University of Houston, Houston, United States Find articles by Nathaniel Z Piety 1, Xiaoxi Yang Xiaoxi Yang 1 Department of Biomedical Engineering, Cullen College of Engineering, University of Houston, Houston, United States Find articles by Xiaoxi Yang 1, Dalia Lezzar Dalia Lezzar 1 Department of Biomedical Engineering, Cullen College of Engineering, University of Houston, Houston, United States Find articles by Dalia Lezzar 1, Alex George Alex George 2 Department of Pediatrics, Section of Hematology-Oncology, Baylor College of Medicine, Houston, United States Find articles by Alex George 2, Sergey S Shevkoplyas Sergey S Shevkoplyas 1 Department of Biomedical Engineering, Cullen College of Engineering, University of Houston, Houston, United States Find articles by Sergey S Shevkoplyas 1 Author information Article notes Copyright and License information 1 Department of Biomedical Engineering, Cullen College of Engineering, University of Houston, Houston, United States 2 Department of Pediatrics, Section of Hematology-Oncology, Baylor College of Medicine, Houston, United States ✉ Corresponding author: Sergey S. Shevkoplyas, Ph.D., University of Houston, Department of Biomedical Engineering, 3605 Cullen Blvd, Houston, TX 77204-5060; phone: +1 (713) 743-5696; fax: +1 (713) 743-0226; sshevkoplyas@uh.edu Issue date 2015 Jun. PMC Copyright notice PMCID: PMC4439389 NIHMSID: NIHMS671313 PMID: 25689370 The publisher's version of this article is available at Am J Hematol Abstract Quantification of sickle hemoglobin (HbS) in patients with sickle cell disease (SCD) undergoing hydroxyurea or chronic transfusion therapy is essential to monitoring the effectiveness of these therapies. The clinical monitoring of %HbS using conventional laboratory methods is limited by high per-test costs and long turnaround times usually associated with these methods. Here we demonstrate a simple, rapid, inexpensive paper-based assay capable of quantifying %HbS in blood samples from patients with SCD. A 20 μL droplet of whole blood and hemoglobin solubility buffer was deposited on chromatography paper. The relative color intensities of regions of the resulting blood stain, determined by automated image analysis, are used to estimate %HbS. We compared the paper-based assay with hemoglobin electrophoresis (comparison method) using blood samples from 88 subjects. The test shows high correlation (R 2 = 0.86) and strong agreement (standard deviation of difference = 7 %HbS) with conventional Hb electrophoresis measurement of %HbS, and closely approximates clinically predicted change in %HbS with transfusion therapy (mean difference 2.6 %HbS, n = 4). The paper-based assay can be completed in less than 35 minutes and has a per-test cost less than $0.25. The assay is accurate across a wide range of HbS levels (10–97%) and hemoglobin concentrations (5.6–12.9 g/dL) and is unaffected by high levels of HbF (up to 80.6%). This study demonstrates the feasibility of the paper-based %HbS assay. The paper-based test could improve clinical care for SCD, particularly in resource-limited settings, by enabling more rapid and less expensive %HbS monitoring. Keywords: sickle, hemoglobin, paper, quantification, paper-fluidic Introduction Sickle cell disease (SCD) is a common recessively inherited blood disorder caused by a point mutation of the β-globin gene.(1) Unlike normal adult hemoglobin (HbA), sickle hemoglobin (HbS) polymerizes and becomes insoluble under hypoxic conditions, causing a number of structural and functional abnormalities in affected red blood cells (RBCs). Sickled RBCs have increased fragility and rigidity that render them prone to early breakdown and occlusion of blood vessels. Patients with SCD experience chronic hemolytic anemia, episodic pain crises and abnormal blood flow to critical organs that cumulatively result in significant illness and shortened lifespans.(2) Although, the severity of the disease varies greatly among patients, for each individual patient the rate of adverse events is strongly correlated with the intraerythrocytic concentration of HbS.(1) The two main therapeutic options for SCD, hydroxyurea and chronic transfusion, both rely primarily on a reduction of the relative HbS level for their beneficial effect. Hydroxyurea induces the expression of hemoglobin F (HbF) in erythrocytes, reducing the intraerythrocytic HbS concentration and making the erythrocyte less likely to sickle in hypoxic conditions. Chronic transfusion therapy does not affect intraerythrocytic HbS concentration but reduces the overall proportion of sickle erythrocytes and partially corrects the anemia of SCD, improving organ perfusion and oxygenation.(3) The main goal of chronic RBC transfusion for SCD patients is to maintain a low level of HbS (<30%) in order to decrease the rate of adverse clinical outcomes, in particular the risk of primary or secondary stroke.(4–6) Generally, the transfusion of 2–3 RBC units or 10–15 mL/kg every 3–5 weeks is sufficient to keep the %HbS below 30% and the [Hb] at 9–10 g/dL and minimize the risk of progressive complications.(7) Both hydroxyurea and chronic transfusion therapy require frequent monitoring of the relative HbS concentration to assess effectiveness of therapy. The monitoring of %HbS in patients on these therapies using conventional laboratory methods (e.g. Hb electrophoresis(8) and HPLC(9)) is impeded by the high cost and long processing time usually associated with these methods.(8) Typically, results are not available rapidly enough to guide clinical decision-making at the time of the clinic visit. Additionally, since HbS levels are not routinely measured during or after transfusions, relatively little is known about the evolution of HbS levels in SCD patients on chronic transfusion therapy. A rapid, inexpensive assay for measuring HbS concentration in patient blood samples could be useful in monitoring the effectiveness of hydroxyurea and transfusion therapy by enabling more frequent testing than currently possible with conventional laboratory methods, thereby providing additional information that could inform clinical decision making. Our research group has recently developed a low-cost paper-based test for diagnosing SCD in resource-limited settings.(10) This test utilizes the separation of HbS and non-sickle hemoglobin by differential wicking in a paper matrix to qualitatively diagnose SCD. We postulated that a similar test could be capable of quantifying the sickle component of total hemoglobin in patient blood samples. In this study, we demonstrate that this paper-based test can be used to rapidly and accurately quantify the HbS content in SCD patients on chronic transfusion and hydroxyurea therapy, as well as in infants with a relatively high endogenous level of Hb F. Methods Blood samples Venous blood samples were collected from patients with SCD, after informed consent following an IRB-approved protocol, during clinic visits (January 2013 – September 2014) at the Texas Children’s Hematology Center (Houston, TX). Healthy, consenting volunteers donated normal venous blood samples. Blood samples were collected in Vacutainer tubes (K 2 EDTA, BD, Franklin Lakes, NJ), stored between 4–25°C and analyzed within 21 days after collection (mean storage time before analysis = 6.4 ± 3.6 days). Sample storage time up to 21 days was only weakly correlated with assay performance (R 2 = 0.09, p < 0.01). Hemoglobin Electrophoresis Hemoglobin A, F, C and S for each sample were measured using conventional Hb electrophoresis performed on the semi-automated Sebia Hydrasys 2 Scan system (Sebia Inc., Norcross, GA). All materials were prepared and run according to manufacturer specifications. Phoresis curve editing software (Sebia Inc., Norcross, GA) was used to calculate hemoglobin composition. For post-transfusion samples, the expected %HbS was calculated using the following method: Post-transfusion %HbS = Total HbS (g) x 100 / (Total post-transfusion Hb (g)); Total HbS = Pre-transfusion [Hb] x pre-transfusion %HbS x Total blood volume (dl); Total post-transfusion Hb = (Pre-transfusion [Hb] x total blood volume (dl)) + (Total transfused volume (dl) x [Hb]Transfused blood). Hemoglobin solubility buffer The Hb solubility buffer was a mixture of a hemolytic agent (saponin), a reducing agent (sodium hydrosulfite) and a high-phosphate buffer. The high-phosphate buffer (2.49M) consisted of 1.24M (169 g/L) monobasic and 1.25M (217 g/L) dibasic potassium phosphate dissolved in deionized water. The saponin (4 g/L) permeabilizes red cell membranes, releasing Hb into the solution. The sodium hydrosulfite (30 g/L) converts the released Hb into deoxy-Hb. Converted HbA, HbE, HbF and HbC remain soluble in the high-phosphate buffer, while deoxy-HbS polymerizes and precipitates.(11, 12) Quantification of the blood stain color intensities The blood stains on patterned chromatography paper were digitized by scanning the paper with a portable flatbed scanner (CanoScan LiDE110, Canon USA Inc, Lake Success, NY). The digitized blood stains were analyzed with a custom image analysis algorithm (MATLAB, The Math Works Inc, Natick, MA). The algorithm automatically isolates features of the blood stain via contrast stretching and thresholding, then calculates the mean color intensity of each feature using RGB color data from the Blue (B) channel of each pixel within the feature (color intensity = 255 – B).(13) The relationship between the quotient of relative mean color intensities of the standardized blood stains and HbS content in patient samples was determined using a series of reconstituted blood samples with %HbS artificially adjusted from 0 – 75%. These calibration samples were created by mixing ABORh-matched, equal-hematocrit normal (HbAA) and SCD (HbSS) blood samples at various ratios to achieve the desired HbS concentrations. %HbS of artificially adjusted samples was verified by Hb electrophoresis. Figure 2b shows this calibration curve relating the quotient of mean color intensities to %HbS, an exponential curve was fit to the data, where estimated %HbS = 0.332 ln(0.639 I C/I R). This calibration curve was used throughout the rest of this study to convert I C/I R to estimated %HbS. Figure 2. Open in a new tab Using a blood stain to estimate %HbS. (a) Schematic illustration of image analysis algorithm. (a-i) Centroid of the blood stain (marked by a cross) is automatically identified by the algorithm using contrast stretching and thresholding. (a-ii) The center spot (area inside dashed circle) and (a-iii) the peripheral ring (area between dashed circles) are extracted from the image through application of binary masks centered on the centroid of the stain. Mean color intensities of the center spot (I C) and the peripheral ring (I R) are extracted from the standardized blood stain. (b) Dependence of I C/I R on %HbS for a series of reconstituted blood (n = 5) samples with artificially adjusted %HbS varying from 0 to 75% (calibration curve, solid line). The values of the quotient of mean color intensities increased exponentially with increasing %HbS. Results The design and operation of the paper-based assay for SCD has been previously described in detail.(10) Briefly, to perform the assay 20 μL of whole blood was added to 200 μL of solubility buffer, mixed by inversion, and allowed to sit for 10 minutes, to allow RBC lysis and hemoglobin deoxygenation. At the end of this period, 20 μL of the lysate was pipetted onto chromatography paper and allowed to wick laterally and dry for 25 minutes (Figure 1a). The entire assay was performed indoors, in ambient conditions (temperature 15°C to 25°C, 20% to 70% humidity, fluorescent lighting). The difference in transport of soluble and insoluble variants of hemoglobin (Hb) through the paper substrate produces a blood stain consisting of two parts: the area of the initial drop where polymerized deoxy-HbS is retained (center spot), and the area where all soluble forms of Hb are wicked laterally (peripheral ring). The color intensity of the center spot relative to that of the peripheral ring is indicative of the HbS level in the blood sample. Figure 1b shows representative blood stains for samples artificially reconstituted with %HbS varying from 0 to 75%. As expected, increasing HbS content results in increased color intensity of the center spot and decreased intensity of the peripheral ring. The absolute color intensity of the stain is dependent on the Hb concentration of the sample, but the relative color intensities of the center spot and peripheral ring are independent of Hb concentration. The resulting blood stain is digitized using a portable flatbed scanner (in reflected light) and analyzed with a custom image analysis algorithm. The entire assay can be performed for up to 20 samples in less than 35 minutes. Figure 1. Open in a new tab Schematic illustration of the paper-based HbS assay and the characteristic blood stains. (a) To perform the assay a 20 μL droplet of blood mixed with Hb solubility buffer (phosphate buffer containing saponin and sodium hydrosulfite) is deposited on chromatography paper, a blood stain is allowed to form (polymerized deoxy-HbS is trapped in the paper mesh and soluble forms of Hb are wicked laterally), and the stain is scanned and analyzed automatically to estimate %HbS in the sample. (b) Characteristic blood stains produced on paper by samples with various %HbS. Scale bar is 1 cm. Figure 2 describes the process of estimating %HbS based on the relative color intensity of the center spot and the peripheral ring areas of the blood stain. First, the custom image analysis algorithm applies a standard contrast stretching and thresholding routine to the digitized image of the blood stain to determine the center of the stain (Figure 2a-i). A standardized binary mask(14) is then used to isolate the appropriate parts of the blood stain image to calculate, I C, the mean color intensity of the center spot (Figure 2a-ii) and, I R, the mean color intensity of the peripheral ring (Figure 2a-iii). For each sample, the quotient of these two values (I C/I R) is then converted to estimated %HbS using the equation (estimated %HbS = 0.332 ln(0.639 I C/I R)) derived from the calibration curve (Figure 2b). To test the accuracy of our paper-based quantification method, we then compared the value of %HbS in individual patient samples measured by conventional Hb electrophoresis (Sebia Hydrasys 2 Scan, Sebia Inc., Norcross, GA) and by the paper-based assay. Of the 88 samples used in this analysis, 55 were from patients on hydroxyurea therapy, 25 were from patients on chronic transfusion therapy, and 8 were from infants. The ages of the patients ranged from one month to 18.8 years, hemoglobin concentrations ranged from 5.6 to 12.9 g/dL, and %HbS ranged from 10 to 97%. Five patients had the SC genotype, 3 patients had the Sβ 0 genotype, and 2 patients had the Sβ+ genotype. As shown in Figure 3a, the values of %HbS estimated using the paper-based assay (Estimated %HbS) and measured using conventional hemoglobin electrophoresis (Actual %HbS) were highly correlated (R 2 = 0.86). When broken down by sample type, the correlation between %HbS measured by electrophoresis and that measured by the paper-based assay remained high, although the correlation was weaker for patients undergoing hydroxyurea therapy (R 2 hydroxyurea = 0.61, R 2 transfusion = 0.87, R 2 infant = 0.83), indicating that the accuracy of the paper-based assay was not affected by significant variations in %HbS and %HbA, and %HbF or by the relative distribution of these hemoglobin complexes within erythrocytes. The lower correlation for patients undergoing hydroxyurea therapy resulted from decreased accuracy of the test at very high %HbS levels (the samples from patients undergoing hydroxyurea therapy tended to have the highest HbS concentrations) rather than from inherent differences between patient groups. Figure 3. Open in a new tab Comparison of paper-based %HbS measurements with a reference method (hemoglobin electrophoresis). (a) %HbS for a group of patient samples (n = 88) was measured using the paper-based assay (Estimated %HbS) and conventional Hb electrophoresis (Actual %HbS). Estimated %HbS is highly correlated (R 2 = 0.86) with actual %HbS (solid line = data trend line; dashed line = perfect correlation). Patient characteristics are indicated by dot color. (b) Bland–Altman plot shows strong agreement (SD difference = 7 %HbS) between estimated and actual %HbS (solid line = mean difference; dashed lines = ± 2 SD difference). The majority of the differences between actual and estimated %HbS (95.5%) are within 2 standard deviations of the mean of the differences. Patient characteristics are indicated by dot color. Figure 3b shows a Bland-Altman plot(15) of the %HbS values obtained using both methods. The values of %HbS for patient samples estimated using the paper-based assay and measured using conventional hemoglobin electrophoresis show strong agreement. The standard deviation of the differences between different %HbS values obtained using both methods was 7 %HbS. The majority (95.5%) of the differences between actual and estimated %HbS are within 2 standard deviations of the mean of the differences. The limits of agreement between the paper-based HbS assay and conventional hemoglobin electrophoresis were −11.5 %HbS and 17.0 %HbS. The paper-based HbS assay agreed with the hemoglobin electrophoresis system within 5 %HbS 58.0% of the time, overestimating %HbS by >5 %HbS in 10.2% of subjects and underestimating %HbS by >5 %HbS in 31.8% of subjects. We tested the repeatability of the paper-based method for quantifying HbS by repeatedly measuring (n = 5) the %HbS for a series of blood samples with HbS content of approximately 10, 20, 30, 40, 50, 60, 70 and 80 %HbS. The standard deviation for the %HbS measurements performed with different droplets of the same blood sample was consistently < 1.5 %HbS for all values of %HbS measured: 0.9 %HbS (CV 6.3%) for the sample with 10 %HbS; 1.0 %HbS (CV 4.8%) for 20 %HbS; 1.3 %HbS (CV 4.1%) for 30 %HbS; 1.1 %HbS (CV 2.6%) for 40 %HbS; 0.7 %HbS (CV 1.3%) for 50 %HbS; 1.3 %HbS (CV 2.1%) for 60 %HbS; 1.1 %HbS (CV 1.6%) for 70 %HbS; and 1.5 %HbS (CV 1.9%) for 80 %HbS. For comparison, the standard deviations for the %HbS measurements performed by the manufacturer of the Sebia Hydrasys 2 Scan system for blood samples with %HbS ranging from 8.8 – 83.6 %HbS was 0.2 – 0.6 %HbS (CV 0.7 – 2.1%). Finally, to assess the utility of the paper-based assay in measuring the change in %HbS with transfusion therapy, we compared the change in %HbS in five patients based on pre- and post-transfusion measurements made using Hb electrophoresis to the same measurements made using the paper-based assay and to an estimate of post-transfusion %HbS made using a formula currently used to guide clinical transfusion practices (based on patient weight, Hb concentration and initial %HbS). As presented in Table 1, the change in %HbS with transfusion measured by Hb electrophoresis was very close to both that estimated by the clinical formula (mean difference 3.3 %HbS) and that measured with the rapid assay (mean difference 3.9 %HbS) for all five patients, indicating that our paper-based assay is capable of accurately determining changes in %HbS following transfusion therapy. Table 1. %HbS information of patients (n = 5) receiving transfusion therapy. Paired blood samples were drawn from each patient before (Initial) and after (Final) transfusion. Initial and final %HbS was measured by Hb electrophoresis and by the paper-based assay. A clinical estimation of final %HbS (based on patient weight, Hb concentration and initial %HbS measured by Hb electrophoresis), used to guide transfusion practices, is shown for comparison. The mean difference of change in %HbS measured by Hb electrophoresis and by the paper-based assay is 3.9 %HbS. The mean difference of change in %HbS measured by Hb electrophoresis and with the clinical estimation is 3.3 %HbS. The clinical estimation overestimated the change in %HbS with respect to Hb electrophoresis in 1 of 5 patients. The decrease in %HbS measured by the paper-based assay was smaller than the decrease in %HbS measured by Hb electrophoresis in 5 of 5 patients. \| \| Hb electrophoresis \| Clinical estimation \| Paper-based assay \| :---: :---: \| \| Patient \| Initial %HbS \| Final %HbS \| Change in %HbS \| Initial %HbS \| Final %HbS \| Change in %HbS \| Initial %HbS \| Final %HbS \| Change in %HbS \| \| 1 \| 41.2 \| 27.8 \| −13.4 \| 41.2 \| 32.1 \| −9.1 \| 34.3 \| 22.8 \| −11.5 \| \| 2 \| 30.5 \| 20.2 \| −10.3 \| 30.5 \| 22.4 \| −8.1 \| 26.3 \| 20.4 \| −5.9 \| \| 3 \| 43.5 \| 30.4 \| −13.1 \| 43.5 \| 34.4 \| −9.1 \| 30.4 \| 24.7 \| −5.7 \| \| 4 \| 33.7 \| 23.2 \| −10.5 \| 33.7 \| 21 \| −12.7 \| 32.3 \| 22.1 \| −10.2 \| \| 5 \| 33.3 \| 19.7 \| −13.6 \| 33.3 \| 23.4 \| −9.9 \| 21.8 \| 13.6 \| −8.2 \| Open in a new tab Discussion In this report, we present a novel paper-based method of quantifying HbS levels in blood samples from patients with sickle cell disease. Our method is based on the previously described qualitative diagnostic test for sickle cell disease that relies on the differential wicking of insoluble HbS and soluble non-sickle hemoglobins on a paper substrate to produce characteristic blood stain patterns. We now extend this method to permit the quantification of sickle hemoglobin using the patterns produced by this diagnostic test. This quantification is effected by digitization of the rapid test paper containing the blood stain and measurement of the relative color intensity of the center spot, composed of trapped, precipitated sickle hemoglobin, and the peripheral ring, composed of free, soluble hemoglobins. As indicated by the data presented above, quantification of %HbS by this method is highly accurate and reproducible relative to conventional hemoglobin electrophoresis. This method is also accurate across a wide range of HbS levels and hemoglobin concentrations and is unaffected by high levels of endogenous HbF or exogenous HbA. As such, it is potentially applicable to monitor any therapeutic modality that alters the relative level of HbS, including acute or chronic transfusion therapy and hydroxyurea therapy. Our paper-based assay offers several advantages over conventional methods for the quantification of HbS. The first and most prominent is the low cost of our assay. The cost of consumable materials and reagents for our test is less than $0.25 per sample, and the initial investment for a computer and scanner for image digitization is less than $400. By contrast, the cost of required supplies for the conventional hemoglobin electrophoresis system used as a reference standard in our laboratory is $1,800 annually for system solutions and controls, plus an additional $5.77 to $12.07 per sample for consumable materials and reagents, and the initial investment for the system is more than $20,000. The estimated cost per sample of both IEF and HPLC at our collaborating center is approximately $60. Our rapid assay thus offers significant cost savings compared to other methods of HbS quantitation. A second advantage of our paper-based assay is the ease of use of the system. Processing of blood samples to develop the blood stain is a simple operation requiring only two steps (mixing and sample deposition). Subsequent digitization and quantification of the blood stain itself is also straightforward and is largely accomplished by the automated image analysis algorithm. The overall process of blood stain development, image scanning, and HbS quantification can therefore be performed with minimal technical training by any operator with basic laboratory skills. A final advantage of the paper-based assay is the rapidity of the assay. Conventional laboratory methods of hemoglobin quantification can be completed rapidly, but due to the limited availability of clinical technicians and costs associated with sample processing, samples are often grouped to be run in large batches, delaying the availability of results. In contrast, our paper-based assay does not require batching and has an extremely low per-test cost, even for single samples. A sample obtained from a patient can be processed, digitized, and quantified within less than 35 minutes, permitting rapid clinical assessment of a patient’s HbS burden and the adjustment of therapeutic interventions. This capability could be especially useful for patients on chronic transfusion therapy since, unlike conventional measures of %HbS, the paper-based test could determine the current %HbS level in time to assist the clinician in making decisions about current transfusion volume and subsequent transfusion frequency to maintain a target HbS level. The paper-based assay for HbS quantification offers obvious advantages for resource-limited settings. The low cost of the assay and its technical simplicity are major advantages in environments in which clinical resources, a reliable stream of specialized supplies for complex medical testing, and training for clinical and laboratory personnel are deficient. This assay is highly portable and can be set up quickly in any clinical setting, including those in which the supply of electricity may be uncertain. A major barrier to the implementation of transfusion and hydroxyurea therapies in resource-limited settings is the inability to monitor their efficacy in a timely manner that could influence patient care. The use of this assay could therefore make chronic transfusion therapy targeting a specific %HbS level, such as for primary or secondary stroke prophylaxis, more feasible in such settings. It can also be used in conjunction with other simple measurements (e.g. hemoglobin concentration and patient clinical status) to indirectly monitor the efficacy of HbF induction with hydroxyurea therapy since the reduction of %HbS in patients on hydroxyurea correlates with the degree of HbF induction. There are a few technical limitations to the paper-based assay. The most prominent of these is that conditions resulting in clotting of blood samples can affect the quantification of HbS in the sample. In our validation cohort of 88 patient samples, the presence of clots in the samples resulted in greater deviation of the value for %HbS determined by the paper-based assay from that measured by hemoglobin electrophoresis. Secondly, the paper-based assay is not as accurate as more technically complex methods of HbS quantitation, especially at very high HbS levels, though this inaccuracy is relatively minor. Thirdly, very high levels of HbF (such as those found in newborns) may interfere with the polymerization of HbS and therefore decrease the accuracy of the assay. Finally, the assay in its current form cannot distinguish between non-pathological soluble hemoglobins, such as HbA, HbA 2, and HbF, and those that can cause SCD in association with HbS, such as HbC or HbD. As such, the utility of this assay is currently limited to patients with homozygous sickle cell disease and Sβ 0 thalassemia. Of note, however, patients with homozygous sickle cell disease and Sβ 0 thalassemia constitute the majority of all sickle cell patients (>70% in the United States and over 95% in many sub-Saharan African countries(16)) and are also the two most severe forms of the disorder with the most pressing need for clinical intervention. In summary, we have developed a new, rapid, inexpensive paper-based assay for the quantification of %HbS in blood samples from patients with sickle cell disease. The assay determines %HbS across a wide range of values with a high degree of accuracy relative to hemoglobin electrophoresis and is unaffected by variations in overall hemoglobin concentration or the presence of HbA or HbF in the patient sample. Our assay has obvious applications in resource-limited settings and could be an important step towards making chronic transfusion therapy and hydroxyurea therapy feasible in such settings. The assay could also be useful in resource-rich settings by permitting more rapid and less expensive monitoring of response to these therapies, thus improving clinical care and reducing costs associated with the management of sickle cell disease. Acknowledgments The authors would like to thank Ms. Norma Estrada and Dr. Bogdan Dinu for coordinating sample collection, and Dr. Tariq Elghetany for critical review of the manuscript. This work was supported in part by a 2012 NIH Director’s Transformative Research Award (NHLBI R01HL117329, PI: Shevkoplyas). SSS, NZP and XY are inventors on a utility PCT application “Paper based diagnostic test” (PCT/US2012/064856, 11/13/2012) claiming priority benefit of US 61/692,994 (8/24/2012) and US 61/558,009 (11/10/ 2011). SSS is a part-owner of Halcyon Biomedical Incorporated, a company which may benefit from commercialization of the paper-based SCD test. All data and described custom-written scripts are available from the authors upon reasonable request. Footnotes All other authors report no potential conflicts of interest. References 1.Stuart MJ, Nagel RL. Sickle-cell disease. Lancet. 2004;364:1343–60. doi: 10.1016/S0140-6736(04)17192-4. [DOI] [PubMed] [Google Scholar] 2.Rees DC, Williams TN, Gladwin MT. Sickle-cell disease. Lancet. 2010;376:2018–31. doi: 10.1016/S0140-6736(10)61029-X. [DOI] [PubMed] [Google Scholar] 3.Simon TL, Snyder EL, Stowell CP, Strauss RG, Solheim BG, Petrides M. Rossi’s principles of transfusion medicine. Wiley; 2011. [Google Scholar] 4.Josephson CD, Su LL, Hillyer KL, Hillyer CD. Transfusion in the patient with sickle cell disease: A critical review of the literature and transfusion guidelines. Transfusion medicine reviews. 2007;21:118–33. doi: 10.1016/j.tmrv.2006.11.003. [DOI] [PubMed] [Google Scholar] 5.Roback JD, Banks AAoB. Technical manual. American Association of Blood Banks; 2008. 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[DOI] [PMC free article] [PubMed] [Google Scholar] ACTIONS View on publisher site PDF (652.8 KB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract Introduction Methods Results Discussion Acknowledgments Footnotes 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
7666	https://www.mathnasium.com/math-centers/mullicahill/news/what-pemdas-mh	Math Centers / Mullica Hill / News / What is PEMDAS? Back To School, Education, General What is PEMDAS? Sep 30, 2022 \| Mullica Hill Mathematics is all around us, and the right mathematical rules can help you solve problems. The PEMDAS rule is one such rule in the mathematical domain. Math basics such as addition, division, subtraction, and multiplication are already familiar to you, but what about the right order to address each of them, were they all in the same equation? Did you know there was an order to solve them? PEMDAS is widely used in applications in mathematics, and computer programming and we use it to solve a wide range of problems. What is PEMDAS? PEMDAS is an acronym used to mention the order of operations to be followed while solving expressions having multiple operations. PEMDAS stands for P- Parentheses, E- Exponents, M- Multiplication, D- Division, A- Addition, and S- Subtraction. There are different acronyms used for the order of operations in different countries. For example, in Canada, the order of operations is stated as BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, and Subtraction). Some people prefer to say BODMAS (B- Brackets, O- Order or Off), while few others call it GEMDAS (G- Grouping). John's Method 5+2×3 = 7×3 = 21 Bob's Method 5+2×3 =5+6 =11 As you can observe, John and Bob reached different answers. There can only be one correct answer to this expression in mathematics! Can you decide who is correct? Don’t worry! PEMDAS is here to help you find the correct answer. PEMDAS Rules These rules start with Parentheses, and then operations are performed on the exponents or powers. Next, we perform operations on multiplication or division from left to right. Finally, operations on addition or subtraction are performed from left to right. You will always get the correct answer if you follow the PEMDAS rule. Let us understand PEMDAS with the help of an example. 4+3[8-2(6-3)]÷2 We will begin with working from the inside of the brackets. We will solve the innermost bracket first and then move outside. Starting with 6 – 3 = 3, we get: 4 + 3[8 – 2(3)] ÷ 2 Next, multiplying 2(3)=6 or 2×3=6, we get: 4 + 3[8 – 6] ÷ 2 There is one bracket left, [8 – 6] = 2, we get: 4 + 3 ÷ 2 Solving 3 or 3 × 2 = 6, we have: 4 + 6 ÷ 2 We can observe that all the expressions in the brackets are solved. Based on PEMDAS, we know that division comes next, hence, 6 ÷ 2 = 3, that is, 4 + 3. And lastly, addition 4 + 3 = 7. Use this rule to solve math questions and soar in your math class. Till then, keep practicing! Related Articles Math Lessons From Ants May 24, 2023 \| Mullica Hill Ants teach us the necessity to plan and look ahead. They store their food during seasons of plenty so that they will have adequate amounts in times of scarcity, all through an intricate system of carefully coordinated calculations to make sure colonies survive. They even plan and calculate time for both work and times for rest to prevent burnout. Effective Ways to Improve Study Habits for Your Child May 16, 2023 \| Mullica Hill Encouraging your children to develop good study habits early is one of the most important things you can do as a parent. Help your child develop a lifelong love of learning by setting up a study space without distractions, getting organized, and helping them to feel positive about their schoolwork. The Math Behind Film Making May 9, 2023 \| Mullica Hill Math is essential to every aspect of filmmaking. It's involved in every step of the moviemaking process, from the design of the video equipment filmmakers use to the production of the film itself. 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7667	https://par.nsf.gov/servlets/purl/10225066	Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. D ISCRETE MATH .c\bigcirc 2019 Society for Industrial and Applied Mathematics Vol. 33, No. 4, pp. 2416–2443 THE TYPICAL STRUCTURE OF GALLAI COLORINGS AND THEIR EXTREMAL GRAPHS \ast J ´OZSEF BALOGH \dagger AND LINA LI \ddagger Abstract. An edge coloring of a graph G is a Gallai coloring if it contains no rainbow triangle. We show that the number of Gallai r-colorings of Kn is ( \bigl( r 2 \bigr) + o(1))2 \Bigl( n 2 \Bigr) . This result indicates that almost all Gallai r-colorings of Kn use only 2 colors. We also study the extremal behavior of Gallai r-colorings among all n-vertex graphs. We prove that the complete graph Kn admits the largest number of Gallai 3-colorings among all n-vertex graphs when n is sufficiently large, while for r ≥ 4, it is the complete bipartite graph K\lfloor n/ 2\rfloor ,\lceil n/ 2\rceil . Our main approach is based on the hypergraph container method, developed independently by Balogh, Morris, and Samotij as well as by Saxton and Thomason, together with some stability results. Key words. Gallai coloring, container method, extremal graph AMS subject classification. 05D99 DOI. 10.1137/19M1253344 Introduction. An edge coloring of a graph G is a Gallai coloring if it contains no rainbow triangle, that is, no triangle is colored with three distinct colors. The term Gallai coloring was firSst introduced by Gy´ arf´ as and Simonyi , but this concept had already occurred in an important result of Gallai on comparability graphs, which can be reformulated in terms of Gallai colorings. It also turns out that Gallai colorings are relevant to generalizations of the perfect graph theorem , and some applications in information theory . There are a variety of papers which consider structural and Ramsey-type problems on Gallai colorings; see, e.g., [14, 16, 17, 18, 25]. Two important themes in extremal combinatorics are to enumerate discrete struc-tures that have certain properties and describe their typical properties. In this paper, we shall be concerned with Gallai colorings from such an extremal perspective. 1.1. Gallai colorings of complete graphs. For an integer r \geq 3, an r-coloring is an edge coloring that uses at most r colors. By choosing two of the r colors and coloring the edges of Kn arbitrarily with these two colors, one can easily obtain that the number of Gallai r-colorings of Kn is at least (1) \biggl( r 2 \biggr) \Bigl( 2(n 2 ) 2 \Bigr) r = \biggl( r 2 \biggr) 2(n 2 ) r(r 2) . If we further consider all Gallai r-colorings of Kn using exactly 3 colors, red, green, \ast Received by the editors March 29, 2019; accepted for publication (in revised form) August 23, 2019; published electronically December 5, 2019. Funding: Research of the first author is partially supported by NSF grants DMS-1500121 and DMS-1764123, and by the Arnold O. Beckman Research Award (UIUC) Campus Research Board 18132 and the Langan Scholar Fund (UIUC). \dagger Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, IL, and Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprodny, Moscow Region, 141701, Russian Federation (jobal@math.uiuc.edu). \ddagger Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 (linali2@illinois.edu). 2416 Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2417 and blue, in which the red color is used only once, the number of them is exactly \biggl( n 2 \biggr) \Bigl( 2(n 2 ) (n 1) 2 \Bigr) . Combining with (1), for n sufficiently large, a trivial lower bound for the number of Gallai r-colorings of Kn is (2) \biggl( \biggl( r 2 \biggr) 2 n \biggr) 2(n 2 ). Motivated by a question of Erd¨ os and Rothchild and the resolution by Alon et al. , Benevides, Hoppen, and Sampaio studied the general problem of counting the number of edge colorings of a graph that avoid a subgraph colored with a given pattern. In particular, they proved that the number of Gallai 3-colorings of Kn is at most 3 2 (n 1)! \cdot 2(n 12 ). At the same time, Falgas-Ravry, O’Connell, and Uzzell provided a weaker upper bound of the form 2 (1+ o(1)) (n 2 ), which is a consequence of the multicolor container theory. Very recently, Bastos et al. improved the upper bound to 7( n+1)2 (n 2 ). Note that the gap between the best upper bound and the trivial lower bound is a linear factor. We show that the lower bound is indeed closer to the truth, and this actually applies for any integer r. Our first main result is as follows. Theorem 1.1. For every integer r \geq 3, there exists n0 such that for all n > n 0,the number of Gallai r-colorings of the complete graph Kn is at most \biggl( \biggl( r 2 \biggr) 2 n 4 log 2n \biggr) 2(n 2 ). Given a class of graphs \scrA , we denote by \scrA n the set of graphs in \scrA of order n.We say that almost all graphs in \scrA have property \scrB if lim n\rightarrow \infty \| { G \in \scrA n : G has property \scrB } \| \| \scrA n\| = 1 . Recall that the number of Gallai r-colorings with at most 2 colors is \bigl( r 2 \bigr) 2(n 2 ) r(r 2). Then the description of the typical structure of Gallai r-colorings immediately follows from Theorem 1.1. Corollary 1.2. For every integer r \geq 3, almost all Gallai r-colorings of the complete graph are 2-colorings. 1.2. The extremal graphs of Gallai colorings. There have been considerable advances in edge coloring problems whose origin can be traced back to a question of Erd¨ os and Rothchild , who asked which n-vertex graph admits the largest number of r-colorings avoiding a copy of F with a prescribed colored pattern, where r is a positive integer and F is a fixed graph. In particular, the study for the extremal graph of Gallai colorings, that is the case when F is a triangle with rainbow pattern, has received attention recently. A graph G on n vertices is Gallai r-extremal if the number of Gallai r-colorings of G is largest over all graphs on n vertices. For r \geq 5, the Gallai r-extremal graph has been determined by Hoppen, Lefmann, and Odermann [19, 20, 21]. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2418 J ´OZSEF BALOGH AND LINA LI Theorem 1.3 (see ). For all r \geq 10 and n \geq 5, the only Gallai r-extremal graph of order n is the complete bipartite graph K\lfloor n/ 2\rfloor ,\lceil n/ 2\rceil . Theorem 1.4 (see ). For all r \geq 5, there exists n0 such that for all n > n 0,the only Gallai r-extremal graph of order n is the complete bipartite graph K\lfloor n/ 2\rfloor ,\lceil n/ 2\rceil . For the cases r \in { 3, 4} , several approximate results were given. Theorem 1.5 (see ). There exists n0 such that the following hold for all n > n 0. (i) For all \delta > 0, if G is a graph of order n, then the number of Gallai 3-colorings of G is at most 2(1+ \delta )n2/2. (ii) For all \xi > 0, if G is a graph of order n, and e(G) \leq (1 \xi )\bigl( n 2 \bigr) , then the number of Gallai 3-colorings of G is at most 2(n 2 ). We remark that part (i) of Theorem 1.5 was also proved in , and the authors further provided an upper bound for r = 4. Theorem 1.6 (see ). There exists n0 such that the following hold for all n > n0. For all \delta > 0, if G is a graph of order n, then the number of Gallai 4-colorings of G is at most 4(1+ \delta )n2/4. The above theorems show that for r \in { 3, 4} , the complete graph Kn is not far from being Gallai r-extremal, while for r = 4, the complete bipartite graph K\lfloor n/ 2\rfloor ,\lceil n/ 2\rceil is also close to be Gallai r-extremal. Benevides, Hoppen, and Sampaio made the following conjecture. Conjecture 1.7 (see ). The only Gallai 3-extremal graph of order n is the complete graph Kn. For the case r = 4, Hoppen, Lefmann, and Odermann believed that K\lfloor n/ 2\rfloor ,\lceil n/ 2\rceil should be the extremal graph. Conjecture 1.8 (see ). The only Gallai 4-extremal graph of order n is the complete bipartite graph K\lfloor n/ 2\rfloor ,\lceil n/ 2\rceil . Using a similar technique to Theorem 1.1, we prove an analogous result for dense noncomplete graphs when r = 3. Theorem 1.9. For 0 < \xi \leq 1 64 , there exists n0 such that for all n > n 0 the following holds. If G is a graph of order n, and e(G) \geq (1 \xi )\bigl( n 2 \bigr) , then the number of Gallai 3-colorings of G is at most 3 \cdot 2e(G) + 2 n 4 log 2n 2(n 2 ). Together with Theorem 1.5 and the lower bound (2), Theorem 1.9 solves Conjec-ture 1.7 for sufficiently large n. Theorem 1.10. There exists n0 such that for all n > n 0, among all graphs of order n, the complete graph Kn is the unique Gallai 3-extremal graph. Our third contribution is the following theorem. Theorem 1.11. For n, r \in N with r \geq 4, there exists n0 such that for all n > n 0 the following holds. If G is a graph of order n, and e(G) > \lfloor n2/4\rfloor , then the number of Gallai r-colorings of G is less than r\lfloor n2/4\rfloor . We remark that for a graph G with e(G) = \lfloor n2/4\rfloor , which is not K\lfloor n/ 2\rfloor ,\lceil n/ 2\rceil , G contains at least one triangle. Therefore, the number of Gallai r-colorings of G is Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2419 at most r(r + 2( r 1)) re(G) 3 < r \lfloor n2/4\rfloor . As a direct consequence of Theorem 1.11 and the above remark, we reprove Theorem 1.4 and, in particular, we show that Conjecture 1.8 is true for sufficiently large n. Theorem 1.12. There exists n0 such that for all n > n 0, among all graphs of order n, the complete bipartite graph K\lfloor n/ 2\rfloor ,\lceil n/ 2\rceil is the unique Gallai 4-extremal graph. 1.3. Overview of the paper. Combining Szemer´ edi’s regularity lemma and the stability method was used in many earlier works on extremal problems, including Erd¨ os–Rothchild-type problems; see, e.g., [1, 2, 9, 20]. However, our main approach relies on the method of hypergragh containers, developed independently by Balogh, Morris, and Samotij as well as by Saxton and Thomason , and some stability results for containers, which may be of independent interest to readers. The paper is organized as follows. First, in section 2, we introduce some important definitions and then state a container theorem which is applicable to colorings. In section 3, we present a key enumeration result on the number of colorings with special restrictions, which will be used repeatedly in the rest of the paper. Then in section 4, we study the stability behavior of the containers for the complete graph, and apply the multicolor container theorem to give an asymptotic upper bound for the number of Gallai r-colorings of the complete graph. In section 5, we deal with the Gallai 3-colorings of dense noncomplete graphs; the idea is the same as in section 4 except that we need to provide a new stability result which is applicable to noncomplete graphs. In the second half of the paper, that is, in section 6, we study the Gallai r-colorings of noncomplete graphs for r \geq 4. When the underlying graph is very dense, that is, close to complete graph, we apply the same strategy as in section 4 for the case r = 4, where we prove a proper stability result for containers. The case r \geq 5 is even simpler, in which we actually prove that the number of Gallai colorings in each container is small enough. When the underlying graph has edge density close to 1 4 , i.e., the edge density of the extremal graph, some new ideas are needed, and we also adopt a result of Bollob´ as and Nikiforov on book graphs. For the rest of the graphs whose edge densities are between 1 4 +o(1) and 1 2 o(1), we use a supersaturation result of triangle-free graphs given by Balogh et al. , and the above results on Gallai r-colorings for both high density graphs and low density graphs. For a positive integer n, we write [ n] = { 1, 2, . . . , n } . For a graph G and a set A \subseteq V (G), the induced subgraph G[A] is the subgraph of G whose vertex set is A and whose edge set consists of all of the edges with both endpoints in A. For two disjoint subsets A, B \subseteq V (G), the induced bipartite subgraph G[A, B ] is the subgraph of G whose vertex set is A \cup B and whose edge set consists of all of the edges with one endpoint in A and the other endpoint in B. Denote by \delta (G) the minimum degree of G, and by ∆( G) the maximum degree of G. For a graph G and a vertex v \in V (G), let NG(v) be the neighborhood of v, i.e., the set of vertices adjacent to v in G, and dG(v) = \| NG(v)\| be the degree of v. For a set A \subseteq V (G), the neighborhood of v restricted to A is the set NG(v, A ) = NG(v) \cap A; the degree of v restricted to A,denoted by dG(v, A ), is the size of NG(v, A ). When the underlying graph is clear, we simply write N (v), d(v), N (v, A ), and d(v, A ) instead. Throughout the paper, we omit all floor and ceiling signs whenever these are not crucial. Unless explicitly stated, all n-vertex graphs are assumed to be defined on the vertex set [ n], and all logarithms have base 2. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2420 J ´OZSEF BALOGH AND LINA LI Preliminaries. 2.1. The hypergraph container theorem. We use the following version of the hypergraph container theorem (Theorem 3.1 in ). Let \scrH be a k-uniform hypergraph with average degree d. The codegree of a set of vertices S \subseteq V (\scrH ) is the number of edges containing S, that is, d(S) = { e \in E(\scrH ) \| S \subseteq e} . For every integer 2 \leq j \leq k, the jth maximum codegree of \scrH is ∆j (\scrH ) = max { d(S) \| S \subseteq V (\scrH ), \| S\| = j} . When the underlying hypergraph is clear, we simply write it as ∆ j . For 0 < \tau < 1, the co-degree function ∆( \scrH , \tau ) is defined as ∆( \scrH , \tau ) = 2 (k 2 ) 1 k \sum j=2 2 (j 12 ) ∆j d\tau j 1 . In particular, when k = 3, ∆( \scrH , \tau ) = 4∆ 2 d\tau + 2∆ 3 d\tau 2 . Theorem 2.1 (see ). Let \scrH be a k-uniform hypergraph on vertex set [N ]. Let 0 < \varepsilon , \tau < 1/2. Suppose that \tau < 1/(200 k!2k) and ∆( \scrH , \tau ) \leq \varepsilon / (12 k!) . Then there exists c = c(k) \leq 1000 k!3k and a collection of vertex subsets \scrC such that (i) every independent set in \scrH is a subset of some A \in \scrC ; (ii) for every A \in \scrC , e(\scrH [A]) \leq \varepsilon \cdot e(\scrH ); (iii) log \| \scrC \| \leq cN \tau log(1 /\varepsilon ) log(1 /\tau ). 2.2. Definitions and multicolor container theorem. A key tool in applying container theory to multicolored structures will be the notion of a template . This notion of template, which was first introduced in , goes back to under the name of “2-colored multigraphs” and later to , where it is simply called “containers.” For more studies about the multicolor container theory, we refer the interested reader to [4, 5, 7, 13, 24]. Definition 2.2 (template and palette). An r-template of order n is a function P : E(Kn) \rightarrow 2[r], associating with each edge e of Kn a list of colors P (e) \subseteq [r]; we refer to this set P (e) as the palette available at e. Definition 2.3 (subtemplate). Let P1, P2 be two r-templates of order n. We say that P1 is a subtemplate of P2 (written as P1 \subseteq P2) if P1(e) \subseteq P2(e) for every edge e \in E(Kn). We observe that for G \subseteq Kn, an r-coloring of G can be considered as an r-template of order n, with only one color allowed at each edge of G and no color allowed at each nonedge. For an r-template P , write RT( P ) for the number of subtemplates of P that are rainbow triangles. We say that P is rainbow-triangle-free if RT( P ) = 0. Using the container method, Theorem 2.1, we obtain the following. Theorem 2.4. For every r \geq 3, there exists a constant c = c(r) and a collection \scrC of r-templates of order n such that Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2421 (i) every rainbow-triangle-free r-template of order n is a subtemplate of some P \in \scrC ; (ii) for every P \in \scrC , RT( P ) \leq n 1/3\bigl( n 3 \bigr) ; (iii) \| \scrC \| \leq 2cn 1/3 log 2 n(n 2 ).Proof. Let \scrH be a 3-uniform hypergraph with vertex set E(Kn) \times { 1, 2, . . . , r } ,whose edges are all triples { (e1, d 1), (e2, d 2), (e3, d 3)} such that e1, e 2, e 3 form a triangle in Kn and d1, d 2, d 3 are all different. In other words, every hyperedge in \scrH corresponds to a rainbow triangle of Kn. Note that there are exactly r(r 1)( r 2) ways to rainbow color a triangle with r colors. Hence, the average degree d of \scrH is equal to d = 3e(\scrH ) v(\scrH ) = 3r(r 1)( r 2) \bigl( n 3 \bigr) r\bigl( n 2 \bigr) = ( r 1)( r 2)( n 2) . For the application of Theorem 2.1, let \varepsilon = n 1 3 /r (r 1)( r 2) and \tau = \surd 72 \cdot 3! \cdot rn 1 3 .Observe that ∆ 2(\scrH ) = r 2, and ∆ 3(\scrH ) = 1. For n sufficiently large, we have \tau \leq 1/(200 \cdot 3! 2 \cdot 3) and ∆( \scrH , \tau ) = 4( r 2) d\tau + 2 d\tau 2 \leq 3 d\tau 2 \leq \varepsilon 12 \cdot 3! . Hence, there is a collection \scrC of vertex subsets satisfying properties (i)–(iii) of Theo-rem 2.1. Observe that every vertex subset of \scrH corresponds to an r-template of order n; every rainbow-triangle-free r-template of order n corresponds to an independent set in \scrH . Therefore, \scrC is a desired collection of r-templates. Definition 2.5 (Gallai r-template). For a graph G of order n, an r-template P of order n is a Gallai r-template of G if it satisfies the following properties: (i) for every e \in E(G), \| P (e)\| \geq 1; (ii) RT( P ) \leq n 1/3\bigl( n 3 \bigr) . For a graph G of order n and a collection \scrP of r-templates of order n, denote by Ga( \scrP , G ) the set of Gallai r-colorings of G which is a subtemplate of some P \in \scrP . If \scrP consists of a single template P , then we simply write it as Ga( P, G ). 2.3. A technical lemma. In this section, we provide a lemma that will be useful to us in what follows. We use a special case of the weak Kruskal–Katona theorem due to Lov` asz . Theorem 2.6 (Lov` asz ). Suppose G is a graph with \bigl( x 2 \bigr) edges for some real number x \geq 2. Then the number of triangles of G is at most \bigl( x 3 \bigr) , with equality if and only if x is an integer and G = Kx. Lemma 2.7. Let n, r \in N with r \geq 3 and 4 n 4 n2 \leq \varepsilon < 1 2 . If G is an r-colored graph of order n, which contains at least (1 \varepsilon )\bigl( n 3 \bigr) monochromatic triangles, then there exists a color c such that the number of edges colored by c is at least e(G) 4r2\varepsilon \bigl( n 2 \bigr) .Proof. We shall prove this lemma by contradiction. Let \delta = 4 r2\varepsilon . Assume that none of the colors is used on at least e(G) \delta \bigl( n 2 \bigr) edges. First, we conclude that e(G) \geq (1 \varepsilon )\bigl( n 2 \bigr) . If not, then by Theorem 2.6, the number of triangles of G is less than \surd 2 3 (1 \varepsilon )3/2 \biggl( n 2 \biggr) 3/2 \leq (1 \varepsilon ) \biggl( n 3 \biggr) , which contradicts the assumption. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2422 J ´OZSEF BALOGH AND LINA LI a bcred blue (a) a b c dred blue (b) Fig. 1 .Two cases of a red-blue pair of edges. By the pigeonhole principle, we can assume without loss of generality that the set of red edges in G, denoted by R( G), satisfies \| R( G)\| \geq (1 \varepsilon )\bigl( n 2 \bigr) /r. By the contradiction assumption, we have \| R( G)\| < e (G) \delta \bigl( n 2 \bigr) . Therefore, the number of nonred edges is greater than \delta \bigl( n 2 \bigr) . Again, without loss of generality, we can assume that the set of blue edges in G, denoted by B( G), satisfies \| B( G)\| \geq \delta \bigl( n 2 \bigr) /r. For an edge in R( G) and an edge in B( G), these two edges either share one endpoint or are vertex disjoint; see Figure 1. In the first case (see Figure 1(a)) the triple abc could not form a monochromatic triangle of G. In the latter case (see Figure 1(b)) at least one of abc and bcd is not a monochromatic triangle of G.Let NT( G) be the family of triples { a, b, c } which does not form a monochromatic triangle of G. The above discussion shows that each pair of red and blue edges generates at least one triple in NT( G). Observe that each triple in NT( G) can be counted in at most 2 + 3( n 3) pairs of red and blue edges. Hence, we obtain that \| NT( G)\| \geq (1 \varepsilon )\bigl( n 2 \bigr) /r \cdot \delta \bigl( n 2 \bigr) /r 2 + 3( n 3) > \delta 4r2 \biggl( n 3 \biggr) = \varepsilon \biggl( n 3 \biggr) , which contradicts the assumption of the lemma. Counting Gallai colorings in r-templates. In this section, we aim to prove the following technical theorem, which will be used repeatedly in the rest of the paper. Theorem 3.1. Let n, r \in N with r \geq 3, and G be a graph of order n. Suppose that \delta = log 11 n and k is a positive constant, which does not depend on n. For two colors i, j \in [r], denote by \scrF = \scrF (i, j ) the set of r-templates of order n which contain at least (1 k\delta )\bigl( n 2 \bigr) edges with palette { i, j } . Then, for n sufficiently large, \| Ga( \scrF , G )\| \leq 2e(G) + 2 n 3 log 2n 2(n 2 ). Fix two colors 1 \leq i < j \leq r, and let S = [ r] { i, j } . For an r-coloring F of G, let S(F ) be the set of edges in G, which are colored by colors in S. From the definition of \scrF , we immediately obtain the following proposition. Proposition 3.2. For every F \in Ga( \scrF , G ), the number of edges in S(F ) is at most k\delta \bigl( n 2 \bigr) . Lemma 3.3. Let \scrF 1 be the set of F \in Ga( \scrF , G ) such that S(F ) contains a match-ing of size \delta n log 2 n. Then, for n sufficiently large, \| \scrF 1\| \leq 2 n2 5 log 9n 2(n 2 ). Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2423 Proof. Let us consider the ways to color G so that the resulting colorings are in \scrF 1. We first choose the set of edges ES which will be colored by the colors in S.Note that ES must contain a matching of size \delta n log 2 n by the definition of \scrF 1. By Proposition 3.2, there are at most \sum i\leq k\delta (n 2 ) \bigl( (n 2 ) i \bigr) choices for such ES , and the number of ways to color them is at most rk\delta (n 2 ). In the next step, take a matching M of size \delta n log 2 n in ES ; the number of ways to choose such a matching is at most \bigl( (n 2 ) \delta n log 2n \bigr) .Let A = V (M ) and B = [ n] \setminus A. Denote by \scrT the set of triangles of Kn with a vertex in B and an edge from M , which contain no edge in ES \cap G[A, B ]. We claim that \| \scrT \| \geq 1 4 \delta n 2 log 2 n as otherwise we would obtain that \| ES \| \geq \| B\| \cdot \delta n log 2 n \| \scrT \| + \| M \| \geq 1 2 \delta n 2 log 2 n 1 4 \delta n 2 log 2 n = 1 4 \delta n 2 log 2 n > k\delta \biggl( n 2 \biggr) , which, by Proposition 3.2, contradicts the fact that F \in Ga( \scrF , G ). Note that if a triangle T in \scrT contains more than one uncolored edge, then they must have the same color as in order to avoid the rainbow triangle. Hence, the number of ways to color the uncolored edges in \scrT is at most 2 \| \scrT \| .There remain at most \bigl( n 2 \bigr) 2\| \scrT \| uncolored edges and they can only be colored by i or j, as edges in ES are already colored. Hence, the number of ways to color the rest of the edges is at most 2 (n 2 ) 2\| \scrT \| . In conclusion, we obtain that \| \scrF 1\| \leq \sum i\leq k\delta (n 2 ) \biggl( \bigl( n 2 \bigr) i \biggr) rk\delta (n 2 ) \biggl( \bigl( n 2 \bigr) \delta n log 2 n \biggr) \cdot 2\| \scrT \| \cdot 2(n 2 ) 2\| \scrT \| \leq 2O(\delta n 2 log n) \cdot 2O(\delta n log 3 n) \cdot 2(n 2 ) 1 4\delta n 2log 2n \leq 2(n 2 ) n2 5 log 9n . Lemma 3.4. For every integer 1 \leq t < \delta n log 2 n, let \scrF (t) be the set of F \in Ga( \scrF , G ), in which the maximum matching of S(F ) is of size t. Then, for n suffi-ciently large, \| \scrF (t)\| \leq 2 n 2 log 2n 2(n 2 ). Proof. For a fixed t, let us count the ways to color G so that the resulting colorings are in \scrF (t). By the definition of \scrF (t), among all edges which will be colored by the colors in S, there exists a maximum matching M of size t. We first choose such a matching; the number of ways is at most \bigl( (n 2 ) t \bigr) . Once we fix the matching M , let A = V (M ) and B = [ n] \setminus A. By the maximality of M , we immediately obtain the following claim. Claim 1. None of the edges in G[B] can be colored by the colors in S. Denote by Cr( S) the set of edges in G[A, B ] which will be colored by the colors in S. For a vertex u \in A, denote by Cr( S, u ) the set of edges in Cr( S) with one endpoint u. Similarly, define Cr( { i, j } , u ) to be the set of edges in G[u, B ] which will be colored by the colors i or j. We shall divide the proof into three cases. Case 1: \| Cr( S)\| \leq nt log 2n . We first color the edges in G[A] and the number of options is at most r(2t 2 ). In the next step, we select and color the edges in Cr( S); by the above inequality, the number of ways is at most \sum i\leq nt log 2n \bigl( 2nt i \bigr) r nt log 2n . By Claim 1, the remaining edges can only use the colors i or j. Let \scrT be the set of triangles of Kn formed by a vertex in B and an edge from M , which contain no edge in Cr( S). Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2424 J ´OZSEF BALOGH AND LINA LI We claim that \| \scrT \| \geq 1 4 nt as otherwise we would obtain \| Cr( S)\| \geq \| B\| t \| \scrT \| \geq 1 2 nt 1 4 nt > nt log 2 n , which contradicts the assumption. If a triangle T in \scrT contains more than one uncol-ored edge, then they must have the same color in order to avoid the rainbow triangle. Hence, the number of ways to color the uncolored edges in \scrT is at most 2 \| \scrT \| .There remain at most \bigl( n 2 \bigr) 2\| \scrT \| \bigl( 2t 2 \bigr) uncolored edges, and they can be colored by i or j. Therefore the number of ways to color the rest of the edges is at most 2(n 2 ) 2\| \scrT \| (2t 2 ). In conclusion, we obtain that the number of r-coloring F \in \scrF (t) with \| Cr( S)\| \leq nt log 2n is at most \biggl( \bigl( n 2 \bigr) t \biggr) \cdot r(2t 2 ) \cdot \sum i\leq nt log 2n \bigl( 2nt i \bigr) r nt log 2n \cdot 2\| \scrT \| \cdot 2(n 2 ) 2\| \scrT \| (2t 2 ) \leq 2O(t log n) \cdot 2O(t2) \cdot 2O( nt log n ) \cdot 2(n 2 ) 1 4nt \leq 2(n 2 ) 1 5nt \leq 2(n 2 ) 1 5n , where the third inequality is given by t2 \leq t \cdot \delta n log 2 n = nt/ log 9 n. Case 2: There exists a vertex u \in A such that (3) \| Cr( S, u )\| \geq n log 4 n and \| Cr( { i, j } , u )\| \geq n log 4 n . We first choose the vertex u, and the number of options is at most 2 t. Moreover, the number of ways to select and color edges in Cr( S, u ) is at most rn2n. In the next step, we color all the uncolored edges in G[A, B ] and G[A], and the number of ways is at most r2nt +(2t 2 ). Let \scrT be the set of triangles T = { uvw } of Kn, in which v, w \in B, uv \in Cr( S, u ), and uw \in Cr( { i, j } , u ). By the relation (3), we have \| \scrT \| \geq n2 log 8n . For every triangle T = { uvw } \in \scrT , if vw is an edge of G, then by Claim 1 it can only be colored by i or j, and must have the same color as uw in order to avoid the rainbow triangle. Therefore, the number of ways to color the uncolored edges in \scrT is 1. There remain at most \bigl( n 2 \bigr) \| \scrT \| uncolored edges in B, as other edges are already colored. By Claim 1, none of the remaining edges in B could use the colors from S.Therefore, the number of ways to color the rest of the edges is at most 2 (n 2 ) \| \scrT \| . In conclusion, we obtain that the number of F \in \scrF (t) which is included in Case 2 is at most \biggl( \bigl( n 2 \bigr) t \biggr) \cdot 2t\cdot rn2n \cdot r2nt +(2t 2 ) \cdot 2(n 2 ) \| \scrT \| \leq 2O(t log n) \cdot 2O(n) \cdot 2O(nt ) \cdot 2(n 2 ) n2 log 8n \leq 2(n 2 ) n2 2 log 8n , where the last inequality is given by the condition that nt \leq n \cdot \delta n log 2 n = n2/ log 9 n. Case 3: \| Cr( S)\| > nt log 2n , and for every vertex u \in A,(4) \| Cr( S, u )\| < n log 4 n or \| Cr( { i, j } , u )\| < n log 4 n . We first color the edges in G[A] and the number of ways is at most r(2t 2 ). By (4), for ev-ery vertex u \in A, the number of ways to select Cr( S, u ) is at most 2 \sum i\leq n/ log 4n \bigl( ni \bigr) \leq 2n/ log 3 n. Therefore, the number of ways to select Cr( S) is at most 2 2nt/ log 3 n. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2425 Subcase 3.1: e(G) \leq \bigl( n 2 \bigr) n2 4 log 6n . The number of ways to color Cr( S) is at most r2nt . By Claim 1, the rest of the edges can only be colored by i or j, and the number of them is at most e(G) \| Cr( S)\| . Hence, the number of F \in \scrF (t) covered in Case 3.1 is at most \biggl( \bigl( n 2 \bigr) t \biggr) \cdot r(2t 2 ) \cdot 2 2nt log 3n \cdot r2nt \cdot 2e(G) \| Cr( S)\| \leq 2O(t log n) \cdot 2O(nt ) \cdot 2(n 2 ) n2 4 log 6n nt log 2n \leq 2(n 2 ) n2 5 log 6n , where the last inequality holds by the condition that nt \leq n \cdot \delta n log 2 n = n2/ log 9 n. Subcase 3.2: e(G) > \bigl( n 2 \bigr) n2 4 log 6n . For u \in A, define NS (u) = { v \in B \| uv \in Cr( S, u )} . Let Gu be the induced subgraph of G on NS (u), and denote by c(Gu) the number of components of Gu. Claim 2. For every u \in A, we have c(Gu) \leq n log 3n .Proof. Suppose that there exists a vertex u in A with c(Gu) > n log 3n . Then the number of nonedges in Gu is at least \bigl( n log 3n 2 \bigr) \geq n2 4 log 6n , which contradicts the assumption of Case 3.2. Claim 3. For every u \in A, the number of ways to color Cr( S, u ) is at most rc(Gu).Proof. Let C be an arbitrary component of Gu. It is sufficient to prove that for every v, w \in V (C), uv and uw must have the same color. Assume that there exist v, w \in V (C) such that uv and uw receive different colors. Since C is a connected component of Gu, there is a path P = { v = v0, v 1, v 2, . . . , v k = w} in Gu, in which uv i is painted by a color in S for every 0 \leq i \leq k. Moreover, since uv and uw receive different colors, there exists an integer 0 \leq j \leq k 1 such that uv j and uv j+1 receive different colors. On the other hand, by Claim 1, vj vj+1 can only be colored by i or j. Therefore, u, v j , v j+1 form a rainbow triangle, which is not allowed in a Gallai r-coloring. By Claims 2 and 3, the number of ways to color Cr( S, u ) is at most r n log 3n , and therefore the total number of ways to color Cr( S) is at most r 2nt log 3n . By Claim 1, the rest of the edges can only be colored by i or j, and the number of them is at most e(G) \| Cr( S)\| . Hence, the number of F \in \scrF (t) included in Case 3.2 is at most \biggl( \bigl( n 2 \bigr) t \biggr) \cdot r(2t 2 ) \cdot 2 2nt log 3n \cdot r 2nt log 3n \cdot 2e(G) \| Cr( S)\| \leq 2O(t log n) \cdot 2O \Bigl( nt log 3n \Bigr) \cdot 2(n 2 ) nt log 2n \leq 2(n 2 ) n 2 log 2n1 . Eventually, we conclude that \| \scrF (t)\| \leq 2(n 2 ) 1 5n 2 (n 2 ) n2 2 log 8n 2 (n 2 ) n 2 log 2n1 \leq 2 n 2 log 2n 2(n 2 )for every 1 \leq t < \delta n log 2 n. Observe that every r-coloring of G using at most 2 colors is a Gallai r-coloring. Then we immediately obtain the following lemma. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2426 J ´OZSEF BALOGH AND LINA LI Lemma 3.5. Let \scrF 0 be the set of F \in Ga( \scrF , G ) such that S(F ) = \emptyset . Then \| \scrF 0\| = 2 e(G). Now, we have all the ingredients to prove Theorem 3.1. Proof of Theorem 3.1 . Applying Lemmas 3.3, 3.4, and 3.5, we obtain that \| Ga( \scrF , G )\| = \| \scrF 1\| + \delta n/ log 2n \sum t=1 \| \scrF (t)\| + \| \scrF 0\| \leq 2e(G) + 2 n 3 log 2n 2(n 2 )for n sufficiently large. Gallai r-colorings of complete graphs. 4.1. Stability of the Gallai r-template of complete graphs. Proposition 4.1. Let n, r \in N with r \geq 3. Suppose P is a Gallai r-template of Kn. Then the number of edges with at least 3 colors in its palette is at most n 1/6n2.Proof. Let E = { e \in E(Kn) : \| P (e)\| \geq 3} and assume that \| E\| > n 1/6n2. Let F be a spanning subgraph of Kn with edge set E. For every i \in [n], denote by di the degree of vertex i of F . Then the number of 3-paths in F is equal to \sum i\in [n] \biggl( di 2 \biggr) \geq n \Biggl( \sum i\in [n] di n 2 \Biggr) \geq n \biggl( 2\| E\| /n 2 \biggr) \geq \| E\| 2 n > 3n 1/3 \biggl( n 3 \biggr) . Observe that if i, j, k is a 3-path in F , then there is at least one rainbow triangle in P with vertex set { i, j, k } since edges ij , jk have at least 3 colors in its palette and edge ik has at least one color in its palette. Therefore, there would be more than n 1/3\bigl( n 3 \bigr) rainbow triangles in P , which contradicts the fact that P is a Gallai r-template. Lemma 4.2. Let n, r \in N with r \geq 3 and n 1/6 \ll \delta \ll 1. Assume that P is a Gallai r-template of Kn with \| Ga( P, K n)\| > 2(1 \delta )(n 2 ). Then the number of triangles T of Kn with \sum e\in T \| P (e)\| = 6 and P (e) = P (e\prime ) for every e, e\prime \in T is at least (1 4\delta )\bigl( n 3 \bigr) .Proof. Let \scrT be the collection of triangles of Kn. We define \scrT 1 = \bigl{ T \in \scrT \| \sum e\in T \| P (e)\| = 6 and P (e) = P (e\prime ) for every e, e \prime \in T \bigr} , \scrT 2 = { T \in \scrT \| \exists e \in T, \| P (e)\| \geq 3} , \scrT 3 = \bigl{ T \in \scrT \setminus (\scrT 1 \cup \scrT 2) \| \sum e\in T \| P (e)\| = 6 \bigr} , \scrT 4 = \bigl{ T \in \scrT \setminus \scrT 2 \| \sum e\in T \| P (e)\| \leq 5\bigr} . Let \| \scrT 1\| = \alpha \bigl( n 3 \bigr) , \| \scrT 2\| = \beta \bigl( n 3 \bigr) , \| \scrT 3\| = \gamma \bigl( n 3 \bigr) . Then \| \scrT 4\| \leq (1 \alpha )\bigl( n 3 \bigr) . By Proposition 4.1, we have \| \scrT 2\| \leq n 1/6n3 and therefore \beta \leq 12 n 1/6. Observe that for every T \in \scrT 3,the template P contains a rainbow triangle with edge set T ; therefore, we obtain that \| \scrT 3\| \leq RT( P ) \leq n 1/3\bigl( n 3 \bigr) , which gives \gamma \leq n 1/3 \leq n 1/6. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2427 Assume that \alpha < 1 4\delta . Then the number of Gallai r-colorings of Kn, which are subtemplates of P , satisfies log \| Ga( P, K n)\| \leq log \biggl( \prod e\in E(Kn) \| P (e)\| \biggr) = log \biggl( \prod T\in \scrT \prod e\in T \| P (e)\| \biggr) 1 n2 \leq log \biggl( \prod T\in \scrT 1 23 \prod T\in \scrT 2 r3 \prod T\in \scrT 3 23 \prod T\in \scrT 4 22 \biggr) \cdot 1 n 2 \leq (3 \alpha + 3 \beta log r + 3 \gamma + 2(1 \alpha )) 1 3 \biggl( n 2 \biggr) \leq \Bigl( 2 + \alpha + (36 log r + 3) n 1/6\Bigr) 1 3 \biggl( n 2 \biggr) < (2 + (1 4\delta ) + \delta ) 1 3 \biggl( n 2 \biggr) = (1 \delta ) \biggl( n 2 \biggr) . This contradicts the assumption that \| Ga( P, K n)\| > 2(1 \delta )(n 2 ). We now prove a stability result for Gallai r-templates of Kn. Theorem 4.3. Let n, r \in N with r \geq 3 and n 1/6 \ll \delta \ll 1. Assume that P is a Gallai r-template of Kn with \| Ga( P, K n)\| > 2(1 \delta )(n 2 ). Then there exist two colors i, j \in [r] such that the number of edges of Kn with palette { i, j } is at least (1 4r4\delta )\bigl( n 2 \bigr) .Proof. Let G be an \bigl( r 2 \bigr) -colored graph with edge set E(G) = { e \in E(Kn) \| \| P (e)\| = 2 } and color set { (i, j ) \| 1 \leq i < j \leq r} , where each edge e is colored by color P (e). By Lemma 4.2, the number of monochromatic triangles in G is at least (1 4\delta )\bigl( n 3 \bigr) .Applying Lemma 2.7 on G, we obtain that there exist two colors i, j such that the number of edges with palette { i, j } is at least e(G) 4 \biggl( r 2 \biggr) 2 \cdot 4\delta \biggl( n 2 \biggr) \geq (1 4\delta ) \biggl( n 2 \biggr) 4 \biggl( r 2 \biggr) 2 \cdot 4\delta \biggl( n 2 \biggr) \geq (1 4r4\delta ) \biggl( n 2 \biggr) . 4.2. Proof of Theorem 1.1. Proof of Theorem 1.1 . Let \scrC be the collection of containers given by Theorem 2.4. We observe that a Gallai r-coloring of Kn can be regarded as a rainbow-triangle-free r-coloring template of order n, with only one color allowed at each edge. Therefore, by property (i) of Theorem 2.4, every Gallai r-coloring of Kn is a subtemplate of some P \in \scrC .Let \delta = log 11 n. We define \scrC 1 = \Bigl{ P \in \scrC : \| Ga( P, K n)\| \leq 2(1 \delta )(n 2 )\Bigr} , \scrC 2 = \Bigl{ P \in \scrC : \| Ga( P, K n)\| > 2(1 \delta )(n 2 )\Bigr} . By property (iii) of Theorem 2.4, we have \| Ga( \scrC 1, K n)\| \leq \| \scrC 1\| \cdot 2(1 \delta )(n 2 ) \leq 2cn 1/3 log 2 n(n 2 ) \cdot 2(n 2 ) log 11 n(n 2 ) \leq 2 n2 4 log 11 n 2(n 2 ). We claim that every template P in \scrC 2 is a Gallai r-template of Kn. First, by property (ii) of Theorem 2.4, we have RT( P ) \leq n 1/3\bigl( n 3 \bigr) . Suppose that there exists an edge e \in E(Kn) with \| P (e)\| = 0. Then we would obtain that Ga( P, K n) = \emptyset Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2428 J ´OZSEF BALOGH AND LINA LI as a Gallai r-coloring of Kn requires at least one color on each edge, which con-tradicts the definition of \scrC 2. Now by Theorem 4.3, we can divide \scrC 2 into classes { \scrF i,j , 1 \leq i < j \leq r} , where \scrF i,j consists of all the r-templates in \scrC 2 which contain at least (1 4r4\delta )\bigl( n 2 \bigr) edges with palette { i, j } . Applying Theorem 3.1 on \scrF i,j , we obtain that \| Ga( \scrF i,j , K n)\| \leq (1 + 2 n 3 log 2n )2 (n 2 ), and therefore \| Ga( \scrC 2, K n)\| \leq \sum 1\leq i<j \leq r \| Ga( \scrF i,j , K n)\| \leq \biggl( r 2 \biggr) \Bigl( 1 + 2 n 3 log 2n \Bigr) 2(n 2 ). Finally, we conclude that \| Ga( \scrC , K n)\| = \| Ga( \scrC 1, K n)\| + \| Ga( \scrC 2, K n)\| \leq \biggl( \biggl( r 2 \biggr) 2 n 4 log 2n \biggr) 2(n 2 ), which gives the desired upper bound for the number of Gallai r-colorings of Kn. Gallai 3-colorings of noncomplete graphs. In this section, we count Gal-lai 3-colorings of dense noncomplete graphs. We shall explore the stability property first, and then follow a somewhat similar strategy as in the proof of Theorem 1.1. The main obstacle is that in a Gallai r-template of a noncomplete graph, a palette of an edge could be an empty set, which leads to a more sophisticated discussion of templates. 5.1. Triangles in r-templates of dense graphs. Let \scrT be the collection of triangles of Kn. For a given r-template P of order n, we partition the triangles into 5 classes. We set an extra class, as a T \in \scrT may not be a triangle in G.(5) \scrT 1(P ) = \bigl{ T \in \scrT \| \sum e\in T \| P (e)\| = 6 and P (e) = P (e\prime ) for every e, e \prime \in T \bigr} , \scrT 2(P ) = { T \in \scrT \| T = { e1, e 2, e 3} , \| P (e1)\| \geq 3, \| P (e2)\| \geq 3, and \| P (e3)\| = 0 } , \scrT 3(P ) = { T \in \scrT \| T = { e1, e 2, e 3} , \| P (e1)\| \geq 3, \| P (e2)\| + \| P (e3)\| \leq 2} , \scrT 4(P ) = \bigl{ T \in \scrT \setminus (\scrT 1 \cup \scrT 2 \cup \scrT 3) \| \sum e\in T \| P (e)\| \geq 6\bigr} , \scrT 5(P ) = \bigl{ T \in \scrT \setminus \scrT 3 \| \sum e\in T \| P (e)\| \leq 5\bigr} . Lemma 5.1. Let n, r \in N with r \geq 4 and 0 < k \leq 1. For 0 < \xi \leq ( k 2+6 k )2, let G be a graph of order n, and e(G) \geq (1 \xi )\bigl( n 2 \bigr) . Assume that P is a Gallai r-template of G. Then, for sufficiently large n, \| \scrT 2(P )\| \leq max \biggl{ k\| \scrT 3(P )\| , 3 + 9 k k n 1 3 \biggl( n 3 \biggr) \biggr} . Proof. Let E = { e \in E(Kn) : \| P (e)\| \geq 3} and F be a spanning subgraph of Kn with edge set E. For every i \in [n], denote by di the degree of vertex i of F . Since \sum ni=1 di = 2 \| E\| , the number of vertices with di > \surd \xi n is less than 2\| E\| \surd \xi n . Therefore, we obtain \| \scrT 2(P )\| \leq n \sum i=1 min \biggl{ \biggl( di 2 \biggr) , \xi \biggl( n 2 \biggr) \biggr} < 2\| E\| \surd \xi n \cdot \xi n 2 2 + \sum di\leq \surd \xi n d2 i 2 \leq 2\| E\| \surd \xi n \cdot \xi n 2 2 + 2\| E\| \surd \xi n \cdot \xi n 2 2 = 2 \| E\| \sqrt{} \xi n \leq k 1 + 3 k n\| E\| , (6) where the third inequality follows from the concavity of the function x2. The rest of the proof is divided into two cases. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2429 Case 1: \| E\| \geq 2+6 k k n 1 3 \bigl( n 2 \bigr) . Consider all triangles of Kn with at least one edge in E. Note that if a triangle has at least one edge in E and belongs to neither \scrT 3(P )nor \scrT 2(P ), then it induces a rainbow triangle in P . Together with (6), we have k\| \scrT 3(P )\| \geq k \biggl( \| E\| (n 2) 2\| \scrT 2(P )\| 3n 1 3 \biggl( n 3 \biggr) \biggr) \geq k \biggl( 1 + k 1 + 3 k n\| E\| 2\| E\| 3n 1 3 \biggl( n 3 \biggr) \biggr) = k 1 + 3 k n\| E\| + k \biggl( k 1 + 3 k n\| E\| 2\| E\| 3n 1 3 \biggl( n 3 \biggr) \biggr) \geq k 1 + 3 k n\| E\| \geq \| \scrT 2(P )\| , where the fourth inequality is given by \| E\| \geq 2+6 k k n 1 3 \bigl( n 2 \bigr) for sufficiently large n. Case 2: \| E\| < 2+6 k k n 1 3 \bigl( n 2 \bigr) . In this case, we have \| \scrT 2(P )\| < 1 2 \| E\| (n 2) < 3 + 9 k k n 1/3 \biggl( n 3 \biggr) . 5.2. Stability of Gallai 3-templates of dense noncomplete graphs. Lemma 5.2. Let 0 < \xi \leq 1 64 and n 1/3 \ll \delta \ll 1. Let G be a graph of order n,and e(G) \geq (1 \xi )\bigl( n 2 \bigr) . Assume that P is a Gallai 3-template of G with \| Ga( P, G )\| > 2(1 \delta )(n 2 ). Then \| \scrT 1(P )\| \geq (1 40 \delta )\bigl( n 3 \bigr) .Proof. Let \| \scrT 1(P )\| = \alpha \bigl( n 3 \bigr) , \| \scrT 2(P )\| = \beta \bigl( n 3 \bigr) , \| \scrT 3(P )\| = \eta \bigl( n 3 \bigr) , and \| \scrT 4(P )\| = \gamma \bigl( n 3 \bigr) .Then \| \scrT 5(P )\| \leq (1 \alpha \beta \eta )\bigl( n 3 \bigr) . Observe that for every T \in \scrT 4(P ), the template P contains a rainbow triangle with edge set T ; therefore, we obtain that \| \scrT 4(P )\| \leq RT (P ) \leq n 1/3\bigl( n 3 \bigr) , which gives \gamma \leq n 1/3.Define for e \in E(Kn) the weight function w(e) = \Biggl{ 1 if P (e) = \emptyset , \| P (e)\| otherwise . Similarly to the proof of Lemma 4.2, the number of Gallai 3-colorings of G which are subtemplates of P satisfies (7) log \| Ga( P, G )\| \leq log \Biggl( \prod e\in Kn \| w(e)\| \Biggr) = log \Biggl( \prod T\in \scrT \prod e\in T \| w(e)\| \Biggr) 1 n2 \leq log \Biggl( \prod T\in \scrT 1 23 \prod T\in \scrT 2 32 \prod T\in \scrT 3 6 \prod T\in \scrT 4 33 \prod T\in \scrT 5 22 \Biggr) \cdot 1 n 2 \leq (3 \alpha + 2 \beta log 3 + \eta log 6 + 3 \gamma log 3 + 2(1 \alpha \beta \eta )) 1 3 \biggl( n 2 \biggr) = \Bigl( 2 + \alpha + (2 log 3 2) \beta + (log 6 2) \eta + 3 n 1/3 log 3 \Bigr) 1 3 \biggl( n 2 \biggr) . Let k = 1. By Lemma 5.1, we have \beta \leq max { \eta , 12 n 1/3} . Assume that \alpha < 1 40 \delta . The rest of the proof shall be divided into two cases. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2430 J ´OZSEF BALOGH AND LINA LI Case 1: \beta \leq \eta . If \eta < 20 \delta , continuing (7) we have log \| Ga( P, G )\| \leq \Bigl( 2 + \alpha + (2 log 3 + log 6 4) \eta + 3 n 1/3 log 3 \Bigr) 1 3 \biggl( n 2 \biggr) \leq (2 + (1 40 \delta ) + 1 .8 \cdot 20 \delta + \delta ) 1 3 \biggl( n 2 \biggr) = (1 \delta ) \biggl( n 2 \biggr) . Otherwise, together with \alpha \leq 1 \beta \eta , continuing (7) we obtain that log \| Ga( P, G )\| \leq \Bigl( 3 + (2 log 3 3) \beta + (log 6 3) \eta + 3 n 1/3 log 3 \Bigr) 1 3 \biggl( n 2 \biggr) \leq \Bigl( 3 + (2 log 3 + log 6 6) \eta + 3 n 1/3 log 3 \Bigr) 1 3 \biggl( n 2 \biggr) \leq (3 0.2 \cdot 20 \delta + \delta ) 1 3 \biggl( n 2 \biggr) = (1 \delta ) \biggl( n 2 \biggr) . Case 2: \beta \leq 12 n 1/3. Together with \eta \leq 1 \alpha and \alpha < 1 40 \delta , continuing (7) we have log \| Ga( P, G )\| \leq \Bigl( 2 + \alpha + 2 log 3 \cdot 12 n 1/3 + (log 6 2)(1 \alpha ) + 3 n 1/3 log 3 \Bigr) 1 3 \biggl( n 2 \biggr) \leq \Bigl( log 6 + (3 log 6) \alpha + 27 n 1/3 log 3 \Bigr) 1 3 \biggl( n 2 \biggr) \leq (log 6 + (3 log 6)(1 40 \delta ) + \delta ) 1 3 \biggl( n 2 \biggr) < (1 \delta ) \biggl( n 2 \biggr) . Both cases contradict our assumption that \| Ga( P, G )\| > 2(1 \delta )(n 2 ). Similarly as in the proof of Theorem 4.3, using Lemmas 2.7 and 5.2, we obtain the following theorem. Theorem 5.3. Let 0 < \xi \leq 1 64 and n 1/3 \ll \delta \ll 1. Let G be a graph of order n and e(G) \geq (1 \xi )\bigl( n 2 \bigr) . Assume that P is a Gallai 3-template of G with \| Ga( P, G )\| > 2(1 \delta )(n 2 ). Then there exist two colors i, j \in such that the number of edges of Kn with palette { i, j } is at least (1 37 \cdot 40 \delta )\bigl( n 2 \bigr) . 5.3. Proof of Theorem 1.9. Proof of Theorem 1.9 . Let \scrC be the collection of containers given by Theorem 2.4 for r = 3. Note that every Gallai 3-coloring of G is a subtemplate of some P \in \scrC . Let \delta = log 11 n. We define \scrC 1 = \Bigl{ P \in \scrC : \| Ga( P, K n)\| \leq 2(1 \delta )(n 2 )\Bigr} , \scrC 2 = \Bigl{ P \in \scrC : \| Ga( P, K n)\| > 2(1 \delta )(n 2 )\Bigr} . Similarly to the proof of Theorem 1.1, applying Theorems 2.4, 3.1, and 5.3, we obtain that \| Ga( \scrC , G )\| = \| Ga( \scrC 1, G )\| + \| Ga( \scrC 2, G )\| \leq 2 n2 4 log 11 n 2(n 2 )+ 3 \cdot \Bigl( 2e(G) + 2 n 3 log 2n 2(n 2 )\Bigr) \leq 3 \cdot 2e(G) + 2 n 4 log 2n 2(n 2 ). Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2431 Gallai r-colorings of noncomplete graphs. Theorem 1.11 is a direct con-sequence of the following three theorems. Theorem 6.1. For n, r \in N with r \geq 4, there exists n0 such that for all n > n 0 the following holds. For a graph G of order n with e(G) \geq (1 log 11 n)\bigl( n 2 \bigr) , the number of Gallai r-colorings of G is strictly less than r\lfloor n2/4\rfloor . Theorem 6.2. Let n, r \in N with r \geq 4, and 0 < \xi \ll 1. For a graph G of order n with \lfloor n2/4\rfloor < e (G) \leq \lfloor n2/4\rfloor + \xi n 2, the number of Gallai r-colorings of G is strictly less than r\lfloor n2/4\rfloor . Theorem 6.3. For n, r \in N with r \geq 4, there exists n0 such that for all n > n 0 the following holds. Let n 1/36 \ll \xi \leq 1 2 log 11 n \ll 1. For a graph G of order n with ( 1 4 3 \xi )n2 \leq e(G) \leq ( 1 2 3\xi )n2, the number of Gallai r-colorings of G is strictly less than r\lfloor n2/4\rfloor . 6.1. Proof of Theorem 6.1 for r ≥ 5. Lemma 6.4. Let n, r \in N with r \geq 5 and 0 < \xi \leq 1 900 . Assume that G is a graph of order n with e(G) \geq (1 \xi )\bigl( n 2 \bigr) , and P is a Gallai r-template of G. Then, for sufficiently large n, \| Ga( P, G )\| \leq r 1 2 (n 2 ) \cdot 2 0.007 (n 2 ). Proof. Let \scrT be the collection of triangles of Kn. For a given r-template P of order n, we again use the partition (5). Let \| \scrT 1(P )\| = \alpha \bigl( n 3 \bigr) , \| \scrT 2(P )\| = \beta \bigl( n 3 \bigr) , \| \scrT 3(P )\| = \eta \bigl( n 3 \bigr) , and \| \scrT 4(P )\| = \gamma \bigl( n 3 \bigr) . Then \| \scrT 5(P )\| \leq (1 \alpha \beta \eta )\bigl( n 3 \bigr) . Note that for every T \in \scrT 4(P ), the template P contains a rainbow triangle with edge set T ;therefore, we obtain that \| \scrT 4(P )\| \leq RT( P ) \leq n 1/3\bigl( n 3 \bigr) , which gives \gamma \leq n 1/3.Define for e \in E(Kn) the weight function w(e) = \Biggl{ 1 if P (e) = \emptyset , \| P (e)\| otherwise . Similarly, as in Lemma 5.2, the number of Gallai r-colorings of G, which is a subtem-plate of P , satisfies (8) log \| Ga( P, G )\| \leq log \Biggl( \prod T\in \scrT 1 23 \prod T\in \scrT 2 r2 \prod T\in \scrT 3 2r \prod T\in \scrT 4 r3 \prod T\in \scrT 5 22 \Biggr) \cdot 1 n 2 \leq (3 \alpha + 2 \beta log r + \eta log 2 r + 3 \gamma log r + 2(1 \alpha \beta \eta )) 1 3 \biggl( n 2 \biggr) \leq \Bigl( 2 + \alpha + (2 log r 2) \beta + (log r 1) \eta + 3 n 1/3 log r \Bigr) 1 3 \biggl( n 2 \biggr) . Let k = 1 /12. By Lemma 5.1, we have \beta \leq max { k\eta , 3+9 k k n 1/3} . The rest of the proof shall be divided into two cases. Case 1: \beta \leq k\eta . Together with \alpha \leq (1 \beta \eta ), continuing (8) we have log \| Ga( P, G )\| \leq \Bigl( 3 + (2 log r 3) \beta + (log r 2) \eta + 3 n 1/3 log r \Bigr) 1 3 \biggl( n 2 \biggr) \leq \Bigl( 3 + ((2 k + 1) log r (3 k + 2)) \eta + 3 n 1/3 log r \Bigr) 1 3 \biggl( n 2 \biggr) . Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2432 J ´OZSEF BALOGH AND LINA LI Note that (2 k + 1) log r (3 k + 2) is positive as r \geq 4. Therefore, together with \eta \leq 1and k = 1 12 , we obtain that log \| Ga( P, G )\| \leq \biggl( 7 6 log r + 3 4 + 3 n 1/3 log r \biggr) 1 3 \biggl( n 2 \biggr) \leq \biggl( 3 2 log r 0.023 + 3 n 1/3 log r \biggr) 1 3 \biggl( n 2 \biggr) \leq 1 2 \biggl( n 2 \biggr) log r 0.007 \biggl( n 2 \biggr) , where the second inequality follows from ( 1 3 log r 3 4 ) \geq 0.023 as r \geq 5. Case 2: \beta \leq 3+9 k k n 1/3. Together with \alpha \leq (1 \eta ), continuing (8) we have log \| Ga( P, G )\| \leq \biggl( 3 + (log r 2) \eta + 2 log r \cdot 3 + 9 k k n 1/3 + 3 n 1/3 log r \biggr) 1 3 \biggl( n 2 \biggr) \leq \biggl( 3 2 log r \biggl( 1 2 log r 1 \biggr) + \biggl( 2 + 6 k k + 1 \biggr) 3n 1/3 log r \biggr) 1 3 \biggl( n 2 \biggr) \leq \biggl( 3 2 log r 0.16 + 0 .01 \biggr) 1 3 \biggl( n 2 \biggr) = 1 2 \biggl( n 2 \biggr) log r 0.05 \biggl( n 2 \biggr) , where the third inequality holds for r \geq 5 and sufficiently large n. Using Lemma 6.4, we prove a stronger theorem for the case r \geq 5. Theorem 6.5. For n, r \in N with r \geq 5 and 0 < \xi \leq 1 900 , there exists n0 such that for all n > n 0 the following holds. If G is a graph of order n, and e(G) \geq (1 \xi )\bigl( n 2 \bigr) ,then the number of Gallai r-colorings of G is less than r 1 2 (n 2 ). Proof. Let \scrC be the collection of containers given by Theorem 2.4. Theorem 2.4 indicates that every Gallai r-coloring of G is a subtemplate of some P \in \scrC and \| \scrC \| \leq 2cn 1/3 log 2 n(n 2 ) for some constant c, which only depends on r. We may assume that all templates P in \scrC are Gallai r-templates of G. By property (ii) of Theorem 2.4, we always have RT( P ) \leq n 1/3\bigl( n 3 \bigr) . Suppose that for a template P there exists an edge e \in E(G) with \| P (e)\| = 0. Then we would obtain \| Ga( P, G )\| = 0 as a Gallai r-coloring of G requires at least one color on each edge. Now applying Lemma 6.4 on every container P \in \scrC , we obtain that the number of Gallai r-colorings of G is at most \sum P\in \scrC \| Ga( P, G )\| \leq \| \scrC \| \cdot r 1 2 (n 2 ) \cdot 2 0.007 (n 2 ) < r 1 2 (n 2 )for n sufficiently large. 6.2. Proof of Theorem 6.1 for r = 4. Given two colors R and B, consider a 4-template P of order n in which every edge of Kn has palette { R, B } . For a constant 0 < \varepsilon \ll 1 and a graph G with e(G) > \bigl( n 2 \bigr) 2\varepsilon n , we can easily check that P is a Gallai 4-template of G and \| Ga( P, G )\| = 2 e(G) > 4 1 2 (n 2 ) \varepsilon n . This indicates that Lemma 6.4 fails to hold when r = 4. Instead, we shall apply the same technique as for 3-colorings: prove a stability result to determine the approximate structure of r-templates, which would contain too many Gallai r-colorings, and then apply this together with Theorem 3.1 to obtain the desired bound. Lemma 6.6. Let n 1/3 \ll \delta \ll 1. Let G be a graph of order n with e(G) \geq (1 \delta )\bigl( n 2 \bigr) . Assume that P is a Gallai 4-template of G with \| Ga( P, G )\| > 2(1 \delta )(n 2 ). Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2433 Then the number of triangles T of Kn with \sum e\in T \| P (e)\| = 6 and P (e) = P (e\prime ) for every e, e\prime \in T is at least (1 16 \delta )\bigl( n 3 \bigr) .Proof. Let \scrT be the collection of triangles of Kn. We define \scrT 1 = \bigl{ T \in \scrT \| \sum e\in T \| P (e)\| = 6 and P (e) = P (e\prime ) for every e, e \prime \in T \bigr} , \scrT 2 = { T \in \scrT \| \exists e \in T, \| P (e)\| = 0 } , \scrT 3 = { T \in \scrT \| T = { e1, e 2, e 3} , \| P (e1)\| = 4 , \| P (e2)\| = \| P (e3)\| = 1 } , \scrT 4 = \bigl{ T \in \scrT \setminus (\scrT 1 \cup \scrT 2 \cup \scrT 3) \| \sum e\in T \| P (e)\| \geq 6\bigr} , \scrT 5 = \bigl{ T \in \scrT \setminus \scrT 2 \| \sum e\in T \| P (e)\| \leq 5\bigr} . Let \| \scrT 1\| = \alpha \bigl( n 3 \bigr) , \| \scrT 2\| = \beta \bigl( n 3 \bigr) , \| \scrT 3\| = \eta \bigl( n 3 \bigr) , and \| \scrT 4\| = \gamma \bigl( n 3 \bigr) . Then \| \scrT 5\| = (1 \alpha \beta \eta \gamma ) \biggl( n 3 \biggr) . Since G satisfies e(G) \geq (1 \delta )\bigl( n 2 \bigr) and P is a Gallai template, we have \| \scrT 2\| \leq \delta \bigl( n 2 \bigr) \cdot n \leq 6\delta \bigl( n 3 \bigr) , and therefore \beta \leq 6\delta . Observe that for every T \in \scrT 4, the template P contains a rainbow triangle with edge set T ; therefore, we obtain that \| \scrT 4\| \leq RT (P ) \leq n 1/3\bigl( n 3 \bigr) ,which gives \gamma \leq n 1/3.Define for e \in E(Kn) the weight function w(e) = \Biggl{ 1 if P (e) = \emptyset , \| P (e)\| otherwise . Assume that \alpha < 1 16 \delta . Similarly, as in Lemma 5.2, the number of Gallai 4-colorings of G which is a subtemplate of P satisfies log \| Ga( P, G )\| \leq log \bigl( \prod T\in \scrT 1 23 \prod T\in \scrT 2 42 \prod T\in \scrT 3 4 \prod T\in \scrT 4 43 \prod T\in \scrT 4 4\bigr) \cdot 1 n2 \leq (3 \alpha + 4 \beta + 2 \eta + 6 \gamma + 2(1 \alpha \beta \eta \gamma )) 1 3 \bigl( n 2 \bigr) = (2 + \alpha + 2 \beta + 4 \gamma ) 1 3 \bigl( n 2 \bigr) < (2 + (1 16 \delta ) + 13 \delta ) 1 3 \bigl( n 2 \bigr) = (1 \delta )\bigl( n 2 \bigr) . This contradicts the assumption that \| Ga( P, G )\| > 2(1 \delta )(n 2 ). Similarly, as in Theorem 4.3, applying Lemmas 2.7 and 6.6, we obtain the follow-ing. Theorem 6.7. Let n 1/3 \ll \delta \ll 1. Let G be a graph of order n with e(G) \geq (1 \delta )\bigl( n 2 \bigr) . Assume that P is a Gallai 4-template of G with \| Ga( P, G )\| > 2(1 \delta )(n 2 ).Then there exist two colors i, j \in such that the number of edges of Kn with palette { i, j } is at least (1 145 \cdot 16 \delta )\bigl( n 2 \bigr) .Proof of Theorem 6.1 for r = 4 . Let \scrC be the collection of containers given by Theorem 2.4 for r = 4. Note that every Gallai 4-coloring of G is a subtemplate of some P \in \scrC . Let \delta = log 11 n. We define \scrC 1 = \Bigl{ P \in \scrC : \| Ga( P, G )\| \leq 2(1 \delta )(n 2 )\Bigr} , \scrC 2 = \Bigl{ P \in \scrC : \| Ga( P, G )\| > 2(1 \delta )(n 2 )\Bigr} . Similarly, as in the proof of Theorem 1.1, applying Theorems 2.4, 3.1, and 6.7, we obtain that \| Ga( \scrC , G )\| = \| Ga( \scrC 1, G )\| + \| Ga( \scrC 2, G )\| \leq 2 n2 4 log 11 n 2(n 2 ) + 6 \Bigl( 2e(G) + 2 n 3 log 2n 2(n 2 )\Bigr) \leq 6 \cdot 2e(G) + 2 n 4 log 2n 2(n 2 ) < 4\lfloor n2/4\rfloor . Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2434 J ´OZSEF BALOGH AND LINA LI 6.3. Proof of Theorem 6.2. A book of size q consists of q triangles sharing a common edge, which is known as the base of the book. We write bk( G) for the size of the largest book in a graph G and call it the booksize of G. Lemma 6.8. Let n, r \in Z+ with r \geq 4, 0 < \alpha , \beta \ll 1, and G be a graph of order n.Assume that there exists a partition V (G) = A \cup B satisfying the following conditions: (i) \delta (G[A, B ]) \geq ( 1 2 \alpha )n; (ii) ∆( G[A]) , ∆( G[B]) \leq \beta n .Then the number of Gallai r-colorings of G is at most r\lfloor n2/4\rfloor . Furthermore, if e(G) \not = \lfloor n2/4\rfloor , then the number of Gallai r-colorings of G is strictly less than r\lfloor n2/4\rfloor .Proof. By condition (i), we have ( 1 2 \alpha )n \leq \| A\| , \| B\| \leq ( 1 2 \alpha )n. Let e(G) = \lfloor n2/4\rfloor + m. Without loss of generality, we can assume that m > 0 and e(G[A]) \geq m 2 .Then there exists a matching M in G[A] of size at least e(G[A]) 2∆( G[A]) 1 \geq m 4\beta n .For two vertices u, v \in A, the number of their common neighbors in B is at least \| B\| 2 ( \| B\| \delta (G[A, B ])) = 2 \delta (G[A, B ]) \| B\| \geq 2 \biggl( 1 2 \alpha \biggr) n \biggl( 1 2 + \alpha \biggr) n \geq n 3 . Then, for every e \in G[A], there exists a book graph Be of size n/ 3 with the base e. Let \scrB = { Be \| e \in M } . Note that M is a matching, and therefore book graphs in \scrB are edge-disjoint. Another crucial fact is that for every B \in \scrB , the number of r-colorings of B without rainbow triangles is at most r (r + 2( r 1)) n/ 3 < r (3 r)n/ 3,since once we color the base edge, each triangle must be colored in the way that two of its edges share the same color. Hence, the number of Gallai r-colorings of G is at most \bigl( r(3 r) n 3 \bigr) \| M \| re(G) \| M \| (1+2 \cdot n 3 )= re(G) (1 log r 3) \| M \| \cdot n 3 \leq r\lfloor n2/4\rfloor +m (1 log r 3) m 4βn \cdot n 3 < r \lfloor n2/4\rfloor , where the last inequality is given by \beta \ll 1. Lemma 6.9. Let n, r \in Z+ with r \geq 4, 0 < \alpha \prime , \beta \ll 1, 0 < \alpha , \gamma , \xi \ll \varepsilon \ll 1, and G be a graph of order n with e(G) \leq \lfloor n2/4\rfloor + \xi n 2. Assume that there exists a partition V (G) = A \cup B \cup C satisfying the following conditions: (i) dGA,B \geq \bigl( 1 2 \alpha \bigr) n for all but at most \gamma n vertices in A \cup B; (ii) \delta (G[A, B ]) \geq \bigl( 1 2 \alpha \prime \bigr) n; (iii) ∆( G[A]) , ∆( G[B]) \leq \beta n ; (iv) 0 < \| C\| \leq \gamma n ; (v) for every v \in C, both d(v, A ), d(v, B ) \geq r\varepsilon n .Then the number of Gallai r-colorings of G is strictly less than r\lfloor n2/4\rfloor .Proof. By condition (i), we have (9) \biggl( 1 2 \alpha \biggr) n \leq \| A\| , \| B\| \leq \biggl( 1 2 + \alpha \biggr) n. For a vertex v, a set S, a set of colors \scrR , and a coloring of G, let N (v, S ; \scrR ) be the set of vertices u \in N (v, S ), such that uv is colored by some color in \scrR . Let d(v, S ; \scrR ) = \| N (v, S ; \scrR )\| . Denote by \scrC 1 the set of Gallai r-colorings of G, in which there exist a vertex v \in C, and two disjoint sets of colors \scrR 1 and \scrR 2, such that both Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2435 d(v, A ; \scrR 1), d(v, B ; \scrR 2) \geq \varepsilon n . Let \scrC 2 be the set of Gallai r-colorings of G, which are not in \scrC 1.We first show that \scrC 1 = o(r\lfloor n2/4\rfloor ). We shall count the ways to color G so that the resulting colorings are in \scrC 1. First, we color the edges in G[C, A \cup B]; the number of ways is at most re(G[C,A \cup B]) . Once we fix the colors of edges in G[C, A \cup B], by the definition of \scrC 1, there exist a vertex v \in C, and two disjoint sets of colors \scrR 1 and \scrR 2, such that d(v, A ; \scrR 1), d(v, B ; \scrR 2) \geq \varepsilon n . We observe that for every edge e = uw between N1 = N (v, A ; \scrR 1) and N2 = N (v, B ; \scrR 2), e either shares the same color with uv , or with vw , as otherwise we would obtain a rainbow triangle uvw . Then the number of ways to color edges in G[N1, N 2] is at most 2 e(G[N1,N 2]) \leq r 1 2e(G[N1,N 2]) .Note that by condition (i), inequality (9), and \alpha , \gamma \ll \varepsilon , we have e(G[N1, N 2]) \geq (\| N1\| \gamma n )( \| N2\| 2\alpha n ) \geq 1 2 \varepsilon 2n2. Hence, we obtain log r \| \scrC 1\| \leq e(G[C, A \cup B]) + 1 2 e(G[N1, N 2]) + ( e(G) e(G[C, A \cup B]) e(G[N1, N 2])) = e(G) 1 2 e(G[N1, N 2]) \leq \lfloor n2/4\rfloor + \xi n 2 1 4 \varepsilon 2n2, which indicates \| \scrC 1\| = o(r\lfloor n2/4\rfloor ) as \xi \ll \varepsilon .It remains to estimate the size of \scrC 2. Recall that for a coloring in \scrC 2, for every vertex v \in C, there are no two disjoint sets of colors \scrR 1 and \scrR 2 such that d(v, A ; \scrR 1), d(v, B ; \scrR 2) \geq \varepsilon n . Claim 4. Let \scrS be a set of r colors. For every coloring in \scrC 2, and every vertex v \in C, there exists a color R \in \scrS , such that both d(v, A ; \scrS \setminus { R} ) < \varepsilon n and d(v, B ; \scrS \setminus { R} ) < \varepsilon n .Proof. We arbitrarily fix a coloring in \scrC 2 and a vertex v \in C. By condition (v), there exists a color R such that d(v, A ; R) \geq \varepsilon n . By the definition of \scrC 2, we obtain that d(v, B ; \scrS \setminus { R} ) < \varepsilon n . Then we also have d(v, B ; R) \geq d(v, B ) d(v, B ; \scrS \setminus { R} ) \geq r\varepsilon n \varepsilon n > \varepsilon n . For the same reason, we obtain that d(v, A ; \scrS \setminus { R} ) < \varepsilon n . By Claim 4, the number of ways to color edges in G[C, A \cup B] is at most \left( r \sum i\leq \varepsilon n \biggl( ni \biggr) \sum i\leq \varepsilon n \biggl( ni \biggr) r2\varepsilon n \right) \| C\| \leq \biggl( 4r \Bigl( ne \epsilon n \Bigr) 2\varepsilon n r2\varepsilon n \biggr) \| C\| \leq r((log r e log r \varepsilon +1)2 \varepsilon n +2) \| C\| < r \| C\| n 3 , where the last inequality is given by (log r e log r \varepsilon + 1) 2 \varepsilon \ll 1 3 as \varepsilon \ll 1. Note that by conditions (ii)–(iv), we have \bullet \delta (G[A, B ]) \geq \bigl( 1 2 \alpha \prime \bigr) n \geq \bigl( 1 2 \alpha \prime \bigr) (\| A\| + \| B\| ); \bullet ∆( G[A]) , ∆( G[B]) \leq \beta n \leq \beta 1\gamma (\| A\| + \| B\| ). Applying Lemma 6.8 on G[A \cup B], we obtain that the number of ways to color edges in G[A \cup B] is at most r (n \| C\| )2 4 . A trivial upper bound for the ways to color the rest of the edges, that is, the edges in G[C] is r(\| C\| 2 ). Hence, we have log r \| \scrC 2\| \leq \| C\| n 3 + (n \| C\| )2 4 + \biggl( \| C\| 2 \biggr) = n2 4 \biggl( n 6 3 4 \| C\| + 1 2 \biggr) \| C\| \leq \lfloor n2/4\rfloor 1 4 , Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2436 J ´OZSEF BALOGH AND LINA LI where the last inequality is given by 0 < \| C\| \leq \gamma n and \gamma \ll 1. Finally, we obtain that the number of Gallai r-colorings of G is \| \scrC 1\| + \| \scrC 2\| \leq o(r\lfloor n2/4\rfloor ) + r\lfloor n2/4\rfloor 1 4 < r \lfloor n2/4\rfloor . Lemma 6.10. Let n, r \in Z+ with r \geq 4, \alpha , \beta , \gamma , \xi \ll 1, and G be a graph of order n with \lfloor n2/4\rfloor < e (G) \leq \lfloor n2/4\rfloor + \xi n 2. Assume that there exists a partition V (G) = A \cup B \cup C satisfying the following conditions: (i) \delta (G[A, B ]) \geq ( 1 2 \alpha )n; (ii) ∆( G[A]) , ∆( G[B]) \leq \beta n ; (iii) 0 < \| C\| \leq \gamma n ; (iv) for every v \in C, d(v) \geq n/ 2.Then the number of Gallai r-colorings of G is strictly less than r\lfloor n2/4\rfloor .Proof. Let \alpha , \gamma , \xi \ll \varepsilon \ll 1. Let C1 = { v \in C \| d(v, A ) < r\varepsilon n } , and C2 = { v \in C \| d(v, B ) < r\varepsilon n } . By conditions (iii) and (iv), for every v \in C1, we have d(v, B ) \geq \bigl( 1 2 \gamma r\varepsilon \bigr) n. Similarly, for every v \in C2, we have d(v, A ) \geq \bigl( 1 2 \gamma r\varepsilon \bigr) n. Define A\prime = A \cup C1, B\prime = B \cup C2, C\prime = C \setminus (C1 \cup C2). If C\prime = \emptyset , then we obtain a new partition V (G) = A\prime \cup B\prime satisfying the following properties: \bullet \delta (G[A\prime , B \prime ]) \geq min { \bigl( 1 2 \alpha \bigr) n, \bigl( 1 2 \gamma r\varepsilon \bigr) n} = \bigl( 1 2 \gamma r\varepsilon \bigr) n; \bullet ∆( G[A\prime ]), ∆( G[B\prime ]) \leq min { (\beta + \gamma )n, (r\varepsilon + \gamma )n} .Together with e(G) > \lfloor n2/4\rfloor , by Lemma 6.8, we obtain that the number of Gallai r-colorings of G is strictly less than r\lfloor n2/4\rfloor . Otherwise, we obtain a new partition V (G) = A\prime \cup B\prime \cup C\prime satisfying the following properties: \bullet dGA\prime ,B \prime \geq \bigl( 1 2 \alpha \bigr) n for all but at most \gamma n vertices in A\prime \cup B\prime ; \bullet \delta (G[A\prime , B \prime ]) \geq \bigl( 1 2 \gamma r\varepsilon \bigr) n; \bullet ∆( G[A\prime ]), ∆( G[B\prime ]) \leq min { (\beta + \gamma )n, (r\varepsilon + \gamma )n} ; \bullet 0 < \| C\prime \| \leq \| C\| \leq \gamma n ; \bullet for every v \in C\prime , both d(v, A \prime ), d(v, B \prime ) \geq r\varepsilon n .Together with e(G) \leq \lfloor n2/4\rfloor + \xi n 2, by Lemma 6.9, the number of Gallai r-colorings of G is strictly less than r\lfloor n2/4\rfloor . Now, we prove a lemma which is crucial to the proof of Theorem 6.2. Lemma 6.11. Let n, r \in Z+ with r \geq 4, \alpha , \beta , \gamma , \xi \ll 1, and G be a graph of order n with \lfloor n2/4\rfloor < e (G) \leq \lfloor n2/4\rfloor + \xi n 2. Assume that there exists a partition V (G) = A \cup B \cup C satisfying the following conditions: (i) \delta (G[A, B ]) \geq ( 1 2 \alpha )n; (ii) ∆( G[A]) , ∆( G[B]) \leq \beta n ; (iii) \| C\| \leq \gamma n .Then the number of Gallai r-colorings of G is strictly less than r\lfloor n2/4\rfloor .Proof. By Lemma 6.8, we can assume that \| C\| > 0 without loss of generality. We begin with the graph G and greedily remove a vertex in C with degree strictly less than \| G\| /2 in G to obtain a smaller subgraph. Let G\prime be the resulting graph when the algorithm terminates, and n\prime = \| V (G\prime )\| . We remark that G\prime is not unique and it depends on the order of removing vertices. Without loss of generality, we can assume that n\prime < n , as otherwise we are done by applying Lemma 6.10 on G. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2437 Let A\prime = A, B\prime = B, and C\prime = V (G\prime ) \cap C. Clearly, we have G\prime = G[A\prime \cup B\prime \cup C\prime ]. Furthermore, by the mechanics of the algorithm, we have (10) e(G) \leq e(G\prime ) + 1 2 \biggl( \biggl( n 2 \biggr) \biggl( n\prime 2 \biggr) \biggr) . We first claim that e(G\prime ) > \lfloor (n\prime )2/4\rfloor , as otherwise we would have e(G) \leq \lfloor (n\prime )2/4\rfloor + 1 2 \biggl( \biggl( n 2 \biggr) \biggl( n\prime 2 \biggr) \biggr) \leq \lfloor n2/4\rfloor , which contradicts the assumption of the lemma. On the other hand, since n\prime \geq (1 \gamma )n, we obtain that e(G) \leq \lfloor n2/4\rfloor + \xi n 2 \leq \lfloor (n\prime )2/4\rfloor + \gamma + 2 \xi 2(1 \gamma )2 (n\prime )2. Let \xi \prime = \gamma +2 \xi 2(1 \gamma )2 . Then we have (11) \lfloor (n\prime )2/4\rfloor < e (G\prime ) \leq \lfloor (n\prime )2/4\rfloor + \xi \prime (n\prime )2. If C\prime = \emptyset , we obtain a vertex partition V (G\prime ) = A\prime \cup B\prime satisfying \bullet \delta (G\prime [A\prime , B \prime ]) \geq ( 1 2 \alpha )n \geq ( 1 2 \alpha )n\prime ; \bullet ∆( G\prime [A]), ∆( G\prime [B]) \leq \beta n \leq \beta 1\gamma n\prime .Together with (11), by Lemma 6.8, we obtain that the number of Gallai r-colorings of G\prime , denoted by \| \scrC (G\prime )\| , is strictly less than r\lfloor n2/4\rfloor . Otherwise, we find the partition V (G\prime ) = A\prime \cup B\prime \cup C\prime satisfying: \bullet \delta (G\prime [A\prime , B \prime ]) \geq ( 1 2 \alpha )n \geq ( 1 2 \alpha )n\prime ; \bullet ∆( G\prime [A]), ∆( G\prime [B]) \leq \beta 1\gamma n\prime ; \bullet 0 < \| C\prime \| \leq \gamma n \leq \gamma 1\gamma n\prime ; \bullet for every v \in C\prime , d(v) \geq n\prime 2 .Together with (11), by Lemma 6.10, we obtain that \| \scrC (G\prime )\| < r \lfloor (n\prime )2/4\rfloor . Combining with (10), we conclude that the number of Gallai r-colorings of G, denoted by \| \scrC (G)\| ,satisfies log r \| \scrC (G)\| \leq log r \| \scrC (G\prime )\| + ( e(G) e(G\prime )) < \lfloor (n\prime )2/4\rfloor + 1 2 \biggl( \biggl( n 2 \biggr) \biggl( n\prime 2 \biggr) \biggr) \leq \lfloor n2/4\rfloor , which completes the proof. Another important tool we need is the stability property of book graphs proved by Bollob´ as and Nikiforov . Theorem 6.12 (see ). For every 0 < \alpha < 10 5 and every graph G of order n with e(G) \geq ( 1 4 \alpha )n2, either bk( G) > \biggl( 1 6 2\alpha 1/3 \biggr) n or G contains an induced bipartite graph G1 of order at least (1 \alpha 1/3)n and with minimum degree \delta (G1) \geq \biggl( 1 2 4\alpha 1/3 \biggr) n. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2438 J ´OZSEF BALOGH AND LINA LI Proof of Theorem 6.2 . Let e(G) = \lfloor n2/4\rfloor +m, where 0 < m \leq \xi n 2. We construct a family \scrB of book graphs by the following algorithm. We start the algorithm with \scrB = \emptyset and G0 = G. In the ith iteration step, if there exists a book graph B of size n 7 in Gi, we let \scrB = \scrB \cup { B} , and Gi = Gi 1 e, where e is the base edge of B. The algorithm terminates when there is no book graph of size n/ 7. Let E0 be the set of base edges of \scrB , and \tau = 7 /(1 log r 3). Suppose that \| \scrB \| \geq 2\tau m . Since \| E0\| = \| \scrB \| \geq 2\tau m , the edge set E0 contains a matching M of size \| E0\| 2( n1) 1 \tau m/n . Let \scrB \prime be the set of book graphs in \scrB whose base edges are in M . Since M is a matching, book graphs in \scrB \prime are edge-disjoint. Note that for every B \in \scrB ‘, the number of r-colorings of B without rainbow triangles is at most r(r + 2( r 1)) n/ 7 < r (3 r)n/ 7. Then the number of Gallai colorings of G is at most \bigl( r(3 r) n 7 \bigr) \| M \| re(G) \| M \| (1+2 \cdot n 7) = r\lfloor n2/4\rfloor +m (1 log r 3) \| M \| n 7 < r \lfloor n2/4\rfloor +m m = r\lfloor n2/4\rfloor . It remains to consider the case for \| \scrB \| < 2\tau m . Without loss of generality, we can assume that there is no matching of size greater than \tau m/n in E0. Let G\prime = G E0.Then we have e(G\prime ) > \lfloor n2/4\rfloor (2 \tau 1) m. Furthermore, by the construction of G\prime , we obtain that bk( G\prime ) < n/ 7. Let \alpha =(2 \tau 1) \xi . By applying Theorem 6.12 on G\prime , we obtain that there is a vertex partition V (G\prime ) = A\prime \cup B\prime \cup C\prime with \| C\prime \| \leq \alpha 1/3n, such that A\prime , B \prime are independent sets, and \delta (G\prime [A\prime , B \prime ]) \geq \biggl( 1 2 4\alpha 1/3 \biggr) n. Let G0 be the spanning subgraph of G with edge set E0. For a small constant \beta with \xi \ll \beta \ll 1, let V0 be the set of vertices in G0 with degree more than \beta n . Since \| E0\| < 2\tau m \leq 2\tau \xi n 2, we have \| V0\| \leq (4 \tau \xi /\beta )n \leq \beta n . Let A = A\prime \setminus V0, B = B\prime \setminus V0,and C = C\prime \cup V0. Then we obtain a vertex partition V (G) = A \cup B \cup C satisfying the following conditions: \bullet \delta (G[A, B ]) \geq ( 1 2 4\alpha 1/3 \beta )n; \bullet ∆( G[A]), ∆( G[B]) \leq \beta n ; \bullet \| C\| \leq (\alpha 1/3 + \beta )n. By Lemma 6.11, we obtain that the number of Gallai r-colorings of G is strictly less than r\lfloor n2/4\rfloor . 6.4. Proof of Theorem 6.3. We say that a graph G is t-far from being k-partite if \chi (G\prime ) > k for every subgraph G\prime \subset G with e(G\prime ) > e (G) t. We will use the following theorem of Balogh et al. . Theorem 6.13 (see ). For every n, k, t \in N, the following holds. Every graph G of order n which is t-far from being k-partite contains at least nk 1 e2k \cdot k! \biggl( e(G) + t \biggl( 1 1 k \biggr) n2 2 \biggr) copies of Kk+1 . Proposition 6.14. Let n \in N and 0 < \varepsilon \leq 1. Every graph F on at least \varepsilon n vertices, which contains at most n 1/3\bigl( n 3 \bigr) triangles, satisfies e(F ) \leq \| F \| 2 4 + e4 6n1/3\varepsilon 3 \| F \| 2. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2439 Proof. Let t = e4 6n1/3\varepsilon 3 \| F \| 2. Assume that e(F ) > \| F \| 2 4 t. Then F is t-far from being bipartite. By Theorem 6.13, the number of triangles in F is at least \| F \| 2e4 \biggl( e(F ) + t \| F \| 2 4 \biggr) \| F \| 2e4 \cdot 2t = 1 6n1/3\varepsilon 3 \| F \| 3 > n 1/3 \biggl( n 3 \biggr) , which gives a contradiction. For an r-template P of order n, we say that an edge e of Kn is an r-edge of P if \| P (e)\| \geq 3. An r-edge e is typical if the number of rainbow triangles containing e is at most n11 /12 . We then immediately obtain the following proposition. Proposition 6.15. For an r-template of order n containing at most n 1/3\bigl( n 3 \bigr) rainbow triangles, the number of r-edges of P , which is not typical, is at most n11 /6. We now prove the following lemma. Lemma 6.16. Let n, r \in N with r \geq 4, and n 1/33 \ll \xi \leq 1 2 log 11 n \ll 1. Assume that G is a graph of order n with ( 1 4 3 \xi )n2 \leq e(G) \leq ( 1 2 3\xi )n2, and P is a Gallai r-template of G. Then, for sufficiently large n, log r \| Ga( P, G )\| \leq n2 4 \xi 3 n2 2 + 4 n23 /12 . Proof. We first construct a subset I of [n] and a sequence of graphs { G0, G 1, . . . , G \ell } by the following algorithm. We start the algorithm with I = \emptyset and G0 = G. In the ith iteration step, we either add a vertex v to I, whose degree is at most ( 1 2 \xi 2)( \| Gi\| 1) in the graph Gi, or add a pair of vertices { u, v } to I, where uv is a typical r-edge satisfying \| NGi (u) \cap NGi (v)\| \geq 2\xi 2(\| Gi\| 2). In both cases, we define Gi+1 = G I. The algorithm terminates when neither of the above types of vertices exists. Assume that the algorithm terminates after \ell steps. Let G\prime = G\ell and k = \| G\prime \| .We now make the following claim. Claim 5. log r \| Ga( P, G )\| \leq \biggl( 1 2 \xi 2 \biggr) \biggl( n2 2 k2 2 \biggr) 3 n23 /12 + log r \| Ga( P, G \prime )\| . Proof. In the ith iteration step of the above algorithm, if we add to I a single vertex v, then the number of ways to color the incident edges of v in Gi satisfies log r \prod e is incident to vin Gi \| P (e)\| \leq dGi (v) \leq ( 1 2 \xi 2)( \| Gi\| 1) . Now we assume that what we add is a pair of vertices { u, v } . For every w \in NGi (u)\cap NGi (v), vertices uvw either span a rainbow triangle in P , or satisfy \| P (uw )\| = \| P (vw )\| = 1. Together with the fact that uv is a typical r-edge, we obtain that the number of ways to color the edges, which are incident to v or u in Gi, satisfies log r \prod e is incident to uor vin Gi \| P (e)\| \leq \| Gi\| 2 \| NGi (u) \cap NGi (v)\| + 2 n11 /12 + 1 \leq (1 2\xi 2)( \| Gi\| 2) + 2 n11 /12 + 1 . From the above discussion, we conclude that the number of ways to color edges in E(G) E(G\prime ) satisfies log r \prod e\in E(G)E(G\prime ) \| P (e)\| \leq \bigl( 1 2 \xi 2\bigr) \Bigl( n2 2 k2 2 \Bigr) n(1 + 2 n11 /12 ), which implies the claim. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2440 J ´OZSEF BALOGH AND LINA LI We now split the proof into several cases. Case 1: k \leq \xi 2n. Then \| Ga( P, G \prime )\| \leq rk2/2 \leq r\xi 4n2/2, and therefore by Claim 5 and \xi \ll 1, we obtain that log r \| Ga( P, G )\| \leq \biggl( 1 2 \xi 2 \biggr) n2 2 + 3 n23 /12 + \xi 4n2/2 \leq n2 4 \xi 2 n2 4 + 3 n23 /12 . Case 2: e(G\prime ) > \bigl( 1 2 2\xi \bigr) k2 and k > \xi 2n. Since 2 \xi \leq log 11 n \leq log 11 k for sufficiently large n, Theorem 6.1 indicates that \| Ga( P, G \prime )\| \leq rk2/4. We claim that k \leq (1 \xi )n, as otherwise we would have e(G) \geq e(G\prime ) > \biggl( 1 2 2\xi \biggr) k2 > \biggl( 1 2 2\xi \biggr) (1 \xi )2n2 \geq \biggl( 1 2 3\xi \biggr) n2, which contradicts the assumption of the lemma. Therefore, by Claim 5, we obtain that log r \| Ga( P, G )\| \leq \biggl( 1 2 \xi 2 \biggr) \biggl( n2 2 k2 2 \biggr) +3 n23 /12 + k2 4 \leq n2 4 \xi 2 n2 2 + \xi 2 k2 2 + 3 n23 /12 \leq n2 4 \xi 2 n2 2 + \xi 2(1 \xi )2 n2 2 + 3 n23 /12 \leq n2 4 \xi 3 n2 2 + 3 n23 /12 . Case 3: e(G\prime ) < \bigl( 1 4 2 \xi \bigr) k2 and k > \xi 2n. Since 2 \xi \ll 1, Theorem 6.2 indicates that \| Ga( P, G \prime )\| \leq rk2/4. We claim that k \leq (1 \xi )n, as otherwise we would have e(G) < \biggl( n2 2 k2 2 \biggr) + \biggl( 1 4 + 2 \xi \biggr) k2 < n2 2 \biggl( 1 4 2\xi \biggr) k2 < n2 2 \biggl( 1 4 2\xi \biggr) (1 \xi )2n2 \leq n2 2 \biggl( 1 4 3\xi \biggr) n2 = \biggl( 1 4 + 3 \xi \biggr) n2, which contradicts the assumption of the lemma. Similarly, as in Case 2, we obtain that log r \| Ga( P, G )\| \leq \biggl( 1 2 \xi 2 \biggr) \biggl( n2 2 k2 2 \biggr) 3 n23 /12 + k2 4 \leq n2 4 \xi 3 n2 2 + 3 n23 /12 . Case 4: ( 1 4 2 \xi )k2 \leq e(G\prime ) \leq ( 1 2 2\xi )k2 and k > \xi 2n. Denote by er (G\prime ) the number of r-edges of P in G\prime . Let A = { v \in V (G\prime ) \| dG\prime (v) \leq \bigl( 1 2 \xi \bigr) k} . Claim 6. All the typical r-edges of G\prime have both endpoints in A.Proof. First, by the construction of G\prime , we have the following two properties: for every v \in V (G\prime ), (12) dG\prime (v) > \biggl( 1 2 \xi 2 \biggr) (k 1) , and for every typical r-edge uv in G\prime ,(13) dG\prime (u) + dG\prime (v) \leq 2 + ( k 2) + \| NGi (u) \cap NGi (v)\| < (1 + 2 \xi 2)k. Suppose that there exists a typical r-edge uv such that u is not in A, i.e., dG\prime (u) >\bigl( 1 2 \xi \bigr) k. Then by (12) and \xi \ll 1, we have dG\prime (u) + dG\prime (v) > \biggl( 1 2 + \xi \biggr) k + \biggl( 1 2 \xi 2 \biggr) (k 1) > (1 + 2 \xi 2)k, which contradicts (13). Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2441 Subcase 4.1: \| A\| \leq \xi k . By Proposition 6.15 and Claim 6, we have er (G\prime ) \leq \biggl( \| A\| 2 \biggr) n11 /6 \leq \xi 2 k2 2 + n11 /6. Therefore, together with the assumption of Case 4, we obtain that log r \| Ga( P, G \prime )\| \leq log r \Bigl( rer (G\prime )2e(G\prime ) er (G\prime )\Bigr) \leq 1 2 (e(G\prime ) + er (G\prime )) \leq 1 2 \biggl( \biggl( 1 2 2\xi \biggr) k2 + \xi 2 k2 2 + n11 /6 \biggr) = k2 4 \biggl( \xi 1 4 \xi 2 \biggr) k2 + 1 2 n11 /6. Then by Claim 5, log r \| Ga( P, G )\| \leq \biggl( 1 2 \xi 2 \biggr) \biggl( n2 2 k2 2 \biggr) 3 n23 /12 + k2 4 \biggl( \xi 1 4 \xi 2 \biggr) k2 + 1 2 n11 /6 \leq n2 4 \xi 2 n2 2 \biggl( \xi 3 4 \xi 2 \biggr) k2 + 4 n23 /12 \leq n2 4 \xi 2 n2 2 + 4 n23 /12 , where the last inequality is given by \xi \ll 1. Subcase 4.2: \| A\| > \xi k . By the definition of A, the number of nonedges of G\prime is at least (14) 1 2 \biggl( k 1 \biggl( 1 2 + \xi \biggr) k \biggr) \| A\| = 1 2 \biggl( \biggl( 1 2 \xi \biggr) k 1 \biggr) \| A\| . We first claim that (15) \| A\| \leq 1 8\xi 1 2\xi k, as otherwise we would obtain that the number of nonedges of G\prime is more than 1 2 \biggl( 1 2 \xi \biggr) k \cdot 1 8\xi 1 2\xi k \| A\| 2 \geq \biggl( 1 4 2\xi \biggr) k2 k 2which contradicts the assumption of Case 4. Inequality (15) implies that (16) (1 2\xi )k \| A\| \geq 4\xi k. By Propositions 6.14 and 6.15, since \| A\| > \xi k > \xi 3n, we have er (G\prime ) \leq e(G\prime [A]) + n11 /6 \leq \| A\| 2 4 + e4 6n1/3\xi 9 \| A\| 2 + n11 /6, as otherwise we would find more than n 1/3\bigl( n 3 \bigr) rainbow triangles, which contradicts the assumption that P is a Gallai r-template of G. Since \xi \gg n 1/33 , we have (17) er (G\prime ) \leq \| A\| 2 4 + \xi 2 2 \| A\| 2 + n11 /6. Combining (14), (16), and (17), we have log r \| Ga( P, G \prime )\| \leq 1 2 (e(G\prime ) + er (G\prime )) \leq 1 2 \biggl( \biggl( k 2 \biggr) 1 2 \biggl( \biggl( 1 2 \xi \biggr) k 1 \biggr) \| A\| + \| A\| 2 4 + \xi 2 2 \| A\| 2 + n11 /6 \biggr) \leq k2 4 \| A\| 8 ((1 2\xi )k \| A\| ) + \xi 2 4 \| A\| 2 + 1 2 n11 /6 \leq k2 4 \xi 2 \| A\| k + \xi 2 4 \| A\| 2 + 1 2 n11 /6. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2442 J ´OZSEF BALOGH AND LINA LI Then by Claim 5 and the assumption of Subcase 4.2, we obtain that log r \| Ga( P, G )\| \leq \biggl( 1 2 \xi 2 \biggr) \biggl( n2 2 k2 2 \biggr) 3 n23 /12 + k2 4 \xi 2 \| A\| k + \xi 2 4 \| A\| 2 + 1 2 n11 /6 < \biggl( 1 2 \xi 2 \biggr) \biggl( n2 2 k2 2 \biggr) 3 n23 /12 + k2 4 \xi 2 2 k2 + \xi 2 4 n2 + 1 2 n11 /6 \leq n2 4 \xi 2 n2 4 + 4 n23 /12 . Proof of Theorem 6.3 . Let \scrC be the collection of containers given by Theorem 2.4. Theorem 2.4 indicates that every Gallai r-coloring of G is a subtemplate of some P \in \scrC and \| \scrC \| \leq 2cn 1/3 log 2 n(n 2 ) for some constant c, which only depends on r. We may assume that all templates P in \scrC are Gallai r-templates of G. By property (ii) of Theorem 2.4, we always have RT( P ) \leq n 1/3\bigl( n 3 \bigr) . Suppose that for a template P there exists an edge e \in E(G) with \| P (e)\| = 0. Then we would obtain \| Ga( P, G )\| = 0 as a Gallai r-coloring of G requires at least one color on each edge. Now applying Lemma 6.16 on every container P \in \scrC , we obtain that the number of Gallai r-colorings of G is at most \sum P\in \scrC \| Ga( P, G )\| \leq \| \scrC \| \cdot r n2 4\xi 3n2 2+4 n23 /12 < r \lfloor n2/4\rfloor , where the last inequality follows from \xi \gg n 1/36 for n sufficiently large. REFERENCES N. Alon, J. Balogh, P. Keevash, and B. Sudakov , The number of edge colorings with no monochromatic cliques , J. Lond. Math. Soc. (2), 70 (2004), pp. 273–288. J. Balogh, B. Bollob´ as, and M. Simonovits , The number of graphs without forbidden sub-graphs , J. Combin. Theory Ser. B, 91 (2004), pp. 1–24. J. Balogh, N. Bushaw, M. Collares, H. Liu, R. Morris, and M. Sharifzadeh , The typical structure of graphs with no large cliques , Combinatorica, 37 (2017), pp. 617–632. J. Balogh, R. Morris, and W. Samotij , Independent sets in hypergraphs , J. Amer. Math. Soc., 28 (2015), pp. 669–709. J. Balogh, R. Morris, and W. Samotij , The method of hypergraph containers , in Proceedings of the International Congress of Mathematicians, Vol. 3, Rio de Janeiro, Brazil, 2018, pp. 3045–3078. J. Balogh and J. Solymosi , On the number of points in general position in the plane , Discrete Anal., 2018 (2018), 16. J. Balogh and A. Z. 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Uzzell , Multicolour containers, extremal entropy and counting , Random Structures Algorithms, 54 (2018), pp. 676–720. J. Fox, A. Grinshpun, and J. Pach , The Erd¨ os–Hajnal conjecture for rainbow triangles , J. Combin. Theory Ser. B, 111 (2015), pp. 75–125. Downloaded 05/01/21 to 130.126.162.126. Redistribution subject to SIAM license or copyright; see Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE TYPICAL STRUCTURE OF GALLAI COLORINGS 2443 T. Gallai , Transitiv orientierbare graphen , Acta Math. Hungar., 18 (1967), pp. 25–66. A. Gy´ arf´ as and G. N. S´ ark¨ ozy , Gallai colorings of non-complete graphs , Discrete Math., 310 (2010), pp. 977–980. A. Gy´ arf´ as, G. N. S´ ark¨ ozy, A. Seb¨ o, and S. Selkow , Ramsey-type results for Gallai col-orings , J. Graph Theory, 64 (2010), pp. 233–243. A. Gy´ arf´ as and G. Simonyi , Edge colorings of complete graphs without tricolored triangles ,J. Graph Theory, 46 (2004), pp. 211–216. C. Hoppen, H. Lefmann, and K. Odermann , A rainbow Erd¨ os–Rothschild problem , Electron. Notes Discrete Math., 49 (2015), pp. 473–480. C. Hoppen, H. Lefmann, and K. Odermann , On graphs with a large number of edge-colorings avoiding a rainbow triangle , European J. Combin., 66 (2017), pp. 168–190. C. Hoppen, H. Lefmann, and K. Odermann , A rainbow Erd¨ os–Rothschild problem , SIAM J. Discrete Math., 31 (2017), pp. 2647–2674. J. K¨ orner and G. Simonyi , Graph pairs and their entropies: Modularity problems , Combina-torica, 20 (2000), pp. 227–240. L. Lov´ asz , Combinatorial Problems and Exercises , North-Holland, Amsterdam, 1979. D. Saxton and A. Thomason , Hypergraph containers , Invent. Math., 201 (2015), pp. 925–992. A. Z. Wagner , Large subgraphs in rainbow-triangle free colorings , J. Graph Theory, 86 (2017), pp. 141–148.
7668	https://www.ncbi.nlm.nih.gov/books/NBK549765/	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. StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2025 Jan-. StatPearls [Internet]. Show details Treasure Island (FL): StatPearls Publishing; 2025 Jan-. Physiology, Bile Acids Isaac Chen; Sebastiano Cassaro. Author Information and Affiliations Authors Isaac Chen1; Sebastiano Cassaro2. Affiliations 1 California Northstate University College of Medicine 2 UN California Irvine / Kaweah Delta Last Update: May 1, 2023. Introduction Bile acids are steroidal acids found in bile. Primary bile acids are steroids produced via the liver, specifically in peroxisomes. There, the acids conjugate/connect to hydrophilic amino acids, namely glycine/taurine (i.e., conjugated bile acids called glycocholic and taurocholic acids, respectively); alongside sodium/potassium, they are termed bile salts. Secondary bile acids, however, are made via colonic bacteria. Of the organic compounds found in bile, bile acids constitute the large majority. Right after being synthesized, bile acids are secreted into bile and concentrated for storage in the gallbladder. Eating then stimulates cholecystokinin release, which causes gallbladder contraction--releasing its bile acids into the duodenum through the sphincter of Oddi. The more bile acid that gets secreted, the faster that bile flows. The primary purpose of bile acids is to facilitate the digestion of fat via its surfactant properties, which emulsify fats into micelles. Hormonally, bile acids are also ligands for the farnesoid X receptor (FXR) and GPBAR1 (TGR5). In sum, the three main functions of bile acids are to (1) emulsify fat, (2) excrete cholesterol, and (3) have an antimicrobial effect. Function Bile acids get conjugated in the liver to increase their water solubility because bile salts have a decreased pK, which favors the basic anionic form in the acidic duodenum. As amphiphiles, conjugated bile salts have a hydrophobic and hydrophilic side that, hence, their function as surfactants. The hydrophobic portion faces lipids, the hydrophilic portion facing the water, which allows them to act as bridges at the lipid/water interface, where they can make micelles when sufficiently concentrated. Micelles, with their bile acids, can support the function of lipases in digesting lipids and also bring them close to the intestine's brush border, augmenting their absorption. Lipids are normally water-insoluble, and the stomach and intestines are full of water. If not for the action of bile, dietary lipids and water would not separate, and water-soluble lipases would inefficiently exert their effects at the small interface between water and lipids. The emulsification of lipids by bile salts thus suspends lipid particles in water to dissolve them, accordingly granting significantly more surface area for lipase action. Other functions of bile acids include removing the body's cholesterol, powering bile flow to remove various metabolites (like bilirubin), and facilitating the removal of bacterial flora of the small bowel and biliary tree (e.g., by disrupting their cellular membranes). Bile acids also hormonally act on the FXR and TGR5 receptors. Control over bile acid concentrations is important because of their potential cytotoxicity, so FXR elicits functions primarily in the liver/intestines that include feedback regulation of bile acid concentration, in addition to regulating triglyceride levels and other biochemical functions. Secondary bile acids, e.g., deoxycholic acid, may have implications in more strongly downregulating bile acid synthesis than primary bile acids when it comes to negative feedback. The TGR5 receptor is involved in the regulation of energy homeostasis by bile acids. In addition to its effects on the above-mentioned hormone receptors, bile acids target other proteins like N-acyl phosphatidylethanolamine-specific phospholipase D, involved in pathways related to stress/pain responses, appetite, and lifespan via crosstalk between lipid amide signals with bile acids. Other miscellaneous functions under the purview of bile acids potentially include the regulation of particular enzymes/ion channels and the synthesis of various substances. Mechanism Altogether, bile salts comprise a broad variety of different molecules, each being a steroid with the following basic components: (1) 4 rings, (2) a 5-/8-carbon side chain that ends with a carboxylic acid, and (3) a number of hydroxyl groups (whose position/number changes among the various salts). The rings are ascribed the letters A, B, C, and D based on their distance from the side chain with the -COOH group, the D ring being the most distant (as well as being 1 C smaller than the other rings). Beta hydroxyl groups face up/out, alpha groups down, and every bile acid has a 3-hydroxyl group that came from their cholesterol precursor. Primary bile acids are made by hepatocytes either by the classic or acidic pathway. The former, which constitutes 95% of bile acid synthesis, is via cytochrome P450-mediated oxidation of cholesterol that requires NADPH and oxygen and occurs in a series of steps, the most important of which is the rate-limiting hydroxylation of the 7th steroid nucleus of cholesterol by cholesterol 7alpha-hydroxylase (CYP7A1) to create 7alpha-hydroxycholesterol. That then gets metabolized into 7alpha-hydroxy-4 cholesten-3-one in the second step. In sum, primary bile acid synthesis requires 14 enzymatic steps. The most common bile acids are cholic and chenodeoxycholic acid (CDCA). Cholic acid (3alpha, 7alpha, 12alpha-trihydroxy-5beta-cholan-24-oic acid) is the commonest bile acid, and CDCA is the prototypical and most basic of bile acids produced and is also known as 3alpha,7alpha-dihydroxy-5beta-cholan-24-oic acid. The alternative/acidic pathway acts via mitochondrial sterol 27-hydroxylase (CYP27A1), found in the liver, macrophages, and other tissues. Before secretion of bile acids, hepatocytes conjugate them with the amino acids taurine or glycine, thereby creating eight different conjugated bile acids/salts (including cholic and chenocholic acids). CYP27A1 significantly adds to bile acid synthesis by facilitating the oxidation of sterol side chains, followed by a peroxisomal cleavage of a 3-C unit to generate a C24 bile acid. Miscellaneous pathways, in addition to these two, include those started by the liver's 25-hydroxylase and the brain's 24-hydroxylase. The purpose of conjugation is so that the bile acids can become water-soluble and thereby emulsify fats. Secondary bile acids are formed from bacterial deconjugation/dehydroxylation and removal of those amino acid groups, creating four more different types of bile acids (including deoxycholic and lithocholic acids). These acids are absorbed through the bloodstream and brought back to the liver via the enterohepatic circulation to then be resecreted. Most triglyceride absorption occurs at the jejunum, but the conjugated bile acids do not get absorbed along with them. Instead, the bile salts remain in the small intestine, where most are later absorbed and recycled by active transport at the terminal ileum (the remaining undergo fecal elimination). The end of the small intestine acting the site of bile salt absorption allows for a high concentration of bile salts to exist throughout the entire organ, maximizing lipid digestion and absorption. Pathophysiology Bile acid synthesis defects comprise about 1 to 2% of the cholestatic diseases found in children. With an autosomal recessive inheritance pattern, these defects have a spectrum of disease severity based on what they affect. Most commonly, these manifest as progressive cholestasis of infancy, in addition to other disorders like an advanced liver disease at birth or neonatal hepatitis; patients may also develop a liver disease later in childhood. Typically, an earlier onset of liver disease occurs with those enzymatic defects that result in a buildup of oxo-bile acids, which tend to be cholestatic. Adult liver disease may also have a connection to inherited defects in bile synthesis, as well. Clinical Significance Impairing enterohepatic circulation of bile acids decreases cholesterol because the liver is driven to use more cholesterol to make more. This state is the basis behind treating hyperlipidemia with bile acid resins like cholestyramine, colestipol, and colesevelam, which bind bile acids in the gut, preventing their reabsorption in the intestines. The bound bile acids are then fecally removed. However, because bile acid sequestrants also interfere with the absorption of fat, the absorption of drugs and fat-soluble vitamins are impaired. Structural/functional abnormalities of the biliary tract can produce an increase in bilirubin, causing jaundice, as well as an increase in serum bile acids. This observed cholestasis (disrupted bile flow to intestines) may also present with primary sclerosing cholangitis, intrahepatic cholestasis of pregnancy, and primary biliary cholangitis, where the buildup of bile acids can lead to pruritus because of the bile salts in skin. Treating these disorders has involved ursodeoxycholic acid (ursodiol), a non-toxic bile acid that increases bile secretion and decreases cholesterol secretion/absorption. Obeticholic acid is a semisynthetic bile acid that can also treat primary biliary cirrhosis by stimulating FXR. Bile that has diluted concentrations of bile acids or phospholipids, in addition to increased cholesterol, leads to a decrease in the solubilization of cholesterol. This situation increases the risk for microcrystal formation from this supersaturated bile--eventually leading to the formation of cholesterol stones. Fibrates like gemfibrozil, bezafibrate, and fenofibrate also increase the risk of cholesterol gallstones by inhibiting cholesterol-7alpha-hydroxylase, resulting in decreased bile acid production. Dissolution or prevention of these gallstones occurs with bile acid administration, in the form of chenodeoxycholic and/or ursodeoxycholic acid. Too much bile acid in the colon may precipitate chronic diarrhea, which may occur in conditions that impair the ileal absorption of bile salts, as seen with surgical removal for Crohn disease (which normally causes fat malabsorption by affecting the terminal ileum). Such malabsorption of bile acids is treatable with bile acid resins. There may also exist a link between bile acids and colorectal cancer. Specifically, low fecal concentrations of bile acids have correlations with lower rates of colorectal cancer. One potential explanation is that high concentrations of deoxycholic acid, a type of bile acid, can increase the generation of reactive oxygen species that can go on to damage DNA. Bile acids have also been used to get rid of undesired fat, i.e., mesotherapy, with deoxycholic acid being one such drug. Review Questions Access free multiple choice questions on this topic. Comment on this article. Figure Recycling of bile Image courtesy: References 1. : Hofmann AF, Hagey LR, Krasowski MD. Bile salts of vertebrates: structural variation and possible evolutionary significance. J Lipid Res. 2010 Feb;51(2):226-46. [PMC free article: PMC2803226] [PubMed: 19638645] 2. : HOFMANN AF. THE FUNCTION OF BILE SALTS IN FAT ABSORPTION. THE SOLVENT PROPERTIES OF DILUTE MICELLAR SOLUTIONS OF CONJUGATED BILE SALTS. Biochem J. 1963 Oct;89(1):57-68. [PMC free article: PMC1202272] [PubMed: 14097367] 3. : Ferdinandusse S, Houten SM. Peroxisomes and bile acid biosynthesis. 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THE INTRALUMINAL PHASE OF FAT DIGESTION IN MAN: THE LIPID CONTENT OF THE MICELLAR AND OIL PHASES OF INTESTINAL CONTENT OBTAINED DURING FAT DIGESTION AND ABSORPTION. J Clin Invest. 1964 Feb;43(2):247-57. [PMC free article: PMC289518] [PubMed: 14162533] 10. : Fiorucci S, Mencarelli A, Palladino G, Cipriani S. Bile-acid-activated receptors: targeting TGR5 and farnesoid-X-receptor in lipid and glucose disorders. Trends Pharmacol Sci. 2009 Nov;30(11):570-80. [PubMed: 19758712] 11. : Kim I, Ahn SH, Inagaki T, Choi M, Ito S, Guo GL, Kliewer SA, Gonzalez FJ. Differential regulation of bile acid homeostasis by the farnesoid X receptor in liver and intestine. J Lipid Res. 2007 Dec;48(12):2664-72. [PubMed: 17720959] 12. : Makishima M, Okamoto AY, Repa JJ, Tu H, Learned RM, Luk A, Hull MV, Lustig KD, Mangelsdorf DJ, Shan B. Identification of a nuclear receptor for bile acids. Science. 1999 May 21;284(5418):1362-5. [PubMed: 10334992] 13. : Parks DJ, Blanchard SG, Bledsoe RK, Chandra G, Consler TG, Kliewer SA, Stimmel JB, Willson TM, Zavacki AM, Moore DD, Lehmann JM. Bile acids: natural ligands for an orphan nuclear receptor. Science. 1999 May 21;284(5418):1365-8. [PubMed: 10334993] 14. : Wang H, Chen J, Hollister K, Sowers LC, Forman BM. Endogenous bile acids are ligands for the nuclear receptor FXR/BAR. Mol Cell. 1999 May;3(5):543-53. [PubMed: 10360171] 15. : Stange EF, Scheibner J, Ditschuneit H. Role of primary and secondary bile acids as feedback inhibitors of bile acid synthesis in the rat in vivo. J Clin Invest. 1989 Jul;84(1):173-80. [PMC free article: PMC303967] [PubMed: 2738150] 16. : Jiao Y, Lu Y, Li XY. Farnesoid X receptor: a master regulator of hepatic triglyceride and glucose homeostasis. Acta Pharmacol Sin. 2015 Jan;36(1):44-50. [PMC free article: PMC4571315] [PubMed: 25500875] 17. : Baxter JD, Webb P. Metabolism: bile acids heat things up. Nature. 2006 Jan 26;439(7075):402-3. [PubMed: 16437098] 18. : Nagahashi M, Takabe K, Liu R, Peng K, Wang X, Wang Y, Hait NC, Wang X, Allegood JC, Yamada A, Aoyagi T, Liang J, Pandak WM, Spiegel S, Hylemon PB, Zhou H. Conjugated bile acid-activated S1P receptor 2 is a key regulator of sphingosine kinase 2 and hepatic gene expression. Hepatology. 2015 Apr;61(4):1216-26. [PMC free article: PMC4376566] [PubMed: 25363242] 19. : Wiemuth D, Sahin H, Falkenburger BH, Lefèvre CM, Wasmuth HE, Gründer S. BASIC--a bile acid-sensitive ion channel highly expressed in bile ducts. FASEB J. 2012 Oct;26(10):4122-30. [PubMed: 22735174] 20. : Magotti P, Bauer I, Igarashi M, Babagoli M, Marotta R, Piomelli D, Garau G. Structure of human N-acylphosphatidylethanolamine-hydrolyzing phospholipase D: regulation of fatty acid ethanolamide biosynthesis by bile acids. Structure. 2015 Mar 03;23(3):598-604. [PMC free article: PMC4351732] [PubMed: 25684574] 21. : Bove KE, Heubi JE, Balistreri WF, Setchell KD. Bile acid synthetic defects and liver disease: a comprehensive review. Pediatr Dev Pathol. 2004 Jul-Aug;7(4):315-34. [PubMed: 15383928] 22. : Stieger B, Zhang J, O'Neill B, Sjövall J, Meier PJ. Differential interaction of bile acids from patients with inborn errors of bile acid synthesis with hepatocellular bile acid transporters. Eur J Biochem. 1997 Feb 15;244(1):39-44. [PubMed: 9063443] 23. : Fischler B, Bodin K, Stjernman H, Olin M, Hansson M, Sjövall J, Björkhem I. Cholestatic liver disease in adults may be due to an inherited defect in bile acid biosynthesis. J Intern Med. 2007 Aug;262(2):254-62. [PubMed: 17645593] 24. : Davidson MH. A systematic review of bile acid sequestrant therapy in children with familial hypercholesterolemia. J Clin Lipidol. 2011 Mar-Apr;5(2):76-81. [PubMed: 21392720] 25. : Pusl T, Beuers U. Intrahepatic cholestasis of pregnancy. Orphanet J Rare Dis. 2007 May 29;2:26. [PMC free article: PMC1891276] [PubMed: 17535422] 26. : Poupon RE, Balkau B, Eschwège E, Poupon R. A multicenter, controlled trial of ursodiol for the treatment of primary biliary cirrhosis. UDCA-PBC Study Group. N Engl J Med. 1991 May 30;324(22):1548-54. [PubMed: 1674105] 27. : Glantz A, Marschall HU, Lammert F, Mattsson LA. Intrahepatic cholestasis of pregnancy: a randomized controlled trial comparing dexamethasone and ursodeoxycholic acid. Hepatology. 2005 Dec;42(6):1399-405. [PubMed: 16317669] 28. : Post SM, Duez H, Gervois PP, Staels B, Kuipers F, Princen HM. Fibrates suppress bile acid synthesis via peroxisome proliferator-activated receptor-alpha-mediated downregulation of cholesterol 7alpha-hydroxylase and sterol 27-hydroxylase expression. Arterioscler Thromb Vasc Biol. 2001 Nov;21(11):1840-5. [PubMed: 11701475] 29. : Danzinger RG, Hofmann AF, Schoenfield LJ, Thistle JL. Dissolution of cholesterol gallstones by chenodeoxycholic acid. N Engl J Med. 1972 Jan 06;286(1):1-8. [PubMed: 5006919] 30. : Thistle JL, Hofmann AF. Efficacy and specificity of chenodeoxycholic acid therapy for dissolving gallstones. N Engl J Med. 1973 Sep 27;289(13):655-9. [PubMed: 4580472] 31. : Pattni S, Walters JR. Recent advances in the understanding of bile acid malabsorption. Br Med Bull. 2009;92:79-93. [PubMed: 19900947] 32. : Degirolamo C, Modica S, Palasciano G, Moschetta A. Bile acids and colon cancer: Solving the puzzle with nuclear receptors. Trends Mol Med. 2011 Oct;17(10):564-72. [PubMed: 21724466] 33. : Reddy BS, Hanson D, Mangat S, Mathews L, Sbaschnig M, Sharma C, Simi B. Effect of high-fat, high-beef diet and of mode of cooking of beef in the diet on fecal bacterial enzymes and fecal bile acids and neutral sterols. J Nutr. 1980 Sep;110(9):1880-7. [PubMed: 7411244] 34. : Hill MJ. Bile flow and colon cancer. Mutat Res. 1990 May;238(3):313-20. [PubMed: 2188127] 35. : Cheah PY. Hypotheses for the etiology of colorectal cancer--an overview. Nutr Cancer. 1990;14(1):5-13. [PubMed: 2195469] 36. : Bernstein H, Bernstein C, Payne CM, Dvorak K. Bile acids as endogenous etiologic agents in gastrointestinal cancer. World J Gastroenterol. 2009 Jul 21;15(27):3329-40. [PMC free article: PMC2712893] [PubMed: 19610133] 37. : Ascher B, Hoffmann K, Walker P, Lippert S, Wollina U, Havlickova B. Efficacy, patient-reported outcomes and safety profile of ATX-101 (deoxycholic acid), an injectable drug for the reduction of unwanted submental fat: results from a phase III, randomized, placebo-controlled study. J Eur Acad Dermatol Venereol. 2014 Dec;28(12):1707-15. [PMC free article: PMC4263247] [PubMed: 24605812] 38. : Wollina U, Goldman A. ATX-101 for reduction of submental fat. Expert Opin Pharmacother. 2015 Apr;16(5):755-62. [PubMed: 25724831] : Disclosure: Isaac Chen declares no relevant financial relationships with ineligible companies. : Disclosure: Sebastiano Cassaro declares no relevant financial relationships with ineligible companies. 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7669	https://allnurseryrhymes.com/as-i-was-going-to-st-ives/	As I Was Going To St Ives \| Nursery Rhymes Skip to content ♬ Home International Songs Christmas Songs Riddles A B C D E F G H I J K L M O P Q R S T W Y Z Main Menu ♬ Home International Songs Christmas Songs Riddles As I was going to St Ives Traditional nursery rhymes “As I was going to St Ives” is a riddle rhyme, having its origins in the 18th century England. Most probably St Ives refers to an old village in Cornwall, England. The riddle is more of a logical problem than a mathematical one; still the answer to the riddle is uncertain: How many were going to St Ives? Actually we need to find out who is finally going to St Ives? Only the narrator is going? The narrator with the man he meets? The narrator together with all the people he meets up? Are also all the animals counted? Or perhaps the riddle is even more complicated: Kits, cats, sacks, wives / how many were going to St Ives? In this case the answer would be none. Bellow will find the modern version of the song, first published in 1825. “As I was going to St Ives” Lyrics As I was going to St Ives I met a man with seven wives Every wife had seven sacks Every sack had seven cats Every cat had seven kits Kits, cats, sacks, wives How many were going to St Ives? Post navigation ← Previous Post Next Post → Copyright © 2024 Nursery Rhymes Home About Feedback Privacy Policy Scroll to Top
7670	https://bnrc.springeropen.com/articles/10.1186/s42269-023-01159-x	Utility of dipstick urinalysis in the diagnosis of urinary tract infections among outpatients in Mwanza and Dar es Salaam regions in Tanzania \| Bulletin of the National Research Centre \| Full Text 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 Advertisement Search Get published Explore Journals Books About Login Menu Get published Explore Journals Books About Login Search all SpringerOpen articles Search Bulletin of the National Research Centre Official International Journal of the National Research Centre of Egypt About Articles Submission Guidelines Submit manuscript Utility of dipstick urinalysis in the diagnosis of urinary tract infections among outpatients in Mwanza and Dar es Salaam regions in Tanzania Download PDF Download PDF Research Open access Published: 02 January 2024 Utility of dipstick urinalysis in the diagnosis of urinary tract infections among outpatients in Mwanza and Dar es Salaam regions in Tanzania Salim S. MasoudORCID: orcid.org/0000-0002-0273-42571, Mtebe Majigo1, Vitus Silago2, Peter Kunambi1, Helmut Nyawale2, Nyambura Moremi3, Erick Komba4, Fauster X. Mgaya1, Emmanuel Magembe3, Mariam M. Mirambo2, Stephen E. Mshana2,4& … Mecky Isaac Matee1,4 Show authors Bulletin of the National Research Centrevolume 48, Article number:6 (2024) Cite this article 3161 Accesses 4 Citations Metrics details Abstract Background Globally, urinary tract infections (UTIs) are the leading cause of hospital visits, especially among women. In many developing countries, especially in lower health facilities, the dipstick urinalysis is the most commonly requested test, as urine culture is not routinely performed. The dipstick test can potentially reduce the number of patients who could be treated with empirical antibiotics and reduce the burden of antibiotic resistance. The current study determined the utility of dipstick urinalysis in diagnosing UTIs using urine culture as a gold standard method. Methods A cross-sectional study was conducted between July and November 2021. The study involved 1327 outpatients attending lower health centres in Mwanza (n = 678) and Dar es Salaam (n = 649) regions. The samples were subjected to dipstick urinalysis (nitrite and leucocytes) and quantitative bacterial culture. Results Of all 1326 patient samples, 808(60.9%) and 48(3.6%) were positive for urinary leucocyte and nitrite, respectively, while significant urine culture was found in 364(27.4%). leucocyte test correctly diagnosed UTI in 283 (77.75%) and nitrite test in 36 (9.89%). The leucocytes and nitrites, in combination, exhibited a sensitivity of 79.40% and NPV of 85.24% but a lower specificity of 44.96% and a lower PPV of 35.29%. Urinary leucocyte test had a higher discrimination ability to detect urinary tract infection than urinary nitrite test (AUC = 0.073, 95% CI 0.043–0.103, p< 0.001); likewise, combined results of urinary nitrite and leucocytes tests had higher discrimination ability to detect UTI than nitrite only (AUC = −0.079, 95% CI −0.107–0.050, p< 0.001). Conclusions In settings where culture is available, dipstick urinalysis can be a helpful screening method for reducing unnecessary urine cultures and related expenses because of its higher negative predictive value. In most low-resource settings where patients' diagnosis solely depends on clinical diagnosis and culture is not easily available, urine dipstick can decrease the risk of overuse of antibiotics. However, the combined (leucocytes and nitrites) dipstick urinalysis has a rather low positive predictive value, with approximately one-third of cases giving positive test results being truly UTI, subjecting over 65% of cases to unnecessary antibiotic treatment. Background Urinary tract infection (UTI) is among the most common infectious diseases, contributing to a large part of the workload in clinical laboratories (Wilson and Gaido 2004F ")). In Tanzania, culture-based laboratory-confirmed UTIs range from 11 to 23% among different populations (Msaki et al. 2012 Prevalence and predictors of urinary tract infection and severe malaria among febrile children attending Makongoro health centre in Mwanza city, North–Western Tanzania. Arch Public Health 70:1–8. "); Chaula et al. 2017 Urinary tract infections and associated factors in HIV infected pregnant women at a tertiary hospital in Lusaka Zambia. Pan Af Med J. "); Gidabayda et al. 2017 Prevalence, aetiology, and antimicrobial susceptibility patterns of urinary tract infection amongst children admitted at Kilimanjaro christian medical centre, Moshi, Tanzania. East Afr Health Res J 1:53–61"); Mwambete and Malaba 2017 High prevalence of antibiotic resistance among bacteria isolated from pregnant women with asymptomatic urinary tract infections in Dar es Salaam. Tanzan Res J Health Sci 5:65. ")). Adoption of urine culture, a gold standard technique, is limited to higher healthcare facilities such as regional and zonal referral hospitals. On the other hand, in lower healthcare facilities (dispensaries, health centres, and district hospitals), dipstick urinalysis is commonly ordered to diagnose UTI. Usually, to initiate UTI treatment based on dipstick urinalysis, one should consider a positive leucocyte esterase and/or nitrite in conjunction with the patient's clinical history (Bates 2013). However, diagnosing UTI using dipstick urinalysis and clinical history is challenging as it can give false-positive results in patients with other underlying illnesses (Wilson and Gaido 2004F ")). Dipstick can reduce the number of patients prescribed antibiotics in settings where diagnosis depends only on the patient's signs and symptoms. Clinical diagnosis without laboratory investigation can lead to unnecessary UTI treatment and healthcare costs from increased over-prescription of antibiotics, which is associated with the emergence and spreading of antimicrobial resistance (Bates 2013 Interpretation of urinalysis and urine culture for UTI treatment. Pharm Fac US Pharm 38(11):65–68")). The biochemical reactions in dipstick urinalysis detect bacteriuria and pyuria by detecting the presence of nitrite and leucocyte esterase in a urine sample, respectively (Wilson and Gaido 2004F ")). Nitrite results from the conversion of nitrate to nitrite by members of the family _Enterobacterales_. On the other hand, leucocyte esterase is associated with the hydrolysis of ester substrate on dipstick strips by esterolytic proteins (esterases) present in both intact and lysed leucocytes in a urine sample (Wilson and Gaido 2004 Laboratory diagnosis of urinary tract infections in adult patients. Clin Infect Dis 38:1150–1158. "); Gordon et al. 2013 Overtreatment of presumed urinary tract infection in older women presenting to the emergency department. J Am Geriatr Soc 61:788–792. "); Marques et al. 2017 Performance of the dipstick screening test as a predictor of negative urine culture. Einstein 15:34. ")a). Dipstick urinalysis is a common clinical practice most frequently requested test, especially at point-of-care, emergency medicine departments, and health centres in the majority of healthcare facilities across the world, including Tanzania, where urine (Lei et al. 20203 ")). However, its utility against urine culture as a gold standard test is not well documented. (Sangeda et al. 2021 Prevalence of urinary tract infections and antibiogram of uropathogens isolated from children under five attending Bagamoyo district hospital in Tanzania: a cross-sectional study. F1000 Research 10: 449 ")). The lack of evidence-based findings limits our recommendation for the appropriate use of dipstick urinalysis in routine diagnosis of UTI in settings where urine culture is unavailable. Therefore, this study aimed to investigate the utility of dipstick urinalysis (leucocyte and nitrite) in diagnosing UTIs in lower-resource clinical settings of health facilities. Methods Study design and population This cross-sectional study was conducted in lower health centres in Mwanza and Dar es Salaam for a period of five months continuously between July and November 2021. This was a nested study under a big study that determined the prevalence of multidrug-resistant bacteria in urine (Silago et al. 20228 ")). Sampling involved participants from the age of two (men and both non-pregnant and pregnant women) who were residents of a given surveillance area and passively presented for health care. All participants were outpatients with signs and symptoms of UTI occurring < 48 h before enrolment (Silago et al. 2022 Multidrug-resistant uropathogens causing community acquired urinary tract infections among patients attending health facilities in Mwanza and Dar es Salaam Tanzania. Antibiotics. ")). Study setting and site selection The study was conducted in four health facilities, two from Dar es Salaam and two from Mwanza regions. Selected surveillance sites were primary health facilities, i.e. dispensaries and health centres located within 50 km of a laboratory where urine cultures were performed. Data and urine sample collection Quantitative data on patients' socio-demographic and clinical characteristics were collected electronically by AfyaData® version 1.4, an open-source software for collecting data from health facilities and submitting it to the main server. On the other hand, laboratory data were captured using the WHONET software 2022, a modernized and expanded version of WHONET 5.6. From each patient, 5 to 10 millilitres of mid-stream urine (MSU) were collected in a sterile urine container and immediately stored in the refrigerator to control the rate of bacterial multiplication that could affect the colony-forming unit (CFU) determination. In all facilities, samples were collected between 0900 and 1200 h and transported in a cool box to the testing laboratory for processing within two hours of collection. Laboratory procedures Urinalysis A urine sample was examined macroscopically for colour and turbidity (e.g. yellow and cloudy, pale yellow and clear) and then by dipstick urinalysis to detect leucocyte esterase and nitrite. The latter was performed and interpreted according to the manufacturer’s instructions (Mission® urinalysis; ACON Laboratories, San Diego, California, USA). A positive dipstick was defined as the presence of nitrites or leucocytes greater than or equal to a trace. A negative dipstick was defined as the absence of any reaction for leucocytes and nitrites. Urine dipstick bottles were stored at room temperature, and expiration dates were checked before use. Urine dipstick test results were interpreted according to the manufacturer’s instructions. Urine culture One microlitre (1 μl) loop was used for quantitative inoculation of urine on MacConkey agar (MCA) with salt (Oxoid, UK), 5% sheep blood supplemented Columbia Blood Agar (BA) (Oxoid, UK), and cysteine–lactose–electrolyte-deficient (CLED) (Oxoid, UK). MCA and CLED plates were incubated aerobically, whereas BA plates were incubated in a candle jar to obtain 5–10% CO 2. All plates were incubated at 35 ± 2°C for 24 h. Plates were examined for growth, colony morphology, and characteristics on a culture medium. Bacterial counts of ≥ 10 5 CFU/mL of one or two species of microorganisms indicated UTI. Contamination was defined as > 10 5 growth of more than two species or < 105 CFU/ml growth. Data analysis Collected data were entered in a Microsoft Excel spreadsheet and transferred to a statistical package for the social sciences (SPSS) version 27 (Armonk, NY: IBM Corp) for analysis. Frequencies and percentages were used for categorical variables, mean (standard deviation (SD)) was used for continuous variables, and descriptive statistics were computed for different attributes. The performance of dipstick urinalysis in detecting UTI cases was evaluated by receiver operating characteristic (ROC) to evaluate the discriminatory power of the urine dipstick test to diagnose UTI. An ROC curve plots the true-positive rate (sensitivity) against the false-positive rate for a diagnostic test's different possible cutoff points. The closer the curve follows the left-hand border and then the top border of the ROC space, the more accurate the test. Results A total of 1327 urine samples were tested for UTI using urine culture and dipstick tests for nitrite and leucocyte detection. Of all, 808(60.9%) and 48(3.6%) were positive for urinary leucocyte and nitrite, respectively (Fig.1). Significant urine culture was found in 364(27.4%). Fig.1 Comparison of urinary leucocyte and nitrite results of the dipstick methods versus growth of significant bacteria in the urine culture Full size image Performance of urinary leucocyte test The comparison of dipstick results with urine culture showed that urinary leucocytes had correctly diagnosed the presence of UTI in 283 (77.75%) of the 364 samples with significant bacterial growth in urine culture. In addition, urinary leucocytes had correctly identified the absence of UTI in 438 (45.48%) samples among the 963 samples with no significant bacterial growth in the culture (Fig.1). The test had a sensitivity (true positives) of 77.75% (95% CI 73.47–82.02), a specificity (true negatives) of 45.48% (95% CI 42.34–48.63%), and positive and negative predictive values of 35.02% (95% CI 31.74–38.31) and 84.39% (81.27–87.52), respectively. Furthermore, urinary leucocytes were found to have positive and negative diagnostic likelihood ratios of 1.43 (95% 1.32–1.54) and 0.49 (95% CI 0.40–0.60), as well as a discriminatory power of detecting the presence of urinary tract infection of 0.616 (95% CI 0.584–0.649), as denoted by the area under the receiver operating curve. (Fig.2A). Fig.2 A. Receiver operating characteristics curve of urinary leucocytes in detecting UTI. B. Receiving operating characteristics curve of urinary nitrite in detecting UTI. C Receiver operating characteristics curve of combined results or urinary leucocytes and nitrites in detecting UTI Full size image Performance of urinary nitrite test Urinary nitrite correctly diagnosed the presence of UTI in 36 (9.89%) of the 364 samples with significant bacterial growth in the culture and correctly identified the absence of UTI in 951 (98.75%) among 963 samples with no significant bacterial growth in the culture. However, urinary nitrites' false positive and negative results were 12 and 328, respectively (Fig.1). The corresponding sensitivity, specificity, positive, and negative predictive values were 9.89% (95% CI 6.82–12.96), 98.75% (95% CI 98.05–99.45), 75.00% (95% CI 62.75–87.25), and 74.35% (95% CI 71.96–76.75), respectively. Furthermore, urinary nitrite was found to have positive and negative diagnostic likelihood ratios of 7.94 (95% CI 4.18–15.08) and 0.91 (95% CI 0.88–0.94), as well as a discriminatory ability to detect the presence of urinary tract infection of 0.543 (95% CI 0.51–0.58, P = 0.015), as denoted by the area under the receiver operating curve (Fig.2B). Performance of leucocyte and nitrite tests Combination of urinary leucocyte and nitrite with the presence of either one of them or both regarded as positive for UTI, the urine dipstick correctly diagnosed 289 (79.40%) among 364 samples with significant bacterial growth in the urine culture and correctly identified the absence of UTI in 433 (44.96%) among 963 samples with no significant bacterial growth in the culture (Fig.1). The combined results of urinary leucocytes and nitrite had sensitivity, specificity, and positive and negative predictive values of 79.40% (95% CI 75.24–83.55), 44.96% (95% CI 41.82–48.11), 35.29 (95% CI 32.01–38.56), and 85.24% (95% CI 82.15–88.32), respectively (Table1). In addition to that, the combined results were found to have positive and negative diagnostic likelihood ratios of 1.44 (95% CI 1.34–1.56) and 0.46 (95% CI 0.37–0.57), respectively, as well as a discriminatory ability to detect the presence of UTI of 0.622 (95%CI 0.59–0.65, p< 0.001), denoted by the area under the receiver operating curve (Fig.2C). Table 1 Sensitivity, specificity, diagnostic accuracy, positive and negative predictive values, and positive and negative diagnostic likelihood ratios of combination urinary leucocyte and nitrite in diagnosing UTI Full size table The urinary leucocyte test has a higher sensitivity (77.75% vs 9.89%) and a higher negative predictive value (84.39% vs 74.35%) than the urinary nitrite test. On the other hand, urinary nitrite has a higher specificity (98.75% vs 45.48%) and a higher positive predictive value (75.00% vs 35.29%) compared to urinary leucocytes. The combined results of urinary leucocytes and nitrite have even higher sensitivity (79.40%) and negative predictive value (85.24%) than either test alone (Table2). Table 2 Comparisons of predictive values and diagnostic likelihood ratios between urinary leucocytes and nitrite in diagnosing UTI Full size table Urinary leucocytes had a higher discrimination ability to detect urinary tract infection than urinary nitrite (difference of AUC = 0.073, 95% CI 0.043–0.103, p< 0.001); likewise, combined results of urinary nitrite and leucocytes had higher discrimination ability to detect UTI than nitrite only (difference of AUC = −0.079, 95% CI −0.107–0.050, p< 0.001). There was no statistically significant difference in the ability to detect UTI between urinary leucocytes and combined results of urinary leucocytes and nitrites (difference of AUC = −0.006, 95% CI −0.013–0.001, p = 0.110) (Fig.2). Discussion This study was part of a surveillance project that aimed to (i) develop a common protocol for surveillance of antimicrobial-resistant bacteria causing community-acquired urinary tract infections in low-income countries and (ii) assess multidrug-resistant bacteria causing community-acquired urinary tract infections. This arm of the study aimed to find the utility of the nitrite/leucocyte dipstick test in screening asymptomatic community-acquired UTIs among suspected adult cases in Tanzania. The dipstick test is usually ‘first-line’ in the diagnosis of UTI because is easy to test, low cost, and can be used as a point-of-care test. Two components of the dipstick, leucocyte esterase, and nitrites are utilized to diagnose UTIs. leucocyte esterase is released by white blood cells (WBCs) when there is an infection in the urinary tract, which correlates with pyuria. Some gram-negative bacteria, such as Escherichia coli, routinely convert urinary nitrates in the urine into nitrite. Therefore, a positive nitrite test on the urine dipstick suggests the presence of these organisms. When using leucocytes alone, we found moderate sensitivity (77.75%) and low specificity (45.8%) of leucocytes, while nitrites had a low sensitivity (9.89%) and a high specificity (98.75%). When the two tests were combined, the sensitivity and specificity were found to be 79.40% and 44.96%, respectively. The leucocyte test had higher sensitivity, while the nitrite test had higher specificity. The high sensitivity and low specificity reported in this study's findings can be attributed to the high number of gram-positive bacteria isolated, evidenced by previously published articles from this research project (Silago et al. 20228 ")). A positive nitrite test result reduces nitrate, a byproduct of some bacteria, mainly those in the _Enterobacterales_ family. It is important to remember that some uropathogens, except those in the family _Enterobacterales_, cannot convert nitrate to nitrite, which might occasionally result in false-negative test findings (Demilie et al. 2014 Diagnostic accuracy of rapid urine dipstick test to predict urinary tract infection among pregnant women in Felege Hiwot referral Hospital, Bahir Dar. North West Ethiopia BMC Res Notes 7:481. "); Marques et al. 2017 Performance of the dipstick screening test as a predictor of negative urine culture. Einstein 15:34. ")b). The high sensitivity reported in this study may also be attributed to leukocyturia, which is not specific to UTIs and may occur with other inflammatory disorders of the genitourinary tract. Also, false positives can occur in contaminated samples, some dietary factors, certain medications like non-steroid anti-inflammatory drugs, and some medical conditions like dehydration. Leukocyturia may persist after bacteriuria has cleared spontaneously or after treatment (Franz and Hörl 1999 Common errors in diagnosis and management of urinary tract infection. I: pathophysiology and diagnostic techniques. Nephrol Dial Transplant 14:2746–2753. "); Schumm and Lam 2008 Types of urethral catheters for management of short-term voiding problems in hospitalized adults: a short version cochrane review. Neurourol Urodyn 27:738–746. ")). False-negative leucocyte results can occur in different conditions like the presence of atypical bacteria, diluted urine, and timing of sample collection. (Early morning sample collection is concentrated compared to later in the day.) (Dadzie et al. 2019 The effectiveness of dipstick for the detection of urinary tract infection. Can J Infect Dis Med Microbiol 23:2019")) The leucocyte urine dipstick exhibited positive and negative predictive values of 35.02% and 84.39%, respectively. This implies that when leucocytes are present, there is a 35.02% chance of a true positive, and on the other hand, the absence of leucocytes indicates an 84.39% likelihood of a true negative outcome. On the other hand, the nitrite tests displayed positive and negative predictive values of 75.00% and 74.35%, respectively. These values signify that a positive nitrite test has a 75.00% probability of being a true positive result, and a negative nitrite test has a 74.35% chance of being a true negative result. Combining the nitrite and leucocyte results showed an increased negative predictive value (NPV) from (urine leucocytes 35.02 and nitrite 74.35 to a combined 85.24%), but a decreased positive predictive value (PPV) (urine leucocytes 84.39% and nitrite 74.35% to combined 35.29%). The combined (leucocytes and nitrites) dipstick urinalysis has a rather low positive predictive value, with 35.29% of cases giving positive test results being truly UTI, subjecting over 64.7% of cases to unnecessary antibiotic treatment. This suggests that the NPV attributed to the combined nitrite, and leucocyte test is a value indicator for ruling out UTI. The increase in NPV is similarly reported in multiple studies showing the importance of urine dipstick screening for community-acquired UTIs (Demilie et al. 20141 "); Marques et al. 2017 Performance of the dipstick screening test as a predictor of negative urine culture. Einstein 15:34. ")b). A screening urine dipstick test with a high NPV rules out the presence of UTI, eliminating the additional urine culture testing and reducing unnecessary tests and associated costs for the patient and the healthcare system. The estimated cost per urine dipstick is $1, and approximately $7 for urine culture, including additional costs of consumables and personnel. The urine dipstick has a low diagnostic accuracy compared to urine culture; however, with a high NPV and a low cost provides a strong rationale for nominating the urine dipstick as a frontline screening test for community-acquired UTIs (Schumm and Lam 20085 "); Mignini et al. 2009 Accuracy of diagnostic tests to detect asymptomatic bacteriuria during pregnancy. Obstet Gynecol 113:346–352. ")). Moreover, combining nitrite and leucocyte esterase tests allows for quicker diagnosis and therapy. When screening tests yield a reliable negative result, healthcare professionals may rule-out UTIs as the underlying cause of the patient's symptoms. The exclusion of UTIs enables healthcare professionals to focus on investigating and addressing other potential conditions, leading to improved patient care and a reduction in unnecessary antibiotic usage (Flokas et al. 20177 "); Childers et al. 2022 Urine testing is associated with inappropriate antibiotic use and increased length of stay in emergency department patients. Heliyon 8:e11049. ")). A similar study suggested that urine dipstick should follow a _rule-out_ strategy as a negative dipstick test is indicative of a low probability of a positive urine culture, making the diagnosis of UTI unlikely, thus reducing the number of samples for culture and the use of empirical antimicrobial therapy (Ginting et al. 2018 Predictive value of the urinary dipstick test in the management of patients with urinary tract infection-associated symptoms in primary care in Indonesia: a cross-sectional study. BMJ Open 8:23051. ")). This study finding reports a high NPV of the urine dipstick, supporting this strategy for setting where culture can be performed. The challenge for the rule-out strategy is false-negative urinalysis results. Still, these patients may be asked to return for further testing if symptoms persist, as in some patients, uncomplicated UTIs are a self-limiting condition. The alternative of the _rule-in_ strategy is the commonly used method in a setting where culture is unavailable. The urine dipstick is the sole diagnosis method and an indication to start antibiotic treatment (Rousham et al. 2019 Overprescribing antibiotics for asymptomatic bacteriuria in older adults: a case series review of admissions in two UK hospitals. Antimicro Resist Infect Control 8(1):1–8")). The _rule-in_ strategy could lead to reduced antimicrobial use compared to clinical diagnosis, where only signs and symptoms are used as criteria to start treating UTI (Ginting et al. 2018 Predictive value of the urinary dipstick test in the management of patients with urinary tract infection-associated symptoms in primary care in Indonesia: a cross-sectional study. BMJ Open 8:23051. ")). This practice, however, has the potential to overuse antibiotics, fostering the emergence of bacterial strains, reducing the effectiveness of crucial antibiotics as culture is not performed, and treating patients according to antibiotic susceptibility results (Rousham et al. 2019 Overprescribing antibiotics for asymptomatic bacteriuria in older adults: a case series review of admissions in two UK hospitals. Antimicro Resist Infect Control 8(1):1–8"); Ginting et al. 2018 Predictive value of the urinary dipstick test in the management of patients with urinary tract infection-associated symptoms in primary care in Indonesia: a cross-sectional study. BMJ Open 8:23051. ")). In countries where culture is not available in most healthcare settings, and most patients are treated empirically by clinical diagnosis, the dipstick test can be applied to reduce the use of antibiotics, given its high NPV. The disadvantage of this test, when used alone without confirmation of culture, is that it has high false-positive results, which can increase the overuse of antibiotics, which may lead to an increase in antibiotic resistance. Conclusions Combining leucocytes and nitrite in a dipstick test has a low specificity and PPV. However, the sensitivity and NPV increase. An actual urinary tract infection (UTI) is confirmed only in around one-third of cases with positive results. This situation leads to over 65% of cases receiving unnecessary antibiotic treatment. An inexpensive urine dipstick is useful for excluding UTIs and minimizing the need for needless urine culture tests and related expenses. This test can be used in lower healthcare settings where clinical diagnosis reduces over-prescription of antibiotics. In higher health care settings where culture can be performed, dipstick can be used as a screening tool to reduce the number of samples for culture. Availability of data and materials All data for this article are presented in this manuscript; more data can be requested from the corresponding author. Abbreviations UTI: Urinary tract infection CFU: Colony-forming unit WBCs: White blood cells PPV: Positive predictive value NPV: Negative predictive value ROC: Receiver operating characteristics References Bates B (2013) Interpretation of urinalysis and urine culture for UTI treatment. Pharm Fac US Pharm 38(11):65–68 Google Scholar Childers R, Liotta B, Brennan J et al (2022) Urine testing is associated with inappropriate antibiotic use and increased length of stay in emergency department patients. Heliyon 8:e11049. ArticleCASPubMedPubMed CentralGoogle Scholar Dadzie I, Quansah E, Puopelle Dakorah M, Abiade V, Takyi-Amuah E, Adusei R (2019) The effectiveness of dipstick for the detection of urinary tract infection. Can J Infect Dis Med Microbiol 23:2019 Google Scholar Demilie T, Beyene G, Melaku S, Tsegaye W (2014) Diagnostic accuracy of rapid urine dipstick test to predict urinary tract infection among pregnant women in Felege Hiwot referral Hospital, Bahir Dar. North West Ethiopia BMC Res Notes 7:481. ArticlePubMedGoogle Scholar Flokas ME, Andreatos N, Alevizakos M et al (2017) Inappropriate management of asymptomatic patients with positive urine cultures: a systematic review and meta-analysis. Open Forum Infect Dis. ArticlePubMedPubMed CentralGoogle Scholar Franz M, Hörl WH (1999) Common errors in diagnosis and management of urinary tract infection. I: pathophysiology and diagnostic techniques. Nephrol Dial Transplant 14:2746–2753. ArticleCASPubMedGoogle Scholar Gidabayda J, Philemon R, Abdallah M et al (2017) Prevalence, aetiology, and antimicrobial susceptibility patterns of urinary tract infection amongst children admitted at Kilimanjaro christian medical centre, Moshi, Tanzania. East Afr Health Res J 1:53–61 ArticlePubMedPubMed CentralGoogle Scholar Ginting F, Sugianli AK, Kusumawati RL et al (2018) Predictive value of the urinary dipstick test in the management of patients with urinary tract infection-associated symptoms in primary care in Indonesia: a cross-sectional study. BMJ Open 8:23051. ArticleGoogle Scholar Gordon LB, Waxman MJ, Ragsdale L, Mermel LA (2013) Overtreatment of presumed urinary tract infection in older women presenting to the emergency department. J Am Geriatr Soc 61:788–792. ArticlePubMedGoogle Scholar Lei R, Huo R, Mohan C (2020) Current and emerging trends in point-of-care urinalysis tests. Expert Rev Mol Diagn 20:69–84. ArticleCASPubMedGoogle Scholar Marques AG, Doi AM, Pasternak J et al (2017) Performance of the dipstick screening test as a predictor of negative urine culture. Einstein 15:34. ArticlePubMedPubMed CentralGoogle Scholar Mignini L, Carroli G, Abalos E et al (2009) Accuracy of diagnostic tests to detect asymptomatic bacteriuria during pregnancy. Obstet Gynecol 113:346–352. ArticlePubMedGoogle Scholar Msaki BP, Mshana SE, Hokororo A et al (2012) Prevalence and predictors of urinary tract infection and severe malaria among febrile children attending Makongoro health centre in Mwanza city, North–Western Tanzania. Arch Public Health 70:1–8. ArticleGoogle Scholar Mukosha M, Nambela L, Mwila C, Chigunta M, Kalungia AC, Lubeya MK, Vwalika B (2020) Urinary tract infections and associated factors in HIV infected pregnant women at a tertiary hospital in Lusaka Zambia. Pan Af Med J. ArticleGoogle Scholar Mwambete KD, Malaba P (2017) High prevalence of antibiotic resistance among bacteria isolated from pregnant women with asymptomatic urinary tract infections in Dar es Salaam. Tanzan Res J Health Sci 5:65. ArticleGoogle Scholar Rousham E, Cooper M, Petherick E, Saukko P, Oppenheim B (2019) Overprescribing antibiotics for asymptomatic bacteriuria in older adults: a case series review of admissions in two UK hospitals. Antimicro Resist Infect Control 8(1):1–8 Google Scholar Sangeda RZ, Paul F, Mtweve DM (2021) Prevalence of urinary tract infections and antibiogram of uropathogens isolated from children under five attending Bagamoyo district hospital in Tanzania: a cross-sectional study. F1000 Research 10: 449 Schumm K, Lam TB (2008) Types of urethral catheters for management of short-term voiding problems in hospitalized adults: a short version cochrane review. Neurourol Urodyn 27:738–746. ArticleCASPubMedGoogle Scholar Silago V, Moremi N, Mtebe M et al (2022) Multidrug-resistant uropathogens causing community acquired urinary tract infections among patients attending health facilities in Mwanza and Dar es Salaam Tanzania. Antibiotics. ArticlePubMedPubMed CentralGoogle Scholar Wilson ML, Gaido L (2004) Laboratory diagnosis of urinary tract infections in adult patients. Clin Infect Dis 38:1150–1158. ArticlePubMedGoogle Scholar Download references Acknowledgements The authors acknowledge research assistants at Muhimbili University of Health and Allied Sciences, Ms Salma Rashid, Mr Elias Boniphace, Mr Ibrahim Mauki and Mr Layson Mwakipesile. The research assistants at the Catholic University of Health and Allied Sciences-Bugando, Mr. Kenan Malindisa and Mr. Abbias Anthon, and the health care workers at all the sites and all patients who participated fully in this study. Funding This research project received a grant from the Fleming Fund Grant number FF49/440 Regional Grants Round 2, provided by the UK Department of Health and Social Care (DHSC). The funder was not involved in forming the idea for the study and manuscript preparation for possible publication consideration. Author information Authors and Affiliations Department of Microbiology and Immunology, School of Medicine, Muhimbili University of Health and Allied Sciences, P.O. Box 65001, Dar es Salaam, 11103, Tanzania Salim S. Masoud,Mtebe Majigo,Peter Kunambi,Fauster X. Mgaya&Mecky Isaac Matee Department of Microbiology and Immunology, Weill Bugando School of Medicine, Catholic University of Health and Allied Sciences, P.O. Box 1464, Mwanza, 33109, Tanzania Vitus Silago,Helmut Nyawale,Mariam M. Mirambo&Stephen E. Mshana Department of Bacteriology, National Public Health Laboratory, Dar es Salaam, United Republic of Tanzania Nyambura Moremi&Emmanuel Magembe SACCIDS Africa Centre of Excellence for Infectious Diseases, the Sokoine University of Agriculture, P.O. Box 3297, Morogoro, 67125, Tanzania Erick Komba,Stephen E. Mshana&Mecky Isaac Matee Authors 1. Salim S. MasoudView author publications Search author on:PubMedGoogle Scholar 2. Mtebe MajigoView author publications Search author on:PubMedGoogle Scholar 3. Vitus SilagoView author publications Search author on:PubMedGoogle Scholar 4. Peter KunambiView author publications Search author on:PubMedGoogle Scholar 5. Helmut NyawaleView author publications Search author on:PubMedGoogle Scholar 6. Nyambura MoremiView author publications Search author on:PubMedGoogle Scholar 7. Erick KombaView author publications Search author on:PubMedGoogle Scholar 8. Fauster X. MgayaView author publications Search author on:PubMedGoogle Scholar 9. Emmanuel MagembeView author publications Search author on:PubMedGoogle Scholar 10. Mariam M. MiramboView author publications Search author on:PubMedGoogle Scholar 11. Stephen E. MshanaView author publications Search author on:PubMedGoogle Scholar 12. Mecky Isaac MateeView author publications Search author on:PubMedGoogle Scholar Contributions M.I.M., S.E.M., S.S.M., M.M., N.M., E.K., M.M.M., and E.M designed the study and supervised the study; S.S.M, V.S., M.M, F.X.M., S.E.M., H.N., and M.M.M. participated in oversee the laboratory work; S.S.M, V.S., and P.K performed data interpretation, cleaning, and analysis; all authors participated in drafting the manuscript, led by S.S.M, P.K, V.S, M.M, M.I.M., and S.E.M. All authors have read and agreed to the published version of the manuscript. Corresponding author Correspondence to Salim S. Masoud. Ethics declarations Ethics approval and consent to participate Ethical clearance to conduct this study was received from the National Institute for Medical Research (NIMR); certificate numbered NIMR/HQ/R.8a/Vol.IX/3580. Permission to conduct the study was requested from the Regional Medical Officers (RMOs) of Mwanza and Dar es Salaam. Medical Officer In-charges (MOIs) of each health facility where sampling was done. Patients were requested to sign written informed consent forms before enrolling after the study's main objective was explained. For patients below 18 years, parents or guardians provided signed consent forms and assent for children aged between 5 and 18 years. Unique identification numbers were used to identify patients throughout this study to maintain confidentiality. Consent for publication Not applicable. Competing interests There were no reported potential conflicts of interest relevant to this article. 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To view a copy of this licence, visit Reprints and permissions About this article Cite this article Masoud, S.S., Majigo, M., Silago, V. et al. Utility of dipstick urinalysis in the diagnosis of urinary tract infections among outpatients in Mwanza and Dar es Salaam regions in Tanzania. Bull Natl Res Cent48, 6 (2024). Download citation Received: 19 September 2023 Accepted: 16 December 2023 Published: 02 January 2024 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 Leucocyte esterase Negative predictive value Nitrite Positive predictive value Sensitivity Specificity Urinalysis dipstick Urine culture Download PDF Associated content Section Medicine and Human Genetics Sections Figures References Abstract Background Methods Results Discussion Conclusions Availability of data and materials Abbreviations References Acknowledgements Funding Author information Ethics declarations Additional information Rights and permissions About this article Advertisement Fig.1 View in articleFull size image Fig.2 View in articleFull size image Bates B (2013) Interpretation of urinalysis and urine culture for UTI treatment. Pharm Fac US Pharm 38(11):65–68 Google Scholar Childers R, Liotta B, Brennan J et al (2022) Urine testing is associated with inappropriate antibiotic use and increased length of stay in emergency department patients. Heliyon 8:e11049. ArticleCASPubMedPubMed CentralGoogle Scholar Dadzie I, Quansah E, Puopelle Dakorah M, Abiade V, Takyi-Amuah E, Adusei R (2019) The effectiveness of dipstick for the detection of urinary tract infection. Can J Infect Dis Med Microbiol 23:2019 Google Scholar Demilie T, Beyene G, Melaku S, Tsegaye W (2014) Diagnostic accuracy of rapid urine dipstick test to predict urinary tract infection among pregnant women in Felege Hiwot referral Hospital, Bahir Dar. North West Ethiopia BMC Res Notes 7:481. ArticlePubMedGoogle Scholar Flokas ME, Andreatos N, Alevizakos M et al (2017) Inappropriate management of asymptomatic patients with positive urine cultures: a systematic review and meta-analysis. Open Forum Infect Dis. ArticlePubMedPubMed CentralGoogle Scholar Franz M, Hörl WH (1999) Common errors in diagnosis and management of urinary tract infection. I: pathophysiology and diagnostic techniques. Nephrol Dial Transplant 14:2746–2753. ArticleCASPubMedGoogle Scholar Gidabayda J, Philemon R, Abdallah M et al (2017) Prevalence, aetiology, and antimicrobial susceptibility patterns of urinary tract infection amongst children admitted at Kilimanjaro christian medical centre, Moshi, Tanzania. East Afr Health Res J 1:53–61 ArticlePubMedPubMed CentralGoogle Scholar Ginting F, Sugianli AK, Kusumawati RL et al (2018) Predictive value of the urinary dipstick test in the management of patients with urinary tract infection-associated symptoms in primary care in Indonesia: a cross-sectional study. BMJ Open 8:23051. ArticleGoogle Scholar Gordon LB, Waxman MJ, Ragsdale L, Mermel LA (2013) Overtreatment of presumed urinary tract infection in older women presenting to the emergency department. J Am Geriatr Soc 61:788–792. ArticlePubMedGoogle Scholar Lei R, Huo R, Mohan C (2020) Current and emerging trends in point-of-care urinalysis tests. Expert Rev Mol Diagn 20:69–84. ArticleCASPubMedGoogle Scholar Marques AG, Doi AM, Pasternak J et al (2017) Performance of the dipstick screening test as a predictor of negative urine culture. Einstein 15:34. ArticlePubMedPubMed CentralGoogle Scholar Mignini L, Carroli G, Abalos E et al (2009) Accuracy of diagnostic tests to detect asymptomatic bacteriuria during pregnancy. Obstet Gynecol 113:346–352. ArticlePubMedGoogle Scholar Msaki BP, Mshana SE, Hokororo A et al (2012) Prevalence and predictors of urinary tract infection and severe malaria among febrile children attending Makongoro health centre in Mwanza city, North–Western Tanzania. Arch Public Health 70:1–8. ArticleGoogle Scholar Mukosha M, Nambela L, Mwila C, Chigunta M, Kalungia AC, Lubeya MK, Vwalika B (2020) Urinary tract infections and associated factors in HIV infected pregnant women at a tertiary hospital in Lusaka Zambia. Pan Af Med J. ArticleGoogle Scholar Mwambete KD, Malaba P (2017) High prevalence of antibiotic resistance among bacteria isolated from pregnant women with asymptomatic urinary tract infections in Dar es Salaam. Tanzan Res J Health Sci 5:65. ArticleGoogle Scholar Rousham E, Cooper M, Petherick E, Saukko P, Oppenheim B (2019) Overprescribing antibiotics for asymptomatic bacteriuria in older adults: a case series review of admissions in two UK hospitals. Antimicro Resist Infect Control 8(1):1–8 Google Scholar Sangeda RZ, Paul F, Mtweve DM (2021) Prevalence of urinary tract infections and antibiogram of uropathogens isolated from children under five attending Bagamoyo district hospital in Tanzania: a cross-sectional study. F1000 Research 10: 449 Schumm K, Lam TB (2008) Types of urethral catheters for management of short-term voiding problems in hospitalized adults: a short version cochrane review. Neurourol Urodyn 27:738–746. ArticleCASPubMedGoogle Scholar Silago V, Moremi N, Mtebe M et al (2022) Multidrug-resistant uropathogens causing community acquired urinary tract infections among patients attending health facilities in Mwanza and Dar es Salaam Tanzania. Antibiotics. ArticlePubMedPubMed CentralGoogle Scholar Wilson ML, Gaido L (2004) Laboratory diagnosis of urinary tract infections in adult patients. Clin Infect Dis 38:1150–1158. ArticlePubMedGoogle Scholar Support and Contact Jobs Language editing for authors Scientific editing for authors Leave feedback Terms and conditions Privacy statement Accessibility Cookies Follow SpringerOpen SpringerOpen Twitter page SpringerOpen Facebook page By using this website, you agree to our Terms and Conditions, Your US state privacy rights, Privacy statement and Cookies policy. Your privacy choices/Manage cookies we use in the preference centre. © 2025 BioMed Central Ltd unless otherwise stated. Part of Springer Nature.
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Register Oct 05 Ace Arithmetic on the GMAT Focus Edition 11:00 AM IST 01:00 PM IST Attend this session to evaluate your current skill level, learn process skills, and solve through tough Arithmetic questions. Save Now! Sep 22 Special Offer: Get 25% Off Target Test Prep GMAT Plans 12:00 PM EDT 11:59 PM EDT The Target Test Prep GMAT Flash Sale is LIVE! Get 25% off our game-changing course and save up to $450 today! Use code FLASH25 at checkout. This special offer expires on September 30, so grab your discount now! Back to Forum Create Topic Reply Each of the nine digits 0, 1, 1, 4, 5, 6, 8, 8, and 9 is used once to parkhydel parkhydel Joined: 03 Jun 2019 Last visit: 27 Mar 2025 Posts: 272 Posts: 272 Post URL27 Apr 2020, 15:32 Show timer 00:00 Start Timer Pause Timer Resume Timer Show Answer a 8% b 9% c 75% d 5% e 3% A B C D E Hide Show History My Mistake Official Answer and Stats are available only to registered users.Register/Login. Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)! Difficulty: 25% (medium) Question Stats: 75% (01:38) correct 25%(01:57) wrong based on 2426 sessions History Date Time Result Not Attempted Yet Each of the nine digits 0, 1, 1, 4, 5, 6, 8, 8, and 9 is used once to form 3 three-digit integers. What is the greatest possible sum of the 3 integers? A. 1,752 B. 2,616 C. 2,652 D. 2,775 E. 2,958 PS34550.02 Show Hide Answer Official Answer Official Answer and Stats are available only to registered users.Register/Login. 7 Kudos Add Kudos 147 Bookmarks Bookmark this Post Most Helpful Reply nick1816 nick1816 Retired Moderator Joined: 19 Oct 2018 Last visit: 11 Aug 2025 Posts: 1,853 Location: India Posts: 1,853 Post URL27 Apr 2020, 15:51 Since we need to maximize the sum, first we will maximize the sum of hundreds digits of the 3 numbers. Hundred digits of 3 numbers will be 8, 8 and 9 in any order. Now we will try to maximize the sum of tens digits of 3 numbers. Tens digits of 3 numbers will be 4, 5 and 6 in any order. Remaining digits can be put in units place in any order. Sum of 3 numbers= 100(8+8+9) + 10(4+5+6) + (0+1+1) = 2500+150+2 = 2652 parkhydel Each of the nine digits 0, 1, 1, 4, 5, 6, 8, 8, and 9 is used once to form 3 three-digit integers. What is the greatest possible sum of the 3 integers? A. 1,752 B. 2,616 C. 2,652 D. 2,775 E. 2,958 PS34550.02 Show more 22 Kudos Add Kudos 14 Bookmarks Bookmark this Post ScottTargetTestPrep ScottTargetTestPrep Target Test Prep Representative Joined: 14 Oct 2015 Last visit: 27 Sep 2025 Posts: 21,497 Status:Founder & CEO Affiliations: Target Test Prep Location: United States (CA) Expert Expert reply Active GMAT Club Expert! Tag them with @ followed by their username for a faster response. Posts: 21,497 Post URL02 May 2020, 17:07 parkhydel Each of the nine digits 0, 1, 1, 4, 5, 6, 8, 8, and 9 is used once to form 3 three-digit integers. What is the greatest possible sum of the 3 integers? A. 1,752 B. 2,616 C. 2,652 D. 2,775 E. 2,958 PS34550.02 Show more Since we want the greatest possible sum of the 3 integers, we should use 8, 8 and 9 as the hundreds digits of the 3 integers, 4, 5 and 6 as the tens digits, and 0, 1 and 1 as the units digits. Therefore, the greatest possible sum of the 3 integers is: (800 + 800 + 900) + (40 + 50 + 60) + (0 + 1 + 1) = 2500 + 150 + 2 = 2652 Answer: C Scott Woodbury-Stewart \| Founder and CEO \| scottwoodburystewart@targettestprep.com Flash Sale: Get 25% Off Our GMAT Plans \| Use Code:FLASH25 Try OnDemand GMAT, the Most Comprehensive Video Course Perfect 5-Star Ratingon GMAT Club See why Target Test Prep is the top rated GMAT course on GMAT Club. Read Our Reviews Signature Read More 7 Kudos Add Kudos 11 Bookmarks Bookmark this Post General Discussion gurmukh gurmukh Joined: 18 Dec 2017 Last visit: 24 Jan 2025 Posts: 260 Posts: 260 Post URL27 Apr 2020, 15:53 Three numbers are 960,851,841 Sum = 2652 Option C is the answer. Posted from my mobile device 3 Kudos Add Kudos Bookmarks Bookmark this Post Nipunh1991 Nipunh1991 Joined: 04 Jan 2016 Last visit: 13 Mar 2024 Posts: 20 Posts: 20 Post URL30 Apr 2020, 12:21 parkhydel Each of the nine digits 0, 1, 1, 4, 5, 6, 8, 8, and 9 is used once to form 3 three-digit integers. What is the greatest possible sum of the 3 integers? A. 1,752 B. 2,616 C. 2,652 D. 2,775 E. 2,958 PS34550.02 Show more The way i prefer solving is making blanks and then filling values: Lets say we have three 3 digit intergers _ _ _ - 1 integer _ _ _ - 2nd integer _ _ _ - 3rd Integer We need to have the highest combined value, with each digit just repeating once in a 3 digit integer. As we know the maximum number will be influenced most by the number placed in the hundreds place followed by tens and then units. Hence ill take the highest 3 digits for the 100s digit which can be 9,8 and 8 (in any order), followed by the next highest digits in the 10s digit which can be 6,5 and 4 (in any order) and finally the units digits 1, 1, 0 (in any order), which will result to 9 5 1 - 1st integer 8 4 1 - 2nd integer 8 6 0 - 3rd integer Had the digits in the hundreds place been in any other order among the 3 integers, and 10s and Units, It would still sum up to be the same number. The sum is 2652. Answer C Hope this helps a little 10 Kudos Add Kudos Bookmarks Bookmark this Post 2021gmat 2021gmat Joined: 06 Apr 2020 Last visit: 15 Jul 2020 Posts: 4 Posts: 4 Post URL30 Apr 2020, 12:47 To maximize the sum, pick the 3 largest integers and place them in hundredths place. 9 8 8 Pick the next three to maximize the sum of 10s place 4 5 6 and then place the remaining numbers in ones place. Do note that the only order that matters in placing the integers in 100s 10s and 1s. 941 851 860 or 960 841 851 both yield 2652 as the sum. Posted from my mobile device 1 Kudos Add Kudos 1 Bookmarks Bookmark this Post itspCv2 itspCv2 Joined: 06 May 2020 Last visit: 18 Jun 2021 Posts: 32 Location: India Schools:HEC '22 Schools:HEC '22 Posts: 32 Post URL15 May 2020, 12:41 parkhydel Each of the nine digits 0, 1, 1, 4, 5, 6, 8, 8, and 9 is used once to form 3 three-digit integers. What is the greatest possible sum of the 3 integers? A. 1,752 B. 2,616 C. 2,652 D. 2,775 E. 2,958 PS34550.02 Show more Given data:In simple terms: each digit can only be used in one integer The largest sum of xyz+abc+Lmn=(x+a+L)100+(y+b+m)10+(z+c+n)1 To make those brackets max: choose: 100(1st max)+10(2nd max)+1 (3rd max)=100(9+8+8)+10(6+5+4)+1(1+1+0)=2,652 Therefore answer is C.2,652 Hope this helps Kudos Add Kudos Bookmarks Bookmark this Post bones2020 bones2020 Joined: 23 Sep 2019 Last visit: 29 Jul 2020 Posts: 6 Posts: 6 Post URL16 May 2020, 09:49 One way to look at this is to look at the 3 highest digits that you have (9,8,8) and place them in the hundreds place since we want to maximize the result of addition and place (0,1,1) in the ones place. You can eliminate D,& E since this would be greater than 900+800+800, and you can eliminate A since 1752 would be to small. Now left with two answer choices, B and C. Since the resultant sum of 0,1,1 in the ones place =2, choose C. Kudos Add Kudos Bookmarks Bookmark this Post CAMANISHPARMAR CAMANISHPARMAR Joined: 12 Feb 2015 Last visit: 13 Mar 2022 Posts: 1,025 Posts: 1,025 Post URL18 May 2020, 20:15 parkhydel Each of the nine digits 0, 1, 1, 4, 5, 6, 8, 8, and 9 is used once to form 3 three-digit integers. What is the greatest possible sum of the 3 integers? A. 1,752 B. 2,616 C. 2,652 D. 2,775 E. 2,958 PS34550.02 Show more Maximize hundreds digit.....use one 9 and the two 8's in the hundreds place.. total should be more than 2500...A is out You would need three 9's in the hundreds place to take the total above 2700; Therefore D and E must be out immediately. You are down to B vs C. Use 6, 5, and 4 in the tens place. and the two 1's in the units place. Option C is the correct answer. Kudos Add Kudos Bookmarks Bookmark this Post bumpbot bumpbot Non-Human User Joined: 09 Sep 2013 Last visit: 04 Jan 2021 Posts: 38,132 Posts: 38,132 Post URL20 Sep 2024, 01:19 Hello from the GMAT Club BumpBot! 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7672	https://www.chemtopper.com/myblog/periodic-trend-of-effective-nuclear-charge-z-eff/	Skip to content Be a Topper in Chemistry Be a Topper in Chemistry Periodic Trend of Effective Nuclear Charge- Z eff Effective nuclear charge (Zeff) periodic trend across the period and down the group. What is effective nuclear charge and how does it vary? The presence of shielding electrons reduces the electrostatic attraction between the positively charged protons in the nucleus and the outer electrons. Moreover, the repulsive forces between electrons in a many-electron atom further offset the attractive force exerted by the nucleus. Net effect of these two forces means less positive charge filtered out and that positive charge felt by an electron is called as Z effective. It is the positive charge from the nucleus felt by an electron. It depends on Shielding effect or screening by inner core electrons and The nuclear charge. How does effective nuclear charge (Zeff) change across the period in the periodic table? Zeff = Z – σ Zeff » Z – number of inner or core electrons 0 < σ < Z (σ= shielding constant) \| \| \| \| \| \| \| --- --- --- \| \| Period no 3 \| No of shells \| Z \| Inner core ( σ) \| Z eff \| Radius of atom(pm) \| \| Na \| 3 \| 11 \| 10 \| 11-10=1 \| 190 \| \| Mg \| 3 \| 12 \| 10 \| 12-10=2 \| 160 \| \| Al \| 3 \| 13 \| 10 \| 13-10=3 \| 143 \| \| Si \| 3 \| 14 \| 10 \| 14-10=4 \| 132 \| The effective nuclear charge (Zeff) increases across a period because: The number of protons in the nucleus increases, strengthening the nuclear attraction. The number of electron shells remains the same, so electrons are not farther from the nucleus. The number of shielding (inner) electrons stays constant, so they do not significantly offset the increased nuclear charge. In 3rd period elements All elements have only three shells and number of inner shell electrons in all elements are 10. However nuclear charge increases by 1 so Zeff increases. This results in more attraction between the nucleus and outermost shell of the atom. Due to increase in attractions, valence shell electrons are more pulled inside closer to the nucleus and decreases the size of the element. Zeff increases across the period. How does effective nuclear charge (Zeff) change down the group in the periodic table? The effective nuclear charge (Zeff) decreases down a group because: There are more electron shells, increasing the distance between the nucleus and the valence electrons. There are more inner (shielding) electrons, which reduce the net attraction felt by the outermost electrons. In 1st group elements How does Zeff change across the periodic table? Effective nuclear charge (Zeff ) and shielding effect explained: As we move down the group, number of shells are increasing and number of electrons screening the valence electron are also increasing. However nuclear charge is also increasing but number of shell and more screening effect outweighs increase in nuclear charge. This results in decrease in effective nuclear charge down the group. As Zeff decrease, the force of attraction between outermost shell and nucleus decreases. Less attraction with nucleus and more repulsion among the electrons results in expansion of the valence shell .Hence atomic size increases with increase in atomic size. \| \| \| \| \| \| --- --- \| Group No 1 \| No of shells \| Z \| Inner core (σ) \| Radius of atom(pm) \| \| Li \| 2 \| 3 \| 2 \| 167 \| \| Na \| 4 \| 11 \| 10 \| 190 \| \| K \| 5 \| 19 \| 18 \| 243 \| \| Rb \| 6 \| 37 \| 36 \| 265 \| Zeff, No of Shells, Shielding Effect and Size of Atom of Group 1. Related posts: Flash Cards- Periodic Trends Coulomb’s Law and Periodic trends. Periodic Trends of Size of Ions Periodic Trends – Comparison Of Sizes of Isoelectronic Species Periodic Trends of Electronegativity – How EN Changes Across the Periodic Table Periodic Trends in Metallic Character of Elements
7673	https://www.mathwords.com/a/absolute_value_rules.htm	\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- \| \| \| \| \| \| \| \| \| \| \| index: click on a letter \| \| A \| B \| C \| D \| E \| \| F \| G \| H \| I \| J \| \| K \| L \| M \| N \| O \| \| P \| Q \| R \| S \| T \| \| U \| V \| W \| X \| Y \| \| Z \| A to Z index \| \| index: subject areas \| \| numbers & symbols \| \| sets, logic, proofs \| \| geometry \| \| algebra \| \| trigonometry \| \| advanced algebra & pre-calculus \| \| calculus \| \| advanced topics \| \| probability & statistics \| \| real world applications \| \| \| multimedia entries \| \| \| \| \| \| --- \| \| www.mathwords.com \| about mathwords \| \| website feedback \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| \| Absolute Value Rules Algebra rules for absolute values are listed below. \| \| \| --- \| \| Piecewise Definition: \| (\left\| a \right\| = \left{ {\begin{array}{{20}{c}}a&{{\rm{if}}\;a \ge 0}\{ - a}&{{\rm{if}}\;a < 0}\end{array}} \right.) \| \| Square root definition: \| (\left\| a \right\| = \sqrt {{a^2}} ) \| \| Rules: \| 1. \|–a\| = \|a\| 2. \|a\| ≥ 0 3. Products: \|ab\| = \|a\|\|b\| 4. Quotients: \|a / b\| = \|a\| / \|b\| 5. Powers: \|an\| = \|a\|n 6. Triangle Inequality: \|a + b\| ≤ \|a\| + \|b\| 7. Alternate Triangle Inequality: \|a – b\| ≥ \|a\| – \|b\| \| \| CAREFUL!! \| Sums: Differences: \| \|a + b\| is not the same as \|a\| + \|b\| \|a – b\| is not the same as \|a\| – \|b\| \| See also Absolute value of a complex number, triangle inequality (geometric) \| \| \| \| \| \| \| --- \| \| this page updated 15-jul-23 Mathwords: Terms and Formulas from Algebra I to Calculus written, illustrated, and webmastered by Bruce Simmons Copyright © 2000 by Bruce Simmons All rights reserved \| \| \|
7674	https://www.csee.umbc.edu/~chang/cs431/dangling-else.shtml	Dangling Else, UMBC CMSC431, Compilers, Fall 2009 UMBC CMSC431, Compiler Design Principles, Fall 2009 Mid-Rule Actions and The Dangling Else The Dangling Else We all know the dangling else problem. The "standard" C-like syntax for if-else statements is ambiguous: if_stmt : IF EXPR THEN statement ELSE statement \| IF EXPR THEN statement ; If you run this bison file (else1.y) through bison, you get a shift/reduce conflict. If you use the -v option, the else1.output file will show you the problem. The parser does not know whether to shift the ELSE token to the stack or use rule 10 to reduce to if_stmt. There are two standard fixes to this problem. First, we can require the programmer to always use braces around the statements in the if part and the else part of an if statement — even when there is only one statement: if_stmt : IF EXPR THEN stmt_block ELSE stmt_block \| IF EXPR THEN stmt_block ; See file else2.y for the full grammar. The other fix is to rewrite the grammar using "matched statements" and "open statements" as described in our textbook: if_stmt : matched_if_stmt \| open_if_stmt ; matched_if_stmt : IF EXPR THEN matched_if_stmt ELSE matched_if_stmt \| other_stmt ; open_if_stmt : IF EXPR THEN if_stmt \| IF EXPR THEN matched_if_stmt ELSE open_if_stmt ; See file else3.y for the full grammar. Mid-Rule Actions A new problem arises if we add mid-rule actions to our grammar: if_stmt : IF EXPR { / ... / } THEN stmt_block ELSE stmt_block { / ... / } \| IF EXPR {/ ... / } THEN stmt_block {/ ... /} ; This creates a reduce/reduce conflict. When bison sees a mid-rule action like this: A : B { / ... / } C bison converts the rule into two separate rules: A1 : B { / ... / } A : A1 C Thus, the mid-rule actions in the productions for if_stmt creates two new rules: if_stmt_1 : IF EXPR { / ... / } if_stmt_2 : IF EXPR { / ... / } One for each form of the if statement. Hence the reduce/reduce conflict. (See file else2a.y for the full grammar and file else2a.output for the parse table.) In this special case, you are likely to want to have the same action after EXPR anyway, so this problem can be resolved by "factoring out" the if part: if_part : IF EXPR { / do something /} ; if_stmt : if_part THEN stmt_block ELSE stmt_block { / do something else / } \| if_part THEN stmt_block {/ do something /} ; (File else2b.y has the full grammar.) Even when we rewrote the grammar with "matched" and "open" if statements, mid-action rules cause reduce/reduce conflicts: matched_if_stmt : IF EXPR { / ... / } THEN matched_if_stmt ELSE { / ... / } matched_if_stmt { / ... / } \| other_stmt ; open_if_stmt : IF EXPR { / ... / } THEN if_stmt { / ... / } \| IF EXPR { / ... / } THEN matched_if_stmt ELSE { / ... / } open_if_stmt { / ... / } ; In this case, we get 2 reduce/reduce conflicts. (File else3a.y has the full grammar and else3a.output has the parse table.) Again, we can factor out to resolve the conflict, since we want to generate the same code in the two mid-rule actions. In this case, we have to factor out twice: once with if_part and once with if_part_then_matched_stmt: if_part : IF EXPR { / ... / } ; if_part_then_matched_stmt : if_part THEN matched_if_stmt { / ... / } ; matched_if_stmt : if_part_then_matched_stmt ELSE matched_if_stmt { / ... / } \| other_stmt ; open_if_stmt : if_part THEN if_stmt { / ... / } \| if_part_then_matched_stmt ELSE open_if_stmt { / ... / } ; (File else3b.y has the full grammar.) Last Modified: 22 Jul 2024 11:27:43 EDT by Richard Changto Fall 2009 CMSC 431 Homepage
7675	https://stewartcalculus.com/data/ESSENTIAL%20CALCULUS%202e/upfiles/instructor/ess_ax_1105.pdf	1–8 ■Use the Chain Rule to ﬁnd or . 1. , , 2. , , 3. , , 4. , , 5. , , 6. , , 7. , , , 8. , , , ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 9–14 ■Use the Chain Rule to ﬁnd and . 9. , , 10. , , 11. , , 12. , , 13. , , 14. , , ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 15–22 ■Use the Chain Rule to ﬁnd the indicated partial derivatives. 15. , , , ; , when , 16. , , , ; , when , 17. , , ; , , when , , v 0 u 1 t 2 z v z u z t y u tv2 x t 2uv z y 2 tan x t 1 s 0 u t u s z t 2 y e st x st u xy yz zx t 0 s 1 w t w s z s sin t y s cos t x st w x 2 y 2 z2 y st 2 x e t z xe y yex y st 2 x s 2t z 2 x3y y se t x t 2 z x tan1xy y set x se t z x 2 3x 2y 3 y s 2 t 2 x s t2 z sin x cos y y 2st x s 2 t 2 z x 2 sin y zt zs z e3t y cos 2t x st w x y y z z 1 e 2t y cos t x sin t w xy 2z 3 y et x te 2t z xs1 y 2 y cos t x e t z 6x 3 3xy 2y 2 y e 2t x cos t z xe xy y 1 st x s1 t z lnx y 2 y 1 st x 1 st z x 2y 3 y 1 t 2 x t 3 z x 2 y 2 dwdt dzdt 18. , , ; , , when , , 19. , , , ; , , 20. , , , ; , , 21. , , ; , , , 22. , , , , ; , ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 23–26 ■Use Equation 6 to ﬁnd . 23. 24. 25. 26. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 27–33 ■Use Equations 7 to ﬁnd and . 27. 28. 29. 30. 31. 32. 33. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 34. The radius of a right cylinder is decreasing at a rate of while its height is increasing at a rate of . At what rate is the volume of the cylinder changing when the radius is and the height is ? 150 cm 80 cm 3 cms 1.2 cms lnx yz 1 xy 2z3 xe y yz ze x 0 xy 2 yz 2 zx 2 3 y 2ze xy sinxyz 0 xy 2z3 x 3y 2z x y z x 2 y 2 z 2 2xy z xy yz xz 0 zy zx 2y 2 s 3 xy 3x 2 17 x cos y y cos x 1 y 5 3x 2y 2 5x 4 12 x 2 xy y 3 8 dydx u y u x s 2xy 32 r x y 4 q x 2y p x 2y u pq p2r 2s w w t w s w r y r 2st cos x rs 2t 3 sin w cosx y t w t v t u z wu y vw x uv t z sec xy u t u r u p z p r t y p r t x p r t u x y y z t 0 s 2 r 1 z t z s z r y rse t x re st z x y SECTION 11.5 THE CHAIN RULE ■ 1 Click here for answers. A Click here for solutions. S THE CHAIN RULE 11.5 Copyright © 2013, Cengage Learning. All rights reserved. Click here for solutions. S 2 ■ SECTION 11.5 THE CHAIN RULE Click here for exercises. E ANSWERS 11.5 1. 6t5 + 4t3 + 4t 2. (1 −t) 1 − √ t 2 √ t −3 (1 −t)2 2 √ t 3. 1 √1 + t + 1 + √ t 1 2√1 + t + 1 + √ t √ t 4. −ecos t/e2t 1 + cos t e2t sin t −2e2t cos2 t e4t 5. 18e2t −3 cos t et + 3et −4 cos t sin t 6. e2t√ 1 + e−2t (1 + 2t) −t √ 1 + e−2t 7. y2z3 (cos t) + 2xyz3 (−sin t) + 3xy2z2 2e2t 8. 1 2y √ t + 2 (sin 2t) x y2 −1 z + 3y z2e3t 9. 4sx sin y + 2tx2 cos y, 4xt sin y + 2sx2 cos y 10. 2 (s −t) cos x cos y −2s sin x sin y, 2 (t −s) cos x cos y + 2t sin x sin y 11. 2x −6xy3 et −9x2y2e−t, 2x −6xy3 set + 9x2y2se−t 12. x2et 1 + x2y2 , tan−1 (xy) + xy 1 + x2y2 (2t) + x2 1 + x2y2 set 13. 2x−3y ln 2 2st −3t2 , 2x−3y ln 2 s2 −6st 14. xey + e−x t2, ey −ye−x et + 2 xey + e−x st 15. 2, 0 16. 3, 2 17. 0, 0, 4 18. 0, −1 4, 1 2 19. −t/ p2 , 0, 1/p 20. sec (xy) [w + vzy tan (xy)], z sec (xy) tan (xy) [yu + xw], sec (xy) [u + vzx tan (xy)] 21. st sin (x −y) 2r cos θ −st2 sin θ , [rt sin (x −y)] r cos θ −2st2 sin θ , [sr sin (x −y)] r cos θ −3st2 sin θ , [−rst sin (x −y)] st2 cos θ + r sin θ 22. 2x −8x2 (x + 2y) (x + y) y13/2 , −8y + (x + 2y) x3y1/2 (5x + 2y) y8 23. y −2x 3y2 −x 25. y sin x −cos y cos x −x sin y 27. z −y y −x, x + z x −y 29. −y2z3 + 3x2y2z −1 3xy2z2 + x3y2 −1, −2xyz3 + 2x3yz −1 3xy2z2 + x3y2 −1 30. z cos (xyz) −yzex+y yex+y −x cos (xyz) , xz cos (xyz) −ex+y 2yz + y2z y2ex+y −xy cos (xyz) 31. −y2 + 2zx 2yz + x2 , −2xy + z2 2yz + x2 32. −ey + zex y + ex , −xey + z y + ex 33. y2z3 (x + yz) −1 y −3xy2z2 (x + yz), 2xyz3 (x + yz) −z y −3xy2z2 (x + yz) 34. −9600π cm3/s 26. 18x −x−2/3y1/3 12y + x1/3y−2/3 28. x −y −z z + x , y −x z + x 24. −6xy2 + 20x3 5y4 + 6x2y Copyright © 2013, Cengage Learning. All rights reserved. SECTION 11.5 THE CHAIN RULE ■ 3 Click here for exercises. E SOLUTIONS 11.5 1. z = x2 + y2, x = t3, y = 1 + t2 ⇒ dz dt = 2xdx dt + 2y dy dt = 2t3 3t2 + 2 1 + t2 (2t) = 6t5 + 4t3 + 4t 2. z = x2y3, x = 1 + √ t, y = 1 − √ t ⇒ dz dt = 2xy3 dx dt + 3x2y2 dy dt = 2xy3 1 2 √ t + 3x2y2 −1 2 √ t = 2 1 + √ t 1 − √ t 3 1 2 √ t + 3 1 + √ t 2 1 − √ t 2 −1 2 √ t = (1 −t) 1 − √ t 2 √ t −3 (1 −t)2 2 √ t 3. z = ln x + y2 , x = √1 + t, y = 1 + √ t ⇒ dz dt = 1 (x + y2) 1 2√1 + t + 1 (x + y2)2y 1 2 √ t = 1 √1 + t + 1 + √ t 1 2√1 + t + 1 + √ t √ t 4. z = xex/y, x = cos t, y = e2t ⇒ dz dt = ex/y 1 + x y (−sin t) −x2y−2ex/y 2e2t = −ecos t/e2t 1 + cos t e2t sin t −2e2t cos2 t e4t 5. z = 6x3 −3xy + 2y2, x = et, y = cos t ⇒ dz dt = 18x2 −3y et + (−3x + 4y) (−sin t) = 18e2t −3 cos t et + 3et −4 cos t sin t 6. z = x 1 −y2, x = te2t, y = e−t ⇒ ∂z ∂t = 1 + y2 e2t + 2te2t + 1 2x 1 + y2−1/2 (2y) −e−t = e2t√ 1 + e−2t (1 + 2t) −t √ 1 + e−2t 7. w = xy2z3, x = sin t, y = cos t, z = 1 + e2t ⇒ dw dt = y2z3 (cos t) + 2xyz3 (−sin t) + 3xy2z2 2e2t 8. w = x y + y z , x = √ t, y = cos 2t, z = e−3t ⇒ dw dt = 1 y 1 2 √ t + −x y2 + 1 z (−2 sin 2t) + −y z2 −3e−3t = 1 2y √ t + 2 (sin 2t) x y2 −1 z + 3y z2e3t 9. z = x2 sin y, x = s2 + t2, y = 2st ⇒ ∂z ∂s = (2x sin y) (2s) + x2 cos y (2t) = 4sx sin y + 2tx2 cos y ∂z ∂t = (2x sin y) (2t) + x2 cos y (2s) = 4xt sin y + 2sx2 cos y 10. z = sin x cos y, x = (s −t)2, y = s2 −t2 ⇒ ∂z ∂s = (cos x cos y) 2 (s −t) −(sin x sin y) (2s) = 2 (s −t) cos x cos y −(2s) sin x sin y ∂z ∂t = (cos x cos y) (−2) (s −t) −(sin x sin y) (−2t) = 2 (t −s) cos x cos y + 2t sin x sin y 11. z = x2 −3x2y3, x = set, y = se−t ⇒ ∂z/∂s = 2x −6xy3 et + −9x2y2 e−t = 2x −6xy3 et −9x2y2e−t ∂z/∂t = 2x −6xy3 set + −9x2y2 −se−t = 2x −6xy3 set + 9x2y2se−t 12. z = x tan−1 (xy), x = t2, y = set ⇒ ∂z ∂s = tan−1 (xy) + x 1 + x2y2 y (0) + x2 1 + x2y2 et = x2et 1 + x2y2 ∂z ∂t = tan−1 (xy) + xy 1 + x2y2 (2t) + x2 1 + x2y2 set 13. z = 2x−3y, x = s2t, y = st2 ⇒ ∂z/∂s = (z ln 2) (2st) + z (−3 ln 2) t2 = 2x−3y ln 2 2st −3t2 ∂z/∂t = (z ln 2) s2 + z (−3 ln 2) (2st) = 2x−3y ln 2 s2 −6st 14. z = xey + ye−x, x = et, y = st2 ⇒ ∂z/∂s = ey −ye−x (0) + xey + e−x t2 = xey + e−x t2 ∂z/∂t = ey −ye−x et + xey + e−x (2st) 15. w = x2 + y2 + z2, x = st, y = s cos t, z = s sin t ⇒ ∂w ∂s = ∂w ∂x ∂x ∂s + ∂w ∂y ∂y ∂s + ∂w ∂z ∂z ∂s = 2xt + 2y cos t + 2z sin t When s = 1, t = 0, we have x = 0, y = 1 and z = 0, so ∂w/∂s = 2 cos 0 = 2. Similarly ∂w/∂t = 2xs + 2y (−s sin t) + 2z (s cos t) = 0 + (−2) sin 0 + 0 = 0 when s = 1 and t = 0. Copyright © 2013, Cengage Learning. All rights reserved. 4 ■ SECTION 11.5 THE CHAIN RULE 16. u = xy + yz + zx, x = st, y = est, z = t2 ⇒ ∂u/∂s = (y + z) t + (x + z) test + (x + y) (0) and ∂u/∂t = (y + z) s + (x + z) sest + (x + y) (2t). When s = 0, t = 1, we have x = 0, y = 1, z = 1, so ∂u/∂s = 2 + 1 + 0 = 3 and ∂u/∂t = 0 + 0 + (1) (2) = 2. 17. z = y2 tan x, x = t2uv, y = u + tv2 ⇒ ∂z/∂t = y2 sec2 x 2tuv + (2y tan x) v2, ∂z/∂u = y2 sec2 x t2v + 2y tan x, ∂z/∂v = y2 sec2 x t2u + (2y tan x) 2tv. When t = 2, u = 1 and v = 0, we have x = 0, y = 1, so ∂z/∂t = 0, ∂z/∂u = 0, ∂z/∂v = 4. 18. z = x y , x = rest, y = rset ⇒ ∂z ∂r = 1 y est + −x y2 set, ∂z ∂s = 1 y rtest −x y2 ret, ∂z ∂t = 1 y rsest −x y2 rset. When r = 1, s = 2 and t = 0, we have x = 1, y = 2, so ∂z/∂r = 1 2 + −1 4 · 2 = 0, ∂z/∂s = 0 −1 4 = −1 4 and ∂z/∂t = 1 2 · 2 −1 4 · 2 = 1 2. 19. u = x + y y + z , x = p + r + t, y = p −r + t, z = p + r −t ⇒ ∂u ∂p = 1 y + z + (y + z) −(x + y) (y + z)2 − x + y (y + z)2 = (y + z) + (z −x) −(x + y) (y + z)2 = 2 z −x (y + z)2 = 2−2t 4p2 = −t p2 ∂u ∂r = 1 y + z + z −x (y + z)2 (−1) − x + y (y + z)2 = 0, and ∂u ∂t = 1 y + z + z −x (y + z)2 + x + y (y + z)2 = 2 y + z (y + z)2 = 2 2p = 1 p 20. t = z sec (xy), x = uv, y = vw, z = wu ⇒ ∂t ∂u = [zy sec (xy) tan (xy)] v + [zx sec (xy) tan (xy)] (0) + [sec (xy)] w = sec (xy) [w + vzy tan (xy)] ∂t ∂v = [zy sec (xy) tan (xy)] u + [zx sec (xy) tan (xy)] w + [sec (xy)] (0) = z sec (xy) tan (xy) [yu + xw] ∂t ∂w = [zy sec (xy) tan (xy)] (0) + [zx sec (xy) tan (xy)] v + [sec (xy)] u = sec (xy) [u + vzx tan (xy)] 21. ∂w ∂r = [−sin (x −y)] s2t3 sin θ + [sin (x −y)] 2rst cos θ = st sin (x −y) 2r cos θ −st2 sin θ ∂w ∂s = [−sin (x −y)] 2rst3 sin θ + [sin (x −y)] r2t cos θ = [rt sin (x −y)] r cos θ −2st2 sin θ ∂w ∂t = [−sin (x −y)] 3rs2t2 sin θ + [sin (x −y)] r2s cos θ = [sr sin (x −y)] r cos θ −3st2 sin θ ∂w ∂θ = [−sin (x −y)] rs2t3 cos θ + [sin (x −y)] −r2st sin θ = [−rst sin (x −y)] st2 cos θ + r sin θ 22. ∂u ∂x = q −2pr2s + p + −2p2rs 1 y4 + −p2r2 2y3/2 = p + q −2pr2s −2p2rs y4 −2p2r2y3/2 = 2x −8x2 (x + 2y) (x + y) y13/2 ∂u ∂y = q −2pr2s (2) + p (−2) + −2p2rs −4x y5 + −p2r2 3xy1/2 = 2 (q −p) −4pr2s + 8p2rsx y5 −3p2r2xy1/2 = −8y + (x + 2y) x3y1/2 (5x + 2y) y8 23. x2 −xy + y3 = 8, so let F (x, y) = x2 −xy + y3 −8 = 0. Then dy dx = −Fx Fy = −(2x −y) −x + 3y2 = y −2x 3y2 −x. 24. y5 + 3x2y2 + 5x4 = 12, so let F (x, y) = y5 + 3x2y2 + 5x4 −12 = 0. Then dy dx = −Fx Fy = −6xy2 + 20x3 5y4 + 6x2y . 25. x cos y + y cos x = 1, so let F (x, y) = x cos y + y cos x −1 = 0. Then dy dx = −cos y −y sin x −x sin y + cos x = y sin x −cos y cos x −x sin y . 26. 2y2 + 3 √xy = 3x2 + 17, so let F (x, y) = 2y2 + 3 √xy −3x2 −17 = 0. Then dy dx = − y 3 (xy)2/3 −6x 4y + x 3 (xy)2/3 = 18x −x−2/3y1/3 12y + x1/3y−2/3 . 27. Let F (x, y, z) = xy + yz −xz = 0. Then ∂z ∂x = −Fx Fz = −y −z y −x = z −y y −x, ∂z ∂y = −Fy Fz = −x + z y −x = x + z x −y . Copyright © 2013, Cengage Learning. All rights reserved. SECTION 11.5 THE CHAIN RULE ■ 5 28. x2 + y2 −z2 = 2x (y + z). Let F (x, y, z) = x2 + y2 −z2 −2x (y + z) = 0. Then ∂z ∂x = −Fx Fz = −2x −2y −2z −2z −2x = x −y −z z + x , ∂z ∂y = −Fy Fz = −2y −2x −2z −2x = y −x z + x 29. xy2z3 + x3y2z = x + y + z. Let F (x, y, z) = xy2z3 + x3y2z −(x + y + z). Then ∂z ∂x = −Fx Fz = −y2z3 + 3x2y2z −1 3xy2z2 + x3y2 −1, ∂z ∂y = −Fy Fz = −2xyz3 + 2x3yz −1 3xy2z2 + x3y2 −1 . 30. Let F (x, y, z) = y2zex+y −sin (xyz) = 0. Then ∂z ∂x = −Fx Fz = −y2zex+y −yz cos (xyz) y2ex+y −xy cos (xyz) = z cos (xyz) −yzex+y yex+y −x cos (xyz) ∂z ∂y = −Fy Fz = xz cos (xyz) −ex+y 2yz + y2z y2ex+y −xy cos (xyz) 31. xy2 + yz2 + zx2 = 3, so let F (x, y) = xy2 + yz2 + zx2 −3 = 0. Then ∂z ∂x = −Fx Fz = −y2 + 2zx 2yz + x2 and ∂z ∂y = −Fy Fz = −2xy + z2 2yz + x2 . 32. Let F (x, y, z) = xey + yz + zex = 0. Then ∂z ∂x = −Fx Fz = −ey + zex y + ex , ∂z ∂y = −Fy Fz = −xey + z y + ex . 33. ln (x + yz) = 1 + xy2z3, so let F (x, y) = ln (x + yz) −1 −xy2z3 = 0. Then ∂z ∂x = −Fx Fz = −1/ (x + yz) −y2z3 y/ (x + yz) −3xy2z2 = y2z3 (x + yz) −1 y −3xy2z2 (x + yz) ∂z ∂y = −Fy Fz = −z/ (x + yz) −2xyz3 y/ (x + yz) −3xy2z2 = 2xyz3 (x + yz) −z y −3xy2z2 (x + yz) 34. dr/ dt = −1.2, dh/ dt = 3, V = πr2h and dV/ dt = 2πrh (dr/dt) + πr2 (dh/dt). Thus when r = 80 and h = 150, dV/ dt = (−28,800) π + (19,200) π = −9600π cm3/s. Copyright © 2013, Cengage Learning. All rights reserved.
7676	https://artofproblemsolving.com/articles/files/MildorfInequalities.pdf?srsltid=AfmBOooU2V5maooF1kAW6UPtU1SFv6b0BIYFA66ZDbp-qfiQNC3oo-qY	Olympiad Inequalities Thomas J. Mildorf December 22, 2005 It is the purpose of this document to familiarize the reader with a wide range of theorems and techniques that can be used to solve inequalities of the variety typically appearing on mathematical olympiads or other elementary proof contests. The Standard Dozen is an exhibition of twelve famous inequalities which can be cited and applied without proof in a solution. It is expected that most problems will fall entirely within the span of these inequalities. The Examples section provides numerous complete solutions as well as remarks on inequality-solving intuition, all intended to increase the reader’s aptitude for the material covered here. It is organized in rough order of difficulty. Finally, the Problems section contains exercises without solutions, ranging from straightforward to quite difficult, for the purpose of practicing techniques contained in this document. I have compiled much of this from posts by my peers in a number of mathematical communities, particularly the Mathlinks-Art of Problem Solving forums, 1 as well as from various MOP lectures, 2 Kiran Kedlaya’s inequalities packet, 3 and John Scholes’ site. 4 I have tried to take note of original sources where possible. This work in progress is distributed for personal educational use only. In particular, any publication of all or part of this manuscript without prior consent of the author, as well as any original sources noted herein, is strictly prohibited. Please send comments - suggestions, corrections, missing information, 5 or other interesting problems - to the author at tmildorf@mit.edu .Without further delay... 1 and respectively, though they have merged into a single, very large and robust group. The forums there are also host to a considerable wealth of additional material outside of inequalities. 2 Math Olympiad Program. Although some people would try to convince me it is the Math Olympiad Summer Program and therefore is due the acronym MOSP, those who know acknowledge that the traditional “MOP” is the preferred appellation. 3 The particularly diligent student of inequalities would be interested in this document, which is available online at Further ma-terial is also available in the books Andreescu-Cartoaje-Dospinescu-Lascu, Old and New Inequalities , GIL Publishing House, and Hardy-Littlewood-P´ olya, Inequalities , Cambridge University Press. (The former is elementary and geared towards contests, the latter is more technical.) 4 where a seemingly inexhaustible supply of Olympiads is available. 5 Such as the source of the last problem in this document. 11 The Standard Dozen Throughout this lecture, we refer to convex and concave functions. Write I and I′ for the intervals [ a, b ] and ( a, b ) respectively. A function f is said to be convex on I if and only if λf (x) + (1 − λ)f (y) ≥ f (λx + (1 − λ)y) for all x, y ∈ I and 0 ≤ λ ≤ 1. Conversely, if the inequality always holds in the opposite direction, the function is said to be concave on the interval. A function f that is continuous on I and twice differentiable on I′ is convex on I if and only if f ′′ (x) ≥ 0 for all x ∈ I (Concave if the inequality is flipped.) Let x1 ≥ x2 ≥ · · · ≥ xn; y1 ≥ y2 ≥ · · · ≥ yn be two sequences of real numbers. If x1 + · · · + xk ≥ y1 + · · · + yk for k = 1 , 2, . . . , n with equality where k = n, then the sequence {xi} is said to majorize the sequence {yi}. An equivalent criterion is that for all real numbers t, \|t − x1\| + \|t − x2\| + · · · + \|t − xn\| ≥ \| t − y1\| + \|t − y2\| + · · · + \|t − yn\| We use these definitions to introduce some famous inequalities. Theorem 1 (Jensen) Let f : I → R be a convex function. Then for any x1, . . . , x n ∈ I and any nonnegative reals ω1, . . . , ω n, ω1f (x1) + · · · + ωnf (xn) ≥ (ω1 + · · · + ωn) f (ω1x1 + · · · + ωnxn ω1 + · · · + ωn ) If f is concave, then the inequality is flipped. Theorem 2 (Weighted Power Mean) If x1, . . . , x n are nonnegative reals and ω1, . . . , ω n are nonnegative reals with a postive sum, then f (r) := (ω1xr 1 · · · + ωnxrn ω1 + · · · + ωn )1 r is a non-decreasing function of r, with the convention that r = 0 is the weighted geometric mean. f is strictly increasing unless all the xi are equal except possibly for r ∈ (−∞ , 0] ,where if some xi is zero f is identically 0. In particular, f (1) ≥ f (0) ≥ f (−1) gives the AM-GM-HM inequality. Theorem 3 (H¨ older) Let a1, . . . , a n; b1, . . . , b n; · · · ; z1, . . . , z n be sequences of nonnegative real numbers, and let λa, λ b, . . . , λ z positive reals which sum to 1. Then (a1 + · · · + an)λa (b1 + · · · + bn)λb · · · (z1 + · · · + zn)λz ≥ aλa 1 bλb 1 · · · zλz 1 · · · + aλz n bλb n · · · zλz n This theorem is customarily identified as Cauchy when there are just two sequences. Theorem 4 (Rearrangement) Let a1 ≤ a2 ≤ · · · ≤ an and b1 ≤ b2 ≤ · · · ≤ bn be two nondecreasing sequences of real numbers. Then, for any permutation π of {1, 2, . . . , n }, we have a1b1 + a2b2 + · · · + anbn ≥ a1bπ(1) + a2bπ(2) + · · · + anbπ(n) ≥ a1bn + a2bn−1 + · · · + anb1 with equality on the left and right holding if and only if the sequence π(1) , . . . , π (n) is de-creasing and increasing respectively. 2Theorem 5 (Chebyshev) Let a1 ≤ a2 ≤ · · · ≤ an; b1 ≤ b2 ≤ · · · ≤ bn be two nondecreas-ing sequences of real numbers. Then a1b1 + a2b2 + · · · + anbn n ≥ a1 + a2 + · · · + an n ·b1 + b2 + · · · + bn n ≥ a1bn + a2bn−1 + · · · + anb1 n Theorem 6 (Schur) Let a, b, c be nonnegative reals and r > 0. Then ar(a − b)( a − c) + br(b − c)( b − a) + cr(c − a)( c − b) ≥ 0 with equality if and only if a = b = c or some two of a, b, c are equal and the other is 0. Remark - This can be improved considerably. (See the problems section.) However, they are not as well known (as of now) as this form of Schur, and so should be proven whenever used on a contest. Theorem 7 (Newton) Let x1, . . . , x n be nonnegative real numbers. Define the symmetric polynomials s0, s 1, . . . , s n by (x + x1)( x + x2) · · · (x + xn) = snxn + · · · + s1x + s0, and define the symmetric averages by di = si/(ni ). Then d2 i ≥ di+1 di−1 Theorem 8 (Maclaurin) Let di be defined as above. Then d1 ≥ √d2 ≥ 3 √d3 ≥ · · · ≥ n √dn Theorem 9 (Majorization) Let f : I → R be a convex on I and suppose that the sequence x1, . . . , x n majorizes the sequence y1, . . . , y n, where xi, y i ∈ I. Then f (x1) + · · · + f (xn) ≥ f (y1) + · · · + f (yn) Theorem 10 (Popoviciu) Let f : I → R be convex on I, and let x, y, z ∈ I. Then for any positive reals p, q, r , pf (x) + qf (y) + rf (z) + (p + q + r)f (px + qy + rz p + q + r ) ≥ (p + q)f (px + qy p + q ) ( q + r)f (qy + rz q + r ) ( r + p)f (rz + px r + p ) Theorem 11 (Bernoulli) For all r ≥ 1 and x ≥ − 1, (1 + x)r ≥ 1 + xr 3Theorem 12 (Muirhead) Suppose the sequence a1, . . . , a n majorizes the sequence b1, . . . , b n.Then for any positive reals x1, . . . , x n, ∑ sym xa1 1 xa2 2 · · · xan n ≥ ∑ sym xb1 1 xb2 2 · · · xbn n where the sums are taken over all permutations of n variables. Remark - Although Muirhead’s theorem is a named theorem, it is generally not favor-ably regarded as part of a formal olympiad solution. Essentially, the majorization criterion guarantees that Muirhead’s inequality can be deduced from a suitable application of AM-GM. Hence, whenever possible, you should use Muirhead’s inequality only to deduce the correct relationship and then explicitly write all of the necessary applications of AM-GM. For a particular case this is a simple matter. We now present an array of problems and solutions based primarily on these inequalities and ideas. 2 Examples When solving any kind of problem, we should always look for a comparatively easy solu-tion first, and only later try medium or hard approaches. Although what constitutes this notoriously indeterminate “difficulty” varies widely from person to person, I usually con-sider “Dumbassing,” AM-GM (Power Mean), Cauchy, Chebyshev (Rearrangement), Jensen, H¨ older, in that order before moving to more clever techniques. (The first technique is de-scribed in remarks after example 1.) Weak inequalities will fall to AM-GM, which blatantly pins a sum to its smallest term. Weighted Jensen and H¨ older are “smarter” in that the effect of widely unequal terms does not cost a large degree of sharpness 6 - observe what happens when a weight of 0 appears. Especially sharp inequalities may be assailable only through clever algebra. Anyway, I have arranged the following with that in mind. 1. Show that for positive reals a, b, c (a2b + b2c + c2a) ( ab 2 + bc 2 + ca 2) ≥ 9a2b2c2 Solution 1. Simply use AM-GM on the terms within each factor, obtaining (a2b + b2c + c2a) ( ab 2 + bc 2 + ca 2) ≥ ( 3 3 √a3b3c3 ) ( 3 3 √a3b3c3 ) = 9 a2b2c2 6The sharpness of an inequality generally refers to the extent to which the two sides mimic each other, particularly near equality cases. 4Solution 2. Rearrange the terms of each factor and apply Cauchy, (a2b + b2c + c2a) ( bc 2 + ca 2 + ab 2) ≥ (√a3b3c3 + √a3b3c3 + √a3b3c3 )2 = 9 a2b2c2 Solution 3. Expand the left hand side, then apply AM-GM, obtaining (a2b + b2c + c2a) ( ab 2 + bc 2 + ca 2) = a3b3 + a2b2c2 + a4bc ab 4c + b3c3 + a2b2c2 a2b2c2 + abc 4 + a3c3 ≥ 9 9 √a18 b18 c18 = 9 a2b2c2 We knew this solution existed by Muirhead, since (4 , 1, 1) , (3 , 3, 0), and (2 , 2, 2) all majorize (2 , 2, 2). The strategy of multiplying out all polynomial expressions and ap-plying AM-GM in conjunction with Schur is generally knowing as dumbassing because it requires only the calculational fortitude to compute polynomial products and no real ingenuity. As we shall see, dumbassing is a valuable technique. We also remark that the AM-GM combining all of the terms together was a particularly weak inequality, but the desired was a multiple of a2b2c2’s, the smallest 6th degree symmetric polynomial of three variables; such a singular AM-GM may not always suffice. 2. Let a, b, c be positive reals such that abc = 1. Prove that a + b + c ≤ a2 + b2 + c2 Solution. First, we homogenize the inequality. that is, apply the constraint so as to make all terms of the same degree. Once an inequality is homogenous in degree d, we may scale all of the variables by an arbitrary factor of k, causing both sides of the inequality to scale by the factor kd. This is valid in that it does not change the correctness of an inequality for any positive k, and if d is even, for any nonzero k. Hence, we need consider a nonhomogenous constraint no futher. In this case, we multiply the left hand side by 3 √abc , obtaining a43 b13 c13 + a13 b43 c13 + a13 b13 c43 ≤ a2 + b2 + c2 As abc = 1 is not homogenous, the above inequality must be true for all nonnegative a, b, c . As (2 , 0, 0) majorizes (4 /3, 1/3, 1/3), we know it is true, and the necessary AM-GM is 2a2 3 + b2 6 + c2 6 = a2 + a2 + a2 + a2 + b2 + c2 6 ≥ 6 √a8b2c2 = a43 b13 c13 Let P (x) be a polynomial with positive coefficients. Prove that if P ( 1 x ) ≥ 1 P (x)5holds for x = 1, then it holds for all x > 0. Solution. Let P (x) = anxn + an−1xn−1 + · · · + a1x + a0. The first thing we notice is that the given is P (1) ≥ 1. Hence, the natural strategy is to combine P (x) and P ( 1 x ) into P (1) in some fashion. The best way to accomplish this is Cauchy, which gives P (x)P ( 1 x ) = (anxn + · · · + a1x + a0) ( an 1 xn + · · · + a1 1 x + a0 ) ≥ (an + · · · + a1 + a0)2 = P (1) 2 ≥ 1as desired. This illustrates a useful means of eliminating denominators - by introducing similar factors weighted by reciprocals and applying Cauchy / H¨ older. 4. (USAMO 78/1) a, b, c, d, e are real numbers such that a + b + c + d + e = 8 a2 + b2 + c2 + d2 + e2 = 16 What is the largest possible value of e? Solution. Observe that the givens can be effectively combined by considering squares: (a − r)2 + ( b − r)2 + ( c − r)2 + ( d − r)2 + ( e − r)2 = (a2 + b2 + c2 + d2 + e2) − 2r(a + b + c + d + e) + 5 r2 = 16 − 16 r + 5 r2 Since these squares are nonnegative, e ≤ √5r2 − 16 r + 16 + r = f (r) for all r. Since equality e = f (r) can be achieved when a = b = c = d = r, we need only compute the smallest value f (r). Since f grows large at either infinity, the minimum occurs when f ′(r) = 1 + 10 r−16 2√5r2−16 r+16 = 0. The resultant quadratic is easily solved for r = 65 and r = 2, with the latter being an extraneous root introduced by squaring. The largest possible e and greatest lower bound of f (r) is then f (6 /5) = 16 /5, which occurs when a = b = c = d = 6 /5 and e = 16 /5. Alternatively, proceed as before except write a = b = c = d = 8−e 4 since the maximum e must occur when the other four variables are equal. The second condition becomes a quadratic, and the largest solution is seen to be e = 16 5 .The notion of equating a, b, c, d is closely related to the idea of smoothing and Jensen’s inequality. If we are working with S1 = f (x1) + · · · + f (xn) under the constraint of a fixed sum x1 + · · · + xn, we can decrease S1 by moving several xi in the same interval I together (that is, replacing xi1 < x i2 with x′ i1 = xi1 + ≤ < x i2 − ≤ = x′ i2 for any sufficiently small ≤) for any I where f is convex. S1 can also be decreased by spreading xi in the same interval where f is concave. When seeking the maximum of S1, we proceed in the opposite fashion, pushing xi on the concave intervals of f together and moving xi on the convex intervals apart. 65. Show that for all positive reals a, b, c, d ,1 a + 1 b + 4 c + 16 d ≥ 64 a + b + c + d Solution. Upon noticing that the numerators are all squares with √1 + √1 + √4 + √16 = √64, Cauchy should seem a natural choice. Indeed, multiplying through by a + b + c + d and applying Cauchy, we have (a + b + c + d) (12 a + 12 b + 22 c + 42 d ) ≥ (1 + 1 + 2 + 4) 2 = 64 as desired. 6. (USAMO 80/5) Show that for all non-negative reals a, b, c ≤ 1, ab + c + 1 + bc + a + 1 + ca + b + 1 + (1 − a)(1 − b)(1 − c) ≤ 1 Solution. Let f (a, b, c ) denote the left hand side of the inequality. Since ∂2 ∂a 2 f = 2b (c+a+1) 3 2c (a+b+1) 3 ≥ 0, we have that f is convex in each of the three variables; hence, the maximum must occur where a, b, c ∈ { 0, 1}. Since f is 1 at each of these 8 points, the inequality follows. Second derivative testing for convexity/concavity is one of the few places where the use of Calculus is not seriously loathed by olympiad graders. It is one of the standard techniques in inequalities and deserves to be in any mental checklist of inequality solving. In this instance, it led to an easy solution. 7. (USAMO 77/5) If a, b, c, d, e are positive reals bounded by p and q with 0 < p ≤ q,prove that (a + b + c + d + e) ( 1 a + 1 b + 1 c + 1 d + 1 e ) ≤ 25 + 6 (√ pq − √ qp )2 and determine when equality holds. Solution. As a function f of five variables, the left hand side is convex in each of a, b, c, d, e ; hence, its maximum must occur when a, b, c, d, e ∈ { p, q }. When all five variables are p or all five are q, f is 25. If one is p and the other four are q, or vice versa, f becomes 17 + 4( pq + qp ), and when three are of one value and two of the other, f = 13 + 6( pq + qp ). pq + qp ≥ 2, with equality if and only if p = q. Clearly, equality holds where p = q. Otherwise, the largest value assumed by f is 13 + 6( pq + qp ), which is obtained only when two of a, b, c, d, e are p and the other three are q, or vice versa. In such instances, f is identically the right hand side. This is a particular case of the Schweitzer inequality, which, in its weighted form, is sometimes known as the Kantorovich inequality. 78. a, b, c, are non-negative reals such that a + b + c = 1. Prove that a3 + b3 + c3 + 6 abc ≥ 14 Solution. Multiplying by 4 and homogenizing, we seek 4a3 + 4 b3 + 4 c3 + 24 abc ≥ (a + b + c)3 = a3 + b3 + c3 + 3 (a2(b + c) + b2(c + a) + c2(a + b)) + 6 abc ⇐⇒ a3 + b3 + c3 + 6 abc ≥ a2(b + c) + b2(c + a) + c2(a + b)Recalling that Schur’s inequality gives a3 +b3 +c3 +3 abc ≥ a2(b+c)+ b2(c+a)+ c2(a+b), the inequality follows. In particular, equality necessitates that the extra 3 abc on the left is 0. Combined with the equality condition of Schur, we have equality where two of a, b, c are 12 and the third is 0. This is a typical dumbass solution. Solution 2. Without loss of generality, take a ≥ b ≥ c. As a+b+c = 1, we have c ≤ 13 or 1 −3c ≥ 0. Write the left hand side as ( a+b)3 −3ab (a+b−2c) = ( a+b)3 −3ab (1 −3c). This is minimized for a fixed sum a + b where ab is made as large as possible. As by AM-GM ( a + b)2 ≥ 4ab , this minimum occurs if and only if a = b. Hence, we need only consider the one variable inequality 2 (1−c 2 )3 + c3 + 6 (1−c 2 )2 c = 14 · (9 c3 − 9c2 + 3 c + 1). Since c ≤ 13 , 3 c ≥ 9c2. Dropping this term and 9 c3, the inequality follows. Particularly, 9c3 = 0 if and only if c = 0, and the equality cases are when two variables are 12 and the third is 0. 9. (IMO 74/5) If a, b, c, d are positive reals, then determine the possible values of aa + b + d + bb + c + a + cb + c + d + da + c + d Solution. We can obtain any real value in (1 , 2). The lower bound is approached by a → ∞ , b = d = √a, and c = 1. The upper bound is approached by a = c → ∞ , b = d = 1. As the expression is a continuous function of the variables, we can obtain all of the values in between these bounds. Finally, these bounds are strict because aa + b + d + bb + c + a + cb + c + d + da + c + d >aa + b + c + d + ba + b + c + d + ca + b + c + d + da + b + c + d = 1 and aa + b + d + bb + c + a + cb + c + d + da + c + d <aa + b + ba + b + cc + d + dc + d = 2 Whenever extrema occur for unusual parameterizations, we should expect the need for non-classical inequalities such as those of this problem where terms were completely dropped. 810. (IMO 95/2) a, b, c are positive reals with abc = 1. Prove that 1 a3(b + c) + 1 b3(c + a) + 1 c3(a + b) ≥ 32 Solution 1. Let x = 1 a , y = 1 b , and z = 1 c . We perform this substitution to move terms out of the denominator. Since abc = xyz = 1, we have 1 a3(b + c) + 1 b3(c + a) + 1 c3(a + b) = x2 y + z + y2 x + z + z2 x + y Now, multiplying through by ( x + y)( y + z)( z + x), we seek x4 + y4 + z4 + x3y + x3z + y3z + xy 3 + xz 3 + yz 3 + x2yz + xy 2z + xyz 2 ≥ 3 √xyz · ( 3xyz + 32 · (x2y + x2z + y2x + xy 2 + xz 2 + yz 2)) which follows immediately by AM-GM, since x2yz +xy 2z+xyz 2 ≥ 3 3 √x4y4z4, x3y+xy 3+x3z 3 ≥ 3 √x7y4z and 7x4+4 y4+z4 12 ≥ 3 √x7y4z - as guaranteed by Muirhead’s inequality. Solution 2. Substitute x, y, z as before. Now, consider the convex function f (x) = x−1 for x > 0. ( f (x) = xc is convex for c < 0 and c ≥ 1, and concave for 0 < c ≤ 1, verify this with the second derivative test.) Now, by Jensen, x2 y + z + y2 z + x + z2 x + y = xf (y + zx ) yf (z + xy ) zf (x + yz ) ≥ (x + y + z)f ((y + z) + ( z + x) + ( x + y) x + y + z ) = x + y + z 2But x + y + z ≥ 3 3 √xyz = 3, as desired. Solution 3. Perform the same substitution. Now, multiplying by ( x + y + z) and applying Cauchy, we have 12 (( y + z) + ( z + x) + ( x + y)) ( x2 y + z + y2 z + x + z2 x + y ) ≥ 12(x + y + z)2 Upon recalling that x+y +z ≥ 3 we are done. Incidentally, the progress of this solution with Cauchy is very similar to the weighted Jensen solution shown above. This is no coincidence, it happens for many convex f (x) = xc. Solution 4. Apply the same substitution, and put x ≥ y ≥ z. Simultaneously, xy+z ≥ yz+x ≥ zx+y . Hence, by Chebyshev, x · ( xy + z ) y · ( yz + x ) z · ( zx + y ) ≥ x + y + z 3 ( xy + z + yx + z + zx + y ) Again, x + y + z ≥ 3. But now we have Nesbitt’s inequality, xy+z + yx+z + zx+y ≥ 32 . This follows immediately from AM-HM upon adding 1 to each term. 911. Let a, b, c be positive reals such that abc = 1. Show that 2(a + 1) 2 + b2 + 1 + 2(b + 1) 2 + c2 + 1 + 2(c + 1) 2 + a2 + 1 ≤ 1 Solution. The previous problem showed the substitution offers a way to rewrite an inequality in a more convenient form. Substitution can also be used to implicity use a given. First, expand the denominators and apply AM-GM, obtaining 2(a + 1) 2 + b2 + 1 = 2 a2 + b2 + 2 a + 2 ≤ 1 ab + a + 1 Now, write a = xy , b = yz , c = zx . We have 1 ab +a+1 = 1 xz+xy+1 = yz xy +yz +zx . It is now evident that the sum of the new fractions is 1. 12. (USAMO 98/3) Let a0, . . . , a n real numbers in the interval (0 , π 2 ) such that tan ( a0 − π 4 ) tan ( a1 − π 4 ) · · · + tan ( an − π 4 ) ≥ n − 1Prove that tan( a0) tan( a1) · · · tan( an) ≥ nn+1 Solution 1. Let yi = tan (x − π 4 ). We have tan( xi) = tan ((xi − π 4 ) + π 4 ) = yi+1 1−yi .Hence, given s = y0 + · · · + yn ≥ n − 1 we seek to prove ∏ni=0 1+ yi 1−yi ≥ nn+1 . Observe that for any a > b and fixed sum a + b, the expression ( 1 + a 1 − a ) · ( 1 + b 1 − b ) = 1 + 2( a + b)(1 − a)(1 − b)can be decreased by moving a and b any amount closer together. Hence, for any sequence y0, . . . , y n, we can replace any yi > sn+1 and yj < sn+1 with y′ i = sn+1 and y′ j = yi + yj − sn+1 , decreasing the product. Now we have n ∏ i=0 1 + yi 1 − yi ≥ ( 1 + sn+1 1 − sn+1 )n+1 ≥ ( 2nn+1 2 n+1 )n+1 = nn+1 Where the last inequality follows from the fact that 1+ x 1−x is an increasing function of x. Solution 2. Perform the same substitution. The given can be written as 1 + yi ≥ ∑ j6=i (1 − yj ), which by AM-GM gives 1+ yn n ≥ ∏ j6=i (1 − yj ) 1 n . Now we have n ∏ i=0 1 + yi n ≥ n ∏ i=0 ∏ j6=i (1 − yj ) 1 n = n ∏ i=0 (1 − yi)as desired. 10 13. Let a, b, c be positive reals. Prove that 1 a(1 + b) + 1 b(1 + c) + 1 c(1 + a) ≥ 31 + abc with equality if and only if a = b = c = 1. Solution. Multiply through by 1+ abc and add three to each side, on the left obtaining 1 + a + ab + abc a(1 + b) + 1 + b + bc + abc b(1 + c) + 1 + c + ac + abc c(1 + a)= (1 + a) + ab (1 + c) a(1 + b) + (1 + b) + bc (1 + a) b(1 + c) + (1 + c) + ac (1 + b) c(1 + a)which is at least 6 by AM-GM, as desired. In particular, this AM-GM asserts the equivalence of (1+ a) a(1+ b) and a(1+ b)1+ a , or that they are both one. Likewise, all of the other terms must be 1. Now, (1 + a)2 = a2(1 + b)2 = a2b2(1 + c)2 = a2b2c2(1 + a)2, so the product abc = 1. Hence, 1+ aa(1+ b) = bc (1+ a)1+ b = bc (1+ a) b(1+ c) so that 1 + b = b + bc = b + 1 a . It is now easy to see that equality holds if and only if a = b = c = 1. 14. (Romanian TST) Let a, b, x, y, z be positive reals. Show that xay + bz + yaz + bx + zax + by ≥ 3 a + b Solution. Note that ( a + b)( xy + yz + xz ) = ( x(ay + bz ) + y(az + bx ) + z(ax + by )). We introduce this factor in the inequality, obtaining (x(ay + bz ) + y(az + bx ) + z(ax + by )) ( xay + bz + yaz + bx + zax + by ) ≥ (x + y + z)2 ≥ 3( xy + yz + xz )Where the last inequality is simple AM-GM. The desired follows by simple algebra. Again we have used the idea of introducing a convenient factor to clear denominators with Cauchy. 15. The numbers x1, x 2, . . . , x n obey −1 ≤ x1, x 2, . . . , x n ≤ 1 and x 31 + x 32 + · · · + x 3 n = 0. Prove that x1 + x2 + · · · + xn ≤ n 3 Solution 1. Substitute yi = x3 i so that y1 + · · · + yn = 0. In maximizing 3 √y1 + · · · + 3 √yn, we note that f (y) = y 13 is concave over [0 , 1] and convex over [ −1, 0], with \|f ′(y1)\| ≥ \| f ′(y2)\| ⇐⇒ 0 < \|y1\| ≤ \| y2\|. Hence, we may put y1 = · · · = yk = −1; −1 ≤ yk+1 < 0, and yk+2 = · · · = yn = k−yk+1 n−k−1 . We first show that yk+1 leads to a maximal sum of 3 √yi if it is -1 or can be made positive. If \|yk+1 \| < \|yk+2 \|, we set 11 y′ k+1 = y′ k+2 = yk+1 +yk+2 2 , increasing the sum while making yk+1 positive. Otherwise, set y′ k+1 = −1 and y′ k+2 = 1 − yk+1 − yk+2 , again increasing the sum of the 3 √yi. Now we may apply Jensen to equate all positive variables, so that we need only show k 3 √−1 + ( n − k) 3 √ kn − k ≤ n 3But we have (n + 3 k)3 − 27( n − k)2k = n3 − 18 n2k + 81 nk 2 = n(n − 9k)2 ≥ 0as desired. Particularly, as k is an integer, equality can hold only if 9 \|n and then if and only if one ninth of the variables yi are -1 and the rest are 1/8. Solution 2. Let xi = sin( αi), and write 0 = x31 + · · · + x3 n = sin 3(α1) + · · · + sin 3(αn) = 14 ((3 sin( α1) − sin(3 α1)) + · · · + (3 sin( αn) − sin(3 αn))). It follows that x1 + · · · + xn =sin( α1) + · · · + sin( αn) = sin(3 α1)+ ··· +sin(3 αn)3 ≤ n 3 . The only values of sin( α) which lead to sin(3 α) = 1 are 12 and -1. The condition for equality follows. 16. (Turkey) Let n ≥ 2 be an integer, and x1, x 2, . . . , x n positive reals such that x21 + x22 + · · · + x2 n = 1. Determine the smallest possible value of x51 x2 + x3 + · · · + xn x52 x3 + · · · + xn + x1 · · · + x5 n x1 + · · · + xn−1 Solution. Observe that ∑ni=1 xi ∑ j6=i xj ≤ n − 1, so that ( n∑ i=1 xi (∑ j6=i xj )) ( n∑ i=1 x5 i ∑ j6=i xi ) ≥ (x31 + · · · + x3 n )2 = n2 (x31 + · · · + x3 n n )2 ≥ n2 (x21 + · · · + x2 n n )3 = 1 n Leads to n∑ i=1 x5 i ∑ j6=i xi ≥ 1 n(n − 1) with equality if and only if x1 = · · · = xn = 1√n .17. (Poland 95) Let n be a positive integer. Compute the minimum value of the sum x1 + x22 2 + x33 3 + · · · + xnn n 12 where x1, x 2, . . . , x n are positive reals such that 1 x1 1 x2 · · · + 1 xn = n Solution. The given is that the harmonic mean of x1, . . . , x n is 1, which implies that the product x1x2 · · · xn is at least 1. Now, we apply weighted AM-GM x1 + x22 2 + x33 3 + · · · + xnn n ≥ ( 1 + 12 + 13 + · · · + 1 n ) 1+ 12 +··· + 1 n √x1x2 · · · xn = 1 + 12 + 13 + · · · + 1 n Prove that for all positive reals a, b, c, d , a4b + b4c + c4d + d4a ≥ abcd (a + b + c + d) Solution. By AM-GM, 23 a4b + 7 b4c + 11 c4d + 10 ad 4 51 ≥ 51 √a102 b51 c51 d51 = a2bcd from which the desired follows easily. Indeed, the most difficult part of this problem is determining suitable weights for the AM-GM. One way is to suppose arbitrary weights x1, x 2, x 3, x 4 for a4b, b 4c, c 4d, ad 4 respectively, and solve the system x1 + x2 + x3 + x4 = 14x1 + x2 = 24x2 + x3 = 14x3 + x4 = 119. (USAMO 01/3) Let a, b, c be nonnegative reals such that a2 + b2 + c2 + abc = 4 Prove that 0 ≤ ab + bc + ca − abc ≤ 2 Solution [by Tony Zhang.] For the left hand side, note that we cannot have a, b, c > Suppose WLOG that c ≤ 1. Then ab +bc +ca −abc ≥ ab +bc +ca −ab = c(a+b) ≥ 0. For the right, 4 = a2 + b2 + c2 + abc ≥ 4( abc )43 =⇒ abc ≤ 1. Since by the pigeon hole principle, among three numbers either two exceed 1 or two are at most 1. Hence, we assume WLOG that ( a − 1)( b − 1) ≥ 0, which gives ab + 1 ≥ a + b ⇐⇒ abc + c ≥ ac + bc ⇐⇒ c ≥ ac + bc − abc . Now, we have ab + bc + ca − abc ≤ ab + c. Either we are done or ab +c > 2. But in the latter case, 4 = ( a2 +b2)+ c(c+2 ab ) > 2ab +2 c = 2( ab +c) > 4, a contradiction. 13 20. (Vietnam 98) Let x1, . . . , x n be positive reals such that 1 x1 + 1998 + 1 x2 + 1998 + · · · + 1 xn + 1998 = 11998 Prove that n √x1x2 · · · xn n − 1 ≥ 1998 Solution. Let yi = 1 xi+1998 so that y1 + · · · + yn = 11998 and xi = 1 yi − 1998. Now n ∏ i=1 xi = n ∏ i=1 ( 1 yi − 1998 ) = e Pni=1 ln “1 yi−1998 ” Hence, to minimize the product of the xi, we equivalently minimize the sum of ln ( 1 yi − 1998 ) .In particular, ddy ( ln ( 1 y − 1998 )) = 1 ( 1 y − 1998 )2 · −1 y2 = −1 y − 1998 y2 d2 dy 2 ( ln ( 1 y − 1998 )) = 1 − 3996 y (y − 1998 y2)2 So ln ( 1 y − 1998 ) is convex on [0 , 1/3996]. If we had 0 < y i ≤ 1/3996 for all i we could apply Jensen. Since yi + yj ≤ 1/1998 for all i, j , we consider ( 1 a − 1998 ) ( 1 b − 1998 ) ≥ ( 2 a + b − 1998 )2 ⇐⇒ 1 ab − 1998 ( 1 a + 1 b ) ≥ 4(a + b)2 − 4 · 1998 a + b ⇐⇒ (a + b)2 − 1998( a + b)3 ≥ 4ab − 4ab (a + b) · 1998 ⇐⇒ (a − b)2 ≥ 1998( a + b)( a − b)2 which incidentally holds for any a + b ≤ 11998 . Hence, any two yi and yj may be set to their average while decreasing the sum in question; hence, we may assume yi ∈ (0 , 13996 ]. Now Jensen’s inequality shows that the minimum occurs when yi = 11998 n for all i, or when xi = 1998( n − 1) for all i. It is easy to see that this yields equality. 21. (Romania 99) Show that for all positive reals x1, . . . , x n with x1x2 · · · xn = 1, we have 1 n − 1 + x1 · · · + 1 n − 1 + xn ≤ 114 Solution. First, we prove a lemma: the maximum of the sum occurs when n − 1 of the xi are equal. Consider f (y) = 1 k+ey for an arbitrary nonnegative constant k. We have f ′(y) = −ey (k+ey)2 and f ′′ (y) = ey (ey −k)(k+ey )3 . Evidently f ′′ (y) ≥ 0 ⇐⇒ ey ≥ k. Hence, f (y) has a single inflexion point where y = ln( k), where f (y) is convex over the interval ((ln( k), ∞). Now, we employ the substitution yi = ln( xi) so that y1 + · · · + yn = 0 and n ∑ i=1 1 n − 1 + xi = n ∑ i=1 f (yi)We take k = n − 1 and write k0 = ln( n − 1). Suppose that y1 ≥ · · · ≥ ym ≥ k0 ≥ ym+1 ≥ · · · xn for some positive m. Then by, Majorization, f (y1) + · · · + f (ym) ≤ (m − 1) f (k0) + f (y1 + · · · + ym − (m − 1) k0)But then, also by Majorization, (m − 1) f (k0) + f (ym+1 ) + · · · + f (yn) ≤ (n − 1) f ((m − 1) k0 + ym+1 + · · · + yn n − 1 ) Otherwise, all of the yi are less than k0. In that case we may directly apply Majorization to equate n − 1 of the yi whilst increasing the sum in question. Hence, the lemma is valid. 7 N Applying the lemma, it would suffice to show kk + x + 1 k + 1 xk ≤ 1Clearing the denominators, ( k2 + kxk ) ( k + x) ≤ k2 + k ( x + 1 xk ) x1−k −xk + x + k ≤ x1−k But now this is evident. We have Bernoulli’s inequality, since x1−k = (1 + ( x − 1)) 1−k ≥ 1 + ( x − 1)(1 − k) = x + k − xk . Equality holds only where x = 1 or n = 2. 22. (Darij Grinberg) Show that for all positive reals a, b, c , √b + ca + √c + ab + √a + bc ≥ 4( a + b + c) √(a + b)( b + c)( c + a) 7This n−1 equal value principle is particularly useful. If a differentiable function has a single inflexion point and is evaluated at narbitrary reals with a fixed sum, any minimum or maximum must occur where some n−1 variables are equal. 15 Solution 1. By Cauchy, we have √(a + b)( a + c) ≥ a + √bc . Now, ∑ cyc √b + ca ≥ 4( a + b + c) √(a + b)( b + c)( c + a) ⇐⇒ ∑ cyc b + ca √(a + b)( a + c) ≥ 4( a + b + c)Substituting our result from Cauchy, it would suffice to show ∑ cyc (b + c) √bc a ≥ 2( a + b + c)WLOG a ≥ b ≥ c, implying b + c ≤ c + a ≤ a + b and √bc a ≤ √ca b ≤ √ab c . Hence, by Chebyshev and AM-GM, ∑ cyc (b + c) √bc a ≥ (2( a + b + c)) (√bc a + √ca b + √ab c ) 3 ≥ 2( a + b + c)as desired. Solution 2. Let x = √b + c, y = √c + a, z = √a + b. Then x, y, z are the sides of acute triangle XY Z (in the typical manner), since x2 + y2 = a + b + 2 c > a + b = z2.The inequality is equivalent to ∑ cyc xy2 + z2 − x2 ≥ x2 + y2 + z2 xyz Recalling that y2 + z2 − x2 = 2 yz cos( X), we reduce this to the equivalent ∑ cyc x2 cos( X) ≥ 2( x2 + y2 + z2)WLOG, we have x ≥ y ≥ z, implying 1cos( X) ≥ 1cos( Y ) ≥ 1cos( Z) , so that applying Chebyshev to the left reduces the desired to proving that the sum of the reciprocals of the cosines is at least 6. By AM-HM, 1cos( X) + 1cos( Y ) + 1cos( Z) ≥ 9cos( X) + cos( Y ) + cos( Z)But recall from triangle geometry that cos( X) + cos( Y ) + cos( Z) = 1 + rR and R ≥ 2r.The desired is now evident. 16 23. Show that for all positive numbers x1, . . . , x n, x31 x21 + x1x2 + x22 x32 x22 + x2x3 + x23 · · · + x3 n x2 n xnx1 + x21 ≥ x1 + · · · + xn 3 Solution. Observe that 0 = ( x1 −x2)+( x2 −x3)+ · · · +( xn −x1) = ∑ni=1 x3 i−x3 i+1 x2 i+xixi+1 +x2 i+1 .Hence, (where xn+1 = x1) n ∑ i=1 x3 i x2 i xixi+1 x2 i+1 = 12 n ∑ i=1 x3 i x3 i+1 x2 i xixi+1 + x2 i+1 But now a3 + b3 ≥ 13 a3 + 23 a2b + 23 ab 2 + 13 b3 = 13 (a + b)( a2 + ab + b2). Hence, 12 n ∑ i=1 x3 i x3 i+1 x2 i xixi+1 + x2 i+1 ≥ 12 n ∑ i=1 xi + xi+3 3 = 13 n ∑ i=1 xi as desired. 24. Let a, b, c be positive reals such that a + b ≥ c; b + c ≥ a; and c + a ≥ b, we have 2a2(b + c) + 2 b2(c + a) + 2 c2(a + b) ≥ a3 + b3 + c3 + 9 abc Solution. After checking that equality holds for ( a, b, c ) = ( t, t, t ) and (2 t, t, t ), it is apparent that more than straight AM-GM will be required. To handle the condition, put a = y + z, b = z + x, c = x + y with x, y, z ≥ 0. Now, the left hand side becomes 4x3 + 4 y3 + 4 z3 + 10 x2(y + z) + 10 y2(z + x) + 10 z2(x + y) + 24 xyz while the right hand side becomes 2 x3 + 2 y3 + 2 z3 + 12 x2(y + z) + 12 y2(z + x) + 12 z2(x + y) + 18 xyz .The desired is seen to be equivalent to x3 + y3 + z3 + 3 xyz ≥ x2(y + z) + y2(z + x) + z2(x + y), which is Schur’s inequality. Equality holds where x = y = z, which gives ( a, b, c ) = ( t, t, t ), or when two of x, y, z are equal and the third is 0, which gives (a, b, c ) ∈ { (2 t, t, t ), (t, 2t, t ), (t, t, 2t)}.25. Let a, b, c be the lengths of the sides of a triangle. Prove that a √2b2 + 2 c2 − a2 + b √2c2 + 2 a2 − b2 + c √2a2 + 2 b2 − c2 ≥ √3 Solution 1. Again write a = y + z, b = z + x, and c = x + y, noting that x, y, z are positive. (Triangles are generally taken to be non-degenerate when used in inequalities.) We have ∑ cyc a √2b2 + 2 c2 − a2 = ∑ cyc y + z √4x2 + 4 xy + 4 xz + y2 + z2 − 2yz 17 Consider the convex function f (x) = 1√x . (As we shall see, Jensen almost always provides a tractable means of eliminating radicals from inequalities.) Put x+y +z = 1. We have ∑ cyc (y + z)f (4x2 + 4 xy + 4 xz + y2 + z2 − 2yz ) ≥ (( y + z) + ( z + x) + ( x + y)) f (∑ cyc (y + z) (4 x2 + 4 xy + 4 xz + y2 + z2 − 2yz )(y + z) + ( z + x) + ( x + y) ) = 2√2 √∑ cyc 4x2(y + z) + (4 xy 2 + 4 xyz ) + (4 xyz + 4 xz 2) + y3 + z3 − y2z − yz 2 Noting that ∑ cyc 4x2(y + z) + (4 xy 2 + 4 xyz ) + (4 xyz + 4 xz 2) + y3 + z3 − y2z − yz 2 = ∑ cyc 2x3 + 7 x2(y + z) + 8 xyz ,8( x + y + z)3 ≥ 3 ∑ cyc 2x3 + 7 x2(y + z) + 8 xyz ⇐⇒ ∑ sym 4x3 + 24 x2y + 8 xyz ≥ ∑ sym 3x3 + 21 x2y + 12 xyz ⇐⇒ 2x3 + 2 y3 + 2 z3 + 3 (x2(y + z) + y2(z + x) + z2(x + y)) ≥ 24 xyz which follows by AM-GM. As a follow up on an earlier mentioned connection, oberserve the similarity of the above application of Jensen and the following inequality (which follows by H¨ older’s inequality) (∑ i αiβi ) ( ∑ i αi 1 √βi )2 ≥ (∑ i αi )3 Solution 2 [by Darij Grinberg.] Let ABC be a triangle of side lengths a, b, c in the usual order. Denote by ma, m b, m c the lengths of the medians from A, B, C respectively. Recall from triangle goemetry that ma = 12 √2b2 + 2 c2 − a2, so that we need only show ama + bmb + cmc ≥ 2√3. But a triangle with side lengths ma, m b, m c, in turn, has medians of length 3a 4 , 3b 4 , and 3c 4 . The desired inequality is therefore equivalent to 43 ma a 43 mb b 43 mc c ≥ 2√3 where we refer to the new triangle ABC . Recalling that 23 ma = AG , where G is the centroid, the desired is seen to be equivalent to the geometric inequality AG a + BG b + CG c ≥ √3. But we are done as we recall from triangle geometry that AM a + BM b + CM c ≥ √3 holds for any point inside triangle ABC .8 8For a complete proof of this last inequality, see post #14. 18 26. (IMO 99/2) For n ≥ 2 a fixed positive integer, find the smallest constant C such that for all nonnegative reals x1, . . . , x n, ∑ 1≤i<j ≤n xixj (x2 i x2 j ) ≤ C ( n∑ i=1 xi )4 Solution. The answer is C = 18 , which is obtained when any two xi are non-zero and equal and the rest are 0. Observe that by AM-GM, (x1 + · · · + xn)4 = ( n∑ i=1 x2 i 2 ∑ 1≤i<j ≤n xixj )2 ≥ 4 ( n∑ i=1 x2 i ) ( 2 ∑ 1≤i<j ≤n xixj ) = 8 ∑ 1≤i<j ≤n xixjn∑ k=1 x2 k But x21 + · · · + x2 n ≥ x2 i x2 j with equality iff xk = 0 for all k 6 = i, j . It follows that (x1 + · · · + xn)4 ≥ 8 ∑ 1≤i<j ≤n xixj (x2 i x2 j ) as desired. 27. Show that for nonnegative reals a, b, c ,2a6 + 2 b6 + 2 c6 + 16 a3b3 + 16 b3c3 + 16 c3a3 ≥ 9a4(b2 + c2) + 9 b4(c2 + a2) + 9 c4(a2 + b2) Solution 1. Consider ∑ cyc (a − b)6 = ∑ cyc a6 − 6a5b + 15 a4b2 − 20 a3b3 + 15 a2b4 − 6ab 5 + b6 ≥ 0and ∑ cyc ab (a − b)4 = ∑ cyc a5b − 4a4b2 + 6 a3b3 − 4a2b4 + ab 5 ≥ 0Adding six times the latter to the former yields the desired result. Solution 2. We shall prove a6 − 9a4b2 + 16 a3b3 − 9a2b4 + b6 ≥ 0. We have a6 − 2a3b3 + b6 = (a3 − b3)2 = ((a − b)( a2 + ab + b2))2 ≥ (a − b)2(3 ab )2 = 9 a4b2 − 18 a3b3 + 9 a2b4 19 As desired. The result now follows from adding this lemma cyclicly. The main difficulty with this problem is the absence of a5b terms on the right and also the presence of a4b2 terms on the right - contrary to where Schur’s inequality would generate them. Evidently AM-GM is too weak to be applied directly, since a6 + 2 a3b3 ≥ 3a4b2 cannot be added symmetrically to deduce the problem. By introducing the factor ( a − b)2,however, we weight the AM-GM by a factor which we “know” will be zero at equality, thereby increasing its sharpness. 28. Let 0 ≤ a, b, c ≤ 12 be real numbers with a + b + c = 1. Show that a3 + b3 + c3 + 4 abc ≤ 932 Solution. Let f (a, b, c ) = a3 + b3 + c3 + 4 abc and g(a, b, c ) = a + b + c = 1. Because f and g are polynomials, they have continuous first partial derivatives. Moreover, the gradient of g is never zero. Hence, by the theorem of Lagrange Multipliers ,any extrema occur on the boundary or where ∇f = λ∇g for suitable scalars λ. As ∇f =< 3a2 + 4 bc, 3b2 + 4 ca, 3c2 + 4 ab > and ∇g =< 1, 1, 1 >, we have λ = 3a2 + 4 bc = 3b2 + 4 ca = 3c2 + 4 ab g(a, b, c ) = a + b + c = 1 We have 3 a2 + 4 bc = 3 b2 + 4 ca or ( a − b)(3 a + 3 b − 4c) = ( a − b)(3 − 7c) = 0 for any permutation of a, b, c . Hence, without loss of generality, a = b. Now, 3 a2 + 4 ac =3c2 + 4 a2 and a2 − 4ac + 3 c2 = ( a − c)( a − 3c) = 0. The interior local extrema therefore occur when a = b = c or when two of {a, b, c } are three times as large as the third. Checking, we have f (13 , 13 , 13 ) = 7 /27 < 13 /49 = f (17 , 37 , 37 ). Recalling that f (a, b, c ) is symmetric in a, b, c , the only boundary check we need is f (12 , t, 12 −t) ≤ 932 for 0 ≤ t ≤ 12 .We solve h(t) = f (12, t, 12 − t ) = 18 + t3 + (12 − t )3 2 t (12 − t ) = 14 + t 4 − t2 2 h(t) is 14 at either endpoint. Its derivative h′(t) = 14 − t is zero only at t = 14 . Checking, h(14 ) = f (12 , 14 , 14 ) = 932 . Since h(t) has a continuous derivative, we are done. (As a further check, we could observe that h′′ (t) = −1 < 0, which guarantees that h(14 ) is a local minimum.) 20 Usage Note. The use of Lagrange Multipliers in any solution will almost certainly draw hostile review, in the sense that the tiniest of errors will be grounds for null marks. If you consider multipliers on Olympiads, be diligent and provide explicit, kosher remarks about the continuous first partial derivatives of both f (x1, . . . , x n) and the constraint g(x1, . . . , x n) = k, as well as ∇g 6 = 0, before proceeding to solve the system ∇f = λ∇g. The main reason this approach is so severely detested is that, given sufficient computational fortitude (if you are able to sort through the relevant algebra and Calculus), it can and will produce a complete solution. The example provided here is included for completeness of instruction; typical multipliers solutions will not be as clean or painless. 9 (Vascile Cartoaje) Let p ≥ 2 be a real number. Show that for all nonnegative reals a, b, c , 3 √ a3 + pabc 1 + p + 3 √ b3 + pabc 1 + p + 3 √ c3 + pabc 1 + p ≤ a + b + c Solution. By H¨ older, (∑ cyc 3 √ a3 + pabc 1 + p )3 ≤ (∑ cyc 11 + p ) ( ∑ cyc a ) ( ∑ cyc a2 + pbc ) But a2 + b2 + c2 ≥ ab + bc + ca (proven by AM-GM, factoring, or a number of other methods) implies that ∑ cyc a2 + pbc ≤ (p + 1) ∑ cyc a2 + 2 bc 3 = p + 1 3 (a + b + c)2 From which we conclude (∑ cyc 3 √ a3 + pabc 1 + p )3 ≤ (a + b + c)3 as desired. 30. Let a, b, c be real numbers such that abc = −1. Show that a4 + b4 + c4 + 3( a + b + c) ≥ a2 b + a2 c + b2 c + b2 a + c2 a + c2 b Solution. First we homogenize, obtaining a4 + b4 + c4 + a3(b + c) + b3(c + a) + c3(a + b) − 3abc (a + b + c) ≥ 0. As this is homogenous in the fourth degree, we can scale a, b, c 9Just how painful can the calculations get? Most multipliers solutions will tend to look more like than this solution. 21 by any real k and hence may now ignore abc = −1. Equality holds at a = b = c = 1, but also at a = b = 1 , c = −2, a = 1 , b = 0 , c = −1, and a number of unusual locations with the commonality that a + b + c = 0. Indeed, c = −a − b is a parametric solution, and we discover the factorization ( a + b + c)2(a2 + b2 + c2 − ab − bc − ca ) ≥ 0. (We are motivated to work with factorizations because there are essentially no other inequalities with a + b + c = 0 as an equality condition.) 31. (MOP 2003) Show that for all nonnegative reals a, b, c , a4(b2 + c2) + b4(c2 + a2) + c4(a2 + b2) +2abc (a2b + a2c + b2c + b2a + c2a + c2b − a3 − b3 − c3 − 3abc ) ≥ 2a3b3 + 2 b3c3 + 2 c3a3 Solution. As was suggested by the previous problem, checking for equality cases is important when deciding how to solve a problem. We see that setting a = b produces equality. As the expression is symmetric, this certainly implies that b = c and c = a are equality cases. Hence, if P (a, b, c ) is the difference LHS - RHS, then ( a − b)( b − c)( c − a)\|P (a, b, c ). Obviously, if the problem is going to be true, ( a−b) must be a double root of P , and accordingly we discover the factorization P (a, b, c ) = ( a − b)2(b − c)2(c − a)2.The result illustrated above was no accident. If ( x−y) divides a symmetric polynomial P (x, y, z ), then ( x − y)2 divides the same polynomial. If we write P (x, y, z ) = ( x − y)Q(x, y, z ), then ( x − y)Q(x, y, z ) = P (x, y, z ) = P (y, x, z ) = ( y − x)Q(y, x, z ), which gives Q(x, y, z ) = −Q(y, x, z ). Hence Q(x, x, z ) = 0, and ( x − y) also divides Q(x, y, z ). 32. (Cezar Lupu) Let a, b, c be positive reals such that a + b + c + abc = 4. Prove that a √b + c + b √c + a + c √a + b ≥√22 · (a + b + c) Solution. By Cauchy (∑ cyc a√b + c ) ( ∑ cyc a √b + c ) ≥ (a + b + c)2 But, also by Cauchy, √(a + b + c) ( a(b + c) + b(c + a) + c(a + b)) ≥ ∑ cyc a√b + c Hence, ∑ cyc a √b + c ≥√22 · (a + b + c) · √ a + b + cab + bc + ca 22 And we need only show a + b + c ≥ ab + bc + ca . Schur’s inequality for r = 1 can be expressed as 9abc a+b+c ≥ 4( ab + bc + ca ) − (a + b + c)2. Now, we suppose that ab + bc + ca > a + b + c, and have 9abc a + b + c ≥ 4( ab + bc + ca ) − (a + b + c)2 (a + b + c) (4 − (a + b + c)) = abc (a + b + c)Hence, a + b + c < 3. But then abc < 1, which implies 4 = a + b + c + abc < 4. Contradiction, as desired. 33. (Iran 1996) Show that for all positive real numbers a, b, c ,(ab + bc + ca ) ( 1(a + b)2 + 1(b + c)2 + 1(c + a)2 ) ≥ 94 Solution. Fearless courage is the foundation of all success. 10 When everything else fails, return to the sure-fire strategy of clearing all denominators. In this case, we obtain 4( a + b)2(b + c)2(c + a)2(ab + bc + ca ) ( 1(a + b)2 + 1(b + c)2 + 1(c + a)2 ) = ∑ sym 4a5b + 8 a4b2 + 10 a4bc + 6 a3b3 + 52 a3b2c + 16 a2b2c2 on the left, and on the right, 9( a + b)2(b + c)2(c + a)2 = ∑ sym 9a4b2 + 9 a4bc + 9 a3b3 + 54 a3b2c + 15 a2b2c2 Canceling like terms, we seek ∑ sym 4a5b − a4b2 + a4bc − 3a3b3 − 2a3b2c + a2b2c2 Sure enough, this is true, since 3a5b+ab 5 4 ≥ a4b2 and a4b2+a2b4 2 ≥ a3b3 by AM-GM, and abc (a3 + b3 + c3 − a2(b + c) + b2(c + a) + c2(a + b) + 3 abc ) ≥ 0 by Schur. 34. (Japan 1997) Show that for all positive reals a, b, c ,(a + b − c)2 (a + b)2 + c2 + (b + c − a)2 (b + c)2 + a2 + (c + a − b)2 (c + a)2 + b2 ≥ 35 10 Found on a fortune cookie by Po-Ru Loh while grading an inequality on 2005 Mock IMO Day 2 that was solved by brutal force. 23 Solution. Put a + b + c = 3 so that equality will hold at a = b = c = 1 and suppose that there exists some k for which (b + c − a)2 (b + c)2 + a2 = (3 − 2a)2 (3 − a)2 + a2 ≥ 15 + ka − k for all positive a, b, c ; such an inequality would allow us to add cyclicly to deduce the desired inequality. As the inequality is parametrically contrived to yield equality where a = 1, we need to find k such that a = 1 is a double root. At a = 1, the derivative on the left is (2(3 −2a)·− 2)((3 −a)2+a2)−((3 −2a)2)(2(3 −a)·− 1+2 a)((3 −a)2+a2)2 = −18 25 . The derivative on the right is k, so we set k = −18 25 . But for this k we find (3 − 2a)2 − (15 + ka − k ) ((3 − a)2 + a2) = 18 25 − 54 a2 25 + 36 a3 25 = 18 25 (a − 1) 2(2 a + 1) ≥ 0as desired. Alternatively, we could have used AM-GM to show a3 + a3 + 1 ≥ 3a2. As hinted at by a previous problem, inequalities are closely linked to polynomials with roots of even multiplicity. The isolated manipulation idea used in this solution offers a completely different approach to the inequalities which work with every term. 35. (MOP 02) Let a, b, c be positive reals. Prove that ( 2ab + c )23 + ( 2bc + a )23 + ( 2ca + b )23 ≥ 3 Solution. Suppose that there exists some r such that ( 2ab + c )23 ≥ 3ar ar + br + cr We could sum the inequality cyclicly to deduce what we want. Since equality holds at a = b = c = 1, we use derivatives to find a suitable r. At the said equality case, on the left, the partial derivative with respect to a is 23 , while the same derivative on the right is 23 r. Equating the two we have r = 1. (This is necessary since otherwise the inequality will not hold for either a = 1 + ≤ or a = 1 − ≤.) 11 Now, 3aa + b + c ≤ 3a 3 3 √ a · (b+c 2 )2 11 Actually, even this is a special case of the general sense that the convexity of one side must exceed the convexity of the other. More precisely, we have the following result: Let fand gfunctions over the domain Dwith continuous partial derivatives. If f(ν)≥g(ν) for all ν∈D, then at every equality case ν0, ∇(f−g)( ν0) = 0and every component of ∇2(f−g) ( ν0) is nonnegative. 24 = a23 (b+c 2 )23 = ( 2ab + c )23 by AM-GM, as desired. 36. (Mildorf) Let n ≥ 2 be an integer. Prove that for all reals a1, a 2, . . . , a n > 0 and reals p, k ≥ 1, ( a1 + a2 + · · · + an ap 1 ap 2 · · · + apn )k ≥ ak 1 ak 2 · · · + akn apk 1 apk 2 · · · + apk n where inequality holds iff p = 1 or k = 1 or a1 = a2 = · · · = an, flips if instead 0 < p < 1, and flips (possibly again) if instead 0 < k < 1. Solution. Taking the kth root of both sides, we see that the inequality is equivalent to n∑ i=1 k √ aki ak 1 ak 2 · · · + akn ≥ n ∑ i=1 k √ apk i apk 1 apk 2 · · · apk n WLOG, suppose that a1 ≥ a2 ≥ · · · ≥ an. We prove a lemma. Let Si = api ap 1+··· +apn and Ti = aqi aq 1+··· +aqn for i = 1 , 2, . . . , n where 0 < q < p . Then the sequence S1, S 2, . . . , S n majorizes the sequence T1, T 2, . . . , T n.To prove the claim, we note that S1 ≥ · · · ≥ Sn and T1 ≥ · · · ≥ Tn and have, for m ≤ n, m ∑ i=1 Si ≥ m ∑ i=1 Ti ⇐⇒ (ap 1 · · · + apm) ( aq 1 · · · + aqn) ≥ (aq 1 · · · + aqm) ( ap 1 · · · + apn) ⇐⇒ (ap 1 · · · + apm) (aqm+1 + · · · + aqn ) ≥ (aq 1 · · · + aqm) (apm+1 + · · · + apn ) ⇐⇒ ∑ (i,j )\| { 1≤i≤m<j ≤n} api aqj − aqi apj ≥ 0Which is obvious. In particular, m = n is the equality case, and the claim is established. But now the desired is a direct consequence of the Majorization inequality applied to the sequences in question and the function f (x) = k √x.37. (Vascile Cartoaje) Show that for all real numbers a, b, c,(a2 + b2 + c2)2 ≥ 3 (a3b + b3c + c3a) 25 Solution. We will be content to give the identity (a2 + b2 + c2)2 − 3( a3b + b3c + c3a) = 12 ∑ cyc (a2 − 2ab + bc − c2 + ca )2 Any Olympiad partipant should be comfortable constructing various inequalities through well-chosen squares. Here, we could certainly have figured we were summing the square of a quadratic that is 0 when a = b = c such that no term a2bc is left uncancelled. A good exercise is to show that equality actually holds iff a = b = c or, for some cyclic permutation, a : b : c ≡ sin 2 (4π 7 ) : sin 2 (2π 7 ) : sin 2 (π 7 ).38. (Anh-Cuong) Show that for all nonnegative reals a, b, c , a3 + b3 + c3 + 3 abc ≥ ab √2a2 + 2 b2 + bc √2b2 + 2 c2 + ca √2c2 + 2 a2 Solution. Upon observing that this inequality is stronger than Schur’s inequality for r = 1, we are inspired to prove a sharp lemma to eliminate the radical. Knowing that √2x2 + 2 y2 ≥ x + y ≥ 2xy x+y , we seek a combination of the latter two that exceeds the former. We find 3x2 + 2 xy + 3 y2 2( x + y) ≥ √2x2 + 2 y2 This follows from algebra, since (3 x2 + 2 xy + 3 y2)2 = 9 x4 + 12 x3y + 22 x2y2 + 12 xy 3 +9y4 ≥ 8x4 + 16 x3y + 16 x2y2 + 16 xy 3 + 8 y4 = 4( x + y)2(2 x2 + 2 y2), so that (3 x2 + 2 xy +3y2)2 − 4( x + y)2(2 x2 + 2 y2) = x4 − 4x3y + 6 x2y2 − 4xy 3 + y4 = ( x − y)4 ≥ 0. Now, ∑ cyc ab √2a2 + 2 b2 ≤ ∑ cyc (3 a2 + 2 ab + 3 b2)ab 2( a + b)So it would suffice to show ∑ cyc a(a − b)( a − c) = ∑ cyc (a3 + abc − ab (a + b)) ≥ ∑ cyc (3 a2 + 2 ab + 3 b2)ab 2( a + b) − ab (a + b)= ∑ cyc 3a3b + 2 a2b2 + 3 ab 3 − 2a3b − 4a2c2 − 2ab 3 2( a + b)= ∑ cyc ab (a − b)2 2( a + b)But ∑ cyc (b + c − a)( b − c)2 = 2 ∑ cyc a(a − b)( a − c)26 so that the desired is ∑ cyc ( b + c − a − bc b + c ) (b − c)2 ≥ 0which is evident, since without loss of generality we may assume a ≥ b ≥ c and find ( a + b − c − ab a + b ) (a − b)2 ≥ 0 ( c + a − b − ac a + c ) ((a − c)2 − (b − c)2) ≥ 0 ( b + c − a − bc b + c ) (b − c)2 + ( c + a − b − ac a + c ) (b − c)2 ≥ 0The key to this solution was the sharp upper bound on the root-mean-square. At first glance our lemma seems rather arbitrary and contrived. Actually, it is a special case of a very sharp bound on the two variable power mean that I have conjectured and proved. Mildorf’s Lemma 1 Let k ≥ − 1 be an integer. Then for all positive reals a and b, (1 + k)( a − b)2 + 8 ab 4( a + b) ≥ k √ak + bk 2 with equality if and only if a = b or k = ±1, where the power mean k = 0 is interpreted to be the geometric mean √ab . Moreover, if k < −1, then the inequality holds in the reverse direction, with equality if and only if a = b. Usage Note. As of early November 2005, I have proven an extension of this lemma to additional values of k.12 Thus, you may rest assured that the result stated above is true. I was unable to get this result published, so I have instead posted the proof here as “ASharpBound.pdf.” However, the proof is rather difficult (or at least so I think, being as though it took me nearly half a year) and the lemma is far from mainstream. Thus, should you require it on an Olympiad, you should prove it for whatever particular value of k you are invoking. This is not terribly difficult if k is a small integer. One simply takes the kth power of both sides and factors the difference of the two sides as (a − b)4 · P (a, b ), etc. For x ≥ y ≥ 1, prove that x √x + y + y √y + 1 + 1 √x + 1 ≥ y √x + y + x √x + 1 + 1 √y + 1 12 In particular, the inequality holds for all kin ( −∞ ,−1) ,{− 1,0,1},(1 ,3/2] ,[2 ,∞) with the signs ≤,≥,≤ ,≥respectively, with equality iff a=bor k=±1. 27 Solution. By observation, equality holds when y = 1 and when x = y. Combining this with the restriction, it makes sense to write x = y + a and y = 1 + b where a, b ≥ 0. Now we can write x − y √x + y + y − 1 √y + 1 + 1 − x √1 + x ≥ 0 ⇐⇒ a √2 + a + 2 b + b √2 + b ≥ a + b √2 + a + b But this is evident by Jensen’s inequality applied to the convex function f (x) = 1√x ,since af (2 + a + 2 b) + bf (2 + b) ≥ (a + b)f (a(2 + a + 2 b) + b(2 + b) a + b ) = (a + b)f ((a + b)2 + 2( a + b) a + b ) = a + b √2 + a + b as desired. 40. (MOP) For n ≥ 2 a fixed positive integer, let x1, . . . , x n be positive reals such that x1 + x2 + · · · + xn = 1 x1 1 x2 · · · + 1 xn Prove that 1 n − 1 + x1 1 n − 1 + x2 · · · + 1 n − 1 + xn ≤ 1 Solution. We will prove the contrapositive. (We are motivated to do this for two good reasons: 1) it is usually difficult the show that the sum of some reciprocals is bounded above, and 2) the given relation in its current form is an abomination.) Take yi = 1 n−1+ xi , and for the sake of contradiction assume y1 + · · · + yn > 1. Since the yi are too large, the xi are too small and we shall prove 1 x1 · · · + 1 xn x 1 + · · · + xn.Since xiyi = 1 − (n − 1) yi, we have (n − 1) yi > (n − 1) ( yi + 1 − n ∑ j=1 yj ) = (n − 1) yi − 1 + n ∑ j=1 (1 − (n − 1) yj )= −xiyi + n ∑ j=1 xj yj (∗)=⇒ n − 1 xi −1 + n ∑ j=1 xj yj xiyi (∗∗ )28 Summing () over i,(n − 1) ( 1 x1 · · · + 1 xn ) n ∑ i=1 xiyi (( n∑ j=1 1 xj yj ) − 1 xiyi ) But by Cauchy and (), we have ( n∑ j=1 1 xj yj ) − 1 xiyi ≥ (n − 1) 2 (∑nj=1 xj yj ) − xiyi (n − 1) 2 (n − 1) yi = n − 1 yi Hence, (n − 1) ( 1 x1 · · · + 1 xn ) n ∑ i=1 xiyi (n − 1 yi ) = ( n − 1)( x1 + · · · + xn)as desired. 41. (Vascile Cartoaje) Show that for positive reals a, b, c ,14a2 − ab + 4 b2 + 14b2 − bc + 4 c2 + 14c2 − ca + 4 a2 ≥ 97( a2 + b2 + c2) Solution. Upon expansion, we see that it is equivalent to ∑ sym 56 a6 − 28 a5b + 128 a4b2 + 44 a3b3 + 95 2 a4bc + 31 a3b2c − 45 2 a2b2c2 ≥ 0We conjure up the following inequalities: ∑ sym a6 − 2a5b + a4bc ≥ 0 (1) ∑ sym a5b − 4a4b2 + 3 a3b3 ≥ 0 (2) ∑ sym a4b2 − a4bc − a3b3 + 2 a3b2c − a2b2c2 ≥ 0 (3) ∑ sym a4bc − 2a3b2c + a2b2c2 ≥ 0 (4) (1) and (4) follow from Schur’s inequality for r = 4 and r = 1 (multiplied by abc )respectively. (2) is the result of expanding ∑ cyc ab (a − b)4 ≥ 0, and (3) is the expanded form of the famous ( a − b)2(b − c)2(c − a)2 ≥ 0. The desired now follows by subtracting 56 times (1), 84 times (2), 208 times (3), 399 2 times (4), and then simple AM-GM to clear the remaining a2b2c2.29 This is about as difficult as a dumbass solution can get. A good general strategy is to work with the sharpest inequalities you can find until you reduce a problem to something obvious, starting with the most powerful (most bunched, in this case ∑ sym a6) term and work your way down to the weak terms while keeping the most powerful term’s coefficient positive. My solution to this problem starts with (1), Schur with r = 4 (Schur is stronger for larger r), which is almost certainly sharper than the inequality in question. Next, inequality (2) is a sharp cyclic sum to use the a5b terms. In particular, it relates terms involving only two of the three variables. Most of the time, the only inequality that can “pull up” symmetric sums involving three variables to stronger ones involving just two is Schur, although it does so at the expense of a very strong term with only one variable. Hence, we made a logical choice. Inequality (3) is extremely sharp, and allowed us to obtain more a4bc and a3b3 terms simultaneously. In particular, it was necessary to cancel the a3b3 terms. I’ll note that this inequality is peculiar to sixth degree symmetry in three variables - it does not belong to a family of similar, nice inequalities. Finally, inequality (4), which is a handy corollary to (3), is another Schur. Every inequality we have used so far is quite sharp, and so it is no surprise that the leftovers are the comparatively weak AM-GM. 42. (Reid Barton, IMO Shortlist 03/A6.) Let n ≥ 2 be a positive integer and x1, x 2, . . . , x n,y1, y 2, . . . , y n a sequence of 2 n positive reals. Suppose z2, z 3, . . . , z 2n is such that z2 i+j ≥ xiyj for all i, j ∈ { 1, . . . , n }. Let M = max {z2, z 3, . . . , z 2n}. Prove that (M + z2 + z3 + · · · + z2n 2n )2 ≥ (x1 + · · · + xn n ) ( y1 + · · · + yn n ) Reid’s official solution. Let max( x1, . . . , x n) = max( y1, . . . , y n) = 1. (We can do this by factoring X from every xi, Y from every yj , and √XY from every zi+j without changing the sign of the inequality.) We will prove M + z2 + · · · + z2n ≥ x1 + x2 + · · · + xn + y1 + y2 + · · · + yn, after which the desired follows by AM-GM. We will show that the number of terms on the left which are greater than r is at least as large as the number of terms on the right which are greater than r, for all r ≥ 0. For r ≥ 1, the claim is obvious, since all terms on the right are at most 1. Now take r < 1. Let A and B denote the set of i for which xi > r and the set of j for which yj > r respectively, and write a = \|A\|, b = \|B\|. Evidently, from our scaling, a, b ≥ 1. Now, xi > r and yj > r implies zi+j ≥ √xiyj ≥ r. Hence, if C is the set of k for which zk > r , we have \|C\| ≥ \| A + B\|, where the set addition is defined by the set of possible values if we take an element of A and add it to an element of B. How-ever, \|A + B\| ≥ \| A\| + \|B\| − 1, since if A and B consist of the values p1 < · · · < p a and q1 < · · · < q b respectively we have all of the values p1 +q1 < . . . < p a +q1 < · · · < p a +qb in A + B. Hence, \|C\| ≥ a + b − 1. Since \|C\| ≥ 1, there is some zk > r , and hence, M > r . Therefore, the left side of the inequality in question has at least a + b terms which exceed r, as desired. • 30 The preponderance of difficulty here stemmed from dealing with the superabundance of givens, especially the mysterious M . Scaling allowed us to introduce some degree of control and, with marked audacity, a profoundly clever idea. As it turned out, the in-equality was no sharper than simple AM-GM! It is my opinion that it is highly unlikely that a problem as staggeringly pernicious as this one will appear on an Olympiad - at least in the foreseeable future. Nevertheless, I have included it here for the purpose of illustrating just how unusual and creative a solution can be. 3 Problems (MOP 04) Show that for all positive reals a, b, c , ( a + 2 ba + 2 c )3 + ( b + 2 cb + 2 a )3 + (c + 2 ac + 2 b )3 ≥ 32. (MOP) Show that if k is a positive integer and a1, a 2, . . . , a n are positive reals which sum to 1, then n∏ i=1 1 − aki aki ≥ (nk − 1)n Let a1, a 2, . . . , a n be nonnegative reals with a sum of 1. Prove that a1a2 + a2a3 + · · · + an−1an ≤ 144. (Ukraine 01) Let a, b, c, x, y, z be nonnegative reals such that x + y + z = 1. Show that ax + by + cz + 2 √(ab + bc + ca )( xy + yz + zx ) ≤ a + b + c Let n > 1 be a positive integer and a1, a 2, . . . , a n positive reals such that a1a2 . . . a n = 1. Show that 11 + a1 · · · + 11 + an ≤ a1 + · · · + an + n (Aaron Pixton) Let a, b, c be positive reals with product 1. Show that 5 + ab + bc + ca ≥ (1 + a)(1 + b)(1 + c)7. (Valentin Vornicu 13 ) Let a, b, c, x, y, z be arbitrary reals such that a ≥ b ≥ c and either x ≥ y ≥ z or x ≤ y ≤ z. Let f : R → R+0 be either monotonic or convex, and let k be a positive integer. Prove that f (x)( a − b)k(a − c)k + f (y)( b − c)k(b − a)k + f (z)( c − a)k(c − b)k ≥ 0 13 This improvement is more widely known than the other one in this packet, and is published in his book, Olimpiada de Matematica... de la provocare la experienta , GIL Publishing House, Zalau, Romania. (In English, “The Math Olympiad... from challenge to experience.”) 31 8. (IMO 01/2) Let a, b, c be positive reals. Prove that a √a2 + 8 bc + b √b2 + 8 ca + c √c2 + 8 ab ≥ 19. (USAMO 04/5) Let a, b, c be positive reals. Prove that (a5 − a2 + 3 ) ( b5 − b2 + 3 ) ( c5 − c2 + 3 ) ≥ (a + b + c)3 (Titu Andreescu) Show that for all nonzero reals a, b, c , a2 b2 + b2 c2 + c2 a2 ≥ ac + cb + ba (IMO 96 Shortlist) Let a, b, c be positive reals with abc = 1. Show that ab a5 + b5 + ab + bc b5 + c5 + bc + ca c5 + a5 + ca ≤ 112. Let a, b, c be positive reals such that a + b + c = 1. Prove that √ab + c + √bc + a + √ca + b ≥ 1 + √ab + √bc + √ca (APMO 2005/2) Let a, b, c be positive reals with abc = 8. Prove that a2 √(a3 + 1) ( b3 + 1) + b2 √(b3 + 1) ( c3 + 1) + c2 √(c3 + 1) ( a3 + 1) ≥ 4314. Show that for all positive reals a, b, c , a3 b2 − bc + c2 + b3 c2 − ca + a2 + c3 a2 − ab + b2 ≥ a + b + c (USAMO 97/5) Prove that for all positive reals a, b, c ,1 a3 + b3 + abc + 1 b3 + c3 + abc + 1 c3 + a3 + abc ≤ 1 abc (Mathlinks Lore) Show that for all positive reals a, b, c, d with abcd = 1, and k ≥ 2, 1(1 + a)k + 1(1 + b)k + 1(1 + c)k + 1(1 + d)k ≥ 22−k (IMO 05/3) Prove that for all positive a, b, c with product at least 1, a5 − a2 a5 + b2 + c2 + b5 − b2 b5 + c2 + a2 + c5 − c2 c5 + a2 + b2 ≥ 032 18. (Mildorf) Let a, b, c, k be positive reals. Determine a simple, necessary and sufficient condition for the following inequality to hold: (a + b + c)k (akbk + bkck + ckak) ≤ (ab + bc + ca )k(ak + bk + ck)19. Let a, b, c be reals with a + b + c = 1 and a, b, c ≥ − 34 . Prove that aa2 + 1 + bb2 + 1 + cc2 + 1 ≤ 910 20. (Mildorf) Show that for all positive reals a, b, c , 3 √4a3 + 4 b3 + 3 √4b3 + 4 c3 + 3 √4c3 + 4 a3 ≤ 4a2 a + b + 4b2 b + c + 4c2 c + a Let a, b, c, x, y, z be real numbers such that (a + b + c)( x + y + z) = 3 , (a2 + b2 + c2)( x2 + y2 + z2) = 4 Prove that ax + by + cz ≥ 022. (Po-Ru Loh) Let a, b, c be reals with a, b, c > 1 such that 1 a2 − 1 + 1 b2 − 1 + 1 c2 − 1 = 1 Prove that 1 a + 1 + 1 b + 1 + 1 c + 1 ≤ 123. (Weighao Wu) Prove that (sin x)sin x < (cos x)cos x for all real numbers 0 < x < π 4 .24. (Mock IMO 05/2) Let a, b, c be positive reals. Show that 1 < a √a2 + b2 + b √b2 + c2 + c √c2 + a2 ≤ 3√2225. (Gabriel Dospinescu) Let n ≥ 2 be a positive integer. Show that for all positive reals a1, a 2, . . . , a n with a1a2 . . . a n = 1, √a21 + 1 2 + · · · + √a2 n 1 2 ≤ a1 + · · · + an 33 26. Let n ≥ 2 be a positive integer, and let k ≥ n−1 n be a real number. Show that for all positive reals a1, a 2, . . . , a n, ( (n − 1) a1 a2 + · · · + an )k + ( (n − 1) a2 a3 + · · · + an + a1 )k · · · + ( (n − 1) an a1 + · · · + an−1 )k ≥ n (Mildorf) Let a, b, c be arbitrary reals such that a ≥ b ≥ c, and let x, y, z be nonnegative reals with x + z ≥ y. Prove that x2(a − b)( a − c) + y2(b − c)( b − a) + z2(c − a)( c − b) ≥ 0and determine where equality holds. 28. (USAMO 00/6) Let n ≥ 2 be an integer and S = {1, 2, . . . , n }. Show that for all nonnegative reals a1, a 2, . . . , a n, b 1, b 2, . . . , b n, ∑ i,j ∈S min {aiaj , b ibj } ≤ ∑ i,j ∈S min {aibj , a j bi} (Kiran Kedlaya) Show that for all nonnegative a1, a 2, . . . , a n, a1 + √a1a2 + · · · + n √a1 · · · an n ≤ n √ a1 · a1 + a2 2 · · · a1 + · · · + an n (Vascile Cartoaje) Prove that for all positive reals a, b, c such that a + b + c = 3, aab + 1 + bbc + 1 + cca + 1 ≥ 3231. (Gabriel Dospinescu) Prove that ∀a, b, c, x, y, z ∈ R+\| xy + yz + zx = 3, a(y + z) b + c + b(z + x) c + a + c(x + y) a + b ≥ 332. (Mildorf) Let a, b, c be non-negative reals. Show that for all real k, ∑ cyc max( ak, b k)( a − b)2 2 ≥ ∑ cyc ak(a − b)( a − c) ≥ ∑ cyc min( ak, b k)( a − b)2 2(where a, b, c 6 = 0 if k ≤ 0) and determine where equality holds for k > 0, k = 0, and k < 0 respectively. 33. (Vascile Cartoaje) Let a, b, c, k be positive reals. Prove that ab + ( k − 3) bc + ca (b − c)2 + kbc + bc + ( k − 3) ca + ab (c − a)2 + kca + ca + ( k − 3) ab + bc (a − b)2 + kab ≥ 3( k − 1) k (Taiwan? 02) Show that for all positive a, b, c, d ≤ k, we have abcd (2 k − a)(2 k − b)(2 k − c)(2 k − d) ≤ a4 + b4 + c4 + d4 (2 k − a)4 + (2 k − b)4 + (2 k − c)4 + (2 k − d)4 34
7677	https://asm.org/asm/media/protocol-images/nitrate-and-nitrite-reduction-test-protocols.pdf?ext=.pdf	American Society for Microbiology © 2016 1 Nitrate and Nitrite Reduction Test Protocols \| \| Created: Tuesday, 01 November 2011 Author • Rebecca Buxton Information History Current tests for nitrate and nitrite reduction are based on the Griess diazotization reaction described in 1858 by Peter Griess. Peter Griess, the son of a blacksmith, was raised on a farm in Prussia, but “…tilling the soil was little to his liking, and on more than one occasion his father found him in a corner of the field, deep in a book, seated on the plough” (30). In his early attempts at higher education, he was far from a model student, spending time in the institution’s prison and eventually expelled for a year. Finally, in his 6th year at university he began to seriously study chemistry. He obtained employment in the coal-tar distillery where the senior chemists discovered and developed the aniline dye industry. Even though the distillery was destroyed by fire, Griess had become obsessed with the chemistry of dye making. He was recommended for a position at the Royal College of Chemistry in Great Britain on the very day that his first article on possible diazo compounds, “A Preliminary Notice on the Influence of Nitrous Acid on Aminonitro- and Aminodinitrophenol,” appeared in print. Griess' first several attempts at diazotization exploded, but his commission at the Royal College was to investigate his new nitrogen intermediates, with the result that diazobenzoic acid was isolated and an entirely new class of compounds was discovered (23, 30). Because many of these compounds were found to be stable and could be used for dying fabric without needing a mordant, Griess is heralded as the father of the modern azo dye industry (8, 13, 34). More colorful details of Griess’ life can be found in articles from the February 1930 and June 1959 Journal of the Society of Dyers & Colourists and April 1958 Journal of Chemical Education (8, 23, 30). In 1879, Griess developed a reagent for the detection of nitrite in solutions. The reagent, an acid solution of sulfanilic acid and alpha-naphthylamine, undergoes a diazotization reaction with nitrites, forming a red azo dye (17). Many variations of the so-called Griess test can be found in chemistry, medicine, and industry, but all are based on the production of an azo dye via the diazotization of nitrite. Crime scene investigation uses one such interesting application of the reaction. The nitrites of gun powder residue can be visualized with a American Society for Microbiology © 2016 2 modified Griess test (33, (Figures 1 and 2 are presented with the permission of J. Scott Doyle, Forensic Scientist Specialist, Kentucky State Police.) FIG. 1. This shirt, from a case investigation, has a bullet entrance hole in the front chest. The shirt has been tested for nitrite and lead residues. FIG. 2. Results of the modified Griess test for the shirt shown in Fig. 1. For many years, adaptations of the Griess test were suggested as a means of testing the urine of asymptomatic patients, especially women during pregnancy, for the presence of nitrites as an indication of bacteriuria (1, 17, 37, 45). Similar chemistry is now employed in commonly-used “dipstick” urine chemistry tests for nitrites (18, 45). The Griess reaction has more recently been employed to detect nitrite and nitrate as products of nitric oxide synthase in human cells and biological systems. These include a constitutive, low-output, endothelial isoform that modulates vascular tone; a constitutive, low-output, neuronal isoform that modulates synaptic plasticity; and a cytokine-inducible, high-output, immune inflammatory isoform that functions as American Society for Microbiology © 2016 3 an effector component of the cell-mediated immune response. Nitric oxide is difficult to quantitate because it is produced in small amounts under most conditions and has a short half-life, however, measuring the accumulation of nitrite and nitrate is a useful way to quantitate nitric oxide synthase activity (22). While all applications of the Griess reaction are interesting background for the student and the instructor (25) including those involving analysis of water (9) and plant physiology (10), the current protocol will focus on the reduction of nitrates and nitrites by bacteria in artificial media. PURPOSE Standard tests for reduction of nitrate, NO3 -, and nitrite, NO2 -, can be useful components of biochemical test batteries for identification of bacteria (15), including separating members of the family Enterobacteriaceae from other gram-negative bacilli, identifying species of Neisseria and separating them from Moraxella and Kingella species (21, 26),and facilitating species identification of Corynebacterium (16) and other asporogenous gram-positive bacilli (36). Nitrate reduction by bacteria is mediated by nitrate reductase and indicates that the organism can use NO3 - as an electron acceptor (2, 44) during anaerobic respiration (2). Nitrite may be reduced to a variety of nitrogen products (44) including NO, N2O, N2, and NH3, depending on the enzyme system of the organism and the atmosphere in which it is growing. Reduction of nitrate often indicates a shift to or facilitation of anaerobic metabolism, as some organisms can use nitrate as an electron acceptor during anaerobic respiration or anaerobic chemolithotrophy (2). THEORY Nitrites react with an acid solution of sulfanilic acid and alpha-naphthylamine to form a red azo dye (1). In each of the test reactions the appearance of the red dye indicates the presence of NO2 - in the test tube, whether as an unreduced primary substrate, a product of the reduction of NO3 - by the test organism, or a product of the forced reduction of NO3 - with a reducing agent (zinc) for control purposes. The essence of each reaction is the ability to detect NO2 -. In the presence of NO2 -, the color reaction begins with the acidification of NO2 - by the acetic acid in the combined reagents A and B to produce HNO2. The reaction below (27, 43) demonstrates the color development that follows: American Society for Microbiology © 2016 4 The -N=N-azo group linkage yields a colored compound via a nitroso reaction. Diazonium dye compounds are formed by coupling through an azo link of an aromatic amine with a phenolic-type compound usually at the paraposition to a hydroxyl (OH) or amino group (NH2). In this case coupling occurs para to an amino group (27). An overview of nitrate reduction and the nitrogen cycle can be found in Richardson’s brief introduction (42). The complexity of nitrate reduction pathways is discussed in depth in Moreno-Vivian’s excellent review (32). RECIPES Several formulations of substrate broth can be found in the literature and are available commercially (3, 7, 19, 38, 41, 46). It is most important to choose a medium that is free from fermentable carbohydrates and in which the organism in question grows well (27). Heart infusion broth with 0.1% KNO3 or KNO2 added is preferred by some authors over the broths described below (11). Nitrate reduction medium Beef (meat) extract 3.0 g Gelatin peptone 5.0 g American Society for Microbiology © 2016 5 Potassium nitrate (KNO3) 1.0 g Deionized water 1,000 ml Nitrite reduction medium Beef (meat) extract 3.0 g Gelatin peptone 5.0 g Potassium nitrite (KNO2) 1.0 g Deionized water 1,000 ml For either broth substrate, carefully weigh the ingredients and heat gently into solution. Dispense into test tubes and add inverted Durham tubes. Autoclave for 15 minutes at 121°C, 15 psi. The pressure of the autoclave will drive the broth into the Durham tube. Cool before use. Refrigerate for storage at 4°C to 10°C. Shelf life is approximately 6 months. Figure 3 shows 4 ml of broth in a 13 mm x 100 mm tube. FIG. 3. The pressure of autoclaving forces broth into the Durham tube. There should be no bubbles visible in the Durham tube when the broth is inoculated. Use a heavy inoculum and incubate overnight before adding reagents. Some strains need up to 5 days for full reduction of the substrates. Reagent A American Society for Microbiology © 2016 6 Several formulations of reagent A are described and available commercially. The one described below is not a proven carcinogen and produces a relatively stable color (12, 20, 22, 27, 39, 40). N,N-Dimethyl-α-naphthylamine 0.6 ml Acetic acid (5N)a 100 ml Note: fresh reagent has a very slight yellowish color. Reagent B Sulfanilic acid 0.8 g Acetic acid (5N)a 100 ml Note: fresh reagent is colorless a5N acetic acid is prepared by adding 287 ml of glacial acetic acid (17.4N) to 713 ml of deionized water. Reagents A and B should be protected from light and stored in the refrigerator. Discard the reagents if they become discolored. FIG. 4. Reagent A, N,N-dimethyl-a-naphthylamine; reagent B, sulfanilic acid. Zinc dust Zinc dust must be nitrate- and nitrite-free. American Society for Microbiology © 2016 7 FIG. 5. Zinc dust will reduce nitrate to nitrite, but will not further reduce nitrite to nitrogen gas or other nitrogenous by-products when used sparingly. PROTOCOL For either substrate, NO3 - or NO2 -, inoculate the medium with a heavy inoculum from well-isolated colonies of the test organism. Incubate at 35°C for 12 to 24 hours. Rarely, incubation up to 5 days may be required. When sufficient growth is observed in the tube, test the broth for reduction of the substrate. For NO3 - substrate 1. Observe for gas production in the Durham tube. 2. Mix two drops each of reagents A and B in a small test tube (12 mm x 75 mm). 3. Add approximately 1 ml of the broth culture to the test tube and mix well. If the test organism has reduced the NO3 - to NO2 -, a red color will usually appear within 2 minutes, indicating the presence of NO2 - in the tube. 2e- + 2H+ + NO3 - → NO2 - + H2O Nitrate reduced to nitrite If no color change is seen within 2 minutes, there are several possible reasons. Either the organism (i) was unable to reduce NO3 - at all, (ii) was capable of reducing NO2 -, or (iii) reduced NO3 - directly to molecular nitrogen. (i) NO3 - Nitrate is unchanged, negative reaction. American Society for Microbiology © 2016 8 (ii) NO3 - → NO2 - → NO → N2O → N2 Nitrate reduced to nitrite to nitric oxide or further to nitrous oxide or further to nitrogen gas; exact pathways vary. (iii) 2NO3 - + 10e- + 12H+ → N2 + 6H2O Nitrate reduced directly to molecular nitrogen. Zinc is a powerful reducing agent. If there is any NO3 - remaining in the tube (option (i) above), a small amount of zinc dust will rapidly reduce it to NO2 -. Therefore the appearance of a red color after the addition of zinc dust to a colorless reaction tube indicates a negative reaction, i.e., the organism has failed to reduce NO3 -. Zinc is added to the tube by dipping a wooden applicator stick in nitrate- and nitrite-free zinc powder, just enough to get the stick dirty, and then dropping it into the tube containing the culture broth and the reagents. If too much zinc is added, the color reaction may fade rapidly. FIG. 6. “Dirty” a wooden stick with zinc dust. American Society for Microbiology © 2016 9 FIG. 7. Drop the zinc-dusted stick into tubes for nitrate reactions that show no change after the addition of reagents. There is no need to add zinc to reactions that began with a nitrite substrate. If the broth remains colorless after the addition of zinc, the organism has also reduced the NO2 - intermediate product to N2 gas or some other nitrogenous product. N2 gas is usually visible in the Durham tube. In the absence of gas, the product is assumed to be other than N2 gas. Occasionally a lighter pink color will appear after the addition of zinc dust (Fig. 16) because of partial reduction, i.e., some of the primary NO3 - substrate remains in the tube. The original tube may be reincubated and retested the following day (Fig. 17). For NO2 - substrate 1. Observe for gas production on the surface and in the Durham tube. 2. Mix two drops each of reagents A and B in a small test tube (12mm x 75 mm). 3. Add approximately 1 ml of the broth culture to the test tube and mix well. If the test organism has reduced the NO2 -, there will be no color change, indicating that all of the original NO2 - is gone, i.e., reduced. Reduction is often confirmed by the presence of N2 gas in the Durham tube or on the surface of the broth, but other nitrogenous products may be produced. Therefore the absence of gas does not rule out reduction of NO2 -. NO2 - → NO → N2O → N2 Nitrite reduced to nitric oxide or further to nitrous oxide or further to nitrogen gas If a red color appears, it indicates the presence of NO2 - and therefore a American Society for Microbiology © 2016 10 negative reaction. Occasionally a lighter pink color will appear because of partial reduction, i.e., some of the primary NO2 - substrate remains in the tube. The original tube may be reincubated and retested the following day. There is no need to add zinc dust to this reaction. EXAMPLES OF RESULTS Nitrate negative and negative controls (uninoculated nitrate broth) FIG. 8. With the addition of reagents to uninoculated nitrate broth (or growth of organisms failing to reduce nitrate), no color change is seen. FIG. 9. The addition of zinc dust to the uninoculated broth in Fig. 8 forces the reduction of the NO3 - to NO2 -. Reagents A and B are already present, therefore the reagents react with NO2 - resulting in a red color change. Nitrite negative and negative controls (uninoculated nitrite broth) American Society for Microbiology © 2016 11 FIG. 10. The appearance of a red color with the addition of reagents A and B to an uninoculated nitrite broth indicates the presence of NO2 -. Reminder: in all cases, a red color change reaction indicates the presence of nitrites in the reaction tube, whether reduced by the organism from nitrate, a result of forced reduction of nitrate by zinc, or as the primary substrate. Reduction of nitrate and nitrite with production of nitrogen gas Pseudomonas aeruginosa American Society for Microbiology © 2016 12 FIG. 11. Growth in both the nitrate and nitrite broth. Gas production is indicated by gas bubbles in the Durham tubes and on the surface of the broth. FIG. 12. Addition of reagents A and B to both the nitrate and nitrite broth results in no color change in either broth. These results indicate reduction of the NO2 -, but whether reduction of NO3 - occurred cannot yet be determined. FIG. 13. Addition of zinc to the NO3 - broth results in no color change. This result indicates reduction of NO3 -. Reduction of nitrate and nitrite without gas production Moraxella catarrhalis FIG. 14. Growth in both the nitrate and nitrite broth. No gas production. FIG. 15. Addition of reagents A and B to both the nitrate and nitrite broth results in no color change in either broth. These results indicate the reduction of the NO2 -, but whether FIG. 16. Addition of zinc to the nitrate broth incubated for 24 hours results in a weak color. This result indicates partial reduction of NO3 -. FIG. 17. Addit zinc to the nit broth incubate 48 hours resu no color chang This result indicates the complete redu of NO3 -. American Society for Microbiology © 2016 13 reduction of NO3 - occurred cannot yet be determined. Reduction of nitrate, but not nitrite Escherichia coli FIG. 18. Growth in both the nitrate and nitrite broth. No gas production. FIG. 19. Addition of reagents A and B to both the nitrate and nitrite broth results in a red color change in both broths. This indicates the presence of NO2 - in both tubes. Nitrate in the nitrate broth has been reduced to NO2 - but NO2 -was not further reduced. Reduction of nitrite but not nitrate Neisseria lactamica American Society for Microbiology © 2016 14 FIG. 20. Growth in both the nitrate and nitrite broth. No gas production. FIG. 21. Addition of reagents A and B to both the nitrate and nitrite broth results in no color change in either broth. These results indicate reduction of the NO2 -, but whether reduction of NO3 - occurred cannot yet be determined. FIG. 22. Addition of zinc to the nitrate broth produces a red color change. This result indicates no reduction of NO3 -. QUALITY CONTROL Pseudomonas aeruginosa reduces NO3 - to N2. Escherichia coli reduces NO3 - to NO2 -. Acinetobacter baumanii does not reduce NO3 - or NO2 -. Acinetobacter baumanii should give the same reaction as an uninoculated broth. Alcaligenes faecalis and Neisseria lactamica reduce NO2 - but do not reduce NO3 -. SAFETY Reagents A and B are poisonous. They may be harmful or fatal if swallowed. They are also corrosive and may cause burns or irritation to skin, eyes, and the respiratory tract. Avoid breathing vapors and having contact with the eyes or skin. In case of contact with eyes, rinse immediately with water and seek medical advice (5, 39). Zinc dust in contact with water liberates extremely flammable gases. Keep container tightly closed and dry. In case of fire use sand, carbon dioxide, or powdered extinguishing agent to put out flames; never use water (3). The ASM advocates that students must successfully demonstrate the ability to explain and practice safe laboratorytechniques. For more information, read the laboratory safety section of the ASM Curriculum Recommendations: Introductory Course in Microbiology and the Guidelines for Biosafety in Teaching Laboratories. COMMENTS AND TIPS 1. Some authors, including those of many commonly used text books (21, 28, 35, 45), prefer adding reagents directly to the primary culture tube, but because some organisms can be slow to reduce the substrates, the small aliquots are preferred to enable testing on a second or third day (6, 36). 2. The original formula for reagent B contained alpha-naphthylamine. Because it is a known carcinogen (14), it is now replaced with N,N-Dimethyl-α-naphthylamine. Fortunately, this formula is also less prone to fading of the color reaction (27). 3. Some authors recommend adding zinc to colorless NO2 - reactions that do not contain gas to make sure that the NO2 has not been oxidized to American Society for Microbiology © 2016 15 NO3 rather than having been reduced to a nitrogen product other than N2 gas (21), but that reaction is rare. 4. Similar procedures can be employed in the identification of some fungi and mycobacteria, but they are not addressed here (24, 29). 5. Because reduction of NO3 - is assumed to be anaerobic, many published procedures warn that the medium needs to be anaerobic or deep enough to support an anaerobic process. However, later experiments have shown that the metabolism on the surface of the broth for most organisms that grow well in the broth will reduce enough dissolving oxygen for the reaction to take place (25, 26). Four to five milliliters of broth in a 13 mm x 100 mm tube provide a sufficiently small surface to volume ratio and sufficient volume to repeat the test if extended incubation is necessary. 6. Filter paper disk tests are commercially available for detecting nitrate reduction by anaerobic species grown on solid-plated media in an anaerobic atmosphere (6). 7. In order to reinforce personal and laboratory safety, the instructor may wish to dispense the zinc dust. This may present an opportune time for the instructor to assess student understanding of the exercise. 8. Be sure to run a negative control, uninoculated broth, to illustrate that the remaining NO2 will be reduced by zinc dust, producing a red color. REFERENCES 1. Aziz, M. K., H. Khan, W. Akhtar, I. Mahsud, and B. Ashiq. 2004. Accuracy of Griess test to predict asymptomatic bacteriuria during pregnancy. Gomal J. Med. Sci. 2:20–23. 2. Balows, A., and B. I. Duerden (ed). 1998. Systematic bacteriology, vol. 2, p. 106, 881. In L. Collier, A. Ballows, and M. Sussman (ed.), Topley & Wilson's microbiology and microbial infections. Oxford University Press, New York, NY. 3. Bayer Healthcare. 2005. Multistix package insert. Bayer Healthcare, Elkhart, IN. 4. Becton, Dickinson and Company. 2006. BBL nitrate broth with Durham tube package insert. Becton, Dickinson and Company, Sparks, MD. %29.pdf 5. Becton, Dickinson and Company. 2010. B D nitrate A, nitrate B, and nitrate C reagent droppers package insert. Becton, Dickenson and Company, Sparks, MD. 6. Becton, Dickinson and Company. 2010. BD BBL taxo differention discs nitrate package insert. Becton, Dickenson and Company, Sparks, MD. 7. Biomerieux. 2009. Nitrate, nitrite media package insert. Biomerieux, American Society for Microbiology © 2016 16 Durham, NC. 8. Boulton, J. 1959. Peter Griess. J. Soc. Dyers Colourists 75:277–278. 9. Campbell, W. H., P. Song, and G. G. Barbier. 2006. Nitrate reductase for nitrate analysis in water. Environ. Chem. Lett. 4:69–73. 10. Campbell, W. H. 1999. Nitrate reductase structure, function, and regulation: bridging the gap between biochemistry and physiology. Annu. Rev. Plant Physiol. Plant Mol. Biol. 50:277–303. 11. Centers for Disease Control and Prevention. 2008. Nitrate reduction test. U.S. Department of Health and Human Services, Atlanta, GA. 12. Chapin, K., and T.-L. Lauderdale. 2007. Reagents, stains, and media: bacteriology, p. 339. In P. R. Murray, E. J. Baron, J. H. Jorgensen, M. L. Landry, and M. A. Pfaller (ed.), Manual of clinical microbiology, 9th ed. ASM Press, Washington, DC. 13. Cliffe, W. H. 1959. The life and times of Peter Griess. J. Soc. Dyers Colourists. 75:278–285. 14. Committee on Laboratory Standards and Practices, American Public Health Association. 1975. Bacterial nitrate reduction test: suggestions for use of alternate (noncarcinogenic) reagents. ASM News 41:225. 15. Conn, H. J., and R. S. Breed. 1919. The use of the nitrate-reduction test in characterizing bacteria. J. Bacteriol.4:267–290. 16. Coyle, M. B., R. B. Leonard, D. J. Nowowiejski, A. Malekniazi, and D. J. Finn. 1993. Evidence of multiple taxa within commercially available reference strains of Corynebacterium xerosis. J. Clin. Microbiol. 31:1788–1793. 17. DeShan, P. W., J. A. Merrill, R. G. Wilkerson, and B. Braden. 1965. The Griess test as a screening procedure for bacteriuria during pregnancy. Obstet. Gynecol. 27:202–205. 18. Eisenstadt, J., and J. A. Washington. 1996. Diagnostic microbiology for bacteria and yeasts causing urinary tract infections, p. 45 (Griess test). In H. L. T. Mobley and J. W. Warren (ed.), Urinary tract infections, molecular pathogenesis and clinical management, ASM Press, Washington, DC. 19. Fluka Analytical. 2008. Ni trate broth package insert. Sigma-Aldrich, Buchs, Switzerland. Fluka/Datasheet/72548dat.Par.0001.File.tmp/72548dat.pdf. 20. Fluka Analytical. 2008. N itrate reducation test package insert. Sigma-Aldrich, Buchs, Switzerland. Fluka/Datasheet/73426dat.Par.0001.File.tmp/73426dat.pdf. 21. Forbes, B. A., D. F. Sahm, and A. Weissfeld (ed). 2002. Bailey and Scott's diagnostic microbiology, 11th ed., p. 277–278. Mosby, St. Louis, MO. 22. Granger, D. L., R. R. Taintor, K. S. Boockvar, and J. B. Hibbs, Jr. 1996. Measurement of nitrate and nitrite in biological samples using nitrate reductase and Griess reaction. Methods Enzymol. 268:142–152. 23. Heines, V. 1958. Peter Griess discoverer of diazo compounds. J. Chem. Educ. 35:187–191. 24. Keen, A. P., and R. G. Mitchell. 1986. Commercial strip test for reduction of nitrate by bacteria. J. Clin. Pathol.39:118. American Society for Microbiology © 2016 17 25. Kefauver, M., and F. E. Allison. 1956. Nitrite reduction by Bacterium denitrificans in relation to oxidation-reduction potential and oxygen tension. J. Bacteriol. 73: 8–14. 26. Knapp, J. S. 1984. Reduction of nitrite by Neisseria gonorrhoeae. Int. J. Syst. Bacteriol. 34:376–377. 27. MacFaddin, J. F. 2000. Biochemical tests for identification of medical bacteria, 3rd ed., p. 348-362. Lippincott Williams & Wilkins, Philadelphia, PA. 28. Mahon, C. R., D. C. Lehman, and G. Manuselis. 2011. Textbook of diagnostic microbiology, p. 191. Saunders, Maryland Heights, MO. 29. Martin, A., S. Panaiotov, F. Portaels, S. Hoffner, J. C. Palomino, and K. Angeby. 2008. The nitrate reductase assay for the rapid detection of isoniazid and rifampin resistance in Mycobacterium tuberculosis: a systematic review and meta-analysis. J. Antimicrob. Chemother. 62:56–64. 30. Mason, F. A. 1930. Johann Peter Griess, 1829-1888. J. Soc. Dyers Colourists 46:33–39. 31. Molecular Probes. 2003. Griess reagent kit for nitrite determination product information. Molecular Probes, Eugene, OR. 32. Moreno-Vivián, C., P. Cabello, M. Martínez-Luque, R. Blasco, and F. Castillo. 1999. Prokaryotic nitrate reduction: molecular properties and functional distinction among bacterial nitrate reductases. J. Bacteriol. 181:6573–6584. 33. National Institute of Justice. 1990. Modified Griess test. U. S. Department of Justice, Washington, DC. 34. Paine, C. 1959. Symposium on "The Diazo Reaction Today," concluding remarks. J. Soc. Dyers Colourists.75:307–308. 35. Pommerville, J. C. (ed.). 2011. Alcamo's fundamentals of microbiology, 9th ed., p. 177. Jones and Bartlett, Sudbury, MA. 36. Pratt-Rippin, K., and M. Pezzlo.1992. Identification of commonly isolated gram-positive bacteria, p.1.20.33 (Nitrate broth). In H. D. Isenberg (ed.), Clinical microbiology procedures handbook. ASM Press, Washington, DC. 37. Rabi, T. 1981. Evaluation of a new sensitive nitrite test as a reliable screening tool for bacteriuria. J. Clin. Pathol.34:723–729. 38. Remel. 2009. Nitrate broth package insert. Remel, Lenexa, KS. (search for R061536). 39. Remel. 2009. Nitrate reagent A package insert. Remel, Lenexa, KS. (search for R21239). 40. Remel. 2009. Nitrate reagent B package insert. Remel, Lenexa, KS. (search for R21242). 41. Remel. 2009. Nitrite broth package insert. Remel, Lenexa, KS. (search for R061552). American Society for Microbiology © 2016 18 42. Richardson, D. J. 2001. Introduction: nitrate reduction and the nitrogen cycle. Cell. Mol. Life Sci. 58:163–164. 43. Vogel, A. I., A. A. R. Tatchell, B. S. Furnis, A. J. Hannaford, P. W. G. Smith. 1996. Vogel's textbook of practical organic chemistry, 5th ed. Pearson/Prentice Hall, Harlow, England. 44. Willey, J. M., L. M. Sherwood, and C. J. Woolverton (ed.). 2011. Prescott's microbiology, 8th ed., p.244-245. 45. Winn, W., Jr., S. Allen, W. Janda, E. Koneman, G. Procop, P. Schreckenberger, G. Woods (ed.). 2006. Koneman's color atlas and textbook of diagnostic microbiology, 6th ed., p. 86, 315. Lippincott Williams & Wilkins, Baltimore, MD. 46. Zimbro, M. J., D. A. Power, S. M. Miller, G. E. Wilson, J. A. Johnson (ed.). 2009. Difco and BBL manual, 2nd ed. Nitrate broth, p. 394. BD Diagnostics, Sparks, MD. owres.pdf. REVIEWERS This resource was peer-reviewed. Participating reviewers: Laura Cathcart University of Maryland, College Park, MD Naowarat Cheeptham Thompson Rivers University, Kamloops, British Columbia, Canada Anne Hanson University of Maine, Orono, ME D. Sue Katz Rogers State University, Claremore, OK Archana Lal Independence Community College, Independence, KS Min-Ken Liao Furman University, Greenville, SC Karen Reiner Andrews University, Berrien Springs, MI Patricia Shields University of Maryland, College Park, MD Erica Suchman Colorado State University, Ft. Collins, CO 2011 AD HOC PROTOCOL REVIEW COMMITTEE Benita Brink Adams State College, Alamosa, CO American Society for Microbiology © 2016 19 Elaine Brunschwig Cuyahoga Community College, Parma, OH Madhusudan Choudary Sam Houston State University, Huntsville, TX Susan Deines Colorado State University, Fort Collins, CO Deborah Harbor College of Southern Nevada, Las Vegas, NV Catherine Hopper University of Maine, Orono, ME Jan Hudzicki Kansas University Medical Center, Kansas City, KS Roxann Karkhoff-Schweizer Colorado State University, Fort Collins, CO Min-Ken Liao Furman University, Greenville, SC Maria MacWilliams University of Wisconsin—Parkside, Kenosha, WI Maria Panec Moorpark College, Moorpark, CA Todd Primm Sam Houston State University, Huntsville, TX Karen Reiner Andrews University, Berrien Springs, MI Jackie Reynolds Richland College, Dallas, TX Amy Siegesmund Pacific Lutheran University, Tacoma, WA CONTRIBUTERS The following contributed to the Comments and Tips section at the ASM Conference for Undergraduate Educators 2011. Participating contributors: Ned Barden Massachusetts College of Pharmacy and Health Sciences, Boston, MA American Society for Microbiology © 2016 20 Carolyn Bouma West Texas A & M University, Canyon, TX Lakshmi Chilukuri University of California—San Diego, La Jolla, CA Thomas Edison dela Cruz University of Santo Tomas, Manila, Philippines Elizabeth Emmert Salisbury University, Salisbury, MD Zoe Hawk Arizona Western College, Yuma, AZ Tamara Marsh Elmhurst College, Elmhurst, IL Jackie Reynolds Richland College, Dallas, TX Nahed Salama SUNY Rockland Community College, Suffern, NY Diana Vullo Universidad Nacional General Sarmiento, Los Polvorines, Argentina Susan Young American International College, Springfield, MA
7678	https://askfilo.com/user-question-answers-smart-solutions/a-copper-specimen-having-length-1-m-width-1-cm-and-thickness-3330383030343630	A copper specimen having length 1 m , width 1 cm and thickness 1 mm is co.. World's only instant tutoring platform Instant TutoringPrivate Courses Tutors Explore TutorsBecome Tutor Login StudentTutor CBSE Smart Solutions A copper specimen having length 1 m , width 1 cm and thickness Question Question asked by Filo student A copper specimen having length 1 m , width 1 cm and thickness 1 mm is conducting a current of 1 ampere along its length. The specimen is kept in magnetic field of 1 Tesla along its thickness. It experiences Hall effect and Hall voltage 0.074 μ V appears along its width. If conductivity of copper is 5.8×107 ( Ohm−m ) - 1 , find the hall coefficient and mobility of electrons in the specimen Views: 5,189 students Updated on: Mar 31, 2025 Not the question you're searching for? Ask your question Ask your question Or Upload the image of your question Get Solution Text solutionVerified Concepts: Hall effect, Hall coefficient, Mobility of electrons, Conductivity Explanation: To find the Hall coefficient (R_H) and mobility (μ) of electrons in the copper specimen, we can use the following formulas: The Hall coefficient is given by: R H=B⋅I V H where V H is the Hall voltage, B is the magnetic field, and I is the current. The mobility of electrons is given by: μ=q R H⋅σ where σ is the conductivity and q is the charge of an electron (approximately 1.6×1 0−19 C). Now, let's calculate: Step 1: Calculate the Hall coefficient (R_H) Given: Hall voltage, V H=0.074 μ V=0.074×1 0−6 V Magnetic field, B=1 T Current, I=1 A Using the formula: R H=1⋅1 0.074×1 0−6=0.074×1 0−6 m 3/C Step 2: Calculate the mobility (μ) Given: Conductivity, σ=5.8×1 0 7(Ohm-m)−1 Charge of an electron, q=1.6×1 0−19 C Using the formula: μ=q R H⋅σ=1.6×1 0−19 0.074×1 0−6⋅5.8×1 0 7 Calculating: μ=1.6 0.074×5.8×1 0 13 m 2/(V-s)μ≈0.2675×1 0 13 m 2/(V-s)=2.675×1 0−1 m 2/(V-s) Final Results: Hall Coefficient, R H=0.074×1 0−6 m 3/C Mobility, μ≈0.2675 m 2/(V-s) Step by Step Solution: Step 1 Calculate the Hall coefficient using the formula R_H = V_H / (B I). Substitute V_H = 0.074 10^{-6} V, B = 1 T, and I = 1 A. Step 2 Calculate the mobility using the formula μ = (R_H σ) / q. Substitute R_H from step 1, σ = 5.8 10^7 (Ohm-m)^{-1}, and q = 1.6 10^{-19} C. Final Answer: Hall Coefficient, R_H = 0.074 10^{-6} m^3/C; Mobility, μ ≈ 0.2675 m^2/(V-s) Ask your next question Or Upload the image of your question Get Solution Get instant study help from an expert tutor 24/7 Download Filo Found 8 tutors discussing this question James Discussed A copper specimen having length 1 m , width 1 cm and thickness 1 mm is conducting a current of 1 ampere along its length. The specimen is kept in magnetic field of 1 Tesla along its thickness. It experiences Hall effect and Hall voltage 0.074 μ V appears along its width. If conductivity of copper is 5.8×107 ( Ohm−m ) - 1 , find the hall coefficient and mobility of electrons in the specimen 10 mins ago Discuss this question LIVE 10 mins ago One destination to cover all your homework and assignment needs Learn Practice Revision Succeed Instant 1:1 help, 24x7 60, 000+ Expert tutors Textbook solutions Big idea maths, McGraw-Hill Education etc Essay review Get expert feedback on your essay Schedule classes High dosage tutoring from Dedicated 3 experts Download AppExplore now Trusted by 4 million+ students Students who ask this question also asked Question 1 Views: 5,382 Well-Formatted Question Q1. What is a sequencing problem? Discuss its applications in at least two areas. Q2. The costs of 5 different jobs on two machines are given: \| Machines \| Job 1 \| Job 2 \| Job 3 \| Job 4 \| Job 5 \| --- --- --- \| \| 1 \| 4 \| 7 \| 3 \| 9 \| 6 \| \| 2 \| 6 \| 9 \| 5 \| 10 \| 6 \| Solve the problem assuming that the objective is to minimize the total cost. Topic: Smart Solutions View solution Question 2 Views: 5,692 Date 13. Following observations were taken with a tacheometer on vertically held sterff. take multiplying constant 50 and additive constant 0.50. find the reduced bel of Δ tation Q. 14. Two straights intersect at a chainage of 2610 m the deflection angle being 3 6∘. A circular curve of 400 m radius is to be set. Calculate the chainages of the tangent paint of right handed curve 15. Two tangents intersect at chaingge of 626.57 m . it is proposed to insert a clrcular carve of radius 300 m the deflection angle is 1 6∘3 8′ Calculate. A. Chainage of tangents points B. Length of curve. C. Tangent length. (1) length of the chord. E. Aper distance or external distance F. Mid ordinate. Topic: Smart Solutions View solution Question 3 Views: 5,061 What are the consonant sounds in English? Topic: Smart Solutions View solution Question 4 Views: 5,348 If A=[a ij] is a square matrix of order 2 such that a ij=1 when i=j, and a ij=0 when i=j, find A 2: (A) [1 00 1] (B) [1 01 0] (C) [1 11 0] (D) [1 00 1] If A and B are invertible square matrices of the same order, which of the following is not correct? (A) (A B)−1=A−1/∣B∣ (B) (A B)−1=1/(∣A∣∣B∣) (C) (A B)−1=B−1 A−1 (D) (A+B)−1=B−1+A−1 If the area of the triangle with vertices (−3,0), (3,0) and (0,k) is 9 sq. units, then the values of k will be: (A) 9 (B) ±3 (C) -9 (D) 6 If f(x)={k x,3,x<0 x≥0 is continuous at x=0, then the value of k is: (A) -3 (B) 0 (C) 3 (D) any real number The lines r=i+j−k+λ(2 i+3 j−6 k) and r=2 i−j−k+μ(6 i+9 j−18 k), where λ,μ are scalars, are: (A) coincident (B) skew (C) intersecting (D) parallel The degree of the differential equation 1+(d x d y)2=d x 2 d 2 y is: (A) 4 (B) 2 3 (C) 2 (D) Not defined The corner points of the bounded feasible region determined by a system of linear constraints are (0,3), (1,1) and (3,0). Let Z=p x+q y, where p,q>0. The condition on p and q so that the minimum of Z occurs at (3,0) and (1,1) is: (A) p=2 q (B) p=2 q (C) p=3 q (D) p=q Topic: Smart Solutions View solution View more Video Player is loading. 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Talk to a tutor now 388 students are taking LIVE classes Question Text A copper specimen having length 1 m , width 1 cm and thickness 1 mm is conducting a current of 1 ampere along its length. The specimen is kept in magnetic field of 1 Tesla along its thickness. It experiences Hall effect and Hall voltage 0.074 μ V appears along its width. If conductivity of copper is 5.8×107 ( Ohm−m ) - 1 , find the hall coefficient and mobility of electrons in the specimen Updated On Mar 31, 2025 Topic All topics Subject Smart Solutions Class Undergraduate Answer Type Text solution:1 Are you ready to take control of your learning? Download Filo and start learning with your favorite tutors right away! 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7679	https://csus.libguides.com/c.php?g=768134&p=7396184	Skip to Main Content Sac State Library Research Guides Physical Sciences, Geography, and Math Math Resources & Test Study Guides Number Theory Math Resources & Test Study Guides Welcome Journals Algebra Calculus Combinatorics Game Theory Geometry and Topology Logic Number Theory Statistics Standardized Tests Toggle Dropdown ACT/SAT CAHSEE CBEST CLEP CSET GED GRE GMAT Other Books & Lecture Notes Number Theory Elementary Number Theory: Primes, Congruences, and Secrets - William Stein Elementary Number Theory - W. Edwin Clark (University of South Florida) A Course on Number Theory - Peter J. Cameron A Computational Introduction to Number Theory and Algebra - Victor Shoup Number Theory: A Contemporary Introduction - Pete L. Clark An Introduction to the Theory of Numbers - Leo Moser Yet Another Introductary Number Theory Textbook - Jonathan A. Poritz Elementary Number Theory - David M. Burton Algebraic Number Theory Introduction to Algebraic Number Theory Introduction to Algebraic Number Theory - F. Oggier Algebraic Number Theory - J.S. Milne Algebraic Number Theory Course Notes - Matthew Baker (Georgia Tech) A Course In Algebraic Number Theory - Robert Ash Analytic Number Theory Introduction to Analytic Number Theory - A.J. Hildebrand (University of Illinois) Elements of Analytic Number Theory - P. S. Kolesnikov, E. P. Vdovin (Novosibirsk) Analytic Number Theory - Otto Forster (LMU Munich) Analytic Number Theory - Lecture Notes based on Davenport’s book - Andreas Strömbergsson Study Resources MITOpenCourseware - Theory of Numbers This course is an elementary introduction to number theory with no algebraic prerequisites. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions. OpenLearn Introduction to Number Theory This free course, Introduction to number theory, is a branch of mathematics concerned with the properties of integers. Section 1 introduces Euclid’s algorithm, which is used to find the HCF of two integers, and the idea of congruences, mathematical statements used to compare remainders when two integers are each divided by another integer. Section 2 introduces modular arithmetic, in which the usual arithmetic operations are applied to congruences. Section 3 introduces multiplicative inverses, which provide a method for division in modular arithmetic, and their use in solving linear congruences which are used in cryptography for disguising information or ciphers. Please note: a Statement of Participation is not issued for this course. AoPSOnline Provides an overview of the concept of number theory, student guides, resources, and related topics. Mathematics Support Centre - Elementary Number Theory From this page you can access a collection of materials dealing with some important areas of Elementary Number Theory. Number Theory Web he Number Theory Web is a collection of links to online information of interest to number theorists. Last Updated: Aug 8, 2025 11:15 AM << Previous: Logic Next: Statistics >> Last Updated: Aug 8, 2025 11:15 AM URL:
7680	https://brainly.com/question/39879658	[FREE] Write each number in expanded form using powers of 10: a. 42,100 b. 5,312 c. 10,034 d. 1,003,145 - brainly.com 1 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 +90k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +13,7k Ace exams faster, with practice that adapts to you Practice Worksheets +5,8k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified Write each number in expanded form using powers of 10: a. 42,100 b. 5,312 c. 10,034 d. 1,003,145 1 See answer Explain with Learning Companion NEW Asked by willidal418 • 10/10/2023 0:01 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 14922604 people 14M 0.0 1 Upload your school material for a more relevant answer The presented numbers are represented in expanded form using powers of 10. This representation is the exponential notation which expresses numbers as a product of two numbers. Explanation To express a number in expanded form using powers of 10, we should identify the place value of each digit and then represent it as multiplication by a power of ten. Here are your numbers in expanded form using powers of 10: 42 100 = 4 x 10^4 + 2 x 10^3 + 1 x 10^2 5312 = 5 x 10^3 + 3 x 10^2 + 1 x 10^1 + 2 x 10^0 10 034 = 1 x 10^4 + 0 x 10^3 + 0 x 10^2 + 3 x 10^1 + 4 x 10^0 1 003 145 = 1 x 10^6 + 0 x 10^5 + 0 x 10^4 + 3 x 10^3 + 1 x 10^2 + 4 x 10^1 + 5 x 10^0 This exponential notation helps to simplify very large numbers making it easier to multiply and divide by reducing the number of zeros one has to deal with. Learn more about Expanded Form Using Powers of 10 here: brainly.com/question/1746885 SPJ4 Answered by TaylorSmith48 •49.2K answers•14.9M people helped Thanks 1 0.0 (0 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 14922604 people 14M 0.0 0 Physics - Boundless DC Electrical Circuit Analysis: A Practical Approach + Lab Manual - James Fiore Basics of General, Organic, and Biological Chemistry - David W. Ball, John W. Hill, Rhonda J. Scott Upload your school material for a more relevant answer To express numbers in expanded form using powers of 10, each digit is multiplied by 10 raised to the power of its position. For example, 42,100 can be expressed as 4 × 10^4 + 2 × 10^3 + 1 × 10^2 + 0 × 10^1 + 0 × 10^0. Each number is broken down similarly based on the place value of each digit. Explanation To write a number in expanded form using powers of 10, we break down each digit in the number according to its place value. Each digit is multiplied by 10 raised to a power that corresponds to its position in the number, starting from the right, which counts as 0. Here’s how you can expand each of the given numbers: For the number 42,100: The digit 4 is in the ten-thousands place: 4 × 10^4 The digit 2 is in the thousands place: 2 × 10^3 The digit 1 is in the hundreds place: 1 × 10^2 The last two digits (0s) represent no value, so they do not appear in the expansion. Therefore, 42,100 in expanded form is: 4×1 0 4+2×1 0 3+1×1 0 2+0×1 0 1+0×1 0 0 For the number 5,312: The digit 5 is in the thousands place: 5 × 10^3 The digit 3 is in the hundreds place: 3 × 10^2 The digit 1 is in the tens place: 1 × 10^1 The digit 2 is in the ones place: 2 × 10^0 Therefore, 5,312 in expanded form is: 5×1 0 3+3×1 0 2+1×1 0 1+2×1 0 0 For the number 10,034: The digit 1 is in the ten-thousands place: 1 × 10^4 The zeros in thousands, hundreds positions do not contribute to the value: 0 × 10^3 and 0 × 10^2 The digit 3 is in the tens place: 3 × 10^1 The digit 4 is in the ones place: 4 × 10^0 Therefore, 10,034 in expanded form is: 1×1 0 4+0×1 0 3+0×1 0 2+3×1 0 1+4×1 0 0 For the number 1,003,145: The digit 1 is in the millions place: 1 × 10^6 The first 0 is in the hundred-thousands place: 0 × 10^5 The second 0 is in the ten-thousands place: 0 × 10^4 The digit 3 is in the thousands place: 3 × 10^3 The digit 1 is in the hundreds place: 1 × 10^2 The digit 4 is in the tens place: 4 × 10^1 The digit 5 is in the ones place: 5 × 10^0 Therefore, 1,003,145 in expanded form is: 1×1 0 6+0×1 0 5+0×1 0 4+3×1 0 3+1×1 0 2+4×1 0 1+5×1 0 0 Examples & Evidence Examples of expanded form include representing 5,312 as 5 × 10^3 + 3 × 10^2 + 1 × 10^1 + 2 × 10^0 and 10,034 as 1 × 10^4 + 0 × 10^3 + 0 × 10^2 + 3 × 10^1 + 4 × 10^0. The breakdown into place value is a standard method in mathematics for expressing numbers. This technique is used widely in teaching to help students understand the value of digits based on their position. Thanks 0 0.0 (0 votes) Advertisement willidal418 has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Mathematics solutions and answers Community Answer .Write the following number in expanded form using powers of 10: 67.938 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. New questions in Mathematics Graph this compound inequality: x<−8 or x≥3. Which are possible values for x? Check all that apply. -10.2, 3, -8, 2.6, 20 Write as a fraction in lowest terms: 0.79=b aa=b= Find the value of x for which 2 lo g 4 x+9 lo g x 4=9 Identify a 1 and r for the geometric series: −4,8,−16,32,… Find the sum of the first six terms of the sequence, S 6. Calculate the income needed based on the following budgeted expenses. \| Month \| Budgeted Amount \| ---: \| \| Rent \| $350 \| \| Utilities \| $100 \| \| Food \| $150 \| \| Entertainment \| $75 \| \| Bus Pass \| $25 \| \| College Savings \| $300 \| A. The income would need to be at least $1000. B. The income would need to be at least $950. C. The income would need to be at least $1125. D. The income would need to be at least $850. Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com
7681	https://www.youtube.com/watch?v=-TQDSJwqSPg	Find the Circumcenter of a Triangle Kim Nelson 4700 subscribers 556 likes Description 78335 views Posted: 15 Jun 2017 This video shows how to construct the circumcenter of a triangle by constructing perpendicular bisectors of each side. The construction uses only a compass and straight edge. Transcript: [Music] the circum center of a triangle is the point where the perpendicular bis sectors of each side of the triangle intersect it can be in the interior or the exterior of the triangle to find the circum Center of this triangle we need to construct the perpendicular bis sectors of all the sides if you need help with perpendicular bis sectors please watch the earlier posted video [Music] oh I've done all three perpendicular bis sectors in different colors so you can see the mark set go along with each and the point right here in the center this point right here is the circum center now the reason that it's called the circum Center is because it is the center of the circumscribed circle for this triangle so if I put the compass point on the circum Center I can circumscribe a circle around the triangle I want you to notice on this one because it's an obtuse triangle the circum Center is actually outside the triangle let's see what the circumscribed circle looks [Music] like even though part of the circle is off the screen you can still see that you do get the center of the circumscribed circle
7682	https://basicmedicalkey.com/hardy-weinberg-equilibrium/	Hardy–Weinberg Equilibrium \| Basicmedical Key Basicmedical Key Fastest Basicmedical Insight Engine Home Log In Register Categories » A-K » ANATOMY BIOCHEMISTRY EMBRYOLOGY GENERAL & FAMILY MEDICINE HISTOLOGY HUMAN BIOLOGY & GENETICS L-Z » MEDICAL DICTIONARY & TERMINOLOGY MICROBIOLOGY PATHOLOGY & LABORATORY MEDICINE PHARMACY PHYSIOLOGY PUBLIC HEALTH AND EPIDEMIOLOGY Gold Membership Contact About Menu Home Log In Register Categories » A-K » ANATOMY BIOCHEMISTRY EMBRYOLOGY GENERAL & FAMILY MEDICINE HISTOLOGY HUMAN BIOLOGY & GENETICS L-Z » MEDICAL DICTIONARY & TERMINOLOGY MICROBIOLOGY PATHOLOGY & LABORATORY MEDICINE PHARMACY PHYSIOLOGY PUBLIC HEALTH AND EPIDEMIOLOGY More References » Abdominal Key Anesthesia Key Basicmedical Key Otolaryngology & Ophthalmology Musculoskeletal Key Neupsy Key Nurse Key Obstetric, Gynecology and Pediatric Oncology & Hematology Plastic Surgery & Dermatology Clinical Dentistry Radiology Key Thoracic Key Veterinary Medicine Gold Membership Contact About Showing most revelant items. Click here or hit Enter for more. Hardy–Weinberg Equilibrium 2.1 Genotype And Allele Frequencies Before getting into the definition, derivation, and application of Hardy–Weinberg equilibrium, some basic ideas for measuring genetic variation in a population need to be clarified, specifically the ideas of genotype frequencies and allele frequencies. 2.1.1 Computing Genotype Frequencies Once again, we use the standard model of a hypothetical locus with two alleles, A and a. As covered in the previous chapter, when we have two alleles, we will have three genotypes: AA, Aa, and aa. To make things easy, let us suppose that the two alleles are codominant, so that we can tell the difference between people having these three genotypes. Imagine that we visit a population and test the genotype of 150 people, and obtain the following numbers: The first thing we would want to do is to figure out the genotype frequencies, which are the proportions of each genotype. To do this, we simply divide each number by the total number of individuals. Thus, because 54 of 150 people have the AA genotype, the genotype frequency is . We can do this for all three genotypes: Although it might seem obvious, it will be important later to see that the sum of the genotype frequencies adds up to 1 (0.36 + 0.48 + 0.16 = 1.0). Although we use genotype frequencies in population genetics, note that we could also express these proportions as percentages by multiplying each proportion by 100: AA = 36%, Aa = 48%, and aa = 16%. Note that these add up to 100%. At this point, it is useful to consider another characteristic of genotype frequencies. Suppose that you were to return to this population and choose a single person at random without knowing that individual’s genotype. What is the probability that this person would have the genotype AA? The answer is 0.36, which is the genotype frequency, because this number also represents the number of times that a specific event occurs (54 people have genotype AA) divided by the total number of events (there are 150 people). 2.1.2 Computing Allele Frequencies You can see that it is easy to estimate out the relative frequency of any genotype; you simply divide the number by the total number of all genotypes. What about the frequency of an allele? What are the number of A alleles and the number of a alleles in the hypothetical population described above? 2.1.2.1 Two Alleles The answer to these questions may not be immediately obvious. I have found that many students understand better the concept of allele frequency using an analogy. Imagine there are 150 people each wearing two socks (one on each foot). For reasons unknown to us (you are welcome to make one up), some of the people are wearing two black socks, some are wearing a black sock and a white sock, and some are wearing two white socks. Suppose that you count them up and come up with the following numbers: Number of people wearing two black socks = 54 Number of people wearing one black sock and one white sock = 72 Number of people wearing two white socks = 24 Now, how many black socks are there in this group of people? If you rush in answering this question, you might come up with 126 people by adding the 54 people, with two black socks and the 72 people with one black sock, but that is actually the number of people having at least one black sock, which is not what I asked. Instead, you have to count the actual number of black socks, keeping in mind that some people are wearing two and some people are wearing only one. In this example, we have 54 people, each with two black socks, and 72 people with only one black sock. When we add these up, we get a total of We can do the same thing for the number of white socks. We have 72 people with one white sock and 24 people with two white socks, giving a total of We have 180 black socks and 120 white socks, for a total of 300 socks, which makes sense because we have a total of 150 people, each with 2 socks [ = 150(2) = 300]. We could now figure out the relative frequency of black socks by dividing the number of black socks by the total number of socks, which is . Likewise, the relative frequency of white socks is . You probably noticed that the numbers I used in the genotype frequency example and the sock example are the same. This was done on purpose to enable you to extend the sock analogy to the concept of allele frequencies. The computation is the same, except that instead of counting socks you are counting alleles. Let us return to the original question. Given the genotype numbers AA = 54, Aa = 72, and aa = 24, what are the relative frequencies of the A and a alleles? Start by counting the number of A alleles, remembering to count the A allele twice for the AA genotype and once for the Aa genotype. There are 54(2) + 72(1) = 180 A alleles. We repeat this procedure for the number of a alleles, getting 72(1) + 24(2) = 120 a alleles. Thus, this hypothetical population has 180 A alleles and 120 a alleles for a total of 300 alleles (which works out since there are 150 people, each with two alleles). Therefore, the relative frequency of the A allele is , and the relative frequency of the a allele is . A simple way to keep all of these calculations straight is to make a table that lists the number of A alleles and the number of a alleles for each of the three genotypes, as shown in Example 2.1. Example 2.1 How to Compute Allele Frequencies In this example, we use the hypothetical example from the text of a single locus with two alleles, A and a. The data are In the following table, we list the number of A and a alleles and the total number of all alleles for each genotype, and then sum each column: There are a total of 180 A alleles and 120 a alleles in this population, for a total of 300 alleles (twice the number of people sampled, because each person has two alleles). As a check, the allele frequencies should add up to 1.0, which they do (0.6 + 0.4 = 1.0). Because we use allele frequencies extensively in population genetics equations, we use symbols to refer to the different allele frequencies. Although you can feel free to create any symbol you wish, the conventional format is to use the symbol p to refer to one allele and the symbol q to refer to the other allele. In Example 2.1, p is the relative frequency of the A allele and q is the relative frequency of the a allele. We will use this notation throughout the rest of the book. Note that in the case of two alleles, there are only two allele frequencies, and these numbers must add up to 1.0. In Example 2.1, this is seen because 0.6 + 0.4 = 1.0. This property allows us a convenient check on our math; if the two allele frequencies do not add up to 1.0 (or very close, in the case of irrational numbers and roundoff), then an error was made in computing the allele frequency. Beware—I have seen this error on homework assignments, so you should always take time to check your answers thoroughly. Mathematically, we can express the sum of the allele frequencies using the following simple formula: 2.1 Note that this relationship allows us to predict one allele frequency if we know the other, because, using some simple algebra, we see that 2.2 These relationships may be intuitively obvious, but as you will see, they allow us to handle much of the mathematics of population genetics in an elegant manner. As another example of computing allele frequencies, I will use a real-world case. In the 1960s, anthropologist Jonathan Friedlaender collected data on a number of genetic markers of the blood in 18 villages on the island of Bougainville in Melanesia. One of the markers collected was the haptoglobin locus, a gene located on chromosome 16. This locus has two alleles, HPA1 and HPA2. In the village of Nupatoro, Friedlaender (1975) collected blood samples from 111 villagers, and found the following genotype numbers: The genotype frequencies are obtained by dividing each number by the total number: We find that there are 109 HPA1 alleles and 113 HPA2 alleles, for a total of 222 alleles (twice the number of people sampled). This gives allele frequencies for HPA1 of and for HPA2 of . The full computation is provided in Example 2.2 for review. Example 2.2 How to Compute Allele Frequencies This example is based on an actual study of human population genetics. Data on a number of genetic markers were collected by Friedlaender (1975) on Bougainville Island in Melanesia. One of these markers was the red blood cell protein haptoglobin protein, a locus with two alleles: HPA1 and HPA2. Genotype numbers and number of alleles are shown below: 2.1.2.2 More than Two Alleles Much of population genetics theory builds on a simple model of a single locus with two alleles. In the real world, however, there are many loci (particularly DNA markers) with more than two alleles. How can we compute allele frequencies in such cases? For loci where all the alleles are codominant, allowing identification of each genotype, the answer is simple—we count the number of alleles in the same manner. Example 2.3 shows computations for a locus with three alleles. When we have three alleles, we typically label the allele frequencies as p, q, and r (when there are more than three alleles, it is common to simply use the letter p with a subscript to refer to different alleles). Example 2.3 How to Compute Allele Frequencies for a Locus with Three Alleles Salzano et al. (1985) collected data from 216 Pacaás Novos Indians of Brazil on the group-specific component locus (GC, also known as the vitamin D binding protein) on chromosome 4, which codes for a protein. A number of early studies indentified two alleles, GC1 and GC2, but later work identified two subtypes of the GC1 allele, known as GC1F and GC1S. Salzano et al. collected data on the six genotypes associated with the three alleles GC1F, GC1S, and GC2: By summing up the columns, we see that there are 195 GC1F alleles, 160 GC1S alleles, and 77 GC2 alleles for a total of 432 alleles. Note that the sum of the three allele frequencies does not add up exactly to 1.0 (0.451 + 0.370 + 0.178 = 0.999). This is because of roundoff error; if you do all the computations of the allele frequencies to four decimal places, you will see that they do add up to 1.0 (0.4514 + 0.3704 + 0.1782 = 1.0). 2.1.2.3 An Alternative Method The method used so far is often called the allele counting method because it relies on counting the number of different alleles over all genotypes. Another method of computing allele frequencies is based on the genotype frequencies. Returning to the first example used in this chapter, we have a locus with two alleles, A and a, in a hypothetical population of 150 people with the following genotypes: 54 AA, 72 Aa, and 24 aa. When we counted the alleles, we found 180 A alleles and 120 a alleles, giving allele frequencies of and . Here is another way to calculate the allele frequencies. First, compute the genotype frequencies as before by dividing the number in each genotype by the total number of genotypes ( = 150). We did these calculations earlier in this chapter and obtained: Note that I have assigned the symbol f to refer to genotype frequency and used subscripts to refer to the different genotypes. Thus f AA refers to the frequency of the AA genotype, f Aa refers to the frequency of genotype Aa, and f aa refers to the frequency of genotype aa. We can now compute the frequency of the A allele by adding the frequency of the AA genotype to half the frequency of the Aa genotype. Likewise, we can compute the frequency of the a allele by adding the frequency of half the Aa genotype to the frequency of the aa genotype. This is easier to express mathematically than in words (which is why we use symbols): 2.3 When we plug in the actual genotype frequencies in our example, we get the same allele frequencies as with the allele counting method: Note that this method may not give the exact same results as the allele counting method if you did not compute the genotype frequencies using enough decimal places to avoid roundoff problems. My rationale for introducing this alternative method will be clear a little bit later. Can you show why equation (2.3) works mathematically? The proof is shown in Appendix 2.1 at the end of this chapter. 2.2 What is Hardy–Weinberg Equilibrium? What exactly is the Hardy–Weinberg law, and why is it so important? It is difficult to explain Hardy–Weinberg equilibrium clearly and comprehensively in a short definition, which is why I have devoted an entire chapter to it. At the most basic level, Hardy–Weinberg is an mathematical expression describing the expected genotype frequencies in a new generation. In the previous chapter, I reviewed how Punnett squares are used to compute the expected genotype distribution in offspring given the parent’s genotypes. Hardy–Weinberg provides a way of extending this idea to an entire population. Imagine that we have a locus with two alleles, A and a, where p is the frequency of the A allele and q is the frequency of the a allele. Both Hardy and Weinberg independently showed that the expected genotype frequencies in the next generation are 2.4 These equations are often familiar to the introductory student in biological anthropology and biology classes. Following the initial math fear experienced by some students, just about everyone soon learns these simple equations. For example, if we know that p = 0.7 and q = 0.3 in the parental gene pool, we can quickly figure out the expected distribution of genotypes in the next generation as The utility of predicting outcomes in the next generation can be illustrated with a more specific type of question. For example, let us imagine that the frequency of a harmful recessive allele is 0.01 and we want to know how many children will be born having two recessives, and hence a genetic disease. In this case, Hardy–Weinberg allows a quick answer: q 2 = (0.01)2 = 0.0001, which is 1 in 10,000 offspring. Likewise, we could compute the expected proportion of heterozygotes that would not get the disease but would be carriers: 2 pq = 2(0.99)(0.01) = 0.0198, which is 198 in 10,000 offspring. Using the Hardy–Weinberg equations is straightforward. It is a second aspect of the Hardy–Weinberg law that causes more confusion. Both Hardy and Weinberg independently showed that, under certain conditions, the genotype and allele frequencies would remain constant from one generation to the next. In other words, Hardy–Weinberg equilibrium states that nothing changes (the very definition of equilibrium). Another way of saying that nothing changes is saying that there is no evolution! At this point in a typical lecture, everyone becomes rightfully perplexed. It is obvious from lab experiments, field studies, and the fossil record that organisms evolve all the time, which makes it difficult to understand why valuable lecture time (and textbook space) is being taken up by something that is clearly incorrect. The answer, which we then give in lecture, is that Hardy–Weinberg equilibrium gives us a baseline; we start with the case where nothing happens to show how something could happen! Although true, the underlying nature and utility of Hardy–Weinberg equilibrium often gets lost in the introductory course. Weiss and Kurland (2007:204) sum the confusing nature of Hardy–Weinberg equilibrium quite nicely, noting, “As one student griped, ‘Let me get this straight. When nothing happens … nothing changes? Duh.’” I think that a major problem with understanding Hardy–Weinberg equilibrium is that it is difficult to reduce the concept and its implications into a simple but understandable definition. The idea instead takes more time and development, which is the goal of the remainder of this chapter. 2.3 The Mathematics of Hardy–Weinberg Equilibrium Given allele frequencies p and q, equation (2.4) allows us to predict the expected genotype frequencies in the next generation. Where did equation (2.4) come from? There are several ways to answer this, and Hardy and Weinberg each used a somewhat different method to answer this question. 2.3.1 A Simple Proof Here, I will start by demonstrating Hardy–Weinberg proportions [equation (2.4)] using a simple model of probability that uses an analogy for allele frequencies. Picture a cup filled with 100 small plastic beads, of which 60 are red and 40 are blue. We can say that the relative frequency of red beads is and the relative frequency of blue beads is . Imagine that all of these beads are mixed together and you pull one bead from the cup at random. What is the probability of getting a red bead? It is 0.6 (and the probability of getting a blue bead is 0.4). Now, imagine something a little more complicated. Pull out a bead, replace it in the cup, mix then beads together, and then pull out another bead. What is the probability of getting a red bead both times? We answer this using the and rule of probability from the previous chapter. The probability of getting two red beads is the probability of getting a red bead (0.6) multiplied by the probability of getting a red bead (0.6), which is 0.6 × 0.6 = 0.36. We can do the same type of computation to answer the question of repeating the same experiment and getting two blue beads. Here, the probability would be 0.4 × 0.4 = 0.16. What about getting one red bead and one blue bead? This is a little more complicated because there are two ways of getting a red bead and a blue bead. The first way is to get a red bead on the first try and a blue bead on the second try, and the second way is the reverse—getting a blue bead on the first try and a red bead on the second try. We can solve this problem by breaking it down into several steps. First, what is the probability of getting a red bead on the first try and a blue bead on the second try? We use the and rule to figure this out and multiply the probability of getting a red bead (0.6) by the probability of getting a blue bead (0.4), which is 0.6 × 0.4 = 0.24. The second step is to determine the probability of the reverse happening, where we get a blue bead on the first try and a red bead on the second. Using the and rule, we get 0.4 × 0.6 = 0.24. Thus, we have a probability of 0.24 of getting a red bead followed by a blue bead, and we have a probability of 0.24 of getting a blue bead followed by a red bead. However, my initial question was only about the overall probability of getting one red bead and one blue bead, and the order does not matter. What is the probability of getting a red bead and then a blue bead or a blue bead and then a red bead? Here, we use the or rule and add the probabilities. The probability of getting one red bead and one blue bead, regardless of the order, is therefore 0.24 + 0.24 = 0.48. We can now summarize the probability distribution of taking a bead from the cup at random, replacing it, and then taking a bead again at random: Probability of getting two red beads = 0.36 Probability of getting one red bead and one blue bead = 0.48 Probability of getting two blue beads = 0.16 Note that the sum of these probabilities is 1 (0.36 + 0.48 + 0.16), because we have listed all possible outcomes. Going back to genetics, we can use the same principle. Instead of an experiment where we take a bead at random from a cup of beads, picture the process of reproduction of an allele that gets into the next generation as equivalent to taking an allele at random from a gene pool. Given two alleles, A and a, the possible genotypes in the next generation are AA, Aa, and aa. How do you get the AA genotype? The answer is when each parent contributes an A allele. To put this another way, what is the probability of a child getting an A allele from one parent and an A allele from the other parent? Given that p represents the frequency of the A allele, and is the probability of a given individual inheriting an A allele, the expected frequency for genotype AA in the next generation is p times p, = p 2. How about the genotype Aa? This can happen in one of two ways. The father can contribute an A allele and the mother can contribute an a allele, or the reverse can occur, where the father contributes an a allele and the mother contributes an A allele. The probability of the first situation occurring is p times q = pq. The probability of the second situation occurring is q times p = qp, which by the commutative law of algebra is the same as pq. Thus, the probability of getting an A allele from the father and an a allele from the mother or the reverse is pq + pq = 2 pq. The final genotype to consider is aa. The probability of obtaining this genotype is computed as the probability of both parents contributing an a allele, which is q times q = q 2. These proportions (AA = p 2, Aa = 2 pq, aa = q 2) are the expected genotype frequencies [shown in equation (2.4)] expected under Hardy–Weinberg equilibrium. Table 2.1 summarizes the logic used here to demonstrate Hardy–Weinberg frequencies. Another method of deriving Hardy–Weinberg frequencies is to consider the genotypes for all possible offspring resulting from all possible random mating of parental genotypes (e.g., AA × AA, AA × Aa). This derivation can be found in most advanced population genetics texts (e.g., Hartl and Clark 2007). Table 2.1 Using Simple Probability to Derive Hardy–Weinberg Proportions a Only gold members can continue reading. Log In orRegister to continue Share this: Click to share on Twitter (Opens in new window) Click to share on Facebook (Opens in new window) Like this: Like Loading... 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7683	https://www.geeksforgeeks.org/maths/product-to-sum-formulas/	Product to Sum Formulas Last Updated : 23 Jul, 2025 Suggest changes 1 Like Product-to-sum formulas are trigonometric identities that convert the product of sine and cosine functions into a sum (or difference) of trigonometric functions. These formulas are particularly useful in simplifying the integrals and solving trigonometric equations. In this article, we will learn about, Product to Sum Formulae, related examples and others in detail. What Are Product To Sum Formulas? Product-to-sum formulas provide a powerful tool for simplifying trigonometric expressions involving products of sines and cosines, and the product to sum formulas are: sin A cos B = (1/2) [sin (A + B) + sin (A - B)] cos A sin B = (1/2) [sin (A + B) - sin (A - B)] cos A cos B = (1/2) [cos (A + B) + cos (A - B)] sin A sin B = (1/2) [cos (A - B) - cos (A + B)] Product to Sum Formulae Product-to-sum identities are used to express the product between sine and/or cosine functions as a sum or difference. The sum and difference formulas of sine and cosine functions are added or subtracted to derive these identities. The product-to-sum identities can be used to simplify the trigonometric expression. The integrals or derivatives of the trigonometric functions can be solved with ease by using these identities. For all possible sine and cosine product combinations, there are four products to sum or difference formulae in total. Product to Sum Identities \| Product of Two Cosine Functions \| cos A cos B = (½) [cos (A + B) + cos (A – B)] \| --- \| \| Product of Cosine and Sine Functions \| cos A sin B = (½) [sin (A + B) - sin (A - B)] \| \| Product of Sine and Cosine Functions \| sin A cos B = (½) [sin (A + B) + sin (A - B)] \| \| Product of Two Sine Functions \| sin A sin B = (½) [cos (A - B) - cos (A + B)] \| Product To Sum Formulas Derivation The product to sum formulae can be derived using the trigonometric sum/difference formulae. The Sum/difference formulae are given below: sin (A + B) = (½) {sin A cos B + cos A sin B}———— (1) sin (A – B) = (½) {sin A cos B – cos A sin B}———— (2) cos (A + B) = (½) {cos A cos B – sin A sin B}———— (3) cos (A – B) = (½) {cos A cos B + sin A sin B}———— (4) cos A cos B Formula (Product of Cosines) To derive the cos A cos B formula add equations (3) and (4) ⇒ cos (A + B) + cos (A – B) = [cos A cos B – sin A sin B] + [cos A cos B + sin A sin B] ⇒ cos (A + B) + cos (A – B) = cos A cos B + cos A cos B ⇒ cos (A + B) + cos (A – B) = 2 cos A cos B Hence, cos A cos B = (½) [cos (A + B) + cos (A – B)] cos A sin B Formula (Product of Sine and Cosine) To derive the cos A sin B formula subtract equations (2) from (1) ⇒ sin (A + B) – sin (A – B) = [sin A cos B + cos A sin B] – [sin A cos B – cos A sin B] ⇒ sin (A + B) – sin (A – B) = sin A cos B + cos A sin B – sin A cos B + cos A sin B ⇒ sin (A + B) – sin (A – B) = 2 cos A sin B Hence, cos A sin B = (½) [sin (A + B) - sin (A - B)] sin A cos B Formula (Product of Sine and Cosine) To derive the sin A cos B formula add equations (1) and (2) ⇒ sin (A + B) + sin (A – B) = [sin A cos B + cos A sin B] + [sin A cos B – cos A sin B] ⇒ sin (A + B) + sin (A – B) = sin A cos B + sin A cos B ⇒ sin (A + B) + sin (A – B) = 2 sin A cos B Hence, sin A cos B = (½) [sin (A + B) + sin (A - B)] sin A sin B Formula (Product of Sines) To derive the sin A sin B formula subtract equations (4) from (3) ⇒ cos (A – B) – cos (A + B) = [cos A cos B + sin A sin B] – [cos A cos B – sin A sin B] ⇒ cos (A – B) – cos (A + B) = cos A cos B + sin A sin B – cos A cos B + sin A sin B ⇒ cos (A – B) – cos (A + B) = 2 sin A sin B Hence, sin A sin B = (½) [cos (A - B) - cos (A + B)] Conclusion Product to sum formulas are useful tools in trigonometry that make it easier to work with sine and cosine functions. By converting products into sums or differences, these formulas help simplify complex trigonometric problems. Learning and using these formulas can help you solve a wide range of math problems more easily. Read More, Trigonometric in Maths Trigonometric Identities Inverse Trigonometric Identities Sample Problems on Product to Sum Formulas Problem 1: Express 6 cos 8x sin 5x as sum/difference. Solution: From one of the product to sum formulas, we have cos A sin B = (½) [sin (A + B) - sin (A - B)] So, by substituting A = 8x and B = 5x in the above formula, we get cos 8x sin 5x = (½) [ sin (8x + 5x) - sin (8x - 5x) ] cos 8x sin 5x = (½) [sin 13x - sin 3x] Now, 6 cos 8x sin 5x = 6 × (½) [sin 13x - sin 3x] Hence, 6 cos 8x sin 5x = 3 [sin 13x - sin 3x] Problem 2: Determine the value of the integral of cos 4x cos 6x. Solution: From one of the product to sum formulas, we have cos A cos B = (½) [cos (A + B) + cos (A - B)] cos 4x cos 6x = ½ [cos (4x + 6x) + cos (4x - 6x)] = (½) [cos 10x + cos (-2x)] = (½) [cos 10x + cos 2x] {Since, cos (-θ) = cos θ} Now, integral of cos 4x cos 6x = ∫ cos 4x cos 6x dx = ∫(½) [cos 10x + cos 2x] dx = (½) [1/10 sin 10x + 1/2 sin 2x] + C {Since, ∫cos(ax) dx = 1/a sin(ax) + c} = 1/20 sin(10x) + 1/2 sin(2x) + C Hence, integral of cos 4x cos 6x = 1/20 sin(10x) + 1/2 sin(2x) + C Problem 3: Determine the value of sin 36° cos 54° without evaluating the sin 36° and cos 54° values. Solution: From one of the product to sum formulas, we have sin A cos B = (½) [sin (A + B) + sin (A - B)] So, sin 36° cos 54° = (½) [sin (36° + 54°) + sin (36° - 54°)] = (½) [sin (90°) + sin (-18°)] = (½) [sin 90° - sin 18°] {Since, sin (-θ) = – sin θ} = (½) [1 - 0.3090] {Since, sin 90° = 1, sin 18° = 0.3090} = 0.3455 Hence, sin 36° cos 54° = 0.3455. Problem 4: Determine the value of the derivative of 4 cos 3x sin 2x. Solution: From one of the product to sum formulas, we have cos A sin B = (½) [sin (A + B) - sin (A - B)] Now, 4 cos 3x sin 2x = 4 × (½) [sin (3x + 2x) - sin (3x - 2x)] = 2 [sin 5x - sin x] Now, derivative of 4 cos 3x sin 2x = d(4 cos 3x sin 2x)/dx = d/(2 [sin 5x - sin x])/dx = 2 [ d(sin 5x)/dx - d(sin x)/dx ] = 2 [5 cos 5x - cos x] {Since, d(sin ax)/dx = a cos ax} Hence, derivative of 4 cos 3x sin 2x = 2 [5 cos 5x - cos x] . Problem 5: Determine the value of sin 15° sin 45° without evaluating the sin 15° and sin 45° values. Solution: From one of the product to sum formulas, we have sin A sin B = (½) [cos (A - B) - cos (A + B)] Now, sin 15° sin 45° = (½)[cos (15° - 45°) - cos (15° + 45°)] = (½) [cos (-30°) - cos (60° )] = (½) [cos 30° - cos 60°] {Since, cos (-θ) = cos θ} = (½) [√3/2 - 1/2] {Since, cos 30° = √3/2 and cos 60° = 1/2} = (½) [(√3 -1)/2] = (√3 -1)/4 Hence, sin 15° sin 45° = (√3 -1)/4. Problem 6: Express 2 cos 9x cos 7x as sum/difference. Solution: From one of the product to sum formulas, we have cos A cos B = (½) [cos (A + B) + cos (A - B)] Now, 2 cos 9x cos 7x = 2 × (½) [cos (9x + 7x) + cos (9x - 7x)] = [cos (16x) + cos (2x)] Hence, 2 cos 9x cos 7x = [cos 16x + cos 2x] K kiran086472 Improve Article Tags : Mathematics School Learning Trigonometry - MAQ Maths-Formulas Explore Maths 4 min read Basic Arithmetic What are Numbers? 15+ min readArithmetic Operations 9 min readFractions - Definition, Types and Examples 7 min readWhat are Decimals? 10 min readExponents 9 min readPercentage 5 min read Algebra Variable in Maths 5 min readPolynomials\| Degree \| Types \| Properties and Examples 9 min readCoefficient 8 min readAlgebraic Identities 14 min readProperties of Algebraic Operations 3 min read Geometry Lines and Angles 9 min readGeometric Shapes in Maths 2 min readArea and Perimeter of Shapes \| Formula and Examples 10 min readSurface Areas and Volumes 10 min readPoints, Lines and Planes 14 min readCoordinate Axes and Coordinate Planes in 3D space 6 min read Trigonometry & Vector Algebra Trigonometric Ratios 4 min readTrigonometric Equations \| Definition, Examples & How to Solve 9 min readTrigonometric Identities 8 min readTrigonometric Functions 6 min readInverse Trigonometric Functions \| Definition, Formula, Types and Examples 11 min readInverse Trigonometric Identities 9 min read Calculus Introduction to Differential Calculus 6 min readLimits in Calculus 12 min readContinuity of Functions 10 min readDifferentiation 2 min readDifferentiability of Functions 9 min readIntegration 3 min read Probability and Statistics Basic Concepts of Probability 7 min readBayes' Theorem 13 min readProbability Distribution - Function, Formula, Table 13 min readDescriptive Statistic 5 min readWhat is Inferential Statistics? 7 min readMeasures of Central Tendency in Statistics 11 min readSet Theory 3 min read Practice NCERT Solutions for Class 8 to 12 7 min readRD Sharma Class 8 Solutions for Maths: Chapter Wise PDF 5 min readRD Sharma Class 9 Solutions 10 min readRD Sharma Class 10 Solutions 9 min readRD Sharma Class 11 Solutions for Maths 13 min readRD Sharma Class 12 Solutions for Maths 13 min read Improvement Suggest Changes Help us improve. 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7684	https://libguides.nwpolytech.ca/Physics/Fundamentals/UnitConversions	Skip to Main Content Physics: Unit Conversions Home Fundamental Concepts Toggle Dropdown Unit Conversions Significant Figures Vectors Motion Toggle Dropdown Linear Motion Motion Vectors Projectile Motion Relative Motion Force Toggle Dropdown Force and Interactions Newton's First Law Newton's Second Law Newton's Third Law Free Body Diagrams Examples Friction Energy Toggle Dropdown Work Kinetic Energy Power Potential Energy Math Help Appointments Unit Conversions SI and Metric Measurement Systems Metric and the International System of Units (Système International or SI) are measurement systems consisting of base units and prefixes. The base units are defined for measurable properties and the prefixes describe varied sizes of units within each base unit. Because of the similarities, the two systems are often used together and regarded as being the same system. However, there are some subtle differences. The definitions of SI base units are based on unchanging fundamental properties of nature while metric base units are less stringently defined. The metric system also includes derived units where SI does not. \| Fun Fact \| \| The metric system was first adopted in France in 1795. SI was derived from the metric system in 1960 to express properties more complicated than length, weight, and volume. \| Quirks of SI Kilogram is the only base unit that has a prefix. There are no base units for area or volume. Area and volume are calculated units determined by the dimensions of a space or object. Base Units SI and metric base units are often used together and there is some overlap in the two systems. Table 1: SI Base Units \| Physical Property \| \| \| \| Amount of a substance \| \| \| \| Electric current \| \| \| \| \| \| \| \| Luminous intensity \| \| \| \| \| \| \| \| \| \| K \| \| Time \| second \| s \| Table 2: Metric Base Units \| Physical Property \| Unit \| Symbol \| \| Length \| metre \| m \| \| Area \| square metre \| m2 \| \| Volume \| cubic metre \| m3 \| \| Weight \| gram \| g \| \| Capacity \| litre \| L \| \| Temperature \| degree Celsius \| ℃ \| Prefixes Prefixes are useful when talking about very large and very small numbers. It would be difficult to measure the distance from Vancouver, Canada to Paris, France using centimetres. Kilometres are more useful in that case. But kilometers or metres would be a poor choice for measuring the dimensions of a book. Table 3: SI and Metric Prefixes \| Name \| Symbol \| Base 10 multiplier \| Decimal \| \| \| peta \| P \| 1015 \| 1000000000000000 \| ⬆ bigger — smaller ⬇ \| \| tera \| T \| 1012 \| 1000000000000 \| \| giga \| G \| 109 \| 1000000000 \| \| mega \| M \| 106 \| 1000000 \| \| kilo \| k \| 103 \| 1000 \| \| hecto \| h \| 102 \| 100 \| \| deca \| da \| 101 \| 10 \| \| — \| — \| 100 \| 1 \| \| deci \| d \| 10−1 \| 0.1 \| \| centi \| c \| 10−2 \| 0.01 \| \| milli \| m \| 10−3 \| 0.001 \| \| micro \| μ \| 10−6 \| 0.000001 \| \| nano \| n \| 10−9 \| 0.000000001 \| \| pico \| p \| 10−12 \| 0.000000000001 \| \| femto \| f \| 10−15 \| 0.000000000000001 \| Converting Units Using a Conversion Diagram Every time you move to the left, move the decimal point to the right the same amount of times, or divide the value by the negative powers of ten. Power in this case is equal to the number of times you moved. Every time you move to the right, move the decimal point to the left however many times you moved, or divide the value by the positive powers of ten. Power in this case is equal to the number of times you moved. Caution! A conversion diagram works well for the prefixes shown in the image above because the base 10 multipliers are consecutive (10-3, 10-2, 10-1, 100 (base unit), 101, 102, 103). When micro (10-6) and mega (106) are added it looks like the decimal should move four spaces, but it actually needs to move six spaces. See the Decimal Conversion section below for a broader application of the principles behind the Conversion Diagram method. Example I Convert 12.52 L to mL To go from L to mL, you have to move three times to theleft. Therefore, move the decimal three times to the right. Method 1 Moving the decimal three places to the right gives: Method 2 Since you moved three spaces to the left to get to ml on the diagram divide the value by ten to the power of minus three (10-3). Have questions about how the ten to the power of three became positive? Check out this guide for more details. Decimal Conversion This method of unit conversion involves moving the decimal point of a measurement, much like using a conversion diagram. Both methods work because the prefixes are related to each other by factors of ten. The decimal conversion method outlined here addresses the limitation of conversion diagrams noted in the Caution! in the section above. Required Knowledge There are two pieces of knowledge required to understand decimal conversion. You will need to know the base 10 multipliers for the prefixes. You will also need to know how to divide exponents. Conversion process \| \| \| \| --- \| Step \| Action \| Example \| \| 1 \| Determine the direction that the decimal needs to move. The relative unit size determines the direction. Is the desired unit smaller or larger than the starting unit? unit getting bigger – move decimal to the left unit getting smaller – move decimal to the right \| Convert 0.274 hectometres (hm) to centimetres (cm). starting prefix: hecto = 102 (larger) desired prefix: centi = 10-2 (smaller) The desired unit size is smaller so the decimal must move to the right. \| \| 2 \| Determine the number of places the decimal must move. The base 10 multipliers determine the decimal places. Divide the base 10 multiplier of the larger unit by the multiplier of the smaller unit. The resulting exponent is the number of places the decimal must move. \| hecto = 102 (larger) centi = 10-2 (smaller) The decimal must move 4 places. \| \| 3 \| Move the decimal and add zeros as needed. \| 0.274 hm = 2740 cm \| \| 4 \| Check: When the unit gets smaller, the value gets larger. When the unit gets larger, the value gets smaller. \| The example went from a larger unit (hm) to a smaller one (cm), so the value for the desired unit should be larger. And it is! 2740 > 0.274 \| Special Cases Area (m2) and Volume (m3) The process to convert the squared or cubic units of area and volume has an additional part to step 2 shown above. Because these units have more than one length parameter (e.g. squared has two lengths), we must account for that in unit conversions. Example: Convert 1000000 cm3 to m3. We can think of this in terms of the three length parameters that make up a cubit centimetre (cm3). 100 cm x 100 cm x 100 cm = 1000000 cm3 If we convert each centimetre length to metres, then multiply, we get cubic meters (m3). 100 cm = 1 m 1 m x 1 m x 1 m = 1 m3 So, 1000000 cm3 = 1 m3. To do this more simply, multiply the value obtained in step 2 above by either 3 (for squared units) or 3 (for cubic units). For the example, metres are larger than centimetres so the decimal must move to the left. With simple units, we would move the decimal 2 places, but these are cubic units so we need to multiply by 3 to determine the number of places to move the decimal. 2 x 3 = 6, so the decimal must move 6 places. So, 1000000 cm3 = 1 m3 Example: Convert 2.382 m3 to cm3. Centimetres are smaller than metres so the decimal must move to the right. These are cubic units, so we need to multiply by 3 to determine the number of places to move the decimal. The decimal must move 6 places. 2.382 m3 = 2382000 cm3 To maintain the four significant digits, the answer can be written as 2.382 x 106 cm3. Quick and Sneaky Tip When converting from any prefixed unit to the base unit, you can simply replace the prefix with its base 10 multiplier and simplify as needed. Example: Convert 4987 cm to m. The base 10 multiplier for centi is 10-2 so 4987 cm = 4987 x 10-2 m Converting Units Using Conversion Factor \| \| \| \| --- \| Units of length in the metric system \| Units of weight in the metric system \| Units of volume in the metric system \| \| 1,000 millimeters (mm) = 1 meter 100 centimeters (cm) = 1 meter 10 decimeters (dm) = 1 meter 1 dekameter (dam) = 10 meters 1 hectometer (hm) = 100 meters 1 kilometer (km) = 1000 meters \| 1 gram = 1,000 milligrams (mg) 1 gram = 100 centigrams (cg) 1 kilogram (kg) = 1,000 grams 1 metric ton (t) = 1,000 kilograms \| 1 cc = 1 cm3 1 milliliter (mL) = 1 cm3 1 liter (L) = 1,000 millimeters (mL) 1 hectoliter (hL) = 100 liters (L) 1 kiloliter (kL) = 1,000 liters (L) \| Example 1 Convert 42.50 m to km Step 1 : Write the conversion factor from the table above. 1 kilometer (km) = 1000 meters (m) Step 2 : Write the conversion factor as a ratio. Since we are converting meters to kilometers, write the km on the top and m on the bottom, as shown below: Step 3: Multiply 42.50m by the above ratio. Then the answer becomes 0.04250 km with four significant digits. Notice how the "m" on the top and the "m" at the bottom cancel out to give "km" as the final unit, as seen below: A unit conversion calculator can be used to verify your answers. Study tips for unit conversion Pick the method that you are most comfortable with and always use it. If you choose to use the conversion factor method, memorize the most important ones used in your courses. If you choose to use a conversion diagram, make sure to know the prefixes and the factors and where they are placed in your diagram. Memorization technique: write down the information or make a chart on a big piece of paper and post it somewhere in your room where you can see it often. Every time you get a chance, even if it's a few minutes, read through and review as many as you can. Want to check your understanding? Try our online Quiz! Attribution Attribution Some of the content of this guide was modeled after a guide originally createdby Openstaxand has been adapted for the GPRC Learning Commons in September 2020. This work is licensed under a Creative Commons BY NC SA 4.0 International License. << Previous: Fundamental Concepts Next: Significant Figures >> Last Updated: Sep 11, 2025 10:12 AM URL: Print Page Login to LibApps Subjects: Science & Technology Tags: Physics, Science Report a problem. 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7685	https://www.youtube.com/watch?v=_8WtlKY02Ro	Counting: Seating People Problems MATHguide 11600 subscribers 4 likes Description 612 views Posted: 10 Feb 2017 This MATHguide video demonstrates how to count problems when people or objects are arranged in a row or a circle using factorials. See our text based lesson at 1 comments Transcript: this is mathguide dcom and my name is Mark kedos in this video we're going to talk about counting problems we're specifically going to address seating people type of problems in our first section we're going to talk about what a factorial is in our second section we're going to talk about how to arrange people in a row in our third section we're going to talk about how to arrange people in a circle all right let's get started let's talk about what a factorial is a factorial is any number or quantity that has an exclamation point after it so in the realm of mathematics if you put some expression or in this case a number with a exclamation point after it this exclamation point means factorial all right well how do you calculate it well it just means 5 4 3 2 1 so you take the first number in all the subsequent numbers that go below it all theend integers that are below that number you start with until you reach one and you find the product of all those numbers so if we did that let's see that would be 20 20 6 it's 120 okay uh likewise I could take four factorial we're just going to stick with some easy numbers so this way I don't have to grab a calculator and I just want to demonstrate what factorial means and how to easily calculate it of course you could plug these into a calculator especially when the numbers get big and the calculator will make these calculations for you so here I'm going get to 4 3 is 12 12 2 is 24 so uh when we do these problems where we seat people uh we'll actually use these factorial to speed up the calculation process let's address the case where we put people in a row here's our first problem all right so here we have four people Anna Blain Clara and Dorian and they're going to be uh sat in a row how many ways are there to arrange them in a row so there's a lot of ways to do this as a matter of fact you could start making a list um I'm just going to use the letters a b c and d and show you how this would work a b c and d all right so I could range him in this order an ablain claran or what I could do is I could switch up Dorian Clara okay well I could instead of putting B second I could put CC uh okay when C is second I could put D and B in that order uh okay well wait I could also put instead of B or C I could put D second and then I would have B and C or maybe I could put a d CB okay now these are all the combinations where a is first so you can see there 1 2 3 4 five six there's six different ways to make arrangements here when uh Anna is first when she's sat first so there's six ways now I don't need to do this again but I could put B first and I can make another Arrangement okay so similarly of six different Arrangements when B is first so think of it this way when a is Anna when Anna sat first there's six ways to do it if Blaine is going to be sat first there's six ways to do it and so on Clara six ways to do it if Dorian is sat first there's six ways to do it and of course I would find the sum and I'm going to get 24 different ways now there's a faster way to do this we would say okay there's four seats and we go hm how many people could I possibly put here well I only put one person but four people are available to sit in that first spot okay there's four possible people I could put there now once that person is sat then how many people could I choose from to sit in this spot now there's three people available that I could choose from and now two people have been set now there's only two people I could choose to possibly put here and then finally there'll be one person left here so the fundamental counting principle says since all these are independent events I'm going to multiply them together and this is going to be also 12 2 or 2 four so in the future I'm not going to do all this work I'm not going to make all these lists and add I'm not even going to multiply like this I'm simply going to take four factorial and I'm going to toss it in the calculator and it's going to give me the answer 24 because I don't want to have to do all this stuff again okay let's try our new technique and let's see if we can make it quick all right so uh just got rid of some stuff let's put on a new problem all right here's our new problem let's say we have seven people and we're going to be seating them in a row how many ways can we do it well let's see seven people seven factorial so I'm going to plug this into a calculator so there's 5,000 40 ways to do it okay so what does this mean in general you know it's nice to have some way to look at this and look at this in general that means if we know there's n people who need to be sat in a row we know the solution the solution is going to be n factorial for an answer okay let's go on to the next section for this section we're going to be seating people in a circle let's check out our first problem all right so here we have the same people that we started off with problem one uh uh from the other section we have Anna Blain CLA and Dorian but this time now we're going to try to seat them at a circular table so just to see how this could be done uh what you could do is put a b c and d now now picture this is a table I'm not going to draw a table but let's say they're around a table and um you know I could draw a little table here just to give you an idea but here you have a b c and d and they're all sitting around a table okay so there's one possible configuration okay you could also do ab DC okay then you could do also a h let's see B was right to A's immediate side so how about I go with C so I get a c b and d uh okay well wait a minute I could also have C after a and then maybe I can go D and B in that order okay so I did all the combinations with B after a and C after a so now I could have D after a so I could also put a d b c okay so you could see that there are if I could separate these I'm going to use some lines here but you could see that there are six different combinations and it turns out that's all there are and then the immediate reaction people have is well wait a minute you you have a first you know why don't I put like uh I don't know B and then maybe I put c a d maybe you know I could put things in there and say hey you didn't count this particular Arrangement and I would say yes I did now if you start with just by looking at a it's a d BC right if I just go clockwise a d BC I'm going to find it a dip right here adbc now you're thinking wait a minute this isn't the same Arrangement as this well yes it is now if you look at a that would be Anna to her left she sees Doran Anna sees Clara to her right okay same thing over here Anna to her right sees claraa Anna to her left sees Dorian and the same thing's true for everything everything else is working out perfectly so even though these Arrangements of people have been slightly rotated remember it's a circular table and I'm looking at each other as they're placed around this table so therefore this Arrangement is the same as this it's a duplicate and I'm not going to count them twice okay so it turns out that this is a a little bit trickier of a situation to count so I'm going to explain how to count that so when we start to consider counting this you have to imagine because of the rotation rotational nature of a table that it doesn't matter where you seat the first person because it's rotationally going to be the same in some other rotated configuration so it it doesn't matter where you place the first person so that means if if I start with four people I really only need to do the calculation as if there are three people because again it doesn't matter where I put the first person down in the table it's how I arrange people around that first person that matters okay so if I start with four people like I I am in this situation we're starting with four people I still do the same calculation I did in the last section with a row but with one fewer person so I'm going to take three people and take the factorial so that means I do 3 2 1 which is six there's six ways to arrange them and there you go there you see them so the mathematics matches now the counting method that I did by L hand it's a shocker I know okay let's try another problem okay so let's say we had six people and we're going to be seating them in a circle how many ways can it be done okay so what do you do You' say it doesn't matter where you sit the first person so you sit the first person down now you have five people left so I want to figure out how many ways to there are to arrange those remaining five people okay so I'd say it's five factorial I throw this in a calculator and it's 120 different ways so I don't want to do these by hand it starts to take way too much time in a lot of organization to make sure I get all 120 it's just easier to calculate okay so in general how does this work what if I were to have to seat n number of people if we were going to seat n people in a circle and I want to know how many ways could I arrange them around that table well well it turns out there's n minus1 factorial ways of doing that and that's how it looks in general this has been mathguide decom talking about seating people make sure you go back to math guide and check out our interactive quizzes our instructional videos and text based lessons for
7686	https://www.reddit.com/r/math/comments/1e10dcs/handwavy_proof_related_to_the_repeated_birthday/	Hand-wavy "proof" related to the repeated birthday problem (23 people in a room, 50% chance two people share a birthday): Expected Value of duplicate values while generating random numbers : r/math Skip to main contentHand-wavy "proof" related to the repeated birthday problem (23 people in a room, 50% chance two people share a birthday): Expected Value of duplicate values while generating random numbers : r/math Open menu Open navigationGo to Reddit Home r/math A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to math r/math r/math This subreddit is for discussion of mathematics. All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics. 3.9M Members Online •1 yr. ago dontwantleague2C Hand-wavy "proof" related to the repeated birthday problem (23 people in a room, 50% chance two people share a birthday): Expected Value of duplicate values while generating random numbers TL;DR: When generating Y values from 1-X, if X is sufficiently large, the expected value of duplicate numbers converges to Y^2/2X. I will prove this with a slightly hand-wavy argument below. You've all heard the idea that if 23 people are in a room, there's a 50% chance of two people sharing a birthday. This got me thinking: if you generate random numbers in a range from 1-X, what is the expected value of duplicates after a certain number of iterations? What I mean by expected value is the mean average, not the probability of at least one duplicate. Let's start with a range from 1-100. The odds of the first number being a duplicate is 0%. The second number: 1%. The third number is 2%. (This is a simplification; since 1% of the time you already have your first duplicate, the third number only gives you a 1.99% chance. However, I'll show later that for sufficiently high numbers, this doesn't matter). Keep adding up, and we have the triangular number sequence. When we add 10%, on the 11th number generation, we go from 45% total to 55% total, which is where our EV is now at 0.5. When we add 14% (the 15th number generation), we get to a value of 104. This indicates that between 14 and 15, we get to an EV of 1. Let's try with X = 10,000 instead. Let's say we want to determine the EV of duplicates in 100 iterations. We start at 1/10,000, then 2/10,000… to 99/10,000. This sum can be simplified by combining opposite terms: 1+99=100, 2+98=100, etc. resulting in 10049+50, or 10049.5. This is approximately equal to 100^2/2. So the EV is (100^2/2)/10000, or 0.5. This shows firstly that at √X, the EV will be approximately 0.5. In this case, it is 0.495. When X is 100, it was only 0.45. At 1,000,000, this EV will be 0.4995. Second, it shows why my assertion makes a lot of sense: when generating Y numbers in the range of 1-X, the EV of duplicates ≈Y^2/2X. Additionally, when Y = √(2X), EV = 1. It, it is important to acknowledge that this isn't completely exact yet. After all, if a duplicate appears, it makes one slightly less likely than we're estimating. However, I can prove that this also converges. Say we assume that when X = 10,000 and Y = 100, the EV of duplicates is approximately 0.5, as we showed before. That means that on attempt 101, instead of there being on average 100 unique numbers chosen, there will only be 99.5, a loss of 0.5 numbers. This means that our EV decreases by 0.5/10,000. Let's say we averaged this out over a range from 90-109. Over this range, we have 20 numbers, and the EV is about 0.5/10000 lower than expected, so in total, we lose about 10/10,000 EV of duplicate numbers, or 1/1,000. This is pretty small but it can get larger as Y increases. What if we go up to X = 1,000,000 and Y = 1,000? At this point, the EV is still about 0.5 duplicate numbers. Let's do the same process and average out over a geometrically equivalent range from 900-1099. We have 200 numbers losing 0.5/1,000,000 EV each, resulting in a total EV loss of 100/1,000,000, or 1/10,000. In other words, the EV loss is proportional to Y/X, whereas the total EV is proportional to Y^2/X. This means that as X increases, if we hold Y^2 to be proportional to X, the impact of having already found duplicates becomes smaller and smaller, while the total EV of duplicate numbers remains the same. Now, my question for you: is this an original idea, or did I explain something that's already been fully figured out? Read more Archived post. 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7687	https://www.youtube.com/watch?v=gMcs6RF_LrQ	L{sin(at)}) - transform of sin(at) \| Laplace transform \| Differential Equations \| Khan Academy Khan Academy 9090000 subscribers 2628 likes Description 922911 views Posted: 4 Sep 2008 Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: Laplace Transform of sin(at) (part 1) Watch the next lesson: Missed the previous lesson? Differential Equations on Khan Academy: Differential equations, separable equations, exact equations, integrating factors, homogeneous equations. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. For free. For everyone. Forever. #YouCanLearnAnything Subscribe to KhanAcademy’s Differential Equations channel:: Subscribe to KhanAcademy: 151 comments Transcript: Let's keep building our table of Laplace transforms. And now we'll do a fairly hairy problem, so I'm going to have to focus so that I don't make a careless mistake. But let's say we want to take the Laplace transform-- and this is a useful one. Actually, all of them we've done so far are useful. I'll tell you when we start doing not-so-useful ones. Let's say we want to take the Laplace transform of the sine of some constant times t. Well, our definition of the Laplace transform, that says that it's the improper integral. And remember, the Laplace transform is just a definition. It's just a tool that has turned out to be extremely useful. And we'll do more on that intuition later on. But anyway, it's the integral from 0 to infinity of e to the minus st, times-- whatever we're taking the Laplace transform of-- times sine of at, dt. And now, we have to go back and find our integration by parts neuron. And mine always disappears, so we have to reprove integration by parts. I don't recommend you do this all the time. If you have to do this on an exam, you might want to memorize it before the exam. But always remember, integration by parts is just the product rule in reverse. So I'll just do that in this corner. So the product rule tells us if we have two functions, u times v. And if I were take the derivative of u times v. Let's say that they're functions of t. These are both functions of t. I could have written u of x times v of x. Then that equals the derivative of the first times the second function, plus the first function times the derivative of the second. Now, if I were to integrate both sides, I get uv-- this should be review-- is equal to the integral of u prime v, with respect to dt-- but I'm just doing a little bit of shorthand now-- plus the integral of uv prime. I'm just trying to help myself remember this thing. And let's take this and subtract it from both sides. So we have this integral of u prime v is going to be equal to this, uv minus the integral of uv prime. And, of course, this is a function of t. There's a dt here and all of that. But I just have to do this in the corner of my page a lot, because I always forget this, and with the primes and the integrals and all that, I always forget it. One way, if you did want to memorize it, you said, OK, the integration by parts says if I take the integral of the derivative of one thing and then just a regular function of another, it equals the two functions times each other, minus the integral of the reverse. Right? Here, when you take the subtraction, you're taking the one that had a derivative, now it doesn't. And the one that didn't have a derivative, now it does. But anyway, let's apply that to our problem at hand, to this one. Well, we could go either way about it. Let's make u prime is equal to-- we'll do our definition-- u prime is equal to e to the minus st, in which case you would be the antiderivative of that, which is equal to minus 1 over s e to the minus st, right? And actually, this is going to be an integration by parts twice problem, so I'm just actually going to define the Laplace transform as y. That'll come in useful later on. And I think I actually did a very similar example to this when we did integration by parts. But anyway, back to the integration by parts. So that's u. And let me do v in a different color. So when v-- if this is u prime, right? This is u prime, then this is v. So v is equal to sine of at. And then what is v prime? Well, that's just a cosine of at, right? The chain rule. And now, we're ready to do our integration. So the Laplace transform, and I'll just say that's y, y is equal to-- y is what we're trying to solve for, the Laplace transform of sine of at-- that is equal to u prime v. I defined u prime in v, right? That's equal to that. The integral of u prime times v. That equals uv. So that's minus 1 over s e to the minus st, times v, sine of at, minus the integral. And when you do the integration by parts, this could be an indefinite integral, an improper integral, a definite integral, whatever. But the boundary stays. And we can still say, from 0 to infinity of uv prime. So u is minus 1 over s e to the minus st, times v prime, times a cosine of at-- fair enough-- dt. Well, now we have another hairy integral we need to solve. So this might involve another integration by parts, and it does. Let's see if we can simplify it at [? all. ?] Let's take the constants out first. Let me just rewrite this. So we get y is equal to minus e to the minus st over s, sine of at. So you have a minus minus plus a over s-- a divided by s, and then these two negative signs cancel out-- times the integral from 0 to infinity, e to the minus st, cosine of at, dt. Let's do another integration by parts. And I'll do this in a purple color, just so you know this is our second integration by parts. Over here. Let's define once again, u prime is equal to e the minus st. So this is u prime. Then u is equal to minus 1 over s e to the minus st. We'll make v equal to cosine of at. The hardest part about this is not making careless mistakes. And then v prime-- I just want it to be on the same row-- is equal to minus a sine of at, right? The chain rule, derivative of cosine is minus sine. So let's substitute that back in, and we get-- this is going to get hairy; actually, it already is hairy-- y is equal to minus e to the minus st over s, sine of at, plus a over s, times-- OK. Integration by parts. uv. So that's minus 1 over s e to the minus st, times v, times cosine at, minus the integral from 0 to infinity. This problem is making me hungry. It's taking so much glucose from my bloodstream. I'm focusing so much not to make careless mistakes. Anyway, integral from 0 to infinity. And now, we have uv prime, so u is minus 1 over s e to the minus st. That's u. And then v prime times minus a. So let's make that minus cancel out with this one. So that becomes a plus. a sine of at, dt. I'm starting to see the light at the end of the tunnel. So then, let's simplify this thing. And, of course, we're going to have to evaluate this whole thing, right? Actually, we're going to have to evaluate everything. Let's just focus on the indefinite integral for now. We're going to have to take this whole thing and evaluate-- let's just say that y is the antiderivative and then evaluate it from infinity to 0. From 0 to infinity. So y is equal to minus e to the minus st over s, sine of at. Now let's distribute this. Minus a over s squared, e to the minus st, cosine of at. Right? OK, now I want to make sure I don't make a careless mistake. OK. Now, let's multiply this times this and take all of the constants out. So we have an a and an s. a over s. There's a minus sign. We have a plus a to the s. So we'll have a minus a squared over s squared, times the integral from 0-- well, I said I'm just worrying about the indefinite integral right now, and we'll evaluate the boundaries later. e to the minus st, sine of at, dt. Now, this is the part, and we've done this before, it's a little bit of a trick with integration by parts. But this expression, notice, is the same thing as our original y. Right? This is our original y. And we're assuming we're doing the indefinite integral, and we'll evaluate the boundaries later. Although we could have kept the boundaries the whole time, but it would have made it even hairier. So we can rewrite this integral as y. That was our definition. And actually, I just realized I'm running out of time, so I'll continue this hairy problem in the next video. See you soon.
7688	https://www.youtube.com/watch?v=WQfqHDsHAZM	Speed of Light in a moving Medium - It's Not What you Think For the Love of Physics 169000 subscribers 419 likes Description 11599 views Posted: 27 May 2021 What is the speed of light in a medium that is in motion? We all know the speed of light in any medium is given by c/n where n is the refractive index of the medium. But what if the medium is moving? What then, will be the speed of a light photon with respect to an observer at rest? We can calculate this by using the Relativistic velocity transformations. Further we can simplify the expression for media travelling at speeds very less compared to the speed of light. It appears that moving medium drags the light photon as it is moving through it. 00:00 Speed of light in moving medium 05:33 If speed of medium is very small SpecialTheoryofRelativity #STR 𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬𓏬 Your financial support provides me an additional incentive to create high quality lecture videos. 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Interval ► Minkowski Spacetime ► Eucledian Space & Minkowski Spacetime ► Spacetime Diagrams ► Four Vectors in Relativity ► Doppler Effect using 4-vectors ► Compton Effect using 4-vectors ► Particle Decay using 4-vectors ► (SHORTS) Does Light experience time ► (SHORTS) Light for moving observer ► (SHORTS) Nothing can travel faster than light ► (SHORTS) What is farther away ► 42 comments Transcript: Speed of light in moving medium hi everyone welcome back to my channel i am divijotidas and in today's video i want to find out the speed of light in a medium that is in motion i want to find out the speed of light in a moving medium so for that i have an example here a tunnel through which water is moving at very very high speeds and there's a light photon which is traveling inside this water medium now you may also deal with other examples like a glass slab which is in motion and there is a light photon in motion within that glass slab so the both the problems are kind of similar so let's suppose that this is some kind of a water medium that is in motion it is traveling at very very high speeds let's suppose the velocity of v and this is the speed of this water medium with respect to some kind of a lab reference frame which i'm going to call as at rest so with respect to some kind of an observer in a lab reference frame this water medium is traveling at a velocity of v so therefore i am going to associate a frame of reference which i am going to call as s and with respect to the lab frame of reference the water body is traveling at a velocity v so therefore i am going to call this water body as the frame of reference of s dash so an observer with respect to this moving medium i am going to associate as s dash which is the reference frame associated with the moving medium now i'm interested in figuring out what is the speed of the light photon with respect to the lab frame because i already know what is the speed of the light photon with respect to the medium any light photon traveling within a medium whether that is a medium of water or medium of glass or any other substance can be written in terms of its refractive index n so if n is the refractive index it simply means the speed of light in vacuum divided by the speed of light in that particular medium so the speed of light in this particular medium can be written as the speed of light in vacuum divided by the refractive index right so the speed of a light in a medium is given by c upon n where c is the speed of light in vacuum which is three to ten per eight meters per second and n is the refractive index since this is the speed of the light photon in the medium what should be the speed of the same light photon with respect to an observer at rest if the medium is in motion with respect to that observer for this i'm going to use the relativistic velocity transformations which i have derived in one of my previous videos the relativistic velocity transformation equations or the relativistic velocity addition equations basically give us a relationship between the measurements of velocities of different objects with respect to two different observers in relative motion so for our case let me define the parameters the velocity of the second observer with respect to the first observer is defined as v and the velocity of the photon with respect to the second observer that is with respect to the s dash frame of reference is defined as u x dash which comes out to be nothing but c upon n and the velocity of the same light photon with respect to the lab frame of reference is defined as ux which we do not know yet we have to calculate it but we know the relationship that exists between u x and u x dash for that i'm going to use the inverse relativistic velocity transformations which says that u x is equal to u x dash plus v upon 1 plus u x dash v upon c squared u x which is the speed of the light photon with respect to the lap frame and u x dash which is the speed of the light photon with respect to the moving frame they are connected by this equation u x is equal to u x dash plus v divided by one plus c h does v upon c square where v is the speed of the medium v is the speed of the s dash frame with respect to the s if you're not familiar with this equation you may go back to one of my previous videos where i have derived the relativistic velocity transformation equations now if i'm going to plug our values into this particular equation i will obtain my answer right so let me plug these values what is u x dash u x dash is nothing but c upon n so i plug in c upon n plus v divided by 1 plus u x dash is c upon n multiplied by v upon c square so c and the square here gets cancelled and let me bring this equation to the other side of the board so let me write it down here so u x which is the speed of the light photon with respect to the lab frame comes out to be c upon v n divided by n in the numerator and in the denominator you have c n plus v divided by c n so here i can cancel out n with this n and i end up getting c and in brackets you have c plus v n divided by c n plus v so this is the speed of the light photon with respect to a lab frame with respect to which the medium is moving and the light photon is traveling at a velocity of c by n in that particular medium i can simplify this expression by taking the n outside so this becomes c upon n and within the bracket you have c plus v n divided by c plus v upon n so this is the formula for the velocity of the light photon in a moving medium If speed of medium is very small now in certain kinds of questions they mentioned that the velocity with which the medium is moving with respect to the lab is actually very very small compared to the speed of light itself so under those situations where v is very very small compared to the speed of light we can still simplify this particular expression as so for v very very less than c i can simplify this expression to simplify it i will still use this expression all right so let me use this expression where c upon n plus v i take inside a bracket and the denominator is 1 plus v upon c n 1 plus v upon c n to the power -1 so if v is very very less compared to c so v by c is actually a very small number so this term actually is a very very tiny number so in that kind of a situation i can expand this term you see i am sure that you must have heard of this kind of an expression where 1 plus x to the power -1 where x is a very small number can simply be written as 1 minus x plus x square minus x cube plus x to the power 4 and on and on and on i can use this kind of an expansion here in this particular expression if i do that this simply becomes c upon n plus v in the first bracket and in the second bracket if i apply this particular expansion it simply becomes 1 minus v upon c n plus v square upon c square n square minus v cube upon c cube n cube and on and on now here these terms v square upon c square is very very tiny because of this given condition so these terms can be neglected they are extremely small so all these higher order terms we will neglect it so in that kind of a situation the only terms that are left are c upon n plus v minus v upon n square minus v square upon c n again this v square upon c term we can neglect given our condition so finally we end up getting the speed of the light photon comes out to be ux is equal to c upon n plus v 1 minus 1 upon n square so this is the formula for the speed of a light photon in a moving medium if the medium is traveling at velocity is much much less compared to the speed of light so this is c upon n plus v into 1 minus 1 upon n square so c upon n is the speed of the light photon in that medium itself but this additional term arises because we are observing that photon with respect to a lab frame of reference so with respect to a lab frame of reference the velocity of the photon is much greater compared to c upon n and it is greater by a factor of v multiplied by 1 minus 1 upon n square so in a way it looks as if the medium is dragging the light photon with an increase in velocity given by this particular factor so as you can see from this video even though the speed of light is a constant in vacuum for all observers in a moving medium however the speed of light is different for different observers and if the medium is at rest the speed of light can be something and if the medium is in motion the speed of light for a rest observer may be something else so that is all for today's video i hope you learned something thank you very much [Music] you
7689	https://en.wiktionary.org/wiki/monotone_function	Jump to content Search Contents Beginning 1 English 1.1 Alternative forms 1.2 Noun 1.2.1 Usage notes 1.2.2 Synonyms 1.2.3 Hypernyms 1.2.4 Hyponyms 1.2.5 Related terms 1.2.6 Translations 1.3 See also 1.4 Further reading monotone function Malagasy தமிழ் Entry Discussion 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 Create a book Download as PDF Printable version In other projects Appearance From Wiktionary, the free dictionary English [edit] Alternative forms [edit] monotonic function Noun [edit] monotone function (plural monotone functions) (calculus) A function f : X→R (where X is a subset of R, possibly a discrete set) that either never decreases or never increases as its independent variable increases; that is, either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x). : Where defined, the first derivative of a monotone function never changes sign, although it may be zero. 2005, Anthony W. Knapp, Basic Real Analysis, Springer, page 357: : Section 1 contains Lebesgue's main tool, a theorem saying that monotone functions on the line are differentiable almost everywhere. 2011, Saminathan Ponnusamy, Foundations of Mathematical Analysis, Springer, page 469: : Monotone functions on have nice properties. For example, they are integrable on and have only a countable number of jump discontinuities. In this section, we shall also show that every monotone function is a function of bounded variation, and hence the class contains the class of monotone functions on . 2013, Donald Yau, A First Course in Analysis, World Scientific, page 104: : We saw in the previous section that monotone functions have some nice properties. For example, a monotone function is continuous except possibly on a countable set. They are also closed under scalar multiplication, and the sum of two increasing functions is increasing. (Exercise (4) on page 103). However, the difference and product of two monotone functions are not necessarily monotone (Exercise (5) on page 103). 2. (order theory, mathematical analysis) A function f : X→Y (where X and Y are posets with partial order "≤") with either: (1) the property that x ≤ y implies f(x) ≤ f(y), or (2) the property that x ≤ y implies f(y) ≤ f(x). 3. (Boolean algebra) A Boolean function with the property that switching any one input variable from 0 to 1 results either in no change in output or a change from 0 to 1. Usage notes [edit] The order theory definition avoids reference to the concepts increasing and decreasing, making it somewhat more generally applicable. Strictly speaking, the partial orders for X and Y need not be related (the notation "≤" is conventional). This case encompasses the possibility that X and Y are multidimensional spaces (e.g. Rn) and f is a mapping between them. In the Boolean algebra case, there is implicit in the definition an intuitively natural partial order "≤" (see product order on Wikipedia) such that, given two input tuples a = (a1, a2,... an) and b = (b1, b2,... bn), a ≤ b means that b can be obtained from a via a series of (zero or more) steps each switching an input from 0 to 1. With this partial order in mind, (only) property (1) of the order theory definition applies. Synonyms [edit] (function that either never decreases or never increases): monotonic function Hypernyms [edit] function Hyponyms [edit] (calculus): (function that never decreases as its independent variable increases): nondecreasing function, monotone nondecreasing function, monotonic nondecreasing function, monotonically nondecreasing function increasing function, monotone increasing function, monotonic increasing function, monotonically increasing function (function that never increases as its independent variable increases): nonincreasing function, monotone nonincreasing function, monotonic nonincreasing function, monotonically nonincreasing function decreasing function, monotone decreasing function, monotonic decreasing function, monotonically decreasing function (order theory): isotone function, order-preserving function anti-monotone function, antitone function, order-reversing function Related terms [edit] monotone monotone operator monotone transformation monotonic Translations [edit] calculus: real function that either never decreases or never increases \| \| \| Bulgarian: монотонна функция f (monotonna funkcija) German: monotone Funktion f Serbo-Croatian: monotona funkcija f Spanish: función monótona f Turkish: monoton fonksiyon \| order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x) \| \| \| Bulgarian: please add this translation if you can German: monotone Funktion f Serbo-Croatian: monotona funkcija f Spanish: función monótona f Turkish: monoton fonksiyon \| Boolean function such that switching an input from 0 to 1 either does not change the output or changes it from 0 to 1 \| \| \| Bulgarian: монотонна функция f (monotonna funkcija) \| See also [edit] absolutely monotonic strictly decreasing strictly increasing weakly decreasing weakly increasing Further reading [edit] Monotonic function on Wikipedia.Wikipedia Bernstein's theorem on monotone functions on Wikipedia.Wikipedia Monotone cubic interpolation on Wikipedia.Wikipedia Monotone function on Encyclopedia of Mathematics Monotonic Function on Wolfram MathWorld Retrieved from " Categories: English lemmas English nouns English countable nouns English multiword terms en:Calculus en:Functions English terms with usage examples English terms with quotations en:Mathematical analysis Hidden categories: Pages with entries Pages with 1 entry Entries with translation boxes Terms with Bulgarian translations Terms with German translations Terms with Serbo-Croatian translations Terms with Spanish translations Terms with Turkish translations Requests for translations into Bulgarian monotone function Add topic
7690	https://people.tamu.edu/~yvorobets/MATH304-2011A/Lect2-07web.pdf	MATH 304 Linear Algebra Lecture 14: Basis and coordinates. Change of basis. Linear transformations. Basis and dimension Deﬁnition. Let V be a vector space. A linearly independent spanning set for V is called a basis. Theorem Any vector space V has a basis. If V has a ﬁnite basis, then all bases for V are ﬁnite and have the same number of elements (called the dimension of V ). Example. Vectors e1 = (1, 0, 0, . . . , 0, 0), e2 = (0, 1, 0, . . . , 0, 0),. . . , en = (0, 0, 0, . . . , 0, 1) form a basis for Rn (called standard) since (x1, x2, . . . , xn) = x1e1 + x2e2 + · · · + xnen. Basis and coordinates If {v1, v2, . . . , vn} is a basis for a vector space V , then any vector v ∈V has a unique representation v = x1v1 + x2v2 + · · · + xnvn, where xi ∈R. The coeﬃcients x1, x2, . . . , xn are called the coordinates of v with respect to the ordered basis v1, v2, . . . , vn. The mapping vector v 7→its coordinates (x1, x2, . . . , xn) is a one-to-one correspondence between V and Rn. This correspondence respects linear operations in V and in Rn. Examples. • Coordinates of a vector v = (x1, x2, . . . , xn) ∈Rn relative to the standard basis e1 = (1, 0, . . . , 0, 0), e2 = (0, 1, . . . , 0, 0),. . . , en = (0, 0, . . . , 0, 1) are (x1, x2, . . . , xn). • Coordinates of a matrix a b c d ∈M2,2(R) relative to the basis 1 0 0 0 , 0 0 1 0 , 0 1 0 0 , 0 0 0 1 are (a, c, b, d). • Coordinates of a polynomial p(x) = a0 + a1x + · · · + an−1xn−1 ∈Pn relative to the basis 1, x, x2, . . . , xn−1 are (a0, a1, . . . , an−1). Vectors u1=(2, 1) and u2=(3, 1) form a basis for R2. Problem 1. Find coordinates of the vector v = (7, 4) with respect to the basis u1, u2. The desired coordinates x, y satisfy v = xu1+yu2 ⇐ ⇒ 2x + 3y = 7 x + y = 4 ⇐ ⇒ x = 5 y = −1 Problem 2. Find the vector w whose coordinates with respect to the basis u1, u2 are (7, 4). w = 7u1 + 4u2 = 7(2, 1) + 4(3, 1) = (26, 11) Change of coordinates Given a vector v ∈R2, let (x, y) be its standard coordinates, i.e., coordinates with respect to the standard basis e1 = (1, 0), e2 = (0, 1), and let (x′, y ′) be its coordinates with respect to the basis u1 = (3, 1), u2 = (2, 1). Problem. Find a relation between (x, y) and (x′, y ′). By deﬁnition, v = xe1 + ye2 = x′u1 + y ′u2. In standard coordinates, x y = x′ 3 1 + y ′ 2 1 = 3 2 1 1 x′ y ′ = ⇒ x′ y ′ = 3 2 1 1 −1 x y = 1 −2 −1 3 x y Change of coordinates in Rn The usual (standard) coordinates of a vector v = (x1, x2, . . . , xn) ∈Rn are coordinates relative to the standard basis e1 = (1, 0, . . . , 0, 0), e2 = (0, 1, . . . , 0, 0),. . . , en = (0, 0, . . . , 0, 1). Let u1, u2, . . . , un be another basis for Rn and (x′ 1, x′ 2, . . . , x′ n) be the coordinates of the same vector v with respect to this basis. Problem 1. Given the standard coordinates (x1, x2, . . . , xn), ﬁnd the nonstandard coordinates (x′ 1, x′ 2, . . . , x′ n). Problem 2. Given the nonstandard coordinates (x′ 1, x′ 2, . . . , x′ n), ﬁnd the standard coordinates (x1, x2, . . . , xn). It turns out that     x1 x2 . . . xn    =     u11 u12 . . . u1n u21 u22 . . . u2n . . . . . . ... . . . un1 un2 . . . unn         x′ 1 x′ 2 . . . x′ n    . The matrix U = (uij) does not depend on the vector x. Columns of U are coordinates of vectors u1, u2, . . . , un with respect to the standard basis. U is called the transition matrix from the basis u1, u2, . . . , un to the standard basis e1, e2, . . . , en. This solves Problem 2. To solve Problem 1, we have to use the inverse matrix U−1, which is the transition matrix from e1, . . . , en to u1, . . . , un . Problem. Find coordinates of the vector x = (1, 2, 3) with respect to the basis u1 = (1, 1, 0), u2 = (0, 1, 1), u3 = (1, 1, 1). The nonstandard coordinates (x′, y ′, z′) of x satisfy   x′ y ′ z′  = U   1 2 3  , where U is the transition matrix from the standard basis e1, e2, e3 to the basis u1, u2, u3. The transition matrix from u1, u2, u3 to e1, e2, e3 is U0 = (u1, u2, u3) =   1 0 1 1 1 1 0 1 1  . The transition matrix from e1, e2, e3 to u1, u2, u3 is the inverse matrix: U = U−1 0 . The inverse matrix can be computed using row reduction. (U0 \| I) =   1 0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 0 1   →   1 0 1 1 0 0 0 1 0 −1 1 0 0 1 1 0 0 1  →   1 0 1 1 0 0 0 1 0 −1 1 0 0 0 1 1 −1 1   →   1 0 0 0 1 −1 0 1 0 −1 1 0 0 0 1 1 −1 1  = (I \| U−1 0 ) Thus   x′ y ′ z′  =   0 1 −1 −1 1 0 1 −1 1     1 2 3  =   −1 1 2  . Change of coordinates: general case Let V be a vector space of dimension n. Let v1, v2, . . . , vn be a basis for V and g1 : V →Rn be the coordinate mapping corresponding to this basis. Let u1, u2, . . . , un be another basis for V and g2 : V →Rn be the coordinate mapping corresponding to this basis. V g1 ւ g2 ց Rn − → Rn The composition g2◦g −1 1 is a transformation of Rn. It has the form x 7→Ux, where U is an n×n matrix. U is called the transition matrix from v1, v2 . . . , vn to u1, u2 . . . , un. Columns of U are coordinates of the vectors v1, v2, . . . , vn with respect to the basis u1, u2, . . . , un. Problem. Find the transition matrix from the basis p1(x) = 1, p2(x) = x + 1, p3(x) = (x + 1)2 to the basis q1(x) = 1, q2(x) = x, q3(x) = x2 for the vector space P3. We have to ﬁnd coordinates of the polynomials p1, p2, p3 with respect to the basis q1, q2, q3: p1(x) = 1 = q1(x), p2(x) = x + 1 = q1(x) + q2(x), p3(x) = (x+1)2 = x2+2x+1 = q1(x)+2q2(x)+q3(x). Hence the transition matrix is   1 1 1 0 1 2 0 0 1  . Thus the polynomial identity a1 + a2(x + 1) + a3(x + 1)2 = b1 + b2x + b3x2 is equivalent to the relation   b1 b2 b3  =   1 1 1 0 1 2 0 0 1     a1 a2 a3  . Problem. Find the transition matrix from the basis v1 = (1, 2, 3), v2 = (1, 0, 1), v3 = (1, 2, 1) to the basis u1 = (1, 1, 0), u2 = (0, 1, 1), u3 = (1, 1, 1). It is convenient to make a two-step transition: ﬁrst from v1, v2, v3 to e1, e2, e3, and then from e1, e2, e3 to u1, u2, u3. Let U1 be the transition matrix from v1, v2, v3 to e1, e2, e3 and U2 be the transition matrix from u1, u2, u3 to e1, e2, e3: U1 =   1 1 1 2 0 2 3 1 1  , U2 =   1 0 1 1 1 1 0 1 1  . Basis v1, v2, v3 = ⇒coordinates x Basis e1, e2, e3 = ⇒coordinates U1x Basis u1, u2, u3 = ⇒coordinates U−1 2 (U1x)=(U−1 2 U1)x Thus the transition matrix from v1, v2, v3 to u1, u2, u3 is U−1 2 U1. U−1 2 U1 =   1 0 1 1 1 1 0 1 1   −1   1 1 1 2 0 2 3 1 1   =   0 1 −1 −1 1 0 1 −1 1     1 1 1 2 0 2 3 1 1  =   −1 −1 1 1 −1 1 2 2 0  . Linear mapping = linear transformation = linear function Deﬁnition. Given vector spaces V1 and V2, a mapping L : V1 →V2 is linear if L(x + y) = L(x) + L(y), L(rx) = rL(x) for any x, y ∈V1 and r ∈R. A linear mapping ℓ: V →R is called a linear functional on V . If V1 = V2 (or if both V1 and V2 are functional spaces) then a linear mapping L : V1 →V2 is called a linear operator. Linear mapping = linear transformation = linear function Deﬁnition. Given vector spaces V1 and V2, a mapping L : V1 →V2 is linear if L(x + y) = L(x) + L(y), L(rx) = rL(x) for any x, y ∈V1 and r ∈R. Remark. A function f : R →R given by f (x) = ax + b is a linear transformation of the vector space R if and only if b = 0. Examples of linear mappings • Scaling L : V →V , L(v) = sv, where s ∈R. L(x + y) = s(x + y) = sx + sy = L(x) + L(y), L(rx) = s(rx) = r(sx) = rL(x). • Dot product with a ﬁxed vector ℓ: Rn →R, ℓ(v) = v · v0, where v0 ∈Rn. ℓ(x + y) = (x + y) · v0 = x · v0 + y · v0 = ℓ(x) + ℓ(y), ℓ(rx) = (rx) · v0 = r(x · v0) = rℓ(x). • Cross product with a ﬁxed vector L : R3 →R3, L(v) = v × v0, where v0 ∈R3. • Multiplication by a ﬁxed matrix L : Rn →Rm, L(v) = Av, where A is an m×n matrix and all vectors are column vectors. Linear mappings of functional vector spaces • Evaluation at a ﬁxed point ℓ: F(R) →R, ℓ(f ) = f (a), where a ∈R. • Multiplication by a ﬁxed function L : F(R) →F(R), L(f ) = gf , where g ∈F(R). • Diﬀerentiation D : C 1(R) →C(R), L(f ) = f ′. D(f + g) = (f + g)′ = f ′ + g ′ = D(f ) + D(g), D(rf ) = (rf )′ = rf ′ = rD(f ). • Integration over a ﬁnite interval ℓ: C(R) →R, ℓ(f ) = Z b a f (x) dx, where a, b ∈R, a < b.
7691	https://www.cuemath.com/ncert-solutions/find-the-value-of-i-30-x-4-1-x-22-ii-2-1-x-4-1-2-2-iii-1-2-2-1-3-2-1-4-2-iv-3-1-4-1-5-10-v-2-3-22/	Find the value of (i) (30 × 4−1) × 22(ii) (2−1 × 4−1) ÷ 2−2 (iii) (1/2)−2 + (1/3)−2 + (1/4)−2 (iv) (3−1 +4−1 + 5−1)0 (v) {(−2/3)−2}2 Solution: The exponent of a number shows how many times the number is multiplied by itself. (i) (30 × 4−1) × 22 According to the rules of exponents, a0 = 1 and a−m = 1/am (30 × 4−1) × 22 = (1 + 1/4) × 22 = [(4 + 1)/4] × 22 = (5/4) × 22 = 5/22× 22 [Since 4 = 2 × 2 = 22] = 5 (ii) (2−1 × 4−1) ÷ 2−2 According to the rules of exponents, (am)n= amn, a−m = 1/am (2−1 × 4−1) ÷ 2−2 = [2−1× {(2)2}−1] ÷ 2−2[Since 4 = 2 × 2 = 22] = (2−1× 2−2) ÷ 2−2 [∵(am)n= amn] = 2−3 ÷ 2−2 [∵am× an= am + n] = 2−3−(−2) [∵am ÷ an = am−n] = 2−3 + 2 = 2−1 = 1/2 [∵a−m= 1/am] (iii) (1/2)−2 + (1/3)−2 + (1/4)−2 According to the rules of exponents, (a/b)−m = (b/a)m (1/2)−2 + (1/3)−2 + (1/4)−2 = (2/1)2 + (3/1)2 + (4/1)2 = (2)2 + (3)2 + (4)2 = 4 + 9 + 16 = 29 (iv) (3−1 +4−1 + 5−1)0 According to the rules of exponents, a0 = 1 and a−m = 1/am (3−1 + 4−1 + 5−1)0 = (1/3 + 1/4 + 1/5)0 [Since a−m = 1/am] = 1 [Since a0 = 1] (v) {(−2/3)−2}2 According to the rules of exponents, a−m = 1/am and (a/b)m = am/bm {(−2/3)−2}2 = {(3/−2)2}2 [Sincea−m = 1/am] ={32/(−2)2}2 [Since(a/b)m = am/bm] =(9/4)2 = 81/16 ☛ Check: NCERT Solutions for Class 8 Maths Chapter 12 Video Solution: Find the value of (i) (3⁰× 4⁻¹) × 2²(ii) (2⁻¹× 4⁻¹) ÷ 2⁻²(iii) (1/2)⁻²+ (1/3)⁻²+ (1/4)⁻²(iv) (3⁻¹+4⁻¹+ 5⁻¹)⁰ (v) {(−2/3)⁻²}² Class 8 Maths NCERT Solutions Chapter 12 Exercise 12.1 Question 3 Summary: The value of the following (i) (30 × 4−1) × 22(ii) (2−1 × 4−1) ÷ 2−2(iii) (1/2)−2 + (1/3)−2 + (1/4)−2(iv) (3−1 +4−1 + 5−1)0(v) {(−2/3)−2}2 are (i) 5 (ii) 1/2 (iii) 29 (iv) 1 and (v) 81/16 respectively. ☛ Related Questions:
7692	https://www.quora.com/What-is-the-center-and-radius-of-the-circle-with-equation-x-2-y-2-2x-4y-11-0	What is the center and radius of the circle with equation, x^2 +y^2 -2x -4y -11 = 0? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Center of Circle Radius (geometry) Standard Form of Circle E... PLANE GEOMETRY Circles Circle (shape) Geometry of Circles Algebra 5 What is the center and radius of the circle with equation, x^2 +y^2 -2x -4y -11 = 0? All related (91) Sort Recommended Haresh Sagar Studied Science&Mathematics (Graduated 1988) · Author has 6.2K answers and 7M answer views ·3y C i r c l e:x 2+y 2−2 x−4 y−11=0 C i r c l e:x 2+y 2−2 x−4 y−11=0 If equation of circle is in general form, x 2+y 2+D x+E y+F=0 x 2+y 2+D x+E y+F=0 there are two methods by which centre and radius of circle can be found. Square method and “last three terms” mehod. Square method : x 2−2 x+y 2−4 y=11⟹x 2−2 x+y 2−4 y=11⟹ x 2−2 x+1+y 2−4 y+4=11+1+4⟹x 2−2 x+1+y 2−4 y+4=11+1+4⟹ (x−1)2+(y−2)2=16(x−1)2+(y−2)2=16 Thus by square method, general equation is rearranged into standard form, (x−h)2+(y−k)2=r 2(x−h)2+(y−k)2=r 2 So the centre is (1,2)(1,2) and radius is 4 4 units. In the “last three terms” method, we'll keep aside first two terms and only use last three terms to determine centre and radius. To determine centre, divide coefficien Continue Reading C i r c l e:x 2+y 2−2 x−4 y−11=0 C i r c l e:x 2+y 2−2 x−4 y−11=0 If equation of circle is in general form, x 2+y 2+D x+E y+F=0 x 2+y 2+D x+E y+F=0 there are two methods by which centre and radius of circle can be found. Square method and “last three terms” mehod. Square method : x 2−2 x+y 2−4 y=11⟹x 2−2 x+y 2−4 y=11⟹ x 2−2 x+1+y 2−4 y+4=11+1+4⟹x 2−2 x+1+y 2−4 y+4=11+1+4⟹ (x−1)2+(y−2)2=16(x−1)2+(y−2)2=16 Thus by square method, general equation is rearranged into standard form, (x−h)2+(y−k)2=r 2(x−h)2+(y−k)2=r 2 So the centre is (1,2)(1,2) and radius is 4 4 units. In the “last three terms” method, we'll keep aside first two terms and only use last three terms to determine centre and radius. To determine centre, divide coefficients of third (h)(h) and fourth (k)(k) term by negative 2 2. C=−1 2(−2,−4)=(1,2)C=−1 2(−2,−4)=(1,2) After getting centre, radius can be derived by, R=√h 2+k 2−F R=h 2+k 2−F R=√1 2+2 2−−11 R=1 2+2 2−−11 R=√16=4 R=16=4 Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Upvote · 9 7 9 2 Sponsored by Grammarly Stuck on the blinking cursor? Move your great ideas to polished drafts without the guesswork. Try Grammarly today! Download 99 34 Related questions More answers below What is the centre and radius of the circle with the equation x²+y²-2x-4y+1=0? What is the center and radius of the circle whose equation is x^2+y^2+4y=32? What is the radius of the circle x 2+y 2−6 y=0 x 2+y 2−6 y=0? What is the equation of a circle with center (2,-3) and passes through the center of the circle x^2+y^2-2x-10y+1=0? What is the equation of the circle whose radius is 5 and which touches the circle x^2 + y^2 -2x - 4y - 20 externally at the point (5,5)? Subramanya R Former Retired Govt Employee, Interested in All Fields · Author has 2K answers and 1.5M answer views ·Feb 1 General equation of the circle is :x 2+y 2−2 a x−2 b y+a 2+b 2−R 2=0 General equation of the circle is :x 2+y 2−2 a x−2 b y+a 2+b 2−R 2=0 Here, (a, b) is coordinates of centre and R is radius.Here, (a, b) is coordinates of centre and R is radius. Given equation is :x 2+y 2−2 x−4 y−11=0 Given equation is :x 2+y 2−2 x−4 y−11=0 Comparing this with general equation we have, a=1 , b=2, hence centre is Comparing this with general equation we have, a=1 , b=2, hence centre is [math]\text{(1, 2), and }a^2+b^2-R^... Upvote · 9 6 9 1 Philip Lloyd Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist. · Author has 6.8K answers and 52.8M answer views ·5y Originally Answered: What is the center and the radius of the circle whose given equation is x^2 + y^2 -6x+8y-25=0? · I will answer this question using clear steps so that a student will be able to follow it and hopefully learn the method! Continue Reading I will answer this question using clear steps so that a student will be able to follow it and hopefully learn the method! Upvote · 9 2 Assistant Bot · 1y To find the center and radius of the circle given by the equation x 2+y 2−2 x−4 y−11=0 x 2+y 2−2 x−4 y−11=0, we can rewrite the equation in standard form (x−h)2+(y−k)2=r 2(x−h)2+(y−k)2=r 2, where (h,k)(h,k) is the center and r r is the radius. Rearranging the equation: x 2−2 x+y 2−4 y=11 x 2−2 x+y 2−4 y=11 2. Completing the square for x x: Take the coefficient of x x (which is −2−2), halve it to get −1−1, and square it to get 1 1. Add and subtract 1 1 inside the equation: (x 2−2 x+1)+y 2−4 y=11+1(x 2−2 x+1)+y 2−4 y=11+1 This simplifies to: (x−1)2+y 2−4 y=12(x−1)2+y 2−4 y=12 3. Completing the square for y y: Take the coefficient of y y (which is −4−4), ha Continue Reading To find the center and radius of the circle given by the equation x 2+y 2−2 x−4 y−11=0 x 2+y 2−2 x−4 y−11=0, we can rewrite the equation in standard form (x−h)2+(y−k)2=r 2(x−h)2+(y−k)2=r 2, where (h,k)(h,k) is the center and r r is the radius. Rearranging the equation: x 2−2 x+y 2−4 y=11 x 2−2 x+y 2−4 y=11 2. Completing the square for x x: Take the coefficient of x x (which is −2−2), halve it to get −1−1, and square it to get 1 1. Add and subtract 1 1 inside the equation: (x 2−2 x+1)+y 2−4 y=11+1(x 2−2 x+1)+y 2−4 y=11+1 This simplifies to: (x−1)2+y 2−4 y=12(x−1)2+y 2−4 y=12 3. Completing the square for y y: Take the coefficient of y y (which is −4−4), halve it to get −2−2, and square it to get 4 4. Add and subtract 4 4: (x−1)2+(y 2−4 y+4)=12+4(x−1)2+(y 2−4 y+4)=12+4 This simplifies to: (x−1)2+(y−2)2=16(x−1)2+(y−2)2=16 4. Identifying the center and radius: Now the equation is in standard form: (x−1)2+(y−2)2=4 2(x−1)2+(y−2)2=4 2 The center (h,k)(h,k) is (1,2)(1,2). The radius r r is 4 4. Final Answer: Center: (1,2)(1,2) Radius: 4 4 Upvote · Related questions More answers below What are the coordinates of the center and the radius of a circle whose equation is x^2+y^ 2+2x-4y-11=0? How do you work out the center of the circle with the equation: x^2+y^2 +2x - 4y - 8 = 0? What is the center and radius of the circle whose equation is (x+1) ^2+ (y-1) ^2= 4? What is the center and radius of a circle, given the equation of the circle as x²+y²-4x-6y+9=0? How do I find the center and radius of a circle with the equation x^2+y^2+8x-2y+15=0? Pramodkumar Tandon Retired as Prof. & Head at Institute of Engineering and Rural Technology (1965–present) · Author has 2.1K answers and 1.2M answer views ·1y Originally Answered: What is the center and the radius of the circle whose given equation is x^2 +y^2-6x+8y-11=0? · .If a circle is given by : x² + y² + 2 g x + 2 f y + c = 0 then, Center = (- g,-f ), and Radius = Sqrt [ f² + g² - c ] Compare the given equation with it . => 2 g = - 6 >>>> g = - 3 and 2 f = 8 >>>>> f = 4 So Center = ( - g, - f ) = ( 3, - 4 ) and Radius = Sqrt [ (-3)² + (4)² - (-11) ] =>Radius = Sqrt ( 36 ) = 6 Continue Reading .If a circle is given by : x² + y² + 2 g x + 2 f y + c = 0 then, Center = (- g,-f ), and Radius = Sqrt [ f² + g² - c ] Compare the given equation with it . => 2 g = - 6 >>>> g = - 3 and 2 f = 8 >>>>> f = 4 So Center = ( - g, - f ) = ( 3, - 4 ) and Radius = Sqrt [ (-3)² + (4)² - (-11) ] =>Radius = Sqrt ( 36 ) = 6 Upvote · Sponsored by All Out Kill Dengue, Malaria and Chikungunya with New 30% Faster All Out. Chance Mat Lo, Naya All Out Lo - Recommended by Indian Medical Association. Shop Now 999 616 Michael Brennan B.A. in Mathematics, St. Thomas University (Graduated 1969) · Author has 5.8K answers and 4.4M answer views ·10mo Given: Circle x.^2 + y^2 - 2x - 4y - 11 = 0 Find: Circle Center and Circle Radius Plan: Convert the given equation into the General Form of a circle’s equation: (x - a)^2 + (y - b)^2 = r^2 where Circle Center (a, b) and Radius (r) x.^2 + y^2 - 2x - 4y - 11 = 0 Group the x terms and y-terms with all number Grouped and combined to the right side. x^2 - 2x + y^2 - 4y = 11 Grouping x and y terms and adding 11 to both sides of the equation. Now we will convert groups by completing the square. (x^2 - 2x + 1) + (y^2 - 4y + 4) = 11 + 1 + 4 (x^2 - 2x + 1) + (y^2 - 4y + 4) = 16 Now factor (x - 1)^2 + (y - 2)^2 = Continue Reading Given: Circle x.^2 + y^2 - 2x - 4y - 11 = 0 Find: Circle Center and Circle Radius Plan: Convert the given equation into the General Form of a circle’s equation: (x - a)^2 + (y - b)^2 = r^2 where Circle Center (a, b) and Radius (r) x.^2 + y^2 - 2x - 4y - 11 = 0 Group the x terms and y-terms with all number Grouped and combined to the right side. x^2 - 2x + y^2 - 4y = 11 Grouping x and y terms and adding 11 to both sides of the equation. Now we will convert groups by completing the square. (x^2 - 2x + 1) + (y^2 - 4y + 4) = 11 + 1 + 4 (x^2 - 2x + 1) + (y^2 - 4y + 4) = 16 Now factor (x - 1)^2 + (y - 2)^2 = 16 ✅ Center(1,2) and Radius(r) = 4 (r = √16) Double Check: Reasonable/Recalculated ✅ ✅ Answers: Center(1,2) and Radius(r) = 4 Upvote · Gordon M. Brown Math Tutor at San Diego City College (2018-Present) · Author has 6.2K answers and 4.3M answer views ·11mo Originally Answered: What is the centre and the radius of the circle of x^2+y^2-6y=0? · The solution is staring you right in the face. Complete the square for the terms of y, and go from there. Continue Reading The solution is staring you right in the face. Complete the square for the terms of y, and go from there. Upvote · 9 3 Sponsored by LPU Online Career Ka Turning Point with LPU Online. 100% Online UGC-Entitled programs with LIVE classes, recorded content & placement support. Apply Now 999 261 B.L. Srivastava Lives in Kanpur, Uttar Pradesh, India (1972–present) · Author has 7.6K answers and 8.1M answer views ·4y Originally Answered: What is the centre and radius of the circle given by the equation x²+ y² +z²-2y-4z-11=0=x+2y+2z-15? · The center of the given sphere S : x²+y² +z² -2y -4z -11 = 0 is C(0, 1, 2) and radius R = √(0²+1²+2²-(-11)) = √16 = 4 . Now, length of the perpendicular (d) from the center C onto the given plane; P : x + 2y + 2z -15 = 0 is given by ; d = \|(0+2+4)-15\|/√9 = 9/3 = 3 . Therefore radius (r) of the circle given by S = 0 = P is r = √R²−d² = √(16−9) = √7 . Next, center M of the circle is a point, on the perpendicular line through C and at a distance d=3 from C. Eq.of perpendicular line CM which has drs as 1/3, 2/3 & 2/3 is ; (x-0)/(1/3)= (y - 1)/(2/3)= (z - 2)/(2/3)= d = 3 ==> point M is given by ; x Continue Reading The center of the given sphere S : x²+y² +z² -2y -4z -11 = 0 is C(0, 1, 2) and radius R = √(0²+1²+2²-(-11)) = √16 = 4 . Now, length of the perpendicular (d) from the center C onto the given plane; P : x + 2y + 2z -15 = 0 is given by ; d = \|(0+2+4)-15\|/√9 = 9/3 = 3 . Therefore radius (r) of the circle given by S = 0 = P is r = √R²−d² = √(16−9) = √7 . Next, center M of the circle is a point, on the perpendicular line through C and at a distance d=3 from C. Eq.of perpendicular line CM which has drs as 1/3, 2/3 & 2/3 is ; (x-0)/(1/3)= (y - 1)/(2/3)= (z - 2)/(2/3)= d = 3 ==> point M is given by ; x = 1, y = 3 & z = 4 i.e. M(x, y, z) = (1, 3, 4) and this point satisfies the equation of the plane P = 0 . Hence it is the required center of the circle. Upvote · Pramodkumar Tandon Retired as Prof. & Head at Institute of Engineering and Rural Technology (1965–present) · Author has 2.1K answers and 1.2M answer views ·2y Originally Answered: Given a circle with equation x^2 + y^2 - 4x + 6y + 1 = 0, what is the center and radius of the circle? · We know that: Any circle can be represented by the following eqn. — x² + y² + 2 g x + 2 f y + c = 0 ………………………… (1) Where : (-g,-f) is center and Radius = Sqrt [ g² + f² - c ] The given eqn of the circle is : x² + y² - 4x + 6y + 1 = 0 …………….. (2) Compare eqn (1) with (2) to get — 2 g = - 4 ====>>>> g = - 2 and 2 f = 6 =====>>>> f = 3 c = 1 Hence center coordinates are (2,-3) Radius = Sqrt [ 4 + 9 + 1 ] = sqrt 14 = 3.74 Continue Reading We know that: Any circle can be represented by the following eqn. — x² + y² + 2 g x + 2 f y + c = 0 ………………………… (1) Where : (-g,-f) is center and Radius = Sqrt [ g² + f² - c ] The given eqn of the circle is : x² + y² - 4x + 6y + 1 = 0 …………….. (2) Compare eqn (1) with (2) to get — 2 g = - 4 ====>>>> g = - 2 and 2 f = 6 =====>>>> f = 3 c = 1 Hence center coordinates are (2,-3) Radius = Sqrt [ 4 + 9 + 1 ] = sqrt 14 = 3.74 Upvote · 9 1 9 2 Sponsored by RedHat Customize AI for your needs, with simpler model alignment tools. Your AI needs context, not common knowledge. Learn More 9 6 Janet Heberling Lives in San Francisco, CA (2022–present) · Upvoted by Richard McConnel , M.S. Aerospace and Aeronautical Engineering & Mathematics, US Naval Postgraduate School (1966) · Author has 21.5K answers and 9.4M answer views ·5y Originally Answered: What is the center and the radius of the circle whose given equation is x^2 + y^2 -6x+8y-25=0? · Upvote · 9 5 9 2 Gordon M. Brown Math Tutor at San Diego City College (2018-Present) · Author has 6.2K answers and 4.3M answer views ·3y You wouldn’t be asking this question if you knew how to rearrange an equation, then complete the squares. x^2 + y^2 - 2x - 4y - 11 = 0 x^2 - 2x _ + y^2 - 4y = 11 + __ + _____ [Prepare to complete the squares] x^2 - 2x + 1 + y^2 - 4y + 4 = 11 + 1 + 4 = 16 [Complete the squares] (x - 1)^2 + (y - 2)^2 = 16 [Make the square expressions more explicit] Now that the equation is in the standard form (x - h)^2 + (y - k )^2 = r^2, you should be able to tell at a glance that the center of the circle is located at (1, 2), and its radius is 4. I’m not gonna lie; I don’t have much patience for que Continue Reading You wouldn’t be asking this question if you knew how to rearrange an equation, then complete the squares. x^2 + y^2 - 2x - 4y - 11 = 0 x^2 - 2x _ + y^2 - 4y = 11 + __ + _____ [Prepare to complete the squares] x^2 - 2x + 1 + y^2 - 4y + 4 = 11 + 1 + 4 = 16 [Complete the squares] (x - 1)^2 + (y - 2)^2 = 16 [Make the square expressions more explicit] Now that the equation is in the standard form (x - h)^2 + (y - k )^2 = r^2, you should be able to tell at a glance that the center of the circle is located at (1, 2), and its radius is 4. I’m not gonna lie; I don’t have much patience for questions of this nature, nor for the people who ask them. These questions almost always signify that the OP isn’t doing the work necessary to succeed in their math course. I haven’t told you everything you need to know to solve this problem successfully, and I’m certainly not going to teach you how to complete the square. You’re on your own there. Upvote · 9 2 9 2 Rick Marcus Former R&D Engineer at Stanford University (1983–1989) · Author has 823 answers and 121.3K answer views ·1y Originally Answered: What is the centre and radius of the circle with the equation x²+y²-2x-4y+1=0? · The standard equation of a circle centered at (x1,y1) with radius R is: equation 1: (x-x1)²+(y-y1)²=R² equation 2: x²+y²-2x-4y+1=0 We need to get this equation 2 into the same form as equation 1. To do this we complete the square as follows: x²-2x+1 = (x-1)² y²-4y+4 = (y-2)² So if we add 4 to each side of equation ... Upvote · Haresh Sagar Studied Science&Mathematics (Graduated 1988) · Author has 6.2K answers and 7M answer views ·4y Originally Answered: What are the coordinates of the center and the radius of a circle whose equation is x^2+y^ 2+2x-4y-11=0? · C i r c l e:x 2+y 2+2 x−4 y−11=0 C i r c l e:x 2+y 2+2 x−4 y−11=0 When circle equation is in general form, x 2+y 2+d x+e y+f=0 x 2+y 2+d x+e y+f=0 centre can be found by negating coefficients of third and fourth terms and dividing them by 2 2 C=(−d 2,−e 2)C=(−d 2,−e 2) C=(−2 2,−−4 2)C=(−2 2,−−4 2) C=(−1,2)C=(−1,2) When centre is found, radius can be found by, R=√h 2+k 2−f R=h 2+k 2−f R=√(−1)2+(2)2−(−11)R=(−1)2+(2)2−(−11) R=√1+4+11=√16=4 R=1+4+11=16=4 units. Continue Reading C i r c l e:x 2+y 2+2 x−4 y−11=0 C i r c l e:x 2+y 2+2 x−4 y−11=0 When circle equation is in general form, x 2+y 2+d x+e y+f=0 x 2+y 2+d x+e y+f=0 centre can be found by negating coefficients of third and fourth terms and dividing them by 2 2 C=(−d 2,−e 2)C=(−d 2,−e 2) C=(−2 2,−−4 2)C=(−2 2,−−4 2) C=(−1,2)C=(−1,2) When centre is found, radius can be found by, R=√h 2+k 2−f R=h 2+k 2−f R=√(−1)2+(2)2−(−11)R=(−1)2+(2)2−(−11) R=√1+4+11=√16=4 R=1+4+11=16=4 units. Upvote · 9 1 Related questions What is the centre and radius of the circle with the equation x²+y²-2x-4y+1=0? What is the center and radius of the circle whose equation is x^2+y^2+4y=32? What is the radius of the circle x 2+y 2−6 y=0 x 2+y 2−6 y=0? What is the equation of a circle with center (2,-3) and passes through the center of the circle x^2+y^2-2x-10y+1=0? What is the equation of the circle whose radius is 5 and which touches the circle x^2 + y^2 -2x - 4y - 20 externally at the point (5,5)? What are the coordinates of the center and the radius of a circle whose equation is x^2+y^ 2+2x-4y-11=0? How do you work out the center of the circle with the equation: x^2+y^2 +2x - 4y - 8 = 0? What is the center and radius of the circle whose equation is (x+1) ^2+ (y-1) ^2= 4? What is the center and radius of a circle, given the equation of the circle as x²+y²-4x-6y+9=0? How do I find the center and radius of a circle with the equation x^2+y^2+8x-2y+15=0? What is the center and radius of the circle (x +2) ^2+ (y + 4) ^2= 16? What is the center of the circle whose equation is x^2+y^2 +4x - 8y + 11 = 0? What are the center and radius of the circle whose equation is x^2+y^2+4x=5? The equation for a circle is x^2 + y^2 + 10y - 7 = 0. What is the center of the circle? What is the center and the radius of the circle whose given equation is x^2 +y^2-6x+8y-11=0? Related questions What is the centre and radius of the circle with the equation x²+y²-2x-4y+1=0? What is the center and radius of the circle whose equation is x^2+y^2+4y=32? What is the radius of the circle x 2+y 2−6 y=0 x 2+y 2−6 y=0? What is the equation of a circle with center (2,-3) and passes through the center of the circle x^2+y^2-2x-10y+1=0? What is the equation of the circle whose radius is 5 and which touches the circle x^2 + y^2 -2x - 4y - 20 externally at the point (5,5)? What are the coordinates of the center and the radius of a circle whose equation is x^2+y^ 2+2x-4y-11=0? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
7693	https://pubmed.ncbi.nlm.nih.gov/17034404/	A review of screening tools for psychiatric comorbidity in headache patients - 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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A review of screening tools for psychiatric comorbidity in headache patients Morris Maizels1,Todd A Smitherman,Donald B Penzien Affiliations Expand Affiliation 1 Department of Family Medicine, Kaiser Permanente, Woodland Hills, CA 91365, USA. PMID: 17034404 DOI: 10.1111/j.1526-4610.2006.00561.x Item in Clipboard Review A review of screening tools for psychiatric comorbidity in headache patients Morris Maizels et al. Headache.2006 Oct. Show details Display options Display options Format Headache Actions Search in PubMed Search in NLM Catalog Add to Search . 2006 Oct:46 Suppl 3:S98-109. doi: 10.1111/j.1526-4610.2006.00561.x. Authors Morris Maizels1,Todd A Smitherman,Donald B Penzien Affiliation 1 Department of Family Medicine, Kaiser Permanente, Woodland Hills, CA 91365, USA. PMID: 17034404 DOI: 10.1111/j.1526-4610.2006.00561.x Item in Clipboard Full text links Cite Display options Display options Format Abstract Psychiatric comorbidity, especially depression and anxiety, has been well documented in patients with primary headache disorders. The presence of psychiatric comorbidity in headache patients is associated with decreased quality-of-life, poorer prognosis, chronification of disease, poorer response to treatment, and increased medical costs. Despite the prevalence and impact, screening for psychiatric disorders in headache patients is not systematically performed, either clinically or in research studies, and there are no guidelines to suggest which patients should be screened or in what manner. We review a variety of screening methods and instruments, focusing primarily on self-report measures and those available in the public domain. Informal verbal screening may be sufficient in a primary care setting, but should include screening for both anxiety and depression. Explicit screening for anxiety is important, as anxiety may have a more significant impact on headache than does depression and may occur in the absence of clinical depression. Formal screening with instruments that can identify a variety of psychiatric disorders is appropriate for patients with daily headache syndromes, patients who are refractory to usual care, and patients referred for specialty evaluation. Limitations of screening instruments include the influence of transdiagnostic symptoms and the need for confirmatory diagnostic interview. The following instruments appear most suitable for use in headache patients: for depression, the Patient Health Questionnaire Depression Module, the Beck Depression Inventory-II, or the Beck Depression Inventory-Primary Care; for anxiety, the Beck Anxiety Inventory and the Generalized Anxiety Disorder 7-item Scale; and for multidimensional psychiatric screening, the Patient Health Questionnaire or Primary Care Evaluation of Mental Disorders. PubMed Disclaimer Similar articles Basic principles and techniques of cognitive-behavioral therapies for comorbid psychiatric symptoms among headache patients.Lipchik GL, Smitherman TA, Penzien DB, Holroyd KA.Lipchik GL, et al.Headache. 2006 Oct;46 Suppl 3:S119-32. doi: 10.1111/j.1526-4610.2006.00563.x.Headache. 2006.PMID: 17034390 Review. Somatic symptoms in headache patients: the influence of headache diagnosis, frequency, and comorbidity.Maizels M, Burchette R.Maizels M, et al.Headache. 2004 Nov-Dec;44(10):983-93. doi: 10.1111/j.1526-4610.2004.04192.x.Headache. 2004.PMID: 15546261 Headache and psychiatric comorbidity: historical context, clinical implications, and research relevance.Lake AE 3rd, Rains JC, Penzien DB, Lipchik GL.Lake AE 3rd, et al.Headache. 2005 May;45(5):493-506. doi: 10.1111/j.1526-4610.2005.05101.x.Headache. 2005.PMID: 15953266 Depression, anxiety, and somatoform disorders: vague or distinct categories in primary care? Results from a large cross-sectional study.Hanel G, Henningsen P, Herzog W, Sauer N, Schaefert R, Szecsenyi J, Löwe B.Hanel G, et al.J Psychosom Res. 2009 Sep;67(3):189-97. doi: 10.1016/j.jpsychores.2009.04.013. Epub 2009 Jun 27.J Psychosom Res. 2009.PMID: 19686874 Clinical Trial. Headache chronification: screening and behavioral management of comorbid depressive and anxiety disorders.Smitherman TA, Maizels M, Penzien DB.Smitherman TA, et al.Headache. 2008 Jan;48(1):45-50. doi: 10.1111/j.1526-4610.2007.00974.x.Headache. 2008.PMID: 18184285 Review. See all similar articles Cited by Screening for depression and anxiety disorder in children with headache.Lee SM, Yoon JR, Yi YY, Eom S, Lee JS, Kim HD, Cheon KA, Kang HC.Lee SM, et al.Korean J Pediatr. 2015 Feb;58(2):64-8. doi: 10.3345/kjp.2015.58.2.64. Epub 2015 Feb 28.Korean J Pediatr. 2015.PMID: 25774198 Free PMC article. Psychometric properties of the stagnation scale in medication overuse headache patients.Innamorati M, Pompili M, Erbuto D, Ricci F, Migliorati M, Lamis DA, Amore M, Girardi P, Martelletti P.Innamorati M, et al.J Headache Pain. 2015;16:1052. doi: 10.1186/1129-2377-16-2. Epub 2015 Feb 19.J Headache Pain. 2015.PMID: 25971238 Free PMC article. Headache secondary to psychiatric disorders.Smitherman TA, Baskin SM.Smitherman TA, et al.Curr Pain Headache Rep. 2008 Aug;12(4):305-10. doi: 10.1007/s11916-008-0051-0.Curr Pain Headache Rep. 2008.PMID: 18625109 Review. Behavioral medicine for migraine and medication overuse headache.Andrasik F, Buse DC, Grazzi L.Andrasik F, et al.Curr Pain Headache Rep. 2009 Jun;13(3):241-8. doi: 10.1007/s11916-009-0041-x.Curr Pain Headache Rep. 2009.PMID: 19457287 Review. Understanding the nature of psychiatric comorbidity in migraine: a systematic review focused on interactions and treatment implications.Dresler T, Caratozzolo S, Guldolf K, Huhn JI, Loiacono C, Niiberg-Pikksööt T, Puma M, Sforza G, Tobia A, Ornello R, Serafini G; European Headache Federation School of Advanced Studies (EHF-SAS).Dresler T, et al.J Headache Pain. 2019 May 9;20(1):51. doi: 10.1186/s10194-019-0988-x.J Headache Pain. 2019.PMID: 31072313 Free PMC article. 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7694	https://math-drills.com/integers/integers_comparing_-25to25_001.php	Comparing Integers from -25 to 25 (A) Comparing Integers from -25 to 25 (A) MenuNews Discover more educational Maths Math Drills math Math-Drills Math Matemáticas Mathematics Education purchase math curriculum materials Welcome to The Comparing Integers from -25 to 25 (A) Math Worksheet from the Integers Worksheets Page at Math-Drills.com. This math worksheet was created or last revised on 2013-11-06 and has been viewed 13 times this week and 885 times this month. It may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math. Teacher s can use math worksheets as test s, practice assignment s or teaching tool s (for example in group work, for scaffolding or in a learning center). Parent s can work with their children to give them extra practice, to help them learn a new math skill or to keep their skills fresh over school breaks. Student s can use math worksheets to master a math skill through practice, in a study group or for peer tutoring. Use the buttons below to print, open, or download the PDF version of the Comparing Integers from -25 to 25 (A) math worksheet. The size of the PDF file is 25118 bytes. Preview images of the first and second (if there is one) pages are shown. If there are more versions of this worksheet, the other versions will be available below the preview images. For more like this, use the search bar to look for some or all of these keywords: math, integers, comparing, negative, positive, numbers, greater, lesser, than. Print Full Version Open Full VersionDownload Full Version Print Student Version Open Student VersionDownload Student Version The Print button initiates your browser's print dialog. The Open button opens the complete PDF file in a new browser tab. The Download button initiates a download of the PDF math worksheet. Teacher versions include both the question page and the answer key. Student versions, if present, include only the question page. The Comparing Integers from -25 to 25 (A) Math Worksheet Page 1 The Comparing Integers from -25 to 25 (A) Math Worksheet Page 2 Discover more Mathematics Maths Math educational Education Math-Drills Matemáticas Math Drills math shop for math teaching resources Other Versions: ABCDEFGHIJAll More Integers Worksheets HomeAddition WorksheetsSubtraction WorksheetsMultiplication Facts WorksheetsLong Multiplication WorksheetsDivision WorksheetsMixed Operations Worksheets Algebra WorksheetsBase Ten Blocks WorksheetsDecimals WorksheetsFact Families WorksheetsFractions WorksheetsGeometry WorksheetsGraph PaperIntegers WorksheetsMeasurement WorksheetsMoney Math WorksheetsNumber Lines WorksheetsNumber Sense WorksheetsOrder of Operations WorksheetsPatterning WorksheetsPercentages WorksheetsPlace Value WorksheetsPowers of Ten WorksheetsStatistics WorksheetsTime Math WorksheetsMath Word Problems Worksheets Halloween Math WorksheetsThanksgiving Math WorksheetsChristmas Math WorksheetsValentine's Day Math WorksheetsSaint Patrick's Day Math WorksheetsEaster Math WorksheetsSeasonal Math Worksheets Math Flash CardsDots Math GameVideo Tutorials Help and FAQTerms of UsePrivacy and Cookie PolicyTour/IntroductionFeedbackTeachersParentsSupport Math-DrillsMath-Drills on Facebook Ejercicios de Matemáticas GratisFiches d'Exercices de Maths Copyright © 2005-2025 Math-Drills.com You may use the math worksheets on this website according to our Terms of Use to help students learn math.
7695	https://arxiv.org/pdf/2110.13305	arXiv:2110.13305v1 [math.CA] 25 Oct 2021 Recurrence equations involving di ff erent orthogonal polynomial sequences and applications A.S. Jooste a, D.D. Tcheutia b, W. Koepf b, ∗ a Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa b Institute of Mathematics, University of Kassel, Heinrich-Plett Str. 40, 34132 Kassel, Germany Abstract Consider {pn}∞ n=0 , a sequence of polynomials orthogonal with respect to w(x) > 0 on ( a, b), and polynomials {gn,k}∞ n=0 , k ∈ N0, orthogonal with respect to ck(x)w(x) > 0 on ( a, b), where ck(x) is a polynomial of degree k in x. We show how Christo ff el’s formula can be used to obtain mixed three-term recurrence equations involving the polynomials pn, pn−1 and gn−m,k, m ∈ { 2, 3, . . . , n−1}.In order for the zeros of pn and Gm−1gn−m,k to interlace (assuming pn and gn−m,k are co-prime), the coe ffi cient of pn−1, namely Gm−1, should be of exact degree m − 1, in which case restrictions on the parameter k are necessary. The zeros of Gm−1 can be considered to be inner bounds for the extreme zeros of the (classical or q-classical) orthogonal polynomial pn and we give examples to illustrate the accuracy of these bounds. Because of the complexity the mixed three-term recurrence equations in each case, algorithmic tools, mainly Zeilberger’s algorithm and its q-analogue, are used to obtain them. Keywords: Classical orthogonal polynomials, q-classical orthogonal polynomials, mixed three-term recurrence equations, interlacing of zeros, bounds for zeros 2010 MSC: 33C05, 33C45, 33F10, 33D15 Introduction A sequence {pn}∞ n=0 of real polynomials, where pn is of exact degree n, is orthogonal with respect to an absolutely continuous measure that can be represented by a real, positive weight function w(x) on the (finite or infinite) interval ( a, b), if ∫ ba pn(x)pm(x)w(x)dx = 0, m , n. ∗ Corresponding author Email addresses: alta.jooste@up.ac.za (A.S. Jooste), duvtcheutia@yahoo.fr (D.D. Tcheutia), koepf@mathematik.uni-kassel.de (W. Koepf) URL: www.mathematik.uni-kassel.de/~koepf (W. Koepf) Preprint submitted to Elsevier October 27, 2021 The classical orthogonal polynomials considered in this paper are defined in terms of the general-ized hypergeometric series p Fq ( a1, . . . , ap b1, . . . , bq ∣∣∣∣∣∣ x ) = ∞ ∑ m=0 (a1)m · · · (ap)m (b1)m · · · (bq)m xm m!, where ( a)m denotes the Pochhammer symbol (or shifted factorial) defined by (a)m = { 1 if m = 0 a(a + 1)( a + 2) · · · (a + m − 1) if m ∈ N. Their q-orthogonal analogues, 0 < q < 1, are given in terms of basic hypergeometric series [ 27 ,Section 1.10] r φs ( a1, . . . , ar b1, . . . , bs ∣∣∣∣∣∣ q; z ) = ∞ ∑ k=0 (a1, . . . , ar; q)k (b1, . . . , bs; q)k ((−1) kq(k 2 ))1+s−r zk (q; q)k , where the q-Pochhammer symbol ( a1, a2, . . . , ak; q)n is defined by (a1, . . . , ar; q)k := (a1; q)k · · · (ar; q)k, with ( ai; q)k =  k−1 ∏ j=0 (1 − aiq j) if k ∈ { 1, 2, 3, . . . } 1 if k = 0. Let the zeros of pn be xn,1 < xn,2 < · · · < xn,n. It is well known that pn satisfies a three-term recurrence equation pn(x) = (x − Cn)pn−1(x) − λn pn−2(x), (1) where Cn and λn do not depend on x, p−1 ≡ 0, p0 ≡ 1 and λn > 0, and that the zeros of pn and pn−1 interlace, i.e., xn,1 < xn−1,1 < xn,2 < · · · < xn,n−1 < xn−1,n−1 < xn,n. It is also known that, if pn and pn−2 do not have a common zero, the n − 1 zeros of ( x − Cn)pn−2(x)interlace with the n zeros of pn [5, Theorem 3], i.e., xn,1 < Cn < xn,n and the point Cn is a natural inner bound for the extreme zeros of pn.By iterating ( 1), Beardon [ 5, Theorem 4] generalised this result and obtained recurrence equa-tions involving polynomials pn−m, pn and pn−1: λnλn−1 . . . λ n−m+2 pn−m(x) = S m−1(x)pn−1(x) + S m−2(x)pn(x), m ≥ 2, (2) where the polynomials S m are of degree m and the n − 1 zeros of S m−1 pn−m interlace with the n zeros of pn, a phenomenon that will be referred to as “completed Stieltjes interlacing”. In this paper we re-phrase [ 5, Theorem 4] and we introduce the associated polynomials of the orthogonal sequence {pn}∞ n=0 , as well as the three-term recurrence satisfied by them (cf. [ 8, 10 , 41 ]). Then we show how Christo ff el’s formula [ 39 , Theorem 2.5] can be used to obtain mixed three-term recurrence equations, similar to ( 2), involving polynomials pn, pn−1, and gn−m,k, m ∈ { 2, 3, . . . }, k an integer, where the polynomial gn−m,k belongs to a related sequence, orthogonal with respect 2to the weight ck(x)w(x) > 0 and ck is a polynomial of degree k. Restrictions on the integer k are necessary, in order for the zeros of gn−m,k and pn to interlace in the Stieltjes sense. Each mixed recurrence equation provides us with a polynomial Gm−1, m ∈ { 2, 3, . . . , n − 1}, whose zeros can be used as inner bounds for the extreme zeros of pn. The bounds obtained in this way are more accurate than the inner bounds obtained using mixed recurrence equations with m = 2, as was done for the extreme zeros of the Jacobi, Laguerre and Gegenbauer polynomials in [ 17 ], Meixner and Krawtchouk polynomials in [ 23 ] and Hahn polynomials in [ 24 ]. In our applications the polynomials gn−m,k, m ∈ { 2, 3, . . . , n − 1}, are typically obtained from the polynomials in the orthogonal sequence {pn}∞ n=0 by making appropriate parameter shifts of (in total) k units. Levit [ 34 ] was the first to study the separation of the zeros of di ff erent sequences of Hahn polynomials and interlacing results for Jacobi polynomials [ 4, 13 ], Krawtchouk polynomials [ 9, 25 ] and Meixner and Meixner-Pollaczek polynomials [ 25 ] followed. Interlacing results for the zeros of di ff erent sequences of q-orthogonal sequences with shifted parameters are given in [ 19 , 26 , 35 , 40 ]. Completed Stieltjes interlacing of zeros of di ff erent orthogonal sequences was done for the Gegenbauer [ 14 ], Laguerre [ 16 ] and Jacobi polynomials [ 15 ] and apart from the papers cited in the previous paragraph, inner bounds for the extreme zeros of Gegenbauer, Laguerre and Jacobi polynomials were also given in [ 2, 6, 21 , 32 , 36 , 39 ]; bounds for the extreme zeros of the discrete orthogonal Charlier, Meixner, Krawtchouk and Hahn polynomials in [ 3, 33 ], for the extreme zeros of the q-Jacobi and q-Laguerre polynomials in [ 21 ] and for the little q-Jacobi polynomials in [ 19 ]. Lower bounds for xn,1 and upper bounds for xn,n can be found in the case of classical continuous and discrete orthogonal polynomials in [ 1, 3, 11 , 12 , 22 , 30 , 32 , 33 , 39 ] and in [ 31 ], non-asymptotic bounds on the extreme zeros of (symmetric) orthogonal polynomials are given in terms of the coe ffi cients of their three-term recurrence equations. In general it is time-consuming to find the zeros of an expanded polynomial of high degree using a computer program such as Maple, since one has to work with high precision. In such cases it is helpful to have a formula for a bound. Our main results are stated in section 2, where we also describe the algorithmic approach used to obtain the mixed recurrence equations necessary to prove Stieltjes interlacing between the zeros of polynomials from di ff erent orthogonal sequences. In the last three sections we display mixed recurrence equations that provide formulas for inner bounds for the extreme zeros of some spe-cific polynomial systems, merely to show the accuracy of the bounds provided by these recurrence equations. In section 3 we show bounds in the generalized Laguerre case. In section 4 we show the quality of bounds obtained for the extreme zeros of the little q-Jacobi and the alternative q Charlier (or q-Bessel) polynomials, where the equation providing the bounds concerns polynomi-als pn(x; α, β ), pn−1(x; α, β ) and either pn−m(x; α + k, β ) or pn−m(x; α, β + k), i.e., only one parameter is shifted. Since we observe that the best possible bounds are obtained when one parameter is shifted optimally, polynomial systems depending on 2 or 3 parameters also belong to this group. In section 5, we provide bounds for the extreme zeros of the Stieltjes-Wigert and the discrete q-Hermite II polynomials, as examples of polynomial systems where no parameters are available to be shifted. On the webpage , we provide optimal (upper and /or lower) inner bounds for the extreme zeros of following polynomial systems: 3(i) The Gegenbauer, Laguerre, Jacobi, Bessel, Meixner and Hahn polynomials, as well as the big q-Jacobi, q-Hahn, q-Meixner, little q-Laguerre (or Wall), little q-Jacobi, q-Krawtchouk, affi ne q-Krawtchouk, q-Laguerre, alternative q-Charlier (or q-Bessel) and q-Charlier poly-nomials. In these cases the bounds were obtained from shifting one parameter optimally. By shifting α to αq−6 and γ to γq−6, respectively, we obtain a lower bound for the largest zero of the q-Charlier polynomials and an upper bound for the smallest zero of the q-Meixner polynomial; (ii) The Hermite, Krawtchouk and Charlier polynomials, as well as the quantum q-Krawtchouk, Stieltjes Wigert, Al-Salam Carlitz I and II and discrete q-Hermite I and II polynomials, where we use equations similar to ( 4) to obtain the bounds. In this way we also find a lower bound for xn,n in the q-Meixner case and an upper bound for the smallest zero of the q-Charlier polynomials. Stieltjes interlacing of zeros of classical orthogonal sequences Lemma 1. (cf. [5, Theorem 4]) Suppose {pn}∞ n=0 is a sequence of polynomials, satisfying (1).Then, given n, there exists a sequence of real orthogonal polynomials S (n) m (x), m ∈ { 0, 1, 2, . . . , n}, of exact degree m, satisfying the three-term recurrence equation S (n) m (x) = (x − Cn−(m−1) )S (n) m−1 (x) − λn−(m−2) S (n) m−2 (x), m ∈ { 1, 2, . . . , n}, (3) with S (n)0 (x) = 1, S (n) −1 (x) = 0, S (n)1 (x) = x − Cn, such that, for m ≥ 2, λnλn−1 . . . λ n−m+2 pn−m(x) = S (n) m−1 (x)pn−1(x) − S (n−1) m−2 (x)pn(x). (4) Proof. For each n we have λn [ pn−2(x) pn−1(x) ] = [ x − Cn −1 λn 0 ] [ pn−1(x) pn(x) ] . Thus λnλn−1 . . . λ n−(m−2) [ pn−m(x) pn−(m−1) (x) ] = [ x − Cn−(m−2) −1 λn−(m−2) 0 ] . . . [ x − Cn−1 −1 λn−1 0 ] [ x − Cn −1 λn 0 ] [ pn−1(x) pn(x) ] (5) = m−2 ∏ j=0 [ x − Cn− j −1 λn− j 0 ] [ pn−1(x) pn(x) ] . Suppose [ Am(x) Bm(x) Cm(x) Dm(x) ] = m−2 ∏ j=0 [ x − Cn− j −1 λn− j 0 ] 4Then [ A2(x) B2(x) C2(x) D2(x) ] = [ x − Cn −1 λn 0 ] and [ A3(x) B3(x) C3(x) D3(x) ] = [ x − Cn−1 −1 λn−1 0 ] [ A2(x) B2(x) C2(x) D2(x) ] , from which it follows that A2(x) = x − Cn and A3(x) = (x − Cn−1)A2(x) − C2(x) = (x − Cn−1)A2(x) − λn. In general, [ Am(x) Bm(x) Cm(x) Dm(x) ] = [ x − Cn−(m−2) −1 λn−(m−2) 0 ] [ Am−1(x) Bm−1(x) Cm−1(x) Dm−1(x) ] , from which we obtain Am(x) = (x − Cn−(m−2) )Am−1(x) − Cm−1(x) and Cm(x) = λn−(m−2) Am−1(x), i.e., Am(x) = (x − Cn−(m−2) )Am−1(x) − λn−(m−3) Am−2(x). Since Am is a polynomial of degree m − 1, we introduce the sequence of polynomials {S (n) m }nm=0, such that Am+1(x) = S (n) m (x), and these polynomials satisfy the three-term recurrence equation S (n) m (x) = (x − Cn−(m−1) )S (n) m−1 (x) − λn−(m−2) S (n) m−2 (x), with S (n)0 (x) = 1, S (n) −1 (x) = 0, S (n)1 (x) = x − Cn.In the same way, we see that Bm(x) is a polynomial of degree m − 2, with B1(x) = 0 and B2(x) = −1, satisfying the equation Bm(x) = (x − Cn−(m−2) )Bm−1(x) − λn−(m−3) Bm−2(x), m ≥ 3, and we can set Bm(x) = −S (n−1) m−2 (x). Equation ( 5) thus becomes λnλn−1 . . . λ n−(m−2) [ pn−m(x) pn−(m−1) (x) ] = [ S (n) m−1 (x) −S (n−1) m−2 (x) λn−(m−2) S (n) m−2 (x) −λn−(m−2) S (n−1) m−3 (x) ] [ pn−1(x) pn(x) ] and the result follows. For completeness, we state the following result, which is proved as part of [ 5, Theorem 4]. Corollary 2. If p n−m(x) and p n(x) do not have any common zeros, the n −1 zeros of S (n) m−1 (x)pn−m(x) interlace with the n zeros of p n(x). The polynomials {S (n) m }nm=0, n ∈ { 0, 1, . . . }, will be called the associated polynomials of the orthogonal sequence {pn}∞ n=0 . These polynomials are completely determined by the coe ffi cients in (1) and, according to Favard’s Theorem [ 8, Theorem 4.4], they are part of an orthogonal sequence. Furthermore, when m = n in ( 4), it follows from Corollary 2 that the n − 1 zeros of S (n) n−1 interlace, in the same way as the n − 1 zeros of pn−1, with the n zeros of pn (cf. [ 8, Theorem 4.1]). 5Remark 3. (i) From ( 4) it is clear that, should p n−m and p n have any common zeros, these will also be zeros of S (n) m−1 and there can be a maximum of m −1 such common zeros (cf. [ 17 , 18 ]). (ii) The polynomials S (n+1) n are the same as the numerator polynomials p (1) n in [ 8, p. 86, Defini-tion 4.1] and the associated polynomials in [ 41 ]. (iii) By expanding and re-writing (3), we obtain a three-term recurrence equation involving the polynomials S (n) m (x), S (n−1) m−1 (x), S (n−2) m−2 (x), m ∈ { 1, 2, . . . , n}, given by S (n) m (x) = (x − Cn)S (n−1) m−1 (x) − λnS (n−2) m−2 (x), (6) from which it is clear that the polynomials S (n) n and p n are the same polynomials. Fix k ∈ N0 and m ∈ { 2, 3, . . . , n − 1} and let {pn}∞ n=0 be a sequence of orthonormal polynomials associated with the weight function w(x) > 0 on the (finite or infinite) interval ( a, b). Then, from Christo ff el’s formula [ 39 , Theorem 2.5], the sequence of polynomials {gn,k}∞ n=0 , orthogonal with respect to ck(x)w(x) > 0 on ( a, b), where ck(x) is a polynomial of degree k, satisfies ck(x)gn−m,k(x) = pn−m(x) pn−m+1(x) . . . pn−m+k(x) pn−m(x1) pn−m+1(x1) . . . pn−m+k(x1) . . . . . . . . . . . . pn−m(xk) pn−m+1(xk) . . . pn−m+k(xk) (7) = k ∑ j=0 U j pn−m+ j(x), where xj, j ∈ { 1, 2, . . . , k} are the zeros of ck and the coe ffi cients U j, j ∈ { 0, 1, . . . , k} are deter-minants. We note that, in the case of a zero xj of multiplicity s > 1, the corresponding rows of the determinant ( 7) are replaced by the derivatives of order 0 , 1, . . . , s − 1 of the polynomials pn−m(x), pn−m+1(x), . . . , pn−m+k(x) at x = xj.Furthermore, using Lemma 1, the ( m − 1) polynomials pn−m+ j, j ∈ { 0, 1, . . . , m − 2}, can be written in terms of the polynomials pn and pn−1, by using λnλn−1 . . . λ n−(m− j−2) pn−(m− j)(x) = S (n) m−(j+1) (x)pn−1(x) − S (n−1) m−(j+2) (x)pn(x). In order to write the rest of the polynomials pn−m+ j, j ∈ { m + 1, m + 2, . . . , k}, in terms of the polynomials pn and pn−1, we use the following iterations of the three-term recurrence equation ( 1) pn+m(x) = S (n+m) m (x)pn(x) − λn+1S (n+m) m−1 (x)pn−1(x), where the polynomials S (n+m) m (x), m ∈ { 1, 2, 3, . . . }, satisfy ( 6), with n replaced by n + m. Or, when we replace m by −m + j, we get pn−m+ j(x) = S (n−m+ j) −m+j (x)pn(x) − λn+1S (n−m+ j) −m+j−1 (x)pn−1(x)6and ( 7) can be written as: ck(x)gn−m,k(x) = ( m−2∑ j=0 U j S (n−1) m−(j+2) (x) λnλn−1 . . . λ n−(m− j−2) Um + k ∑ j=m+1 U j S (n−m+ j) j−m (x) ) pn(x) + ( m−2∑ j=0 U j S (n) m−(j+1) (x) λnλn−1 . . . λ n−(m− j−2) Um−1 − λn+1 k ∑ j=m+1 U j S (n−m+ j) j−m−1 (x) ) pn−1(x) = R(x)pn(x) + G(x)pn−1(x)Christo ff el’s formula thus provides us with a mixed three-term recurrence equation, involving polynomials gn−m,k(x), pn and pn−1 and the coe ffi cients in the mixed three-term recurrence equation are determined by the coe ffi cients in ( 1) and the determinants U j, j ∈ { 0, 1, . . . , k}. It is clear that the coe ffi cient G(x) of pn−1(x) is of degree max {m − 1, k − m − 1} and the coe ffi cient R(x) of pn(x)is of degree max {m − 2, k − m}.In order for the zeros of G(x)gn−m,k(x) to fully interlace with the zeros of pn(x), the polynomial G(x)gn−m,k(x) must be of exact degree n − 1, which means that G(x) must be of exact degree m − 1, i.e., max {m − 1, k − m − 1} = m − 1, i.e., k − m − 1 ≤ m − 1, from which we can deduce k ≤ 2m.We have thus proved the following result. Theorem 4. Let {pn}∞ n=0 be a sequence of polynomials orthogonal on the (finite or infinite) interval (a, b) with respect to the weight function w (x) > 0. Let k ∈ N0 be fixed. Suppose {gn,k}∞ n=0 is a sequence of polynomials orthogonal with respect to c k(x)w(x) > 0 on (a, b), where c k(x) is a polynomial of degree k. Then, for m ∈ { 2, 3, . . . , n − 1} and k ∈ { 0, 1, . . . , 2m}, there exist on (a, b) a polynomial G m(x) of degree m, and a polynomial a k−m that is of degree m − 2 when k − m ∈ {− m, −m + 1, . . . , m − 2} and of degree k − m whenever k − m ∈ { m − 1, m, m + 1, . . . }, such that Anck(x)gn−m,k(x) = ak−m(x)pn(x) − Gm−1(x)pn−1(x), n ∈ { 3, 4, . . . }. (8) Furthermore, (i) if g n−m,k and p n are co-prime, the n − 1 real, simple zeros of G m−1(x)gn−m,k interlace with the zeros of p n, the smallest zero of G m−1 is an upper bound for the smallest zero of p n, and the largest zero of G m−1 is a lower bound for the largest zero of p n;(ii) if g n−m,k and p n are not co-prime and have r common zeros counting multiplicity, then a) r ≤ min {m, n − m − 1};b) these r common zeros are simple zeros of G m−1;c) no two successive zeros of p n, nor its largest or smallest zero can be a zero of G m−1;d) the n − 2r − 1 zeros of G m−1gn−m,k(x), none of which is a zero of p n, together with the r common zeros of g n−m,k and p n, interlace with the n − r non-common zeros of p n;e) the smallest zero of G m−1 is an upper bound for the smallest zero of p n, and the largest zero of G m−1 is a lower bound for the largest zero of p n. For the proofs of (i) and (ii) above, we refer the reader to [ 17 , Theorem 2.1, Corollary 2.2]. 7Remark 5. In [ 23 ] the authors use equations like (8), with m = 2, to obtain inner bounds for the Meixner polynomials. In the proof of [ 23 , Theorem 2.1], the assumption is made that if q is any polynomial, such that ∫ ba q(x)pn(x)w(x)dx = 0, n ∈ { 3, 4, . . . }, then deg (q(x)) ≤ n − 1, which is not always true. In the following sections we will assume that the polynomials gn−m,k and pn involved in ( 8) do not have any common zeros, in order to obtain inner bounds for the extreme zeros of pn. Due to their complexity, we use computer algebra to find equations like ( 8) and, following the approach in [ 7, 29 ], we write a procedure that applies the Gosper [ 20 , 29 ] and Zeilberger [ 28 , 29 , 37 , 42 ]algorithms. Gosper’s algorithm deals with finding, for j an integer, an anti-di ff erence s j, for given a j, i.e., a sequence s j for which a j = ∆ s j = s j+1 − s j, in the particular case that s j is a hypergeometric term, i.e., s j+1 s j ∈ Q( j). Given the hypergeometric term F(n, j, α ) with respect to variables n, j and α, Zeilberger’s algo-rithm provides a recurrence equation (with polynomial coe ffi cients) for sn = ∞ ∑ j=−∞ F(n, j, α ). Suppose we have the particular case where a j = F(n − m, j, α + k) + J ∑ i=0 σi(n)F(n − i, j, α ), with undetermined variables σi(n), i ∈ { 0, 1, . . . , J}, and sn := pn(x; α) = ∞ ∑ j=−∞ F(n, j, α ). We apply Gosper’s algorithm to a j and, if successful, an anti-di ff erence g(n, j) of a j is found, i.e., a j = g(n, j + 1) − g(n, j) and at the same time σi(n), i ∈ { 0, 1, . . . , J}. By summation, we obtain ∞ ∑ j=−∞ a j = ∞ ∑ j=−∞ F(n − m, j, α + k) + J ∑ i=0 σi(n)F(n − i, j, α )  = pn−m(x; α + k) + J ∑ i=0 σi(n)pn−i(x; α) = ∞ ∑ j=−∞ (g(n, j + 1) − g(n, j)) = 0, since the last sum is telescoping. It follows that pn(x; α) satisfies a recurrence equation of type (8). We refer the reader to [ 29 , Chapters 5–7] and references there-in for more details about the 8algorithms of Gosper and Zeilberger implemented in the Maple hsum package. The q-analogues of Gosper’s and Zeilberger’s algorithms are implemented in the Maple qsum package [ 29 ]. We apply an adaption of the sumdiffeq [29 , p. 210] and qsumdiffeq [29 , p. 219] procedure of the hsum and qsum packages, respectively, in order to obtain two procedures to derive recurrence equations of type ( 8) for the polynomial systems considered in the sequel. The hsum and the qsum packages can be downloaded from , as well as the two Maple codes used to obtain our mixed three-term recurrence equations. The first program called Mixedrec (F, k, S (n), s0, a, s1, b, s2) finds a recurrence equation of the form S (n − s0, a + s1, b + s2) = J ∑ j=0 σ jS (n − j, a, b), J ∈ { 1, 2, . . . }, where S (n, a, b) = ∞ ∑ k=−∞ F, F is a function of k, n, a and b, and s0, s1, s2 are integers. The second one denoted by qMixedrec (F, k, S (n), s0, a, s1, b, s2) is the q-analogue of the first one and finds a recurrence equation of the form S (n − s0, aq s1 , bq s2 ) = J ∑ j=0 σ jS (n − j, a, b), J ∈ { 1, 2, . . . }. We use this algebraic method to obtain mixed three-term recurrence equations involving poly-nomials pn(x; α, β ) and pn−1(x; α, β ), belonging to the same sequence that is orthogonal on an interval ( a, b) with respect to a measure w(x; α, β ), and a polynomial from a related sequence, ob-tained by integer shifts of the parameters α and β, namely pn−m(x; α + s, β + t), m ∈ { 2, 3, . . . , n − 1},which is orthogonal with respect to w(x; α + s, β + t) = cs+t(x; α, β )w(x; α, β ) > 0on ( a, b), where ck(x; α, β ) is a polynomial of degree k in x. If the sequence is q-orthogonal with respect to the weight w(x; α, β ), the equations involve the polynomials pn(x; α, β ) and pn−1(x; α, β ), and pn−m(x; αqs, β qt), m ∈ { 2, 3, . . . , n − 1} and the latter polynomial is orthogonal with respect to w(x; αqs, β qt) = cs+t(x; α, β )w(x; α, β ) > 0on ( a, b). From Theorem 4 we know that for positive integers s and t such that s+t ∈ { 0, 1, . . . , 2m},the polynomial coe ffi cient of pn−1(x; α, β ) in this mixed recurrence equation, is of exact degree m − 1 and will be denoted by G(α+s, β +t) m−1 (x). Bounds for the extreme zeros of the classical orthogonal polynomials The generalized Laguerre polynomials L(α) n (x) = (α + 1) n n! 1F1 ( −n α + 1 ∣∣∣∣∣∣ x ) , α > −19are orthogonal with respect to w(x; α) = xαe−x on (0 , ∞). Since optimal bounds are obtained (cf. [17 ]) when we shift α optimally, we use mixed three-term recurrence equations involving the polynomials L(α) n (x), L(α) n−1 (x) and L(α+2m) n−m (x) for m = 3 and m = 4, where L(α+2m) n−m (x) is orthogonal on (0 , ∞) with respect to w(x; α + 2m), where w(x; α + 2m) w(x; α) = x2m = c2m(x), to illustrate the quality of our newly found bounds in the classical case. When we take m = 3, the mixed three-term recurrence equation x6 L(α+6) n−3 (x) = −na 3(x)L(α) n (x) + (n + α)G(α+6) 2 (x)L(α) n−1 (x), (9) with a3(x) = (n − 1) ( n − 2) ( x3 + 3 ( α + 3) x2) − (α + 2) 3 (α + 4 n − 3) x + (α + 1) 5 and G(α+6) 2 (x) = (α + 3)(3 n(n + α + 1) + (α + 1) 2)x2 − 2( α + 2) 3(α + 2n + 1) x + (α + 1) 5. is clearly in the form of ( 8) and from Theorem 4 we know that the n − 1 zeros of G(α+6) 2 (x)L(α+6) n−3 (x)interlace with the n zeros of L(α) n (x). The zeros of G(α+6) 2 (x) are B(α+6) 3 = (α + 2)( α + 4)( α + 2n + 1) 3n(n + α + 1) + (α + 1) 2 (10) ± √ (α + 2)( α + 4) ( ( α2 + 6 α + 17 ) (n2 + αn + n) − (α + 2) ( α + 1) 2 ) 3n(n + α + 1) + (α + 1) 2 and they are inner bounds for the extreme zeros of L(α) n (x). By computing the values of these bounds, we find that the smallest value in ( 10 ) is an accurate upper bound for xn,1, however, by substituting m = 4, k = 2m = 8 in ( 8), we obtain x8 L(α+8) n−4 (x) = a4(x)L(α) n (x) + (n + α)G(α+8) 3 (x)L(α) n−1 (x), where a4(x) = (n − 3) 4 (x + 4α + 16) x3 − n (α + 3) 3 (5 n(2 n + α − 4) + α2 − 2 α + 12 )x2 2 n (α + 2) 5 (α + 3 n − 2) x − n (α + 1) 7 and G(α+8) 3 (x) = − (α + 4) (2 n + α + 1) (α2 + 2 α n + 2 n2 + 5 α + 2 n + 6) x3 (11) (α + 3) 3 (3( α + 1) 2 + 10 n (n + α + 1)) x2 − 3 ( α + 2) 5 (2 n + α + 1) x + (α + 1) 7 and the smallest zero of G(α+8) 3 (x) is an even more accurate bound than the bound obtained from (9). In Table 1 we show examples indicating the accuracy of this value, computed numerically, 10 as upper bound for the lowest zero of L(α) n (x). To find a lower bound for the largest zero, we let m = 4 and k = 0 in ( 8), i.e., we don’t consider any parameter shifts, and we obtain the recurrence equation L(α) n−4 (x) = −n (x2 − 2 ( α + 2n − 4) x + α2 + 3 α n + 3 n2 − 6 α − 12 n + 11 ) L(α) n (x) (α − 3 + n)3 G(α)3 (x)L(α) n−1 (x) (α − 3 + n)3 with G(α)3 (x) = −x3 + 3 ( α + 2 n − 3) x2 − (3 α2 + 10 α n + 10 n2 − 15 α − 30 n + 18 ) x (12) (α − 3 + 2 n) (α2 + 2 α n + 2 n2 − 3 α − 6 n + 2) . The largest zero of G(α)3 (x) is a lower bound for the largest zero of L(α) n (x). Here we use a zero of a third degree polynomial, that can easily be computed numerically, to approximate a zero of a polynomial of much higher degree. We show the quality of these bounds in Table 1 and we compare our upper bound for xn,1 with the upper bound BGM = (α + 1) ( α + 2) ( α + 4) (2 n + α + 1) (α + 1) 2 (α + 2) + (5 α + 11) n (n + α + 1) (13) obtained in [ 21 , Equation 2.11]. We also provide the lower bound for the largest zero, as given in [32 ] for values of n ≥ 30. This bound is more precise than the bound obtained from ( 12 ), however, the recurrence equation involving polynomials L(α) n , L(α) n−1 and L(α) n−7 , which is too big to display, provides us with a polynomial G(α)6 (x) and the largest zero of this polynomial, that can be computed numerically, is a more accurate bound than the bound in [ 32 ]. The equation can be found in the accompanying Maple file. Similar results can be obtained for the Jacobi, Gegenbauer, Hermite, Table 1: Examples to show the quality of the inner bounds for the extreme zeros of L(α) n(x) for di ff erent values of n and α. n α xn,1 Bound from ( 11 ) BGM (13 ) Bound from ( 12 ) Bound in [ 32 ] xn,n 10 -0.5 0.06019206315 0.06019206332 0.060269 28.469 n/a 29.025 10 10.0 3.50805 3.51556 4.0168 44.945 n/a 46.365 100 10.0 0.49692 0.49863 0.5746 353.225 387.960 394.294 100 -0.5 0.00615313229 0.00615313230 0.0061611 335.472 367.816 374.006 Bessel, Meixner and Hahn polynomials and the recurrence equations providing the bounds, as well as the bounds, are also available in the Maple file. Remark 6. (i) The Charlier polynomials Cn(x; a) = 2F0 ( −n, −x − ∣∣∣∣∣∣ −1 a ) , a > 0, are orthogonal on (0 , ∞) with respect to w (x; a) = ax x! and when we shift the parameter a by k units, we obtain the polynomial C n(x; a + k), which is orthogonal with respect to w (x; a + k)11 on (0 , ∞) and since w(x;a+k) w(x;a) = (a+k)x ax is not a polynomial of degree k in x, these polynomials do not satisfy the conditions of Theorem 4 and we use Lemma 1 and Corollary 2 to obtain bounds for the extreme zeros of these polynomials. (ii) We also use Lemma 1 in the case of the Krawtchouk polynomials Kn(x; p, N) = 2F1 ( −n, −x −N ∣∣∣∣∣∣ 1 p ) , n ∈ { 0, 1, . . . , N}, N ∈ N, that are orthogonal for 0 < p < 1, since integer shifts of the parameter p will result in the parameter moving outside the interval where orthogonality is guaranteed. Bounds for the extreme zeros of q-orthogonal polynomials obtained by shifting one pa-rameter In this section we illustrate the method by obtaining new bounds for the extreme zeros of the little q-Jacobi and the alternative q-Charlier (or q-Bessel) polynomials. 4.1. The little q-Jacobi polynomials The little q-Jacobi polynomials (cf. [ 27 , Section 14.12]) pn(qx; α, β \|q) = 2φ1 ( q−n, αβ qn+1 αq ∣∣∣∣∣∣ q; qq x ) , 0 < α q < 1, β q < 1are discrete orthogonal with respect to the weight w(x; α, β ) = (βq; q)x (q; q)x (αq)x on the interval (0 , 1) . The polynomial pn(qx; αq4, β \|q) is orthogonal on (0 , 1) with respect to ( qx)4w(x; α, β ). By replacing qx with x, we obtain α (qn − q)( αβ q2 n − 1)( βqn − q)( α β qn+1; q)2(α qn+1; q)2 (α q; q)4 x4 pn−2(x; αq4, β \|q) = a2(x)pn(x; α, β \|q) + q3 n−5 (1 − αq2) G(αq4 , β )(x)pn−1(x; α, β \|q), (14) with a2(x) = α (qn − q)( βqn − q) (( αβ q2 n − 1) q2 n−3 x + (q + 1) (αq2 − 1) q3 n−5) x + (αq; q)3q4 n−6 and G(αq4 , β )(x) is a linear function with zero B(αq4 , β )2 = (α q3 − 1) (α q − 1) qn−1 (α β q2 n+1 + 1)( αq2 + 1) − α qn+1(β + 1)( q + 1) . (15) In Table 2 we show the quality of B(αq4, β )2 as bound and we compare it to bounds given in [ 21 ,Equation 4.3] when n = 10 , 30 and in [ 21 , Equation 4.2] for n = 100 (see Remark 7). We also observe that the value of q is an upper bound for the zero xn,n−1. The accuracy of this upper bound decreases for q in the vicinity of 1. Furthermore, we observe that xn, j+1 xn,j ≈ 1 q for j ∈ { 1, 2, . . . , n − 1} and again this is less accurate when q → 1. 12 Table 2: Examples to show the quality of the inner bounds for the extreme zeros of pn(x; 0 .5, β \|q) for di ff erent values of n, β and q. n β q xn,1 B(αq4, β )2 in ( 15 ) Bound in [ 21 ] xn,n−1 xn,n 10 1 0.6 0.005216 0.005359 0.021776 0.600000 1.000000 30 -10 0.6 1.8642 · 10 −7 1.9497 · 10 −7 7.9382 · 10 −7 0.600000 1.000000 100 -10 0.9 0.0000059 0.0000073 0.0000131 0.8999999 1.000000 Remark 7. (i) The little q-Laguerre (or Wall) polynomials are obtained from the little q-Jacobi polynomials, by letting β = 0 and the bound B (αq4, β )2 in (15 ), with β = 0, can be used as an upper bound for the smallest zero of the little q-Laguerre polynomial. (ii) In [ 21 ], Gupta and Muldoon provide bounds for the smallest zeros of the little q-Jacobi polynomial p n((1 − q)x; α, β \|q) and the q-Laguerre polynomial L (α) n ((1 − q)x; q), α > −1. Using a suitable comparison we observe that, for both these systems, the upper bounds for the smallest zeros obtained by our method are more accurate than the upper bounds obtained in [ 21 ]. 4.2. The alternative q-Charlier or q-Bessel polynomials The alternative q-Charlier polynomials (cf. [ 27 , Section 14.22]) yn(qx; α, q) = 2φ1 ( q−n, −αqn 0 ∣∣∣∣∣∣ q; q q x ) , α > 0, are orthogonal with respect to the weight function w(x; α) = αx (q; q)x q(x+12 ) on (0 , 1) and yn(qx; αq4, q)are orthogonal on (0 , 1) with respect to ( qx)4w(x; α). By replacing qx with x, these polynomials satisfy (−α qn, q)2 (qn − q) (αq2 n + q) (αq4)x4yn−2(x; αq4, q) = q2 n (α (αq3 n+1 − αq2 n+2 + qn+2 − q3) x2 − αqn (q2 − qn+1 − qn + q) x + qn) yn(x; α, q) − q2 n (( αq2 n+1 − αqn+2 − αqn+1 − q) x + qn) yn−1(x; α, q). From the coe ffi cient of yn−1, we obtain B(αq4)2 = qn−1 1 − αqn(qn + q + 1) , (16) which is a relatively sharp upper bound for the lowest zero of yn, as shown in Table 3. In the recurrence equation involving yn−3(x; αq6, q), yn−1(x; α, q) and yn(x; α, q), the coe ffi cient of yn−1 is the quadratic polynomial G(αq6 )2 (x) = q3 (1 + (q4 n + (q2 n+2 − q3 n)( q2 + q + 1) ) α2 + (qn + qn+2 − q2 n + qn+1) α) x2 qn+1 (q + 1) (αq2 n − αqn+2 − α qn − 1) x + q2 n 13 and the lowest zero of this polynomial is B(αq6)3 = −b − √b2 − 4ac 2a , (17) with c = G2(0), b = G′ 2 (0) and a = G′′ 2(0) 2 , which is a more accurate upper bound for xn,1, as shown in Table 3. In Table 3 we also illustrate the quality of the bound Bn = −qn+1 (q2 nα − α qn+1 − α qn − q2) (q2 nα + q) ( q2 nα + q3) , (18) obtained from the three-term recurrence equation satisfied by these polynomials, which increases for large values of n. Table 3: Examples to show the quality of the upper bound for the lowest zero of yn(x;α, q) for di ff erent values of n,α and q. n q α xn,1 B(αq6)3 in ( 17 ) B(αq4)2 in ( 16 ) Bn in ( 18 ) 10 0.55 0.5 0.0045925 0.0045925 0.0045964 0.004635 10 0.99 10 0.06207968 0.06796546 0.08444314 0.115245 10 0.45 100 0.00071318 0.00071318 0.00072109 0.000941 70 0.45 10 1.179406 · 10 −24 1.179406 · 10 −24 1.179406 · 10 −24 1.179406 · 10 −24 70 0.8 100 2.05671 · 10 −7 2.05671 · 10 −7 2.05682 · 10 −7 2.05783 · 10 −7 The value of q is an upper bound for the zero xn,n−1 of yn(x; α, q), as shown in Table 4 below and the accuracy of this upper bound decreases for q in the vicinity of 1. Furthermore, xn, j+1 xn,j ≈ 1 q for j ∈ { 1, 2, . . . , n − 1} and again this is less accurate for the values of q in the vicinity of 1. Table 4: Examples to show the quality of qas upper bound for xn,n−1for di ff erent values of n,αand q. n q α xn,n−1 xn,n 10 0.55 0.50 0.54999999 1.00000000 10 0.99 0.10 0.9735504 0.9935188 10 0.10 1000 0.099999999 1.00000000 70 0.70 1000 0.700000000 0.99999999 70 0.70 0.10 0.700000000 0.99999999 Remark 8. Finding inner bounds by using Theorem 4 is not possible for the following polynomial systems, since (i) the quantum q-Krawtchouk polynomials K (qtm ) n (q−x; p, N; q) are orthogonal on [0 , N] for p > q−N and shifting p causes a change in the interval of orthogonality; (ii) the Al-Salam Carlitz I polynomials U (α) n (x; q) are orthogonal for α < 0 on (α, 1) and shifting α to αqk will result in a change in the interval of orthogonality; 14 (iii) the Al-Salam Carlitz II polynomials V (α) n (q−x; q) are orthogonal for 0 < α q < 1 on (0 , 1) with respect to w(x; α) = qx2 αx (q; q)k(αq; q)k and w(x; αq−k) w(x; α) = ( 1 α q−x; q) k ( 1 α ; q) k = ck(q−x; α) is a polynomial of degree k in the variable q −x. However, when we substitute α with αq−k, k ∈{1, 2, . . . }, the condition 0 < α q < 1 is not satisfied. We find bounds for the zeros of these systems by using Corollary 2. Bounds for the extreme zeros of q-orthogonal polynomials obtained without any parame-ter shifts In this section we obtain inner bounds for the extreme zeros of the Stieltjes-Wigert and dis-crete q-Hermite II polynomials, where no parameter shifts are involved, in order to illustrate the accuracy of the bounds obtained by Corollary 2, for m = 3 and /or m = 4. 5.1. The Stieltjes-Wigert polynomials The Stieltjes-Wigert polynomials (cf. [ 27 , Section 14.27]) S n(x; q) = 1 (q; q)n 1 φ1 ( q−n 0 ∣∣∣∣∣∣ q; −qn+1 x ) are orthogonal on (0 , ∞). When we let m = 3, k = 0 in ( 8), we obtain the equation q6S n−3 (x; q) = −q (qn − 1) (xq 2 n + qn+2 − q3 − q4) S n (x; q) (19) (q4n x2 + q2n+1 (q + 1) (qn − q2 − 1) x + q3 (q2 n + (q − qn)( q2 + q + 1) )) S n−1 (x; q) . The coe ffi cient of S n−1 provides us with the two inner bounds B3 = (q2 − qn + 1)( q + 1) ± √q6 − 2 qn+4 + 2 q5 + q2n+2 − q4 − 2 q2n+1 + q2n − q2 − 2 qn + 2 q + 1 2q2n−1 . In Table 5 we show the quality of the largest of these two bounds, which is a good lower bound for the largest zero of S n(x; q). A more accurate lower bound is obtained from q12 S n−4(α, q) = q (qn − 1) ( q4 n x2 + (q3 n+2 − q2 n+5 − q2 n+3)( q + 1) x q2 n+5 + (q8 − qn+6)( q2 + q + 1)) S n(α, q) − A3(x)S n−1(α, q), with A3(x) = q6 n x3 + (q5 n+1 − q4 n+4 − q4 n+1) ( q2 + q + 1) x2 (20) (( q2 n+8 + q2 n+6 + q2 n+4 + q4 n+3 − q3 n+6 − q3 n+5 − q3 n+4) ( q2 + q + 1) − q3 n+3(q + 1) ) x − q9(q3 + q2 + q + 1) − q2 n+6(q3 + q2 + q + 1) + q3 n+6 + qn+7(q2 + 1)( q2 + q + 1) . The values of the zeros of A3(x) can be found numerically and we also show the value of the largest zero of A3(x) in Table 5.15 Table 5: Examples to show the quality of the inner bound for the extreme zeros of S n(x; q) for di ff erent values of n and q. n q B3 from ( 19 ) Lowest zero of A3 (20 ) xn,n 10 0.5 8.3925988 · 10 5 8.3946795 · 10 5 8.3946799 · 10 5 10 0.9 15.3689887 16.0730951 16.1508699 70 0.5 1.116350296 · 10 42 1.1166280 · 10 42 1.1166281 · 10 42 70 0.9 5.9402911 · 10 6 6.2658907 · 10 6 6.3132591 · 10 6 5.2. The discrete q-Hermite II polynomials The discrete q-Hermite II polynomials (cf. [ 27 , Section 14.29]) ˜hn(x; q) = i−nq−(n 2 )2φ0 ( q−n, ix − ∣∣∣∣∣∣ q; −qn ) , are orthogonal on the real line. The zeros of these polynomials are symmetric about the origin with a simple zero at the origin when n is odd. We get the recurrence equation (q−n+1; q) 4 q14 ˜hn−5(x; q) = − (qn+3 + qn+4 − q5 − q7 + q2 n x2) q2 n x˜hn(x; q) + G4(x)˜ hn−1(x; q)with G4(x) = q4 n x4 + (q2 + qn − q − q3) ( q2 + q + 1) x2q2 n+2 + q6 (qn − q) (qn − q3) . The largest zero of G4(x) is B5 = (21) ( (q + q3 − q2 − qn)( q2 + q + 1) + √(q2 + qn − q − q3)2(q2 + q + 1) 2 − 4( qn − q3)( qn − q)q2 2q2n−2 )1 2 . We show the quality of this bound in Table 6. Table 6: Examples to show the quality of the inner bound for the extreme zeros of ˜ hn(x;q) for di ff erent values of n and q. n q B5 in ( 21 ) xn,n 10 0.5 406 .3811 406 .4079 10 0.98 0.72968 0.76655 100 0.5 5.0368 · 10 29 5.0372 · 10 29 100 0.98 10 .7735 12 .1420 Acknowledgments This work has been supported by a TWAS / DFG fellowship for A. Jooste and the Institute of Mathematics of the University of Kassel (Germany) for D.D. Tcheutia. All these institutions receive our sincere thanks. 16 References References I. Area, D.K. Dimitrov, E. Godoy, A. 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Jordaan, Stieltjes interlacing of zeros of Jacobi polynomials from di ff erent sequences, Electron. Trans. Numer. Anal. 38 (2011), 317-326. K. Driver and K. Jordaan. Stieltjes interlacing of zeros of Laguerre polynomials from di ff erent sequences, Indag. Math. 21 (2011), 204-211. K. Driver and K. Jordaan, Bounds for extreme zeros of some classical orthogonal polynomials, J. Approx. Theory 164 (2012), 1200–1204. PC Gibson, Common zeros of two polynomials in an orthogonal sequence, J. Approx. Theory 105 (2000), 129-132. P. Gochhayat, K. Jordaan, K. Raghavendar and A. Swaminathan, Interlacing properties and bounds for zeros of 2 φ1 hypergeometric and little q-Jacobi polynomials, Ramanujan J. 40 (2016), 45–62. R.W. Gosper Jr, Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978), 40–42. D.E. Gupta and M.E. Muldoon, Inequalities for the smallest zeros of Laguerre polynomials and their q-analogues, J. Inequal. Pure Appl. Math. 8 (2007). M.E.H. 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7696	https://www.youtube.com/watch?v=DerrI1FstO4	Reflection Across the X and Y Axis mrmaisonet 67500 subscribers 5466 likes Description 456735 views Posted: 18 Feb 2020 Review how to reflect objects across the x and y axis on the coordinate plane by following simple rules. This lesson is given by Taina Maisonet. Download over 1,000 math resources at my website, including transformations resources. Visit my Teachers Pay Teachers store at Please subscribe to my YouTube channel! You can also follow me at You can also follow me at 399 comments Transcript: Introduction welcome to maisonette Matt this is ty Anna maisonette and what we're going to do in this tutorial is practice reflecting across the X and the y axis on the coordinate plane when reflecting anything across the X or Y axis there is a simple rule that we can follow whenever reflecting across the x axis the X values of the original coordinates stay the same and the Y values become opposite and when reflecting across the y axis the original Y values of your coordinates stay the same and your x values become opposite alright let's go ahead and demonstrate Reflection Across the X Axis the rules of reflection with two examples one reflection across the x-axis and one across the y-axis first we're going to reflect the triangle ABC across the x-axis I'd like to start by recording where our original points are located point a is located at negative 8 positive to point B is located at negative 3 positive 2 and Point C is located at negative 3 positive 8 because we are reflecting our triangle across the x-axis we know that our X values are going to stay the same and our Y values will become opposite so we're going to write every x value the same and every Y value opposite of its original now that we have our new coordinates let's go ahead and plot them point a is located at negative 8 negative 2 point B is located at negative 3 negative 2 and Point C is located at negative 3 negative 8 now that we have plotted our points we can now see where our object will be reflected to notice our x value stayed exactly the same and our Y values our opposite values alright let's go ahead and reflect Reflection Across the Y Axis triangle ABC across the y-axis a reflection across the y-axis requires making all y-values the same and the X values opposite so we're going to keep our Y values the same and turn all the X values which are negative values and the positive values now that we have our new points let's go ahead and plot our points point a is going to be reflected to positive 8 positive to point B is going to be reflected to positive 3 positive 2 and Point C will be reflected to positive 3 positive 8 all right now we have the location of our new points we can reflect our triangle across the y axis we can see that the Y values of the reflected objects are exactly the same but the X values are opposite all right thank you for checking out this video please don't forget to subscribe and enable notifications or follow me at any of my social media links shown on the screen if you need further assistance with other types of transformations such as translations rotations or dilations we have plenty of more examples that you can find by searching this channel until next time this is taina maisonette for mason at math [Music]
7697	https://dspace.mit.edu/bitstream/handle/1721.1/153312/table.pdf?sequence=8	TABLE OF KISSING NUMBER BOUNDS HENRY COHN The kissing problem asks how many spheres can be arranged tangent to a given sphere, if they all have the same size and their interiors cannot overlap. The maximum such number in n dimensions is called the n-dimensional kissing number. Equivalently, we can ask how many points can be arranged on the surface of a sphere such that no two distinct points form an angle of less than 60◦with the center of the sphere. See for more information about the kissing problem. Table 1 shows the best upper and lower bounds known for the kissing numbers in dimensions up to 48 as well as 72. It was originally based on a table of lower bounds that was part of the Nebe-Sloane Catalogue of Lattices, and before that Tables I.2(a) and I.2(b) in . The columns of the table list the following information: Dimension Number of dimensions. Lower bound The best construction currently known. Upper bound The best upper bound currently known. Ratio Ratio of the upper and lower bounds, rounded up. References References for the construction and the upper bound. There are also three supplementary files: kissingbounds.txt Plain text file containing the lower and upper bounds from this table. dimensions1-24.txt Explicit coordinates for kissing configurations in up to 24 dimensions, including alternate configurations constructed in [3, 5, 14, 15, 26]. dimensions32-47.txt Details of how configurations in dimensions 32 through 47 can be constructed using the approach of and constant weight codes from . In the file dimensions1-24.txt, the inner product coefficients are a sequence c1, . . . , cn that defines the inner product of two vectors x and y via ⟨x, y⟩= Pn i=1 cixiyi. In many cases this avoids the need for irrational coordinates, since it amounts to rescaling the i-th coordinate by a factor of √ci with the usual inner product. However, there are cases in which irrational coordinates are still needed, such as 13 dimensions. Note also that the listed vectors may not all have the same norm; again, this approach minimizes the need for irrational numbers. References E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, J. Austral. Math. Soc. 1 (1959), 47–63, doi:10.1017/S1446788700025064. A. E. Brouwer, Bounds for binary constant weight codes, H. Cohn, Y. Jiao, A. Kumar, and S. Torquato, Rigidity of spherical codes, Geom. Topol. 15 (2011), 2235–2273, doi:10.2140/gt.2011.15.2235, arXiv:1102.5060. H. Cohn and A. Li, Improved kissing numbers in seventeen through twenty-one dimensions, preprint, 2024, arXiv:2411.04916. H. Cohn and I. Rajagopal, Variations on five-dimensional sphere packings, preprint, 2024, arXiv:2412.00937. J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, third edition, Grundlehren der Mathematischen Wissenschaften 290, Springer-Verlag, New York, 1999, doi:10.1007/978-1-4757-6568-7. Y. Edel, E. M. Rains, and N. J. A. Sloane, On kissing numbers in dimensions 32 to 128, Electron. J. Combin. 5 (1998), Research Paper 22, 5 pp., doi:10.37236/1360, arXiv:math/0207291. M. Ganzhinov, Highly symmetric lines, preprint, 2022, arXiv:2207.08266. K. Kallal, T. Kan, and E. Wang, Improved lower bounds for kissing numbers in dimensions 25 through 31, SIAM J. Discrete Math. 31 (2017), 1895–1908, doi:10.1137/16M1095810, arXiv:1608.07270. A. Korkine and G. Zolotareff, Sur les formes quadratiques, Math. Ann. 6 (1873), 366–389, doi:10.1007/BF01442795. D. de Laat and N. Leijenhorst, Solving clustered low-rank semidefinite programs arising from polynomial optimization, preprint, 2022, arXiv:2202.12077. D. de Laat, N. Leijenhorst, and W. H. H. de Muinck Keizer, Optimality and uniqueness of the D4 root system, preprint, 2024, arXiv:2404.18794. J. Leech, Notes on sphere packings, Canadian J. Math. 19 (1967), 251–267, doi:10.4153/CJM-1967-017-0. J. Leech, Five dimensional non-lattice sphere packings, Canad. Math. Bull. 10 (1967), 387–393, doi:10.4153/CMB-1967-037-5. J. Leech, Six and seven dimensional non-lattice sphere packings, Canad. Math. Bull. 12 (1969), 151–155, doi:10.4153/CMB-1969-014-1. Date: Last updated August 16, 2025. 1 2 HENRY COHN Table 1. Kissing number bounds. Dimension Lower bound Upper bound Ratio References 1 2 1 2 6 1 3 12 1 4 24 1 [24, 20] 5 40 44 1.100 [10, 19] 6 72 77 1.070 [10, 12] 7 126 134 1.064 [10, 19] 8 240 1 [10, 17, 23] 9 306 363 1.187 [16, 18] 10 510 553 1.085 [8, 18] 11 593 868 1.464 [22, 11] 12 840 1355 1.614 [16, 11] 13 1154 2064 1.789 [27, 11] 14 1932 3174 1.643 [8, 11] 15 2564 4853 1.893 [16, 11] 16 4320 7320 1.695 [1, 11] 17 5730 10978 1.916 [4, 11] 18 7654 16406 2.144 [4, 11] 19 11692 24417 2.089 [4, 11] 20 19448 36195 1.862 [4, 11] 21 29768 53524 1.799 [4, 11] 22 49896 80810 1.620 [13, 11] 23 93150 122351 1.314 [13, 11] 24 196560 1 [13, 17, 23] 25 197048 265006 1.345 [9, 11] 26 198512 367775 1.853 [9, 11] 27 199976 522212 2.612 [9, 11] 28 204368 752292 3.682 [9, 11] 29 208272 1075991 5.167 [9, 11] 30 219984 1537707 6.991 [9, 11] 31 232874 2213487 9.506 [9, 11] 32 345408 3162316 9.156 [2, 11] 33 360640 4494570 12.47 [2, 11] 34 380868 6422593 16.87 [2, 11] 35 409548 9162403 22.38 [2, 11] 36 484568 13017098 26.87 [2, 11] 37 494312 18498316 37.43 [2, 11] 38 566652 26496684 46.77 [2, 11] 39 755988 37826766 50.04 [2, 11] 40 1064368 53589200 50.35 [2, 11] 41 1170384 76287040 65.19 [2, 11] 42 1250676 108404055 86.68 [2, 11] 43 1745692 153813582 88.12 [2, 11] 44 2948552 220788272 74.89 [7, 11] 45 3047160 316735249 104.0 [2, 11] 46 5318060 441900184 83.10 [2, 11] 47 9741412 621658419 63.82 [2, 11] 48 52416000 867897072 16.56 [16, 11] 72 6218175600 2545617287927 409.4 [21, 11] TABLE OF KISSING NUMBER BOUNDS 3 J. Leech and N. J. A. Sloane, Sphere packings and error-correcting codes, Canadian J. Math. 23 (1971), 718–745, doi:10.4153/CJM-1971-081-3. V. I. Levenˇ ste˘ ın, On bounds for packings in n-dimensional Euclidean space (Russian), Dokl. Akad. Nauk SSSR 245 (1979), 1299–1303; English translation in Soviet Math. Dokl. 20 (1979), 417–421. F. C. Machado and F. M. de Oliveira Filho, Improving the semidefinite programming bound for the kissing number by exploiting polynomial symmetry, Exp. Math. 27 (2018), 362–369, doi:10.1080/10586458.2017.1286273, arXiv:1609.05167. H. D. Mittelmann and F. Vallentin, High-accuracy semidefinite programming bounds for kissing numbers, Experiment. Math. 19 (2010), 175–179, doi:10.1080/10586458.2010.10129070, arXiv:0902.1105. O. R. Musin, The kissing number in four dimensions, Ann. of Math. (2) 168 (2008), 1–32, doi:10.4007/annals.2008.168.1, arXiv:math/0309430. G. Nebe, An even unimodular 72-dimensional lattice of minimum 8, J. Reine Angew. Math. 673 (2012), 237–247, doi:10.1515/crelle.2011.175, arXiv:1008.2862. A. Novikov, N. Vu, M. Eisenberger, E. Dupont, P.-S. Huang, A. Z. Wagner, S. Shirobokov, B. Kozlovskii, F. J. R. Ruiz, A. Mehrabian, M. P. Kumar, A. See, S. Chaudhuri, G. Holland, A. Davies, S. Nowozin, P. Kohli, and M. Balog, AlphaEvolve: A coding agent for scientific and algorithmic discovery, Google DeepMind white paper, 2025, DeepMind.com/Blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/AlphaEvolve.pdf. A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. Combin. Theory Ser. A 26 (1979), 210–214, doi:10.1016/0097-3165(79)90074-8. L. Schl¨ afli, Theorie der vielfachen Kontinuit¨ at , Denkschriften der Schweizerischen naturforschenden Gesellschaft, vol. 38, J. H. Graf, Bern, 1901. K. Sch¨ utte and B. L. van der Waerden, Das Problem der dreizehn Kugeln, Math. Ann. 125 (1953), 325–334, doi:10.1007/BF01343127. F. Sz¨ oll˝ osi, A note on five-dimensional kissing arrangements, Math. Res. Lett. 30 (2023), no. 5, 1609–1615, doi:10.4310/MRL.2023.v30.n5.a13, arXiv:2301.08272. V. A. Zinoviev and T. Ericson, New lower bounds for contact numbers in small dimensions (Russian), Problemy Peredachi Informatsii 35 (1999), 3–11; English translation in Problems Inform. Transmission 35 (1999), 287–294.
7698	https://en.wikipedia.org/wiki/List_of_fallacies	Jump to content Search Contents (Top) 1 Formal fallacies 1.1 Propositional fallacies 1.2 Quantification fallacies 1.3 Formal syllogistic fallacies 2 Informal fallacies 2.1 Improper premise 2.2 Faulty generalizations 2.3 Questionable cause 2.3.1 Statistical fallacies 2.4 Relevance fallacies 2.4.1 Red herring fallacies 3 See also 4 References 4.1 Citations 4.2 Sources 5 Further reading 6 External links List of fallacies العربية বাংলা Български الدارجة Español فارسی Հայերեն हिन्दी 日本語 پښتو Português Romnă 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 A fallacy is the use of invalid or otherwise faulty reasoning in the construction of an argument. All forms of human communication can contain fallacies. Because of their variety, fallacies are challenging to classify. They can be classified by their structure (formal fallacies) or content (informal fallacies). Informal fallacies, the larger group, may then be subdivided into categories such as improper presumption, faulty generalization, error in assigning causation, and relevance, among others. The use of fallacies is common when the speaker's goal of achieving common agreement is more important to them than utilizing sound reasoning. When fallacies are used, the premise should be recognized as not well-grounded, the conclusion as unproven (but not necessarily false), and the argument as unsound. Formal fallacies [edit] Main article: Formal fallacy A formal fallacy is an error in the argument's form. All formal fallacies are types of non sequitur. Appeal to probability – taking something for granted because it would probably be the case (or might possibly be the case). Argument from fallacy (also known as the fallacy fallacy) – the assumption that, if a particular argument for a "conclusion" is fallacious, then the conclusion by itself is false. Base rate fallacy – making a probability judgment based on conditional probabilities, without taking into account the effect of prior probabilities. Conjunction fallacy – the assumption that an outcome simultaneously satisfying multiple conditions is more probable than an outcome satisfying a single one of them. Non sequitur fallacy – where the conclusion does not logically follow from the premise. Masked-man fallacy (illicit substitution of identicals) – the substitution of identical designators in a true statement can lead to a false one. Propositional fallacies [edit] A propositional fallacy is an error that concerns compound propositions. For a compound proposition to be true, the truth values of its constituent parts must satisfy the relevant logical connectives that occur in it (most commonly: [and], [or], [not], [only if], [if and only if]). The following fallacies involve relations whose truth values are not guaranteed and therefore not guaranteed to yield true conclusions. Types of propositional fallacies: Affirming a disjunct – concluding that one disjunct of a logical disjunction must be false because the other disjunct is true; A or B; A, therefore not B. Affirming the consequent – the antecedent in an indicative conditional is claimed to be true because the consequent is true; if A, then B; B, therefore A. Denying the antecedent – the consequent in an indicative conditional is claimed to be false because the antecedent is false; if A, then B; not A, therefore not B. Quantification fallacies [edit] A quantification fallacy is an error in logic where the quantifiers of the premises are in contradiction to the quantifier of the conclusion. Types of quantification fallacies: Existential fallacy – an argument that has a universal premise and a particular conclusion. Formal syllogistic fallacies [edit] Syllogistic fallacies – logical fallacies that occur in syllogisms. Affirmative conclusion from a negative premise (illicit negative) – a categorical syllogism has a positive conclusion, but at least one negative premise. Fallacy of exclusive premises – a categorical syllogism that is invalid because both of its premises are negative. Fallacy of four terms (quaternio terminorum) – a categorical syllogism that has four terms. Illicit major – a categorical syllogism that is invalid because its major term is not distributed in the major premise but distributed in the conclusion. Illicit minor – a categorical syllogism that is invalid because its minor term is not distributed in the minor premise but distributed in the conclusion. Negative conclusion from affirmative premises (illicit affirmative) – a categorical syllogism has a negative conclusion but affirmative premises. Fallacy of the undistributed middle – the middle term in a categorical syllogism is not distributed. Modal fallacy – confusing necessity with sufficiency. A condition X is necessary for Y if X is required for even the possibility of Y. X does not bring about Y by itself, but if there is no X, there will be no Y. For example, oxygen is necessary for fire. But one cannot assume that everywhere there is oxygen, there is fire. A condition X is sufficient for Y if X, by itself, is enough to bring about Y. For example, riding the bus is a sufficient mode of transportation to get to work. But there are other modes of transportation – car, taxi, bicycle, walking – that can be used. Modal scope fallacy – a degree of unwarranted necessity is placed in the conclusion. Informal fallacies [edit] Main article: Informal fallacy Informal fallacies – arguments that are logically unsound for lack of well-grounded premises. Argument from incredulity – when someone can't imagine something to be true, and therefore deems it false, or conversely, holds that it must be true because they can't see how it could be false. Argument to moderation (false compromise, middle ground, fallacy of the mean, argumentum ad temperantiam) – assuming that a compromise between two positions is always correct. Continuum fallacy (fallacy of the beard, line-drawing fallacy, sorites fallacy, fallacy of the heap, bald man fallacy, decision-point fallacy) – improperly rejecting a claim for being imprecise. Correlative-based fallacies Suppressed correlative – a correlative is redefined so that one alternative is made impossible (e.g., "I am not fat because I am thinner than John."). Divine fallacy (argument from incredulity) – arguing that, because something is so phenomenal or amazing, it must be the result of superior, divine, alien or paranormal agency. Double counting – counting events or occurrences more than once in probabilistic reasoning, which leads to the sum of the probabilities of all cases exceeding unity. Equivocation – using a term with more than one meaning in a statement without specifying which meaning is intended. Ambiguous middle term – using a middle term with multiple meanings. Definitional retreat – changing the meaning of a word when an objection is raised. Often paired with moving the goalposts (see below), as when an argument is challenged using a common definition of a term in the argument, and the arguer presents a different definition of the term and thereby demands different evidence to debunk the argument. Motte-and-bailey fallacy – conflating two positions with similar properties, one modest and easy to defend (the "motte") and one more controversial (the "bailey"). The arguer first states the controversial position, but when challenged, states that they are advancing the modest position. Fallacy of accent – changing the meaning of a statement by not specifying on which word emphasis falls. Persuasive definition – purporting to use the "true" or "commonly accepted" meaning of a term while, in reality, using an uncommon or altered definition. (cf. the if-by-whiskey fallacy) Ecological fallacy – inferring about the nature of an entity based solely upon aggregate statistics collected for the group to which that entity belongs. Etymological fallacy – assuming that the original or historical meaning of a word or phrase is necessarily similar to its actual present-day usage. Fallacy of composition – assuming that something true of part of a whole must also be true of the whole. Fallacy of division – assuming that something true of a composite thing must also be true of all or some of its parts. False attribution – appealing to an irrelevant, unqualified, unidentified, biased or fabricated source in support of an argument. Fallacy of quoting out of context (contextotomy, contextomy; quotation mining) – selective excerpting of words from their original context to distort the intended meaning. False authority (single authority) – using an expert of dubious credentials or using only one opinion to promote a product or idea. Related to the appeal to authority. False dilemma (false dichotomy, fallacy of bifurcation, black-or-white fallacy) – two alternative statements are given as the only possible options when, in reality, there are more. False equivalence – describing two or more statements as virtually equal when they are not. Feedback fallacy – believing in the objectivity of an evaluation to be used as the basis for improvement without verifying that the source of the evaluation is a disinterested party. Historian's fallacy – assuming that decision-makers of the past had identical information as those subsequently analyzing the decision. This is not to be confused with presentism, in which present-day ideas and perspectives are anachronistically projected into the past. Historical fallacy – believing that certain results occurred only because a specific process was performed, though said process may actually be unrelated to the results. Baconian fallacy – supposing that historians can obtain the "whole truth" via induction from individual pieces of historical evidence. The "whole truth" is defined as learning "something about everything", "everything about something", or "everything about everything". In reality, a historian "can only hope to know something about something". Homunculus fallacy – using a "middle-man" for explanation; this sometimes leads to regressive middle-men. It explains a concept in terms of the concept itself without explaining its real nature (e.g.: explaining thought as something produced by a little thinker – a homunculus – inside the head simply identifies an intermediary actor and does not explain the product or process of thinking). Inflation of conflict – arguing that, if experts in a field of knowledge disagree on a certain point within that field, no conclusion can be reached or that the legitimacy of that field of knowledge is questionable. If-by-whiskey – an argument that supports both sides of an issue by using terms that are emotionally sensitive and ambiguous. Incomplete comparison – insufficient information is provided to make a complete comparison. Intentionality fallacy – the insistence that the ultimate meaning of an expression must be consistent with the intention of the person from whom the communication originated (e.g. a work of fiction that is widely received as a blatant allegory must necessarily not be regarded as such if the author intended it not to be so). Kettle logic – using multiple, jointly inconsistent arguments to defend a position. Ludic fallacy – failing to take into account that non-regulated random occurrences unknown unknowns can affect the probability of an event taking place. Lump of labour fallacy – the misconception that there is a fixed amount of work to be done within an economy, which can be distributed to create more or fewer jobs. Magic fallacy – Profit derived by any means besides the production of physical goods (e.g. an artisan cabinet, a loaf of bread) must be made by either exploitation (modern) or magic (classical). McNamara fallacy (quantitative fallacy) – making an argument using only quantitative observations (measurements, statistical or numerical values) and discounting subjective information that focuses on quality (traits, features, or relationships). Mind projection fallacy – assuming that a statement about an object describes an inherent property of the object, rather than a personal perception. Moralistic fallacy – inferring factual conclusions from evaluative premises in violation of fact–value distinction (e.g.: inferring is from ought). Moralistic fallacy is the inverse of naturalistic fallacy. Moving the goalposts (raising the bar) – argument in which evidence presented in response to a specific claim is dismissed and some other (often greater) evidence is demanded. Nirvana fallacy (perfect-solution fallacy) – solutions to problems are rejected because they are not perfect. Package deal – treating essentially dissimilar concepts as though they were essentially similar. Proof by assertion – a proposition is repeatedly restated regardless of contradiction; sometimes confused with argument from repetition (argumentum ad infinitum, argumentum ad nauseam). Prosecutor's fallacy – a low probability of false matches does not mean a low probability of some false match being found. Proving too much – an argument that results in an overly generalized conclusion Psychologist's fallacy – an observer presupposes the objectivity of their own perspective when analyzing a behavioral event. Referential fallacy – assuming that all words refer to existing things and that the meaning of words reside within the things they refer to, as opposed to words possibly referring to no real object (e.g.: Pegasus) or that the meaning comes from how they are used (e.g.: "nobody" was in the room). Reification (concretism, hypostatization, or the fallacy of misplaced concreteness) – treating an abstract belief or hypothetical construct as if it were a concrete, real event or physical entity (e.g.: saying that evolution selects which traits are passed on to future generations; evolution is not a conscious entity with agency). Retrospective determinism – believing that, because an event has occurred under some circumstance, the circumstance must have made the event inevitable (e.g.: because someone won the lottery while wearing their lucky socks, wearing those socks made winning the lottery inevitable). Slippery slope (thin edge of the wedge, camel's nose) – asserting that a proposed, relatively small, first action will inevitably lead to a chain of related events resulting in a significant and negative event and, therefore, should not be permitted. Special pleading – the arguer attempts to cite something as an exemption to a generally accepted rule or principle without justifying the exemption (e.g.: an orphaned defendant who murdered their parents asking for leniency). Improper premise [edit] Begging the question (petitio principii) – using the conclusion of the argument in support of itself in a premise (e.g.: saying that smoking cigarettes is deadly because cigarettes can kill you; something that kills is deadly). Loaded label – while not inherently fallacious, the use of evocative terms to support a conclusion is a type of begging the question fallacy. When fallaciously used, the term's connotations are relied on to sway the argument towards a particular conclusion. For example, in an organic foods advertisement that says "Organic foods are safe and healthy foods grown without any pesticides, herbicides, or other unhealthy additives", the terms "safe" and "healthy" are used to fallaciously imply that non-organic foods are neither safe nor healthy. Circular reasoning (circulus in demonstrando) – the reasoner begins with what they are trying to end up with (e.g.: The defendant is guilty because they failed a lie detector test. How do you know the test is accurate? Because convicts fail it.). Fallacy of many questions (complex question, fallacy of presuppositions, loaded question, plurium interrogationum) – someone asks a question that presupposes something that has not been proven or accepted by all the people involved. This fallacy is often used rhetorically so that the question limits direct replies to those that serve the questioner's agenda. (E.g., "Have you or have you not stopped beating your wife?".) Faulty generalizations [edit] Faulty generalization – reaching a conclusion from weak premises. Accident – an exception to a generalization is ignored. No true Scotsman (aka appeal to purity) – makes a generalization true by changing the generalization to exclude a counterexample. Cherry picking (suppressed evidence, incomplete evidence, argument by half-truth, fallacy of exclusion, card stacking, slanting) – using individual cases or data that confirm a particular position, while ignoring related cases or data that may contradict that position. Nut-picking (suppressed evidence, incomplete evidence) – using individual cases or data that falsify a particular position, while ignoring related cases or data that may support that position. Survivorship bias – a small number of successes of a given process are actively promoted while completely ignoring a large number of failures. False analogy – an argument by analogy in which the analogy is poorly suited. Hasty generalization (fallacy of insufficient statistics, fallacy of insufficient sample, fallacy of the lonely fact, hasty induction, secundum quid, converse accident, jumping to conclusions) – basing a broad conclusion on a small or unrepresentative sample. Argument from anecdote – a fallacy where anecdotal evidence is presented as an argument; without any other contributory evidence or reasoning. Inductive fallacy – a more general name for a class of fallacies, including hasty generalization and its relatives. A fallacy of induction happens when a conclusion is drawn from premises that only lightly support it. Misleading vividness – involves describing an occurrence in vivid detail, even if it is an exceptional occurrence, to convince someone that it is more important; this also relies on the appeal to emotion fallacy. Overwhelming exception – an accurate generalization that comes with qualifications that eliminate so many cases that what remains is much less impressive than the initial statement might have led one to assume. Thought-terminating cliché – a commonly used phrase, sometimes passing as folk wisdom, used to quell cognitive dissonance, conceal lack of forethought, move on to other topics, etc. – but in any case, to end the debate with a cliché rather than a point. Questionable cause [edit] Questionable cause is a general type of error with many variants. Its primary basis is the confusion of association with causation, either by inappropriately deducing (or rejecting) causation or a broader failure to properly investigate the cause of an observed effect. Cum hoc ergo propter hoc (Latin for 'with this, therefore because of this'; correlation implies causation; faulty cause/effect, coincidental correlation, correlation without causation) – a faulty assumption that, because there is a correlation between two variables, one caused the other. Post hoc ergo propter hoc (Latin for 'after this, therefore because of this'; temporal sequence implies causation) – X happened, then Y happened; therefore X caused Y. Wrong direction (reverse causation) – cause and effect are reversed. The cause is said to be the effect and vice versa. The consequence of the phenomenon is claimed to be its root cause. Ignoring a common cause Fallacy of the single cause (causal oversimplification) – it is assumed that there is one, simple cause of an outcome when in reality it may have been caused by a number of only jointly sufficient causes. Furtive fallacy – outcomes are asserted to have been caused by the malfeasance of decision makers. Magical thinking – fallacious attribution of causal relationships between actions and events. In anthropology, it refers primarily to cultural beliefs that ritual, prayer, sacrifice, and taboos will produce specific supernatural consequences. In psychology, it refers to an irrational belief that thoughts by themselves can affect the world or that thinking something corresponds with doing it. Statistical fallacies [edit] The observational interpretation fallacy occurs when associations identified in observational studies are misinterpreted as causal relationships. Regression fallacy – ascribes cause where none exists. The flaw is failing to account for natural fluctuations. It is frequently a special kind of post hoc fallacy. Gambler's fallacy – the incorrect belief that separate, independent events can affect the likelihood of another random event. If a fair coin lands on heads 10 times in a row, the belief that it is "due to the number of times it had previously landed on tails" is incorrect. Inverse gambler's fallacy – the inverse of the gambler's fallacy. It is the incorrect belief that on the basis of an unlikely outcome, the process must have happened many times before. p-hacking – belief in the significance of a result, not realizing that multiple comparisons or experiments have been run and only the most significant were published. Garden of forking paths fallacy – incorrect belief that a single experiment can not be subject to the multiple comparisons effect. Sunk costs fallacy - refusal to leave a situation because you have already put large amounts of time or effort into it- for example trying to jump over a wall you physically cannot jump over because you have already spent an hour trying to jump. Relevance fallacies [edit] Appeal to the stone (argumentum ad lapidem) – dismissing a claim as absurd without demonstrating proof for its absurdity. Invincible ignorance (argument by pigheadedness) – where a person simply refuses to believe the argument, ignoring any evidence given. Argument from ignorance (appeal to ignorance, argumentum ad ignorantiam) – assuming that a claim is true because it has not been or cannot be proven false, or vice versa. Argument from incredulity (appeal to common sense) – "I cannot imagine how this could be true; therefore, it must be false." Argument from repetition (argumentum ad nauseam or argumentum ad infinitum) – repeating an argument until nobody cares to discuss it any more and referencing that lack of objection as evidence of support for the truth of the conclusion; sometimes confused with proof by assertion. Argument from silence (argumentum ex silentio) – assuming that a claim is true based on the absence of textual or spoken evidence from an authoritative source, or vice versa. Ignoratio elenchi (irrelevant conclusion, missing the point) – an argument that may in itself be valid, but does not address the issue in question. Red herring fallacies [edit] A red herring fallacy, one of the main subtypes of fallacies of relevance, is an error in logic where a proposition is, or is intended to be, misleading in order to make irrelevant or false inferences. This includes any logical inference based on fake arguments, intended to replace the lack of real arguments or to replace implicitly the subject of the discussion. Red herring – introducing a second argument in response to the first argument that is irrelevant and draws attention away from the original topic (e.g.: saying "If you want to complain about the dishes I leave in the sink, what about the dirty clothes you leave in the bathroom?"). In jury trial, it is known as a Chewbacca defense. In political strategy, it is called a dead cat strategy. See also irrelevant conclusion. Ad hominem – attacking the arguer instead of the argument. (Note that "ad hominem" can also refer to the dialectical strategy of arguing on the basis of the opponent's own commitments. This type of ad hominem is not a fallacy.) Circumstantial ad hominem – stating that the arguer's personal situation or perceived benefit from advancing a conclusion means that their conclusion is wrong. Poisoning the well – a subtype of ad hominem presenting adverse information about a target person with the intention of discrediting everything that the target person says. Appeal to motive – dismissing an idea by questioning the motives of its proposer. Tone policing – focusing on emotion behind (or resulting from) a message rather than the message itself as a discrediting tactic. Traitorous critic fallacy (ergo decedo, 'therefore I leave') – a critic's perceived affiliation is portrayed as the underlying reason for the criticism and the critic is asked to stay away from the issue altogether. Easily confused with the association fallacy (guilt by association) below. Bulverism (psychogenetic fallacy) – inferring why an argument is being used, associating it to some psychological reason, then assuming it is invalid as a result. The assumption that if the origin of an idea comes from a biased mind, then the idea itself must also be a falsehood. Appeal to authority (argument from authority, argumentum ad verecundiam) – an assertion is deemed true because of the position or authority of the person asserting it. Appeal to accomplishment – an assertion is deemed true or false based on the accomplishments of the proposer. This may often also have elements of appeal to emotion see below. Courtier's reply – a criticism is dismissed by claiming that the critic lacks sufficient knowledge, credentials, or training to credibly comment on the subject matter. Appeal to consequences (argumentum ad consequentiam) – the conclusion is supported by a premise that asserts positive or negative consequences from some course of action in an attempt to distract from the initial discussion. Appeal to emotion – manipulating the emotions of the listener rather than using valid reasoning to obtain common agreement. Appeal to fear – generating distress, anxiety, cynicism, or prejudice towards the opponent in an argument. Appeal to flattery – using excessive or insincere praise to obtain common agreement. Appeal to pity (argumentum ad misericordiam) – generating feelings of sympathy or mercy in the listener to obtain common agreement. Appeal to ridicule – mocking or stating that the opponent's position is laughable to deflect from the merits of the opponent's argument. Appeal to spite – generating bitterness or hostility in the listener toward an opponent in an argument. Judgmental language – using insulting or pejorative language in an argument. Pooh-pooh – stating that an opponent's argument is unworthy of consideration. Style over substance – embellishing an argument with compelling language, exploiting a bias towards the esthetic qualities of an argument, e.g. the rhyme-as-reason effect Wishful thinking – arguing for a course of action by the listener according to what might be pleasing to imagine rather than according to evidence or reason. Appeal to nature – judgment is based solely on whether the subject of judgment is 'natural' or 'unnatural'. (Sometimes also called the "naturalistic fallacy", but is not to be confused with the other fallacies by that name.) Appeal to novelty (argumentum novitatis, argumentum ad antiquitatis) – a proposal is claimed to be superior or better solely because it is new or modern. (opposite of appeal to tradition) Appeal to poverty (argumentum ad Lazarum) – supporting a conclusion because the arguer is poor (or refuting because the arguer is wealthy). (Opposite of appeal to wealth.) Appeal to tradition (argumentum ad antiquitatem) – a conclusion supported solely because it has long been held to be true. Appeal to wealth (argumentum ad crumenam) – supporting a conclusion because the arguer is wealthy (or refuting because the arguer is poor). (Sometimes taken together with the appeal to poverty as a general appeal to the arguer's financial situation.) Argumentum ad baculum (appeal to the stick, appeal to force, appeal to threat) – an argument made through coercion or threats of force to support position. Argumentum ad populum (appeal to widespread belief, bandwagon argument, appeal to the majority, appeal to the people) – a proposition is claimed to be true or good solely because a majority or many people believe it to be so. Association fallacy (guilt by association and honor by association) – arguing that because two things share (or are implied to share) some property, they are the same. Logic chopping fallacy (nit-picking, trivial objections) – Focusing on trivial details of an argument, rather than the main point of the argumentation. Ipse dixit (bare assertion fallacy) – a claim that is presented as true without support, as self-evidently true, or as dogmatically true. This fallacy relies on the implied expertise of the speaker or on an unstated truism. Chronological snobbery – a thesis is deemed incorrect because it was commonly held when something else, known to be false, was also commonly held. Fallacy of relative privation (also known as "appeal to worse problems" or "not as bad as") – dismissing an argument or complaint due to what are perceived to be more important problems. First World problems are a subset of this fallacy. Genetic fallacy – a conclusion is suggested based solely on something or someone's origin rather than its current meaning or context. I'm entitled to my opinion – a person discredits any opposition by claiming that they are entitled to their opinion. Moralistic fallacy – inferring factual conclusions from evaluative premises, in violation of fact–value distinction; e.g. making statements about what is, on the basis of claims about what ought to be. This is the inverse of the naturalistic fallacy. Naturalistic fallacy – inferring evaluative conclusions from purely factual premises in violation of fact–value distinction. Naturalistic fallacy (sometimes confused with appeal to nature) is the inverse of moralistic fallacy. Is–ought fallacy – deduce a conclusion about what ought to be, on the basis of what is. Naturalistic fallacy fallacy (anti-naturalistic fallacy) – inferring an impossibility to infer any instance of ought from is from the general invalidity of is-ought fallacy, mentioned above. For instance, is does imply ought for any proposition , although the naturalistic fallacy fallacy would falsely declare such an inference invalid. Naturalistic fallacy fallacy is a type of argument from fallacy. Straw man fallacy – refuting an argument different from the one actually under discussion, while not recognizing or acknowledging the distinction. Texas sharpshooter fallacy – improperly asserting a cause to explain a cluster of data. Tu quoque ('you too' – appeal to hypocrisy, whataboutism) – stating that a position is false, wrong, or should be disregarded because its proponent fails to act consistently in accordance with it. Two wrongs make a right – assuming that, if one wrong is committed, another wrong will rectify it. Vacuous truth – a claim that is technically true but meaningless, in the form no A in B has C, when there is no A in B. For example, claiming that no mobile phones in the room are on when there are no mobile phones in the room. See also [edit] Lists portal Philosophy portal Cognitive distortion – Exaggerated or irrational thought pattern List of cognitive biases List of common misconceptions List of memory biases List of paradoxes Outline of public relations – Overview of and topical guide to public relations Map–territory relation – Relationship between an object and a representation of that object (confusing map with territory, menu with meal) Mathematical fallacy – Certain type of mistaken proof Sophistical Refutations – Text by Aristotle on logical fallacies, in which Aristotle presented thirteen fallacies Straight and Crooked Thinking – Book by Robert H. Thouless (book) References [edit] Citations [edit] ^ Hornby, A. S. (2010). "sophist". Oxford Advanced Learner's Dictionary of Current English (8th ed.). Oxford University Press. ISBN 9780194799003. ^ Bunnin & Yu 2004, "formal fallacy". ^ Bennett, Bo. "Appeal to possibility". Logically Fallacious. Retrieved 2023-03-17. ^ Carrier, Richard (2012). Proving History: Bayes's Theorem and the Quest for the Historical Jesus. Prometheus Books. pp. 26–29. ISBN 9781616145590. ^ Curtis, "Fallacy Fallacy". ^ "Base Rate Fallacy". Psychology Glossary. AlleyDog.com. Archived from the original on 2011-07-07. Retrieved 2011-02-01. ^ Straker, David. "Conjunction Fallacy". ChangingMinds.org. Archived from the original on 2011-08-13. Retrieved 2011-02-01. ^ Bennett, Bo. "Non Sequitur". logicallyfallacious. Archived from the original on 2022-07-01. 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Archived from the original on 4 February 2020. Retrieved 30 January 2020. ^ Shackel, Nicholas (2005). "The Vacuity of Postmodernist Methodology" (PDF). Metaphilosophy. 36 (3): 295–320. doi:10.1111/j.1467-9973.2005.00370.x. Archived (PDF) from the original on 2020-10-13. Retrieved 2020-09-06. For my purposes the desirable but only lightly defensible territory of the Motte and Bailey castle, that is to say, the Bailey, represents a philosophical doctrine or position with similar properties: desirable to its proponent but only lightly defensible. The Motte is the defensible but undesired position to which one retreats when hard pressed ... ^ Shackel, Nicholas (5 September 2014). "Motte and Bailey Doctrines". Practical Ethics: Ethics in the News. Cardiff University / University of Oxford. Archived from the original on 14 May 2019. Retrieved 23 May 2019. Some people have spoken of a Motte and Bailey Doctrine as being a fallacy and others of it being a matter of strategic equivocation. Strictly speaking, neither is correct. ^ Fischer 1970, pp. 119–120. ^ Gula 2002, p. 70. ^ Pirie 2006, p. 31. ^ Pirie 2006, p. 53. ^ Gula 2002, p. 97. ^ "Fallacy – False Dilemma". Nizkor. The Nizkor Project. Archived from the original on 2015-09-23. Retrieved 2011-02-01. ^ Marcus Buckingham; Ashley Goodall. "The Feedback Fallacy". Harvard Business Review. No. March–April 2019. Archived from the original on 2019-05-27. Retrieved 2019-03-11. ^ Fischer 1970, pp. 209–213. ^ "The Reflex Arc Concept in Psychology", John Dewey, The Psychological Review, Vol. III. No. 4. July 1896. p. 367 ^ Fischer 1970, pp. 4–8. ^ Bunnin & Yu 2004, "Homunculus". ^ a b "A List Of Fallacious Arguments". Retrieved 6 October 2012. ^ Bennett, Bo. "Inflation of Conflict". ^ Wimsatt, William K.; Beardsley, Monroe C. (1946). "The Intentional Fallacy". Sewanee Review. 54: 468–488. Revised and republished in Wimsatt, W.K. (1954). The Verbal Icon: Studies in the Meaning of Poetry. University Press of Kentucky. pp. 3–18. 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Archived from the original on 2019-03-10. Retrieved 2016-02-24. ^ "Begging the Question". txstate.edu. Archived from the original on 2015-09-28. Retrieved 2016-02-24. ^ Ramage, John D.; Bean, John C.; Johnson, June (2016). Writing Arguments: A Rhetoric with Readings, Concise Edition (MLA Update ed.). Pearson Education. p. 275. ISBN 9780134586496. Archived from the original on 2020-02-20. Retrieved 2018-04-03. ^ Clure, Kristen (October 30, 2023). "Circular Reasoning (29 Examples + How to Avoid)". Practical Psychology. Retrieved 2025-07-09. ^ Pirie 2006, p. 5. ^ Flew 1984, "No-true-Scotsman move". ^ Hurley 2007, p. 155. ^ Bennett, Bo. "Cherry Picking". logicallyfallacious. Archived from the original on 2022-07-01. ^ Damer 2009, p. 151. ^ Hurley 2007, p. 134. ^ Fischer 1970, p. 127. ^ Pirie 2006, p. 41. ^ Damer 2009, p. 180. ^ Gula 2002, p. 135. ^ Damer 2009, p. 178. ^ Damer 2009, p. 186. ^ Patey, Douglas Lane (1986). "Johnson's Refutation of Berkeley: Kicking the Stone Again". Journal of the History of Ideas. 47 (1): 139–145. doi:10.2307/2709600. JSTOR 2709600. ^ "Invincible Ignorance" by Bruce Thompson, Department of Humanities (Philosophy), Cuyamaca College ^ Damer 2009, p. 165. ^ "Argument from personal incredulity – Toolkit For Thinking". www.toolkitforthinking.com. Archived from the original on 2015-07-05. Retrieved 2013-11-08. ^ "Repetition". changingminds.org. Archived from the original on 2016-03-04. Retrieved 2016-02-24. ^ "Ad nauseam – Toolkit For Thinking". toolkitforthinking.com. Archived from the original on 2016-03-04. Retrieved 2016-02-24. ^ "Argument from silence – Toolkit For Thinking". toolkitforthinking.com. Archived from the original on 2016-03-04. Retrieved 2016-02-24. ^ Copi & Cohen 1990, pp. 105–107. ^ Curtis, Red Herring. ^ "Logical Fallacies". logicalfallacies.info. Archived from the original on 2016-02-21. Retrieved 2016-02-24. ^ Damer 2009, p. 208. ^ Nizkor. "Circumstantial Ad Hominem". Archived from the original on July 13, 2015. Retrieved September 5, 2018. ^ Walton 2008, p. 187. ^ Clark & Clark 2005, pp. 13–16. ^ Walton 1997, p. 28. ^ Walton 2008, p. 27. ^ Damer 2009, p. 111. ^ "Appeal to Fear". changingminds.org. Archived from the original on 2014-02-22. Retrieved 2014-02-11. ^ Gula 2002, p. 12. ^ Walton 2008, p. 128. ^ "Appeal to Ridicule". changingminds.org. Archived from the original on 2014-02-22. Retrieved 2014-02-11. ^ "Appeal to Spite". changingminds.org. Archived from the original on 2014-02-22. Retrieved 2014-02-11. ^ Munson, Ronald; Black, Andrew (2016). The Elements of Reasoning. Cengage Learning. p. 257. ISBN 9781305886834. Archived from the original on 2016-01-18. Retrieved 2015-11-13. ^ Bennett, Bo. "Style over Substance". logicallyfallacious. Retrieved 2022-07-06. ^ Damer 2009, p. 146. ^ Curtis, Appeal to Nature. ^ Pirie 2006, p. 116. ^ Pirie 2006, p. 104. ^ Pirie 2006, p. 14. ^ Pirie 2006, p. 39. ^ Damer 2009, p. 106. ^ "Appeal to Widespread Belief". Archived from the original on 13 June 2011. Retrieved 6 October 2012. ^ Curtis, Guilt by Association. ^ Bennett, Bo. "Logic Chopping". LogicallyFallacious. Archived from the original on 2022-07-01. Retrieved 2021-05-29. ^ Byerly, Henry (1973). A primer of logic. Harper & Row. ISBN 0060411139. ^ Whitney, William Dwight; Smith, Benjamin Eli, eds. (1897). The Century Dictionary and Cyclopedia. Vol. IV. New York: The Century Co. pp. 3179–3180. Archived from the original on December 21, 2023. Retrieved December 21, 2023. ^ Westbrook, Robert B. (1991). John Dewey and American Democracy. Cornell University Press. p. 359. ISBN 978-0-8014-8111-6. ^ VanderMey, Randall; Meyer, Verne; Van Rys, John; Sebranek, Patrick (2012). COMP. Cengage Learning. ISBN 9781133307747. Archived from the original on 2021-10-16. Retrieved 2017-12-12. Bare assertion. The most basic way to distort an issue is to deny that it exists. This fallacy claims, 'That's just how it is.' ^ "Encyclopedia Barfieldiana". davidlavery.net. Archived from the original on 2013-05-20. Retrieved 2014-02-11. ^ "Chronological snobbery – Summa Bergania". Archived from the original on February 5, 2012. Retrieved February 11, 2014. ^ Turkel, Bruce (2016). All about Them: Grow Your Business by Focusing on Others. Da Capo Press. ISBN 9780738219202. Archived from the original on 5 August 2021. Retrieved 15 November 2020 – via Google Books. ^ Bennett, Bo. "Relative privation". Archived from the original on 2019-11-05. Retrieved 2019-12-30 – via Logically Fallacious. ^ Damer 2009, p. 93. ^ Dowden 2010, "Naturalistic". ^ "Naturalistic fallacy". TheFreeDictionary.com. Archived from the original on 2013-06-04. Retrieved 2013-07-01. ^ Dowden 2010, "Is-Ought". ^ Searle, John R. (January 1964). "How to Derive "Ought" From "Is"". The Philosophical Review. 73 (1): 43–58. doi:10.2307/2183201. ISSN 0031-8108. JSTOR 2183201. Archived from the original on 2017-02-07. Retrieved 2017-09-03. ^ Walter, Alex (2006). "The Anti-naturalistic Fallacy: Evolutionary Moral Psychology and the Insistence of Brute Facts". Evolutionary Psychology. 4 (1) 147470490600400102: 34–48. doi:10.1177/147470490600400102. ISSN 1474-7049. ^ Downes, Stephen. "The Logical Fallacies". Archived from the original on 3 March 2016. Retrieved 25 February 2016. ^ Curtis, "The Texas Sharpshooter Fallacy". ^ Pirie 2006, p. 164. ^ Johnson & Blair 1994, p. 122. Sources [edit] Bunnin, Nicholas; Yu, Jiyuan, eds. (2004). The Blackwell Dictionary of Western Philosophy. Blackwell. ISBN 9781405106795. Clark, Jef; Clark, Theo (2005). Humbug! The Skeptic's Field Guide to Spotting Fallacies in Thinking. Nifty Books. ISBN 0646444778. Archived from the original on 2016-11-21. Retrieved 2016-10-04. Also available as an ebook Archived 2016-03-06 at the Wayback Machine. Copi, Irving M.; Cohen, Carl (1990). Introduction to Logic (8th ed.). Macmillan. ISBN 9780023250354. Curtis, Gary N. "Logical Fallacies: The Fallacy Files". Archived from the original on 2015-10-01. Retrieved 2011-04-23. Damer, T. Edward (2009). Attacking Faulty Reasoning: A Practical Guide to Fallacy-free Arguments (6th ed.). Wadsworth. ISBN 9780495095064. Archived from the original on 16 November 2016. Retrieved 30 November 2010. Dowden, Bradley (December 31, 2010). "Fallacy". The Internet Encyclopedia of Philosophy. ISSN 2161-0002. Archived from the original on 2008-09-13. Retrieved 2011-04-22. Fischer, David Hackett (1970). Historians' Fallacies: Toward a Logic of Historical Thought. HarperCollins. ISBN 9780061315459. Flew, Antony (1984). A Dictionary of Philosophy (Revised 2nd ed.). Macmillan. ISBN 978-0312209230. Archived from the original on 2016-11-21. Retrieved 2016-10-04. Gula, Robert J. (2002). Nonsense: Red Herrings, Straw Men and Sacred Cows: How We Abuse Logic in Our Everyday Language. Axios Press. ISBN 9780975366264. Archived from the original on 2016-11-21. Retrieved 2016-10-04. Hurley, Patrick J. (2007). A Concise Introduction to Logic (10th ed.). Cengage. ISBN 9780495503835. Archived from the original on 2016-11-21. Retrieved 2016-10-04. Johnson, Ralph H.; Blair, J. Anthony (1994). Logical Self-Defense. Idea. ISBN 9781932716184. Archived from the original on 2016-11-21. Retrieved 2016-10-04. Pirie, Madsen (2006). How to Win Every Argument: The Use and Abuse of Logic. Continuum International Publishing Group. ISBN 0826490069. Archived from the original on 2016-11-21. Retrieved 2016-10-04. Wilson, W. Kent (1999). "Formal fallacy". In Audi, Robert (ed.). The Cambridge Dictionary of Philosophy (2nd ed.). Cambridge University Press. pp. 316–317. ISBN 9780511074172. Walton, Douglas (1997). Appeal to Expert Opinion: Arguments from Authority. Pennsylvania State University. ISBN 0271016949. Archived from the original on 2016-11-21. Retrieved 2016-10-04. Walton, Douglas (2008). Informal Logic: A Pragmatic Approach (2nd ed.). Cambridge University Press. ISBN 9780511408786. Further reading [edit] : The following is a sample of books for further reading, selected for a combination of content, ease of access via the internet, and to provide an indication of published sources that interested readers may review. The titles of some books are self-explanatory. Good books on critical thinking commonly contain sections on fallacies, and some may be listed below. DiCarlo, Christopher (2011). How to Become a Really Good Pain in the Ass: A Critical Thinker's Guide to Asking the Right Questions. Prometheus Books. ISBN 9781616143978. Engel, S. Morris (1994). Fallacies and Pitfalls of Language: The Language Trap. Dover Publications. ISBN 0486282740. Retrieved 30 November 2010. Hamblin, C. L. (2004). Fallacies. Methuen & Co. ISBN 0416145701. Hughes, William; Lavery, Jonathan (2004). Critical Thinking: An Introduction to the Basic Skills (4th ed.). Broadview Press. ISBN 1551115735. Retrieved 30 November 2010. Paul, Richard; Elder, Linda (2006). Thinker's Guide to Fallacies: The Art of Mental Trickery. Foundation for Critical Thinking. ISBN 9780944583272. Archived from the original on 5 July 2014. Retrieved 30 November 2010. Sinnott-Armstrong, Walter; Fogelin, Robert (2010). Understanding Arguments: An Introduction to Informal Logic (8th ed.). Wadsworth Cengage Learning. ISBN 9780495603955. Retrieved 30 November 2010. Thouless, Robert H. (1953). Straight and Crooked Thinking (PDF). Pan Books. Retrieved 30 November 2010. Tindale, Christopher W. (2007). Fallacies and Argument Appraisal. Critical Reasoning and Argumentation. Cambridge University Press. ISBN 9780521842082. Retrieved 30 November 2010. Wilson, David Carl (2020). A Guide to Good Reasoning: Cultivating Intellectual Virtues (2nd ed.). University of Minnesota Libraries. ISBN 9781946135667. Creative Commons Attribution-Non-Commercial 4.0 International License External links [edit] Logical Fallacies, Literacy Education Online Informal Fallacies, Texas State University page on informal fallacies Stephen's Guide to the Logical Fallacies (mirror) Visualization: Rhetological Fallacies, Information is Beautiful Master List of Logical Fallacies, University of Texas at El Paso Fallacies, Internet Encyclopedia of Philosophy \| v t e Common fallacies (list) \| \| Formal \| \| \| \| --- \| \| In propositional logic \| Affirming a disjunct Affirming the consequent Conflation Denying the antecedent Argument from fallacy Masked man Mathematical fallacy \| \| In quantificational logic \| Existential Illicit conversion Proof by example Quantifier shift \| \| Syllogistic fallacy \| Affirmative conclusion from a negative premise Negative conclusion from affirmative premises Exclusive premises Existential Necessity Four terms Illicit major Illicit minor Undistributed middle \| \| \| Informal \| \| \| \| --- \| \| Equivocation \| Equivocation False equivalence False attribution Moral equivalence Conflation Quoting out of context Loki's Wager No true Scotsman Reification + Map–territory relation \| \| Question-begging \| Circular reasoning / Begging the question Loaded language + Leading question Compound question / Loaded question / Complex question No true Scotsman \| \| Correlative-based \| False dilemma + Perfect solution Denying the correlative Suppressed correlative \| \| Illicit transference \| Composition Division Ecological \| \| Secundum quid \| Accident Converse accident \| \| Faulty generalization \| Anecdotal evidence Sampling bias + Cherry picking + McNamara Base rate / Conjunction Double counting False analogy Slothful induction Overwhelming exception \| \| Ambiguity \| Accent False precision Moving the goalposts Quoting out of context Slippery slope Sorites paradox Syntactic ambiguity \| \| Questionable cause \| Animistic + Furtive Correlation implies causation + Cum hoc + Post hoc Gambler's + Inverse Regression Single cause Slippery slope Texas sharpshooter \| \| Appeals \| Law/Legality Stone / Proof by assertion \| \| \| --- \| \| Consequences \| Argumentum ad baculum Wishful thinking \| \| Emotion \| Children Fear Flattery Novelty Pity Ridicule In-group favoritism Invented here / Not invented here Island mentality Loyalty Parade of horribles Spite Stirring symbols Wisdom of repugnance \| \| \| Genetic fallacy \| \| \| \| --- \| \| Ad hominem \| Appeal to motive Association + Reductio ad Hitlerum + Reductio ad Stalinum Bulverism Poisoning the well Tone Tu quoque Whataboutism \| \| Authority + Accomplishment + Ipse dixit + Poverty / Wealth Etymology Nature Tradition / Novelty + Chronological snobbery \| \| \| Other fallacies of relevance \| \| \| \| --- \| \| Arguments \| Ad nauseam + Sealioning Argument from anecdote Argument from silence Argument to moderation Argumentum ad populum \| \| Cliché The Four Great Errors I'm entitled to my opinion Ignoratio elenchi Invincible ignorance Moralistic / Naturalistic Motte-and-bailey fallacy Psychologist's fallacy Rationalization Red herring + Two wrongs make a right Special pleading Straw man \| \| \| \| Category \| \| v t e Philosophical logic \| \| Critical thinking andinformal logic \| Analysis Ambiguity Argument Belief Bias Credibility Dialectic + Antithesis, Socratic method, Unity of opposites Evidence Explanation Explanatory power Fact Fallacy + List of fallacies Hypothesis Inquiry Opinion Parsimony (Occam's razor) Premise Propaganda Prudence Razor Reasoning Relevance Rhetoric Rigor Theory Vagueness \| \| Theories of deduction \| Constructivism Dialetheism Fictionalism Finitism Formalism Intuitionism Logical atomism Logicism Nominalism Platonic realism Pragmatism Realism \| \| Category \| \| v t e \| \| \| \| Major fields \| Computer science Formal semantics (natural language) Inference Philosophy of logic Proof Semantics of logic Syntax \| \| \| --- \| \| Logics \| Classical Informal + Critical thinking + Reason Mathematical Non-classical Philosophical \| \| Theories \| Argumentation Metalogic Metamathematics Set \| \| \| Foundations \| Abduction Analytic and synthetic propositions Antecedent Consequent Contradiction + Paradox + Antinomy Deduction Deductive closure Definition Description Dichotomy Entailment + Linguistic Form Induction Logical truth Name Necessity and sufficiency Premise Probability Proposition Reference Statement Substitution Truth Validity \| \| Lists \| \| \| \| --- \| \| Topics \| Mathematical logic Boolean algebra Set theory \| \| Other \| Logicians Rules of inference Paradoxes Fallacies Logic symbols \| \| \| Category Outline Portal WikiProject changes \| Retrieved from " Categories: Fallacies Rhetoric Logic-related lists Hidden categories: CS1 errors: ISBN date Webarchive template wayback links Articles with short description Short description is different from Wikidata Articles containing Latin-language text Pages that use a deprecated format of the math tags List of fallacies Add topic
7699	https://www.expii.com/t/temperature-dependence-of-equilibrium-constant-overview-8708	Expii Temperature Dependence of Equilibrium Constant — Overview - Expii If ΔG for a reaction is negative, increasing the temperature decreases the equilibrium constant (K). If ΔG is positive, increasing the temperature increases K. Explanations (2) Eric Sears Text 1 Previously, we studied how Gibbs free energy relates to the equilibrium constant. But, we know that Gibbs free energy is temperature-dependent. So, we expect that the equilibrium constant is also temperature-dependent. Last time, we derived the formula, lnKeq=−ΔGoRT. We used the formula to relate ΔG and the equilibrium constant's magnitude. Now we can use it to study how temperature affects equilibrium. The Van't Hoff Equation We want to relate the equilibrium constant and temperature. But first, we need to derive the Van't Hoff equation. We'll start with the equation we found last time. lnKeq=−ΔGoRT We know that ΔGo=ΔHo−TΔSo. Let's start by replacing ΔGo. So, lnKeq=−ΔHoRT+ΔSoR. Now we can do something strange. Let's subtract two equilibrium constants lnK1 at T1 and lnK2 at T2. They are for the same reaction but at different temperatures. We can assume that the standard enthalpy change and standard entropy change are temperature-independent. So, lnK2−lnK1=(−ΔHoRT2+ΔSoR)−(−ΔHoRT1+ΔSoR). Now we can apply the rule of natural logs. We know that ln(x)−ln(y)=lnxy. So, lnK2−lnK1=lnK2K1. We can also simplify the other side of the equation. (−ΔHoRT2+ΔSoR)−(−ΔHoRT1+ΔSoR)→−ΔHoRT2−−ΔHoRT1+ΔSoR−ΔSoR→ΔHoR×(1T1−1T2). So, we derived the Van't Hoff equation, lnK2K1=ΔHoR×(1T1−1T2). Van't Hoff Plots Let's return to the second step in our derivation. lnKeq=−ΔHoRT+ΔSoR. The equation has the form y=mx+b. y=Keq m=−HoR x=1T b=ΔSoR So, if we plot Keq against 1T, we get a linear graph. Keq and Enthalpy What happens when are reactions are endothermic or exothermic? We can plug in the enthalpy change into Van't Hoff plot equation. Endothermic ΔH=+ The slope is negative. So as 1T increases, Keq decreases. As the temperature increases, Keq gets larger. Exothermic ΔH=− The slope is positive. So as 1T increases, Keq increases. As the temperature increases, Keq gets smaller. Using the Van't Hoff Equation Let's use the Van't Hoff equation to confirm our conclusions. Let's start with a generic endothermic reaction. At 25oC its Keq=0.00045 and ΔH=15.0Jmol. What is its Keq at 32oC? Remember, this is an endothermic reaction. So, we expect Keq to increase. lnK2K1=ΔHoR(1T1−1T2)→lnK20.00045=15.0Jmol8.314JmolK(1298.15K−1305.15K)→lnK20.00045=1.80→K2=0.0027 K2>K1 like we expected. Now, let's say we have a very exothermic reaction. At 25oC its Keq=0.00045 and ΔH=−50,000.0Jmol. What is its Keq at 32oC? For an exothermic reaction, we expect Keq to decrease. lnK2K1=ΔHoR(1T1−1T2)→lnK20.00045=−50000.0Jmol8.314JmolK(.0000771K)→lnK20.00045=.097→K2=0.000044 K2<K1, like we expected. We used the Van't Hoff equation to confirm our conclusion! Report Share 1 Like Related Lessons Gibbs Free Energy and Equilibrium Constant — Calculation Coupled Reactions — Overview & Importance Standard Free Energy of Formation — Definition & Overview Precipitation Reactions — Overview & Examples View All Related Lessons Eric Sears Video 1 (Video) Free Energy, Equilibrium Constants and Temperature Webcast-legacy Departmental This quick video will show you the relationship between the equilibrium constant and temperature. The video starts by connecting the Gibbs free energy to the equilibrium constant. Then, they display the graphs of K vs. T and ln[(K)] vs. 1T. So, you get to visualize the relationship between equilibrium constants and temperatures. Report Share 1 Like You've reached the end TOP How can we improve? General Bug Feature Send Feedback

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