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2100	https://bio.libretexts.org/Bookshelves/Genetics/Introduction_to_Genetics_(Singh)/03%3A_The_Cell_Cycle_and_Mitosis/3.06%3A_Compare_and_Contrast_Mitosis_and_Meiosis	Skip to main content 3.6: Compare and Contrast Mitosis and Meiosis Last updated : Mar 1, 2024 Save as PDF 3.5: Appearance of a Typical Nuclear Chromosome During the Cell Cycle 3.7: Chapter 3 Summary Page ID : 132157 Natasha Ramroop Singh Thompson Rivers University ( \newcommand{\kernel}{\mathrm{null}\,}) 3.6 Compare and Contrast Mitosis and Meiosis Here, we outline the differences between mitosis and meiosis in humans (Diploid #46). Knowing the differences between these fundamental cell processes is an important foundation in your understanding of genetics for the rest of the course. Table 3.6.1 Differences between Mitosis and Meiosis in Humans (Diploid #46) \| Cell Processes \| Mitosis \| Meiosis \| \| Creates \| all the cells in your body except sex cells \| sex cells only; Female egg cells or Male sperm cells \| \| Definition \| process of cell division that forms two new cells (daughter cells), each of which has the same number of chromosomes \| process in cell division during which four new cells are created each with half the original number of chromosomes, which results in the production of sex cells \| \| End Products \| 2 daughter cells \| 4 daughter cells \| \| Steps \| Interphase, Prophase, Metaphase, Anaphase, Telophase \| Interphase, Prophase I, Metaphase I, Anaphase I, Telophase I, Prophase II, Metaphase II, Anaphase II, Telophase II \| \| Type of Reproduction \| asexual \| sexual \| \| Are they identical to the parent cell? \| Yes, they are identical \| No, they are different since they have half the number of chromosomes as the original \| \| When does cytokinesis occur? \| occurs in Telophase \| occurs in Telophase I, and Telophase II \| \| How many times does the parent cell divide? \| 1 \| 2 \| \| What happens to the number of chromosomes at the end of each process? Are they in pairs or individual chromosomes? \| Identical to parent; Individual chromosomes \| ½ chromosomes as parent; individual chromosomes \| \| Why is each important? \| Needed to repair damaged body, create new body cells, for growth, and to replace cells that have died \| Needed to create sex cells required for sexual reproduction to create a new organism, and for variation within a population \| \| How many chromosomes do human body cells and human sex cells have after they go through each process? \| 46 \| 23 \| Take a look at the following video, Mitosis vs. Meiosis, by Beverly Biology (2014) on YouTube, which compares and contrasts mitosis and meiosis. A YouTube element has been excluded from this version of the text. You can view it online here: Reference Beverly Biology. (2014, May 3). Mitosis vs. Meiosis (video file). YouTube. 3.5: Appearance of a Typical Nuclear Chromosome During the Cell Cycle 3.7: Chapter 3 Summary
2101	https://www.youtube.com/watch?v=xpzt0wqMT6Y	Unit conversion word problem: roadtrip \| Introduction to algebra \| Algebra I \| Khan Academy Khan Academy 9090000 subscribers 106 likes Description 200334 views Posted: 4 Aug 2015 The challenge in a word problem sometimes is knowing which numbers to choose when creating a simple equation. In this unit conversion word problem we'll answer the common question: "How much will a road trip cost me in gas?" Practice this lesson yourself on KhanAcademy.org right now: Watch the next lesson: Missed the previous lesson? Algebra I on Khan Academy: Algebra is the language through which we describe patterns. Think of it as a shorthand, of sorts. As opposed to having to do something over and over again, algebra gives you a simple way to express that repetitive process. It's also seen as a "gatekeeper" subject. Once you achieve an understanding of algebra, the higher-level math subjects become accessible to you. Without it, it's impossible to move forward. It's used by people with lots of different jobs, like carpentry, engineering, and fashion design. In these tutorials, we'll cover a lot of ground. Some of the topics include linear equations, linear inequalities, linear functions, systems of equations, factoring expressions, quadratic expressions, exponents, functions, and ratios. 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 Khan Academy’s Algebra channel: Subscribe to Khan Academy: 7 comments Transcript: 0 seconds Your car gets 25 miles per gallon, and you want to go on a 400-mile road trip. Right now, gas costs $3 per gallon. How much will the gas for your road trip cost? So let's see. They tell us they're going on a 400-mile road trip. 400 miles. So the first thing that I'd want to think about is, well, how many gallons am I using? And then once I know how many gallons I'm using, I know it's $3 per gallon, so I can multiply the number of gallons by $3. So to figure out the number of gallons, would I want to multiply 400 miles-- would I want to multiply that times the miles per gallon, which is 25, or would I want to multiply by the gallons per mile? Well, if I multiply by the gallons per mile, and I multiply that times 400, then I would get the number of gallons. So let's just think about that. I want to multiply that times-- and I'll write the units first-- the gallons per mile. And what are the gallons per mile? Well, we have 25 miles per gallon. We have 25 miles for every 1 gallon. Or you could say we have 1 gallon for every 25 miles. So I really just took the reciprocal of 25 miles per gallon and made it 1/25 gallons per mile. Now what do we get when we multiply these two things? The whole purpose was to figure out how many gallons we're going to use. Well, we see that our miles cancel out. Miles cancel out with miles. And then I have 400 times 1/25 gallons, which is the same thing as 400 divided by 25 gallons. So this is equal to 400/25 gallons, which is the same thing as 400 divided by 25 is equal to 16 gallons. Now it's always important to do a reality check here, not just to try to blindly cancel out units. Does this actually makes sense that 16 is a much lower number than 400? Well, sure it does. And actually, if you have any experience filling up a car, you would sense that, OK, well, that's about the size. On around 16 gallons, a car tends to go 300 or 400 miles if it gets pretty good fuel mileage. So that just makes sense from experience. And it also make sense based on how it's stated. You get 25 miles per gallon. So you're going to need fewer gallons than you're going to need miles. So this all seems to make sense so far. But we haven't answered their question. They want to know, how much will my trip cost? Right now, we've just figured out how much fuel we're going to use. So then we could take our 16 gallons. And to figure out the dollar cost, are we going to multiply it by dollars per gallon or gallons per dollar? Well, if we're thinking just about unit conversion, we want to multiply times the dollars per gallon. So I could write it like this. I could write it like dollars per gallon. Actually, let me just write out the word "dollar." Dollars per gallon. The units will cancel out. And it also makes sense. Whatever number of dollars per gallon, I multiply it times the number of gallons, and that's going to tell me how much it's going to cost. This happens at the fuel pump every day. Hey, it's $3 per gallon. I'm going to fill up 16 gallons. Hey, 3 times 16. So let's do that. So it's $3 per gallon. We see the gallons cancel out. And we are left with 16 times $3, which is the same thing as $48. And we are done. 3 minutes, 46 seconds
2102	https://pubmed.ncbi.nlm.nih.gov/10037250/	Trans-sphenoidal surgery for microprolactinoma: an acceptable alternative to dopamine agonists? - 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. Log inShow account info Close Account Logged in as: username Dashboard Publications Account settings Log out Access keysNCBI HomepageMyNCBI HomepageMain ContentMain Navigation Search: Search AdvancedClipboard User Guide Save Email Send to Clipboard My Bibliography Collections Citation manager Display options Display options Format Save citation to file Format: Create file Cancel Email citation Email address has not been verified. Go to My NCBI account settings to confirm your email and then refresh this page. 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H E Turner1,C B Adams,J A Wass Affiliations Expand Affiliation 1 Department of Endocrinology, Radcliffe Infirmary, Oxford, UK. PMID: 10037250 DOI: 10.1530/eje.0.1400043 Item in Clipboard Comparative Study Trans-sphenoidal surgery for microprolactinoma: an acceptable alternative to dopamine agonists? H E Turner et al. Eur J Endocrinol.1999 Jan. Show details Display options Display options Format Eur J Endocrinol Actions Search in PubMed Search in NLM Catalog Add to Search . 1999 Jan;140(1):43-7. doi: 10.1530/eje.0.1400043. Authors H E Turner1,C B Adams,J A Wass Affiliation 1 Department of Endocrinology, Radcliffe Infirmary, Oxford, UK. PMID: 10037250 DOI: 10.1530/eje.0.1400043 Item in Clipboard Cite Display options Display options Format Abstract Aims: Reported cure rates following trans-sphenoidal surgery for microprolactinoma are variable and recurrence rates in some series are high. We wished to examine the cure rate of trans-sphenoidal surgery for microprolactinoma, and to assess the long-term complications and recurrence rate. Design: A retrospective review of the outcome of trans-sphenoidal surgery for microprolactinoma, performed by a single neurosurgeon at a tertiary referral centre between 1976 and 1997. Patients: All thirty-two patients operated on for microprolactinoma were female, with a mean age of 31 years (range 16-49). Indications for surgery were intolerance of dopamine agonists in ten (31%), resistance in six (19%) and resistance and intolerance in four (12.5%). Two patients were from countries where dopamine agonists were unavailable. Results: The mean pre-operative prolactin level was 2933 mU/l (range 1125-6000). All but 1 had amenorrhoea or oligomenorrhoea, with galactorrhoea in 15 (46.9%). Twenty-five (78%) were cured by trans-sphenoidal surgery, as judged by a post-operative serum prolactin in the normal range. During a mean follow-up of 70 months (range 2 months to 16 years) there was one recurrence at 12 years. Post-operatively, one patient became LH deficient, two patients became cortisol deficient and two became TSH deficient. Out of 21 patients tested for post-operative growth hormone deficiency, 6 (28.6%) were deficient. Five patients developed post-operative diabetes insipidus which persisted for greater than 6 months. There were no other complications of surgery. The estimated cost of uncomplicated trans-sphenoidal surgery, and follow-up over 10 years, was similar to that of dopamine agonist therapy. Conclusion: In patients with hyperprolactinaemia due to a pituitary microprolactinoma, transsphenoidal surgery by an experienced pituitary surgeon should be considered as a potentially curative procedure. The cost of treatment over a 10 year period is similar in uncomplicated cases to long-term dopamine agonist therapy. PubMed Disclaimer Similar articles The outcome of hypophysectomy for prolactinomas in the era of dopamine agonist therapy.Soule SG, Farhi J, Conway GS, Jacobs HS, Powell M.Soule SG, et al.Clin Endocrinol (Oxf). 1996 Jun;44(6):711-6. doi: 10.1046/j.1365-2265.1996.738559.x.Clin Endocrinol (Oxf). 1996.PMID: 8759184 The effect of quinagolide and cabergoline, two selective dopamine receptor type 2 agonists, in the treatment of prolactinomas.Di Sarno A, Landi ML, Marzullo P, Di Somma C, Pivonello R, Cerbone G, Lombardi G, Colao A.Di Sarno A, et al.Clin Endocrinol (Oxf). 2000 Jul;53(1):53-60. doi: 10.1046/j.1365-2265.2000.01016.x.Clin Endocrinol (Oxf). 2000.PMID: 10931080 Clinical Trial. Outcomes of transsphenoidal surgery in prolactinomas: improvement of hormonal control in dopamine agonist-resistant patients.Primeau V, Raftopoulos C, Maiter D.Primeau V, et al.Eur J Endocrinol. 2012 May;166(5):779-86. doi: 10.1530/EJE-11-1000. Epub 2012 Feb 2.Eur J Endocrinol. 2012.PMID: 22301915 The use of surgery for the treatment of prolactinomas.van't Verlaat JW.van't Verlaat JW.Acta Endocrinol (Copenh). 1993 Jul;129 Suppl 1:34-7.Acta Endocrinol (Copenh). 1993.PMID: 8103958 Review. Treatment of prolactinomas.Colao A, Annunziato L, Lombardi G.Colao A, et al.Ann Med. 1998 Oct;30(5):452-9. doi: 10.3109/07853899809002486.Ann Med. 1998.PMID: 9814831 Review. See all similar articles Cited by Long-term Outcome of Microscopic Transsphenoidal Surgery for Prolactinomas as an Alternative to Dopamine Agonists.Park K, Park KH, Park HR, Lee JM, Kim YH, Kim DY, Won TB, Kong SH, Kim JH, Shin CS, Paek SH.Park K, et al.J Korean Med Sci. 2021 Apr 19;36(15):e97. doi: 10.3346/jkms.2021.36.e97.J Korean Med Sci. 2021.PMID: 33876586 Free PMC article. Are dopamine agonists still the first-choice treatment for prolactinoma in the era of endoscopy? A systematic review and meta-analysis.Cai X, Zhu J, Yang J, Tang C, Cong Z, Ma C.Cai X, et al.Chin Neurosurg J. 2022 Apr 8;8(1):9. doi: 10.1186/s41016-022-00277-1.Chin Neurosurg J. 2022.PMID: 35395837 Free PMC article. Surgical outcomes in hyporesponsive prolactinomas: analysis of patients with resistance or intolerance to dopamine agonists.Hamilton DK, Vance ML, Boulos PT, Laws ER.Hamilton DK, et al.Pituitary. 2005;8(1):53-60. doi: 10.1007/s11102-005-5086-1.Pituitary. 2005.PMID: 16411069 Pituitary surgery for small prolactinomas as an alternative to treatment with dopamine agonists.Babey M, Sahli R, Vajtai I, Andres RH, Seiler RW.Babey M, et al.Pituitary. 2011 Sep;14(3):222-30. doi: 10.1007/s11102-010-0283-y.Pituitary. 2011.PMID: 21170594 Free PMC article. Clinical factors involved in the recurrence of pituitary adenomas after surgical remission: a structured review and meta-analysis.Roelfsema F, Biermasz NR, Pereira AM.Roelfsema F, et al.Pituitary. 2012 Mar;15(1):71-83. doi: 10.1007/s11102-011-0347-7.Pituitary. 2012.PMID: 21918830 Free PMC article.Review. 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2103	https://www.ramauniversity.ac.in/online-study-material/faculty_sciences/bscbiotechnology/iiisemester/molecularbiology/lecture-9.pdf	FACULTY OF ENGINEERING &TECHNOLOGY DEPARTMENT OF BIOTECHNOLOGY Variations in DNA • Most of the DNA is in the classic Watson-Crick model simply called as B-DNA or B-form DNA. • In certain condition ,different forms of • In certain condition ,different forms of DNAs are found to be appeared like A-DNA,Z-DNA,C- DNA,D-DNA,E-DNA. • This deviation in forms are based on their structural diversity. DIFFERENT FORMS OF DNA • X-ray analysis of DNA crystals at atomic resolution have revealed that DNA exhibits much more structural diversity than formly envisaged. Such variations are: B-DNA: Most common ,originally deduced from X-ray diffraction of sodium salt of DNA fibres at 92% relative humidity. humidity. A-DNA: Originally identified by X-ray diffraction of analysis of DNA fibres at 75% relative humidity . Z-DNA: Left handed double helical structure winds to the left in a zig- zag pattern . C-DNA: Formed at 66% relative humidity and in presence of Li+ and Mg2+ ions. D-DNA: Rare variant with 8 base pairs per helical turn ,form in structure devoid of guanine . E- DNA: Extended or eccentric DNA. B-DNA • Described by James D. Watson & Francis crick. • Commonly found in DNA. • DNA molecule consists of 2 helical polynucleotide chains coiled around common axis. • 2 helices are wound in such a way so as to produce 2 interchain spacing or groove – spacing or groove – • Major/wide groove(width 12A°,depth 8.5A°) • Minor /narrow groove(width 6A°,depth 7.5A°) • These grooves provide surface with which proteins,chemicals,drugs can interact. • 2 helical wind along the molecules. • 2 chains run in opposite direction ,they are antiparallel ,the plane of bases are perpendicular to helix axis. • Right handed twisting . • Right handed twisting . • Uniform diameter(20A°) • Complementary base pairing. • Base pair per turn is 10.4. • Rise per base pair is 3.4A°. A-DNA • A-DNA is one of the possible double helical structure which DNA can adopt along with other two biologically active helix structure(B-DNA,Z-DNA). • Right handed double helix. • Short and fat compared to B-DNA. • Occur only in dehydrated sample of DNA ,Such those used in crystallographic experiments. • A-DNA was originally identified by X-ray diffraction analysis of DNA fibres at 75% relative humidity. • The grooves are not as deep in B-DNA. • The bases are more tilted (to 20A°). • The base pairs per turn is 11. • Rise per base pair is 2.3A°. Z-DNA • One of the many possible DNA double helix structure. • Left handed double helix structure winds to left in zig-zag manner.(DNA backbone were in zig-zag manner)so they are termed as Z-DNA. • Discovered by Rich, Nordheim &Wang in 1984. • It has antiparallel strands as B-DNA. • It is long and thin as compared to B-DNA. • It is long and thin as compared to B-DNA. A B Z HELIX RIGHT HANDED RIGHT HANDED LEFT HANDED WIDTH WIDEST INTERMEDIATE NARROWEST PLANE OF BASE PERPENDICULAR TO HELIX AXIS PERPENDICULAR TO HELIX AXIS PERPENDICULAR TO HELIX AXIS BASE CENTRAL AXIS 6A° HOLES ALONG HELIX AXIS TINY CENTRAL NO INTERNAL SPACE MAJOR AXIS NARROW & DEEP WIDE &DEEP NO MAJOR GROOVE WIDE AND NARROW & DEEP NARROW & DEEP
2104	https://fuelgreatminds.com/teaching-inverse-operations-challenging-learners/	Skip to content Teaching Inverse Operations to Challenging Learners Last weekend I was looking ahead at the math skills that I had to teach my 3rd graders and based on what I saw I had to teach them how recognize that multiplication and division were inverses of one another. So, yesterday I was teaching teaching my students how to subtract 2 digit numbers with regrouping using a place value chart while teaching my lesson I had an AHA! moment. First, I realized that I could introduce the concept of inverse operations while teaching subtraction because subtraction is the inverse of addition. Modeling the inverse concept was very simple and didn’t take much time. While I was modeling trading a ten from the tens place and regrouping it as 10 ones, I asked my students this question,”In addition problems how do we trade?” Their answers were, ” We trade 10 ones for a group of ten.” At that moment I introduced the term inverse operation. I then went on to explain that an inverse operation is when we do the opposite for subtraction than we do for addition. One of my students responded as if a light turned on, “It’s backwards?” I said yes it’s backwards! When he said that I instantly realized that he was attempting to make the connection and put it into his own words.I was so excited because I had finally found a picture and a conceptual approach that I could use to explain what an inverse operation looks like. After I introduced the concept of inverse operation, I then used it as an anchor to teach the students how to trade a 10 from the tens place and trade/regroup it as a group of 10 ones. Teaching complex concepts to my students is very challenging because the they lack the literacy proficiency that is needed for them to grasp the concept. When I know that I have to introduce a skill or concept that will be difficult for my students to understand I follow these steps: Connect a previous lesson to the new skill or front load my instruction with the foundation skill 2. Teach the new skill from simple to complex (begin with conceptual then move to the abstract) 3. Use a think aloud to explain my thought process 4. Look for patterns in the process that will support acquisition of the new skill I often see the fact family concept taught to students in 1st and 2nd grade with the numbers, but I sometimes wonder how the teacher introduced the concept to their students. Fact families are an example of inverse operations on a simpler level. The facts 2+3=5 and 5-3=2 are inverses of each other because the student has to put 2 parts plus 3 parts to equal 5 parts and in the subtraction fact a student would have to begin with the total (which is the opposite) and then subtract 3 parts. Teaching the inverse operation concept to struggling learners can be a challenge for any teacher, but if you remember to begin with a picture it will make your life much easier! If you are in need of a lesson to teach multiplication and division inverses I have a complete lesson plan in my store. PrevPreviousExplicit Teaching and Struggling Learners NextHow to Find the Right Intervention for Your Struggling LearnersNext Leave a Comment Cancel Reply I help math teachers who teach high need students to effectively manage their classrooms and deliver high quality math instruction. Learn more about me CONNECT WITH ME Facebook Twitter Instagram Youtube Join our free resource library for free math instruction lesson plans and materials. JOIN TODAY Facebook Twitter Instagram Youtube GET STARTED Join our free resource library for free math lesson plans and materials. JOIN TODAY IgnitED \| All rights reserved © 2025 IgnitED Create a new list
2105	https://pmc.ncbi.nlm.nih.gov/articles/PMC8593030/	Formulas and algorithms for the length of a Farey sequence - 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. PMC Search Update PMC Beta search will replace the current PMC search the week of September 7, 2025. Try out PMC Beta search now and give us your feedback. 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Learn more: PMC Disclaimer \| PMC Copyright Notice Sci Rep . 2021 Nov 15;11:22218. doi: 10.1038/s41598-021-99545-w Search in PMC Search in PubMed View in NLM Catalog Add to search Formulas and algorithms for the length of a Farey sequence Vladimir Sukhoy Vladimir Sukhoy 1 Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA Find articles by Vladimir Sukhoy 1, Alexander Stoytchev Alexander Stoytchev 1 Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA Find articles by Alexander Stoytchev 1,✉ Author information Article notes Copyright and License information 1 Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA ✉ Corresponding author. Received 2021 Apr 23; Accepted 2021 Sep 24; Collection date 2021. © The Author(s) 2021 Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit PMC Copyright notice PMCID: PMC8593030 PMID: 34782723 Abstract This paper proves several novel formulas for the length of a Farey sequence of order n. The formulas use different trade-offs between iteration and recurrence and they range from simple to more complex. The paper also describes several iterative algorithms for computing the length of a Farey sequence based on these formulas. The algorithms are presented from the slowest to the fastest in order to explain the improvements in computational techniques from one version to another. The last algorithm in this progression runs in 𝑂⁡(𝑛 2/3) time and uses only 𝑂⁡(√𝑛) memory, which makes it the most efficient algorithm for computing\|𝐹 𝑛\| described to date. With this algorithm we were able to compute the length of the Farey sequence of order 1 0 1 8. Subject terms: Mathematics and computing, Computer science Introduction Farey sequences1,2 are related to the theory of prime numbers and they show up in many different scientific disciplines. Their fundamental properties, e.g., the mediant property, can be described with basic algebra. At the same time, Farey sequences are linked to unsolved mathematical mysteries, e.g., the Riemann hypothesis3. These sequences are also tied to the Stern–Brocot tree, which could be used to find the best rational approximations for irrational numbers4. Recently, the elements of a Farey sequence have been linked to the singularities5 of the Inverse Chirp Z-Transform(ICZT), which is a generalization6 of the Inverse Fast Fourier Transform(IFFT). A Farey sequence of order n, which is denoted by , is a sequence formed by all irreducible fractions between 0 and 1 for which the denominator q is between 1 and n, i.e., . By convention, it is assumed that the elements of the sequence are sorted in increasing order. For example, the first five Farey sequences are: If , , and are any three adjacent elements of a Farey sequence, then the middle fraction is equal to the mediant2,7 of its neighbors and , i.e., This property has been known to mathematicians for centuries8,9, but it received a name only after Farey stated it formally in a paper1 that he published in 1816. Previously, Haros2,9 had used the mediant property in 1802 to generate the tables of irreducible fractions between 0 and 1 for which the denominator was less than 100 (see Ref.2,p.36). Cauchy published a formal proof of the mediant property10 for all n in 1816. As the order n of the Farey sequence increases, the length of grows as a quadratic function of n. More specifically, , where denotes the length of [2, p. 268] [7, p. 156] [4, p. 139]. However, no formula for computing the exact value of in O(1) time is known. The length of can be computed by enumerating its elements. Algorithm S1 in Supplementary SectionS1 gives the pseudo-code for an algorithm11 that uses the mediant property to enumerate all elements of a given Farey sequence. The algorithm also counts the elements and returns the sequence length. The computational complexity of this approach is , which makes it too slow and impractical for computing the value of for large n. The fast algorithms described in this paper do not use enumeration. Their computational complexities are summarized in Supplementary Sect.S2. Increasing the value of n only adds new elements to the Farey sequence without removing any of them. That is, is a subsequence of . To see this, consider the elements of the first five Farey sequences shown below, which are arranged with extra spacing between them. In this view the identical fractions are stacked vertically. The new elements that are added when the order is increased by 1 are highlighted in red. The length of each Farey sequence is shown at the end of each row. Each element of the set is a fraction that is irreducible and lies between 0 and 1. Thus, the difference between and is equal to the number of integers between 1 and n that are coprime with n. By definition, this number is equal to the value of Euler’s totient function . Therefore, an algorithm12 for computing the length of can also be used to calculate the sums of the values of Euler’s totient function for all integers between 1 and n. Figure1 lists the values of Euler’s totients , , , , and expresses the values of , , , , and in terms of these totients. This example also illustrates the following well-known formula for computing that is proven in Supplementary Sect.S3: 1 This equation implies that the value of can be added to to obtain , i.e., . Figure 1. Open in a new tab Visualization of the link between Euler’s totient function and the length of the Farey sequence . The value of can also be expressed using the following recursive formula13: 2 Supplementary Sections S4andS5 prove this formula using basic algebra, mathematical induction, and a property of Euler’s totient function that was proven by Gauss14. Algorithm S11 in Supplementary Sect.S9 implements formula(2) using recursion and optimizes the repeated recursive calls using memoization15. That algorithm runs in time, which is still too slow for large values of n. The algorithms described in this paper build upon these two formulas, extend them, and combine them in different ways. The next section summarizes the key mathematical insights that were used to derive the formulas. The rest of the paper describes the algorithms, proves their properties, and evaluates their run-time performance. Overview and formulas This paper describes five different algorithms for computing the length of the Farey sequence, where n is a parameter that specifies the order. The algorithms will be denoted with the letters A, B, C, D, and E. These letters will also be used as suffixes to form the names of the algorithms, e.g., F arey L ength A or F arey L ength B. This section briefly summarizes the formulas for the length of on which the algorithms are based. Algorithm A is based on the following well-known formula: 3 which was illustrated with an example in the introduction. The algorithm uses a modified linear sieve that returns a list of prime numbers and an array with the smallest prime factor for each integer. From these values, another function computes Euler’s totients, which are then summed up to compute the length of. This algorithm runs in O(n) time and uses O(n) memory. The formula for algorithm B is derived from(3) by splitting the sum into three separate sums: 4 where S is the set of -smooth integers in the interval and is the set of integers that are not -smooth in the same interval. Two helper algorithms are introduced to process the smooth and the non-smooth integers, which are formally defined in one of the next sections. The overall algorithm still runs in O(n) time, but uses only memory. Algorithm C is based on the following novel formula for the length of: 5 where . The formula works for all . Supplementary Sect.S6 proves this formula by splitting the sum in formula(2) at u(n) and then expressing the second sum in a different way. This leads to an algorithm that runs in time and uses memory. Algorithm D can be derived from formula(5) by splitting the first sum at as follows: 6 Reversing the direction of the last sum and changing the index variable from k to i we get: 7 where . To continue the derivation, let be the order of the Farey sequence in the last sum in formula(7), i.e., . Then, 8 This sum can be expressed using Euler’s totients for all integers in the intervals , where , , and for . To derive this result, the value of can be expressed as the value of plus the sum of Euler’s totients for all integers in the interval. That is, 9 Therefore, formula(8) can be expressed as follows: 10 Plugging the right-hand side of the last equation into formula(6) and that result into(5) leads to the following formula: 11 where , , and . Also, and . Furthermore, and for . The last term in(11) expands the value of as a sum of totients, which suggests how it can be computed iteratively. Because the index k in the last sum in(11) goes up to , the last value of is equal to . If the last sum is processed first, then the computed value of can be used to bootstrap the calculation of in the middle sum, e.g., see formula(9). These insights lead to algorithm D, which runs in time and uses memory. Algorithm E was inspired by a modified version of formula(11) that is shown below: 12 In other words, this modification splits one of the sums in(11) as follows: 13 where is the set of -smooth integers in the interval and is the set of integers that are not -smooth in the same interval. Thus, algorithm E uses a similar approach to summing the totients as in formula(4) that is used by algorithm B. In this case, however, the smooth and the non-smooth numbers are processed separately but the values of their corresponding totients are accumulated in the array B. This change is sufficient to reduce the memory complexity from to. The time complexity remains unchanged, i.e.,. Related work A prime sieve is an efficient algorithm for generating all prime numbers in some interval16. The sieve of Eratosthenes is probably the oldest and the most well-known prime sieve algorithm4,16,17. It can generate all prime numbers in the interval [1,n] in time. Our algorithms, however, do not use the sieve of Eratosthenes. Instead, they use the linear sieve18 or the sieve of Atkin19. These two sieves do not have the factor in their time complexity, i.e., they run in O(n) time. Algorithms A,B,D,and E use the linear sieve algorithm18, making it possible to compute Euler’s totients for all integers between 1 and n in O(n) time. In addition to the linear sieve, Algorithms B and E also use the sieve of Atkin19 to reduce the space complexity. Algorithm C does not use a sieve. The sieve of Atkin generates all prime numbers between 1 and N in O(N) time19. It uses memory19. Algorithm E uses this sieve with to enumerate numbers that are not -smooth in the interval. This sub-routine and other optimizations reduce its space complexity down to. This makes it practical to compute for very large values of n by using the computer’s memory more efficiently. Algorithm E enumerates the smooth numbers in separately from the non-smooth numbers, which are complementary to them. Smooth numbers20–22 are often used in computational number theory for primality testing, integer factorization, and computing discrete logarithms. Computing the length of the Farey sequence is related to several other problems. For example, by subtracting 1 from one can obtain12 the sum of all Euler’s totients between 1 and n. Some authors have described approaches for attacking the order statistics problem23,24 and the rank problem25 on Farey sequences. Their work is related to methods for summing the values of the Möbius function26, which is equivalent to computing the Mertens function. The time complexity class of these methods is. More recently, the best time complexity for computing the Mertens function was estimated27 at, i.e., slightly worse than . The time complexity of our most efficient algorithm for computing is exactly, i.e., without any additional small factors that depend on n. Its space complexity is. All estimates use the same computational model, which assumes that any arithmetic or storage operation on any integer runs in O(1)time and storing any integer requires O(1) memory. In digital signal processing, the Inverse Chirp Z-Transform (ICZT) is a generalization6 of the Inverse Fast Fourier Transform (IFFT). This transform is parametrized by the complex numbers A and W. They define a logarithmic spiral contour formed by the sampling points where and n is the size of the transform. For the special case when the magnitudes of both A and W are equal to 1, the contour is restricted to lie on the unit circle in the complex plane. In this case, the ICZT has a singularity5 if and only if the polar angle of W can be expressed as where is an element of. Consequently, the numerical error profile for the ICZT of size n is determined5 by the elements of. Therefore, the number of possible values of the parameter W for which the transform is singular is equal to the length of . Algorithm A Algorithm 1 shows the pseudo-code for the first algorithm that computes without enumerating the sequence elements. On line 2 the algorithm uses a linear sieve to compute a list P of all prime numbers in the interval [1,n]. The linear sieve also returns an array of size n such that its k-th element is equal to the smallest prime factor of k. On line 3 the elements of are used to compute an array that contains the values of Euler’s totient function . The rest of the code (i.e., lines 4–7) uses formula(3) to compute the value of, i.e., it sums up the n totients and adds 1 to the sum s. Supplementary SectionS3 gives the pseudo-code for the linear sieve algorithm and for the function C ompute T otients. The appendix also proves a property of Euler’s totient function that makes it possible to compute from the array in O(n) time. Thus, the computational complexity of Algorithm 1 is O(n). It uses O(n) memory. This algorithm is fairly easy to describe, but it is not very fast. It also uses a lot of memory. It is important to understand how it works, however, as some of the more efficient algorithms use the same approach to solve a part of the problem. Algorithm B This section describes another algorithm for computing that also runs in O(n) time, but uses only slightly more than memory. The main idea is to split all integers in the interval into two disjoint sets, S and , such that their elements and their corresponding Euler’s totients can be computed using memory instead of O(n). The set S is the set of -smooth numbers and the set is the set of numbers that are not -smooth. Both smooth and non-smooth numbers are integers. They are defined as follows: A positive integer N is B-smooth if its largest prime factor p does not exceed B, i.e., . A positive integer N is not B-smooth if its largest prime factor p is strictly greater than B, i.e., . For example, the number 100 is 5-smooth because . It is also 6-smooth, 7-smooth, 8-smooth, etc. But it is not k-smooth for . To give another example, the number 84 is 7-smooth because . It is also 8-smooth, 9-smooth, etc. But it is not 5-smooth because its largest prime factor is 7, which is greater than 5. In fact, it is not k-smooth for . Algorithm 2 starts by computing exactly using the method28 described in Supplementary Sect.S7. Next, it implements the summation of Euler’s totients in a way that avoids allocating an array of length n to store them. In fact, the algorithm uses only slightly more than memory. This is achieved by running the linear sieve only for . Starting from , the sum in formula(3) is split into three separate sums as follows: 14 where S is the set of -smooth numbers in the interval and is the set of numbers that are not -smooth in the same interval. A similar technique is used in Algorithm 8 to reduce its space complexity from to . The smooth numbers are processed with Algorithm 3. It enumerates the -smooth numbers in the interval. More specifically, the algorithm implements a depth-first traversal of the search space formed by all integers in this interval that have prime factorizations that include only prime factors from the array P, which contains the prime numbers in the interval . The algorithm calls the visitor function for each -smooth number m in the interval and its Euler’s totient, which is computed using formula(22) that is described in Supplementary Sect.S3. Algorithm 3 runs in time and uses memory. It traverses no more than values of m and performs O(1) operations for each of them. The space complexity of this algorithm is determined by the maximum depth of the stack, which cannot exceed because 2 is the smallest possible prime factor in the list P. The non-smooth numbers are handled by Algorithm 4. It traverses the numbers that are not -smooth in the interval and calls the visitor function for each non-smooth number and for all its integer multiples that fit in the interval. The computational complexity of this algorithm is determined by the sieve of Atkin, which runs in time and uses memory19. In other words, Algorithm 4 performs O(1) operations for each non-smooth number that it visits, which doesn’t change its time complexity class. Thus, the algorithm runs in time and uses memory. To give a concrete example, let . Figure2a shows the search tree for the -smooth numbers in the interval[5,20]. Algorithm 3 calls the visitor function for the pairs where k is 4-smooth (i.e., ) and is its totient. The totients are shown in Fig.2b. The algorithm calls the visitor function only for , i.e., only for the circled integers and totients shown in the figure. The tree in this case is binary because the list contains only two prime numbers. Thus, all left branches multiply the parent node by 2 and all right branches multiply it by 3. The order of enumeration is: 1, 2, 4, 8, 16, 12, 6, 18, 3, 9 (i.e., pre-order traversal). Figure 2. Open in a new tab Traversal of the 4-smooth numbers in the interval[5,20] by Algorithm 3 when . The tree of -smooth numbers is shown in(a). Their corresponding totients are shown in(b). The numbers for which the visitor function is called are circled. The list of prime numbers in this case is . Thus, each left branch in(a) multiplies the parent node by 2 and each right branch multiplies it by 3. If the product exceeds 20, then the corresponding branch or leaf is excluded from the tree. Figure3 visualizes the enumeration of the numbers that are not -smooth for . Algorithm 4 calls the sieve of Atkin to generate the prime numbers in the interval [5,20], i.e., 5,7,11,13,17,19. This is done one-at-a-time, i.e., without storing the list of primes in memory. The algorithm also calls the visitor function for each pair , where k is each prime number or its integer multiples that fit in the interval. The enumeration order here is: 5,10,15,20,7,14,11,13,17, and 19. Figure 3. Open in a new tab Enumeration of not 4-smooth numbers in the interval [5,20] by Algorithm 4 when . The tree of non-smooth numbers is shown in(a), their corresponding totients are shown in(b). The visitor function is called for all circled numbers. Table1 shows the values of Euler’s totient function for each k between 1 and 20. The sum of these totients is equal to 128. Therefore, . To compute this result, Algorithm 2 first adds . Then, it calls Algorithm 3, which computes the sum of all circled numbers in Fig.2b (i.e., ) and adds it to 7. Finally, it calls Algorithm 4, which computes the circled numbers in Fig.3b and adds their sum (i.e., 92) to the previous result. Thus, the return value for is . Table 1. Values of Euler’s totient function for all integers k between 1 and 20. k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16 6 18 8 Open in a new tab Algorithm C This section describes another algorithm for computing. Unlike algorithms A and B, it does not sum Euler’s totients. Instead, it uses formula(5), which expresses in terms of values of for . The algorithm uses this formula multiple times and stores the computed values for in a lookup table, which is described in Supplementary Sect. S8. Algorithm 5 gives the pseudo-code for algorithm C. First, the code sets the variables r and u to the upper limits of the two summations in formula(5), which are and u(n), respectively. Next, it creates an empty lookup table and initializes its first entry, which corresponds to . Then, the two for-loops fill the required entries in the lookup table in an order that makes it possible to eventually compute . The first for-loop computes in that order. The second for-loop computes . The value of is computed in the last iteration when the value of the variable j is equal to 1. Each iteration through line 7 or line 10 corresponds to an instance of formula(5). A helper function implements this computation. Its pseudo-code is listed in Algorithm 6. It has two for-loops that correspond to the two summations in formula(5). Each call to the helper function stores the computed value of in the lookup table so that it is available later on. Supplementary SectionS10 proves that the order in which the algorithm fills the lookup table entries is correct. It shows that computing each subsequent entry requires accessing only those lookup table elements that have already been set in previous iterations. The appendix also proves that Algorithm 5 runs in time and uses memory. Supplementary SectionS11 gives a recursive version of this algorithm. Figure4 visualizes the computation performed by algorithm C (i.e., Algorithm 5) for . This process uses formula(5) three times. Each tree shown in this figure corresponds to one call to the helper function. The root node is the term that the helper function stores in the lookup table. The three branches show the terms in formula(5) that the function adds to compute the value of. The first for-loop in algorithm C computes and stores its value in the lookup table. Figure4a visualizes the corresponding instance of formula(5) in which the middle branch is empty. The second for-loop computes the values of and . The corresponding trees are shown in Fig.4b and c, respectively. Note that there is no tree for because the algorithm stores its value in the lookup table during initialization, before the first for-loop. Also, note that the tree for uses the previously computed values for and . Figure 4. Open in a new tab Visualizations of the three instances of formula(5) in the execution of algorithm C when it is called with . In this case, the first for-loop of the algorithm invokes the formula to compute as shown in(a). The second for-loop computes and . The trees for these two terms are shown in(b)and(c), respectively. The result is . Supplementary SectionS10 proves that for each , every term that appears in a branch of some tree also appears as a root node for one of the preceding trees. In other words, the order in which the algorithm fills the lookup table is correct. Algorithm D This section describes the fourth algorithm for computing. Algorithm D can be viewed as an optimized version of algorithm C that runs faster, i.e., instead of time. This run-time improvement, however, is achieved at the expense of using more memory than algorithm C, i.e., instead of memory. The pseudo-code is given in Algorithm 7. The first for-loop in algorithm C computes the values of using the helper function, which implements formula(5). This process requires time. Because the orders of these Farey sequences form a contiguous range of integers, however, we can compute their lengths a bit faster. That is, they can be computed in time by adding Euler’s totients . These totients, in turn, can be generated in time using Algorithms S2 and S3. Thus, the computation performed by the first for-loop in algorithm C can be implemented so that it runs in instead of time. This optimization alone won’t affect the overall computational complexity because the second for-loop would still run in time. It cannot be optimized in the same way as the first for-loop because the orders of the Farey sequences do not form a contiguous range of integers. In addition to the optimization described above, it is also possible to change the split point between the two stages of computation so that the first stage computes more than entries of the lookup table. That is, instead of running the sieve-based approach up to, we can let it run for some number of additional iterations beyond that. Correspondingly, the starting value of the loop variable j in the second for-loop can be adjusted to be smaller than . Supplementary SectionS12 shows that the optimal running time is achieved when the first stage runs up to and when the last for-loop starts from instead of u(n). Algorithm D is the result of making these changes to algorithm C. The loop on lines 10–14 computes by using the linear sieve to generate an array of totients and then summing them similarly to what algorithm A does. The for-loop on lines 15–23 implements formula(10) to compute by adding the totients from the array. The last value is equivalent to because . The last for-loop (i.e., lines 24–26) computes using formula(5) that is implemented by the helper function. Supplementary AppendixS13 gives an alternative, shorter version of this algorithm in which the second for-loop is removed (i.e., lines 15–23). In that version, the first for-loop runs up to , instead of . This fills more entries in the lookup table than necessary for computing , but the space complexity remains in . The time complexity also remains in. Algorithm 7 was selected for presentation in the main paper because it is easier to map it to the formulas and because in this form it is easier to understand the optimization performed by algorithm E, which is discussed next. Algorithm E This section describes algorithm E, which computes the length of a Farey sequence of order n. Its pseudo-code is shown in Algorithm 8. Its time complexity is, which is the same as that of algorithm D. However, algorithm E uses only memory. This improvement makes it the most efficient algorithm for computing described in this paper. Algorithm E can be viewed as a modified version of algorithm D. It changes how the entries of the lookup table are computed for indices that fall in the interval , where and . The process of computing all other entries of the lookup table is similar for both algorithms. That is, both algorithms use the linear sieve to compute the totients and then add them to compute . Also, both algorithms use the helper function to compute the entries of the lookup table for indices that fall in the interval , including n. For indices in the interval, algorithm D uses the linear sieve. Because the value of is in, its memory complexity is . In contrast, algorithm E uses the linear sieve only in the interval and is in. For each integer m in, algorithm E calls the visitor function with arguments m and . More specifically, Algorithm 3 is used to process all -smooth numbers in that interval and Algorithm 4 is called to process all numbers that are not -smooth in the same interval. Algorithms 3 and 4 use only memory because is in . The visitor function determines the index i of the corresponding interval in formula(13) where m falls and then adds to the array element B[i]. After processing all integers in, the element B[i] becomes equal to the sum of Euler’s totients for all integers in . Then, the algorithm computes the values of by adding the elements of the array B to . The length of the array B is equal to , which is in. This reduces the amount of memory required to process all integers in the interval. Therefore, the space complexity of algorithm E is. Supplementary SectionS14 gives a formal proof for its time and space complexity. To give a concrete example, let’s look at how algorithm E computes for . In this case, . First, on lines 8–9 the algorithm uses the linear sieve to compute the values of . Supplementary SectionS3 describes this process and gives the pseudo-code for the two algorithms that implement it. Second, the loop on lines 10–14 adds these totients to compute the values of. Essentially, this is an implementation of formula(1). Next, on line 24, the algorithm processes the smooth numbers by invoking Algorithm 3. It calls a visitor function for each 10-smooth number in the interval , which in this case is equal to [11,24]. Figure5 visualizes this process using two trees. The circled nodes in Fig.5a show the 10-smooth numbers that the algorithm visits; the circled nodes in Fig.5b show their Euler’s totients. The nodes that are not circled correspond to the 10-smooth numbers that are less than 11 and for which the visitor function is not called in Algorithm 3. The Euler’s totients shown in Fig.5b are computed using formula(22). The algorithm visits the numbers in lexicographic order with respect to their prime factorizations, i.e., , , , , , , , and . Figure 5. Open in a new tab Enumeration of 10-smooth numbers and their Euler’s totients in the interval [11,24]. On line 25, the algorithm processes the non-smooth numbers by calling Algorithm 4, which visits the numbers that are not 10-smooth in the interval [11,24] and their Euler’s totients. Figure6 visualizes this process. Algorithm 4 uses the sieve of Atkin to generate the prime numbers in this interval, i.e., 11, 13, 17, and 23. For each of them the algorithm iterates over their integer multiples that fit in the interval. In this example, this results in visiting after 11, but before 13. Figure6b shows the totients for each of the non-smooth numbers in Fig.6a. Their values are computed using formula(22). Figure 6. Open in a new tab Enumeration of numbers that are not 10-smooth and their Euler’s totients in the interval [11,24]. The visitor function adds each circled totient from Figs.5b and 6b to the corresponding element of the array B. After visiting all integers in [11,24], the elements of B become equal to the sums of Euler’s totients in the corresponding sub-interval. Then, the algorithm sums them with to compute , , , , , and . More formally, Finally, on lines 34–36, the algorithm computes , , , and using the helper function as shown in Fig.7. The colors in this figure indicate the three different methods that the algorithm uses to compute the lengths of Farey sequences and store them in the lookup table. That is, green corresponds to line 13, purple to line 31, and red to line 35. Figure 7. Open in a new tab Visualization of the last stage of algorithm E in which the helper function is used to compute , , , and when the algorithm is called with . The three colors correspond to the three different methods for computing intermediate results and storing them in the lookup table. That is, green terms are set on line 13. Purple terms are processed on line 31. Finally, the red terms are computed on line 35, during the call to U pdate L ookup T able. Results We evaluated the performance of C and Python implementations of algorithms C,D,and E (i.e., Algorithms 5, 7, and 8). The value of n was varied between and in increments of 0.5 on the decimal logarithm scale. For each of the 29 values of n, each of the three algorithms, and each of the two implementations, the evaluation program ran the code 10 times while measuring its run time, memory usage, and the number of CPU instructions (see “Methods”). We also ran algorithm E for n between and and recorded the corresponding values of . Figure8 visualizes the results of the experiments. Fig.8a,b give the run time plots for C and Python, respectively. Fig.8c,d show the corresponding memory usage plots. All of them are on a log scale. The vertical coordinate of each point is obtained by averaging the results of the 10 code runs. Figure8e lists the slopes and intercepts for lines fitted to each of the 12 curves in (a)–(d) using least squares. The fitting process used the region between and . Finally, Fig.8f gives the values of for n between and computed using algorithm E. Figure 8. Open in a new tab Evaluation results for C and Python implementations of algorithms C, D, and E. The run time curves plotted in(a) and the memory usage curves plotted in(c) are for the C code. The run time and memory usage plots for the Python code are shown in (b)and(d). The plots use the log scale for both axes and each point represents the average of 10 runs. The table in(e) gives the slopes and intercepts for lines that were fitted to the twelve curves in (a)–(d) in the region between and . The table in(f) shows the lengths of computed with algorithm E for several large values of n. The experiments confirm the theoretical time complexities of the algorithms. For each of the three algorithms and for each of the two implementations, the slopes are close to the theoretical predictions, i.e., 0.75 for algorithm C and for algorithms D and E. The slope for algorithm C agrees with the theory up to the third digit after the decimal point. The slopes for the other two algorithms are close to the theoretical predictions, but slightly higher by about 0.01–0.02. In other words, as n increases, these algorithms slow down slightly more than the theory predicts. Even though this difference is small, we investigated it further by measuring the number of CPU instructions in addition to the run time. As described in Supplementary SectionS15, that metric agrees with the theory very well. The slight deviation of the run time from the theory is due to practical aspects of code execution, including cache misses and inaccurate branch predictions. On average, this leads to slower execution of CPU instructions. The number of executed instructions, however, agrees with the theory as confirmed by Supplementary Fig.S4. The reason for this is that for our algorithms the total number of executed instructions is not affected by the need to access the main memory more frequently due to more cache misses. This number is also not affected by the decreased efficiency of the instruction pipeline that results from more frequent branch mispredictions. For memory usage, the theoretically predicted slope is equal to 0.5 for algorithms C and E and for algorithm D. Figure8 shows that there is a good match between the empirical results and the theoretical predictions for these algorithms. The slopes deviate from the theoretical predictions by less than 0.01 and there does not appear to be a significant positive or negative bias. The slight jump in time and memory usage of the C version of algorithm E at is due to the implementation of the sieve of Atkin. For small n it is optimized to return values from a hard-coded list of small prime numbers. That is, in this case there is no sieving. In our experiments, more computationally intensive sieving starts only when n reaches . For the Python version, the jump in run time occurs earlier, i.e., between and . The reason for this is that our Python code runs the sieve of Atkin in a separate process. This process is started only when , which leads to a run time spike at that point. For smaller n, algorithm E does not call the sieve of Atkin and the Python code does not start the process. Conclusion This paper introduced several novel formulas for the length of a Farey sequence of order n. They extend two classic results and combine them in different ways to achieve various trade-offs between iteration and recurrence. The paper also studied the problem of how to efficiently compute. It described several algorithms that implement the formulas in ways that reduce both the computational time and the memory usage requirements. Our most efficient algorithm runs in time and uses only memory. These properties make it the most efficient algorithm for computing that has been described so far. Algorithm E is based on formula(12). Even though this formula is long, it leads to the fastest algorithm. It combines the computational optimizations and approaches used by the other algorithms described in the paper. More specifically, it uses the linear sieve to help compute the lengths of Farey sequences of orders up to. Next, it enumerates smooth and non-smooth numbers in the interval , where , , and . The sieve of Atkin is used to enumerate the non-smooth numbers separately from the smooth numbers in order to improve the run time and memory usage while computing the values of in that interval. In its final stage, algorithm E uses formula(5) several times to compute using previously computed values of for . This paper also showed that the empirical time and memory usage of the algorithms agree with the corresponding theoretical time and space computational complexities. The experiments also showed that with algorithm E it is possible to compute the length of the Farey sequence of order. In other words, this paper makes it possible to explore the properties of for larger n than was previously possible, given the same amount of computational resources. Future work could explore the applicability of other prime sieves29,30, some of which may be faster and more compact than the sieves used in our algorithms. However, the time and space complexities of our most efficient algorithm are not tied directly to the prime sieves. They result from a combination of theoretical insights and computational techniques. In other words, merely switching to a more efficient prime sieve may not result in better time or space complexity without other changes. Methods The experiments evaluated the run time and memory usage of algorithms C, D, and E (i.e., Algorithms 5,7,and 8). This was done for two different implementations of the algorithms and their dependencies: one written in C and another in Python. We used version 4.4.7 of GCC, the default C compiler on the experimental platform, to build the C code. It generated native binaries using the 64-bit version of the x86 instruction set. The performance of our Python code was measured with version 3.9.0 of the CPython interpreter31, which serves as the reference implementation of the Python language. Run time measurements We used the same high-level Python script to run the evaluation and collect the time and memory usage measurements in all experiments. Each instance of each algorithm ran in a separate process that the evaluation script spawned before running the algorithm. This process exited after completing the run and transmitting the time and memory measurements to the evaluation script. For the Python implementation of the algorithms, this process called the corresponding Python function directly. For the C implementation, we compiled the C code into a shared library from which the evaluation process invoked the corresponding function using Python’s ‘ctypes’ module. In all cases, the run time of an algorithm was measured using the function ‘time.process_time’ in Python 3, i.e., by subtracting the return value of this function recorded right before launching the algorithm from its value returned right after the algorithm’s completion. We disabled Python’s garbage collector by calling ‘gc.disable()’ before starting an evaluation run. In other words, the run time measurements don’t include the time that Python would normally spend on garbage collection, because the garbage collector was disabled. Memory usage measurements The memory usage of an algorithm was measured by subtracting the peak amount of physical memory used by the spawned process (after initialization but before launching the algorithm) from the peak amount of physical memory recorded after its completion. The memory usage plots report the value of this difference in kilobytes. The script obtained these values from the ‘VmHWM’ record in the special file ‘/proc/[pid]/status’ provided32 by our GNU/Linux system, where ‘[pid]’ is the process identifier. The evaluation script also monitored the number of virtual memory pages transferred from the physical RAM to the designated swap storage on the computer. We used the ‘pswpout’ record in the file ‘/proc/vmstat’ made available by the OS. During the experiments, its value remained the same before and after running each instance of each algorithm, which implies that all virtual memory pages that our program used during these time intervals remained resident in RAM. Counting CPU instructions We used the perf-stat utility33 provided by the OS to count the number of central processing unit (CPU) instructions that the code executed. Each process spawned by the evaluation script launched the command ‘perf stat -e instructions -p [pid]’, where ‘[pid]’ was its process identifier. The standard output of perf-stat was redirected to a temporary file. The monitoring process started after initialization but before running the designated algorithm. After the algorithm finished, the spawned process sent the SIGINT signal34 to the perf-stat process, which wrote the measurements to its standard output before exiting. The printed text included the number of instructions executed by the spawned process while perf-stat was monitoring. This information, together with the run time and memory usage statistics, was stored for subsequent analysis. Averaging results from multiple runs The logarithms of the run time, the memory usage, and the number of instructions were averaged over 10 independent runs. This was done for each of the two implementations (i.e., C and Python) of the three algorithms (i.e., C, D, and E). The order of the 29 values of n between and used for the plots was randomized independently in each run. Multiple instances of the algorithms ran in parallel on our server to the extent that they could fit in the available memory without swapping. The number of simultaneously running algorithm instances never exceeded 15. The number of cores on the machine was 32. In other words, there were at least 2 cores available for each algorithm at run time. Experimental platform All results were computed on a 32-core Dell PowerEdge R720 server with 315 gigabytes of RAM. The processor on this machine was 2.20 GHz Intel Xeon E5-2660. The operating system was Red Hat Enterprise Linux (RHEL) version 6.10. Native and long integers in Python Python uses a unified implementation of integer arithmetic that automatically switches from native to long integers when necessary to avoid overflow35. On the experimental platform, the size of a native integer was equal to 64 bits. In other words, the Python interpreter automatically switched to long integers whenever it encountered an integer less than or greater than. This switch occurs for . 128-bit integers in C Our C code used 128-bit integers for all integer values that would not fit in 64 bits for large n. More specifically, we built the C code using GCC 4.4.7 and used the ‘__uint128_t’ unsigned 128-bit integer type provided by this compiler. Computational model The theoretical model used for estimating the computational complexity of the algorithms assumes that adding, subtracting, multiplying, dividing, and storing any integer requires O(1) time. Similarly, the model assumes that the size of an integer is also in O(1). The 128-bit integers used in the C code were sufficiently large to avoid overflow in all experiments. The Python interpreter automatically switched to larger integers when necessary35. The sieve of Atkin The experiments used a C implementation of the sieve of Atkin from the ‘primegen’ package36, version 0.97, which supports generating prime numbers in any interval where . This bound was sufficiently large for each of our experiments. This implementation uses a relatively small fixed-size static memory buffer instead of dynamic memory allocation. That is, even though in theory the sieve of Atkin requires memory, in practice the memory usage of this particular implementation was constant. For the memory usage for computing was dominated by the arrays used by our algorithms, i.e., the memory used by the sieve was only a tiny fraction of all memory used by the code. Supplementary Information Supplementary Information. (339.5KB, pdf) Supplementary Source Code. (102.8KB, zip) Author contributions V.S. developed the algorithms, wrote the evaluation code, and generated the tables and the figures. A.S. designed the scope of the study and the structure of the paper. A.S. advised on all experiments and supervised the work. Both authors wrote the paper. Data availability All data and procedures are described in the main paper or in the supplementary information. Competing interests The authors declare no competing interests. Footnotes Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Supplementary Information The online version contains supplementary material available at 10.1038/s41598-021-99545-w. References 1.Farey J. On a curious property of vulgar fractions. Philos. Mag. 1816;47:385–386. doi: 10.1080/14786441608628487. [DOI] [Google Scholar] 2.Hardy G, Wright E. An Introduction to the Theory of Numbers. 4. Oxford University Press; 1975. [Google Scholar] 3.Borwein P, Choi S, Rooney B, Weirathmueller A. The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike. Springer; 2007. pp. 48–49. [Google Scholar] 4.Graham R, Knuth D, Patashnik O. Concrete Mathematics. 2. Addison-Wesley; 1994. [Google Scholar] 5.Sukhoy V, Stoytchev A. Numerical error analysis of the ICZT algorithm for chirp contours on the unit circle. Sci. Rep. 2020;10:4852. doi: 10.1038/s41598-020-60878-7. [DOI] [PMC free article] [PubMed] [Google Scholar] 6.Sukhoy V, Stoytchev A. Generalizing the inverse FFT off the unit circle. Sci. Rep. 2019;9:14443. doi: 10.1038/s41598-019-50234-9. 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Articles from Scientific Reports are provided here courtesy of Nature Publishing Group ACTIONS View on publisher site PDF (2.8 MB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract Introduction Overview and formulas Related work Algorithm A Algorithm B Algorithm C Algorithm D Algorithm E Results Conclusion Methods Supplementary Information Author contributions Data availability Competing interests Footnotes Supplementary Information References Associated Data 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
2106	https://www.euclideanspace.com/maths/discrete/groups/categorise/finite/triangle/index.htm	Maths - Equilateral Triangle Transforms Dihedral Group D3 In this example lets take the mapping of the equilateral triangle to itself, that we looked at in the introduction, but this time we will also include reflections. So we end up with the identity, 2 possible rotations and 3 possible reflections. This is all the possible permutations so this is a symmetry group: \| \| \| \| \| --- --- \| \| \| identity (0°) \| anticlockwise (120°) \| clockwise (240°) \| \| permutation \| \| \| \| \| cycle notation \| (1)(2)(3) \| (1 2 3) \| (3 2 1) \| \| \| reflect in 1 - 23 \| reflect in 2 - 13 \| reflect in 3 -12 \| \| permutation \| \| \| \| \| cycle notation \| (1)(2 3) \| (2)(1 3) \| (3)(1 2) \| So, what is the minimum set of generators to generate this group? We can generate the rotations by combining two reflections: \| \| \| \| \| \| \| \| --- --- --- \| \| = \| \| \| = \| \| \| \| \| = \| \| \| = \| \| \| So the 3 reflections will generate the group, but we don't need 3 because once we have one rotation we can generate the other reflection. So 2 permutations are needed to generate the whole group, we can either use: 2 different reflections, or 1 reflection and 1 rotation however we can't use 2 rotations because rotations form a self contained subset of this group. So, as an example, lets choose the generators to be 'a' and 'b' where: a = ma reflection through in 1 - 23 b = ma reflection through in 2 - 13 We can generate all the permutations as follows: e = 1 ma = a mb = b mc = aba ra = ba rb = ab Cayley Table The complete table is: \| \| \| \| \| \| \| \| --- --- --- \| \| 1 \| a \| b \| aba \| ba \| ab \| \| 1 \| \| \| \| \| \| \| \| a \| \| \| \| \| \| \| \| b \| \| \| \| \| \| \| \| aba \| \| \| \| \| \| \| \| ba \| \| \| \| \| \| \| \| ab \| \| \| \| \| \| \| or we can show this completely in terms of the generators: \| \| \| \| \| \| \| \| --- --- --- \| \| 1 \| a \| b \| aba \| ba \| ab \| \| 1 \| 1 \| a \| b \| aba \| ba \| ab \| \| a \| a \| 1 \| ab \| ba \| aba \| b \| \| b \| b \| ba \| 1 \| ab \| a \| aba \| \| aba \| aba \| ab \| ba \| 1 \| b \| a \| \| ba \| ba \| b \| aba \| a \| ab \| 1 \| \| ab \| ab \| aba \| a \| b \| 1 \| ba \| Ways to define the structure of this group What is the minimum information required to define the structure of this group? We could: Specify each permutation in the group by using the 3 arrows as shown above for each permutation. Specify each permutation in the group by using the cycle notation for each permutation. By the Cayley table for the group. Another way to define the group is the 'Presentation of the group', as explained on this page, for this example it is: Γ< a,b \| aa,bb,bab=aba> We could have set all relations to be equal to unity by replacing bab=aba with baba-1b-1a-1=1 Can we use this to generate all the entries in the table and thus competely define the structure? Lets take an example of multiplying 'ab' by 'ab': ab ab = abab [associativity] = aaba [since bab=aba] = ba [since aa=1] Doing this for all the other entries in the Cayley table shows that this competely defines the group. How can we work out the minimum set of relations that will define the group? Cayley Graph Upto now we have chosen to use two reflections for the generators but it is more common to use 1 rotation and 1 reflection because can be more easily generalised to an 'n' sided dihedral. This is given by one cycle going clockwise and another going anticlockwise connected by the reflections. Here are Cayley graphs for the case where the generator is 1 rotation + 1 reflection and the case where the generator is 2 reflections. \| Generator \| 2 reflections \| 1 rotation + 1 reflection \| --- \| Presentation \| < a,b \| aa,bb,bab=aba> \| < a,b \| aa,bbb,a-1ba=b-1> \| \| Graph \| \| \| Given two Cayley graphs can we find if they represent the same group? Sandwich Product The form y x y-1 has some interesting effects: For instance if y is a rotation and x is a reflection, then we get a different reflection \| \| \| \| \| \| --- --- \| \| \| \| = \| \| Or if x is a rotation and y is a reflection, then we get the inverse rotation. \| \| \| \| \| \| --- --- \| \| \| \| = \| \| So: m r m -1 = r -1 EuclideanSpace home Group Theory Rotating a Cube Duality Successive Rotations Transforms \| \| \| --- \| \| metadata block \| \| \| \| \| \| see also: \| To find out more about this subject search for 'Combinatorial Group Theory' (don't confuse with combinatorics which is a different branch of mathematics). \| \| Correspondence about this page \| \| \| Book Shop - Further reading. Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them. \| \| \| \| --- \| \| \| The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you. \| \| \| Terminology and Notation Specific to this page here: \| \| This site may have errors. Don't use for critical systems. Copyright (c) 1998-2023 Martin John Baker - All rights reserved - privacy policy.
2107	https://www.youtube.com/watch?v=NkGvw18zlGQ	Two-sample t test for difference of means \| AP Statistics \| Khan Academy Khan Academy 9090000 subscribers 4030 likes Description 562342 views Posted: 29 Mar 2018 Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: Hypothesis test for the difference between means of two populations. View more lessons or practice this subject at AP Statistics on Khan Academy: Meet one of our writers for AP¨ Statistics, Jeff. A former high school teacher for 10 years in Kalamazoo, Michigan, Jeff taught Algebra 1, Geometry, Algebra 2, Introductory Statistics, and AP¨ Statistics. Today he's hard at work creating new exercises and articles for AP¨_ Statistics. Khan Academy is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. 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Donate here: Volunteer here: 127 comments Transcript: kaito grows tomatoes in two separate fields when the tomatoes are ready to be picked he is curious as to whether the sizes of his tomato plants differ between the two fields he takes a random sample of plants from each field and measure the and measures the heights of the plants here is a summary of the results so what i want you to do is pause this video and conduct a two sample t test here and let's assume that all of the conditions for inference are met the random condition the normal condition and the independence condition and let's assume that we are working with a significance level of 0.05 so can pause the video and conduct the two sample t test here to see whether there's evidence that the sizes of tomato plants differ between the fields all right now let's work through this together so like always let's first construct our null hypothesis and that's going to be the situation where there is no difference between the mean sizes so that would be that the mean size in field a is equal to the mean size in field b now what about our alternative hypothesis well he wants to see whether the sizes of his tomato plants differ between the two fields he's not saying whether a is bigger than b or whether b is bigger than a and so his alternative hypothesis would be around his suspicion that the mean of a is not equal to the mean of b that they differ and to do this two sample t test now we assume the null hypothesis we assume our null hypothesis and remember we're assuming that all of our conditions for inference are met and then we want to calculate a t statistic based on this sample data that we have and our t statistic is going to be equal to the differences between the sample means all of that over our estimate of the standard deviation of the sampling distribution of the difference of the sample means this will be the sample standard deviation from sample a squared over the sample size from a plus the sample standard deviation from the b sample squared over the sample size from b and let's see we have all the numbers here to calculate it this numerator is going to be equal to 1.3 minus 1.6 1.3 minus 1.6 all of that over the square root of let's see the standard deviation the sample standard deviation from the sample from field a is 0.5 if you square that you're going to get 0.25 and then that's going to be over the sample size from field a over 22 plus 0.3 squared so that is 0.3 squared is 0. all of that over the sample size from field b all that over 24. the numerator is just going to be negative 0.3 negative 0.3 divided by the square root of .25 divided by 22 plus .09 divided by 24 and that gets us negative 2.44 approximately negative 2.44 and so if you think about a t distribution and we'll use our calculator to figure out this probability so this is a t-distribution right over here this would be the assumed mean of our t-distribution and so we got a result that is negative we get a t statistic of negative 2.44 so we're right over here so this is negative 2.44 and so we want to say what is the probability from this t distribution of getting something at least this extreme so it would be this area and it would also be and it would also be this area if we got 2.44 above the mean it would also be this area and so what i could do is i'm going to use my calculator to figure out this probability right over here and then i'm just going to multiply that by 2 to get this one as well so the probability of getting a t value i guess i could say where it's absolute value is greater than or equal to 2.44 is going to be approximately equal to i'm going to go to second distribution i'm going to go to the cumulative distribution function for our t distribution click that and since i want to think about this tail probability here and then i'm just going to multiply by 2. the lower bound is a very very very negative number you could view that as functionally negative infinity the upper bound is negative 2.44 negative 2.44 and now what's our degrees of freedom well if we take the conservative approach it'll be the smaller of the two samples minus one well the smaller of the two samples is 22 and so 22 minus 1 is 21. so put 21 in there 2 21 and now i can paste and i get that number right over there and if i multiply that by 2 because this just gives me the probability of getting something lower than that but i also want to think about the probability of getting something 2.44 or more above the mean of our t distribution so times 2 is going to be equal to approximately 0.024 so approximately 0.024 and what i want to do then is compare this to my significance level and you can see very clearly this right over here this is equal to our p-value our p-value in this situation our p-value in this situation is clearly less than our significance level and because of that we said hey assuming the null hypothesis is true we got something that's a pretty low probability below our threshold so we are going to reject reject our null hypothesis which tells us that there is so this suggests this suggests the alternative hypothesis that there is indeed a difference between the sizes of the tomato plants in the two fields
2108	https://www.statpearls.com/nursepractitioner/ce/activity/97464/?specialty=specialty	CE Activity \| Superior Vena Cava Syndrome \| NPs PhysicianPhysician Board ReviewsPhysician Associate Board ReviewsCMELifetime CMEFree CMEMATE and DEA ComplianceCPD StudentUSMLE® Step 1USMLE® Step 2USMLE® Step 3COMLEX-USA® Level 1COMLEX-USA® Level 2COMLEX-USA® Level 396 Medical School ExamsStudent Resource Center NCLEX - RNNCLEX - LPN/LVN/PN24 Nursing Exams Nurse PractitionerAPRN/NP Board ReviewsCNS Certification ReviewsCE - Nurse PractitionerFREE CE NurseRN Certification ReviewsCE - NurseFREE CE PharmacistPharmacy Board Exam PrepCE - Pharmacist AlliedAllied Health Exam PrepDentist ExamsCE - Social WorkerCE - Dentist Point of Care Institutional Sales Student Resources Search Sign-upLogin PhysicianPhysician Board ReviewsPhysician Associate Board ReviewsCMELifetime CMEFree CMEMATE and DEA ComplianceCPD StudentUSMLE® Step 1USMLE® Step 2USMLE® Step 3COMLEX-USA® Level 1COMLEX-USA® Level 2COMLEX-USA® Level 396 Medical School ExamsStudent Resource Center NCLEX - RNNCLEX - LPN/LVN/PN24 Nursing Exams Nurse PractitionerAPRN/NP Board ReviewsCNS Certification ReviewsCE - Nurse PractitionerFREE CE NurseRN Certification ReviewsCE - NurseFREE CE PharmacistPharmacy Board Exam PrepCE - Pharmacist AlliedAllied Health Exam PrepDentist ExamsCE - Social WorkerCE - Dentist Point of Care Free CME/CE Superior Vena Cava Syndrome Home Nurse Practitioner CE Superior Vena Cava Syndrome Overview 4.8 out of 5 (56 Reviews) Credits 1.50 Pharmacology Credits 0.5 Post-Assessment Questions 10 Start Date 1 Sep 2023 Last Review Date 1 Sep 2023 Expiration Date 31 Aug 2026 Estimated Time To Finish 90 Minutes Start This Activity Unlimited Nurse Practitioner CE Stay up to date on the latest medical knowledge with 6633 CE activities. In these online self-assessment activities, read our reference articles and test your knowledge with more than 7901.5 hours of CE. Single Activity Take this single activity $73.50 1 activity Buy Now 6 Month Unlimited Nurse Practitioner CE Access to all the Unlimited Nurse Practitioner CE activities in all specialties. $299 per half year per user Buy Now 1 Year Unlimited Nurse Practitioner CE Access to all the Unlimited Nurse Practitioner CE activities in all specialties. $479 per 1 year per user Buy Now Need Help?If you have a system or content concerns, please contactsupport@statpearls.com Activity Description Superior vena cava (SVC) syndrome is a collection of clinical signs and symptoms resulting from partial or complete obstruction of blood flow through the SVC. This obstruction is most commonly a result of thrombus formation or tumor infiltration of the vessel wall. The superior vena cava is formed by the junction of the left and right innominate (brachiocephalic) veins and is tasked with returning blood from the head, neck, upper extremities, and torso to the heart. Today, this syndrome is most commonly seen secondary to malignancy, although there has been a more recent rise in benign etiologies. This activity describes the causes, pathophysiology, and presentation of superior vena cava syndrome and highlights the role of the interprofessional team in its management. Target Audience This activity has been designed to meet the educational needs of physicians, physician associates, nurses, pharmacists, and nurse practitioners. Learning Objectives At the conclusion of this activity, the learner will be better able to: Evaluate the causes of superior vena cava syndrome. Identify the pathophysiology of superior vena cava syndrome. Assess the treatment options for superior vena cava syndrome. 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2109	https://ocw.mit.edu/courses/18-700-linear-algebra-fall-2013/	Browse Course Material Course Info Instructor David Vogan Departments Mathematics As Taught In Fall 2013 Level Undergraduate Topics Mathematics Linear Algebra Learning Resource Types assignment Problem Sets Download Course search GIVE NOW about ocw help & faqs contact us 18.700 \| Fall 2013 \| Undergraduate Linear Algebra Course Description This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with 18.06 Linear Algebra, more emphasis is placed on theory and … This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with 18.06 Linear Algebra, more emphasis is placed on theory and proofs. Course Info Instructor David Vogan Departments Mathematics Topics Mathematics Linear Algebra Learning Resource Types assignment Problem Sets Gram–Schmidt process which is a method for orthonormalising a set of vectors in an inner product space. (Image from public domain.) Download Course Over 2,500 courses & materials Freely sharing knowledge with learners and educators around the world. Learn more © 2001–2025 Massachusetts Institute of Technology Accessibility Creative Commons License Terms and Conditions Proud member of: © 2001–2025 Massachusetts Institute of Technology You are leaving MIT OpenCourseWare Please be advised that external sites may have terms and conditions, including license rights, that differ from ours. MIT OCW is not responsible for any content on third party sites, nor does a link suggest an endorsement of those sites and/or their content. Continue
2110	https://www.sciencedirect.com/topics/immunology-and-microbiology/antiarrhythmic-activity	Antiarrhythmic Activity - an overview \| ScienceDirect Topics Skip to Main content Journals & Books Antiarrhythmic Activity In subject area:Immunology and Microbiology Antiarrhythmic activity refers to the effects of beta-blockers that contribute to the prevention of arrhythmias in patients with myocardial infarction, which may be due to their ability to reduce myocardial oxygen demand and prolong the diastolic period, thereby enhancing myocardial perfusion. AI generated definition based on:Muscle, 2012 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: Biochemistry, Genetics and Molecular Biology Medicine and Dentistry Pharmacology, Toxicology and Pharmaceutical Science Discover other topics 1. On this page On this page Definition Chapters and Articles Related Terms Recommended Publications On this page Definition Chapters and Articles Related Terms Recommended Publications Chapters and Articles You might find these chapters and articles relevant to this topic. Chapter Antiarrhythmic Drugs 2017, Pharmacology and Therapeutics for Dentistry (Seventh Edition)Frank J. Dowd Antiarrhythmic Drugs Antiarrhythmic drugs are used to modify or restore to normal, aberrant electrophysiologic properties of cardiac muscle. Arrhythmias may result from various disease conditions or drug treatments. In all arrhythmias, some facet of the normal electrophysiologic system that governs cardiac contraction is behaving abnormally. Several methods of treating arrhythmias are used today. Nonpharmacologic interventions for cardiac arrhythmias include electrical cardioversion, automatic implantable cardioversion/defibrillation devices, ablation therapy, and pacemakers. The type of arrhythmia that presents is a major factor in the selection of an antiarrhythmic drug. Enhanced impulse generation can be reduced by drugs that slow phase 4 depolarization by reducing the inward Na+ current or the inward Ca 2+ current. The treatment of reentry includes drugs that reduce Na+ channel and Ca 2+ channel activity, which in turn slows conduction velocity. Drugs that block K+ channels, prolonging repolarization and the refractory period, may also be useful. Drugs used in the treatment of cardiac arrhythmias are not easily classified because they often have more than one action. Drugs within each class vary in their magnitudes of action or types of effects produced. The most common scheme classifies drugs according to certain specific properties. Type I drugs, such as quinidine, lidocaine, and flecainide, depress Na+ current. The type I agents are subdivided further according to their relative effects on phase 0 depolarization, conduction velocity, and action potential duration. Na+ channels exist in at least three states: closed, open, and inactivated. At resting membrane potentials, the Na+ channels are closed except for a Na+ “leak” associated with phase 4 depolarization in the cells that display automaticity. During rapid depolarization (phase 0, especially in Purkinje fibers and ventricular muscle), the Na+ channels are open. The Na+ channels then convert to the inactivated state before returning to the resting, closed state. The inactivated state occurs mostly in phases 2 and 3 of the action potential. Class IA and IC drugs bind more selectively to the open state of the channel. Class IB drugs bind more selectively to the inactivated state of the channel. Because the Purkinje fibers and ventricular myocardial cells have longer plateau phases (phase 2), class IB drugs are able to block Na+ channels more effectively in these tissues since the Na+ channels remain in an inactivated state longer during systole. Inasmuch as ischemic ventricular tissue is more depolarized, it, too, is especially sensitive to Na+ channel blockade by class IB drugs. Quinidine-like, or class IA, drugs depress phase 0 depolarization at all heart rates. They prolong the action potential duration of the ventricle because they also inhibit K+ (chiefly Kr) channels (Fig. 19-6). Class IB agents, such as lidocaine, block Na+ channels more selectively, but the rapid onset and recovery of Na+ channel blockade results in little accumulated lidocaine effect on phase 0 and conduction velocity in healthy tissue at normal heart rates. In damaged or rapidly firing cells, lidocaine causes a frequency-dependent or use-dependent block to reduce the slope of phase 0 and lowers the slope of phase 4 in ectopic pacemakers and in Purkinje fibers under high sympathetic tone. The faster the heart rate, the greater is the effect of lidocaine. (Use-dependent block is discussed in Chapter 14.) In contrast to other class I agents, lidocaine and related class IB antiarrhythmics may actually shorten the action potential duration. Flecainide and other class IC antiarrhythmics are characterized by their profound depression of phase 0 depolarization and slowing of conduction in the atria, AV node, and ventricles at normal heart rates. This pronounced effect results from their slow dissociation from Na+ channels and accumulation of the channel-blocking effect over several contraction cycles. There is little or no prolongation of the action potential duration. Metoprolol and related β-adrenergic–blocking agents constitute class II drugs and inhibit cardiac stimulation brought on by β-adrenergic agonists. They depress the slope of phase 4 depolarization (see Fig. 19-6). The class III group drugs, including amiodarone and sotalol, block K+ channels (chiefly Kr channels) and prolong the action potential duration by delaying phase 3 repolarization. Diltiazem and Amlodipine and other class IV drugs selectively block Ca 2+ channels (L type) and depress slow fiber conduction (phase 0 of the SA and AV nodes) and depress the slope of phase 4 depolarization (see Fig. 19-6). Drugs that cannot be classified by the Vaughan-Williams scheme include digitalis and adenosine. Table 19-1 outlines the various categories of antiarrhythmic agents. The drugs vary widely in their clinical usefulness. Class IA drugs are almost never used today, partly because of the introduction of class IC and class III drugs. In Table 19-1, the action responsible for the classification of each drug, which is usually its major action, is enclosed in a box. In the discussion of individual agents that follows, reference should also be made to Table 19-2 for the electrophysiologic actions of representative antiarrhythmic drugs. The net effects of the relevant drug classes and various action potentials in the heart are shown in Figure 19-6. Pharmacokinetic data for specific drugs are given in Table 19-3. The use of digoxin for certain kinds of arrhythmias is discussed in Chapter 20. Quinidine Although rarely used today, Quinidine is a useful drug to discuss as an example of class IA antiarrhythmic drugs. Historically, quinidine has been effective in the treatment of some atrial and (to a lesser extent) ventricular tachyarrhythmias. It was discovered that during treatment of patients with malaria, the reversal of atrial fibrillation was noted in some of them. Quinidine, the d isomer of quinine, is found in the bark of the cinchona tree, which is indigenous to certain regions of South America. Pharmacologic effects Quinidine reduces automaticity and conduction velocity and increases refractoriness. Quinidine tends to inhibit reentrant pathways. Automaticity is depressed through a more positive threshold potential and a decrease in the slope of spontaneous diastolic depolarization (phase 4) in pacemaker fibers, particularly at sites other than the SA node. Quinidine has the potential to slow or abolish tachyarrhythmias. Quinidine decreases the slope of phase 0 depolarization and decreases conduction velocity in cells such as those of the AV node and ventricular myocardium (see Fig. 19-6). Quinidine influences automaticity and conduction velocity by blocking Na+ channels, particularly channels in the open state. The rate of recovery from quinidine block is intermediate between class IB and IC antiarrhythmic drugs (see Table 19-1). Lidocaine Lidocaine has been used as a local anesthetic for more than half a century. In contrast to procaine, however, lidocaine has also long been a primary drug for arresting and preventing certain ventricular arrhythmias in emergency situations. For additional discussion of its pharmacologic characteristics, see Chapter 14. Pharmacologic effects Lidocaine decreases automaticity but is devoid of antimuscarinic activity. Lidocaine preferentially influences ventricular function. Lidocaine acts by blocking Na+ channels, particularly inactivated Na+ channels. This effect on Na+ channels is rapidly reversed, which restricts its use-dependent blocking effect to patients with rapid heart rates. Because lidocaine has a preferential effect on Na+ channels in the inactivated state, it selectively inhibits automaticity in ischemic tissue where membrane depolarization or an enhanced frequency of excitation occurs, such as in the His-Purkinje system (see Table 19-2). Lidocaine also reduces delayed afterdepolarizations seen with digoxin toxicity. Lidocaine does not slow repolarization but instead may hasten it. The drug has little effect on conduction velocity and phase 0. Lidocaine is usually administered intravenously for the treatment of ventricular ectopic rhythms. Because lidocaine must be administered parenterally, it is largely restricted to emergency situations and hospital settings. Its use is contraindicated in supraventricular arrhythmias because it is largely ineffective against these arrhythmias and excessive ventricular rates may result. Absorption, fate, and excretion After intravenous administration, the plasma concentration initially decreases rapidly, followed by a slower decline. For this reason, various loading regimens are used to achieve therapeutic plasma concentrations quickly. Lower constant perfusion rates are subsequently used. Lidocaine is broken down in the liver to various metabolites. Adverse effects Lidocaine exhibits only minor effects on the autonomic nervous system. Arterial pressure is not depressed. After acutely high dosages or prolonged infusion, lidocaine may cause convulsions and respiratory depression. Note that these reactions rarely occur with the dosages and routes of administration used in dentistry, especially if proper precautions are taken to avoid intravenous administration. Cardiac arrest may occur if lidocaine is administered to a patient with higher degrees of heart block. Flecainide A third category of type I antiarrhythmics (class IC) is represented by drugs that are newer to clinical use than the drugs in classes IA and IB. Flecainide (Fig. 19-7) belongs to this class (see Table 19-1). Flecainide is indicated for the prevention of supraventricular arrhythmias including atrial fibrillation and for sustained life-threatening ventricular arrhythmias unresponsive to other medications. Pharmacologic effects Although possessing some similarity to lidocaine, flecainide and related class IC drugs significantly depress conduction velocity by strongly reducing Na+ conductance during phase 0 of the action potential. This effect is felt throughout the heart but is especially strong in the atrium and His-Purkinje system. It results from the slow association of drug with, and dissociation from, Na+ channels, especially channels in the open configuration. Recovery from Na+ channel blockade is protracted. Flecainide does not selectively reduce phase 0 in diseased tissues. Rather, it inhibits phase 0 more or less uniformly in diseased and healthy tissues and tends to be effective on reentry mechanisms. Flecainide widens the QRS complex. Flecainide also reduces conduction velocity in the AV node but to a lesser degree than in ventricular muscle (see Fig. 19-6 and Table 19-2). Absorption, fate, and excretion Flecainide is not significantly metabolized on its first pass through the liver and has good bioavailability after oral administration. Approximately 75% of a flecainide dose is eventually metabolized to inactive products. The large range in reported half-lives for the drug stems partly from genetically determined variations in the rate of hepatic metabolism by cytochromeCYP2D6. As discussed in Chapter 4, some individuals lack this enzyme or have a CYP2D6 variant that is less able to metabolize the drug. The resulting potential differences in patient response necessitate careful monitoring of drug effects. Adverse effects Central nervous system toxicity is the most common adverse effect. Dizziness, blurred vision, tremor, paresthesia, and headache may occur. Nausea and a metallic taste have been reported for flecainide. Flecainide administration to patients with recent myocardial infarction shows twofold to threefold increase in mortality. An arrhythmogenic effect of these drugs is suspected despite the fact that they suppressed premature ventricular depolarizations in these patients. These data have led to cautions concerning the use of flecainide after myocardial infarction. Similar concern exists for the other class IC drugs. Propafenone Propafenone is classified as a class IC antiarrhythmic because of its strong tendency to depress the maximum rate of depolarization and conduction velocity. The drug is indicated for life-threatening ventricular arrhythmias and is prescribed for atrial fibrillation and other types of supraventricular arrhythmias. The structure of propafenone is depicted in Figure 19-7. Pharmacologic effects Propafenone exerts several actions on the heart. In addition to blocking Na+ channels, it blocks Ca 2+ channels and exerts β-adrenergic receptor–blocking effects (see Table 19-1). The drug reduces the slope of phase 0, prolongs the PR and QRS intervals, and suppresses ectopic pacemakers (see Table 19-2). Negative inotropic effects are possible but usually occur only with high doses. Absorption, fate, and excretion Propafenone is well absorbed orally, but approximately 80% of a given dose is destroyed in the first pass through the liver. The same genetic predisposition for inefficient or slow metabolism by CYP2D6 of the drug exists as for flecainide, and the half-life may be prolonged in patients who are slow metabolizers. At least one major metabolite is pharmacologically active. Adverse effects Adverse effects include dizziness, blurred vision, dysgeusia, and gastrointestinal symptoms. CNS toxicity seems to be more likely with slow metabolizers. Asthma may be exacerbated in susceptible individuals. Untoward cardiac signs include SA nodal dysfunction, AV nodal block, and worsening of heart failure. The arrhythmogenic potential of the drug must be considered in light of the problems documented for other class IC agents. Competition for metabolism by CYP2D6 is the basis for interactions involving propafenone and other drugs. Propafenone may increase the anticoagulant effect of warfarin given concurrently. β-Adrenergic Receptor–Blocking Drugs Since the introduction of propranolol in 1968 for clinical use in the United States, a number of β-adrenergic receptor–blocking agents have been approved. (The β-adrenergic antagonists are discussed in Chapters 9, 21, and 23 Chapter 9 Chapter 21 Chapter 23.) Two β-adrenergic blockers—propranolol and esmolol—are the primary class II antiarrhythmic drugs discussed here (see Table 19-1). Sotalol, a third drug with ability to block the β-adrenergic receptor, is discussed under class III drugs. Propranolol is reviewed here as the prototypic agent; special features of the other β blockers are also noted. Pharmacologic effects Propranolol, the prototypic type II antiarrhythmic, has two types of effects on the heart: indirect effects as a consequence of blockade of β-adrenergic receptors and “membrane-stabilizing” effects similar to those of quinidine. The membrane-stabilizing effects of propranolol have antiarrhythmic effects in addition to those due to β-adrenergic receptor blockade. Propranolol decreases automaticity and conduction velocity and increases refractoriness. The drug’s greatest effects are on SA nodal automaticity, AV node refractoriness, and (if it exists) His-Purkinje automaticity (see Fig. 19-6 and Table 19-2). Activation of the sympathetic nervous system leading to β receptor stimulation enhances automaticity by increasing the slope of phase 4 depolarization, speeds conduction velocity, and shortens the ERP (especially in the AV node). By blocking β receptors, propranolol can produce opposite effects proportional to the sympathetic input to the heart at the time of administration. In addition to decreasing automaticity in the SA node (and decreasing the heart rate), propranolol variably reduces automaticity and conduction velocity in the atria, AV node, His-Purkinje system, and ventricles. Increased refractoriness in the AV node is an especially important manifestation of blockade. The direct actions of propranolol include decreasing the slope of phase 0 and phase 4 depolarization and prolonging the ERP. The β-adrenergic blockers, with the exception of sotalol, do not appreciably affect repolarization. The major antiarrhythmic indication for propranolol is in the management of supraventricular tachyarrhythmias in which protection of the ventricles (by interfering with AV transmission) is the major clinical objective. Propranolol is also useful in suppressing paroxysmal supraventricular tachycardia and in treating afterdepolarizations and other ventricular arrhythmias in which catecholamine stimulation is involved. Because propranolol reduces the ratio of oxygen demand to oxygen supply, arrhythmias caused by myocardial ischemia may also be relieved. β Blockers have been shown to reduce the incidence of heart attack and death in patients with previous myocardial infarction. The mechanism is not established, but it may relate to an antiarrhythmic mechanism. Absorption, fate, and excretion Propranolol is readily absorbed after oral administration, but more than two-thirds of the drug is destroyed in its first pass through the liver. Peak plasma concentrations are reached in 1 to 2 hours. The rate of metabolism of propranolol, which involves CYP2D6, varies considerably among individuals, so plasma titers may differ markedly with long-term therapy. Propranolol is metabolized by hydroxylation, deamination, and glucuronide conjugation. Adverse effects The important adverse effects of propranolol can be explained by its antagonism of β-adrenergic receptors. Heart rate and myocardial contractility are reduced, at least initially, during therapy. Congestive heart failure and AV block are the major severe cardiac side effects. After large doses, severe bradycardia or asystole may occur. Sudden withdrawal of the drug in patients prone to angina pectoris may lead to anginal attacks or myocardial infarction. Bronchoconstriction is a predictable side effect and may be significant in susceptible individuals, such as asthmatics. Propranolol inhibits the glycogenolytic and lipolytic actions of endogenous catecholamines released in response to hypoglycemia and complicates therapy of diabetic patients. β 1-Selective blockers Metoprolol is used in chronic therapy for the same indications as propranolol. Esmolol is a very short-acting selective β 1-adrenergic receptor blocker that is metabolized by plasma esterases. It is used intravenously for short-term β-adrenergic receptor blockade. The adverse effects of these drugs resemble the adverse effects of propranolol. Despite their selectivity for β 1-adrenergic receptors, these drugs as well as other β blockers should be avoided if possible in asthmatic patients. This drug is primarily used by anesthesiologists to give short-acting beta blockade in patients undergoing general anesthesia. Sotalol Sotalol, a β-adrenergic–blocking drug, also has properties of and is classified as a class III drug. It increases the ERP in addition to its β-adrenergic–blocking activity. The relative importance of its β-blocking properties and its class III antiarrhythmic effects has yet to be determined. Sotalol is well absorbed when taken orally, with a bioavailability of nearly 100%. Sotalol is useful in supraventricular arrhythmias such as atrial fibrillation and in certain cases of ventricular tachycardia. It has been shown to be effective in preventing recurrences of ventricular tachyarrhythmias. Amiodarone Amiodarone, a benzofuran derivative resembling thyroid hormone (Fig. 19-8), was originally introduced in Europe as a coronary vasodilator for the treatment of angina pectoris. It is now widely used for various acute and chronic arrhythmias. It has an interesting mix of blocking several ion channels, including sodium, potassium, and calcium. Pharmacologic effects Amiodarone’s major action is to increase the ERP by slowing the rate of repolarization. Blockade of K+ channels is a major mechanism of action (see Table 19-1). Repolarization is slowed in the His-Purkinje system and in ventricular and atrial myocardium. The Q-T interval is increased (see Fig. 19-6). In addition to blocking K+ channels, amiodarone blocks Na+ and Ca 2+ channels. Inhibition of these latter channels probably prevents much of the inward depolarizing current that can trigger early afterdepolarizations. Amiodarone decreases automaticity in the SA node and in ectopic pacemakers, but it has little effect on automaticity elsewhere in the heart. Conduction velocity is slowed in the AV node by Na+ and Ca 2+ channel blockade (see Table 19-1), and the ERP in the AV node is lengthened. Conduction velocity in the His-Purkinje system and ventricular muscle is also slowed. The ventricular fibrillation threshold is increased. Amiodarone has an active metabolite, desethylamiodarone, which also contributes to the antiarrhythmic effect. Desethylamiodarone binds to cellular thyroid hormone receptors, inhibiting thyroid hormone–induced gene expression. Of the numerous genes normally induced by thyroid hormone, several support the synthesis of certain K+ channels. This finding is consistent with the fact that amiodarone’s effects on K+ channels are generally delayed compared with its effects on Na+ and Ca 2+ channels. Long-term therapy with amiodarone is more likely to generate a class III antiarrhythmic effect (K+ channel block), largely because of reduction in the number of Kr channels. Short-term therapy is more likely to limit effects to Na+ channels, Ca 2+ channels, and β-adrenergic receptors. The resulting cardiac effects with short-term use seem to be different from the effects of long-term therapy and avoid many of the adverse effects seen with long-term administration, such as pulmonary fibrosis and hypothyroidism. Amiodarone is a vasodilator, noncompetitively inhibiting the vascular effect of catecholamines. The cardiac effects of catecholamines are likewise inhibited, and coronary arterial resistance is decreased, resulting in increased coronary blood flow. Amiodarone is used for various arrhythmias, including ventricular extrasystoles, tachycardia, and fibrillation. It is also efficacious in some atrial arrhythmias, including atrial fibrillation and flutter, in which it may have unique effects on the adverse remodeling changes in the atria, which are thought to be partly responsible for the genesis of atrial fibrillation. Absorption, fate, and excretion When administered orally, amiodarone’s bioavailability is low (20% to 50%). Amiodarone is also administered intravenously. A highly lipophilic drug, amiodarone is sequestered in tissues, yielding an apparent volume of distribution of approximately 60 L/kg. It is highly bound to protein in plasma. Because the drug would normally take weeks to reach a steady-state concentration after the initiation of therapy, loading doses are routinely used. Amiodarone is extensively metabolized by the liver; a desethyl derivative, which has antiarrhythmic properties as indicated previously, has been identified. When the drug is withdrawn, the tissue concentrations decrease only gradually as the drug is eliminated. Plasma determinations of amiodarone may not reflect tissue concentrations. Adverse effects Sinus arrest may occur if amiodarone is given with β-adrenergic–blocking drugs or other antiarrhythmics, and AV nodal conduction abnormalities may be exacerbated. Because amiodarone has negative inotropic properties, its use may be associated with a decrease in cardiac function. Some preexisting arrhythmias may be worsened by the drug. Non-cardiac adverse reactions are common and occasionally life-threatening. The primary concerns are pulmonary fibrosis and pneumonitis, which may become clinically evident in a significant percentage of patients with long-term use, and can be lethal. Amiodarone also commonly causes CNS disturbances (ataxia, dizziness), photosensitivity, and hepatic dysfunction as indicated by an increase of liver enzymes in the blood. The skin may take on a unique blue-gray hue. Corneal microdeposits occur routinely but usually do not interfere with vision and disappear after withdrawal of the drug. Changes in thyroid function (hyperthyroidism and especially hypothyroidism) have been reported, and these may be related to the aforementioned facts that amiodarone resembles thyroid hormone and influences thyroid hormone actions. Amiodarone can reduce the action of thyroid hormone by binding to the thyroid hormone receptor and blocking its cellular effects. It has also been shown that amiodarone can inhibit the action of thyroid-stimulating hormone, which could also contribute to a hypothyroid effect by amiodarone. The drug inhibits the conversion of thyroxine to triiodothyronine in peripheral tissues and causes the buildup of reverse triiodothyronine (see Chapter 29). Finally, amiodarone inhibits several cytochrome P450 enzymes and can cause a number of drug–drug interactions. Dronedarone Dronedarone is structurally similar to amiodarone, but it lacks the iodine atoms and is more water-soluble, giving it a half-life of about 20 hours. It is similar in its pharmacodynamics to amiodarone and is considered to be a class III antiarrhythmic, although, like amiodarone, it has effects on several ion channels. Lacking the iodine atoms, it does not have the thyroid-related adverse effects associated with amiodarone. Pulmonary fibrosis and pneumonitis are also not associated with dronedarone. Dronedarone can cause bradycardia, an increase in the Q-T interval, and GI symptoms. It also can interact with other drugs because of its ability to inhibit and be metabolized by CYP3A4. Ibutilide and Dofetilide Ibutilide is classified as a class III drug because it delays repolarization. It blocks Kr channels and causes the opening of Ca 2+ channels, which promote Na+ influx through slow channels, extending phase 2 of the action potential. The drug is administered intravenously for atrial fibrillation and atrial flutter. It can be used to convert these arrhythmias, especially of recent onset, to normal sinus rhythm. Hypotension and arrhythmias are adverse effects. Dofetilide is considered a “pure” class III drug, blocking the Kr channel selectively. It is used for acute conversion of atrial fibrillation and atrial flutter and for short-term maintenance of normal sinus rhythm. Dofetilide is available for intravenous and oral use. The structures for ibutilide and dofetilide are shown in Figure 19-9. Ca 2+ Channel Blockers Ca 2+ channel blockers, represented by verapamil, diltiazem, and amlodipine, are used for the treatment of certain cardiovascular diseases. Verapamil and diltiazem are prescribed primarily for their antianginal (see Chapter 21) and antiarrhythmic effects. Amlodipine, which has a greater effect on vascular smooth muscle, is a major antihypertensive drug (see Chapter 23) and is used as an antiarrhythmic drug. In each case, the drugs are selective for potential-dependent Ca 2+ channels rather than receptor-operated channels. The potential-dependent Ca 2+ channels are of at least three types: L, N, and T. These channels are distinguished by their electrical properties and anatomic location. The L (long-lasting) channels are selectively inhibited by these drugs. The fact that they are the predominant Ca 2+ channels in the heart and vascular smooth muscle is consistent with the major effects of Ca 2+ channel blockers on these organs. The N (neuronal) and T (transient) channels are not affected by these channel blockers to a major degree, although the T-type channels play a role in phase 2 of action potentials in the heart (see Fig. 19-2). By interfering with the slow inward current in pacemaker cells, these drugs depress the rate of phase 4 depolarization; automaticity in the SA node and the AV node is decreased. The major direct cardiac effect is to reduce conduction velocity and to increase the refractory period of the AV node (see Fig. 19-6). Verapamil and diltiazem are useful in treating supraventricular arrhythmias such as atrial fibrillation and flutter. This helps indirectly to control ventricular rate, which can be affected by the aberrant supraventricular activity. Both drugs have also been used successfully in preventing attacks of atrial tachycardia. Verapamil and diltiazem have a negative inotropic effect. Other aspects of the pharmacologic features of the Ca 2+ channel blockers are discussed in Chapters 21 and 23 Chapter 21 Chapter 23. Adenosine The endogenous purine nucleoside, adenosine, is approved for terminating attacks of supraventricular tachycardia. It does not match the profile of any other antiarrhythmic. The structure of adenosine is shown in Figure 19-10. Pharmacologic effects Adenosine stimulates the A 1 adenosine receptor in the heart. This receptor is linked to the G protein G i, which, when activated, increases K+ conductance and decreases Ca 2+ channel activity, leading to hyperpolarization. Adenosine also reduces the release of norepinephrine from nerve endings. The net effect on the heart is to reduce automaticity in the SA node and Purkinje fibers and reduce the AV nodal conduction rate (see Fig. 19-6 and Table 19-2). The drug is useful for short-term treatment of supraventricular tachycardia involving reentry with rapid ventricular response. Adenosine also dilates coronary vessels and reduces contractility. Selective adenosine receptor agonists offer promise for future drug development. Adverse effects Adenosine must be injected intravenously as a rapid bolus with a saline flush behind it because of its extremely short plasma half-life, due to its rapid transport into tissues. This is followed by incorporation into purine biosynthetic pathways. Many patients have transient flushing and dyspnea with the drug. Arrhythmias, including heart block and transient asystole, may also occur immediately after injection. Patients are warned they may feel a metallic taste and may become dizzy or black out as the drug is injected. Because of the drug’s rapid uptake into tissues, however, therapeutic and adverse responses are normally short-lived. Digoxin Digoxin is used in treating certain supraventricular arrhythmias. Its use is discussed in Chapter 20. Magnesium Magnesium sulfate is used intravenously to overcome drug-induced torsades de pointes. It may be effective even in the absence of hypomagnesemia. Indications for Antiarrhythmic Drugs Table 19-4 presents the general indications of the various agents discussed in this chapter in treating some of the most commonly encountered arrhythmias. It is not intended to be a comprehensive listing of applications of these drugs. Drugs administered orally are largely used to prevent the recurrence of arrhythmias, whereas drugs administered parenterally are usually given to treat acute disorders. Drug Interactions Antiarrhythmic drugs can participate in a wide variety of drug interactions. Because the margin of safety with these drugs as a group is narrow, clinically significant interactions may develop whenever the activity or plasma concentration of an antiarrhythmic agent is altered. The following discussion provides an illustrative but not exhaustive list of interactions involving these drugs. Drugs that slow AV conduction, such as the β-adrenergic–blocking drugs and amiodarone, can exaggerate the AV conduction effects of drugs with similar actions, possibly leading to bradycardia and heart block. Drugs that have negative inotropic effects (e.g., class IA drugs, Ca 2+ channel blockers, and β blockers) may precipitate heart failure, especially in the presence of other negative inotropic agents. Metoprolol and related drugs prevent the tachycardia that normally results from hypoglycemic drugs. The metabolism of flecainide and propafenone is catalyzed by CYP2D6 in the liver. Drugs that share this same pathway of metabolism increase each other’s elimination half-life. Cimetidine, erythromycin, amiodarone, quinidine, and several other drugs that inhibit CYP2D6 increase the plasma concentrations of flecainide and propafenone. Drugs that induce CYP2D6 decrease the half-life of flecainide and propafenone. Propranolol and cimetidine reduce lidocaine clearance, whereas induction of hepatic microsomal enzymes by drugs such as phenobarbital increases lidocaine clearance. Amiodarone increases the effect of warfarin and procainamide. There may be more than one mechanism accounting for these interactions. In addition to its effects on metabolism, amiodarone may reduce renal clearance of these drugs by inhibiting the transport function of P-glycoprotein in the kidney. Verapamil, propafenone, amiodarone, flecainide and quinidine have been reported to increase plasma digoxin concentrations. These interactions may also involve the P-glycoprotein. Show more View chapterExplore book Read full chapter URL: Book 2017, Pharmacology and Therapeutics for Dentistry (Seventh Edition)Frank J. Dowd Chapter Standard Antiarrhythmic Drugs 2014, Cardiac Electrophysiology: From Cell to Bedside (Sixth Edition)Dawood Darbar Historical Perspective on Current Antiarrhythmic Drugs and Their Limitations Drugs that are currently marketed as antiarrhythmics were initially developed in whole-animal models by screening for their effects on normal cardiac electrophysiology. However, decades elapsed before the initial use of older antiarrhythmic drugs in humans and more importantly the definition of their molecular mechanisms of action. Antiarrhythmic drugs are only partly effective and have many adverse effects, the most important of which is the potential to generate new life-threatening arrhythmias (proarrhythmia). The increasing appreciation of ventricular arrhythmias as a marker of underlying heart disease and, therefore, a potential drug target, led to the development of multiple new antiarrhythmic drugs in the late 1970s and early 1980s. An early animal model that was used to screen for antiarrhythmic activity was the “Harris dog,” which demonstrates extremely high-frequency ventricular ectopic activity after two-stage coronary artery ligation. Ventricular arrhythmias are especially sensitive to sodium channel blockers; therefore, the first wave of new antiarrhythmic drugs were derived from existing structures and demonstrated activity in this and other models (i.e., primarily sodium channel blockers). The Cardiac Arrhythmia Suppression Trial (CAST) represents a landmark trial result, not only for antiarrhythmic drug development but for drug development in general.4 The CAST tested the prevalent hypothesis that because ventricular ectopy following myocardial infarction (MI) is a risk factor for SCD, suppression of ventricular ectopy would reduce the incidence of SCD. Mortality among patients randomized to ventricular ectopic suppression with encainide or flecainide in the CAST was approximately threefold higher than that of patients randomized to receive placebo. The study also emphasized the importance of a randomized, placebo-controlled trial with a primary “hard” endpoint, such as death, as opposed to a surrogate end point, such as ectopic beat suppression, to determine whether a drug was beneficial. The drugs tested in CAST were potent sodium channel–blocking agents. For basic and clinical scientists interested in arrhythmia mechanisms, the CAST result provided a strong impetus to further work that defined the way in which loss of sodium channel function was arrhythmogenic. It is thought that blocking sodium channels increases the risk of SCD by slowing conduction or increasing the heterogeneity of repolarization, both of which can be arrhythmogenic, especially in diseased hearts.5,6 Multiple lines of evidence support the hypothesis that loss of sodium current (I Na) slows cardiac conduction and may be arrhythmogenic; such loss can be structural (e.g., mutations in the cardiac sodium channel gene SCN5A leading to loss of peak I Na),7-9 pharmacologic,7,10 or reflect activation of second messenger systems. For example, protein kinase A (PKA) activation has been reported to increase peak I Na, at least in part because of altered trafficking,11 whereas protein kinase C activation decreases it.12 Another consequence of CAST was that development of sodium channel–blocking drugs came to a rapid halt and QT-prolonging agents then assumed the limelight. Just as procainamide and lidocaine provided the structural starting point of a range of sodium channel–blocking molecules, two compounds with prominent action potential (AP)-prolonging properties formed the framework here: N-acetyl procainamide (NAPA) and sotalol. Both compounds also did not block the sodium channel and gave rise to a generation of QT-prolonging drugs, such as d-sotalol, dofetilide, almokalant, and sematilide. Although the screening assay to develop these agents usually was APD in a guinea pig papillary muscle, and as with newer sodium channel blockers, these compounds were not synthesized to interact with a specific predefined molecular target. Nevertheless, subsequent studies identified block of one specific potassium current, termed I Kr (the rapid component of the cardiac delayed rectifier potassium current), as the major mechanism underlying QT prolongation by antiarrhythmic drugs. I Kr is encoded by KCNH2 (also known as HERG). However, blocking I Kr carries the risk of marked QT interval prolongation and a distinctive form of polymorphic ventricular tachycardia, termed torsades de pointes. Indeed, potassium-channel blocking antiarrhythmic drugs have been tested in CAST-like trials and did not prevent more deaths than did placebo.13-15 Moreover, the same mechanism underlies the development of torsades de pointes in response to noncardiovascular drugs, such as certain antihistamines, antibiotics, and antipsychotics.16 Show more View chapterExplore book Read full chapter URL: Book 2014, Cardiac Electrophysiology: From Cell to Bedside (Sixth Edition)Dawood Darbar Chapter Cardiovascular Pharmacology 2018, Kaplan's Essentials of Cardiac Anesthesia (Second Edition)Roger L. Royster MD, ... Thomas F. Slaughter MD Pharmacotherapy for Cardiac Arrhythmias The most widely used electrophysiologic and pharmacologic classification of antiarrhythmic drugs is that proposed by Vaughan Williams (Table 8.6). There is, however, substantial overlap in pharmacologic and electrophysiologic effects of specific agents among the classes, and the linkage between observed electrophysiologic effects and the clinical antiarrhythmic effect is often tenuous. Likewise, especially in class I, there may be considerable diversity within a single class. Other antiarrhythmic drugs are not included in this classification, such as digitalis, the classic antiarrhythmic for chronic AF, or adenosine, a drug with potent antiarrhythmic effects mediated by a specific class of membrane receptors. Although the class I and especially subclass IC agents are most commonly known for their proarrhythmic effects, the other classes are not devoid of this side effect. For the first week after initiation of sotalol, a nonspecific β-adrenergic blocker that is considered a class III arrhythmic agent, there is an increased incidence of torsades de pointes. The proarrhythmic effects appear to be increased in the setting of hypokalemia, bradycardia, CHF, and a history of sustained ventricular dysfunction (Box 8.4). Chronic antiarrhythmic therapy should be initiated only after careful evaluation of the risks and benefits of the intervention. The appropriate use of intravenous antiarrhythmic agents with sudden-onset arrhythmias is not clear. Life-threatening ventricular arrhythmias must be treated. High-risk patients may be treated more safely in some cases by implantation of internal cardioverter-defibrillators. Class I Antiarrhythmic Drugs: Sodium Channel Blockers Class I drugs inhibit the fast inward depolarizing current carried by sodium ions. Because of the diversity of other effects of the class I drugs, a subgroup of the class has been proposed (Table 8.7). Whether the depression of fast inward current of the sodium channel produces the primary antiarrhythmic effect of all class I drugs is controversial. Class IA Drugs Procainamide Electrophysiologic effects of procainamide include decreased maximal velocity (V max) and amplitude during phase 0, decreased rate of phase 4 depolarization, and prolonged effective refractory period (ERP) and action potential duration (APD). Clinically, procainamide prolongs conduction and increases the ERP in atrial and His-Purkinje portions of the conduction system, which may prolong PR interval and QRS complex durations. Procainamide is used to treat ventricular arrhythmias and to suppress atrial premature beats to prevent the occurrence of AF and atrial flutter. It has been useful for chronic suppression of premature ventricular contractions (PVCs). Procainamide reduces the frequency of the short-coupling interval (<400 ms) PVCs and thereby reduces the frequency of ventricular tachycardia (VT) or ventricular fibrillation (VF) created by the R-on-T phenomenon. Administered intravenously, procainamide is an effective emergency treatment for ventricular arrhythmias, especially after lidocaine failure, but amiodarone has become a more popular drug for intravenous suppression of ventricular arrhythmias. Dosage is 100 mg, or approximately 1.5 mg/kg given at 5-minute intervals until the therapeutic effect is obtained or a total dose of 1 g or 15 mg/kg is given (Tables 8.8 and 8.9). Arterial pressure and the electrocardiogram should be monitored continuously during loading and administration stopped if significant hypotension occurs or if the QRS complex is prolonged by 50% or more. Maintenance infusion rates are 2 to 6 mg/min to maintain therapeutic plasma concentrations of 4 to 8 µg/mL. Class IB Drugs Lidocaine First introduced as an antiarrhythmic drug in the 1950s, lidocaine has become the clinical standard for the acute intravenous treatment of ventricular arrhythmias except those precipitated by an abnormally prolonged QT interval. Lidocaine may be one of the most useful drugs in clinical anesthesia because it has local and general anesthetic properties in addition to an antiarrhythmic effect. The direct electrophysiologic effects of lidocaine produce virtually all of its antiarrhythmic action. Lidocaine depresses the slope of phase 4 diastolic depolarization in Purkinje fibers and increases the VF threshold. The clinical pharmacokinetics of lidocaine are well described. The distribution and elimination half-lives of lidocaine are short, approximately 60 seconds and 100 minutes, respectively. Hepatic extraction of lidocaine is about 60% to 70%, and essentially all lidocaine is metabolized because the urine contains negligible amounts of unchanged lidocaine. Therapeutic plasma levels of lidocaine range from 1.5 to 5 µg/mL; signs of toxicity are common with concentrations greater than 9 µg/mL. An initial bolus dose of 1 to 1.5 mg/kg should be followed immediately by a continuous infusion of 20 to 50 µg/kg per minute to prevent the therapeutic hiatus produced by the rapid redistribution half-life of lidocaine. Class IC Drugs Propafenone Propafenone blocks the fast sodium current in a use-dependent manner. Propafenone blocks β-receptors and is a weak potassium channel blocker. This drug usually slows conduction and prolongs refractoriness of most cardiac conduction system tissue. Propafenone is indicated for life-threatening ventricular arrhythmias, various supraventricular arrhythmias, and AF. In one study, a single 600-mg oral dose of propafenone converted 76% of patients in AF. Propafenone was more effective than placebo in preventing atrial tachyarrhythmias after cardiac surgery with combined intravenous and oral therapy. Propafenone is well absorbed orally and is highly protein bound, with an elimination half-life of 6 to 8 hours. Therapeutic serum levels are 0.2 to 1.5 µg/mL. The metabolites of propafenone are active and demonstrate significant action potential and β-blocking effects. The drug has few proarrhythmic problems, likely because of the β-blocking effects, which tend to decrease arrhythmic traits of antiarrhythmic drugs. Class II Drugs: β-Adrenergic Receptor Antagonists Propranolol Propranolol was the first major β-receptor–blocking drug to be used clinically. Propranolol is very potent but is nonselective for β 1- and β 2-receptor subtypes. It possesses essentially no ISA. Because it interferes with the bronchodilating actions of epinephrine and the sympathetic stimulating effects of hypoglycemia, propranolol is less useful in patients with diabetes or bronchospasm. These difficulties with propranolol stimulated the search for β-receptor–blocking drugs with receptor subtype specificity, such as metoprolol, esmolol, and atenolol. The electrophysiologic effects of β-receptor antagonism are decreased automaticity, increased APD (primarily in ventricular muscle), and a substantially increased ERP in the atrioventricular (AV) node. β-Blockade decreases the rate of spontaneous (phase 4) depolarization in the sinoatrial (SA) node; the magnitude of this effect depends on the background sympathetic tone. Although resting heart rate is decreased by β-blockade, inhibition of the increase of heart rate in response to exercise or emotional stress is much more marked. Automaticity in the AV node and more distal portions of the conduction system is also depressed. β-Blockade affects VF threshold variably, but it consistently reverses the fibrillation threshold–lowering effect of catecholamines. An appropriate intravenous dose for acute control of arrhythmias is 0.5 to 1.0 mg titrated to therapeutic effect up to a total of 0.1 to 0.15 mg/kg. Stable therapeutic plasma concentrations of propranolol can be obtained with a continuous intravenous infusion. However, with the availability of esmolol, a propranolol infusion is no longer necessary. Metoprolol Metoprolol is a relatively selective β-receptor antagonist. The potency of metoprolol for β 1-receptor blockade is equal to that of propranolol, but metoprolol exhibits only 1% to 2% of the effect of propranolol at β 2-receptors. Metoprolol is useful for treating supraventricular and ventricular arrhythmias that are adrenergically driven. The primary advantage of metoprolol is its relative lack of most of the bronchoconstrictive effects in patients with chronic obstructive pulmonary disease. Acute intravenous dosage is 1.0 mg titrated to therapeutic effect up to 0.1 to 0.2 mg/kg. Esmolol Esmolol is a cardioselective β 1-receptor antagonist with an extremely brief duration of action. Electrophysiologic effects of esmolol are those of β-adrenergic receptor antagonism. Esmolol is rapidly metabolized in blood by hydrolysis of its methyl ester linkage. Its half-life in whole blood is 12.5 to 27.1 minutes in humans. The acid metabolite possesses a slight degree (1500 times less than esmolol) of β-antagonism. Esmolol is not affected by plasma cholinesterase; the esterase responsible is located in erythrocytes and is not inhibited by cholinesterase inhibitors. Of importance to clinical anesthesia, no metabolic interactions between esmolol and other ester molecules are known. In a multicenter trial that compared esmolol with propranolol for the treatment of paroxysmal supraventricular tachyarrhythmia (PSVT), esmolol was equally efficacious and had the advantage of a much faster termination of the β-blockade. Esmolol has become a useful agent in controlling sinus tachycardia in the perioperative period, a time when a titratable and brief β-blockade is highly desirable. Dosing begins at 25 µg/kg per minute and is titrated to effect up to 250 µg/kg per minute. Doses greater than this may cause significant hypotension because of reduced CO in patients. Esmolol is especially effective in treating acute-onset AF or flutter perioperatively, and it results in acute control of the ventricular response and conversion of the arrhythmia to sinus rhythm. Class III Drugs: Potassium Channel Blockers and Agents That Prolong Repolarization Amiodarone Amiodarone is a benzofuran derivative that was initially introduced as an antianginal drug and was subsequently found to have antiarrhythmic effects. The drug has a wide spectrum of effectiveness, including supraventricular, ventricular, and preexcitation arrhythmias. It also may be effective against VT and VF refractory to other treatment. Amiodarone has been approved by the AHA as the first-line antiarrhythmic in cardiopulmonary resuscitation. Amiodarone may be effective prophylactically in preventing AF after surgery. It also can decrease the number of shocks in patients who have implantable cardioverter-defibrillators compared with other antiarrhythmic drugs. Amiodarone increases the amount of electric current required to elicit VF (ie, increase in the VF threshold). In most patients, refractory VT is suppressed by acute intravenous use of amiodarone. This effect has been attributed to a selectively increased activity in diseased tissue, as has been seen with lidocaine. Hemodynamic effects of intravenous amiodarone (10 mg/kg) include decreased LV dP/dt, maximal negative dP/dt, mean aortic pressure, heart rate, and peak LV pressure after coronary artery occlusion in dogs. CO was increased despite the negative inotropic effect as a result of the more marked decrease in LV afterload. A 5-mg/kg intravenous dose during cardiac catheterizationdecreased BP, LV end-diastolic pressure, and SVR and increased CO, but it did not affect heart rate. Chronic amiodarone therapy is not associated with clinically significant depression of ventricular function in patients without LV failure. In acute situations with stable patients, a 150-mg intravenous bolus is followed by a 1.0-mg/min infusion for 6 hours and then 0.5 mg/min thereafter. In cardiopulmonary resuscitation (CPR), a 300-mg intravenous bolus is given and repeated with multiple boluses as needed if defibrillation is unsuccessful. Despite relatively widespread use of amiodarone, anesthetic complications infrequently have been reported. In case reports, bradycardia and hypotension were prominent. The slow decay of amiodarone in plasma and tissue makes such adverse reactions possible long after discontinuing its administration. Epinephrine is more effective than dobutamine or isoproterenol in reversing amiodarone-induced cardiac depression. An RCT of amiodarone administered 6 days before and 6 days after cardiac surgery demonstrated significant reductions in atrial tachyarrhythmias and ventricular arrhythmias in patients of different ages and in different types of cardiac surgical procedures. There were no differences in hospital mortality rates between groups. Sotalol Sotalol is classified as a class III agent, but it also has class II β-adrenergic–blocking properties. It prolongs refractoriness in atrial and ventricular tissues because of blockade of the delayed rectifier potassium current. The β-blocking effects result in decreased heart rate and increased refractory periods at the atrial and ventricular levels. It is indicated for life-threatening ventricular arrhythmias and AF. Sotalol has been used to treat supraventricular and ventricular tachyarrhythmias. It was found to be superior to class I agents in preventing the recurrence of ventricular arrhythmias. Sotalol administration is associated with increased risk for torsades de pointes and QT-interval prolongation. Female patients and patients with renal failure are at increased risk for the proarrhythmic side effects. Class IV Drugs: Calcium Channel Antagonists Although the principal direct electrophysiologic effects of the three main chemical groups of calcium antagonists (ie, verapamil, a benzoacetonitrite; nifedipine, a DHP; and diltiazem, a benzothiazepine) are similar, verapamil and diltiazem are the primary antiarrhythmics. Verapamil and Diltiazem Verapamil and diltiazem have been used extensively in the treatment of supraventricular arrhythmias, AF, and atrial flutter. They are especially effective at preventing or terminating PSVT by blocking impulse transmission through the AV node by prolonging AV nodal conduction and refractoriness. They are also useful in the treatment of AF and atrial flutter by slowing AV nodal conduction and decreasing the ventricular response. The effect on ventricular response is similar to that of the cardiac glycosides, although the onset is more rapid and acutely effective for control of tachycardia in patients. In the perioperative period, verapamil is a useful antiarrhythmic agent. In one study of anesthetized patients, it successfully controlled a variety of supraventricular and ventricular arrhythmias. However, verapamil should be used with caution intraoperatively because significant cardiac depression may occur in conjunction with inhalation anesthetics. Verapamil dosage for acute intravenous treatment of PSVT is 0.07 to 0.15 mg/kg over 1 minute, with the same dose repeated after 30 minutes if the initial response is inadequate (10 mg maximum). Because the cardiovascular depressant effects of the inhalation anesthetics involve inhibition of calcium-related intracellular processes, the interaction of verapamil and these anesthetics is synergistic. Diltiazem in doses of 0.25 to 0.30 mg/kg administered intravenously and followed by a titratable intravenous infusion of 10 to 20 mg/h is rapid acting and efficacious in controlling ventricular response rate in new-onset AF and atrial flutter. The prophylactic use of intravenous diltiazem can reduce the incidence of postoperative supraventricular arrhythmias after pneumonectomy and cardiac surgery. Diltiazem also may have a role in treating ventricular arrhythmias. Other Antiarrhythmic Agents Digoxin The primary therapeutic use of digitalis drugs is to slow the ventricular response during AF or atrial flutter, which occurs because of a complex combination of direct and indirect actions on the AV node. The primary direct pharmacologic effect of digitalis is inhibition of the membrane-bound Na+/K+-ATPase. This enzyme provides the chemical energy necessary for the transport of sodium (out) and potassium (in) during repolarization. The glycosides bind to the enzyme in a specific saturable way that inhibits enzyme activity and impairs the active transport of sodium and potassium. The net result is a slight increase in intracellular sodium and a corresponding decrease in intracellular potassium concentration. The sodium exchanges for calcium, resulting in a relatively weak inotropic effect. The main preparation of cardiac glycosides available is digoxin. Digoxin reaches peak effects in 1.5 to 2 hours but has a significant effect within 5 to 30 minutes. For undigitalized patients, the initial dose is 0.5 to 0.75 mg of digoxin, with subsequent doses of 0.125 to 0.25 mg. The usual total digitalizing dose is 0.75 to 1.0 mg administered by the intravenous route. Adenosine The important cardiac electrophysiologic effects of adenosine are mediated by the A 1 receptor and consist of negative chronotropic, dromotropic, and inotropic actions. Adenosine decreases SA node activity, AV node conductivity, and ventricular automaticity. In many ways, these effects mimic those of acetylcholine. For clinical use, adenosine must be administered by a rapid intravenous bolus in a dose of 100 to 200 µg/kg, although continuous intravenous infusions of 150 to 300 µg/kg per minute have been used to produce controlled hypotension. For practical purposes, in adults, a dose of 3 to 6 mg is given by intravenous bolus followed by a second dose of 6 to 12 mg after 1 minute if the first dose was not effective. This therapy rapidly interrupts narrow-complex tachycardia caused by AV nodal reentry. Comparison with verapamil has shown adenosine to be equally effective as an antiarrhythmic, but with the advantages of fewer adverse hemodynamic effects, a faster onset of action, and a more rapid elimination so that undesired effects are short-lived. Potassium Because of the close relationship between extracellular pH and potassium, the primary mechanism of pH-induced arrhythmias may be alteration of potassium concentration. Hypokalemia and hyperkalemia are associated with cardiac arrhythmias, but hypokalemia is more common perioperatively in cardiac surgical patients and is more often associated with arrhythmias. Decreasing the extracellular potassium concentration increases the peak negative diastolic potential, which appears to decrease the likelihood of spontaneous depolarization. However, because the permeability of the myocardial cell membrane to potassium is directly related to extracellular potassium concentration, hypokalemia decreases cellular permeability to potassium. This prolongs the action potential by slowing repolarization, which slows conduction, increases the dispersion of recovery of excitability, and predisposes to the development of arrhythmias. ECG correlates of hypokalemia include appearance of a U wave and increased P-wave amplitude. The arrhythmias most commonly associated with hypokalemia are premature atrial contractions, atrial tachycardia, and supraventricular tachycardia. Hypokalemia also accentuates the toxicity of cardiac glycosides. With chronic potassium deficiency, the plasma level poorly reflects the total body deficit. Because only 2% of total body potassium is in plasma and total body potassium stores may be 2000 to 3000 mEq, a 25% decline in serum potassium from 4 to 3 mEq/L indicates an equilibrium total body deficiency of 500 to 800 mEq, replacement of which should be undertaken slowly. Acute hypokalemia frequently occurs after CPB as a result of hemodilution, urinary losses, and intracellular shifts, with the latter perhaps relating to abnormalities of the glucose-insulin system seen with nonpulsatile hypothermic CPB. With frequent assessment of serum potassium concentrations and continuous ECG monitoring, potassium infusion at rates of up to 10 to 15 mEq/h may be administered to treat serious hypokalemia. Magnesium Magnesium deficiency is a relatively common electrolyte abnormality in critically ill patients, especially in chronic situations. Hypomagnesemia is associated with a variety of cardiovascular disturbances, including arrhythmias. Functionally, magnesium is required for the membrane-bound Na+/K+-ATPase, which is the principal enzyme that maintains normal intracellular potassium concentration. Not surprisingly, the ECG findings seen with magnesium deficiency mimic those seen with hypokalemia: prolonged PR and QT intervals, increased QRS duration, and ST-segment abnormalities. As with hypokalemia, magnesium deficiency predisposes to the development of the arrhythmias produced by cardiac glycosides. Magnesium is effective as an adjuvant in the treatment of patients with a prolonged QT syndrome and torsades de pointes. Arrhythmias induced by magnesium deficiency may be refractory to treatment with antiarrhythmic drugs and electrical cardioversion or defibrillation. Adjunctive treatment of refractory arrhythmias with magnesium has been advocated even when magnesium deficiency has not been documented. Magnesium deficiency is common in cardiac surgery patients because of the diuretic agents these patients are often receiving and because magnesium levels decrease with CPB because of hemodilution of the pump. Magnesium lacks a counterregulatory hormone to increase magnesium levels during CPB, in contrast to the hypocalcemia that is corrected by parathyroid hormone. The results of magnesium administration trials involving CABG have been conflicting. Some studies have shown a benefit, and others have not in regard to reducing the incidence of postoperative arrhythmias. Show more View chapterExplore book Read full chapter URL: Book 2018, Kaplan's Essentials of Cardiac Anesthesia (Second Edition)Roger L. Royster MD, ... Thomas F. Slaughter MD Chapter Standard Antiarrhythmic Drugs 2018, Cardiac Electrophysiology: From Cell to Bedside (Seventh Edition)Dawood Darbar Classification of Antiarrhythmic Drugs The earliest schemes classified drugs based on shared efficacies and toxicities. Some AADs share important electrophysiological properties, notably block of Na+ channels or of the K+ current I Kr, and these can provide the basis for predicting shared or “class” actions and toxicities. Block or enhancement of other ionic currents, such as I Ks, the transient outward current (I TO), or acetylcholine-activated current (I K-Ach) may also contribute to clinical drug actions in some cases, and are not usually considered in broad classification schemes. A more modern approach and one that was presciently adopted by the “Sicilian Gambit” investigators is to classify drugs based on key electrophysiological mechanisms involved in arrhythmogenesis, thereby allowing specific drugs to be chosen to target these. This approach is nicely exemplified by the definition of new molecular mechanisms in congenital arrhythmia syndromes, such as AF and the congenital long QT syndrome (LQTS), and the way in which these then lead naturally to mechanism-based therapies, as discussed earlier. As our understanding of the molecular and cellular basis of arrhythmias evolves, a more rational choice of key molecules(s) whose targeting is likely to be safe and effective in the therapy of a particular arrhythmia holds much promise. The term “antiarrhythmic drugs” has traditionally been used to include drugs targeting ion channels in cardiac myocytes (Na+channel blockers, Ca 2+ channel blockers, and QT-prolonging drugs, generally K+ channel blockers), β-adrenergic receptor blockers, and a series of drugs with diverse mechanisms used primarily for the therapy of arrhythmias, such as digoxin, amiodarone, magnesium, and adenosine. However, recent studies have demonstrated that other widely used cardiovascular therapies, such as ACE and HMG Co-A reductase inhibitors may also exert important antiarrhythmic effects. Such effects may include not only reduction in SCD, an arrhythmic event that represents the final common pathway for many potential disease pathways, but also prevention of AF. These studies not only provide novel potential therapies but also implicate new signaling pathways in the pathogenesis of arrhythmias and therefore as potential targets for the development of effective antiarrhythmic interventions. View chapterExplore book Read full chapter URL: Book 2018, Cardiac Electrophysiology: From Cell to Bedside (Seventh Edition)Dawood Darbar Chapter Amiodarone Hydrochloride 2016, Saunders Handbook of Veterinary Drugs (Fourth Edition)Mark G. Papich DVM, MS, DACVCP Pharmacology and mechanism of action Antiarrhythmic drug, Class III. Antiarrhythmic effects are primarily caused by blocking the outward potassium channel in cardiac tissues. Amiodarone prolongs the action potential, delays myocardial repolarization, and delays the refractory period in cardiac tissues. It also may have some alpha-adrenergic receptor, beta-adrenergic receptor, and calcium-channel blocking properties. Half-life is several days in duration, and in some animals the half-life may be as long as 100 days with chronic therapy. In horses the terminal half-life was 38-84 hours. The intravenous formulation of amiodarone uses Polysorbate 80 to enhance solubility, which may be responsible for some of the adverse reactions. There is a new noniodinated derivative, dronedarone (Multaq), that is less lipophilic, has a shorter half-life, and may be safer, but there has been no use reported in animals. View chapterExplore book Read full chapter URL: Book 2016, Saunders Handbook of Veterinary Drugs (Fourth Edition)Mark G. Papich DVM, MS, DACVCP Chapter Innovations in Antiarrhythmic Drug Therapy 2018, Cardiac Electrophysiology: From Cell to Bedside (Seventh Edition)Paulus Kirchhof, Larissa Fabritz Treatment of Heart Failure Although treating the underlying heart disease can prevent arrhythmias, treatment of heart failure, unfortunately, is often symptomatic only and can facilitate proarrhythmia, for example, by potassium depletion with use of diuretics. Avoiding proarrhythmic effects of medications is an important part of preventing arrhythmias. Proarrhythmia induced by inotropic substances is an important clinical problem and the use of inotropes, especially in heart failure, faces the dilemma that increased contractility often augments arrhythmic activity as well. Fortunately, there are several recent examples of “heart failure” medications with potential antiarrhythmic effects. A randomized, placebo-controlled clinical trial of human recombinant relaxin-2 for treatment of acute heart failure showed relaxin-2 to be safe and effective.83 Of note in this context, the positive effect on mortality was primarily driven by prevention of stroke (possibly related to atrial fibrillation) and sudden death,84 suggesting that this substance may have antiarrhythmic effects. The antifibrotic hormone relaxin has been shown to suppress experimental atrial fibrillation, improving conduction from a combination of reversal of fibrosis and hypertrophy and by increasing sodium current.85 This provides an example of a “substrate modifier” that can have antiarrhythmic properties. View chapterExplore book Read full chapter URL: Book 2018, Cardiac Electrophysiology: From Cell to Bedside (Seventh Edition)Paulus Kirchhof, Larissa Fabritz Chapter Disorders of Cardiac Rhythm 2011, Pediatric Critical Care (Fourth Edition)Frank A. Fish, ... James A. Johns Antiarrhythmic Agents The addition of pharmacologic agents following adenosine administration should be guided by the clinical situation, known or suspected tachycardia mechanism, and response to adenosine administration. In some cases (such as in patients with wide QRS tachycardia or in hemodynamically compromised patients), it may be most appropriate to proceed directly to pacing termination or cardioversion if adenosine is unsuccessful in restoring sinus rhythm. In other instances, acute antiarrhythmic drug therapy may be warranted.26 Antitachycardia drugs usually are classified according to their surface ECG effects, which often correlate closely with their cellular electrophysiologic effects. The Vaughan Williams classification divides drugs into those that block cardiac sodium channels (class I), block β-adrenoreceptors (class II), prolong repolarization (class III), and block calcium channels (class IV). Digoxin and adenosine, which are not included in this classification scheme, exert their primary antiarrhythmic effects on the AV node. Magnesium also has depressant effects on the AV node and suppresses early and late afterdepolarizations (triggered activity). Many of the available drugs manifest properties of more than one class, which contribute collectively to their antiarrhythmic action.27 In general, class I drugs (particularly IA and IC) slow conduction in atrial, ventricular, or accessory pathway tissue and class III drugs increase refractoriness in these same tissues. Class IA drugs usually accomplish both effects. β-Adrenergic antagonists, calcium channel antagonists, digoxin, and adenosine act primarily by slowing AV nodal conduction or inhibiting abnormal automaticity. Thus the latter group of drugs is primarily used for reciprocating tachycardias using the AV node (ART, ORT, AVNRT) or to induce second-degree AV block during a primary atrial tachycardia. In contrast, class IA, IC, and III drugs may be more effective in terminating or directly suppressing primary atrial tachycardias and may be effective for reciprocating tachycardias.5,26 Despite the various antiarrhythmic agents available for chronic therapy, relatively few are suitable for acute administration to the critically ill patient either because the drugs are not available in intravenous formulation or they have significant negative inotropic effects when administered intravenously (Table 28-3). This discussion is limited to agents suitable for acute and short-term parenteral administration. All antiarrhythmic agents have the potential for producing bradycardia, particularly when administered acutely, and most have negative inotropic and/or hypotensive effects. Careful observation is required during initial administration and subsequent infusion of all intravenous antiarrhythmic agents. Although many are contraindicated in cases of heart failure or hypotension, therapy may be necessary if the arrhythmia is contributing significantly to the patient’s hemodynamic compromise. View chapterExplore book Read full chapter URL: Book 2011, Pediatric Critical Care (Fourth Edition)Frank A. Fish, ... James A. Johns Chapter Impact of Nontraditional Antiarrhythmic Drugs on Sudden Cardiac Death 2014, Cardiac Electrophysiology: From Cell to Bedside (Sixth Edition)Leonard Ilkhanoff, ... Jeffrey J. Goldberger Assessing the Impact of Nontraditional Antiarrhythmic Drugs on Sudden Cardiac Death One difficulty in interpreting the results of clinical studies on SCD is the diversity of potential underlying mechanisms of death, with arrhythmic SCD representing a large fraction of underlying etiologies. This is further complicated by variable definitions of SCD. In some studies, all deaths that occurred up to 24 hours after the onset of symptoms were considered SCDs. This definition is not specific for sudden arrhythmic death because many patients who die from myocardial infarction (MI) within the first 24 hours of symptom onset do so from a cause unrelated to a primary arrhythmia. Thus, in evaluating the effect of nontraditional antiarrhythmic drugs on the incidence of SCD, we will also consider the effects on total mortality, with the understanding that it is possible that some of the mortality reduction may be the result of a decreased incidence of SCD. Studies have suggested that the major mechanisms responsible for sudden arrhythmic death are ventricular tachycardia (VT) and ventricular fibrillation (VF). However, there has been a decreased incidence of VF as a cause of out-of-hospital cardiac arrest and a concomitant increase in non-VF causes.3,4 Therefore, evaluating how therapies alter the occurrence of VT and VF provides more specific insight into the pathophysiological action of these agents to reduce SCD. Broadly, the risk of sudden death stems from several key pathophysiological components: (1) left ventricular dysfunction; (2) myocardial scar or substrate; (3) autonomic nervous system effects; and (4) triggers, such as premature ventricular complexes. In theory, therapies that directly ameliorate these components can indirectly reduce the incidence of SCD. For example, one of the strongest predictors of arrhythmic sudden death is left ventricular dysfunction. The risk of VT and VF varies inversely with left ventricular function. Therefore, therapies that may improve myocardial structure and function and decrease disease progression may also reduce the risk of VT and VF and the risk of SCD. In addition, left ventricular dilatation may contribute to arrhythmogenesis by altering cardiac electrophysiological properties through contraction-excitation feedback. Thus, pharmacologic agents that modulate cardiac hemodynamics may be antiarrhythmic by altering ventricular function and size, even if direct antiarrhythmic activities are not present. Despite extensive research in the field of sudden death, a coherent pathophysiological framework to identify the series of events or conditions necessary and sufficient to trigger sudden death has not been fully delineated. Although the majority of patients who experience SCD have coronary artery disease, patients with nonischemic cardiomyopathy also have substantial risk for SCD. In addition, in the minority of patients with coronary artery disease who have SCD, acute MI is the immediate cause. Transient autonomic and metabolic factors may also contribute to SCD. Thus, a pathophysiological framework underlying even arrhythmic SCD needs to broadly include many contributing factors. One potential model of the pathophysiology of sudden death is shown in Figure 113-1. In this construct, transient factors or triggers interact with spontaneous arrhythmias on an abnormal, underlying anatomic substrate that contributes to electrophysiological conditions leading to SCD. This model recognizes that the triggering beats may be caused by reentry, abnormal automaticity, or triggered activity, and that the electrophysiological substrate could consist of either functional or anatomic reentry. Thus, transient metabolic and autonomic factors that alter electrophysiological properties can modulate the anatomic substrate to heighten the occurrence of sudden death. To target the electrophysiological substrate that forms part of the model, it is necessary to identify a specific electrophysiological mechanism and to identify specific ion channels or other targets that are present in the region of the myocardium that predisposes to sudden death. At present, antiarrhythmic drugs target sodium or potassium channels present throughout the myocardium and therefore are not specific for the arrhythmic mechanism (focal or reentrant) or abnormal myocardial region. Drugs such as β-blockers also operate on global regions of the myocardium, but their lack of specificity does not appear to be harmful, but rather is protective. Given the overall safety profile of nontraditional drugs compared with traditional antiarrhythmic drugs, it should not be surprising that nontraditional drugs that act on more global mechanisms of arrhythmogenesis are more effective in preventing sudden death than are drugs that indiscriminately block ion channels throughout the myocardium. Potential target sites for therapies that might reduce sudden death discussed in this chapter are shown in Figure 113-1. Table 113-1 shows interventions that may reduce mortality in patients with heart disease and the drugs that may reduce the incidence of sudden death. Show more View chapterExplore book Read full chapter URL: Book 2014, Cardiac Electrophysiology: From Cell to Bedside (Sixth Edition)Leonard Ilkhanoff, ... Jeffrey J. Goldberger Chapter Cardiac arrhythmias 2018, Medical Pharmacology and Therapeutics (Fifth Edition)Derek G. Waller BSc (HONS), DM, MBBS (HONS), FRCP, Anthony P. Sampson MA, PhD, FHEA, FBPhS Drug Treatment of Arrhythmias Arrhythmias can be asymptomatic or can produce a variety of consequences that range from mild symptoms to a life-threatening reduction of cardiac output. The probability of developing symptoms depends on several factors, the most important of which are the rate of an abnormal rhythm and the presence of underlying heart disease. The range of consequences of rhythm disturbances includes: ▪ awareness of palpitation, ▪ dizziness, ▪ syncope, ▪ precipitation of angina or heart failure, ▪ sudden death. Treatment may not be necessary for brief self-terminating arrhythmias with minimal symptoms; reassurance may be all that is required. For some arrhythmias it may be possible to remove or treat an underlying cause, such as an electrolyte disturbance. The choice of treatment depends on the situation. With most tachyarrhythmias, sinus rhythm should be restored if possible. Direct current (DC) cardioversion is used to achieve this in severe, life-threatening, or drug-resistant arrhythmias. Drug therapy is used if there is less need for an immediate effect or to control the ventricular rate if the abnormal rhythm cannot be terminated. Radiofrequency ablation of an arrhythmogenic focus or pathway is increasingly used to prevent arrhythmia. This is carried out after intracardiac electrophysiological studies, using a cardiac catheter. Long-term drug treatment for bradyarrhythmias is not possible; an implanted pacemaker may be necessary. Supraventricular Arrhythmias Atrial premature beats Atrial premature beats are very common and usually benign, but sometimes they arise as a consequence of digoxin toxicity; frequent multifocal atrial ectopics can also result from organic heart disease. Other than treatment of an underlying cause, specific drug therapy is rarely needed. Some people are disturbed by a postectopic pause followed by a more forceful beat when sinus rhythm recommences. If treatment is required, a β-adrenoceptor antagonist or a calcium channel blocker such as verapamil or diltiazem can be used to suppress the ectopics. Atrial tachycardia Atrial tachycardia (AT) is an infrequent rhythm disturbance usually arising from an automatic focus that produces an atrial rate of 150–250 beats/minute. There is often an associated AV conduction block with faster heart rates, since the AV node will still be in its refractory period when some of the impulses reach it. This prevents all of the atrial impulses reaching the ventricle and results in a slower ventricular rate. AT is not usually associated with significant cardiac disease but can be a manifestation of digitalis toxicity. If drug therapy is necessary, an AV nodal blocking agent such as a β-adrenoceptor antagonist or calcium channel blocker (verapamil or diltiazem) will control the ventricular rate but rarely restores sinus rhythm. Sinus rhythm can be achieved with a class Ic antiarrhythmic agent such as flecainide given with an AV nodal blocking drug (flecainide alone increases the risk of 1 : 1 AV nodal conduction and a paradoxical increase in the ventricular rate if the atrial rate slows sufficiently, but sinus rhythm is not restored). Sotalol or amiodarone can also be used to maintain sinus rhythm. Ablation of the initiating focus may also be considered. A less common form of AT is multifocal AT arising from several ectopic foci, usually in people with severe pulmonary disease. If treatment is needed, calcium channel blockers are usually used for ventricular rate control. Atrial flutter In atrial flutter, the atrial rate is usually 250–350 beats/minute and the impulses are conducted to the ventricles with 2 : 1 or greater degrees of AV block. Flutter waves may be obvious on the ECG or appear if the ventricular rate is slowed by transiently increasing the degree of AV block using vagal stimulation (such as carotid sinus massage) or the administration of adenosine. Atrial flutter usually arises from a macro reentrant circuit in the right atrium. Underlying causes include recent cardiac surgery, cor pulmonale and congenital heart disease, but it can arise for no obvious reason. It may be paroxysmal and can degenerate into AF. Drug therapy is relatively unsuccessful for restoring sinus rhythm, and DC cardioversion (synchronised to discharge on the R wave of the ECG) or rapid ‘overdrive’ electrical pacing to capture the ventricle followed by a gradual reduction in the paced rate may be required. Class Ic and III antiarrhythmic agents, such as sotalol or amiodarone, can prevent recurrence of paroxysmal atrial flutter. If a class Ic agent such as flecainide or propafenone is used, an AV nodal blocking drug (such as a β-adrenoceptor antagonist or a calcium channel blocker) should be given concurrently, since the atrial rate could slow with 1 : 1 AV conduction, causing an unacceptably high ventricular rate (see also AT, discussed earlier). Control of the ventricular rate in atrial flutter can be achieved in a similar manner to that in AF (see below), but treatment is often less successful. For this reason, radiofrequency ablation of the reentrant pathway via a cardiac catheter is becoming increasingly popular. Anticoagulant prophylaxis against thromboembolism should be given, similar to the treatment for AF. Atrial fibrillation Apart from ectopic beats, AF is the most common rhythm disturbance in clinical practice. It has a variety of underlying causes (Box 8.1), some of which may be treatable. In younger people, AF often occurs without any obvious underlying cause; it is then called ‘lone’ AF. The arrhythmia usually arises from multiple microreentry circuits in the atria, although a rapid ectopic focus in a pulmonary vein may be responsible for triggering the reentry circuits in paroxysmal AF. The ventricular rate will depend on AV nodal function, so that when the AV node conducts well, AF produces a rapid but irregular ventricular rate. AF predisposes to left atrial thrombus formation that begins in the left atrial appendage and can lead to systemic emboli, which most commonly cause stroke. Clinically, three forms of AF are recognised: paroxysmal (intermittent self-limiting episodes), persistent (present for more than 7 days but less than 1 year) and permanent (present for more than 1 year after unsuccessful attempts to maintain sinus rhythm or if a decision has been made not to attempt this). Management of AF has four components: ▪ To identify and treat an underlying cause whenever possible. ▪ To restore or maintain sinus rhythm in paroxysmal or persistent AF (Box 8.2). It is desirable to attempt to restore sinus rhythm in younger people or those who tolerate the rhythm disturbance poorly. In these individuals, symptoms and exercise tolerance are usually improved when they are in sinus rhythm, but the risk of stroke is not removed (see below). There is less justification for restoring sinus rhythm in older people who tolerate the rhythm well because there is no reduction in the risk of thromboembolic events (see below) and their quality of life may not improve. Restoration of sinus rhythm is usually possible in lone AF or when a precipitating factor has been treated. It can be achieved with drugs (pharmacological or chemical cardioversion), especially if the rhythm disturbance is of recent onset (there is a 40–80% success rate if the arrhythmia is of less than 7 days' duration). QRS-synchronised DC cardioversion is usually required for a more prolonged duration. Pharmacological cardioversion is most rapidly achieved (within a few hours) by using a single oral dose of a class Ic drug such as flecainide or propafenone. These drugs should be avoided if there is underlying structural heart disease, when intravenous amiodarone is preferred, although this takes longer to restore sinus rhythm. Recurrence of AF is most frequent during the first 3–6 months after restoration of sinus rhythm. Drugs are not always recommended for prophylaxis to maintain sinus rhythm after a first cardioversion because of their proarrhythmic effects. However, if there is a high risk of recurrence, sinus rhythm can be maintained with a class Ic drug, sotalol, amiodarone, or dronedarone (see the compendium at the end of this chapter). Amiodarone is the most successful single drug for long-term prevention of recurrence; although it maintains sinus rhythm in only about 75% of people at 1 year, it is superior to the 40% maintenance of sinus rhythm with the other drugs. Digitalis glycosides are ineffective for restoring or maintaining sinus rhythm in paroxysmal AF and should be avoided. Radiofrequency isolation of a pulmonary vein trigger area via a cardiac catheter is becoming increasingly used for younger people with paroxysmal or persistent AF. Other curative procedures are infrequently used. ▪ To control a rapid ventricular response in persistent or permanent AF. For ventricular rate control both at rest and on exercise, a drug that slows AV nodal conduction – such as a β-adrenoceptor antagonist, verapamil, or diltiazem – should be used. Rate control at rest can be achieved with digoxin, but a rapid heart rate often still occurs during exercise (see Chapter 7), so it is only used alone for sedentary people. A β-adrenoceptor antagonist, verapamil, diltiazem, or amiodarone can be used together with digoxin if rate control is difficult to achieve. Sotalol has no additional benefit in sustained AF and should be avoided because of its greater proarrhythmic activity compared with that of other β-adrenoceptor antagonists. Dronedarone is not used for rate control due to evidence of increased mortality. If drug combinations do not provide satisfactory rate control, AV nodal ablation with insertion of a pacemaker can be considered. ▪ To reduce thromboembolism by long-term anticoagulation (see Chapter 11). Anticoagulation with warfarin is the treatment of choice in AF associated with rheumatic heart disease or thyrotoxicosis, which have a high risk of embolism. In nonrheumatic AF, the risk of embolism is related to the number of associated risk factors according to the CHA 2 DS 2–Vasc score (Table 8.3). Most people with AF, whether sustained or paroxysmal, should take thromboprophylaxis. Warfarin, maintaining the international normalised ratio (INR) between 2 and 3 (see Chapter 11), reduces the risk of thromboembolism by about two-thirds. Non-vitamin K-dependent oral anticoagulants (NOACs) have similar efficacy to warfarin in nonvalvular AF. These include dabigatran, apixaban, edoxaban and rivaroxaban (see Chapter 11) and are increasingly preferred to warfarin because they pose a lower risk of intracranial bleeding and there is no need for monitoring. Oral anticoagulation has little advantage in those at very low risk, when the increased risk of bleeding outweighs the benefit. Even after restoration of sinus rhythm in paroxysmal or persistent AF, people at high risk of thromboembolic events should continue to take thromboprophylaxis, since the risk of stroke does not decrease. This may reflect the high risk of recurrence (often asymptomatic) of AF. Aspirin is ineffective for thromboprophylaxis and should not be used for this purpose. For those unable to take an anticoagulant, occlusion of the left atrial appendage with a mechanical device can be considered. Junctional (nodal) tachycardias Junctional tachycardias usually arise from a reentry circuit and are often initiated by an ectopic beat. A macroreentry circuit can form within the AV node if there are two functional intranodal pathways with different recovery times (AV nodal reentrant tachycardia or AVNRT). Such circuits account for 60% of supraventricular tachycardias (SVTs) excluding AF/flutter and are not usually associated with structural cardiac disease. Alternatively, a macroreentry circuit may involve an accessory AV pathway connecting the atria and ventricles, as in Wolff–Parkinson–White syndrome (AV reentrant tachycardia or AVRT), which accounts for 30% of SVTs excluding AF/flutter. Termination of an acute attack of nodal tachycardia can often be achieved with vagotonic manoeuvres such as carotid sinus massage or by a bolus injection of adenosine. For AVNRT, β-adrenoceptor antagonists, diltiazem or verapamil can be used to treat acute episodes or for prophylaxis. However, if there is an accessory AV pathway (AVRT) diltiazem, verapamil and digoxin should be avoided because selective blockade of the AV node by these drugs can predispose to rapid conduction of atrial arrhythmias through the accessory pathway. AVRTs often respond well to flecainide, sotalol or amiodarone. Radiofrequency ablation of the reentry circuit via a cardiac catheter is being employed increasingly for both AVNRTs and AVRTs. Immediate management of narrow-complex tachycardia of uncertain origin If the rhythm is regular, it is often not possible to determine from the ECG whether a narrow-complex tachycardia has an atrial or nodal origin. If vagotonic manoeuvres are unsuccessful and the person is haemodynamically stable, intravenous adenosine should be given. This often converts a junctional tachycardia to sinus rhythm or can slow the ventricular rate sufficiently to identify the origin of the rhythm on an ECG (e.g. revealing flutter waves). If there is a history of severe asthma, intravenous verapamil may be preferred. DC cardioversion should be considered if there is haemodynamic instability. Ventricular Tachyarrhythmias Ventricular ectopic beats Ventricular ectopic beats often occur in healthy individuals or in association with a variety of cardiac disorders such as ischaemic heart disease and heart failure. They are often asymptomatic. Frequent ventricular ectopic beats after myocardial infarction predict a poorer long-term outcome; however, suppressing such ectopics with class I antiarrhythmic drugs increases mortality and should be avoided. In contrast, β-adrenoceptor antagonists after myocardial infarction may suppress the ectopic beats and reduce the risk of sudden death (see Chapter 5). A β-adrenoceptor antagonist can also suppress ventricular ectopic beats induced by stress or anxiety. In the absence of ischaemic heart disease, symptomatic ventricular ectopic beats that do not respond to a β-adrenoceptor antagonist can be suppressed by a class I drug such as flecainide. Radiofrequency catheter ablation is considered for multiple ventricular ectopic beats that disturb the person's well-being and are not controlled by drugs. Ventricular tachycardia VT presents with broad QRS complexes on the ECG (broad-complex tachycardia). Although broad complexes can arise with SVTs (when there is bundle branch block), broad-complex tachycardia is usually treated on the assumption that it is VT. VT is often associated with serious underlying heart disease, such as ischaemic heart disease or heart failure, and is more common following myocardial infarction. It can be either sustained or nonsustained. Sustained VT can be associated with a minimal or absent cardiac output (‘pulseless’ VT) when it is treated in the same way as VF (see below). Polymorphic or incessant VTs can arise as a complication of antiarrhythmic drug therapy (see above) and with other drugs that prolong the Q–T interval on the ECG. For sustained VTs, drug options include class Ib antiarrhythmic agents such as lidocaine (especially after myocardial infarction), and amiodarone. Ablation or an implantable cardioverter defibrillator (ICD) is often appropriate to avoid sudden cardiac death. A β-adrenoceptor antagonist can be used with an ICD to reduce the frequency of shocks. Sustained VT is often associated with a poor long-term outlook in ischaemic heart disease, and coronary revascularisation with or without an ICD is often used. During and after the acute phase of myocardial infarction, a β-adrenoceptor antagonist is the treatment of choice to suppress nonsustained VTs. Polymorphic or incessant VTs do not respond well to conventional treatments. Withdrawal of a precipitant drug, correction of electrolyte imbalance and intravenous magnesium sulphate are the therapies of choice. Temporary transvenous overpacing at a rate of 90–110 beats/minute may prevent recurrence. In the congenital form of long QT syndrome a β-adrenoceptor antagonist is the mainstay of treatment. Ventricular fibrillation VF is a potentially lethal arrhythmia that constitutes one form of ‘cardiac arrest’. An algorithm for the management of cardiac arrest is regularly updated by the Resuscitation Council (UK) and is shown in Fig. 8.6. The important principles of prolonged resuscitation are the maintenance of adequate cardiac output by external chest compression and oxygenation by artificial inflation of the lungs while attempting to restore sinus rhythm. VF is the commonest arrhythmia in cardiac arrest and it should be assumed to be present unless an ECG is available to show otherwise. It should be treated with immediate DC cardioversion. Adrenaline (epinephrine; see Chapter 4) may be given to vasoconstrict the peripheries and thus maintain pressure in the central arteries perfusing the heart and brain. For recurrent VF, an automatic ICD is usually required. Suppression by long-term use of antiarrhythmic drugs such as sotalol or amiodarone is less commonly used. Bradycardias Sinus bradycardia Treatment with atropine may be necessary if sinus bradycardia is causing symptoms (e.g. after myocardial infarction or an overdose of a β-adrenoceptor antagonist). Hypotension precipitated by drugs such as streptokinase (see Chapter 11) is often associated with vagally mediated bradycardia, which will respond to atropine. Atrioventricular block (‘heart block’) AV block can be congenital or may accompany a variety of heart diseases. When it occurs after myocardial infarction, it is usually temporary if the infarct is inferior but is often permanent after anterior infarction. First-degree heart block (prolongation of the P–R interval on the ECG but with all P waves conducted to the ventricles) or Wenckebach (Mobitz type 1) second-degree heart block (progressive P–R prolongation until there is a nonconducted P wave) rarely requires treatment. Higher degrees of second-degree block (Mobitz type 2 with regular nonconducted P waves without a preceding P–R prolongation) and third-degree heart or complete heart block (when all P waves are nonconducted and there is a slow escape rhythm arising from tissues below the AV node) should be treated. If complete AV block arises suddenly, loss of consciousness (Stokes–Adams attack) or death can occur. If the onset of heart block is acute, atropine should be given intravenously to increase AV conduction. Alternatively, an intravenous infusion of the nonselective β-adrenoceptor agonist isoprenaline can be used (see Chapter 7). However, external or temporary transvenous electrical cardiac pacing is usually required in an emergency. If the AV block is permanent, the implantation of a permanent electrical cardiac pacemaker is almost always necessary. Compendium: drugs used to treat cardiac arrhythmias \| Drug \| Characteristics \| --- \| \| Drugs are listed by their main effect on the cardiac action potential according to the Vaughan Williams classification, although some antiarrhythmic drugs have multiple sites and mechanisms of action. The types of arrhythmias commonly treated are AF, AT, SVT, VA, VF and VT. \| \| Class I drugs: sodium channel blockers (‘membrane stabilisers’) \| \| Disopyramide \| Class Ia drug. Used for SVT, VF and VT but rarely in the United Kingdom. Given orally or by slow intravenous injection (over at least 5 min) or intravenous infusion. Half-life: 4–10 h \| \| Flecainide \| Class Ic drug. Used for AF, SVT. Given orally or by slow intravenous injection (over 10–30 min) or intravenous infusion (for resistant ventricular tachyarrhythmias). Half-life: 12–30 h \| \| Lidocaine \| Class Ib drug. Used for VT in haemodynamically stable patients and for VF and pulseless VT in cardiac arrest refractory to defibrillation. Given by intravenous injection or intravenous infusion. Half-life: 2 h \| \| Mexelitine \| Class 1b drug. Available in the United Kingdom only from special-order manufacturers or specialist importers; used for life-threatening VA. Given orally or intravenously. Half-life: 10–12 h \| \| Procainamide \| Class Ia drug. Available in the United Kingdom only from special-order manufacturers or specialist importers; used for VA. Given by slow intravenous injection or intravenous infusion. Half-life: 6–9 h \| \| Propafenone \| Class Ic drug; also has some β-adrenoceptor antagonist (class II) activity. Used for VA and some SVTs. Given orally. Half-life: 4–17 h \| \| Class II drugs: β-adrenoceptor antagonists \| \| β-Adrenoceptor antagonists are used in a wide variety of indications and are listed in the drug compendium at the end of Chapter 5 . The β-adrenoceptor antagonists most commonly used in arrhythmia are atenolol, bisoprolol, esmolol and propranolol. See also sotalol in class III below. \| \| Class III drugs: potassium channel blockers \| \| Amiodarone \| Blocks potassium channels but also has class Ib, class II and class IV actions. Used for paroxysmal SVT, VT, AF and VF, with treatment usually initiated in hospital or under specialist supervision. Given orally or by intravenous injection (over 3 min) for VF. Half-life: 50–60 days \| \| Dronedarone \| Multichannel blocker; related to amiodarone but has less complicated dosing due to shorter half-life. Used for maintenance of sinus rhythm after cardioversion in those who are clinically stable with paroxysmal or persistent AF when alternative treatments are unsuitable. Given orally. Half-life: 25–30 h \| \| Sotalol \| Blocks potassium channels but is also a nonselective β-adrenoceptor antagonist (class II). Used for life-threatening VT and prophylaxis of SVT and AF. Given orally or by intravenous injection (over 10 min). Half-life: 7–18 h \| \| Class IV drugs: calcium channel blockers \| \| Dihydropyridine calcium channel blockers (see Chapter 5) have no antiarrhythmic activity. \| \| Diltiazem \| Nondihydropyridine calcium channel blocker. Not licensed for arrhythmia in the United Kingdom. Given orally. Half-life: 2–5 h \| \| Verapamil \| Nondihydropyridine calcium channel blocker. Used for SVT. Given orally or by intravenous injection. Half-life: 2–7 h \| \| Other drugs \| \| Adenosine \| Purine nucleoside. Used as the treatment of choice for terminating paroxysmal SVT. Given intravenously. Half-life: 10 s \| \| Atropine \| Muscarinic receptor antagonist. Used for bradycardia, especially if complicated by hypotension. Given intravenously. Half-life: 2–5 h \| \| Digoxin \| Cardiac glycoside (see Chapter 7). Used for AF. Given orally or intravenously. Half-life: 20–50 h \| \| Magnesium sulphate \| Magnesium salt. Used for emergency treatment of serious arrhythmias, especially in presence of hypokalaemia, hypomagnesaemia or torsade de pointes. Given intravenously over 10–15 min \| AF, Atrial fibrillation; AT, atrial tachycardia; SVT, supraventricular tachycardia; VA ,ventricular arrhythmias; VF,ventricular fibrillation; VT,ventricular tachycardia. Show more View chapterExplore book Read full chapter URL: Book 2018, Medical Pharmacology and Therapeutics (Fifth Edition)Derek G. Waller BSc (HONS), DM, MBBS (HONS), FRCP, Anthony P. Sampson MA, PhD, FHEA, FBPhS Chapter Pharmacologic Bases of Antiarrhythmic Therapy 2014, Cardiac Electrophysiology: From Cell to Bedside (Sixth Edition)Juan Tamargo, Eva Delpón Antiarrhythmic Effects of Class I AADs Considering their mechanisms of action, class I AADs are able to terminate arrhythmias generated by both focal ectopic activity and reentry. Class I AADs slow or suppress automatic activity in atrial and ventricular ectopic pacemaker cells that generate Na+-dependent APs, because they slow the spontaneous diastolic depolarization and shift the threshold voltage to less negative potentials. They can also suppress DADs by decreasing Na+-dependent Ca i 2+ overload (I Na [class I drugs] and I NaL [amiodarone, flecainide, ranolazine] inhibitors). Describing the ultimate mechanisms responsible for the antiarrhythmic effects of class I AADs in reentry is a big challenge. Indeed, the underlying mechanism responsible for reentry itself is a matter of debate. Class I AADs markedly slow conduction and prolong refractoriness. The leading-circle theory of reentry predicts that because Na+channel blockers slow conduction they should, if anything, favor reentry by decreasing the wavelength (product of refractory period and conduction velocity), resulting in proarrhythmic effects. The proarrhythmia risk increases with drugs that produce marked conduction slowing in depolarized-ischemic cardiac tissues. Experimental and clinical evidence, however, shows that class I AADs can terminate reentrant arrhythmias, such as AF, without increasing the wavelength. The most characteristic electrophysiological change that they produce before AF termination is an increase in the temporal excitable gap.13,14 According to the multiple wavelets theory of the origin of AF, this widening will lower the chance that reentrant waves encounter areas of partially refractory tissue, so that slowing of conduction and fractionation of wavelets will occur less frequently. This widening decreases the number of fibrillation waves by promoting their fusion, which increases the chance to terminate the arrhythmia. However, there are clinical and experimental data demonstrating that reentry is maintained by the periodic activity of one or a small number of functional reentrant sources with a wavefront rotating around a central core (“rotor”).15,16 The waves emerging from the rotors undergo spatially distributed fragmentation and give rise to fibrillatory conduction. The size of the spiral wave is determined by tissue excitability and refractoriness, so that the rotor will turn faster and in a more stable position, resulting in higher excitability and shorter refractoriness.15,16 In this context, Na+ channel blockers terminate reentry as they (1) enlarge center of rotation, so that the rotor cannot any longer be accommodated by the substrate; (2) decrease anchoring to functional obstacles, increasing meander and extinction at boundaries; and (3) reduce the number of daughter waves that could provide new primary rotors.15,16 View chapterExplore book Read full chapter URL: Book 2014, Cardiac Electrophysiology: From Cell to Bedside (Sixth Edition)Juan Tamargo, Eva Delpón Related terms: Amiodarone Repolarization Calcium Channel Myocarditis Uncaria Vagus Nerve Stellate Ganglion Receptor Potential Metabolite Action Potential View all Topics Recommended publications Heart RhythmJournal Journal of the American College of CardiologyJournal Free Radical Biology and MedicineJournal The American Journal of CardiologyJournal Browse books and journals About ScienceDirect Remote access Advertise Contact and support Terms and conditions Privacy policy Cookies are used by this site. 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2111	https://ocw.mit.edu/courses/7-016-introductory-biology-fall-2018/resources/lecture-14-genetics-32014linkage-crossing-over/	Lecture 14: Genetics 3—Linkage, Crossing Over \| Introductory Biology \| Biology \| MIT OpenCourseWare Browse Course Material Syllabus Calendar Readings Lecture Videos Recitations Assignments Exams Study Materials Course Info Instructors Prof. Barbara Imperiali Prof. Adam Martin Dr. Diviya Ray Departments Biology As Taught In Fall 2018 Level Undergraduate Topics Health and Medicine Cancer Functional Genomics Science Biology Biochemistry Cell Biology Genetics Microbiology Molecular Biology Virology Learning Resource Types assignment_turned_in Problem Sets with Solutions grading Exams with Solutions theaters Lecture Videos Download Course menu search Give Now About OCW Help & Faqs Contact Us searchGIVE NOWabout ocwhelp & faqscontact us 7.016 \| Fall 2018 \| Undergraduate Introductory Biology Menu More Info Syllabus Calendar Readings Lecture Videos Recitations Assignments Exams Study Materials Lecture Videos Lecture 14: Genetics 3—Linkage, Crossing Over Description In his third lecture on genetics, Professor Martin picks up from the last lecture on eye color in fruit flies, and then continues with Mendelian inheritance. He then talks about linkage, crossing over, and gene mapping. Instructor: Adam Martin Video Player is loading. Play Video Play Mute Current Time 0:00 / Duration 49:02 Loaded: 0.00% 00:00 Stream Type LIVE Seek to live, currently behind live LIVE Remaining Time-49:02 1x Playback Rate Chapters Chapters Descriptions descriptions off, selected Captions captions settings, opens captions settings dialog captions off, selected English Captions Audio Track Picture-in-Picture download button Download video Fullscreen playback speed Playback speed 0.25 0.5 0.75 1, selected 1.25 1.5 This is a modal window. Beginning of dialog window. Escape will cancel and close the window. Text Color Transparency Background Color Transparency Window Color Transparency Font Size Text Edge Style Font Family Reset restore all settings to the default values Done Close Modal Dialog End of dialog window. Transcript Download video Download transcript 0:17 ADAM MARTIN: And so I wanted to start today's lecture 0:20 by continuing what we were talking 0:22 about in the last lecture. 0:25 So I'm just going to hide this real quick. 0:28 And so we're talking about the fruit fly and the white gene 0:34 and the white mutant, which results in white-eyed flies. 0:38 And we talked about how if you take females that have red eyes 0:46 and cross them to males, the white-eyed male, 0:54 then 100% of the progeny has red eyes in the F1 generation. 1:01 And so I asked you guys, would you 1:04 get the same results if you did the reciprocal cross? 1:10 So what if we took white-eyed females 1:13 and mated them to red-eyed males? 1:17 So what about this? 1:23 Actually, I'm going to move this over to over 1:27 here so that maybe it's more visible. 1:30 So what if we have white-eyed females 1:34 and crossed this to red-eyed males? 1:41 So let's unpack this sort of a little bit at a time. 1:44 So what's the genotype of these white-eyed females here? 1:50 Miles? 1:51 AUDIENCE: So if you designate the eye gene as the letter A, 1:57 a female would be X lowercase a, X lowercase a. 2:00 ADAM MARTIN: Yes. 2:01 So Miles is exactly right. 2:05 So the dominant phenotype is red eyes, 2:08 because the gene encodes for an enzyme that's 2:12 important for the production of the red pigment. 2:15 And so X lowercase a here would be a recessive mutant that 2:26 lacks the pigment. 2:30 And because it's a recessive allele-- 2:32 because you need only one copy of this gene 2:36 to produce the pigment. 2:37 So the recessive allele results in the white phenotype. 2:42 Therefore, this has to be homozygous recessive. 2:46 How about this red-eyed male? 2:49 Yeah, Ory? 2:51 AUDIENCE: Wouldn't you have a Y and then an X capital A? 2:53 ADAM MARTIN: Yes. 2:54 So this would be this phenotype, right, 2:56 where capital A is the gene that produces-- 3:01 is a normal functioning gene that produces the pigment. 3:07 So then in your F1 here, are you going 3:11 to see something similar to this or something different? 3:15 AUDIENCE: Something different. 3:17 ADAM MARTIN: Different, great. 3:17 Who said different? 3:18 Javier? 3:19 Do you want to propose what you might see? 3:22 AUDIENCE: Yeah. 3:22 For the males, they're going to inherit the Y 3:25 gene from the father and the [INAUDIBLE].. 3:28 ADAM MARTIN: Exactly. 3:29 So the males are going to get the Y from the father, 3:33 and they're going to get one X from their mother. 3:36 So all the males are going to be of this genotype 3:39 here, which means they're going to have what color eye? 3:44 Javier is exactly right. 3:46 That means they're going to have white eyes. 3:49 So all the males will have white eyes. 3:58 And what about the females? 4:02 AUDIENCE: [INAUDIBLE] 4:04 ADAM MARTIN: What's that? 4:05 AUDIENCE: Red eyes. 4:06 ADAM MARTIN: Yeah. 4:07 So Ory is saying the males are going to get red eyes, right, 4:10 because they're-- 4:11 or the females are going to have red eyes, because they're 4:14 going to get the X chromosome from their father, which 4:17 has the dominant gene that produces the red pigment. 4:24 So all the females are going to be heterozygous, 4:27 but have a functional copy of this gene. 4:29 So all of the females will have red eyes. 4:38 OK, does everyone see how-- 4:41 now, how would this compare with Mendel's crosses and pea color? 4:45 Would there be a difference in Mendel's crosses 4:47 if you switch the male versus the female 4:50 if these were autosomal traits? 4:52 Ory is shaking his head no, and he's right, right? 4:55 In that case, it doesn't matter. 4:57 You can do the reciprocal crosses, 4:59 you get the same result. But because this is sex linked, 5:03 which one is the male and which is the female is relevant. 5:07 And this actually relates to something 5:10 that we just saw on the MIT news. 5:14 I just got this email this morning, 5:15 but it came out I think yesterday, 5:18 which is that biology-related research in the mechanical 5:23 engineering department-- specifically, the CAM lab-- 5:27 they've been able to design a 3D sort of model 5:32 for ALS disease, which is also known as Lou Gehrig's disease. 5:37 And so what they've done in the CAM lab 5:39 is to take cells from either patients that have ALS 5:44 or from normal individuals, and they coax 5:47 these cells to become neurons. 5:49 Here, you're seeing a neuron in blue and green here. 5:52 And you see the neurites extend from this neuron. 5:56 And they have a model where this neuron can then 5:58 synapse with a muscle. 6:00 And so they're using this 3D sort of tissue model 6:06 to model ALS and to look for drugs that might affect 6:12 ALS, potentially curing ALS. 6:17 And so last night, I started reading about ALS 6:20 and was pleased to find that there's actually 6:24 a very rare X-linked, dominant form of the disease that 6:28 can be passed on from generation to generation. 6:31 And the inheritance pattern of this X-linked, dominant version 6:36 of ALS would have an inheritance pattern 6:39 that's similar to what we observe for the white mutant 6:43 in the fruit fly, right? 6:45 Whereas, if you have an affected father, 6:49 and this is a dominant mutant on the X chromosome, 6:52 then all of his daughters will get that X chromosome 6:56 and be affected, whereas the sons will all be unaffected. 7:00 However, if you have the reciprocal situation, 7:04 where you have an affected mother and an unaffected 7:07 father, then the sons and daughters 7:10 get the disease randomly. 7:13 So this is a sort of form of inheritance, which is relevant 7:18 if you're considering human disease and some forms of it. 7:22 Most versions of ALS, well, are sporadic, 7:26 but inherited forms are usually autosomal dominant. 7:29 So this is a rare case here. 7:33 But I thought it was interesting in that it's relevant to what 7:36 we've been talking about. 7:38 So now, just to recap-- 7:41 here, I'll throw this up so everything's up. 7:47 So in the last lecture, we talked 7:49 about Mendelian inheritance. 7:58 And we talked about when you take two parents that 8:02 differ in two traits and you perform a cross, 8:06 you get a hybrid individual that is heterozygous for both genes. 8:19 And now this is the F1 individual. 8:21 Let's say we want to know what types of gametes 8:24 this F1 individual produces. 8:29 We can perform a type of a cross known as a test cross, where 8:34 we cross this individual to another individual that 8:40 is homozygous recessive for both these genes, which means 8:48 that you know exactly which alleles 8:51 are coming from this parent. 8:52 And they're both recessive, so you can see whether or not 8:56 the gamete produced by this individual 8:59 has either the dominant or the recessive allele. 9:05 Let me see. 9:05 I'll boost this up. 9:12 So now we can consider the different types of progeny 9:15 that result from this test cross. 9:18 And some will have the two dominant alleles 9:22 from this parent and will, therefore, 9:25 be heterozygous for the A and B gene. 9:30 And it would exhibit the dominant A and B phenotype. 9:35 I think that is what I'm showing here. 9:39 So if the chromosomes, during meiosis I, 9:42 align like this, then the two dominant alleles 9:46 segregate together, and you get AB gametes. 9:50 And you also, reciprocally, get these lowercase a 9:54 and b gametes, as well. 9:57 So that's the other class here. 10:00 So you can get these two classes of progeny. 10:05 And the phenotypes of these two classes 10:08 will resemble the parents, right? 10:11 So these are known as parental gametes. 10:16 So these are the parentals. 10:20 But you know because Mendel showed 10:23 that if you have genes and their alleles 10:26 on separate chromosomes, they can assert independently 10:29 of each other. 10:30 So an alternatively likely scenario 10:36 is that the chromosomes align like this, 10:38 where now the dominant allele of B 10:41 is on the other side of the spindle. 10:44 And therefore, these chromosomes are 10:47 going to segregate like this during the first meiotic 10:50 division. 10:52 And that gives rise to gametes that 10:55 have a different combination of alleles than the parents. 11:02 So you have some that look like this. 11:07 So each of these would be different classes of progeny. 11:12 And you have one last class that would look like this. 11:18 And so neither of these look like the original parents, 11:23 and so they're known as non-parental. 11:30 And so if these two genes are behaving according 11:35 to Mendel's second law, where there's 11:37 independent assortment-- 11:43 if you have independent assortment, 11:46 what's going to be the ratio of parental to non-parental? 11:55 Rachel? 11:55 AUDIENCE: One to one. 11:56 ADAM MARTIN: Yeah, Rachel says one to one, 11:58 and I think a number of others also said one to one. 12:00 So you have 50% parental, 50% non-parental, right? 12:05 Because it's equally likely to get 12:07 either of those alignments of the homologous chromosomes 12:11 during meiosis I. 12:15 So now I'm going to basically break 12:17 the rules I just explained to you in the last lecture 12:21 and tell you about an exception, which is known as linkage. 12:28 Gesundheit. 12:29 And in the abstract sense, linkage 12:34 is simply when you have two traits that 12:37 tend to be inherited together. 12:40 So just considering probability. 12:43 So you have traits inherited together. 12:46 They're exhibiting what is known as linkage. 12:52 But that's an abstract way to think about it. 12:54 It's just based on probability, right? 12:57 So a physical model for what linkage is, 13:01 is that you have chromosomes. 13:03 The genes are on the chromosomes. 13:05 And for two genes to be linked, those genes 13:09 are physically near each other on the chromosome. 13:12 So the physical model is that two genes 13:16 are near each other on the chromosome. 13:30 OK, so let's consider again these generic genes, 13:33 A and B. If A and B resulting from this cross-- 13:38 if these two happen to be on the same chromosome, 13:42 now they're physically coupled to each other. 13:45 Then they're going to tend to be inherited together. 13:49 No matter how these align, they're 13:52 always going to go together during 13:54 the first meiotic division. 13:56 And that's just going to only give you the parental gametes. 14:00 So if there's linkage, you're going to have-- 14:04 let's consider the case where you have complete linkage. 14:10 If you have complete linkage, 100% of the gametes 14:14 are going to be parentals, and you're 14:16 going to have 0% non-parental. 14:23 That's if the genes are really, really, 14:26 really close to each other, and maybe you 14:28 don't count so many progeny. 14:30 You won't see any mixing between the two. 14:33 But there is a phenomenon that can separate these genes, 14:37 and it's known as crossing over. 14:42 And another term to describe it is recombination. 14:46 So the alleles are getting recombined 14:48 between the chromosomes. 14:53 Recombination. 14:57 And what crossing over or recombination is, is it's 15:01 a mixing of the chromosomes, if you will. 15:08 Or it's an exchange of DNA. 15:11 So there's a physical exchange of DNA from one 15:14 of the homologous chromosomes to the other, OK? 15:18 So you can think of this as an exchange of DNA 15:26 between the homologous chromosomes, OK? 15:30 And that's important. 15:31 It's not an exchange between the sister chromatids, 15:35 but between the homologous chromosomes 15:37 that have the different alleles. 15:43 And what's shown here is a micrograph showing you 15:47 a picture of the process of crossing over. 15:50 You can see the centromeres are the dark structures there. 15:54 And you can see how the homologous chromosomes 15:57 intertwine. 15:59 And there are regions where it looks like there's a cross. 16:02 Those are the homologous chromosomes crossing over 16:06 and exchanging DNA such that one part of that chromosome 16:11 gets attached to the other centromere. 16:15 So I'll just show you in sort of my silly cartoon form 16:20 how this is, just to make it clear. 16:23 So let's say, again, you have these A and B genes, 16:25 and they're physically linked on the chromosome. 16:28 During crossing over, you can get 16:31 an exchange of these alleles, such as a bit of one chromosome 16:36 goes to the other homologous chromosome and vice versa, OK? 16:40 So now you have the dominant A allele 16:42 with the recessive b allele and vice versa. 16:45 So now during meiosis I-- 16:50 after meiosis II, this will give rise 16:52 to two types of gametes, one of which is non-parental. 16:57 And the same for the one down here. 17:00 You get two types of gametes. 17:02 One is non-parental-- the lowercase a, uppercase B 17:06 gamete. 17:11 So this happens if there's incomplete linkage. 17:14 That means there can be a recombination event that 17:17 separates the two genes. 17:22 And I'm going to give you an example of a case where 17:27 data was collected with what fraction of each class there 17:30 is. 17:31 So now we're considering an example 17:33 where you have a linkage. 17:35 So A and B are on the same chromosome. 17:40 And so we'll consider a case where, in this class, 17:43 there are 165 members. 17:47 For this one, there is 191. 17:49 So I'm kind of-- 17:51 line down like this. 17:54 And then for the first recombinant class, 17:56 23 individuals. 17:58 And for the last, there are 21 individuals. 18:02 So you can see there are many more of the parental class 18:05 than the recombinant class, but we 18:07 can calculate a frequency, or recombination frequency, 18:15 between these two genes. 18:23 And in this case, the recombination frequency 18:26 is 44/400, which is equal to 11%, OK? 18:34 So 11% of the progeny from this cross 18:38 had some sort of crossing over between the A and B alleles. 18:42 It would've been up here. 18:50 Now, this frequency is interesting, 18:55 because it is proportional to the distance that 19:02 separates these two genes. 19:05 So this recombination frequency is 19:07 proportional to the linear distance along the chromosome 19:12 between the genes. 19:26 Now, it also depends on the recombination frequency 19:29 in a given organism or in a given part of the chromosome. 19:33 So when you're comparing recombination frequencies 19:37 between different organisms, there's 19:39 actually differences in the real different-- 19:42 they're not equivalent. 19:43 You can't compare them, basically. 19:46 And also, there are regions of the chromosome 19:48 where recombination happens less frequently than others. 19:51 And so, again, you can't compare distances along those. 19:56 But overall, you can use this as a distance in order 20:00 to map genes along the linear axis of a chromosome. 20:08 And maps are useful, because you can see where stuff is, right? 20:13 So in this example here, I'll highlight a couple of places. 20:17 Here's Rivendell. 20:19 Here's Lonely Mountain. 20:21 Here's Beorn's house. 20:24 So let's say we are able to determine 20:27 the distance between Rivendell and Lonely Mountain, 20:31 and the distance between Lonely Mountain and Beorn's house, 20:35 and the distance between Rivendell and Beorn's house. 20:38 You'd be able to get a relative picture of where 20:42 all of these places are in relation to each other. 20:47 So this is a two-dimensional map I'm showing here. 20:49 It's not one dimensional, but chromosomes 20:51 are one dimensional, so it's a bit more accurate, OK? 20:56 So this idea that recombination frequency can 21:01 be used to measure distances between genes 21:04 and that this could be used to generate a map 21:08 is an idea that an undergraduate had 21:12 while working in Thomas Hunt Morgan's lab back in 1911. 21:16 And what I find fascinating about the story 21:19 is this guy basically blew off his homework 21:22 to produce the first genetic map in any organism. 21:25 So the person who did it was Alfred Sturtevant, 21:29 and he was an undergraduate at Columbia 21:32 working for Thomas Hunt Morgan. 21:34 And I'll just paraphrase this quote here. 21:37 In 1911, he was talking with his advisor, Morgan, 21:40 and he realized that the variations 21:42 in the strength of linkage attributed by Morgan 21:45 to differences in the separation of genes-- 21:49 so Morgan had already made this connection, 21:52 that the recombination frequency reflects 21:55 the distance between the genes. 21:58 But then Sturtevant realized that this 22:00 offered the possibility of determining sequences 22:03 in the linear dimension of the chromosome between the genes, 22:07 OK? 22:08 So then-- this is my favorite part-- 22:11"I went home and spent most of the night, 22:13 to the neglect of my undergraduate homework, 22:15 in producing the first chromosome map." 22:19 And this is it. 22:20 So the first chromosome map was of the Drosophila X chromosome, 22:24 which we've been talking about. 22:25 There's the white gene, which we've 22:27 been talking about in the context of eye color. 22:30 There's a yellow body gene here. 22:31 There's vermilion, miniature, rudimentary, right? 22:34 These are all visible phenotypes that you can see in the fly. 22:39 And you can measure recombination 22:41 between various alleles of these different genes. 22:48 All right, so now I want to go through with you 22:51 an example of how you can make one of these genetic maps. 22:57 And it's essentially the same conceptually 22:59 as to what Sturtevant did. 23:02 And it involves what is known as a three-point cross. 23:07 So a three-point cross. 23:14 So there are going to be three genes, all of which 23:17 are going to be hybrid, and I'll start 23:19 with the parental generation that 23:21 is little a, capital B, capital D. 23:25 And we'll cross this fly or organism 23:28 to an organism that is capital A, lower case b, lowercase d. 23:34 Yes, Carmen? 23:35 AUDIENCE: So when you write the gametes up there, 23:37 does that imply that they were analogous parents? 23:40 ADAM MARTIN: So what I'm writing here 23:42 is the phenotype, basically. 23:44 And so these are homozygous for each of these, yes. 23:50 I could also write this as-- 23:57 but I'm not going to draw the chromosomes, because it 24:00 kind of gets more confusing. 24:04 I'll draw the chromosomes here in F1, 24:07 because we have now, basically, a tri-hybrid with one 24:12 chromosome that looks like this, right? 24:14 They got that chromosome from this individual here. 24:17 And another chromosome will look like this. 24:22 See? 24:23 So this F1 fly is heterozygous for these three genes, 24:30 and it has these two parental chromosomes. 24:37 So now we can look at the gametes that result from this 24:40 fly by doing a test cross, just like we did before. 24:47 And so we want to cross this to a fly that's 24:49 homozygous recessive for each of these genes. 24:56 And now we can look at the progeny. 24:59 And just by looking at the phenotype, 25:01 we're going to know the genotype, because we 25:03 know all of the flies from this cross 25:07 have a chromosome from this individual that 25:11 has recessive alleles for each gene. 25:15 So we can consider now this first one here. 25:20 That's one potential class of progeny. 25:24 Another class would be this one. 25:27 And these two, you can see, resemble the parents, right? 25:31 So these are the parental classes of the progeny. 25:36 So this is parental. 25:41 All right, now you can consider all other combinations 25:45 of alleles. 25:47 And so I'll quickly write them down. 25:50 You could have something-- 25:52 progeny that look like this and this. 25:56 These are just kind of reciprocal from each other. 25:59 You could have progeny that look like this and this. 26:05 And the last class would be this and this. 26:14 So all of these progeny that I drew down here 26:17 are recombinant, because they don't 26:18 resemble the parents, right? 26:22 Because there are three genes, now there's 26:25 many more ways to get recombinant progeny, as opposed 26:29 to having just two genes, right? 26:32 So you can have many different combinations 26:34 of these different alleles. 26:39 And so now I'm going to give you data 26:41 from a cross with three such genes. 26:44 So you might get 580 individuals that 26:48 look like this, 592 like this, 45 and 40, 89, 94, 3, and 5. 26:59 So this is data that is, I believe, from fly genes. 27:04 I've just ignored the fly nomenclature, 27:06 because it's confusing, and just given them lettered names, OK? 27:11 But this reflects data from some cross somewhere. 27:16 So now we want to know-- 27:20 let's go back to our map. 27:22 We want to make a map, OK? 27:25 And so to make our map, we're going 27:26 to want to consider all pairwise distances 27:30 between different genes. 27:33 So we'll start with the A and B gene. 27:36 I'll write over here. 27:38 So let's consider the A/B distance. 27:42 And remember, to get a distance, we're 27:44 looking at the number of the frequency which 27:47 there is recombination between these two genes, OK? 27:51 So we now have to look through all of these recombinant class 27:55 of progeny and figure out the ones that 27:58 have had a recombination between A and B, right? 28:01 So on the parent chromosome, you see little lowercase a 28:05 started out with capital B and vice versa. 28:11 So any case where we don't have lowercase 28:13 a paired with capital B, there's been some type of exchange. 28:19 So here, lowercase a's with lowercase b. 28:21 So that's a recombinant. 28:23 Here, uppercase A's with uppercase B. 28:26 That's a recombinant, too. 28:28 So we have to add all these up. 28:30 So 45 plus 40. 28:33 How about here? 28:34 Recombination here or no? 28:37 Yes. 28:37 I'm hearing yes. 28:38 That's correct. 28:39 Here, recombination, yes or no? 28:43 Carmen's shaking her head no. 28:44 She's exactly right. 28:46 So we just have to-- these are all the recombinants between A 28:49 and B. So it's 45 plus 40 plus 89 28:54 plus 94, which equals 268 over a total progeny of 1,448. 29:04 And that gives you a map distance of 18.5%. 29:09 Because this method was developed in Morgan's lab, 29:12 this measurement is also known as a centimorgan. 29:15 It was named in honor of Morgan. 29:18 So that's what I refer to when I have lowercase c capital 29:21 M. That's a centimorgan. 29:23 So you can also use centimorgan here. 29:28 All right, so that's A and B, but now we 29:30 have to consider other distances. 29:33 So how about the A/D distance? 29:39 And again, we have to go through and figure out 29:42 where the alleles for A and D have been recombined, OK? 29:48 So little a is with capital D, and upper case A 29:54 is with lowercase d. 29:56 So we have to find all the cases where that's not the case. 29:59 Here, this is lowercase a with capital D 30:02 and capital A with lowercase d. 30:05 This is parental from the respect of just the A and D 30:08 genes, but all the rest of these guys are recombinants, OK? 30:13 So this is 89 plus 94 plus 3 plus 5, which 30:18 comes out to be 191 over 1,448. 30:26 And this is 13.2 centimorgans. 30:30 So that's the distance between A and D. Distance between A-B, 30:39 distance between A-D. 30:42 So the last combination, then, is just B and D. 30:48 So if we consider the B/D distance, 30:53 again, we have to look for all cases in which lowercase 30:57 b and d become separated and uppercase 31:00 B and D become separated. 31:03 Here, they're separated. 31:05 Here, they're separated. 31:08 Wait, no, not here, sorry. 31:10 Here, they're not separated. 31:12 Here, they are separated. 31:15 Everyone see how I'm doing this? 31:18 Are there any questions about it? 31:22 You can just shout it out if you have a question? 31:27 So this distance is 6.4 centimorgans. 31:34 Everyone see how I'm considering every pairwise combination 31:38 of genes and then just ignoring the other one 31:40 and looking for where there's been 31:42 a recombination in the progeny? 31:46 So now that we have our distances, 31:49 we can make our map, right? 31:51 So the two genes that are farthest apart are A and B, OK? 31:57 So that's kind of like here, Rivendell and Lonely Mountain. 32:01 Those are the two genes that are at the extremities. 32:04 So I'll draw this out. 32:07 It doesn't matter which way you put it. 32:10 We're just mapping these genes relative to each other. 32:14 But B and A are the farthest apart from each other. 32:18 Now if we consider the distance between B and D, 32:24 that's 6.4 centimorgans. 32:28 So it appears that D is closer to B 32:31 than it is to A, because it's 13 centimorgans away from A, OK? 32:37 So D is kind of like Beorn's house here. 32:40 It's closer to Rivendell than it is from Lonely Mountain. 32:47 So we'll put that in there. 32:49 This distance is 6.4 centimorgans. 32:52 And then the distance here is 13.2 centimorgans. 32:59 As far as I know, no one of the field of genetic mapping 33:03 has been assaulted by large spiders, 33:05 but the field is still young. 33:08 So one thing that should be maybe bothering you right now 33:13 is if you add up the distance between B and D 33:17 and D and A, you don't, in fact, get 18.4 centimorgans. 33:22 Instead, you get 19.6 centimorgans. 33:28 This is 19.6 centimorgans. 33:32 So it seems like, somehow, we are underestimating 33:37 this distance here, OK? 33:39 So we seem to be underestimating this. 33:47 So why is it that we are underestimating this distance? 33:52 Well, to consider that, you have to sort of look 33:55 at how all of these classes were generated. 34:00 So now I'm going to go through each class, 34:02 and we'll look at how it was generated. 34:05 I'm also going to-- 34:07 well, I'll just draw new chromosomes. 34:10 So we have to draw this order now. 34:12 We have B, D, a. 34:14 So the first chromosome is B, D, a. 34:22 That's right, B, D, a. 34:25 The other chromosome is b, d, A. So now we 34:33 look and see how this recombinant class was 34:36 generated. 34:38 So this is lowercase a. 34:40 We'll start with b-- lowercase b, lowercase a, but capital D. 34:48 So it's capital D, lowercase a. 34:52 So that recombinant results from this chromosome here, where 34:55 there's crossing over, and little b gets 34:58 hooked up to big D and little a. 35:01 See how I did that? 35:02 And then if we consider this class, 35:05 this is uppercase B, lowercase d, capital A. So these two 35:13 classes of progeny result from a single crossover 35:17 between B and D, OK? 35:19 So this is a single crossover between B and D genes. 35:24 And now we can go through and look at how this is generated. 35:29 So to get all recessive alleles on the same chromosome, 35:32 there would be a crossover here. 35:35 And so this is a single crossover between D and A. 35:42 So a single crossover between D and A. Now, 35:49 these last couple classes of progeny 35:51 are interesting in that they're the least frequent class. 35:55 And so when we consider how they're generated, 35:59 we'll start with uppercase B. 36:02 Let's see if I can get rid of this. 36:08 So uppercase B, lowercase d, lowercase a. 36:16 And so what this last class is, is actually a double crossover. 36:26 So this is a double crossover. 36:28 And it's least frequent because there's 36:31 a lower probability of getting two crossovers in this region. 36:35 But now you see that even though it doesn't 36:38 look like there was recombination between A and D, 36:42 in fact, there was. 36:43 There were two crossovers, and it just 36:46 looks like there was no recombination, if you didn't 36:48 see the behavior of gene D. 36:55 So if we take into account that there are actually 36:57 double crossovers between B and A, 37:03 then if we add that into our calculation 37:05 here, where you add in 3 plus 5 multiplied by 2 37:11 because these are both double crossover events, then 37:16 you get the 19.6 centimorgans that you 37:19 would expect by adding up the other recombinations, OK? 37:27 How is that? 37:28 Is that clear to everybody? 37:30 You're going to have to make maps like this on the problem 37:35 set and possibly the test. 37:38 So make sure you can given-- yeah, Ory? 37:40 AUDIENCE: I realized that you immediately [INAUDIBLE] 37:43 overestimated the difference between A and B 37:45 and not overestimated B to D or D to A? 37:48 ADAM MARTIN: It's because when you 37:50 have two genes that are very far apart, 37:53 you can have multiple crossovers. 37:56 And when you have sort of crossovers 38:00 that are in pairs of two, then it's 38:02 going to go from one strand back to the other, 38:05 and so you're not going to see a recombination between the two 38:08 alleles. 38:10 So it's an underestimate, because if you have multiples 38:14 of two in terms of crossovers, you're 38:16 going to miss the recombination events. 38:20 You see what I mean? 38:25 You understand that you can miss the double crossover events? 38:28 AUDIENCE: Yeah, I get that. 38:28 ADAM MARTIN: Yeah, right? 38:30 So then you're going to underestimate 38:31 the number of crossovers that actually happened 38:34 in that genetic region. 38:38 All right, now I want to end with an experiment that, again, 38:45 makes the point that genes are these entities that 38:51 are on chromosomes. 38:54 So just like you can have linkage 38:56 between two genes on the same chromosome, 38:59 you can also have linkage between genes 39:02 and physical structures on chromosomes, 39:05 like the centromere. 39:09 So you could have genes like A and B 39:12 here that are present on the chromosomes 39:14 and present very near the centromere 39:16 of those chromosomes, OK? 39:20 So they could be right on top of the centromere, OK? 39:24 And to show you how this manifests itself, 39:27 I have to tell you about another organism, which 39:31 is a unicellular organism called yeast. 39:35 And yeast is special and that it can 39:38 exist in both a haploid and a diploid form. 39:41 So it has a lifecycle that involves 39:44 it going both as a haploid and as a diploid. 39:50 And so you can take yeast-- 39:54 and we'll take two haploid yeast cells. 39:58 And much like gametes, these can fuse to form a zygote. 40:03 So, in this case, I'm taking-- 40:05 again, we'll consider two generic genes, A and B. 40:09 And we'll make a diploid yeast cell that is heterozygous, 40:14 or hybrid, for A and B. 40:17 And what's great and special about yeast 40:20 and why I'm telling you about this is because as opposed 40:25 to flies and us and other organisms, 40:30 the product of a single meiosis is 40:33 packaged in this single package, if you will. 40:40 So the yeast can undergo meiosis, 40:44 and the product of a single meiosis 40:47 is present in this case, where each of these 40:51 would represent a haploid cell that then can divide and make 40:59 many cells. 41:00 But this is the product of a single meiosis in a package, 41:04 OK? 41:05 So you can actually see the direct result 41:09 of a meiotic division, a single meiotic division. 41:12 So this is the product of a single meiotic division. 41:22 And that's special because when we make gametes, 41:25 we have individual cells. 41:27 All the products of meiosis are split up, 41:30 and then just one randomly finds an egg and fertilizes it. 41:33 So you don't know which of the gametes 41:35 are from the product of a single meiosis. 41:41 And so being able to see the product of a single meiosis 41:45 allows us to see things like genes being 41:47 linked to physical structures on the chromosome 41:50 like the centromere. 41:52 So if we consider this case, these two genes 41:55 are both linked to the centromere. 41:58 And during metaphor phase of meiosis I, 41:59 they could align like this, in which case 42:03 you would get spores that are parental 42:07 for both dominant alleles or parental 42:10 for both recessive alleles. 42:12 So each of these cells is known as a spore. 42:15 So I'll label spore numbers here. 42:18 So this is spore number. 42:20 And in this case, you get two spores 42:27 that are dominant for both alleles and two spores that 42:30 are recessive for both alleles. 42:33 Because there are two types, it's known as a ditype. 42:36 And this is a parental ditype, because you have two types 42:42 of spores . 42:43 And they are both parental 42:53 An alternative scenario is that these chromosomes 42:57 would align differently, right? 42:58 So you get parental spores there. 43:02 However, alternatively, you could have this configuration, 43:06 where this is now flipped. 43:08 And during meiosis I here, these chromosomes move together. 43:13 And they, again, produce two types of spores, 43:16 so it's a ditype. 43:17 But in this case, all the spores are non-parental. 43:23 So another scenario is you get this. 43:30 And because there are two types and they're non-parental, 43:34 this is known as non-parental ditype. 43:45 That's a non-parental ditype. 43:49 And if these genes are linked to the centromere completely, 43:55 then you can only get these two classes of packages, OK? 44:02 So if these genes are unlinked-- 44:04 so the two genes are unlinked, but both 44:10 linked to the centromere, then you get parental ditype-- 44:22 50% parental ditype, type 50% non-parental ditype. 44:28 So I'm abbreviating parental ditype PD 44:32 and non-parental ditype NPD. 44:37 So what has to happen to get another type of spore? 44:43 And another type of spore would be-- 44:45 you could have spores that are all different genotypes 44:48 from each other and that you have A cap dominant A/B; 44:54 dominant A, recessive b; recessive a, dominant B; 44:59 and lowercase a and b. 45:01 And this is known as tetratype, because there are four types. 45:08 So how do you get this tetratype? 45:10 Anyone have an idea? 45:15 Yeah, Jeremy? 45:16 AUDIENCE: You're crossing over. 45:18 So one of A and B would switch in one of the [INAUDIBLE].. 45:26 ADAM MARTIN: Where would the crossing over happen? 45:29 AUDIENCE: Between the two [INAUDIBLE] one [INAUDIBLE].. 45:41 ADAM MARTIN: Between the allele and what? 45:46 AUDIENCE: Sorry? 45:47 ADAM MARTIN: The crossing over would occur between the gene-- 45:50 AUDIENCE: Oh, and the centromere. 45:51 ADAM MARTIN: And the centromere, exactly. 45:52 Jeremy is exactly right. 45:54 So Jeremy said that in order to get a tetratype, 45:57 you have to have a recombination event, 45:59 but this time, not between two genes, 46:02 but between a gene and the centromere. 46:05 So at least one of the genes has to be 46:07 unlinked to the centromere. 46:12 And in that case, now you get a meiotic event 46:16 that gives rise to four spores. 46:18 And there are four different ways to get this. 46:21 So if you have two genes unlinked and at least one 46:31 is unlinked to the centromere, then you 46:40 get a pattern where you have a 1 to 1 46:43 to 4 ratio between all these different events. 46:46 So you have a 1 to 1 to 4 ratio between parental ditype, 46:51 non-parental ditype, and tetratype. 46:56 And we can see this in yeast. 46:58 If you have two genes that are linked to the centromere, 47:01 you only get parental ditypes and non-parental ditypes, 47:05 where virtually everything else gives rise to tetratypes, 47:08 except if they're linked. 47:10 What happens if the two genes are 47:11 linked to each other, irregardless of the centromere? 47:17 If you have two genes that are linked, what's going to be-- 47:20 what are your progeny going to look like? 47:23 AUDIENCE: Parentals. 47:25 ADAM MARTIN: You're only going to get the parentals, 47:26 or you're going to get a lot of parentals. 47:28 Javier is exactly right, right? 47:30 If the two genes are linked, the parental ditypes 47:33 are going to be much greater than any of the other classes. 47:41 Now, this might seem esoteric, but I like the idea 47:44 that you can have linkage between a gene and something 47:48 that's just the place on the chromosome that's 47:50 getting physically pulled. 47:52 It all makes it much more physical, 47:54 which I think is nice to think about. 47:59 All right, we're almost done. 48:01 I just have-- yes, Natalie? 48:03 AUDIENCE: Can you go over what the PD [INAUDIBLE]?? 48:08 ADAM MARTIN: Yes. 48:09 So PD is parental ditype. 48:11 So this is the parental ditype. 48:13 It's a class of product here where 48:18 you get four spores that are each of these genotypes, OK? 48:22 So each of these 1, 2, 3, and 4 would 48:25 represent one of these cells from a single meiotic event. 48:31 Does that make sense, Natalie? 48:33 Does everyone see what I did there? 48:39 So these 1 2 and 3 are the spores of the meiotic event 48:44 right here. Course Info Instructors Prof. Barbara Imperiali Prof. Adam Martin Dr. Diviya Ray Departments Biology As Taught In Fall 2018 Level Undergraduate Topics Health and Medicine Cancer Functional Genomics Science Biology Biochemistry Cell Biology Genetics Microbiology Molecular Biology Virology Learning Resource Types assignment_turned_in Problem Sets with Solutions grading Exams with Solutions theaters Lecture Videos Download Course Over 2,500 courses & materials Freely sharing knowledge with learners and educators around the world. Learn more © 2001–2025 Massachusetts Institute of Technology Accessibility Creative Commons License Terms and Conditions Proud member of: © 2001–2025 Massachusetts Institute of Technology You are leaving MIT OpenCourseWare close Please be advised that external sites may have terms and conditions, including license rights, that differ from ours. 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2112	https://www.quora.com/A-traingle-is-inscribed-in-a-circle-Whats-the-probability-that-the-center-of-the-circle-is-inside-the-triangle	Something went wrong. Wait a moment and try again. Probability (statistics) Inscribed Figure Mathematical Problems Inscribed Problems Probability Theory Geometric Probability 5 A traingle is inscribed in a circle. What's the probability that the center of the circle is inside the triangle? David Schulman Amateur mathematician · Author has 3K answers and 2.8M answer views · Updated 2y Consider this sketch. If we arbitrarily place the triangle’s first two vertices and along the circumference of the circle, we can extend imaginary diameters (red) through its center. Note that, whatever the placement of and might be, this effectively divides the circle into four “quadrants” (not necessarily equal). I’ve numbered them 1 through 4. Note that if the triangle’s third vertex is located in any quadrant but 3 in my sketch, the triangle will not contain the circle’s center. From this, I conclude that the desired probability is . Related questions A triangle with three equal sides is inscribed inside a circle. A point is selected at random inside the circle. What is the probability that a point selected is inside the triangle? What does a circle inside a circle mean? What is the probability of hitting outside the circle but inside the square, assuming the circle is enclosed in a square? What does the circle in the triangle mean? What is the symbol that has a large circle with a triangle inside and an inverted triangle inside that with a vertical line dividing the circle? Bill Crean Solved the Holey Cube problem without calculus. · Author has 5.1K answers and 6.4M answer views · Updated 5y I was attracted to this question because I did not have a clue how to do it. A chance to do some original thinking, for a change. So here goes: First thought is that there are an infinite number of inscribed triangles, which do not contain the circle’s centre. There are also an infinite number of inscribed triangles, which do contain the circle’s centre. By that token, then, there are equal numbers I was attracted to this question because I did not have a clue how to do it. A chance to do some original thinking, for a change. So here goes: First thought is that there are an infinite number of inscribed triangles, which do not contain the circle’s centre. There are also an infinite number of inscribed triangles, which do contain the circle’s centre. By that token, then, there are equal numbers of triangles for each set and so it could be said that the probability is 0.5. No. Mathematicians on Quora would not let me get away with that use of infinity. It might be possible to do an estimate of the probability. Put 8 equi-spaced points on the circumference of the circle. Starting at one of the points move anti-clockwise, say, and choose the adjacent two points and draw a straight line from the starting point to the second and third points. you have, now one triangle, which does not contain the centre. Now take in the 4th point and draw a straight line from it to the starting point and then from 4th point to 3rd point and then 3rd point to starting point. This forms another triangle which does not contain the centre. Now take in the 5th point and do similar constructions. Similarly go through all the 8 points. You will find ... David Joyce Professor Emeritus of Mathematics at Clark University · Author has 9.9K answers and 68.3M answer views · 6y Originally Answered: What is the probability that the center of the circle is contained within the triangle? · Which circle? Which triangle? You’ll need to specify a distribution on the set of all triangles in order to answer this question. In other words, what is the process for choosing a random triangle? Bertrand emphasized the importance of specifying a distribution for triangles before answering any probabilistic question about triangles. His so-called paradox shows how different distributions give different answers. Footnotes Bertrand paradox (probability) - Wikipedia Robby Goetschalckx I work with statistics daily. · Upvoted by Terry Moore , M.Sc. Mathematics, University of Southampton (1968) and David Joyce , Ph.D. Mathematics, University of Pennsylvania (1979) · Author has 6.4K answers and 9.2M answer views · 6y Originally Answered: What is the probability that the center of the circle is contained within the triangle? · What circle and what triangle? If you mean an arbitrary circle and the triangle formed by 3 uniformly randomly sampled points, this is a really fun problem, so I assume that’s what you meant. See here: Mathematics: If three points are chosen at random on a circle, and a triangle is formed by taking the three chosen points as its vertices, what is the probability that the triangle thus formed contains the center of the circle within it? for more information. Promoted by Coverage.com Johnny M Master's Degree from Harvard University (Graduated 2011) · Updated Sep 9 Does switching car insurance really save you money, or is that just marketing hype? This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take. But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious. They claimed I could save over $1,000 per year. I ended up exceeding that number and I cut my monthly premium by over $100. That’s over $1200 a year. For the exact same coverage. No phone tag. No junk emails. Just a better deal in less time than it takes to make coffee. Here’s the link to two comparison sites - the one I used and an alternative that I also tested. If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying. Related questions What does a circle inside of a triangle mean? What does an equalateral triangle, inside a circle, vertices on the circle, symbolize? If a point is inside a circle, then what is the probability that it will be near the centre of a circle? Is it possible for all shapes to fit inside a circle? Why or why not? What is the average area of a triangle constructed from 3 points chosen randomly on a unit circle? Assistant Bot · 1y To determine the probability that the center of a circle (the circumcenter) is inside a triangle inscribed in that circle, we can use some geometric considerations. Key Points: Circumcenter Definition: The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is the center of the circle that passes through the triangle's vertices. Condition for the Circumcenter to be Inside the Triangle: The circumcenter of a triangle is inside the triangle if the triangle is acute. If the triangle is obtuse, the circumcenter lies outside the triangle, and if the tri To determine the probability that the center of a circle (the circumcenter) is inside a triangle inscribed in that circle, we can use some geometric considerations. Key Points: Circumcenter Definition: The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is the center of the circle that passes through the triangle's vertices. Condition for the Circumcenter to be Inside the Triangle: The circumcenter of a triangle is inside the triangle if the triangle is acute. If the triangle is obtuse, the circumcenter lies outside the triangle, and if the triangle is right, the circumcenter lies on the triangle. Probability Estimation: Random Triangle Selection: When we select a triangle randomly by choosing three points on the circumference of the circle, we can analyze the type of triangle formed. Types of Triangles: Acute Triangle: All angles are less than 90 degrees. Obtuse Triangle: One angle is greater than 90 degrees. Right Triangle: One angle is exactly 90 degrees. Known Result: It is known from geometric probability that if you randomly choose three points on the circumference of a circle, the probability that the triangle formed by these points is an acute triangle is [math]\frac{1}{4}[/math] or [math]25\%[/math]. Consequently, the probability that the circumcenter is inside the triangle (i.e., the triangle is acute) is also [math]\frac{1}{4}[/math]. Conclusion: Thus, the probability that the center of the circle is inside the triangle formed by three random points on the circle is: [math]\text{Probability} = \frac{1}{4} \text{ or } 25\%[/math] Gopal Menon B Sc (Hons) in Mathematics, Indira Gandhi National Open University (IGNOU) (Graduated 2010) · Author has 10.2K answers and 15.1M answer views · 4y Related A regular 9-sided polygon is inscribed in a circle. 3 of the polygon's vertices are chosen at random, and the triangle formed by connecting these vertices is drawn. What is the probability that the center of the circle lies inside the triangle? A regular 9-sided polygon is inscribed in a circle. 3 of the polygon's vertices are chosen at random, and the triangle formed by connecting these vertices is drawn. What is the probability that the center of the circle lies inside the triangle? Draw a regular nonagon and number the vertices from [math]1[/math] to [math]9.[/math] Let us see how many triangles can be drawn where vertex [math]1[/math] is not only present in all the triangles but also the centre of the circle lies within the triangle. Let [math]1-2[/math] be one side of the triangle. Then only vertex [math]6[/math] would meet our requirement. Let [math]1-3[/math] be one side of the triangle. Then vertices [math]6[/math] and A regular 9-sided polygon is inscribed in a circle. 3 of the polygon's vertices are chosen at random, and the triangle formed by connecting these vertices is drawn. What is the probability that the center of the circle lies inside the triangle? Draw a regular nonagon and number the vertices from [math]1[/math] to [math]9.[/math] Let us see how many triangles can be drawn where vertex [math]1[/math] is not only present in all the triangles but also the centre of the circle lies within the triangle. Let [math]1-2[/math] be one side of the triangle. Then only vertex [math]6[/math] would meet our requirement. Let [math]1-3[/math] be one side of the triangle. Then vertices [math]6[/math] and [math]7[/math] would meet our requirement. Let [math]1-4[/math] be one side of the triangle. Then vertices [math]6,7[/math] and [math]8[/math] would meet our requirement. Let [math]1-5[/math] be one side of the triangle. Then vertices [math]6,7,8[/math] and [math]9[/math] would meet our requirement. By taking [math]1-9,1-8,1-7[/math] and [math]1-6[/math] as one side of the triangle what we get are repetitions of the above triangles. Thus, it can be seen that we can get [math]10[/math] triangles in all of which vertex [math]1[/math] is present and in which the centre lies inside the triangle. Similarly we can get [math]10[/math] triangles for each of the other [math]8[/math] vertices also. Thus, we get [math]90[/math] triangles in all. However, since each triangle has three vertices, in these [math]90[/math] triangles, each triangle would be repeated thrice. Hence we see that there are only [math]30[/math] distinct triangles meeting our requirement. The total number of ways in which the [math]9[/math] vertices of the nonagon can be selected to form triangles is [math]C(9,3)=84.[/math] Hence, the probability that the centre of the circle lies inside the triangle [math]\frac{30}{84}=\frac{5}{14}.[/math] Edit: My sincere thanks to Glenn Clemens for pointing out the typo in the last line, which has now been corrected. Vinod Chandrasekaran Have some experience teaching high-school Mathematics · Author has 182 answers and 448.6K answer views · Updated 10y Related Mathematics: If three points are chosen at random on a circle, and a triangle is formed by taking the three chosen points as its vertices, what is the probability that the triangle thus formed contains the center of the circle within it? Basically, you want to determine the probability that three randomly chosen points on a circle form an acute angled triangle (because only an acute angled triangle will always have its circumcentre within it). This reduces to a geometric probability problem whose answer I am shamelessly going to copy from cut-the-knot.org - Fix point C. The positions of points A and B are then defined by arcs α and β extending from C in two directions. A priori we know that 0 < α + β < 2π. The favorable for our problem values of α a Basically, you want to determine the probability that three randomly chosen points on a circle form an acute angled triangle (because only an acute angled triangle will always have its circumcentre within it). This reduces to a geometric probability problem whose answer I am shamelessly going to copy from cut-the-knot.org - Fix point C. The positions of points A and B are then defined by arcs α and β extending from C in two directions. A priori we know that 0 < α + β < 2π. The favorable for our problem values of α and β (as subtending acute angles satisfy) 0 < α < π and 0 < β < π. Their sum could not be less than π as this would make angle C obtuse, therefore, α + β > π. The situation is presented in the following diagram where the square has the side 2π. Region D is the intersection of three half-planes: 0 < α,0 < β, and α + β < 2π. This is the big triangle in the above diagram. The favorable events belong to the shaded triangle which is the intersection of the half-planes α < π, β < π, and α + β > π. The ratio of the areas of the two is obviously 1/4. Promoted by Grammarly Grammarly Great Writing, Simplified · Aug 18 Which are the best AI tools for students? There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you? That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners. Here are five AI tools inside Grammarly’s document editor that are worth checking out: Do There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? 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Sabarish Vadarevu Turbulence researcher · Author has 1.3K answers and 1.9M answer views · 10y Related Mathematics: If three points are chosen at random on a circle, and a triangle is formed by taking the three chosen points as its vertices, what is the probability that the triangle thus formed contains the center of the circle within it? Define a polar coordinate system so that the first point is located at [math]\theta = 0[/math] The probability of the second point being in some math [/math] is [math]P_2 = d\theta/2\pi[/math]. For any second point at [math]\theta \in (0,\pi) [/math], any third point that lies between [math]\pi, \pi + \theta[/math] would give rise to a triangle that contains the centre. So, the probability of a triangle forming that contains the centre is [math]P_{3\|2} = \theta/2\pi[/math] Since the points are chosen independently, to get the total probability, multiply the two given above and integrate from 0 to [math]\pi[/math], and then multiply by 2. The Define a polar coordinate system so that the first point is located at [math]\theta = 0[/math] The probability of the second point being in some math [/math] is [math]P_2 = d\theta/2\pi[/math]. For any second point at [math]\theta \in (0,\pi) [/math], any third point that lies between [math]\pi, \pi + \theta[/math] would give rise to a triangle that contains the centre. So, the probability of a triangle forming that contains the centre is [math]P_{3\|2} = \theta/2\pi[/math] Since the points are chosen independently, to get the total probability, multiply the two given above and integrate from 0 to [math]\pi[/math], and then multiply by 2. The answer would be 1/4. Terry Moore PhD in statistics · Author has 16.5K answers and 29.3M answer views · 6y Related Three points are chosen randomly and independently on a circle. What is the probability that the triangle with vertices at these three points contains the center of the circle? Three points are chosen randomly and independently on a circle. What is the probability that the triangle with vertices at these three points contains the center of the circle? Let’s try an intuitive answer that avoids too much mathematics. There are two interpretations: are the points on the circumference of the circle (that’s the usual meaning of circle)? or are they anywhere within the disc (which is what we often call the set of points inside the circle)? Take the first interpretation. There are some symmetries. The first point can be anywhere on the circumference. So we can rotate the circle Three points are chosen randomly and independently on a circle. What is the probability that the triangle with vertices at these three points contains the center of the circle? Let’s try an intuitive answer that avoids too much mathematics. There are two interpretations: are the points on the circumference of the circle (that’s the usual meaning of circle)? or are they anywhere within the disc (which is what we often call the set of points inside the circle)? Take the first interpretation. There are some symmetries. The first point can be anywhere on the circumference. So we can rotate the circle so that that point is in any position we like. We can reflect the circle about the diameter that contains the first point. So we can choose either side of the diameter. If the third point is on the same side as the second the three points don’t form a triangle containing the centre. So the probability is less than [math]0.5[/math]. Now draw the diameter through the second point. For the triangle to contain the centre, the third point must be in the region between the two diameters, but on the opposite side to the first two points. The probability of this event is [math]\frac{\theta}{2\pi}[/math] where [math]\theta[/math] is the angle between the two diameters. At this point I can’t avoid a bit of mathematics. We have to integrate over the possible positions of the second point on the semicircle. The density of its position is uniform [math]\frac1{\pi}[/math]. Therefore the probability is [math]\frac{\pi^2}{4\pi^2}=\frac14[/math]. This suggests that there is a simple proof. I’ll leave the second interpretation to you. Note: there is a possibility that the first two points are the same. In that case the three points lie on a diameter. But with a continuous distribution this possibility has zero probability and can therefore be ignored. Promoted by Savings Pro Mark Bradley Economist · Updated Aug 14 What are the stupidest money mistakes most people make? Where do I start? I’m a huge financial nerd, and have spent an embarrassing amount of time talking to people about their money habits. Here are the biggest mistakes people are making and how to fix them: Not having a separate high interest savings account Having a separate account allows you to see the results of all your hard work and keep your money separate so you're less tempted to spend it. Plus with rates above 5.00%, the interest you can earn compared to most banks really adds up. Here is a list of the top savings accounts available today. Deposit $5 before moving on because this is one of th Where do I start? I’m a huge financial nerd, and have spent an embarrassing amount of time talking to people about their money habits. Here are the biggest mistakes people are making and how to fix them: Not having a separate high interest savings account Having a separate account allows you to see the results of all your hard work and keep your money separate so you're less tempted to spend it. Plus with rates above 5.00%, the interest you can earn compared to most banks really adds up. 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Here are the links to get started: Have a separate savings account Stop overpaying for car insurance Finally get out of debt Start investing with a free bonus Fix your credit Pavel Juranek Technician, Programmer, Analyst, Consultant · Author has 3.7K answers and 1.3M answer views · Updated 4y Related A regular 9-sided polygon is inscribed in a circle. 3 of the polygon's vertices are chosen at random, and the triangle formed by connecting these vertices is drawn. What is the probability that the center of the circle lies inside the triangle? Hard to say. There are insribed polygons where circle center is outer point, wanted probability is zero. For regular polygons having all sides of equal length, You may probability find easy. Fix M as the middle from 3 points, 2 points A,B varies. Make a table of possible A,B positions and underlying N = 9 - A - B = count of seen edges (from point M) Count C as line count and A as aplicable Line count where N<5 i.e. 4 < (A+B) P=A/C is wanted probability Proof Edges can be seen from 360/9=40 deg from centre, from 20 deg from M When N=4, AB is seen from 80 deg from M, 160 deg from centre in MAB When Hard to say. There are insribed polygons where circle center is outer point, wanted probability is zero. For regular polygons having all sides of equal length, You may probability find easy. Fix M as the middle from 3 points, 2 points A,B varies. Make a table of possible A,B positions and underlying N = 9 - A - B = count of seen edges (from point M) Count C as line count and A as aplicable Line count where N<5 i.e. 4 < (A+B) P=A/C is wanted probability Proof Edges can be seen from 360/9=40 deg from centre, from 20 deg from M When N=4, AB is seen from 80 deg from M, 160 deg from centre in MAB When N=5, AB is seen from 100 deg from M, 200 deg from centre out of MAB A, B positions and applicables 1+1, 1+2, 2+1, 2+2, 1+3, 3+1, 2+3, 3+2, 3+3, 1+4, 4+1, 2+4, 4+2, 3+4, 4+3, 4+4 and All next C=7+6+5+4+3+2+1=28 A=28–6=22 P=22/28=11/14 Works at Carnegie Mellon University · 9y Related Mathematics: If three points are chosen at random on a circle, and a triangle is formed by taking the three chosen points as its vertices, what is the probability that the triangle thus formed contains the center of the circle within it? An easier way to think about this problem is that this question is equivalent as 1- P(three point on the circle are within a semicircle). When the three points are within the semicircle, then the center will not be included in the triangle. Let's now compute the probability part. Pick any point, say A, label the rest B,C. The probability that B and C are within the clockwise semicircle is 1/4. And we can pick A, B, and C as our starting point. Hence in total, the desired probability we would like to compute is 3/4. And the answer is 1-3/4 = 1/4. Steve Brown MSc in Applied Mathematics (minor in Computer Science) · Author has 4.1K answers and 4.3M answer views · Updated 6y Related Three points are chosen randomly and independently on a circle. What is the probability that the triangle with vertices at these three points contains the center of the circle? Three points are chosen randomly and independently on a circle. What is the probability that the triangle with vertices at these three points contains the center of the circle? Answer: 50%. Let's call the 3 points on the circle A, B, and C. Since the points are arbitrary, we can select them in any order. So let's select them alphabetically. By symmetry, it doesn't matter where we place the first point, A. So, let's just place it on the top. Also by symmetry, it doesn't matter whether be place point B on the left side of the circle or the right side. So, let's place it somewhere on the left side. Three points are chosen randomly and independently on a circle. What is the probability that the triangle with vertices at these three points contains the center of the circle? Answer: 50%. Let's call the 3 points on the circle A, B, and C. Since the points are arbitrary, we can select them in any order. So let's select them alphabetically. By symmetry, it doesn't matter where we place the first point, A. So, let's just place it on the top. Also by symmetry, it doesn't matter whether be place point B on the left side of the circle or the right side. So, let's place it somewhere on the left side. As shown on the diagram below, no matter where on the left side we place point B, for triangle ABC to contain the center point of the circle, point C must be placed within the green part of the circle, which of course is the same percentage of the circumference of the circle as from point A to point B. Now, on average, as B is moved from the top of the circle to the bottom of the circle on the left side, the green area will make up 50% of half of the circumference of the circle. Therefore, since it's the same on the right side of the circle, the probability that triangle ABC will contain the center of the circle is 50% In reality, the probability would be slightly less than 50% because the two or all of the points might be the same or two of the points might be directly opposite each other. Anil de Silva Former Retired Civil Engineer (1980–2021) · Author has 474 answers and 266.6K answer views · 6y Related What is the probability that 3 points chosen at random inside a circle form an acute triangle? Related questions A triangle with three equal sides is inscribed inside a circle. A point is selected at random inside the circle. What is the probability that a point selected is inside the triangle? What does a circle inside a circle mean? What is the probability of hitting outside the circle but inside the square, assuming the circle is enclosed in a square? What does the circle in the triangle mean? What is the symbol that has a large circle with a triangle inside and an inverted triangle inside that with a vertical line dividing the circle? What does a circle inside of a triangle mean? What does an equalateral triangle, inside a circle, vertices on the circle, symbolize? If a point is inside a circle, then what is the probability that it will be near the centre of a circle? Is it possible for all shapes to fit inside a circle? Why or why not? What is the average area of a triangle constructed from 3 points chosen randomly on a unit circle? On a bmw523i what does the Exclamation mark inside a circle inside a triangle mean? Why is a circle called a circle? Upside down triangle in a circle means what? Mathematics: If three points are chosen at random on a circle, and a triangle is formed by taking the three chosen points as its vertices, what is the probability that the triangle thus formed contains the center of the circle within it? The vertices of a triangle are in random locations inside of a circle. If a point is placed randomly inside the circle, what is the probability that it is also inside the triangle? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
2113	https://math.stackexchange.com/questions/688032/function-inversion-and-the-horizontal-shift	algebra precalculus - function inversion and the horizontal shift - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more function inversion and the horizontal shift Ask Question Asked 11 years, 7 months ago Modified4 years, 11 months ago Viewed 1k times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. I am currently doing inverse functions and graphing radical equations of the form y=a x−h−−−−−√+k y=a x−h+k with my algebra class and one of my students asked me the following question. "Why is it that we shift left when x+h−−−−−√x+h and into the negative x-values but we shift up when x−−√+k x+k and into the positive y-values?" I explained to them that part of it can be seen as a result of the inversion process of some functions/relations. So I recalled the vertex form of a parabola, something they studied at the beginning of the year and I put it on the board in a specific function, as y=x 2+3 y=x 2+3 . Then I graphed it and showed them that it was a parabola shifted up three units. Then I inverted it by switching y y and x x and solving for y y. So x=y 2+3⇒x−3=y 2⇒y=±x−3−−−−−√x=y 2+3⇒x−3=y 2⇒y=±x−3 Now this is the inverse of our original function (although in its current form it is NOT a function) and notice since they are inverses they are symmetrical along the y=x y=x line. Now look at the graphs and notice that even though our argument under the radical is (x−3)(x−3) we shift right into the positive reals. I don't feel like this is reason enough to justify the why. I'm also not sure that I satisfied the asker. They saw both functions, they saw the symmetry on the y=x y=x line, so after learning about inverses, I think they sort of understood the relationship, but hopefullly someone can provide me with a bit of better insight as to why. Please remember this question is ultimately for me and I do understand math well. If I understand, then I can better teach. EDIT: I'm specifically looking for teachers who have taught this material before who have maybe had to explain this. algebra-precalculus graphing-functions education inverse Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Feb 25, 2014 at 0:54 Eleven-ElevenEleven-Eleven asked Feb 24, 2014 at 3:01 Eleven-ElevenEleven-Eleven 8,989 9 9 gold badges 44 44 silver badges 82 82 bronze badges Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 4 Save this answer. Show activity on this post. I explain the "horizontal shift" this way: when we graph y=f(x+h),y=f(x+h), we are composing f(x)f(x) on the function x+h.x+h. This is to say that we are first adding h h to x,x, evaluating the function f f at x+h,x+h, and then plotting the result at x.x. This has the effect of reading off values of the function that lie to the right of x x and moving them back to the left by h h units. Similarly, f(x−h)f(x−h) moves the graph of f(x)f(x)to the right by h h units, as we are reading off values of the function that lie to the left of x,x, in order to plot them at x.x. Horizontal shifts are definitely the more puzzling of the two. I rarely see anyone have difficulties understanding what adding or subtracting a value directly to f(x)f(x) is going to do to the graph. I also don't find anything "wrong" with your explanation in terms of inverse functions (a sort of "diagonal mirror" argument), but I think a student would typically need more experience with graphing functions (like so many things, a skill even university students don't have nearly enough practice with) to appreciate your description. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Feb 25, 2014 at 22:11 answered Feb 24, 2014 at 3:19 colormegonecolormegone 11k 6 6 gold badges 23 23 silver badges 50 50 bronze badges Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. I like to think of it this way. Imagine that you have drawn a function say y=x−−√y=x on a grid. But then you decide to "magically" take your grid, including the origin and move it down by 2 units. You are replacing every y y-value with y−2 y−2 What will happen to the graph if you have "moved" your origin down 2 units. The graph will appear to be two units "higher" The output y y-values on the function will now be y=f(x)+k y=f(x)+k where k=2 k=2 By replacing y y-values with y−2 y−2, you are moving the plane down, and in effect moves the function "up" When we notice it this way, what happens to x x and y y is really the same. f(x−h)f(x−h) is like moving the origin to the left h h units, which, in turn, moves the function right by h h-units. So I see the replacements acting the same way. y→y−k y→y−k is thinking y=f(x)+k y=f(x)+k moving k k units up x→x−h x→x−h is moving h h units right. In relation to your original function. y=x−−√+k y=x+k comes from the parent function y=x−−√y=x but with a replacement of y→y−k y→y−k This will result in the function appearing up. y=x+h−−−−−√y=x+h comes from the parent function y=x−−√y=x but is a replacement of x→x+h x→x+h (moving the plane right) so results in the function appearing left. In summary, I think we are transforming the plane and the function coordinates respond to the change in the coordinate system and scale. I wish I could find research to support this. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Oct 3, 2020 at 3:50 E.YuE.Yu 123 4 4 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions algebra-precalculus graphing-functions education inverse See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Related 1Graphing inverse of parabola, should the domain be restricted or not? 1Why can't this mixed function be inverted? 0Graph of the function in the form: sqrt(-x+a) through manipulation of the function sqrt(x)? 1Inversion of Trigonometric Equations 1If both f(x)f(x) and g(x)g(x) are one to one functions and g(f(x))=f(g(x))=x g(f(x))=f(g(x))=x, why does this prove they are inverse of each other? 3Seeing why horizontal shifts are reversed? 1Difficulty understanding the inverse of a function 4What is a phase shift in trigonometry, and how can I determine them given a graph? 2How do I apply horizontal stretches to logarithmic functions? 1If x=a x=a is not a function, nor an equation—as in its graph is not the set of all points which satisfy that equation—, then what is it? 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2114	https://www.conservationphysics.org/strstr/newton.html	Published Time: Mon, 26 Nov 2018 19:56:53 GMT Conservation physics: The Newton: SI unit of force The Newton: SI unit of force 1 2 3 4 5 6 7 8 newton Index The Newton The Newton is the SI unit of force. Strictly defined: the force that will give a mass of 1 kilogram an acceleration of 1 meter per second per second. It is not in everyday use. When did you last read a recipe that required you to add one newton of flour to a cake mixture? There is no logical reason for not doing so. A force of 1 Newton on the balance pan is approximately the force exerted downwards by 100 grams of flour on the pan. So why not use kilograms per square metre also for the stress strain diagram? The pompous answer is that using Newtons allows this diagram to be used on a distant planet with a different mass. Weight is the expression of the force exerted by the earth on the mass of an object. The mass is the fundamental quantity, the weight, even on earth, varies with the position of the object. On the moon, 100 grams of flour would make a very big cake, if you measured it with a spring balance. 1 Newton of flour would work just like on earth. Back to the main story This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License. 1 2 3 4 5 6 7 8 newton Index
2115	https://www.thesaurus.com/e/grammar/between-vs-among/	Previous Verb Moods In English Grammar: What Are They? Next Novel vs. Book: When to Use Each Word Between vs. Among: When To Use Each One Among Vs. Between Examples ⚡ Quick summary The prepositions among and between are used similarly to refer to relationships. Typically, between refers to relationships that involve two people or things or to relationships in which every member is specifically mentioned. Among is typically used to refer to relationships of two or more people or things that aren’t individually mentioned or that are mentioned generally. The words between and among are two prepositions that are often used very similarly. Sometimes, they are even used to mean the same thing, depending on the context. However, they often can’t be used interchangeably, so it is important to know the differences between these two words. In this article, we will break down the difference between among and between, explain how they are typically used, state how they are often distinguished from each other, and provide examples that show how we typically use them in sentences. among vs. between The words among and between are prepositions used to describe relationships of two or more people or things. They each have several meanings. Among is most commonly used to mean “surrounded by” and “in the midst of.” For example: She slept among the tulips. The spy was a sheep among wolves. Between is most commonly used to mean “in the space separating” and “distinguishing one from another.” For example: He was standing between Todd and Becky. Is there really any difference between ivory and eggshell white? Although between and among are both used to describe relationships involving multiple people or things, they’re generally used distinctly in two ways based on context. The first distinction is based on how many people or things are being specifically referred to. Between is typically used to refer to a relationship involving two people or things, while among is used to refer to a relationship involving more than two people or things. For example: Two things: I can’t choose between vanilla and chocolate cake. Two things: We’re stuck between a rock and a hard place. More than two things: Among all fifty states, Alaska is the biggest in size. More than two things: I have ten puppies, and Princess is the cutest among them. The second distinction involves how the members of a relationship are discussed. The word between is typically preferred when referring to relationships in which each individual person or thing is distinctly mentioned, even if there are more than two members. Among is preferred if the members of a relationship are only mentioned generally or vaguely, even if there might be only two members. For example: Specific mention: The money was divided between Robin, Tuck, Marion, and John. General mention: The money was divided among the band of merry men. Specific mention: There is bad blood between the elves and dwarves. General mention: There is bad blood among the fey nations. Generally speaking, most language resources do not recommend using among and between interchangeably and will recommend one of these distinctions. In informal usage, though, between and among can be used interchangeably in some situations. Between is often used to refer to relationships or comparisons of more than two people or things, even when speaking generally. For example: There shouldn’t be any secrets between/among the four of us. There are many similarities between/among the seven basketball teams. Examples of between and among used in a sentence Let’s prove our grammar knowledge is among the best in the world by looking at how we use between and among in sentences. The store is open between 9 a.m. and 9 p.m. There is an imposter among us. I think Sara would win in a race between Patrick and her. Among all of the runners, Aroline is the fastest. She sat among the sheep in a field between the river and forest. #### Speaking of "between," should you say "between you and me" or "between you and I"? Previous Verb Moods In English Grammar: What Are They? Next Novel vs. Book: When to Use Each Word RELATED POSTS Why Is “Ain’t” Such A Controversial Word? Grammar 2. “Fathers Day” vs. “Father’s Day”: What’s The Correct Way For Celebrating Dad? Grammar 3. Is the Semicolon Just Plain Silly? Grammar YOU MIGHT ALSO LIKE Thickest Thieves or Thick as Thieves: Which One Is Correct? Ways To Say How to Use Wordplay to Elevate Your Poetry Writing 2. How To Use Wordplay To Elevate Your Poetry Writing Come Together For 14 Agreeable Words About Peace Ways To Say 2. How to Write a Haiku: Tips and Examples Writing 3. Hip-Hop: The World’s Most Popular Poetry (And How It Works) Ways To Say The Power of Poetry: A Beginner’s Guide to Writing Poetry Writing 9 Types Of Poems To Spark Your Creativity Writing 2. Which Is Correct? “It Is I” vs. “It Is Me” Grammar 3. Lust Vs. Love: The Heart Of The Difference Grammar Synonym of the day Choose the synonymfor guess
2116	https://bayanebartar.org/file-dl/library/GRE/Essential-Words-for-the-GRE-4th-edition.pdf	GRE BARRON’S The Leader in Test Preparation 4TH EDITION Philip Geer, Ed.M. ESSENTIAL WORDS FOR THE ® ®Graduate Record Exam and GRE are registered trademarks of the Educational Testing Service (ETS). This publication is not endorsed or approved by the ETS. Master our 800 high-frequency GRE words and ace the GRE Verbal Reasoning Test! BARRON’S FOURTH EDITION Philip Geer, Ed.M. GRE ESSENTIAL WORDS FOR THE ® ® Graduate Record Exam and GRE are registered trademarks of Educational Testing Service (ETS). This publication is not endorsed or approved by ETS. The excerpts and sample sentences included in this book are for educational purposes only. They should not be taken to reflect the views of either the author or Barron’s Educational Series, Inc. About the Author Philip Geer (Ed.M.) has been teaching English and preparing students for the GRE and SAT for over thirty years in high schools and colleges in the United States and abroad. He is the author of a number of test preparation books, including Barron’s GRE Verbal Workbook and Barron’s 6 Practice Tests for the New SAT. © Copyright 2016, 2013, 2010, 2007 by Barron’s Educational Series, Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without the written permission of the copyright owner. All inquiries should be addressed to: Barron’s Educational Series, Inc. 250 Wireless Boulevard Hauppauge, NY 11788 www.barronseduc.com ISBN: 978-1-4380-0749-6 Library of Congress Catalog Card No.: 2016930771 PRINTED IN CANADA 9 8 7 6 5 4 3 2 1 Contents INTRODUCTION: MASTERING ADVANCED VOCABULARY FOR THE GRE 1–9 PRETEST 10–19 SUGGESTED STUDY PLANS 20–23 300 HIGH-FREQUENCY GRE WORDS 24–26 ESSENTIAL WORDS FOR THE GRE 27–283 REVIEW: 300 HIGH-FREQUENCY GRE WORDS 284–288 REVIEW: ESSENTIAL WORDS FOR THE GRE 289–296 300 HIGH-FREQUENCY WORD ROOTS 297–384 COMMON SUFFIXES 385–387 POSTTEST 388–397 ANSWER KEYS ESSENTIAL WORDS FOR THE GRE 398–411 ROOT ROUNDUP 411–414 ROOT ROUNDUP REVIEW 415–416 INDEX 417–418 ACKNOWLEDGMENTS I would like to thank Susan Geer for her invaluable assistance in the preparation of this book. I am grateful to the following for permission to reproduce copyright material: Singapore Press Holdings for the article “Delving into the Mind of a Great Thinker, Chomsky” from The Straits Times Guide to Good English and Greater Knowledge, © copyright 2005. Harper San Francisco for an extract from The Historical Jesus, The Life of a Mediterranean Jewish Peasant, John Dominic Crossan, © copyright 1992 by John Dominic Crossan. Reprinted by permission of HarperCollins publishers. 1 Introduction: Mastering Advanced Vocabulary for the GRE This is how the makers of the GRE General Test, the Educational Test ing Service, describe the verbal reasoning section of the test: The Verbal Reasoning section measures your ability to: • analyze and draw conclusions from discourse; reason from incomplete data; identify author’s assumptions and/or perspective; understand multiple levels of meaning, such as literal, figurative, and author’s intent • select important points; distinguish major from minor or relevant points; summarize text; understand the structure of a text • understand the meanings of words, sentences, and entire texts; understand relationships among words and among concepts What does this mean for you as a student preparing for the verbal reasoning section of the GRE? It means you must make sure you have highly developed skills in all these areas. To make sure you have such skills, you should work through the skills section and practice tests in a reputable GRE General Test guidebook, such as Barron’s GRE Verbal Workbook. You also need to read widely, thinking critically about what you read. Read high quality books, both fiction and nonfiction, on a wide range of subjects and in a variety of genres. It is also important to read well-written periodicals, such as The New York Times, The Christian Sci ence Monitor, The Atlantic, The New Yorker, Time, The Economist, and Scientific American. Such reading will improve your skills in all of the areas described above. If you look up words you do not know as you read in a good dictionary, you will also improve your vocabulary. Speaking of vocabulary, is knowledge of advanced vocabulary impor tant on the GRE? The answer is yes, very much so. Advanced words appear regularly in all three types of questions: reading comprehension, text completion, and sentence equivalence. Let’s examine some of these types of questions and answer choices to see how important vocabulary is on the revised GRE. ESSENTIAL WORDS FOR THE GRE SENTENCE-EQUIVALENCE AND TEXT-COMPLETION QUESTIONS In the sentence-equivalence question below, you must choose two answers that could correctly complete the sentence. It would be difficult to imagine two more different personalities— Liz is shy and taciturn, while Stan is outgoing and _. 🄰 salubrious 🄱 laconic 🄲 specious 🄳 loquacious 🄴 doctrinaire 🄵 talkative Can you arrive at the correct answers to this question without know ing the meaning of the advanced word in the sentence, taciturn (incom municative, not inclined to speak much), and the meanings of the five advanced words that appear in the answer choices? If you don’t know the meanings of salubrious, laconic, specious, loquacious, and doctri naire, you will be forced to guess one of the correct answers to this question, (D) loquacious, which means (F) talkative, the second cor rect answer. Not all sentence-equivalence questions are so vocabulary dependent but you can expect quite a few to require knowledge of very advanced words. Let’s take a look at another type of question that requires knowledge of advanced words. In the first text-completion question below, you must choose one answer to fill in the blank. In most industrial countries, government intervenes in the economy by changing fiscal and monetary policy to _ the negative effects of the business cycle, despite the fact that there exists no theory supported by conclusive evidence to explain the underlying cause of the business cycle. remonstrate exacerbate understand establish mitigate 2 INTRODUCTION: MASTERING ADVANCED VOCABULARY FOR THE GRE If you don’t know the definition of mitigate (to cause to become less harsh, severe, or painful), you will not be able to answer this question correctly. Also, the more words you know in the other answer choices, the more confident you can be in answering the question. Here’s a question that requires you to choose words to fill in two blanks. Some scholars deny that there is a direct correlation between the scientific theory of relativity and intellectual fashions in the arts, pointing out that many important modernist works, such as Igor Stravinsky’s (i) _ symphony The Rites of Spring (ii) _ the theory of relativity. Blank (i) Blank (ii) seminal presage syllogistic antecede ephemeral subsume In this question, all of the answer choices are advanced words. It is not possible to answer this question correctly without knowing the mean ings of these advanced words, especially the meanings of the correct answer choices, (i) seminal (containing the seeds of later development) and (ii) antecede (precede). The final question below requires you to fill in three blanks. The phrase “It’s a matter of (i) _” is often used to indicate that the real meaning of a statement is being lost in verbiage, often with the implication that there is (ii) _ or (iii) _. Blank (i) Blank (ii) Blank (iii) definition exculpation peculation semantics meritriciousness equivocation debate obfuscation vacillation Two key words in the sentence—verbiage and implication—are advanced words. It would be impossible to figure out the central meaning of the sentence without knowing the meaning of verbiage (an excess of words for the purpose)—and if you don’t know the meaning of implication (that which is hinted or suggested), you will be hard-pressed to follow the logic of the sentence. 3 ESSENTIAL WORDS FOR THE GRE Advanced vocabulary also plays a central part in the answer choices given for this question. Once again, this question would be impossible to answer without knowledge of the meanings of these difficult words. If you know that answer choice (i) semantics means “the meaning and interpretation of words,” answer choice (ii) obfuscation means “the act of confusing or obscuring,” and answer choice (iii) equivocation means “the intentional use of vague language,” the sentence makes good sense. READING QUESTIONS Next, let’s consider a GRE-level reading passage and questions. The passage below uses a lot of fairly advanced vocabulary (indelible, domin ion, retroaction, ensuing) as well as some very advanced vocabulary (effi cacious, plastic, sanction, precipitate, vicissitudes). You might already be familiar with the first group of words, while you may not know the more advanced words. Not knowing any of these words would make it difficult to understand this passage. The word plastic is especially important because it expresses one of the central concepts discussed by the author—the human ability to consciously shape nature. Do you understand the difficult words in the passage below and in the questions that follow it? To chop a stick, to catch a fly, to pile a heap of sand, is a satisfying action; for the sand stays for a while in its novel arrangement, proclaiming to the surrounding level that we have made it our instrument, while the fly will never stir nor the stick grow together again in all eternity. If the impulse that has thus left its indelible mark on things is constant in our own bosom, the world will have been permanently improved and humanized by our action. Nature cannot but be more favorable to those ideas which have once found an efficacious champion. Plastic impulses find in this way an immediate sanction in the sense of victory and dominion which they carry with them; it is so evident a proof of power in ourselves to see things and animals bent out of their habitual form and obedient instead to our idea. But a far weightier sanction immediately follows. Man depends on things for his experience, yet by automatic action he changes these very things so that it becomes possible that by his action he should promote his welfare. He may, of course, no less readily precipitate his ruin. The animal is more subject to vicissitudes than the plant, which makes no effort to escape them or to give chase to what it feeds upon. The greater perils of 4 (5) (10) (15) (20) INTRODUCTION: MASTERING ADVANCED VOCABULARY FOR THE GRE action, however, are in animals covered partly by fertility, partly by adaptability, partly by success. The mere possibil-ity of success, in a world governed by natural selection, is an earnest of progress. Sometimes, in impressing the environment, a man will improve it: which is merely to say that a change may sometimes fortify the impulse which brought it about. As soon as this retroaction is perceived and the act is done with knowledge of its ensuing benefits, plastic impulse becomes art, and the world begins actually to change in obedience to reason. —George Santayana (1906) The first question on the passage requires you to select the best answer choice. According to the author Ⓐ whenever nature is shaped by a human being, art is produced. Ⓑ art can only come into existence when the plastic impulse ceases to exist in a person. Ⓒ art is created when a human being acts on a plastic impulse with awareness of its effect on the world. Ⓓ a human creation can only be called art when it is in accord with nature. Ⓔ actions prompted by the plastic impulse are always beneficial to human beings. The second question asks you to consider each of the three choices separately and select all that apply. Based on the information in the passage, which of the following statements would the author be likely to agree with? 🄰 Art is an inherently irrational process. 🄱 Any viable theory of aesthetics must take into account man’s relationship to nature. 🄲 All living things are subject to the process of natural selection. 5 (25) (30) ESSENTIAL WORDS FOR THE GRE The third question asks you to identify a sentence in the passage that best meets the description given in the question. Select the sentence that provides examples of the plastic impulse at work in human beings. _________ How did you do on the questions? The correct answer to Question 1 is C. To answer this question correctly, you must understand the meaning of the final sentence of the passage: “As soon as...obedience to reason” (lines 29–32). And to comprehend this sentence, you must understand the meaning of that key word in the passage, plastic, which was discussed above. The correct answer to Question 2 is B and C. Here, it is necessary to know advanced vocabulary in the answer choices: inherently, viable, aesthetics. Finally, the correct answer to Question 3 is “To chop a...in all eter-nity.” (lines 1–5) As with Question 1, you must know the meaning of the word plastic to answer this question correctly. From our analysis, it is clear that vocabulary plays a critical part in answering GRE Verbal Reasoning questions. Therefore, to give yourself a good chance of correctly answering questions with a lot of advanced vocabulary, you should learn the words that are most likely to appear in such questions. You may say that you can figure out a word’s, meaning from context and use elimination. This is true, but only to an extent. Also, guessing words from context and using a process of elimination consumes valu-able time that could be better used for figuring out tricky reading com-prehension and other questions. HOW ESSENTIAL WORDS WILL HELP YOU DO WELL ON THE GRE Now turn to the list of 300 High-Frequency GRE Words on page 24. These words appear with high frequency on the GRE and, thus, can be learned specifically for the test. They, as well as an additional 500 important words that commonly appear on the GRE, will be taught to you in Essential Words for the GRE. Each of the 300 High-Frequency GRE Words appears with an asterisk in the main section of the book (pages 27–283). Each of the 800 advanced words in Essential Words for the GRE has been carefully selected through reference to published GRE lists, past tests, and on the basis of my many years of experience in preparing stu-dents for the test. These 800 words also appear in Barron’s GRE Master 6 INTRODUCTION: MASTERING ADVANCED VOCABULARY FOR THE GRE 7 Word List and High-Frequency Word List. The principle behind this book is that the best way to learn new words for the GRE is to see how these words are used in complex sentences and practice on exercise material that is similar in content, structure, and level of difficulty to that which appears on the actual test. Practicing on such material will improve your skills in understanding complex sentences and arguments and increase your familiarity with important ideas that appear on the GRE. THE SYSTEMATIC STUDY OF GRE WORDS Essential Words for the GRE teaches the important words you need to know. Words are taught in units of ten words, along with their parts of speech, most commonly used definitions on the GRE, and illustrative sentences showing how words are used. The content, style, and tone of the example sentences are consistent with that of sentences appearing in the actual GRE and generally deal with topics in the arts, sciences, and social sciences. Comprehensive exercises at the end of each unit ensure that you know the words and provide practice in their correct use. Many of the words taught in a given unit reappear in subsequent units, both in illustrative sentences and in exercises, providing sys tematic reinforcement of learning. New to this edition, after every ten units an Applying Your Knowledge exercise explains how to use the words you have learned to tackle GRE-style text-completion questions. To further build your knowledge of GRE vocabulary, definitions and background information on important terms mentioned in the illus trative sentences appear in highlighted boxes labeled Terms from the Arts, Sciences, and Social Sciences. The example below shows how this works. In the example, the illustrative sentence for the word literati refers to two important terms, the First Amendment and Philistines. These two important terms are explained in the highlighted box, help ing you to gain a firm understanding of the context in which the word literati is used and expanding your knowledge of important terms. literati n. scholarly or learned persons “Any test that turns on what is offensive to the community’s stan dards is too loose, too capricious, too destructive of freedom of expres sion to be squared with the First Amendment. Under that test, juries can censor, suppress, and punish what they don’t like, provided the matter relates to ‘sexual impurity’ or has a tendency ‘to excite lustful thoughts.’ This is community censorship in one of its worst forms. It creates a regime where in the battle between the literati and the Philistines, the Philistines are certain to win.” —U.S. Supreme Court Justice William O. Douglas, dissenting in the case of Roth v. United States, 1957. ESSENTIAL WORDS FOR THE GRE 8 Terms from the Arts, Sciences, and Social Sciences First Amendment: a part of the United States Bill of Rights pro hibiting the federal legislature from making laws that establish a state religion or prefer a certain religion, prevent free exercise of religion, infringe the freedom of speech; infringe the freedom of the press; limit the right to assemble peaceably; limit the right to petition the government for a redress of grievances. Philistines: people considered to be ignorant of the value of cul tures and smug and conventional in their thinking. It is recommended that you keep a good college dictionary handy as you work through this book. This will allow you to explore additional meanings of words you learn and fine-tune your understanding of nuances in meaning between similar words. Two of the best college dictionaries are The American Heritage College Dictionary (Fourth Edi tion) and Merriam-Webster’s Collegiate Dictionary (Eleventh Edition). If you prefer to use an online dictionary, the excellent American Heritage Dictionary of the English Language, 4th edition is available for free at Bartelby.com. MASTERING HIGH-FREQUENCY WORD ROOTS Other than learning the difficult words likely to appear on the GRE, how can you improve your chances of doing well on the GRE Verbal Reasoning test? The answer is simple: learning important word roots. Essential Words for the GRE features extensive information on word roots, prefixes, and suffixes in the High-Frequency Word Roots section. This section contains a list of the 300 most important Latin and Greek roots that commonly appear in English words. It also gives you hun dreds of derivative words, and includes exercises to improve your ability to make use of roots in remembering words you have learned in this book and in deciphering the meaning of unknown words. In addition to being useful for students taking the GRE, a knowledge of word roots provides a great foundation for the student embarking on a career in fields that use many specialized terms based on Latin and Greek, such as law, science, and medicine. GETTING STARTED So, what do you do now? First, take the Pretest on the following pages. This will tell you how well prepared you are to deal with the advanced vocabulary that appears on the GRE. After that, you should make a INTRODUCTION: MASTERING ADVANCED VOCABULARY FOR THE GRE study plan based on how many words you have to learn and how much time you have before the test. On pages 20–23 you will find some suggestions to help you plan how to use this book effectively. It is best to study regularly for a relatively short amount of time (30 minutes, for example), rather than intermit tently for longer amounts of time. 9 Pretest It’s time to test your readiness for graduate-level reading. Don’t worry. If you aren’t familiar with the GRE words in this test, you can be sure you will learn them in Essential Words for the GRE. Fill in the blank in each sentence by selecting two answer choices that fit the overall meaning of the sentence and produce completed sentences that are equivalent in meaning. Answers that are not fully correct will receive no credit. 1. The belief that music is the _ for the other arts and best exemplifies the power of art to express subtle feelings, was expressed by the critic Walter Pater: “All art constantly aspires towards the condition of music.” 🄰 aesthetic 🄱 precursor 🄲 paradigm 🄳 tome 🄴 sensitivity 🄵 model 2. Some scholars believe that the impetus for the building of the Great Pyramid of Giza was for reasons other than the purely functional one of providing imposing and secure burial chambers for pharaohs; these experts see _ meaning behind its design. 🄰 an arcane 🄱 an insipid 🄲 a nefarious 🄳 a sportive 🄴 an obscure 🄵 a desultory 10 PRETEST 3. Many statements are not true unless they are __ by the use of words and phrases such as sometimes, frequently, in many cases, and most. 🄰 sanctioned 🄱 limited 🄲 superseded 🄳 embellished 🄴 qualified 🄵 supported 4. In the final stage in the impeachment process of an American president, the chief justice of the Supreme Court presides over the Senate, which sits as a body _ to a jury to decide whether to convict the president. 🄰 comparable 🄱 commensurate 🄲 inherent 🄳 analogous 🄴 extraneous 🄵 impermeable Fill in the blank in each sentence by selecting one entry from the corresponding column of choices in the way that best completes the text. 5. _, law can be used either to tyrannize the populace, denying them liberty, or to protect their rights, enabling them to live as free citizens expressing their views and doing as they wish. Enigmatically Paradoxically Purportedly Felicitously Preternaturally 11 ESSENTIAL WORDS FOR THE GRE 6. Steeped in mysticism and allegory, alchemy has been a favorite subject of people speculating about the existence of an occult tradition concealed behind _ facade. a hermetic a sedulous an ephemeral a mundane an iconoclastic 7. In 1787, when the U.S. Constitution was being framed, it was proposed that slavery be abolished, but opponents of the measure forced a compromise whereby slavery would not be _ until early in the next century. disparaged stipulated proscribed allowed exculpated Fill in all of the blanks in the sentences by selecting one entry from the corresponding column of choices in the way that best com pletes the text. Answers that are not fully correct will not receive any credit. 8. The statement “India has recently made great progress in (i) _ poverty” should be seen in the context of India’s vast population of one billion, of which 320 million remain in (ii) _ poverty. Blank (i) Blank (ii) satiating equivocal vitiating abject alleviating intransigent 12 PRETEST 9. Language purists pounce on errors as though they were ghastly offenses against the natural order; however, it is wise to remember that language is a wonderfully (i) __ tool that is some times at its best when it is most (ii) _ . Blank (i) Blank (ii) malleable limpid labile compliant demotic unfettered 10. According to the view of the nineteenth-century apologist for capitalism, (i) _ was an unfortunate but unavoidable (ii) __ of both capitalism and of the natural order of the world. Blank (i) Blank (ii) misogyny concomitant abnegation transgression indigence miscellany 11. It is interesting to (i) _ the bromide “Haste makes waste” and the (ii) _____ “Better safe than sorry.” Blank (i) Blank (ii) juxtapose homily delineate platitude belie epithet A bromide is a commonplace remark or idea. 13 ESSENTIAL WORDS FOR THE GRE 12. Quantum theory (i) _that waves and particles possess a dual nature, with one aspect (ii) _ in some situations and the other becoming (iii) _ in other situations. Blank (i) Blank (ii) Blank (iii) refutes vacillating pellucid queries vitiating salient postulates predominating obtuse 13. Dr. Gupta’s hundreds of hoaxes (i) _ the valid research of many of the scientists with whom he had collaborated; fortunately, such (ii) __ behavior as Dr. Gupta’s is (iii) _ in the history of science. Blank (i) Blank (ii) Blank (iii) honed obsequious an anomaly tainted edifying a paragon substantiated unconscionable a metamorphosis 14. The sociologist Stanley Milgram theorized that the rise of mass society has made it easy for individuals to ignore moral responsibility because the individual is frequently “an intermediate link in a chain of evil,” making it possible for a person to (i) _ his actions by saying, “I was only carrying out my social responsibilities as given in the orders of my superiors; it is not for me to decide the ultimate morality of socially (ii) __ actions, even if they appear (iii) _ .” Blank (i) Blank (ii) Blank (iii) rationalize maligned execrable stipulate venerated Machiavellian repudiate sanctioned sycophantic 14 PRETEST Read the passages below, and then answer the questions that fol low them based on the information in the passages themselves and in any introductory material or notes. The correct answer may be either stated or merely suggested in the passages. The term “the arts,” when used to classify a group of academic disciplines at schools and universities, subsumes the study of lan guages, history, and literature, while universities use the term “fine arts” to refer to painting and sculpture as an object of study. Else where the term “the arts” includes painting and sculpture (usu ally grouped together as “the plastic arts”), music, and literature, and is often extended to embrace dance, mime, and cinema (the word “artist” is used to refer to a practitioner of any of these). Art is commonly opposed to science (as “subjective,” whereas science is “objective”) and its sense is distinguished from etymologically related words such as “artifact,” “artificial,” and “artisan.” Select one answer choice for each of the following questions. 15. The word “subsumes” as it is used in line 2 most nearly means Ⓐ assumes to be true Ⓑ makes inferior to Ⓒ undermines Ⓓ includes in a less comprehensive category Ⓔ incorporates in a more comprehensive category 16. Based on the information in lines 8–11 (“Art . . . artisan”), which statement would the author be most likely to agree with? Ⓐ Two words can be related etymologically but have different meanings. Ⓑ The word “art” has only an accidental and tangential similar ity to the words “artifact,” “artificial,” and “artisan,” because it originated from a different word. Ⓒ Artifacts can only be studied subjectively. Ⓓ “Artisan” is an older word than the modern word “artist” is. Ⓔ The words “artifact,” “artificial,” and “artisan” are not cognate. 15 (5) (10) ESSENTIAL WORDS FOR THE GRE The following is a review of the book Understanding Power: The Indis pensable Chomsky by Noam Chomsky. You may have heard the cynic’s version of the golden rule, namely, he who has the gold makes the rules. That is the thrust of this treatise on power by 75-year-old Noam Chomsky, a professor of linguistics at the Massachusetts Institute of Technology (MIT). He has written more than 30 books on linguistics and current affairs, including the best-selling Language and Politics (1990), Manufacturing Consent (1994) and 911 (2001). Once called “argu ably the most important intellectual alive” by The New York Times, this very politically incorrect academic has taught at MIT since 1955, immediately after graduating with a PhD in linguistics from the University of Pennsylvania. He is currently one of MIT’s insti tute professors, which means he can teach in any department of the university. But, as he notes wryly in this book: “If I even get near Political Science, you can feel the bad vibes starting.” The book is an edited collection of his lectures and tutorials from 1989 to 1999. Published as a book for the first time, his talks offer high-definition snapshots of the ills of the twentieth century, even as he slices through history to serve up unpalatable truths—like how America’s founding fathers actually loathed the idea of democ racy, why the United States hires rogue states to fight its wars and why nation-states are the wrong political model for a post-modern world. He reserves one of his biggest knives for the media, which he takes methodical stabs at for being dictated by the desires of the elite. With the same vigor, the gleeful iconoclast tells his students why there is not only no such thing as a free lunch, but also no such thing as a free market. As he puts it: “Of course, the ‘free market’ ideology is very useful—it’s a weapon against the general population (in the U.S.), because it’s an argument against social spending, and it’s a weapon against poor people abroad, because we can hold it up to them and say, ‘You guys have to follow these rules,’ then just go ahead and rob them.” He also sees the world’s current economic star, China, and its people as “brutal,” and so finds no profit in cozying up to either. Still, conceding that its ascendance to power is unstoppable, he says: “I don’t think we should be asking the question, ‘How do we improve relations with China?’ We should be asking other ques tions like, What kind of relations do we want to have with China?’” The saddest cautionary tale in this book is that of Princeton University graduate Norman Finkelstein, a bright young man who committed career suicide by exposing best-selling historian Joan Peters—whose book From Time Immemorial said Palestinians never 16 (5) (10) (15) (20) (25) (30) (35) (40) PRETEST existed—as a charlatan. The problem was that her work had been embraced by most of America’s finest intellectuals—including writer Saul Bellow and historian Barbara Tuchman—so Mr. Finkel stein’s expose was akin to calling them frauds. Eyebrow-raisers aside, the question-and-answer format of this book captures the rhythm of intellectual repartee between Chom sky and his audience but, more importantly, breaks the monotony of what would otherwise be his marathon soliloquy on the world’s ills. Indeed, his mind is such a ragbag of ideas that it is not above pondering such things as the validation of vegetarianism. Yet, in the end, his brilliance falls prey to a certain kind of intellectual snobbery, the sort which asserts that heroes are not to be found “mentioned in the newspapers.” As he puts it: “If they’re there, you know probably they’re not heroes, they’re anti-heroes.” Still, love him or hate him, there are not many thinkers around who can proffer credible alternative perspectives on how power cor rupts today. This book is as much an antidote to apathy as it is a counterweight to elitist thought. As in this paradox he surfaces: “You’ll see that so long as power remains privately concentrated, everybody, everybody, has to be committed to one overriding goal: To make sure that the rich folk are happy—because unless they are, nobody else is going to get anything. “So, if you’re a homeless person sleeping in the streets of Manhattan, let’s say, your first concern must be that the guys in the mansions are happy—because if they’re happy, then they’ll invest, and the economy will work, and things will function, and then maybe something will trickle down to you somewhere along the line. But if they’re not happy, everything’s going to grind to a halt . . . basically, that’s a metaphor for the whole society.” Select one answer choice for each of the following questions. 17. The phrase unpalatable truths as it is used in line 18 most nearly means Ⓐ theories that have not been conclusively proven Ⓑ facts that many people don’t like to accept as true Ⓒ facts that are not accepted as true by experts Ⓓ facts about a wide range of topics Ⓔ information that is not widely disseminated 17 (45) (50) (55) (60) (65) (70) ESSENTIAL WORDS FOR THE GRE 18. Based on the information in the passage, which of the following terms would Noam Chomsky be most likely to apply to the present American economic-political system? Ⓐ socialistic Ⓑ anarchic Ⓒ bureaucratic Ⓓ plutocratic Ⓔ theocratic Consider each of the three choices separately and select all that apply. 19. Which of the following are reasons that the author describes Noam Chomsky as an “iconoclast” (line 25)? 🄰 Noam Chomsky does not accept the presupposition of most historians and political scientists that social phenomena are subject to objective scientific analysis. 🄱 Noam Chomsky is a maverick among intellectuals because of his view that the media does not play a major role in American society. 🄲 Noam Chomsky frequently attacks widely held cherished beliefs. Identify the sentence by writing its first three words and last three words on the line below. Select the sentence that gives the author’s opinion of the validity of Noam Chomsky’s political views. 20. _________ End of Pretest 18 PRETEST PRETEST ANSWERS 1. C, F 2. A, E 3. B, E 4. A, D 5. Paradoxically 6. a mundane 7. proscribed 8. alleviating/abject 9. malleable/unfettered 10. indigence/concomitant 11. juxtapose/platitude 12. postulates/predominating/salient 13. tainted/an anomaly/unconscionable 14. rationalize/sanctioned/execrable 15. E 16. A 17. B 18. D 19. C 20. “Still, love him . . . power corrupts today” (lines 57–59). YOUR PRETEST SCORE 1–2 CORRECT ANSWERS: VERY POOR 3–5 CORRECT ANSWERS: POOR 6–9 CORRECT ANSWERS: BELOW AVERAGE 10–13 CORRECT ANSWERS: AVERAGE 14–16 CORRECT ANSWERS: GOOD 17–18 CORRECT ANSWERS: VERY GOOD 19–20 CORRECT ANSWERS: EXCELLENT 19 Suggested Study Plans PRETEST SCORE 1 TO 9 (VERY POOR TO BELOW AVERAGE) ONE-MONTH (4 WEEKS) STUDY PLAN c 3 Weeks Learn the 300 High-Frequency GRE Words listed on pages 24–26 and asterisked on pages 27–283. Study about 100 words per week. c 1 Week Go over all 300 High-Frequency GRE Words, concentrating on words you have trouble with. Do the Review: 300 High-Frequency GRE Words on pages 284–288. THREE-MONTHS (12 WEEKS) STUDY PLAN c 10 Weeks Learn all of the Essential Words for the GRE on pages 27–283 by carefully reading all the material and doing all the exercises. Do about eight units (80 words) per week. c 1 Week Go over all of the Essential Words for the GRE, concentrating on words you have trouble with. Do the Review: Essential Words for the GRE on pages 289–296 and the Posttest on pages 388–397. c 1 Week Study the 300 High-Frequency GRE Words listed on pages 24–26 and asterisked on pages 27–283. Do the Review: 300 High-Frequency GRE Words on pages 284–288. SIX-MONTHS (26 WEEKS) STUDY PLAN c 20 Weeks Learn all of the Essential Words for the GRE on pages 27–283 by carefully reading all the material and doing all the exercises. Do about four units (40 words) per week. 20 c 3 Weeks Go over all of the Essential Words for the GRE, concentrating on words you have trouble with. c 1 Week Do the Review: Essential Words for the GRE on pages 289–296 and the Posttest on pages 388–397. c 2 Weeks Study the 300 High-Frequency GRE Words listed on pages 24–26 and asterisked on pages 27–283. Do the Review: 300 High-Frequency GRE Words on pages 284–288. PRETEST SCORE 10 TO 16 (AVERAGE TO GOOD) ONE-MONTH (4 WEEKS) STUDY PLAN c 2 Weeks Learn the 300 High-Frequency GRE Words listed on pages 24–26 and asterisked on pages 27–283. Study about 150 words per week. c 1 Week Do as much of the Essential Words for the GRE on pages 27–283 as you can. c 1 Week Go over the 300 High-Frequency GRE Words, concentrating on words you have trouble with. Do the Review: 300 High-Frequency Words on pages 284–288. Do the Review: Essential Words for the GRE on pages 289–296, learning as many additional new words as possible by referring to the main text for words you don’t know. THREE-MONTHS (12 WEEKS) STUDY PLAN c 8 Weeks Learn all of the Essential Words for the GRE on pages 27–283 by carefully reading all the material and doing all the exercises. Do about ten units (100 words) per week. c 2 Weeks Go over all of the Essential Words for the GRE, concentrating on words you have trouble with. c 1 Week Do as many as possible of the 300 High-Frequency Word Roots on pages 297–384. 21 SUGGESTED STUDY PLANS ESSENTIAL WORDS FOR THE GRE 22 c 1 Week Do the Review: Essential Words for the GRE on pages 289–296 and the Posttest on pages 388–397. Go over the 300 High-Frequency GRE Words and do the Review: 300 High-Frequency GRE Words on pages 284–288. SIX-MONTHS (26 WEEKS) STUDY PLAN c 20 Weeks Learn the Essential Words for the GRE on pages 27–283 by care-fully reading all the material and doing all the exercises. Do about four units (40 words) per week. c 2 Weeks Go over all the Essential Words for the GRE, concentrating on words you have trouble with. c 2 Weeks Do the 300 High-Frequency Word Roots on pages 297–384. c 1 Week Do the Review: Essential Words for the GRE on pages 289–296 and the Posttest on pages 388–397. c 1 Week Go through the 300 High-Frequency GRE Words listed on pages 24–26 and asterisked on pages 27–283 and do the Review: 300 High-Frequency GRE Words on pages 284–288. PRETEST SCORE 17 TO 20 (VERY GOOD TO EXCELLENT) ONE-MONTH (4 WEEKS) STUDY PLAN c 1 Week Go through the 300 High-Frequency GRE Words listed on pages 24–26. Study words you don’t know by referring to pages 27–283 in which the high-frequency words are asterisked. Do the Review: 300 High-Frequency GRE Words on pages 284–288. c 2 Weeks Go through the Essential Words for the GRE on pages 27–283, concentrating on words you don’t know. c 1 Week Do the Review: Essential Words for the GRE on pages 289–296. Do as much of the 300 High-Frequency Word Roots on pages 297–384 as possible. 23 SUGGESTED STUDY PLANS THREE-MONTHS (12 WEEKS) STUDY PLAN c 8 Weeks Learn the Essential Words for the GRE on pages 27–283 by carefully reading all the material and doing all the exercises. Do about ten units (80 words) per week. c 2 Weeks Do the 300 High-Frequency Word Roots on pages 297–384. c 2 Weeks Go over the Essential Words for the GRE, concentrating on words you have trouble with. Do the Review: Essential Words for the GRE on pages 289–296. Go over the 300 High-Frequency GRE Words listed on pages 24–26 and asterisked on pages 27–283 and do the Review: 300 High-Frequency GRE Words on pages 284–288. SIX-MONTHS (26 WEEKS) STUDY PLAN c 10 Weeks Learn the Essential Words for the GRE on pages 27–283 by care-fully reading all the material and doing all the exercises. Do about eight units (80 words) per week. c 6 Weeks Go over all of the Essential Words for the GRE, concentrating on words you have trouble with. c 6 Weeks Do the 300 High-Frequency Word Roots on pages 297–384. c 1 Week Do the Review: Essential Words for the GRE on pages 289–296 and the Posttest on pages 388–397. c 2 Weeks Go through the 300 High-Frequency GRE Words asterisked on pages 27–283, making sure you know them all. c 1 Week Do the Review: 300 High-Frequency GRE Words on pages 284–288. 300 High-Frequency GRE Words 24 aberrant abeyance abstemious aesthetic alacrity alleviate amalgamate ambiguous ambivalence ameliorate anachronism analogous anarchy anomalous antipathy apathy apprise approbation appropriate arcane arduous artless ascetic aspersion assiduous attenuate audacious austere aver banal belie beneficent bombastic boorish burgeon burnish buttress cacophonous cant capricious castigation catalyst causality chicanery coagulate commensurate compendium complaisant conciliatory concomitant confound contentious conundrum conventional convoluted cosmology craven credence decorum deference delineate demotic demur denigrate denouement derivative desiccate desultory diatribe dichotomy diffidence diffuse disabuse discordant discrepancy discrete disingenuous disinterested dismiss disparage disparate dissemble disseminate dissolution dissonance doctrinaire dogmatic ebullient eclectic effete efficacy effrontery elegy elicit embellish empirical emulate endemic enervate The following words appear with the most frequency on the GRE and are asterisked on pages 27–283 of this book. 300 HIGH-FREQUENCY GRE WORDS ephemeral equanimity equivocate erudite esoteric euphemism exacerbate exacting exculpate execrable exigency existential extant extraneous extrapolation facetious fallacious fatuous felicitous fledgling foment forestall fractious frugality fulminate gainsay garrulous grandiloquent gregarious guileless gullible harangue heterodox histrionic homily homogeneous hyperbole iconoclastic ideological idolatry igneous immutable impassive impermeable imperturbable impervious implacable implicit inadvertently inchoate incongruity indeterminate indigence indolent ineluctable inert ingenuous inherent innocuous insensible insinuate insipid insularity intractable intransigence inundate inured invective irascible irresolute juxtapose laconic lassitude laud lethargic levity limpid loquacious lucid magnanimity malinger malleable maverick mendacious meretricious metamorphosis metaphysical meticulous misanthrope misogynist mitigate mollify morose mundane neophyte obdurate obsequious obviate occlude officious onerous opprobrium oscillate ostentatious paragon partisan pathological paucity pedantic pellucid penchant penury perfidious perfunctory permeable pervasive phlegmatic piety placate plasticity platitude plethora pragmatic 25 ESSENTIAL WORDS FOR THE GRE precarious precipitate precursor presumptuous prevaricate pristine probity problematic prodigal profound proliferate propensity propitiate propriety proscribe qualified quiescent rarefied recalcitrant recant recondite refractory refute relegate reproach reprobate repudiate rescind resolute resolved reticent reverence sage salubrious salutary sanction sartorial satiate sensual sensuous sentient skeptic solicitous soporific specious sporadic stigma stipulate stolid striated substantiate subsume supersede supposition tacit taciturn tangential tenuous tirade torpor tortuous tractable transgression truculence turgid untenable vacillate vapid venerate veracious verbose viable viscous vitiate vituperative volatile whimsical zealot 26 27 Essential Words for the GRE UNIT 1 abate v. to decrease; reduce NASA announced that it would delay the launch of the manned spacecraft until the radiation from the solar flares abated. abdicate v. to give up a position, right, or power Romulus Augustus, the last Western Roman emperor, was forced to abdicate the throne in 476 a.d., and the Germanic chieftain Odovacar became the de facto ruler of Italy. The appeals judge has abdicated his responsibility to review the findings of the high court. Terms from the Arts, Sciences, and Social Sciences de facto: in fact, whether by right or not; exercising power without being legally established (Latin: from the fact) aberrant adj. deviating from what is normal When a person’s behavior becomes aberrant, his or her peers may become concerned that the individual is becoming a deviant. Aberration is a noun meaning something different from the usual or normal. For centuries, solar eclipses were regarded as serious aberrations in the natural order. Terms from the Arts, Sciences, and Social Sciences deviant: a person whose behavior differs from the accepted stan-dards of society ESSENTIAL WORDS FOR THE GRE abeyance n. temporary suppression or suspension A good judge must hold his or her judgment in abeyance until all the facts in a case have been presented. abject adj. miserable; pitiful John Steinbeck’s novel The Grapes of Wrath portrays the abject poverty of many people during the Great Depression. abjure v. to reject; abandon formally Most members of the Religious Society of Friends (commonly known as the Quakers or Friends) abjure the use of violence to settle disputes between nations. For a foreigner to become a U.S. citizen, he or she must take an oath abjuring allegiance to any other country and pledging to take up arms to defend the United States. abscission n. the act of cutting; the natural separation of a leaf or other part of a plant Two scientists, Alan G. Williams and Thomas G. Whitham, have hypothesized that premature leaf abscission is an adaptive plant response to herbivorous attack. The verb abscise means to cut off or away. The surgeon abscised a small growth on the patient’s hand. Terms from the Arts, Sciences, and Social Sciences hypothesized: form a hypothesis, that is a proposition put forward as a starting point for further investigation adaptive: relating to adaptation, an alteration in structure or habits by which a species improves its condition in relationship to its environment herbivorous: feeding mainly on plants abscond v. to depart secretly A warrant is out for the arrest of a person believed to have absconded with three million dollars. abstemious adj. moderate in appetite Some research suggests that people with an abstemious lifestyle tend to live longer than people who indulge their appetites. 28 ESSENTIAL WORDS FOR THE GRE abstinence n. the giving up of certain pleasures The monk’s vow of abstinence includes all intoxicating substances. REVIEW 1 The correct answers are given on page 398. Matching Match each word with its definition. 1. abate (A) to abandon formally 2. abdicate (B) temporary suppression 3. aberrant (C) to give up a position or power 4. abeyance (D) giving up of certain pleasures 5. abject (E) to depart secretly 6. abjure (F) miserable; pitiful 7. abscission (G) to decrease 8. abscond (H) moderate in appetite 9. abstemious (I) the act of cutting 10. abstinence (J) deviating from what is normal Fill-ins Choose the best word to fill in the blank in each sentence. abate abdicated aberrations abeyance abject abjured absconded abscission abstemious abstinence 1. The 90-year-old monarch __ the throne to allow his son to become king. 2. Psychotherapy relies on psychological rather than physiological approaches to curing mental _ . 3. Implementation of the new plan has been held in _ pending an investigation of its effectiveness to date. 4. Ms. Johnson’s __ lifestyle helped her to amass a fortune. 5. The crew of the vessel waited for the storm to _ before going on deck to make repairs. 6. The alcoholic’s physician recommended total _ from liquor for her patient. 7. The documentary filmmaker was accused of using misleading foot age to make it appear that nearly everyone in the country lived in __ conditions. 8. The judge said he would reduce the convicted woman’s sentence if she _ all association with those convicted of treason. 29 ESSENTIAL WORDS FOR THE GRE 9. The senior surgeon performed the difficult _ . 10. The audit of the bank’s financial records led investigators to sus pect that someone had __ with $100,000. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The doctor decided to let her patient’s fever abate before ordering further clinical tests. 2. The university’s plans for expansion have been put in abeyance until the economic outlook is more favorable. 3. Ruth’s abstemious appetite has caused her to put on ten pounds in the last month. _ 4. The senator announced that he formally accepted and abjured all of his past statements on the issue. 5. The judge instructed the members of the jury that they would be abdicating their responsibilities if they did not reach a verdict in the case. __ UNIT 2 abysmal adj. very bad The abysmal failure of the free market system in Russia has led some people to argue that the planned economy of the Soviet Union, while not perfect, was better suited to Russia’s history and culture than Western-style capitalism. Terms from the Arts, Sciences, and Social Sciences free market: an economic market in which the demand and supply of goods and services is either not regulated or is slightly regulated planned economy: an economic system in which the production, allo-cation, and consumption of goods and services is planned in advance. Another term for planned economy is “command economy.” capitalism: an economic and political system in which a country’s industry and trade are controlled by private owners rather than the government 30 ESSENTIAL WORDS FOR THE GRE accretion n. growth in size or increase in amount In the 1960s, the American geophysicist Harry Hess conceived the idea of sea-floor spreading, a process in which the new crust in the ocean is continually generated by igneous processes at the crests of the mid-oceanic ridges, causing a steady accretion of the crust. Terms from the Arts, Sciences, and Social Sciences geophysicist: one who specializes in the physics of the earth and its environment igneous: in geology, relating to the formation of rocks by solidifica-tion from a molten state. The word igneous is from Latin ignis (fire). accrue v. to accumulate; grow by additions Regulating the growth of large companies when they begin to become monopolistic is a difficult task for government in a capital ist country; if it limits monopolies too much, the nation’s firms could become less competitive than foreign companies that enjoy the advan tages accruing from greater monopolies. Terms from the Arts, Sciences, and Social Sciences monopolistic: having exclusive control over a commercial activity adamant adj. uncompromising; unyielding Despite widespread opposition to his plan, the political party’s leader is adamant that the party must move to the center to appeal to moderate voters. adjunct n. something added, attached, or joined Speed walking, cross-country running, and marathons are normally regarded as adjuncts of track and field athletics since races in these sports are not normally held on a track. admonish v. to caution or reprimand The judge admonished the jury to discount testimony that had been ruled inadmissible. 31 ESSENTIAL WORDS FOR THE GRE adulterate v. to corrupt or make impure The unscrupulous company sells an adulterated version of the drug, and doesn’t inform consumers that they are getting a less effi cacious drug than they think they are getting. aesthetic adj. relating to beauty or art Members of the English aesthetic movement, such as Oscar Wilde, were proponents of the doctrine of art for art’s sake, which is the belief that art cannot and should not be useful for any purpose other than that of creating beauty. Aesthetic is also a noun that means a conception of what is artistically beautiful. The Gothic aesthetic dominated European art and architecture from approximately the twelfth to the fifteenth century. Aesthetics is the conception of what is beautiful; it is also a branch of philosophy dealing with beauty and art, and standards in judging them. An aesthete is someone who cultivates a special sensitivity to beauty; often the word refers to a person whose interest in beauty and art is regarded as excessive or superficial. Terms from the Arts, Sciences, and Social Sciences Gothic: a style of architecture that was very popular in the late Mid-dle Ages characterized by such features as pointed arches, soaring spaces, and light. In literature the term refers to a genre of fiction that was popular in the eighteenth and early nineteenth centuries. Gothic novels have an atmosphere of gloom, mystery, and horror. affected adj. pretentious, phony It has been argued that the emphasis on so-called “proper English” leads to unnatural and affected speech. affinity n. fondness; liking; similarity The female students in the class felt an affinity for the ancient Greek playwright Euripides because he sympathized with women, slaves, and other despised members of his society. 32 ESSENTIAL WORDS FOR THE GRE REVIEW 2 The correct answers are given on page 398. Matching Match each word with its definition. 1. abysmal (A) grow by additions 2. accretion (B) very bad 3. accrue (C) relating to beauty or art 4. adamant (D) something added 5. adjunct (E) to corrupt or make impure 6. admonish (F) increase in amount 7. adulterate (G) pretentious 8. aesthetic (H) fondness 9. affected (I) uncompromising 10. affinity (J) to caution Fill-ins Choose the best word to fill in the blank in each sentence. abysmal accretion accrued adamant adjunct admonished adulterated aesthetic affected affinity 1. The film is marred by the actor’s _ English accent. 2. In Emily Brontë’s Wuthering Heights the characters Heathcliff and Catherine feel such an __ for each other that they almost literally cannot live without each other. 3. Over the years the university’s computer system has grown so much by _ that no one person has a complete under-standing of it. 4. The committee on education reform recommended that the school introduce more art courses to develop students’ _ awareness. 5. The poet __ the critic for failing to appreciate the subtle changes in his poem’s meter. 6. Tom’s savings account has _ $3,000 in interest over the last ten years. 7. The band’s playing was so _ that they were booed off stage. 8. The English teacher is __ about one thing: students must correct all the errors in written work that she returns to them. 9. Over the last 20 years or so consumers have increasingly demanded food that is not _ with additives. 10. Nearly half of the college courses in America are taught by _ professors. 33 ESSENTIAL WORDS FOR THE GRE Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. In many ways Aristotle’s aesthetic dictums have never been surpassed. _ 2. Beth is so adamant about the plan that she is willing to give it up at the first opportunity. 3. The waiters in the expensive restaurant were told to affect a French accent to impress customers. 4. Most students love to be admonished for their good work. _ 5. State law requires that whole milk be 100 percent adulterated. UNIT 3 aggrandize v. to make larger or greater One of the concerns of the framers of the U.S. Constitution was that one branch of government would try to aggrandize itself at the expense of the others. aggregate adj. amounting to a whole; total The aggregate wealth of a country includes private as well as public resources and possessions. Aggregate is also a verb meaning to collect into a mass. Portals are Web sites designed to aggregate information and are used as a starting point on the Web. Aggregate is also a noun meaning collective mass or sum. alacrity n. cheerful willingness; eagerness; speed The football coach was pleased to see the team get to work on the task of improving its tackling skills with alacrity. alchemy n. medieval chemical philosophy based on changing metal into gold; a seemingly magical power or process of transmutation. Alchemy was the forerunner of the modern science of chemistry. None of their friends could understand the mysterious alchemy that caused two people as different from one another as Rob and Barbara to fall in love. 34 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences alchemy: Modern scientists believe alchemy was not a true science since there’s no evidence that anyone succeeded in turning a base metal into gold. Interestingly, however, the word for the modern science of “chemistry” is derived directly from the word “alchemy.” allay v. to lessen; ease; soothe Improvements in antivirus software have allayed many people’s fears of having their computers “infected” with malicious software. alleviate v. to relieve; improve partially According to some commentators, one of the weaknesses of capitalism is that, although it is very efficient at increasing absolute wealth, it is not as successful at alleviating relative poverty; thus, a person living in a slum in America may be reasonably well off by historical standards, but he might perceive himself to be poor compared to members of the bourgeoisie, whom he sees regularly buying luxury goods that he is not able to afford. Terms from the Arts, Sciences, and Social Sciences bourgeoisie: the social order dominated by the property-owning class. The term is associated with Marxism, the political and economic philosophy of Karl Marx and Friedrich Engels, but today it is often used disparagingly to suggest materialism and philis-tinism (an unenlightened and smug attitude toward culture). alloy n. a combination; a mixture of two or more metals Scientists formulate alloys to create properties that are not possessed by natural metals or other substances. allure n. the power to entice by charm Political groups in the United States often lobby Congress to use the allure of America’s vast market as an incentive for countries to pursue policies in accordance with American policies. 35 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences lobby: a group whose members share certain goals and work to bring about the passage, modification, or defeat of laws that affect these goals Allure is also a verb meaning to entice by charm. The adjective is alluring. The idea of a clockwork universe is very alluring to some people because it explains how the universe was created, yet allows human beings to live in it without believing in supernatural intervention. Terms from the Arts, Sciences, and Social Sciences clockwork universe: a theory of the origin of the universe that com-pares the universe to a mechanical clock created by God. Accord-ing to this theory, once created, the universe continues to run according to the laws of nature and does not require further Divine intervention. This idea was very popular in the Enlightenment, an eighteenth-century philosophical movement that emphasized the use of reason to examine accepted beliefs and traditions. amalgamate v. to combine into a unified whole In early 1999, six municipalities were amalgamated into an enlarged city of Toronto, Canada. ambiguous adj. unclear or doubtful in meaning The gender of the Mahayana Buddhist deity Avalokitesuara, the god of infinite mercy, is ambiguous in both China and Japan, where the god is sometimes called a goddess. Terms from the Arts, Sciences, and Social Sciences Mahayana Buddhist: one of the three major traditions of Buddhism. It regards the historical Buddha as a manifestation of the celestial Buddha. 36 ESSENTIAL WORDS FOR THE GRE REVIEW 3 The correct answers are given on page 398. Matching Match each word with its definition. 1. aggrandize (A) cheerful willingness 2. aggregate (B) a combination 3. alacrity (C) the power to entice by charm 4. alchemy (D) to make larger or greater 5. allay (E) to combine into a unified whole 6. alleviate (F) to lessen; ease 7. alloy (G) amounting to a whole 8. allure (H) to relieve; improve partially 9. amalgamate (I) unclear or doubtful in meaning 10. ambiguous (J) medieval chemical philosophy Fill-ins Choose the best word to fill in the blank in each sentence. aggrandize aggregate alacrity alchemy allay alleviate alloys allure amalgamate ambiguous 1. The _ of France is great; millions of people around the world study its language and culture. 2. With the organic chemistry test coming up soon, Maria knew she had to start studying for it with __ . 3. The computer manufacturer donated one hundred computers to the inner-city school to _ the problem of children not having access to the Internet. 4. The corporation’s CEO claimed that his purchase of a personal jet airplane was not meant to personally _ him. 5. By what remarkable artistic __ did the interior decorator transform the drab living room into a room of vibrant color and light? 6. John’s role in the affair is _ ; it is not clear whether he took an active part in it or was merely an advisor. 7. The final plan is an __ of the ideas of everyone in the class. 8. Modern _ have helped make cars lighter and more resistant to corrosion. 9. To _ the public’s fears that his health was failing, the prime minister played tennis every day and invited reporters to be present. 37 ESSENTIAL WORDS FOR THE GRE 10. Now separate entities, the twelve colleges will __ to create a single university. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The allure of gold is so great that hardly anyone is buying it. 2. The governor is concerned that one agency of government is aggrandizing itself at the expense of other agencies. 3. The judge instructed the witness to make an ambiguous state ment so that everyone could be clear about what she meant. _ 4. The alacrity of John Milton’s Paradise Lost makes it one of the great epic poems in English. 5. The charity was set up to alleviate the suffering of the poor. UNIT 4 ambivalence n. the state of having conflicting emotional attitudes. John felt some ambivalence about getting married before finishing college. The adjective is ambivalent. In public opinion surveys in the United States, scientists rank second only to physicians in public esteem, yet much of the public is increasingly ambivalent about some of the implications for society of “Big Science” and its related technology. ambrosia n. something delicious; the food of the gods The combination of flavors in the Moroccan baked eggplant was pure ambrosia. The adjective is ambrosial. The food critic praised the chef for preparing what he called an “ambrosial meal.” Terms from the Arts, Sciences, and Social Sciences In Greek mythology, ambrosia and nektar were the delicious and fragrant food and drink of the gods that gave them immortality. The English word nectar (from Greek nektar) means a sweet liquid secreted by flowers, or a delicious drink. 38 ESSENTIAL WORDS FOR THE GRE ameliorate v. to improve Knowing they could not stop the spread of a contagion in a few days, health authorities worked to inhibit its spread and to amelio rate its effects by issuing warnings to the public and initiating immu nization programs. amenable adj. agreeable; cooperative; suited The young writer is amenable to suggestions for improving her prose style to make it more interesting. amenity n. something that increases comfort Many amenities considered normal and necessary by people in developed countries, such as indoor plumbing, were luxuries only a few generations ago. amulet n. ornament worn as a charm against evil spirits The early Christian Church forbade the use of amulets, which had become common in the Roman Empire at the time the Christian Church began to develop. anachronism n. something out of the proper time Some experts regard the retirement age of 65 as an anachronism at a time when people in the developed world have much longer life expectancies than previously. analgesic n. medication that reduces or eliminates pain Aspirin (the trademark of the drug acetylsalicylic acid) is a power ful analgesic that was introduced in 1899 and is still one of the most effective medicines available to alleviate pain, fever, and inflammation. analogous adj. comparable The psychology researcher’s experiment postulates that the brain is analogous to a digital computer. Analogy is a noun meaning a similarity in some ways between things that are otherwise dissimilar. The idea of evolution in nature is sometimes misconstrued and applied by analogy to other areas in which there is scant evidence for its existence; a notable example of this is Social Darwinism, in which it is argued that society is like nature, and thus people, like animals, are competing for survival, with those who are genetically superior at surviving and reproducing. 39 ESSENTIAL WORDS FOR THE GRE Analog is a noun meaning something that is comparable to something else. Some commentators have posited the existence of an analog to the Protestant work ethic in Chinese culture, which they call the “Confucian work ethic,” to explain the economic success of some countries with large Chinese populations. Terms from the Arts, Sciences, and Social Sciences Social Darwinism: a theory in sociology that individuals or groups achieve advantage over others as the result of genetic or biological superiority Protestant work ethic: a view of life that encourages hard work and a rational view of the world as a way to achieve material success Confucian: a system or ethics based on the teachings of the ancient Chinese sage Confucius. It places a high value on family relationships. anarchy n. absence of government; state of disorder The American philosopher Robert Nozick does not advocate anarchy; rather, he argues for the merits of a minimal state that would not violate the natural rights of individuals. The adjective anarchic means lacking order or control. The student of mythology speculated that Dionysos was created as a projection of the pleasure-loving, anarchic aspect of human nature. Terms from the Arts, Sciences, and Social Sciences Dionysos: known as Bacchus to the Romans, Dionysos was the son of Zeus and Selene. He was the Greek god of agriculture, fertility, wine, and ecstasy, and later regarded as a patron of the arts. Dio-nysos was worshipped by an emotional cult that held secret rites called Bacchanalia—wild orgies of frenzied revelry, drunkenness, and debauchery. The noun anarchism refers to the theory that all forms of government are oppressive and should be abolished. It also means the advocacy of this theory or the attempt to bring about anarchism. Most political scientists do not believe anarchism to be a tenable theory of government. 40 ESSENTIAL WORDS FOR THE GRE REVIEW 4 The correct answers are given on page 398. Matching Match each word with its definition. 1. ambivalence (A) agreeable; cooperative 2. ambrosia (B) medication that reduces pain 3. ameliorate (C) the state of having conflicting emotional attitudes 4. amenable (D) absence of government 5. amenity (E) ornament worn as a charm against evil spirits 6. amulet (F) something out of the proper time 7. anachronism (G) to improve 8. analgesic (H) comparable 9. analogous (I) something delicious 10. anarchy (J) something that increases comfort Fill-ins Choose the best word to fill in the blank in each sentence. ambivalent ambrosia ameliorate amenable amenities amulet anachronism analgesic analogy anarchy 1. Many people have an __ attitude to war: it causes great suffering, yet appears at times to be the only solution to a serious problem. 2. During the revolution the country began to slip toward _ . 3. The soldier attributed his survival through three battles to an _ he had found in Borneo. 4. After fasting for 24 hours, Wayne said that his first bite of steak tasted like __ . 5. “I’m afraid all I can do for your headache is prescribe an _ to relieve the pain,” the doctor told her patient. 6. The governor drew an _ between a family and soci ety, pointing out that both need a leader if they are to function smoothly. 7. The antithesis of the principle of art for art’s sake is social real ism, which feels a heavy responsibility to identify, and even __ , social ills. 8. The history professor is _ to student suggestions for the topic of the term paper. 41 ESSENTIAL WORDS FOR THE GRE 9. The editor discovered an _ in the script; set in 1944, it contained a reference to the atomic bombing of Hiroshima in 1945. 10. Many modern tourists like to have all the __ of home when they travel. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Amulets have been used for centuries to generate electric power. 2. The economist drew an analogy between a family spending beyond its means and a government running a deficit. 3. Although the government program was intended to help the poor, in reality it has only ameliorated their situation. _ 4. John is ambivalent about whether to apply to graduate school or look for a job after receiving his degree. 5. A group of anarchists called for the overthrow of the government. __ UNIT 5 anodyne n. something that calms or soothes pain Some people use alcohol as an anodyne to numb their emotional pain. Anodyne is an adjective that means relaxing, or capable of soothing pain. The public relations officer is remarkably anodyne; all he does is mouth comforting, politically correct platitudes, saying nothing of substance. anomalous adj. irregular; deviating from the norm The psychologist discounted the anomalous behavior of the sol dier, saying it was merely a short-term effect of the stress of battle. The noun is anomaly. A moral dilemma that arises with humanity’s ability to clone is posed in the following hypothetical scenario: a pig that produces much more meat than a normal pig can be cloned, but the pig’s life span would be cut in half because of anomalies in the cloning pro cess: Is it right to clone such an animal? 42 ESSENTIAL WORDS FOR THE GRE antecedent n. something that comes before Historical factors, such as the increased emphasis on the indi vidual, the invention of printing, and the rise of the bourgeoisie, con tributed to make the Reformation, which had its antecedents in the reform movement within the Roman Catholic Church, into a much broader phenomenon that created powerful churches that grew to rival the original church. Terms from the Arts, Sciences, and Social Sciences Reformation: a sixteenth-century movement aimed at reforming abuses in the Roman Catholic Church. It led to the establishment of new churches. antediluvian adj. prehistoric Most of our knowledge of antediluvian times has been built up as a result of one of humanity’s grandest collaborative endeavors—the gathering, identification, dating, and categorization of fossils as they are discovered. antipathy n. dislike; hostility Heathcliff, the protagonist of Emily Brontë’s novel Wuthering Heights, feels great antipathy for Edgar Linton, the man who marries the woman he loves. Terms from the Arts, Sciences, and Social Sciences protagonist: the main character in a work of literature apathy n. indifference Apathy was high in the election because there was no major con troversy or issue to arouse voter interest. The adjective is apathetic. One criticism of the welfare state is that it makes people overly reli ant on government, with the result that democracy is gradually weak ened as citizens take a more apathetic and detached view of politics. Terms from the Arts, Sciences, and Social Sciences welfare state: the provision of welfare services by the state (that is, the government) 43 ESSENTIAL WORDS FOR THE GRE apex n. the highest point In English literature, classicism reached its apex in the poetry of Alexander Pope and the other Augustans. Terms from the Arts, Sciences, and Social Sciences classicism: an aesthetic tradition that values simplicity, elegance, restraint, and order Augustans: a period of English literature from around 1700 to 1789. Satire was a feature of the writing of many authors of this period. Two notable authors of the Augustan Age were Alexander Pope and Jonathan Swift. apogee n. the point in an orbit most distant from the body being orbited; the highest point The Ottoman Empire reached its apogee in the seventeenth century, when it controlled a territory running from Budapest to North Africa. Terms from the Arts, Sciences, and Social Sciences Ottoman Empire: an empire that arose in Anatolia (which corre-sponds to the Asian portion of modern Turkey) in the fourteenth century, destroying the Byzantine Empire. By the early sixteenth century it controlled much of Persia, Arabia, Hungary, the Balkans, Syria, and Egypt. apothegm n. a terse, witty saying (pronounced AP-uh-them and also spelled apophthegm) One of the best-known political apothegms was written by the British historian Lord Acton: “Power tends to corrupt and absolute power corrupts absolutely.” appease v. to calm; pacify; placate Many historians have criticized British Prime Minister Neville Chamberlain for trying to appease Adolf Hitler in the 1930s. 44 ESSENTIAL WORDS FOR THE GRE REVIEW 5 The correct answers are given on page 398. Matching Match each word with its definition. 1. anodyne (A) indifference 2. anomalous (B) prehistoric 3. antecedent (C) something that comes before 4. antediluvian (D) to pacify 5. antipathy (E) hostility 6. apathy (F) point in orbit most distant from body being orbited 7. apex (G) something that soothes pain 8. apogee (H) the highest point 9. apothegm (I) irregular 10. appease (J) terse, witty saying Fill-ins Choose the best word to fill in the blank in each sentence. anodyne anomalous antecedents antediluvian antipathy apathy apex apogee apothegms appease 1. The transistor was the result of a collaborative effort by research ers at Bell Laboratories in New Jersey, one of the world’s most advanced scientific and technological laboratories, which had its __ in the great laboratories created in the late nineteenth century by people like Thomas Edison. 2. In “Strange Meeting,” one of Wilfred Owen’s poems about World War I, the speaker says that he has no _ for the foe he killed in battle. 3. The students are trying to overcome public _ on the issue by setting up exhibitions about it in shopping centers. 4. The scientist asked the lab technician to check the __ results again. 5. To _ angry voters the legislature approved a tax cut. 6. The English teacher showed his class the classic film On the Beach, but many of the students had trouble appreciating it because of what one student called its “_ black and white film technology.” 45 ESSENTIAL WORDS FOR THE GRE 7. The eighteenth-century British writer Samuel Johnson is famous for his sage __ , such as “If you are idle, be not solitary; if you are solitary, be not idle.” 8. The pastor’s comforting words at the child’s funeral were an _ for the grieving family. 9. When the spacecraft reaches its _ in its orbit around Earth, another craft will be launched from it on a voyage to Mars. 10. Many religions view human beings as standing at the __ of creation. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The simple electric circuit consists of a battery and an anodyne. 2. Although the scientist’s career as a researcher had long since reached its apex, she continued to give valuable guidance to younger scientists. 3. The gun is capable of firing deadly apothegms that can rip enemy soldiers apart. _ 4. The anomalous test results mean that the rocket is ready to launch. 5. Anthropologists believe that the tribe used animal sacrifice to appease the angry gods. UNIT 6 appellation n. name The discovery of the bones of a person with the appellation Ken newick Man in the state of Washington in 1996 has raised important questions about who the earliest people to populate America were. apposite adj. strikingly appropriate and relevant The writer searched two dictionaries and a thesaurus before find ing the perfectly apposite word he was looking for. apprise v. to inform Nadine Cohodas’s biography of the blues singer Dinah Washington keeps the reader apprised of the racism black Americans had to endure. 46 ESSENTIAL WORDS FOR THE GRE approbation n. praise; approval The Congressional Medal of Honor is the highest approbation an American soldier can receive. appropriate v. to take possession for one’s own use; confiscate The pronunciation is uh-PROH-pree-ayt. The adjective appropriate is pronounced uh-PROH-pree-it. The invading army appropriated supplies from the houses of the local people. apropos adj. relevant Apropos of nothing, the speaker declared that the purpose of life is to love. arabesque n. ornate design featuring intertwined curves; a ballet posi tion in which one leg is extended in back while the other supports the weight of the body The ballerina stunned the audience with her perfectly executed arabesque. archeology n. the study of material evidence of past human life Carbon-14 dating is of great use in archeology because it can determine the age of specimens as old as 35,000 years, but it is of less use in geology because most of the processes studied in this field occurred millions of years ago. Terms from the Arts, Sciences, and Social Sciences Carbon-14 dating: determining the actual or relative age of an object, of a natural phenomenon, or of a series of events through the use of the isotope carbon-14, which occurs naturally ardor n. great emotion or passion The twentieth-century American poet Wallace Stevens said, “It is the unknown that excites the ardor of scholars, who, in the known alone, would shrivel up with boredom.” arduous adj. extremely difficult; laborious The task of writing a research paper is arduous, but if it is broken down into logical steps it becomes less daunting. 47 ESSENTIAL WORDS FOR THE GRE REVIEW 6 The correct answers are given on page 399. Matching Match each word with its definition. 1. appellation (A) relevant 2. apposite (B) confiscate 3. apprise (C) great emotion or passion 4. approbation (D) ornate design 5. appropriate (E) name 6. apropos (F) laborious 7. arabesque (G) strikingly appropriate and relevant 8. archeology (H) praise 9. ardor (I) the study of material evidence of past human life 10. arduous (J) to inform Fill-ins Choose the best word to fill in the blank in each sentence. appellation apposite apprised approbation appropriated apropos arabesque archeology ardor arduous 1. Some people felt the remarks were out of place, but others thought they were perfectly __ . 2. The president ordered his chief of staff to keep him _ of any changes in the situation. 3. The English professor has started on the _ task of writing book-length commentaries on all thirty-seven of William Shakespeare’s plays. 4. During an economic “bubble” there is a great __ for speculative investing. 5. _ provides anthropologists with important information about prehistoric cultures. 6. The city _ private land to build low-cost housing. 7. The __ is one of the fundamental ballet poses. 8. Former U.S Supreme Court justice Byron White was given the _ “Whizzer” when he played football in college. 9. The fashion book contains the perfect, _ image to represent one hundred famous designers. 10. The young scientist is working 80 hours a week to gain the __ of her peers. 48 ESSENTIAL WORDS FOR THE GRE Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The judge rejected the witness’ testimony because nothing in it was apropos to the case. 2. One of the major questions in modern archaeology is whether God exists or not. 3. A comet with the appellation Shoemaker-Levy 9 collided spectacu larly with Jupiter in July 1994. _ 4. The group plans to make the arduous ascent of Mt. Everest without oxygen supplies. 5. “Jack hit Jim” has an apposite meaning from “Jim was hit by Jack.” UNIT 7 argot n. a specialized vocabulary used by a group Writers of crime fiction often use the argot of criminals and detec tives to create a realistic atmosphere. arrest v. to stop; to seize Temporary arrest of the patient’s respiration made it easier for the doctor to perform surgery on him. artifact n. item made by human craft Marxists contend that appreciation of art has declined because capitalism has trained people to perceive human artifacts as commodities, and has alienated people from nature, their true humanity, and their creations. Terms from the Arts, Sciences, and Social Sciences Marxist: a follower of Marxism, the political and economic philoso-phy of Karl Marx and Friedrich Engels. In Marxism the concept of class struggle plays a central role in understanding society’s inevi-table development from bourgeois oppression under capitalism to a socialist and ultimately classless society. artless adj. guileless; natural The source of the meaning of artless as guileless is the poet John Dryden, who wrote of William Shakespeare in 1672: “Such artless beauty lies in Shakespeare’s wit. . . .” 49 ESSENTIAL WORDS FOR THE GRE ascetic n. one who practices self-denial Muslim ascetics consider the internal battle against human passions a greater jihad than the struggle against infidels. Ascetic is also an adjective meaning self-denying or austere. The writer’s ascetic lifestyle helped her to concentrate on finishing her novel. The noun is asceticism. One tradition of asceticism derives from the belief that the body is fundamentally bad and must be subjugated to the soul. Terms from the Arts, Sciences, and Social Sciences jihad: the religious duty of Muslims to defend their religion (Islam) by war or spiritual struggle against nonbelievers asperity n. severity; harshness; irritability In his autobiography Gerald Trywhitt, the British writer, composer, artist, and aesthete, recounts a humorous incident: “Many years later, when I was sketching in Rome, a grim-looking Englishwoman came up to me and said with some asperity, ‘I see you are painting MY view.’” aspersion n. slander; false rumor The Republic of Singapore is a young democracy, and its leaders often respond strongly to journalists and others who cast aspersions on their integrity. assiduous adj. diligent; hard-working The assiduous people of Hong Kong live in a territory with one of the highest per capita incomes in the world. assuage v. to make less severe On November 21, 1864, during the Civil War, President Abraham Lincoln wrote the following in a letter to Mrs. Bixby of Boston, who had lost five sons in battle: “I pray that our Heavenly Father may assuage the anguish of your bereavement and leave you only the cherished memory of the loved and lost, and the solemn pride that must be yours to have laid so costly a sacrifice upon the altar of freedom.” astringent adj. harsh; severe Bob tends to nick himself when he shaves, so he uses an astringent aftershave to stop the bleeding. 50 ESSENTIAL WORDS FOR THE GRE REVIEW 7 The correct answers are given on page 399. Matching Match each word with its definition. 1. argot (A) guileless; natural 2. arrest (B) slander 3. artifact (C) item made by human craft 4. artless (D) to make less severe 5. ascetic (E) to stop; seize 6. asperity (F) hard-working 7. aspersion (G) harshness 8. assiduous (H) specialized vocabulary used by a group 9. assuage (I) harsh; severe 10. astringent (J) one who practices self-denial Fill-ins Choose the best word to fill in the blank in each sentence. argot arrest artifact artless ascetic asperity aspersions assiduously assuage astringent 1. The young actor’s brilliant portrayal of the __ young boy was the result, paradoxically, of many hours of careful rehearsal. 2. Considering that the two men had been such good friends for so long, we were surprised by the _ of their attacks on each other. 3. In his book Confessions, Saint Augustine tells of his sinful life before he was converted to Christianity and began to live an _ and virtuous life. 4. Mate, a popular beverage in South America, is similar to tea but is less __ and often contains more caffeine. 5. Some scholars have argued that the idea of romantic love is an _ of culture, unique to the West, with its origin in the European tradition of courtly love; however, sociological research has shown that romantic love exists in most cultures. 6. The new drug is able to _ the development of cancerous cells. 7. China’s leaders talked with a group of American congressional representatives to __ fears that China plans to threaten American military preeminence. 51 ESSENTIAL WORDS FOR THE GRE 8. The study’s conclusion is that more females attend college than males because girls tend to apply themselves more _ to their studies than boys. 9. Much of the _ from the field of information technology that previously was familiar only to experts in the field is now used in everyday conversation (“Internet Service Provider,” for example). 10. The report in the newspaper cast __ on the candidate. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The hungry people gratefully received the aspersions from the charity. 2. The crew of the submarine bid their families farewell before setting off on their long argot. 3. The poet Walt Whitman quotes John Burroughs’s comment on the writer Henry David Thoreau: “He improves with age—in fact, it requires age to take off a little of his asperity, and fully ripen him.” _ 4. “My comments on your book were merely astringent, not venomous,” the critic said to the author. 5. The writer James Boswell is so famous for his assiduous record ing of Samuel Johnson’s words that the word “Boswell” now refers to a person who admires another so greatly that he or she records their words and deeds. UNIT 8 asylum n. place of refuge or shelter The Stoic, accused of seeking asylum in the consolations of phi losophy, rebutted this charge, saying that Stoicism is simply the most prudent and realistic philosophy to follow. Terms from the Arts, Sciences, and Social Sciences Stoic: follower of Stoicism, a pantheistic philosophy emphasizing submission to divine will and freedom from emotion atavism n. in biology, the reappearance of a characteristic in an organism after several generations of absence; individual or a part that exhibits atavism; return of a trait after a period of absence Some modern political theorists reject nationalism as a tribal atavism. 52 ESSENTIAL WORDS FOR THE GRE attenuate v. to weaken Modern digital radio equipment allows even signals that have been greatly attenuated to be transmitted by one station and received by another station. audacious adj. bold; daring The German army commander Erwin Rommel was known as the “Desert Fox” as a result of his audacious surprise attacks on Allied forces in World War II. austere adj. stern; unadorned Deism is an austere belief that reflects the predominant philoso phy of the Age of Enlightenment: a universe symmetrical and gov erned by rationality. Terms from the Arts, Sciences, and Social Sciences Deism: the belief in a God who created the universe and then aban-doned it, assuming no control over life or natural phenomena, and giving no supernatural revelation Age of Enlightenment: a period in European philosophy during the eighteenth century that emphasized reason autonomous adj. self-governing; independent Some biologists have theorized that our belief in our ability to act as autonomous agents is in conformity with the theory of evolution because it gives us a sense of meaning and purpose in our lives that helps us to survive. Terms from the Arts, Sciences, and Social Sciences theory of evolution: the theory that living things originate from other similar organisms and that differences between types of organisms are due to modifications in successive generations. A central tenet of Darwinian evolution is that surviving individuals of a species vary in a way that enables them to live longer and reproduce, thus passing this advantage to future generations (Natural Selection). avarice n. greed Successful investment bankers are sometimes accused of avarice; their defenders, however, say that they are simply very good at what they do and should be rewarded accordingly. 53 ESSENTIAL WORDS FOR THE GRE aver v. to affirm; declare to be true Yogis aver that everyone has a guru, whether it be a person, God, or the experiences of the world, that helps him or her practice the yoga that is in accordance with his or her nature, and assists on the path toward enlightenment. Terms from the Arts, Sciences, and Social Sciences yogis: a yogi is the Sanskrit (an ancient Indian language) name for a man who practices yoga. A woman who practices yoga is a yogini. guru: a personal spiritual teacher. The term is also used to refer to a trusted advisor or an authority. yoga: spiritual practices in the Hindu and Buddhist religions that are believed to help one to attain higher awareness and union with God. avocation n. secondary occupation Dan became so proficient at his avocation—computer programming—that he is thinking of giving up his job as a teacher to do it full time. avuncular adj. like an uncle, benevolent and tolerant Walter Cronkite, who was the anchorman of CBS News during much of the 1970s and 1980s, had an avuncular manner that made him one of America’s most trusted personalities. REVIEW 8 The correct answers are given on page 399. Matching Match each word with its definition. 1. asylum (A) stern; unadorned 2. atavism (B) return of a trait after a period of absence 3. attenuate (C) greed 4. audacious (D) secondary occupation 5. austere (E) bold; daring 6. autonomous (F) like an uncle 7. avarice (G) self-governing; independent 8. aver (H) place of refuge 9. avocation (I) to affirm 10. avuncular (J) to weaken 54 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. asylum atavism attenuate audacious austere autonomous avarice avers avocation avuncular 1. It is important to have an __ judiciary so that laws can be interpreted free of political influence and considerations. 2. The monks live in _ quarters. 3. Scientists examining the whale discovered an _ : it had two legs. 4. The __ teacher is popular with students. 5. Many people prefer to pursue an _ that is very different from their occupation. 6. The United States and Britain have long histories of offering _ to victims of persecution. 7. The plan to eliminate hunger in the world is an __ one, but it can be achieved if all the nations of the world cooperate. 8. A criticism that has been made of capitalism is that it encourages _ . 9. Materialism is a philosophy that _ that matter is the only reality and denies the existence of idealism and spiritualism. 10. Aspirin has the power to __ a fever. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The doctor is considering surgery because of the atavism in the patient’s left eye. 2. Before giving testimony at a trial, each witness must solemnly aver that he or she is telling the truth. 3. Shortly after taking office in 1977, President Jimmy Carter fulfilled his campaign promise to pardon young men who had sought asylum in Canada because of their opposition to the war in Vietnam and to the military draft. _ 4. There is a strong tradition that physicians should practice medicine to ease human suffering rather than be motivated by avarice. 5. The singer’s voice was so attenuated by the PA system that she could be heard even outside the stadium. 55 ESSENTIAL WORDS FOR THE GRE UNIT 9 axiomatic adj. taken for granted In nineteenth-century geology, uniformitarianism was the antith-esis of catastrophism, asserting that it was axiomatic that natural law and processes do not fundamentally change, and that what we observe now is essentially the same as what occurred in the past. The noun axiom means established principle. Terms from the Arts, Sciences, and Social Sciences uniformitarianism: a geological theory popular in the nineteenth century. Uniformitarianism holds that geological processes have slowly shaped the Earth and continue to do so. catastrophism: a theory that was a rival to uniformitarianism. It pos-tulates an Earth formed in a series of unique, catastrophic events. bacchanalian adj. pertaining to riotous or drunken festivity; pertain-ing to revelry. For some people New Year’s Eve is an occasion for bacchanalian revelry. Terms from the Arts, Sciences, and Social Sciences Bacchus: known as Dionysos to the ancient Greeks, Bacchus was the god of agriculture, fertility, and wine. He was worshipped by an emotional cult that held secret rites called Bacchanalia—wild orgies of frenzied revelry, drunkenness, and debauchery. Bacchanalian is derived from Bacchanalia. banal adj. commonplace; trite The writer has a gift for making even the most banal observation seem important and original. banter n. playful conversation The governor engaged in some banter with reporters before getting to the serious business of the news conference. bard n. poet The great bards of English literature have all been masters of the techniques of verse. 56 ESSENTIAL WORDS FOR THE GRE bawdy adj. obscene Geoffrey Chaucer’s Canterbury Tales is the story of a group of Christian pilgrims who entertain one another with stories, ranging from the holy to the bawdy, on their journey to Canterbury Cathedral. beatify v. to sanctify; to bless; to ascribe a virtue to In the year 2000 Pope John Paul II traveled to Fatima in Portugal to beatify two of the three children who said they saw the appearance of the Virgin Mary there in 1917. Beatification is the noun. Beatification is the second and next to last step on the path to sainthood. bedizen v. to dress in a vulgar, showy manner Paul went to the costume party bedizened as a seventeenth-century French aristocrat. behemoth n. huge creature; anything very large and powerful In the 1980s and 1990s, the trend in American business was toward increased privatization of government industries (such as power generation), partly because it was believed that private indus try is more efficient and partly because foreign private companies were becoming commercial behemoths, outstripping government-owned companies in competitiveness. belie v. to contradict; misrepresent; give a false impression The boxer’s childlike face belies the ferocity with which he can attack opponents in the ring. 57 ESSENTIAL WORDS FOR THE GRE REVIEW 9 The correct answers are given on page 399. Matching Match each word with its definition. 1. axiomatic (A) playful conversation 2. bacchanalian (B) give a false impression 3. banal (C) pertaining to riotous or drunken activity 4. banter (D) dress in a vulgar, showy manner 5. bard (E) obscene 6. bawdy (F) commonplace 7. beatify (G) huge creature 8. bedizen (H) taken for granted 9. behemoth (I) poet 10. belie (J) to bless; sanctify Fill-ins Choose the best word to fill in the blank in each sentence. axiomatic bacchanalian banal banter bard bawdy beatification bedizen behemoths belie 1. The comedian dropped the __ jokes from his routine for his appearance on national television. 2. The _ Ted Hughes was appointed Britain’s Poet Laureate in 1984. 3. In Jack London’s novel The Sea Wolf, one of the characters says, “The sacredness of life I had accepted as _ .” 4. At first, college seemed to __ all the good things Steve had heard about it in high school; gradually, however, he came to like it. 5. First IBM, next Microsoft and then Google became the _ of the computer industry. 6. The TV show’s producer tries to steer a middle path between making a typical _ program and being so original that much of the audience is lost. 7. The queen decided to __ herself with expensive jewelry for the ball. 8. In the Roman Catholic Church, the final stage in the path to sainthood is canonization, which occurs after _ . 9. The college’s annual spring break party in Florida is a/an _ affair. 58 ESSENTIAL WORDS FOR THE GRE 10. The world leaders enjoyed some friendly __ before getting down to the serious business of the negotiations. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The prisoner was beatified by the jury and sentenced to ten years imprisonment. 2. The band’s backstage crew often exchange banter during long rehearsals. 3. The principle that every person has certain fundamental rights is regarded by most people as axiomatic. _ 4. The ascetic monks regularly hold bacchanalian parties. 5. The question of whether the Greek bard Homer was a single individual or the name given to several authors is still hotly debated by scholars. UNIT 10 beneficent adj. kindly; doing good The theologian discussed the question of why a beneficent and omnipotent God allows bad things to happen to good people. bifurcate v. to divide into two parts Contemporary physicists generally bifurcate their discipline into two parts—classical physics and modern physics; the former are the fields of study that were already well developed before the momen tous breakthroughs of the early twentieth century by scientists such as Albert Einstein, Niels Bohr, and Werner Heisenberg, which inaugu rated the age of modern physics. Bifurcation is the noun. Some people regard the Hindu-Buddhist philosophy on animals as more in accordance with the modern scientific view than the tradi tional Western view, since it does not posit a radical bifurcation of man and nature. blandishment n. flattery Despite the salesperson’s blandishments, Donna did not buy the car. Blandish is the verb, meaning to coax with flattery. 59 ESSENTIAL WORDS FOR THE GRE blasé adj. bored because of frequent indulgence; unconcerned We were amazed by John’s blasé attitude toward school; he seems to have made it a rule never to open a book. bolster v. to give a boost to; prop up; support The president has visited the state several times to bolster his sagging popularity there. bombastic adj. pompous; using inflated language Nearly lost in the senator’s long, bombastic speech were several sensible ideas. boorish adj. rude; insensitive Bob apologized for his boorish behavior at the party, saying he hadn’t realized that it was such a formal occasion. bovine adj. cowlike Following the slow-moving group of students up the long path to the school’s entrance, the word “bovine” popped into the English teacher’s mind. brazen adj. bold; shameless The brazen student irritated his teacher by saying that he could learn more from a day spent “surfing” the World Wide Web than a day spent in school. broach v. to mention for the first time Steve’s boss knew that she couldn’t put off warning him about his poor performance and decided to broach the subject the next time she saw him. 60 ESSENTIAL WORDS FOR THE GRE REVIEW 10 The correct answers are given on page 399. Matching Match each word with its definition. 1. beneficent (A) flattery 2. bifurcate (B) cowlike 3. blandishment (C) kindly; doing good 4. blasé (D) bold 5. bolster (E) bored because of frequent indulgence 6. bombastic (F) mention for the first time 7. boorish (G) give a boost to 8. bovine (H) rude; insensitive 9. brazen (I) divide into two parts 10. broach (J) pompous; using inflated language Fill-ins Choose the best word to fill in the blank in each sentence. beneficence bifurcation blandishments blasé bolstered bombastic boorish bovine brazen broached 1. The coach warned the lacrosse team not to become __ even though they had won a school record twenty matches the previous season. 2. There is a _ in American politics between a tradition that believes that interference in the affairs of other countries is imprudent, and an idealistic streak that seeks to use American power to help other countries. 3. Bill Gates showed his _ by setting up with his wife Melinda a foundation to provide financial help to, among other things, fight disease in the third world. 4. Amanda went out with her boyfriend for two years before she __ the subject of marriage. 5. The audience listened to the boring speech with _ expressions on their faces. 6. Russian historians have shown how the Soviet leader Joseph Sta lin used a mixture of arguments, _ , and threats to overcome resistance to his repressive policies among his fellow Politburo members. 7. The small company startled investors by its __ takeover of a company with three times its assets. 61 ESSENTIAL WORDS FOR THE GRE 8. Many people in the audience were annoyed at the _ behavior of the two men who talked loudly to each other through the entire movie. 9. The president’s speechwriter told him that she was doing her best to write a speech that was serious and solemn but not _ . 10. The prosecutor’s case was __ by the new testimony of a credible witness. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Satan is often portrayed as the embodiment of malevolence and beneficence. 2. The actor is admired by everyone for her boorish behavior. 3. The brazen student demanded that the teacher postpone the test so she would have more time to study for it. _ 4. The general visited the troops to bolster their morale before the crucial battle. 5. Jim is so blasé about the upcoming GRE test that he is studying six hours a day for it. __ APPLYING YOUR KNOWLEDGE—1–10 1. The popular conception of the oppressed worker at the barricades on strike is now __, considering that the average American worker today owns a home and property, and many are themselves “capitalists” in that they own shares in the companies that employ them. an anachronism an analogy an anomaly an anodyne true Explanation: An important clue to the answer to this question are the words “now” and “today,” which signal a contrast between today’s reality and the popular conception that is now an anachronism (some thing out of the proper time). The correct answer is anachronism. 62 ESSENTIAL WORDS FOR THE GRE 2. An important (i) _ in law is “Justice must not only be done, it must be seen to be done,” which means that the legal system must not only reach fair verdicts, but also must in the process (ii) _ the fairness and efficiency of the system. Blank (i) Blank (ii) deterrent give citizens confidence in propriety undermine axiom satirize Explanation: An axiom in this context is an “established principle,” which is an appropriate definition of the quotation in the sentence. An important clue is the word “seen” in the quotation, which suggests that the judicial process is open to public scrutiny. Such scrutiny of a fair judicial system will create public confidence in it. The correct answers are axiom and give citizens confidence in. UNIT 11 bucolic adj. characteristic of the countryside; rustic; pastoral The south end of Toronto’s beautiful High Park is a bucolic expanse of land that is perfect for anyone wanting a quiet walk. burgeon v. to flourish After World War II, the increased speed of industrialization and the burgeoning world population resulted in such an increase in pollu tion that it began to be recognized by some people as a threat to the human habitat, Earth. burnish v. to polish The poet T. S. Eliot burnished his reputation as one of the master poets of the twentieth century with Four Quartets, four long poems published between 1936 and 1942. buttress v. to reinforce; support Some critics of the American legal system argue that the require ment of proving guilt “beyond a reasonable doubt” is too difficult a criterion to use, and buttress their case by citing the fact that objec tive studies suggest that only a very small number of criminals are successfully prosecuted. 63 ESSENTIAL WORDS FOR THE GRE cacophonous adj. unpleasant or harsh-sounding The dissonant harmonies of the great jazz pianist and composer Thelonious Monk might seem cacophonous to some listeners, but to many jazz aficionados they are sublime. A cacophony is a jarring, unpleasant noise. Terms from the Arts, Sciences, and Social Sciences aficionados: people who are enthusiastic admirers or followers cadge v. to beg; sponge An enduring image of the Great Depression in America is the out-of-work man cadging money with the line, “Hey, mister, can you spare a dime for a cup of coffee?” Terms from the Arts, Sciences, and Social Sciences Great Depression: a very large economic decline that began in 1929. Major industrial nations such as Great Britain, Japan, and the United States were greatly affected by declines in nearly all measures of economic prosperity (such as employment and profits). callous adj. thick-skinned; insensitive Jim’s terrible experiences in the war have made him callous about the suffering of others. calumny n. false and malicious accusation; slander “Be thou chaste as ice, as pure as snow, thou shalt not escape calumny.” —William Shakespeare, Hamlet Act III, Scene 1 (Hamlet addressing Ophelia) canard n. false, deliberately misleading story Most politicians do not want to be associated with the old canard that big government in Washington can solve all of America’s problems. canon n. an established principle; a basis or standard for judgment; a group of literary works Canons of aesthetic taste vary over the years; the Rococo period, for example, valued ornate art. 64 ESSENTIAL WORDS FOR THE GRE The sixty-volume Great Books of the Western World is an attempt to gather the central canon of Western civilization into one collection. Canon is also an adjective. The system of civil law originated in the Roman Empire and was kept alive in the Middle Ages in the canon law of the Church. Canonical is an adjective meaning belonging to a group of literary works. The English professor is trying to persuade the chairperson of her department to let her teach some writers that are not canonical. Terms from the Arts, Sciences, and Social Sciences Rococo: refers to a style of architecture in eighteenth-century Europe that made use of elaborate curved forms. The word is often used to refer to something that is excessively ornate. REVIEW 11 The correct answers are given on page 399. Matching Match each word with its definition. 1. bucolic (A) to polish 2. burgeon (B) to reinforce 3. burnish (C) to beg 4. buttress (D) false, misleading story 5. cacophonous (E) established principle 6. cadge (F) characteristic of the countryside 7. callous (G) insensitive 8. calumny (H) to flourish 9. canard (I) unpleasant or harsh-sounding 10. canon (J) false, malicious accusation Fill-ins Choose the best word to fill in the blank in each sentence. bucolic burgeoning burnish buttress cacophonous cadge callous calumny canard canon 1. The __ of modern communications has made fiber optics nearly indispensable because of its ability to transmit vast amounts of information. 65 ESSENTIAL WORDS FOR THE GRE 2. The link between economic boom and war is used by Marxists to _ their view that capitalism thrives on war, and to some degree, encourages it in periods of low economic activity. 3. How many times have you heard the old _ “Those who can, do; those who can’t, teach”? 4. A traditional olive farm is a __ sight: big trees spaced fairly far apart providing good cover for grass and grazing animals. 5. The student is well known for his tendency to _ money from his friends. 6. The company’s new advertising campaign is intended to _ its image as a dynamic, forward-looking firm. 7. The movie star sued the newspaper for printing __ about him. 8. The task the poultry farm worker looked forward to the least was going into the _ hen yard at feeding time. 9. The public relations director’s comments that the inmates had hanged themselves as a public relations stunt was widely regarded as showing a _ disregard for life. 10. The nineteenth-century French composer Hector Berlioz has become a central figure in the Western musical __ . Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. It may sound odd, but I actually enjoy the cacophonous sound of an orchestra tuning up. 2. The artist is painting a bucolic rush hour scene in Manhattan. 3. We sailed our canard around the world last year. _ 4. Government officials in China are concerned about the burgeoning numbers of old people in their country. 5. Steve buttressed his position in the pharmaceutical company by earning a Ph.D. in chemistry. __ 66 ESSENTIAL WORDS FOR THE GRE UNIT 12 cant n. insincere talk; language of a particular group Many of the beat artists of the 1950s reacted against what they regarded as the cant of bourgeois society. Terms from the Arts, Sciences, and Social Sciences bourgeois: both bourgeois and bourgeoisie come from Old French burgeis, citizen of a bourg (town). Bourgeois refers to a person who belongs to the middle class or has middle-class attitudes. It can be used in a neutral way. However, it is frequently used to suggest that someone is not sophisticated. cantankerous adj. irritable; ill-humored Many of us have in our mind the stereotype of the cantankerous old man who is constantly complaining about something or other. capricious adj. fickle The rule of law is regarded by many historians as one of human ity’s great achievements because since its inception citizens are no longer subject to capricious decisions and penalties of rulers. Caprice is a noun meaning an inclination to change one’s mind compulsively. Styles in high fashion seem governed by caprice as much as any thing else. captious adj. faultfinding; intended to entrap, as in an argument The pedantic and captious critic seems incapable of appreciating the merits of even the most highly regarded books. cardinal adj. of foremost importance The cardinal rule of any weight-loss diet must be limiting the intake of calories. carnal adj. of the flesh or body; related to physical appetites The yogi’s goal is to achieve nirvana through, among other things, the overcoming of carnal desires. 67 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences nirvana: ideal condition of rest, harmony, or joy. Nirvana is from Sanskrit nirvanam (a blowing out), as in the blowing out of a flame. According to Buddhism and Hinduism, in order to reach nirvana one must extinguish the fire fueled by the ego, which causes suffering, ignorance, delusion, and greed. carping v. to find fault; complain Cost-benefit analysis owes much of its origin to utilitarian thought; despite the carping of critics that such analysis is based on faulty premises, the technique has proved useful in many areas. Terms from the Arts, Sciences, and Social Sciences Cost-benefit analysis: the process of weighing the total expected costs against the total expected benefits of one or more actions in order to choose the best option utilitarian: the ethical philosophy that human activity should be aimed at achieving the greatest good for the greatest number. Jeremy Bentham was the founder of the theory and his student John Stuart Mill was its most famous proponent. Mill used the theory to argue for social reform and increased democracy. cartography n. science of making maps Satellites in Earth orbit take pictures of topography that have greatly aided cartography. caste n. any of the hereditary social classes of Hindu society; social stratification The dalits, formerly known as untouchables, are at the bottom of the thousands of castes that make up Indian society. Caste is also an adjective. Most modern corporations employ a sort of caste system, with senior executives at the top and ordinary workers at the bottom. 68 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences There are four main castes or heredity groups (and thousands of subcastes) in Hindu society that restrict the occupations of members and limit their interaction with members of other castes. There are four main castes: Brahmans (priests and teachers) Ksatriyas (noblemen) Vaisyas (merchants and traders) Sudras (laborers) A fifth group, called “Harijans” or “untouchables” are considered impure and are discriminated against. They toil in lowly occupa-tions such as cleaning up waste and leatherworking. castigation n. punishment; chastisement; criticism Many British writers recall with loathing the castigation they received at school. REVIEW 12 The correct answers are given on page 400. Matching Match each word with its definition. 1. cant (A) of foremost importance 2. cantankerous (B) fickle 3. capricious (C) science of making maps 4. captious (D) insincere talk 5. cardinal (E) of the flesh or body 6. carnal (F) hereditary social class 7. carping (G) complaining 8. cartography (H) punishment 9. caste (I) irritable 10. castigation (J) faultfinding 69 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. cant cantankerous capricious captious cardinal carnal carping cartography caste castigated 1. The military employs a type of __ system with generals at the top and privates at the bottom. 2. Commentators dismissed the speech as the mere _ of someone desperately trying to be reelected. 3. The English teacher is so pedantic and _ in her marking that her students have become discouraged. 4. It is a postulate of science that the laws of nature are not __ and that the universe is not chaotic. 5. According to this book the _ rule of good writing is to be clear. 6. Rococo painting often reflects the great pleasure the French aristocracy took in all things _ . 7. The ability of modern __ to produce very accurate maps of the Earth’s surface has been a boon to navigators. 8. The boss _ the worker for losing the important client’s file. 9. The band decided to continue to play in their new style despite the _ of critics who said it was a sell-out to commercial interests. 10. The __ old woman is always getting into arguments with people. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The judge is respected by legal scholars for her capricious rulings. 2. The cantankerous old man seems to like nothing better than arguing with the checkout clerk in the supermarket. 3. Cartography has helped scientists gain a good understanding of the fundamental workings of the human brain. _ 4. A cardinal rule of hiking is to never hike alone. 5. The painting’s caste gives it the mood of a Rembrandt. 70 ESSENTIAL WORDS FOR THE GRE UNIT 13 cataclysm n. a violent upheaval that causes great destruction and change The French Revolution of 1789 was a cataclysm whose effects are still felt today. catalyst n. something causing change Among the catalysts of the Romantic movement were the libertar ian ideals of the French Revolution. Terms from the Arts, Sciences, and Social Sciences Romantic movement: a late eighteenth- and nineteenth-century movement in literature and the arts. The movement was a very var-ied one, and so is not easily described in a few words. Romanticism was a revolt against classicism and reason and emphasized the indi-vidual and the emotional. The Romantics also stressed the inherent goodness of man and nature and valued freedom highly. Important Romantic poets in England include William Blake, John Keats, Wil-liam Wordsworth, and P. B. Shelley. Famous Romantic composers include Hector Berlioz, Franz Liszt, and Frederic Chopin. libertarian: libertarians place great importance on individual free-dom. They believe that no limitations should be placed on a person’s freedom unless that person’s actions limit the freedom of others. French Revolution: a crucial period (1789–1799) in French, and more generally, Western civilization. France’s absolute monarchy was replaced by republicanism. It is regarded by most historians as a major turning point in Western civilization, ushering in the era of citizens as the major force in politics. categorical adj. absolute; without exception Although incest is categorically forbidden by every state, recent evidence that marriage between cousins is no more likely to produce abnormal offspring than “normal” marriages may allow the constitu tionality of bans on marriage between cousins to be challenged. caucus n. smaller group within an organization The workers formed an informal caucus to discuss their difficulties. causal adj. involving a cause The philosopher Plato believed there is a causal relationship between income inequality, on the one hand, and political discontent and crime, on the other hand: in his Laws he quantified his argument, contending that the income of the rich should be no more than five 71 ESSENTIAL WORDS FOR THE GRE times that of the poor, and he proposed policies to limit extremes of wealth and poverty. caustic adj. sarcastically biting; burning The columnist’s caustic comments on government policy did not win her any friends among government officials. celestial adj. concerning the sky or heavens; sublime Astronomers make use of the Doppler effect to measure the veloci ties and distance from Earth of stars and other celestial objects. Terms from the Arts, Sciences, and Social Sciences Doppler effect: change in the wavelength and frequency of a wave as a result of the motion of either the source or receiver of the waves centrifugal adj. moving away from a center As the empire expanded, there was an ever-increasing centrifugal stress as remote colonies sought autonomy. centripetal adj. moving or directed toward a center Astronomers calculate that the centripetal force exerted by the Earth’s gravity on the Moon will keep the Moon in orbit around the Earth for billions of years. champion v. to defend or support Robin Hood is famous for championing the underdogs of England. REVIEW 13 The correct answers are given on page 400. Matching Match each word with its definition. 1. cataclysm (A) involving a cause 2. catalyst (B) absolute 3. categorical (C) concerning the sky or heavens 4. caucus (D) moving toward a center 5. causal (E) violent upheaval 6. caustic (F) moving away from a center 7. celestial (G) defend or support 8. centrifugal (H) something causing change 9. centripetal (I) sarcastically biting 10. champion (J) smaller group within an organization 72 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. cataclysm catalysts categorical caucus causal caustic celestial centrifugal centripetal championed 1. Since its founding in 1966, the National Organization for Women (NOW) has __ the rights of women. 2. A study finds that people who exercise more tend to be healthier: Its authors raise the question, “Are these individuals healthier because they exercise—a _ link—or do they exercise more because they are healthier to begin with?” 3. “My position is _ ,” the CEO said. “I will not allow this company to be bought out.” 4. Biochemical __ , called enzymes, occur naturally in cells, changing one molecule into another. 5. Scientists say that the impact of a large meteor with the Earth would cause a _ that might end all life on our planet. 6. Japanese sociologists are studying the _ effects of a homogenous population on society. 7. Theoretically, a space station could be rotated to create artificial gravity as a result of __ force. 8. Gothic cathedrals place a great importance on light and a sense of space that seems to lift one toward the _ . 9. The parliament’s minority _ issued a report condemn ing government policy. 10. Wear protective gloves when working with __ substances in the laboratory. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Laboratory centrifuges make use of centrifugal force to separate substances according to their relative masses. 2. A study in Western Australia established a causal relationship between talking on a cell phone while driving and an increased likelihood of having an accident. 3. Catalysts for change on the school board blocked attempts to implement reforms. _ 4. In 1054, ancient Chinese astronomers recorded their observation of many important celestial events, such as the supernova that created the Crab Nebula. 73 ESSENTIAL WORDS FOR THE GRE 5. The newly discovered species of caucus is remarkable for its ability to survive for months with almost no water. __ UNIT 14 chasten v. to correct by punishment or reproof; to restrain or subdue The child’s behavior improved after she had been chastened by punishment. chicanery n. trickery; fraud The governor ordered an audit to investigate alleged financial chicanery. chivalry n. the qualities idealized by knighthood such as bravery and gallantry toward women Chivalry was rooted in Christian values, and the knight was bound to be loyal to Christian ideals; the Crusades enhanced this idea, as knights vowed to uphold Christianity against heathens. Terms from the Arts, Sciences, and Social Sciences chivalry: The code of chivalry dictated how a knight should act; this code enjoined the knight to defend the Church, make war against infidels, perform scrupulously feudal duties, and in general cham-pion virtue against evil. Crusades: military expeditions by Christians in the Middle Ages to win the Holy Land from the Muslims churlish adj. rude; boorish According to the chivalric code, a knight was never supposed to be churlish, especially toward noble ladies, to whom he was supposed to be unfailingly gentle and courteous. circuitous adj. roundabout According to Hindu philosophy, some souls take a circuitous path through many births to reach God. clairvoyant n. one who can predict the future; psychic Edgar Cayce was a famous clairvoyant who some people believe was able to go into a trance during which he was in touch with a spiritual realm. 74 ESSENTIAL WORDS FOR THE GRE clamor n. noisy outcry Over the past 12 years or so the clamor for better protection of the Earth’s rain forests has increased dramatically. Clamor is also a verb meaning to cry out noisily. The crowd clamored their disapproval of the plan. clique n. a small, exclusive group The principal of the high school is concerned that one clique of students is dominating the student council. cloister v. to confine; seclude The writer cloistered herself in a country house to finish her novel. The adjective cloistered means shut away from the world. The journalist described the large American philanthropic founda tions as arrogant, elitist, and cloistered. The noun cloister means a monastery or convent. coagulate v. thicken; congeal In normal individuals, blood begins to coagulate about 20 seconds after a wound is sustained, thus preventing further bleeding. REVIEW 14 The correct answers are given on page 400. Matching Match each word with its definition. 1. chasten (A) rude; boorish 2. chicanery (B) to confine 3. chivalry (C) trickery; fraud 4. churlish (D) roundabout 5. circuitous (E) correct by punishment 6. clairvoyant (F) noisy outcry 7. clamor (G) thicken; congeal 8. clique (H) qualities idealized by knighthood 9. cloister (I) small, exclusive group 10. coagulate (J) psychic 75 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. chastened chicanery chivalric churlish circuitous clairvoyant clamor clique cloistered coagulates 1. The idea of the gentleman is derived from the __ ideal that a man should be honorable, courteous, brave, and loyal, especially to women. 2. Egg white _ when heated. 3. The college newspaper is dominated by a _ of students who seem to be interested mainly in sports. 4. Over the last few years there has been a __ in the media about increased global warming. 5. After robbing the store, the thief took a _ route back to his house in case anyone was following him. 6. Mr. Jones tends to be _ before he has had breakfast. 7. The government’s budget deficit was covered up by __ ; several items were moved off-budget and unrealistically high revenues were projected. 8. The dictator of the small country was _ by the great power’s show of naval strength in the harbor of his country’s capi tal city. 9. We all said that Claire must be _ after she predicted the exact score of the football game. 10. The scholar lives a __ life among his books. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The chivalric code commanded knights to be brave, generous, and faithful. 2. Blood is coagulating around the wound. 3. The United States is an open, cloistered society that prides itself on being tolerant of a wide range of views. _ 4. Herb’s boss chastened him with a raise of $10,000 dollars a year. 5. Stage magicians often use clever tricks to make it appear that they are clairvoyant. __ 76 ESSENTIAL WORDS FOR THE GRE UNIT 15 coalesce v. to cause to become one President John F. Kennedy said that Americans must be vigilant so that the interests of business and the military do not coalesce and thus undermine those of society as a whole. coda n. concluding part of a literary or musical composition; some thing that summarizes or concludes The coda of the Danish composer Per Norgard’s Sixth Symphony seems to return to the serene sounds of the opening. codify v. to systematize The state legislature voted to codify regulations governing banking fraud. Codification is the noun. The most influential codification of civil law was the Napoleonic Code in France, which became the paradigm for law in the non-English-speaking countries of Europe and had a generally civilizing influence on most of the countries in which it was enacted. Codified is the adjective. Common law is the system of laws that originated in England; it is based on court decisions and on customs rather than on codified written laws. Terms from the Arts, Sciences, and Social Sciences civil law: a system of law developed from Roman law that is used in continental Europe, the U.S. state of Louisiana, and several other places. The basis of civil law is statute rather than custom and prec-edent, which are the basis of common law. Napoleonic Code: French legal code enacted by Napoleon in 1804. It made uniform the private law of France. common law: body of law that includes many nonstatutory laws based on many years of precedent derived from rulings by judges cognizant adj. informed; conscious; aware O. Henry’s “The Gift of the Magi” is a simple evocation of a young couple’s love for one another, a story in which a husband and wife in straitened circumstances each sacrifices to buy a Christmas present for the other, not cognizant of what the other is doing. 77 ESSENTIAL WORDS FOR THE GRE collage n. artistic composition of materials pasted over a surface; an assemblage of diverse elements The cubist Juan Gris is noted for his use of collage to create trompe l’oeil effects—the illusion of photographic reality. Terms from the Arts, Sciences, and Social Sciences cubist: a movement in art in the twentieth century that represented subjects from several points of view rather than from a single per-spective. Pablo Picasso and Georges Braques were the two most influential cubist artists. trompe l’oeil: a French term meaning “deceive the eye.” It refers to a style of portraying objects in a way that deceives the observer into believing it is the object itself. commensurate adj. proportional In the United States, malpractice suits have raised the cost of medicine because doctors must pay more for insurance, and thus increase their fees commensurately. compendium n. brief, comprehensive summary The Mozart Compendium: A Guide to Mozart’s Life and Music by H. C. Robbins Landon is a convenient reference for finding information about the life and music of Wolfgang Amadeus Mozart. complacent adj. self-satisfied Although Tom received an “A” on his midterm exam, Professor Donovan warned him not to become complacent since the work in the second term would be harder. complaisant adj. overly polite; willing to please; obliging Although France and Germany have a close relationship, neither would consider the other a complaisant ally. complement n. something that completes or makes up a whole Some people envision chess developing into a game between teams of humans and computers, each complementing the other and providing investigators with insight into the cognitive processes of each. 78 ESSENTIAL WORDS FOR THE GRE REVIEW 15 The correct answers are given on page 400. Matching Match each word with its definition. 1. coalesce (A) to systematize 2. coda (B) cause to become one 3. codify (C) assemblage of diverse elements 4. cognizant (D) overly polite 5. collage (E) proportional 6. commensurate (F) self-satisfied 7. compendium (G) something that makes up a whole 8. complacent (H) something that summarizes 9. complaisant (I) brief, comprehensive summary 10. complement (J) informed; conscious Fill-ins Choose the best word to fill in the blank in each sentence. coalesced coda codification cognizant collage commensurate compendium complacent complaisant complement 1. The final chapter of the scientist’s book is a __ in which the author reflects on her life and the important role science played in it. 2. The former chain-smoker describes herself as “Now a _, passive nonsmoker.” 3. A recent theory of how the Earth got its moon is that a very large object collided with the Earth about 4.5 billion years ago, produc ing iron-free material that gradually _ into the Moon. 4. One of the cornerstones of capitalism is the conviction that a work er’s rewards should be __ with his or her contribution. 5. Another important _ of modern civil law in addition to the Napoleonic Code is the German Civil Code (German Bürgerliches Gesetzbuch) that went into effect in the German Empire in 1900. 6. When one is studying a complex novel, it is helpful to have a _ that gives information about characters, setting, plot, etc. 7. Traditionally, white wine is considered a good __ to fish, whereas red wine is considered to be more suitable for meat. 8. The company’s CEO is worried that this quarter’s record profits will make his employees _ . 79 ESSENTIAL WORDS FOR THE GRE 9. It is important that a person accused of a crime be _ of his or her legal rights. 10. Modern Singapore is a multiethnic __ of Malays, Indians, Chinese, and many other groups. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. After three months of training, the battalion coalesced into a formidable fighting force. 2. The collage portrays the university’s history since its founding in 1766. 3. A widely held belief is that an employee’s pay should be commen surate with his or her qualifications and experience. _ 4. It is unwise to become complaisant and assume you will do well on the GRE just because you did well on the SAT or ACT. 5. During World War II German naval commanders radioed codas to their submarines containing the location of Allied ships. UNIT 16 compliant adj. yielding The young negotiator is trying to learn the skill of being open to proposals by the other side without seeming too compliant. compunction n. uneasiness caused by guilt The American psychiatrist Frank Pittman said, “Men who have been raised violently have every reason to believe it is appropriate for them to control others through violence; they feel no compunction over being violent to women, children, and one another.” concave adj. curving inward Concave lenses are used in glasses to compensate for myopia (nearsightedness). conciliatory adj. overcoming distrust or hostility The leader of the country made conciliatory statements assuring the world that his country did not intend to acquire nuclear weapons. concoct v. to invent The various human cultures have concocted a great many explanations to describe the beginning of the Earth, life, and humanity. 80 ESSENTIAL WORDS FOR THE GRE concomitant n. existing concurrently A rebuttal of the argument that homo sapiens’s higher cognitive functions could not be the result solely of evolution is that such abilities arose as concomitants of language, which gave early hominids a tremendous advantage over other species. Terms from the Arts, Sciences, and Social Sciences hominids: a hominid is any member of the biological family Hominidae (the “great apes”), which include, chimpanzees, gorillas, orangutans, and humans. condone v. to overlook voluntarily; forgive Mahatma Gandhi believed in the principle of ahimsa and refused to condone violence of any kind, even if used in a just cause. Terms from the Arts, Sciences, and Social Sciences Mahatma Gandi: twentieth-century Indian political leader who was instrumental in India’s gaining independence. Gandi is widely revered for his championing of nonviolence. ahimsa: a belief in Hinduism, Jainism, and Buddhism that advocates noninjury to all living beings confound v. to baffle; perplex; mix up Everyone but astrophysicists seems to be confounded by the question, “What happened before the Big Bang?” Terms from the Arts, Sciences, and Social Sciences Big Bang: a model of the origin of the universe stating that it began as infinitely compressed and has been expanding since then congenial adj. similar in tastes and habits; friendly; suited to The physicist Freeman Dyson has expressed his awe at how congenial the universe is to intelligent life and consciousness. conjugal adj. pertaining to marriage agreement The goal of the Bennett sisters in Jane Austen’s Pride and Prejudice is to find a suitable man to marry with whom they can live in conjugal happiness. 81 ESSENTIAL WORDS FOR THE GRE REVIEW 16 The correct answers are given on page 400. Matching Match each word with its definition. 1. compliant (A) curving inward 2. compunction (B) yielding 3. concave (C) baffle; perplex 4. conciliatory (D) to invent 5. concoct (E) to overlook voluntarily 6. concomitant (F) pertaining to marriage agreement 7. condone (G) overcoming distrust or hostility 8. confound (H) similar in tastes or habits 9. congenial (I) uneasiness caused by guilt 10. conjugal (J) existing concurrently Fill-ins Choose the best word to fill in the blank in each sentence. compliant compunction concave conciliatory concocted concomitant condoned confounded congenial conjugal 1. It appears that bureaucracies are today a necessary evil, a __ of modern society. 2. Amateur radio operators must be _ with federal laws as administered by the Federal Communications Commission. 3. The novel’s plot centers around a woman’s search for _ bliss. 4. One of the main goals of military training is to train soldiers to kill without __ . 5. The dating service matches men and women with _ interests. 6. For centuries, Fermat’s last theorem _ mathematicians. 7. The story Bud __ about having been abducted by Vegans in search of Earth’s greatest knowledge was not deemed by his professor an acceptable excuse for not handing in his term paper. 8. A lens with two _ surfaces is called a biconcave lens. 9. Some people believe that the use of nuclear weapons should never be _ . 10. After ten years of feuding with her neighbor, Mrs. Clampett decided enough was enough: as a __ gesture, she baked a cake and brought it over to her neighbor. 82 ESSENTIAL WORDS FOR THE GRE Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Since the enemy had made a number of conciliatory gestures, we had no option but to attack them in return. 2. The prison allows conjugal visits on weekends. 3. This amateur radio transceiver’s complex menu system has me completely confounded. _ 4. How the mind concocts new ideas is still a mystery to both psychologists and philosophers. 5. The optical telescope’s complex system of lenses contains both concave and convex lenses. __ UNIT 17 connoisseur n. a person possessing expert knowledge or training; a person of informed and discriminating taste The art connoisseur selected works by Van Gogh, Rembrandt, and Picasso for the exhibition. conscript n. person compulsorily enrolled for military service The position of NOW (The National Organization for Women) is that having male-only conscripts violates the principle of gender equality. Conscript is also a verb meaning to enroll a person for military service The French writer Andre Breton was conscripted into the artillery and had to put his medical studies in abeyance for the duration of World War I. Conscription is the noun. During the War of 1812, American political leaders considered national conscription to augment state militias, but Daniel Webster successfully argued before Congress that such a measure would be unconstitutional and thus the proposal was rejected. Terms from the Arts, Sciences, and Social Sciences War of 1812: a war fought between the British Empire and the United States from 1812 to 1815. The war ended in a stalemate. conscription: forced enlistment of people in the military. Modern conscription originated during the French Revolution. Daniel Webster: American lawyer and political leader during the period before the Civil War, which he tried to avert 83 ESSENTIAL WORDS FOR THE GRE consecrate v. to declare sacred In his Gettysburg Address, President Abraham Lincoln said of the soldiers who died in the Battle of Gettysburg in July, 1863: “We have come to dedicate a portion of that field, as a final resting place for those who here gave their lives that that nation might live... But, in a larger sense, we cannot dedicate—we cannot consecrate—we cannot hallow—this ground. The brave men, living and dead, who struggled here, have consecrated it, far above our poor power to add or detract.” contend v. to assert One of the most famous philosophers to argue for ethical relativ ism was the German Friedrich Nietzsche, who contended that the rightness of a particular action is dependent on the circumstances of the time and culture in which it occurs. Contention is a noun meaning an assertion. The study’s contention is that obesity is America’s biggest health problem. Terms from the Arts, Sciences, and Social Sciences ethical relativism: the view that there is no objective truth in issues of what is right or wrong Friedrich Nietzsche: nineteenth-century German philosopher. Nietzsche is best known for his doctrine of “the Superman,” which held that superior people should reject the “slave morality” of traditional Christianity in favor of a new morality centered on the individual. contentious adj. quarrelsome; causing quarrels When genetic engineering began in the 1970s, there was a contentious, and sometimes acrimonious, debate among scientists themselves about its dangers. Terms from the Arts, Sciences, and Social Sciences genetic engineering: the use of various methods to manipulate the DNA (genetic material) of cells to change hereditary traits or produce biological products 84 ESSENTIAL WORDS FOR THE GRE contiguous adj. touching; neighboring; connecting without a break There are forty-eight contiguous states in the United States of America. continence n. self-control; abstention from sexual activity Saint Augustine’s famous line “Give me chastity and continence, but not just now” is sometimes used to highlight the idea that action is desirable at some point, but not at present. contrite adj. very sorrowful for a wrong In sentencing the convicted man to a life sentence, the judge took into consideration the fact that he did not seem to be at all contrite about his crime. contumacious adj. disobedient; rebellious In the late eighteenth century, Great Britain tried unsuccessfully to put down the uprising against their rule by contumacious Americans, leading eventually to the establishment of a separate nation. conundrum n. riddle; puzzle with no solution The paradoxical statement “This statement is false” presents us with a conundrum. REVIEW 17 The correct answers are given on page 400. Matching Match each word with its definition. 1. connoisseur (A) self-control 2. conscript (B) to declare sacred 3. consecrate (C) touching; neighboring 4. contend (D) disobedient; rebellious 5. contentious (E) person compulsorily enrolled for military service 6. contiguous (F) quarrelsome 7. continence (G) person of informed and discriminating taste 8. contrite (H) puzzle with no solution 9. contumacious (I) very sorry for a wrong 10. conundrum (J) to assert 85 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. connoisseur conscripted consecrated contends contentious contiguous continence contrite contumacious conundrums 1. The appropriate function of literary criticism is a __ issue, even among critics themselves. 2. The art critic _ that the art of what are called less sophisticated cultures has an immediacy that is often lacking in civilized art, perhaps because it is less self-conscious, intellectual, and stylized. 3. In Israel, women as well as men are _ into the armed forces; however, men can be made to serve in combat, whereas women serve in a noncombat capacity. 4. One of the great __ in economics is how to achieve full employment without high inflation. 5. The monk pledged himself to a life of _ . 6. The dream holiday of the wine _ is a trip to France to visit famous chateaux in the region of Bordeaux. 7. The king ordered his army to quell the rebellion by his __ subject. 8. The landowner had the abandoned house _ to his house torn down. 9. The Cardinal _ the cathedral in 1676. 10. The __ sinner prayed every day for God to forgive her. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. There is nothing that a connoisseur of fine cigars hates more than smoking the finest Cuban cigars. 2. Conscription is seen by many people as a last resort to be used when an army must absolutely be raised and sufficient forces cannot be provided by a volunteer army. 3. Canada and the United States are contiguous. _ 4. The meeting was so contentious that the proposal was passed in a few minutes and without objection. 5. Cynthia contends that jogging is the best way to keep physically fit. __ 86 ESSENTIAL WORDS FOR THE GRE UNIT 18 convention n. practice widely observed in a group; custom; accepted technique or device The work of French artist Henri Rousseau demonstrates a naiveté that many people find more attractive than the sophistication of highly complex works that make use of all the conventions of their genre. Conventional is an adjective meaning customary or commonplace Guerrilla war presents a dilemma for framers of rules of war: should guerrilla fighters be subject to the same rules as those imposed on soldiers who fight conventional wars? Terms from the Arts, Sciences, and Social Sciences Henri Rousseau: French painter (1844–1910) famous for his paint-ings, often of jungles, done in a Primitive manner guerrilla war: a war involving small groups of soldiers that are flex-ible and mobile. In guerrilla war there is no front line as there is in conventional war. converge v. to approach; come together; tend to meet Although the People’s Republic of China and India are rivals in many ways, in certain areas their interests converge. convex adj. curved outward The term for a lens with one convex and one concave side is “convex-concave.” convivial adj. sociable One of the jobs of an ambassador is to provide a convivial atmo-sphere for diplomats to meet. convoluted adj. twisted; complicated Unraveling the convoluted genetic code is one of the great achievements of modern science. copious adj. abundant; plentiful The copious rainfall was welcomed by farmers in the parched land. 87 ESSENTIAL WORDS FOR THE GRE coquette n. woman who flirts After she had played the part of a coquette in the college play, Pam’s boyfriend felt that he needed to remind her that real life was quite different from the theater. cornucopia n. horn overflowing with fruit and grain; state of abundance The U.S. economy has produced a cornucopia of employment opportunities. cosmology n. study of the universe as a totality; theory of the origin and structure of the universe Albert Einstein downplayed the strength of the evidence for quantum theory because a universe governed by laws that are inconsistent in their application was not congruent with his personal cosmology. Cosmos is a noun meaning the physical universe regarded as a totality. Shakespeare embodies the incredible confidence and vitality of Renaissance artists and writers, depicting the entire cosmos, not intimidated by its vastness. Cosmic is an adjective meaning relating to the physical universe, especially as distinct from Earth, and suggests infinite vastness. The gods of ancient Greece were concerned not only with cosmic events, but also with the ordinary events of everyday life. Terms from the Arts, Sciences, and Social Sciences quantum theory: a theory in physics based on the principle that matter and energy have the properties of both particles and waves Renaissance: the period of revival in art and learning that occurred in Europe during the fourteenth to the seventeenth century covert adj. hidden; secret The CIA gathers information about foreign intelligence through many means, including covert ones. 88 ESSENTIAL WORDS FOR THE GRE REVIEW 18 The correct answers are given on page 401. Matching Match each word with its definition. 1. convention (A) complicated 2. converge (B) curved outward 3. convex (C) study of the universe 4. convivial (D) to approach; come together 5. convoluted (E) horn overflowing with fruit and grain 6. copious (F) practice widely observed in a group 7. coquette (G) abundant 8. cornucopia (H) woman who flirts 9. cosmology (I) hidden; secret 10. covert (J) sociable Fill-ins Choose the best word to fill in the blank in each sentence. conventions converges convex convivial convoluted copious coquette cornucopia cosmology covert 1. One need not know anything of medieval Christian __ to appreciate the great Gothic cathedrals, edifices that are a supreme legacy of that age. 2. A work of art may seem contrived to a person who is unfamiliar with the _ of the form of art he is observing. 3. Fyodor Dostoevsky’s The Possessed has a fascinating, though _ , plot. 4. __ lenses are used to correct farsightedness. 5. Politicians are often _ individuals who are comfortable with a wide variety of people. 6. Stella takes _ notes in all of her classes. 7. In Robert Frost’s famous poem “The Road Not Taken” the speaker must choose which path to take after the one he is on __ with another. 8. The plainclothes detective took part in a _ operation. 9. Sarah has a reputation as a bit of a _ . 10. Tropical rain forests contain a __ of plant substances that have proven to be effective medicines. 89 ESSENTIAL WORDS FOR THE GRE Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Every Sunday morning Steve buys half a dozen fresh coquettes at the corner bakery. 2. After the technician replaced the damaged convex lens in my 50 reflector telescope, it worked fine. 3. The government launched the covert operation amid great fanfare. _ 4. The convivial host helped make sure everyone enjoyed the party. 5. Cosmology has helped poetry to become more popular among the public. UNIT 19 covetous adj. desiring something owned by another The astronomer is covetous of the time that his colleague gets for research using the Hubble Space Telescope. Covet is the verb. The latest model cell phone is designed to make people covet it so much that they go out and buy it even though their present phone is perfectly adequate. cozen v. to mislead by trick or fraud; deceive The writer H. L. Mencken pointed out that a common strategy of politicians is to cozen the people by exaggerating the seriousness of a problem and then offering a solution that, conveniently, only they can provide. craven adj. cowardly In the Hindu epic poem the Bhagavad-Gita, Lord Krishna warns the hero, who is reluctant to fight, that refusing to fight would be a craven act. credence n. acceptance of something as true One of the lessons in Aesop’s fable “The Shepherd Boy and the Wolf ” is that if a person “cries wolf ” too many times without real danger being present (that is, raises too many false alarms) people will be less likely to give credence to future alarms raised by that person. 90 ESSENTIAL WORDS FOR THE GRE credo n. statement of belief or principle; creed The credo of Google is “Don’t be evil.” daunt v. to discourage; intimidate; dishearten Do not let the difficulty of learning the 800 words in Essential Words for the GRE daunt you. Daunting is an adjective that means discouraging or disheartening. Earning a Ph.D. is a daunting task, but it can be done. The adjective dauntless means fearless. dearth n. scarcity In his book The Affluent Society, published in 1958, the economist J. K. Galbraith pointed out that in America affluence is located disproportionately in the private sector, leaving a dearth of resources available for the public sector. debauchery n. corruption The prince lived a life of debauchery until he discovered a spiritual dimension to life. decorum n. proper behavior When addressing the nation, the president generally has an air of decorum. The adjective is decorous. defame v. to malign; harm someone’s reputation The ancient Greek philosopher Socrates was defamed as a teacher who corrupted the morals of his students. 91 ESSENTIAL WORDS FOR THE GRE REVIEW 19 The correct answers are given on page 401. Matching Match each word with its definition. 1. covetous (A) cowardly 2. cozen (B) scarcity 3. craven (C) to intimidate; discourage 4. credence (D) desiring something owned by another 5. credo (E) acceptance of something as true 6. daunt (F) to harm someone’s reputation 7. dearth (G) corruption 8. debauchery (H) to mislead by trick or fraud 9. decorum (I) statement of belief or principle; creed 10. defame (J) proper behavior Fill-ins Choose the best word to fill in the blank in each sentence. covets cozens craven credence credo daunting dearth debauchery decorum defaming 1. Because so many young men were killed in the war, there is a __ of potential husbands for the young women of the village. 2. The general called his advisor’s suggestion that he surrender “the _ proposal of a coward.” 3. The sales pitch _ potential customers by omitting the fact that the product has been superseded by far superior products available at the same price. 4. To make the task of writing the book less __ , the author broke the task into a number of small tasks he could do one at a time. 5. I admit that the professor’s statement is baffling; however, it should be given some _ because of his towering reputation in the field. 6. The principal reminded the students to conduct themselves with _ during the guests’ visit. 7. The amateur radio operator __ a new ICOM 7800 high-frequency transceiver costing more than $10,000, but his wife says he can afford only the Kenwood 570D costing about $1,000. 8. The novelist follows the _ that plot proceeds from character. 9. The students went to Fort Lauderdale for a week of _ . 92 ESSENTIAL WORDS FOR THE GRE 10. The journalist was sued for __ a police officer in his article. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The unscrupulous newspaper regularly defames public figures to boost circulation. 2. The candidate was elected governor on a platform of moral purity and debauchery. 3. The company’s credo is “The buyer is king.” _ 4. No one called the boxer craven after he asked the referee to stop the bout; he had been knocked down six times and was bleeding profusely. 5. A trial should be conducted with decorum. __ UNIT 20 default v. to fail to act Economists have pointed out the danger of using government money to help banks in danger of defaulting on a loan: such help might encourage banks to take excessive risks on the future, knowing they will be “bailed out” by the government. deference n. respect; regard for another’s wish There was a movement to condemn slavery among some of the writers of the Declaration of Independence, but despite many misgivings, the proposal was dropped in deference to the objections of a number of people. The verb defer means to submit to the wishes of another due to respect or recognition of the person’s authority or knowledge. The young lawyer deferred to the view of the senior partner in the law firm. defunct adj. no longer existing Skeptics have been prognosticating that Moore’s Law, which says computer processing power doubles every 18 months, will soon become defunct, but the ingenuity of engineers, coupled with commercial incentives, has so far succeeded in preventing the law from being invalidated. 93 ESSENTIAL WORDS FOR THE GRE delineate v. to represent or depict Quantum theory led to the formulation of the uncertainty principle, which was delineated in 1937 by Werner Heisenberg. Terms from the Arts, Sciences, and Social Sciences uncertainty principle: the statement in quantum mechanics stating that it is impossible to measure accurately two properties of a quan-tum object, such as its position and momentum demographic adj. related to population balance Demographic trends in many European countries indicate that in the next generation there will be relatively fewer working people to support retired people. Demography is the study of human population. Demography makes use of the knowledge of other fields such as geography and statistics. A demographer is one who studies human population. If, beginning in the mid-twentieth century, many governments in the world had not taken steps to promote birth control among their citizens, causing a diminution in the birth rate, demographers say the world would now have a much greater population than it does. demotic adj. pertaining to people Walt Whitman is considered by many to be a quintessentially American poet, a poet who celebrated the glory of the ordinary person; one critic praised him as a poet who was able to “make the demotic sing.” demur v. to express doubt The Supreme Court’s decision was not unanimous; one justice demurred, saying that the majority decision used specious reasoning. denigrate v. to slur someone’s reputation According to a recent biography of Napoleon Bonaparte, the famous leader felt a need to denigrate women. denizen n. an inhabitant; a regular visitor The U.S. Census Bureau has the responsibility of collecting information about the denizens of the United States. 94 ESSENTIAL WORDS FOR THE GRE denouement n. outcome; unraveling of the plot of a play or work of literature The book tells the story of what was for Europe a rather embar rassing denouement to the Crusades. REVIEW 20 The correct answers are given on page 401. Matching Match each word with its definition. 1. default (A) to express doubt 2. deference (B) respect; regard for another’s wish 3. defunct (C) an inhabitant 4. delineate (D) relating to the study of human population 5. demographic (E) to slur someone’s reputation 6. demotic (F) failure to act 7. demur (G) to represent or depict 8. denigrate (H) outcome 9. denizen (I) pertaining to people 10. denouement (J) no longer existing Fill-ins Choose the best word to fill in the blank in each sentence. default deference defunct delineated demographic demotic demurred denigrated denizens denouement 1. The __ of a novel by crime writer Mickey Spillane is generally very violent. 2. Data gathered in the census provides planners with important _ information. 3. On his first scuba dive, Kenny was happy to find that the _ of the sea did not appear to be hostile. 4. Solid-state electronic equipment has made vacuum tube equip ment __ in most areas other than very specialized applications. 5. The political science professor _ a plan to reorganize the United Nations to make it better reflect the realities of the contemporary world. 95 ESSENTIAL WORDS FOR THE GRE 6. The chairperson asked for a vote on the proposal; since no one _ , it passed unanimously. 7. The professor never watches movies, which he calls “__ entertainment for the semiliterate.” 8. Rather than _ on her car loan payments after losing her job, Ruth worked out an agreement that allowed her to make lower monthly payments. 9. In Victorian times servants were expected to show great _ to their employers. 10. In many societies women have been __ as inferior to men. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The denouement at the beginning of the story really caught my interest. 2. The book delineates the characters clearly. 3. The demotic king was deposed in the popular uprising. _ 4. Demographic trends in Japan show that the proportion of old people to young people is increasing. 5. Since Singapore became an independent country in 1965, its denizens have become among the wealthiest in the world. APPLYING YOUR KNOWLEDGE—11–20 1. During World War II, President Truman faced (i) _ when he had to choose between two morally (ii) _____ alternatives: allow the war against Japan to continue at the cost of hundreds of thousands of lives, or bring it to a swift end by inflicting catastrophic damage on Japan with the atomic bomb, forcing it to (iii) _. Blank (i) Blank (ii) Blank (iii) a conundrum repugnant attack a cataclysm relative capitulate a canard analogous implode Explanation: An important clue is the phrase “had to choose,” which suggests that President Truman faced a difficult choice. This makes conundrum (puzzle with no solution) an appropriate choice for Blank (i). 96 ESSENTIAL WORDS FOR THE GRE Repugnant (distasteful; offensive) is a good choice for Blank (ii) because it helps create a sentence describing the difficult choice between two alternatives, both of which presented serious moral difficulties. For Blank (iii) capitulate (surrender) is the best choice. This is signaled by the phrase “bring it [the war] to a swift end.” The correct answers are a conundrum, repugnant, and capitulate. 2. NASA’s Apollo missions in the late 1960s and early 1970s may have played a significant role in (i) _ the environmental movement by helping to put Earth’s ecosystem in a (ii) _ perspective: photographs of (iii) __ Earth from space per-haps encouraged many people to reflect on the folly of disrupting such a beautiful and intricately interrelated biosphere. Blank (i) Blank (ii) Blank (iii) hastening the demise of hypothetical a pristine-looking alleviating problems in cosmic an apposite crystallizing support for canonical an antediluvian Explanation: The independent clause after the colon describes photographs of Earth from space. This signals that the best choice for Blank (ii) is cosmic (relating to the universe as distinct from Earth). An important clue to Blank (iii) is “such a beautiful and intricately inter-related biosphere,” which suggests that Earth was pristine-looking. The answer to Blank (i) can now be selected. The sentence says that pho-tographs of Earth from space may have encouraged people to “reflect on the folly of disrupting such a beautiful and intricately interrelated biosphere” of Earth, so it makes sense that the Apollo missions had a part in crystallizing support for the environmental movement. The correct answers are crystallizing support for, cosmic, and a pristine-looking. UNIT 21 deride v. to mock Innovation often requires challenges to orthodox thinking; for example, in the late 1960s, scientists from the U.S. Department of Defense’s Advanced Research Projects Agency presented their idea of a vast network of computers to leading scientists from IBM and AT&T—companies with innumerable research breakthroughs to their credit—and were derided as impractical visionaries. 97 ESSENTIAL WORDS FOR THE GRE derivative n. something derived; unoriginal The drug morphine—considered by doctors to be one of the most effective analgesics—is the principal derivative of opium, which is the juice in the unripe seed pods of the opium poppy. Derivative is also an adjective. The critic dismissed the new novel as dull and derivative. The verb derive means obtained from another source. One of the attempts to create a lingua franca resulted in Esperanto, a synthetic language whose vocabulary is created by adding various affixes to individual roots and is derived from Latin and Greek, as well as Germanic and Romance languages. Terms from the Arts, Sciences, and Social Sciences lingua franca: a language used as a medium of communication between peoples of different languages affixes: word elements that are affixed to the beginning (prefixes) or the end (suffixes) of words to refine the meaning or change the word’s grammatical form Romance languages: the Romance languages, or Indo-European lan-guages that descended from Vulgar Latin, include Italian, Romanian, Spanish, Portuguese, French, their many dialects, and the pidgins and creoles (mixed languages) that developed from them. The term “Romance” is from Vulgar Latin romanice loqui (vulgar languages derived from Latin). desiccate v. to dry completely The dry desert air caused the bodies of the dead animals to desiccate quickly. desuetude n. state of disuse NASA is considering a plan to refurbish booster rockets from the Apollo Program that have fallen into desuetude. desultory adj. random; disconnected; rambling The jury had difficulty following the witnesses’ desultory testimony. deterrent n. something that discourages or hinders During the Cold War, the United States maintained a large number of nuclear weapons as a deterrent to aggression by the Soviet Union and its allies. 98 ESSENTIAL WORDS FOR THE GRE detraction n. the act of taking away; derogatory comment on a person’s character The writer responded in a letter to the critic’s long list of detractions about his book. diaphanous adj. transparent; fine-textured; insubstantial; vague In World War II, many soldiers went to war with diaphanous dreams of glory, but found instead horror and death. diatribe n. bitter verbal attack The speaker launched into a diatribe against what he called “the evils of technology.” dichotomy n. division into two usually contradictory parts The philosopher is a dualist who argues that there is a dichotomy between the mind and physical phenomena. Terms from the Arts, Sciences, and Social Sciences dualist: one who believes in dualism, the theory that two basic enti-ties constitute reality (such as mind and matter or good and evil) REVIEW 21 The correct answers are given on page 401. Matching Match each word with its definition. 1. deride (A) something that discourages 2. derivative (B) bitter verbal attack 3. desiccate (C) to dry completely 4. desuetude (D) random; disconnected 5. desultory (E) the act of taking away 6. deterrent (F) unoriginal 7. detraction (G) division into two contradictory parts 8. diaphanous (H) state of disuse 9. diatribe (I) transparent; fine-textured 10. dichotomy (J) to mock 99 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. derided derivative desiccated desuetude desultory deterrent detraction diaphanous diatribe dichotomy 1. In his book Supernature the British biologist Lyall Watson argues that the __ between nature and the supernatural exists more in the human mind than in reality. 2. The two areas of the room are separated only by a _ curtain. 3. Scientists are studying the _ bones to see if they are the remains of a person. 4. Some studies suggest that capital punishment is a __ against murder. 5. The critics _ the movie as “a waste of $100 million dollars.” 6. The poet describes his work as _ because it draws on the work of many other poets. 7. The two men walked along the beach, engaged in __ conversation. 8. The rise of Irish nationalism has probably helped bring the Irish language back from the _ it was falling into in the nineteenth century. 9. The prime minister’s _ against foreign influence in the country lasted three hours. 10. The only __ from the excellence of the climate is the rainy winter. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. In many traditional societies women wear diaphanous clothing to hide their bodies. 2. Early attempts to communicate by the use of electromagnetic waves were derided by many people as ridiculous. 3. The book is a long, desultory narrative recounting its author’s life. _ 4. The senator’s speech was a diatribe against the increasing influence of government in everyday life. 5. South Korea maintains a large military force as a deterrent against North Korean aggression. 100 ESSENTIAL WORDS FOR THE GRE UNIT 22 diffidence n. shyness; lack of confidence As a result of the strength of his opposition to the Vietnam War Senator Eugene McCarthy overcame his diffidence and ran against President Lyndon Johnson for the Democratic nomination for president. diffuse v. to spread out The idea of equality and liberty diffused through society after the French Revolution. Diffuse is also an adjective meaning wordy; rambling; spread out. This essay is so diffuse it is difficult to follow its central argument. digression n. act of straying from the main point The novel Zen and the Art of Motorcycle Maintenance by Robert M. Pirsig contains many fascinating digressions from the main story that discuss topics such as Platonic philosophy. Terms from the Arts, Sciences, and Social Sciences Platonic: refers to the philosophy of Plato, an ancient Greek philoso-pher who held that both actual things and ideas such as beauty and truth are copies of transcendent ideas The adjective platonic (with a small “p”) means spiritual, without sen-sual desire, or theoretical. dirge n. funeral hymn The music critic described the movement of the symphony portray ing the hero’s last days as “dirgelike.” disabuse v. to free from a misconception The chairman of the Federal Reserve used his testimony before Congress to disabuse his audience of the idea that the business cycle had been eliminated by the unprecedented period of prosperity. discerning adj. perceptive; exhibiting keen insight and good judgment Discerning movie critics have praised the work of producer Stanley Kubrick, who produced such excellent films as 2001, Dr. Strangelove, A Clockwork Orange, and Lolita. Discern is a verb that means to perceive something obscure. 101 ESSENTIAL WORDS FOR THE GRE Superficially, expressionism can appear to be unrealistic because of its extreme distortion of reality, but upon closer examination, an inner psychological reality can often be discerned. Terms from the Arts, Sciences, and Social Sciences expressionism: an artistic style in which the artist expresses emo-tional experience as opposed to his or her view of the external world. Expressionists often use distortion and exaggeration. El Greco, Van Gogh, and Edward Munch are examples of expressionist artists. discomfit v. to make uneasy; disconcert The young man was discomfited being the only male in the play. discordant adj. not in tune In a pluralistic society there exists a cacophony of discordant voices, each shouting to be heard. discredit v. to dishonor; disgrace; cause to be doubted The candidate’s attempt to discredit his opponent by spreading damaging rumors about him failed. discrepancy n. difference between The book studies the discrepancy in values and outlook between men who fought in the war, whether voluntarily or not, and those who remained civilians. REVIEW 22 The correct answers are given on page 401. Matching Match each word with its definition. 1. diffidence (A) to free from a misconception 2. diffuse (B) to spread out 3. digression (C) to make uneasy 4. dirge (D) act of straying from the main point 5. disabuse (E) difference between 6. discerning (F) shyness 7. discomfit (G) not in tune 8. discordant (H) funeral hymn 9. discredit (I) to dishonor; disgrace 10. discrepancy (J) exhibiting keen insight and good judgment 102 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. diffidence diffuse digressions dirge disabuse discerning discomfited discordant discredited discrepancy 1. One year of medical school was enough to __ Steve of the idea that medical school is a “piece of cake.” 2. Auditors are investigating the _ between the company’s stated earnings and its projected earnings based on sales. 3. The band played a _ at the soldier’s funeral. 4. Some readers are annoyed by the long __ on geology and other scientific subjects in Kim Stanley Robinson’s Mars trilogy; other readers, however, find them fascinating, illuminating and beautifully written. 5. Historians of science study theories that have become accepted by modern science as well as those that have been _ . 6. One of the aims of the English literature course is to help students become _ readers. 7. The study suggests that women do not find __ in men to be an attractive quality. 8. The intravenous drug will _ through the patient’s body in about 20 minutes. 9. Many people are _ by the idea of their own death. 10. The governor traveled around the state listening to the __ views on the controversial issue. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. A woman of discerning taste in literature, Jane mainly reads works by William Shakespeare, William Blake, Henry James, and Willa Cather. 2. There is a great discrepancy between the performance of the best student in the class and the worst student. 3. Good writers often use digression to help keep their discussion on the main topic. _ 4. The opposition party tried to discredit the leadership of the ruling party by charging it with corruption. 5. A week doing the house cleaner’s chores disabused Cindy of the idea that the house cleaner has an easy job. __ 103 ESSENTIAL WORDS FOR THE GRE UNIT 23 discrete adj. constituting a separate thing; distinct Like the physicist, the abstract artist strives to identify the discrete elements of reality and to understand how they interact. discretion n. quality of showing self-restraint in speech or actions; circumspection; freedom to act on one’s own In nineteenth-century Britain gentlemen were expected to behave with discretion. disingenuous adj. not candid; crafty When a person starts a sentence, “I don’t mean to appear disingenuous,” one might be tempted to suspect that the person is being just that. disinterested adj. unprejudiced; objective The newspaper reporter looked for disinterested witnesses to the events so that she could get an objective account of what had happened. disjointed adj. lacking order or coherence; dislocated The technique of telling a story through a disjointed narrative is a technique best left to masters of the modern novel such as James Joyce and William Faulkner. dismiss v. put away from consideration; reject Investigators dismissed the man’s account of a visit to another planet aboard an alien spacecraft as the product of an overactive imagination. disparage v. to belittle Though sometimes disparaged as merely an intellectual game, philosophy provides us with a method for inquiring systematically into problems that arise in areas such as medicine, science, and technology. disparate adj. dissimilar Many technological projects are interdisciplinary, requiring a knowl edge of fields as disparate as physics and biology. Disparity is a noun meaning the condition of being unequal or unlike. The huge income disparity in the world is clearly illustrated by the fact that the assets of the world’s two hundred richest people exceed the combined income of 41 percent of the world’s population. 104 ESSENTIAL WORDS FOR THE GRE dissemble v. to pretend; disguise one’s motives “Miss,” the prosecutor said, “I believe you are dissembling. I want you to tell me the whole truth about what happened that night.” disseminate v. to spread; scatter; disperse While belief in reincarnation appeared as doctrine first in India and was disseminated throughout Asia by Buddhism, it is interest ing that it was accepted by the most influential philosophy of the West, Platonism, and by some important early Christian thinkers, such as the theologian Origen. Terms from the Arts, Sciences, and Social Sciences Platonism: the philosophy of Plato, which holds that both actual things and ideas such as beauty and truth are copies of transcendent ideas REVIEW 23 The correct answers are given on page 401. Matching Match each word with its definition. 1. discrete (A) lacking order or coherence 2. discretion (B) unprejudiced; objective 3. disingenuous (C) dissimilar 4. disinterested (D) to belittle 5. disjointed (E) to spread; disperse 6. dismiss (F) circumspection 7. disparage (G) to pretend 8. disparate (H) not candid; crafty 9. dissemble (I) constituting a separate thing; distinct 10. disseminate (J) to reject Fill-ins Choose the best word to fill in the blank in each sentence. discrete discretion disingenuous disinterested disjointed dismissed disparaged disparate dissembled disseminated 1. The historian tries to take a __ view of how the United States got involved in the Vietnam War. 105 ESSENTIAL WORDS FOR THE GRE 2. The great increase in travel in modern times makes it difficult to determine how and where a disease originated, as well as how it was _ , so that measures can be taken to mitigate its effects. 3. The novel’s narrative is so _ that many readers have trouble following it. 4. Scientific laws identify a common fundamental element in seem ingly __ phenomena. 5. The historian describes her method as “not so much the study of _ events but rather the study of relationships between those events.” 6. The judge _ the evidence as not relevant to the case at hand. 7. The school lets its teachers use considerable __ in designing lessons for students. 8. The investigating committee ruled that the governor “had been _” in not providing important information to them. 9. The noted director Stanley Kubrick, who turned down the chance to go to college when he was seventeen, _ formal education, saying, “I never learned anything at all at school.” 10. The girl __ when her date asked if she had ever been kissed. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The technician dissembled the computer to find out what was wrong with it. 2. The battalion’s commander told his men to use their own discretion in selecting enemy targets. 3. Diplomats must be discrete to do their job effectively. _ 4. The disingenuous student must work harder than other students to make up for his lack of ability. 5. The writer was proud to have her work disparaged by leading critics. __ UNIT 24 dissident n. person who disagrees about beliefs, etc. Some of the most notorious concentration camps in history were the Gulag camps used by the Soviet Union to control dissidents. 106 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences Gulag: forced-labor prison camps in the Soviet Union. Established in the 1920s, the Gulag system had 476 camps throughout the country used to imprison people considered a threat to the state. dissolution n. disintegration; debauchery Some philosophers maintain that the dissolution of the body does not mean the destruction of the mind. dissonance n. discord; lack of harmony In psychology, the term “cognitive dissonance” refers to a conflict resulting from inconsistency between one’s beliefs and one’s actions. For example, a soldier who believes that all killing is immoral but is forced to kill by his superiors might experience cognitive dissonance. distend v. to expand; swell out People in an advanced stage of starvation often have distended bellies. distill v. extract the essential elements In his book Men of Ideas: Some Creators of Contemporary Philosophy, Bryan Magee manages to distill the essence of leading thinkers such as W. V. Quine, John Searle, Iris Murdoch, and Noam Chomsky. distrait adj. inattentive; preoccupied The chairperson became distrait because his secretary was not sitting in her usual position on his right. diverge v. to vary; go in different directions from the same point A famous line in American poetry is from Robert Frost’s “The Road Not Taken”: Two roads diverged in a wood, and I— I took the one less traveled by .... Divergence is the noun. Psychological tests show that there is a wide divergence between citizens of different countries in how much importance they place on the virtue of justice, on the one hand, and the virtue of mercy, on the other hand. 107 ESSENTIAL WORDS FOR THE GRE divest v. to strip; deprive; rid The candidate for secretary of defense pledged to divest himself of the shares he held in defense-related companies. divulge v. to make known something that is secret Under the Geneva Conventions, prisoners of war cannot be tortured and forced to divulge information. doctrinaire adj. relating to a person who cannot compromise about points of a theory or doctrine; dogmatic; unyielding The doctrinaire Marxists say that capitalism is merely a temporary phenomenon on the road to socialism. REVIEW 24 The correct answers are given on page 402. Matching Match each word with its definition. 1. dissident (A) unyielding; dogmatic 2. dissolution (B) to extract the essential elements 3. dissonance (C) disintegration 4. distend (D) to strip; deprive 5. distill (E) to expand; swell out 6. distrait (F) to go in different directions from the same point 7. diverge (G) inattentive; preoccupied 8. divest (H) to make known something secret 9. divulge (I) person who disagrees about beliefs 10. doctrinaire (J) lack of harmony Fill-ins Choose the best word to fill in the blank in each sentence. dissidents dissolution dissonance distended distill distrait diverged divested divulge doctrinaire 1. How the poet John Keats was able to __ so much beauty and wisdom into his poetry remains a mystery. 2. The members’ vote of no confidence in the ruling government led to the _ of parliament. 3. The man who ate more than fifty hot dogs to win the hot dog eating competition gained seven pounds and had a _ belly for a few days. 108 ESSENTIAL WORDS FOR THE GRE 4. According to the child psychologist, __ between family and school is normal. 5. The psychologist’s patient _ himself of the secrets he had been carrying within for 30 years. 6. During World War I many people in the United States considered conscientious objectors to be radical _ . 7. Pam’s life __ from Bob’s after they graduated from college in 1971; he was drafted and sent to fight in Vietnam and she went to Paris to do a Ph.D. in French literature. 8. The guest seemed to be melancholy and _ , so I asked him what was troubling him. 9. Companies that are not publicly listed and have no major debt normally do not need to _ much about their sales and other matters to financial markets. 10. “If the world is lucky enough to enjoy peace, it may even one day make the discovery, to the horror of __ free-enterprisers and doctrinaire planners alike, that what is called capitalism and what is called socialism are both capable of working quite well.” (J. K. Galbraith, American economist) Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Peter and Paul disagree on most political issues, but their views diverge on religion. 2. During the holiday season the story distends its opening hours to accommodate customers. 3. The Bhagavad-Gita, one of the holy books of the Hindus, is a long poem that distills much of the teachings of Hinduism. _ 4. The foreign government gave dissidents in the country support in the hope of destabilizing the country. 5. The moderate urged her political party to adopt a less doctrinaire position on the issue. UNIT 25 document v. to provide with written evidence to support The insurance company asked Debbie to document her claim with letters from the doctors who treated her for her condition. 109 ESSENTIAL WORDS FOR THE GRE doggerel n. poor verse In his book Poetic Meter and Poetic Form, the literary critic Paul Fussell quotes this bit of doggerel from a U.S. Army latrine during World War II: Soldiers who wish to be a hero Are practically zero. But those who wish to be civilians, Jesus, they run into millions. dogmatic adj. stating opinions without proof Since every case is unique, jurists must not be dogmatic in applying precedents to make their decision, but instead must base their decision on a combination of such precedents and the facts of the case at hand. Dogma is a noun meaning a belief asserted on authority without evidence. Religions whose dogma specifies a time of the creation of the world have found difficulty in reconciling their view of creation with that of modern science. Terms from the Arts, Sciences, and Social Sciences The original meaning of dogma was “that which seems good.” In Christian theology it came to mean truths known by divine revelation and taught by the Church. dormant adj. inactive There is a considerable body of evidence showing that many diseases, such as ulcers, asthma, and hypertension have a large psychological component; the working hypothesis is that they represent manifestations of dormant emotional disturbances. dross n. waste; worthless matter; trivial matter One of the ways the dross among blogs on the Internet are filtered out from the worthwhile ones is through links good blogs provide to other good blogs. Terms from the Arts, Sciences, and Social Sciences blog: short for weblog, an online journal 110 ESSENTIAL WORDS FOR THE GRE dupe v. to deceive; trick “In friendship, as well as in love, the mind is often duped by the heart.” (Philip Dormer Stanhope) ebullient adj. exhilarated; enthusiastic The ebullient candidate for president appeared before his supporters to announce that he had won in a landslide. eclectic adj. selecting from various sources Neo-Platonism—an eclectic third-century synthesis of Platonic, Pythagorean, Aristotelian, Stoic, and Jewish philosophy—was an essentially mystical belief that a person can achieve spiritual emanci pation through union of the soul with the ultimate source of existence. Terms from the Arts, Sciences, and Social Sciences Platonic: refers to the philosophy of Plato, an ancient Greek philoso-pher who held that both actual things and ideas such as beauty and truth are copies of transcendent ideas Pythagorean: refers to the philosophy of Pythagoras, a sixth-century b.c. philosopher and mathematician. Pythagoras described reality in terms of arithmetical relationships. Aristotelian: refers to the philosophy of Aristotle, an ancient Greek sci-entist and philosopher whose teaching had a great influence on West-ern thought, especially in the areas of logic, metaphysics, and science Stoic: refers to Stoicism, a philosophy of ancient Greece that taught that the highest good is virtue, which is based on knowledge. The Stoics believed that the wise live in harmony with Divine Reason that governs nature and are indifferent to suffering and the chang-ing fortunes of life. mystical: related to mysticism, the practice of putting oneself into direct relation with God, the absolute, or any unifying principle of life effervescence n. state of high spirits or liveliness; the process of bubbling as gas escapes Effervescence occurs when hydrochloric acid is added to a block of limestone. The adjective is effervescent. A person who believes himself to be physically unattractive might develop an effervescent personality as a compensation for his perceived deficiency. 111 ESSENTIAL WORDS FOR THE GRE effete adj. depleted of vitality; overrefined; decadent In 1969, U.S. Vice President Spiro T. Agnew denounced people protesting against the Vietnam War: “A spirit of national masochism prevails, encouraged by an effete corps of impudent snobs who characterize themselves as intellectuals.” REVIEW 25 The correct answers are given on page 402. Matching Match each word with its definition. 1. document (A) to deceive 2. doggerel (B) poor verse 3. dogmatic (C) provide with written evidence to support 4. dormant (D) state of high spirits 5. dross (E) selecting from various sources 6. dupe (F) stating opinions without proof 7. ebullient (G) exhilarated 8. eclectic (H) inactive 9. effervescence (I) depleted of vitality 10. effete (J) worthless matter Fill-ins Choose the best word to fill in the blank in each sentence. documented doggerel dogmatic dormant dross duped ebullient eclectic effervescent effete 1. Clinical psychologists provide treatment for psychological disor ders, and today can choose from an array of psychotherapies; often they are __ , choosing elements of therapies best suited to each particular case. 2. Police investigators _ the case with photographs and recorded interviews. 3. The country’s leaders _ the people into thinking it was necessary to declare war. 4. Even the __ of a great poet like John Milton is interest ing. 5. It is interesting to observe how some traditions remain strong, while others gradually become _ . 112 ESSENTIAL WORDS FOR THE GRE 6. The philosopher Bertrand Russell once observed that people are often most _ about things that it is least possible to be certain about. 7. Julia’s __ personality makes her one of the college’s most popular students. 8. The doctor suspected that the patient had once contracted malaria, but that the disease was now _ . 9. One of the traditional functions of literary critics is to help separate the _ from the worthwhile among the many books published every year. 10. Oregon State baseball fans were __ after their team captured the College World Series in June 2006. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The poet’s sonnets are superb, but it is his doggerel that has made him immortal. 2. The student duped the teacher into thinking she had written the paper herself. 3. The historian was happy to find several sources documenting the same event. _ 4. The bride and groom looked ebullient after the priest pronounced them man and wife. 5. The dogmatic philosopher has a well-deserved reputation for being open-minded. UNIT 26 efficacy n. efficiency; effectiveness A cardinal rule of medicine is that the efficacy of a treatment should be measured against the seriousness of its side effects. The adjective is efficacious. In a situation where some subjects are benefiting while others are not, a researcher is likely to have ambivalent feelings, since he or she is in a “no-win” situation. In such a situation, the experimenter must choose between, on the one hand, getting more conclusive results by continuing the experiment and, on the other hand, stopping it and administering the drug that has proven efficacious to those who have not received it. 113 ESSENTIAL WORDS FOR THE GRE effrontery n. shameless boldness; presumptuousness In her essay the student had the effrontery to argue that school is largely a waste of time. egoism n. the tendency to see things in relation to oneself; self-centeredness The beginning of philosophy has been described as a moving away from egoism to an understanding of the larger world. egotistical adj. excessively self-centered; conceited The critics accused the writer of being egotistical since she wrote only about herself. elegy n. poem or song expressing lamentation Adonais is a pastoral elegy written by Percy Bysshe Shelley in the spring of 1821 after he learned of the death of his friend and fellow poet John Keats. elicit v. to provoke; draw out The Socratic method is designed to elicit responses that guide the student toward understanding. Terms from the Arts, Sciences, and Social Sciences Socratic method: a method of seeking the truth about a subject through systematic questioning. Often it results in the questioning of assumptions. The Socratic method is attributed to the ancient Greek philosopher Socrates. elixir n. a substance believed to have the power to cure ills The doctor said that her prescription would help to alleviate my condition but that I could not expect it to be an elixir. Elysian adj. blissful; delightful In Book VI of Virgil’s Aeneid, the hero Aeneas descends to the Underworld where he meets the soul of his dead father, Anchises, in the Elysian fields and learns from him the future of the Roman race. 114 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences Elysian fields or Elysium: in Greek and Roman mythology this refers to an otherworld where the spirits of the virtuous and heroic dwell after being transported there without experiencing death Elysium is described in Homer’s Odyssey as a place of eternal spring where the souls of heroes and others who are blessed by the gods wander blissfully. Homer placed Elysium at the western edge of the Earth near the stream of Oceanus, while other ancient Greek poets, such as Hesiod and Pindar, placed it in the Isles of the Blessed, or the Fortunate Islands, of the Western Ocean. Later, in the Aeneid, Virgil describes it as being located in the realms of the dead under the Earth. emaciated adj. thin and wasted The prisoner was emaciated after being fed only bread and water for three months. embellish v. to adorn; decorate; enhance; make more attractive by adding details The story he had been told was so powerful that the writer felt no need to embellish it. REVIEW 26 The correct answers are given on page 402. Matching Match each word with its definition. 1. efficacy (A) blissful; delightful 2. effrontery (B) song expressing lamentation 3. egoism (C) substance that cures ills 4. egotistical (D) seeing things in relation to oneself 5. elegy (E) shameless boldness 6. elicit (F) excessively self-centered 7. elixir (G) thin and wasted 8. Elysian (H) to provoke; draw out 9. emaciated (I) efficiency 10. embellish (J) to adorn; enhance 115 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. efficacious effrontery egoism egotistical elegy elicit elixirs Elysian emaciated embellish 1. In the nineteenth century, snake oil salesmen traveled around America selling __ to gullible people. 2. One theory of child development is that the infant moves from _ to an increased ability to understand the viewpoint of other people. 3. Nothing the teacher could say was able to _ a response from the bored students. 4. The aid program provides emergency food to feed the __ people of the drought-stricken country. 5. Some critics consider the artist _ because he does only self-portraits. 6. It seems to be almost a natural human trait to _ a good story to make it an even better story. 7. The politician has found a grassroots approach to garnering support to be most __ . 8. The teachers were shocked when the student council had the _ to pass a motion stating that teachers were using outdated methods of instruction. 9. The novel portrays an _ world in which suffering and death have been eliminated. 10. The poet wrote an __ for the soldiers who had given their lives for their country. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. No one could solve the efficacious math problem. 2. In her second account of events, the witness emaciated her story. 3. Some people are considered egotistical simply because they are not conceited. _ 4. The poem contains an allusion to an elixir that was believed to make a person immortal. 5. The scientist embellished the results of his experiment with data that had not been subjected to rigorous testing. __ 116 ESSENTIAL WORDS FOR THE GRE UNIT 27 emollient adj. soothing; mollifying The politician’s speech is filled with emollient phrases to make his message more palatable. Emollient is also a noun that means an agent that soothes or makes more acceptable. empirical adj. derived from observation or experiment Some people erroneously cite the theory of relativity as support for ethical relativism, whereas in reality the former is a scientific theory, while the latter is a moral issue, and thus by its nature is not subject to empirical verification. Empiricism is a noun meaning the view that experience is the only source of knowledge. It can also mean the employment of empirical methods, as in science. Terms from the Arts, Sciences, and Social Sciences theory of relativity: the theory of the relative as opposed to the absolute character of motion and mass, and the interdependence of matter, space, and time emulate v. to imitate; copy Bionics uses technology to emulate nature, but sometimes a simi lar process occurs in reverse, in which scientists use technology as a heuristic tool to better understand natural processes. Terms from the Arts, Sciences, and Social Sciences Bionics: the application of biological principles to the design of electrical or engineering systems heuristic: relating to a speculative formulation guiding the investiga-tion or solution of a problem; educational method in which students learn from their own investigations encomium n. a formal expression of praise The prime minister asked her speechwriter to compose an encomium for the retiring general. 117 ESSENTIAL WORDS FOR THE GRE endemic adj. inherent; belonging to an area Malaria, once endemic to the area, has now been largely eradicated. enervate v. to weaken During World War II Russian commanders counted on the bitter cold to enervate German soldiers invading their country. engender v. to cause; produce Freudians believe that the traumatic events of infancy often engender repression that creates neuroses. Terms from the Arts, Sciences, and Social Sciences Freudians: followers of Sigmund Freud, the nineteenth-century physician who pioneered the study of the unconscious mind. Some central ideas of Freudian psychology are given below. repression: a psychological process by which desires and impulses are kept out of the conscious mind and kept in the subconscious mind neuroses: a mental disease that causes distress but does not inter-fere with a person’s ability to function in everyday life. In Freudian psychology, a neurosis results from an ineffectual strategy adopted by the Ego to resolve conflict between the Id and the Superego. Ego: in Freudian psychology, the part of the mind that tries to match the desires of the Id with what is required by reality Id: in Freudian psychology, the part of the mind that is the source of psychic energy that comes from instinctual drives and needs Superego: in Freudian psychology, the part of the mind that opposes the desires of the Id. It is based on the childhood process by which a person makes the values of society part of his or her personality. enhance v. to increase; improve Although it is widely believed that the primary objective of the researchers developing the Internet was to secure the American nuclear missile system, in fact their main goal was to foster science by enhancing the ability of technology to disseminate information among scientists. 118 ESSENTIAL WORDS FOR THE GRE entomology n. the scientific study of insects Considering that there are approximately 925,000 species of insects (more than all other species combined), entomology is a vast field of study. enunciate v. to pronounce clearly In everyday speech the sounds of many words are not enunciated clearly. REVIEW 27 The correct answers are given on page 402. Matching Match each word with its definition. 1. emollient (A) to increase; improve 2. empirical (B) inherent; belonging to an area 3. emulate (C) agent that soothes or makes more acceptable 4. encomium (D) scientific study of insects 5. endemic (E) derived from observation or experiment 6. enervate (F) to cause; produce 7. engender (G) to pronounce clearly 8. enhance (H) to weaken 9. entomology (I) to imitate; copy 10. enunciate (J) formal expression of praise Fill-ins Choose the best word to fill in the blank in each sentence. emollient empirical emulated encomiums endemic enervating engendered enhance entomologist enunciate 1. As technology developed at a prodigious rate in the nineteenth and twentieth centuries, technologists increasingly __ the professionalization and methodology of science by establishing, for example, professional associations and publications that published peer-reviewed articles. 2. The dream of many Internet users is the building of a net work connected entirely by optical cable, which would greatly _ the ability of the system to cope with the vast amount of data that it carries. 119 ESSENTIAL WORDS FOR THE GRE 3. It has been said that Charles Darwin, virtually single-handedly, emancipated science from the ideologies of philosophy and religion by being fiercely independent in his thinking, rejecting all prevail ing dogmas as to the immutability of species, and relying solely on _ evidence. 4. Many people who travel to tropical countries find the heat __ . 5. There is a tendency in casual conversation for speakers to not _ each word clearly. 6. Faced with _ high unemployment, the government lowered taxes on foreign investment to encourage economic growth. 7. Much of the tragedy of the Holocaust can be attributed to the fanatical racism __ by the Nazis. 8. _ to Pope Paul II began to be published in newspapers around the world shortly after his death in 2005. 9. The veteran mediator is famous for his _ approach that rarely fails to find a way to bring opposing sides together. 10. The eminent Harvard biologist Edward O. Wilson is an __ specializing in ants. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Knowing the entomology of a difficult word can help you remember it. 2. Carrying the fifty-pound pack in the 95° Fahrenheit heat enervated the infantryman. 3. The Supreme Court ruling has engendered new debate on the controversial issue. _ 4. When learning to speak a new language, it is a good idea to enunciate words clearly. 5. It is advisable to see a doctor before traveling to countries in which malaria or other infectious diseases are endemic. __ UNIT 28 ephemeral adj. short-lived; fleeting Impressionist painters such as Claude Monet share with the Romantics an affinity for nature, but the Impressionists took a more scientific interest in it, attempting to accurately depict ephemeral phenomena such as the play of light on water. 120 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences Impressionist: refers to Impressionism, a movement in art that began in France in the late nineteenth century. Impressionism seeks to portray the visual effects of light reflected on subjects. Claude Monet is one of the most famous Impressionist painters. The term can also be used to refer to literature that tries to convey a general impression of a subject rather than a detailed one and to musical compositions that create impressions and moods. epistemology n. branch of philosophy that examines the nature of knowledge A major question in epistemology is whether the mind can ever gain objective knowledge, limited as it is by its narrow range of sense experience. equable adj. steady; unvarying; serene Throughout the crisis the president remained equable. Do not confuse equable with equitable, which means fair, or just, or impartial. Much of modern economic history can be seen as a dialectic between advocates of laissez-faire policies, who want to leave the market free to create wealth untrammeled by restrictions (believing it will “trickle down” to all members of the society), and exponents of redistribution of wealth, who want to ensure that the fruits of capitalism are shared equitably. Terms from the Arts, Sciences, and Social Sciences dialectic: in this context, dialectic refers to the action of opposing forces in society laissez-faire: in economics and politics, doctrine that an economic system functions best when there is no interference by government. It is based on the belief that the natural economic order tends, when undisturbed by artificial stimulus or regulation, to secure the maximum well-being for the individual and therefore for the community as a whole. equanimity n. composure; calmness Emergency room doctors and nurses are trained to maintain their equanimity when treating patients. 121 ESSENTIAL WORDS FOR THE GRE equivocate v. to intentionally use vague language The businessperson has earned a reputation as someone who never equivocates and can be trusted to do exactly what he promises. The noun is equivocation. The saying “It’s a matter of semantics” is often used to indicate that the real meaning of something is being lost in verbiage, often with the implication that there is obfuscation or equivocation. Terms from the Arts, Sciences, and Social Sciences semantics: interpretation of a word, sentence, or other language form errant adj. mistaken; straying from the proper course The pitcher’s errant fastball struck the batter on the shoulder. erudite adj. learned; scholarly Frederick Copleston, author of the nine-volume History of Philosophy, was undoubtedly one of the most erudite people who ever lived. The noun is erudition. Great erudition does not necessarily mean that a person is sagacious. esoteric adj. hard to understand; known only to a few Epidemiologists, using esoteric statistical analyses, field investi gations, and complex laboratory techniques, investigate the cause of a disease, its distribution (geographic, ecological, and ethnic), method of spread, and measures for preventing or controlling it. Terms from the Arts, Sciences, and Social Sciences Epidemiologists: experts in the branch of medicine that deals with the study of the causes, distribution, and control of disease in populations essay v. to make an attempt; subject to a test The composer began work on a sonata, a form she had not previ ously essayed. 122 ESSENTIAL WORDS FOR THE GRE estimable adj. admirable; possible to estimate Alistair Cooke’s book Six Men contains character studies of estimable modern figures including H. L. Mencken, Humphrey Bogart, and Adlai Stevenson. REVIEW 28 The correct answers are given on page 402. Matching Match each word with its definition. 1. ephemeral (A) admirable 2. epistemology (B) branch of philosophy that examines the nature of knowledge 3. equable (C) hard to understand 4. equanimity (D) steady; unvarying; serene 5. equivocate (E) to intentionally use vague language 6. errant (F) mistaken 7. erudite (G) to make an attempt 8. esoteric (H) short-lived; fleeting 9. essay (I) learned 10. estimable (J) composure; calmness Fill-ins Choose the best word to fill in the blank in each sentence. ephemeral epistemology equable equanimity equivocate errant erudition esoteric essayed estimable 1. Much slang originates in a specific group as a sort of argot that allows that group to share something __ . 2. Although most slang is _ , there are many examples of slang that endures and even comes to be accepted as legitimate. 3. Swami Vivekananda, the founder of the Ramakrishna Math, an Indian order of monks, counseled that one should try to maintain one’s _ , even in trying circumstances. 4. The __ missile had to be destroyed after it veered off course. 5. Members of the Society of Jesus (often called Jesuits), are famous for their _ , which they believe should be used in the service of God. 6. The cognitive sciences are providing _ with new insights into how the mind acquires knowledge. 123 ESSENTIAL WORDS FOR THE GRE 7. “Don’t __ ; tell me if you want to marry me or not,” Ruth said to Seth. 8. The infant _ walking up a stairs for the first time in her life. 9. Perth, Australia is often cited as a pleasant place to live because of its _ climate. 10. Chris Evert was an __ tennis player who won three Wimbledon titles. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. One of the important disciplines that a doctor must master to become a brain surgeon is epistemology. 2. The young history Ph.D. candidate is not as erudite as his supervising professor, who appears to know just about everything that happened in history. 3. The literary critic essayed the new novel in her review. _ 4. In view of the fact that journalism is so often ephemeral, the reporter was pleased when some of her work was published in book form. 5. The logic of the argument is so errant we cannot help but agree with it. UNIT 29 ethnocentric adj. based on the attitude that one’s group is superior The words “primitive” and “savage” reflect an ethnocentric bias in Western culture that regards societies that do not have Western science and technology as inferior because they have not achieved as much material success as Western societies. The noun is ethnocentrism. During certain periods of Chinese history, foreigners were consid ered to be “barbarians”; perhaps this ethnocentrism made it difficult for the Chinese to accept innovations from other countries. etiology n. causes or origins The etiology of mental illness is complex because of the diversity of factors—social, biological, genetic, and psychological—that contribute to many disorders. 124 ESSENTIAL WORDS FOR THE GRE etymology n. origin and history of a word The origin of the word “barbarian” reflects the ethnocentrism of the ancient Greeks; its etymology is that it comes (through Latin and French words) from the Greek word barbaros, meaning non-Greek, foreign. eugenics n. study of factors that influence the hereditary qualities of the human race and ways to improve these qualities The science fiction novel describes a military eugenics program designed to create a race of “super-soldiers” possessing intelligence, strength, and other qualities far in advance of the ordinary person. eulogy n. high praise, especially of a person who has recently died After the death of Abraham Lincoln, many eulogies of him appeared in newspapers throughout America. euphemism n. use of agreeable or inoffensive language in place of unpleasant or offensive language An illustration of the tendency toward euphemism is the change (reflecting the political concerns of the day) in the accepted appella-tion of poor countries from the unambiguous poor, to undeveloped, to underdeveloped, to less developed, to developing. euphoria n. a feeling of extreme happiness There was euphoria in the professor’s house after it was learned that she had received the Nobel Prize for Chemistry. euthanasia n. mercy killing Modern medicine’s ability to prolong life has raised ethical questions, such as “Is euthanasia ever morally justifiable?” evince v. to show plainly; be an indication of The student’s response to the teacher’s question evinced his ignorance of the subject. evocative adj. tending to call to mind or produce a reaction Somerset Maugham’s short stories are often evocative of exotic places such as Pago-Pago and Gibraltar. Evocation is the noun. Some literary critics believe that Charles Dickens’ use of caricature makes his characters one-dimensional, but others see these characters as evocations of universal human types that resonate powerfully with readers’ experiences of real people. 125 ESSENTIAL WORDS FOR THE GRE The verb is evoke. The terms “loaded language” and “charged language” are used to specify language that has so many connotations for most readers that it is difficult for a writer to use it without evoking myriad associations, which will distract attention from the topic under discussion. Terms from the Arts, Sciences, and Social Sciences one-dimensional: relating to a portrayal of a character that lacks depth REVIEW 29 The correct answers are given on page 402. Matching Match each word with its definition. 1. ethnocentric (A) origins 2. etiology (B) high praise 3. etymology (C) based on attitude that a person or group is superior 4. eugenics (D) feeling of extreme happiness 5. eulogy (E) tending to produce a reaction 6. euphemism (F) use of inoffensive language in place of unpleasant language 7. euphoria (G) origin and history of a word 8. euthanasia (H) mercy killing 9. evince (I) study of factors that influence heredi tary qualities 10. evocative (J) to show plainly Fill-ins Choose the best word to fill in the blank in each sentence. ethnocentrism etiology etymology eugenics eulogy euphemisms euphoria euthanasia evinces evocative 1. “Folk __ ” is the term used by linguists to refer to popular theories of how words originated or changed their meaning. 2. The book describes the _ among Allied soldiers after Japan surrendered in 1945. 126 ESSENTIAL WORDS FOR THE GRE 3. The Oxford Dictionary of the English Language _ the scholarship of a large team of dedicated scholars. 4. The diversity of factors involved in triggering cancers makes it difficult to be certain of the __ of a particular case of cancer. 5. Alexander Graham Bell advocated a form of _ ; from his research, he concluded that deafness was hereditary and in 1881 he recommended that deaf people be prohibited from getting married. 6. The novel includes many descriptions _ of New England in winter. 7. The captain’s __ of the dead soldier described his bravery in battle. 8. In order to discourage _ the college requires students to take three courses dealing with other cultures. 9. Advances in medical technology have made the question of whether _ is morally justifiable an important issue in many countries. 10. Modern warfare has produced __ such as antipersonnel mines for mines that rip soldiers’ bodies into shreds with bits of metal and collateral damage for noncombatants killed as a result of war. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The phrase “domestic helper” can be considered a euphemism for “maid.” 2. The patient was given euthanasia before undergoing major surgery. 3. The euphoria in the stadium rose to a fever pitch as the seconds ticked down on the college football team’s 12th straight victory. _ 4. The eulogy talked only about the many flaws in the dead man’s character. 5. The ethnocentric villagers have no interest in anything outside their own little world. __ 127 ESSENTIAL WORDS FOR THE GRE UNIT 30 exacerbate v. to aggravate; make worse The release of carbon dioxide from the burning of fossil fuels has increased the amount of this gas in the atmosphere, exacerbating the naturally occurring “greenhouse effect” that has predominated in Earth’s recent past. Terms from the Arts, Sciences, and Social Sciences greenhouse effect: the process by which a planet’s atmosphere warms the planet exact v. to force the payment of; demand and obtain by authority The conquering rulers exacted a tax of 10% from every adult male in the country. The adjective exacting means extremely demanding. Early in his career the English writer Aldous Huxley made this comment: “What occupation is pleasanter, what less exacting, than the absorption of curious literary information?” exculpate v. to clear of blame; vindicate The report exculpated the FBI of any wrongdoing in its handling of the investigation. execrable adj. detestable; abhorrent When folk artists such as Bob Dylan began to use rock instruments, many folk music traditionalists considered it an execrable travesty. exhort v. to urge by strong appeals In 1943 U.S. General George S. Patton exhorted American troops about to invade Hitler’s Europe, saying that victory was assured because American soldiers were more virile and courageous than their German counterparts. exigency n. crisis; urgent requirements Astronauts must be prepared for exigencies such as damage to their spacecraft’s life support system. 128 ESSENTIAL WORDS FOR THE GRE existential adj. having to do with existence; based on experience; having to do with the philosophy of existentialism Existential writers such as Jean-Paul Sartre have argued that human beings are free, but that this freedom entails a burden of responsibility that makes them anxious. Terms from the Arts, Sciences, and Social Sciences Existentialism is a philosophical movement that stresses individual experience in relation to the world. Existential thought is very varied, but often concerns itself with the ideas of freedom, responsi-bility, and the isolation of the individual self. exorcise v. to expel evil spirits; free from bad influences A modern parallel to the shaman is the psychiatrist, who helps the patient exorcise personal demons and guides him toward mental wholeness. Terms from the Arts, Sciences, and Social Sciences shaman: a tribal healer who is believed to be able to enter the world of good and evil spirits. Shamans often enter a trance and practice divination. expatiate v. to speak or write at length Every year the book club invites a famous author to come to expatiate on the art of writing. expatriate v. to send into exile The pronunciation is ek-SPAY-tree-ayt. People seeking asylum in another country are sometimes expatriated. Expatriate is also a noun meaning a person living outside his or her own land. The pronunciation is ek-SPAY-tree-it. The adjective is also expatriate. 129 ESSENTIAL WORDS FOR THE GRE REVIEW 30 The correct answers are given on page 403. Matching Match each word with its definition. 1. exacerbate (A) crisis; urgent requirements 2. exact (B) to clear of blame 3. exculpate (C) relating to existence 4. execrable (D) to make worse 5. exhort (E) speak or write at length 6. exigency (F) to urge by strong appeals 7. existential (G) to force the payment of 8. exorcise (H) to send into exile 9. expatiate (I) to free from bad influences 10. expatriate (J) detestable Fill-ins Choose the best word to fill in the blank in each sentence. exacerbating exacting exculpated execrable exhorted exigency existential exorcises expatiate expatriate 1. The Boy Scout motto, “Be Prepared,” is a concise reminder to be ready for any __ . 2. In E. M. Forster’s A Passage to India, Miss Quested, one of the novel’s important characters, _ what she calls her psychological “bothers” by coming to terms with their underlying cause. 3. In India, small farmers are increasingly abandoning their farms to live in urban centers, _ the problems faced by already overcrowded cities with insufficient infrastructure and services. 4. Amateur radio equipment generally is not built to the __ standards that professional and military radio equipment is. 5. The eminent poet T. S. Eliot was born in the United States in 1888 and lived in England as an _ from 1914 until 1927, when he became a British subject. 6. The principal _ the students to study hard for the final exams. 7. The literature student was amazed that the professor could __ for an hour on a poem containing only twelve words. 8. The people living in the slums of Mexico City live in _ conditions. 130 ESSENTIAL WORDS FOR THE GRE 9. _ writers such as Albert Camus and Jean-Paul Sartre tend to focus on the individual human condition as opposed to human social interaction. 10. The defendant’s attorney brought forward new evidence that __ her of the crime. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The builder exculpated the ground to build a foundation for the house. 2. The football fans exhorted their team’s defense to keep the opposi tion from scoring a touchdown. 3. The expedition to Antarctica brought equipment to help deal with any exigency. _ 4. The philosopher’s existential approach stresses an objective, rational approach to seeking truth. 5. The expatriate loves her country so much that she has never set foot on foreign soil. APPLYING YOUR KNOWLEDGE—21–30 1. Today’s technology is so diverse and complex that no one person—not even a highly trained engineer with __ interests— can understand the inner workings of many of the devices used in modern professional equipment. ephemeral latent existential eclectic liberal Explanation: An important clue to the answer to this question is the word “diverse” that describes today’s technology. Because of this diversity, even an engineer with eclectic (selecting from various sources) interests cannot understand the processes involved in the operation of many modern devices. The correct answer is eclectic. 2. In Beyond Freedom and Dignity, B.F. Skinner maintains that (i) _ the source of all behavior is located in the environment 131 ESSENTIAL WORDS FOR THE GRE and not in people, it follows that modifications of the environment will in turn change behavior. He terms this indirect (ii) _ of people “behavioral technology.” Skinner proposes that in order to produce and maintain a “good” society, it is necessary to con struct an environment in which desirable behaviors are rewarded and in this way reinforced, while undesirable behaviors go unre warded and consequently (iii) ____. Blank (i) Blank (ii) Blank (iii) although manipulation disappear unless evocation continue to exist because admonishment become desiccated Explanation: Blank (i): The clue to the answer is the phrase “it follows that.” Because is correct because it creates a clause that states the cause that produces consequences—that is, modifications in the environment changing behavior—that follow from the cause. Blank (ii): The phrase “behavioral technology” signals that manipula tion (shrewd management) of people is involved. Blank (iii): The word “while” signals a contrast between behaviors that are “reinforced” and those that disappear. The correct answers are because, manipulation, and disappear. UNIT 31 expiate v. to atone for The pilgrims undertook their long journey to expiate their sins. Expiation is the noun. explicate v. to explain; interpret; clarify The literature exam requires students to explicate three poems they studied in class and one they have not studied. Explication is the noun. expository adj. explanatory There is no one model of expository prose that a student can emu late, since each piece of good writing is unique. extant adj. in existence; not lost Unfortunately for Bible scholars, there are no extant writings of Jesus Christ. 132 ESSENTIAL WORDS FOR THE GRE extemporaneous adj. unrehearsed I enjoyed the speaker’s extemporaneous remarks more than her prepared speech, because they gave me insight into her personality that helped me understand the decisions she made during her time as a federal judge. extirpate v. to root up; to destroy The new federal prosecutor promised voters that he would extirpate corruption in the state. extraneous adj. not essential The encyclopedia editors worked hard to cut out extraneous material so that readers could find information easily on a given subject. extrapolation n. the act of estimation by projecting known informa tion The economist’s extrapolation suggests that the economy will grow by 4 percent next year. The verb is extrapolate. Strict determinists believe that it is possible, at least theoretically, to extrapolate the future movement of every atom in the universe based on present conditions. Terms from the Arts, Sciences, and Social Sciences determinists: followers of the belief that all events are determined by causes external to the will extrinsic adj. not inherent or essential The experiment is designed to exclude factors that are extrinsic to the phenomenon. facetious adj. humorous The comedian’s facetious comments about prominent politicians kept the audience amused. 133 ESSENTIAL WORDS FOR THE GRE REVIEW 31 The correct answers are given on page 403. Matching Match each word with its definition. 1. expiate (A) unrehearsed 2. explicate (B) act of estimation by projecting known information 3. expository (C) to root up; to destroy 4. extant (D) in existence; not lost 5. extemporaneous (E) humorous 6. extirpate (F) to explain; interpret 7. extraneous (G) not inherent or essential 8. extrapolation (H) explanatory 9. extrinsic (I) to atone for 10. facetious (J) not essential Fill-ins Choose the best word to fill in the blank in each sentence. expiate explication expository extant extemporaneous extirpate extraneous extrapolating extrinsic facetious 1. Joan’s comments are so subtle some of us have trouble telling whether she is being __ or not. 2. If you would like to read a profound _ of English Romantic poetry, a good book to read is Harold Bloom’s The Visionary Company. 3. To solve the mystery of who had committed the crime, the detective systematically eliminated _ evidence. 4. Three modern masters of __ writing are Bertrand Russell, C. S. Lewis, and Lewis Thomas. 5. The book contains all the _ writings of Edgar Allan Poe. 6. The students were assigned to give a/an _ talk on a subject of their choice. 7. Many of the comic book heroes of the 1950s pledged to __ evil wherever they found it. 8. Being born to a wealthy family can be considered a/an _ advantage to a person. 9. The priest advised the man to perform penance to _ his sins. 10. __ from present trends, scientists predict that the star will explode 100 million years from now. 134 ESSENTIAL WORDS FOR THE GRE Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Upon investigation, we found that the extant of the problem was not as great as we had feared. 2. The two events that occurred in 1969 were extemporaneous. 3. Using complex mathematical extrapolations, astronomers predict that the asteroid will pass by the Earth at a distance of 400,000 miles. _ 4. The book contains clear explications of twenty difficult poems. 5. The new ruler made it a priority to extirpate gangs of criminals. __ UNIT 32 facilitate v. to make less difficult The Internet—together with the availability of relatively inexpensive personal computers—has greatly facilitated the ability of ordinary people to conveniently exchange information with one another and with large computer systems. factotum n. a person who does all sorts of work; a handyman In Shakespeare’s play Twelfth Night, the character Malvolio aspires to become more than merely a factotum in the house of Lady Olivia. fallacious adj. based on a false idea or fact; misleading The belief of the Nazis that they could create a “master race” was based on the fallacious premise that some races are inherently superior to others. The noun fallacy means an incorrect idea. Critics of the “strong” anthropic principle argue that its proponents are guilty of a logical fallacy: on the basis of one known case of intelligent life, they extrapolate the existence of a multitude of such cases. Terms from the Arts, Sciences, and Social Sciences anthropic principle: the theory that only a limited number of pos-sible universes are favorable to the creation of life and that of these only some have intelligent observers. Since humankind exists, it follows that the universe is suited to the evolution of intelligence. 135 ESSENTIAL WORDS FOR THE GRE fallow adj. plowed but not sowed; uncultivated At the beginning of each school year the teacher looks out at the new students and thinks of a fallow field, ready to be cultivated. fatuous adj. foolishly self-satisfied The student could not understand why no one took seriously his fatuous comments. fauna n. animals of a period or region When humans introduce fauna from one habitat into another habitat, the ecological balance is upset. fawning adj. seeking favor by flattering The boss has a reputation for hiring fawning employees. felicitous adj. suitably expressed; appropriate; well-chosen The Gettysburg Address is full of felicitous phrases such as “government of the people, by the people, and for the people.” feral adj. existing in a wild or untamed state Feral dogs returning to an untamed state after domestication sometimes form packs, becoming a threat to humans. fervor n. warmth and intensity of emotion American soldiers were welcomed back to the United States with fervor after the end of World War II. The adjective fervent means full of strong emotion, or impassioned. The fervent libertarian believed that government is a necessary evil that should be constrained from excessive interference in the affairs of individuals. 136 ESSENTIAL WORDS FOR THE GRE REVIEW 32 The correct answers are given on page 403. Matching Match each word with its definition. 1. facilitate (A) foolishly self-satisfied 2. factotum (B) existing in a wild state 3. fallacious (C) to make less difficult 4. fallow (D) suitably expressed 5. fatuous (E) based on a false idea or fact 6. fauna (F) plowed but not sowed 7. fawning (G) person who does all sorts of work 8. felicitous (H) seeking favor 9. feral (I) animals of a period or region 10. fervor (J) warmth and intensity of emotion Fill-ins Choose the best word to fill in the blank in each sentence. facilitate factotum fallacious fallow fatuous fauna fawning felicitous feral fervor 1. The __ of Australia includes quite a number of species introduced from Europe. 2. The bishop’s secretary tries to be respectful of his superior’s office without being _ . 3. _ dogs have become a problem in the more rural areas of Hong Kong, where people buy dogs as pets only to later abandon them. 4. The general’s aide-de-camp functions as the general’s __ . 5. President John F. Kennedy expressed the idea of duty to the country in these _ words: “Ask not what your country can do for you; ask what you can do for your country.” 6. Carbon-14 dating is predicated on the assumption that the amount of carbon-14 in the atmosphere remains constant, but recently this has been proved _ . 7. The football team’s leading running back blocks and runs with equal __ . 8. The teacher was becoming tired of her students’ _ response to literature. 137 ESSENTIAL WORDS FOR THE GRE 9. The black box on commercial airliners, which records flight and engineering data, is usually painted a bright color to _ finding it after a crash. 10. The farmer could not afford to let any of his fields lie __ . Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The chairperson of the investigative committee announced, “I will not make a decision until all the factotums in the case have been discovered. 2. Vegetarians eat only fauna. 3. The president’s chief speechwriter is admired for his felicitous style. _ 4. The statement “George Washington was the first president of the United States” is fallacious. 5. The farmer let his field lie fallow for three years. UNIT 33 fetid adj. having a bad smell Many people find the smell of Limburger cheese fetid. fetter v. to bind; confine The poet William Blake believed that each person creates “mind-forged manacles,” fettering his or her natural instincts and spirit. The noun fetter means something that restricts or restrains. The adjective fettered means bound or confined. fiat n. arbitrary order; authorization The dictator rules almost entirely by fiat. fidelity n. loyalty; exact correspondence Monks joining the Franciscan Order pledge fidelity to the ideals and rules of the Order. filibuster n. use of obstructive tactics in a legislature to block passage of a law The senator threatened that his filibuster would include a full reading of his eight-volume autobiography. 138 ESSENTIAL WORDS FOR THE GRE finesse v. to handle with a deceptive or evasive strategy; to use finesse, that is, refinement in performance Engineers decided that the problem could be finessed by using lighter materials. fissure n. crevice Geologists measure the width of the fissure regularly to monitor movement of the Earth’s plates in the area. flag v. to droop; grow weak Noticing that the students’ attention was flagging, the professor gave them a short break. fledgling n. beginner; novice The coach said that some of the team’s fledglings would play in Saturday’s game. The adjective fledgling means immature or inexperienced. flora n. plants of a region or era Singapore’s Botanical Gardens contain an extensive collection of the flora of Southeast Asia. REVIEW 33 The correct answers are given on page 403. Matching Match each word with its definition. 1. fetid (A) use of obstructive tactics in a legislature to block passage of a law 2. fetter (B) crevice 3. fiat (C) arbitrary order 4. fidelity (D) to droop; grow weak 5. filibuster (E) loyalty 6. finesse (F) to bind; confine 7. fissure (G) plants of a region or era 8. flag (H) to handle with deceptive strategy 9. fledgling (I) having a bad smell 10. flora (J) beginner; novice 139 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. fetid fettered fiat fidelity filibuster finesse fissures flag fledgling flora 1. In the U.S. Senate, a two-thirds vote is required to break a __ . 2. Mosquitoes are breeding in the _ pond. 3. _ to one’s spouse is one of the most important require ments for a successful marriage. 4. The country’s prime minister reflected how much easier it would be to rule by __ than by seeking consensus. 5. The marathon runner began to _ about two miles from the finish line. 6. The _ reporter was assigned to cover mundane events such as school board meetings. 7. Botanists at the university have carried out a comprehensive survey of the __ of the region. 8. He refused to be _ by the conventions of society. 9. The boxer is known for relying more on _ than strength. 10. The appearance of __ in the rock suggested to geologists a movement in the Earth’s crust. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. We all enjoyed the fetid smell of the meal being cooked. 2. Members of the minority party in the Senate were so much against the legislation that they threatened to filibuster. 3. The libertarian believes that modern democratic governments place unacceptable fetters on individual liberty. _ 4. The president gave a speech to rally flagging public support for the war. 5. The fledgling soldiers gradually became accustomed to army life. __ 140 ESSENTIAL WORDS FOR THE GRE UNIT 34 florid adj. ruddy; reddish; flowery As he grew older, the novelist eschewed the florid, ostentatious style of his youth in favor of a more direct and sparse style. flourish n. an embellishment or ornamentation The Sophists often gave interminable speeches full of rhetorical flourishes. Flourish is also a verb meaning to grow vigorously, or to thrive. Capitalism flourished in the eighteenth century in Europe and the United States as the industrial revolution created a prodigious amount of wealth that, for the first time in history, was in the hands of land owners. Terms from the Arts, Sciences, and Social Sciences Sophists: fifth-century b.c. Greek philosophers (Sophistes meant expert or deviser) who speculated on theology, science, and meta-physics. Many people came to dislike the Sophists, accusing them of dishonest reasoning. The word sophistry means reasoning that is subtle and seemingly true but is actually incorrect. flout v. to treat scornfully In his book Poetic Meter and Poetic Form the distinguished literary critic Paul Fussel discusses the dangers poets face when they flout poetic conventions. flux n. flowing; a continuous moving In some cultures time is conceptualized as a flux moving in one direction. foment v. to incite; arouse The government accused the newspaper of fomenting unrest in the country. forbearance n. patience The president warned that great courage and forbearance would be required to see the war through to a successful conclusion. forestall v. to prevent; delay The government took steps to forestall an economic downturn by increasing government spending. 141 ESSENTIAL WORDS FOR THE GRE formidable adj. menacing; threatening By the middle of the nineteenth century the United States had become a formidable economic and military power. forswear v. renounce; repudiate When she became a U.S. citizen, Julia forswore allegiance to all other countries and pledged to defend the United States if called upon to do so. founder v. to sink; fail; collapse Most attempts to create advanced new technology by government fiat founder, probably because of the difficulty in anticipating changes in the fluid world of high technology. REVIEW 34 The correct answers are given on page 403. Matching Match each word with its definition. 1. florid (A) an embellishment or ornamentation 2. flourish (B) menacing; threatening 3. flout (C) patience 4. flux (D) a continuous moving 5. foment (E) fail; collapse 6. forbearance (F) to treat scornfully 7. forestall (G) to renounce; repudiate 8. formidable (H) to prevent; delay 9. forswear (I) ruddy; reddish 10. founder (J) to incite; arouse Fill-ins Choose the best word to fill in the blank in each sentence. florid flourishes flouts flux foment forbearance forestall formidable forswear foundered 1. Rhetorical __ are generally frowned upon under the canons of modern English. 2. The negotiations _ when agreement could not be reached on the central issue. 142 ESSENTIAL WORDS FOR THE GRE 3. The head football coach and his staff spent the week devising a way to break down the _ defense of the next week’s opponent. 4. Peace activists are working to get governments to __ the use of nuclear weapons. 5. The education system is in a state of _ , as administra tors struggle to keep up with changes in society. 6. Negotiators worked frantically to _ the outbreak of hostilities. 7. The country accused the neighboring country of employing agents to __ revolution. 8. The student’s essay _ the rules of written English. 9. The governor urged the people of the state to show _ during the crisis. 10. A __ style is generally best avoided when one is writing a business letter or report. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Good Scottish whiskey must be fomented for at least 12 years. 2. The company was foundered by a Scot who came to America in 1828. 3. “If you insist on flouting the law,” the warden told the prisoner, “you’ll be spending a lot more time behind bars.” _ 4. The U.S. Navy’s Seventh Fleet, with its more than 50 ships and 350 aircraft, possesses a formidable amount of firepower. 5. The teacher took steps on the first day of school to forestall discipline problems in the class. __ UNIT 35 fracas n. a loud quarrel; brawl The police were called in to break up a fracas that had erupted in the bar. fractious adj. quarrelsome; unruly; rebellious In an effort to unify their divided party, its leaders decided to first placate the party’s most fractious elements. 143 ESSENTIAL WORDS FOR THE GRE fresco n. a painting done on plaster The Italian Renaissance was the greatest period of fresco painting, as seen in the work of artists such as Michelangelo, Raphael, and Giotto. frieze n. ornamental band on a wall One of the best-known friezes, on the outer wall of the Parthenon in Athens, is a 525-foot depiction of the Panathenaic procession honoring Athena. Terms from the Arts, Sciences, and Social Sciences Parthenon: the chief temple of the goddess Athena on the Acropolis in Athens Panathenaic: relating to the Panathenaea, an Athenian festival held in honor of the Greek goddess Athena, the patron goddess of Athens froward adj. stubbornly contrary; obstinately disobedient The teacher had no choice but to send the froward child to the vice-principal for disciplining. frugality n. thrift In these days of credit card and installment plan buying, frugality seems to have become a rarely practiced virtue. fulminate v. to attack loudly; denounce The senator fulminated against what he termed “foreign meddling in America’s business.” fulsome adj. so excessive as to be disgusting The actor was embarrassed by the fulsome praise he received after winning the Academy Award for best actor. fusion adj. union; synthesis A hydrogen bomb requires tremendous heat to trigger the fusion reaction, which is provided by the detonation of a fission bomb. 144 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences fusion: In physics, nuclear fusion is the process by which multiple nuclei join together to form a heavier nucleus, resulting in the release of energy. fission: splitting into two parts. In physics, nuclear fission is a process where a large nucleus is split into two smaller nuclei. In biology, binary fission refers to the process whereby a prokaryote (a single-celled organism lacking a membrane-bound nucleus) reproduces by cell division futile adj. ineffective; useless; fruitless To some non-philosophers, the discipline seems frivolous and futile because it produces no tangible benefits. REVIEW 35 The correct answers are given on page 403. Matching Match each word with its definition. 1. fracas (A) painting done on plaster 2. fractious (B) so excessive as to be disgusting 3. fresco (C) stubbornly contrary 4. frieze (D) useless 5. froward (E) quarrelsome; unruly 6. frugality (F) ornamental band on a wall 7. fulminate (G) to denounce 8. fulsome (H) synthesis 9. fusion (I) loud quarrel 10. futile (J) thrift Fill-ins Choose the best word to fill in the blank in each sentence. fracas fractious fresco frieze froward frugality fulminated fulsome fusion futile 1. The philosopher’s conclusion is that it is __ to try to understand the ultimate meaning of existence. 2. The genesis of the computer revolution lay, to a large extent, in a _ of science and technology. 145 ESSENTIAL WORDS FOR THE GRE 3. A _ broke out on the field after the pitcher hit a third batter in a row. 4. Many people find __ a difficult virtue to practice. 5. The _ horse resisted every effort of its rider to make it follow the path. 6. Archaeologists are studying the _ , which they hope will give them a better understanding of life in ancient Greece. 7. The guest of honor at the banquet warned her hosts that she would leave if speakers began to heap __ praise on her for her work for the poor. 8. _ elements within the party have prevented a consensus from being reached on the issue. 9. The reformer _ against a society in which wealth is distributed so unequally. 10. The earliest form of __ in history was Egyptian wall paintings in tombs. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The development of modern friezing techniques allows us to enjoy foods from all over the world. 2. We had to fulminate the house to kill the insects that had infested it. 3. The froward child refuses to go to bed when he’s told to. _ 4. The country’s leader urged citizens to practice frugality to help reduce private debt. 5. There were quite a few futile attempts at manned flight before the Wright brothers. UNIT 36 gainsay v. to deny; dispute; oppose No one can gainsay the fact that she put great effort into the project. gambol v. to frolic; leap playfully The children gamboled on the lawn while their parents ate lunch. The noun gambol means frolicking about. 146 ESSENTIAL WORDS FOR THE GRE garrulous adj. very talkative; wordy The garrulous houseguest made it difficult for us to get much work done on the project. gauche adj. coarse and uncouth; clumsy What is considered gauche in one culture might not be considered gauche in another culture; for example, burping is considered rude in America but is acceptable in China. geniality n. cheerfulness; kindliness; sociability Hosts of television talk shows are generally people who possess a great deal of geniality. The adjective genial means having a pleasant or friendly disposition. gerrymander v. to divide an area into voting districts in a way that favors a political party An argument against the practice of gerrymandering is that it tends to make it difficult for the party that is out of power to regain power. glib adj. fluent in an insincere way; offhand Sharon’s parents were not satisfied by her glib explanation of why she had not been able to study for the exam. goad v. to prod; urge on Goaded by his friends into trying out for the football team as a walk-on, Jeff went on to become an all-American linebacker. gossamer adj. sheer; light and delicate, like cobwebs Some experts in NASA believe that what they call a gigantic “gossamer spacecraft” could be constructed in space using extremely lightweight materials. gouge v. to tear out; scoop out; overcharge The store is able to gouge its customers because it is the only store in the area that carries that particular line of merchandise. 147 ESSENTIAL WORDS FOR THE GRE REVIEW 36 The correct answers are given on page 404. Matching Match each word with its definition. 1. gainsay (A) to tear out; overcharge 2. gambol (B) to prod; urge on 3. garrulous (C) to deny; dispute 4. gauche (D) very talkative 5. geniality (E) sheer; light and delicate, like cobwebs 6. gerrymander (F) to frolic; leap playfully 7. glib (G) fluent in an insincere way 8. goad (H) cheerfulness; kindliness 9. gossamer (I) coarse and uncouth 10. gouge (J) to divide into voting districts so that a political party is favored Fill-ins Choose the best word to fill in the blank in each sentence. gainsay gambol garrulous gauche geniality gouged glib goaded gossamer gerrymandering 1. The __ witness keeps digressing from his account of the incident to tell amusing anecdotes. 2. Semi-tame deer _ in the lush green field. 3. The host’s _ impressed everyone at the party. 4. The suspect’s explanation sounded suspiciously __ to the detective. 5. The political scientist suggested that _ be prohibited so that political districts would remain the same over the years. 6. Jim’s friends _ him into joining the Marines. 7. The pilot assured me that the glider’s __ wings would support the aircraft just fine, but I still had my doubts. 8. The protagonist of the novel is a shy woman who becomes flustered and _ in formal social situations. 9. Engineers _ a new channel for the stream to follow. 10. No one can __ the fact that China has made great progress in improving the lives of its people over the past half century. 148 ESSENTIAL WORDS FOR THE GRE Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Ted gamboled away his savings in Atlantic City. 2. The river gerrymanders through Ocean County. 3. After goading on the problem for several days, the mathematician hit on a solution. _ 4. The garrulous baseball announcer told a record twenty-six anecdotes in the course of a single game. 5. The dean applauded the students for their gauche, decorous behavior. UNIT 37 grandiloquent adj. pompous; bombastic The orator abandoned grandiloquent phrases and instead uses simple and direct language. gregarious adj. sociable A recent anthropological theory is that human beings are gregarious creatures that are comfortable living in groups of around 150 individuals. grouse v. to complain Instead of grousing about the policy, do something about it: write to your congressional representative. Grouse is also a noun. The lieutenant told his men “If you have any grouses, take them to the captain.” guileless adj. free of cunning or deceit; artless One of the charms of the novel is that the guileless hero manages to defeat the scheming villain. Guile is a noun meaning deception or trickery. Playing poker well requires guile as well as skill. guise n. outward appearance; false appearance; pretense In Greek mythology, the god Zeus often appeared to mortal women to whom he was attracted in strange guises: as a swan, he made love to Leda of Sparta; with other women he took on the form of a shower of gold, or a bull, or thunder and lightning. 149 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences Zeus, known to the Romans as Jupiter, was the head of the Olympian pantheon and the god of weather. An amorous god, his liaisons with goddesses, nymphs, and mortal women produced many offspring, including Perseus, Heracles, Hermes, Ares, the Fates, and the Muses. Leda was the wife of King Tyndareus of Sparta. Her union with Zeus produced Helen and Polydeuces. gullible adj. easily deceived Gullible members of the audience believed the young performer’s claim that he had composed “Hey, Jude.” gustatory adj. affecting the sense of taste According to scientists, our gustatory sense depends to a large extent on our olfactory sense. halcyon adj. calm and peaceful; happy; golden; prosperous The movie evokes the halcyon years immediately after World War II when America was at peace and the economy was booming. As a noun, halcyon is a genus of kingfisher. It also is the name of a mythological bird identified with the kingfisher that symbolizes life and renewal. Terms from the Arts, Sciences, and Social Sciences In folklore the halcyon (kingfisher) is a bird that brings peace and calm to the ocean waves for several days around the time of winter solstice, when it builds its nest on the sea and lays its eggs there. The expressions halcyon days and halcyon years describes periods of time that are tranquil and happy. The origins of the halcyon myth can be traced back to ancient Greece and the story of the queen Alcyone (Halcyone) who threw herself into the sea when she saw the dead body of her husband Ceyx, the King of Thessaly, who had drowned in a shipwreck. Pitying Alcyone, the gods transformed both her and Ceyx into kingfishers (halcyon), and they remained in the sea where they mated and had young. While Alcyone laid her eggs and brooded over the nest on the sea, Aeolus, keeper of the sea winds, restrained these winds so that the ocean surface would remain calm and peaceful. 150 ESSENTIAL WORDS FOR THE GRE hallowed adj. holy; sacred The questioning of scientific and religious orthodoxy by scientists such as Charles Lyell and Charles Darwin led to stupendous advances in both geology and biology, as these fields freed them selves from the fetters of hallowed, but fallacious, assumptions about the age and development of the Earth and life. harangue n. long, pompous speech; tirade The football team sat silently listening to their coach’s half-time harangue about poor tackling, dropped passes, and lost opportunities to score. REVIEW 37 The correct answers are given on page 404. Matching Match each word with its definition. 1. grandiloquent (A) free of deceit 2. gregarious (B) affecting the sense of taste 3. grouse (C) long, pompous speech 4. guileless (D) easily deceived 5. guise (E) calm and peaceful 6. gullible (F) pompous; bombastic 7. gustatory (G) outward appearance 8. halcyon (H) to complain 9. hallowed (I) holy; sacred 10. harangue (J) sociable Fill-ins Choose the best word to fill in the blank in each sentence. grandiloquent gregarious grouse guileless guises gullible gustatory halcyon hallowed harangue 1. “Anyone with a __ about my marking can see me in my office after class,” the law professor told her class. 2. Researchers have found that many primates—such as chimpanzees and humans, for example—are _ , while others, like the orangutan, live largely solitary lives. 3. The field in France is _ by the graves of the brave soldiers who fought and died for their country. 151 ESSENTIAL WORDS FOR THE GRE 4. Abraham Lincoln’s famous adage—“You can fool some of the people all the time, and all of the people some of the time, but you cannot fool all of the people all the time.”—can be paraphrased: “There are a lot of __ people in the electorate, but there are also some people who insist on knowing the truth.” 5. The president governs with the adage “ _ phrases don’t house the homeless” always in mind. 6. The restaurant critic called the dish “a _ triumph.” 7. According to Hindu belief, God appears throughout history in many __ . 8. In Somerset Maugham’s story “The Facts of Life” a _ young man triumphs over a crafty, worldly-wise young woman who tries to steal his money. 9. In retrospect, the prosperous 1950s seem like _ years to many Americans. 10. The professor finished his __ about student tardiness with the words, “The next time any of you are late, don’t bother coming to my class.” Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Gustatory winds made it difficult to sail the yacht back to port. 2. The con man is always on the lookout for guileless individuals. 3. The poem harkens back to an imagined halcyon Golden Age. _ 4. Many of America’s greatest thinkers and leaders have passed through the hallowed halls of Harvard University. 5. The computer dating service helps people too gregarious to mingle with others at social functions to find a partner. __ UNIT 38 harrowing adj. extremely distressing; terrifying The journey “inward” to explore the unconscious mind has been described as more harrowing than the most dangerous voyage to explore the Earth. herbivorous adj. relating to a herbivore, an animal that feeds mainly on plants Most researchers now believe that the common ancestor of apes and humans was a strongly herbivorous animal. 152 ESSENTIAL WORDS FOR THE GRE hermetic adj. tightly sealed; magical Scholars have traced many of the hermetic traditions of ancient Greece to Egypt. heterodox adj. unorthodox; not widely accepted The orthodox view among scientists is that the ancestors of the great apes and humans evolved solely in Africa; however, recently a competing, heterodox view has arisen theorizing that they also may have evolved in Euroasia. hieroglyphics n. a system of writing in which pictorial symbols represent meaning or sounds; writing or symbols that are difficult to decipher; the symbols used in advanced mathematics The deciphering of hieroglyphics on the Rosetta Stone in 1822 was a great step forward in understanding hieroglyphics. Terms from the Arts, Sciences, and Social Sciences Rosetta Stone: a granite stone inscribed with the same passage of writing in two Egyptian languages and one in classical Greek. Comparative translation helped scholars to gain a much better understanding of hieroglyphics. hirsute adj. covered with hair One of the most obvious differences between humans and closely related species such as chimpanzees is that the latter are hirsute, while the former have relatively little hair. histrionic adj. relating to exaggerated emotional behavior calculated for effect; theatrical arts or performances Whenever the star of the movie does not get her way on the set, she flies into a histrionic fit. The noun histrionics means emotional behavior done for effect. “Cut the histrionics and tell me how you really feel,” the woman said to her angry husband. homeostasis n. automatic maintenance by an organism of normal temperature, chemical balance, etc. within itself An example of homeostasis in mammals is the regulation of glucose levels in the blood, which is done mainly by the liver and insulin secreted by the pancreas. homily n. sermon; tedious moralizing lecture; platitude The pastor’s homilies have been published in an anthology. 153 ESSENTIAL WORDS FOR THE GRE homogeneous adj. composed of identical parts; uniform in composition Pluralists in America argue that the country’s institutions can withstand great diversity, and even be strengthened by it, while those who argue for a more homogeneous society believe that such a situation results in unhealthy contention and animosity between groups. Terms from the Arts, Sciences, and Social Sciences Pluralists: followers of pluralism, the belief that it is beneficial to have a variety of distinct ethnic and cultural groups in society REVIEW 38 The correct answers are given on page 404. Matching Match each word with its definition. 1. harrowing (A) unorthodox 2. herbivorous (B) extremely distressing 3. hermetic (C) relating to exaggerated emotional behavior calculated for effect 4. heterodox (D) composed of identical parts 5. hieroglyphic (E) tightly sealed; magical 6. hirsute (F) covered with hair 7. histrionic (G) sermon 8. homeostasis (H) ability of a cell to maintain its internal equilibrium 9. homily (I) relating to a herbivore, an animal that feeds on plants 10. homogeneous (J) relating to a system of writing using pictorial symbols Fill-ins Choose the best word to fill in the blank in each sentence. harrowing herbivorous hermetic heterodox hieroglyphics hirsute histrionic homeostatic homily homogeneous 1. This Sunday’s __ deals with the parable of the Good Samaritan. 2. The “ _ tradition” refers to a number of interrelated subjects such as alchemy, magic, and astrology. 154 ESSENTIAL WORDS FOR THE GRE 3. The theologian’s _ conclusions were censured by the Church. 4. Many primatologists believe that early human beings were __, living on fruit, seeds, and nuts. 5. J. R. R. Tolkien’s story The Lord of the Rings recounts Frodo Baggin’s _ journey to carry the One Ring from Rivendell to the Crack of Doom and destroy it before the evil Sauron could get his hands on it. 6. Some educators believe it is best to group students according to their ability, while others prefer _ grouping. 7. Anthropologists believe that early human beings were __ . 8. The removal of waste products by excretory organs such as the lungs and kidneys is an important _ process in mammals. 9. The UFO researcher claims to have found writings inscribed on the side of an alien craft that resemble _ . 10. Most mothers are astute at judging whether their child’s tears are genuine or merely __ . Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Stan’s herbivorous diet consists mainly of hamburgers and steaks. 2. In the seventeenth century, a voyage by ship from London to New York was a harrowing experience. 3. Many patients are turning to homeostasis as an alternative to traditional medicine. _ 4. The heterodox pastor teaches only doctrines approved by his church. 5. Hieroglyphics on the Egyptian pot indicate it was used to store records of the pharaoh’s accounts. __ UNIT 39 hyperbole n. purposeful exaggeration for effect The American tradition of the tall tale uses hyperbole to depict a world in which the inhabitants and their deeds are larger than life, as befitting a people inhabiting a vast landscape. iconoclastic adj. attacking cherished traditions The linguist and political commentator Noam Chomsky has been described as gleefully iconoclastic because of the zeal with which he attacks many of the central beliefs of American society. 155 ESSENTIAL WORDS FOR THE GRE An icon is an image or representation. The internal combustion engine is a ubiquitous feature of modern industrial society, helping the automobile to become an icon of the twentieth century, loved by many people but loathed by environmentalists. Terms from the Arts, Sciences, and Social Sciences The icons of the Eastern Orthodox Church are usually portraits of holy men and women that worshipers use as a help to focus their prayers. A person who smashes such an object is an iconoclast, which comes from the Greek word eikonoklastes meaning “break-ing of an image.” Iconoclastic has come to be used more generally to refer to an attack on any cherished belief. ideological adj. relating to ideology, the set of ideas that form the basis of a political or economic system Recent social science research suggests that a person’s psychologi cal makeup plays a large part in determining his or her ideological leanings. idolatry n. idol worship; blind or excessive devotion During the Protestant Reformation images in churches were felt to be a form of idolatry and were banned and destroyed. igneous adj. produced by fire; volcanic The presence of igneous rocks on the beach suggests that there was a volcanic eruption in the area millions of years ago. imbroglio n. complicated situation; an entanglement The plot of many of Somerset Maugham’s stories consists of an unraveling of an imbroglio in which the main character finds himself. immutable adj. unchangeable If humanity colonizes Mars, it will become a tabula rasa on which we will inscribe our immutable values and beliefs in a new environment. Terms from the Arts, Sciences, and Social Sciences tabula rasa: something that is new and not marked by external influence. Tabula rasa is from Latin, meaning “scraped tablet” (a tablet from which the writing has been erased). 156 ESSENTIAL WORDS FOR THE GRE The noun is immutability. The dogma of creation and the immutability of species was endorsed virtually unanimously by the leading anatomists, botanists, and zoologists of Charles Darwin’s day. impassive adj. showing no emotion The judge sat, impassive, listening to the man’s emotional account of the crime. impecunious adj. poor; having no money The businessman’s biography tells how he went from being an impecunious student in the 1980s to one of the richest people in America. impede v. to hinder; block The development of the western region of China has been impeded by a lack of trained workers. REVIEW 39 The correct answers are given on page 404. Matching Match each word with its definition. 1. hyperbole (A) complicated situation 2. iconoclastic (B) relating to the set of ideas that form the 3. ideological basis of a political or economic system 4. idolatry (C) purposeful exaggeration for effect 5. igneous (D) worshipping idols 6. imbroglio (E) to hinder 7. immutable (F) unchangeable 8. impassive (G) attacking cherished traditions 9. impecunious (H) poor 10. impede (I) volcanic (J) showing no emotion Fill-ins Choose the best word to fill in the blank in each sentence. hyperbole iconoclastic ideological igneous imbroglio immutable idolatry impassive impecunious impeded 1. It would be __ to say that scientists have gained a per fect understanding of the process of human evolution; however, it is fair to say that over the last century and a half a reasonably clear idea of it has emerged. 157 ESSENTIAL WORDS FOR THE GRE 2. The _ artist is applying for a grant so that she can continue painting full-time. 3. Anthropologists, mindful of the danger of ethnocentrism, avoid the use of emotionally charged words such as “ _ .” 4. The president warned Congress that the United States should not become involved in the diplomatic __ . 5. This week’s essay topic is “War has _ human progress.” 6. The philosopher searches for _ truths, striving to gain a comprehensive view of reality. 7. __ rocks are formed when molten rock cools and solidifies. 8. The _ book debunks the belief that all of America’s Founding Fathers believed fervently in democracy. 9. The judge sat _ through the entire murder trial, carefully considering the evidence presented. 10. The leader of the political party urged members to stop their endless __ debates and concentrate instead on achieving realistic goals. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Politicians often use hyperbole to embellish their achievement so that the electorate will vote for them. 2. Modern biologists regard evolution to be an immutable law of nature. 3. Gorillas are an igneous species in which a single male usually dominates a family unit. _ 4. The doctor warned her patient that alcohol would impede the action of the antibiotics that she had prescribed. 5. Since he regularly questioned conventional wisdom, the philosopher Socrates can be described as an iconoclast. UNIT 40 impermeable adj. impossible to penetrate The virus protection software is said to be impermeable to attacks by malicious software sent over the Internet. imperturbable adj. not easily disturbed Buddha counseled that one should try to remain imperturbable through life’s vicissitudes. 158 ESSENTIAL WORDS FOR THE GRE impervious adj. impossible to penetrate; incapable of being affected We were amazed how Laura could sit at the noisy party studying organic chemistry, impervious to the noise around her. impinge v. to strike; encroach Scientists have found chimpanzees to be a territorial species; individuals that are not members of a group impinging on the territory of that group are normally met with aggression. implacable adj. inflexible; incapable of being pleased Once an implacable foe of capitalism, the People’s Republic of China in recent years seems, in practice if not in principle, to have embraced it. implausible adj. unlikely; unbelievable To say that Napoleon Bonaparte achieved what he did merely because he was compensating for his shortness is simplistic, reductionistic, and implausible. Terms from the Arts, Sciences, and Social Sciences reductionistic: attempting to explain complex phenomena by simple principles implicit adj. implied; understood but not stated Implicit in the review is the idea that the writing of serious litera ture is a moral undertaking. An implication is that which is hinted at or suggested. The guiding principle of common law is that decisions of previous courts should be followed unless there are compelling reasons for ruling differently, which by implication would invalidate the earlier rulings. implode v. collapse inward violently The building was imploded in order to make way for the construc tion of a new apartment complex. The noun is implosion. imprecation n. curse The convicted man was taken away by court officers, uttering imprecations against the jury that had found him guilty. 159 ESSENTIAL WORDS FOR THE GRE impute v. to relate to a particular cause or source; attribute the fault to; assign as a characteristic Primatologists generally impute relatively high intelligence to chimpanzees based on, among other things, the ability of chimpanzees to recognize themselves in a mirror. REVIEW 40 The correct answers are given on page 404. Matching Match each word with its definition. 1. impermeable (A) unlikely 2. imperturbable (B) to encroach 3. impervious (C) curse 4. impinge (D) to collapse inward violently 5. implacable (E) implied 6. implausible (F) impossible to penetrate 7. implicit (G) to attribute the fault to 8. implode (H) incapable of being affected 9. imprecation (I) inflexible 10. impute (J) not easily disturbed Fill-ins Choose the best word to fill in the blank in each sentence. impermeable implicit impinging implacable implausible impervious imprecations impute imperturbable implosions 1. It seems __ to some people that a complex organ such as the human eye developed purely as a result of the process of evolution through natural selection. 2. Sometimes seen as _ foes of science, many theologians are working to reconcile divergent views of science and religion. 3. _ in the idea of democracy is the notion of individual liberty. 4. Submarines are pressurized to prevent catastrophic __ due to the pressure of water on the hull. 5. When you look at a star that is 50 light-years away, the light that is _ on your retina forms an image of the star as it was 50 years in the past. 6. The plastic coating on the table’s surface makes it _ to water. 160 ESSENTIAL WORDS FOR THE GRE 7. Joe, __ to reason, insisted on trying to swim to the island alone. 8. An important attribute of a leader is the ability to remain _ in a crisis. 9. People often _ great cleverness to cats. 10. Frustrated by his inability to gain revenge on his enemies, all George could do was hurl __ at them. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The young soldiers were amazed how their captain sat, imperturb able, through the heavy enemy bombardment, chatting and play ing cards. 2. Cornered by the police, the fleeing suspect began to utter imprecations. 3. Before the development of radio, the idea that people could speak to each other over thousands of miles was generally regarded as implausible. _ 4. Everyone in the class likes Professor Wilson because of her fair, flexible, and implacable marking. 5. The first mate warned the captain of the submarine that implosion was imminent. APPLYING YOUR KNOWLEDGE—31–40 1. The development of antidepressant, antipsychotic, anti-anxiety, and other (i) __ drugs has allowed treatment of conditions that in the past were (ii) __ to therapy. Blank (i) Blank (ii) hermetic precursors ineffective impervious psychotropic analogous Explanation: Psychotropic (having an altering effect on perception or behavior) is the correct choice because the three types of drugs men tioned all affect perception or behavior. If you didn’t know the mean ing of psychotropic, you could still arrive at the correct answer if you knew the meaning of hermetic and ineffective, which don’t make sense in context. Also, the word “psychotropic” contains the Greek root psych (mind), showing that the word has something to do with the mind. 161 ESSENTIAL WORDS FOR THE GRE Clues to the answer to Blank (ii) are the words “development” and “past.” These words tell us that drugs developed recently have made possible treatment of conditions that in the past were impervious (inca pable of being affected) to treatment. The correct answers are psychotropic and impervious. 2. The French writer Alexis de Tocqueville noted in his Democracy in America (1835) that there seems to be a (i) _ in American society: that bastion of individualism produces a profusion of pub lic institutions requiring (ii) __ to create. Blank (i) Blank (ii) conundrum tremendous ingenuity weakness a great collective effort paradox admirable altruism Explanation: The clue to the answer for Blank (ii) is “public institu tions.” It makes sense that public institutions require a great collective effort to create. Knowing the answer to Blank (ii) allows us to select the answer for Blank (i) because “collective effort” and “individualism” are antithetical. The paradox (contradiction, incongruity) is that a society that values individualism produces many institutions requiring groups of people to work together. The correct answers are paradox and a great collective effort. UNIT 41 inadvertently adv. carelessly; unintentionally The songwriter says that it is easy to inadvertently use the melody of another song when composing. incarnate adj. having bodily form Christians believe that Jesus Christ was God incarnate. inchoate adj. imperfectly formed or formulated In his book Chronicles, Bob Dylan describes the process of how some of his songs went from an inchoate state to finished, well-produced songs. incongruity n. state of not fitting There is an incongruity between the poem’s solemn tone and its light-hearted theme. 162 ESSENTIAL WORDS FOR THE GRE The adjective is incongruous. The assumptions underlying Jonathan Swift’s definition of literary style—“The proper words in the proper order”—recognize that there are many effective styles, but that the effectiveness of each is dependent on the context within which it is found: for example, the rambling, exuberant style of Walt Whitman’s poem “Song of Myself” would be incongruous in Alexander Pope’s The Rape of the Lock, with its dependence on sustained wit and irony. Terms from the Arts, Sciences, and Social Sciences Jonathan Swift: Anglo-Irish writer (1667–1745) known today mainly for his prose satires such as Gulliver’s Travels Walt Whitman: American poet (1819–1892) widely regarded as one of the nation’s greatest writers. His most famous work is Leaves of Grass Alexander Pope: English poet (1688–1744) known today mainly for his satirical poetry, most notably The Rape of the Lock inconsequential adj. insignificant; unimportant The meeting of the two women seemed inconsequential at the time, but in retrospect it led to one of literature’s great collaborations. incorporate v. introduce something into another thing already in existence; combine According to Bob Dylan in his autobiography, Chronicles, he systematically tried to incorporate what he learned about life and music into the songs he wrote. incursion n. sudden invasion At first, the Native Americans were not too concerned about the incursions of European settlers, but their anxiety grew with the relentless flow of people, until, finally, calamitous wars were fought between the two sides. indeterminate adj. uncertain; indefinite The novel describes the main character as “being of an indeterminate age, somewhere between 50 and 60.” indigence n. poverty Most economists believe that the best way to prevent indigence is to expand employment opportunities. 163 ESSENTIAL WORDS FOR THE GRE The adjective is indigent. For approximately 20 percent of the world’s population, nearly all of whom are indigent, malnutrition is the main impediment to achiev ing good health. indolent adj. habitually lazy; idle An argument against welfare is that it encourages people to be indolent. REVIEW 41 The correct answers are given on page 404. Matching Match each word with its definition. 1. inadvertently (A) imperfectly formed 2. incarnate (B) to introduce something into another thing already in existence; combine 3. inchoate (C) insignificant; unimportant 4. incongruity (D) sudden invasion 5. inconsequential (E) habitually lazy; idle 6. incorporate (F) carelessly; unintentionally 7. incursion (G) poverty 8. indeterminate (H) having bodily form 9. indigence (I) uncertain; indefinite 10. indolent (J) state of not fitting Fill-ins Choose the best word to fill in the blank in each sentence. inadvertently incarnate inchoate incongruous inconsequential incorporates incursions indeterminate indigent indolent 1. In view of the fact that in most elections fewer than half the eli gible voters cast their ballot, it would appear that many citizens consider their vote to be __ . 2. In societies that place a high value on hard work, people who spend most of the day sitting around chatting are often considered to be _ . 3. During an ice age, the polar ice caps make _ into regions that are temperate at other times. 4. The study of human evolution __ the latest research from primatology, anthropology, and related fields. 164 ESSENTIAL WORDS FOR THE GRE 5. The writer is approaching that _ age at which one cannot accurately be described either as young or middle-aged. 6. In retrospect, it seems _ that a country founded on the principle of liberty condoned slavery. 7. Astronomers believe that the solar system formed out of an __ mass of dust and gas. 8. The typesetter _ omitted a line from the poem. 9. Many people consider Adolf Hitler to have been evil _ . 10. The new welfare program is targeted to help the truly __ in the population. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. “The method you use to memorize the information is inconsequen tial,” the teacher told her class, “as long as it works.” 2. The book The Historical Jesus by John Dominic Crossan incor porates the methodology of and insights of a number of fields, including anthropology, history, and theology. 3. Military intelligence indicates that the enemy has been making incursions into our territory. _ 4. The president hailed the unprecedented economic growth as “ushering in a new era of industry and indigence.” 5. The poem is the writer’s attempt to articulate an inchoate vision of the future that was beginning to form in her mind. __ UNIT 42 ineluctable adj. not to be avoided or escaped; inevitable No one can escape the ineluctable truth that every creature that is born will one day die. inert adj. unable to move; sluggish The teacher was frustrated by his inability to get an answer to his question from his inert class. The noun is inertia, meaning disinclination to action or change. The fact that industrialization occurred in Europe hundreds of years before it did in China, which had reached a similar level of technology, is perhaps attributable to cultural factors such as bureaucratic inertia in China and a culture that placed a high value on the status quo. 165 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences status quo: the existing state of affairs (Latin, state in which) ingenuous adj. naive and trusting; lacking sophistication The conman could not bring himself to take advantage of the ingenuous boy. inherent adj. firmly established by nature or habit Some studies of random numbers generated by computers suggest that an inherent order exists in nature, since certain patterns appear that one would not expect in a random system, but skeptics dismiss such patterns as either artifacts of imperfectly designed experiments, or as the attempt of the human mind to impose a pattern where there is no intrinsic order. innocuous adj. harmless The bodyguard looked innocuous enough, but under his jacket were several weapons that could kill an attacker in seconds. insensible adj. unconscious; unresponsive The gas is intended to render enemy soldiers insensible. insinuate v. to suggest; say indirectly; imply If you read his speech carefully you will see that the senator is insinuating that his party has taken the wrong path. insipid adj. lacking in flavor; dull Ironically, the book about how to write lively, engaging prose is an insipid piece of writing. insouciant adj. indifferent; lacking concern or care Considering the gravity of the situation, Nancy’s colleagues could not understand her insouciant attitude. insularity n. narrow-mindedness; isolation The insularity of many tribes in New Guinea allows anthropolo gists to study cultures that have been relatively uninfluenced by the modern world. 166 ESSENTIAL WORDS FOR THE GRE REVIEW 42 The correct answers are given on page 405. Matching Match each word with its definition. 1. ineluctable (A) to suggest; say indirectly 2. inert (B) indifferent; lacking concern 3. ingenuous (C) unable to move 4. inherent (D) unconscious; unresponsive 5. innocuous (E) lacking in flavor; dull 6. insensible (F) not to be avoided or escaped 7. insinuate (G) firmly established by nature or habit 8. insipid (H) narrow-mindedness; isolation 9. insouciant (I) naive and trusting 10. insularity (J) harmless Fill-ins Choose the best word to fill in the blank in each sentence. ineluctable inert ingenuous inherent innocuous insensible insinuating insipid insouciance insularity 1. The referee stopped the bout after one boxer was rendered __ . 2. The country’s _ makes it difficult for its people to accept ideas from different cultures. 3. Indonesians who travel to America sometimes find the food so _ that they add chili to it. 4. The Internet “scam” relies on __ people to sign up and spend money for which they get essentially nothing in return. 5. Scientists are still studying the question of how life arose from _ matter. 6. The lawyer apologized to the judge for _ that she was biased. 7. The “cool” look that many fashion models affect seems meant to convey a look of __ . 8. The judicious doctor knows that sometimes the best therapy is not physical but emotional, reassuring the patient that the illness will run its course as a result of the body’s _ powers of self-healing. 9. We cannot escape the _ truth that someone in the group has betrayed our cause. 167 ESSENTIAL WORDS FOR THE GRE 10. The toxic chemical is present in the drug in such minute amounts that it is __ . Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The mathematician has devised an ingenuous solution to the problem. 2. Innocuous weapons such as the hydrogen bomb are capable of killing millions of people in an instant. 3. The professor’s comment on the student’s essay read, “An insensible and incoherent piece of writing.” _ 4. Spicy, insipid dishes are popular throughout Southeast Asia. 5. In today’s interconnected world, countries that remain insular face the risk of falling behind technologically. UNIT 43 insuperable adj. insurmountable; unconquerable Attempts by the United States to develop an antiballistic missile system have met with limited success because of the almost insuperable difficulties presented by the speed of the approaching warhead that must be intercepted. intangible adj. not material When considering what occupation to pursue it is prudent to consider intangible rewards as well as financial ones. interdict v. to forbid; prohibit; to confront and halt the activities, advance, or entry of Under U.S. law, interdicted goods can be seized by customs officials. internecine adj. deadly to both sides The U.S. Civil War (1861–1865) was an internecine conflict that lead to the deaths of 620,000 soldiers out of the 2.4 million who fought in the war. interpolate v. to insert; change by adding new words or material The book The Five Gospels was produced by having leading Bible scholars vote on which sayings of Jesus they believe to be authentic and which they believe to have been interpolated by other writers. 168 ESSENTIAL WORDS FOR THE GRE interregnum n. interval between reigns; gap in continuity Those who believe that Western culture represents the culmination of history are not disheartened by considering the fall of previous dominant civilizations, believing that these were merely interregnums in the march of humanity from the cave to a united world founded on Western principles. intimate adj. marked by close acquaintance Intimate is pronounced IN-tuh-mit. During the 1990s Bob Dylan and Jerry Garcia became good, though not intimate, friends. The noun is intimacy. The American artist Grandma Moses, although considered by art experts to be deficient in technique, achieved an admirable intimacy with her subject matter. The verb intimate means to make known subtly and indirectly. It is pronounced IN-tuh-mayt. The editor intimated that substantial changes would have to be made in the book. intractable adj. not easily managed General practitioners are equipped to deal with most psychoso matic disorders, but in intractable cases a psychiatrist is consulted. Terms from the Arts, Sciences, and Social Sciences psychosomatic disorder: a disease with physical symptoms believed to be caused by emotional or psychological factors intransigence n. stubbornness; refusal to compromise Each side in the negotiations accused the other of intransigence, so talks broke down. introspective adj. contemplating one’s own thoughts and feelings In many ways William Wordsworth’s great poem The Prelude is an introspective work, retrospectively exploring his thoughts and feelings as he matured. 169 ESSENTIAL WORDS FOR THE GRE REVIEW 43 The correct answers are given on page 405. Matching Match each word with its definition. 1. insuperable (A) stubbornness 2. intangible (B) insurmountable 3. interdict (C) not easily managed 4. internecine (D) not material 5. interpolate (E) deadly to both sides 6. interregnum (F) marked by close acquaintance 7. intimate (G) interval between reigns 8. intractable (H) contemplating one’s own thoughts and feelings 9. intransigence (I) to forbid 10. introspective (J) to insert Fill-ins Choose the best word to fill in the blank in each sentence. insuperable intangible interdicting internecine interpolated interregnum intimate intractable intransigence introspection 1. The __ of both sides means that there will be no prog ress in the peace talks. 2. Over the years the boss and her assistant have become _ friends as well as colleagues. 3. Since, according to the theory of relativity, an object traveling at the speed of light would have infinite mass, astronauts traveling at that speed would, presumably, face ___ difficulties. 4. Military intelligence officers played a major role in __ spies attempting to pass top-secret intelligence to the enemy. 5. In addition to providing a salary, a job often provides _ benefits such as camaraderie with colleagues. 6. Scholars disagree on whether the text is entirely the work of the original author or contains passages _ by later writers. 7. The book analyzes the __ struggles within Christianity throughout its history. 8. The injunction “Know Thy Self,” which was inscribed over the sanctuary of Apollo at Delphi, suggests that for spiritual advance ment it is necessary to engage in _ . 9. The __ between the two empires was a period of near anarchy. 170 ESSENTIAL WORDS FOR THE GRE 10. Although the majority of Americans are members of what has been called the “affluent society,” poverty remains an _ problem, with a sizable minority of people living below what is considered to be an acceptable standard of living. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The king’s interregnum lasted 22 years, during which time he presided over a happy and peaceful kingdom. 2. Greater intransigence on the part of both sides will increase the chance of an agreement. _ 3. The problem seemed intractable at first, but after we analyzed it as being the result of a number of smaller problems, we were able to solve it. 4. The old text contains a number of interpolations by a rival group seeking to justify their views. 5. Many African countries are beset by internecine conflict between rival tribes. _ UNIT 44 inundate v. to cover with water; overwhelm Farmers in the arid areas called for the government to build a dam to provide water to irrigate their crops and provide hydroelectric power; however, this plan was opposed by environmentalists, who dislike inundation of land because it would have an adverse effect on wildlife. inured v. hardened; accustomed; used to After 20 years in the army, the chaplain had not become inured to the sight of men dying on the battlefield. invective n. verbal abuse The debate judge cautioned participants not to engage in invective, but rather in reasoned and decorous discourse. inveigh v. to disapprove; protest vehemently The conservative writer inveighed against the school board’s decision to exclude moral education from the curriculum. 171 ESSENTIAL WORDS FOR THE GRE inveigle v. to win over by flattery or coaxing The students inveigled their professor into postponing the test for a week. inveterate adj. confirmed; long-standing; deeply rooted The columnist is an inveterate iconoclast who continually questions conventional wisdom. invidious adj. likely to provoke ill will; offensive Most publications in the United States prohibit their writers from making invidious comparisons between racial groups. irascible adj. easily angered The irascible old man complains every time someone makes a little noise. irresolute adj. unsure of how to act; weak The president admonished Congress, saying that although it faced difficult choices it must not be irresolute. itinerant adj. wandering from place to place; unsettled According to state law, companies hiring itinerant workers must provide adequate housing for them. REVIEW 44 The correct answers are given on page 405. Matching Match each word with its definition. 1. inundate (A) to disapprove; protest vehemently 2. inured (B) hardened; accustomed 3. invective (C) wandering from place to place 4. inveigh (D) to overwhelm 5. inveigle (E) verbal abuse 6. inveterate (F) confirmed; long-standing 7. invidious (G) unsure of how to act; weak 8. irascible (H) likely to provoke ill will 9. irresolute (I) easily angered 10. itinerant (J) to win over by flattery 172 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. inundated inured invective inveigh inveigle inveterate invidious irascible irresolute itinerant 1. The talk show host uses __ to anger his guests so that they say things they ordinarily would not. 2. The _ young man gets into a fight practically every weekend. 3. The book makes _ comparisons between French and American culture. 4. The writer spent his twenties as a/an __ salesperson traveling throughout the Midwest. 5. The country’s leaders regularly _ against “the corrupting influence of Western decadence.” 6. Some developing countries argue that they lack the capacity to compete in a completely free world market, and that in such a situation their domestic market would be __ with foreign goods to the detriment of local manufacturers. 7. An _ gambler, every year Tom offers his family a choice of two vacation destinations—Las Vegas, Nevada, or Atlantic City, New Jersey. 8. War has raged for so long in the country that people have become _ to violence. 9. The president warned the nation that we must not be __ in our determination to prevent terrorism. 10. I was amazed how Charlie, Doris, and Marcia managed to _ Fred into playing bridge, a game he finds completely boring. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Sam inured himself for one million dollars before going on the dangerous expedition. 2. Every summer, the apple orchard hires itinerant workers to pick the apples. _ 3. The educators are concerned that students are being inundated with so much information that they have trouble making sense of it. 173 ESSENTIAL WORDS FOR THE GRE 4. Medical researchers are working on a cure for various types of invective. __ 5. The pastor warned his congregation that they must not be irresolute in facing evil. _____ UNIT 45 itinerary n. route of a traveler’s journey We planned our itinerary to be flexible, so that if we especially enjoyed a particular place we could stay there longer. jaundiced adj. having a yellowish discoloration of the skin; affected by envy, resentment, or hostility Norman’s experience as an infantryman during the war has given him a jaundiced view of human nature. The noun jaundice refers to a medical condition often due to liver disease and characterized by yellowness of the skin. jibe v. to be in agreement The auditor checked the company’s account books to make sure that they jibed with the tax return it filed. jocose adj. fond of joking; jocular; playful The English words jocose, jocular, and joke all come from deriva tives of the Latin noun jocus, which means “jest” or “joke,” but the etymology of the word jocund is unrelated to these. Jocose (fond of joking; jocular; playful) is from Latin jocosus (humorous, merry, sportive), from jocus. Jocular (fond of joking; playful; speaking in jest) is from Latin jocularis (jocular; laughable), also from jocus. Jocund (mirthful; merry; light-hearted; delightful) is from jocundus (pleasant, agreeable), from juvare (to delight). juggernaut n. huge force destroying everything in its path Some people in Britain regard American English as a juggernaut sweeping through the British Isles, destroying British English. junta n. group of people united in political intrigue The country’s ruling junta consists of a general, an admiral, and the mayor of the capital city. 174 ESSENTIAL WORDS FOR THE GRE juxtapose v. place side by side To illustrate their case, opponents of functionalism juxtapose the products of modern architecture and those of classical architecture, such as the Parthenon, or those of medieval architecture, such as the Cathedral of Notre-Dame. The noun juxtaposition means a side-by-side placement. Terms from the Arts, Sciences, and Social Sciences functionalism: twentieth-century aesthetic doctrine in architecture. Functionalists believe that the outward form of a structure should follow its interior function. kudos n. fame; glory; honor Kudos won by Bob Dylan include an honorary doctorate in music from Princeton University. labile adj. likely to change Blood pressure in human beings is, to varying degrees, labile. laconic adj. using few words The laconic actor seemed to be a good choice to play the strong, silent hero in the western. REVIEW 45 The correct answers are given on page 405. Matching Match each word with its definition. 1. itinerary (A) to be in agreement 2. jaundiced (B) to place side by side 3. jibe (C) fond of joking; jocular 4. jocose (D) likely to change 5. juggernaut (E) having a yellowish discoloration of the skin 6. junta (F) fame; glory 7. juxtapose (G) group of people united in political intrigue 8. kudos (H) route of a traveler’s journey 9. labile (I) using few words 10. laconic (J) huge force destroying everything in its path 175 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. itinerary jaundiced jibe jocose juggernaut junta juxtaposed kudos labile laconic 1. During the first several years of World War II, the German army was a/an __ , easily defeating any force that tried to stop it. 2. A military _ seized power in the country in 1988. 3. Dr. Taylor’s considerable girth and _ manner made him the obvious choice to play Santa Claus in the faculty Christmas play. 4. The __ for our visit to Edinburgh, Scotland included a visit to Edinburgh University and Edinburgh Castle. 5. The psychologist’s diagnosis was that Eric was emotionally _ . 6. The textual scholar _ the two translations in order to compare them. 7. Infectious hepatitis is a viral form of hepatitis that causes fever and makes a person’s skin __ . 8. Most scientists regard the Noble Prize as the highest _ they can receive. 9. It is difficult for a person who tends to be _ to learn how to speak a new language. 10. Listening to the witness’ testimony, the judge discovered that it did not __ with the account of the incident he had given to the police. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The young jazz trumpeter decided he should learn to “talk the jibe.” 2. The juggernauts performed amazing feats of legerdemain that had the children laughing all afternoon. 3. In a healthy individual body temperature is not labile. _ 4. The host has decided to seat people at the formal dinner so that people who tend to be laconic sit next to individuals that are more garrulous. 5. Thirty years on the police force has given Captain Lucas a jaundiced view of life. __ 176 ESSENTIAL WORDS FOR THE GRE UNIT 46 lambaste v. to thrash verbally or physically The critic lambasted the movie in her column, calling it “the most insipid, jejune film made in our generation.” lascivious adj. lustful The court ruled that the movie could be censored because its sole aim was to promote lascivious thoughts. lassitude n. lethargy; sluggishness After the death of his wife, Steven suffered a three-month period of lassitude and depression. latent adj. present but hidden; potential Some experts in human psychology believe that we are just begin ning to explore the latent powers of the human mind. laud v. to praise The literary critic lauded Jane Austen’s Pride and Prejudice, calling it a novel that “explores the tension between a person’s life as a social being and his or her individual consciousness.” lethargic adj. inactive After the 18-hour flight from New York to Singapore, the passengers were lethargic. levee n. an embankment that prevents a river from overflowing An extensive system of levees is the only way to prevent the river from flooding the area during periods of heavy rain. levity n. light manner or attitude The comedian has a gift for finding an element of levity in the most serious of subjects. liberal adj. tolerant; broad-minded; generous; lavish Bankruptcy laws should not be too stringent, or not enough people will venture their capital; on the other hand, they should not be too liberal, or entrepreneurs will take unreasonable risks and waste capital. libertine n. one without moral restraint Don Juan is a legendary, archetypal libertine whose story has been told by many poets, such as Lord Byron. 177 ESSENTIAL WORDS FOR THE GRE REVIEW 46 The correct answers are given on page 405. Matching Match each word with its definition. 1. lambaste (A) embankment that prevents a river from overflowing 2. lascivious (B) to thrash verbally or physically 3. lassitude (C) to praise 4. latent (D) lustful 5. laud (E) inactive 6. lethargic (F) tolerant 7. levee (G) present but hidden; potential 8. levity (H) light manner or attitude 9. liberal (I) person without moral restraint 10. libertine (J) lethargy; sluggishness Fill-ins Choose the best word to fill in the blank in each sentence. lambasted lascivious lassitude latent lauded lethargic levee levity liberal libertine 1. Engineers worked to reinforce the __ after the prediction of an unprecedented amount of rain. 2. To everyone’s surprise, the 14-point underdog _ the reigning champions 42–0. 3. Suddenly overcome by _ in the afternoon, Jill decided to take a nap. 4. The former president was __ for his indefatigable efforts to bring peace to the war-torn area. 5. In the view of some commentators, a paradox of modern _ democracy is that although people have more free dom than ever, they often are unable to use this freedom to find meaningful values and goals. 6. The goal of the course is to help people develop their _ abilities. 7. After the long winter layoff, many of the baseball players were __ at the first day of spring training. 8. The bikini-clad young woman attracted _ stares from a group of men. 9. The speaker decided to tell a joke to introduce some _ into the solemn occasion. 178 ESSENTIAL WORDS FOR THE GRE 10. James Boswell, the eighteenth-century Scottish writer best remembered for his biography of the eminent literary figure Samuel Johnson, was a heavy drinker and a __ . Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The captain lauded his troops into battle. 2. The psychologist suggested that the patient take life less seriously and try to introduce some levity into her life every day. 3. The picnickers were overcome by lassitude after eating a heavy lunch. _ 4. Carol discovered late in life that she had a latent ability for mathematics. 5. The women’s rights group condemned the swimsuit part of the Miss Galaxy contest “designed solely to appeal to men’s lascivious impulses.” __ UNIT 47 libido n. sexual desire According to psychologists, the libido of human males peaks at around the age of 18. Lilliputian adj. extremely small Microbiologists study Lilliputian organisms. limn v. to draw; describe The artist based his painting on a sketch he had limned several years earlier. limpid adj. clear; transparent At the bottom of the limpid pond we could see hundreds of fish swimming. linguistic adj. pertaining to language Humans are at the acme of their linguistic proficiency in the first several years of life, during which they master thousands of complex grammatical operations. Linguistics is the scientific study of language. A linguist is someone who studies language. 179 ESSENTIAL WORDS FOR THE GRE Linguists such as Noam Chomsky believe that what people come to know and believe depends on experiences that evoke a part of the cognitive system that is latent in the mind. litany n. lengthy recitation; repetitive chant The student listened intently to his teacher’s litany of the grammatical errors committed by the class. literati n. scholarly or learned persons “Any test that turns on what is offensive to the community’s standards is too loose, too capricious, too destructive of freedom of expression to be squared with the First Amendment. Under that test, juries can censor, suppress, and punish what they don’t like, provided the matter relates to ‘sexual impurity’ or has a tendency ‘to excite lustful thoughts.’ This is community censorship in one of its worst forms. It creates a regime where in the battle between the literati and the Philistines, the Philistines are certain to win.” —U.S. Supreme Court justice William O. Douglas, dissenting in the case of Roth v. United States, 1957. Terms from the Arts, Sciences, and Social Sciences First Amendment: a part of the United States Bill of Rights prohibit-ing the federal legislature from making laws that establish a state religion or prefer a certain religion, prevent free exercise of religion, infringe the freedom of speech; infringe the freedom of the press; limit the right to assemble peaceably; limit the right to petition the government for a redress of grievances Philistines: people considered to be ignorant of the value of cultures and smug and conventional in their thinking litigation n. legal proceedings The radio amateur’s neighbor resorted to litigation in an attempt to have her neighbor dismantle his 100-foot-high antenna tower. log n. record of a voyage; record of daily activities Although no longer required to do so by the Federal Communications Commission, many amateur radio operators nevertheless keep a meticulous record of stations they communicate with, logging the details of each contact. 180 ESSENTIAL WORDS FOR THE GRE loquacious adj. talkative Eighty meters is a portion of the radio spectrum where a shortwave listener can often hear loquacious “hams” chatting (“chewing the rag” in amateur radio parlance) for hours. REVIEW 47 The correct answers are given on page 405. Matching Match each word with its definition. 1. libido (A) transparent 2. Lilliputian (B) sexual desire 3. limn (C) legal proceedings 4. limpid (D) to draw; describe 5. linguistic (E) talkative 6. litany (F) extremely small 7. literati (G) lengthy recitation 8. litigation (H) scholarly or learned persons 9. log (I) record of a voyage 10. loquacious (J) pertaining to language Fill-ins Choose the best word to fill in the blank in each sentence. libido Lilliputian limning limpid linguistics litany literati litigation logs loquacious 1. The study’s hypothesis is that the low birthrate is a result of a reduction in many people’s __ . 2. According to the historian Richard J. Hofstadter, there has been a strong feeling of suspicion of the _ throughout American history. 3. The _ of the eighteenth-century ships’ captains provide an interesting perspective on that time. 4. The judge warned the __ attorney to stop digressing and “cut to the chase.” 5. The critic praised the novel for its _ prose and original characters. 6. The United Nations Human Rights Commission outlined a _ of the rights regularly being abused in the country. 7. The threat of __ was enough to induce the company to settle the claim against it. 181 ESSENTIAL WORDS FOR THE GRE 8. The writer Somerset Maugham had a gift for _ a character perfectly in a few paragraphs. 9. Applied _ takes the findings of theoretical linguistics and applies them to such areas as language learning. 10. After his experiences in the war, the problems Howard encountered in civilian life seemed positively __ . Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. If ants can perceive human beings, we must appear Lilliputian to them. 2. Exhaustive litigation has proven that gravity exists throughout the universe. 3. To the unaided eye the liquid appears limpid, but in reality it contains millions of microscopic organisms. _ 4. Magazines read regularly by most members of the New York literati include The New Yorker and the New York Review of Books. 5. The judge warned the witness not to use the occasion to give a litany of his personal grievances. __ UNIT 48 lucid adj. bright; clear; intelligible The eminent surgeon Dr. Christian Barnard, who performed the first human heart-transplant operation in 1967, made his views on euthanasia clear in this lucid injunction: “The prime goal is to alleviate suffering, and not to prolong life. And if your treatment does not alleviate suffering, but only prolongs life, that treatment should be stopped.” lucre n. money or profits Many religions regard the pursuit of lucre for what it can do to help others as laudable. luminous adj. bright; brilliant; glowing The Moon is the most luminous object in the night sky. The noun is luminosity. A supernova can suddenly increase its luminosity to as much as a billion times its normal brightness. 182 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences supernova: a rare astronomical event in which most of the mate-rial in a star explodes, resulting in the emission of vast amounts of energy for a short period of time lustrous adj. shining On the clear night we gazed up in awe at the lustrous stars. Machiavellian adj. crafty; double-dealing One theory of the evolution of high intelligence in primates is that it evolved largely as a result of Machiavellian calculations on the part of apes. Terms from the Arts, Sciences, and Social Sciences Machiavelli: Niccolo Machiavelli (1469–1527) was an Italian philoso-pher known for his writings on how a ruler should govern, notably by favoring expediency over principles. machinations n. plots or schemes The mayor resorted to behind-the-scenes machinations to try to win his party’s nomination for governor. maelstrom n. whirlpool; turmoil Nearly everyone in Europe was caught up in the maelstrom that was World War II. magnanimity n. generosity; nobility The senator showed his magnanimity when he conceded defeat to his opponent in the disputed election, saying that further uncertainty would be harmful to public confidence in the political system. malign v. to speak evil of Lawyers are sometimes maligned as greedy and dishonest. malinger v. to feign illness to escape duty In order to discourage malingering, the company decided to require employees taking sick leave to produce a doctor’s certification of their illness. 183 ESSENTIAL WORDS FOR THE GRE REVIEW 48 The correct answers are given on page 406. Matching Match each word with its definition. 1. lucid (A) bright; brilliant; glowing 2. lucre (B) money or profits 3. luminous (C) generosity; nobility 4. lustrous (D) plots or schemes 5. Machiavellian (E) to feign illness to escape duty 6. machinations (F) whirlpool; turmoil 7. maelstrom (G) clear; intelligible 8. magnanimity (H) to speak evil of 9. malign (I) crafty; double-dealing 10. malinger (J) shining Fill-ins Choose the best word to fill in the blank in each sentence. lucid lucre luminous lustrous Machiavellian machinations maelstrom magnanimity maligned malingering 1. The magazine Scientific American can be relied on to provide _ discussions of complex scientific topics. 2. We could only imagine the _ maneuvering that allowed Stan to replace his boss as the company’s manager. 3. The Sun is by far the most __ object in the daytime sky. 4. Tired of being _ as a coach who “can’t win the big games,” Coach Butler resolved that his team would be ready for the Super Bowl. 5. The lure of _ draws many people to speculate in the stock market. 6. Harriet Beecher Stowe described saintliness as “a certain quality of __ and greatness of soul that brings life within the circle of the heroic.” 7. The soldiers marched toward battle under the _ Moon. 8. The book tells the story of a young British soldier thrust into the _ of the Napoleonic Wars. 9. One of a military commander’s most difficult tasks is to separate soldiers who are seriously battle-stressed from those who are merely __ . 184 ESSENTIAL WORDS FOR THE GRE 10. No one outside a few powerful party leaders could say by what _ they had managed to have their crony nominated to run for governor. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Several of us malingered late at the party, discussing politics. 2. The dual pursuits of lucre and adventure have been the motivation of many explorers throughout history. _ 3. The mechanic maligned my tires, so I took my car to another mechanic. 4. Most offices seem to have at least one Machiavellian schemer, ready to do almost anything to get ahead. 5. Eric proposed to Wendy, calling her eyes “as lustrous as this diamond that will soon be on your finger.” _ UNIT 49 malleable adj. capable of being shaped by pounding; impressionable Behaviorists such as B. F. Skinner believe that human nature is malleable, and that people’s behavior can be changed by changing their environment. Terms from the Arts, Sciences, and Social Sciences Behaviorists: followers of behaviorism, the school of psychology that seeks to explain behavior entirely in terms of observable responses to environmental stimuli maverick n. dissenter Bernie Sanders of Vermont has a reputation as a maverick; he is one of only two members of the United States Congress who is independent (that is, not a member of the Republican or Democratic Party). megalomania n. delusions of power or importance In his farewell speech the retiring trial judge warned his colleagues to beware of megalomania as they exercise their power in the courtroom. 185 ESSENTIAL WORDS FOR THE GRE menagerie n. a variety of animals kept together Linda seems to take home every abandoned pet in the town; she now has an incredible menagerie of dogs, cats, turtles, rabbits, and other animals. mendacious adj. dishonest The judge ruled the testimony inadmissible because he considered it mendacious. mendicant n. beggar In Thailand it is traditional for young men to become monks for a year, a period during which they become mendicants. meretricious adj. gaudy; plausible but false; specious One of the allures of jargon is that it can make a poor idea appear worthwhile, or something meretricious easier to accept because it is dressed in fancy language. mesmerize v. to hypnotize The audience sat, mesmerized, listening to the retired soldier’s account of hand-to-hand combat against the Japanese in New Guinea during World War II. metamorphosis n. change; transformation In recent years, many areas of China have been undergoing a metamorphosis, transforming themselves from predominantly agricultural areas to industrial ones. metaphysics n. a branch of philosophy that investigates the ultimate nature of reality To skeptics, metaphysics is an arbitrary search for a chimerical truth. Metaphysical is an adjective meaning pertaining to metaphysics. Some critics of evolution object to its implication that human thought is reduced to a peripheral phenomenon; they find it implausible that the ability to conceptualize—to write a sonnet, a symphony, a metaphysical treatise—would have evolved in early hominids solely as a secondary effect. Metaphysician is a noun meaning a person who is an expert in metaphysics. Whether we are aware of it or not, we are all metaphysicians in the sense that we all have beliefs about what things are the most real; for example, a person who believes in God may believe that God is the “ultimate reality.” 186 ESSENTIAL WORDS FOR THE GRE REVIEW 49 The correct answers are given on page 406. Matching Match each word with its definition. 1. malleable (A) dissenter 2. maverick (B) variety of animals kept together 3. megalomania (C) transformation 4. menagerie (D) beggar 5. mendacious (E) delusions of power 6. mendicant (F) branch of philosophy that examines the nature of reality 7. meretricious (G) hypnotize 8. mesmerize (H) impressionable 9. metamorphosis (I) gaudy 10. metaphysics (J) dishonest Fill-ins Choose the best word to fill in the blank in each sentence. malleable mavericks megalomania menagerie mendacious mendicant meretricious mesmerized metamorphosed metaphysical 1. Realist novelists such as Charles Dickens seem to have had little interest in _ questions; rather, they seem to have been interested mainly in analyzing social and psychological reality. 2. We were amazed when we saw Lionel after ten years; he had _ from a lazy, carefree young man into a hard-working and responsible member of the community. 3. Tom spent one year as a __ monk before becoming a priest. 4. It is hard to escape the feeling that it requires at least a touch of _ to run for the office of President of the United States. 5. For many years the prevailing view among social scientists was that human nature is essentially __ ; however, recent thinking in the field has placed more emphasis on the part played by genes in human nature. 6. The World Wide Web has made it easier for _ to have their views on controversial issues heard. 7. The judge ruled that the defendant’s argument was rejected as disingenuous and _ . 8. The students, __ by the professor’s fascinating lecture, did not realize the class had run overtime. 187 ESSENTIAL WORDS FOR THE GRE 9. The writer’s biographer could not escape the conclusion that her subject had given _ testimony on various occasions. 10. The local SPCA shelter has a _ of animals—parrots, cats, dogs, and many others. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. One thing that no one disputes is that metaphysics does more than any other area of human pursuit to put food on the table. _ 2. Many people consider it unfair that approximately two hundred super-wealthy mendicants control 60 percent of the country’s wealth. 3. In four years, Leonard Rice has metamorphosed from a gangling 140-pound freshman third-string football player into a 210-pound All-State tailback. 4. The party leader can always count on the vote of a group of loyal party mavericks. _ 5. The teacher regards her students as malleable clay that she can mold into fine, intelligent young people. UNIT 50 meteorological adj. concerned with the weather Some experts believe that reports of UFOs are attributable to natural astronomical or meteorological phenomena. Meteorology is a science that deals with weather and atmospheric phenomena. Meteorologists are those who study meteorology or forecast weather conditions. The term “butterfly effect” to refer to the process driving chaotic systems was first used in 1979 by meteorologist E. M. Lorenz in an address entitled, “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” meticulous adj. very careful; fastidious Science is an empirical field of study based on the belief that the laws of nature can best be discovered by meticulous observation and experimentation. 188 ESSENTIAL WORDS FOR THE GRE mettle n. courage; endurance In many cultures, young men are expected to test their mettle by performing difficult and dangerous tasks. mettlesome adj. full of courage and fortitude; spirited The mettlesome young officer was well regarded by all the senior officers. Do not confuse mettlesome with meddlesome, which means “inclined to interfere.” microcosm n. a small system having analogies to a larger system; small world For many years the atom was seen as a sort of microcosm of the larger universe, with electrons—analogous to the planets of a solar system—orbiting the nucleus, or “sun.” militate v. to work against The manager asked all of his employees to think of any factors that might militate against the project’s success. minatory adj. threatening; menacing Intelligence information suggests minatory troop concentrations on the border. minuscule adj. very small Ancient geological processes are beyond the scope of carbon-14 dating (which is at most 120,000 years) because the amount of carbon-14 in material from such processes that has not decayed is minuscule. minutia n. petty details President Ronald Reagan said that a president should concentrate on the formulation and execution of broad policy and leave the minutia of running the country to subordinates. misanthrope n. one who hates humanity One of the most famous misanthropes in literature is the protago nist of the seventeenth century French writer Moliere’s play Le Misanthrope (The Misanthrope). 189 ESSENTIAL WORDS FOR THE GRE REVIEW 50 The correct answers are given on page 406. Matching Match each word with its definition. 1. meteorological (A) courage; endurance 2. meticulous (B) very small 3. mettle (C) very careful; fastidious 4. mettlesome (D) to work against 5. microcosm (E) one who hates humanity 6. militate (F) a small system having analogies to a larger system 7. minatory (G) full of courage and fortitude; spirited 8. minuscule (H) concerned with the weather 9. minutia (I) threatening 10. misanthrope (J) petty details Fill-ins Choose the best word to fill in the blank in each sentence. meteorological meticulous mettle mettlesome microcosm militates minatory minuscule minutia misanthropic 1. After a month of inter-squad scrimmage, the members of the football team were eager to test their _ against another team. 2. __ data collected from around the world helps scientists to get an accurate picture of the world’s weather patterns. 3. In many of Arthur Conan Doyle’s Sherlock Holmes stories the detective reveals quite strong _ tendencies. 4. The student’s laziness _ strongly against the likelihood of his success. 5. In the retired general’s memoirs, he says that most of the battles he fought were won through a combination of courage on the part of soldiers, __ planning, and luck. 6. The student stood silent as the teacher scolded him, her hand making _ gestures. 7. Political pollsters keep a close watch on the town because they view it as a representative _ of American society. 8. The __ horse can only be controlled by a very skillful rider. 190 ESSENTIAL WORDS FOR THE GRE 9. Engineers decided that the anomaly was so _ that it could safely be ignored. 10. The general’s factotum deals with the _ of everyday life, leaving him free to do his job as commander of the Third Division. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The poison is so powerful that even minuscule amounts of it can cause harm. _ 2. The diary contains a meticulous record of the events of the poet’s life when she traveled to France in 1888. 3. “Stop being mettlesome and mind your own business,” we told the busybody. 4. “Not only do I not like human beings in the abstract, I don’t like even one individual member of the human race,” the misanthrope declared. _ 5. The scientist’s meteorological record deals exclusively with meteors and comets in orbit around the Sun. APPLYING YOUR KNOWLEDGE—41–50 1. Labor unions sometimes (i) _ the use of confrontational tactics and (ii) _ in negotiation as necessitated by the entrenched power of big business. Blank (i) Blank (ii) justify compliance abjure intransigence criticize submissiveness Explanation: An important clue to the answer to this question is the phrase “confrontational tactics,” which signals that extreme methods are being discussed. The phrase “as necessitated” signals that labor unions sometimes see such tactics as necessary, so it makes sense that they would justify (demonstrate to be right) such tactics. Because the word for Blank (ii) is paired with the words “confrontational tactics,” it should have a similar meaning, so intransigence (being uncompromis ing) is the correct answer. The correct answers are justify and intransigence. 191 ESSENTIAL WORDS FOR THE GRE 2. The name “impressionist” is a misnomer. Every painter is an impressionist insofar as he or she records his or her impressions, and all art is impressionistic. What Manet, the leader of the origi-nal movement, meant to say was that nature should not be painted (i) _, but as it “impresses” the painter. He and his few followers tried to change the name to “independents,” but the original name has clung to them. Manet was extravagant in method and disposed toward low life for a subject, which has always (ii) _ his popularity; but he was a very important man for his technical discoveries regarding the relations of light and shadow, the flat appearance of nature, and the exact value of color tones. Blank (i) Blank (ii) at all militated against inadvertently added to as it actually is sanctioned Explanation: The words “but as it ‘impresses’ the painter” signal that the correct answer for Blank (i) is as it actually is. Blank (ii): What is said about Manet in the sentence after the semi-colon is positive, so what is said about Manet before the semicolon must be negative because of the word “but.” Militated against (exerted influence against) is the correct answer because this creates a negative statement about Manet. The correct answers are as it actually is and militated against. UNIT 51 miscellany n. mixture of writings on various subjects The book is a fascinating miscellany collected from the writer’s life work. miscreant n. villain; criminal The public execution of miscreants was common in Great Britain in the eighteenth century. misogynist n. one who hates women Some people have called the philosopher Freidrich Nietzsche a misogynist because of the numerous negative comments he made about women. 192 ESSENTIAL WORDS FOR THE GRE mitigate v. to cause to become less harsh, severe, or painful; alleviate Although the Supreme Court under the leadership of Chief Justice Warren Burger did not rescind any of the fundamental rulings of the Warren Court that preceded it, its decisions did mitigate the effects of some of the rulings of the Warren Court. Terms from the Arts, Sciences, and Social Sciences Warren Court: Earl Warren was named chief justice of the Supreme Court in 1953, and served on the Court until 1969. Under his leadership the Supreme Court tended to interpret the Constitution boldly, frequently with the result that disadvantaged people were helped. Mitigation is a noun meaning the act of reducing the severity or pain fulness of something. Before sentencing the woman, the judge asked if she had anything to say in mitigation. mnemonic adj. related to memory; assisting memory In the introduction to a collection of poetry, By Heart, the British poet Ted Hughes says that “the more absurd, exaggerated, grotesque” the images used as a mnemonic device to help remember a poem, the easier it will be to recall. Mnemonics is a system that develops and improves the memory. Symbolic languages—the second generation of computer lan guages—were developed in the early 1950s, making use of mne monics such as “M” for “multiply,” which are translated into machine language by a computer program. modicum n. limited quantity The scientist Carl Sagan wrote about astronomy and other scien tific subjects in a way that enabled a reader with even a modicum of knowledge of science to understand what he was saying. mollify v. to soothe The prime minister tried to mollify people protesting the tax increase with a promise that she would order a study of other means to raise revenue. 193 ESSENTIAL WORDS FOR THE GRE monolithic adj. solid and uniform; constituting a single, unified whole In the fifteenth century, there was a significant movement to revitalize the Church from within; however, it had become so monolithic over the centuries and contained so many vested interests that piecemeal reform was difficult and ineffective. morose adj. ill-humored; sullen The assessment of some skeptical critics of existentialism is that it is generally a view of life created by a group of thinkers whose distinguishing characteristic is that they are morose. motley adj. many colored; made up of many parts The new political party is made up of a motley group of people who are unhappy with the existing parties. REVIEW 51 The correct answers are given on page 406. Matching Match each word with its definition. 1. miscellany (A) solid and uniform 2. miscreant (B) villain 3. misogynist (C) limited quantity 4. mitigate (D) ill humored; sullen 5. mnemonic (E) mixture of writings on various subjects 6. modicum (F) one who hates women 7. mollify (G) related to memory 8. monolithic (H) many colored; made up of many parts 9. morose (I) to alleviate 10. motley (J) to soothe Fill-ins Choose the best word to fill in the blank in each sentence. miscellany miscreant misogynist mitigate mnemonic modicum mollify monolithic morose motley 1. The writer was able to offer constructive criticism of the feminist movement without being called a _ . 2. To __ war “hawks,” the president ordered a one-week bombing campaign against the country. 194 ESSENTIAL WORDS FOR THE GRE 3. Socialists tend to view big business as _ ; however, many large corporations are in direct competition with one another, and thus collusion is usually not to their advantage. 4. Many people find it useful to use _ devices to memorize information. 5. The volume contains a __ of the writings of Walt Whitman. 6. Mr. Samuels was _ for over a month following the death of his beloved wife. 7. In the nineteenth century, accurate prognosis based on the history of disease began to be possible, but it was not until the twentieth century that doctors were able to actually cure a number of diseases rather than merely _ their effects. 8. “I’m not looking for adulation, just a __ of respect,” the angry teacher told his class. 9. The judge said she had no alternative but to sentence the _ to 20 years imprisonment. 10. The protest began with a _ group of people from virtu ally all occupations. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Mnemonic devices currently supply nearly 20 percent of the country’s electric power. _ 2. Anyone with even a modicum of common sense could see that the plan had little chance of success. 3. Hindus believe that one should not be morose as one approaches death, since physical death means only the death of the body and not the soul. 4. The speaker’s misogynist comments drew the ire of several women’s rights groups. _ 5. The president ordered the creation of a commission to study ways to mitigate the effects of unemployment on the poor. UNIT 52 multifarious adj. diverse Modern technology is so complex and multifarious that it requires thousands of specialists to devise and operate; thus, even a brilliant engineer could not by himself fabricate a sophisticated radio or computer without the help of existing black boxes and expertise. 195 ESSENTIAL WORDS FOR THE GRE mundane adj. worldly as opposed to spiritual; concerned with the ordinary Fundamentalists contend that the Bible’s account of the creation is literally true, while others believe that it is the retelling of a powerful myth current in the Middle East that sought to explain the mundane in spiritual language. Terms from the Arts, Sciences, and Social Sciences Fundamentalists: those who stress adherence to a set of basic beliefs, especially in religion. Specifically, fundamentalism refers to the movement in Protestantism stressing a literal interpretation of the Bible. necromancy n. black magic Television might seem like necromancy to a time traveler from the fifteenth century. negate v. to cancel out; nullify The soldiers’ poor treatment of the prisoners negated the goodwill they had built up among the population. neologism n. new word or expression The word “anesthesia” was the neologism of the American physician and poet Oliver Wendell Holmes, who used it in 1846 in a letter to Dr. William Morton, who had recently demonstrated the use of ether; the word is derived from the Latin word anaisthesia, meaning “lack of sensation.” neophyte n. novice; beginner The school provides extensive support and guidance for neophyte teachers. nexus n. a means of connection; a connected group or series; a center Wall Street is the nexus of America’s financial system. nonplussed adj. bewildered The members of the football team were nonplussed by the presence of a female reporter in the locker room. nostalgia n. sentimental longing for a past time The product’s marketing is centered on nostalgia for the 1950s. 196 ESSENTIAL WORDS FOR THE GRE The adjective is nostalgic. The idea of an extended family existing in nineteenth-century Amer-ica consisting of loving uncles and doting aunts has been shown to be largely a product of a nostalgic and romanticized view of the past. nostrum n. medicine or remedy of doubtful effectiveness; supposed cure Although there are many nostrums urged on obese consumers, the only effective remedy for this condition is prosaic but nonetheless valid: eat less and exercise more. REVIEW 52 The correct answers are given on page 406. Matching Match each word with its definition. 1. multifarious (A) to cancel out 2. mundane (B) novice 3. necromancy (C) black magic 4. negate (D) diverse 5. neologism (E) sentimental longing for a past time 6. neophyte (F) new word or expression 7. nexus (G) bewildered 8. nonplussed (H) remedy of doubtful effectiveness 9. nostalgia (I) a connected group or series 10. nostrum (J) worldly as opposed to spiritual Fill-ins Choose the best word to fill in the blank in each sentence. multifarious mundane necromancy negated neologisms neophyte nexus nonplussed nostalgia nostrums 1. A number of commentators have argued that the benefits offered by television are _ by its narcotic effect on viewers. 2. Some theologians regard attempts to prove God’s existence logically valuable largely as pointers toward God, helping to turn a person’s attention from the __ to the spiritual. 3. Even the normally unflappable police officer was _ when confronted by the armed suspect. 4. Many _ for “correcting” English to make it more consistent and “rational” have been proposed, but the language is robust and has survived such attempts. 197 ESSENTIAL WORDS FOR THE GRE 5. Although intelligence agents have identified parts of the terrorist organization around the world, they are still working to locate its __ . 6. Dr. Robert Burchfield, chief editor of the Oxford English Dictionary, has estimated that approximately 90 percent of English _ originate in the United States. 7. The head football coach at a Division I college has _ duties, such as supervising the coaching staff, recruiting players, and talking to the media. 8. The advertisement is based on __ for an America that probably never existed. 9. The _ novelist was fortunate to have the advice of an established older writer. 10. A colorful term used to belittle something regarded as nonsense is “voodoo”; another one is “_ .” Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Mrs. Morrison was nonplussed when she discovered that her husband was a humanoid creature from the planet Varga, a small planet in a nearby galaxy. _ 2. After suffering through ten losing football seasons in a row, the president of the college’s alumni association suggested—somewhat sarcastically, no doubt—hiring a necromancer to replace the current head coach. 3. It is generally advisable to avoid neologisms such as “like” and “and” when writing. 4. After running the giant corporation for 30 years, the retiring CEO found himself looking forward to a simple life doing mundane tasks around his house. _ 5. The speaker mounted the nostrum to give the keynote speech of the convention. UNIT 53 nugatory adj. trifling; invalid The historian has a knack for focusing on information that appears nugatory but that, upon examination, illuminates the central issue. obdurate adj. stubborn Coach Knight is obdurate about one thing: the offensive line is the heart of his football team. 198 ESSENTIAL WORDS FOR THE GRE obsequious adj. overly submissive Tom’s tendency to submit meekly to any bullying authority is so great that his wife suggested he overcome this obsequiousness by taking an assertiveness training course. obsequy n. funeral ceremony (often used in the plural, obsequies) Solemn obsequies were held for President John F. Kennedy follow ing his assassination on November 22, 1963. obviate v. to make unnecessary; to anticipate and prevent An experienced physician can often discern if a patient’s symptoms are psychosomatic, thus obviating the need for expensive medical tests. occlude v. to shut; block One of the primary uses of solar cells is in spacecraft to provide electric power; this is because space is an environment uniquely suited to these devices since it has no weather to occlude the Sun and it is not susceptible to interruptions in sunlight caused by the rotation of the Earth. occult adj. relating to practices connected with supernatural phenomena In his book Supernature the biologist Lyall Watson explores what he regards as phenomena on the border between natural and occult phenomena. odyssey n. a long, adventurous voyage; a quest Steve’s quest for enlightenment took him on a spiritual odyssey that helped him to gain an understanding of many philosophers and religions. officious adj. too helpful; meddlesome Some of us on the tour found the guide officious, but others thought she was helpful and courteous. olfactory adj. concerning the sense of smell Wine connoisseurs say that the olfactory senses play as impor tant a part in appreciating good wine as the sense of taste. 199 ESSENTIAL WORDS FOR THE GRE REVIEW 53 The correct answers are given on page 406. Matching Match each word with its definition. 1. nugatory (A) too helpful 2. obdurate (B) overly submissive 3. obsequious (C) stubborn 4. obsequy (D) a long voyage 5. obviate (E) to shut; block 6. occlude (F) funeral ceremony 7. occult (G) trifling; invalid 8. odyssey (H) practices connected with supernatural phenomena 9. officious (I) concerning the sense of smell 10. olfactory (J) to make unnecessary Fill-ins Choose the best word to fill in the blank in each sentence. nugatory obdurate obsequious obsequies obviated occludes occult odyssey officious olfactory 1. The assertiveness-training course helped Jeremy go from being __ to being assertive and confident. 2. Nuclear power has _ the needs for submarines to refuel frequently, allowing long undersea voyages. 3. Sometimes a/an _ stimulus can trigger a memory associated with that particular smell. 4. The director of the government agency encouraged workers to provide efficient service without being __ . 5. The _ has been described as what does not fit into a rationalistic view of the world. 6. Astronomers welcome an eclipse of the Sun because when the Moon _ the light of the Sun, observation of that body becomes easier. 7. The president is __ about the issue; he will not negotiate with terrorists. 8. In the television show Star Trek: The Next Generation, the Enterprise embarks on a/an _ to explore the Universe. 200 ESSENTIAL WORDS FOR THE GRE 9. After the judge ruled the evidence he had presented to the court to be _ , the lawyer muttered jocularly to his partner, “Negatory.” 10. Solemn __ were held for Pope John Paul II after his death in 2005. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Modern refinement in olfactory processes have made it possible to mass-produce complex electronic circuits. 2. Since the Sun was occluded by clouds, the sailor could not use it to determine his position. 3. After the couple retired they went on an odyssey around the world. _ 4. Science is concerned primarily with the study of occult phenomena. 5. The obdurate student refused to study despite repeated warnings that he would fail if he did not start to work in the course. __ UNIT 54 oligarchy n. form of government in which power belongs to only a few leaders In 411 b.c., democratic government was overthrown in Athens and a conservative oligarchy called the Four Hundred came to power. onerous adj. burdensome The duty the judge considers most onerous is sentencing convicted criminals. onomatopoeia n. formation or use of words that imitate sounds of the actions they refer to One theory of the origin of language is that it began as a sort of onomatopoeia as early humans imitated sounds they heard. opprobrium n. disgrace; contempt It is difficult to imagine the opprobrium heaped on a person who is a traitor to his or her group. ornithologist n. scientist who studies birds Ornithologists believe that there currently exist only about twenty individuals of a bird called the Balinese sparrow. 201 ESSENTIAL WORDS FOR THE GRE oscillate v. to move back and forth The teacher oscillates between a student-centered approach to teaching and a subject-centered approach. ostentatious adj. showy; trying to attract attention; pretentious A member of the bourgeoisie might purchase a vacation home on Maui or Cape Cod that some would regard as an ostentatious display of wealth, but that the person regards as simply a pleasant place to go on vacation. overweening adj. presumptuous; arrogant; overbearing The ancient Greeks believed that overweening pride—what they called hubris—would be punished, eventually, by the gods. paean n. song of joy or triumph; a fervent expression of joy Fundamentally, the poem is a paean of joy, celebrating the coming of democracy to the country. paleontology n. study of past geological eras through fossil remains Primatology, together with anthropology, paleontology, and several other fields, has given scientists a fairly accurate picture of the evolution of homo sapiens. A paleontologist is an expert in the field of paleontology. The attempts of the Jesuit priest and paleontologist Teilhard de Chardin to reconcile evolution and the Catholic dogma of original sin were regarded by Church authorities as nearly heretical, and he had to abandon his position in 1926. Terms from the Arts, Sciences, and Social Sciences Primatology: the branch of zoology that deals with the study of primates (that is, mammals belonging to any of the suborders of primates: Anthropoides (humans, great apes, and several others), Prosimi (lemurs and several others), and Tarsiodea. Primates are characterized by a high level of social interaction, flexible behavior, and use of hands. 202 ESSENTIAL WORDS FOR THE GRE REVIEW 54 The correct answers are given on page 407. Matching Match each word with its definition. 1. oligarchy (A) disgrace; contempt 2. onerous (B) showy 3. onomatopoeia (C) burdensome 4. opprobrium (D) song of joy or triumph 5. ornithologist (E) government by a few leaders 6. oscillate (F) to move back and forth 7. ostentatious (G) presumptuous; arrogant 8. overweening (H) scientist who studies birds 9. paean (I) study of past geological eras through fossil remains 10. paleontology (J) formation of words that imitate sounds of actions they refer to Fill-ins Choose the best word to fill in the blank in each sentence. oligarchy onerous onomatopoeia opprobrium ornithologists oscillating ostentatious overweening paeans paleontologists 1. After the end of the war, churches across the country rang out __ of joy. 2. The country is ruled by an _ consisting of senior military officers. 3. Over the last few days, the weather has been _ between sunny and cloudy. 4. __ are studying a bird that can fly without stopping from Scotland to Africa. 5. The physician faced the _ task of telling the patient that the disease was terminal. 6. The system of gathering, identifying, dating, and categorizing fossils allows _ to place newly discovered fossils in their proper place, making their picture of the past progressively more accurate. 7. An argument for the wearing of school uniforms is that it discourages __ displays of wealth through the wearing of expensive jewelry and clothing. 203 ESSENTIAL WORDS FOR THE GRE 8. The manager’s _ ambition led her to do something she regretted for the rest of her life: she told a lie about a vice-president to help her get his job. 9. The country incurred global _ for its poor treatment of prisoners of war. 10. The word “ping-pong” arose from __ ; the sound of the words is similar to the sound of a table tennis ball hitting first one paddle and then another. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Onomatopoeia helps scientists to understand the nature of the atom. 2. The paeans live a basic existence, subsisting mostly on rice and vegetables. 3. Ornithologists are concerned that Canadian geese migrating south no longer have enough places to rest and feed along the way. _ 4. Geologists called in a paleontologist to examine fossils they had uncovered. 5. When it was discovered that the scientist had published a paper based on data he knew was falsified, he received the opprobrium of the scientific community. __ UNIT 55 pallid adj. lacking color or liveliness Archeological evidence indicates that women have been using makeup to give color to a pallid face for millennia. panegyric n. elaborate praise; formal hymn of praise Many panegyrics were written to Abraham Lincoln in the years after his death, and he has become one of the most revered figures in American history. paragon n. model of excellence or perfection The epic poet Homer was regarded by the ancient Greeks as a paragon of literary excellence. 204 ESSENTIAL WORDS FOR THE GRE partisan adj. one-sided; committed to a party, group, or cause; prejudiced Supporters of constitutional monarchy believe that while in this system, as it is generally practiced today, virtually all power is vested in popularly elected assemblies, the institution of the monarchy continues to serve a purpose as a focus of national unity above the furor of partisan politics. pathological adj. departing from normal condition People sometimes confound psychology and psychiatry: the former is the science that studies cognitive and affective functions, both normal and pathological, in human beings and other animals, whereas the latter is a branch of medicine that deals with mental disorders. Pathology is the noun. Some of the most spectacular examples of spin-off in the twentieth century are the advances that have been made in medicine as an unforeseen result of pure biological research; an example of this is diagnostic testing for defective genes that predispose a person to certain pathologies. Pathos is a quality that causes a feeling of pity or sorrow. It is pronounced PAY-thahs. patois n. a regional dialect; nonstandard speech; jargon In Singapore the lingua franca is increasingly becoming Singapore English, widely regarded as a patois. paucity n. scarcity An argument sometimes advanced for euthanasia is that the amount of money spent on prolonging a person’s life for several months is exorbitant in relation to the paucity of funds available for preventive health programs and child health, both of which are highly cost-effective. pedantic adj. showing off learning The Sophists have acquired a reputation as being learned but rather pedantic entertainers who gave didactic talks on every sub ject under the Sun; the truth, however, is that some of the Sophist philosophers (notably Protagoras) were very able thinkers. The noun pedant means an uninspired, boring academic. 205 ESSENTIAL WORDS FOR THE GRE pellucid adj. transparent; translucent; easily understood Two writers often mentioned as having an admirably pellucid style are Bertrand Russell and George Orwell. penchant n. inclination Sue has a penchant for science, while her brother is more inter ested in the arts. REVIEW 55 The correct answers are given on page 407. Matching Match each word with its definition. 1. pallid (A) regional dialect; nonstandard speech 2. panegyric (B) one-sided 3. paragon (C) showing off learning 4. partisan (D) departing from normal condition 5. pathological (E) inclination 6. patois (F) transparent; easily understood 7. paucity (G) model of excellence 8. pedantic (H) lacking color or liveliness 9. pellucid (I) scarcity 10. penchant (J) elaborate praise Fill-ins Choose the best word to fill in the blank in each sentence. pallid panegyric paragons partisan pathology patois paucity pedantic pellucid penchant 1. Subtle differences in symptoms between one patient and another one with a similar condition allow a competent doctor to diagnose the nature of the underlying _ . 2. The textbook was so well written and edited that students describe it as “wonderfully __ .” 3. Academic writing should be erudite without being _ . 4. The job of political scientists is the objective study of government and politics; thus they are expected to be aloof from _ politics. 5. The people of the area speak a __ based on English, Spanish, and French. 206 ESSENTIAL WORDS FOR THE GRE 6. According to archeologists, Roman tiles were not the _ objects we see today; rather, they were painted a variety of vivid colors. 7. In his later years Lewis was able to indulge the _ for performing music that he had as a young man. 8. The business professor assigned her students to select the three firms they would consider __ for other companies to imitate. 9. No funeral _ for the slain general was as eloquent as the looks of grief on the faces of the mourners at his funeral. 10. The historian is unable to reach a definite conclusion about when the battle began because of a _ of evidence. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Every weekend the Scott family has a gathering on the patois. _ 2. The museum has an exhibition of elaborately carved penchants. 3. There is a paucity of specialist doctors in many rural areas of the United States. 4. The class became bored listening to the pedantic, long-winded professor. _ 5. Steve’s penchant for collecting things when he was a child led his mother to speculate that he might become a museum curate. UNIT 56 penury n. extreme poverty The autobiography tells the story of the billionaire’s journey from penury to riches beyond his imagining. peregrination n. a wandering from place to place Swami Vivekananda’s peregrinations took him all over India. peremptory adj. imperative; leaving no choice The general’s words were spoken in the peremptory tone of a man who is used to having his commands obeyed without question. 207 ESSENTIAL WORDS FOR THE GRE perennial adj. present throughout the years; persistent Perennial warfare has left most of the people of the country in poverty. perfidious adj. faithless; disloyal; untrustworthy The novel tells the story of the hero’s perfidious lover. perfunctory adj. superficial; not thorough; performed really as a duty The perfunctory inspection of the airplane failed to reveal structural faults in the wing. perigee n. point in an orbit that is closest to the Earth The Earth observation satellite reaches a perigee of 320 miles above the Earth’s surface. permeable adj. penetrable Wetsuits, used by divers in cold water, are permeable to water but designed to retain body heat. perturb v. to disturb greatly; make uneasy or anxious; cause a body to deviate from its regular orbit The findings that violence is increasing in schools greatly perturbed government officials. The noun perturbation means disturbance. Scientists believe that the Earth has undergone alternating periods of relatively cooler and warmer climate, and that this is due largely to fluctuations in the intensity of the greenhouse effect and perturbations in the Earth’s orbit around the Sun. pervasive adj. spread throughout every part It is a plausible hypothesis that the atheistic and materialistic philosophy of Marxism was readily accepted in China because of its similarities with Confucian views on spiritual matters, which had a pervasive influence in China for many centuries. The noun is pervasiveness. An indicator of the pervasiveness of psychotropic drugs in American society is the fact that approximately 50 percent of adults have used tranquilizers at some time in their lives. The verb is pervade. 208 ESSENTIAL WORDS FOR THE GRE REVIEW 56 The correct answers are given on page 407. Matching Match each word with its definition. 1. penury (A) penetrable 2. peregrination (B) superficial 3. peremptory (C) point in an orbit closest to body being orbited 4. perennial (D) present throughout the years 5. perfidious (E) to disturb greatly 6. perfunctory (F) extreme poverty 7. perigee (G) imperative 8. permeable (H) spread throughout every part 9. perturb (I) a wandering from place to place 10. pervasive (J) faithless; disloyal Fill-ins Choose the best word to fill in the blank in each sentence. penury peregrinations peremptory perennial perfidious perfunctory perigee permeable perturbed pervasive 1. Scientists calculate that the satellite will have a __ of 120 miles from Earth. 2. Our well draws water from a _ rock layer (an aquifer) in which the water is under pressure, so we generally do not have to use a pump. 3. Caricature is _ in the work of the English novelist Charles Dickens. 4. Once again, Congress debated the __ problem of the budget deficit. 5. While its diplomats were negotiating a peace settlement with the enemy, its _ leaders were planning a full-scale invasion. 6. The great expense of his continual legal battles has practically reduced the man to _ . 7. A proverb says that time heals everything; it might be com mented, however, that its healing is rarely complete and is often __ . 8. The rock band’s _ have taken it to over fifty cities around the world. 209 ESSENTIAL WORDS FOR THE GRE 9. Military leaders were _ by the report that important classified information had fallen into enemy hands. 10. The boss dismissed her employee’s suggestion with a __ laugh. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The consumer group accused the bank of using penury to amass vast profits. 2. The poet laureate wrote a perigee condemning the nation’s king as an incompetent ruler. 3. Astronomers believe that the distant star’s orbit is being perturbed by some unknown body. _ 4. The dictator was used to having his peremptory commands obeyed. 5. Typhoons are a perennial problem in the coastal areas of Southeast China during the late summer and early autumn. __ UNIT 57 petulant adj. rude; peevish The boy’s father worried that his disobedient and petulant child would grow up to be a bitter and annoying man. phlegmatic adj. calm in temperament; sluggish “Phlegmatic natures can be inspired to enthusiasm only by being made into fanatics.” (Friedrich Nietzsche) phoenix n. mythical, immortal bird that lives for 500 years, burns itself to death, and rises from its ashes; anything that is restored after suffering great destruction The captain believed the battalion had been destroyed by the enemy and was amazed to see it arise, phoenix-like, its men still fighting valiantly. physiognomy n. facial features The art teacher assigned her students to make drawings of people with a wide variety of physiognomy. 210 ESSENTIAL WORDS FOR THE GRE piety n. devoutness Saint Bernard of Clairvaux was a medieval French monk revered for his piety. piquant adj. appealingly stimulating; pleasantly pungent; attractive Many of the guests enjoyed the piquant barbecue sauce, but others found it too spicy for their taste. pique n. fleeting feeling of hurt pride Sally left the restaurant in a fit of pique after her date called to say he couldn’t come because he was working late. As a verb, pique means to provoke or arouse. The geologist’s curiosity was piqued by the unusual appearance of the rock formation. placate v. to lessen another’s anger; to pacify After his team’s third consecutive winless season, the Big State football coach opened his address to the irate alumni with a barrage of clichés and euphemisms to try to placate them: “Gentlemen, it is not my intention today to pull the wool over your eyes. Heaven only knows I have given my all. I have truly made the old college try. Unfortunately, however, by any reasonable criteria we have been less than completely successful in our endeavors, but I assure you that hope springs eternal in the human breast and next year we will rise to the occasion, put our noses to the grindstone and emerge triumphant in the face of adversity. I certainly admit that we have had a run of bad luck but that is nothing that can’t be cured by true grit and determination.” placid adj. calm We were amazed how the monk was able to remain placid despite the fire that was raging through the building. plaintive adj. melancholy; mournful After the battle all that could be heard was the plaintive cries of women who had lost their husbands. 211 ESSENTIAL WORDS FOR THE GRE REVIEW 57 The correct answers are given on page 407. Matching Match each word with its definition. 1. petulant (A) calm 2. phlegmatic (B) calm in temperament; sluggish 3. phoenix (C) rude; peevish 4. physiognomy (D) art of judging character from facial features 5. piety (E) mournful 6. piquant (F) mythical, immortal bird 7. pique (G) fleeting feeling of hurt pride 8. placate (H) pleasantly pungent 9. placid (I) to pacify 10. plaintive (J) devoutness Fill-ins Choose the best word to fill in the blank in each sentence. petulant phlegmatic phoenix physiognomy piety piquant piqued placated placid plaintive 1. Rebecca is a quiet person, but beneath a _ exterior lies a continual ferment of emotion. 2. The monk is admired for his __ . 3. The only sound after the battle was the _ cry of a soldier who had been disemboweled. 4. The teacher _ the students’ interest in geology by taking them on a field trip to look at rock formations. 5. Studies show that a person’s __ has an effect on his or her life; for example, people considered to have attractive features are more likely to be successful than those considered to be unattractive. 6. The _ child will not stop complaining that he does not like the present he has been given. 7. Japan rose like a _ from the destruction of World War II to become one of the world’s leading industrial nations. 8. The restaurant manager apologized for the poor service and __ the customer by saying that the meal was on the house. 9. The chef is known throughout Texas for his wonderfully _ sauces. 212 ESSENTIAL WORDS FOR THE GRE 10. The emergency room doctor trained herself to be _ despite the great suffering she witnessed every day. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The chef has prepared a range of plaintive desserts for our enjoyment. _ 2. People stare at the man because of his unusual physiog nomy. 3. After being destroyed by an atomic bomb in 1945, the Japanese city of Hiroshima rose like a phoenix to become once again one of Japan’s major cities. 4. Tom, with his phlegmatic and excitable personality, is not the person I would like to see in charge during a crisis. _ 5. The pastor urged the members of his congregation to show their piety by attending church every week. UNIT 58 plasticity n. condition of being able to be shaped or formed; pliability The sociologist is continually amazed by the plasticity of social institutions. platitude n. stale, overused expression Though Sarah’s marriage didn’t seem to be going well, she took comfort in the platitude that the first six months of a marriage were always the most difficult. platonic adj. spiritual; without sensual desire; theoretical Gradually what had been a platonic relationship between Tim and Kyoko became a romantic one. plethora n. excess; overabundance Because it deals with death and grieving, the funeral business has produced a plethora of euphemisms such as “slumber room” for the place where the corpse is placed for viewing. 213 ESSENTIAL WORDS FOR THE GRE plumb v. to determine the depth; to examine deeply A recurrent theme of mystical experience is “the dark night of the soul,” in which a person plumbs the depths of despair before finding a transcendent reality that brings the person closer to what he or she regards as God. The pronunciation of plumb is PLUM. Do not confuse plumb with the verb plume, which means to congratulate oneself in a self-satisfied way. John plumed himself on his ability to read both Sanskrit and Greek. plummet v. to fall; plunge The fighter jet, struck by an enemy missile, plummeted to earth. plutocracy n. society ruled by the wealthy It has been argued that modern democracies are plutocracies to the extent that wealth allows certain people to have a disproportion ately large influence on political decision-making. porous adj. full of holes; permeable to liquids If you go camping, make sure to spend enough money to buy a tent with a roof that is not porous. poseur n. person who affects an attitude or identity to impress others The critic labeled the writer a poseur who was more interested in getting the public’s attention than in writing good books. pragmatic adj. practical The cult of romantic love was a major factor in making a marriage for love, rather than for more pragmatic reasons, a ubiquitous phenomenon in the West by the nineteenth century. Pragmatism means a practical way of approaching situations or solving problems. Pragmatism is similar to Positivism in rejecting lofty metaphysical conceptions and in asserting that the main role of philosophy is to help clarify phenomena experienced. A pragmatist is someone who approaches situations in a practical way. The word “pragmatist” is often used to refer to someone who is willing to sacrifice his principles to expediency. 214 ESSENTIAL WORDS FOR THE GRE REVIEW 58 The correct answers are given on page 407. Matching Match each word with its definition. 1. plasticity (A) overused expression 2. platitude (B) full of holes; permeable to liquids 3. platonic (C) practical 4. plethora (D) excess 5. plumb (E) to fall; plunge 6. plummet (F) spiritual; without sensual desire 7. plutocracy (G) pliability 8. porous (H) society ruled by the wealthy 9. poseur (I) to examine deeply 10. pragmatic (J) person who affects an identity to impress others Fill-ins Choose the best word to fill in the blank in each sentence. plasticity platitudes platonic plethora plumbed plummet plutocracy porous poseur pragmatic 1. The _ of excellent rock bands makes it difficult for new bands to gain an audience. 2. The __ clay allows the track to dry quickly. 3. Scholars are not certain whether Socrates’ relation with his student Plato was only _ . 4. The poet William Wordsworth _ his own psyche in his masterpiece, The Prelude, or Growth of a Poet’s Mind. 5. The motivational speaker is full of __ , such as “Nothing succeeds like success.” 6. The members of the stage club finally realized that Anthony was a _ who enjoyed acting like an actor more than doing all the work necessary to be a real actor. 7. Some commentators have likened the United States more to a _ than a democracy because of the great power held by the rich. 8. A __ leader is not constrained by ideological preconcep tions and continually adjusts his plans to conform to reality. 215 ESSENTIAL WORDS FOR THE GRE 9. A compelling body of evidence has been built up by scientists suggesting that the _ of human nature is more limited than was generally believed by social scientists for much of the twentieth century. 10. Scientists predict that the orbit of the satellite will decay over the next few days and it will _ to Earth. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Anthropologists and sociologists tend to stress the plasticity of human nature, whereas biologists emphasize the role of genes. _ 2. State law forbids platonic relationships between members of the same family. 3. The tennis court is designed to be porous enough to dry thoroughly in a few hours. 4. The coach told the press, “It might be a platitude, but I really mean it: We’re taking the season one game at a time.” _ 5. Some of his friends consider Morris to be a bit of a poseur: he loves to hang out at the café, sipping an espresso and acting as if he were America’s most famous writer. UNIT 59 prate v. to talk idly; chatter The “talk radio” program allows people to call in and prate about their pet peeves. prattle n. meaningless, foolish talk The sociologist theorizes that what may seem like prattle often has an important social function: what might be labeled “gossip” is an important means for people to communicate valuable information about themselves and others. preamble n. preliminary statement Along with the opening words of the Declaration of Independence and the Gettysburg Address, the preamble to the Constitution of the United States contains some of the most memorable language in American history: “We the People of the United States, in order to form a more perfect Union, establish justice, insure domestic tranquil ity, provide for the common defense, promote the general welfare, and secure the blessings of liberty, to ourselves and our posterity, do ordain and establish this Constitution for the United States of America.” 216 ESSENTIAL WORDS FOR THE GRE precarious adj. uncertain The prime minister’s precarious hold on power ended when she lost a vote of confidence in Parliament. precept n. principle; law A good precept to follow in writing is to avoid redundancies such as “track record” (unless the record was set on a racecourse), “revert back,” “free gift,” and “general consensus.” precipitate v. to cause to happen; throw down from a height Full-scale American entry into World War II remained unpopular with the vast majority of Americans until a declaration of war was precipitated by the Japanese attack on the naval base at Pearl Harbor, a day that President Roosevelt predicted, in a memorable phrase, would “live in infamy.” precipitate adj. rash; hasty; sudden The secretary of state advised the president not to take precipitate action. Precipitous is another adjective meaning hasty; quickly with too little caution. Precipitation is water droplets or ice particles from atmospheric water vapor that falls to Earth. It would be helpful if the atmosphere could be induced to deposit its precipitation more evenly over the Earth’s surface, so that some land areas are not inundated while others remain arid. precursor n. forerunner; predecessor The precursor to the theory of plate tectonics was the theory of continental drift. Terms from the Arts, Sciences, and Social Sciences plate tectonics: geological theory stating that the outer part of the Earth’s interior is composed of two layers, one of which “floats” on the other. According to this theory, which is widely accepted by scientists, ten major plates move in relation to one another, creating such phenomena as earthquakes and mountain building along the boundaries of the plates. continental drift: the theory that the continents shift their positions over time 217 ESSENTIAL WORDS FOR THE GRE preempt v. to supersede; appropriate for oneself The movie was preempted for the president’s emergency address to the nation. prehensile adj. capable of grasping Many more animals in South America have prehensile tails than those in Southeast Asia and Africa, possibly because the greater density of the forest there favored this adaptation over the ability to glide through the trees. REVIEW 59 The correct answers are given on page 407. Matching Match each word with its definition. 1. prate (A) capable of grasping 2. prattle (B) talk idly 3. preamble (C) preliminary statement 4. precarious (D) cause to happen 5. precept (E) meaningless talk 6. precipitate (adj.) (F) supersede 7. precursor (G) principle; law 8. preempt (H) rash; hasty 9. prehensile (I) uncertain 10. precipitate (v.) (J) forerunner Fill-ins Choose the best word to fill in the blank in each sentence. prated prattle preamble precarious precepts precipitate precursor preempted prehensile precipitated 1. Thomas Edison’s famous laboratory in Menlo Park, New Jersey, was a _ to the great laboratories later created by corporations such as AT&T and IBM, out of which have poured a torrent of new techniques and devices. 2. Moral __ vary from society to society, but all societies have sanctions against certain acts, such as murder. 3. _ tails help many arboreal animals to find and eat food as they move through the trees. 218 ESSENTIAL WORDS FOR THE GRE 4. Steve earns a _ living as a part-time waiter. 5. Tired of the gossip’s __ , Alicia said she was late for an appointment so she could end the conversation. 6. The _ to the bill describes the background of the legislation and explains how it relates to existing laws. 7. The increased tariffs in the 1930s _ a collapse in world trade, exacerbating the Great Depression. 8. All TV and radio broadcasts have been __ by an emergency announcement by the president. 9. The commander said he would not be pressured into making a _ decision. 10. The retired couple _ all evening about their latest trip to Europe, oblivious to the fact that no one had the slightest interest in what they were talking about. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Scientists have shown that the precursor to birds was a flying dinosaur. _ 2. The Democrats have a precarious majority in the state senate. 3. The audience of distinguished scientists listened intently as the Nobel Prize-winning physicist prated eloquently about her latest discovery. 4. The man studied the religion’s precepts so that he could be accepted as a convert. _ 5. A preamble to the official report describes its rationale and how the commission gathered its information. UNIT 60 premonition n. forewarning; presentiment Shortly after his reelection in 1864, President Abraham Lincoln had a premonition of his impending death, and on April 14, 1865, he was shot and died the next day. presage v. to foretell; indicate in advance The English poet William Blake believed his work presaged a new age in which people would achieve political, social, psychological, and spiritual freedom. 219 ESSENTIAL WORDS FOR THE GRE presumptuous adj. rude; improperly bold; readiness to presume The new employee did not offer her advice to her boss because she was afraid he might consider it presumptuous for a recent graduate to make a suggestion to someone with 30 years experience in the field. The verb presume means assume or act with impertinent boldness. Proponents of the view presume that there exist only two antitheti cal positions, with no middle ground between their opponent’s view and their own (eminently more reasonable) position. The noun is presumption. Anti-Semitism originated in the presumption that Jews were responsible for Jesus’ crucifixion, and was responsible for periodic persecutions such as the expulsion of Jews from Spain in 1492. preternatural adj. beyond the normal course of nature; supernatural Most scientists believe that putative preternatural phenomena are outside the scope of scientific inquiry. prevaricate v. to quibble; evade the truth Journalists accused government leaders of prevaricating about the progress of the war. primordial adj. original; existing from the beginning Scholars are divided as to whether polytheism represents a degen eration from a primordial monotheism, or was a precursor to a more sophisticated view, monotheism. Terms from the Arts, Sciences, and Social Sciences polytheism: belief in the existence of more than one god monotheism: belief in the existence of one god pristine adj. untouched; uncorrupted The bank’s hermetically sealed vault has kept the manuscript in pristine condition for 50 years. probity n. honesty; high-mindedness No one questioned the probity of the judge being considered for elevation to the U.S. Supreme Court; what was at issue was his controversial views on several important issues. 220 ESSENTIAL WORDS FOR THE GRE problematic adj. posing a problem; doubtful; unsettled The idea of the universe originating at a certain point in time seems problematic to many scientists. prodigal adj. wasteful; extravagant; lavish Betty warned her husband that he must stop his prodigal spend ing on sports cars and expensive clothing. REVIEW 60 The correct answers are given on page 408. Matching Match each word with its definition, 1. premonition (A) rude 2. presage (B) doubtful 3. presumptuous (C) beyond the normal course of nature 4. preternatural (D) existing from the beginning 5. prevaricate (E) forewarning 6. primordial (F) honesty 7. pristine (G) to foretell 8. probity (H) wasteful 9. problematic (I) to quibble 10. prodigal (J) untouched Fill-ins Choose the best word to fill in the blank in each sentence. premonition presage presumptuous preternatural prevaricating primordial pristine probity problematic prodigal 1. Scientists are investigating Edna’s claim to having a _ ability to predict the future. 2. Air strikes against military bases __ a full-scale invasion. 3. Ruth’s dream contained a _ that war would break out. 4. The museum exhibition allows visitors to experience what a _ forest was like. 5. The president told the senator to stop __ on the issue and give him her decision by Monday on whether she had his support. 6. Tom keeps his pride and joy, a 1966 Triumph, in _ condition in his temperature-controlled garage. 221 ESSENTIAL WORDS FOR THE GRE 7. One of the considerations that makes a return to a military draft _ is that gender equality would almost certainly require the equal participation of males and females. 8. Bruce’s __ spending on luxuries left him nearly bankrupt. 9. The math student decided that it would be _ of her to correct the error in the eminent mathematics professor’s calculations. 10. The senator’s unquestioned _ and incisive intelligence made her a unanimous choice to lead the sub-committee investi gating official misconduct. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. A primordial number is an integer divisible only by itself or one. _ 2. The premonition to the play introduces us to the main characters and the setting. 3. Some people believe that prevaricating helps to develop charac ter because it encourages a person to make up his or her mind quickly. 4. The chairperson of the finance committee warned that the state’s prodigal spending would have to stop. _ 5. The brain researcher believes that what may appear to be preternatural occurrences are actually the result of the activation of certain areas of the brain. APPLYING YOUR KNOWLEDGE—51–60 1. Art critics praised the work of the black modernist painter Jacob Lawrence, whose paintings were quite restrained despite the _ of his subject matter—ghettos, race riots, prisons, and lynchings. diversity pathos paucity insularity plasticity 222 ESSENTIAL WORDS FOR THE GRE Explanation: The best clue to the answer to this question is “ghettos, race riots, prisons, and lynchings,” all of which are subjects that arouse pathos (feelings of pity and sorrow). The correct answer is pathos. 2. The denotation of the word lady is “A well-mannered and consider ate woman with high standards of proper behavior;” however, some women object to the use of the word because it has the connotation of _. probity determination obsequiousness diligence piety Explanation: The words “some women object to the use of the word” signal that a word with a negative meaning is required. Obsequiousness (being overly submissive) is thus the correct choice. The correct answer is obsequiousness. UNIT 61 profound adj. deep; not superficial There is an adage in philosophy that everyone is born either a Platonist or an Aristotelian, meaning that everyone has a predisposition to believing either that reality is completely “here and now,” or that there exists a more profound, hidden reality. The noun profundity means the quality of being profound. prohibitive adj. so high as to prevent the purchase or use of; prevent ing; forbidding Most people in poor countries are unable to purchase a computer because of its prohibitive price. Prohibition is the noun. The word taboo was taken from Polynesia (tabu in Tongan) and broadened to mean any culture’s prohibition of a particular object or activity. Note: Aristotle was Plato’s student; in contrast to Plato, he believed that there exist no entities separate from matter. 223 ESSENTIAL WORDS FOR THE GRE proliferate v. to increase rapidly With the pervasive influence of American culture, “fast-food” restaurants are proliferating in many countries. Proliferation is the noun. A problem with the proliferation of jargon is that it impedes communication between different fields of knowledge. propensity n. inclination; tendency There is a natural propensity to stress the importance of what one is saying by exaggerating it. propitiate v. to win over; appease M.E.W. Sherwood, an author alive at the time of the U.S. Civil War, eloquently expressed the sacrifice made by soldiers on both sides of that great conflict: “But for four years there was a contagion of nobility in the land, and the best blood of North and South poured itself out a libation to propitiate the deities of Truth and Justice. The great sin of slavery was washed out, but at what a cost!” propriety n. correct conduct; fitness Judges are expected to conduct themselves with propriety, especially in the courtroom. proscribe v. to condemn; forbid; outlaw The expert in English believes that since the tendency to use hyperbole is natural and often enriches the language, it should not be proscribed. The adjective proscriptive means relating to prohibition. Proponents of the view that dictionaries should be proscriptive, dictating what correct usage is, believe that without such guides the standard of language will decline; however, advocates of descriptive dictionaries argue that dictionary makers have no mandate to dictate usage and therefore should merely record language as it is used. provident adj. providing for future needs; frugal Most people have heard the story of the prodigal grasshopper and the provident ant that spends the summer saving food for the winter. 224 ESSENTIAL WORDS FOR THE GRE puissant adj. powerful The article analyzes the similarities and differences between the Roman Empire and the British Empire when each was at its most puissant. The noun is puissance. punctilious adj. careful in observing rules of behavior or ceremony. The prime minister reminded his staff that they must be punctilious in following protocol during the visit by the foreign head of state. REVIEW 61 The correct answers are given on page 408. Matching Match each word with its definition. 1. profound (A) correct conduct 2. prohibitive (B) powerful 3. proliferate (C) preventing; forbidding 4. propensity (D) to condemn 5. propitiate (E) not superficial 6. propriety (F) frugal 7. proscribe (G) inclination; tendency 8. provident (H) careful in observing rules of behavior 9. puissant (I) to win over 10. punctilious (J) to increase rapidly Fill-ins Choose the best word to fill in the blank in each sentence. profound prohibitive proliferating propensity propitiated propriety proscribes provident puissant punctilious 1. In 1972, the United States Supreme Court voided all state and federal laws specifying the death penalty on the basis that they are unconstitutional, since they violate the eighth amendment of the Constitution, which _ “cruel and unusual punishment.” 2. As Russ grew older, he found his intellectual interests __ rather than narrowing, as he had expected. 225 ESSENTIAL WORDS FOR THE GRE 3. Sharon is _ in doing her homework; every evening she reviews all of the day’s classes and carefully completes the written tasks. 4. American cultural influence in the world has been described as a force more _ than any army. 5. __ in that country demands that young single women be accompanied in public by an adult female. 6. Defenders of philosophy say that, far from being a superfluous and self-indulgent activity, it is one of the most _ of human enterprises, having given humankind such useful fields of thought as science, and conceived of such noble ideas as freedom, democracy, and human rights. 7. In her article the anthropologist suggests that homo sapiens is a species with an innate _ for violence. 8. A belief in angry gods who must be __ to prevent them from venting their wrath on human beings is pervasive in human cultures. 9. According to some scientists, the technology exists for establishing a base on Mars, but the cost of doing so would be _ . 10. The _ housekeeper insists on buying everything when it is on sale. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The letter argues that the city council must take measures to control the proliferation of wild dogs. _ 2. No one could blame the passengers on the jetliner for being a bit puissant after a UFO was sighted flying off their plane’s wing. 3. Throughout the priest’s writings is a profound regard for the dignity and sanctity of human life. 4. The chief of protocol planned every official function so that propriety was strictly observed. _ 5. The prohibitive cost of many modern medical therapies makes them unsuitable for patients in poor countries. 226 ESSENTIAL WORDS FOR THE GRE UNIT 62 pungent adj. strong or sharp in smell or taste; penetrating; caustic; to the point Slang frequently expresses an idea succinctly and pungently. purport v. to profess; suppose; claim The United States is generally considered to be a secular society in which church and state are separate; however, religion plays a large role, since nearly everyone purports to believe in God and many people are members of churches. Purport is also a noun. Its definition is meaning intended or implied. pusillanimous adj. cowardly Traditionally, a ship captain is considered pusillanimous if he abandons his ship before everyone else has. The noun is pusillanimity, which means cowardice. quagmire n. marsh; difficult situation The federal government’s antitrust suit in the 1990s against Micro soft created a legal quagmire. quail v. to cower; lose heart The defendant quailed when the judge entered the room to announce the sentence. qualified adj. limited; restricted In Indian philosophy a position between monism at one extreme and dualism at the other is qualified nondualism, a philosophy in which reality is considered to have attributes of both dualism and monism. Terms from the Arts, Sciences, and Social Sciences monism: the belief that reality is a unified whole consisting of one fundamental principle dualism: the theory that two basic entities constitute reality (e.g. mind and matter or good and evil) Qualification is a noun meaning limitation or restriction. So many qualifications had been added to the agreement that Sue was now reluctant to sign it. The verb qualify means to modify or limit. 227 ESSENTIAL WORDS FOR THE GRE qualm n. sudden feeling of faintness or nausea; uneasy feeling about the rightness of actions The judge had no qualms about sentencing the thief to five years imprisonment. query v. to question Until widespread industrialization caused massive pollution in the nineteenth and twentieth centuries, the ability of the biosphere to dissipate and assimilate waste created by human activity was not queried. Query is also a noun meaning a question. The history professor answered the student’s interesting query about the influence of Arabic thought on Western civilization. quibble v. to argue over insignificant and irrelevant details The lawyers spent so much time quibbling over details that they made little progress in reaching an agreement on the central issue. Quibble is also a noun. quiescent adj. inactive; still Although malignant tumors may remain quiescent for a period of time, they never become benign. The noun is quiescence. REVIEW 62 The correct answers are given on page 408. Matching Match each word with its definition. 1. pungent (A) difficult situation 2. purport (B) argument over insignificant details 3. pusillanimous (C) to profess; suppose 4. quagmire (D) inactive 5. quail (E) strong or sharp in smell or taste 6. qualified (F) limited 7. qualm (G) cowardly 8. query (H) to question 9. quibble (I) lose heart 10. quiescent (J) uneasy feeling 228 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. pungent purported pusillanimous quagmire quailed qualified qualms query quibble quiescent 1. The Nissan Patrol sank halfway into the _ . 2. The __ alien craft turned out to be an experimental aircraft performing unusual maneuvers. 3. During our tennis match we smelled the _ odor of lamb curry being cooked. 4. The bank teller _ as the masked robber threatened her with a gun. 5. The soldier said he has no __ about killing the enemy since it was his duty. 6. The fortune-teller answered her customer’s _ with an ambiguous “It will come about if Fate wills it.” 7. The student’s essay asserts that “Humanity made great progress in the twentieth century”; however, when her teacher asked her what she meant by “progress” she _ her statement by specifying that she meant that humanity made great economic and scientific progress. 8. The senator argued that it would be __ for Congress to simply rubber-stamp every bill proposed by the president. 9. When asked by reporters which of the starting pitchers he thought was better, the manager replied, “I’m not going to _ about which is better. They’re both superb.” 10. The patient’s emotional disturbance appeared to be _ , but the psychologist feared that it would manifest itself again soon. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The discovery was purported to be the most important technologi cal breakthrough of the modern age. _ 2. The head football coach called spring practice a qualified success because the conditioning program had gone well but there had been only limited progress in other areas. 3. The quiescent volcano is spewing out lava that is threatening to destroy the nearby town. 229 ESSENTIAL WORDS FOR THE GRE 4. “Let’s accept the report’s conclusion and not quibble over inconse quential details,” the manager told his workers. _ 5. U.S. military leaders are leery of becoming involved in a quagmire that would drain resources and limit their forces’ effectiveness in other theatres. UNIT 63 quorum n. number of members necessary to conduct a meeting The U.S. Senate’s majority leader asked three members of his party to be available to help form a quorum. raconteur n. witty, skillful storyteller Former president Bill Clinton is known as an accomplished raconteur who can entertain guests with amusing anecdotes about politics all evening. rail v. to scold with bitter or abusive language The critic of globalization railed against its effect on the poor people of the world. raiment n. clothing It took two hours for the princess’ handmaidens to help her put on her splendid raiment for her coronation as queen. ramification n. implication; outgrowth; consequence The full ramification of the invention of the laser did not become apparent for many years; now it is used in a great variety of applica tions, from DVD players to surgery. rarefied adj. refined Many scholars flourish in the rarefied intellectual atmosphere of the Institute for Advanced Studies in Princeton, New Jersey. The verb rarefy means to make thinner, purer, or more refined. rationale n. fundamental reason The philosophy of “enlightened self-interest” justifies acting in one’s own interest by asserting that this is not selfish or motivated by a “beggar thy neighbor” rationale, but is simply the best way to ensure the welfare of the entire community. 230 ESSENTIAL WORDS FOR THE GRE rebus n. puzzle in which pictures or symbols represent words Egyptian writing uses the principle of the rebus, substituting pictures for words. recalcitrant adj. resisting authority or control The officer had no choice but to recommend that the recalcitrant soldier be court-martialed. recant v. to retract a statement or opinion The bishop told the theologian that he must recant his heretical teaching or risk excommunication. REVIEW 63 The correct answers are given on page 408. Matching Match each word with its definition. 1. quorum (A) fundamental reason 2. raconteur (B) implication 3. rail (C) refined 4. raiment (D) clothing 5. ramification (E) witty, skillful storyteller 6. rarefied (F) resisting authority or control 7. rationale (G) to retract a statement or opinion 8. rebus (H) to scold with bitter or abusive language 9. recalcitrant (I) puzzle in which pictures or symbols represent words 10. recant (J) number of members necessary to conduct a meeting Fill-ins Choose the best word to fill in the blank in each sentence. quorum raconteur rails raiment ramifications rarefied rationale rebus recalcitrant recant 1. A counselor was called in to talk to the __ student. 2. Carl Sagan’s novel Contact explores the _ for humanity of contact with an advanced alien civilization. 3. The _ offered for invading the country was that it posed a threat to peace in the region. 231 ESSENTIAL WORDS FOR THE GRE 4. As a girl Sheila dreamed of being dressed in the golden __ of a princess. 5. Every week the newspaper columnist _ against what he calls the “unprecedented stupidity of our age.” 6. Unable to obtain a _ , leaders of the majority party had no choice but to postpone the vote on the legislation. 7. The __ was the life of the party, telling hilarious jokes long into the evening. 8. Saint Thomas Aquinas combined an acute, practical intellect and the most _ spirituality. 9. The fourth-grade class project was to design a _ incor porating pictures of animals. 10. The company said it would drop its lawsuit for defamation if the journalist agreed to publicly __ his false statement about its products. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The witch cast a raiment on the man, turning him into a tree. 2. Scientists had to destroy the rebus because they were afraid it would break out of the lab and infect the population of the city. 3. The speaker railed against profligate government spending. _ 4. The raconteur has a repertoire of over three hundred jokes, all of which he can tell with perfect timing. 5. Fans questioned the rationale for the coach’s decision to go for a risky two-point conversion after the touchdown rather than a nearly certain one-point conversion. __ UNIT 64 recluse n. person who lives in seclusion and often in solitude The monk spent three years of his life as a recluse, praying and meditating. The adjective is reclusive. John is a reclusive person who enjoys reading more than anything else. 232 ESSENTIAL WORDS FOR THE GRE recondite adj. abstruse; profound Many classical and biblical references known to educated nineteenth-century readers are now considered recondite by most readers. redoubtable adj. formidable; arousing fear; worthy of respect As a result of winning 95 percent of her cases, the prosecutor has earned a reputation as a redoubtable attorney. refractory adj. stubborn; unmanageable; resisting ordinary methods of treatment The general practitioner called in specialists to help determine the cause of the patient’s refractory illness. The verb refract means to deflect sound or light. Intermittently the ionosphere refracts radio waves of certain frequencies, allowing transmissions between distant points on the Earth. refulgent adj. brightly shining; resplendent On the queen’s neck was a necklace of jewels, in the middle of which was a large, refulgent diamond. refute v. to contradict; disprove The eighteenth-century English author Samuel Johnson claimed to have refuted the philosophy of idealism by kicking a large stone. The noun is refutation. Fundamentalism arose in Protestantism as a refutation of the liberal theology of the early twentieth century, which interpreted Christianity in terms of contemporary scientific theories. Terms from the Arts, Sciences, and Social Sciences Idealism: the belief that everything that exists is fundamentally mental in nature regale v. to entertain Former U.S. presidents Lyndon Johnson, Ronald Reagan, and Bill Clinton often regaled visitors with amusing political anecdotes. 233 ESSENTIAL WORDS FOR THE GRE relegate v. to consign to an inferior position Idealist philosophers are a common target of satire; however, instead of relegating them all to the garbage can, one should reflect that thinkers such as Plato and Kant have given humanity some of its most profound ideas. Terms from the Arts, Sciences, and Social Sciences Idealist: refers to the followers of the philosophy of Idealism, which holds that the object of external perception consists of ideas. Immanuel Kant (1724–1804): German philosopher who held that the mind shapes the world as it perceives it and that this world takes the form of space and time remonstrate v. to object or protest Minority members of the committee remonstrated with the major ity members, saying that the proposal was unjust; nevertheless, it was approved. renege v. to go back on one’s word Generally, if one party to an agreement reneges on its contractual obligations, it must provide appropriate compensation to the other party. REVIEW 64 The correct answers are given on page 408. Matching Match each word with its definition. 1. recluse (A) brightly shining 2. recondite (B) to entertain 3. redoubtable (C) abstruse; profound 4. refractory (D) to object or protest 5. refulgent (E) to contradict; disprove 6. refute (F) person who lives in seclusion 7. regale (G) stubborn; unmanageable 8. relegate (H) to go back on one’s word 9. remonstrate (I) arousing fear 10. renege (J) to consign to an inferior position 234 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. recluse recondite redoubtable refractory refulgent refute regaled relegated remonstrated reneged 1. The guest speaker __ the audience with hilarious anecdotes from her childhood. 2. The school has announced plans to deal with the _ students. 3. Students of religion have discerned a pattern in many religions in which some gods gradually attain prominence and others are _ to an inferior status. 4. Tim __ on his bet with Harry, claiming it had just been a joke. 5. Astronomers are studying the _ object that suddenly appeared in the sky. 6. Edith’s friends are concerned that she is becoming a _ ; she does not go out with them anymore and rarely leaves her house. 7. The book God and the New Physics by the Australian physicist Paul Davies succeeds in making __ areas of physics more comprehensible to the general public. 8. The prospect of being interviewed for admission by the _ dean of the law school was a daunting one. 9. The conservative and liberal _ with each other over the issue long into the night. 10. One way to __ an argument is to show that one or more of the premises on which it is based is false. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. When learning a new subject, it is wise to start with straight-forward, recondite topics first. 2. The retired football coach regaled the young coaches with sto ries from his playing days with the Green Bay Packers in the 1960s. 3. In the English professional soccer league, a team can be relegated from the “premier” division to a lower division because of poor performance. _ 4. The debate coach reminded his team to refute every argument made by the opposing team. 5. The recluse has many friends at his house every night. __ 235 ESSENTIAL WORDS FOR THE GRE UNIT 65 reparation n. amends; compensation The judge said she would not sentence the man to jail on the condition that he pay full reparation to the family hurt by his crime. repine v. fret; complain The president told the congressional representative he should stop repining over the lost opportunity and join the majority in exploring new ones. reprise n. repetition, especially of a piece of music The standing ovation at the end of the set meant that the band had little choice but to reprise a few of their most popular tunes. The verb is also reprise. reproach v. to find fault with; blame The speaker in Andrew Marvell’s poem “To His Coy Mistress” reproaches his beloved for ignoring the passing of time and for not being willing to physically express her love for him. Reproach is also a noun. reprobate n. morally unprincipled person The social worker refused to give up hope of reforming the criminal who was generally regarded as a reprobate. repudiate v. to reject as having no authority In the 1960s, many black leaders such as Malcolm X and Stokely Carmichael repudiated integration and nonviolence in favor of black separatism and passive resistance in the fight for civil rights. rescind v. to cancel The salesperson said he would rescind his offer to sell the goods at a 10 percent discount unless he received full payment within 24 hours. resolution n. determination; resolve Fred’s resolution to succeed is unshaken despite the many setbacks he has suffered. 236 ESSENTIAL WORDS FOR THE GRE resolve n. determination; firmness of purpose President Abraham Lincoln displayed remarkable resolve in preventing the Confederate states from seceding. The verb is also resolve. reticent adj. not speaking freely; reserved; reluctant Many people in the west are reticent to criticize science, which in the view of many has become a sacred cow. Terms from the Arts, Sciences, and Social Sciences sacred cow: something that is so greatly respected that it is beyond question, e.g., “The virtue of free trade is a sacred cow of modern economic theory.” REVIEW 65 The correct answers are given on page 408. Matching Match each word with its definition. 1. reparation (A) to blame 2. repine (B) to fret 3. reprise (C) determination 4. reproach (D) firmness of purpose 5. reprobate (E) to reject as having no authority 6. repudiate (F) morally unprincipled person 7. rescind (G) amends 8. resolution (H) reserved 9. resolve (I) repetition 10. reticent (J) to cancel Fill-ins Choose the best word to fill in the blank in each sentence. reparations repine reprise reproached reprobate repudiated rescinded resolution resolved reticent 1. Janet _ her friend for being lazy. 2. John __ to study hard so he would get an “A” in chemistry. 237 ESSENTIAL WORDS FOR THE GRE 3. The gangster _ all his past associations with criminals in the city. 4. The company _ its job offer when it was found that the candidate had provided falsified documents. 5. Every year Joanne makes a firm __ to work harder. 6. The court ordered the convicted woman to make _ to the family that she had done so much harm to. 7. The counselor was finally able to get the _ boy to talk about the problems in his family. 8. The employee did not __ at being assigned to do the arduous task, but rather, accepted it as a challenge. 9. The judge warned the convicted man that he was beginning to consider him a hopeless _ who should be kept in prison away from innocent people. 10. The New Year’s Eve revelers demanded a _ of “Auld Lang Syne.” Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The burden of war reparations plunged the country into a finan cial crisis. _ 2. The counselor is encouraging the reticent patient to talk about his feelings. 3. The teacher reproached the student for her sloppy work. 4. The gangster pledged to start a new life and repudiate his past involvement with criminals. _ 5. The couple’s grandchildren decided to reprise them with a 30th anniversary party. UNIT 66 reverent adj. expressing deep respect; worshipful The biologist Loren Eisely had what could be described as a reverent attitude toward nature. The verb is revere. riposte n. a retaliatory action or retort The commander decided that the enemy attack must be countered with a quick riposte. 238 ESSENTIAL WORDS FOR THE GRE rococo adj. excessively ornate; highly decorated; style of architecture in eighteenth-century Europe In music, the Rococo period (1730 –1780) comes between the preceding Baroque period and the subsequent Classical period. The highly ornamented style of the Rococo period created new forms of dissonance that to listeners in previous eras would have sounded cacophonous. The noted authors Lawrence Durrell and Vladimir Nabokov often wrote in a rich, almost rococo style. Terms from the Arts, Sciences, and Social Sciences Rococo: a style of architecture that made use of elaborate curved forms. Examples of the Rococo in architecture are the extremely ornate court and opera buildings of Mannheim and Stuttgart in Germany rubric n. title or heading; category; established mode of procedure or conduct; protocol The data from the experiment was so diverse that the scientist decided to design a new rubric to organize it. rue v. to regret The judge told the convicted man that he would come to rue his decision to commit the crime. ruse n. trick; crafty stratagem; subterfuge In July, 1999, a group of Christians from the United Kingdom traveled to various countries in which Crusaders had massacred people to apologize; however, many of the Moslems spurned this overture, believing it to be another Crusade in the form of a ruse. sage adj. wise Samuel Johnson gave this sage, albeit hard, advice to writers wishing to improve their style: “Read over your compositions, and whenever you meet with a passage that you think is particularly fine, strike it out.” Sage is also a noun meaning a wise older person. salacious adj. lascivious; lustful The school board decided that the book is too salacious to be in the school library. 239 ESSENTIAL WORDS FOR THE GRE salubrious adj. healthful The salubrious effects of exercise on both physical and mental health have been well documented. salutary adj. expecting an improvement; favorable to health “The system of universal education is in our age the most promi nent and salutary feature of the spirit of enlightenment. . . .” —President Benjamin Harrison, 1892 REVIEW 66 The correct answers are given on page 409. Matching Match each word with its definition. 1. reverent (A) crafty stratagem 2. riposte (B) lustful 3. rococo (C) wise older person 4. rubric (D) excessively ornate 5. rue (E) expecting an improvement 6. ruse (F) expressing deep respect 7. sage (G) to regret 8. salacious (H) retaliatory action 9. salubrious (I) favorable to health 10. salutary (J) title or heading Fill-ins Choose the best word to fill in the blank in each sentence. revere riposte rococo rubric rue ruse sage salacious salubrious salutary 1. In Chinese culture children are expected to __ their parents. 2. The talk show host is always ready with a clever _ to the barbs of her guests. 3. The defendant told the members of the jury that they would _ the day they had convicted him. 4. As a __ , the president’s press secretary opened the news conference with the statement that the government would guarantee everyone in America a minimum salary of $100,000 per year. 240 ESSENTIAL WORDS FOR THE GRE 5. The ancient Greek philosopher Socrates was a _ who believed that everyone must engage in his or her own search for truth. 6. The movie was given an “R” rating because of its _ content. 7. Many people from the Midwest retire to Arizona because of the __ climate. 8. Advocates of Prohibition believed that it would have a _ effect on people who enjoyed drinking alcoholic beverages. 9. The author decided to discuss forced sterilization under the _ of eugenics. 10. The __ furniture seems out of place in the ultramodern building. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The debater prepared clever ripostes for the arguments she expected her opponent to make. 2. Some readers find the writer’s straightforward, rococo style boring. 3. Confucius was a Chinese sage revered for his wisdom. _ 4. The fraternity brother who came up with the best ruse was told he would get a date with the homecoming queen. 5. To have your article published in the chemistry journal, you must carefully follow the rubric provided by its editor. __ UNIT 67 sanction v. to approve; ratify; permit The establishment of the state of Israel from Palestinian territory in 1948 was the realization of a hallowed dream for Zionists, but for many Palestinians it meant the sanctioning of continued domination of their land by Europeans. Sanction is also a noun meaning approval; ratification; permission. In the West, the institution of marriage is traditionally given formal sanction by both the Church and the State, which has the social function of reinforcing its importance and the seriousness of the duties it entails. 241 ESSENTIAL WORDS FOR THE GRE The noun sanction can also mean penalization. The United Nations has the power to compel obedience to interna tional law by sanctions or even war, but there must be unanimity for such action among the five permanent members of the Security Council. The verb sanction can also mean to penalize. sardonic adj. cynical; scornfully mocking Satire that is too sardonic often loses its effectiveness. sartorial adj. pertaining to tailors Off-screen, the glamorous actress’ sartorial style runs more to jeans and T-shirts than to elaborate gowns. satiate v. to satisfy The bully satiated his fury by pummeling the helpless little boy. saturate v. to soak thoroughly; imbue throughout The writer’s recollection of her childhood is saturated with sun shine and laughter. saturnine adj. gloomy When the long list of casualties from the battle were announced, the mood in the room was saturnine. satyr n. a creature that is half-man, half-beast with the horns and legs of a goat; it is a follower of Dionysos; a lecher One of the best-known satyrs is Pan, the god of the woods in Greek mythology. savor v. to enjoy; have a distinctive flavor or smell The coach gave his team a day off practice to savor their big victory. schematic adj. relating to or in the form of an outline or diagram The engineer outlined the workings of the factory in schematic form. secrete v. produce and release substance into organism The pancreas gland secretes a fluid that helps fat, carbohydrates, and protein to be digested in the small intestine. 242 ESSENTIAL WORDS FOR THE GRE REVIEW 67 The correct answers are given on page 409. Matching Match each word with its definition. 1. sanction (A) pertaining to tailors 2. sardonic (B) half-man, half-beast 3. sartorial (C) relating to a diagram 4. satiate (D) to approve; ratify 5. saturate (E) to produce and release substance into organism 6. saturnine (F) to satisfy 7. satyr (G) cynical 8. savor (H) gloomy 9. schematic (I) to enjoy 10. secrete (J) to soak thoroughly Fill-ins Choose the best word to fill in the blank in each sentence. sanctions sardonic sartorial satiate saturated saturnine satyr savored schematic secrete 1. Celebrating the end of her diet, Tina _ every mouthful of the ice cream sundae. 2. A fried chicken dinner should be enough to __ the hungry student’s appetite. 3. June is one of those people whose mood can suddenly become _ and then just as quickly become sunny and cheerful. 4. The company decided to try to sell another product because the market for personal computers had become _ . 5. Economic __ against the country have made life difficult for its people; even everyday necessities are becoming scarce. 6. The book claims to give advice that solves men’s _ problems easily and cheaply. 7. Hugh has a reputation as a bit of a _ among the women in the office. 8. The electrical engineer made a __ diagram of the circuit. 9. Cells in the mucous membrane of the stomach _ hydro chloric acid to help in the digestion of food. 243 ESSENTIAL WORDS FOR THE GRE 10. The satirist’s unremittingly _ tone left the reviewer feeling that here was a man of great talent who had, sadly, retreated to a bitterly cynical, even misanthropic attitude toward the world. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The novel is a satyr on human nature. _ 2. We satiated our appetite for science fiction novels by reading twenty of them on summer vacation. 3. Not everyone appreciates the comedian’s sardonic commentary on modern life. 4. Twelve hours of heavy rain left the field saturated. _ 5. I suggest you savor the food, not just gobble it down. UNIT 68 sedition n. behavior prompting rebellion The federal prosecutor argued that the journalist’s article could be interpreted as an act of sedition since it strongly suggested that the government should be overturned. sedulous adj. diligent The Nobel Prize-winning scientist attributed his success to what he termed “curiosity, a modicum of intelligence, and sedulous application.” seismic adj. relating to earthquakes; earthshaking The study of seismic waves enables scientists to learn about the Earth’s structure. sensual adj. relating to the senses; gratifying the physical senses, especially sexual appetites The yogi teaches his students that attachment to sensual pleasure is one of the great hindrances to spiritual advancement. sensuous adj. relating to the senses; operating through the senses The American painter Georgia O’Keeffe is known especially for her sensuous paintings of plants and flowers and for her landscapes. 244 ESSENTIAL WORDS FOR THE GRE sentient adj. aware; conscious; able to perceive Charles Darwin regarded many animals as being sentient and as having intelligence. The noun is sentience. An analgesic relieves pain but unlike an anesthetic, does not cause loss of sensation or sentience. servile adj. submissive; obedient None of the dictator’s servile citizens dared question his decree. sextant n. navigation tool that determines latitude and longitude Because it enabled precise determination of position, the sextant quickly became an essential tool in navigation after its invention in 1731. shard n. a piece of broken glass or pottery Archeologists were able to reconstruct the drinking vessel from shards found around the ancient campsite. sidereal adj. relating to the stars A sidereal year is longer than a solar year by 20 minutes and 23 seconds. REVIEW 68 The correct answers are given on page 409. Matching Match each word with its definition. 1. sedition (A) operating through the senses 2. sedulous (B) navigation tool 3. seismic (C) behavior prompting rebellion 4. sensual (D) piece of broken glass or pottery 5. sensuous (E) gratifying the physical senses 6. sentient (F) aware 7. servile (G) diligent 8. sextant (H) relating to the stars 9. shard (I) submissive 10. sidereal (J) relating to earthquakes 245 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. sedition sedulous seismic sensual sensuous sentient servile sextant shards sidereal 1. Most of the population of the occupied country behaved in a __ manner toward the foreign soldiers. 2. _ is treated so seriously because it is a threat to the very existence of the state. 3. The detective was _ in collecting evidence to prove his client’s innocence. 4. According to geologists, in its early history the Earth was continu ally shaken by massive __ disturbances. 5. _ found at the site suggest that there was human habitation in the area 5,000 years ago. 6. Because it is not dependent on electricity for power, the _ is still used as a backup navigation tool on many ships. 7. The science fiction novel describes a __ adventure. 8. The book explores the question of how _ beings that evolved differently from humans would regard the world. 9. The book describes a society almost entirely dedicated to _ delight. 10. The philosopher Plato believed that a process of reason, indepen dent of __ information, could help a man arrive at the true nature of reality. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The French Revolution was a momentous event that sent seismic shocks through Western civilization. 2. Sidereal surveillance of the suspect provided police with enough evidence to make an arrest. 3. One of the goals of artificial intelligence is to produce a machine that an unbiased observer judges to be sentient. _ 4. The police captain warned the protesters that they were in danger of crossing the line between lawful public protest and sedition. 5. The invention of the magnetic compass and the sextant were two of the major developments in navigation. __ 246 ESSENTIAL WORDS FOR THE GRE UNIT 69 simian adj. apelike; relating to apes Many people in the nineteenth century denied the evolutionary significance of the simian characteristics of human beings. simile n. comparison of one thing with another using “like” or “as” In his autobiographical book Chronicles, Volume 1, Bob Dylan uses two similes in succession to try to convey the experience of writing a song: “A song is like a dream, and you try to make it come true. They’re like strange countries you have to enter.” sinecure n. well-paying job or office that requires little or no work The company established the high-paying position of senior advisor as a sinecure for the man who had been instrumental in the company’s success for so many years. singular adj. unique; extraordinary; odd The defendant’s singular appearance made it easy for the witness to identify him as the person at the scene of the crime. sinuous adj. winding; intricate; complex The students had trouble following the philosopher’s sinuous line of reasoning. skeptic n. one who doubts Like the nihilist, a comprehensive philosophic skeptic can be a difficult person to debate: if you tell him you know you exist, he is likely to ask you to prove it—and that can be harder than it first appears. The adjective is skeptical. A good scientist is skeptical about inferences made from data; however, he must not be dogmatic about the possible implications the data might have. Terms from the Arts, Sciences, and Social Sciences nihilist: one who believes that existence and all traditional values are meaningless sobriety n. seriousness The student approaches her studies with commendable sobriety. 247 ESSENTIAL WORDS FOR THE GRE sodden adj. thoroughly soaked; saturated The sodden field makes it difficult for the soccer players to move effectively. solicitous adj. concerned; attentive; eager The nurse is extremely solicitous of the health of every patient in the ward. soliloquy n. literary or dramatic speech by one character, not addressed to others The nineteenth-century English poet Robert Browning used the dramatic monologue—which is essentially a soliloquy in a poem— successfully in many of his poems. REVIEW 69 The correct answers are given on page 409. Matching Match each word with its definition. 1. simian (A) well-paying job requiring little work 2. simile (B) seriousness 3. sinecure (C) comparison of one thing with another using “like” or “as” 4. singular (D) thoroughly soaked 5. sinuous (E) unique 6. skeptic (F) one who doubts 7. sobriety (G) dramatic speech by one character 8. sodden (H) concerned 9. solicitous (I) apelike 10. soliloquy (J) winding Fill-ins Choose the best word to fill in the blank in each sentence. simian similes sinecure singular sinuous skeptic sobriety sodden solicitous soliloquy 1. The judge recommended her law clerk for the position in the law firm as “a young person of probity and __ .” 2. “Money is a _ thing. It ranks with love as man’s greatest source of joy. And with death as his greatest source of sorrow.” —John Kenneth Galbraith 248 ESSENTIAL WORDS FOR THE GRE 3. Mary complains that when they were young her husband was very _ of her, but now he practically ignores her. 4. The __ argued that the purported exhibition of occult powers was created by the use of conjurer’s tricks. 5. We often use _ in expressions like “as old as the hills” and “as sharp as a tack” without being consciously aware that they are similes. 6. The governor awarded his advisor with a _ as a reward for 20 years of service to the party and the state. 7. The __ road curves along the mountainside. 8. In Act III of Hamlet, Shakespeare has Hamlet speak a _ on the question of “To be, or not to be.” 9. Looking at the _ field, the football coach realized he would have to adapt his game plan to wet conditions. 10. Before Charles Darwin proved the close biological relation between human beings and apes, many people saw human __ characteristics as comical and inconsequential. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The philosopher Bertrand Russell was skeptical of Idealist philosophies, believing they are based on false assumptions about knowledge. 2. The philosophy student compared following the treatise’s long, subtle argument to following the path of a sinuous river for thousands of miles. 3. The poem’s central simile is that the nation’s leader is like a captain of a ship. _ 4. Italian mothers are famous for being so solicitous of their sons that they spend most of the day cooking for them. 5. The farmers are hoping for rain after the long period of hot and sodden weather. __ UNIT 70 solvent adj. able to meet financial obligations During the financial crisis several large banks had difficulty remaining solvent. somatic adj. relating to or affecting the body; corporeal A psychosomatic disorder is a malady caused by a mental distur bance that adversely affects somatic functioning. 249 ESSENTIAL WORDS FOR THE GRE soporific adj. sleep producing For some people the best soporific is reading a boring book. sordid adj. filthy; contemptible and corrupt The Monica Lewinsky scandal, which led to President Bill Clinton’s impeachment in 1998, must certainly rank as one of the most sordid affairs in American history. specious adj. seeming to be logical and sound, but not really so The article systematically rebuts the specious argument advanced by the so-called expert in the field. spectrum n. band of colors produced when sunlight passes through a prism; a broad range of related ideas or objects The political science course deals with the whole spectrum of political ideologies. spendthrift n. person who spends money recklessly A Chinese proverb describes a paradox: Rich spendthrifts never save enough, but the poor always manage to save something. The adjective spendthrift means wasteful and extravagant. Tom’s spendthrift habits resulted in his accumulating a huge amount of credit card debt. sporadic adj. irregular Despite the ceasefire, there have been sporadic outbreaks of violence between the warring factions. squalor n. filthy, wretched condition The family lives in squalor in the slums of Mexico City. staccato adj. marked by abrupt, clear-cut sounds We listened to the staccato steps of the woman in high heels running down the street. 250 ESSENTIAL WORDS FOR THE GRE REVIEW 70 The correct answers are given on page 409. Matching Match each word with its definition. 1. solvent (A) filthy; corrupt 2. somatic (B) broad range 3. soporific (C) irregular 4. sordid (D) able to meet financial obligations 5. specious (E) person who spends recklessly 6. spectrum (F) seeming to be logical and sound, but not so 7. spendthrift (G) filthy, wretched condition 8. sporadic (H) affecting the body 9. squalor (I) marked by abrupt, clear-cut sounds 10. staccato (J) sleep producing Fill-ins Choose the best word to fill in the blank in each sentence. solvent somatic soporific sordid specious spectrum spendthrift sporadic squalor staccato 1. Newspapers sometimes publish stories with _ claims to increase sales. 2. A __ most of his life, Alex has only recently begun to save for his retirement. 3. Many towns have an area where people live in _ . 4. The salesperson has a sort of machine-gun way of speaking, fast and _ . 5. The various portions of the electromagnetic __ are allocated to broadcasters, commercial operators, amateur hobbyists, and other users. 6. _ outbreaks of violence marred the ceasefire. 7. Economists are concerned that some of the poorest countries will have difficulty remaining _ as interest rates rise and the amount of their debt repayments increase. 8. The long car ride was a __ for the family’s small children; soon they were fast asleep in the back of the car. 9. In recent years, medicine has placed greater emphasis on how psychological factors contribute to _ disorders such as heart disease and cancer. 10. The governor issued a complete and public apology to put the _ affair behind him. 251 ESSENTIAL WORDS FOR THE GRE Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The novels of Mickey Spillane portray the sordid world of criminals. _ 2. In the logic class, students were asked to identify specious lines of reasoning in several arguments. 3. If you absolutely have to stay awake you should take a soporific. 4. The bank’s president warned its directors that it could not remain solvent if it kept making bad loans. _ 5. What the tourist brochure described as “local color” was called “squalor” by a plain-speaking member of the tour group. APPLYING YOUR KNOWLEDGE—61–70 1. In intellectual discourse, statements often must be (i) _ so that they are true; for example, the statement “The cause of war is economic competition between countries” is almost certainly too sweeping and could be (ii) _ by being restated as “One of the major causes of war is economic competition between countries.” Blank (i) Blank (ii) refuted rendered more credible sanctioned made more problematic qualified embellished Explanation: The statement in quotes about the cause of war is described as “sweeping” (too inclusive), so the best choice for Blank (i) is qualified (limited). The second statement in quotes is more believable than the first statement because it has been qualified. Thus, the best choice for Blank (ii) is rendered more credible (made more believable). The correct answers are qualified and rendered more credible. 2. Albert Einstein expressed dismay about quantum mechanics because he felt it introduced a troubling (i) _ into nature, which he felt should be able to be described by laws that, (ii) _, are fully testable and verifiable. 252 ESSENTIAL WORDS FOR THE GRE Blank (i) Blank (ii) capriciousness no matter how recondite dilemma however unscientific aesthetic consideration although empirically-based Explanation: An important clue to the answers to this question is the phrase “laws that... are fully testable and verifiable,” which tells what kind of laws of nature Einstein preferred. Capriciousness (being subject to sudden change) is the best choice for Blank (i) because this quality would make laws of nature difficult to test and verify. Another clue to this answer is the word “troubling;” Einstein found this capriciousness of nature troubling. The answer to Blank (ii) can be found by elimination of however unscientific and although empirically-based, which make no sense in context. No matter how recondite is a good choice because it makes sense that it would not matter if laws of nature were recondite (abstruse) as long as they could be tested and verified. The correct answers are capriciousness and no matter how recondite. UNIT 71 stanch v. to stop or check the flow of The country’s government has put controls on currency movement to stanch the flow of money out of the country. stentorian adj. extremely loud The stentorian speaker prefers not to use a microphone so that the audience can appreciate what he calls “the full effect of my powerful oratory.” stigma n. mark of disgrace or inferiority A problem with giving formal psychological treatment to a child who is believed to be poorly adjusted to society is that he may acquire a stigma as a result of officially being labeled as deviant, and he may act to corroborate society’s expectation. The verb is stigmatize. The civil rights movement helped to stigmatize racism, augmenting legal efforts to desegregate American society. 253 ESSENTIAL WORDS FOR THE GRE stint v. to be sparing Stinting on funding for education strikes many people as shortsighted. Stint is also a noun meaning a period of time spent doing something Isaac Asimov did a short involuntary stint in the army as a conscript during the 1950s. stipulate v. to specify as an essential condition The president’s lawyer stipulated that he would appear before the investigative committee, but would answer only questions directly relevant to the issue at hand. The noun is stipulation. Stipulations in a contract should be clear in order to obviate the need for parties to resort to litigation. stolid adj. having or showing little emotion Behind the professor’s stolid appearance is a fun-loving, gregari ous character. stratified adj. arranged in layers One of the implications of an increasingly stratified economy for America might be increased social unrest. The noun stratum means a layer. In the English-speaking world many members of the upper classes historically have had a deprecatory attitude toward slang, a form of language they regard as indecorous and thus suitable only for the lowest stratum of society. The plural of stratum is strata. As it matured as a science, geology began to complement biology, a process that helped it to gain a more comprehensive view of the history of life on Earth by allowing fossils to be dated and identified (paleontology), often using knowledge gained from stratigraphy—the study of the deposition, distribution, and age of rock strata. The noun stratification is used in the sociological term social strati fication. It refers to the hierarchical arrangement of individuals in a society into classes or castes. striated adj. marked with thin, narrow grooves or channels The striated surface suggested to the geologist that he was walk ing over an area in which there once had been a torrent of water. 254 ESSENTIAL WORDS FOR THE GRE Striation is the noun. The geologist examined striations in the rock to learn about the glacier that had made them 10,000 years ago. stricture n. something that restrains; negative criticism As professionals, lawyers are expected to abide by a set of ethical strictures in their practice of the law. strident adj. loud; harsh; unpleasantly noisy Calls for the prime minister’s resignation became more strident after it was discovered that he had strong connections to organized crime. REVIEW 71 The correct answers are given on page 409. Matching Match each word with its definition. 1. stanch (A) to be sparing 2. stentorian (B) arranged in layers 3. stigma (C) something that restrains 4. stint (D) to specify as an essential condition 5. stipulate (E) unpleasantly noisy 6. stolid (F) showing little emotion 7. stratified (G) marked with thin, narrow grooves 8. striated (H) extremely loud 9. stricture (I) to stop or check the flow of 10. strident (J) mark of disgrace Fill-ins Choose the best word to fill in the blank in each sentence. stanch stentorian stigma stint stipulate stolid stratified striated strictures strident 1. The baseball stadium’s ground rules _ that a bat ter who hits a ball that bounces off the ground into the left field bleachers gets a double. 2. Luke was one of those __ individuals who rarely show their feelings. 3. The geologists examined _ rocks left by the retreating glaciers. 255 ESSENTIAL WORDS FOR THE GRE 4. Modern societies tend to be _ into classes determined by such factors as wealth and occupation. 5. They sat silently in the room, listening to the telephone’s __ ringing. 6. The medic used a tourniquet to _ the woman’s bleeding wound. 7. The speaker’s _ voice rang through the hall. 8. A two-year __ in the navy allowed Janet to visit 22 countries. 9. Perhaps the central paradox of poetry is that the _ imposed by form on a poet of talent can help produce works of great power. 10. In most societies there is a _ attached to mental illness. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The young doctor learned a lot about both medicine and human nature during her stint in the emergency room. _ 2. The contract stipulates that the agreement will remain in force unless both sides agree to cancel it. 3. The banker deliberately cultivated his image as a careful, stolid, conservative person. 4. The Greeks and Persians fought a stentorian battle at Thermopy lae in 480 b.c. _ 5. Anthropologists believe that the society is stratified by occupation, with warriors at the top and workers at the bottom. UNIT 72 strut v. to swagger; display to impress others The star quarterback strutted around campus the entire week after he led his team to a 42–0 win over the county’s top-ranked team. stultify v. to impair or reduce to uselessness The professor of education believes that overreliance on rote learning stultifies students’ creativity. stupefy v. to dull the senses of; stun; astonish After drinking three glasses of wine, Linda was stupefied. 256 ESSENTIAL WORDS FOR THE GRE stygian adj. dark and gloomy; hellish Wilfred Owens’s famous poem “Dulce Et Decorum Est” describes an unfortunate soldier who was unable to get his gas mask on in time, seen through the stygian gloom of poison gas: GAS! Gas! Quick, boys!—An ecstasy of fumbling, Fitting the clumsy helmets just in time; But someone still was yelling out and stumbling And floundering like a man in fire or lime.— Dim, through the misty panes and thick green light As under a green sea, I saw him drowning. subpoena n. notice ordering someone to appear in court The judge issued a subpoena for the man but the prosecutor had little hope that he would appear because he was living abroad. subside v. to settle down; grow quiet Army personnel told the civilians to wait for the violence to subside before reentering the town. substantiate v. to support with proof or evidence The validity of fossil identification is substantiated by data from geology and carbon-14 dating. substantive adj. essential; pertaining to the substance The judge cautioned the attorney to present only information that was substantive to the case at hand. subsume v. to include; incorporate The philosopher described his work as an attempt to arrive at a final generalization that will subsume all previous generalizations about the nature of logic. subversive adj. intended to undermine or overthrow, especially an established government The verb is also subvert. Anything that subverts the market mechanism is believed to cause anomalies in prices, making the economy less efficient. Subversive is also a noun meaning a person intending to undermine something. 257 ESSENTIAL WORDS FOR THE GRE REVIEW 72 The correct answers are given on page 410. Matching Match each word with its definition. 1. strut (A) dark and gloomy 2. stultify (B) to support with proof 3. stupefy (C) to dull the senses of 4. stygian (D) intended to undermine or overthrow 5. subpoena (E) to display to impress others 6. subside (F) to settle down 7. substantiate (G) notice ordering someone to appear in court 8. substantive (H) to include; incorporate 9. subsume (I) to impair or reduce to uselessness 10. subversive (J) essential Fill-ins Choose the best word to fill in the blank in each sentence. strutted stultifying stupefied stygian subpoenaed subside substantiate substantive subsumes subversive 1. Several people at the party were _ from overdrinking. 2. The experiment provided such __ evidence for the new theory that most scientists now accept it. 3. The drill team _ into the stadium to perform the half-time show. 4. The scientist was able to formulate a general principle that _ five more specific principles. 5. Businesses complained that government regulations are __ free competition and innovation. 6. The critic called Emily Bronte’s novel Wuthering Heights _ because it attacks capitalist beliefs. 7. The news that the country was being invaded plunged it into a _ gloom. 8. The prosecution __ three witnesses it considered vital to its case. 9. The engineers waited for the floodwaters to _ before assessing the damage. 10. Advocates of the theory that Atlantis existed more than 6,000 years ago sometimes use evidence of dubious authenticity to _ their claims. 258 ESSENTIAL WORDS FOR THE GRE Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Prosecutors obtained a subpoena to require the witness to testify. _ 2. The old miser is so stygian he refuses to buy his grandchildren birthday presents. 3. After the excitement of the election subsided, the new administra tion settled down to the serious business of governance. 4. The theory was substantiated by new evidence, so scientists were forced to abandon it. _ 5. The Army–McCarthy hearings of the 1950s investigated many citizens alleged to be engaged in subversive activities. UNIT 73 succor n. relief; help in time of distress or want The woman was accused of providing succor to the enemy in the form of food and medical help. suffrage n. the right to vote The pivotal feminist goal of suffrage was not obtained in the United States until 1920, and in Britain not until 1928. sundry adj. various The main character in the novel returns home safely after his sundry adventures. supersede v. to replace, especially to displace as inferior or antiquated Malay was the lingua franca of the Malay peninsula for centuries, but in many parts of that region it is being superseded in that role by a European interloper, English. supine adj. lying on the back; marked by lethargy The captured robbery suspects were held supine on the floor. supplant v. to replace; substitute The “Frankenstein monster” fear of some people is that AI machines will eventually supplant biological life forms, making such life redundant or even subservient. 259 ESSENTIAL WORDS FOR THE GRE suppliant adj. beseeching The worshippers raised their suppliant voices to God, praying for forgiveness. supplicant n. one who asks humbly and earnestly The mother of the man sentenced to be executed appeared as a supplicant before the governor, asking him to grant her son clemency. supposition n. the act of assuming to be true or real Science proceeds on the supposition that knowledge is possible. syllogism n. a form of deductive reasoning that has a major premise, a minor premise, and a conclusion The following syllogism is often taught in logic courses: “All Xs are Ys, all Ys are Zs; therefore, all Xs are Zs.” REVIEW 73 The correct answers are given on page 410. Matching Match each word with its definition. 1. succor (A) beseeching 2. suffrage (B) various 3. sundry (C) lying on the back 4. supersede (D) one who asks humbly and earnestly 5. supine (E) to replace, especially as inferior or antiquated 6. supplant (F) a form of deductive reasoning 7. suppliant (G) relief 8. supplicant (H) act of assuming to be true 9. supposition (I) to replace; substitute 10. syllogism (J) the right to vote Fill-ins Choose the best word to fill in the blank in each sentence. succor suffrage sundry superseded supine supplanted suppliant supplicants supposition syllogism 1. Some experts predict that books made from paper will one day be _ by electronic books. 2. The book tells the story of the protagonist’s __ adventures in Africa over the last 20 years. 260 ESSENTIAL WORDS FOR THE GRE 3. The _ approached the king, begging him to forgive their offences. 4. The depressed man found _ by going inside the church to pray. 5. After eating our picnic lunch, we all lay __ on the ground, looking at the clouds. 6. The logic instructor asked her class to consider whether the following _ was true: Some A are B, some B are C. Therefore, some A are C. 7. The astronomers searching for extraterrestrial life are proceeding on the _ that life requires water. 8. The Twenty-sixth Amendment to the United States Constitution extended __ to both men and women from the age of 18 years, largely because of the fact that many men younger than 21 were being conscripted to fight in the Vietnam War but had no vote. 9. The first generation of digital computers based on vacuum tube technology were _ by a second generation of transistorized computers in the late 1950s and 1960s that could perform millions of operations a second. 10. The painter portrays a _ sinner begging for forgiveness. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The political scientist predicts that by the year 2050 China will supplant Japan as Asia’s most powerful nation. _ 2. The president ordered a halt to the bombing to end the suffrage of the people. 3. The astronomer’s theory makes several suppositions about the nature of the early universe that are not well supported by the evidence. 4. The poem makes use of sophisticated figurative language, notably syllogism. _ 5. The science fiction novel speculates that human beings will one day be superseded by a race of specially bred superintelligent cyborgs. UNIT 74 sylvan adj. related to the woods or forest The house’s sylvan setting provides the family with beauty and tranquility. 261 ESSENTIAL WORDS FOR THE GRE tacit adj. silently understood; implied During the Cold War, there was a tacit assumption on the part of both the Soviet Union and the United States that neither side would launch an unprovoked nuclear attack against the other side. Terms from the Arts, Sciences, and Social Sciences Cold War: the ideological, geopolitical, and economic conflict between capitalist nations (led by the United States) and commu-nist nations (led by the Soviet Union) from around 1947 to 1991 taciturn adj, habitually untalkative The teacher couldn’t get the taciturn child to tell her what activi-ties he enjoyed during recess. talisman n. charm to bring good luck and avert misfortune The soldier’s mother gave him a talisman to protect him from harm during battle. tangential adj. peripheral; digressing The judge ruled that the evidence had only a tangential bearing on the case and directed the lawyer to present only a brief summary of it. tautology n. unnecessary repetition Unless the phrase “repeat again” is being used to refer to some-thing that has occurred more than twice, it is a tautology. taxonomy n. science of classification; in biology, the process of clas-sifying organisms in categories In the late seventeenth century and the eighteenth century accurate observation of organisms developed, leading to the development of the sciences of taxonomy and morphology (the study of the form and structure of organisms.) tenet n. belief; doctrine In his novel Walden II, the psychologist B. F. Skinner depicts a brave new world based on the tenets of a behavioral psychology that frees human beings from the inhibitions and preconceptions of traditional society. 262 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences behavioral psychology: the school of psychology that seeks to explain behavior entirely in terms of observable responses to environmental stimuli tenuous adj. weak; insubstantial Study of the historical evidence has shown that there is only a tenuous connection between the country Plato describes in The Republic and the legendary land of Atlantis. theocracy n. government by priests representing a god All Islamic fundamentalists are opposed to secularism, and some of them support theocracy. Terms from the Arts, Sciences, and Social Sciences secularism: a political movement that advocates making society less religious Secularization is a process by which society gradually changes from close identification with the institutions of religion to a greater separation of religion from the rest of social life. REVIEW 74 The correct answers are given on page 410. Matching Match each word with its definition. 1. sylvan (A) science of classification 2. tacit (B) implied 3. taciturn (C) government by priests 4. talisman (D) weak; insubstantial 5. tangential (E) digressing; diverting 6. tautology (F) related to the woods or forest 7. taxonomy (G) unnecessary repetition 8. tenet (H) habitually untalkative 9. tenuous (I) charm to bring good luck 10. theocracy (J) belief; doctrine 263 ESSENTIAL WORDS FOR THE GRE Fill-ins Choose the best word to fill in the blank in each sentence. sylvan tacit taciturn talismans tangential tautologies taxonomy tenet tenuous theocracy 1. By _ agreement no one in the group talked about the controversial subject of the war. 2. The judge asked everyone involved in the hearing to avoid intro ducing information __ to the main issue. 3. The poet lives in _ seclusion, writing about the beauty of nature. 4. Alice is _, whereas Amy is garrulous. 5. The aim of the revolutionaries was to establish a __ in the country run by senior clergy. 6. Archeologists have discovered objects they believe were used as _ by warriors to ward off death. 7. A central _ of democracy is that the law should treat everyone equally, regardless of his or her race, gender, or social status. 8. Linnaean __ , used in biology, classifies living things into a hierarchy, assigning each a unique place in the system. 9. The study has established a relationship, albeit a _ one, between brain size in mammals and intelligence. 10. The English teacher asked the class to consider whether the phrases “past history” and “old adage” are _ . Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The landlord went to court to evict his tenets. _ 2. Research has demonstrated only a tenuous connection between the two phenomena. 3. Most successful politicians are not taciturn. 4. Members of the tribe believe that the talisman protects them from the evil spirits of the dead. _ 5. Religious leaders are arguing that the only way to save the country is to establish a theocracy. 264 ESSENTIAL WORDS FOR THE GRE UNIT 75 thespian n. an actor or actress Every year the Edinburgh Festival in Scotland gives thespians from around the world the opportunity to perform before a diverse audience. timbre n. the characteristic quality of sound produced by a particular instrument or voice; tone color The audience was delighted by the rich timbre of the singer’s soprano. tirade n. long, violent speech; verbal assault The students had no choice but to sit and wait for the principal’s tirade about poor discipline to end. toady n. flatterer; hanger-on; yes-man The boss had no respect for the employee because he considered him a toady who would do anything he said. tome n. book, usually large and academic Despite being an abridged edition of the twenty-volume Oxford English Dictionary, the Shorter Oxford English Dictionary consists of two tomes that define over half a million words. torpor n. lethargy; dormancy; sluggishness After returning home from his coast-to-coast trip, the truck driver sank into a peaceful torpor, watching TV and dozing. torque n. a turning or twisting force; the moment of a force; the measure of a force’s tendency to produce twisting or turning and rotation around an axis Internal combustion engines produce useful torque over a rather circumscribed range of rotational speeds (normally from about 1,000 rpm to 6,000 rpm). tortuous adj. having many twists and turns; highly complex Only the world’s leading mathematicians are able to follow the tortuous line of reasoning used by the English mathematician Andrew Wiles to prove Fermat’s Last Theorem via the Taniyama-Shimura conjecture. 265 ESSENTIAL WORDS FOR THE GRE tout v. to promote or praise energetically The critic touted Moby Dick as the greatest book in American literature. tractable adj. obedient; yielding The country’s leader found that the people became more tractable when he made them believe there was a great threat facing them that only he could overcome. REVIEW 75 The correct answers are given on page 410. Matching Match each word with its definition. 1. thespian (A) long, violent speech 2. timbre (B) a turning or twisting force 3. tirade (C) to promote 4. toady (D) actor or actress 5. tome (E) obedient; yielding 6. torpor (F) flatterer 7. torque (G) combination of qualities of a sound that distinguish it from others 8. tortuous (H) large, academic book 9. tout (I) having many twists and turns 10. tractable (J) lethargy; sluggishness Fill-ins Choose the best word to fill in the blank in each sentence. thespians timbre tirade toady tome torpor torque tortuous touts tractable 1. The musician has a special affinity for the guitar because of its beautiful _ . 2. The college __ plan to perform three of Shakespeare’s comedies this year. 3. The café _ its cappuccino as the best in town. 4. The violent prisoner became _ after he was given a sedative. 266 ESSENTIAL WORDS FOR THE GRE 5. Every day the talk show host launches into a __ against the failings of modern society. 6. In his Malayan Trilogy, the British novelist Anthony Burgess describes the _ induced by hot Malaysian afternoons. 7. The book describes the author’s _ journey from cynicism and despair to faith and hope. 8. The diesel model of the Nissan Patrol is popular in Australia because it develops sufficient __ to drive through steep, muddy terrain. 9. This 800-page _ called Biology contains most of the information students need to learn for the introductory biology course. 10. Yes, the _ won his promotion, but at what cost to his self-respect? Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Many high church officials are interred in tomes in the cathedral. _ 2. The farmer leased 100 acres of tractable land to grow corn. 3. The enemy launched a tirade of artillery and missiles against our position. 4. Timbre in the forests of most of the developed countries is self-sustaining. _ 5. The group of experts working on the space probe includes mechanical engineers, electrical engineers, physicists, and thespians. UNIT 76 transgression n. act of trespassing or violating a law or rule The teacher made it clear on the first day of the term that she would not countenance any transgression of classroom rules. The verb is transgress. Western medicine transgressed Hippocrates’ prescriptions for medicine when doctors debilitated patients through the administration of purges and bloodletting. 267 ESSENTIAL WORDS FOR THE GRE Terms from the Arts, Sciences, and Social Sciences Hippocrates: ancient Greek physician who is often called “the father of medicine.” He believed that medicine should stress prevention rather than cure of illness and that a regimen of a good diet and a sensible lifestyle is healthy, building a person’s ability to withstand disease. transient adj. temporary; short-lived; fleeting A hypothesis to explain the fact that American states in which the population is composed of a large number of recently settled people (California, for example) tend to have high rates of crime, suicide, divorce, and other social problems is that anomie is higher in transient populations than in more stable populations, resulting in more antisocial behavior. Terms from the Arts, Sciences, and Social Sciences anomie: a social condition marked by a breakdown of social norms translucent adj. partially transparent The architect decided to install a translucent door in the room to allow outside light to shine in. travail n. work, especially arduous work; tribulation; anguish America’s early pioneers endured great travail, but persevered and eventually settled much of the vast continent. Travail is also a verb meaning to work strenuously. travesty n. parody; exaggerated imitation; caricature The playwright complained that the musical comedy version of his play was a travesty of his work. treatise n. article treating a subject systematically and thoroughly The thesis of the philosopher’s treatise is that reality is, ulti mately, opaque to human understanding. tremulous adj. trembling; quivering; frugal; timid One of the most famous poems in English literature is Matthew Arnold’s “Dover Beach,” in which the speaker listens to the “tremulous cadence slow” of waves on the shore. 268 ESSENTIAL WORDS FOR THE GRE trepidation n. fear and anxiety John tried to hide his trepidation when he proposed to Susie, the girl he loved. truculence n. aggressiveness; ferocity The principal warned the student that his truculence might one day land him in jail. tryst n. agreement between lovers to meet; rendezvous In his novel The Mayor of Casterbridge, Thomas Hardy describes an ancient Roman amphitheater where lovers often arranged secret trysts. REVIEW 76 The correct answers are given on page 410. Matching Match each word with its definition. 1. transgression (A) article treating a subject systematically 2. transient (B) partially transparent 3. translucent (C) fear and anxiety 4. travail (D) temporary; fleeting 5. travesty (E) exaggerated imitation; parody 6. treatise (F) aggressiveness 7. tremulous (G) arduous work 8. trepidation (H) act of violating a law 9. truculence (I) rendezvous 10. tryst (J) quivering; fearful Fill-ins Choose the best word to fill in the blank in each sentence. transgressed transient translucent travails travesty treatise tremulous trepidation truculence tryst 1. The pastor urged the members of his congregation to face life’s _ cheerfully. 2. The gang has such a reputation for __ that even the police approach its members with great caution. 3. This afternoon’s solar eclipse will be a _ phenomenon, so make sure you are ready to observe it as soon as it begins. 269 ESSENTIAL WORDS FOR THE GRE 270 4. The philosophic _ deals with Spinoza’s metaphysics. 5. The soldier, his voice __ , begged his captor not to kill him. 6. A prism is a _ piece of glass or crystal that creates a spectrum of light separated according to colors. 7. The judge in the most recent of the many times Dr. Jack Kervorkian was tried for murder for assisting a terminally ill person to kill himself held that the law is sacrosanct and cannot be _ by an individual, even for reasons of conscience. 8. Bill and Sue arranged a __ for Saturday afternoon. 9. The defense attorney called the trial of the soldier accused of war crimes a _ of justice since the judges were all citizens of the nation that had defeated the country for which her defen dant had been fighting. 10. The young scholar approached the problem with considerable _ , knowing that it had been thoroughly discussed by many of the great thinkers through the ages. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Beth’s father said he would prefer that she wore the opaque top, but her mother said she could wear the translucent one. _ 2. The transient nature of the phenomenon makes it difficult for scientists to study. 3. The professor’s treatise on the influence of structuralism on modern thought was published last year. 4. The principal congratulated the student for successfully trans gressing every school regulation. _ 5. The doctor in the soap opera spends so much of her time arrang ing trysts with her lover one wonders how she has time left to practice medicine. UNIT 77 tumid adj. swollen; distended The prose of writers discussing lofty subjects sometimes becomes tumid. turbid adj. muddy; opaque; in a state of great confusion The poem captures the restless and turbid state of the soldier’s mind the night before the decisive battle was set to begin. ESSENTIAL WORDS FOR THE GRE 271 turgid adj. swollen; bloated; pompous The professor’s editor advised him to change his writing style so that it was less pedantic and turgid if he wanted to appeal to a mass audience. tutelary adj. serving as a guardian or protector Most of the people of ancient Rome believed in the existence of tutelary spirits. Terms from the Arts, Sciences, and Social Sciences tutelary spirits: gods who are guardians of a particular area or person uncanny adj. mysterious; strange Some people believe that the psychic has an uncanny ability to accurately predict the future. undulating adj. moving in waves The undulating terrain of the area has made it difficult for engineers to build roads there. unfeigned adj. not false; not made up; genuine The child smiled in unfeigned delight when she opened the Christmas present. untenable adj. indefensible Skeptics are inclined to regard arguments for God’s existence from design as meaningless, since they rely on a logically untenable position that assumes the conclusion of their argument—God’s existence. Terms from the Arts, Sciences, and Social Sciences design: The argument from design is a philosophical argument for God’s existence stating that God must exist because the universe is too complex to have been created any other way. untoward adj. not favorable; troublesome; adverse; unruly Police were called in to investigate whether anything untoward had happened to the missing man. ESSENTIAL WORDS FOR THE GRE 272 usury n. practice of lending money at exorbitant rates In the 1980s, Delaware Governor Pierre S. Du Pont succeeded in having the state’s usury laws liberalized, with the result that many large New York banks set up subsidiaries in Delaware. The adjective is usurious. The consumer advocate’s group complained about the bank’s usurious interest rates. REVIEW 77 The correct answers are given on page 410. Matching Match each word with its definition. 1. tumid (A) serving as a guardian 2. turbid (B) moving in waves 3. turgid (C) swollen; distended 4. tutelary (D) not made up; genuine 5. uncanny (E) mysterious 6. undulating (F) practice of lending money at exorbitant rates 7. unfeigned (G) muddy; opaque; in a state of great confusion 8. untenable (H) not favorable; adverse; troublesome 9. untoward (I) swollen; bloated; pompous 10. usury (J) indefensible Fill-ins Choose the best word to fill in the blank in each sentence. tumid turbid turgid tutelary uncanny undulating unfeigned untenable untoward usury 1. The student looked up with _ astonishment—“You mean I got a perfect score on the GRE?” 2. The prime minister’s position became __ after he lost the support of his own party, so he resigned from office. 3. The consumer organization accused the credit card company of _ after it raised its interest rate to 22 percent per year. 4. The head of the commission said that she did not want the report written in the _ prose too often found in official documents. ESSENTIAL WORDS FOR THE GRE 273 5. The British writer George Orwell often satirized __ political prose. 6. The commander told his troops that _ circumstances had prevented victory, but that if they fought on valiantly, victory would be achieved eventually. 7. The orbiting spacecraft sent a manned vehicle down to the Martian surface, where it explored the area’s _ surface. 8. Steve’s __ ability to predict the outcome of college basketball games has helped him to win a lot of money on bets. 9. Many people believe that they have a guardian angel, a/an _ being that guides and protects them. 10. After the storm the river was _ because of all the soil that had flowed into it from the nearby stream. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. When chess grand masters find themselves in an untenable position they generally resign. _ 2. The tumid weather has made it difficult for the soccer team to train. 3. Despite its entertaining plot, the novel’s turgid prose makes it rather difficult to enjoy. 4. The professor holds an extra tutelary class every Saturday morning. _ 5. The loan shark’s usurious interest rates have attracted the attention of the district attorney. UNIT 78 vacillate v. to waver; oscillate The senator’s position keeps vacillating between remaining neutral and lending his support to the proposal. vacuous adj. empty; void; lacking intelligence; purposeless In Jane Austen’s novel Pride and Prejudice, the youngest of the five Bennett daughters, Lydia, is portrayed as a vacuous young woman with few interests other than having fun. valedictory adj. pertaining to a farewell The 80-year-old actor came out of retirement to give a valedictory performance on Broadway. ESSENTIAL WORDS FOR THE GRE 274 vapid adj. tasteless; dull To relax in the evening the judge likes to watch vapid situation comedies on TV. variegated adj. varied; marked with different colors Botanists are still working to catalog the variegated species of the tropical rain forest. vaunt v. to boast; brag The head coach warned her players not to vaunt their undefeated record. Vaunted is an adjective meaning boasted about. Since every human activity depends on the integrity and proper functioning of the biological system, its destruction through pollution would cause our vaunted technological and economic systems to founder. venal adj. bribable; mercenary; corruptible The depressing though inescapable conclusion the journalist reached is that the mayor went into politics for motives that were almost entirely venal. vendetta n. prolonged feud marked by bitter hostility The judge warned both families that the vendetta between them had to end at once. venerate v. to adore; honor; respect Mother Teresa is venerated for her compassion for the poor people of India. Venerable is an adjective meaning respected because of age, character, or position. In the plain-language edition of the venerable Merck Manual of Diagnosis and Therapy the original definition of a hangnail—“Acute or chronic inflammation of the periungual tissues”—is transmogrified into “An infection around the edge of a fingernail or toenail.” veracious adj. truthful; accurate The witness’ testimony appeared to be veracious at first, but under cross-examination, several inconsistencies appeared. ESSENTIAL WORDS FOR THE GRE 275 REVIEW 78 The correct answers are given on page 411. Matching Match each word with its definition. 1. vacillate (A) bribable; corruptible 2. vacuous (B) varied 3. valedictory (C) to waver; oscillate 4. vapid (D) to boast; brag 5. variegated (E) truthful; accurate 6. vaunt (F) tasteless; dull 7. venal (G) to adore; honor 8. vendetta (H) pertaining to a farewell 9. venerate (I) prolonged feud 10. veracious (J) empty; lacking intelligence Fill-ins Choose the best word to fill in the blank in each sentence. vacillating vacuous valedictory vapid variegated vaunted venal vendetta venerated veracious 1. The saint is _ for her compassion toward all living things. 2. It is a mystery to critics how the writer went from producing __ and sentimental stories to turning out some of the best stories ever written in America. 3. The jury’s decision was based largely on the testimony of a single witness they believed to be _ . 4. The historian’s book describes America’s allies in Vietnam during the 1960s and 1970s as _ and corrupt. 5. The booster club held a __ breakfast for the football team. 6. The plot of Romeo and Juliet is centered around a _ between two noble families, the Capulets and the Montagues. 7. Despite its _ high-tech weapons, the invading army could not defeat the peasants, who were armed only with rifles. 8. The actress, a highly intelligent and well-educated young woman, plays the stereotyped part of the __ “bimbo” in the film. 9. From odd bits of material the artist has achieved _ effects. 10. Philip is _ between going to medical school and law school. ESSENTIAL WORDS FOR THE GRE 276 Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. A veracious reader, Heather is planning to read five of Joseph Conrad’s novels this month. _ 2. The president of the university has prepared some valedictory remarks for the Commencement ceremony. 3. The district attorney was elected mayor largely on his promise to prosecute venal government officials whenever possible. 4. The plot of the movie centers around a family’s vendetta against another family that they believed had disgraced them. _ 5. The editor knew that the reporter’s claim could not be true because it was clearly veracious. UNIT 79 verbose adj. wordy The skillful editor cut 20 percent of the words from the verbose manuscript without appreciably altering its meaning. vertigo n. dizziness The physician diagnosed the patient’s vertigo as being caused by an acute anxiety attack. vexation n. irritation; annoyance; confusion; puzzlement Some people have the ability to prosper and live happily despite life’s inevitable vexations. viable adj. practicable; capable of developing Since the early 1950s, government planners have faced a dilemma: Spend a great deal of money to keep cities viable by rebuilding decrepit infrastructure, or allow them to decay. The noun is viability. According to the historian Arnold Toynbee, there is a strong rela tionship between a society’s view of itself relative to other societies and its continued viability. vindictive adj. spiteful; vengeful; unforgiving The Treaty of Versailles, which concluded World War I, was delib erately vindictive, imposing tremendous penalties on Germany. ESSENTIAL WORDS FOR THE GRE 277 virtuoso n. someone with masterly skills; expert musician The British guitar virtuoso John Williams has entertained thou sands of people during his long career. Virtuoso is also the adjective. Raymond is a virtuoso pianist. visage n. countenance; appearance; aspect The infant studied its mother’s visage intently. viscous adj. thick, syrupy, and sticky The maple syrup is so viscous we had trouble pouring it. vitiate v. to impair the quality of; corrupt morally; make inoperative Unfortunately, one error in the study’s methodology vitiates the entire body of work. vituperative adj. using or containing harsh, abusive censure The young music critic’s vituperative comments aroused the wrath of nearly every serious composer. The verb is vituperate. REVIEW 79 The correct answers are given on page 411. Matching Match each word with its definition. 1. verbose (A) thick, syrupy, and sticky 2. vertigo (B) to impair the quality of 3. vexation (C) spiteful; vengeful 4. viable (D) countenance; appearance 5. vindictive (E) practicable; capable of developing 6. virtuoso (F) wordy 7. visage (G) someone with masterly skills 8. viscous (H) using or containing abusive censure 9. vitiate (I) dizziness 10. vituperative (J) irritation; annoyance ESSENTIAL WORDS FOR THE GRE 278 Fill-ins Choose the best word to fill in the blank in each sentence. verbose vertigo vexations viable vindictive virtuoso visage viscous vitiated vituperative 1. The judge cautioned the attorney not to use his summing up as an opportunity to make __ remarks about imperfections in the criminal justice system. 2. Heathcliff, the protagonist of Wuthering Heights, is _ in seeking revenge against those he believes have harmed him. 3. Many people experience _ when they stand near the edge of a cliff. 4. Sergei Rachmaninoff, a distinguished Russian-born composer, was also a __ pianist who is famous for his interpretations of late romantic composers. 5. On the night before the battle, the soldier had a dream in which he saw the smiling _ of his beloved mother. 6. The candidate’s advisor warned her not to make her acceptance speech _ . 7. The engineer designed the motor to be lubricated with very __ oil. 8. The congressional committee is trying to work out a _ plan to give every American access to affordable, high-quality medical care. 9. The effectiveness of the new government will probably be _ by factors beyond its control. 10. Returning home after the war, the soldier reflected that the __ of daily civilian life would seem like nothing compared to the suffering he had endured as a conscript on the front line. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. The verbose speaker kept digressing to tell anecdotes about her life. 2. The government is studying the plan to provide universal health care to see whether it is economically viable. 3. According to the English professor, virtuosos of the novel form include Nathaniel Hawthorne, Henry James, Willa Cather, and Joseph Conrad. _ ESSENTIAL WORDS FOR THE GRE 279 4. The children enjoyed their visage to their uncle’s house during the summer vacation. 5. Attacks of vertigo can be a symptom of a serious underlying malady. __ UNIT 80 vivisection n. dissection, surgery, or painful experiments performed on a living animal for the purpose of scientific research The book Animal Rights by the philosopher Tom Regan contains a long discussion of vivisection. vogue n. prevailing fashion or practice Although protectionist policies are not in vogue today, great capi talist democracies, such as Great Britain and the United States, flourished for long periods of their histories under protectionist trade policies that were nearly mercantilist—policies that imposed high tariffs on many foreign goods to promote domestic production. volatile adj. tending to vary frequently; fickle Volatility is the noun. Some contemporary economists believe that advances in the understanding of the business cycle virtually preclude a recurrence of the crash of 1929, because governments can take steps to forestall depression. However, others worry that new factors are developing that are, to a significant extent, beyond the control of governments: notably, the ability of investors to quickly switch capital into and out of markets, a situation that could lead to volatility in prices and destabilize markets. vortex n. whirlpool; whirlwind; center of turbulence Inexorably, the country was drawn into the vortex of war. warranted adj. justified The book argues that a new investigation into Marilyn Monroe’s death is warranted by new evidence released by the FBI under the Freedom of Information Act. Warrant is a verb meaning to attest to the accuracy or quality; justify; grant authorization Throughout most of America, procedures in criminal law cases are essentially the same: The government, through a prosecutor, presents its case against a suspect to a grand jury, which decides if there is sufficient evidence to warrant a full trial. ESSENTIAL WORDS FOR THE GRE 280 wary adj. careful; cautious According to psychologists, human beings are naturally wary of strangers. welter v. to wallow or roll; toss about; be in turmoil The pigs weltered about happily in the mud. whimsical adj. fanciful; unpredictable Many children appreciate Dr. Seuss’ whimsical stories. The noun whimsy means a playful or fanciful idea. Despite its rigorous and systematic methodology, there is still considerable room in science for imagination and even whimsy. wistful adj. vaguely longing; sadly thoughtful The poem casts a wistful look back at a way of life that has vanished forever. zealot n. one who is fanatically devoted to a cause The Crusades of the eleventh to thirteenth centuries were conceived of by Christian zealots as a way to drive the Islamic interlopers from the Holy Land. Zealotry is a noun meaning fanaticism. The fact that the judicial branch is relatively undemocratic com pared to the other two branches of government is justified by some theorists of democracy on the grounds that it serves as a check not only on the legislative branch and executive branch, but also on democratic zealotry. The adjective zealous means enthusiastically devoted to a cause. It is heretical to suggest to a zealous capitalist that free enterprise is not the only conceivable realistic economic system. ESSENTIAL WORDS FOR THE GRE 281 REVIEW 80 The correct answers are given on page 411. Matching Match each word with its definition. 1. vivisection (A) tending to vary frequently 2. vogue (B) dissection performed on a living animal for scientific research 3. volatile (C) fanciful 4. vortex (D) one who is fanatically devoted to a cause 5. warranted (E) whirlpool; center of turbulence 6. wary (F) to wallow or roll; be in turmoil 7. welter (G) prevailing fashion 8. whimsical (H) careful; cautious 9. wistful (I) sadly thoughtful 10. zealot (J) justified Fill-ins Choose the best word to fill in the blank in each sentence. vivisections vogue volatile vortex warranted wary welter whimsical wistful zealot 1. Brad is such a party _ that he has never even consid ered voting for a candidate who does not belong to his party. 2. Joan’s friend said that she should be __ of the man loitering around campus. 3. Lost in the _ of conflicting information was the fact that there was no decisive proof of the theory’s validity. 4. In James Boswell’s Life of Johnson (1775), Samuel Johnson comments that the American colonists are “a race of convicts;” Boswell, however, expresses a contrary view: “I had now formed a clear and settled opinion, that the people of America were well _ to resist a claim that their fellow subjects in the mother country should have the entire command of their fortunes, by taxing them without their consent.” 5. Militant feminism reached its zenith in the 1960s, and since then a less confrontational approach to asserting women’s rights has been in __ . 6. Steve advised his friend not to invest in the stock market until it became less _ . ESSENTIAL WORDS FOR THE GRE 282 7. The animal rights group organized a protest against the _ being performed in the university biology laboratory. 8. The Swiss painter Paul Klee is famous for his humorous, personal, and often __ paintings. 9. All of the people visiting the war memorial had _ looks on their faces. 10. The young people of the country were drawn steadily into the _ of revolutionary activity. Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 1. Mini-skirts were in vogue in the 1960s. _ 2. The panel is considering the legal and moral implications of vivisection. 3. The coach decided that his team’s excellent performance in the game warranted a day off practice. 4. The psychologist excels in helping patients learn to understand their volatile emotions. _ 5. The English novelist Evelyn Waugh was a practicing Roman Catholic, though hardly a zealot. APPLYING YOUR KNOWLEDGE—71–80 1. Describing an argument as “sophistical” means that you believe it is _—that is, misleading and false. jocular judicious specious euphemistic conventional Explanation: A sophistical (relating to deceptive reasoning or argu mentation) argument is by definition specious (seeming to be logical and sound but not really so). Thus, the correct answer is specious. The correct answer is specious. ESSENTIAL WORDS FOR THE GRE 283 2. The philosopher Karl Popper has given a clear example to illustrate the process of induction: an observer seeking to establish the color of swans could observe thousands of white swans and reasonably conclude that all swans are white; however, when the first black swan appeared, the __ that all swans are white would be disproved. stipulation syllogism tautology hypothesis platitude Explanation: Hypothesis is the correct answer because the sentence describes an attempt to reach a conclusion about the color of all swans by observation. A hypothesis is a tentative explanation for an observation. The correct answer is hypothesis. ESSENTIAL WORDS FOR THE GRE 284 Review: 300 High-Frequency GRE Words Matching Match each word with its definition. Part A 1. aberrant (A) causing quarrels 2. aesthetics (B) known only to a few 3. anomaly (C) intentionally use vague language 4. arcane (D) the conception of what is beautiful 5. bombastic (E) pertaining to people 6. contentious (F) attacking cherished traditions 7. demotic (G) deviating from what is normal 8. disparage (H) using inflated language 9. equivocate (I) belittle 10. iconoclastic (J) irregularity Matching Match each word with its definition. Part B 11. immutable (A) talkative 12. implacable (B) using few words 13. intractable (C) disgrace 14. juxtapose (D) incapable of being pleased 15. laconic (E) too helpful 16. laudable (F) transparent 17. loquacious (G) not easily managed 18. officious (H) praiseworthy 19. opprobrium (I) unchangeable 20. pellucid (J) place side by side REVIEW: 300 HIGH-FREQUENCY GRE WORDS Matching Match each word with its definition. Part C 21. plethora (A) wordy 22. prevaricate (B) excess 23. propitiate (C) inactive 24. quiescent (D) resisting control 25. reprobate (E) evade the truth 26. salubrious (F) not inclined to speak much 27. taciturn (G) morally unprincipled person 28. recalcitrant (H) containing harsh censure 29. verbose (I) appease 30. vituperative (J) healthful Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 31. Perhaps because it had become too insular in its outlook, the monarchy obdurately refused to heed growing calls for liberalization. 32. An example of Earth’s suitability for life is the vapid combination of temperature and pressure that allows large quantities of water to remain in the liquid phase. _ 33. Historians agree that the executive branch of government claimed more power for itself largely in response to the exigencies of the modern world. 34. A collective endeavor requires some degree of apathy because in order to work together people must to some extent sacrifice self-interest. _ 35. Aesop’s fables are fatuous, teaching profound moral lessons through symbolism. 36. Studies of various types of psychotherapy in use show that they do not vary appreciably in their efficacy as measured by empirically verifiable means. _ 37. The existence of planets in other star systems is deduced from indirect evidence gathered from meticulous research on the effects of these planets on their sun’s orbit. 38. In the view of some legal experts, if every individual acted according to his or her conscience the polity would break down into anarchy. _ 39. Defenders of intelligence tests say that they measure a quality, which although elusive, is none the less real, because scores on intelligence tests belie academic and career success. 285 ESSENTIAL WORDS FOR THE GRE 40. Human behavior can be regarded as a result of the interaction between instincts, on the one hand, and, on the other hand, the effects of socialization and individual turgidity. _ Fill-ins Choose the best word to fill in the blank in each sentence. 41. _ a list of phobias is a good way to learn some Latin and Greek roots; for example, agora in Greek means “marketplace,” giving us agoraphobia—fear of public places. Ⓐ Burnishing Ⓑ Exacting Ⓒ Dissembling Ⓓ Perusing Ⓔ Diffusing 42. The __ librarian merely raised her eyebrows when a fire broke out in the reference section. Ⓐ effete Ⓑ heterodox Ⓒ phlegmatic Ⓓ pedantic Ⓔ ambivalent 43. Many pacifists believe that because people are adept at rationalizing violence _ by the state, warfare will not end until it is deemed an unacceptable option for pursuing national policy. Ⓐ sanctioned Ⓑ repudiated Ⓒ feigned Ⓓ apprised Ⓔ vitiated 44. Although most Americans were reluctant to enter World War II, many were persuaded that “fortress America”—a bulwark against antidemocratic forces—would become _ if Germany triumphed in Europe and Japan dominated Asia. Ⓐ refractory Ⓑ untenable Ⓒ incompatible Ⓓ perfidious Ⓔ pervasive 286 REVIEW: 300 HIGH-FREQUENCY GRE WORDS 45. Some third world leaders argue that the West, given its advanced development, can afford to give precedence to political rights, but that poor countries must be __ and give priority to economic rights such as the right to eat and have a job, even if this means circumscribing political freedom. Ⓐ whimsical Ⓑ magnanimous Ⓒ abstemious Ⓓ pragmatic Ⓔ diffident 46. At one extreme of poetic form is the sonnet, a poem of fourteen lines demanding _ rigorous _ governing meter and form, while at the other extreme is free verse, which is composed of variable, unrhymed lines that have no fixed metrical pattern. Ⓐ concurrence with..platitudes Ⓑ adherence to..conventions Ⓒ conformity with..discrepancies Ⓓ deference to..anomalies Ⓔ approbation of..stigmas 47. There is an ongoing debate about whether watching violent programs on television makes people more violent, or whether it actually purges, or at least __, the violent tendencies already _ in people; unfortunately, there is no conclusive evidence yet for either view. Ⓐ tempers..inherent Ⓑ mitigates..discordant Ⓒ reprises..innate Ⓓ inculcates..pernicious Ⓔ avers..oblique 48. Some observers worry that the trend toward globalization may perpetuate poverty in developing countries and _ the __ between the rich and poor nations. Ⓐ aggrandize...transgressions Ⓑ assuage..discrepancy Ⓒ palliate..divergence Ⓓ exacerbate..disparity Ⓔ corroborate...propriety 287 ESSENTIAL WORDS FOR THE GRE 288 49. An argument can appear plausible due to the author’s ability to embellish a(n) _ argument so that its _ reasoning is concealed beneath a beguiling exterior. Ⓐ tenuous..specious Ⓑ doctrinaire..convoluted Ⓒ disingenuous..rarefied Ⓓ desultory..meretricious Ⓔ inchoate..lucid 50. The remarkable conductivity of fiber-optic cables has __ the need for expensive repeaters to boost signals that in _ cable become _ over long distances. Ⓐ precipitated..analogous..amalgamated Ⓑ exacerbated..pristine..mitigated Ⓒ proscribed..recondite..ameliorated Ⓓ subsumed..ordinary..occluded Ⓔ obviated..conventional..attenuated End of Review REVIEW: 300 HIGH-FREQUENCY GRE WORDS ANSWERS 1. G 11. I 21. B 31. S 41. D 2. D 12. D 22. E 32. N 42. C 3. J 13. G 23. I 33. S 43. A 4. B 14. J 24. C 34. N 44. B 5. H 15. B 25. G 35. N 45. D 6. A 16. H 26. J 36. S 46. B 7. E 17. A 27. F 37. S 47. A 8. I 18. E 28. D 38. S 48. D 9. C 19. C 29. A 39. N 49. A 10. F 20. F 30. H 40. N 50. E 289 Review: Essential Words for the GRE Matching Match each word with its definition. Part A 1. desuetude (A) growth in size 2. extraneous (B) unrehearsed 3. bifurcate (C) self-control 4. fulminate (D) divide into two parts 5. continence (E) complain 6. extemporaneous (F) moving away from the center 7. accretion (G) denounce 8. centrifugal (H) state of disuse 9. grouse (I) moving toward the center 10. centripetal (J) not essential Matching Match each word with its definition. Part B 11. impervious (A) fleeting feeling of hurt pride 12. labile (B) not candid 13. affinity (C) impossible to appease 14. plethora (D) to waver; oscillate 15. implacable (E) fondness; similarity 16. vitiate (F) overabundance 17. disingenuous (G) likely to change 18. pique (H) incapable of being affected 19. subsume (I) to include; incorporate 20. vacillate (J) to impair the quality of ESSENTIAL WORDS FOR THE GRE 290 Sense or Nonsense Indicate whether each sentence makes good sense or not. Put S (SENSE) if it does, and put N (NONSENSE) if it does not. 21. Artwork of the classical period can seem austere to the uninitiated, but to the discerning audience it is satisfying to have feelings expressed in this form because they are transmuted in the crucible of art into a more stygian form. _ 22. Although he is remembered chiefly as a novelist, D.H. Lawrence also had an enervation for writing verse. 23. In his poetry and novels, writer Thomas Hardy often portrayed a contumacious God who interfered almost maliciously in human affairs. 24. The nineteenth-century British satirist Thomas Love Peacock lampooned the metaphysical speculation of thinkers like Samuel Taylor Coleridge as pretentious and limpid. _ 25. Skeptics believe that the Green Revolution can only mitigate the effects of a rapidly increasing demand for food, and that in the long run starvation will reappear when pestilence and other disasters decrease food supplies. 26. The scientist is in an ethical quandary about whether he should repudiate his past involvement in developing a weapon of mass destruction. 27. In 1787, when the U.S. Constitution was being framed, it was proposed that slavery be abolished, but opponents of the measure forced a compromise whereby slavery would not be prescribed until early in the next century. _ 28. Evolution is a process that results in the overall improvement of life; paradoxically, however, that process of improvement is driven by aberrations in the process of DNA’s self-replication. 29. In the so-called “clockwork universe” of the Deists, God is relegated to the role of a “clockmaker” who creates the cosmos and then withdraws to allow man autonomous action. 30. In his argument against conscription, Joseph conceded that there might be rarefied situations in which it is justified, but warned that allowing it in these cases might be a step down the slippery slope to totalitarianism. _ REVIEW: ESSENTIAL WORDS FOR THE GRE 291 Fill-ins Choose the best word or set of words to fill in the blanks in each sentence. 31. The Hubble Space Telescope—in orbit around Earth to offer observations not _ by Earth’s atmosphere—has been a boon to astronomers; it is one of the finest astronomical instruments ever developed, greatly expanding man’s gaze into space. Ⓐ attenuated Ⓑ mitigated Ⓒ imploded Ⓓ subsumed Ⓔ intimated 32. The literary critic Susan Sontag uttered a famous _ dictum: “Taste has no system and no proofs”—by which she meant that artistic taste is subjective, since there are no unbiased criteria for assessing art. Ⓐ desultory Ⓑ aesthetic Ⓒ existential Ⓓ linguistic Ⓔ capricious 33. __ of primitivism is that there is no progress in art, and thus the art of so-called “primitive” cultures is as _ as that of so-called “high” civilization. Ⓐ An exigency..plastic Ⓑ A precept..felicitous Ⓒ A credo..gauche Ⓓ A supposition..whimsical Ⓔ A tenet..evocative ESSENTIAL WORDS FOR THE GRE 292 34. In 1787, when the United States Constitution was written, it was proposed that slavery be abolished, but opponents of the measure forced a compromise whereby slavery would not be _ until early in the next century. Ⓐ admonished Ⓑ emulated Ⓒ proscribed Ⓓ interpolated Ⓔ obviated 35. The following __ , called Olber’s paradox, long puzzled astronomers: If the universe is infinite in extent and age, and filled with stars, why is the sky dark at night? Ⓐ apothegm Ⓑ stricture Ⓒ valedictory Ⓓ conundrum Ⓔ vendetta 36. _ was an academic discipline at many universities in the early twentieth century, and was supported by such _ figures as Winston Churchill and George Bernard Shaw until it became closely associated with abuses of the Nazis of the 1940s in Germany, who carried out atrocities such as the extermination of undersized population groups. Ⓐ Epistemology..impassive Ⓑ Eugenics..redoubtable Ⓒ Necromancy..discerning Ⓓ Hieroglyphics..beneficent Ⓔ Cartography..avuncular 37. __ generally believe that determinism is incompatible with human dignity, and _ attempts to limit man’s freedom. Ⓐ Libertines..aver Ⓑ Existentialists..disparage Ⓒ Neophytes..repudiate Ⓓ Conscripts..aggrandize Ⓔ Anarchists..admonish REVIEW: ESSENTIAL WORDS FOR THE GRE 293 38. Scholars are sometimes tempted into _ off the main topic to discuss esoteric areas of interest to them, but which are regarded by many readers as __ display of _ . Ⓐ diatribes..a formidable..miscellany Ⓑ soliloquies..a megalomaniacal..propriety Ⓒ digressions..a pedantic..erudition Ⓓ homilies..an egotistical..sagacity Ⓔ expositions..a bombastic..chivalry 39. When the word “gay” began to be widely adopted to refer to homosexuals, some commentators, presumably unaware of the word’s complex history and long association with homosexuality, _ it as a __ with connotations of merriment that was being foisted by homosexuals on the heterosexual majority. Ⓐ defamed..tautology Ⓑ denigrated..syllogism Ⓒ derided..euphemism Ⓓ disparaged..neologism Ⓔ maligned..mnemonic 40. _ student of literature remembers that literary terms are notoriously _ in that their meanings are ever shifting depending on the premises of the writer using them and the nature of the work under discussion. Ⓐ A jejune..inevitably Ⓑ A craven..poignantly Ⓒ A judicious..impetuously Ⓓ An astute..querulously Ⓔ A sagacious..plastic 41. The fact that social welfare programs existed only in embryonic form during the Great Depression __ the effects of that depression because there was virtually no mechanism for coping with sudden and _ unemployment. Ⓐ negated..perennial Ⓑ forestalled..precipitate Ⓒ alleviated..ineluctable Ⓓ exacerbated..pervasive Ⓔ impeded..substantive ESSENTIAL WORDS FOR THE GRE 294 42. _ believe that because people are __ agents, they should not allow themselves to be circumscribed by the restrictions of the state. Ⓐ Misanthropes..covert Ⓑ Mavericks..complaisant Ⓒ Iconoclasts..viable Ⓓ Zealots..guileless Ⓔ Anarchists..autonomous 43. Because it is very quickly destroyed by ordinary matter, antimatter has _ existence in our locality of the universe. Ⓐ a derivative Ⓑ an intangible Ⓒ a viable Ⓓ an ephemeral Ⓔ a poignant 44. To make your writing _, it is a good idea to read what you have written from a reader’s perspective, looking for any language that is __ . Ⓐ execrable..bombastic Ⓑ banal..convoluted Ⓒ pellucid..equivocal Ⓓ discordant..tangential Ⓔ amenable..disjointed 45. Modern _ uses _ techniques involving methodologies such as photogrammetry, which utilizes photographs taken from airplanes and satellites to measure topography with extreme accuracy. Ⓐ meteorology..salutary Ⓑ paleontology..hermetic Ⓒ entomology..audacious Ⓓ ornithology..recondite Ⓔ cartography..esoteric REVIEW: ESSENTIAL WORDS FOR THE GRE 295 46. The __, “Women are more intelligent than men” needs to be _, because not all women are smarter than all men. Ⓐ axiom..jibed Ⓑ stricture..refuted Ⓒ extrapolation..queried Ⓓ contention..qualified Ⓔ credo..vitiated 47. The concept of the biosphere has helped to _ the idea of life on earth as a fragile and interdependent system that humanity disrupts at its peril. Ⓐ supplant Ⓑ bifurcate Ⓒ burnish Ⓓ disseminate Ⓔ amalgamate 48. It seems likely that herd mentality plays a part in depressions; as an economy slumps, some people panic, others __ this panic, and something akin to mass hysteria ensues. Ⓐ arrest Ⓑ foreswear Ⓒ impede Ⓓ subsume Ⓔ emulate 49. In burning fossil fuels so _ , humanity is squandering a legacy from _ times. Ⓐ ostentatiously..indeterminate Ⓑ presumptuously..antediluvian Ⓒ precipitously..sidereal Ⓓ prodigally..primordial Ⓔ abstemiously..anachronistic ESSENTIAL WORDS FOR THE GRE 50. Because of its political problems and __ inflation for long periods after World War II, some economists have cited Argentina as a developed country that was nearly _ to the rank of an underdeveloped country. Ⓐ insuperable..divested Ⓑ intractable..relegated Ⓒ ineluctable..accrued Ⓓ implacable..goaded Ⓔ nugatory..interpolated End of Review REVIEW: ESSENTIAL WORDS FOR THE GRE ANSWERS 1. H 11. H 21. N 31. A 41. D 2. J 12. G 22. N 32. B 42. E 3. D 13. E 23. N 33. E 43. D 4. G 14. F 24. N 34. C 44. C 5. C 15. C 25. S 35. D 45. E 6. B 16. J 26. S 36. B 46. D 7. A 17. B 27. N 37. B 47. D 8. F 18. A 28. S 38. C 48. E 9. E 19. I 29. S 39. C 49. D 10. I 20. D 30. N 40. E 50. B 296 297 300 High-Frequency Word Roots 300 Essential Roots and Prefixes 1,500 English Derivatives MASTERING HIGH-FREQUENCY WORD ROOTS TO FURTHER EXPAND YOUR VOCABULARY You have learned 800 very important words. You now have a good vocabulary for the GRE. However, no book could possibly teach every single word that might appear on the test. What is the most efficient way to learn even more words? The answer—as was already mentioned in the introduction—is to learn important root words and build up your knowledge of advanced words based on these roots. This section teaches you 300 high-frequency roots and prefixes, as well as 1,500 words derived from them. Learning these roots will give you a solid understanding of the building blocks of English words. It will also reinforce your learning of many of the words covered earlier in the book. Most importantly, it will give you the tools to decipher tens of thousands of the sort of academic words that appear in high-level read ing material—and therefore could be included on the GRE. HOW ROOTS WORK—AND HOW THEY CAN WORK FOR YOU Most English words were created from Anglo-Saxon, Greek, Latin, French, Italian, and German roots and stems, and certain affixes—word elements that are affixed to words as prefixes or suffixes to refine the meaning or change a word’s grammatical form. Of the more than one million words in English, approximately 60 percent come from Latin and Greek roots. This means that knowledge of Latin and Greek roots that frequently appear in English words will help you to gain a better understanding of the origin and meaning of many words. For example, in Unit 30 you learned that the word exacerbate means “to aggravate; make worse,” but do you know the origin of this word? Exacerbate was formed from a combination of the following: ex (an intensive prefix) + the Latin root acer (harsh, bit ter), and the suffix ate (make, do). You also learned the word aberrant, ESSENTIAL WORDS FOR THE GRE which means “deviating from the expected or normal course.” It is from ab (away from) + errare (to stray). Therefore, if you knew the meaning of the root err (to stray), and all of the suffixes attached to that root, such as or (a quality or condition), ous (full of), and ant (state of being), would you be able to decipher the meanings (or at least part of the meaning) of the following words? Try it: err error errant erratic erroneous You would be correct if you said: err means to make a mistake an error is a mistake errant means mistaken, or straying from the proper course erratic means deviating from the customary course erroneous means mistaken Let’s expand on this exercise to demonstrate how you can put roots to work to help decode very advanced words. The following ten words were created from one or more Latin and Greek roots and certain com mon suffixes: acuminate ergatocracy orthotropism neonate noctilucent osseous paleography sacrosanct sequatious somniloquy Write down what you think are the meanings of each word. If you do not know the exact definition, jot down your best guess. Then check to see if your answers are correct, or at least partially correct. Give your self 10 points for each completely correct answer and 5 points for each partially correct answer. Here are the correct definitions: acuminate = make sharp; taper to a point ergatocracy = government by workers orthotropism = vertical growth neonate = newborn child noctilucent = shining at night osseous = bony; composed of, or containing bone paleography = study of ancient writings sacrosanct = extremely sacred; inviolable sequatious = disposed to follow another somniloquy = the act of talking in one’s sleep Total your score. If your score is below 50 percent, your root skills for the GRE are low and you will benefit tremendously from an intensive study of the major roots and prefixes that are listed in 300 High-Frequency Word Roots. If your score is between 50 percent and 80 percent, you are strongly advised to review these Greek and Latin roots for a superior score on the GRE. 298 300 HIGH-FREQUENCY WORD ROOTS So, exactly how were the words in our exercise formed from Greek and Latin roots? acuminate is from acu (sharp) + ate (verb suffix meaning make) ergatocracy is from erg (work) + cracy (government) orthotropism is from ortho (straight, upright, correct) + tropo (turning, change) neonate is from neo (new) + natur (born) noctilucent is from nocti (night) + luc (light, shine) osseous is from oss (bone) + suffix ending ous (full of; characterized by) paleography is from paleo (ancient) + graph (write) sacrosanct is from sacrum (religious rite) + sanctus (to consecrate) sequatious is from sequi (follow) + suffix ending ous (full of; characterized by) somniloquy is from somni (sleep) + loqui (speak) You may say that it is very unlikely you will have to know words like orthotropism or sequatious. This may be true if you become a plumber. However, if you do a postgraduate degree and go onto a career in aca demia or a professional career in medicine, law, and other fields you will be seeing plenty of such words. After you complete this unit, you will be able to attack words like this that you do not know, and steadily keep expanding your vocabulary. Of course, your learning of roots should not stop here. You should keep adding new roots to your knowledge. You can do this by regularly consulting a dictionary. When you meet a word you don’t know and you cannot decipher it, look it up. The dictionary will give you the word’s etymology (origin and history). Make sure to use a good dictionary such as the American Heritage College Dictionary or Merriam-Webster’s Col legiate Dictionary. Let’s take an example to show how it works. Below is reproduced part of the entry for the word “aberration” from the American Heritage College Dictionary, fourth edition. The etymology is given at the end of the entry in brackets [ ]. [Lat. aberratio, aberration-, diversion < aberratus, p. part. of aberrare, to go astray: ab-, away from; see AB- + errare, to stray.] Now let’s begin our study of 300 High-Frequency Word Roots. Major roots and prefixes appear as headings in each “Root Roundup” on the following pages, and there is a list of common suffixes provided at the end of the section. After mastering all the words and roots in each “Root Roundup,” do the “Root Work” exercise to make sure you have learned everything. Comprehensive “Root Roundup” reviews are provided after every five units. The correct answers are given on pages 411–416. 299 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP 1 • A/AN (WITHOUT, ABSENSE OF, NOT) Greek atheist = person who does not believe in the existence of a god agnostic = person who is doubtful about something anarchy = absence of political authority anemia = deficiency in the part of the blood that carries oxygen anachronism = something out of the proper time • AB/ABS (FROM, AWAY, OFF) Latin abduct = carry away by force; kidnap aberrant = deviating away from the expected or normal course abrade = wear away by friction; erode abdicate = formally relinquish power or responsibility abstinence = refraining from something • ACER/ACID/ACRI (HARSH, BITTER, SOUR) Latin acrid = sharp or bitter to the taste or smell; sharp in language or tone acrimonious = bitter and sharp in language and tone acerbate = annoy acerbity = sourness or bitterness of taste, character, or tone exacerbate = increase bitterness; make worse • ACT/AG (DRIVE, DO, LEAD, ACT, MOVE) Latin active = being in physical motion actuate = put into motion; activate agenda = list or program of things to be done agency = condition of being in action agitation = act of causing to move with violent force • ACU (SHARP) Latin acumen = keenness of judgment acuminate = tapering to a point; make sharp; taper acupuncture = therapeutic technique that uses needles to relieve pain aculeate = having a stinger; having sharp prickles acuity = sharpness of perception or vision 300 300 HIGH-FREQUENCY WORD ROOTS Root Work 1 Match each word with its definition. 1. agency (A) formally relinquish power 2. exacerbate (B) sharpness of vision 3. actuate (C) deficiency in the blood 4. abstinence (D) increase bitterness 5. anarchy (E) condition of being in action 6. acuity (F) make sharp; taper 7. abdicate (G) sharp or bitter to the taste or smell 8. anemia (H) absence of political authority 9. acrid (I) put into motion 10. acuminate (J) refraining from something (Answers are on page 411.) ROOT ROUNDUP 2 • AD (TO, TOWARD) AC/AF/AG/AL/AN/AP/AR/AS/AT before consonants Latin accord = cause to agree; bring into harmony acquiesce = consent quietly to something advent = arrival or coming aggregate = amounting to a whole; total appease = bring peace or calm to; to soothe • AEV/EV (AGE, ERA) Latin primeval = belonging to the earliest age medieval = belonging to the Middle Ages medievalism = devotion to the ideas of the Middle Ages coeval = existing during the same era longevity = long life; long duration • AGOG (LEADER) Greek pedagogue = teacher; a dogmatic teacher synagogue = place of meeting for worship in the Jewish faith emmenagogue = agent that induces menstrual flow anagogy = mystical interpretation that detects allusions to the afterlife hypnagogic = inducing sleep • AGR (FIELD) Latin agribusiness = farming done as a large-scale business agriculture = farming agrarian = relating to farming or rural matters agritourism = form of tourism that lets people experience life on a farm agronomy = application of science to farming 301 ESSENTIAL WORDS FOR THE GRE • ALI (ANOTHER) Latin alien = characteristic of another place or society; strange alienation = emotional isolation or disassociation inalienable = not capable of being surrendered alibi = fact of absence from the scene of a crime alienage = official status as an alien Root Work 2 Match each word with its definition. 1. agrarian (A) application of science to farming 2. primeval (B) from another place or society 3. inalienable (C) existing during the same era 4. pedagogue (D) relating to farming 5. agronomy (E) inducing sleep 6. appease (F) belonging to the earliest age 7. coeval (G) arrival or coming 8. advent (H) bring peace or calm to 9. hypnagogic (I) teacher; dogmatic leader 10. alien (J) not capable of being surrendered (Answers are on page 411.) ROOT ROUNDUP 3 • ALIM (SUPPORT, NOURISH, CHERISH) Latin aliment = something that nourishes alimony = allowance for support to a divorced person by the former chief provider alible = nourishing alimentary = concerned with nutrition or food alimentation = giving or receiving of nourishment • ALTER (OTHER) Latin alter = change; modify; become different alternate = proceed by turns alternative = one of two mutually exclusive possibilities alter ego = second self or another side of oneself altercate = argue vehemently • ALT (HIGH, DEEP) Latin altar = elevated structure before which religious ceremonies are performed exaltation = condition of being raised up in rank altimeter = instrument that measures elevation altiplano = high plateau altitude = height of something above a certain reference level 302 300 HIGH-FREQUENCY WORD ROOTS • AM (LOVE, LIKING) Latin amiable = friendly; likeable enamored = captivated amicable = friendly amity = friendship amatory = inclined toward love • AMB/AMBUL (TO GO, TO WALK) Latin ambulate = walk from place to place amble = walk slowly ambulance = vehicle to transport injured people perambulate = walk about ambulatory = capable of walking Root Work 3 Match each word with its definition. 1. perambulate (A) walk slowly 2. alter ego (B) friendly 3. amatory (C) become different 4. alter (D) nourishing 5. amicable (E) high plateau 6. altiplano (F) walk about 7. alimentary (G) another side of oneself 8. exaltation (H) concerned with nutrition 9. amble (I) being raised up in rank 10. alible (J) inclined toward love (Answers are on page 411.) ROOT ROUNDUP 4 • AMBI (AROUND, ON BOTH SIDES) Latin ambient = surrounding ambidextrous = able to use both hands well ambivalent = having conflicting feelings ambiguous = doubtful or unclear ambiversion = personality trait that combines both introversion and extroversion • AMPH/AMPHI (AROUND, DOUBLE, ON BOTH SIDES) Greek amphibian = animal that can live both on land and in water amphora = two-handled Greek or Roman jar amphitheater = round structure with levels of seats rising upward from central area amphidiploid = having a diploid set of chromosomes from each parent amphibolous = having a grammatical structure that allows two interpretations 303 ESSENTIAL WORDS FOR THE GRE • ANIM (LIFE, BREATH, SPIRIT) Latin animal = multicellular organism of the kingdom Animalia animation = enthusiasm; excitement animism = belief that individual spirits inhabit natural phenomena animosity = hostility; hatred inanimate = not exhibiting life • ANNU/ANNI/ENNI (YEARLY) Latin annuity = yearly income payment anniversary = yearly recurring date of an event that occurred in the past biennial = happening every two years perennial = lasting throughout the year or for several years millennium = thousand-year period • ANT/ANTE (BEFORE) Latin antecedent = something that comes before antediluvian = extremely old; happening before the Flood antedate = come before in time anterior = placed before; earlier antler = bony growth on the head of a deer Root Work 4 Match each word with its definition. 1. inanimate (A) enthusiasm 2. ambiguous (B) having conflicting feelings 3. antediluvian (C) yearly income payment 4. animation (D) animal able to live on land or in water 5. millennium (E) not exhibiting life 6. amphibian (F) doubtful; unclear 7. ambivalent (G) something coming before 8. amphora (H) two-handled Greek or Roman jar 9. annuity (I) thousand-year period 10. antecedent (J) extremely old (Answers are on page 411.) ROOT ROUNDUP 5 • ANT/ANTI (AGAINST, OPPOSITE) Greek antibiotic = substance that can kill microorganisms antiseptic = substance that can kill disease-causing organisms antipathy = dislike antithesis = opposite of antagonistic = hostile 304 300 HIGH-FREQUENCY WORD ROOTS • ANTHROP (MANKIND, HUMAN BEING) Greek anthropic = related to the human race anthropoid = resembling human beings anthropology = study of man misanthropy = hatred of humanity anthropocentric = regarding human beings as the center of the universe • ANTIQU (OLD, ANCIENT) Latin antiquated = too old to be useful or fashionable antique = belonging to an earlier period antiquity = ancient times; an object from ancient times antiquarian = relating to the study of antiquities antiquate = make old-fashioned or obsolete • APPELL (NAME, CALL UPON) Latin appellation = name or title appellative = relating to the assignment of names appeal = earnest or urgent request appellant = relating to an appeal appellate = having the power to hear court appeals • APT/EPT (SKILL, ABILITY) Latin inept = not suitable; having a lack of judgment or reason apt = exactly suitable unapt = not suitable adapt = make suitable to a specific situation aptitude = inherent ability; a talent Root Work 5 Match each word with its definition. 1. antiquated (A) exactly suitable 2. misanthropy (B) name or title 3. inept (C) hatred of humanity 4. antithesis (D) relating to the assignment of names 5. appellative (E) dislike 6. anthropic (F) not suitable; lacking judgment 7. appellation (G) belonging to an earlier period 8. antipathy (H) related to the human race 9. apt (I) too old to be useful 10. antique (J) the opposite of (Answers are on page 411.) 305 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP REVIEW 1–5 Match It Match each of the following roots to its meaning. 1. ANTHROP ___ (A) harsh, bitter, sour 2. ACER/ACID/ACRI ___ (B) skill, ability 3. ACT/AG ___ (C) leader 4. AMBI ___ (D) to, toward 5. ANTE ___ (E) other 6. A/AD ___ (F) human being 7. ALTER ___ (G) around, on both sides 8. AP/EPT ___ (H) love, liking 9. AM ___ (I) drive, do, lead, act, move 10. AGOG ___ (J) before Fill-ins Fill in the blanks with the word that fits the definition. aliment anarchy annuity acumen pedagogue unapt antiquate agrarian inanimate perambulate 1. teacher; a dogmatic teacher __ 2. walk about __ 3. make old-fashioned or obsolete __ 4. something that nourishes ___ 5. not exhibiting life ___ 6. absence of political authority __ 7. not suitable __ 8. yearly income payment __ 9. relating to farming or rural matters __ 10. keenness of judgment __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. An anachronism is something that is in tune with the times. 2. Things that are coeval are equally evil. _ 3. An altimeter is an instrument that measures elevation. 4. An amphora is a Greek or Roman jar with two handles. 5. Antagonistic people do not usually display hostility. _ (Answers are on page 415.) 306 300 HIGH-FREQUENCY WORD ROOTS ROOT ROUNDUP 6 • AQU/AQUA (WATER) Latin aquarium = tank for holding fish and sea plants aqueduct = large pipe or canal that carries water to large communities aquatic = relating to things that occur in or on water: aquatic plants or sports subaqueous = created or existing under water aquifer = underground rock formation that bears water; where water flows underground • ARCH (FIRST, CHIEF, RULE, SUPERIOR) Greek archangel = chief angel archaic = out of date patriarchy = family or community governed by men archeology = study of material evidence of past human life archetype = original model after which others are patterned • ARM/ARMA (WEAPONS) Latin armistice = truce; temporary stop to fighting armada = fleet of warships disarmament = reduction of a nation’s weapons and military forces armor = covering that protects one’s body against weapons armadillo = burrowing mammal that has armorlike long plates • ART (ART) Latin artisan = craftsperson artifact = object made by human craft art nouveau = late nineteenth-century style of art artificial = made by human action artifice = artful expedient • ASTR/ASTER (STAR) Greek asterisk = the sign astral = relating to stars astronaut = person who travels in space astrology = study of the influence of the stars and planets on human beings astronomy = scientific study of the stars and other bodies in the universe 307 ESSENTIAL WORDS FOR THE GRE Root Work 6 Match each word with its definition. 1. aqueduct (A) craftsperson 2. patriarchy (B) fleet of warships 3. archetype (C) existing underwater 4. artisan (D) community or family governed by men 5. armada (E) scientific study of the stars and other bodies in the universe 6. astronomy (F) reduction of a nation’s weapons and military forces 7. subaqueous (G) object made by human craft 8. astral (H) relating to the stars 9. disarmament (I) original model after which others are patterned 10. artifact (J) canal that carries water to communities (Answers are on page 411.) ROOT ROUNDUP 7 • AUD/AUDI/AUS (BOLD, DARING, LISTEN, HEAR) Latin auditorium = part of a theater where the audience sits audible = capable of being heard audacious = bold, daring audacity = fearless, daring, and adventurousness auscultation = listening to the heart or other organs • AUG/AUX (INCREASE) Latin augment = make greater inaugurate = begin or start officially august = dignified; awe-inspiring augur = foretell auxiliary = supplementary • AUTO (SELF) Greek automatic = self-acting or self-regulating autograph = person’s signature autonomic = occurring involuntarily autonomous = self-governing autobiography = self-written account of one’s own life • BE (THOROUGLY, OVER) Old English befuddled = confused; perplexed beguile = delude; deceive by guile besmirched = stained; soiled bedecked = adorned in a showy manner bedizen = dress in a showy manner 308 300 HIGH-FREQUENCY WORD ROOTS • BEL/BELL (WAR) Latin rebel = carry out armed resistance to the government bellicose = aggressive; warlike belligerent = hostile; tending to fight antebellum = existing before a war postbellum = existing after a war Root Work 7 Match each word with its definition. 1. autonomic (A) make greater 2. august (B) existing before a war 3. audacious (C) self-governing 4. antebellum (D) confused; perplexed 5. auscultation (E) hostile; tending to fight 6. bedizen (F) bold; daring 7. befuddled (G) occurring involuntarily 8. autonomous (H) ornament or dress in a showy manner 9. belligerent (I) dignified; awe-inspiring 10. augment (J) listening to the heart or other organs (Answers are on page 411.) ROOT ROUNDUP 8 • BEN/BON (WELL, GOOD, FAVORABLE) Latin beneficent = kindly; doing good benediction = blessing benevolent = generous; charitable benign = harmless; kind bonanza = large amount • BI (TWO, TWICE, DOUBLE) Latin bicycle = light-framed vehicle mounted on two wheels biannual = happening twice each year bifurcate = divide into two parts bicuspid = having two points bivalve = having a shell composed of two valves • BIO (LIFE) Greek biologist = scientist who studies life biosphere = part of the Earth’s surface and atmosphere in which life exists bionics = science concerned with applying biological systems to engineering problems biotic = produced by living organisms symbiotic = relating to a relationship of mutual benefit or dependence 309 ESSENTIAL WORDS FOR THE GRE • BREV (SHORT) Latin abbreviate = make shorter abbreviation = act or product of shortening brevity = state of briefness in duration breve = symbol over a vowel to indicate a short sound breviary = book containing hymns and prayers for canonical hours • CAP/CAPT/CEPT/CIP (HOLD, SEIZE, TAKE) Latin capable = having ability or capacity intercept = interrupt the course of captious = faultfinding; intending to entrap, as in an argument precept = principle that prescribes a course of action capture = take captive; to seize Root Work 8 Match each word with its definition. 1. biotic (A) interrupt the course of 2. capture (B) happening twice a year 3. bicuspid (C) shortness in duration 4. breve (D) harmless; kind 5. biannual (E) produced by living organisms 6. intercept (F) having two points 7. benign (G) symbol over a vowel that indicates a short sound 8. biosphere (H) generous; charitable 9. benevolent (I) part of the Earth’s surface and atmosphere in which life exists 10. brevity (J) seize (Answers are on page 411.) ROOT ROUNDUP 9 • CAP/CAPIT (HEAD) Latin per capita = per unit of population capitol = building in which a state legislature meets decapitate = behead capitulate = surrender captain = someone who commands others • CARD/CORD (HEART) Latin cardiac = relating to the heart cardiology = branch of medicine concerned with the heart cordial = warm and sincere concord = harmony; agreement discordant = disagreeable in sound; conflicting 310 300 HIGH-FREQUENCY WORD ROOTS • CARN (FLESH, BODY) Latin carnal = of the flesh or body carnation = perennial plant with showy flowers carnivore = animal or plant that feeds on flesh incarnate = give bodily form to carnage = massive slaughter, as in war • CATA (DOWN, DOWNWARD) Greek catalyst = something causing change cataract = high waterfall; a great downpour catapult = ancient military machine for hurling missiles cataclysm = violent upheaval catastrophic = relating to a great calamity • CED/CEED/CESS (YIELD, SURRENDER, MOVE, GO) Latin cede = surrender; yield accede = agree to precede = go before antecedent = something that comes before incessant = never ceasing Root Work 9 Match each word with its definition. 1. concord (A) warm and sincere 2. incarnate (B) animal or plant that feeds on flesh 3. captain (C) high waterfall; great downpour 4. antecedent (D) relating to a great calamity 5. cordial (E) surrender; yield 6. cataract (F) something that comes before 7. decapitate (G) harmony; agreement 8. catastrophic (H) behead 9. cede (I) someone who commands others 10. carnivore (J) give bodily form to (Answers are on page 412.) ROOT ROUNDUP 10 • CELER (SWIFT) Latin accelerate = increase speed accelerant = substance used as a catalyst celerity = swiftness; speed deceleration = decrease the velocity of accelerando = musical direction for a dual quickening in time 311 ESSENTIAL WORDS FOR THE GRE • CENTR (CENTER) Latin concentric = having a common center centrifugal = moving or directed away from a center centripetal = moving or directed toward a center concentrate = direct toward a center centric = situated near or at the center of something • CENT (HUNDRED) Latin centimeter = unit of length equal to one hundredth of a meter centenary = relating to a 100-year period centenarian = one who is 100 years old or more century = period of 100 years centennial = relating to a period of 100 years; occurring every 100 years • CERN (PERCEIVE) Latin concern = regard for or interest in discern = perceive; detect discerning = showing good judgment; perceptive indiscernible = difficult to perceive unconcernedly = in a way that is unworried • CERT (CERTAIN) Latin certify = confirm formally as genuine certificate = document confirming the truth of something certainty = state or fact of being certain certitude = state of being certain; sureness of occurrence ascertain = discover with certainty Root Work 10 Match each word with its definition. 1. centrifugal (A) moving toward a center 2. certitude (B) swiftness; speed 3. centennial (C) substance used as a catalyst 4. centenarian (D) discover with certainty 5. indiscernible (E) relating to a period of 100 years 6. celerity (F) showing good judgment; perceptive 7. discerning (G) moving away from a center 8. accelerant (H) difficult to perceive 9. centripetal (I) state of being certain; sureness of occurrence 10. ascertain (J) someone 100 years old or more (Answers are on page 412.) 312 300 HIGH-FREQUENCY WORD ROOTS ROOT ROUNDUP REVIEW 6–10 Match It Match each of the following roots to its meaning. 1. ASTR/ASTER ___ (A) short 2. ARM/ARMA ___ (B) flesh, body 3. AUG/AUS ___ (C) weapons 4. BE ___ (D) swift 5. BIO ___ (E) down 6. BREV ___ (F) perceive 7. CARN ___ (G) thoroughly, over 8. CATA ___ (H) star 9. CELER ___ (I) increase 10. CERN ___ (J) life Fill-ins Fill in the blanks with the word that fits the definition. benediction augur symbiotic archaic captious catapult incessant centripetal centrifugal subaqueous 1. relating to a relationship of mutual benefit or dependence __ 2. foretell __ 3. created or existing underwater __ 4. out of date __ 5. never ceasing __ 6. moving away from a center __ 7. a blessing __ 8. an ancient military machine for hurling missiles __ 9. moving toward a center __ 10. faultfinding; intending to entrap, as in argument __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. Antebellum refers to a period after a war. 2. Archeology is the study of the influence of stars and planets on human life. _ 3. Auscultation means speaking clearly and fluently. 4. A patriarchy is a family or community governed by men. 5. Centennial relates to a period of 1,000 years. _ (Answers are on page 415.) 313 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP 11 • CHRON (TIME, A LONG TIME) Greek chronic = constant; prolonged chronicle = record of historical events chronometer = instrument that measures time anachronism = something out of the proper time chronology = arrangement in order of occurrence • CID/CIS (CUT, KILL) Latin homicide = killing of one person by another scissors = cutting instrument with two blades exorcise = expel evil spirits excision = remove by cutting abscission = natural separation of flowers, leaves, etc. from plants • CIRCU/CIRCUM (AROUND) Latin circumvent = avoid; get around circumflex = curving around circuitous = taking a roundabout course circumlocution = indirect way of saying something circumscribe = limit • CIT/CITAT (CALL, START) Latin cite = mention as illustration or proof; to quote as an example citable = able to be brought forward as support or proof citation = the act of citing; a quotation recite = say aloud before an audience something rehearsed recitative = having the character of a recital • CIVI (CITIZEN) Latin civil = relating to a citizen or citizens; of ordinary citizens or ordinary community life civic = relating to a city, a citizen, or citizenship civilize = raise from barbarism to civilization; educate in matters of culture civilian = citizen who is not an acting member of the military or police civility = courteous behavior 314 300 HIGH-FREQUENCY WORD ROOTS Root Work 11 Match each word with its definition. 1. excision (A) constant; prolonged 2. circumflex (B) relating to a city or a citizen 3. cite (C) say aloud before an audience something rehearsed 4. anachronism (D) natural separation of flowers and leaves from plants 5. recite (E) mention as an illustration 6. civic (F) avoid; get around 7. abscission (G) citizen who is not a member of the military 8. civilian (H) curving around 9. chronic (I) something out of the proper time 10. circumvent (J) remove by cutting (Answers are on page 412.) ROOT ROUNDUP 12 • CLAM/CLAIM (CALL OUT, SHOUT) Latin exclaim = cry out suddenly; utter vehemently exclamation = an abrupt forceful utterance; an outcry clamor = a loud outcry reclaim = demand the return of something acclamation = shout of enthusiastic approval • CLEMEN (MILD, KIND, MERCIFUL) Latin clemency = disposition to show mercy; merciful act; mildness clement = inclined to be merciful; mild inclement = stormy; showing no mercy inclemency = state of showing no mercy inclemently = in a way that shows no mercy • CLAUD/CLAUS/CLOS/CLUD/CLUS (SHUT, CLOSE) Latin clause = a stipulation or provision in a document exclude = keep out; reject; put out seclusion = isolation; solitude recluse = person who lives in seclusion occlude = cause to become closed; obstruct • CLI/CLIN/CLIV (LEANING, INCLINED, SLOPED) Latin climax = point of greatest intensity in an ascending progression inclination = a tendency toward a certain condition disinclination = lack of inclination; reluctance synclinal = sloping downward from opposite directions and meeting in a common point proclivity = tendency; inclination 315 ESSENTIAL WORDS FOR THE GRE • CO/COL/COM/CON/COR (TOGETHER, WITH) Latin coherent = understandable; sticking together collaborate = work together communication = exchange of thoughts and information conformity = harmony; agreement corroborate = confirm Root Work 12 Match each word with its definition. 1. collaborate (A) person who lives in seclusion 2. clamor (B) shout of enthusiastic approval 3. occlude (C) inclined to be merciful 4. conformity (D) harmony; agreement 5. acclamation (E) work together 6. disinclination (F) cause to become closed 7. clement (G) tendency; inclination 8. inclement (H) lack of inclination 9. proclivity (I) loud outcry 10. recluse (J) stormy; showing no mercy (Answers are on page 412.) ROOT ROUNDUP 13 • COD (BOOK) Latin code = systematic, comprehensive collection of laws; system of symbols used for sending messages that require secrecy decode = convert from code into text codify = reduce to a code codex = manuscript volume of a classic work codicil = an appendix to a will • COGNI/GNO (LEARN, KNOW) Latin/Greek cognition = mental process by which knowledge is acquired incognito = in disguise; concealing one’s identity diagnosis = process of determining the nature and cause of a disease prognosticate = predict on the basis of present conditions agnosia = loss of the ability to interpret sensory stimuli • CONTRA/CONTRO (AGAINST, OPPOSITE) Latin contradict = speak against contrary = opposed contravene = act contrary to; to violate contraindicate = indicate the inadvisability of the use of a medicine controversy = dispute between sides holding opposing views 316 300 HIGH-FREQUENCY WORD ROOTS • CORP (BODY) Latin corpse = a dead body corpulent = excessively fat corporeal = concerned with the body corpus = a large collection of writings incorporate = unite one thing with something else already in existence • COSM (UNIVERSE) Greek cosmic = relating to the universe; infinite; vast microcosm = a small system having analogies to a larger system cosmology = study of the physical universe cosmos = the universe as a harmonious whole cosmopolitan = common to or having elements from all over the world Root Work 13 Match each word with its definition. 1. cognition (A) manuscript volume of a classic work 2. contradict (B) concerned with the body 3. codex (C) study of the physical universe 4. cosmos (D) mental process by which knowledge is acquired 5. corporeal (E) speak against 6. code (F) predict on the basis of present conditions 7. contravene (G) the universe as a harmonious whole 8. cosmology (H) act contrary to 9. corpus (I) systematic, comprehensive collection of laws 10. prognosticate (J) a large collection of writings (Answers are on page 412.) ROOT ROUNDUP 14 • CRACY/CRAT (GOVERNMENT, RULE, STRENGTH) Greek aristocracy = hereditary ruling class bureaucracy = administration of a government or a large complex plutocracy = society ruled by the wealthy theocracy = government by priests technocrat = strong believer in technology • CREA (BRING FORTH, CREATE) Latin create = bring into being creature = something created; a living being recreate = give fresh life to; refresh mentally or physically procreation = the conceiving of offspring; producing or creating miscreate = make or shape badly 317 ESSENTIAL WORDS FOR THE GRE • CRED (BELIEVE, TRUST) Latin credo = statement of belief or principle; creed credentials = evidence concerning one’s right to confidence or authority credible = believable; plausible credence = acceptance of something as true incredulous = skeptical; doubtful • CRE/CRESC/CRET/CRU (RISE, GROW) Latin accrue = increase; come about as a result of growth crescent = increasing; waxing, as the moon crescendo = in music, a gradual increase in the volume or intensity of sound increment = something added; process of increasing increscent = waxing; growing; showing a surface that is ever larger and lighted • CRIT (SEPARATE, JUDGE) Greek critical = inclined to judge severely; characterized by careful judgment criterion = a standard on which a judgment can be made hypocritical = professing beliefs that one does not possess; false criticism = a critical comment or judgment critique = a critical review or commentary Root Work 14 Match each word with its definition. 1. technocrat (A) believable; plausible 2. incredulous (B) producing or creating 3. accrue (C) government by priests 4. miscreate (D) waxing; growing 5. critical (E) standard on which a judgment can be made 6. procreation (F) strong believer in technology 7. criterion (G) increase; come about as a result of growth 8. increscent (H) inclined to judge severely 9. theocracy (I) make or shape badly 10. credible (J) skeptical; doubtful (Answers are on page 412.) ROOT ROUNDUP 15 • COUR/CUR (RUN, COURSE) Latin concurrence = agreement in opinion; simultaneous occurrence courier = a messenger curriculum = the courses offered by an educational institution precursor = a forerunner or predecessor current = a steady, smooth, onward movement 318 300 HIGH-FREQUENCY WORD ROOTS • CUR/CURA (CARE) Latin curator = someone who oversees a museum collection curé = a parish priest curette = surgical instrument that removes growths from a body cavity curative = tending to cure curate = a cleric who is in charge of a parish • CYCL/CYCLO (CIRCLE, WHEEL, CYCLE) Greek cyclical = characterized by cycles; moving in cycles cyclosis = rotary motion of protoplasm within a cell cycloid = resembling a circle Cyclops = in Greek mythology, any of a race of one-eyed giants cyclothymia = affective disorder characterized by alternating periods of depression and elation • DE (INTENSIVE PREFIX; FROM, DOWN, AWAY, AGAINST, THOROUGHLY) Latin demolish = tear down completely deplore = disapprove of; regret deride = mock denounce = condemn deprecate = belittle; express disapproval • DEC/DECA (TEN) Greek Decalogue = the Ten Commandments decimate = destroy a large part of; inflict great destruction on decade = a period of ten years decahedron = a polyhedron with ten faces decapod = a crustacean having ten legs Root Work 15 Match each word with its definition. 1. deride (A) tear down completely 2. decahedron (B) crustacean having ten legs 3. curative (C) one who oversees a museum collection 4. cyclical (D) forerunner 5. precursor (E) resembling a circle 6. concurrence (F) simultaneous occurrence 7. decapod (G) mock 8. cycloid (H) characterized by cycles 9. curator (I) polyhedron with ten faces 10. demolish (J) tending to cure (Answers are on page 412.) 319 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP REVIEW 11–15 Match It Match each of the following roots to its meaning. 1. CID/CIS ___ (A) body 2. CIRCU/CIRCUM ___ (B) call out, shout 3. CLAM/CLAIM ___ (C) believe, trust 4. CLEMEN ___ (D) book, writing 5. COD ___ (E) run, course 6. CORP ___ (F) mild, kind, merciful 7. CRED ___ (G) cut, kill 8. CRIT ___ (H) from, down, away, against, thoroughly 9. COUR/CUR ___ (I) around 10. DE ___ (J) separate, judge Fill-ins Fill in the blanks with the word that fits the definition. criterion synclinal cosmopolitan civil coherent corpulent cyclothymia miscreate deprecate cite 1. relating to a citizen or citizens; of ordinary citizens or ordinary community life __ 2. common to or having elements from all over the world __ 3. sticking together; understandable ___ 4. mention as illustration or proof; to quote as an example __ 5. a standard on which a judgment can be made __ 6. excessively fat ___ 7. make or shape badly __ 8. belittle; to express disapproval __ 9. sloping downward from opposite directions and meeting in a common point ___ 10. an affective disorder characterized by alternating periods of depres sion and elation __ 320 300 HIGH-FREQUENCY WORD ROOTS True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. A chronometer is an instrument that measures wind speed. 2. A proclivity is a tendency or inclination. _ 3. To prognosticate is to predict based on present conditions. 4. Something increscent is growing or showing a surface that is ever larger and lighted. 5. Decahedrons are polyhedrons with nine faces. _ (Answers are on page 415.) ROOT ROUNDUP 16 • DEI/DIV (GOD) Latin divine = having the nature of a god divinity = the state of being divine deify = raise to the condition of a god deism = belief that a God has created the universe, but exerts no control or influence on it deific = making divine; characterized by a godlike nature • DEMI (PARTLY) Latin demigod = the male offspring of a god and a mortal; a minor god demimonde = a group whose respectability is questionable demirelief = structural relief having modeled forms projecting halfway from a background demirep = person whose reputation is doubtful demitasse = a small cup of espresso • DEM (COMMON PEOPLE) Greek demographic = related to population balance epidemic = a widespread disease that affects many people at the same time pandemic = spread over a whole area or country demagogue = leader who appeals to emotion or prejudice democratic = of or for the people; popular • DERM (SKIN) Greek dermatology = branch of medicine concerned with pathology of the skin dermatitis = inflammation of the skin epidermis = the outer layer of the skin taxidermist = one who works in the art of stuffing and mounting skins of dead animals pachyderm = a thick-skinned hoofed animal like the elephant or hippopotamus 321 ESSENTIAL WORDS FOR THE GRE • DI (TWO, DOUBLE) Greek diphase = having two phases dichotomy = division into two usually contradictory parts dilemma = situation necessitating a choice between two unsatisfactory options dibromide = chemical compound having two bromine atoms dihedral = two-sided Root Work 16 Match each word with its definition. 1. demigod (A) leader who appeals to emotion or prejudice 2. pachyderm (B) one who works stuffing and mounting dead animal skin 3. deific (C) chemical compound having two bromine atoms 4. diphase (D) minor god 5. taxidermist (E) person whose reputation is doubtful 6. demirep (F) raise to the condition of a god 7. deify (G) making divine; having a godlike nature 8. demagogue (H) spread over a whole area or country 9. pandemic (I) thick-skinned hoofed animal 10. dibromide (J) having two phases (Answers are on page 412.) ROOT ROUNDUP 17 • DIA (ACROSS, THROUGH, BETWEEN) Greek diagram = drawing that explains the relationship between parts of a whole diachronic = concerned with phenomena as they change through time diatribe = an abusive denunciation dialogue = a conversation between two or more people diaphanous = so fine as to be almost transparent or translucent • DIC/DICT (SAY, SPEAK, PRONOUNCE) Latin edict = a formal command benediction = blessing indict = charge with a crime malediction = curse dictum = authoritarian statement 322 300 HIGH-FREQUENCY WORD ROOTS • DIF/DIS (APART, AWAY, NOT) Latin diffuse = spread out disparity = difference dissuade = to persuade someone to alter intentions dispassionate = impartial; unaffected by emotion disseminate = to spread; scatter • DON (GIVE) Latin donation = the act of giving to a cause or charity donor = one who contributes a donation to a cause or charity donee = one who receives a gift donary = a votive offering; a gift donatio mortis causa = gift by reason of death • DOC/DOCT (TEACH, PROVE) Latin doctrinaire = relating to a person who cannot compromise about points of a theory or doctrine docent = lecturer doctrine = principle or system presented for acceptance or belief indoctrinate = instruct in a body of principles docile = willing to be taught; yielding to supervision Root Work 17 Match each word with its definition. 1. dispassionate (A) conversation between two or more people 2. dictum (B) lecturer 3. donatio mortis causa (C) spread out 4. malediction (D) authoritarian statement 5. donary (E) willing to be taught 6. diaphanous (F) gift by reason of death 7. diffuse (G) unaffected by emotion 8. docent (H) so fine as to be almost transparent or translucent 9. dialogue (I) curse 10. docile (J) a gift; a votive offering (Answers are on page 412.) 323 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP 18 • DOG/DOX (OPINION, BELIEF, PRAISE) Greek dogmatic = characterized by an authoritarian assertion of unproved principles orthodox = adhering to what is commonly accepted or traditional paradox = self-contradictory assertion based on valid deduction from acceptable premises heterodox = not in agreement with accepted beliefs doxology = an expression of praise to God • DOM/DOMIN (MASTER, LORD) Latin dominate = control by authority or power dominion = control; sovereignty predominant = having greatest authority, influence, or force domination = control or power over another indomitable = unconquerable; not able to be subdued • DORM (SLEEP) Latin dormant = inactive; asleep dormitory = room used for sleeping quarters for many people dormient = sleeping; dormant; latent dormitive = causing sleep dormouse = a squirrel-like rodent • DROM/DROME (RUN, STEP, ARENA) Greek dromedary = one-humped camel aerodrome = an airport; military air base hippodrome = arena for equestrian shows dromond = a medieval sailing ship catadromous = inhabiting fresh water but migrating to the ocean to breed • DUC/DUCT (LEAD, PULL) Latin induce = bring about seduce = lead away from duty or proper conduct abduct = carry off by force viaduct = series of arches used to carry a road over a valley or other roads ductile = easily drawn into wire; easily molded 324 300 HIGH-FREQUENCY WORD ROOTS Root Work 18 Match each word with its definition. 1. indomitable (A) control; sovereignty 2. dormitive (B) not in agreement with accepted beliefs 3. dromond (C) arena for equestrian shows 4. doxology (D) asleep; inactive 5. ductile (E) lead away from duty or proper conduct 6. dominion (F) unconquerable 7. hippodrome (G) medieval sailing ship 8. seduce (H) expression of praise to God 9. heterodox (I) causing sleep 10. dormant (J) easily molded (Answers are on page 412.) ROOT ROUNDUP 19 • DUPL (DOUBLE, TWO) Latin duplicity = double-dealing; being twofold; deceptiveness duplex = twofold; double duplicate = identically copied from an original duplicator = machine that copies printed material duple = consisting of two; double • DUR (HARD, LASTING) Latin durable = able to withstand wear and tear duration = persistence in time; a period of existence endure = carry on through hardships; bear tolerantly duress = constraint by threat; forcible confinement obdurate = hardened; hardhearted; inflexible • DYN/DYNAM (POWER, ENERGY) Greek dynamite = a class of powerful explosives dynamo = a generator that produces current; an energetic person dynasty = succession of rulers from the same line dynamic = marked by intensity and vigor heterodyne = having alternating currents with two different frequencies • DYS (BAD, IMPAIRED, ABNORMAL) Greek dysfunctional = functioning abnormally dyslexia = learning disorder causing impairment of the ability to read dystopia = an imaginary place in which life is bad dysentery = disorder of the lower intestinal tract dyspepsia = indigestion 325 ESSENTIAL WORDS FOR THE GRE • E/EX (INTENSIVE PREFIX; APART, ABOVE, AWAY, BEYOND, FROM, OUT) Latin emit = send out enervate = weaken extricate = free from exhale = breathe out exotic = unusual Root Work 19 Match each word with its definition. 1. dynasty (A) imaginary place where life is bad 2. obdurate (B) being twofold; deceptiveness 3. emit (C) energetic person 4. endure (D) hardhearted 5. dystopia (E) breathe out 6. duplicity (F) succession of rulers from the same line 7. dysentery (G) consisting of two 8. exhale (H) disorder of lower intestinal tract 9. dynamo (I) send out 10. duple (J) carry on through hardships (Answers are on page 412.) ROOT ROUNDUP 20 • ECTO (OUTSIDE, EXTERNAL) Greek ectogenous = able to develop outside a host ectoplasm = outer part of the cytoplasm of a cell ectopic pregnancy = development of an ovum outside the uterus ectoderm = outermost germ layer of an embryo ectopia = abnormal location of an organ or body part • EGO (I, SELF) Latin egocentric = self-centered egomania = extreme egocentrism egotistical = excessively self-centered egoist = person devoted to his or her own interests super ego = the part of the mind that opposes the desires of the id (the subconscious source of instinctual impulses) 326 300 HIGH-FREQUENCY WORD ROOTS • ENDO (WITHIN, INSIDE) Greek endomorph = a mineral enclosed inside another mineral endogenous = growing from within; produced inside an organism endoscope = instrument for viewing the inside of an organ of the body endobiotic = living as a parasite within a host endocardial = relating to a membrane that lines the heart’s interior • EPI (UPON, OVER, NEAR) Greek epidermis = outer layer of skin covering the dermis epidemic = widely prevalent epigeal = living in or near the surface of the ground epitaph = inscription on a tombstone epicenter = point in the Earth directly above the center of an earthquake • EQU (EQUAL) Latin equator = imaginary circle around the Earth, which is equidistant from the poles equation = statement asserting the equality of two mathematical expressions equivocal = ambiguous; misleading equanimity = composure inequity = unfairness Root Work 20 Match each word with its definition. 1. egoist (A) able to develop outside a host 2. ectoderm (B) instrument to look inside an organ of the body 3. epigeal (C) excessively self-centered 4. endobiotic (D) composure 5. epitaph (E) inscription on a tombstone 6. ectogenous (F) unfairness 7. equanimity (G) outermost germ layer of an embryo 8. inequity (H) living near the surface of the ground 9. egotistical (I) living as a parasite within a host 10. endoscope (J) person devoted to his or her own interests (Answers are on page 412.) 327 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP REVIEW 16–20 Match It Match each of the following roots to its meaning. 1. DEI/DIV ___ (A) teach, prove 2. DEMI ___ (B) power, energy 3. DON ___ (C) sleep 4. DOC/DOCT ___ (D) within 5. DOM/DOMIN ___ (E) apart, above, beyond, from, intensive prefix 6. DORM ___ (F) I, self 7. DYN/DYNAM ___ (G) God 8. ENDO ___ (H) give 9. EGO ___ (I) master, lord 10. E/EX ___ (J) partly Fill-ins Fill in the blanks with the word that fits the definition. dichotomy obdurate malediction duple ductile inequity pandemic dermatitis epigeal donatio mortis causa 1. spread over an entire area ___ 2. consisting of two __ 3. hardened; inflexible; hardhearted __ 4. gift by reason of death ___ 5. unfairness ___ 6. division into two usually contradictory parts __ 7. living near or in the surface of the ground __ 8. easily molded __ 9. inflammation of the skin __ 10. curse __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. A taxidermist is a thick-skinned hoofed animal. 2. Diachronic means concerned with phenomena of a particular time. _ 3. An ectoderm is the outermost germ layer of an embryo. 4. Doxology is adherence to unorthodox beliefs. 5. A dystopia is an imaginary place in which life is bad. _ (Answers are on page 415.) 328 300 HIGH-FREQUENCY WORD ROOTS ROOT ROUNDUP 21 • ERG (WORK) Greek ergatocracy = government by workers ergonomics = science of the design of equipment for maximizing productivity erg = a unit of work ergograph = instrument for measuring work capacity of a muscle while contracting synergic = working together • ERR (WANDER, MISTAKE) Latin errant = mistaken; straying from the proper course erratic = lacking regularity; deviating from the customary course erroneous = mistaken err = make a mistake erratum = mistake in writing or printing • ETH/ETHOS (CHARACTER) Greek ethos = character peculiar to a person, people, or culture bioethics = study of ethical implications of scientific discoveries, as in genetic engineering ethic = set of principles of correct conduct; system of moral values ethics = rules that govern conduct of people or members of a profession ethology = the study of human ethos • EU (GOOD, WELL) Greek eulogy = high praise euphemism = use of inoffensive language in place of unpleasant language eugenics = a philosophy that advocates the improvement of human traits through various means euphoria = feeling of extreme happiness euphony = pleasant and harmonious sound • EXTRA/EXTRO (BESIDES, BEYOND, OUTSIDE OF, MORE) Latin extraordinary = beyond the ordinary extracurricular = outside of the regular curriculum extraterrestrial = outside Earth extraneous = not essential extroversion = behavior directed outside one’s self 329 ESSENTIAL WORDS FOR THE GRE Root Work 21 Match each word with its definition. 1. euphony (A) character peculiar to a person or people 2. erratic (B) high praise 3. ethos (C) working together 4. erg (D) set of principles of correct conduct 5. ethic (E) deviating from the customary course 6. erroneous (F) outside Earth 7. extraterrestrial (G) a unit of work 8. eulogy (H) harmonious sound 9. synergic (I) mistaken 10. extraneous (J) not essential (Answers are on page 412.) ROOT ROUNDUP 22 • FAC/FIC/FEC (DO, MAKE) Latin beneficent = performing acts of kindness manufacture = make or process confection = act of making a sweet preparation; a sweet preparation soporific = something that produces sleep facile = done with little effort; easy • FALL/FALS (DECEPTIVE/FALSE/ERRONEOUS) Latin false = untrue; mistaken; wrong; misleading fallacious = based on a false idea or fact; misleading falsify = state untruthfully; misrepresent fallible = capable of making an error infallible = incapable of making a mistake • FED (LEAGUE, PACT) Latin federal = related to a system of government in which power is divided between a central government and constituent states federation = the act of joining into a league or federal union; a federal group of states federacy = an alliance; a confederacy federative = forming, belonging to, or having the nature of a federation confederate = a member of a league; an ally • FER (BEAR, CARRY) Latin coniferous = pertaining to needle-leaved cone-bearing trees such as pines or firs aquifer = stratum of permeable rock that bears water pestiferous = bearing moral contagion; pestilent; deadly vociferous = loud, vocal, and noisy sporiferous = producing spores 330 300 HIGH-FREQUENCY WORD ROOTS • FID (FAITH, TRUST) Latin confide = tell in confidence fidelity = loyalty; exact correspondence confidence = trust or faith in someone or something perfidious = faithless; disloyal; untrustworthy confidante = a person to whom one’s private affairs or thoughts are disclosed Root Work 22 Match each word with its definition. 1. fidelity (A) loud, vocal, and noisy 2. facile (B) based on a false idea 3. vociferous (C) loyalty 4. beneficent (D) done with little effort 5. federation (E) pertaining to cone-bearing trees 6. fallacious (F) a federal group of states 7. federacy (G) performing acts of kindness 8. coniferous (H) deliberate breach of trust 9. infallible (I) an alliance 10. perfidy (J) incapable of making a mistake (Answers are on page 412.) ROOT ROUNDUP 23 • FIN (END) Latin finite = limited; impermanent finale = concluding part of a musical composition definitive = conclusive; authoritative; precisely defined fin de siècle = end of a century ad infinitum = forever; again and again • FLECT/FLEX (BEND) Latin flexible = capable of being bent reflex = bent or thrown back; reflected reflection = the act of bending back or throwing back from a surface deflect = turn aside; bend; deviate circumflex = curving around • FLU/FLUCT/FLUX (FLOW) Latin fluctuate = vary irregularly; rise and fall in waves influent = flowing into confluence = a flowing together; a juncture of two or more streams flux = flowing; a continuous moving reflux = a flowing back; an ebbing 331 ESSENTIAL WORDS FOR THE GRE • FORT (BRAVE, STRONG, CHANCE) Latin fortify = strengthen; reinforce forte = a person’s strong point fortitude = quality that enables a person to face pain and suffering with courage fortuitous = occurring by chance fortress = a large military stronghold • FRAC/FRAG (BREAK) Latin fractional = very small; being in fractions or pieces refract = deflect sound or light infringe = transgress; violate fractious = unruly; rebellious fragmentary = consisting of small disconnected parts Root Work 23 Match each word with its definition. 1. fortuitous (A) curving around 2. definitive (B) a person’s strong point 3. fractious (C) a flowing together 4. confluence (D) conclusive 5. forte (E) deflect sound or light 6. circumflex (F) end of a century 7. deflect (G) occurring by chance 8. refract (H) rise and fall in waves 9. fluctuate (I) unruly; rebellious 10. fin de siècle (J) turn aside; bend (Answers are on page 412.) ROOT ROUNDUP 24 • FRAT (BROTHER) Latin fraternity = a social organization of men students confraternity = an association of persons united in a common purpose fraternal = brotherly fraternize = mingle on friendly terms fratricide = the killing of one’s brother or sister • FUNC (PERFORM, DISCHARGE) Latin function = assigned duty or activity defunct = no longer existing perfunctory = performed really as a duty; superficial functionary = someone who performs a particular function malfunction = failure to work 332 300 HIGH-FREQUENCY WORD ROOTS • GAM (MARRIAGE) Greek polygamous = having more than one wife or husband at a time endogamy = marriage within a particular group exogamy = marriage outside a social unit gamic = requiring fertilization to reproduce monogamous = relating to marriage to one person at a time • GEN (BIRTH, CLASS, DESCENT, RACE, GENERATE) Latin engender = cause, produce genesis = beginning; origin genetics = branch of biology that deals with heredity gentry = people of standing; class of people just below nobility genre = type, class; distinct literary or artistic category • GEO (EARTH) Greek geology = science that studies the structure and composition of the Earth geography = science that studies the Earth and the distribution of life on it geocentric = having the Earth as center geothermal = produced by the heat in the Earth’s interior geophysics = the physics of the Earth Root Work 24 Match each word with its definition. 1. polygamy (A) cause; produce 2. fratricide (B) relating to marriage to one person at a time 3. defunct (C) type; class 4. geothermal (D) no longer existing 5. fraternal (E) the physics of the Earth 6. monogamous (F) killing of one’s brother or sister 7. engender (G) produced by heat in the Earth’s interior 8. genre (H) performed really as a duty 9. geophysics (I) having more than one wife or husband at a time 10. perfunctory (J) brotherly (Answers are on page 412.) ROOT ROUNDUP 25 • GLOSS/GLOT (LANGUAGE, TONGUE) Latin glossa = the tongue polyglot = speaker of many languages glossolalia = fabricated and meaningless speech associated with trance states 333 ESSENTIAL WORDS FOR THE GRE glossary = list of words and their meanings, usually at the back of a book glottis = opening between the vocal chords and the larynx • GRAD/GRESS (STEP) Latin regress = move backward; revert to an earlier state progressive = going step-by-step; favoring progress egress = an exit ingress = an entrance graduate = advance to a new level of skill or achievement • GRAPH/GRAM (WRITE, DRAW, RECORD) Greek graphology = study of handwriting, particularly for the purpose of character analysis biographical = relating to facts and events of a person’s life gramophone = a record player epigram = short and witty saying grammar = the system of rules of a language • GRAT (PLEASING) Latin gratify = please gratitude = thankfulness gratuitous = free; voluntary persona non grata = a person who is not acceptable or welcome ex gratia = done voluntarily, out of kindness or grace • GRAV/GRIEV (SERIOUS, HEAVY, HARMFUL) Latin gravity = seriousness grave = requiring serious thought gravitas = seriousness in demeanor or treatment grievous = causing grief or pain aggrieved = afflicted; distressed Root Work 25 Match each word with its definition. 1. epigram (A) please 2. progressive (B) entrance 3. polyglot (C) free; voluntary 4. gratify (D) the tongue 5. gravity (E) afflicted; distressed 6. ingress (F) short and witty saying 7. aggrieved (G) study of handwriting 8. graphology (H) speaker of many languages 9. glossa (I) going step-by-step 10. gratuitous (J) seriousness (Answers are on page 412.) 334 300 HIGH-FREQUENCY WORD ROOTS ROOT ROUNDUP REVIEW 21–25 Match It Match each of the following roots to its meaning. 1. ERR ___ (A) league, pact 2. EU ___ (B) earth 3. FER ___ (C) language, tongue 4. FED ___ (D) flow 5. FLECT/FLEX ___ (E) pleasing 6. FLU/FLUCT/FLUX ___ (F) good, well 7. FRAT ___ (G) bear, carry 8. GEO ___ (H) brother 9. GRAT ___ (I) wander, mistake 10. GLOSS/GLOT ___ (J) bend Fill-ins Fill in the blanks with the word that fits the definition. genre gravitas soporific egress perfidious extraneous fractious perfunctory euphony fin de siècle 1. seriousness in demeanor ___ 2. faithless; disloyal; untrustworthy __ 3. unruly; rebellious __ 4. type; class ___ 5. performed really as a duty ___ 6. pleasant and harmonious sound __ 7. an exit __ 8. end of a century __ 9. not essential __ 10. producing sleep __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. Endogamy refers to marriage outside a particular group. 2. Something fallacious is based on a false idea or fact. _ 3. An erg is a mistake in writing or printing. 4. Fortitude enables one to face pain and suffering with courage. 5. Ethos is a feeling of extreme happiness. _ (Answers are on page 415.) 335 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP 26 • GREG (FLOCK, HERD) Latin aggregate = collective mass or sum; total congregation = a gathering gregarious = sociable gregarine = various parasitic protozoans in the digestive tracts of invertebrates segregate = separate from a main body or group • GYN (WOMAN) Greek misogynist = one who hates women gynecology = branch of medicine dealing with women’s health care gynarchy = government by women gynophobia = fear of women gynecoid = characteristic of a woman • HELIO (SUN) Greek heliocentric = having the Sun as a center heliolatry = Sun worship heliotaxis = an organism’s movement in response to the Sun’s light heliotrope = kind of plant that turns toward the Sun heliotherapy = therapy based on exposure to sunlight • HEMO (BLOOD) Greek hemorrhage = heavy bleeding hemoglobin = respiratory pigment in red blood cells hemophilia = blood coagulation disorder hemoptysis = the expectoration of blood hemophobia = fear of blood • HERB (VEGETATION) Latin herbal = relating to or containing herbs herbicide = chemical that destroys plants or weeds herbaceous = characteristic of an herb herbivorous = feeding mainly on plants herbalism = herbal medicine 336 300 HIGH-FREQUENCY WORD ROOTS Root Work 26 Match each word with its definition. 1. herbaceous (A) blood coagulation disorder 2. hemophobia (B) kind of plant that turns toward the Sun 3. aggregate (C) characteristic of a herb 4. heliotrope (D) collective mass or sum 5. gregarious (E) an organism’s movement in response to the Sun’s light 6. gynarchy (F) feeding mainly on plants 7. hemophilia (G) sociable 8. herbivorous (H) fear of blood 9. heliotaxis (I) characteristic of a woman 10. gynecoid (J) government by women (Answers are on page 412.) ROOT ROUNDUP 27 • HETERO (DIFFERENT, MIXED, UNLIKE) Greek heterosexual = sexually oriented to persons of the opposite sex heterodox = unorthodox, not widely accepted heterogeneous = composed of unlike parts, different, diverse heterodyne = having alternating currents of two different frequencies producing two new ones heterochromatic = characterized by different colors • HOMO (SAME, ALIKE) Greek homologous = similar in value or function homonym = word identical in pronunciation and sometimes spelling to one or more other words but different in meaning homogeneous = composed of identical parts; uniform in composition homocentric = having the same center homogenize = make uniform in consistency • HOM/HOMO/HUMAN (MAN, HUMANITY) Latin humane = characterized by kindness or compassion humanity = humans as a group humanism = system of thought focusing on humans, their values, and capacities humanitarian = relating to the promotion of human welfare hominoid = belonging to the family Hominidae, which includes apes and man • HYDR/HYDRA/HYDRO (WATER) Greek hydroelectric = producing electricity through action of falling water hydroponics = science of growing plants in water reinforced with nutrients 337 ESSENTIAL WORDS FOR THE GRE hydrant = large pipe for drawing water dehydrate = remove water from hydrophyte = a water plant • HYPER (ABOVE, EXCESSIVE, OVER) Greek hyperbole = purposeful exaggeration for effect hyperactive = excessively active hypertension = high blood pressure hypercritical = excessively critical hyperventilate = to breathe abnormally fast Root Work 27 Match each word with its definition. 1. homologous (A) characterized by kindness or compassion 2. dehydrate (B) composed of identical parts 3. heterodox (C) breathe abnormally fast 4. heterochromatic (D) characterized by different colors 5. hyperbole (E) remove water from 6. humane (F) similar in value or function 7. hydroponics (G) purposeful exaggeration for effect 8. humanity (H) not widely accepted 9. hyperventilate (I) science of growing plants in water reinforced with nutrients 10. homogeneous (J) humans as a group (Answers are on page 413.) ROOT ROUNDUP 28 • HYPO (BENEATH, LOWER, UNDER) Greek hypothetical = based on assumptions or hypotheses hypothermia = abnormally low body temperature hypoglycemia = abnormally low glucose level in the blood hypochondria = unfounded belief that one is likely to become ill hypoplasia = arrested development of an organ • IG/IL/IM/IN/IR (CAUSATIVE PREFIX; NOT, WITHOUT) Latin ignominious = disgraceful and dishonorable impecunious = poor; having no money impoverish = make poor or bankrupt intractable = not easily managed irrelevant = not applicable; unrelated • IN (INTENSIVE PREFIX; IN, ON, UPON, NOT) Latin incite = arouse to action incarnate = having bodily form indigenous = native, occurring naturally in an area 338 300 HIGH-FREQUENCY WORD ROOTS inclusive = tending to include all incongruity = state of not fitting • INTER (AMONG, BETWEEN, WITHIN, MUTUAL) Latin intervene = come between interpose = insert; intervene interregnum= interval between reigns intersperse = distribute among; mix with internecine = deadly to both sides • INTRA/INTRO (INTO, INWARD, WITHIN) Latin intraocular = occurring within the eyeball intravenous = within a vein intramural = within an institution such as a school introvert = someone given to self-analysis introspective = contemplating one’s own thoughts and feelings Root Work 28 Match each word with its definition. 1. incongruous (A) within an institution 2. interregnum (B) occurring naturally in an area 3. irrelevant (C) not easily managed 4. introspective (D) not fitting 5. indigenous (E) not applicable; unrelated 6. hypothermia (F) unfounded belief that one is likely to become ill 7. intramural (G) contemplating one’s own thoughts and feelings 8. hypochondria (H) insert; intervene 9. interpose (I) abnormally low body temperature 10. intractable (J) interval between reigns (Answers are on page 413.) ROOT ROUNDUP 29 • JAC/JACT/JECT (THROW, FLING) Latin reject = refuse to accept or consider; deny eject = throw out trajectory = path taken by a projectile interject = interpose; insert ejaculate = eject abruptly • JUR/JUS/JUD (SWEAR, LAW, JUDGE, JUST) Latin judicious = wise; sound in judgment perjure = tell a lie under oath jurisdiction = power to interpret and apply law; control jurisprudence = philosophy of law justice = quality of being honorable and fair 339 ESSENTIAL WORDS FOR THE GRE • LECT/LEG/LIG (READ, CHOOSE) Latin legible = readable select = make a choice lector = someone who reads scriptural passages in a church service election = ability to make a choice predilection = preference; liking • LEG (LAW) Latin legacy = a gift made by a will illegal = prohibited by law legalese = abstruse vocabulary of the legal profession legitimate = in accordance with established standards; genuine; reasonable legislation = laws, decrees, mandates • LEV (LIGHT) Latin levity = light manner or attitude levitate = rise in the air or cause to rise lever = a means of accomplishing something alleviate = relieve; improve partially leverage = power to act effectively Root Work 29 Match each word with its definition. 1. legitimate (A) laws, decrees, mandates 2. jurisprudence (B) to relieve, improve partially 3. legislation (C) tell a lie under oath 4. levity (D) readable 5. perjure (E) preference, liking 6. legible (F) throw out 7. eject (G) in accordance with established standards 8. trajectory (H) path taken by a projectile 9. alleviate (I) light manner or attitude 10. predilection (J) philosophy of law (Answers are on page 413.) ROOT ROUNDUP 30 • LEX (WORD) Greek lexicon = dictionary; list of words lexicography = process of compiling a dictionary lexis = vocabulary; set of words in a language lexical = relating to the words of a language lexeme = basic unit of the lexicon of a language 340 300 HIGH-FREQUENCY WORD ROOTS • LIBER (FREE) Latin liberal = tolerant, broad-minded; generous, lavish liberation = freedom, emancipation libertine = one without moral restraint illiberal = bigoted; narrow-minded libertarian = one who believes in unrestricted freedom • LIBRAR/LIBR (BOOK) Latin library = place where books are kept; collection of books librarianship = specialization in library work libel = defamatory statement; act of writing something that smears a person’s character libretto = text of a dramatic musical work librettist = author of a libretto • LITER (LETTER) Latin literati = scholarly or learned persons literature = a body of written works illiterate = unable to read and write literation = letter for letter literal = limited to the most obvious meaning of a word; word for word • LOC/LOG/LOQU (WORD, SPEAK, TALK) Latin loquacious = talkative colloquial = typical of informal speech soliloquy = literary or dramatic speech by one character, not addressed to others circumlocution = indirect way of saying something epilogue = short speech at the end of a play Root Work 30 Match each word with its definition. 1. lexis (A) talkative 2. literate (B) indirect way of saying something 3. libertarian (C) word for word 4. loquacious (D) tolerant; broadminded 5. lexicon (E) believer in unrestricted freedom 6. libretto (F) able to read and write 7. circumlocution (G) list of words 8. libel (H) writing something to smear a character 9. literal (I) vocabulary 10. liberal (J) text of a musical work (Answers are on page 413.) 341 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP REVIEW 26–30 Match It Match each of the following roots to its meaning. 1. HERB ___ (A) free 2. HEMO ___ (B) law 3. HETERO ___ (C) same, alike 4. HOMO ___ (D) into, inward, within 5. IN ___ (E) light 6. INTRA/INTRO ___ (F) blood 7. LEG ___ (G) in, on, upon, not, intensive prefix 8. LEV ___ (H) letter 9. LIBER ___ (I) vegetation 10. LITER ___ (J) different, mixed, unlike Fill-ins Fill in the blanks with the word that fits the definition. internecine literati hominoid impecunious hyperbole lector lexical misogynist jurisprudence hemoptysis 1. relating to the words of a language ___ 2. expectoration of blood __ 3. one who reads scriptural passages in a church service __ 4. purposeful exaggeration for effect __ 5. scholarly or learned persons __ 6. deadly to both sides __ 7. philosophy of law __ 8. hater of women __ 9. having no money __ 10. belonging to the family that includes apes and man __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. A homonym is a word identical in pronunciation and sometimes spelling to one or more other words but different in meaning. 2. Hypothermia is an abnormally high body temperature. _ 342 300 HIGH-FREQUENCY WORD ROOTS 3. Heliotaxis refers to an organism’s movement in response to the Sun’s light. 4. Circumlocution is an extremely direct way of saying something. 5. To interject is to throw out or delete. _ (Answers are on page 415.) ROOT ROUNDUP 31 • LUC/LUM/LUS (LIGHT) Latin lucid = bright; clear; intelligible translucent = partially transparent elucidation = clarification pellucid = transparent; translucent; easily understood luminous = bright; brilliant; glowing • LUD (PLAY, GAME) Latin allude = make an indirect reference to ludicrous = laughable; ridiculous prelude = an introductory performance preceding the principal matter interlude = an entertainment between acts of a play elude = evade • MACRO (LARGE, LONG) Greek macrocosm = the universe macroeconomics = study of the overall workings of the economy macroscopic = large enough to be seen with the naked eye macronutrient = an element needed in large proportion for a plant’s growth macrocyte = an abnormally large red blood cell • MAGN (GREAT) Latin magnify = enlarge magna cum laude = with high honors magnate = powerful person magnitude = extent; greatness of size magnanimity = generosity; nobility • MAL (BAD) Latin malign = speak evil of malaise = feeling of discomfort; general sense of depression malicious = full of animosity and hatred malefactor = doer of evil malfeasance = misconduct 343 ESSENTIAL WORDS FOR THE GRE Root Work 31 Match each word with its definition. 1. pellucid (A) large enough to be seen with the naked eye 2. elucidation (B) powerful person 3. macroscopic (C) the universe 4. malfeasance (D) evade 5. magnate (E) clarification 6. malefactor (F) make an indirect reference to 7. allude (G) misconduct 8. macrocosm (H) transparent; translucent 9. magnanimous (I) generous; noble 10. elude (J) doer of evil (Answers are on page 413.) ROOT ROUNDUP 32 • MAN (HAND) Latin emancipate = liberate manipulate = operate or control by the hands manubrium = a body part that is shaped like a handle maniable = easy to handle; flexible quadrumanus = having four feet with the first digits being opposable • MAND (ORDER) Latin mandate = authoritative order or instruction mandatory = commanded by authority commandment = a command mandamus = an order issued by a superior court to a lower court reprimand = a strong formal rebuke • MANIA (OBSESSION) Greek mania = mental disorder characterized by excessive gaiety; wild enthusiasm maniac = someone who has excessive enthusiasm for something; an insane person megalomania = delusions of power or importance maniacal = characterized by excessive enthusiasm; marked by insanity manic-depressive = affective disorder marked by alternating periods of mania and depression • MAR/MARI (SEA) Latin maritime = relating to the sea marine = native to the sea; relating to the sea marina = a boat basin for small boats aquamarine = pale blue to light greenish blue submarine = undersea 344 300 HIGH-FREQUENCY WORD ROOTS • MATER/MATR (MOTHER) Latin maternal = relating to or characteristic of a mother matron = a mother of mature age and social position matrix = the womb matrilineal = tracing ancestry through the mother’s line matriarchy = a family or community governed by women Root Work 32 Match each word with its definition. 1. reprimand (A) characterized by excessive enthusiasm 2. matriarchy (B) body part shaped like a handle 3. submarine (C) easy to handle 4. mandate (D) community governed by women 5. mania (E) undersea 6. maritime (F) wild enthusiasm 7. matrilineal (G) an authoritative order 8. maniable (H) tracing ancestry through the mother’s line 9. manubrium (I) relating to the sea 10. maniacal (J) a strong formal rebuke (Answers are on page 413.) ROOT ROUNDUP 33 • MEGA (GREAT, LARGE) Greek megaphone = device used to amplify the voice megalomania = delusions of power or importance megalith = huge stone used in prehistoric structures megalopolis = vast city megalophonous = having a loud voice • META (CHANGE, AT A LATER TIME, BEYOND) Greek metaphor = figure of speech that compares two different things metamorphosis = change, transformation metaplasia = change of one kind of tissue into another kind metanoia = spiritual conversion metaphysical = pertaining to speculative philosophy • METER/METR (MEASURE) Greek barometer = instrument used in weather forecasting for measuring atmospheric pressure perimeter = the outer limits of an area micrometer = device that measures small distances or objects ammeter = instrument that measures electric current in amperes metrology = study of weights and measures • MICRO (SMALL) Greek microbiota = the microscopic life of an area 345 ESSENTIAL WORDS FOR THE GRE micrology = excessive devotion to small details microclimate = the climate of a small area microcosm = a small system having analogies to a larger system; small world microdont = having small teeth • MIN (SMALL) Latin diminution = lessening; reduction diminutive = small minute = very small minutia = petty details minuscule = very small Root Work 33 Match each word with its definition. 1. microcosm (A) huge stone used in prehistoric structures 2. megalomania (B) very small 3. metamorphosis (C) study of weights and measures 4. micrometer (D) spiritual conversion 5. metanoia (E) device that measures small distances or objects 6. minuscule (F) having small teeth 7. microdont (G) small world 8. metrology (H) delusions of importance or power 9. megalith (I) petty details 10. minutia (J) transformation (Answers are on page 413.) ROOT ROUNDUP 34 • MIS (HATRED, BAD, IMPROPER, WRONG) Greek misconstrue = misunderstand misapprehension = a misunderstanding misnomer = incorrect name misanthropy = hatred of humanity misogynist = hater of women • MISS/MIT (MOVE, SEND) Latin transmit = send from one place to another; cause to spread remittance = something sent as payment missive = a letter dismiss = put away from consideration; reject mittimus = writ that commits one to prison 346 300 HIGH-FREQUENCY WORD ROOTS • MOB/MOT/MOV (MOVE) Latin immobile = fixed; motionless mobility = state of being capable of moving automobile = passenger vehicle having four wheels and an engine emotive = appealing to or expressing emotion movie = sequence of images on a screen so rapid that they create the illusion of movement • MONO (ONE, SINGLE) Greek monogamy = marriage to one person at a time monologue = speech performed by one actor monocline = in geology, a single upward fold monochromatic = having one color monolithic = constituting a single, unified whole • MON/MONIT (WARN) Latin monitor = maintain continuous observation of admonish = caution or reprimand admonition = mild reproof premonition = forewarning; presentiment remonstrate = object or protest Root Work 34 Match each word with its definition. 1. premonition (A) speech performed by one actor 2. misogynist (B) something sent as payment 3. movie (C) exciting emotion 4. misnomer (D) a letter 5. monologue (E) incorrect name 6. monochromatic (F) images on a screen that give the illusion of movement 7. missive (G) forewarning 8. emotive (H) caution 9. remittance (I) hater of women 10. admonish (J) having one color (Answers are on page 413.) ROOT ROUNDUP 35 • MORI/MORT (DEATH) Latin moribund = dying mortorio = sculpture of the dead Christ immortal = not subject to death; everlasting mortification = shame or humiliation mortician = undertaker 347 ESSENTIAL WORDS FOR THE GRE • MORPH (FORM, SHAPE) Greek morphous = having definite form amorphous = lacking definite form morphometry = measurement of form anthropomorphic = attributing human qualities to nonhumans morphology = the form and structure of an organism • MULTI (MANY) Latin multipara = mother of two or more children multifaceted = made up of many parts multifarious = diverse multiplicity = state of being numerous multeity = state of being many • MUT (CHANGE) Latin mutative = in grammar, expressing change of state or place mutation = significant genetic change transmutation = change in appearance, shape, or nature immutable = unchangeable mutable = changeable • NAS/NAT (BIRTH, BE FROM, SPRING FORTH) Latin nationality = state of belonging to a particular nation by birth or by naturalization nativity = place or circumstances of birth natal = relating to birth native = an original inhabitant of a particular place nascent = starting to develop, coming into existence Root Work 35 Match each word with its definition. 1. multifarious (A) having definite form 2. natal (B) significant genetic change 3. nascent (C) mother of two or more children 4. moribund (D) undertaker 5. immutable (E) the form and structure of an organism 6. morphous (F) dying 7. morphology (G) diverse 8. mortician (H) unchangeable 9. mutation (I) starting to develop; come into existence 10. multipara (J) relating to birth (Answers are on page 413.) 348 300 HIGH-FREQUENCY WORD ROOTS ROOT ROUNDUP REVIEW 31–35 Match It Match each of the following roots to its meaning. 1. LUD ___ (A) sea 2. MAGN ___ (B) small 3. MANIA ___ (C) one, single 4. MAR/MARI ___ (D) great 5. METER/METRE ___ (E) change 6. MICRO ___ (F) death 7. MON/MONIT ___ (G) play 8. MONO ___ (H) obsession 9. MORI/MORT ___ (I) warn 10. MUT ___ (J) measure Fill-ins Fill in the blanks with the word that fits the definition. mandamus morphology matrilineal missive diminution nascent misogynist macrocosm metanoia malfeasance 1. a letter ___ 2. starting to develop __ 3. lessening; reduction __ 4. misconduct ___ 5. hater of women ___ 6. tracing ancestry through the mother’s line __ 7. the form and structure of an organism __ 8. an order issued by a superior court to a lower court __ 9. spiritual conversion __ 10. the universe __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. Pellucid means cloudy, vague, or unintelligible. 2. Manubrium refers to a body part that is shaped like an egg. _ 3. A person’s nativity refers to his or her place of birth. 4. Metrology means urban planning. 5. In geology, a monocline is a single upward fold. _ (Answers are on page 416.) 349 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP 36 • NAV (SHIP) Latin naval = relating to ships or shipping navigation = theory and practice of charting a ship’s course navarch = commander of a fleet naviform = boat-shaped circumnavigate = to go completely around • NEC/NIC/NOC/NOX (KILL, DEATH, HARM) Latin internecine = deadly to both sides pernicious = very harmful nocent = causing injury innocuous = not harmful noxious = injurious to health • NECRO (DEAD, CORPSE) Greek necromancy = divination through communicating with spirits necrophobia = fear of dead bodies necrotype = extinct species necrophilia = intercourse with dead bodies necromorphous = feigning death • NEG (NO) Latin negate = nullify; cancel out negative = indicating opposition renege = go back on one’s word negligent = careless; inattentive negligible = not worth considering • NEO (NEW, RECENT) Greek neologism = new word or expression; an existing word or expression used in a new way neophyte = novice, beginner neonate = newborn child neoplasia = formation of new tissue neolithic = New Stone Age; period in the development of technology at end of the Stone Age 350 300 HIGH-FREQUENCY WORD ROOTS Root Work 36 Match each word with its definition. 1. necrotype (A) boat-shaped 2. negligible (B) deadly to both sides 3. navarch (C) nullify 4. pernicious (D) fear of dead bodies 5. neonate (E) novice 6. necrophobia (F) commander of a fleet 7. neophyte (G) newborn child 8. internecine (H) very harmful 9. naviform (I) not worth considering 10. negate (J) extinct species (Answers are on page 413.) ROOT ROUNDUP 37 • NEUR (NERVE) Greek neurology = study of the nerves and the brain neurosis = disorder of the nervous system neuroid = nervelike neurergic = pertaining to nerve action neuralgia = pain along nerve • NEUT/NEUTR (NOT EITHER) Latin neutral = belonging to neither side in a war or controversy neutralize = make neutral; counterbalance the effect of neutrality = state of being neutral neutron = an electrically neutral subatomic particle neuter = neither masculine nor feminine • NOCT/NOX (NIGHT) Latin nocturnal = pertaining to night; active at night equinox = either of two times in a year when the Sun crosses the celestial equator noctambulant = walking in one’s sleep noctivagant = wandering around at night noctilucous = shining at night • NOM/NOMEN/NYM (NAME) Latin, Greek nominal = existing in name only nom de guerre = war name; pseudonym misnomer = incorrect name ignominious = disgraceful and dishonorable nomenclature = terms used in a particular science or discipline 351 ESSENTIAL WORDS FOR THE GRE • NON (NOT) Latin nonplussed = bewildered nonchalant = casual, unconcerned non licet = not lawful non sequitur = conclusion not following from apparent evidence nondescript = lacking interesting or distinctive qualities; dull Root Work 37 Match each word with its definition. 1. noctambulant (A) counterbalance the effect of 2. misnomer (B) shining at night 3. nonplussed (C) pain along nerve 4. neuroid (D) not lawful 5. nom de guerre (E) neither masculine nor feminine 6. neuralgia (F) nervelike 7. non licet (G) incorrect name 8. noctilucous (H) walking in one’s sleep 9. neuter (I) bewildered 10. neutralize (J) war name (Answers are on page 413.) ROOT ROUNDUP 38 • NOV (NEW) Latin renovate = restore to an earlier condition novice = apprentice, beginner novel = new or original novitiate = state of being a beginner or novice innovation = something newly introduced • NUM (NUMBER) Latin numismatics = coin collecting numeral = a symbol that represents a number numerate = count enumerate = count off; list numerology = study of mystical meanings in numbers • OB/OC/OF/OP (AGAINST) Latin obdurate = stubborn occlude = shut; block obliterate = destroy completely opprobrious = disgraceful; contemptuous obfuscate = obscure; confuse 352 300 HIGH-FREQUENCY WORD ROOTS • OLIG (FEW, LITTLE) Greek oligopoly = situation with only a few sellers so that action by any one of them will affect price oligarchy = government by only a few oligodontous = having few teeth oligophagous = eating only a few kinds of food oligosyllable = a word with only a few syllables • OMNI (ALL, EVERY) Latin omnipotent = having unlimited power omnivorous = eating everything; absorbing everything omnipresent = present everywhere omniscient = having infinite knowledge omneity = state of including all things Root Work 38 Match each word with its definition. 1. obdurate (A) shut; block 2. novice (B) coin collecting 3. oligarchy (C) eating everything 4. innovation (D) stubborn 5. oligodontous (E) study of mystical meanings in numbers 6. occlude (F) state of including all things 7. omnivorous (G) something newly introduced 8. numerology (H) government by only a few 9. omneity (I) having few teeth 10. numismatics (J) beginner (Answers are on page 413.) ROOT ROUNDUP 39 • OPER (WORK) Latin opera = musical work opere citato = already mentioned in the work operon = a group of genes that operate as a unit operative = functioning; working operose = working hard • OPTI/OPTO (EYE, VISION) Latin optimistic = looking on the positive side optician = someone who makes eyeglasses optometry = measuring and testing of vision optogram = an image fixed on the retina optical = of or relating to sight 353 ESSENTIAL WORDS FOR THE GRE • ORTHO (STRAIGHT, CORRECT) Latin orthodox = traditional; conservative orthopraxy = correct action orthodontics = correction of irregularity of teeth orthognathism = condition of having straight jaws orthopedic = correcting physical deformities • OS/OSS/OST/OSTEO (BONE) Latin, Greek osseous = bony ossify = turn into bone osteitis = bone inflammation ostosis = formation of bone osteoma = bone tumor • PAL/PALEO (ANCIENT) Greek paleontology = study of past geological eras through fossil remains paleoethnic = relating to the earliest races of man paleography = study of ancient writings paleology = study of antiquities paleogenetic = of past origin Root Work 39 Match each word with its definition. 1. optician (A) bony 2. paleogenetic (B) correction of teeth irregularity 3. orthopraxy (C) functioning 4. operon (D) bone formation 5. paleography (E) study of ancient writings 6. optometry (F) testing of vision 7. ostosis (G) correct action 8. orthodontics (H) group of genes operating as a unit 9. operative (I) of past origin 10. osseous (J) maker of eyeglasses (Answers are on page 413.) ROOT ROUNDUP 40 • PAC (PEACE) Latin pact = a treaty pacifist = person opposed to war or violence between nations pacify = to restore calm, bring peace pacific = calm; peaceful pacification = appeasement 354 300 HIGH-FREQUENCY WORD ROOTS • PAN (ALL, EVERY) Greek panorama = broad view; comprehensive picture panacea = cure-all pantheon = all the gods of a people; group of highly regarded persons panoply = impressive array pandemic = spread over a whole area • PAR (EQUAL) Latin parable = simple story that teaches a lesson parity = equality par = equality of status or value disparity = difference apartheid = a system of discrimination based on race that formerly existed in South Africa • PARA (BEYOND, RELATED, ALONGSIDE) Greek paradigm = model; example; pattern paradisiacal = heavenly; wonderful paramount = supreme; primary paragon = model of excellence parasite = person or animal that lives at another’s expense • PAS/PATH (FEELING, DISEASE, SUFFERING) Greek apathy = indifference antipathy = dislike pathos = pity, compassion pathogen = agent that causes disease dispassionate = impartial; unaffected by emotion Root Work 40 Match each word with its definition. 1. paradigm (A) all the gods of a people 2. pantheon (B) restore calm 3. paramount (C) dislike 4. disparity (D) comprehensive picture 5. pacify (E) difference 6. pathos (F) equality 7. panorama (G) person opposed to war 8. antipathy (H) supreme 9. parity (I) model 10. pacifist (J) pity; compassion (Answers are on page 413.) 355 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP REVIEW 36–40 Match It Match each of the following roots to its meaning. 1. NECRO ___ (A) kill, death, harm 2. NEC/NIC/NOC/NOX ___ (B) against 3. NEUR ___ (C) peace 4. NOCT/NOX ___ (D) ancient 5. NOV ___ (E) nerve 6. OB/OC/OF/OP ___ (F) dead, corpse 7. OPER ___ (G) new 8. PAL/PALEO ___ (H) feeling, disease, suffering 9. PAC ___ (I) work 10. PAS/PATH ___ (J) night Fill-ins Fill in the blanks with the word that fits the definition. paramount noctivagant oligophagous omniscient navarch pernicious apartheid ignominious orthopraxy osteoma 1. correct action ___ 2. disgraceful and dishonorable __ 3. supreme; primary __ 4. bone tumor___ 5. having infinite knowledge ___ 6. commander of a fleet __ 7. wandering around at night __ 8. eating only a few kinds of food __ 9. a system of discrimination based on race that formerly existed in South Africa __ 10. very harmful __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. To renege is to repeat an offer of negotiation. 2. A neologism can be an existing word or expression used in a new way. _ 3. A neutron is an electrically neutral subatomic particle. 4. Numerology is the study of mystical meanings in natural phenomena. 5. A non sequitor is a conclusion that follows from apparent evidence. _ (Answers are on page 416.) 356 300 HIGH-FREQUENCY WORD ROOTS ROOT ROUNDUP 41 • PATER/PATR (FATHER) Latin paternity = fatherhood; descent from father’s ancestors patronize = condescend to; disparage; buy from patronage = support of a sponsor or benefactor, as for a cause or an institution patricide = murder of one’s father patrimony = inheritance or heritage derived from one’s father • PED (CHILD) Greek pedant = uninspired, boring academic who makes a display of his or her learning pedantic = showing off learning pedagogue = teacher pedodontics = dentistry dealing with the treatment of children’s teeth encyclopedia = reference work that contains articles on a broad range of subjects • PED/POD (FOOT) Greek pediform = shaped like a foot pedestrian = commonplace pedate = having feet pedometer = a device that measures distance by the number of steps of a walker podiatry = the diagnosis and treatment of diseases of the foot • PEL/PULS (DRIVE, URGE) Latin pulse = a regular or rhythmical beating pulsate = beat; to vibrate repellant = something that repels or drives back repulse = drive back; repel propellant = something that provides thrust • PER (THROUGH, COMPLETELY) Latin peregrinate = wander through percutaneous = effecting something through the skin permeable = penetrable pervasive = spread throughout every part permeate = diffuse through 357 ESSENTIAL WORDS FOR THE GRE Root Work 41 Match each word with its definition. 1. pedate (A) fatherhood 2. pedodontics (B) murder of one’s father 3. repellant (C) something that provides thrust 4. patricide (D) dentistry dealing with children’s teeth 5. propellant (E) shaped like a foot 6. pedantic (F) having feet 7. peregrinate (G) something that drives back 8. paternity (H) diffuse through 9. permeate (I) showing off learning 10. pediform (J) wander through (Answers are on page 413.) ROOT ROUNDUP 42 • PERI (AROUND, NEAR) Greek periosteal = around a bone peripatetic = moving about or from place to place perihelion = the point in orbit nearest the Sun perigee = the point in orbit nearest Earth periphrasis = circumlocution • PET (SEEK, REQUEST, ASSAIL) Latin petition = a request to a superior authority competition = the act of striving against others to attain a goal petulant = contemptuous; peevish repetitive = given to the act of repeating petitio principli = begging the question • PHIL (LOVE, FONDNESS, PREFERENCE) Greek philanthropist = lover of mankind; doer of good technophile = lover of technology philogynist = lover of women philhelline = lover of things Greek philtre = love potion • PHOBOS (FEAR) Greek phobia = abnormal, irrational fear of a situation or thing arachnophobia = abnormal fear of spiders agoraphobia = fear of places that are public or open claustrophobic = fear of being in enclosed spaces hydrophobia = fear of water 358 300 HIGH-FREQUENCY WORD ROOTS • PHON (SOUND, VOICE) Greek microphone = an instrument that changes sound waves into electric current phonogram = a symbol that represents sound phonic = relating to sound phonetics = study of speech sounds cacophony = jarring, unpleasant noise Root Work 42 Match each word with its definition. 1. petition (A) lover of mankind 2. philtre (B) begging the question 3. perihelion (C) moving about 4. hydrophobia (D) relating to sound 5. cacophony (E) request to a superior authority 6. peripatetic (F) love potion 7. phonic (G) the point in orbit nearest the Sun 8. petitio principli (H) fear of water 9. philanthropist (I) fear of places that are public or open 10. agoraphobia (J) unpleasant, jarring sound (Answers are on page 413.) ROOT ROUNDUP 43 • PHOS/PHOT (LIGHT) Greek photosensitive = sensitive to light or radiant energy photograph = an image recorded by a camera and reproduced on a photosensitive surface phototaxis = growth directed by light photophile = loving light photometry = measurement of the properties of light • PHYS/PHYSIO (NATURE) Greek physiology = the function of a living organism physical = relating to the body physiolatry = worship of nature physiocracy = government that is in accord with the operation of natural laws physiognomy = divination of character from a person’s face • PICT (PAINT) Latin picture = an image rendered on a flat surface depict = represent in a picture pictograph = a picture that represents a word or idea picturesque = of a picture; quaintly attractive pictorial = relating to or composed of pictures 359 ESSENTIAL WORDS FOR THE GRE • PLAC (PLEASE) Latin placid = calm placate = lessen another’s anger; pacify implacable = inflexible, incapable of being appeased complacent = self-satisfied placebo = something given to please or quiet • POLI (CITY, STATE, CITIZEN) Greek politics = the art of governing a state and the control of its affairs metropolitan = relating to a major city cosmopolitan = common to the whole world political = relating to the affairs of the state polity = political organization of a state Root Work 43 Match each word with its definition. 1. implacable (A) something given to please 2. physiology (B) growth directed by light 3. metropolitan (C) worship of nature 4. polity (D) incapable of being appeased 5. phototaxis (E) functions of a living organism 6. depict (F) of a picture 7. placebo (G) loving light 8. physiolatry (H) relating to a major city 9. picturesque (I) represent in a picture 10. photophile (J) political organization of a state (Answers are on page 413.) ROOT ROUNDUP 44 • POLY (MANY) Greek polyphony = use of one symbol for many sounds polymorphic = having many forms polyglot = speaker of many languages polygamy = having more than one wife or husband at a time polytheist = one who believes in more than one god • PON/POS (PUT, PLACE) Latin compose = constitute or form component = an element or ingredient composite = made up of components oppose = place so as to be opposite something else repose = place; to lay down 360 300 HIGH-FREQUENCY WORD ROOTS • POPUL (THE PEOPLE) Latin populate = supply with inhabitants; people popular = reflecting the taste of the people at large populace = the masses population = all of the people who inhabit an area populous = containing many inhabitants • PORT (CARRY, GATE) Latin portage = the act of transporting or carrying portal = an entrance or gate portable = able to be carried easily deport = expel from a country portfolio = case to carry papers • POST (AFTER, BEHIND) Latin posterity = future generations; all of a person’s descendants posterior = bottom, rear postdiluvian = after the flood posthumous = after a person’s death post factum = after the event Root Work 44 Match each word with its definition. 1. repose (A) future generations 2. posterity (B) having many forms 3. populace (C) people 4. polytheist (D) after the event 5. portal (E) entrance or gate 6. populate (F) place; to lay down 7. oppose (G) place so as to be opposite something else 8. portfolio (H) one who believes in more than one god 9. polymorphic (I) case to carry papers 10. post factum (J) the masses (Answers are on page 414.) ROOT ROUNDUP 45 • PRE (BEFORE, EARLIER) Latin prenatal = before birth preclude = make impossible in advance premise = proposition upon which an argument is based precept = principle; law precedent = a model for something that follows 361 ESSENTIAL WORDS FOR THE GRE • PREHEND/PREHENS (SEIZE, GRASP) Latin reprehend = censure; reprove comprehend = take in the meaning; to grasp apprehension = act of seizing; understanding prehensile = able to grasp prehension = act of grasping • PRIM (BEFORE, FIRST) Latin prima facie = at first sight; on the face of it primapara = a woman having her first child primeval = ancient, primitive primordial = original, existing from the beginning primogeniture = state of being the eldest child • PRO (IN FAVOR OF) Latin pro re nata = for an emergency (for the thing born) proponent = a supporter prodigy = highly gifted child; marvel propensity = inclination, tendency proclivity = tendency, inclination • PROTO (FIRST, EARLIEST) Greek protogenic = formed at the beginning prototype = an original model protomorphic = primitive protoplast = an original ancestor protolithic = relating to the first Stone Age Root Work 45 Match each word with its definition. 1. prehension (A) a supporter 2. primordial (B) at first sight 3. precedent (C) highly gifted child 4. protoplast (D) original ancestor 5. prima facie (E) able to grasp 6. prodigy (F) a model for something that follows 7. proponent (G) act of grasping 8. protogenic (H) original 9. prenatal (I) formed at the beginning 10. prehensile (J) before birth (Answers are on page 414.) 362 300 HIGH-FREQUENCY WORD ROOTS ROOT ROUNDUP REVIEW 41–45 Match It Match each of the following roots to its meaning. 1. PATER ___ (A) seek, request, assail 2. PEL/PULS ___ (B) put, place 3. PET ___ (C) father 4. PHIL ___ (D) in favor of 5. PHOS/PHOT ___ (E) city, state, citizen 6. POLI ___ (F) before, first 7. PON/POS ___ (G) drive, urge 8. POPUL ___ (H) light 9. PRIM ___ (I) love, fondness, preference 10. PRO ___ (J) the people Fill-ins Fill in the blanks with the word that fits the definition. pedagogue prehensile placate depict percutaneous protogenic cacophony portage postdiluvian agoraphobia 1. able to grasp ___ 2. represent in a picture__ 3. fear of places that are open or public __ 4. teacher __ 5. formed at the beginning __ 6. affecting something through the skin __ 7. after the flood __ 8. lessen another’s anger in; pacify __ 9. jarring, unpleasant noise __ 10. act of transporting or carrying __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. The perigee is the point in orbit farthest from Earth. 2. A physiocracy is a government that is in accord with the operation of natural laws. _ 3. Pro re nata means for an emergency. 4. Polyphony refers to the use of one symbol for many sounds. 5. A pedometer is a device that measures distance by the number of steps of a walker. _ (Answers are on page 416.) 363 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP 46 • PSEUDO (FALSE) Greek pseudonym = pen name; fictitious or borrowed name pseudopsia = an optical illusion pseudodox = false doctrine pseudomorph = false or irregular form pseudocyesis = false pregnancy • PSYCH (MIND) Greek psyche = the mind psychic = perceptive of nonmaterial, spiritual forces; originating in the mind psychiatrist = a doctor who treats disorders of the mind psychedelic = mind-expanding psychology = study of the mind • PUB (THE PUBLIC) Latin public = concerning the community or the people republic = a political order in which a body of citizens has supreme power publication = communication of information to the public publicity = act of communicating information to attract public interest publish = announce; bring to the attention of the public • PUNCT (POINT, PRICK) Latin punctilious = strictly attentive to small details of form in conduct punctilio = a fine point of etiquette punctual = prompt puncture = a hole made by a sharp object punctate = like a point; ending in a point • PUT/PUTAT (THINK, CALCULATE) Latin putative = supposed repute = consider; suppose reputed = supposed to be such reputation = state of being held in high esteem compute = determine an amount or number 364 300 HIGH-FREQUENCY WORD ROOTS Root Work 46 Match each word with its definition. 1. psychic (A) concerning the people 2. punctate (B) the mind 3. compute (C) bring to the attention of the public 4. public (D) fine point of etiquette 5. putative (E) an optical illusion 6. pseudodox (F) determine an amount 7. punctilio (G) false doctrine 8. psyche (H) supposed 9. publish (I) originating in the mind 10. pseudopsia (J) like a point (Answers are on page 414.) ROOT ROUNDUP 47 • QUER/QUES/QUIR/QUIS (ASK/SEEK) Latin query = a question inquest = an investigation; court or legal proceeding inquisition = an investigation; act of inquiring inquirer = one who asks a question requisite = required • RE (BACK AGAIN, REPEAT) Latin recant = retract a statement or opinion rebut = refute by evidence or argument retract = withdraw; take back recurrence = repetition redundant = exceeding what is necessary; unnecessarily repetitive • RECT (STRAIGHT) Latin erect = fix in an upright position erectile = able to be raised to an upright position rectitude = moral uprightness rectilinear = bounded by straight lines recto = right-hand page of a book • RECT/REG/REGN (RULE, GOVERN) Latin rector = an Anglican cleric in charge of a parish correct = remove the errors from; punish for the purpose of improving regime = a government in power regulation = a law to govern conduct regular = conforming with fixed procedure or discipline 365 ESSENTIAL WORDS FOR THE GRE • RETRO (BACKWARD) Latin retrospect = review or contemplation of the past retrograde = having a backward motion or direction retroactive = applying to an earlier time retroject = throw back retrovirus = a virus that synthesizes DNA from RNA instead of the reverse Root Work 47 Match each word with its definition. 1. retrograde (A) applying to an earlier time 2. redundant (B) a question 3. rectitude (C) right-hand page of a book 4. query (D) having a backward motion 5. recto (E) law to govern conduct 6. retroactive (F) government in power 7. regulation (G) required 8. regime (H) unnecessarily repetitive 9. requisite (I) moral uprightness 10. retract (J) take back (Answers are on page 414.) ROOT ROUNDUP 48 • RID/RIS (LAUGH) Latin deride = mock ridicule = words that evoke contemptuous laughter at a person derision = ridicule ridiculous = deserving ridicule; absurd ridibund = easily moved to laughter • ROG/ROGAT (ASK) Latin interrogatory = asking a question prerogative = a special right or privilege rogatory = requesting information rogation = solemn prayer derogatory = disparaging; belittling • SACR/SANCT (SACRED, HOLY) Latin sanctuary = haven, retreat sanctify = set apart as holy; consecrate sanction = approval; ratification; permission sacrosanct = extremely sacred; beyond criticism sanctimonious = pretending to be pious or righteous 366 300 HIGH-FREQUENCY WORD ROOTS • SCI (KNOW) Latin scibile = something that is possible to know sciolism = conceited and shallow knowledgeability conscientious = careful and thorough; governed by conscience prescient = having foresight nescience = absence of knowledge; ignorance • SCOP (EXAMINE, OBSERVE, WATCH) Greek scopic = visual telescope = device used to observe distant objects periscope = optical instrument used to see things from a position not in a direct line of sight microscopic = too small to be seen with the naked eye colonoscope = long, flexible instrument used to visually examine the colon Root Work 48 Match each word with its definition. 1. prescient (A) absence of knowledge 2. ridibund (B) set apart as holy 3. sacrosanct (C) mock 4. rogatory (D) visual 5. telescope (E) having foresight 6. interrogatory (F) extremely sacred 7. nescience (G) easily moved to laughter 8. deride (H) device used to observe distant objects 9. sanctify (I) asking a question 10. scopic (J) requesting information (Answers are on page 414.) ROOT ROUNDUP 49 • SCRIB/SCRIP (WRITE) Latin ascribe = attribute to a cause or source circumscribe = limit; confine prescribe = set down a rule; recommend a treatment manuscript = a document written by hand nondescript = lacking interesting or distinctive qualities; dull • SE (AWAY, APART) Latin secede = withdraw from membership in an alliance select = pick out; choose seclusion = act of keeping apart from social contact seduce = lead away from accepted principles; attract secretive = not open or frank 367 ESSENTIAL WORDS FOR THE GRE • SEC/SECT/SEGM (CUT) Latin sectile = relating to a mineral that can be cut smoothly by a knife dissect = cut apart intersection = the process or result of cutting across or through secant = a straight line that intersects a curve at two or more points segmented = divided into parts • SECU/SEQU (FOLLOW) Latin prosecutor = one who initiates a civil or criminal court action sequel = something that follows inconsequential = insignificant; unimportant obsequious = overly submissive sequatious = disposed to follow another • SEMI (HALF, PARTLY) Latin semiterrestrial = partially living on land semiannual = occurring twice a year semiaquatic = not entirely adapted for living in water semicircle = half of a circle semitaur = in mythology, a creature that is half-man and half-bull Root Work 49 Match each word with its definition: 1. secede (A) act of keeping apart from social contact 2. obsequious (B) a document written by hand 3. sectile (C) something that follows 4. sequel (D) cut apart 5. manuscript (E) withdraw from membership in an alliance 6. semitaur (F) half-man and half-bull 7. seclusion (G) set down a rule 8. semiaquatic (H) not entirely adapted for living in water 9. dissect (I) overly submissive 10. prescribe (J) relating to a mineral that can be cut smoothly by a knife (Answers are on page 414.) ROOT ROUNDUP 50 • SENS/SENT (FEEL, BE AWARE) Latin sensate = perceived by the senses insensible = unconscious; unresponsive sentiment = a view based on emotion rather than reason sentient = aware, conscious, able to perceive sentisection = vivisection performed without the use of anesthesia 368 300 HIGH-FREQUENCY WORD ROOTS • SOL (SUN) Latin solarium = room exposed to sunlight solarize = expose to sunlight solstice = point or time when the Sun is furthest from the Equator soliterraneous = relating to the meteorological effect of Sun and Earth solarimeter = device that measures the flux of the Sun’s radiation • SOLV/SOLU (FREE, LOOSEN, DISSOLVE) Latin dissolve = make something pass into solution; melt; dispel soluble = able to be dissolved; possible to solve solute = a substance dissolved in another one dissolution = disintegration; debauchery irresolute = undecided • SOMN (SLEEP) Latin somnambulance = walking in one’s sleep somniloquence = talking in one’s sleep insomnia = inability to fall asleep or remain asleep somnolent = sleepy somniferous = inducing sleep • SOPH (WISE, SKILLFUL, SHREWD) Greek sophist = one who is skilled in deceptive argumentation sophistry = plausible but misleading argumentation sophisticate = make more worldly; refine sophomoric = showing lack of judgment and immaturity philosophy = love of wisdom Root Work 50 Match each word with its definition. 1. solarium (A) able to perceive 2. somniferous (B) able to be dissolved or solved 3. philosophy (C) inducing sleep 4. dissolution (D) point when the Sun is furthest from the Equator 5. sentiment (E) sleepy 6. soluble (F) disintegration; debauchery 7. somnolent (G) refine; make more worldly 8. sophisticate (H) a view based on emotion 9. solstice (I) room exposed to sunlight 10. sentient (J) love of wisdom (Answers are on page 414.) 369 ESSENTIAL WORDS FOR THE GRE ROOT REVIEW 46–50 Match It Match each of the following roots to its meaning. 1. PSYCH ___ (A) sacred, holy 2. PUB ___ (B) backward 3. RECT ___ (C) know 4. RETRO ___ (D) free, loosen, dissolve 5. SCI ___ (E) the public 6. SACR/SANCT ___ (F) follow 7. SE ___ (G) straight 8. SECU/SEQU ___ (H) feel, be aware 9. SENS/SENT ___ (I) mind 10. SOLV/SOLUT ___ (J) away, apart Fill-ins Fill in the blanks with the word that fits the definition. retrospect secant punctilio ridibund pseudodox somniloquence sophistry sequacious recant rogation 1. easily moved to laughter ___ 2. plausible but misleading argumentation __ 3. contemplation of the past __ 4. straight line that intersects a curve at two or more points ___ 5. talking in one’s sleep ___ 6. false doctrine __ 7. retract a statement or opinion __ 8. solemn prayer __ 9. a fine point of etiquette __ 10. disposed to follow another __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. A punctilious person is inattentive to small details of form in conduct. 2. Rectilinear means bounded by straight lines. _ 3. Sectile relates to a mineral that can be cut smoothly by a knife. 4. A semitaur is a mythological creature that is half-man and half-lion. 5. Soliterraneous refers to the meteorological effect of Moon and Earth. _ (Answers are on page 416.) 370 300 HIGH-FREQUENCY WORD ROOTS ROOT ROUNDUP 51 • SPEC/SPECT/SPIC (SEE, LOOK AT) Latin specimen = a representative of a class or whole; a sample specter = an apparition speculate = take something as true based on insufficient evidence retrospective = review of the past perspicacious = shrewd, astute, keen-witted • SPIR (BREATH, ENERGY, ANIMATION) Latin respiration = breathing aspiration = expulsion of breath in speaking spirit = animating force within living things spirited = animated; courageous spirograph = device that records the movements of breathing • STAS/STAT (STAND, BEING IN A PLACE, POSITION) Greek stationary = not moving static = having no motion; fixed; stationary state = condition of being status = position or standing in relation to that of others status quo = existing state of affairs • STRICT/STRING (TIGHT, DRAWN TOGETHER) Latin restrict = keep within limits stricture = a limit or restriction stringent = imposing rigorous standards; constricted; tight strict = precise; within narrow limits; rigorous in discipline constrict = squeeze or compress; restrict the scope of • STRUCT (BUILD) Latin destruction = act of destroying structure = something built construct = build instruction = act of methodically providing with knowledge indestructible = impossible to destroy 371 ESSENTIAL WORDS FOR THE GRE Root Work 51 Match each word with its definition. 1. spirited (A) impossible to destroy 2. stricture (B) existing state of affairs 3. retrospective (C) limit or restriction 4. stringent (D) device that records the movements of breathing 5. constrict (E) restrict the scope of 6. status quo (F) review of the past 7. spirograph (G) fixed; stationary 8. indestructible (H) imposing rigorous standards; constricted 9. static (I) an apparition 10. specter (J) animated (Answers are on page 414.) ROOT ROUNDUP 52 • SUB/SUC/SUF/SUG/SUP/SUS (BELOW, UNDER, LESS) Latin subtle = hard to detect or describe subterfuge = trick or tactic used to avoid something subsume = include; incorporate suppress = put down by force; restrain suspend = defer, interrupt; dangle, hang • SUPER/SUR (OVER, ABOVE) Latin superior = higher than another in rank or authority supersede = take the place of insuperable = insurmountable, unconquerable supernal = celestial; heavenly surtax = additional tax • SYL/SYM/SYN/SYS (TOGETHER, WITH) Greek syllogism = argument with a conclusion deduced from two premises synchronous = occurring at the same time; moving at the same rate syndicate = association of people who undertake a duty or transact business synthesis = blend, combination system = group of interrelated elements that form a whole • TACT/TANG (TOUCH) Latin contact = a coming together or touching tactile = relating to the sense of touch tactus = sense of touch tangible = able to be touched tangent = digression, diversion 372 300 HIGH-FREQUENCY WORD ROOTS • TELE (DISTANCE, FAR) Greek telemeter = an instrument that measures distance telemetry = science of transmitting data from someplace remote to a distant receiving station telecommunication = science of communicating over distances by electronic transmission teleseism = tremor from a distant earthquake telephony = sound transmission between distant stations Root Work 52 Match each word with its definition. 1. suspend (A) a blend 2. supernal (B) sense of touch 3. suppress (C) heavenly 4. tactus (D) interrupt; dangle 5. synchronous (E) able to be touched 6. teleseism (F) insurmountable 7. synthesis (G) instrument that measures distance 8. telemeter (H) put down by force 9. tangible (I) tremor from a distant earthquake 10. insuperable (J) occurring at the same time (Answers are on page 414.) ROOT ROUNDUP 53 • TEMPOR (TIME) Latin contemporary = belonging to the same time period temporal = related to time temporize = act evasively to gain time, avoid an argument, or postpone a decision extemporaneous = unrehearsed temporality = being bounded in time • TEN/TAIN (HOLD) Latin detain = delay; keep from proceeding pertain = relate to tenacious = stubborn, holding firm content = something that is contained tenure = a period during which something is held • TEND/TENS/TENT (STRETCH, STRIVE) Latin tension = act of stretching tight contend = strive in opposition; to struggle contention = act of striving in controversy tense = tightly stretched tensor = a muscle that stretches a body part 373 ESSENTIAL WORDS FOR THE GRE • TERM (END, LIMIT) Latin terminal = concluding, final; fatal midterm = middle of an academic term terminate = end interminable = endless termless = having no limits; unending • TERR (LAND, THE EARTH) Latin terraqueous = consisting of land and water terraceous = earthen terrestrial = earthly terra firma = dry land terra incognita = unknown land Root Work 53 Match each word with its definition. 1. interminable (A) earthen 2. terminal (B) act evasively to gain time 3. tensor (C) stubborn; holding firm 4. contend (D) consisting of land and water 5. temporal (E) endless 6. tenacious (F) a muscle that stretches a body part 7. detain (G) final 8. terraqueous (H) strive in opposition 9. temporize (I) related to time 10. terraceous (J) delay (Answers are on page 414.) ROOT ROUNDUP 54 • TERTI (THIRD) Latin tertial = relating to the third row of flight feathers on the basal section of a bird’s wing tertian = recurring after three days tertiary = third in degree or rank tertium quid = a third thing of indeterminate character tertiary color = a color that results from mixing two secondary colors • THE/THEI/THEO (GOD) Greek atheist = person who does not believe in the existence of God theocracy = government by priests representing a god theology = study of God and religion apotheosis = glorification; glorified ideal theogamy = marriage of gods 374 300 HIGH-FREQUENCY WORD ROOTS • THERAP (ATTEND, TREAT) Greek therapy = treatment of illness therapeutic = having healing powers therapist = specialist in a particular therapy biotherapy = treatment of disease with preparations synthesized from living organisms physiotherapy = treatment of physical injury with therapeutic exercise • THERM/THERMO (WARM, HOT) Greek thermochemistry = the chemistry of heat thermal = relating to, or caused by heat thermoduric = able to survive high temperatures thermodynamic = resulting from heat conversion thermolabile = subject to change or destruction by heating • TOM/TOME/TOMY (CUT, SECTION) Greek tome = one book in a work of many volumes dermatome = instrument used to cut slices of the skin in skin grafts microtome = instrument that cuts specimens into slices for examination with microscopes gastrotomy = surgical incision into the stomach vasectomy = surgical removal of a duct that carries semen Root Work 54 Match each word with its definition. 1. tertian (A) marriage of gods 2. therapy (B) surgical incision into the stomach 3. thermoduric (C) having healing powers 4. apotheosis (D) recurring after three days 5. thermal (E) one book in a work of many volumes 6. tertium quid (F) caused by heat 7. gastrotomy (G) glorified ideal 8. therapeutic (H) able to survive high temperatures 9. tome (I) a third thing of an indeterminate character 10. theogamy (J) treatment of illness (Answers are on page 414.) ROOT ROUNDUP 55 • TOP/TOPO (PLACE, REGION) Greek topography = the configuration of a land surface topology = regional anatomy toponym = the name of a place topophobia = fear of certain places topos = a traditional theme 375 ESSENTIAL WORDS FOR THE GRE • TORS/TORT/TORQU (TWIST, TURN) Latin retort = quick caustic reply that turns the first speaker’s words to one’s own advantage torsion = act of twisting or turning contort = twist out of shape torque = turning or twisting force tortuous = winding; twisting; circuitous • TRACT (DRAG, PULL) Latin attract = cause to draw near tractor = vehicle for pulling machinery tractable = easy to manage or control; easy to manipulate; easily remedied intractable = not easily managed protract = draw out; prolong • TRAN/TRANS (ACROSS, THROUGH) Latin transcend = rise above, go beyond transmute = change in appearance, shape or nature transgression = the exceeding of a limit or boundary translucent = clear; lucid transmogrify = change into a different shape or form • TREM/TREMU (TREMBLE, SHAKE) Latin tremendous = enormous; able to make one tremble tremulous = marked by shaking or trembling tremble = shake involuntarily; quake; feel fear tremor = a shaking movement; a trembling or quivering tremolo = a quivering effect produced by quickly repeating a single tone Root Work 55 Match each word with its definition. 1. tortuous (A) turning or twisting force 2. attract (B) name of a place 3. topophobia (C) go beyond 4. protract (D) change in appearance or nature 5. transmute (E) cause to draw near 6. torque (F) winding; twisting 7. tremulous (G) fear of certain places 8. toponym (H) a shaking movement 9. tremor (I) marked by trembling 10. transcend (J) draw out; prolong (Answers are on page 414.) 376 300 HIGH-FREQUENCY WORD ROOTS ROOT ROUNDUP REVIEW 51–55 Match It Match each of the following roots to its meaning. 1. SPIR ___ (A) far 2. STRUCT ___ (B) end, limit 3. TACT/TANG ___ (C) place, region 4. TELE ___ (D) attend, treat 5. TEMPOR ___ (E) warm, hot 6. TERM ___ (F) build 7. THERAP ___ (G) touch 8. THERM/THERMO ___ (H) time 9. TOP/TOPO ___ (I) drag, pull 10. TRACT ___ (J) breath, energy, animation Fill-ins Fill in the blanks with the word that fits the definition. supernal tenacious theogamy synchronous stricture thermoduric toponym torque status quo contention 1. marriage of gods ___ 2. act of striving in controversy __ 3. able to survive high temperatures __ 4. existing state of affairs___ 5. stubborn, holding firm ___ 6. occurring at the same time; moving at the same rate __ 7. twisting or turning force __ 8. celestial; heavenly __ 9. a limit or restriction __ 10. name of a place __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. To transmogrify is to change into a different shape or form. 2. To subsume is to regard something as assumed. _ 3. A spirograph is an instrument used to measure wind direc tion. 4. Tertial refers to the second row of flight feathers on the basal section of a bird’s wing. 5. A tremolo is a quivering effect produced by quickly repeating a single tone. _ (Answers are on page 416.) 377 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP 56 • TURB (SPINNING, CONFUSION, DISTURBANCE) Greek turbid = in a state of turmoil turbine = machine that converts the energy of moving fluid to rotary mechanical power turbojet = jet engine with a turbine-driven compressor turbulent = violently agitated disturbance = a commotion • TYP/TYPE (TYPE) Greek typal = relating to or serving as a type typical = conforming to a type typify = represent as a typical example of; symbolize typology = the study or classification of types stereotype = a formulaic conception or image • ULTRA (BEYOND, EXCESSIVE, ON THE OTHER SIDE OF) Latin ultrasonic = relating to acoustic frequencies above the ear’s audible range ultramodern = extremely modern in style or ideas ultraconservative = conservative to an extreme ultranationalism = extreme nationalism ultramundane = extending beyond the world or the universe • UN (INTENSIVE PREFIX; NOT, REVERSE, UNDO, REMOVE) Latin unfeigned = not feigned; not made up; genuine; real untenable = not viable; indefensible unyielding = firm, resolute unequivocal = absolute, certain unfetter = free from restraints; liberate • UNI (ONE) Latin unique = one of a kind universal = characterizing or affecting all; present everywhere unipolar = having a single magnetic or electric pole unicorn = mythological creature, usually represented as a horse, with a single horn projecting from its forehead unanimity = state of total agreement or unity 378 300 HIGH-FREQUENCY WORD ROOTS Root Work 56 Match each word with its definition. 1. unfeigned (A) a commotion 2. unipolar (B) extending beyond the world or universe 3. unanimity (C) in a state of turmoil 4. typify (D) not made up; genuine 5. turbid (E) serving as a type 6. ultramodern (F) not viable 7. typal (G) state of total agreement 8. untenable (H) represent as an example 9. disturbance (I) extremely modern in ideas or style 10. ultramundane (J) having a single magnetic or electric pole (Answers are on page 414.) ROOT ROUNDUP 57 • URB (CITY) Latin urban = related to a city suburb = residential area outlying a city urbane = refined, sophisticated, suave urbanite = city dweller urbanologist = specialist in city life • US/UT (USE/USEFUL) Latin usage = act of using usurp = seize by force usury = lending money at exorbitant rates utilitarian = concerned with usefulness rather than beauty utile = useful • VAC/VACA/VACU (EMPTY) Latin vacate = empty of occupants vacuum = space empty of matter vacuous = empty; void; lacking intelligence; purposeless vacuity = emptiness of mind; lack of ideas vacuole = small cavity in cell cytoplasm • VEH/VECT (CARRY) Latin vehicular = relating to vehicles vehemently = vigorously; energetically vector = a course or direction vectoring = guiding by radio communication according to vectors convection = transmission 379 ESSENTIAL WORDS FOR THE GRE • VEN/VENT (COME) Latin invent = produce or contrive by ingenuity prevention = act of impeding; a hindrance conventional = customary circumvent = avoid contravene = act contrary to; to violate Root Work 57 Match each word with its definition. 1. vacuity (A) act of using 2. urbanite (B) vigorously 3. conventional (C) concerned with usefulness rather than beauty 4. usage (D) city dweller 5. vehemently (E) small cavity in cell cytoplasm 6. utilitarian (F) avoid 7. convection (G) specialist in city life 8. urbanologist (H) customary 9. circumvent (I) emptiness of mind; lack of ideas 10. vacuole (J) transmission (Answers are on page 414.) ROOT ROUNDUP 58 • VER/VERAC/VERI (TRUE) Latin veritable = real; genuine verity = truthfulness aver = to affirm; declare to be true veracity = accuracy, truthfulness verisimilitude = quality of appearing true or real • VERB (WORD) Latin verbatim = corresponding word for word verbal = associated with words proverbial = widely referred to verbiage = an excess of words; wordiness verbose = wordy; long-winded • VERS/VERT (TURN, CHANGE) Latin versatile = adaptable, all-purpose reversion = return to an earlier stage avert = turn away; prevent extrovert = person whose psychological energy is directed outward toward other people vertigo = dizziness 380 300 HIGH-FREQUENCY WORD ROOTS • VIA (WAY, ROAD) Latin via = by way of via media = middle way or course viaduct = series of spans that carry a road over another road or a valley viatical = relating to a road or traveling viaticum = traveling provisions • VICT/VINC/VANQ (CONQUER) Latin victor = one who defeats an adversary victory = defeat of an enemy; triumph evict = force out; expel invincible = impossible to overcome or defeat vanquish = conquer in battle; subjugate Root Work 58 Match each word with its definition. 1. verbose (A) an excess of words 2. aver (B) turn away; prevent 3. via media (C) relating to a road or traveling 4. verisimilitude (D) declare to be true 5. vanquish (E) adaptable 6. avert (F) wordy; long-winded 7. viatical (G) conquer in battle; subjugate 8. verbiage (H) the middle way or course 9. invincible (I) quality of appearing true or real 10. versatile (J) impossible to defeat (Answers are on page 414.) ROOT ROUNDUP 59 • VID/VIS (SEE, LOOK) Latin video = relating to televised images visible = perceptible to the eye supervisor = one who is in charge vista = a distant view or prospect visage = appearance; aspect; countenance • VIR (MAN) Latin virile = having the characteristics of an adult male virulent = very harmful; poisonous; hostile virago = woman who is noisy and scolding, or domineering virility = manly characteristic; potency virilism = male sexual characteristics in a female 381 ESSENTIAL WORDS FOR THE GRE • VIT (LIFE) Latin vital = characteristic of or relating to life vitalize = endow with life; invigorate curriculum vitae = summary of a person’s education and professional life vitamin = organic substance needed for normal growth and body activity vitality = capacity to live and grow • VIV/VIVA (ALIVE, LIVELY, ANIMATED) Latin survive = remain alive vivacious = lively conviviality = sociable; merry vivarium = enclosure where living things are raised for observation and research vivisection = the practice of cutting into or otherwise injuring living animals, especially for the purpose of scientific research • VOC/VOKE (CALL) Latin advocate = recommend; to plead for equivocal = ambiguous; misleading irrevocable = conclusive, irreversible vociferous = loud, vocal, and noisy evoke = to produce a reaction Root Work 59 Match each word with its definition. 1. virulent (A) capacity to live and grow 2. conviviality (B) appearance; countenance 3. vista (C) endow with life 4. advocate (D) having the characteristics of an adult male 5. vitalize (E) a distant view 6. virile (F) loud, vocal, and noisy 7. vociferous (G) lively 8. vitality (H) very harmful; poisonous 9. visage (I) plead for 10. vivacious (J) sociable; merry (Answers are on page 414.) ROOT ROUNDUP 60 • VOL/VOLI (WISH, CHOICE) Latin voluntary = done of one’s own free will volition = act of making a conscious choice benevolent = characterized by doing good 382 300 HIGH-FREQUENCY WORD ROOTS malevolent = showing ill will; wishing harm to others volitive = relating to the will; expressing a wish • VOLV/VOLU (ROLL, TURN) Latin evolution = changing of a thing into a more complex or better form devolve = pass on or transfer to another revolution = a turning around an axis volution = a turn or twist around a center volvulus = abnormal twisting of the intestine • VOR/VORAC (DEVOUR, GREEDY) Latin voracious = having an insatiable appetite; ravenous voracity = condition of being eager to consume great amounts of food carnivore = flesh-eating animal herbivore = animal that feeds mainly on plants omnivorous = eating both animals and plants • XEN/XENO (STRANGER, FOREIGNER) Greek xenophobe = person who is afraid of strangers or foreigners xenophile = one who is attracted to foreigners xenobiotic = foreign to living organisms xenocryst = foreign crystal in an igneous rock xenogenesis = production of children that are very different from either parent • ZO/ZOO (ANIMAL) Greek zoology = study of the structure and classification of animals zoolatry = animal worship zoogenic = produced by animals zoonosis = animal disease that can be transmitted to human beings zoophilia = affection for animals Root Work 60 Match each word with its definition. 1. volvulus (A) foreign to living organisms 2. malevolent (B) relating to the will 3. herbivore (C) animal worship 4. xenobiotic (D) a turning around an axis 5. revolution (E) abnormal twisting of the intestine 6. zoogenic (F) wishing harm to others 7. xenophile (G) ravenous 8. volitive (H) animal that feeds on plants 9. zoolatry (I) produced by animals 10. voracious (J) one who is attracted to foreigners (Answers are on page 414.) 383 ESSENTIAL WORDS FOR THE GRE ROOT ROUNDUP REVIEW 56–60 Match It Match each of the following roots to its meaning. 1. TURB ___ (A) life 2. UNI ___ (B) word 3. VEH/VECT ___ (C) one 4. VEN/VENT ___ (D) conquer 5. VERB ___ (E) devour, greedy 6. VICT/VINC/VANQ ___ (F) carry 7. VIT/VITA ___ (G) spinning, confusion, disturbance 8. VOC/VOKE ___ (H) stranger, foreigner 9. VOR/VORAC ___ (I) come 10. XEN/XENO ___ (J) call Fill-ins Fill in the blanks with the word that fits the definition. invincible vacuous virago convection ultramundane usurp urbane unfeigned viatical curriculum vitae 1. extending beyond the world or the universe ___ 2. empty; void __ 3. impossible to defeat or overcome __ 4. sophisticated; refined; suave __ 5. woman who is domineering __ 6. not made up; genuine; real __ 7. relating to a road or traveling __ 8. seize by force __ 9. summary of a person’s education and professional life __ 10. transmission __ True or False If the statement is correct, put (T) True; if it is incorrect, put (F) False. 1. Unanimity is a state of total chaos. 2. A utilitarian is concerned with aesthetics rather than useful ness. _ 3. The quality of appearing real or true is verisimilitude. 4. Xenogenesis is the production of children that are very similar to one parent. __ 5. Zoophilia is an abnormal fear of animals. _ (Answers are on page 416.) 384 385 Common Suffixes Below is a list of common suffixes with examples of how they are added to roots: able, ible capable of, subject to, prone to; worthy of, deserving of (impeccable, incorrigible, irrefutable, mutable, feasible, affable, gullible, laudable, reprehensible, culpable) ac relating to; person affected with (ammoniac, celiac, maniac, cardiac, hypochondriac) age relationship; condition; action or result; place (parentage, bondage, carnage, anchorage) al of, pertaining to; the act of (logical, ephemeral, equivocal, glacial, peripheral, polemical, prodigal, provincial, rhetorical, satirical, superficial, terrestrial, whimsical, denial, rehearsal) an, ian belonging to, related to, characteristic of, resembling, one that is (Canadian, Freudian, reptilian, civilian, antediluvian, subterranean, authoritarian, partisan, artisan) ance, ence action or process; state of being (emergence, dependence, arrogance, compliance, vigilance, exuberance, impudence, nonchalance, opulence, quiescence, reticence) ant, ent causing or performing something; state of being; one who does or undergoes (document, flagrant, ardent, benevolent, indifferent, inherent, munificent, strident, virulent, contestant, pedant) ar, ary relating to; connected to (solar, polar, jocular, arbitrary, exemplary, mercenary, centenary) ate act upon; having; characterized by (obliterate, mitigate, deprecate, emulate, debilitate, extricate, facilitate, instigate, perpetuate, truncate; placate; intimidate, repudiate, ornate, innate, articulate) cy state of being; quality (ascendancy, bankruptcy, lunacy, dependency, complacency) dom domain; rank; state of being; collective office (fiefdom, boredom, martyrdom, officialdom) eer, er, or person who does something (auctioneer, engineer, contender, director, executor, orator) ESSENTIAL WORDS FOR THE GRE ery a place for; the act of; state of; qualities of (bakery, bribery, chicanery, slavery, snobbery) escent becoming; beginning to be; characterized by (crescent, nascent, evanescent, phosphorescent) ferous producing; carrying (coniferous, vociferous, aquiferous, calciferous, carboniferous) fic making; causing (terrific, horrific, beatific, prolific, soporific, benefic, malefic) fy make; cause to become (falsify, magnify, exemplify, ratify, rectify, personify, purify, mortify) ia abnormal condition; relating to (anorexia, toxemia, septicemia, memorabilia, personalia) ial relating to; characterized by (colloquial, glacial, terrestrial, inconsequential, superficial, cordial) ic having to do with; one characterized by (cosmic, hedonistic, caustic, aesthetic, altruistic, archaic, ascetic, bombastic, cryptic, dogmatic, eclectic, ironic, soporific, sporadic, lunatic, heretic) ide group of related chemical compounds; binary compound; chemical element with properties that are similar to another (diglycer ide, monosaccharide, sodium chloride, potassium bromide, boride) il, ile pertaining to; capable of being (puerile, ductile, infantile, senile, servile, tensile, versatile) ine having the nature of; relating to; resembling; made of; chemical substance (divine, feline, marine, leonine, saturnine, opaline, crystalline, tourmaline, incarnadine, gasoline) ion, tion, ation state or condition; the result of (criterion, oblivion, limitation, adulation, affirmation, apprehension, aversion, conviction, degradation, disinclination, innovation, sanction, seclusion) ise, ize make; become like (surmise, maximize, scrutinize, vaporize, hypothesize, cauterize) ism belief; doctrine; devotion to; act of (ethnocentrism, egotism, fanaticism, criticism, witticism) ist one who does something; one who believes or adheres to; an expert (opportunist, cartoonist, ventriloquist, altruist, pacifist, nihilist, prohibitionist, linguist, geologist, psychiatrist, scientist) ite make, do; inhabitant or native of; descendant of; adherent of (ignite, Israelite, Luddite) 386 COMMON SUFFIXES itis inflammatory disease (dermatitis, phlebitis, appendicitis, tendonitis, osteoarthritis) ity, ty state of; quality (animosity, paucity, reality, uniformity, similarity, enmity, duplicity, depravity, insularity, notoriety, novelty, integrity, virility, tenacity, veracity) ive tending toward an action; belonging, quality of (argumentative, introspective, collective, comprehensive, derivative, elusive, exhaustive, furtive, inclusive) let small one; small object worn on the body (eaglet, islet, piglet, ringlet, amulet, rivulet, pamphlet) logy, ology expression; theory; science or study of (eulogy, phraseology, ideology, geology) ly like; to the extent of, recurring at specified intervals; in a specified way (miserly; daily, slowly) ment an act; state; means (entertainment, admonishment, abatement, detachment, instrument) oid resembling; relating to (android, humanoid, planetoid, asteroid, spheroid, paranoid) or a person or thing that does something; a quality or condition (inspector, progenitor, incisor) ory relating to; characterized by; a place used for (obligatory, conciliatory, cursory, observatory) ose full of; characterized by; a form of sugar (verbose, lachrymose, jocose, sucrose, dextrose) osis condition; disease (apotheosis, metamorphosis, morphosis, apoptosis, neurosis, psychosis) ous full of, characterized by (assiduous, autonomous, capricious, contentious, erroneous, fastidious, gregarious, ingenious, innocuous, nefarious, pretentious, querulous, raucous, scrupulous) tude state of (magnitude, solitude, solicitude, verisimilitude, lassitude, pulchritude, turpitude) 387 388 Posttest It’s time to put your new knowledge of words and roots to the test. If you studied carefully and did the exercises diligently, you should see a significant improvement in your score compared to your score on the Pretest. Good luck! Fill in the blank in each sentence by selecting two answer choices that fit the overall meaning of the sentence and produce completed sentences that are equivalent in meaning. Answers that are not fully correct will receive no credit. 1. The advent of immunization helped to _ the spread of many communicable diseases, but more stringent public health standards probably were the major cause of their diminution. 🄰 precipitate 🄱 forestall 🄲 enhance 🄳 reprise 🄴 prevent 🄵 augment 2. The bourgeoisie is often __ as conformist and materialistic, but they played an important role in the emergence of democracy in Europe. 🄰 abjured strongly 🄱 mocked harshly 🄲 lauded openly 🄳 lampooned 🄴 patronized widely 🄵 averred POSTTEST 3. As a scientist, Sigmund Freud believed that nothing happens __ and that all behavior is governed by laws. 🄰 capriciously 🄱 jocularly 🄲 latently 🄳 myopically 🄴 unpredictably 🄵 morbidly 4. Opponents of didacticism argue that writers should be free to write as they want, exploring human nature unfettered by the expectations of society and unafraid of the __ of conservative readers and critics. 🄰 contempt 🄱 clichés 🄲 opprobrium 🄳 chagrin 🄴 puissance 🄵 mettle Fill in the blank in each sentence below by selecting one entry from the corresponding column of choices in the way that best completes the text. 5. In his book Knowledge and Wisdom, the distinguished twentieth-century philosopher Bertrand Russell said, “Although our age far surpasses all previous ages in knowledge, there has been no _ increase in wisdom.” correlative articulate analogous prodigious imminent 389 ESSENTIAL WORDS FOR THE GRE 6. The phrase “true fact” may prompt one to _ whether a fact can be untrue. gainsay foreswear jibe query juxtapose 7. The statement “Men can run faster than women” is not true because it is an overgeneralization since some women can run faster than some men. The statement could be made valid by __ it: “Many men can run faster than many women.” substantiating rescinding sanctioning distilling qualifying Fill in all of the blanks in the sentences by selecting one entry from the corresponding column of choices in the way that best completes the text. Answers that are not fully correct will not receive any credit. 8. The English expert regards concern about slight redundancies as (i) _ and senseless (ii) _ . Blank (i) Blank (ii) meretricious metaphysics pedantic argot fractious quibbling 390 POSTTEST 9. The field of parapsychology is a (i) __ one, with some experimenters reporting relatively small but statistically significant extrasensory perception phenomena, whereas other experimenters have been unable to replicate these results and thus (ii) _ them. Blank (i) Blank (ii) nascent refute moribund corroborate controversial abjure 10. Until his death in 2004 John Mack, who was a professor of psychiatry at Harvard Medical School, had a reputation as (i) ___ investigator who believed that many scientists are reluctant to investigate reports of humans being abducted by aliens because such events are incompatible with the prevailing western materialist and (ii) _ worldview. Blank (i) Blank (ii) an iconoclastic salacious a saturnine anthropocentric a garrulous egotistical 11. Countries with strong elements of communism have existed in history, but modern communism formed in response to the rise of capitalism and industrialization, which created a new class of people living and working in (i) _ poverty of a type (ii) _ in history. Blank (i) Blank (ii) demotic tangential complaisant unprecedented abject problematic 391 ESSENTIAL WORDS FOR THE GRE 12. Literary critics are sometimes (i) __ as parasites of art, but it can be argued that they serve the (ii) _ function of distinguishing literary works that are (iii) _ from ones that are to become central components of culture. Blank (i) Blank (ii) Blank (iii) mollified indispensible implausible maligned tacit nugatory rhapsodized innocuous ephemeral 13. The (i) __ tradition in America has its roots in the Christian churches, such as the Quakers and Brethren, whose (ii) _ holds that war is a transgression of Christian principles, and that men should practice conscientious objection and refuse (iii) _ . Blank (i) Blank (ii) Blank (iii) sartorial treatise reparations platonic dogma minutia pacifist tautology conscription 14. In Africa, the European colonial powers often imposed borders (i) ___ , and so when colonies later achieved autonomy, they found themselves to be nations that were (ii) ___ and thus difficult to govern due to (iii) __ disputes. Blank (i) Blank (ii) Blank (iii) arbitrarily defunct internecine warily democratic international equably heterogeneous mundane 392 POSTTEST Read the passages below, and then answer the questions that follow them based on the information in the passages themselves and in any introductory material or notes. The correct answer may be either stated or merely suggested in the passages. “You cannot plumb the depths of the human heart, nor find what a man is thinking; how do you expect to search out God, who made all these things, and find out his mind or comprehend his thoughts?” —Apocrypha, Judith 8:14 Experience has repeatedly confirmed that well-known maxim of Bacon’s, that “a little philosophy inclineth man’s mind to atheism, but depth in philosophy bringeth men’s minds about to religion.” In every age the most comprehensive thinkers have found in the religion of their time and country something they could accept, interpreting and illustrating that religion so as to give it depth and universal application. Even the heretics and atheists, if they have had profundity, turn out after a while to be forerunners of some new orthodoxy. What they rebel against is a religion alien to their nature; they are atheists only by accident, and relatively to a convention which inwardly offends them, but they yearn mightily in their own souls after the religious acceptance of a world interpreted in their own fashion. So it appears in the end that their atheism and loud pro testation were in fact the hastier part of their thought, since what emboldened them to deny the poor world’s faith was that they were too impatient to understand it. Indeed, the enlightenment common to young wits and worm-eaten old satirists, who plume themselves on detecting the scientific ineptitude of religion—something which the blindest half see—is not nearly enlightened enough; it points to notorious fact incompatible with religious tenets literally taken, but it leaves unexplored the habits of thought from which those tenets sprang; their original meaning, and their true function. Such studies would bring the skeptic face to face with the mystery and pathos of mortal existence. They would make him understand why religion is so profoundly moving and in a sense so profoundly just. There must needs be something humane and necessary in an influence that has become the most general sanction of virtue, the chief occasion for art and philosophy, and the source, perhaps, of the best human happiness. —Reason in Religion, George Santayana 393 (5) (10) (15) (20) (25) (30) ESSENTIAL WORDS FOR THE GRE Select one answer choice for each of the following questions. 15. The phrase “the poor world’s faith” (line 16) refers to Ⓐ the moribund belief systems of an earlier stage of history Ⓑ religions practiced in developing countries Ⓒ prevailing religious orthodoxy Ⓓ unfounded spiritual beliefs Ⓔ the need of indigent people to have faith that they will have a better life in the hereafter 16. The most accurate term to apply to the word “perhaps” as it is used in line 29 is Ⓐ irony Ⓑ tongue-in-cheek Ⓒ qualification Ⓓ hyperbole Ⓔ oxymoron 17. The phrase “the scientific ineptitude of religion” as it is used in line 19 refers to the Ⓐ failure of theologians to create a systematically structured belief system Ⓑ failure of religion to provide verifiable explanations for natural phenomena Ⓒ reluctance of orthodox religions to accept the discoveries of modern science Ⓓ disinclination of religious leaders to question church dogma Ⓔ inability of religion to provide a solid rational and empirical basis for its beliefs In the following passage the author refers to articles by several scholars in two collections of papers on Mediterranean anthropology. “Mediterranean honor,” according to David Gilmore’s introduc tion, “is a ‘libidinized’ social reputation; and it is this eroticized aspect of honor—albeit unconscious or implicit—that seems to make the Mediterranean variant distinctive.” Again: “Mediter ranean . . . unity is at least partly derived from the primordial values of honor and shame, and these values are deeply tied up with sexuality and power, with masculine and gender relations.” Again: “If a gender-based honor-and-shame moral system defines a 394 (5) POSTTEST Mediterranean World, then this category emerges not simply as an example of butterfly collecting, but as a mutually intelligible frame work of moral choices by which people communicate and gain an identity both with and within the group.” That same specification is underlined in the collection’s concluding essay by Stanley Brandes. “It is this pervasive sexuality that is particularly characteristic of Mediterranean value systems, of Mediterranean codes of honor and shame. In this, the codes may be distinguished from parallel moral systems elsewhere, in Japan, for example.” In his 1977 survey of Mediterranean anthropology, John Davis claims, “There are three main forms of stratification which have been observed in the Mediterranean: bureaucracy, class, and honor. Each of them is related to the distribution of wealth, more or less directly. They are, for the purpose of analysis, ideal types, distinct elementary forms which, in substantive politics, are inter twined, mixed in varying degrees, variously important. Each is associated with an appropriate mode of political representation— again, ideal types, elementary forms, which in the hugger-mugger of actual political activity have variable importance. These are: insistence on citizen’s rights; class struggle; patronage.” Those three stratification systems are exemplified very clearly in John G. Peristiany’s essay. His fieldwork concentrated on the Greek Cypriot mountain village of Alona in the middle 1950’s. Stratifica tion by bureaucracy and power is clear when the villager has to approach a government civil servant and when “in the impersonal interaction between citizen and civil servant the only claim upon the latter’s philotimo is that of his sectional interests, and these call for the assertion of his administrative dignity, for arrogance and the marking of social distance.” Stratification by wealth and class is clear “when the returned expatriate who had achieved suc cess in a city environment wished to trade on this success as a means of achieving immediate recognition in the village . . . [and] considered . . . further that his financial success raised him above the confines of the village hierarchy.” But neither of those other stratifications is confused by the villagers with their own hierarchy of honor and shame. Peristiany concludes by comparing honor and honesty: “The punctiliousness of honor must be referred to the code of an exclusive and agonistic microsociety: that of honesty to an inclusive, egalitarian macrosociety. Duty, in the first instance, is to those with whom one shares honor. In the second, the un-Greek macrosociety, one’s duty is to all fellow citizens or, even further, 395 hugger-mugger: disorderly confusion philotimo: a Greek word meaning “love of honor” agonistic: combative (10) (15) (20) (25) (30) (35) (40) (45) (50) ESSENTIAL WORDS FOR THE GRE to all humans. . . . Honor is active. Here insecurity and the daily reevaluation of one’s standing breed constant self-assertion and even heroism. The ideals of honesty and equality breed passive conformity and are more congenial to a conception of duty wide in its application, but more accommodating in its expectation.” —The Historical Jesus, The Life of a Mediterranean Jewish Peasant, John Dominic Crossan Select one answer choice for each of the following questions. 18. According to David Gilmore, honor and shame are Ⓐ analogous values Ⓑ anachronistic moral codes found only in Mediterranean culture Ⓒ original values of Mediterranean culture Ⓓ immutable values in all cultures studied by anthropologists Ⓔ perennial truths 19. The phrase “punctiliousness of honor” as it is used in line 46 most nearly refers to Ⓐ the uniqueness of the Mediterranean variant of honor Ⓑ the subjective nature of the concept of honor Ⓒ carefulness in observing rules governing honor Ⓓ the fact that honor is a universal concept found in all cultures Ⓔ the fact that the idea of honor is inextricably linked to the idea of honesty Identify the sentence by writing its first three words and last three words on the line below. Identify the sentence in which it is asserted that in the Mediterranean the moral code governing honesty is not as rigid as the moral code governing honor. 20. _________ End of Posttest 396 (55) POSTTEST POSTTEST ANSWERS 1. B, E 2. B, D 3. A, E 4. A, C 5. correlative 6. query 7. qualifying 8. pedantic/quibbling 9. controversial/corroborate 10. an iconoclastic/anthropocentric 11. abject/unprecedented 12. maligned/indispensable/ephemeral 13. pacifist/dogma/conscription 14. arbitrarily/heterogeneous/internecine 15. C 16. C 17. E 18. C 19. C 20. “The ideals of . . . in its expectation.” (lines 53–55) YOUR POSTTEST SCORE 1–2 CORRECT ANSWERS: VERY POOR 3–5 CORRECT ANSWERS: POOR 6–9 CORRECT ANSWERS: BELOW AVERAGE 10–13 CORRECT ANSWERS: AVERAGE 14–16 CORRECT ANSWERS: GOOD 17–18 CORRECT ANSWERS: VERY GOOD 19–20 CORRECT ANSWERS: EXCELLENT 397 Answer Keys ESSENTIAL WORDS FOR THE GRE UNIT 1 Matching 1. G 2. C 3. J 4. B 5. F 6. A 7. I 8. E 9. H 10. D Fill-ins 1. abdicated 2. aberrations 3. abeyance 4. abstemious 5. abate 6. abstinence 7. abject 8. abjured 9. abscission 10. absconded Sense or Nonsense 1. S 2. S 3. N 4. N 5. S UNIT 2 Matching 1. B 2. F 3. A 4. I 5. D 6. J 7. E 8. C 9. G 10. H Fill-ins 1. affected 2. affinity 3. accretion 4. aesthetic 5. admonished 6. accrued 7. abysmal 8. adamant 9. adulterated 10. adjunct Sense or Nonsense 1. S 2. N 3. S 4. N 5. N UNIT 3 Matching 1. D 2. G 3. A 4. J 5. F 6. H 7. B 8. C 9. E 10. I Fill-ins 1. allure 2. alacrity 3. alleviate 4. aggrandize 5. alchemy 6. ambiguous 7. aggregate 8. alloys 9. allay 10. amalgamate Sense or Nonsense 1. N 2. S 3. N 4. N 5. S UNIT 4 Matching 1. C 2. I 3. G 4. A 5. J 6. E 7. F 8. B 9. H 10. D Fill-ins 1. ambivalent 2. anarchy 3. amulet 4. ambrosia 5. analgesic 6. analogy 7. ameliorate 8. amenable 9. anachronism 10. amenities Sense or Nonsense 1. N 2. S 3. N 4. S 5. S UNIT 5 Matching 1. G 2. I 3. C 4. B 5. E 6. A 7. H 8. F 9. J 10. D Fill-ins 1. antecedents 2. antipathy 3. apathy 4. anomalous 5. appease 6. antediluvian 7. apothegms 8. anodyne 9. apogee 10. apex Sense or Nonsense 1. N 2. S 3. N 4. N 5. S 398 ANSWER KEYS UNIT 6 Matching 1. E 2. G 3. J 4. H 5. B 6. A 7. D 8. I 9. C 10. F Fill-ins 1. apropos 2. apprised 3. arduous 4. ardor 5. archeology 6. appropriated 7. arabesque 8. appellation 9. apposite 10. approbation Sense or Nonsense 1. S 2. N 3. S 4. S 5. N UNIT 7 Matching 1. H 2. E 3. C 4. A 5. J 6. G 7. B 8. F 9. D 10. I Fill-ins 1. artless 2. asperity 3. ascetic 4. astringent 5. artifact 6. arrest 7. assuage 8. assiduously 9. argot 10. aspersions Sense or Nonsense 1. N 2. N 3. S 4. S 5. S UNIT 8 Matching 1. H 2. B 3. J 4. E 5. A 6. G 7. C 8. I 9. D 10. F Fill-ins 1. autonomous 2. austere 3. atavism 4. avuncular 5. avocation 6. asylum 7. audacious 8. avarice 9. avers 10. attenuate Sense or Nonsense 1. N 2. S 3. S 4. S 5. N UNIT 9 Matching 1. H 2. C 3. F 4. A 5. I 6. E 7. J 8. D 9. G 10. B Fill-ins 1. bawdy 2. bard 3. axiomatic 4. belie 5. behemoths 6. banal 7. bedizen 8. beatification 9. bacchanalian 10. banter Sense or Nonsense 1. N 2. S 3. S 4. N 5. S UNIT 10 Matching 1. C 2. I 3. A 4. E 5. G 6. J 7. H 8. B 9. D 10. F Fill-ins 1. blasé 2. bifurcation 3. beneficence 4. broached 5. bovine 6. blandishments 7. brazen 8. boorish 9. bombastic 10. bolstered Sense or Nonsense 1. N 2. N 3. S 4. S 5. N UNIT 11 Matching 1. F 2. H 3. A 4. B 5. I 6. C 7. G 8. J 9. D 10. E Fill-ins 1. burgeoning 2. buttress 3. canard 4. bucolic 5. cadge 6. burnish 7. calumny 8. cacophonous 9. callous 10. canon Sense or Nonsense 1. S 2. N 3. N 4. S 5. S 399 ESSENTIAL WORDS FOR THE GRE UNIT 12 Matching 1. D 2. I 3. B 4. J 5. A 6. E 7. G 8. C 9. F 10. H Fill-ins 1. caste 2. cant 3. captious 4. capricious 5. cardinal 6. carnal 7. cartography 8. castigated 9. carping 10. cantankerous Sense or Nonsense 1. N 2. S 3. N 4. S 5. N UNIT 13 Matching 1. E 2. H 3. B 4. J 5. A 6. I 7. C 8. F 9. D 10. G Fill-ins 1. championed 2. causal 3. categorical 4. catalysts 5. cataclysm 6. centripetal 7. centrifugal 8. celestial 9. caucus 10. caustic Sense or Nonsense 1. S 2. S 3. N 4. S 5. N UNIT 14 Matching 1. E 2. C 3. H 4. A 5. D 6. J 7. F 8. I 9. B 10. G Fill-ins 1. chivalric 2. coagulates 3. clique 4. clamor 5. circuitous 6. churlish 7. chicanery 8. chastened 9. clairvoyant 10. cloistered Sense or Nonsense 1. S 2. S 3. N 4. N 5. S UNIT 15 Matching 1. B 2. H 3. A 4. J 5. C 6. E 7. I 8. F 9. D 10. G Fill-ins 1. coda 2. complaisant 3. coalesced 4. commensurate 5. codification 6. compendium 7. complement 8. complacent 9. cognizant 10. collage Sense or Nonsense 1. S 2. S 3. S 4. N 5. N UNIT 16 Matching 1. B 2. I 3. A 4. G 5. D 6. J 7. E 8. C 9. H 10. F Fill-ins 1. concomitant 2. compliant 3. conjugal 4. compunction 5. congenial 6. confounded 7. concocted 8. concave 9. condoned 10. conciliatory Sense or Nonsense 1. N 2. S 3. S 4. S 5. S UNIT 17 Matching 1. G 2. E 3. B 4. J 5. F 6. C 7. A 8. I 9. D 10. H Fill-ins 1. contentious 2. contends 3. conscripted 4. conundrums 5. continence 6. connoisseur 7. contumacious 8. contiguous 9. consecrated 10. contrite Sense or Nonsense 1. N 2. S 3. S 4. N 5. S 400 ANSWER KEYS UNIT 18 Matching 1. F 2. D 3. B 4. J 5. A 6. G 7. H 8. E 9. C 10. I Fill-ins 1. cosmology 2. conventions 3. convoluted 4. convex 5. convivial 6. copious 7. converges 8. covert 9. coquette 10. cornucopia Sense or Nonsense 1. N 2. S 3. N 4. S 5. N UNIT 19 Matching 1. D 2. H 3. A 4. E 5. I 6. C 7. B 8. G 9. J 10. F Fill-ins 1. dearth 2. craven 3. cozens 4. daunting 5. credence 6. decorum 7. covets 8. credo 9. debauchery 10. defaming Sense or Nonsense 1. S 2. N 3. S 4. S 5. S UNIT 20 Matching 1. F 2. B 3. J 4. G 5. D 6. I 7. A 8. E 9. C 10. H Fill-ins 1. denouement 2. demographic 3. denizens 4. defunct 5. delineated 6. demurred 7. demotic 8. default 9. deference 10. denigrated Sense or Nonsense 1. N 2. S 3. N 4. S 5. S UNIT 21 Matching 1. J 2. F 3. C 4. H 5. D 6. A 7. E 8. I 9. B 10. G Fill-ins 1. dichotomy 2. diaphanous 3. desiccated 4. deterrent 5. derided 6. derivative 7. desultory 8. desuetude 9. diatribe 10. detraction Sense or Nonsense 1. N 2. S 3. S 4. S 5. S UNIT 22 Matching 1. F 2. B 3. D 4. H 5. A 6. J 7. C 8. G 9. I 10. E Fill-ins 1. disabuse 2. discrepancy 3. dirge 4. digressions 5. discredited 6. discerning 7. diffidence 8. diffuse 9. discomfited 10. discordant Sense or Nonsense 1. S 2. S 3. N 4. S 5. S UNIT 23 Matching 1. I 2. F 3. H 4. B 5. A 6. J 7. D 8. C 9. G 10. E Fill-ins 1. disinterested 2. disseminated 3. disjointed 4. disparate 5. discrete 6. dismissed 7. discretion 8. disingenuous 9. disparaged 10. dissembled Sense or Nonsense 1. N 2. S 3. N 4. N 5. N 401 ESSENTIAL WORDS FOR THE GRE UNIT 24 Matching 1. I 2. C 3. J 4. E 5. B 6. G 7. F 8. D 9. H 10. A Fill-ins 1. distill 2. dissolution 3. distended 4. dissonance 5. divested 6. dissidents 7. diverged 8. distrait 9. divulge 10. doctrinaire Sense or Nonsense 1. N 2. N 3. S 4. S 5. S UNIT 25 Matching 1. C 2. B 3. F 4. H 5. J 6. A 7. G 8. E 9. D 10. I Fill-ins 1. eclectic 2. documented 3. duped 4. doggerel 5. effete 6. dogmatic 7. effervescent 8. dormant 9. dross 10. ebullient Sense or Nonsense 1. N 2. S 3. S 4. S 5. N UNIT 26 Matching 1. I 2. E 3. D 4. F 5. B 6. H 7. C 8. A 9. G 10. J Fill-ins 1. elixirs 2. egoism 3. elicit 4. emaciated 5. egotistical 6. embellish 7. efficacious 8. effrontery 9. Elysian 10. elegy Sense or Nonsense 1. N 2. N 3. N 4. S 5. S UNIT 27 Matching 1. C 2. E 3. I 4. J 5. B 6. H 7. F 8. A 9. D 10. G Fill-ins 1. emulated 2. enhance 3. empirical 4. enervating 5. enunciate 6. endemic 7. engendered 8. encomiums 9. emollient 10. entomologist Sense or Nonsense 1. N 2. S 3. S 4. S 5. S UNIT 28 Matching 1. H 2. B 3. D 4. J 5. E 6. F 7. I 8. C 9. G 10. A Fill-ins 1. esoteric 2. ephemeral 3. equanimity 4. errant 5. erudition 6. epistemology 7. equivocate 8. essayed 9. equable 10. estimable Sense or Nonsense 1. N 2. S 3. N 4. S 5. N UNIT 29 Matching 1. C 2. A 3. G 4. I 5. B 6. F 7. D 8. H 9. J 10. E Fill-ins 1. etymology 2. euphoria 3. evinces 4. etiology 5. eugenics 6. evocative 7. eulogy 8. ethnocentrism 9. euthanasia 10. euphemisms Sense or Nonsense 1. S 2. N 3. S 4. N 5. S 402 ANSWER KEYS UNIT 30 Matching 1. D 2. G 3. B 4. J 5. F 6. A 7. C 8. I 9. E 10. H Fill-ins 1. exigency 2. exorcises 3. exacerbating 4. exacting 5. expatriate 6. exhorted 7. expatiate 8. execrable 9. existential 10. exculpated Sense or Nonsense 1. N 2. S 3. S 4. N 5. N UNIT 31 Matching 1. I 2. F 3. H 4. D 5. A 6. C 7. J 8. B 9. G 10. E Fill-ins 1. facetious 2. explication 3. extraneous 4. expository 5. extant 6. extemporaneous 7. extirpate 8. extrinsic 9. expiate 10. extrapolating Sense or Nonsense 1. N 2. N 3. S 4. S 5. S UNIT 32 Matching 1. C 2. G 3. E 4. F 5. A 6. I 7. H 8. D 9. B 10. J Fill-ins 1. fauna 2. fawning 3. feral 4. factotum 5. felicitous 6. fallacious 7. fervor 8. fatuous 9. facilitate 10. fallow Sense or Nonsense 1. N 2. N 3. S 4. N 5. S UNIT 33 Matching 1. I 2. F 3. C 4. E 5. A 6. H 7. B 8. D 9. J 10. G Fill-ins 1. filibuster 2. fetid 3. fidelity 4. fiat 5. flag 6. fledgling 7. flora 8. fettered 9. finesse 10. fissures Sense or Nonsense 1. N 2. S 3. S 4. S 5. S UNIT 34 Matching 1. I 2. A 3. F 4. D 5. J 6. C 7. H 8. B 9. G 10. E Fill-ins 1. flourishes 2. foundered 3. formidable 4. forswear 5. flux 6. forestall 7. foment 8. flouts 9. forbearance 10. florid Sense or Nonsense 1. N 2. N 3. S 4. S 5. S UNIT 35 Matching 1. I 2. E 3. A 4. F 5. C 6. J 7. G 8. B 9. H 10. D Fill-ins 1. futile 2. fusion 3. fracas 4. frugality 5. froward 6. frieze 7. fulsome 8. fractious 9. fulminated 10. fresco Sense or Nonsense 1. N 2. N 3. S 4. S 5. S 403 ESSENTIAL WORDS FOR THE GRE UNIT 36 Matching 1. C 2. F 3. D 4. I 5. H 6. J 7. G 8. B 9. E 10. A Fill-ins 1. garrulous 2. gambol 3. geniality 4. glib 5. gerrymandering 6. goaded 7. gossamer 8. gauche 9. gouged 10. gainsay Sense or Nonsense 1. N 2. N 3. N 4. S 5. N UNIT 37 Matching 1. F 2. J 3. H 4. A 5. G 6. D 7. B 8. E 9. I 10. C Fill-ins 1. grouse 2. gregarious 3. hallowed 4. gullible 5. grandiloquent 6. gustatory 7. guises 8. guileless 9. halcyon 10. harangue Sense or Nonsense 1. N 2. S 3. S 4. S 5. N UNIT 38 Matching 1. B 2. I 3. E 4. A 5. J 6. F 7. C 8. H 9. G 10. D Fill-ins 1. homily 2. hermetic 3. heterodox 4. herbivorous 5. harrowing 6. homogeneous 7. hirsute 8. homeostatic 9. hieroglyphics 10. histrionic Sense or Nonsense 1. N 2. S 3. N 4. N 5. S UNIT 39 Matching 1. C 2. G 3. B 4. D 5. I 6. A 7. F 8. J 9. H 10. E Fill-ins 1. hyperbole 2. impecunious 3. idolatry 4. imbroglio 5. impeded 6. immutable 7. igneous 8. iconoclastic 9. impassive 10. ideological Sense or Nonsense 1. S 2. S 3. N 4. S 5. S UNIT 40 Matching 1. F 2. J 3. H 4. B 5. I 6. A 7. E 8. D 9. C 10. G Fill-ins 1. implausible 2. implacable 3. implicit 4. implosions 5. impinging 6. impermeable 7. impervious 8. imperturbable 9. impute 10. imprecations Sense or Nonsense 1. S 2. S 3. S 4. N 5. S UNIT 41 Matching 1. F 2. H 3. A 4. J 5. C 6. B 7. D 8. I 9. G 10. E Fill-ins 1. inconsequential 2. indolent 3. incursions 4. incorporates 5. indeterminate 6. incongruous 7. inchoate 8. inadvertently 9. incarnate 10. indigent Sense or Nonsense 1. S 2. S 3. S 4. N 5. S 404 ANSWER KEYS UNIT 42 Matching 1. F 2. C 3. I 4. G 5. J 6. D 7. A 8. E 9. B 10. H Fill-ins 1. insensible 2. insularity 3. insipid 4. ingenuous 5. inert 6. insinuating 7. insouciance 8. inherent 9. ineluctable 10. innocuous Sense or Nonsense 1. N 2. N 3. N 4. N 5. S UNIT 43 Matching 1. B 2. D 3. I 4. E 5. J 6. G 7. F 8. C 9. A 10. H Fill-ins 1. intransigence 2. intimate 3. insuperable 4. interdicting 5. intangible 6. interpolated 7. internecine 8. introspection 9. interregnum 10. intractable Sense or Nonsense 1. N 2. N 3. S 4. S 5. S UNIT 44 Matching 1. D 2. B 3. E 4. A 5. J 6. F 7. H 8. I 9. G 10. C Fill-ins 1. invective 2. irascible 3. invidious 4. itinerant 5. inveigh 6. inundated 7. inveterate 8. inured 9. irresolute 10. inveigle Sense or Nonsense 1. N 2. S 3. S 4. N 5. S UNIT 45 Matching 1. H 2. E 3. A 4. C 5. J 6. G 7. B 8. F 9. D 10. I Fill-ins 1. juggernaut 2. junta 3. jocose 4. itinerary 5. labile 6. juxtaposed 7. jaundiced 8. kudos 9. laconic 10. jibe Sense or Nonsense 1. N 2. N 3. S 4. S 5. S UNIT 46 Matching 1. B 2. D 3. J 4. G 5. C 6. E 7. A 8. H 9. F 10. I Fill-ins 1. levee 2. lambasted 3. lassitude 4. lauded 5. liberal 6. latent 7. lethargic 8. lascivious 9. levity 10. libertine Sense or Nonsense 1. N 2. S 3. S 4. S 5. S UNIT 47 Matching 1. B 2. F 3. D 4. A 5. J 6. G 7. H 8. C 9. I 10. E Fill-ins 1. libido 2. literati 3. logs 4. loquacious 5. limpid 6. litany 7. litigation 8. limning 9. linguistics 10. Lilliputian Sense or Nonsense 1. N 2. N 3. S 4. S 5. S 405 ESSENTIAL WORDS FOR THE GRE UNIT 48 Matching 1. G 2. B 3. A 4. J 5. I 6. D 7. F 8. C 9. H 10. E Fill-ins 1. lucid 2. Machiavellian 3. luminous 4. maligned 5. lucre 6. magnanimity 7. lustrous 8. maelstrom 9. malingering 10. machinations Sense or Nonsense 1. N 2. S 3. N 4. S 5. S UNIT 49 Matching 1. H 2. A 3. E 4. B 5. J 6. D 7. I 8. G 9. C 10. F Fill-ins 1. metaphysical 2. metamorphosed 3. mendicant 4. megalomania 5. malleable 6. mavericks 7. meretricious 8. mesmerized 9. mendacious 10. menagerie Sense or Nonsense 1. N 2. N 3. S 4. N 5. S UNIT 50 Matching 1. H 2. C 3. A 4. G 5. F 6. D 7. I 8. B 9. J 10. E Fill-ins 1. mettle 2. meteorological 3. misanthropic 4. militates 5. meticulous 6. minatory 7. microcosm 8. mettlesome 9. minuscule 10. minutia Sense or Nonsense 1. S 2. S 3. N 4. S 5. N UNIT 51 Matching 1. E 2. B 3. F 4. I 5. G 6. C 7. J 8. A 9. D 10. H Fill-ins 1. misogynist 2. mollify 3. monolithic 4. mnemonic 5. miscellany 6. morose 7. mitigate 8. modicum 9. miscreant 10. motley Sense or Nonsense 1. N 2. S 3. S 4. S 5. S UNIT 52 Matching 1. D 2. J 3. C 4. A 5. F 6. B 7. I 8. G 9. E 10. H Fill-ins 1. negated 2. mundane 3. nonplussed 4. nostrums 5. nexus 6. neologisms 7. multifarious 8. nostalgia 9. neophyte 10. necromancy Sense or Nonsense 1. S 2. S 3. N 4. S 5. N UNIT 53 Matching 1. G 2. C 3. B 4. F 5. J 6. E 7. H 8. D 9. A 10. I Fill-ins 1. obsequious 2. obviated 3. olfactory 4. officious 5. occult 6. occludes 7. obdurate 8. odyssey 9. nugatory 10. obsequies Sense or Nonsense 1. N 2. S 3. S 4. N 5. S 406 ANSWER KEYS UNIT 54 Matching 1. E 2. C 3. J 4. A 5. H 6. F 7. B 8. G 9. D 10. I Fill-ins 1. paeans 2. oligarchy 3. oscillating 4. ornithologists 5. onerous 6. paleontologists 7. ostentatious 8. overweening 9. opprobrium 10. onomatopoeia Sense or Nonsense 1. N 2. N 3. S 4. S 5. S UNIT 55 Matching 1. H 2. J 3. G 4. B 5. D 6. A 7. I 8. C 9. F 10. E Fill-ins 1. pathology 2. pellucid 3. pedantic 4. partisan 5. patois 6. pallid 7. penchant 8. paragons 9. panegyric 10. paucity Sense or Nonsense 1. N 2. N 3. S 4. S 5. S UNIT 56 Matching 1. F 2. I 3. G 4. D 5. J 6. B 7. C 8. A 9. E 10. H Fill-ins 1. perigee 2. permeable 3. pervasive 4. perennial 5. perfidious 6. penury 7. perfunctory 8. peregrinations 9. perturbed 10. peremptory Sense or Nonsense 1. N 2. N 3. S 4. S 5. S UNIT 57 Matching 1. C 2. B 3. F 4. D 5. J 6. H 7. G 8. I 9. A 10. E Fill-ins 1. placid 2. piety 3. plaintive 4. piqued 5. physiognomy 6. petulant 7. phoenix 8. placated 9. piquant 10. phlegmatic Sense or Nonsense 1. N 2. S 3. S 4. N 5. S UNIT 58 Matching 1. G 2. A 3. F 4. D 5. I 6. E 7. H 8. B 9. J 10. C Fill-ins 1. plethora 2. porous 3. platonic 4. plumbed 5. platitudes 6. poseur 7. plutocracy 8. pragmatic 9. plasticity 10. plummet Sense or Nonsense 1. S 2. N 3. S 4. S 5. S UNIT 59 Matching 1. B 2. E 3. C 4. I 5. G 6. H 7. J 8. F 9. A 10. D Fill-ins 1. precursor 2. precepts 3. prehensile 4. precarious 5. prattle 6. preamble 7. precipitated 8. preempted 9. precipitate 10. prated Sense or Nonsense 1. S 2. S 3. N 4. S 5. S 407 ESSENTIAL WORDS FOR THE GRE UNIT 60 Matching 1. E 2. G 3. A 4. C 5. I 6. D 7. J 8. F 9. B 10. H Fill-ins 1. preternatural 2. presage 3. premonition 4. primordial 5. prevaricating 6. pristine 7. problematic 8. prodigal 9. presumptuous 10. probity Sense or Nonsense 1. N 2. N 3. N 4. S 5. S UNIT 61 Matching 1. E 2. C 3. J 4. G 5. I 6. A 7. D 8. F 9. B 10. H Fill-ins 1. proscribes 2. proliferating 3. punctilious 4. puissant 5. propriety 6. profound 7. propensity 8. propitiated 9. prohibitive 10. provident Sense or Nonsense 1. S 2. N 3. S 4. S 5. S UNIT 62 Matching 1. E 2. C 3. G 4. A 5. I 6. F 7. J 8. H 9. B 10. D Fill-ins 1. quagmire 2. purported 3. pungent 4. quailed 5. qualms 6. query 7. qualified 8. pusillanimous 9. quibble 10. quiescent Sense or Nonsense 1. S 2. S 3. N 4. S 5. S UNIT 63 Matching 1. J 2. E 3. H 4. D 5. B 6. C 7. A 8. I 9. F 10. G Fill-ins 1. recalcitrant 2. ramifications 3. rationale 4. raiment 5. rails 6. quorum 7. raconteur 8. rarefied 9. rebus 10. recant Sense or Nonsense 1. N 2. N 3. S 4. S 5. S UNIT 64 Matching 1. F 2. C 3. I 4. G 5. A 6. E 7. B 8. J 9. D 10. H Fill-ins 1. regaled 2. refractory 3. relegated 4. reneged 5. refulgent 6. recluse 7. recondite 8. redoubtable 9. remonstrated 10. refute Sense or Nonsense 1. N 2. S 3. S 4. S 5. N UNIT 65 Matching 1. G 2. B 3. I 4. A 5. F 6. E 7. J 8. C 9. D 10. H Fill-ins 1. reproached 2. resolved 3. repudiated 4. rescinded 5. resolution 6. reparations 7. reticent 8. repine 9. reprobate 10. reprise Sense or Nonsense 1. S 2. S 3. S 4. S 5. N 408 ANSWER KEYS UNIT 66 Matching 1. F 2. H 3. D 4. J 5. G 6. A 7. C 8. B 9. I 10. E Fill-ins 1. revere 2. riposte 3. rue 4. ruse 5. sage 6. salacious 7. salubrious 8. salutary 9. rubric 10. rococo Sense or Nonsense 1. S 2. N 3. S 4. S 5. S UNIT 67 Matching 1. D 2. G 3. A 4. F 5. J 6. H 7. B 8. I 9. C 10. E Fill-ins 1. savored 2. satiate 3. saturnine 4. saturated 5. sanctions 6. sartorial 7. satyr 8. schematic 9. secrete 10. sardonic Sense or Nonsense 1. N 2. S 3. S 4. S 5. S UNIT 68 Matching 1. C 2. G 3. J 4. E 5. A 6. F 7. I 8. B 9. D 10. H Fill-ins 1. servile 2. sedition 3. sedulous 4. seismic 5. shards 6. sextant 7. sidereal 8. sentient 9. sensual 10. sensuous Sense or Nonsense 1. S 2. N 3. S 4. S 5. S UNIT 69 Matching 1. I 2. C 3. A 4. E 5. J 6. F 7. B 8. D 9. H 10. G Fill-ins 1. sobriety 2. singular 3. solicitous 4. skeptic 5. similes 6. sinecure 7. sinuous 8. soliloquy 9. sodden 10. simian Sense or Nonsense 1. S 2. S 3. S 4. S 5. N UNIT 70 Matching 1. D 2. H 3. J 4. A 5. F 6. B 7. E 8. C 9. G 10. I Fill-ins 1. specious 2. spendthrift 3. squalor 4. staccato 5. spectrum 6. sporadic 7. solvent 8. soporific 9. somatic 10. sordid Sense or Nonsense 1. S 2. S 3. N 4. S 5. S UNIT 71 Matching 1. I 2. H 3. J 4. A 5. D 6. F 7. B 8. G 9. C 10. E Fill-ins 1. stipulate 2. stolid 3. striated 4. stratified 5. strident 6. stanch 7. stentorian 8. stint 9. strictures 10. stigma Sense or Nonsense 1. S 2. S 3. S 4. N 5. S 409 ESSENTIAL WORDS FOR THE GRE UNIT 72 Matching 1. E 2. I 3. C 4. A 5. G 6. F 7. B 8. J 9. H 10. D Fill-ins 1. stupefied 2. substantive 3. strutted 4. subsumes 5. stultifying 6. subversive 7. stygian 8. subpoenaed 9. subside 10. substantiate Sense or Nonsense 1. S 2. N 3. S 4. N 5. S UNIT 73 Matching 1. G 2. J 3. B 4. E 5. C 6. I 7. A 8. D 9. H 10. F Fill-ins 1. superseded 2. sundry 3. supplicants 4. succor 5. supine 6. syllogism 7. supposition 8. suffrage 9. supplanted 10. suppliant Sense or Nonsense 1. S 2. N 3. S 4. N 5. S UNIT 74 Matching 1. F 2. B 3. H 4. I 5. E 6. G 7. A 8. J 9. D 10. C Fill-ins 1. tacit 2. tangential 3. sylvan 4. taciturn 5. theocracy 6. talismans 7. tenet 8. taxonomy 9. tenuous 10. tautologies Sense or Nonsense 1. N 2. S 3. S 4. S 5. S UNIT 75 Matching 1. D 2. G 3. A 4. F 5. H 6. J 7. B 8. I 9. C 10. E Fill-ins 1. timbre 2. thespians 3. touts 4. tractable 5. tirade 6. torpor 7. tortuous 8. torque 9. tome 10. toady Sense or Nonsense 1. N 2. N 3. N 4. N 5. N UNIT 76 Matching 1. H 2. D 3. B 4. G 5. E 6. A 7. J 8. C 9. F 10. I Fill-ins 1. travails 2. truculence 3. transient 4. treatise 5. tremulous 6. translucent 7. transgressed 8. tryst 9. travesty 10. trepidation Sense or Nonsense 1. S 2. S 3. S 4. N 5. S UNIT 77 Matching 1. C 2. G 3. I 4. A 5. E 6. B 7. D 8. J 9. H 10. F Fill-ins 1. unfeigned 2. untenable 3. usury 4. turgid 5. tumid 6. untoward 7. undulating 8. uncanny 9. tutelary 10. turbid Sense or Nonsense 1. S 2. N 3. S 4. N 5. S 410 ANSWER KEYS UNIT 78 Matching 1. C 2. J 3. H 4. F 5. B 6. D 7. A 8. I 9. G 10. E Fill-ins 1. venerated 2. vapid 3. veracious 4. venal 5. valedictory 6. vendetta 7. vaunted 8. vacuous 9. variegated 10. vacillating Sense or Nonsense 1. N 2. S 3. S 4. S 5. N UNIT 79 Matching 1. F 2. I 3. J 4. E 5. C 6. G 7. D 8. A 9. B 10. H Fill-ins 1. vituperative 2. vindictive 3. vertigo 4. virtuoso 5. visage 6. verbose 7. viscous 8. viable 9. vitiated 10. vexations Sense or Nonsense 1. S 2. S 3. S 4. N 5. S UNIT 80 Matching 1. B 2. G 3. A 4. E 5. J 6. H 7. F 8. C 9. I 10. D Fill-ins 1. zealot 2. wary 3. welter 4. warranted 5. vogue 6. volatile 7. vivisections 8. whimsical 9. wistful 10. vortex Sense or Nonsense 1. S 2. S 3. S 4. S 5. S ROOT ROUNDUP ROOT WORK 1 1. E 2. D 3. I 4. J 5. H 6. B 7. A 8. C 9. G 10. F ROOT WORK 2 1. D 2. F 3. J 4. I 5. A 6. H 7. C 8. G 9. E 10. B ROOT WORK 3 1. F 2. G 3. J 4. C 5. B 6. E 7. H 8. I 9. A 10. D ROOT WORK 4 1. E 2. F 3. J 4. A 5. I 6. D 7. B 8. H 9. C 10. G ROOT WORK 5 1. I 2. C 3. F 4. J 5. D 6. H 7. B 8. E 9. A 10. G ROOT WORK 6 1. J 2. D 3. I 4. A 5. B 6. E 7. C 8. H 9. F 10. G ROOT WORK 7 1. G 2. I 3. F 4. B 5. J 6. H 7. D 8. C 9. E 10. A ROOT WORK 8 1. E 2. J 3. F 4. G 5. B 6. A 7. D 8. I 9. H 10. C 411 ESSENTIAL WORDS FOR THE GRE ROOT WORK 9 1. G 2. J 3. I 4. F 5. A 6. C 7. H 8. D 9. E 10. B ROOT WORK 10 1. G 2. I 3. E 4. J 5. H 6. B 7. F 8. C 9. A 10. D ROOT WORK 11 1. J 2. H 3. E 4. I 5. C 6. B 7. D 8. G 9. A 10. F ROOT WORK 12 1. E 2. I 3. F 4. D 5. B 6. H 7. C 8. J 9. G 10. A ROOT WORK 13 1. D 2. E 3. A 4. G 5. B 6. I 7. H 8. C 9. J 10. F ROOT WORK 14 1. F 2. J 3. G 4. I 5. H 6. B 7. E 8. D 9. C 10. A ROOT WORK 15 1. G 2. I 3. J 4. H 5. D 6. F 7. B 8. E 9. C 10. A ROOT WORK 16 1. D 2. I 3. G 4. J 5. B 6. E 7. F 8. A 9. H 10. C ROOT WORK 17 1. G 2. D 3. F 4. I 5. J 6. H 7. C 8. B 9. A 10. E ROOT WORK 18 1. F 2. I 3. G 4. H 5. J 6. A 7. C 8. E 9. B 10. D ROOT WORK 19 1. F 2. D 3. I 4. J 5. A 6. B 7. H 8. E 9. C 10. G ROOT WORK 20 1. J 2. G 3. H 4. I 5. E 6. A 7. D 8. F 9. C 10. B ROOT WORK 21 1. H 2. E 3. A 4. G 5. D 6. I 7. F 8. B 9. C 10. J ROOT WORK 22 1. C 2. D 3. A 4. G 5. F 6. B 7. I 8. E 9. J 10. H ROOT WORK 23 1. G 2. D 3. I 4. C 5. B 6. A 7. J 8. E 9. H 10. F ROOT WORK 24 1. I 2. F 3. D 4. G 5. J 6. B 7. A 8. C 9. E 10. H ROOT WORK 25 1. F 2. I 3. H 4. A 5. J 6. B 7. E 8. G 9. D 10. C ROOT WORK 26 1. C 2. H 3. D 4. B 5. G 6. J 7. A 8. F 9. E 10. I 412 ANSWER KEYS ROOT WORK 27 1. F 2. E 3. H 4. D 5. G 6. A 7. I 8. J 9. C 10. B ROOT WORK 28 1. D 2. J 3. E 4. G 5. B 6. I 7. A 8. F 9. H 10. C ROOT WORK 29 1. G 2. J 3. A 4. I 5. C 6. D 7. F 8. H 9. B 10. E ROOT WORK 30 1. I 2. F 3. E 4. A 5. G 6. J 7. B 8. H 9. C 10. D ROOT WORK 31 1. H 2. E 3. A 4. G 5. B 6. J 7. F 8. C 9. I 10. D ROOT WORK 32 1. J 2. D 3. E 4. G 5. F 6. I 7. H 8. C 9. B 10. A ROOT WORK 33 1. G 2. H 3. J 4. E 5. D 6. B 7. F 8. C 9. A 10. I ROOT WORK 34 1. G 2. I 3. F 4. E 5. A 6. J 7. D 8. C 9. B 10. H ROOT WORK 35 1. G 2. J 3. I 4. F 5. H 6. A 7. E 8. D 9. B 10. C ROOT WORK 36 1. J 2. I 3. F 4. H 5. G 6. D 7. E 8. B 9. A 10. C ROOT WORK 37 1. H 2. G 3. I 4. F 5. J 6. C 7. D 8. B 9. E 10. A ROOT WORK 38 1. D 2. J 3. H 4. G 5. I 6. A 7. C 8. E 9. F 10. B ROOT WORK 39 1. J 2. I 3. G 4. H 5. E 6. F 7. D 8. B 9. C 10. A ROOT WORK 40 1. I 2. A 3. H 4. E 5. B 6. J 7. D 8. C 9. F 10. G ROOT WORK 41 1. F 2. D 3. G 4. B 5. C 6. I 7. J 8. A 9. H 10. E ROOT WORK 42 1. E 2. F 3. G 4. H 5. J 6. C 7. D 8. B 9. A 10. I ROOT WORK 43 1. D 2. E 3. H 4. J 5. B 6. I 7. A 8. C 9. F 10. G 413 ESSENTIAL WORDS FOR THE GRE ROOT WORK 44 1. F 2. A 3. J 4. H 5. E 6. C 7. G 8. I 9. B 10. D ROOT WORK 45 1. G 2. H 3. F 4. D 5. B 6. C 7. A 8. I 9. J 10. E ROOT WORK 46 1. I 2. J 3. F 4. A 5. H 6. G 7. D 8. B 9. C 10. E ROOT WORK 47 1. D 2. H 3. I 4. B 5. C 6. A 7. E 8. F 9. G 10. J ROOT WORK 48 1. E 2. G 3. F 4. J 5. H 6. I 7. A 8. C 9. B 10. D ROOT WORK 49 1. E 2. I 3. J 4. C 5. B 6. F 7. A 8. H 9. D 10. G ROOT WORK 50 1. I 2. C 3. J 4. F 5. H 6. B 7. E 8. G 9. D 10. A ROOT WORK 51 1. J 2. C 3. F 4. H 5. E 6. B 7. D 8. A 9. G 10. I ROOT WORK 52 1. D 2. C 3. H 4. B 5. J 6. I 7. A 8. G 9. E 10. F ROOT WORK 53 1. E 2. G 3. F 4. H 5. I 6. C 7. J 8. D 9. B 10. A ROOT WORK 54 1. D 2. J 3. H 4. G 5. F 6. I 7. B 8. C 9. E 10. A ROOT WORK 55 1. F 2. E 3. G 4. J 5. D 6. A 7. I 8. B 9. H 10. C ROOT WORK 56 1. D 2. J 3. G 4. H 5. C 6. I 7. E 8. F 9. A 10. B ROOT WORK 57 1. I 2. D 3. H 4. A 5. B 6. C 7. J 8. G 9. F 10. E ROOT WORK 58 1. F 2. D 3. H 4. I 5. G 6. B 7. C 8. A 9. J 10. E ROOT WORK 59 1. H 2. J 3. E 4. I 5. C 6. D 7. F 8. A 9. B 10. G ROOT WORK 60 1. E 2. F 3. H 4. A 5. D 6. I 7. J 8. B 9. C 10. G 414 ANSWER KEYS ROOT ROUNDUP REVIEW ROOT ROUNDUP REVIEW 1–5 Match It 1. F 2. A 3. I 4. G 5. J 6. D 7. E 8. B 9. H 10. C Fill-ins 1. pedagogue 2. perambulate 3. antiquate 4. aliment 5. inanimate 6. anarchy 7. unapt 8. annuity 9. agrarian 10. acumen True or False 1. F 2. F 3. T 4. T 5. F ROOT ROUNDUP REVIEW 6–10 Match It 1. H 2. C 3. I 4. G 5. J 6. A 7. B 8. E 9. D 10. F Fill-ins 1. symbiotic 2. auger 3. subaqueous 4. archaic 5. incessant 6. centrifugal 7. benediction 8. catapult 9. centripetal 10. captious True or False 1. F 2. F 3. F 4. T 5. F ROOT ROUNDUP REVIEW 11–15 Match It 1. G 2. I 3. B 4. F 5. D 6. A 7. C 8. J 9. E 10. H Fill-ins 1. civil 2. cosmopolitan 3. coherent 4. cite 5. criterion 6. corpulent 7. miscreate 8. deprecate 9. synclinal 10. cyclothymia True or False 1. F 2. T 3. T 4. T 5. F ROOT ROUNDUP REVIEW 16–20 Match It 1. G 2. J 3. H 4. A 5. I 6. C 7. B 8. D 9. F 10. E Fill-ins 1. pandemic 2. duple 3. obdurate 4. donatio mortis causa 5. inequity 6. dichotomy 7. epigeal 8. ductile 9. dermatitis 10. malediction True or False 1. F 2. F 3. T 4. F 5. T ROOT ROUNDUP REVIEW 21–25 Match It 1. I 2. F 3. G 4. A 5. J 6. D 7. H 8. B 9. E 10. C Fill-ins 1. gravitas 2. perfidious 3. fractious 4. genre 5. perfunctory 6. euphony 7. egress 8. fin de siècle 9. extraneous 10. soporific True or False 1. F 2. T 3. F 4. T 5. F ROOT ROUNDUP REVIEW 26–30 Match It 1. I 2. F 3. J 4. C 5. G 6. D 7. B 8. E 9. A 10. H Fill-ins 1. lexical 2. hemoptysis 3. lector 4. hyperbole 5. literati 6. internecine 7. jurisprudence 8. misogynist 9. impecunious 10. hominoid True or False 1. T 2. F 3. T 4. F 5. F 415 ESSENTIAL WORDS FOR THE GRE 416 ROOT ROUNDUP REVIEW 31–35 Match It 1. G 2. D 3. H 4. A 5. J 6. B 7. I 8. C 9. F 10. E Fill-ins 1. missive 2. nascent 3. diminution 4. malfeasance 5. misogynist 6. matrilineal 7. morphology 8. mandamus 9. metanoia 10. macrocosm True or False 1. F 2. F 3. T 4. F 5. T ROOT ROUNDUP REVIEW 36–40 Match It 1. F 2. A 3. E 4. J 5. G 6. B 7. I 8. D 9. C 10. H Fill-ins 1. orthopraxy 2. ignominious 3. paramount 4. osteoma 5. omniscient 6. navarch 7. noctivagant 8. oligophagous 9. apartheid 10. pernicious True or False 1. F 2. T 3. T 4. F 5. F ROOT ROUNDUP REVIEW 41–45 Match It 1. C 2. G 3. A 4. I 5. H 6. E 7. B 8. J 9. F 10. D Fill-ins 1. prehensile 2. depict 3. agoraphobia 4. pedagogue 5. protogenic 6. percutaneous 7. postdiluvian 8. placate 9. cacophony 10. portage True or False 1. F 2. T 3. T 4. T 5. T ROOT ROUNDUP REVIEW 46–50 Match It 1. I 2. E 3. G 4. B 5. C 6. A 7. J 8. F 9. H 10. D Fill-ins 1. ridibund 2. sophistry 3. retrospect 4. secant 5. somniloquence 6. pseudodox 7. recant 8. rogation 9. punctilio 10. sequatious True or False 1. F 2. T 3. T 4. F 5. F ROOT ROUNDUP REVIEW 51–55 Match It 1. J 2. F 3. G 4. A 5. H 6. B 7. D 8. E 9. C 10. I Fill-ins 1. theogamy 2. contention 3. thermoduric 4. status quo 5. tenacious 6. synchronous 7. torque 8. supernal 9. stricture 10. toponym True or False 1. T 2. F 3. F 4. F 5. T ROOT ROUNDUP REVIEW 56–60 Match It 1. G 2. C 3. F 4. I 5. B 6. D 7. A 8. J 9. E 10. H Fill-ins 1. ultramundane 2. vacuous 3. invincible 4. urbane 5. virago 6. unfeigned 7. viatical 8. usurp 9. curriculum vitae 10. convection True or False 1. F 2. F 3. T 4. F 5. F 417 Index adaptive, 28 affixes, 98, 297 aficionados, 64 Age of Enlightenment, 53 ahimsa, 81 alchemy, 12, 34, 35, 37, 154 ambrosia, 38, 41 anomie, 268 anthropic principle, 135 Aristotelian, 111, 223 Augustans, 44 Bacchanalia, 40, 56, 58, 59 Bacchus, 40, 56 behavioral psychology, 262, 263 Behaviorists, 185 Big Bang, 81 Bionics, 117, 309 blog, 110 bourgeois, 49, 67 bourgeoisie, 35, 43, 67, 202, 388 capitalism, 13, 30, 35, 49, 55, 66, 79, 108, 109, 121, 141, 159, 391 Carbon-14 dating, 47, 137, 189, 257 castes, 68, 69, 254 catastrophism, 56 chivalry, 74, 75, 293 civil law, 65, 77, 79 classicism, 44, 71 clockwork universe, 36, 290 Cold War, 98, 262 common law, 77, 159 Confucian, 40, 208 conscription, 83, 86, 290, 392 continental drift, 217 Cost-benefit analysis, 68 Crusades, 74, 95, 280 cubist, 78 de facto, 27 Deism, 53, 321 design, 271 determinists, 133 deviant, 27, 253 dialectic, 121 Dionysos, 40, 56, 242 dogma, 110, 120, 157, 202, 392, 394 Doppler effect, 72 dualism, 223 dualist, 99, 227 Ego, 68, 118, 302, 303, 326, 328 Elysian fields, 114, 115 Epidemiologists, 122 ethical relativism, 84, 117 Existentialism, 129, 194 expressionism, 102 First Amendment, 78, 180 fission, 144, 145 free market, 16, 30 French Revolution, 71, 83, 101, 246 Freudians, 118 functionalism, 175 Fundamentalists, 196, 263 fusion, 144, 145 Gandhi, Mahatma, 81 genetic engineering, 84, 329 geophysicist, 31 Gothic, 32, 73, 89 Great Depression, 28, 64, 219, 293 greenhouse effect, 128, 208 guerrilla war, 87 Gulag, 106, 107 guru, 54 halcyon, 150, 151, 152 herbivorous, 28, 152, 154, 155, 336, 337 heuristic, 117 Hippocrates, 267, 268 hominids, 81, 186 hypothesized, 28 iconoclast, 16, 18, 156, 158, 172, 294 iconoclastic, 12, 25, 155, 156, 157, 284, 391 icons, 156 Id, 118 Idealism, 55, 233, 234 ESSENTIAL WORDS FOR THE GRE Idealist, 234, 249 igneous, 25, 31, 156, 157, 158, 383 Impressionist, 120, 121, 192 jihad, 50 Kant, Immanuel, 234 laissez-faire, 121 Leda, 150 libertarian, 71, 136, 140, 341 lingua franca, 98, 205, 259 lobby, 35, 36 Machiavelli, 14, 183, 184, 185 Mahayana Buddhist, 36 Marxist, 49, 66, 108 monism, 227 monopolistic, 31 monotheism, 220 mystical, 111, 214, 301, 352, 353, 356 Napoleonic Code, 77, 79 nectar, 38 neuroses, 118 Nietzsche, Friedrich, 84, 192, 210 nihilist, 247, 386 nirvana, 67, 68 one-dimensional, 125, 126 Ottoman Empire, 44 Panathenaic, 144 Parthenon, 144, 145 Philistines, 78, 180 planned economy, 30 plate tectonics, 217 Platonic, 101, 111, 213, 215, 216, 392 Platonism, 105, 111 Pluralists, 154 polytheism, 220 Pope, Alexander, 44, 163 posttest, 388–396 pretest, 10–18 Primatology, 164, 202 protagonist, 43, 148, 189, 260, 278 Protestant work ethic, 40 psychosomatic disorder, 169, 249 Pythagorean, 111 quantum theory, 49, 88, 94 reductionistic, 159 Reformation, 43, 156 Renaissance, 88, 144 repression, 118 Rococo, 64, 65, 70, 239, 240, 241 Romance languages, 98 Romantic movement, 71 Rosetta Stone, 153 Rousseau, Henri, 87 sacred cow, 237 secularism, 263 Secularization, 263 semantics, 2, 4, 122 shaman, 129 Social Darwinism, 39, 40 Socratic method, 114 Sophists, 141, 205 status quo, 165, 166, 371, 372, 377 Stoic, 52, 111 study plans, 20–23 suffixes, 98, 297–299, 385–387 Superego, 118 supernova, 73, 182, 183 Swift, Jonathan, 44, 163 tabula rasa, 156 theory of evolution, 53 theory of relativity, 3, 117, 170 trompe l’oeil, 78 tutelary spirits, 271 uncertainty principle, 94 uniformitarianism, 56 utilitarian, 68, 379, 380, 384 War of 1812, 83 Warren Court, 193 Webster, Daniel, 83 welfare state, 43 Whitman, Walt, 52, 94, 163, 195 word roots, 297–384 yoga, 54 yogis, 54 Zeus, 40, 149, 150 418 Notes Barron’s GRE® with Optional CD-ROM, 22nd Ed. Sharon Weiner Green, M.A., and Ira K. Wolf, Ph.D. ISBN 978-1-4380-0915-5, Paperback, $26.99, Can$33.99 Barron’s Essential Words for the GRE®, 4th Ed. Philip Geer, Ed.M. ISBN 978-1-4380-0749-6, Paperback, $14.99, Can$17.99 Pass Key to the GRE®, 9th Ed. Sharon Weiner Green, M.A., and Ira K. Wolf, Ph.D. ISBN 978-1-4380-0912-4, Paperback, $9.99, Can$12.50 Barron’s 6 GRE® Practice Tests, 2nd Ed. David Freeling and Vince Kotchian ISBN 978-1-4380-0629-1, Paperback, $14.99, Can$17.99 Barron’s GRE® Psychology, 7th Ed. Laura Freberg, Ph.D., Edward L. Palmer, Ph.D., and Sharon L. Thompson-Schill, Ph.D. ISBN 978-1-4380-0573-7, Paperback, $19.99, Can$23.99 Barron’s GRE® Math Workbook, 3rd Ed. Blair Madore, Ph.D., and David Freeling ISBN 978-1-4380-0632-1, Paperback, $14.99, Can$17.99 Barron’s GRE® Verbal Workbook, 2nd Ed. Philip Geer, Ed.M. 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2117	https://criticalpoint.co.za/wp-content/uploads/2016/10/DOH-enteral-nutrition-guidelines.pdf	1 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS A long and Healthy Life for All South Africans NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS A long and Healthy Life for All South Africans 2 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS Copyright - 2016 national Department of Health, South Africa This publication is intended to support nutrition activities and may be freely quoted, reproduced and distributed, provided that the source is acknowledged. Distribution for remuneration is not permitted. Permission from the copyright holder is required for changes to the format of this publication. Prepared and obtainable free of charge from: Directorate: Nutrition national Department of Health Private Bag X828 Pretoria 0001 Tel: (012) 395 9621 Fax: 086 632 8484 Department of Health – 2016 www.health.gov.za 3 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS TABLE OF CONTENTS Acknowledgements................................................................................................................................... 5 Abbreviations............................................................................................................................................. 6 1. Scope and purpose....................................................................................................................... 8 2. Background and motivations....................................................................................................... 8 3. Nutritional assessment................................................................................................................. 8 3.1. Anthropometric methods................................................................................................................. 8 3.2. Biochemical methods...................................................................................................................... 9 3.3. Clinical methods.............................................................................................................................. 9 3.4. Diet methods if applicable............................................................................................................... 9 3.5. Medication....................................................................................................................................... 9 4. Nutritional intervention................................................................................................................. 9 4.1 Medical nutrition therapy goals....................................................................................................... 9 4.1.1 Indications for the use of enteral tube feeding................................................................................ 10 4.1.2 Access routes................................................................................................................................. 11 4.1.3 Enteral product formulations........................................................................................................... 12 4.1.4 Initiation of enteral feeding and risk of re-feeding syndrome.......................................................... 13 4.1.5 Infusion methods............................................................................................................................ 17 4.1.6 Safety.............................................................................................................................................. 17 4.2 Dietary and nutritional recommendations.................................................................................. 19 4.2.1 Macronutrients................................................................................................................................ 19 4.2.2 Indirect calorimetry.......................................................................................................................... 19 4.2.3 Micronutrients.................................................................................................................................. 20 4.2.4 Pharmaconutrition........................................................................................................................... 20 4.3 Disease related conditions........................................................................................................... 22 4.4 Stopping a tube feed..................................................................................................................... 27 5. Monitoring...................................................................................................................................... 27 5.1. Nutritional assessment and monitoring and follow-up..................................................................... 27 5.2. Complications of enteral feeding ..................................................................................................... 28 5.3. Medicine nutrient interaction........................................................................................................... 29 6. Home based enteral nutrition...................................................................................................... 32 6.1. Patient education............................................................................................................................ 32 6.2. Tube care........................................................................................................................................ 32 6.3. Formulations................................................................................................................................... 32 6.3.1. Home based.................................................................................................................................... 32 6.3.2. Commercial..................................................................................................................................... 32 7. References..................................................................................................................................... 33 8. Annexures Annexure 1: Access routes............................................................................................................. 35 Annexure 2: Immunonutrition recommendations............................................................................ 35 Annexure 3: Checklist for the intensive care unit (ICU) setting....................................................... 36 4 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS List of tables Table1: Adjustment of desirable body weight for amputees............................................................ 8 Table 2: Indications and contra-indications for enteral feeding....................................................... 9 Table 3: Enteral product formulation definitions.............................................................................. 11 Table 4: Refeeding regime for patients at risk of refeeding syndrome............................................ 16 Table 5: Comparison of feeding methods ....................................................................................... 17 Table 6: Enteral nutrition related safety measures.......................................................................... 18 Table 7: Macronutrient requirements of general critically ill patients............................................... 19 Table 8: Interpreting respiratory quotient (RQ) value of indirect calorimetry................................... 19 Table 9: Suggested enteral vitamin supplementation in the critically ill........................................... 20 Table 10: Suggested trace element supplementation in the critically ill.......................................... 20 Table 11: Indications, contra-indications and recommended dosages of specialised nutrients........................................................................................................................... 21 Table 12: Disease specific requirements......................................................................................... 22 Table 13: Monitoring the patient receiving enteral nutrition............................................................. 27 Table 14: Risk factors for feeding intolerance.................................................................................. 27 Table 15: Complications related to enteral nutrition......................................................................... 27 Table 16: Medicines affecting gastrointestinal (GI) function that are used in the critically ill........... 29 Table 17: Special considerations for medicine administration via enteric tube................................ 29 Table 18: Specific medicines – enteral nutrition interactions........................................................... 31 List of figures Figure 1: Route of administration algorithm..................................................................................... 10 Figure 2: Access route algorithm..................................................................................................... 11 Figure 3: Diagram illustrating the method of choosing an enteral feed........................................... 12 Figure 4: Diagram illustrating the initiation of enteral feeding.......................................................... 13 Figure 5: Gastric test feed guideline................................................................................................ 14 Figure 6: Small bowel test feed guideline........................................................................................ 15 Figure 7 Prevention/management of refeeding syndrome.............................................................. 16 5 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS Acknowledgements The National Adult Enteral Nutrition Practice Guidelines for Public Health Establishments will assist in providing standardised and quality nutrition services to patients attending public health institutions. The provision of adequate and appropriate nutrition in hospitals is imperative in building and maintaining individual’s nutritional status and thus decreasing hospital length of stay. The Department of Health would like to express its sincere gratitude to all national and provincial departments for their contribution to the development of these guidelines. Special thanks are extended to the core clinical working group for their technical input, commitment and dedication, which contributed to the development of this document. The following members were instrumental in this process: Engela Francis – Dietitian: Steve Biko Academic Hospital Vanessa Kotze – Lecturer: University of Pretoria Caida MacDougall – Lecturer: Sefako Makgatho Medical University Nolene Naicker – Assistant Director: national Department of Health Andiswa Ngqaka – Independent Consultant Hanlie Pohl – Dietitian: Independent Practice Frances Van Schalkwyk – Dietitian: Kalafong Hospital Representatives from the University of Pretoria, the Directorate: Affordable Medicines and the National Essential Medicines List Committee, the Critical Care Society of Southern Africa, the South African Society for Parenteral and Enteral Nutrition in South Africa and provincial nutrition units contributed to the development of these guidelines and we thank them for their time and technical inputs. MP MATSOSO DIRECTOR GENERAL: HEALTH 6 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS ACRONYMS ABW actual body weight ALI acute lung injury ARDS acute respiratory distress syndrome ARF acute renal failure ATI abdominal trauma index BEE basal energy expenditure BCAA branched chain amino acid BMI body mass index BMR basal metabolic rate BW body weight CHO carbohydrate COAD chronic obstructive airway disease COPD chronic obstructive pulmonary disease CRF chronic renal failure CRP c - reactive protein CRRT chronic renal replacement therapy CVI cerebrovascular incident CVP central venous pressure DM diabetes mellitus DRI dietary reference intake Ecg electrocardiogram EN enteral nutrition FR French GFR glomerular filtration rate GI gastrointestinal GIT gastrointestinal tract GL glycaemic load GRV gastric residual volume HACCP hazards analysis and critical control points HD haemodialysis HOB head of bed IBD inflammatory bowel disease IBW ideal body weight ICU intensive care unit IHD ischemic heart disease ISS injury severity score IU international unit IV intravenous IVF intravenous fluid KCL potassium chloride Kcal kilocalories LFT liver function test MCT medium chain triglyceride MODS multiple organ dysfunction syndrome MOF multi-organ failure MSG monosodium-l glutamate MVO2 myocardial oxygen consumption MUFA mono-unsaturated fatty acids Ω-3 FA omega-3 fatty acid NCJ needle catheter jejunostomy NGT nasogastric tube NJT nasojejunal tube NPE non-protein energy NPO nil per os PEG percutaneous endoscopic gastrostomy PH potential hydrogen PMV prolonged mechanical ventilation (>21 days for at least six hours per day) PN parenteral nutrition PO post-operative REE resting energy expenditure RNA ribonucleic acid RRT renal replacement therapy RV residual volume RQ respiratory quotient SCFAs short chain fatty acids SBS short bowel syndrome SOFA sequential organ failure assessment TB tuberculosis 7 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS TBSA total body surface area TE total energy TEN total enteral nutrition TIBC total iron binding capacity TPN total parenteral nutrition U&E urea and electrolytes VAP ventilator associated pneumonia Vit B Co vitamin B complex VO2 oxygen uptake VCO2 carbon dioxide production 8 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 1 Scope and purpose The goal of this document is to provide guidelines and suggest practical strategies for the implementation of a successful enteral feeding regime in adult patients at public health facilities. Enteral nutrition is defined as the administration of nutrients via the gastro-intestinal tract orally or by tube, catheter or stoma distal to the oral cavity. 1 The method used for enteral feeding will be determined by the patient’s adaptability and ability to tolerate the method of feeding as well as by the site of the damage to the gastro-intestinal tract. This is but one example of the application of medical nutrition therapy to improve patient health outcome, improve quality of life and reduce patient care costs. The prescription, composition, preparation and the method of administration requires special care. The content of the guideline document focuses on assessing nutritional status, indications for the use of enteral feeding, contra-indications, nutritional requirements, formulating the enteral feeding regimen, handling of complications and monitoring and evaluation of enteral therapy. 2 Background and motivations Historically, starvation was an accepted approach in the treatment of ill patients. However, currently it is said that providing adequate and appropriate fluid and nutrients to sick patients is a basic duty.2 Adequate nutrition is a vital part of successful treatment, and should be sold as such. 3 Malnutrition is said to occur in about 15-70 per cent of hospital patients. In addition, malnutrition is often undiagnosed in about 70 per cent of patients admitted to hospital. It is of further concern that 70-80 per cent of admitted malnourished patients are discharged from hospital without receiving any nutritional support. A patient’s disease state, coupled with the length of hospital stay further worsens malnutrition and is often associated with death. Weight loss during hospitalisation is mainly due to reduced food intake, malabsorption, modified metabolism, increased nutrient and energy requirements, lack of early nutritional assessment and treatment, medicine-nutrient interactions, mechanical reasons and the actual disease condition. Thus, nutritional status screening, assessment and monitoring is essential in reducing morbidity and mortality amongst hospitalised patients. 3 A multidisciplinary approach in providing nutritional support is critical in ensuring effective assessment and treatment interventions. Active nutritional support programmes implemented by a nutritional support team can prevent malnutrition and weight loss. This support team consists of multi-disciplinary healthcare workers i.e. medical doctors, professional nurses and dietitians. The team to provide nutritional support may utilise different technical approaches, such as oral, enteral and parenteral nutrition, in a complementary fashion to one another. A registered dietitian with a competency in nutritional support is qualified to assume responsibility for the assessment, planning, implementing and monitoring of enteral, parenteral and specialised oral therapies associated with patient care. 4 3 Nutritional assessment Nutritional status assessment is the first step to ensure the successful outcome of enteral feeding. A comprehensive nutritional assessment consists of a combination of the following methods: 3.1 Anthropometric methods It is the measurement of the physical dimension and gross composition of the body. The methods include: • height/ recumbent length/ knee height/ arm span/ demi span/ ulna length - actual body weight or ideal body weight • mid-upper arm circumference • skinfold thickness Note: Ideal body weight must be adjusted downward to compensate for missing limbs or paralysis as outlined in Table 1 below. TABLE 1: Adjustment of desirable body weight for amputees 5 Body segment Average % of body weight Lower arm and hand 2.3 Trunk with extremities 50.0 Entire arm 5.0 Hand 0.7 Entire lower leg 16.0 Below knee including foot 5.9 Foot 1.5 Estimated weight = 100 - % amputation x IBW for original height 100 In case of presence of oedema and/or ascites: Use IBW for calculations. 9 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 3.2 Biochemical methods It is measuring a nutrient or its metabolite in blood, faeces or urine or measuring a variety of other components in blood and other tissues that have a relationship to nutritional status. The methods include: • serum protein • liver function tests • calcium, magnesium, phosphate test • haemoglobin • serum ferritin (iron levels) • serum lipid and lipid profile levels • general electrolytes • urea and creatinine • Total proteins • glomerular filtration rate (GFR) • c-reactive protein (CRP) • glucose • albumin (should not be used as an independent criterion) 3.3 Clinical methods Include the detection of signs and symptoms that indicate malnutrition. Methods include: • medical history • physical examination 3.4 Dietary methods if applicable 6 Generally involve surveys measuring the quantity of the individual foods and beverages consumed during the course of one to several days or assessing the pattern of food use during the previous several months. These can provide data on intake of nutrients or specific classes of foods. Methods include: • 24-hour recall • food record or diary • food frequency questionnaire • diet history 3.5 Medication Find relevant medicines listed under Annexure 1 of the document. Pay special attention to any medicines that may affect the gastrointestinal tract. 4 NUTRITIONAL INTERVENTION 4.1 MEDICAL NUTRITION THERAPY GOALS 4.1.1 Indications and contra-indications for the use of enteral tube feeding are indicated in Table 2. TABLE 2: Indications and contra-indications for enteral feeding 7 INDICATIONS CONTRA-INDICATIONS Decreased food intake: • Inability to consume sufficient food (< 80% of TE) • Neurological disorders, e.g. coma, meningitis, cardiovascular incident/episode (CVI) • Psychiatric conditions, e.g. severe depression, Anorexia Nervosa • Senility • Cachexia • Dysphagia (difficulty in swallowing) • Severe existing malnutrition • Anorexia • Adequate oral intake (>80% of TE) Mechanical GIT disorders: • Facial, mandible or dental injuries • Head, neck or mouth trauma or malignancy • Obstruction of the esophagus or upper duodenum • Severe stomatitis or mucosal damage (Stevens Johnson Syndrome or mucositis) • Delayed gastric emptying or short bowel syndrome (SBS) • Radiation to head and neck • Inability to swallow, coma • Incomplete bowel obstruction • Entero-cutaneous fistula with output (<500 ml per day) • Complete intestinal obstruction (except if able to feed distal to the obstruction) • Intestinal perforation • High output fistula (> 500 ml per day) 10 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS Gastrointestinal dysfunction: • Inflammatory bowel disease (IBD) including ulcerative colities (UC) and Crohn’s Disease • SBS • Pancreatitis • Abdominal radiation therapy • Some intestinal surgery • Impaired ability to digest and absorb nutrients/ malabsorption syndrome • Sprue, enteritis (e.g. radiation therapy, chemotherapy) • Non-specific diarrhoea associated with malnutrition • Biliary tract disease • Chronic vomiting and infectious intestinal diseases • Gastroparesis • Upper GIT haemorrhage • Intractable vomiting and diarrhoea • Fresh uncertain anastomoses • Severe acute pancreatitis • Risk for aspiration (except if jejunostomy tube is in place for feeding) • Shock, haemodynamically unstable • Paralytic ileus Hypermetabolic conditions: • Severe trauma • Septicaemia • Major surgery • Neurologic disorders e.g.multiple sclerosis • from TPN to normal food • Major burns • Ventilated patients • Cancer therapy and bone marrow transplantation • Adequate food intake • Not haemodynamically stable Adapted from Zaloga G.P. Timing and route of nutritional support. In: Zaloga G.P editors. Nutrition in Critical Care. St. Louis, M.O: Mosby; 1994;p. 267-330 Post-operative ileus is not a contraindication. Feeding directly into the small intestine with semi-elemental short-peptide formulas is recommended N.B the above guidelines are relative and decisions should be based on individual presentations. 4.1.2 Enteral nutrition route of administration algorithm FIGURE 1: Route of administration algorithm 8: Patient Assessment Candidate for Nutrition Support Contraindications to Enteral Nutrition? Enteral Nutrition Parenteral Nutrition Intestinal obstruction Ileus Peritonitis Bowel ischemia Intractable vomiting and Diarrhoea Short-term No central access Anticipated long-term need for concentrated PN solution Central PN Return of GI function Peripheral PN Oral intake indicated Advance to oral feeding GI function Normal Compromised Standard formula Specialized formula Adequate Inadequate Supplemention with PN Consider oral feeding Adequate Progress to enteral feeding Advance to oral feeding Feeding tolarence Yes No Yes No Yes No Short-term Nasogastric Nasoduodenal Nasojejunal Long-term Gastrostomy Jejunostomy Adapted from: Ukleja A, Freeman KL, Gilbert K, Kochevar M, Kraft MD, Russel MK, Shuster MH, and Task Force on Standards for Nutrition Support: Adult hospitalized patients, and the American Society for Parenteral and Enteral Nutrition Board of Directors. Nutrition Clinical Practice 2010; 25: 403-414 Indication for enteral feed 11 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 4.1.2 Access routes Figure 2: Access route algorithm 9 Adapted from: European society of enteral and parenteral nutrition, 1998. ESPEN conference report. Nice, France: 16-19 Note: For further information on access routes refer to Annexure 1 4.1.3 Enteral Product Formulations 10 Enteral products formulations are indicated in Table 3. TABLE 3: Enteral product formulation definitions Enteral product formulations Definition Standard formulas Enteral formulas of which the composition reflects the reference values for macronutrients and micronutrients for a healthy individual. Most standard formulations contain whole or complete proteins, lipids and fibre. (Non-fibre compositions also exist) Disease-specific formulas These formulas include those with adapted macro- and micronutrient compositions to meet the needs of a specific disease (including digestive and metabolic disorders) Immune modulating formulas (immunonutrition) These formulas contain substrates to modulate immune functions Low energy formulas These formulas provide less than 0.9kcal/ml Normal energy formulas These formulas provide 0.9 – 1.2kcal/ml High energy formulas These formulas provide more than 1.2kcal/ml High protein formulas These formulas contain 20% or more of total energy from protein Whole/complete protein formulas (polymeric) These formulas contain intact proteins Peptide-based formulas (oligomeric) These formulas contain protein predominantly in peptide form Free amino acid formulas (monomeric) These formulas contain single amino acids as the protein source High lipid formulas These formulas contain more than 40% of total energy from lipids High mono-unsaturated fatty acids (MUFA) formulas These formulas contain 20% or more of total energy from MUFA 12 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS FIGURE 3: Diagram illustrating the method of choosing an enteral feed 11 13 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS FIGURE 4: Diagram illustrating the initiation of enteral feeding 12,13 4.1.4 Initiation of enteral feeding and risk of refeeding syndrome 14 • Enteral delivery method, initiation and advancement of EN regimens should be based on patient condition, age, enteral route (gastric vs. small bowel), nutrition requirements, and GI status. • Full strength, isotonic formulas for initial feeding regimen should be chosen. The mnemonic “CAN WE FEED” can assist in planning an enteral feeding regime and initiate early enteral feeding. The following checklist could be used in an ICU setting (Refer to Annexure 3). 14 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS FIGURE 5: Gastric test feed guideline 15,16,17 The following test guideline was developed by the enteral nutrition clinical working group and was found to be practically accepted for use at public health facilities. 15 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS FIGURE 6: Small bowel test feed guideline 18 Small bowel feeding is associated with a reduction in pneumonia in critically ill patients when compared to gastric feeding. Thus, if feasible, it is recommended for use in patients with a high risk for intolerance to EN, risk for regurgitation, aspiration, patients that repeatedly demonstrate high gastric residuals. The following test guideline was adopted and was found to be practically accepted for use at public health facilities. NOTE: Should the target rate not be reached in 48 hours then supplemental TPN should be considered. 16 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS FIGURE 7: Prevention / management of refeeding syndrome 19 Adapted from NICE and BAPEN guidelines. Mehanna H, Nankivell PC, Moledina J, Travis J. Head and Neck Oncology 2009; 1(4) TABLE 4: Refeeding regime for patients at risk of refeeding syndrome 20 Day Calorie intake (all feeding routes) Supplements Day 1 -10 kCal/kg/day -For extreme cases (BMI < 14kg/m2 or no food > 15 days): 5 kCal/kg/day -Carbohydrate: 50-60% -Fat: 30-40% -Protein: 15-20% -Prophylactic supplement -PO4 2-: 0.5-0.8mmol/kg/day -K+: 1-3mmol/kg/day -Mg2+: 0.3-0.4 mmol/kg/day -Na+: <1 mmol/kg/day (restricted) -IV fluids: restricted, maintain “zero” balance -IV thiamine + vitamin B complex 30 minutes prior to feeding Day 2-4 -Increase by 5 kCal/kg/day If low or no tolerance stop or keep minimal feeding regime -Check all biochemistry and correct any abnormality -Thiamine and vitamin B complex orally or IV till day 3 -Monitoring as required Day 5-7 -20-30 kCal/kg/day -Check electrolytes, renal and liver functions and minerals -Fluid: maintain zero balance -Consider iron supplement from day 7 Day 8-10 -30 kCal/kg/day or increase to full requirement -Monitor as required 4.1.5 Infusion methods There are three types of feeding options that can be chosen from. These are bolus feeding, intermittent feeding and continuous feeding. See Table 5 for a comparison of the feeding methods. 17 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS TABLE 5: Comparison of feeding methods 21,22 BOLUS FEEDING INTERMITTENT FEEDING CONTINUOUS FEEDING Description The intermittent, rapid feeding of large volumes of formula divided into six/eight regular daytime feedings administered 3 or 4 hourly Continuous controlled delivery of a feed, by either gravity or pump-assisted method with a rest period of about 4-6 hours daily Continuous controlled delivery of a feed over 24 hours without interruption, either by gravity or pump-assisted method Volume/rate ±250-350ml per feed, depending on requirements and tolerance Usually between 50-125ml/hr Usually between 50-125ml/hr Special considerations/precautions Give special attention to minimising risk of bacterial contamination Consider alternative infusion method if very high quantities of feeds are required Monitor for vomiting and aspiration in prevention of ventilator associated pneumonia (VAP) and other complications Monitor for vomiting and aspiration in prevention of VAP and other complications Monitor for vomiting and aspiration in prevention of VAP and other complications Note: Patients` heads should be elevated to at least 30 to 45 degrees during feeding to prevent micro-aspiration 4.1.6 Safety Serious harm and death may occur due to adverse events occurring throughout the process of ordering, administering and monitoring. These include: • enteral misconnections • metabolic abnormalities • broncho-pulmonary aspiration • mechanical tube complications • enteral access device misplacements • GI intolerance related to formula contamination Promoting patient safety in enterally fed patients is dependent on continuous surveillance and recognition of potential areas of harm and medical errors. Table 6 provides information on safety measures that should be observed in relation to enteral nutrition. 18 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS TABLE: 6: Enteral nutrition related safety measures 14 AREA SAFETY MEASURES Enteral nutrition formulas • Clear and accurate labeling of formulas which include: - patient demographics - formula type - delivery site/device - administration method and rate - time and date the formula is prepared and hung - “Not for IV use” label to decrease risk of enteral misconnections • Prevent contamination in preparation/storage/administration: - implementation of quality control measures and corrective actions - critical points should be documented – use HACCP - controlled environment - aseptic techniques are essential - hand hygiene is very important – wash and use alcohol rub - if formulas are not used after preparation - refrigerate - unused open formulas must be discarded • Hang time: - 8 hours: Sterile formula in OPEN system - 12 hours: Sterile formula in OPEN system at home - 24 hours: Sterile formula in CLOSED system - reconstituted formula should not be exposed to room temperature for longer than 4 hours • Stability of the products: - important to maintain product integrity - conditions of storage and composition of container can influence stability, which includes: o light o temperature o oxygen exposure - degree of fatty acid oxidation increases with storage time - vitamin losses found in formulas stored > 9 months Administration sets • Flushes - recommended: Water - water to use: o tap water/bottled water - healthy, immune competent patients o purified water -acute or chronically ill patients o saline - flush feeding tubes with: o 30 ml of water every 4 hours during continuous feeding or o before and after intermittent feeding • Change administration sets every 24 hours • Enteral misconnections – how to resolve problem: - colour-coded enteral set tips - luer adaptors - training staff to connect lines - trace line back to their origins to ensure safe insertion - label feeds: “WARNING for enteral use ONLY” Enteral feeding pumps • Periodic calibration is needed to ensure: - proper function - proper delivery within 10% of prescribed amount of formula • Calibration of pumps are done according to the manufacturing company’s requirements Patient • Positioning: - head-of-bed at 30 - 45º to prevent aspiration and pneumonia but contraindicated when: o hemodynamically unstable o unstable spine o prone positioning o certain medical procedures - strategies to increase use of an elevated HOB position: o medical orders o staff education o reverse trendelenberg (head up) position • Maintenance considerations of feeding devices - useful methods to confirm tube placement: o determine external length of tube since time of placement o observing for negative pressure when attempting to withdraw fluid from the tube o observing unexpected changes in residual volume o measuring pH of feeding tube aspirates 19 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 4.2 DIETARY AND NUTRITIONAL RECOMMENDATIONS 4.2.1 Macronutrients Macronutrient requirements of critically ill patients are reflected on Table 7. TABLE 7: Macronutrient requirements of general critically ill patients 13,23,24 Nutrient Krause’s ASPEN ESPEN Energy: BMI <18.5 25–30 total kcal/kg ABW/day Normal body weight 25 – 30 kal / IBW 25 – 30 kCal / kg (IBW) Initial phase: 20-25kcal/kg/d (ABW) Recovery phase:25 – 30 kCal/kg (ABW)/d Hypocaloric feeding (obese patient) 18 – 20 kCal /IBW 11-14 kCal/ kg (ABW)or 22 – 25 kCal/kg ( IBW) Protein 0.8 – 2.0 g BMI < 30: 1.2–2.0 g/kg (ABW) BMI 30-40: ≥2.0 g/kg (IBW) BMI >40: ≥2.5 g/kg (IBW) Target of 1.5 g/kg 1.3-1.5g/kg (IBW) Carbohydrates 60 – 70 % TE Fats 15 – 40 % TE Fluid 30-35ml/Kg 30-35ml/Kg 4.2.2 Indirect calorimetry Steps to improve accuracy in measuring indirect calorimetry: • 30 minute bed rest prior to measuring • TEN / TPN at same rate during measuring • ventilator settings should not be changed 90 minutes before taking the measurement • avoid anxiety in the patient • try to avoid interruptions by healthcare professionals while measuring • one reading takes about 30 minutes • REE varies within 24 hours TABLE 8: Interpreting RQ value of indirect calorimetry 25 RQ VALUE INTERPRETATION >1 Hyperventilation Lipogenesis/overfeeding 1 CHO oxidation 0.85 (Optimal) Mixed substrate oxidation Mixed diet 0.82 Protein oxidation 0.7 Fat oxidation/underfeeding <0.7 Gluconeogenesis (muscle wasting) 4.2.3 Micro-nutrients Current recommendations indicate that the daily administration of reference values for both vitamins and trace elements are adequate.  Electrolyte requirements: Electrolytes should be replaced according to the clinical situation. The following may however be used as a guideline per day: • Sodium 1 – 2 mmol/kg • Potassium 0,7 – 1mmol/kg • Calcium 0,1mmol/kg • Magnesium 0,1mmol/kg • Phosphorous 0,4 mmol/kg 2 It has been documented that vitamin and mineral requirements are increased in the following conditions: Stress (vitamins B2, B6, pantothenic acid, C and Zn); for an increased demand on the immune system function (vitamins A, D, E, B6, pantothenic acid; C and folic acid and Zn); during wound healing (vitamins A, B2, C and selenium); and for the prevention of free radical/ peroxidative injury (vitamins C and E). Many medicines have been shown to increase vitamin and mineral requirements. Varying degrees of mal-absorption must also be considered in the critically ill 2. 20 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS There are no specific guidelines for general enteral nutrition however, in specific circumstances Table 9 can be used: TABLE 9: Suggested enteral vitamin supplementation in the critically ill 26 Vitamin Recommendations for the uncomplicated critically ill patient Vitamin A 25 000 IU β Carotene 15 - 50 mg Vitamin D 400 IU/day Vitamin E 400 IU/day Vitamin K 1,5 µg/kg/day Vitamin B1 10 mg/day Riboflavin 10 mg/day Niacin 200 mg/day Pantothenic Acid 100 mg/day Vitamin B1 20 µg/day Biotin 5 mg/day Folic Acid 2 mg/day Vitamin C 1 000 mg/day TABLE 10: Suggested trace element supplementation in the critically ill 2 Trace element Recommendations for the enteral supplementation of the uncomplicated critically ill patient Selenium 100 µg/day Zinc 25 – 50 mg/day Manganese 5 - 7 mg/kg/day Chromium > 50 – 200 µg /day Molybdenum 0,2 – 0,5 mg/day 21 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 4.2.4 Pharmaconutrition Immune-system enhancing nutrients are those that have been demonstrated to have measurable effects on the immune system. Of these the most important are:  Arginine  Omega-3 fatty acids  Nucleotides TABLE 11: Indications, contra-indications and recommended dosages of specialised nutrients 27,28,29,30,31,32,33,34,35 Specialised nutrient Indication Recommendation Arginine • Should be considered in trauma and surgery patients to improve wound healing Contra-indications:  Should be avoided in patients with systemic sepsis  Should not be used in critically ill patients • Although specific recommendations are not available, dosages of 20g per day have been proposed or 9% of the protein energy intake • Optimal levels is not yet determined, but dosages of 15 – 30g per day in enteral fed critically ill patients appears to be safe Omega-3 fatty acids (Fish oils, borage oils and antioxidants) • Omega-3 fatty acid supplementation recommended in patients with ALI and ARDS • Doses of up to 5g per day of omega-3 fatty acids have been used in critically ill patients • Omega-6:Omega-3 ratio:2:1 to 4:1 Dietary fibre Dietary fibre has many benefits in critical patient care. • It delays gastric emptying and small intestinal transit time. • Improves salt and water absorption. • Improves the integrity of the gut mucosa and increases intestinal bulk. • Dietary fibre also absorbs carcinogens, modifies bile salt secretion and influences intestinal micro-flora. Short-chain fatty acids (SCFAs) are produced by dietary fibre and are responsible for:  absorption of sodium and water  mucosal energy  mucosal blood flow  mucosal cell proliferation  mucosal cell differentiation  mucus release  prevention of colitis • 25 – 30 g/day Fructo-oligosaccharides (FOS) • Fructo-oligosaccharides are highly soluble, with a low viscosity that results in the reduction of constipation and diarrhea. • It improves liver function and reduces cholesterol and triglyceride levels. • 5 – 10 g Medium-chain triglycerides • Medium-chain triglycerides are useful when fat mal-absorption is involved. • They also may have a greater protein-sparing effect than long-chain triglycerides. Short bowel syndrome: 20 – 60 g MCT /d Nucleotides and antioxidants • RNA-nucleotides keep the gut mucosa barrier intact and it stimulates the immune system. • A reduce oxidative stress. Probiotics • Should be considered in critically ill to reduce VAP incidences – reduce colonisation of the respiratory tract with pseudomonas aeruginosa • Antibiotic associated diarrhoea (AAD) – antibiotics most commonly associated with AAD are Aminopenicillins with or without Clavulanic acid, Cephalosporins and Clindamycin Contra-indications:  Saccharomyces Boulardii should be avoided in ICU patients  Probiotics should be use with caution in severe acute pancreatitis Saccharomyces Boulardii appears to be most effective for preventing AAD. Lactobacillus rhamnosus GG has also proven effective 22 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 4.3 DISEASE RELATED CONDITIONS TABLE 12: Disease specific requirements 45,46,47,48,49,50,51,52,53,54,55,56,57,58 LIVER DISEASE Diseases Energy Protein Carbohydrate Fat Fluid Other  Alcoholic steatohepatitis 1.3 x BMR Without ascites: Actual body weight With ascites: Ideal body weight EN: 35-40kcal/ kg/d Well nourished, moderately malnourished: 1.2 g/kg/d Severely malnourished: 1.5 g/kg/d EN: polymeric protein formula 50 – 60% NPE Give as glucose 40-50% NPE See general recommendations Water soluble vitamins: thiamin (vit B1), pyridoxine (vit B6), nicotinamide, folic acid. Thiamin prior to commencement due to high risk for Wernicke encephalopathy Fat soluble vitamins: All, look at Vit K if jaundice with fat-malabsoprtion Minerals and trace elements:  Liver Cirrhosis Weight to be used: Without ascites: Actual body weight With ascites: Ideal body weight 25-40kcal/kg/d Stable and malnourished: REE x 1.2-1.4 Without encephalopathy: REE x 1.2-1.4 Acute encephalopathy: REE x 1.2-1.4 PN: 30 – 35 kcal/kg dry body weight EN: 35-40kcal/ kg/d Ascites: energy dense formula Without encephalopathy: 1 – 1.5 g/kg/d Compensated cirrhosis: 1.2 g/kg/d (no malnutrition) Decompensated cirrhosis with severe malnutrition: 1.5 g/kg/d Acute encephalopathy: 0.6-0.8g/kg/d (short term until cause determined and treated) EN: polymeric protein formula 50 – 60% NPE Give as glucose In case of hyperglycaemia: 2 – 3 g/kg/d + IV insulin infusion 40 – 50% NPE Lower in omega 6 See general recommendations Water, electrolytes, water – and fat soluble vitamins, trace elements If ascites: Na < 2000mg  Liver transplant and surgery 124 x BMR 1.3 X REE (NPE) EN: 1.2 – 1.5 g/ kg/d EN: polymeric protein formula See general recommendations  Acute liver failure 1.2 – 1.3 x REE 0.8 – 1.2 g/kg/d 2- 3 g/kg/d 0.8 – 1.2 g/k/g/d See general recommendations 23 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS RENAL DISEASES Diseases Energy Protein Carbohydrate Fat Fluid Other  Acute kidney injury 25 – 30 kcal/kg/d (total energy) Hidden energy sources: lactate, citrate, glucose from treatment 20-30kcal/kg No catabolism, no RRT: 0.8 – 1.0 g/kg/d Moderate catabolism, on RRT: 1.2 – 1.5 g/ kg/d Severe catabolism, on RRT e.g CRRT: 1.7 – 2.0 g/kg/d or 1.8 – 2.5 g/ kg/d On IHD: 1.5 – 2.0 g/kg/d RRT – renal replacement therapy 3-5g/kg (max 7g/ kg) 0.8-1.0g/kg Thiamin Vit C Se and Cu  Chronic kidney disease < 60 years-35Kcal/kg/Day >60 years-30-35Kcal/kg/Day If GFR- <25ml/ min + not on dialysis- 0.6g/kg/ day If 25-55ml/min -0.6g/kg/day If >55ml/min-0.8g/kg/day If Stable on HD- 1.2g/kg/day If Stable on PD-1.2-1.3g/kg/day Acute Illness-1.2-1.3g/kg/day 50-60% of TE 25-35% of TE See general recommendations Limit Sodium to 2-3g/day PULMONARY DISEASES Diseases Energy Protein Carbohydrate Fat Fluid Other Prolonged mechanical ventilation (PMV) REE: (V02 x 3.941)+((VCO2 x 1.11) x 1440 (Weir equation) 20 – 30kcal/kg/d Penn state, Mifflin st jeor Indirect calorimetry 1.2 – 1.5g/kg ABW/d Use general recommendation-literature is inconclusive ARDS: Omega-3 fatty acids (fish oils, borage oils) Use general recommendation-literature is inconclusive See general recommendations Dietary Fiber: 21-38g/d Vitamin D, phosphate, routine supplementation antioxidants COPD 94% to 146% of predicted requirements 1.2 – 1.7 g/kg dry body weight / d (15-20% of total energy) 40 – 55% Total energy 30 – 45% Total energy See general recommendations 24 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS INJURIES Diseases Energy Protein Carbohydrate Fat Fluid Other Traumatic brain injury (TBI) Penn states that mortality improves with every 10kcal/ kg/d but plateau at 25 kcal/kg/d 140% (range: 120 -250%) of basal energy expenditure (BEE) using predictive equation If sedated or on barbiturates: max 120% BEE ESPEN: 25-30kcal/ kg desirable weight/d ASPEN: 20 – 25 kcal/kg desirable weight /d 35 – 45 kcal/kg/d First 2 weeks: 1 – 1.5g./kg/d There-after: 1.5 – 2 g/kg/d BCAA recommended 2 – 2.5 g/kg/d Glycaemic control: First 2 weeks: 8.3 to 8.9 mmol/L See general recommendations See general recommendations See general recommendations Early EN (within 24 hrs) – 50% of energy and 1–1.5 g/ kg protein requirements SPINAL CORD INJURY Energy Protein Carbohydrate Fat Fluid Other Spinal cord injury Acute phase: Predictive equation + stress factor of 1.2 + activity factor of 1.1 weight = (admission weight) Rehabilitation phase: 22.7kcal/ kg body weight/d (quadriplegic) (Total energy intake) 27.9kcal/ per kg weight/d (paraplegic) The higher the injury, the lower energy requirements If pressure ulcers present: 30kcal to 40kcal/ kg body weight/day or Harris-Benedict times stress factor (1.2 for stage II ulcer, 1.5 for stage III and IV ulcers). Acute phase: 2 g/kg/d Rehabilitation phase: 0.8-1 g/ kg/d If pressure ulcers present: 1.2g to 1.5g of protein per kg body weight per day (Stage II pressure ulcers) 1.5g to 2.0g of protein per kg body weight per day (Stage III and IV pressure ulcers). 45 - 65% TE 20 – 35% TE (recommended: 30% TE) Min of 1.5 L / day 1mL/1 kcal 15 g fiber / day Zinc, vit A and C, B-complex 25 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS OTHER DISEASES Diseases Energy Protein Carbohydrate Fat Fluid Other Stroke Total energy: 110-115% TEE No difference between acute and chronic: 25 – 45kcal/kg/d 1-1.5g/kg/d 1.2 – 1.5g/kg/d 30-35 ml/kg Energy Protein Carbohydrate Fat Fluid Other Burns Curreri formula Xie et al Toronto formula _- 4343 + (10.5 x % TBSA) + (0.23 x caloric intake) + (0.84 x REE by Harris-Benedict ) + (114 x t˚) - (4.5 x days after injury) 1.5-2.0g/kg/d Major burns: 2 – 2.5g/kg/day Optimal NPE:N ratio: 100:1 1.5-3g/kg/d 55-60 % of NPE Max 5mg/kg/min glc infusion rate Monitor and maintain serum glucose levels as close as possible to normal levels 50-60% < 35%TE 20-30% Zinc, Cu, Se, Vit B1, C, D, E Vit C 25mg/ml IV in first 24 hrs Energy Protein Carbohydrate Fat Fluid Other Congestive cardiac failure 25kcal/kg/day to 31-35kcal/kg/day If Cardiac Cachexia 160-180% of REE 1.3-1.5g/kg/day See general recommendations Minimum of 1g omega 3 per day See general recommendations Sodium limited to 1200-2400mg/ day Energy Protein Carbohydrate Fat Fluid Other Pancreatitis 25–35 kcal/kg/d 1.2–1.5 g protein/ kg/d 50% TE 30% TE In case of steatorrhea, decrease fat intake to 0.5g/kg/d, if steatorrhea persists change fat source to MCT See general recommendations Chronic pancreatitis – need supps of fat soluble vits, Ca, Mg, Zn, thiamine and folic acid. See attached reference. 26 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS Energy Protein Carbohydrate Fat Fluid Other Gastrointestinal Tract Failure (Inflammatory Bowel Disease) See general recommendations 1.3-1.5g/kg/day See general recommendations See general recommendations Short Bowel Syndrome- MCT- 20-60g/day See general recommendations Vitamin B6 and B12 If patient has diarrhea-Supplement with zinc, selenium and potassium If on corticosteroids- give calcium and Vitamin D Diseases Energy Protein Carbohydrate Fat Fluid Other Oncology Ambulate Patients-30-35kcal/kg/day-TE Bed Ridden Patients-20-25kcal/kg/day of TE Weight Gain-30-40kcal/kg/day of TE Hypermetabolic/ Stressed 35kcal/kg/day of TE Haemopoietic cell transplant-30-35kcal/kg/day of TE Non Stressed- 1-1.2g/kg/day Hypercatabolic-1.2-1.6g/kg/day Haemopoietic cell transplant-1.5-2g/ kg/day Severe Stress-1.5-2.5g/kg/day 60% NPE 40% NPE See general recommendations Omega 3 fatty acids are beneficial Micronutrients -100% DRI 4.4 STOPPING A TUBE FEED (exit criteria) A tube feed may be stopped: • When nutritional assessment by a dietitian confirms that the patient is able to orally ingest at least 80% of his/her nutritional requirements for 3 consecutive days. • In selected cases of swallowing difficulties; the speech therapist needs to confirm swallowing capability and adequate oral motor skills before oral intake can commence. • When complications develop and require further nutrition intervention such as total parenteral nutrition (TPN). Removing the tube: • In the hospital setting the tube may be removed by a health professional preferably a registered nurse. • Button and Percutanous endoscopic gastrostomies (PEGs) need to be removed surgically • A gastrostomy tube may be removed by the stoma therapy nurse or a nurse at the ward or clinic level Please note • It is imperative that both the dietitian and medical team be informed and/or involved in the decision before any feeding tube is removed • The patient’s weight and height needs to be recorded on the day that the tube is removed and regular follow- ups scheduled to monitor progress. 27 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 5 Monitoring 5.1 NUTRITIONAL ASSESSMENT MONITORING AND FOLLOW-UP TABLE 13: Monitoring the patient receiving enteral nutrition 13,23 Parameter Frequency Abdominal distention and discomfort Daily Fluid intake and output Daily Gastric residuals Every 4 hours where appropriate Signs and symptoms of oedema or dehydration Daily Stool output and consistency Daily Weight At least 3 x per week Nutritional intake adequacy Daily Serum electrolytes, blood urea nitrogen, creatinine At least 2-3 x per week Calcium, magnesium, phosphorous Weekly, or as ordered Serum glucose 4-6 hourly Note: Once tolerance of feeds is established it is not recommended to monitor gastric residuals frequently as this may lead to inappropriate interruption of enteral feeding. GIT function and tolerance should be assessed daily to determine the initiation of appropriate feeding and tolerance of feeding. GIT function should be assessed in terms of bowel sounds, tympani, nasogastric/fistula drainage and abdominal distension (measure circumference), intra-abdominal pressures, abdominal x-ray/sonar, failure to pass flatus/stool, vomiting and diarrhoea (test for C. difficle) or constipation. Clearly it is important to identify the patient at risk of enteral feeding intolerance as indicated in Table 14: TABLE 14: Risk factors for feeding intolerance 3 Admission diagnosis Head injury/spinal cord injury, central nervous system diseases, major surgery, pancreatitis, sepsis, burns Biochemical abnormalities Hyperglycaemia, hypokalaemia, hypophosphatemia Clinical history Diabetes mellitus, renal insufficiency, endocrine diseases, prior GIT surgery Formula related issues Osmolality, large volume/rapid infusion of formula, , formula pH, infusion of very cold formula, high-fat formula/type of fat, bacterial or fungal infection of formula, inappropriate formula Others Pain, anxiety, infection Medicines Opioids (particularly pentobarbital), hypnotics, inotropes, sedatives, analgesics, anticholinergics 28 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 5.2 COMPLICATIONS OF ENTERAL FEEDING TABLE 15: Complications related to enteral nutrition 23,59 Problem Effects Management Tube-related (access or administration problems) Presence of tube Damage to the nose, pharynx, or oesophagus Sinusitis Early placement of small bore polyurethane tube strongly recommended Pharyngostomy or orogastric tube placement is recommended Blockage of tube lumen Inadequate feeding Flush with luke warm water before and after medication Misplacement of a nasogastric tube intracranially Brain trauma, infection, base of skull fracture, severe facial fractures Use an orogastric placement Misplacement or migration of a nasogastric or orogastric tube in the tracheobronchial tree Pneumonia Check placement of tube using radiography before initiation of feeding Dislodgement of a gastrostomy or jejunostomy tube; leakage Peritonitis After being dislodged, a tube may be replaced into the peritoneal cavity. If tubes were originally placed using invasive techniques, replacement is more difficult and more likely to cause complications Formula-related leading to gastrointestinal complications Intolerance of one of the formula’s main nutrient components Diarrhoea, GI discomfort, nausea, vomiting, mesenteric ischemia (occasionally), constipation, distention, bloating, maldigestion, malabsorption, micronutrient deficiencies If bolus feeding- change to continuous feeding If on polymeric feed-change to semi-elemental Consider supplemental TPN if requirements cannot be met using EN If malabsorption occurs due to pancreatic insufficiency- add pancreatic enzymes Osmotic diarrhoea Frequent, loose stools Monitor tolerance to the feed given and change accordingly Monitor the osmolarity of the feed and adjust accordingly Nutrient imbalances Electrolyte disturbances, hyperglycaemia, volume overload, hyperosmolarity Body weight and blood levels of electrolytes, glucose, Mg, and phosphate should be frequently monitored (daily during the first week) Other Reflux of tube feedings or difficulty with oropharyngeal secretions Aspiration Flex upper body to an angle of 30-45 degrees Delayed gastric emptying High gastric residuals Should be checked 4 hourly If using polymeric feeds –change to semi-elemental feeds Consider prokinetic medicines Consider supplemental TPN if requirements cannot be met using EN Jejunal access if possible Metabolic complications Medicine-nutrient interactions Refer to Table 16 Refeeding syndrome Refer to Table 3 GI discomfort may have other causes, including reduced compliance of the stomach due to shrinkage caused by lack of feeding, distension due to volume of feeding, and decreased gastric emptying due to dysfunction of the pylorus. 29 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 5.3 MEDICINE-NUTRIENT INTERACTION TABLE 16: Medicines affecting GI function that are used in the critically ill Medicine type Medicine Effect Medicine-nutrient interactions Medicines affecting GI perfusion Adrenaline Affect predominantly α-receptors at high doses with resultant splanchnic vasoconstriction Hypokalemia, nausea, vomiting Dopamine Low doses act on β2 and DA1 receptors, relaxing smooth muscle. DA1 effect may be protective even during α-receptor stimulation. High doses cause intense vasoconstriction via α-receptor stimulation Digoxin Constricts mesenteric vasculature Medicines affecting GI flora Antibiotics, particularly broad spectrum, e.g. cephalosporins and ampicillin Disturbs the balance of commensal to pathogenic flora, allowing proliferation of pathogens Medicines used in prevention of GI bleeding Adrenalin Potent vasoconstrictor, reducing splanchnic blood flow and hepatoportal pressure Somatostatin Reduces gastric and pancreatic secretion, GI blood flow, small intestine transit and nutrient absorption Anti-diarrhoeal agents Loperamide Reduces GI motility and secretions by interacting with opioid and cholinergic receptors Codeine phosphate Opioid action inhibiting non-adrenergic and non-cholinergic nerves and exciting cholinergic nerves, reducing peristalsis Prokinetic agents Metoclopramide Increases gastric emptying, duodenal/jejunal motility and gastro-oesophageal tone Enteral feed/nutrition interaction Phenytoin The pharmacological action of phenytoin is reduced Long term therapy: folate deficiency, and rarely megaloblastic anaemia, interference with vit D metabolism Medicines reducing GI motility Opiates, e.g. morphine Delayed gastric emptying, reduced biliary and pancreatic secretions, diminished propulsive contractions in small and large intestine Medicines promoting osmotic diarrhoea Sorbitol containing oral syrups, e.g. KCL syrup Excess amounts can increase intraluminal osmolarity producing movement of fluid into the gut and causing diarrhoea Laxative agents Osmotic laxatives, e.g. lactulose, sorbitol Lactulose is metabolised to lactate and other organic acids by colonic bacteria. These substances exert an osmotic effect and increase stool water Lactulose contraindicated in galactosemia Stimulant laxatives e.g. senna These stimulate the myenteric plexus, inducing increased smooth muscle contraction 30 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS Table 17: Special considerations for medicine administration via enteric tubes Problem/interaction Effect/consequence Examples Solution/ recommendation Changes in pH after mixing EN and pharmaceutical agents together Acidic preparations (such as syrups) cause the greatest problems, with increased clumping of the EN formula or enteral tube obstruction from precipitate formation Ferrous sulfate liquid frequently clog the feeding tube when mixed directly into the EN formulation Use the oral route whenever possible. Consider alternative routes (i.e., buccal, nebulized, rectal, intravenous, transdermal). If a feeding tube must be used for medication administration, oral liquid dosage forms are preferred. • Elixirs and suspensions are preferable to syrups • Dilute hyperosmolar solutions in at least 30 mL of water Components of the EN formula itself can influence the risk for an interaction • Protein in the form of hydrolyzed or free amino acids appears to have a higher compatibility with medicines than intact protein products • Enteral products with fiber generally are not compatible with medications Do not mix medications directly into EN formulations. Give each medication separately and flush feeding tube with 30ml water between medications Medication administration devices (i.e., tubing) can interact with medicines Complexation, altering final medicine potency and causing a therapeutic failure from suboptimal medication delivery. Phenytoin absorption may be reduced by up to 70%, thus decreasing serum medicine levels Adherence of phenytoin and carbamazepine suspensions to the walls of PVC enteric tubes can result in inadequate medicine delivery to patients Diluting and irrigating the tubes prior to administration of these oral suspensions significantly improved medicine recovery and the final amount received by the patient. Complexation of medications with components of EN formulations can occur, reducing the efficacy of the agent. Decreased bioavailability from proposed binding with divalent cations in the EN formulations has resulted in increased time to peak concentrations and decreased peak concentrations of fluoroquinolones Fluoroquinolone antibiotics, including ciprofloxacin, levofloxacin, and ofloxacin To ensure proper medicine delivery: • Parenteral administration of fluoroquinolones in patients with intravenous access • The solid dosage form (i.e., tablet) should be crushed into a fine powder and mixed in 30 mL of water. if enteral administration cannot be avoided • Flush the feeding tube with 30 mL of water following administration to clear any residual medication The manufacturing processes for certain medications are specialised Crushing a tablet or opening the contents of a capsule, alters the intended dosage form and the medication may not act as intended Enteric-coated tablets, sustained-release or extended release coated capsules or tablets, sublingual and buccal tablets, and microencapsulated products Never open or crush in order to administer through a feeding tube 31 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS Table 18: Specific medicine-enteral nutrition interactions60,61 MEDICINE CAUSES SOLUTION CAUTION Phenytoin suspension Possible explanations include: • binding of phenytoin to the protein source (calcium caseinates), binding to divalent cations (calcium, magnesium) • binding to the feeding tube • Hold EN one hour before and one hour after phenytoin administration • Using the capsule formulation (versus the suspension) as the powder from the capsules appears less likely to bind • Change to a bolus feeding regimen (e.g., 240 mL given four times per day) and administer phenytoin between boluses • Administer intravenous phenytoin via the feeding tube, as the bioavailability is unchanged, but the maximum concentration of phenytoin is significantly greater and the time to maximum concentration is significantly shorter when compared with the suspension formulation • Some institutions prefer to not hold EN at all, administer higher doses of phenytoin suspension, and closely monitor serum phenytoin concentrations • In general, phenytoin suspension given through a feeding tube, should be diluted with 20–60 ml of water to enhance absorption and increase the dissolution rate Can cause underfeeding due to insufficient feeding time or feed intolerance due to high infusion rate. To minimise the amount of time that the feedings are held, phenytoin suspension should be given twice daily rather than more often if possible. • Phenytoin dosages will require adjustment if the feeding regimen is discontinued or temporarily held to prevent toxic levels Proton pump inhibitors e.g. Lansoprazole Formulated as delayed-release capsules containing enteric-coated granules. When ingested by mouth, the delayed-release capsule protects the base-labile granules until they reach the alkaline pH of the duodenum, at which time the granules dissolve and the medicine is absorbed. Crushing the enteric-coated granules can result in tube clogging and dissolving the granules in water can destroy the medication before it reaches the absorption site (i.e., small intestine). • Mix intact granules with an acidic medium (e.g., apple or orange juice) and flush with the acidic medium after administered down a gastric feeding tube. • If the feeding tube terminates in the small bowel (i.e., jejunum), alkaline liquids should be used to dissolve the medicine granules prior to administration. Dissolve intact granules in sodium bicarbonate 8.4% solution. Pour suspension down feeding tube, flush with 30ml water and hold feeds for at least one hour. • Enteric coated, delayed-release tablets cannot be crushed and should not be administered via gastric or jejunal feeding tubes • Products available as a packet of granules that is reconstituted with water to form a suspension, however, has been reported to clog feeding tubes as it contains xanthan gum 32 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 6. Home based enteral nutrition 6.1 Patient education 62,63 Bolus feeding is most often the preferred way of feeding, when using a PEG tube. Care must be taken to flush the PEG tube, with clean water, before and after administration of food or medication. Usually at least 30 milliliters of water is used to flush the tube. Feeds are administrated by using a feeding syringe; 60ml catheter tip syringe. Care must be taken not to push feeds through the PEG, but to make use of gravity or slowly push down the plunger of the syringe. The family/caregiver of the patient must be educated to care for the patient at home. Person responsible for education: • At primary level: A nurse or the community dietitian must reinforce information. • At tertiary and secondary level: The hospital dietitian and/or nurse at discharge. The patient discharged on tube feeding and caretaker should be taught: • Safe preparation of tube feeding. • To emphasise the importance of feeding rate according to prescription to maintain nutritional requirements. • To recognise symptoms of intolerance. • How to manage simple problems. • Where to call for guidance and when to call the doctor or the dietitian. • To maintain social contact at mealtimes. 6.2 TUBE CARE 58,59 • Check the length of the PEG tube daily. • Check that the lower disk of the PEG tube fits tightly to the skin daily. • Flush the tube regularly with lukewarm water to prevent clogging of the tube. If clogged, unclog the tube with bicarbonate of soda mixed in warm water. • Clean the skin around the stoma twice daily. • Take care to keep the skin around the stoma dry, check daily for any redness around the stoma. • Turn the tube daily. 6.3 FORMULATIONS 6.3.1 Home based Blenderised or pureed food can be used if a tube greater than 20 French is inserted. Take care to prevent clogging of the tube; this can be done by thinning the puree with nutritious liquids e.g. milk. Meal plan must be calculated by a dietitian and additional supplementation provided if required. 62 6.3.2 Commercial Any commercially available formula can be used, as long as it fulfills the patient’s diet prescription as calculated by a dietitian. Take fluid recommendations into consideration when using commercially prepared formulations 62 Note: Patients should be monitored at least monthly. 33 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS REFERENCES 1. 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Canadian Clinical Practice Guidelines 2013, Section 5.2: Strategies to Optimize Delivery and Minimize Risks of EN: Motility Agents 18. Canadian Clinical Practice Guidelines 2013, Section 5.3: Strategies to Optimize Delivery and Minimize Risks of EN: Small Bowel Feeding vs. Gastric 19. Adapted from NICE and BAPEN guidelines. Mehanna H, Nankivell PC, Moledina J, Travis J. Head and Neck Oncology 2009; 1(4) 20. Khan LUR, Ahmed J, Khan S, MacFie J. Refeeding Syndrome: A literature review. Gastroenterology Practice and Practice 2011:1-6 21. Yu-Chin Chen. Critical Analysis of factors associated with Enteral Feeding in Preventing VAP: A Systematic Review. 2009 22. Barrett JS, Sheperd SJ and Gibson PR. ASPEN: Strategies to Manage Gastrointestinal Symptoms Complicating Enteral Feeding. 2009 23. Mahan LK, Escott-Stump S, Raymond JL. Krause’s Food and Nutrition Care Process. 13th Ed. 2012. Elsevier 24. Kreymann KG, Berger MM, Deutz NEP, Hiesmayr M, Jolliet P, Kazandjiev G, et al. ESPEN guidelines on enteral nutrition: Intensive care. Clinical nutrition 2006; 25(1): 210-33 25. Indirect Calorimetry: Principles and Applications for Managing Critically Ill Patients. Terry L. Forrette, MHS, RRT. Published 11/09/2005, cme.medscape.com 26. Adapted from: Visser J, Labadarios D, Blaauw R. Micronutrient supplementation for critically ill adults: A systemic review and meta-analysis. Nutrition 2011; 27:745-758 27. Prins A, Visser J. Immunonutrition: A South African perspective.. S Afr J Clin Nutr 2012;25(3), 94-111 28. Canadian Clinical Practice Guidelines 2013: Section 4.1c. Composition of EN: Glutamine 29. Canadian Clinical Practice Guidelines 2013: Section 4.1a. EN Composition: Diets Supplemented with Arginine and selected Other Nutrients 30. Canadian Clinical Practice Guidelines 2013: Section 4.1b (i). Composition of EN: Fish Oils, Borage Oils and Antioxidants 31. Canadian Clinical Practice Guidelines 2013: Section 6.2. Enteral Nutrition (Other): Probiotics 32. Lochs H, Dejong C, Hammarqvist F et al. ESPEN Guidelines on Enteral Nutrition: Gastoenterology. Clinical Nutrition (2006)25,260-274) 33. Zhou M, Martindale RG. Arginine in the critical care setting. Journal of Nutrition 2007 June 01;137(6):1687S-1692S 34 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS 34. Morrow Lee E, Gogineni Vijaya, Malesker Mark A. Probiotic, prebiotic and symbiotic use in critically ill patients. Curr Opin Crit Care 2012, 18:186-191. 35. Morrow Lee E, Gogineni Vijaya, Malesker Mark A. Synbiotics and probiotics in the critically ill after the PROPATRIA trail. Curr Opin Clin Nutr Meatb Care 2012, 15:147 - 150. 36. Chadalavada R, Biyyani RSS, Maxwell J, Mullen K. Nutrition in Hepatic encephalopathy. Nutr in CLin Prac. 2010;25:257-264 37. Plauth M, Cabre E, Riggio O, Assis-Camilo M, Pirlich M, Kondrup J, et al. ESPEN guidelines on enteral nutrition: liver disease. Clin Nutr,2006;25:285-294 38. Cabral Cm, Burns DL. 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Cook AM, Peppard A, Magnuson B. Nutrition considerations in traumatic brain injury. Nutr Clin Prac.2008;23:608-620 45. Escott-Stump S. Nutrition and Diagnosis Related Care, thirteenth edition 2012. 234 – 236 46. American Dietetic Association: Spinal Cord Injury and nutrition guideline. Accessed 17/04/2012 from adaevidencelibrary.com/topic.cfm?format_tables=0&cat=3486 47. Khalil RE, Gorgey AS, Janisko M, Dolbow DR, Moore JR, Gater DR. The role of nutrition in health status after spinal cord injury. Aging and disease.2013;4(1):14-22 48. Magnuson B, peppard A, Flomenhoft DA. Hypocaloric considerations in patinets with potentially hypometabolic disease states. Nutr Clin Prac.2011;26:253-260 49. Escott-Stump S. Nutrition and Diagnosis Related Care, seventh edition 2012. 259-261 50. Rousseau A, Losser M, Ichai C, Berger MM. ESPEN endorsed recommendations: nutritional therapy in major burns. Clin Nutr.2013;32:497-502 51. Escott-Stump S. Nutrition and Diagnosis Related Care, seventh edition 2012. 838-842 52. Anne-Françoise Rousseau a, Marie-Reine Losser b, Carole Ichai c, Mette M. Berger d,. ESPEN endorsed recommendations: Nutritional therapy in major Burns. Clinical Nutrition 32 (2013) 497-502 53. Anker SD, Laviano A, Filipattos G, John M, Paccagnelia, Pnoikowski P. ESPEN guidelines on parenteral nutrition: On cardiology and pneumology. Clin nutr.2009;28:3455-460 54. Mirtallo M, Forbes A, McClave SA, Jensen GL, Watzberg DL, Davies AR. International consensus guidelines for nutrition therapy in pancreatitis. J Parent Enteral nutr.2012;36(3):284-291 55. Grant BL, Hamilton KK. Medical nutrition therapy for cancer prevention, treatment and recovery. In: Mahan LK, Escott-Stump S, editors. Krause’s food, nutrition, and diet therapy. 13th Ed. St Louis: Elsevier Saunders; 2012. p.832 – 863 56. Bozetti F. Nutritional support of the oncology patient. Critical Reviews in Oncology/Hematology.2013;87:172–200) 57. Huhmann MB. Review of American Society for Parenteral and Enteral Nutrition (A.S.P.E.N.) Clinical guidelines for Nutrition Support in Cancer Patients: Nutrition Screening and Assessment. NCP.2008; 23(2):182-188 58. Kidney Disease: Improving Global Outcomes (KDIGO) Acute Kidney Injury Work Group. KDIGO Clinical Practice Guideline for Acute Kidney Injury. Kidney inter., Suppl. 2012; 2: 1–138. 59. Elsevier; Adapted from Merck Manual Online. Accessed 17 February 2014 from professional/nutritional_disorders/nutritional_support/enteral_tube_nutrition.html#v883394 60. Gordon S. Sacks Drug-Nutrient Considerations in Patients Receiving Parenteral and Enteral Nutrition Practical Gastroenterology. July 2004: 38 – 48 61. Nancy Toedter Williams. Medication Administration Through Enteral Feeding Tubes. Am J Health Syst Pharm. 2008;65(24):2347-2357 62. How to use and care for your PEG tube – Care Guide. Drugs.com 63. Tube Feeding Using The Bolus Method. Brown Patricia A, Quesada Ofelia, Scott Faye, Patient Information Fact Chart 2009, Memorial Sloan-Kettering Cancer Centre 64. Enteral Nutrition practice recommendation task force. Bankhead, Robin, et al. ASPEN Enteral Nutrition Practice Recommendations. JPEN. 2009; January on line first as doi: 10.1177/0148607108330314 Bockus, Sherry. Troubleshooting your tube feedings. American Jnl of Nursing. May 1991: 24 – 28 65. Wischmeyer P. Nutritional pharmacology in surgery and critical care: “you must unlearn what you have learned”. Curr Opin Anesthesiol 2011; 24:381-388 35 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS ANNEXURES ANNEXURE 1: ACCESS ROUTES 4,23,64 Access Route Indication Advantages Disadvantages Comment Gastric feeding Orogastric or nasogastric Pharyngostomy • Patients requiring short term enteral nutrition (3-4 weeks) • Orogastric tubes not suitable for awake patients • Easy to place, less expertise required for placement • Normal digestive, hormonal and bacteriocidal processes in stomach • Increased protection against stress ulcers due to increased blood flow to the stomach and the buffering effect • More feed options available • Higher risk for aspiration • Increased risk for sinusitis (NG tubes) • Incompetence of lower oesophageal sphincter predisposing patient to regurgitation and aspiration Default route for most patients. Recommended tube types: • Polyurethane or silicone • 8 – 10Fr • Length of 91 – 120cm Verification of tube position • Aspirating gastric content • Auscultation of air insufflation into stomach • Radiographic confirmation of tube tip (Gold standard) Note: If Oesophageal Varices is present, place tube cautiously. Small bowel feeding • Nasoduodenal (post pyloric) • Nasojejunal (post ligament of treitz) • Intolerance to gastric feeding (abdominal distension & discomfort, vomiting, persistant high GRV (>400ml) • Gastroparesis • Potentially lower risk of aspiration • Reduced stimulation of pancreatic secretions Placement is more resource & skill dependant Placement methods: • Intra-operative • Endoscopic or fluoroscopic guidance • Spontaneous that depends on tube migrating into the duodenum by peristalsis Enterostomies Gastrostomy or PEG Jejunostomy or PEJ (Both can be done surgically or via percutaneous endoscopy) • Patient requires long term enteral nutrition (>4-6 weeks) e.g. neurologically impaired • Evaluation by multidisciplinary team is indicated to establish if a. Benefit outweighs risk of placement b. Insertion near end of life is warranted c. Patient is close to achieving oral feeding CONTRA INDICATIONS • Ascites • Haemocoagulation disorders • Larger bore feeding tubes can be used (16 – 24Fr) resulting in less clogging of tubes • Improved comfort and mobility for patients • Prevent complications associated with nasal tubes • Use of buttons (skin level device) can be more aesthetic – no bulkiness under clothing. Also appropriate for active patients. • Buttons also useful in patients who pull out tubes e.g. dementia patients • General anaestetic and laparotomy required for surgical placement. More invasive method. • Skin may break down from leakage • Peritonitis can occur with premature removal – within the first few weeks after insertion • Tube dislodgement • Tube migration • Knowledge of GI anatomy, and motility, prior GI surgery, patency of upper GIT, intended use and intended length of therapy must be part of decision making process. • Patient’s potential risk for anaesthesia, effect of pre-existing co-morbidities and expected patient outcomes must be assessed before placement. • Avoid placement of catheters not intended for use as enteral devices e.g. urinary catheters or drainage tubes. Use of such tubes leads to enteral misconnection as well as tube migration which can cause obstruction ANNEXURE 2: IMMUNONUTRITION RECOMMENDATIONS 65 Patients CCCPG ESPEN ASPEN/SCCM General Arginine (no benefit) No recommendations Antioxidants (possible benefit) No recommendations Arginine (possible benefit) Antioxidant (benefit) Elective surgery No recommendations Arginine (benefit) Arginine (benefit) Trauma Arginine (no benefit) No recommendations Arginine (benefit) No recommendations Arginine (benefit) Antioxidant (benefit) Burns Arginine (no benefit) No recommendations No recommendations Antioxidant (benefit) Arginine (benefit) Antioxidant (benefit) Sepsis Arginine (harm) Arginine (harm) Arginine (harm) Antioxidant (benefit) ALI/ARDS Ω-3 FA and GLA (benefit) Arginine (no benefit) No recommendations Ω-3 FA (benefit) No recommendations Antioxidant (benefit) Ω-3 FA and GLA (benefit) No recommendations Antioxidant (benefit) Enteral feeding intolerance Whole-protein formulas for most patients, hydrolysed protein formulas for patients with GI dysfunction Whole-protein formulas for most patients, hydrolysed protein formulas for patients with GI dysfunction Hydrolysed protein formulas for patients with GI dysfunction 36 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS ANNEXURE 3: CHECKLIST FOR THE ICU SETTING C= Commodities Severity of illness: Calculate score, as indicated by patient ICU admitting diagnosis APACHE II Ranson Criteria SOFA ISS Preexisting conditions: Diabetes Mellitus? Liver dysfunction? Renal dysfunction? Alcohol abuse and related malnutrition? Medicine abuse and related malnutrition Yes If yes, carefully target glucose control If yes, carefully monitor protein control If yes, carefully monitor for malnutrition refeeding syndrome If yes, carefully monitor for malnutrition refeeding syndrome If yes, carefully monitor for malnutrition refeeding syndrome Yes Yes Yes Yes Age Age ___ N=Nutrition Risk Screening Initial screening BMI <20.5? Weight loss in the last 3 months? Reduced dietary intake in past week? Severe illness? Final screening (Complete if “yes” to any of the above questions) Part I Impaired Nutrition Status Absent Mild Moderate Severe Score 0: Score 1: Score 2: Score 3: Normal Nutrition status Weight loss >5% in 3 months or food intake <50-70% normal in last week Weight loss >5% in last 2 months OR BMI 18.5-20.5 OR food intake <25-50% normal in last week Weight loss >5% in last months (>15% in last 3 months) OR BMI <18.5 + impaired general condition OR food intake 0-25% normal in last week Part II Severity of the disease Absent Mild Moderate Severe Score 0: Score 1: Score 2: Score 3: Normal nutrition requirements Hip fracture, chronic patients (HD, DM, cancer, cirrhosis, COPD) with acute complication Major abdominal surgery, stroke, severe pneumonia, hematologic malignancy Head injury, bone marrow transplantation, APACHE III score >10 Part I Impaired NS score + Part II Severity of Disease Score _ + 1 (if age >70) _ = __ Total (If total of 3 or more, nutrition support is indicated) W=Wait for resuscitation Proceed with caution if answer if YES. Hold feeding if answer is NO. Fluid resuscitation complete? Mean arterial pressure ≥65mmHg? Stable pressor agents for 24 hours? CVO2 ≥ 70% or MVO2 ≥ 65%? CVP 8-12mmHg? Serum lactate <22mg/dl? Base excess < 5mEq? Yes No Yes No Yes No Yes No Yes No Yes No Yes No E= Estimate energy requirements Calories BMI <30: use 25-30 kcal/kg ABW/d BMI ≥30: use 11-14 kcal/kg ABW/d Protein BMI <30: use 1.2-2g/kg ABW/d BMI 30-40: use ≥2g/kg ABW/d BMI <40: use 2.5g/k ABW/d F= Formula section Candidate for arginine-containing pharmaconutrition formula Yes No Criteria: Major surgery, trauma (ATI score >20), burns (TBSA>30%), cancer, critically ill on mechanical ventilation Candidate for Anti-inflammatory pharmaconutrition formula Yes No Criteria: ARDS or ALI Candidate for Malassimilation formula: Small peptide? MCT oil Yes No Fibre containing Yes No If no to all above, then candidate is for standard enteral formula 37 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS E=Enteral access Access site Nasoenteric Oroenteric (concern for sinusitis) Percutaneous (anticipate feeds > 4 weeks) Yes No Yes No Yes No Level of infusion Stomach Postpyloric Below ligament of Treitz Yes No Yes No Yes No Need for simultaneous gastric decompression (aspirate/feed tube Yes No E=Efficacy Days NPO Initial rate Goal rate Cumulative Caloric Balance Rapid ramp-up rate Goal volume/day Initiate volume-based feeds Yes No Gastric (max 280ml/hr) Postpyloric (max 150ml/hr D= Determine tolerance 1st GRV >500ml Continue current infusion, recheck in 4 hrs Normalize serum electrolytes Initiate metoclopromide 10mg IV q 6 hrs Initiate Narcan 8 mg in saline per tube q 6 hrs Elevate head of bed Turn patient to right lateral decubitis position 2nd GRV>500ml Hold enteral infusion Restart infusion once GRV <500ml Recheck GRV in 2 hrs. Part II Severity of the disease S-glucose 80-150 mg/dl or 4.4-8.3 mmol/L Yes No Passage of stool/gas Yes No Diarrhoea (>250 ml/day stool output per rectum OR 1000ml/day output ileostomy) Remove sorbitol from oral/enteric medications Obtain stool cultures/axin assay to rule out infectious diarrhoea Initiate opiates once infectious etiology ruled out (lamotil, immodium, paregoric) Consider fibre-containing formula and/or small peptide/MCT formula Provide fibre additive 38 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS Notes 39 NATIONAL ENTERAL NUTRITION PRACTICE GUIDELINES FOR ADULTS National Department of Health Civitas Building Cnr Thabo Sehume and Struben Streets Pretoria 0001 Switchboard: 012 395 8000
2118	https://dermnetnz.org/images/pemphigus-vulgaris-images	Are you a healthcare professional GO TO DERMNET PRO Main menu Common skin conditions NEWS Join DermNet PRO Read more Quick links Skin checker Try our skin symptom checker Images Pemphigus vulgaris images Pemphigus vulgaris images Go to the pemphigus vulgaris topic page Erosions and flaccid blisters in pemphigus vulgaris (PV-patient1) Abdominal erosions in pemphigus vulgaris (PV-patient1) Oozing scalp plaques in pemphigus vulgaris (PV-patient1) An intact blister and erosions in pemphigus vulgaris An intact blister on the palate in pemphigus vulgaris Eroded plaques in pemphigus vulgaris over the trunk (PV-patient2) Erosions over the neck and upper back in pemphigus vulgaris (PV-patient2) Arcuate erosions and an occassional intact blister in pemphigus vulgaris (PV-patient3) Erosions and blisters in pemphigus vulgaris (PV-patient3) Erosions in pemphigus vulgaris (PV-patient3) Scalp macceration and crusting in pemphigus vulgaris (PV-patient3) Chronic lip erosion in pemphigus vulgaris (PV-patient3) Vulval erosions in pemphigus vulgaris Histopathology images of pemphigus vulgaris Direct immunofluorescence of perilesional skin in pemphigus vulgaris - immunostaining with IgG producing intercellular "chicken wire" fluorescence Suprabasal intraepidermal blister formation with acantholytic keratinocytes and "tombstone" basal keratinoytes in pemphigus vulgaris Direct immunofluorescence of perilesional skin in pemphigus vulgaris - immunostaining with IgG producing intercellular "chicken wire" fluorescence Suprabasal intraepidermal blister formation with acantholytic keratinocytes and "tomb stone" basal keratinocytes in pemphigus vulgaris Go to the pemphigus vulgaris topic page Pemphigus vulgaris Pemphigus vulgaris Pemphigus vulgaris Pemphigus vulgaris Pemphigus vulgaris Pemphigus vulgaris pemphigus vulgaris pemphigus vulgaris pemphigus vulgaris pemphigus vulgaris Pemphigus vulgaris pemphigus vulgaris Mucosal pemphigus vulgaris Oral pemphigus Ocular pemphigus vulgaris pemphigus vulgaris pemphigus vulgaris pemphigus vulgaris pemphigus vulgaris Go to the pemphigus vulgaris topic page Join our newsletter RESOURCES Skin checker PO-PASI scoring AI image dataset Quizzes Glossary CONTACT Contact us Website feedback Volunteer Donate ABOUT About DermNet Editorial process Website terms Image licence FAQ Privacy settings Privacy policy © DermNet® 2025 IMPORTANT NOTICE: DermNet does not provide a free online consultation service. If you have any concerns with your skin or its treatment, see a dermatologist for advice.
2119	https://www.jstor.org/stable/13475	Mutator Strains of Escherichia coli, mutD and dnaQ, with Defective Exonucleolytic Editing by DNA Polymerase III Holoenzyme on JSTOR Free online reading for over 10 million articles Save and organize content with Workspace Link account to institutional access Continue with Google Continue with Microsoft Find my institution or Username or email address Password SHOW Stay logged in Forgot password? Log in Don't have an account? Register for free Skip to main content Have library access? Log in through your library Get support Help logging in Contact us Log in Register Workspace Advanced Search By subject By title Publishers Collections Images Get support Help logging in Contact us All Content Images Advanced Search Search all content Register Log in Browse By subject Journals and books By title Journals and books Publishers Collections Images Workspace Your Artstor image groups were copied to Workspace. The Artstor website will be retired on Aug 1st. journal article Mutator Strains of Escherichia coli, mutD and dnaQ, with Defective Exonucleolytic Editing by DNA Polymerase III Holoenzyme Harrison Echols, Chi Lu and Peter M. J. Burgers Proceedings of the National Academy of Sciences of the United States of America Vol. 80, No. 8, [Part 1: Biological Sciences] (Apr. 15, 1983), pp. 2189-2192 (4 pages) Published By: National Academy of Sciences Cite Read and download Log in through your school or library Read Online Read 100 articles/month free Subscribe to JPASS Unlimited reading + 10 downloads This is a preview. Log in through your library . Abstract The closely linked mutD and dnaQ mutations confer a vastly increased mutation rate on Escherichia coli and thus might define a gene with a central role in the fidelity of DNA replication. To look for the biochemical function of the mutD gene product, we have measured the 3′→ 5′ exonucleolytic editing activity of polymerase III holoenzyme from mutD5 and dnaQ49 mutants. The editing activities of the mutant enzymes are defective compared to wild type, as judged by two assays: (i) decreased excision of a terminal mispaired base from a copolymer substrate and (ii) turnover of dTTP to dTMP during replication with a phage G4 DNA template. Thus, the mutD (dnaQ) gene product is likely to control the editing (proofreading) capacity of polymerase III holoenzyme. Journal Information PNAS is the world's most-cited multidisciplinary scientific serial. It publishes high-impact research reports, commentaries, perspectives, reviews, colloquium papers, and actions of the Academy. In accordance with the guiding principles established by George Ellery Hale in 1914, PNAS publishes brief first announcements of Academy Members' and Foreign Associates' more important contributions to research and of work that appears to a Member to be of particular importance. Publisher Information The National Academy of Sciences (NAS) is a private, nonprofit organization of the country’s leading researchers. The NAS recognizes and promotes outstanding science through election to membership; publication in its journal, PNAS; and its awards, programs, and special activities. Through the National Academies of Sciences, Engineering, and Medicine, the NAS provides objective, science-based advice on critical issues affecting the nation. Rights & Usage This item is part of a JSTOR Collection. 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2120	https://www.lessonplanet.com/teachers/nctm-illuminations-fraction-and-percent-tool	Nctm: Illuminations: Fraction and Percent Tool Interactive for 9th - 10th Grade \| Lesson Planet Search Search educational resources Search Menu Sign InTry It Free AI Teacher Tools Discover - [x] Discover Resources Search reviewed educational resources by keyword, subject, grade, type, and more Curriculum Manager (My Content) Manage saved and uploaded resources and folders To Access the Curriculum Manager Sign In or Join Now Browse Resource Directory Browse educational resources by subject and topic Curriculum Calendar Explore curriculum resources by date Lesson Planning Articles Timely and inspiring teaching ideas that you can apply in your classroom About - [x] Our Story Frequently Asked Questions Testimonials Contact Us Pricing School Access Your school or district can sign up for Lesson Planet — with no cost to teachers Learn More Sign In Try It Free Hi, What do you want to do? Create a lesson plan Generate resources with AI teacher tools Search 2 million educational videos Find a teaching resource Publisher National Council of Teachers of Mathematics Resource Details Curator Rating Educator Rating Not yet Rated Rate this Resource Grade 9th - 10th SubjectsMath1 more... Resource TypeInteractives AudiencesFor Administrator Use2 more... Lexile Measures1260L Interactive Nctm: Illuminations: Fraction and Percent Tool Curated by ACT This tool allows you to explore fractions, decimals and percents by creating fractions with denominators up to 100. 3 Views 0 Downloads CCSS:Adaptable Concepts decimals, converting fractions Show MoreShow Less Additional Tags nctm: illuminations: fraction and percent tool, circle, rectangle, set Show MoreShow Less Classroom Considerations Knovation Readability Score: 4 (1 low difficulty, 5 high difficulty) Common Core 7.EE.B.3 See similar resources: Interactive #### Nctm: Illuminations: Isometric Drawing Tool National Council of Teachers of Mathematics This site includes an interactive math tool to create drawings on isometric dot paper. You can draw figures using edges, faces, or cubes. Rotate and color your drawings, and view them in 3D or 2D. 9th - 10th Math Interactive #### Nctm: Illuminations: Fraction Models National Council of Teachers of Mathematics An interactive game to learn fractions less than one, greater than or equal to 1, which are written as mixed numbers and their equivalent decimal and percentage forms 9th - 10th Math CCSS:Adaptable Interactive #### Nctm: Illuminations: Mirror Tool National Council of Teachers of Mathematics This interactive tool helps student visualize symmetry. The exploration provided by the site helps students explore reflective symmetry. [Requires Java.] 9th - 10th Math Interactive #### Nctm: Illuminations: Fractal Tool National Council of Teachers of Mathematics This computer tool offers studnets a chance to see fractals being made step by step, including their own, and view the mathematical data each produce. 9th - 10th Math Activity #### Nctm: Illuminations: Igd: Area of a Triangle National Council of Teachers of Mathematics Use this interactive tool to learn how to find the area of a triangle. Click on "Instructions" for details on how to use this tool. Click on "Explore" for more ideas. [Requires Java.] 9th - 10th Math Interactive #### Nctm: Illuminations: Histogram Tool National Council of Teachers of Mathematics This tool can be used to create a histogram for analyzing the distribution of a data set using data that you enter or using pre-loaded data that you select. 9th - 10th Math CCSS:Adaptable Activity #### Nctm: Illuminations: Computing Pi National Council of Teachers of Mathematics This resource investigates approximating the value of pi through the calculation of perimeters of inscribed or circumscribed polygons. The interactive tool allows learners to examine and compare both methods. 9th - 10th Math CCSS:Adaptable Activity #### Nctm: Illuminations: Diagonals to Quadrilaterals Ii National Council of Teachers of Mathematics This resource explores the bisectors of quadrilaterals. The interactive tool allows learners to draw quadrilaterals with perpendicular bisectors, congruent perpendicular bisectors, and congruent bisectors. 9th - 10th Math CCSS:Adaptable Activity #### Nctm: Illuminations: Diagonals to Quadrilaterals I National Council of Teachers of Mathematics This resource explores the diagonals of quadrilaterals. The interactive tool allows learners to examine intersecting line segments in a variety of ways. Topics discussed are perpendicular diagonals, congruent diagonals, and bisected... 9th - 10th Math Interactive #### Nctm: Illuminations: Geometric Solids National Council of Teachers of Mathematics The instructional tool explores various geometric solids and their properties. Student can manipulate and color the different shapes. Learners examine the edges, faces, and vertices of the different solid figures. 9th - 10th Math CCSS:Adaptable Try It Free © 1999-2025 Learning Explorer, Inc. 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2121	https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/AC_Electrical_Circuit_Analysis%3A_A_Practical_Approach_(Fiore)/06%3A_Nodal_and_Mesh_Analysis/6.2%3A_Nodal_Analysis	Skip to main content 6.2: Nodal Analysis Last updated : May 22, 2022 Save as PDF 6.1: Introduction 6.3: Mesh Analysis Page ID : 25273 James M. Fiore Mohawk Valley Community College ( \newcommand{\kernel}{\mathrm{null}\,}) Nodal analysis can be considered a universal solution technique as there are no practical circuit configurations that it cannot handle. It does not matter if there are multiple sources or if there are complex configurations that cannot be reduced using series-parallel simplification techniques, nodal analysis can handle them all. Further, nodal analysis tends to “give us what we want”, namely, a set of node voltages for the circuit. Once the node voltages are obtained, finding any branch currents or component powers becomes an almost trivial exercise. Nodal analysis relies on the application of Kirchhoff's current law to create a series of node equations that can be solved for node voltages. These equations are based on Ohm's law and will be of the form , or more generally, We will examine two variations on the theme; first, a general version that can be used with both voltage and current sources, and a second somewhat quicker version that can be used with circuits only driven by current sources. General Method Consider the circuit shown in Figure . We begin by labeling connection nodes. We are interested in identifying current junctions, that is, places where currents can combine or split. These are also known as summing nodes and are circled in blue on the figure. We do not concern ourselves with points where just two components connect without any other connection, such as points and . Once the proper nodes are identified, reference current directions are assigned. The reference current directions are chosen arbitrarily and for convenience. They may be the opposite of reality. This is not a problem. If we assign directions that are reversed, we'll simply wind up with a current version of a double negative, and the computed node voltages will work out just fine. One node is chosen as the reference. This is the point to which all other node voltages are measured against. Typically, the reference node is ground, although it does not have to be. We now write a current summation equation for each summing node, except for the reference node. In this circuit there is only one node where currents combine (other than ground) and that's node . Points and are places where components connect, but they are not summing nodes, so we can ignore them for now. Using KCL on node we can say: Now we'll describe these currents in terms of the source and node voltages, and associated components. For example, is the node voltage divided by while is the voltage across divided by . This voltage is . Noting that and , with a little algebra this can be reduced to: All quantities are known except for and thus it is easily found with a little more algebra. If there had been more nodes, there would have been more equations and more unknowns, one for each node. As we shall see, this conductance-voltage product format turns out to be a convenient way of writing these equations. Also, note that the first two terms on the left reduce to fixed current values. For current sources, the process is similar but a bit more direct. Consider the circuit of Figure . We start as before, identifying nodes and labeling currents. We then write the current summation equations at each node (except for ground). We consider currents entering a node as positive and exiting as negative. There are two nodes of interest here, and thus, two equations each with two unknowns will be generated. Node : Node : , and rearranging in terms of the fixed source, Node : The currents are then described by their Ohm's law equivalents: Node : Node : Expanding and collecting terms yields: Node : Node : As the impedance values and currents are known, simultaneous equation solution techniques may be used to solve for the node voltages. Once again, there are as many equations as node voltages. One practical point before continuing: It is very important that the coefficients for the various node voltage terms “line up” when the final system of equations is written out. That is, there should be a column for the terms, a column for the terms, and so on. They should not be written out in random order, but rather following the style shown in Figure . This format will make it much it easier to enter the coefficients into a calculator or solve manually. Further, the set of coefficients must show diagonal symmetry. That is, if we draw a major diagonal from upper-left to lower-right (red), whatever coefficients are above-right from the diagonal should be mirrored below-left of the diagonal (blue, purple, green). If the set of values does not show diagonal symmetry, an error has been made. You must go back and recheck the original node summations. Simple as that. Even the simple 2x2 of Figure shows this symmetry (namely, the coefficient of for in the first equation and in the second). Example Determine the voltage across the inductor in the circuit of Figure . Source one is volts RMS and source two volts RMS. Other than ground, there is only one current summing node in this circuit, and that's the junction at the top of the inductor. We will refer to this junction as node . Following the outline of Figure , we define three currents; entering from the left, entering from the right, and exiting down through the inductor. Next, these currents are described in terms of the voltages and components. We'll number the resistors from left to right. This can be rearranged as: Populate with values: This simplifies to: Solving for the unknown, we find that volts RMS. Example In the circuit of Figure , determine and . is volts peak while is amps peak. The system frequency is 2 kHz. There are two nodes of interest here, other than ground. This means we will generate two equations with two unknowns ( and ). Using the standard reactance formulas, the inductive and capacitive reactances are found to be and , respectively. If we assume the reference direction for current is from node to node , and that the current flow through the two center resistors is downward, the equations are: Node : Node : Expanding and collecting terms yields: Node : Node : These are simplified, ready for manipulation (note diagonal symmetry). After solving the system of equations, we see that volts and volts. Computer Simulation To verify the results of the preceding example, the circuit of Figure is captured in a simulator as shown in Figure . A transient analysis is performed on the circuit. Node voltages 1, 3 and 4 are plotted, corresponding to the voltage source and nodes and , respectively, in Figure . The amplitudes are just as computed. Node voltage appears to be nearly in phase with the voltage source, as expected. Node voltage lags the source by between one-quarter to one-third of a division, or some 30 microseconds. For a 2 kHz source, this translates to around −22 degrees, verifying the calculated result. Example For the circuit of Figure , find and . The system frequency is 1 kHz. A and A. We will generate two equations with two unknowns, and . The reactance formulas yield and for the inductor and capacitor. If we assume the reference direction for current is from node to node , and that the current flow through the capacitor and inductor is from nodes and downward, the equations are: Node : Node : Expanding and collecting terms yields (note diagonal symmetry): The results are: volts and volts. Computer Simulation To verify the results of the preceding example, the circuit of Figure is captured in a simulator as shown in Figure . A transient analysis is performed on the circuit. Node voltages 1 and 2 (i.e., nodes and , respectively) are plotted in Figure . The simulation results agree nicely with the computed values in terms of both amplitude and phase. Inspection Method The system of equations can be obtained directly through inspection if the circuit contains current sources and no voltage sources. Let's take another look at the equations developed in the preceding example. For convenience, the circuit is reproduced in Figure with reactance values. A and A. The top equation was built around a current summation at node while the bottom was built around a summation at node . The first thing that might be apparent is that on the left of the equals signs are the current sources connected to these nodes. Positive means the current is entering while negative denotes an exiting current. The second thing is that, for the node of interest (node for the top equation, node for the bottom), the coefficients represent the items connected to that particular node. For example, in the top equation, the components connected to node are the 10 resistor and the reactance. Likewise, in the bottom equation, the components connected to node are the 10 resistor and the impedance. The third thing is that the remaining coefficients consist of the components that are in common between the node of interest and the other node (i.e., 10 connects to for the first equation, and also connects to for the second equation). These other connections always show up as negative. The reason for this should be apparent if you examine the structure of the original equations from Example . If there is no bridging element between a node and the node of interest, then that coefficient will be zero. If voltage sources exist in the circuit, source conversions can be used to obtain an equivalent circuit that uses only current sources. The huge advantage of the inspection method is that it cuts out a time consuming and error prone section of the process, namely converting the original KCL summations into a set of simplified equations with coefficients for each unknown. The inspection method generates the equations directly. To further speed the process, it can be useful to turn each impedance value into a corresponding admittance value before creating the equations. In this way, the reciprocals are computed once for each item rather than multiple times in multiple equations. Finally, remember that the resulting set of equations must exhibit diagonal symmetry, as shown back in Figure . The inspection method is summarized as follows: Verify that the circuit uses only current sources and no voltage sources. If voltage sources exist, they must be converted to current sources before proceeding. Find all of the current summing nodes and number (or letter) them. Also decide on the reference node (usually ground). To generate an equation, locate the first node. This is the node of interest and the next few steps will be associated with it. Sum the current sources feeding the node of interest. Entering is deemed positive while exiting is deemed negative. The sum is placed on one side of the equals sign. Next, find all of the impedances connected to the node of interest and write them as a sum of admittances on the other side of the equals sign, the group being multiplied by this node's voltage (e.g., ). That makes one term. Now for the other terms. Find all of the admittances that are connected between the node of interest and the next node (e.g., node 2). Sum these together and multiply the group by this other node's voltage (e.g., ). Subtract that product from the equation built so far. Repeat this process until all of the other nodes have been examined (except ground). If there are no common impedances between the node of interest and the other node, use zero for the coefficient of that node's voltage. Once all other nodes are considered, this equation is finished. Go to the next node and treat this as the new node of interest. Repeat steps 4 through 7 until all nodes have been treated as the node of interest. Each iteration creates a new equation. There will be as many equations as there are nodes, less the reference node. Check for diagonal symmetry and solve. The inspection method is best observed in action, and is used in the following example. Example Write the system of equations for the circuit of Figure . A and A. We begin at node , the first node of interest. Find all of the current sources connected to this node. All we have is . It is exiting, and thus negative. Now find all of the items connected to this node and create a sum of admittances. Include the terms that are common between node and node . This is negative. And finally, include the terms common between node and node . This is also negative. The first equation is done. We now make node the node of interest and repeat the process. Find all of the current sources connected to this node. There are none. Find all of the items connected to this node and create a sum of admittances. Include the terms that are common between node and node . This is negative and goes into the lead (a before b) to keep everything nicely lined up. Now include the terms common between node and node . This is also negative and is inserted at the tail (c after b). The second equation is finished. We now make node the node of interest and repeat the process for the final time. Find all of the current sources connected to node . We have both and entering. This current is equivalent to . Now find all of the items connected to this node and create a sum of admittances. Include the terms that are common between node and node . This is negative and goes into the lead (a before c). Now include the terms common between node and node . This is also negative and is inserted in the middle (b before c). The third and final equation is finished. The completed set of equations is: Note that the set exhibits diagonal symmetry and that all coefficient groups are negative except for those along the major diagonal. Consequently, the coefficient groups may now be simplified to obtain single coefficients for the unknowns, and the equations are ready for solution. The results are: volts, volts and volts. These values can be crosschecked by using them to find the currents through each component, and then verifying KCL for each node. Although this example may appear to be somewhat long winded, with a little practice the process will become second nature. At that point, the set of equations can be created quickly and with little possibility of error, even for large circuits with many nodes. Using Source Conversions As mentioned previously, given circuits with voltage sources, it may be easier to convert them to current sources and then apply the inspection technique rather than using the general approach outlined initially. There is one trap to watch out for when using source conversions: the voltage across or current through a converted component will most likely not be the same as the voltage or current in the original circuit. This is because the location of the converted component will have changed. For example, the circuit of Figure (Example ) could be solved using the inspection method of nodal analysis by converting the voltage source and its associated impedance of the 100 \Omega resistor in series with the 10 mH inductor into a current source. Although the associated impedance still connects to the converted source, the other end no longer connects to node . Rather, it would connect to ground. Therefore, the voltage drop across this impedance in the converted circuit is not likely to equal the voltage drop seen across it in the original circuit (the only way they would be equal is if the voltage source turned out to be 0). Supernode From time to time you may see a circuit utilizing an ideal voltage source like the one shown in Figure . That is, this voltage source does not have a series impedance associated with it. Without that impedance, it becomes impossible to create an expression for the current passing through the source using the general method, and impossible to convert the voltage source into a current source in order to use the inspection method. There are a few of ways out of this quandary. The first way is to recognize that all realistic sources have some internal impedance, so we simply add a very small resistor in series with the source so that a source conversion is possible. Of course, not just any resistor will do. In order to maintain accuracy, the newly added resistance has to be much smaller than any surrounding resistances or reactances. A reduction by two orders of magnitude generally yields a variation smaller than that produced by component tolerances in all but high precision circuits and will usually do the trick. Still smaller values will further increase accuracy. Another way out is to use a supernode. A supernode is, in effect, the combination of two nodes. It relies on a simple observation. If we examine the circuit of Figure , the path of the voltage source produces identical currents flowing into and out of nodes and . As a consequence, if we treat the two nodes as one big node, then when we write a KCL summation, these two terms will cancel. To see just how this works, refer to Figure . In this version we have replaced the voltage source with its ideal internal impedance; a short. We have also labeled the two nodes of interest, and , and labeled the currents, drawn with convenient reference directions. The specific choice of direction will not matter, just use whatever scheme seems appropriate. Due to the shorted voltage source, nodes and are now the same node. Take a look at the currents entering and exiting this combined “super” node. On the left side (formerly node ) we see a constant current entering while , and are exiting. On the right side (formerly node ) we see the entering along with and , and exiting we see . At this point we'll create an expression where all of currents entering the super node are on the left side of the equals sign and all of the exiting currents are on the right: This can be simplified to: Writing this in terms of Ohm's law we have: We also know that from the original circuit. We know this because the reference polarity of the source is + toward the node and − toward the node. Therefore it must be and not . Assuming all sources and components are known, that makes two equations with two unknowns, solvable using simultaneous equation techniques. This is illustrated in the following example. Example Find and for the circuit of Figure . volts, amps and amps. The circuit is redrawn in Figure with nodes and currents labeled. We short the 16 volt source and write a current summation at the supernode: This can be simplified to: Writing this in terms of Ohm's law we have: We also know that volts. Therefore volts. Substituting this into the prior equation yields: We know that is volts below , and thus after subtracting, we find volts. To verify, we will perform a KCL summation at each node. For node , assuming exits as drawn: Doing likewise for node , and assuming enters as drawn: Other than the small deviation due to accumulated rounding, these currents match. That means that the current through the voltage source is verified to be the same at both terminals, as it must be. An alternative to the basic supernode technique is to recognize that the two nodes on either side of the voltage source are effectively locked together by the source voltage. That is, if one of the node voltages is found, then the other may be determined by adding or subtracting the source voltage to or from the known node voltage, depending on the reference polarity. This idea is exploited by simply describing one node voltage in terms of the other at the outset. This will reduce the total number of unknowns by one and reduce the system of equations by one. The technique is illustrated in the example following. Example Find node voltages , and for the circuit of Figure . The sources are: volts and amps. Once again we have a situation of a voltage source lacking a series impedance which makes a source conversion impossible. Without having to short it and thus treating nodes and as an explicit supernode, we can take an alternate route. We begin by noting that the currents entering and exiting the voltage source must be identical. The circuit is redrawn in Figure with the nodes and convenient current directions labeled. The circuit also uses equivalent conductances and susceptances in place of the original resistances and reactances in order to speed the process of simplifying the equations. Unlike the basic supernode technique, this time the voltage source is left in. The key observation is that V. In other words, is locked to and if we find one of them, we can determine the other. Therefore, instead of writing three equations using three unknowns, we shall instead refer to node in reference to node . In other words, wherever we need we instead shall write V. Thus, this three node circuit will only need two equations. We begin at node and apply KCL as usual. This is expanded using Ohm's law and we solve for : On to node : And finally node : The equations for nodes and both equal , thus they equal each other. The final equations are: The solution is volts and volts. As is volts less than , then volts. KCL summations at each of the three nodes will verify these values. 6.1: Introduction 6.3: Mesh Analysis
2122	https://mathworld.wolfram.com/CyclicPermutation.html	Cyclic Permutation -- from Wolfram MathWorld TOPICS AlgebraApplied MathematicsCalculus and AnalysisDiscrete MathematicsFoundations of MathematicsGeometryHistory and TerminologyNumber TheoryProbability and StatisticsRecreational MathematicsTopologyAlphabetical IndexNew in MathWorld Discrete Mathematics Combinatorics Permutations Cyclic Permutation A permutation which shifts all elements of a set by a fixed offset, with the elements shifted off the end inserted back at the beginning. For a set with elements , , ..., , a cyclic permutation of one place to the left would yield , ..., , , and a cyclic permutation of one place to the right would yield , , , .... The mapping can be written as for a shift of places. A shift of places to the left is implemented in the Wolfram Language as RotateLeft[list, k], while a shift of places to the right is implemented as RotateRight[list, k]. See also Permutation Explore with Wolfram\|Alpha More things to try: 3-color code 1086 d/dx[ x f(x^2) ] last digit of 9^9^9 Cite this as: Weisstein, Eric W. "Cyclic Permutation." From MathWorld--A Wolfram Resource. Subject classifications Discrete Mathematics Combinatorics Permutations About MathWorld MathWorld Classroom Contribute MathWorld Book wolfram.com 13,278 Entries Last Updated: Sun Sep 28 2025 ©1999–2025 Wolfram Research, Inc. Terms of Use wolfram.com Wolfram for Education Created, developed and nurtured by Eric Weisstein at Wolfram Research Created, developed and nurtured by Eric Weisstein at Wolfram Research
2123	https://brainly.com/question/24793921	[FREE] Design an algorithm that generates even numbers between 1000 and 2000, prints them to the standard output, - brainly.com 5 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +61k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +25,2k Ace exams faster, with practice that adapts to you Practice Worksheets +6,4k Guided help for every grade, topic or textbook Complete See more / Computers and Technology Textbook & Expert-Verified Textbook & Expert-Verified Design an algorithm that generates even numbers between 1000 and 2000, prints them to the standard output, and calculates the total sum of all the even numbers. Represent your algorithm with corresponding pseudocode and a flowchart. Pseudocode: 1. Initialize sum as 0 2. For number from 1000 to 2000 (inclusive): a. If number is even: i. Print number ii. Add number to sum 3. Print total sum Flowchart: Start \| v Initialize sum = 0 \| v For number = 1000 to 2000 \| v Is number even? \| \ \| \ Yes No \| \| v v Print number Continue loop \| v Add number to sum \| v Continue loop \| v Print total sum \| v End 1 See answer Explain with Learning Companion NEW Asked by Ayuub926 • 09/26/2021 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 415973526 people 415M 4.5 13 Upload your school material for a more relevant answer Algorithms are simply the methods and procedures to solve a problem They can be in graphical form (i.e. flowcharts) They can also be in form of texts (i.e. pseudocodes) Algorithms do not conform to programming rules, and they do not have definite syntax. The pseudocode that represents the required algorithm is as follows: Sum = 0 For even = 1000 to 2000 step = 2 print even sum = sum + even print sum The first line of the pseudocode intializes sum to 0 The next line iterates through even numbers between 1000 and 2000 Within the iteration, the even numbers are printed and the sum is calculated After the iteration, the sum of even numbers is printed The last line ends the pseudocode The above explanation is also applicable to the flowchart (see attachment) Read more about algorithms, flowchart and pseudocode at: brainly.com/question/21220060 Answered by MrRoyal •54.5K answers•416M people helped Thanks 13 4.5 (14 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 415973526 people 415M 4.6 12 Python for Informatics: Exploring Information - Charles Severance A Primer for Computational Biology - Shawn T. O’Neil Computer Organization - Sukjin Lee Upload your school material for a more relevant answer The algorithm to generate even numbers from 1000 to 2000 involves iterating through this range, checking for even numbers, and summing them. Pseudocode and a flowchart can effectively illustrate this algorithm. The final output will include the even numbers and their total sum. Explanation To design an algorithm that generates even numbers between 1000 and 2000 and calculates their sum, we can use pseudocode and a flowchart to represent the procedure. Pseudocode: Initialize sum as 0 For number from 1000 to 2000 (inclusive): a. If number is even, then: i. Print number ii. Add number to sum 3. Print total sum This pseudocode initializes a variable sum to 0, then iterates through each number from 1000 to 2000. For every number, it checks if the number is even. If it is, it prints the number and adds it to the sum. After the loop finishes, it prints the total sum of all the even numbers collected. Flowchart: Start Initialize sum = 0 For number = 1000 to 2000: Check if number is even: If Yes: a. Print number b. Add number to sum If No: a. Continue the loop After the loop, print the total sum End The flowchart visually represents the steps of the algorithm, showing the initialization, looping, conditional checks, and the final output. This method of outlining the process makes it easier to understand how the algorithm works before it is implemented in code. Examples & Evidence For instance, if we look at the range from 1000 to 2000, the even numbers include 1000, 1002, 1004, and so on up to 2000. The algorithm will print each of these numbers and add them to the total sum, which will be 999000 in the end. The method described is a standard programming approach for generating and summing even numbers using basic iteration and conditional checks. Thanks 12 4.6 (12 votes) Advertisement Ayuub926 has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Computers and Technology solutions and answers Community Answer 1. Design an algorithm and a flowchart which generates even numbers between 1000 and 2000 and then prints them in the standard output. It should also print total sum. Community Answer 5.0 2: Design an algorithm which generates even numbers between 1000 and 2000 and then prints them in the standard output. It should also print total sum Community Answer 5.0 1 What can quantum computers do more efficiently than regular computers? Community Answer 1 You are experiencing issues when trying to transfer files between two computers using FTP. What could be the potential cause(s) of the issue Community Answer 4.5 117 describe what happens at every step of our network model, when a node on one network establishes a TCP connection with a node on another network. You can assume that the two networks are both connected to the same router. Your submission must include a detailed explanation of the following: Physical layer Data link layer Network layer Transport layer MAC address IP address TCP port Checksum check Routing table TTL Community Answer 19 To return the value of the cell D8, the formula should be OFFSETA1=__. New questions in Computers and Technology Which part of the Windows desktop interface is the file management system? A. finder B. start C. file explorer D. dock Which of the following forms of messaging would be the best substitute for a voice call? A. Text B. E-mail C. Memo The impact of Moore's Law is that: A. analog media becomes less expensive. B. processing power becomes faster and cheaper. C. computers become larger and quicker. D. digital media becomes more costly. What is one benefit of cloud computing? A. Your internet connection is faster. B. You can access your files from any device. C. You automatically have better security. D. You can use digital files instead of analog. Which two devices are associated with the shift from mobile phones to smartphones? A. iPad and Kindle B. laptop and desktop C. cell phone and iPad D. Blackberry and iPhone 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
2124	https://openstax.org/books/biology-2e/pages/23-2-characteristics-of-protists	23.2 Characteristics of Protists - Biology 2e \| OpenStax This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising purposes. Privacy Notice Customize Reject All Accept All Customize Consent Preferences We use cookies to help you navigate efficiently and perform certain functions. You will find detailed information about all cookies under each consent category below. The cookies that are categorized as "Necessary" are stored on your browser as they are essential for enabling the basic functionalities of the site. ...Show more For more information on how Google's third-party cookies operate and handle your data, see:Google Privacy Policy Necessary Always Active Necessary cookies are required to enable the basic features of this site, such as providing secure log-in or adjusting your consent preferences. 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Reject All Save My Preferences Accept All Skip to ContentGo to accessibility pageKeyboard shortcuts menu Log in Biology 2e 23.2 Characteristics of Protists Biology 2e23.2 Characteristics of Protists Contents Contents Highlights Table of contents Preface The Chemistry of Life The Cell Genetics Evolutionary Processes Biological Diversity 21 Viruses 22 Prokaryotes: Bacteria and Archaea 23 Protists Introduction 23.1 Eukaryotic Origins 23.2 Characteristics of Protists 23.3 Groups of Protists 23.4 Ecology of Protists Key Terms Chapter Summary Visual Connection Questions Review Questions Critical Thinking Questions 24 Fungi 25 Seedless Plants 26 Seed Plants 27 Introduction to Animal Diversity 28 Invertebrates 29 Vertebrates Plant Structure and Function Animal Structure and Function Ecology A \| The Periodic Table of Elements B \| Geological Time C \| Measurements and the Metric System Index Search for key terms or text. Close Learning Objectives By the end of this section, you will be able to do the following: Describe the cell structure characteristics of protists Describe the metabolic diversity of protists Describe the life cycle diversity of protists There are over 100,000 described living species of protists, and it is unclear how many undescribed species may exist. Since many protists live as commensals or parasites in other organisms and these relationships are often species-specific, there is a huge potential for protist diversity that matches the diversity of their hosts. Because the name "protist" serves as a catchall term for eukaryotic organisms that are not animal, plant, or fungi, it is not surprising that very few characteristics are common to all protists. On the other hand, familiar characteristics of plants and animals are foreshadowed in various protists. Cell Structure The cells of protists are among the most elaborate of all cells. Multicellular plants, animals, and fungi are embedded among the protists in eukaryotic phylogeny. In most plants and animals and some fungi, complexity arises out of multicellularity, tissue specialization, and subsequent interaction because of these features. Although a rudimentary form of multicellularity exists among some of the organisms labelled as “protists,” those that have remained unicellular show how complexity can evolve in the absence of true multicellularity, with the differentiation of cellular morphology and function. A few protists live as colonies that behave in some ways as a group of free-living cells and in other ways as a multicellular organism. Some protists are composed of enormous, multinucleate, single cells that look like amorphous blobs of slime, or in other cases, like ferns. In some species of protists, the nuclei are different sizes and have distinct roles in protist cell function. Single protist cells range in size from less than a micrometer to three meters in length to hectares! Protist cells may be enveloped by animal-like cell membranes or plant-like cell walls. Others are encased in glassy silica-based shells or wound with pellicles of interlocking protein strips. The pellicle functions like a flexible coat of armor, preventing the protist from being torn or pierced without compromising its range of motion. Metabolism Protists exhibit many forms of nutrition and may be aerobic or anaerobic. Those that store energy by photosynthesis belong to a group of photoautotrophs and are characterized by the presence of chloroplasts. Other protists are heterotrophic and consume organic materials (such as other organisms) to obtain nutrition. Amoebas and some other heterotrophic protist species ingest particles by a process called phagocytosis, in which the cell membrane engulfs a food particle and brings it inward, pinching off an intracellular membranous sac, or vesicle, called a food vacuole (Figure 23.7). In some protists, food vacuoles can be formed anywhere on the body surface, whereas in others, they may be restricted to the base of a specialized feeding structure. The vesicle containing the ingested particle, the phagosome, then fuses with a lysosome containing hydrolytic enzymes to produce a phagolysosome, and the food particle is broken down into small molecules that can diffuse into the cytoplasm and be used in cellular metabolism. Undigested remains ultimately are expelled from the cell via exocytosis. Figure 23.7 Phagocytosis. The stages of phagocytosis include the engulfment of a food particle, the digestion of the particle using hydrolytic enzymes contained within a lysosome, and the expulsion of undigested materials from the cell. Subtypes of heterotrophs, called saprobes, absorb nutrients from dead organisms or their organic wastes. Some protists can function as mixotrophs, obtaining nutrition by photoautotrophic or heterotrophic routes, depending on whether sunlight or organic nutrients are available. Motility The majority of protists are motile, but different types of protists have evolved varied modes of movement (Figure 23.8). Some protists have one or more flagella, which they rotate or whip. Others are covered in rows or tufts of tiny cilia that they beat in a coordinated manner to swim. Still others form cytoplasmic extensions called pseudopodia anywhere on the cell, anchor the pseudopodia to a substrate, and pull themselves forward. Some protists can move toward or away from a stimulus, a movement referred to as taxis. For example, movement toward light, termed phototaxis, is accomplished by coupling their locomotion strategy with a light-sensing organ. Figure 23.8 Locomotor organelles in protists. Protists use various methods for transportation. (a) Paramecium waves hair-like appendages called cilia to propel itself. (b) Amoeba uses lobe-like pseudopodia to anchor itself to a solid surface and pull itself forward. (c) Euglena uses a whip-like tail called a flagellum to propel itself. Life Cycles Protists reproduce by a variety of mechanisms. Most undergo some form of asexual reproduction, such as binary fission, to produce two daughter cells. In protists, binary fission can be divided into transverse or longitudinal, depending on the axis of orientation; sometimes Paramecium exhibits this method. Some protists such as the true slime molds exhibit multiple fission and simultaneously divide into many daughter cells. Others produce tiny buds that go on to divide and grow to the size of the parental protist. Sexual reproduction, involving meiosis and fertilization, is common among protists, and many protist species can switch from asexual to sexual reproduction when necessary. Sexual reproduction is often associated with periods when nutrients are depleted or environmental changes occur. Sexual reproduction may allow the protist to recombine genes and produce new variations of progeny, some of which may be better suited to surviving changes in a new or changing environment. However, sexual reproduction is often associated with resistant cysts that are a protective, resting stage. Depending on habitat of the species, the cysts may be particularly resistant to temperature extremes, desiccation, or low pH. This strategy allows certain protists to “wait out” stressors until their environment becomes more favorable for survival or until they are carried (such as by wind, water, or transport on a larger organism) to a different environment, because cysts exhibit virtually no cellular metabolism. Protist life cycles range from simple to extremely elaborate. Certain parasitic protists have complicated life cycles and must infect different host species at different developmental stages to complete their life cycle. Some protists are unicellular in the haploid form and multicellular in the diploid form, a strategy employed by animals. Other protists have multicellular stages in both haploid and diploid forms, a strategy called alternation of generations, analogous to that used by plants. Habitats Nearly all protists exist in some type of aquatic environment, including freshwater and marine environments, damp soil, and even snow. Several protist species are parasites that infect animals or plants. A few protist species live on dead organisms or their wastes, and contribute to their decay. PreviousNext Order a print copy Citation/Attribution This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax. Attribution information If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution: Access for free at If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution: Access for free at Citation information Use the information below to generate a citation. We recommend using a citation tool such as this one. 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2125	https://www.examples.com/physics/units-of-wavelength.html	Units of Wavelength- Examples, Definition, Units, Conversion Loading [MathJax]/jax/output/HTML-CSS/config.js Subjects English Maths Chemistry Physics Biology Docs Design Business Education Resources English Maths Chemistry Physics Biology Docs Design Business Education AP Practice Tests AP English AP English Language AP English Literature AP Sciences AP Chemistry AP Biology AP Environmental Science AP Physics 1 AP® Physics 2: Algebra-Based AP® Physics C: Electricity and Magnetism AP® Physics C: Mechanics AP® Psychology AP Math and Computer Science AP Calculus AB AP Calculus BC AP® Computer Science A AP® Computer Science Principles AP® Precalculus AP® Statistics AP History and Social Sciences AP® European History AP® Human Geography AP® Macroeconomics AP® Microeconomics AP® United States Government and Politics AP® United States History AP® World History: Modern Accounting CPA CMA CIA Digital Sat Finance CFA CMT More Grad School MCAT Legal MBE Other ACT Digital Sat HomeUnits of Wavelength Physics Units and Measurements – 20+ Examples, Types, Uses Force – 20+ Examples, Formula, Units, Types, Characteristics Inertia – 19+ Examples, Formula, Units, Types Light Energy – 30+ Examples, Formula, Units, Types, Uses Ampere – Definition, Units, Conversion, Types, FAQ’S Units of Electric Flux – Examples, Definition, Units, Conversion Units of Electric Field – Examples, Definition, Units, Conversion Units of Displacement – Examples, Definition, Units, Conversion Units of Vibration – Examples, Definition, Units, Conversion Chart Units of Distance – Definition, Units, Formula, Conversion Units of Moment of Inertia – Definition, Formula, Units, Examples, Conversion Units of Torque – Examples, Definition, Units, Conversion Chart Units of Sound – Definition, Characteristics, Units, Conversion Units of Inductance – Definition, Formula, Units, Conversion Units of Radioactivity – Definition, Formula, Units, Conversion Units of Viscosity – Definition, List of Units, Conversion Chart Units of Current – Examples, Definition, Units, Conversion Chart Units of Magnetic Flux – Definition, List of Units, Conversion Chart Units of Temperature – Examples, Definition, Units, Conversion Units of Acceleration – Definition, List of Units, Conversion Chart Home Physics Units of Wavelength- Examples, Definition, Units, Conversion Last Updated:July 3, 2024 Notes 1 save Save Notes Units of Wavelength- Examples, Definition, Units, Conversion Wavelength, the distance between successive peaks or troughs of a wave, is crucial in various scientific fields such as physics, optics, and telecommunications. Understanding the units used to measure wavelength allows for precise communication and calculation in these disciplines. What are Units of Wavelength? Wavelength, a key measure in the fields of physics, optics, and communications, quantifies the distance between consecutive points of a wave cycle. Understanding these units is essential for accurately describing wave properties across various applications. SI Unit of Wavelength meter (m) The meter (m) is the official SI unit for measuring wavelength. This unit is versatile, encompassing wavelengths that span from the very small, such as ultraviolet light, to the comparatively large, like radio waves. The meter’s universal applicability ensures that it is essential in various scientific fields, including physics, chemistry, and astronomy. CGS Unit of Wavelength centimeter (cm) In the Centimeter–Gram–Second(CGS) system, the centimeter (cm) is the standard unit for measuring wavelength. This unit is particularly useful for recording wavelengths of electromagnetic waves such as infrared radiation, where precision is crucial. Formula of Wavelength The formula for wavelength (𝜆) is: 𝜆=𝑣^𝑓 𝑣 is the speed of the wave, 𝑓 is the frequency of the wave. This equation calculates the distance over which the wave’s shape repeats. List of Wavelength Units Here is the table format for different units of wavelength, organized similar to your example of energy units: \| Unit \| Symbol \| --- \| \| Meter \| m \| \| Centimeter \| cm \| \| Nanometer \| nm \| \| Angstrom \| Å \| \| Micrometer \| µm \| \| Millimeter \| mm \| Here’s the detailed format for different units of wavelength, similar to the format used for units of energy: Meter (m) 1 m The meter is the SI unit of wavelength used to measure the distance over which a wave’s shape repeats, such as in light or sound waves. It is universally utilized in scientific research, telecommunications, and optics. Centimeter (cm) 1 cm = 0.01 m Centimeters are often used to describe the wavelengths of microwaves, infrared radiation, and other types of electromagnetic waves that require precision on a smaller scale. Nanometer (nm) 1 nm = 1 x 10^-9 m Nanometers measure the wavelengths of light in the visible spectrum and other forms of electromagnetic radiation such as ultraviolet and infrared light, crucial in fields like photonics and materials science. Angstrom (Å) 1 Å = 1 x 10^-10 m Angstroms are used specifically in the fields of chemistry and crystallography to measure the scale of atoms and the spacing between elements in a crystal structure. Micrometer (μm) 1 μm = 1 x 10^-6 m Micrometers are used to measure wavelengths of infrared radiation and various other types of light that are slightly longer than the visible spectrum. Millimeter (mm) 1 mm = 0.001 m Millimeters are suitable for measuring radio wavelengths and other longer waves, useful in radar and other wireless communications technologies. Conversion of Wavelength Units Here is a table format for different units of wavelength, showing conversion factors among common units like meters, nanometers, angstroms, and micrometers: \| Wavelength Unit \| Meter (m) \| Nanometer (nm) \| Angstrom (Å) \| Micrometer (µm) \| --- --- \| Meter (m) \| 0 \| 1E9 \| 1E10 \| 1E6 \| \| Nanometer (nm) \| 1E-9 \| 0 \| 10 \| 1E-3 \| \| Angstrom (Å) \| 1E-10 \| 0.1 \| 0 \| 1E-4 \| \| Micrometer (µm) \| 1E-6 \| 1E3 \| 1E4 \| 0 \| Meter to Nanometer Conversion: 1 meter = 1 x 10^9 nanometers. Example: To convert 0.0001 meters to nanometers: 0.0001 m × 1 x 10^9 = 100,000 nm Meter to Angstrom Conversion: 1 meter = 1 x 10^10 angstroms. Example: To convert 0.0001 meters to angstroms: 0.0001 m × 1 x 10^10 = 1,000,000,000 Å Meter to Micrometer Conversion: 1 meter = 1 x 10^6 micrometers. Example: To convert 0.001 meters to micrometers: 0.001 m × 1 x 10^6 = 1,000 µm Nanometer to Meter Conversion: 1 nanometer = 1 x 10^-9 meters. Example: To convert 500 nanometers to meters: 500 nm × 1 x 10^-9 = 0.0000005 m Nanometer to Angstrom Conversion: 1 nanometer = 10 angstroms. Example: To convert 500 nanometers to angstroms: 500 nm × 10 = 5,000 Å Nanometer to Micrometer Conversion: 1 nanometer = 0.001 micrometers. Example: To convert 800 nanometers to micrometers: 800 nm × 0.001 = 0.8 µm Angstrom to Meter Conversion: 1 angstrom = 1 x 10^-10 meters. Example: To convert 1,000,000 angstroms to meters: 1,000,000 Å × 1 x 10^-10 = 0.0001 m Angstrom to Nanometer Conversion: 1 angstrom = 0.1 nanometers. Example: To convert 1,000,000 angstroms to nanometers: 1,000,000 Å × 0.1 = 100,000 nm Angstrom to Micrometer Conversion: 1 angstrom = 1 x 10^-4 micrometers. Example: To convert 10,000,000 angstroms to micrometers: 10,000,000 Å × 1 x 10^-4 = 1,000 µm Micrometer to Meter Conversion: 1 micrometer = 1 x 10^-6 meters. Example: To convert 2000 micrometers to meters: 2000 µm × 1 x 10^-6 = 0.002 m Micrometer to Nanometer Conversion: 1 micrometer = 1,000 nanometers. Example: To convert 1 micrometer to nanometers: 1 µm × 1,000 = 1,000 nm Micrometer to Angstrom Conversion: 1 micrometer = 10,000 angstroms. Example: To convert 1 micrometer to angstroms: 1 µm × 10,000 = 10,000 Å FAQ’S What is the SI unit and CGS unit of wavelength? The SI unit of wavelength is the meter (m), and the CGS unit of wavelength is also the centimeter (cm), reflecting smaller-scale measurements. What is the SI unit of wavelength and frequency? The SI unit of wavelength is the meter (m), and the SI unit of frequency is the hertz (Hz), crucial for characterizing waves accurately. What is symbol and SI unit of wavelength? The symbol for wavelength is λ (lambda), and its SI unit is the meter (m), essential for specifying wave distances in science. Share : Set content Set content Set content Set content Set content Save Download Units of Wavelength- Examples, Definition, Units, Conversion Wavelength, the distance between successive peaks or troughs of a wave, is crucial in various scientific fields such as physics, optics, and telecommunications. Understanding the units used to measure wavelength allows for precise communication and calculation in these disciplines. What are Units of Wavelength? Wavelength, a key measure in the fields of physics, optics, and communications, quantifies the distance between consecutive points of a wave cycle. Understanding these units is essential for accurately describing wave properties across various applications. SI Unit of Wavelength meter (m) The meter(m) is the official SI unit for measuring wavelength. This unit is versatile, encompassing wavelengths that span from the very small, such as ultraviolet light, to the comparatively large, like radio waves. The meter’s universal applicability ensures that it is essential in various scientific fields, including physics, chemistry, and astronomy. CGS Unit of Wavelength centimeter (cm) In the Centimeter–Gram–Second(CGS) system, the centimeter (cm) is the standard unit for measuring wavelength. This unit is particularly useful for recording wavelengths of electromagnetic waves such as infrared radiation, where precision is crucial. Formula of Wavelength The formula for wavelength (𝜆) is: 𝜆=𝑣^𝑓 𝑣 is the speed of the wave, 𝑓 is the frequency of the wave. This equation calculates the distance over which the wave’s shape repeats. List of Wavelength Units Here is the table format for different units of wavelength, organized similar to your example of energy units: \| Unit \| Symbol \| --- \| \| Meter \| m \| \| Centimeter \| cm \| \| Nanometer \| nm \| \| Angstrom \| Å \| \| Micrometer \| µm \| \| Millimeter \| mm \| Here’s the detailed format for different units of wavelength, similar to the format used for units of energy: Meter (m) 1 m The meter is the SI unit of wavelength used to measure the distance over which a wave’s shape repeats, such as in light or sound waves. It is universally utilized in scientific research, telecommunications, and optics. Centimeter (cm) 1 cm = 0.01 m Centimeters are often used to describe the wavelengths of microwaves, infrared radiation, and other types of electromagnetic waves that require precision on a smaller scale. Nanometer (nm) 1 nm = 1 x 10^-9 m Nanometers measure the wavelengths of light in the visible spectrum and other forms of electromagnetic radiation such as ultraviolet and infrared light, crucial in fields like photonics and materials science. Angstrom (Å) 1 Å = 1 x 10^-10 m Angstroms are used specifically in the fields of chemistry and crystallography to measure the scale of atoms and the spacing between elements in a crystal structure. Micrometer (μm) 1 μm = 1 x 10^-6 m Micrometers are used to measure wavelengths of infrared radiation and various other types of light that are slightly longer than the visible spectrum. Millimeter (mm) 1 mm = 0.001 m Millimeters are suitable for measuring radio wavelengths and other longer waves, useful in radar and other wireless communications technologies. Conversion of Wavelength Units Here is a table format for different units of wavelength, showing conversion factors among common units like meters, nanometers, angstroms, and micrometers: \| Wavelength Unit \| Meter (m) \| Nanometer (nm) \| Angstrom (Å) \| Micrometer (µm) \| --- --- \| Meter (m) \| 0 \| 1E9 \| 1E10 \| 1E6 \| \| Nanometer (nm) \| 1E-9 \| 0 \| 10 \| 1E-3 \| \| Angstrom (Å) \| 1E-10 \| 0.1 \| 0 \| 1E-4 \| \| Micrometer (µm) \| 1E-6 \| 1E3 \| 1E4 \| 0 \| Meter to Nanometer Conversion: 1 meter = 1 x 10^9 nanometers. Example: To convert 0.0001 meters to nanometers: 0.0001 m × 1 x 10^9 = 100,000 nm Meter to Angstrom Conversion: 1 meter = 1 x 10^10 angstroms. Example: To convert 0.0001 meters to angstroms: 0.0001 m × 1 x 10^10 = 1,000,000,000 Å Meter to Micrometer Conversion: 1 meter = 1 x 10^6 micrometers. Example: To convert 0.001 meters to micrometers: 0.001 m × 1 x 10^6 = 1,000 µm Nanometer to Meter Conversion: 1 nanometer = 1 x 10^-9 meters. Example: To convert 500 nanometers to meters: 500 nm × 1 x 10^-9 = 0.0000005 m Nanometer to Angstrom Conversion: 1 nanometer = 10 angstroms. Example: To convert 500 nanometers to angstroms: 500 nm × 10 = 5,000 Å Nanometer to Micrometer Conversion: 1 nanometer = 0.001 micrometers. Example: To convert 800 nanometers to micrometers: 800 nm × 0.001 = 0.8 µm Angstrom to Meter Conversion: 1 angstrom = 1 x 10^-10 meters. Example: To convert 1,000,000 angstroms to meters: 1,000,000 Å × 1 x 10^-10 = 0.0001 m Angstrom to Nanometer Conversion: 1 angstrom = 0.1 nanometers. Example: To convert 1,000,000 angstroms to nanometers: 1,000,000 Å × 0.1 = 100,000 nm Angstrom to Micrometer Conversion: 1 angstrom = 1 x 10^-4 micrometers. Example: To convert 10,000,000 angstroms to micrometers: 10,000,000 Å × 1 x 10^-4 = 1,000 µm Micrometer to Meter Conversion: 1 micrometer = 1 x 10^-6 meters. Example: To convert 2000 micrometers to meters: 2000 µm × 1 x 10^-6 = 0.002 m Micrometer to Nanometer Conversion: 1 micrometer = 1,000 nanometers. Example: To convert 1 micrometer to nanometers: 1 µm × 1,000 = 1,000 nm Micrometer to Angstrom Conversion: 1 micrometer = 10,000 angstroms. Example: To convert 1 micrometer to angstroms: 1 µm × 10,000 = 10,000 Å FAQ’S What is the SI unit and CGS unit of wavelength? The SI unit of wavelength is the meter (m), and the CGS unit of wavelength is also the centimeter (cm), reflecting smaller-scale measurements. What is the SI unit of wavelength and frequency? The SI unit of wavelength is the meter (m), and the SI unit of frequency is the hertz (Hz), crucial for characterizing waves accurately. What is symbol and SI unit of wavelength? The symbol for wavelength is λ (lambda), and its SI unit is the meter (m), essential for specifying wave distances in science. AI Generator Text prompt Add Tone Select a Tone Friendly Formal Casual Instructive Professional Empathetic Humorous Serious Optimistic Neutral Generate 10 Examples of Public speaking 20 Examples of Gas lighting Practice Test What is the unit of wavelength in the SI system? Choose the correct answer Meter Hertz Newton Coulomb of 10 Which of the following units is commonly used to measure wavelengths of visible light? Choose the correct answer Kilometer Centimeter Nanometer Gigameter of 10 How many nanometers are there in one meter? Choose the correct answer 10 100 1000 1,000,000,000 of 10 The wavelength of a radio wave is 3 meters. What is this wavelength in centimeters? Choose the correct answer 3 cm 30 cm 300 cm 3000 cm of 10 What is the typical unit used to measure the wavelength of X-rays? Choose the correct answer Meter Millimeter Micrometer Angstrom of 10 If the wavelength of a sound wave is 0.5 meters, what is its wavelength in millimeters? Choose the correct answer 0.5 mm 5 mm 50 mm 500 mm of 10 Which unit is larger, a micrometer or a nanometer? Choose the correct answer Micrometer Nanometer They are the same It depends on the context of 10 The wavelength of a particular color of light is 700 nm. What is this wavelength in micrometers? Choose the correct answer 0.07 µm 0.7 µm 7 µm 70 µm of 10 Which of the following units is NOT used to measure wavelength? Choose the correct answer Meter Hertz Angstrom Nanometer of 10 What is the wavelength in nanometers of ultraviolet light with a wavelength of 0.2 micrometers? 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2126	https://courses.lumenlearning.com/louisville-wm-physics/chapter/putting-it-together-vector-multiplication-and-subtraction/	Putting It Together: Scalar Multiplication and Vector Subtraction \| Waymaker Physics (Louisville) Skip to main content Waymaker Physics (Louisville) Module 5: Scalar Multiplication and Vector Subtraction Search for: Putting It Together: Scalar Multiplication and Vector Subtraction In practice, the similarities between working a vector addition problem and a vector subtraction problem should be striking. In both cases, you will make a vector diagram, choose a coordinate system, and break your vectors up into components. The only real difference between the two is whether you add the components together or subtract them. However you combine them, remember that you only want to combine x components with x components and y components with y components. The other difference between vector addition problems and vector subtraction problems that can cause mistakes is the order of your vectors. In a vector addition problem, the order doesn’t matter. →A+→B A→+B→ is the same as →B+→A B→+A→. However, order does matter in vector subtraction. The vector →A−→B A→−B→points in the opposite direction as the vector →B−→A B→−A→. When you are calculating the change in a vector quantity, make sure you always subtract the initial vector from the final vector, and not the other way around. Candela Citations CC licensed content, Original Putting It Together: Scalar Multiplication and Vector Subtraction. Authored by: Raymond Chastain. Provided by: University of Louisville, Lumen Learning. License: CC BY: Attribution Licenses and Attributions CC licensed content, Original Putting It Together: Scalar Multiplication and Vector Subtraction. Authored by: Raymond Chastain. Provided by: University of Louisville, Lumen Learning. License: CC BY: Attribution PreviousNext
2127	https://www.nist.gov/pml/owm/si-units-mass	SI Units - Mass \| NIST 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. 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Credit: ©Robert Rathe The primary standard of mass for this country is United States Prototype Kilogram 20, which is a platinum-iridium cylinder kept at NIST. The kilogram, originally defined as the mass of one cubic decimeter of water at the temperature of maximum density, was known as the Kilogram of the Archives. It was replaced after the International Metric Convention in 1875 by the International Prototype Kilogram which became the unit of mass without reference to the mass of a cubic decimeter of water or to the Kilogram of the Archives. Each country that subscribed to the International Metric Convention was assigned one or more copies of the international standards; these are known as National Prototype Meters and Kilograms. Learn more about the history and current definition of the kilogram. Among the SI base units, the kilogram (kg) is the only one whose name and symbol, for historical reasons, include a prefix. "Kilo" the SI prefix for 1000 or 10 3. Names and symbols for decimal multiples and submultiples of the unit of mass are formed by attaching prefix names to the unit name "gram," and prefix symbols to the unit symbol "g." Learn more about this historical quirk. \| Units of Mass \| \| 10 milligrams (mg) \| = \| 1 centigrams (cg) \| \| 10 centigrams \| = \| 1 decigrams (dg) = 100 milligrams \| \| 10 decigrams \| = \| 1 gram (g) \| \| 10 decigrams \| = \| 1000 milligrams \| \| 10 grams \| = \| 1 dekagrams (dag) \| \| 10 dekagrams \| = \| 1 hectogram (hg) \| \| 10 dekagrams \| = \| 100 grams \| \| 10 hectograms \| = \| 1 kilogram (kg) \| \| 10 hectograms \| = \| 1000 grams \| \| 1000 kilograms \| = \| 1 megagram (Mg) or 1 metric ton (t) \| Return to Top of Page Resources for Students and Teachers Teachers:Top 10 Tips for Teaching the Metric System. (NIST) Credit: ©2020 U.S. Secretary of Commerce. All Rights Reserved. Resources for K-2 Not sure how to tell that scientific minded person in your life how you feel? Maybe ourSuperhero Valentine's Day cards will do the trick. Download these fun coloring pages forMonsieur Kilogramand our SI Superheroes. The Metric Meal (option 2) (NIST) focuses exclusively on length measurements and uses a multidisciplinary approach to build measurement skills. Balancing Act (PhET, Simulation) - This interactive simulation lets students explore balance, torque, and rotational equilibrium by placing objects on a plank, predicting its motion, and solving puzzles using proportional reasoning. Credit: ©2020 U.S. Secretary of Commerce. All Rights Reserved. Resources for Grade 3-5 League of SI Superheroes- Monsieur Kilogram This comic book-style video animation series has been developed to help middle school students learn about the 7 SI base measurement units. SI Units Card Deck (SP 1297) (NIST) -This activity offers a fun way to enhance understanding of the International System of Units. Metric Estimation Game (SP 1336) (NIST) - This fun, hands-on activity helps participants become familiar with SI measurements by practicing estimation skills. During this activity, participants will become familiar with using the metric system (International System of Units, SI) measurements on common, everyday items. Metric Kitchen - Cook and Bake with a Kitchen Scale with Grams (NIST) - For young chefs, these activities offer an opportunity to refine fine motor skills and coordination, practice selecting appropriate cooking tools, opening containers, mixing, pouring without spilling, counting, weighing, as well as reading and interpreting a digital scale readout or oven thermometer. Credit: B. Hayes/NIST Resources for Grade 6-8 Metric Estimation Game (SP 1336) (NIST) - This fun, hands-on activity helps participants become familiar with SI measurements by practicing estimation skills. During this activity, participants will become familiar with using the metric system (International System of Units, SI) measurements on common, everyday items. SI Base Units Relationship Poster (SP 1247) (NIST) - A colorful poster illustrating the relationships of the International System of Units (SI) derived units with special names and symbols and the seven traditional base units. How Do You Measure the Strength of Gravity? (NIST HDYMI Series) - Gravity is one of the fundamental forces of the universe. It pulls us toward the center of the Earth and keeps our feet on the ground. Understanding the force of gravity makes it possible to predict the orbits of planets and moons in our solar system, enabling us to aim spacecraft precisely enough to land on them. U.S. Coin Specifications(U.S. Mint) - How much does a dime weigh? What are pennies made of? Find out in the table below, which gives specifications for U.S. Mint legal tender coins presently in production for annual sets. What is the Planck Constant(NIST) - The Planck constant is a fundamental number in quantum mechanics that describes how energy is measured in very small amounts called quanta. Learn more about its role in modern science and technology. Everyday Estimation - Mass (NIST) - This section shows examples of common objects and their masses, helping you understand the weight of everyday items from a small car to a cotton ball. Credit: NIST Resources for Grade 9+ Metric Estimation Game (SP 1336) (NIST) - This fun, hands-on activity helps participants become familiar with SI measurements by practicing estimation skills. During this activity, participants will become familiar with using the metric system (International System of Units, SI) measurements on common, everyday items. Pirates of the Caribbean (Metric Edition) (NIST) - In 1794, a French scientist’s capture by pirates may have delayed U.S. adoption of the metric system by nearly a century. Learn more about this surprising twist of history The NIST Do-It-Yourself Kibble Balance (NIST) - The NIST "D.I.Y." Kibble Balance is a classroom science project that teaches the principles behind the effort to redefine the world's basic unit of mass, using LEGO® bricks and some basic electronics. Mass Calibration Brochure (NIST) - This brochure explains when and how to recalibrate weights following the new kilogram definition, what to expect from NIST calibrations, and answers common questions about uncertainty, stability, and traceability. Elemental Composition of the Human Body. (Visual Capitalist) - The human body is a miraculous, well-oiled, and exceptionally complex machine. Learn more to know what elements makeup the human body. Kibble Balance Principles (NIST) - Learn how the Kibble balance uses electricity and magnetism to precisely measure mass, replacing the physical kilogram standard by linking it to fundamental constants like the Planck constant. Kilogram: Introduction) (NIST) - For over 100 years, the kilogram was defined by a small metal cylinder kept in a vault in France. Learn more about how it was recently redefined using a constant of nature for greater accuracy. (Former) International Prototype Kilogram(BIPM) - This explains a platinum-iridium artefact that served as the global standard for the kilogram from 1889 until its redefinition based on physical constants in 2019. Return to Top of Page Videos Pirates Plunder the Kilogram- NISTory Operating Principles of the NIST-4 Kibble Balance How to Build Your NIST D.I.Y. Watt Balance NIST in 90: Measuring Planck’s Constant FAQ What is the difference between the terms "mass" and "weight"? The mass of a body is a measure of its inertial property or how much matter it contains. The weight of a body is a measure of the force exerted on it by gravity or the force needed to support it. Gravity on earth gives a body a downward acceleration of about 9.8 m/s 2. In common parlance, weight is often used as a synonym for mass in weights and measures. For instance, the verb “to weigh” means “to determine the mass of” or “to have a mass of.” The incorrect use of weight in place of mass should be phased out, and the term mass used when mass is meant. The SI unit of mass is the kilogram (kg). In science and technology, the weight of a body in a particular reference frame is defined as the force that gives the body an acceleration equal to the local acceleration of free fall in that reference frame. Thus, the SI unit of the quantity weight defined in this way (force) is the newton (N). Return to Top of Page League of SI Superheroes - Monsieur Kilogram This comic book-style video animation series has been developed to help middle school students learn about the 7 SI base measurement units. With his immensely strong arms, Monsieur Kilogram is the master of mass. The kilogram is a cylinder of special metal about 39 millimeters wide by 39 millimeters tall that serves as the world's mass standard. Navigate to more SI base unit information SI Units Fundamental Physical Constants Length – meter (m) Time – second (s) Amount of substance – mole (mole) Electric current – ampere (A) Temperature – kelvin (K) Luminous intensity – candela (cd) Credit: J. Wang and B. Hayes/NIST ©2020 U.S. Secretary of Commerce. All Rights Reserved. Return to Top of Page EXPLORE THE METRIC PROGRAM Becoming Familiar with SI \| Everyday Estimation \| U.S. Metrication \| U.S. Metrication FAQs \| Prefixes \| Metric Kitchen \| Metric Estimation Game \| SI Education and Training \| SI Publications \| Understanding Metric \| Writing with Metric Units \| National Metric Week \| NEST-R (STEM Registry) \| NIST Education Resources Metrology, Metric and Weights and measures Contacts Metric Program [email protected] Created March 30, 2011, Updated July 28, 2025 Was this page helpful? 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2128	https://brainly.com/question/25892537	[FREE] Octane has a density of 0.703 g/mL. Calculate the mass of $\text{CO}_2(g)$ produced by burning one gallon - brainly.com Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +83,1k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +17,6k Ace exams faster, with practice that adapts to you Practice Worksheets +6,1k Guided help for every grade, topic or textbook Complete See more / Chemistry Textbook & Expert-Verified Textbook & Expert-Verified Octane has a density of 0.703 g/mL. Calculate the mass of CO 2(g) produced by burning one gallon of octane. There are 3.79 liters in one gallon. 2 See answers Explain with Learning Companion NEW Asked by aherrera295 • 12/09/2021 0:02 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 106486884 people 106M 0.0 0 Upload your school material for a more relevant answer The mass of CO₂ gas produced during the combustion of one gallon of octane is 8.21 kg. The given parameters: Density of the octane, ρ = 0.703 g/ml Volume of octane, v = 3.79 liters The mass of the octane burnt is calculated as follows; m=ρ V m=0.703 m l g×3.79 L L 1000 m lm=2,664.37 g The combustion reaction of octane is given as; 2 C 8H 18+25 O 2−−>16 C O 2+18 H 2O From the reaction above: 228.46 g of octane -------------------> 704 g of CO₂ gas 2,664.37 of octane --------------------> ? of CO₂ gas =228.46 2,664.37×704=8,210.3 g=8.21 k g Thus, the mass of CO₂ gas produced during the combustion of one gallon of octane is 8.21 kg. Learn more about combustion of organic compounds here: brainly.com/question/13272422 Answered by onyebuchinnaji •20.4K answers•106.5M people helped Thanks 0 0.0 (0 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 106486884 people 106M 0.0 0 Introductory Chemistry Structure and Properties - Tro Chemistry 2e - Paul Flowers, Klaus Theopold, Richard Langley, William R. Robinson Upload your school material for a more relevant answer The mass of carbon dioxide produced by burning one gallon of octane is approximately 8.22 kg, based on the combustion reaction and calculations considering the density of octane and molar masses. By converting volume to mass, then to moles, and applying the stoichiometry from the balanced equation, we arrive at the final mass. This emphasizes the significance of stoichiometry in chemical reactions. Explanation To calculate the mass of carbon dioxide (CO₂) produced by burning one gallon of octane, let's follow these steps: Understand the Reaction. The combustion of octane (C₈H₁₈) can be represented by the following balanced chemical equation: 2 C 8H 18+25 O 2→16 C O 2+18 H 2O This equation shows that 2 moles of octane produce 16 moles of carbon dioxide. Find the Mass of Octane in One Gallon. The density of octane is given as 0.703 g/mL and there are 3.79 liters in one gallon. First, we convert liters to milliliters: 3.79 L×1000 L m L=3790 mL Now, we calculate the mass of octane: m=density×volume m=0.703 m L g×3790 mL=2664.37 g Calculate the Number of Moles of Octane. The molar mass of octane (C₈H₁₈) is approximately 114.22 g/mol. Now, we convert the mass of octane to moles: n=molar mass m=114.22 m o l g2664.37 g≈23.36 mol Determine the Mass of CO₂ Produced. Using the mole ratio from the balanced equation, for every 2 moles of octane, we produce 16 moles of CO₂: Moles of C O 2=23.36 mol C 8H 18×2 mol C 8H 1816 mol C O 2≈186.88 mol C O 2 Now, we find the mass of CO₂ (molar mass of CO₂ is approximately 44.01 g/mol): m C O 2=186.88 mol×44.01 m o l g≈8222.23 g≈8.22 kg Thus, the mass of CO₂ produced by burning one gallon of octane is approximately 8.22 kg. Examples & Evidence For example, if you were to burn 2 gallons of octane instead of one, you would simply double the amount of CO₂ produced, leading to approximately 16.44 kg of CO₂. This illustrates the direct relationship between the amount of octane burned and the mass of CO₂ produced. The calculations are based on established chemical principles including stoichiometry derived from balanced equations and the conversion factors from volume to mass using density. Thanks 0 0.0 (0 votes) Advertisement Community Answer This answer helped 881713 people 881K 5.0 0 The combustion of one gallon of octane, which weighs about 2.662 kg, produces approximately 8.19 kg of CO2. This calculation involves understanding the chemical reaction of octane with oxygen, yielding CO2 and H2O, and considering the mass ratios. Explanation To calculate the mass of CO2 produced by burning one gallon of octane, a multi-step process is required. Determine the mass of octane in one gallon. Given that 1 gallon is 3.79 liters, and with a density of 0.703 g/ml (or 0.703 g/cm3, or 0.703 kg/l) for octane, you can calculate that one gallon of octane would weigh 3.79 0.703 = 2.662 kg (or 2662 g). Understand the chemical reaction of octane (C8H18) burning in air, resulting in CO2 and H2O: 2 C8H18 + 25 O2 -> 16 CO2 + 18 H2O. From this reaction, 2 moles of octane produces 16 moles of CO2. Since 1 mole of octane has a mass of 114g (based on its molar mass), 228g (2 moles) of octane produces 1644g (based on CO2 molar mass) = 704g CO2. Calculate the amount of CO2 produced by 1g of octane by taking the ratio of 704g CO2 to 228g octane, which results in about 3.08g of CO2 produced per gram of octane. Finally, multiply the amount of CO2 produced by 1 g of octane by the actual mass of octane in one gallon, which is approximately 3.08 2662 = 8194 g, or 8.19 kg of CO2 produced by burning 1 gallon of octane. Learn more about Chemical Reaction here: brainly.com/question/34137415 SPJ11 Answered by DemetriaGene •3.7K answers•881.7K people helped Thanks 0 5.0 (1 vote) Advertisement ### Free Chemistry solutions and answers Community Answer 5.0 2 What mass of CO2 is produced from the combustion of 2.00 gallons of gas? Assume the gas is all octane (C8H18) with a density of 0.703 g/mL. 1 gallon = 3785 mL Community Answer The complete combustion of octane, a component of gasoline, is represented by the equation: 2 C8H18(l) + 25 O2(g) →16 CO2(g) + 18 H2O(l) How many liters of CO2(g), measured at 63.1°C and 688 mmHg, are produced for every gallon of octane burned? (1 gal = 3.785 L; density of C8H18(l) = 0.703 g/mL) Community Answer 5.0 How many pounds of carbon dioxide are produced from burning one gallon of gasoline? Assume gasoline is 100% octane, and the density of octane is 0.78 g / ml. Show the balanced reaction with your calculations. Community Answer 4.2 19 A drink that contains 4 1/2 ounces of a proof liquor… approximately how many drinks does this beverage contain? Community Answer 5.0 7 Chemical contamination is more likely to occur under which of the following situations? When cleaning products are not stored properly When dishes are sanitized with a chlorine solution When raw poultry is stored above a ready-to-eat food When vegetables are prepared on a cutting board that has not been sanitized Community Answer 4.3 189 1. Holding 100mL of water (ebkare)__2. Measuring 27 mL of liquid(daudgtear ldnreiyc)____3. Measuring exactly 43mL of an acid (rtube)____4. Massing out120 g of sodium chloride (acbnela)____5. Suspending glassware over the Bunsen burner (rwei zeagu)____6. Used to pour liquids into containers with small openings or to hold filter paper (unfenl)____7. Mixing a small amount of chemicals together (lewl letpa)____8. Heating contents in a test tube (estt ubet smalcp)____9. Holding many test tubes filled with chemicals (estt ubet karc) ____10. Used to clean the inside of test tubes or graduated cylinders (iwer srbuh)____11. Keeping liquid contents in a beaker from splattering (tahcw sgasl)____12. A narrow-mouthed container used to transport, heat or store substances, often used when a stopper is required (ymerereel kslaf)____13. Heating contents in the lab (nuesnb bneurr)____14. Transport a hot beaker (gntos)____15. Protects the eyes from flying objects or chemical splashes(ggloges)____16. Used to grind chemicals to powder (tmraor nda stlepe) __ Community Answer Food waste, like a feather or a bone, fall into food, causing contamination. Physical Chemical Pest Cross-conta Community Answer 8 If the temperature of a reversible reaction in dynamic equilibrium increases, how will the equilibrium change? A. It will shift towards the products. B. It will shift towards the endothermic reaction. C. It will not change. D. It will shift towards the exothermic reaction. Community Answer 4.8 52 Which statements are TRUE about energy and matter in stars? Select the three correct answers. Al energy is converted into matter in stars Only matter is conserved within stars. Only energy is conserved within stars. Some matter is converted into energy within stars. Energy and matter are both conserved in stars Energy in stars causes the fusion of light elements Community Answer 4.5 153 The pH of a solution is 2.0. Which statement is correct? Useful formulas include StartBracket upper H subscript 3 upper O superscript plus EndBracket equals 10 superscript negative p H., StartBracket upper O upper H superscript minus EndBracket equals 10 superscript negative p O H., p H plus P O H equals 14., and StartBracket upper H subscript 3 upper O superscript plus EndBracket StartBracket upper O upper H superscript minus EndBracket equals 10 to the negative 14 power. New questions in Chemistry Use what you've learned in physics and chemistry to summarize the energy transformation that occurs when we burn a fuel in air. Provide one or two examples, including relevant chemical equations. Using those examples, state in four or five sentences how burning a large quantity of fuel might affect the quality of air. Use the activity series to predict the products of each of the reactions. You do not need to balance the equations. Li\ \textgreater \ K\ \textgreater \ Ba\ \textgreater \ Sr\ \textgreater \ Ca\ \textgreater \ Na\ \textgreater \ Mg\ \textgreater \ Al\ \textgreater \ Mn\ \textgreater \ Zn\ \textgreater \ Cr\ \textgreater \ Fe\ \textgreater \ Cd\ \textgreater \ Co\ \textgreater \ Ni\ \textgreater \ Sn\ \textgreater \ Pb\ \textgreater \ H\ \textgreater \ Sb\ \textgreater \ Bi\ \textgreater \ Cu\ \textgreater \ Ag\ \textgreater \ Pd\ \textgreater \N a 2C O 3+K→? A. no reaction B. K 2C O 3+N a C. N a K+C O 3Z n C l 2+C u→? D. no reaction E. C u Z n+C l 2 F. C u C l 2+Z n Use the activity series to predict the products of each of the reactions. You do not need to balance the equations. Activity Series: L i>K>B a>S r>C a>N a>M g>A l>M n>Z n>C r>F e>C d>C o>N i>S n>P b>H>S b>B i>C u>A g>P d F 2>C l 2>B r 2>I 2 1. For the reaction CC l 4+B r 2→?, predict the products:A. B r Cl+C B. CB r 4+C l 2 C. No reaction 2. For the reaction F e S O 4+N a→?, predict the products:A. N a 2S O 4+F e B. No reaction C. F e N a+S O 4 What is the concentration (M) of a 23.6 mL solution containing 0.3996 g of NaCl? Molar mass of NaCl = 58.44 g/mol. Round your answer to 3 decimal places. Do not include units. Which of the following is an appropriate mass of sodium chloride to measure out? A. 0.0700 g B. 0.0300 g C. 0.2000 g D. 0.1000 g What is the volume of 2.50 lb (pounds) of mercury? The density of mercury is 13.546 g/m L, and 1 l b=453.59 g. 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2129	https://arxiv.org/pdf/1009.2480	arXiv:1009.2480v2 [physics.class-ph] 1 Dec 2010 Generalized second-order partial derivatives of 1/r V Hnizdo National Institute for Occupational Safety and Health, Morgantown, West Virginia 26505, USA E-mail: vhnizdo@cdc.gov Abstract. The generalized second-order partial derivatives of 1 /r , where r is the radial distance in 3D, are obtained using a result of the potential theory of classical analysis. Some non-spherical-regularization alternatives to the standard spherical-regularization expression for the derivatives are derived. The utility of a spheroidal-regularization expression is illustrated on an example from classical electrodynamics. Introduction The expression for the Laplacian of 1 /r , ∇2 1 r = −4πδ (r), (1) where r = \|r\| = ( x21 + x22 + x23)1/2 is the magnitude of a vector r = ( x1, x 2, x 3), the Laplacian ∇2 is the differential operator ∂2/∂x 21 + ∂2/∂x 22 + ∂2/∂x 23 and δ(r) = δ(x1)δ(x2 )δ(x3) is the 3D delta function, is well-known and its correct use involves only the elementary rules of the delta-function formalism. The Laplacian (1) is useful in innumerable calculations of electrodynamics, but sometimes the 2nd-order partial derivatives ( ∂2/∂x i∂x j )(1 /r ) themselves are needed. Examples are the calculation of the electromagnetic fields of point dipoles and solving the Poisson equation for the difference between the Coulomb- and Lorenz-gauge vector potentials of a uniformly moving point charge . While the expression for ( ∂2/∂x i∂x j )(1 /r ) has been known for some time, its correct use is rather more intricate than that of ∇2(1 /r ). This expression is usually written as ∂2 ∂x i∂x j 1 r = 3xixj − r2δij r5 − 4π 3 δ(r)δij , (2) which hides an important fact that an integration of the product of the first term on the right-hand side and a ‘well-behaved’ test function over a 3-dimensional domain that includes the origin r = 0 still has to be regularized because of the term’s 1 /r 3 behaviour at the origin ‡. The regularization assumed in (2) is of a specific, ‘spherical’ ‡ An expression for the (generalized) 2nd-order derivative of 1 /r that takes this circumstance into account explicitly seems to have been given first in , p 28. Generalized second-order partial derivatives of 1/r 2kind, which can be effected in many equivalent forms, e.g. 3xixj − r2δij r5 → wlim ǫ→0 3xixj − r2δij (r2 + ǫ2)5/2 (3) → wlim ǫ→0 3xixj − r2δij r5 Θ( r − ǫ). (4) Here, the symbol wlim indicates the weak limit § and Θ( ·) is the Heaviside step function. The form (4) is implemented automatically when spherical coordinates are used in integration and the angular integration is done first. Relation (2) was derived as a delta-function identity by Frahm . In the present paper, we obtain it easily as a generalized (distributional) derivative using a result of the potential theory of classical analysis. We also derive some ‘non-spherical’ alternatives to (2); while integration in spherical coordinates provides a straightforward implementation of the spherical regularization implied in (2), the use of a non-spherical regularization may be more advantageous computationally in some applications. An example from classical electrodynamics where a spheroidal regularization is useful is given in Appendix. Apart from the utility, awareness of non-spherical alternatives is important for the correct use of the standard, ‘spherical’ expression. Our treatment assumes no knowledge of the theory of generalized functions and generalized (distributional) derivatives beyond the elementary delta-function formalism, but it should help elucidate the operational meaning of some essentially non-classical mathematical objects ‖. Presenting the 2nd-order partial derivatives of 1/r from the very beginning as generalized (distributional) derivatives should help avoid the pitfalls that would await anyone attempting to use the expression (2), or its very recent ‘generalization’ (remarked on in the last section), in an integration in non-spherical coordinates. This topic and its treatment are suitable for graduate and advanced undergraduate courses of electrodynamics. A generalized-function treatment of the singularities that may arise at the origin r = 0 of a spherical (polar) coordinate system has been presented recently in this Journal by Gsponer . In contrast to Gsponer, our approach does not rely on any specific choice of coordinates. Derivation using derivatives of the potential of an extended density Let φ(r) = ∫ d3r′ ρ(r′) R , R = \|r − r′\| (5) be the ‘potential’ created by a density ρ(r) that is assumed to be a ‘well-behaved’ localized function of r. While the 2nd-order partial derivatives ∂2φ(r)/∂x i∂x j perfectly exist at any point r if the density ρ(r) is sufficiently ‘smooth’ ¶, they cannot be calculated by a straightforward differentiating inside the integral that defines the §wlim a→a0ga(r) = g(r) iff lim a→a0 ∫d3r f (r)ga(r) = ∫d3r f (r)g(r), where g(r) is in general a generalized function (distribution) and f(r) is any well-behaved test function. ‖For a general-function theoretic underpinning of our approach, the interested reader is directed to the canonical regularization method of Gelfand and Shilov and the completeness theorem of generalized functions . ¶If the density ρa(r) has a sharp surface, the 2nd-order derivatives of the potential are discontinuous at that surface. Generalized second-order partial derivatives of 1/r 3potential because the resulting integrand is not integrable at the point r′ = r. Acorrect way of performing here the differentiation under the integral sign is given by the following formula: ∂2φ(r) ∂x i∂x j = lim ǫ→0 ∫ R>ǫ d3r′ ρ(r′) ∂2 ∂x i∂x j 1 R − 4π 3 ρ(r)δij = lim ǫ→0 ∫ d3r′ ρ(r′) 3( xi − x′ i )( xj − x′ j ) − R2δij R5 Θ( R − ǫ) − 4π 3 ρ(r)δij . (6) Here, the integration domain excludes a ball of radius ǫ and centre at r′ = r; the 2nd line is a more explicit transcription of the 1st line, with the integration domain R = \|r′ − r\| > ǫ expressed by the Heaviside step-function factor Θ( R − ǫ) in the integrand. An instructive derivation of formula (6) is given in the well-known text of Tikhonov and Samarskii (they consider the most involved case i = j, but the generalization to any i, j = 1 , 2, 3 is straightforward). This derivation uses the divergence theorem with a function that has only a 1 /R singularity at the point R = 0 and thus, unlike the informal proof in , is fully legitimate in classical analysis +.Note that the regularization of the integral on the right-hand side of (6) is explicitly of the spherical kind. If the excluded integration domain was assumed to have a non-spherical shape, the ensuing regularization would in general yield a different value for the integral, resulting in an incorrect value for the derivative in question. The reason why different regularizations yield in general different values of the regularized integral is that, because of the factor ( ∂2/∂x i∂x j )(1 /R ), the integrand goes through large positive and negative variations near the point R = 0. The integral thus can be made convergent only conditionally . Defining a generalized (distributional) derivative ¯∂2 ∂x i∂x j 1 R ≡ wlim ǫ→0 3( xi − x′ i )( xj − x′ j ) − R2δij R5 Θ( R − ǫ) − 4π 3 δ(r − r′)δij (7) (denoted by a bar to distinguish it from a classical derivative [4, 10, 11]), formula (6) can be written simply as ∂2φ(r) ∂x i∂x j = ∫ d3r′ρ(r′) ¯∂2 ∂x i∂x j 1 R . (8) The generalized derivative (7) can thus be seen as the mathematical operation using which the 2nd-order differentiation with respect to components of r of a function defined as the integral with respect to r′ of an integrand involving the factor 1/R = 1 /\|r − r′\| may be performed under the integral sign: ∂2 ∂x i∂x j ∫ d3r′ ρ(r′) R = ∫ d3r′ ρ(r′) ¯∂2 ∂x i∂x j 1 R , (9) where the ‘density’ ρ(r) now plays the role of a well-behaved test function. Expression (2) is a special case r′ = 0 (so that R ≡ r − r′ = r) of the generalized derivative (7), ¯∂2 ∂x i∂x j 1 r = wlim ǫ→0 3xixj − r2δij r5 Θ( r − ǫ) − 4π 3 δ(r)δij . (10) +As the informal proof in , informal derivations of the full Laplacian relation (1) typically use the divergence theorem with a function that has a 1 /r 2singularity at r= 0; for a more rigorous alternative see . Generalized second-order partial derivatives of 1/r 4Adding expressions (10) with i = j = 1 , 2, 3, the non-delta-function terms cancel out, and we obtain the full Laplacian (1) as ¯∇2 1 r = −4πδ (r), (11) where ¯∇2 = ¯∂2 ∂x 21 ¯∂2 ∂x 22 ¯∂2 ∂x 23 (12) is now the generalized Laplacian operator. The generalized derivative (10) is the weak limit a → 0 of the corresponding classical derivative of the potential φa(r) = ∫ d3r′ ρa(r′) R , R = \|r − r′\| (13) that is due to a localized density ρa(r) the spatial extension of which depends on a parameter a so that lim a→0 ∫ d3r f (r)ρa(r) = f (0) , (14) where f (r) is any well-behaved test function; this condition is transcribed formally as wlim a→0 ρa(r) = δ(r). (15) This can be shown by calculating the limit a → 0 of the integral of the product of (6) with φ(r) = φa(r) and a well-behaved function f (r). It suffices to consider the case i = j = 1: lim a→0 ∫ d3r f (r) ∂2φa(r) ∂x 21 = lim a→0 ∫ d3rf (r) lim ǫ→0 ∫ d3r′ρa(r′) 3( x1−x′ 1 )2−R2 R5 Θ( R−ǫ) − 4π 3 lim a→0 ∫ d3rf (r)ρa(r)= lim a→0 lim ǫ→0 ∫ d3r f (r) ∫ d3r′ ρa(r′) 3( x1 − x′ 1 )2 − R2 R5 Θ( R − ǫ) − 4π 3 ∫ d3r f (r)δ(r)= lim a→0 lim ǫ→0 ∫ d3r′ ρa(r′) ∫ d3r f (r) 3( x1 − x′ 1 )2 − R2 R5 Θ( R − ǫ) − 4π 3 f (0) = lim a→0 ∫ d3r′ ρa(r′)F (r′) − 4π 3 f (0) , (16) where F (r′) = lim ǫ→0 ∫ d3r f (r) 3( x1 − x′ 1 )2 − R2 R5 Θ( R − ǫ). (17) The function F (r′) is well-behaved, and thus lim a→0 ∫ d3r′ ρa(r′)F (r′) = ∫ d3r′ δ(r′)F (r′)= F (0) = lim ǫ→0 ∫ d3r f (r) 3x21 − r2 r5 Θ( r − ǫ). (18) The algebraic manipulations in (16) are legitimate operations of moving limits without changing their sequential order and of interchanging the orders of integration. Generalized second-order partial derivatives of 1/r 5Equations similar to (16) and (18) obviously hold for all the derivatives ∂2φa(r)/∂x i∂x j , i, j = 1 , 2, 3. We thus have that, for any well-behaved test function f (r), lim a→0 ∫ d3r f (r) ∂2φa(r) ∂x i∂x j = lim ǫ→0 ∫ d3r f (r) 3xixj − r2δij r5 Θ( r − ǫ) − 4π 3 f (0) δij , (19) which is expressed formally as wlim a→0 ∂2φa(r) ∂x i∂x j = wlim ǫ→0 3xixj − r2δij r5 Θ( r − ǫ) − 4π 3 δ(r)δij = ¯∂2 ∂x i∂x j 1 r . (20) Since the limit a → 0 of the potential φa(r) itself is the potential 1 /r of a point density δ(r), this result is the formal underpinning of a natural interpretation of the generalized derivative ( ¯∂2/∂x i∂x j )(1 /r ) as the 2nd-order derivative of the potential of a point source. Note that while the regularization used in (20) is of the spherical kind, the extended density ρa(r) that generates the potential φa(r) does not have to have any particular symmetry as a function of r. Using the powerful methods and results of the theory of generalized functions and derivatives, the result (20) can be obtained almost immediately (see , Appendix), but the approach adopted here required only a little more effort. Non-spherical regularizations 3.1. Spheroidal regularization Let us replace the spherical excluded integration domain used in (6), (7) and (10) by a spheroid of semiaxes ǫ/γ , ǫ and ǫ along the x1, x2 and x3 axes, respectively, with the parameter γ given by γ = 1 √1 − v2 , 0 < \|v\| < 1. (21) Regularization using such a domain may be suitable in applications involving effects of special relativity, according to which a spherical charge of radius ǫ contracts to an oblate spheroid of this geometry when it is set in motion with a speed vc along the x1 axis. To find the modification of expression (10) for the generalized derivative ( ¯∂2/∂x i∂x j )(1 /r ) that such spheroidal regularization entails, we only need to evaluate the difference lim ǫ→0 ∫ γ2x21+x22+x23>ǫ 2 d3r f (r) 3xixj − r2δij r5 − lim ǫ→0 ∫ r>ǫ d3r f (r) 3xixj − r2δij r5 = lim ǫ→0 ∫ Uǫ d3r f (r) 3xixj − r2δij r5 , (22) where f (r) is again a well-behaved test function and the integration domain Uǫ is the region delimited by the oblate surface γ2x21 + x22 + x23 = ǫ2 and the spherical surface r2 = x21 + x22 + x23 = ǫ2: Uǫ = {(x1, x 2, x 3); γ2x21 + x22 + x23 > ǫ 2 ∩ x21 + x22 + x23 < ǫ 2}. (23) Generalized second-order partial derivatives of 1/r 6As ǫ tends to zero, the integration domain Uǫ gets progressively smaller and closer to the origin r = 0 so that, for any r ∈ U ǫ, f (r) → f (0) as ǫ → 0. The right-hand side of (22) can therefore be written as lim ǫ→0 ∫ Uǫ d3r f (r) 3xixj − r2δij r5 = f (0) lim ǫ→0 ∫ Uǫ d3r 3xixj − r2δij r5 . (24) Here, the integral on the right-hand side can be evaluated easily in spherical coordinates. With x1 axis as the polar axis and cos θ = ξ, we obtain for i = j = 1: ∫ Uǫ d3r 3x21 − r2 r5 = 2 π ∫ 1 −1 dξ (3 ξ2 − 1) ∫ ǫǫ/ √1+( γ2−1) ξ2 dr r = π ∫ 1 −1 dξ (3 ξ2 − 1) ln[1 + ( γ2 − 1) ξ2]= 2 π ( 2 v2 − 2 arcsin v γv 3 − 2 3 ) . (25) The case i = j = 2 gives ∫ Uǫ d3r 3x22 − r2 r5 = 1 2 ∫ 1 −1 dξ ∫ 2π 0 dφ [3(1 − ξ2) cos 2 φ − 1] ln[1 + ( γ2 − 1) ξ2]= π 2 ∫ 1 −1 dξ (1 − 3ξ2) ln[1 + ( γ2 − 1) ξ2]= 2 π ( 1 3 − 1 v2 + arcsin v γv 3 ) , (26) and the same result obviously will be obtained for i = j = 3. The mixed cases i 6 = j will all yield zero on account of the integration with respect to the azimuthal angle φ. A notable feature of the results (25) and (26) is that they are independent of ǫ.Collecting all these results, equation (24) can be written as lim ǫ→0 ∫ Uǫ d3r f (r) 3xixj − r2δij r5 = 2 π[gij (v) − 2 3 δij ]f (0) , (27) where gij (v) =  2/v 2 − (2 /γv 3) arcsin v i = j = 1 1 − 1/v 2 + (1 /γv 3) arcsin v i = j = 2 , 30 i 6 = j. (28) This result establishes a generalized-function identity wlim ǫ→0 3xixj − r2δij r5 [Θ( γ2x21 + x22 + x23 − ǫ2) − Θ( r − ǫ)] = 2 π[gij (v) − 2 3 δij ]δ(r), (29) using which the difference (22) with any well-behaved function f (r) can be evaluated immediately. According to this identity wlim ǫ→0 3xixj − r2δij r5 Θ( r − ǫ)= wlim ǫ→0 3xixj − r2δij r5 Θ( γ2x21 + x22 + x23 − ǫ2) − 2π[gij (v) − 2 3 δij ]δ(r) (30) and thus the generalized derivative (10) can be re-written as ¯∂2 ∂x i∂x j 1 r = wlim ǫ→0 3xixj − r2δij r5 Θ( γ2x21 + x22 + x23 − ǫ2) − 2πg ij (v)δ(r), (31) Generalized second-order partial derivatives of 1/r 7which is a spheroidal-regularization alternative to the standard, spherical-regularization expression (10) for the generalized derivative ( ¯∂2/∂x i∂x j )(1 /r ); (31) reduces to (10) in the limit v → 0 since lim v→0 γ = 1 and lim v→0 gij (v) = 2 3 δij . An electrodynamic example in which expression (31) is useful is given in Appendix. 3.2. Cylindrical regularization In some applications, cylindrical coordinates are natural to the problem and the requisite integrations are performed most easily in these coordinates. We can find the expression for ( ¯∂2/∂x i∂x j )(1 /r ) that employs a cylindrical regularization by evaluating the difference lim ǫ→0 ∫ r>ǫ d3r f (r) 3xixj − r2δij r5 − lim ǫ→0 ∫ Tǫ,κ d3r f (r) 3xixj − r2δij r5 = lim ǫ→0 (∫ V(1) ǫ,κ d3r f (r) 3xixj − r2δij r5 − ∫ V(2) ǫ,κ d3r f (r) 3xixj − r2δij r5 ) . (32) Here, the integration domain Tǫ,κ is defined as Tǫ,κ = {(x1, x 2, x 3); \|x1\| > κǫ ∪ x22 + x23 > ǫ 2}, (33) which is the complement of a cylinder of base radius ǫ and half-height κǫ , parallel to the x1 axis and centered at the origin r = 0, and the integration domains V(1) ǫ,κ and V(2) ǫ,κ are defined as V(1) ǫ,κ = {(x1, x 2, x 3); x21 + x22 + x23 > ǫ 2 ∩ x22 + x23 < ǫ 2 ∩ \| x1\| < κǫ } (34) V(2) ǫ,κ = {(x1, x 2, x 3); x21 + x22 + x23 < ǫ 2 ∩ x22 + x23 < ǫ 2 ∩ \| x1\| > κǫ }, (35) which are regions delimited by the surface of the cylinder and the spherical surface r2 = x21 + x22 + x23 = ǫ2. The region V(2) ǫ,κ is nonempty only when κ < 1, in which case the integral over V(2) ǫ,κ has to be subtracted from that over the region V(1) ǫ,κ .As ǫ tends to zero, the regions V(1 ,2) ǫ,κ progressively shrink and collapse onto the origin r = 0 so that, for any r ∈ V (1 ,2) ǫ,κ , f (r) → f (0) as ǫ → 0, and thus lim ǫ→0 [∫ V(1) ǫ,κ d3r − ∫ V(2) ǫ,κ d3r ] f (r) 3xixj − r2δij r5 = f (0) lim ǫ→0 [∫ V(1) ǫ,κ d3r − ∫ V(2) ǫ,κ d3r ] 3xixj − r2δij r5 , (36) where the large brackets are used to denote the difference of the indicated integrals. The integrals on the right-hand side are evaluated easily in cylindrical coordinates s, φ, x 1. The case i = j = 1 gives [∫ V(1) ǫ,κ d3r − ∫ V(2) ǫ,κ d3r ] 3x21 − r2 r5 = 4 π ∫ ǫ 0 s ds ∫ κǫ √ǫ2−s2 dx1 2x21 − s2 (x21 + s2)5/2 = 4 π ( κ √1 + κ2 − 2 3 ) . (37) The cases i = j = 2 , 3 give [∫ V(1) ǫ,κ d3r − ∫ V(2) ǫ,κ d3r ] 3x22,3−r2 r5 = 2 ∫ ǫ 0 s ds ∫ κǫ √ǫ2−s2 dx1 ∫ 2π 0 dφ (3 cos 2 φ − 1) s2−x21 (x21 + s2)5/2Generalized second-order partial derivatives of 1/r 8= 2 π ∫ ǫ 0 s ds ∫ κǫ √ǫ2−s2 dx1 s2 − 2x21 (x21 + s2)5/2 = 2 π ( 2 3 − κ √1 + κ2 ) , (38) and the mixed cases i 6 = j yield zero because of the integration with respect to φ.Similarly to the establishing of identity (30), these results now establish a generalized-function identity wlim ǫ→0 3xixj − r2δij r5 Θ( r − ǫ)= wlim ǫ→0 3xixj − r2δij r5 [Θ( \|x1\| − κǫ ) + Θ( x22 + x23 − ǫ2)Θ( κǫ − \| x1\|)] − 2π[hij (κ) − 2 3 δij ]δ(r), (39) where hij (κ) =  2 − 2κ/ √1 + κ2 i = j = 1 κ/ √1 + κ2 i = j = 2 , 30 i 6 = j. (40) Using identity (39), the generalized derivative (10) can be written as ¯∂2 ∂x i∂x j 1 r = wlim ǫ→0 3xixj − r2δij r5 [Θ( \|x1\| − κǫ ) + Θ( x22 + x23 − ǫ2)Θ( κǫ − \| x1\|)] − 2πh ij (κ)δ(r). (41) This is a cylindrical-regularization alternative to (10). In the limit κ → 0, the delta-function term in (41) simplifies to −4πδ (r) for i = j = 1 (, p 29) and to 0 otherwise; such regularization is implemented automatically by using the cylindrical coordinates s, φ, x 1 and performing the requisite integration over the whole space R3, but so that the integration with respect to the variable x1 is done last: ∫ d3r f (r) ¯∂2 ∂x i∂x j 1 r = ∫ ∞−∞ dx1 (∫ ∞ 0 s ds ∫ 2π 0 dφ f (s, φ, x 1) dij (s, φ, x 1 ) (x21 + s2)5/2 ) − 4πf (0) δi1δj1, (42) where dij (s, φ, x 1) is the function obtained by the transformation of 3 xixj − r2δij from the Cartesian to the cylindrical coordinates (e.g., d11 (s, φ, x 1) = 2 x21 − s2). Concluding remarks In reference , the equivalence of the spherical- and spheroidal-regularization expressions for the generalized 2nd-order partial derivatives of 1 /r was illustrated by a relatively laborious explicit calculation of the weak limit a → 0 of the derivatives of the Coulomb potential of a charged conducting spheroid of finite extension a. The derivation of the generalized derivatives given here, together with the presented results on non-spherical regularization, ensures that any similar explicit calculation must yield the same result. A point worth making is that while the standard, spherical-regularization expression (10) and the non-spherical-regularization expressions (31) and (41) for the generalized derivatives ( ¯∂2/∂x i∂x j )(1 /r ) are guaranteed to yield the same results in an integral with a well-behaved function, care should be taken in numerical work to Generalized second-order partial derivatives of 1/r 9use an integration grid that is compatible with the kind of regularization employed . Very recently, Frahm’s formula (2) has been criticized as being valid only when averaged over smooth functions, and, to remedy that, an expression ∂2 ∂x i∂x j 1 r = 3xixj − r2δij r5 − 4π xixj r2 δ(r) (43) has been proposed . However, our analysis shows that the only ‘flaw’ of Frahm’s formula is that it does not indicate explicitly the spherical regularization that it assumes. We note that expression (43) still suffers from the lack of an appropriate regularization of the non-delta function term. Moreover, the delta-function term as it stands there is ill-defined; it would become meaningful in an integration in spherical coordinates and the replacement of the 3D delta function δ(r) by the radial equivalent δ(r)/(4 πr 2 ). Clearly, the ‘general’ expression (43) will yield correct results only in an integration in spherical coordinates, with the angular integration of the term involving the non-delta-function part done first. Acknowledgments The author thanks F Farassat of NASA Langley Research Center for useful discussions on generalized derivatives and helpful comments on a draft. This paper is written by the author in his private capacity. No official support or endorsement by the Centers for Disease Control and Prevention is intended or should be inferred. Appendix We shall employ here a spheroidal-regularization generalized derivative in a calculation of the difference A(C) x1 − A(L) x1 between the x1-components of the Coulomb- and Lorenz-gauge vector potentials of a unit point charge moving uniformly with a velocity v along the x1 axis. The difference A(C) x1 − A(L) x1 satisfies an inhomogeneous wave equation : [A(C) x1 (r, t ) − A(L) x1 (r, t )] = −v ∂2 ∂x 21 φ(C) (r, t ), (A.1) where = ∇2 − ∂2/∂t 2 is the d’Alembertian operator (we use Gaussian units with the speed of light c = 1) and φ(C) (r, t ) = 1 √(x1 − vt )2 + x22 + x23 (A.2) is the Coulomb-gauge scalar potential of the charge. But an inhomogeneous wave equation f = s(x1 − vt, x 2, x 3), (A.3) whose source term is ‘moving’ with a constant velocity v along the x1 axis, can be simplified by a simple transformation of the variables to a Poisson equation, the vanishing-at-infinity solution of which is given in terms of the original variables by f (r, t ) = − γ 4π ∫ d3r′ s(x′ 1 − vt, x ′ 2 , x ′ 3 ) √γ2(x1 − x′ 1 )2 + ( x2 − x′ 2 )2 + ( x3 − x′ 3 )2 , (A.4) Generalized second-order partial derivatives of 1/r 10 where γ = (1 − v2)−1/2 (see also [2, 14]). The wave equation (A.1) is therefore solved by A(C) x1 (r, t ) − A(L) x1 (r, t ) = vγ 4π ∫ d3r′ ( ¯∂2/∂x ′ 12 ) φ(C) (r′, t ) √γ2(x1 − x′ 1 )2 + ( x2 − x′ 2 )2 + ( x3 − x′ 3 )2 . (A.5) Note that, in order that the integral is defined properly, the 2nd-order derivative of the point-charge potential φ(C) must be here a generalized one. However, instead of using the standard spherical-regularization expression for the derivative, it will be seen that the evaluation of the integral on the right-hand side of (A.5) is facilitated greatly when a spherodial-regularization expression is used. Following (31), this is given by ¯∂2φ(C) (r, t ) ∂x 21 = wlim ǫ→0 3( x1 − vt )2 − R2 R5 Θ( R∗ − ǫ) − 4π ( 1 v2 − arcsin v γv 3 ) δ(x1 − vt )δ(x2 )δ(x3), (A.6) where R = √ (x1 − vt )2 + x22 + x23 and R∗ = √ γ2(x1 − vt )2 + x22 + x23. (A.7) The integration in (A.5) is now done in two steps. First, the term involving the delta-function term of (A.6) is integrated readily and a transformation γ(x′ 1 −vt ) → x′ 1 is performed in the remaining integrand. This gives A(C) x1 (r, t ) − A(L) x1 (r, t ) = G(r, t ) − ( γ v − arcsin v v2 ) 1 R∗ , (A.8) where G(r, t ) = lim ǫ→0 v 4π ∫ d3r′ \|R∗ − r′\| 2x′ 12 /γ 2 − x′ 22 − x′ 32 (x′ 12 /γ 2 + x′ 22 x′ 32 )5/2 Θ( r′ − ǫ). (A.9) Here, R∗ is a vector with components γ(x1 − vt ), x 2 , x 3. The expansion of the factor 1/\|R∗ − r′\| in Legendre polynomials now can be used to separate angular and radial integrations; moreover, the use of spherical coordinates r′, θ ′, φ ′ with the angular integration done first implements the regularization limit ǫ → 0 automatically. Thus G(r, t ) = v 2R∗∞∑ l=1 (4 l + 1) P2l(ξ∗) 2l(2 l + 1) ∫ 1 −1 dξ′ (3 − 2v2)ξ′2 − 1 (1 − v2ξ′2)5/2 P2l(ξ′), (A.10) where ξ∗ = γ(x1 − vt )/R ∗, ξ′ = x′ 1 /r ′ = cos θ′, and the result ∫ ∞ 0 dr ′ r′ r2l< r2l+1 = 4l + 1 2l(2 l + 1) 1 R∗ , l ≥ 1, (A.11) where r< (r>) is the lesser (greater) of r′ and R∗, is used. The summation in (A.10) runs only over Legendre polynomials of even non-zero order since the integration with respect to ξ′ yields zero when the Legendre-polynomial order is zero or odd. Second, the v-dependent part of the integrand is expanded in powers of v2 to facilitate the integration with respect to ξ′,(3 − 2v2)ξ′2 − 1 (1 − v2ξ′2)5/2 = ∞ ∑ n=0 (2 n + 1)!! (2 n)!! [(2 n + 3) ξ′2 − 2n − 1] ξ′2nv2n. (A.12) Generalized second-order partial derivatives of 1/r 11 Using this expansion in (A.10), we obtain after interchanging the orders of summation and integrating term by term with respect to ξ′: G(r, t ) = v 2√π R ∗∞∑ n=0 [Γ( n + 3 2 )] 2v2nn+1 ∑ l=1 (4 l + 1) P2l(ξ∗) Γ( n − l + 2)Γ( n + l + 5 2 )= 1 vR ∗∞∑ n=1 [(2 n − 1)!!] 2 (2 n + 1)! [(2 n + 1)( vξ ∗)2n − v2n]= ( 1 √1 − v2ξ∗2 − arcsin v v ) 1 vR ∗ . (A.13) Here, in the 1st line, the series over l terminates at l = n+1 since all its l > n +1 terms vanish; in the 2nd line, the terminated series is summed and the resulting series over n re-arranged so that it has an overall multiplier 1 /v ; and, in the 3rd line, the series over n is summed using the well-known expansions of the functions (1 + x)−1/2 and arcsin x in powers of x. Using (A.13) in (A.8) and then the definition ξ∗ = γ(x1 − vt )/R ∗ with the definition (A.7) of R∗, we obtain finally A(C) x1 (r, t ) − A(L) x1 (r, t ) = ( 1 √1 − v2ξ∗2 − γ ) 1 vR ∗ = 1 v ( 1 √(x1 − vt )2 + x22 + x23 − 1 √(x1 − vt )2 + ( x22 + x23)/γ 2 ) . (A.14) The same closed-form expression for the difference between the x1 components of the Coulomb- and Lorenz-gauge vector potentials of a point charge moving uniformly along the x1 axis was obtained in by calculating the requisite gauge function for the transformation between the Lorenz and Coulomb gauges using a formula derived by Jackson . References Griffiths D J 1999 Introduction to Electrodynamics 3rd edn (Upper Saddle River, NJ: Prentice Hall) Problem 3.42 Hnizdo V 2004 Potentials of a uniformly moving point charge in the Coulomb gauge Eur. J. Phys. 25 351–60 Frahm C P 1983 Some novel delta-function identities Am. J. Phys. 51 826–29 Farassat F 1996 Introduction to generalized functions with applications in aerodynamics and aeroacoustics NASA Technical Paper 3248 (Hampton, Virginia: NASA Langley Research Center) Gelfand I M and Shilov G E 1964 Generalized Functions vol 1 (New York: Academic Press) Franklin J 2010 Comment on “Some novel delta-function identities” by Charles P. Frahm [Am. J. Phys. 51, 826–829 (1983)] Am. J. Phys. 78 1225–6 Gsponer A 2007 Distributions in spherical coordinates with applications to classical electrodynamics Eur. J. Phys. 28 267–75 Corrigendum Eur. J. Phys. 28 1241 Tikhonov A N and Samarskii A A 1990 Equations of Mathematical Physics (New York: Dover) chapt IV sect 5.5 Hnizdo V 2000 On the Laplacian of 1 /r Eur. J. Phys. 21 L1–3 Kanwal R P 2004 Generalized Functions, Theory and Applications 3rd edn (Boston: Brikh¨ auser) Estrada R and Kanwal R P 1995 The appearance of nonclassical terms in the analysis of point-source fields Am. J. Phys. 63 278–78 Hnizdo V 2006 Regularization of the second-order partial derivatives of the Coulomb potential of a point charge e-print arxiv:physics/0409072 Panofsky W K H and Phillips M 1962 Classical Electricity and Magnetism 2nd edn (Reading, MA: Addison-Wesley) sect 19.3 Generalized second-order partial derivatives of 1/r 12 Jefimenko O D 1996 Retardation and relativity: new integrals for electric and magnetic potentials for time-independent charge distributions moving with constant velocity Eur. J. Phys. 17 258–64 Jackson J D 2002 From Lorenz to Coulomb and other explicit gauge transformations Am. J. Phys. 70 917–28
2130	https://edu.rsc.org/experiments/measuring-the-amount-of-vitamin-c-in-fruit-juices/539.article	Measuring the amount of vitamin C in fruit juices \| Class experiment \| RSC Education Skip to main content Skip to navigation Resources Practical Experiments Mast navigation Register Sign In Search our site All All Resources Articles Search our site Search Menu Close menu Home I am a … Back to parent navigation item I am a … Primary teacher Secondary/FE teacher Early career or student teacher Technician HE teacher Student Resources Back to parent navigation item Resources Primary Secondary Higher education Curriculum support Practical Analysis Literacy in science teaching Periodic table Back to parent navigation item Periodic table Interactive periodic table Climate change and sustainability Careers Resources shop Collections Back to parent navigation item Collections Remote teaching support Starters for ten Screen experiments Assessment for learning Microscale chemistry Faces of chemistry Classic chemistry experiments Nuffield practical collection Anecdotes for chemistry teachers Literacy in science teaching More … Climate change and sustainability Alchemy On this day in chemistry Global experiments PhET interactive simulations Chemistry vignettes Context and problem based learning Journal of the month Chemistry and art Back to parent navigation item Chemistry and art Techniques Art analysis Pigments and colours Ancient art: today's technology Psychology and art theory Art and archaeology Artists as chemists The physics of restoration and conservation Cave art Ancient Egyptian art Ancient Greek art Ancient Roman art Classic chemistry demonstrations In search of solutions In search of more solutions Creative problem-solving in chemistry Solar spark Chemistry for non-specialists Health and safety in higher education Analytical chemistry introductions Exhibition chemistry Introductory maths for higher education Commercial skills for chemists Kitchen chemistry Journals how to guides Chemistry in health Chemistry in sport Chemistry in your cupboard Chocolate chemistry Adnoddau addysgu cemeg Cymraeg The chemistry of fireworks Festive chemistry Education in Chemistry Teach Chemistry Events Teacher PD Back to parent navigation item Teacher PD Courses Back to parent navigation item Courses Self-led Live online Selected PD articles PD for primary teachers PD for secondary teachers What we offer Chartered Science Teacher (CSciTeach) Teacher mentoring Enrichment Back to parent navigation item Enrichment UK Chemistry Olympiad Back to parent navigation item UK Chemistry Olympiad Who can enter? 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How does it work? Resources and past papers FAQs Top of the Bench Schools' Analyst How to prepare for the Chemistry Olympiad Chemistry Olympiad worked answers Chemistry Olympiad past papers Our work Regional support Education coordinators RSC Yusuf Hamied Inspirational Science Programme Policy Science Education Policy Alliance RSC Education News Supporting teacher training Interest groups More navigation items Measuring the amount of vitamin C in fruit juices No comments In this experiment, students will explore their favourite fruit juices, and through the process of titration discover the vitamin C content Have you ever poured a tall glass of a fruit juice, and wondered just how much vitamin C it contains? Well, wonder no longer, with this titration experiment students discover how much, or little, vitamin C is in different juices. This experiment should take 20 minutes. Equipment Apparatus Eye protection Student worksheet Microscale titration apparatus (see our apparatus and techniques for microscale chemistry guidance) Pipette (glass), 1 cm 3 Pipette (glass), 2 cm 3 Pipette filler Beaker, 25 cm 3 Measuring cylinder, 5 cm 3 Beaker (for filling titration apparatus), 10 cm 3 Chemicals Solutions should be contained in plastic pipettes. See the accompanying guidance on apparatus and techniques for microscale chemistry, which includes instructions for preparing a variety of solutions. Sodium thiosulfate Potassium iodate Potassium iodide Starch solution (freshly made) Sulfuric acid, 1 mol dm 3 Sample(s) of fruit juice Health, safety and technical notes Read our standard health and safety guidance. Wear eye protection. Sulfuric acid, 1 mol dm–3 is a skin/eye irritant (see CLEAPSS Hazcard HC098a). Sodium thiosulfate, 0.010 mol dm–3, potassium iodate, 0.001 mol dm–3 and potassium iodide, 0.005 mol dm–3 solutions are of low hazard, as are the starch solution and fruit juices (see CLEAPSS Hazcards HC095a, HC080, HC047b). Procedure Set up the microscale titration apparatus. Fill the apparatus with sodium thiosulfate solution. Using the glass pipette, add 2 cm 3 of potassium iodate solution to the beaker. Measure, using the measuring cylinder, 3 cm 3 of potassium iodide solution, then add this to the beaker. (Note: the potassium iodide solution is added in slight excess.) Add three drops of sulfuric acid. A yellow-brown colour appears due to iodine. Add a few drops of starch solution. A deep blue-black colour forms. Using the glass pipette, add 1 cm 3 of the fruit juice to the beaker and swirl gently. Titrate the remaining iodine in the beaker against the sodium thiosulfate solution. (The beaker can be swirled very gently to mix the chemicals. Alternatively, the tip of a plastic pipette can be used as a mini stirring rod.) The disappearance of the deep blue-black colour marks the end-point. Do a duplicate titration and check the agreement between the two titres. If it is acceptable, take the mean value of the two titres and use it for your calculations. Observations The titre volume should be in the range 0.5–1 cm 3, the disappearance of the blue-black colour marking the end-point. This experiment offers possibilities for assessing students’ abilities in following instructions and/or processing results. A survey of a range of fruit juices (and maybe other products containing vitamin C) could form the basis of a class project or as an activity for a school or college chemistry club. Note The reaction to generate the iodine is based on using an accurately known volume of the potassium iodate solution (the concentration of which is accurately known). The potassium iodide solution and the sulphuric acid are added in slight excess, and thus the concentrations of these solutions is not critical. Instead of generating the iodine in situ, it is possible to use standard iodine solution in this procedure. This would need to be diluted to give an aliquot containing 7.2 x 10–6 moles of iodine for each determination. Downloads Measuring the amount of vitamin C in fruit drinks – teacher notes PDF, Size 0.21 mb #### Measuring the amount of vitamin C in fruit drinks – teacher notes Word, Size 0.11 mb #### Measuring the amount of vitamin C in fruit drinks – student sheet PDF, Size 0.18 mb #### Measuring the amount of vitamin C in fruit drinks – student sheet Word, Size 0.12 mb Download all References S. W. Breuer, Microscale practical organic chemistry. Lancaster: Lancaster University, 1991. Additional information This resource is part of our Microscale chemistry collection, which brings together smaller-scale experiments to engage your students and explore key chemical ideas. The resources originally appeared in the book Microscale chemistry: experiments in miniature, published by the Royal Society of Chemistry in 1998. © Royal Society of Chemistry Health and safety checked, 2018 No comments Level 14-16 years 16-18 years Use Practical experiments Download Category Biological chemistry Analysis Specification Republic of Ireland Junior Cycle Science The nature of science Investigating science 2. Recognise questions that are appropriate for scientific investigation, pose testable hypotheses, and evaluate and compare strategies for investigating hypotheses. 3. Design, plan and conduct investigations; explain how reliability, accuracy, precision, fairness, safety, ethics, and the selection of suitable equipment have been considered. 4. 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2131	https://www.kenhub.com/en/library/anatomy/thalamic-nuclei	Connection lost. Please refresh the page. #1 platform for learning anatomy Login Register Success stories Anatomy Basics Upper limb Lower limb Spine and back Thorax Abdomen Pelvis and perineum Head and neck Neuroanatomy Cross sections Radiological anatomy+ Histology Types of tissues Body systems+ Physiology Introduction Muscular system Nervous system Anatomy Basics Upper limb Lower limb Spine and back Thorax Abdomen Pelvis and perineum Head and neck Neuroanatomy Cross sections Radiological anatomy+ Histology Types of tissues+ Physiology Nervous system Get help How to study What's new? Kenhub in... Deutsch Português Español Français What's new? 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ArticlesAnatomyNeuroanatomyDiencephalonThalamic nuclei Thalamic nuclei Author: Gordana Sendić, MD • Reviewer: Jana Vasković, MD Last reviewed: November 03, 2023 Reading time: 26 minutes Recommended video: Thalamic nuclei [27:34] Main nuclei of the thalamus. The thalamic nuclei are the clusters of densely packed neuronal cell bodies that comprise the thalamus. The thalamus is an ovoid, paired gray matter structure, found in the center of the brain, just superior to the brainstem. Each side of the thalamus contains six groups of nuclei; Anterior nuclei of thalamus Lateral nuclei of thalamus Medial nuclei of thalamus Intralaminar nuclei of thalamus Paraventricular (midline) nuclei of thalamus Reticular nucleus of thalamus The thalamic nuclei relay and modulate information incoming from the periphery to the cerebral cortex. Basically, almost all ascending neural pathways first synapse within a thalamic nucleus, where the information is sorted, integrated, and analysed by the thalami before they are sent further to the cerebral cortex. This fact makes the thalamus a so-called “gateway” to the cerebral cortex for limbic, motor, and all sensory modalities besides olfaction, including vision, hearing, taste, and somatic sensation. Key facts about the thalamic nuclei \| Functions \| Relaying limbic, sensory and motor signals to the cerebral cortex, Regulating consciousness, sleep, alertness \| \| Anterior nuclei of thalamus \| Anteroventral, anterodorsal, anteromedial nuclei \| \| Lateral nuclei of thalamus \| Medial and lateral geniculate nuclei, dorsal group (lateral dorsal, lateral posterior, pulvinar), ventral group (ventral anterior, ventral lateral, ventral posterior) \| \| Medial nuclei of thalamus \| Dorsomedial nucleus (parvocellular, magnocellular parts) \| \| Intralaminar nuclei of thalamus \| Anterior group: Central medial, paracentral, central lateral nucleiPosterior group: Centromedian, parafascicular nuclei \| \| Periventricular nuclei of thalamus \| Rhomboid, reuniens, parataenial nuclei \| Contents Gross anatomy Anterior nuclei of thalamus Lateral thalamic nuclei Ventral anterior nucleus Ventral lateral nucleus Ventral posterior nucleus Medial geniculate nucleus Lateral geniculate nucleus Lateral dorsal nucleus Lateral posterior nucleus Pulvinar Medial thalamic nuclei Dorsomedial nucleus Intralaminar nuclei Anterior (rostral) group Posterior (caudal) group Periventricular nuclei Reticular nucleus Connections and functions Clinical aspects Sources Show all Gross anatomy Thalamus 1/4 Synonyms: Dorsal thalamus, Thalamencephalon , show more... The thalamus is an egg shaped, bilateral, symmetrical mass of grey matter which forms a major part of the diencephalon, along with the hypothalamus and epithalamus. There are two thalami in total, one situated on each side of the third ventricle. They are connected by a band of grey matter called the interthalamic adhesion. Each side of the thalamus is divided into three main areas that each contain a collection of nuclei which are explained in the sections to follow; Anterior part, that contains the anterior nuclei of thalamus. Lateral part, that contains the lateral nuclei of thalamus. Medial part, that contains the medial nuclei of thalamus. These groups of nuclei are separated by a Y-shaped vertical sheet of white matter called the internal medullary lamina. The internal medullary lamina itself contains a group of nuclei called the intralaminar nuclei. In addition, the thalamus contains an area of thin, paraventricular nuclei of thalamus, scattered in the periventricular gray matter that separates the medial part of the thalamus from the ependyma of the third ventricle. Lastly, the thalamus contains another nucleus called the thalamic reticular nucleusthat envelopes each lateral aspect of the thalamus. In addition to the just mentioned grouping by their anatomic location, the thalamic nuclei can also be classified according to their function into the: Relay nuclei, which receive information (e.g. visual, acoustic, primary somatosensory, motor input) that is very specific and well-defined, and project this information to very specific, functionally discrete cortical areas. These nuclei include the ventral anterior (VA), ventral lateral (VL), ventral posterior (VP), medial geniculate and lateral geniculate nuclei. Association nuclei, which receive most of their information from the cerebral cortex itself, and then project back to the “association areas” in the cortex to regulate the integration and interpretation of the sensory information. These nuclei include the anterior nucleus, pulvinar, and dorsomedial nucleus. Nonspecific nuclei, which show broad and diffuse projections through the cerebral cortex, and thus may be involved in general functions such as consciousness and attention. These nuclei include the reticular nucleus, and intralaminar and midline nuclei. The thalamic nuclei are a pretty complicated topic. Make sure you don't make any of these common learning mistakes when it comes to revising them! Review and consolidate your knowledge about the thalamus with our quiz below! You can also customize it and refine the selection to fit your needs. Learn faster Custom quiz: Thalamus Start quiz Anterior nuclei of thalamus Anterior nuclei of thalamus Nuclei anteriores thalami 1/3 Synonyms: none Anterior thalamic nuclei, also called the anterior nuclear group, is a collection of neuronal cell bodies on the anterior part of the thalamus, situated between the limbs of the Y-shaped internal medullary lamina. There are three subdivisions of the anterior part of the thalamus: anteroventral, anteromedial and anterodorsal nuclei. These nuclei receive information from the limbic system, thus having important functions and influence upon emotional states, such as attention and alertness and memory acquisition. Specifically, the anterior thalamic nuclei receive afferent fibers input from the mammillary bodies via the mammillothalamic tract. The medial mammillary nucleus, that projects onto the ipsilateral anteroventral and anteromedial thalamic nuclei. The lateral mammillary nucleus, that projects bilaterally onto the anterodorsal nucleus. After acquiring the information from the mammillary bodies, the anterior thalamic nuclei in turn project efferent fibers to the cerebral cortex. These pathways are reciprocal (bidirectional) and are projected to the following cortical structures: Cingulate gyrus Anterior limbic area Parahippocampal gyrus Dorsolateral prefrontal and posterior areas of neocortex Lateral thalamic nuclei Lateral nuclei of thalamus Nuclei laterales thalami Synonyms: none Lateral thalamic nuclei, also referred to as lateral nuclear group, are a collection of neuronal cell bodies found laterally to the internal medullary lamina. These are the largest division of the thalamic nuclei, divided into dorsal and ventral tiers of nuclei. The ventral tier nuclei are the ventral anterior (VA), ventral lateral (VL) and ventral posterior (VP) nuclei. The dorsal tier nuclei are the medial geniculate nucleus (MGN), lateral geniculate nucleus (LGN), lateral dorsal (LD) nucleus, lateral posterior (LP) nucleus and pulvinar. Ventral anterior nucleus Ventral anterior nucleus Nucleus ventralis anterior Synonyms: VA nucleus The ventral anterior (VA) nucleus lies on the anterior pole of the ventral group of nuclei. It is bordered by the reticular nucleus anteriorly, the ventral lateral nucleus posteriorly, and the internal and external medullary laminae bilaterally. It consists of two parts; a principal part and a magnocellular part. The ventral anterior nucleus is located on the path between the basal ganglia and the motor areas of the premotor cortex, relaying information from both. Specifically, the afferent fibers to the ventral anterior nucleus stem from the following: Internal segment of the globus pallidus, that projects onto the principal part of the ventral anterior nucleus. Substantia nigra, that projects onto the magnocellular part of the ventral anterior nucleus. In turn, the ventral anterior nucleus sends efferent fibers to Brodmann area 6 and area 8 of the premotor cortex. These fibers are reciprocal. Additional projections from the ventral anterior nucleus are directed to the intralaminar thalamic nuclei, the frontal lobe and anterior parietal complex. Through these connections, the ventral anterior nucleus modulates and relays the signals from the basal ganglia to the premotor cortex, thus playing a role in planning and initiating movements. It is also involved in transmitting the cortical “recruiting response”. Ventral lateral nucleus Ventral lateral nucleus Nucleus ventralis lateralis Synonyms: VL nucleus This subgroup has two major divisions; pars oralis, located anteriorly, and pars caudalis, located posteriorly. The afferent fibers to the ventral lateral nucleus stem from the following: Ipsilateral internal globus pallidus projects onto the pars oralis via the thalamic fasciculus. Contralateral deep cerebellar nuclei project onto the pars caudalis. Additional projections to both pars oralis and pars caudalis stem from the spinothalamic tract, vestibular nuclei, and precentral motor cortex. The ventral lateral nucleus projects efferent fibers to the following areas: Pars oralis: Supplementary motor cortex, lateral premotor cortex Pars caudalis: Area 4 of primary motor cortex The ventral lateral nucleus is active during both passive and active movements of the contralateral part of the body. It also has a role in relaying motor feedback from the cerebellum to the cerebral cortex. Ventral posterior nucleus Ventral posteromedial nucleus Nucleus ventralis posteromedialis 1/2 Synonyms: VPM nucleus The ventral posterior nucleus is the main relay nucleus for the somatosensory pathways. It is subdivided into two parts: ventral posteromedial (or VPM) and ventral posterolateral (or VPL). This nucleus establishes several neuronal pathways with different regions within the brain.The afferent fibers to the ventral posterior nucleus stem from the following: Medial lemniscal and spinothalamic pathways project their fibers onto the ventral posterolateral (VPL) nucleus Trigeminothalamic pathway projects onto the ventral posteromedial (VPM) nucleus The trigeminothalamic pathway (ventral trigeminal tract) carries temperature, conscious proprioception, crude touch and pain sensations from the face, head and neck. In turn, the medial lemniscal and spinothalamic pathways carry sensory information from the skin and joints. Neurons receiving the inputs are organised into curved lamellae, with one lamella usually representing one region of the body. Within one lamella, different regions receive inputs distinguished by the level of sensorial “depth”. For instance, when referring to the hand, the anterodorsal part receives information from deep stimuli such as joint, tendon and muscle movement. The ventral part receives more superficial stimuli from tapping, while the neurons located between these two parts receive the most superficial inputs, such as cutaneous touch. The ventral posterior nucleus projects efferent fibers to the following areas: Brodmann areas 1 and 3b of the primary somatosensory cortex Secondary somatosensory cortex Insular cortex The efferent fibers running to the somatosensory cortex pass through the posterior limb of the internal capsule and corona radiata. This illustrates their involvement in relaying sensory sensations for conscious identification. Medial geniculate nucleus Medial geniculate body Corpus geniculatum mediale 1/3 Synonyms: Medial geniculate nucleus The medial geniculate nucleus is found within the medial geniculate body, an ovoid projection of the ventrolateral surface of the thalamus, right underneath the pulvinar. It should be noted that some authors differ on opinion whether the medial and lateral geniculate bodies are part of the thalamus, or constitute a part of the dorsal diencephalon. The medial geniculate nucleus represents the thalamic relay station of the auditory pathway, between the inferior colliculus and the auditory cortex, which receives various inputs from other auditory nuclei and, in turn, projects its fibers via the acoustic radiation to auditory centers of the temporal cortex. The afferent fibers to the medial geniculate nucleus stem from the following: Central nucleus of the ipsilateral and contralateral inferior colliculi Pericentral nucleus of the inferior colliculus Brainstem nuclei of auditory pathways Reticular nucleus of the thalamus Inferior and superior colliculi The medial geniculate nucleus projects efferent fibers to the following areas: Primary auditory cortex Auditory areas around the primary auditory cortex Insular and opercular cortical fields Medial geniculate nucleus seems to be responsible for tonotopy (organization of sound frequencies along an axis or structure) and detection of sound duration and intensity. Sound tones trigger specific cortical patterns which in turn project back to the medial geniculate body. This reciprocal relationship provides a feedback control system of ascending auditory input. Lateral geniculate nucleus Lateral geniculate body Corpus geniculatum laterale 1/4 Synonyms: Lateral geniculate nucleus The lateral geniculate nucleus is located within the lateral geniculate body, an ovoid projection of the posterior aspect of the thalamus. The lateral geniculate nucleus represents the thalamic relay station of the visual pathway. The nucleus consists of six to eight laminae, which are separated by interlaminar zones. The laminae receive the fibers from the optic tract that carries visual information from the ganglionic cell layer of the retina of both eyes. In turn, the lateral geniculate nucleus projects to the visual cortex around the calcarine fissure via the geniculocalcarine tract. The afferent fibers to the lateral geniculate nucleus stem from the following: Contralateral nasal hemiretina project via the optic tract to laminae 1, 4 and 6 Ipsilateral temporal hemiretina project via the optic tract to laminae 2, 3 and 5 The lateral geniculate nucleus projects efferent fibers to the following areas: Area 17 of primary visual cortex Extrastriate visual areas in the occipital lobe Lateral dorsal nucleus Lateral dorsal nucleus Nucleus dorsalis lateralis Synonyms: LD nucleus The lateral dorsal nucleus is the most anterior of the dorsal group of nuclei, situated in the splitting of the internal medullary lamina. This nucleus receives afferent fibers from the pretectum and the superior colliculus. Additional reciprocal connections of the lateral dorsal nucleus are with the cingulate gyrus, parahippocampal gyrus, parietal cortex and presubiculum of the hippocampal formation. Lateral posterior nucleus Lateral posterior nucleus Nucleus lateralis posterior Synonyms: LP nucleus The lateral posterior nucleus lies dorsally to the ventral posterior nucleus. It receives afferent fibers from the superior colliculus and has reciprocal connections to the superior parietal lobe. Additional connections of the lateral posterior nucleus are the inferior parietal lobe, cingulate gyrus, and medial parahippocampal cortex. To master the anatomy of the thalamic nuclei, take a look at the videos, articles, illustrations and quizzes included in the following study unit: Learn faster Thalamic nuclei Explore study unit Pulvinar Pulvinar of thalamus Pulvinar thalami 1/4 Synonyms: none The pulvinar is an expansion found on the posterior aspect of the thalamus. The pulvinar is subdivided into the medial, lateral and inferior pulvinar nuclei. The pulvinar receives inputs from the nuclei of the visual pathway and projects efferent fibers to the visual cortex, as well as to association areas of the parietal and temporal cortices. The exact function of the pulvinar is unknown, but due to its connections, it is suggested that it takes part in modulating higher functions (perception, cognition, memory) in regards to vision. Specifically, the afferent fibers to the pulvinar stem from the following: Superior colliculus Retina The pulvinar nuclei each project their own efferent fibers to the following areas: The medial pulvinar nucleus projects onto the parietotemporal complex, inferior parietal cortex, posterior cingulate gyrus and the temporal lobe. The lateral pulvinar nucleus projects onto the visual areas in the occipital and posterior temporal lobes, occipital cortex, temporal association cortex, parietal cortex and rostromedial prefrontal cortex. The inferior pulvinar nucleus projects onto the visual areas in posterior temporal lobes and the striate and extrastriate cortex in occipital lobe. Medial thalamic nuclei The medial thalamic nuclei are clusters of neuronal cell bodies located medial to the internal medullary lamina and lateral to the midline nuclei of the thalamus. This group contains one component called the dorsomedial nucleus. Dorsomedial nucleus Mediodorsal nucleus Nucleus mediodorsalis Synonyms: MD nucleus, Dorsomedial nucleus , show more... The dorsomedial nucleus, also called the medial dorsal (MD) nucleus, is subdivided into two parts: anteromedial magnocellular and posterolateral parvocellular part. The anteromedial magnocellular part communicates with several regions of the brain and projects its fibers mainly to the frontal cortex. Some fibers are purely a one-way communication (afferent/efferent), while others are two way (reciprocal). The anteromedial magnocellular division establishes the following connections: It receives olfactory input (afferent fibers) from the piriform cortex and the adjacent olfactory areas, the ventral pallidum and the amygdala. It establishes reciprocal connections with the anterior and medial prefrontal cortex, mainly to the olfactory areas on the orbital surface of the frontal lobe; the ventromedial cingulate cortex and the inferior parietal cortex and anterior insula. The posterolateral parvocellular division establishes the following connections: It establishes reciprocal connections with the dorsolateral and dorsomedial prefrontal cortex, the anterior cingulate gyrus and the supplementary motor area. It projects efferent fibers to the posterior parietal cortex. Through all these connections, the dorsomedial nucleus is responsible for integrating sensory, motor, visceral and olfactory information and subsequently relating it to the individual’s emotional state. Its overall functions are similar to those of the prefrontal cortex. Intralaminar nuclei Intralaminar nuclei of thalamus Nuclei intralaminares thalami Synonyms: Nuclei of internal medullary lamina These are collections of nerve cells within the internal medullary lamina, the “Y” shaped white matter sheet discussed before. There are two groups of nuclei involved: the anterior (rostral) and posterior (caudal). The intralaminar nuclei play a role in the activation of the cortex from the brainstem reticular formation. It is also involved in sensory-motor integration. The posterior intralaminar nuclei also seem to be important in speaking and motivation. Anterior (rostral) group The anterior group consists of subdivisions, namely the central medial, paracentral and central lateral nuclei. These nuclei have widespread reciprocal connections with the following cortical areas of the brain: The central lateral nucleus connects mainly to parietal and temporal association areas. The paracentral nucleus connects to the occipitotemporal and prefrontal cortex. The central medial nucleus connects to the orbitofrontal and prefrontal cortex and to the cortex on the medial surface. These nuclei also establish purely one-way connections with the following: All anterior group nuclei receive afferent fibers from the brainstem reticular formation, superior colliculus, and pretectal nuclei Central lateral nucleus receives afferent fibers from the spinothalamic tract. All anterior group nuclei project efferent fibers to the striatum. Central medial nucleus projects efferent fibers to the orbitofrontal and prefrontal cortex and to the cortex on the medial surface. Posterior (caudal) group The posterior group is subdivided into centromedian and parafascicular nuclei. The centromedian nucleus is related to the globus pallidus, deep cerebellar nuclei and motor cortex. The smaller parafascicular nucleus lies more medially within the lamina. The posterior group of the intralaminar nuclei receives afferent fibers from the following: Internal segment of globus pallidus Pars reticulata of substantia nigra Deep cerebellar nuclei Pedunculopontine nucleus of midbrain Spinothalamic tract The posterior group projects efferent fibers to the striatum, and establishes reciprocal connections with the motor, premotor and supplementary motor areas. Periventricular nuclei Periventricular nuclei of thalamus Nuclei periventriculares thalami Synonyms: Midline nuclei of thalamus, Median nuclei of thalamus , show more... The periventricular nuclei, also commonly referred to as the midline nuclei, are clusters of neuronal cell bodies that are found in the periventricular gray matter underneath the ependyma of the third ventricle and within the interthalamic adhesion. The entire group is located ventrally to the central medial nucleus. The midline nuclei can be subdivided into three nuclei; rhomboid, reuniens and parataenial nuclei. The midline nuclei receive afferent fibres from the following: Hypothalamus Periaqueductal grey matter of the midbrain Spinothalamic tract Medullary reticular formation Pontine reticular formation Efferent fibers from the midline nuclei pass to the following areas: Hippocampal formation Amygdala Nucleus accumbens The midline nuclei also establish reciprocal connections with the cingulate gyrus and possibly the orbitofrontal cortex. As illustrated by their connections, the midline nuclei are important in the limbic system. In addition, they may also play a role in memory and arousal. Reticular nucleus Thalamic reticular nucleus Nucleus reticularis thalami Synonyms: TRN, NRT The reticular nucleus is a narrow, shell-shaped nucleus that wraps around the lateral margin of the thalamus and extends from the rostral pole to the pulvinar and the lateral geniculate nucleus. It is found adjacent to the lateral thalamic nuclei, separated from them by the external medullary lamina. The reticular nucleus is the only nucleus that does not project to the cortex. This nucleus receives inputs from other thalamic nuclei and the cerebral cortex and sends out projections that circle back to the thalamus. Thus, it is believed that this nucleus is involved in gating and modulating information that is relayed through the thalamus, thereby regulating the activity of the thalamus itself. In addition, it is believed that the reticular nucleus receives input from the reticular formation and in turn projects to other thalamic nuclei, regulating the flow of information through these to the cortex. This would imply that the reticular nucleus plays a role in regulating alertness and attention. The reticular nucleus receives afferent fibers from the following: Nucleus cuneiformis of the midbrain Frontal cortex Temporal cortex Occipital cortex The efferent fibers that pass from the reticular nucleus are directed to the following: Body of the thalamus Contralateral dorsal thalamus Connections and functions The thalamus has multiple functions. Generally, it acts as a relay station, exchanging and modulating information between the body and the brain. In addition, the thalamus plays a role in regulating consciousness and alertness. As we’ve seen throughout this article, the thalamic nuclei have many connections. Each nucleus receives unique afferent pathways as inputs that mainly stem from subcortical structures including the diencephalon, limbic structures and the basal ganglia. These signals are first analyzed, sorted and integrated by the thalamic nuclei, and only then projected further as efferent pathways to the cerebral cortex. Thereby, the thalamic nuclei act as a hub that filter and modulate all motor, limbic and most sensory information incoming from the subcortical structures to the cerebral cortex. This fact makes the thalamus the defining factor of how the subcortical structures communicate with and influence the cerebral cortex. This is particularly the case with ascending sensory pathways that pass from the periphery to the cerebral cortex. Namely, every sensory modality besides olfaction has a thalamic nucleus that receives, analyses, and selects the information to be sent to an associated cortical area. To recap the previous sections, here is a list of the main thalamic nuclei that process the incoming sensory information: The lateral geniculate nucleus receives visual sensory information from the retina and relays it to the visual cortex of the occipital lobe. The medial geniculate nucleus receives auditory sensory information from the inferior colliculus and relays it to the primary auditory cortex of the temporal lobe. The ventral posterolateral nucleus receives information for pain, temperature and crude touch via the spinothalamic tract The ventral posteromedial nucleus receives sensory information from the face via the trigeminal nerve. Some thalamic nuclei are also involved in processing information necessary for motor control, by relaying inputs from the cerebellum and basal ganglia to the cerebral cortex. The ventral lateral nucleus is involved in relaying cerebellar signals to the primary motor area and plays a role in fine motor control. The ventral anterior nucleus is involved in relaying the basal ganglia signals to the premotor cortex and plays a role in planning and initiating movements. Finally, the thalamus plays a role in regulating consciousness, sleep and alertness. This is due to the fact that the thalamic nuclei have strong reciprocal connections with the cerebral cortex, forming thalamo-cortico-thalamic circuits that are thought to be involved with consciousness. It is also because certain thalamic nuclei have connections with the brainstem and basal forebrain such as the reticular nucleus and intralaminar nuclei. Now that you learned everything about the thalamic nuclei, put that knowledge to the test with our quiz below! Clinical aspects Thalamic lesions can paradoxically present with different non-sensory clinical symptoms despite being a central sensory hub, thus complicating diagnostic problems. Dejerine-Roussy syndrome (thalamic pain syndrome) is caused by a stroke (ischemic or hemorrhagic). This condition is most commonly preceded by numbness in the affected side. Subsequently, numbness is replaced by burning, tingling sensations, or even allodynia (pain from a stimulus that normally would not cause pain), widely varying in degree of severity across patients. The reticular nucleus of the thalamus plays an important role in the rhythmic activation of the cerebral cortex, which some studies consider it being the cause of idiopathic generalized epilepsy. Korsakoff syndrome is a syndrome seen in alcoholic patients due to the damage (by thiamine B1 deficiency) of their mamillary bodies which can extend into the thalamus through the mammillothalamic fasciculus. Fatal familial insomnia is a hereditary condition where patients gradually lose their ability to sleep which invariably leads to death. Patients can have associated psychiatric symptoms such as panic attacks, paranoia, phobias and even hallucinations. Rapid weight loss follows with dementia and mutism until death occurs. It is caused by a mutation in the PRNP gene located on the 20th chromosome. Sources All content published on Kenhub is reviewed by medical and anatomy experts. The information we provide is grounded on academic literature and peer-reviewed research. Kenhub does not provide medical advice. You can learn more about our content creation and review standards by reading our content quality guidelines. References: Haines, D. E., Mihailoff, G. A., Cunningham, W., Schenk, M., Armstrong, G. W., & Runyan, C. P. (2018). Fundamental neuroscience for basic and clinical applications. Philadelphia, PA: Elsevier. Kahle, W., & Frotscher, M. (2015). Color atlas and textbook of human anatomy: Volume 3. Stuttgart: Thieme. Mancall, E. L., & Brock, D. G. (2011). Gray's clinical neuroanatomy: the anatomic basis for clinical neuroscience. Philadelphia, PA: Elsevier Saunders. Blumenfeld, H. (2021). Neuroanatomy through clinical cases. New York, NY: Sinauer Associates/Oxford University Press. Torrico TJ, Munakomi S. Neuroanatomy, Thalamus. [Updated 2020 Jul 31]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2020 Jan-. Sheridan N, Tadi P. Neuroanatomy, Thalamic Nuclei. [Updated 2020 Jul 31]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2020 Jan-. Illustrators: Thalamic nuclei (overview) - Paul Kim Thalamic nuclei: want to learn more about it? Our engaging videos, interactive quizzes, in-depth articles and HD atlas are here to get you top results faster. What do you prefer to learn with? Videos Quizzes Both “I would honestly say that Kenhub cut my study time in half.” – Read more. Kim Bengochea, Regis University, Denver © Unless stated otherwise, all content, including illustrations are exclusive property of Kenhub GmbH, and are protected by German and international copyright laws. All rights reserved. Register now and grab your free ultimate anatomy study guide! Learning anatomy isn't impossible.We're here to help. 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2132	https://testbook.com/question-answer/find-the-coordinates-of-the-focus-of-the-parabola--5f7d90b81a1049193e18a9e0	[Solved] Find the coordinates of the focus of the parabola y2= Get Started ExamsSuperCoachingTest SeriesSkill Academy More Pass Skill Academy Free Live Classes Free Live Tests & Quizzes Previous Year Papers Doubts Practice Refer & Earn All Exams Our Selections Careers English Hindi Home Mathematics Parabola, Ellipse and Hyperbola Parabola Question Download Solution PDF Find the coordinates of the focus of the parabola y 2= 8x ? (2, 0) (- 2, 0) (0, 2) (0, - 2) Answer (Detailed Solution Below) Option 1 : (2, 0) Crack with India's Super Teachers FREE Demo Classes Available Explore Supercoaching For FREE Free Tests View all Free tests > Free Army Havildar SAC - Quick Quiz 2 K Users 5 Questions 10 Marks 6 Mins Start Now Detailed Solution Download Solution PDF CONCEPT: The following are the properties of a parabola of the form: y 2= 4ax where a > 0 Focus is given by (a, 0) Vertex is given by (0, 0) Equation of directrix is given by: x = - a Equation of axis is given by: y = 0 Length of latus rectum is given by: 4a Equation of latus rectum is given by: x = a CALCULATION: Given: Equation of parabola is y 2= 8x The given equation can be re-written as: y 2= 4⋅2⋅ x---------(1) Now by comparing the equation (1), with y 2= 4ax we get ⇒ a = 2 As we know that, focus of the parabola of the form y 2= 4ax is given by: (a, 0) So, the focus of the given parabola is (2, 0) Hence, option A is the correct answer. 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India’s #1 Learning Platform Start Complete Exam Preparation Daily Live MasterClasses Practice Question Bank Mock Tests & Quizzes Get Started for Free Trusted by 7.6 Crore+ Students More Parabola Questions Q1.Area above -axis enclosed by the parabola and the circle is Q2.If the straight lines and be coplanar then Q3.If the straight line be normal to the parabola then value is Q4.A tangent to the parabola y2 = 4x is inclined at an angle 45° deg with the positive direction of x-axis. What is the point of contact of the tangent and the parabola? Q5.Consider the points P(4k, 4k) and Q(4k, -4k) lying on the parabola y2 = 4kx. If the vertex is A, then what is ∠PAQ equal to? Q6.If vertex of a parabola is (2, −1) and the equation of its directrix is 4x − 3y = 21, then the length of its latus rectum is Q7.The locus of the mid-point of the line segment joining the focus of the parabola y2= 4ax to a moving point of the parabola, is another parabola whose directrix is Q8.If P is a point on the parabola y = x2+ 4 which is closest to the straight line y = 4x − 1, then the co-ordinates of P are : Q9.Let R be the focus of the parabola y2 = 20x and the line y = mx + c intersect the parabola at two points P and Q. Let the point G(10, 10) be the centroid of the triangle PQR. If c – m = 6, then (PQ)2 is Q10.Let PQ be a chord of the parabola y2 = 12x and the midpoint of PQ be at (4, 1). Then, which of the following point lies on the line passing through the points P and Q ? More Parabola, Ellipse and Hyperbola Questions Q1.Area above -axis enclosed by the parabola and the circle is Q2.If the straight lines and be coplanar then Q3.If the straight line be normal to the parabola then value is Q4.For points on the conic , minimum radius vector is Q5.If any point on an ellipse is (3sin α, 5cos α), then what is the eccentricity of the ellipse? Q6.. What is the distance between the two foci of the hyperbola 25x2 - 75y 2= 225 ? Q7.A tangent to the parabola y2 = 4x is inclined at an angle 45° deg with the positive direction of x-axis. What is the point of contact of the tangent and the parabola? Q8.Consider the points P(4k, 4k) and Q(4k, -4k) lying on the parabola y2 = 4kx. If the vertex is A, then what is ∠PAQ equal to? Q9.The foci of the ellipse 4x2 + 9y2 = 1 are at Q and R. If P(x, y) is any point on the ellipse, then what is PQ + PR equal to? Q10.Eccentricity of ellipse x 2 a 2+y 2 b 2=1, if it passes through point (9, 5) and (12, 4) is Suggested Test Series View All > NCERT XI-XII Math Foundation Pack Mock Test 117 Total Tests with 2 Free Tests Start Free Test Physics for Defence Examinations Mock Test 138 Total Tests with 0 Free Tests Start Free Test Suggested Exams Army Havildar SAC Army Havildar SAC Important Links More Mathematics Questions Q1.If f(z)={u(x,y)+i v(x,y),for z≠0 0,for z=0 where u(x,y)=x 3−y 3 x 2+y 2,v(x,y)=x 3+y 3 x 2+y 2 then the value of lim z→0 f(z)−f(0)z−0 along y = x wiil be Q2.The characteristic roots of a Hermitian matrix are Q3.If the roots of the equation(a−b)x 2+(c−a)x+(b−c)=0 are equal, then a, b and c are in Q4.For the equation \|x2\| +\|x\| - 6 =0 Q5.If f(x−1 x)=x 3−1 x 3 then the value of f(1) is Q6.if A is a 2×2 matrix such that trace(A) = 6 , \|A\| = 12 hen trace(A−1) is Q7.The system of equations x+2 y+3 z=1 2 x+y+3 z=2 x+2 y+3 z=3 has Q8.Consider the following statements: I. If A is a skew-symmetric matrix, then A2 is symmetric. II. Trace of a skew-symmetric matrix of an odd order is always zero. Which of the above statements is/are true? Q9.if the function f:R→R is defined by f(x)=x 2+x then the function f is Q10.The Least order of non- Abelian group is Important Exams SSC CGLSSC CHSLSSC JESSC CPO IBPS POIBPS ClerkIBPS RRB POIBPS RRB Clerk IBPS SOSBI POSBI ClerkCUET UGC NETRBI Grade BRBI AssistantUPSC IAS UPSC CAPF ACUPSC CDSUPSC IESUPSC NDA RRB NTPCRRB Group DRRB JERRB SSE LIC AAOLIC AssistantNABARD Development AssistantSEBI Grade A Super Coaching UPSC CSE CoachingBPSC CoachingAE JE electrical CoachingAE JE mechanical Coaching AE JE civil Coachingbihar govt job CoachingGATE mechanical CoachingSSC Coaching CUET CoachingGATE electrical CoachingRailway CoachingGATE civil Coaching Bank Exams CoachingCDS CAPF AFCAT CoachingGATE cse CoachingGATE ece Coaching CTET State TET CoachingCTET CoachingUPTET CoachingREET Coaching MPTET CoachingJTET Coaching Exams Territorial ArmyArmy ClerkIndian Army TGCIndian Army BSc Nursing Indian Army GDIndian Army TESIndian Army JAGIndian Army Artillery Indian Army TradesmanIndian Army TechnicalIndian Army Group CArmy Havildar SAC Indian Army CEEIndian Army AgniveerIndian Army 36 FADGATE ece Coaching CTET State TET Coaching Test Series Territorial Army Mock TestIndian Army BSc Nursing MNS Mock TestArmy Havildar SAC Mock TestAgniveer Army Clerk/Store Keeper Mock Test Agniveer Army Technical (All Arms) Mock Test Previous Year Papers Territorial Army Previous Year PapersArmy Clerk Previous Year PapersIndian Army GD Previous Year PapersIndian Army Tradesman Previous Year Papers Indian Army Technical Previous Year PapersIndian Army CEE Previous Year PapersIndian Army Agniveer Previous Year PapersIndian Army Nursing Assistant Previous Year Papers Objective Questions Microsoft Word MCQDeterminers MCQArtificial Intelligence MCQEngineering Mechanics MCQ English MCQGeography MCQHistory MCQJavascript MCQ Software Testing MCQBiology MCQCompany Law MCQE Commerce MCQ Networking MCQSql MCQWeb Technologies MCQTransformers MCQ Compiler Design MCQCurrent Affairs MCQComputers MCQConstitution Of India MCQ Ms Powerpoint MCQRespiration In Plant MCQWater Resource MCQChemistry In Everyday Life MCQ Embedded Systems MCQ Testbook Edu Solutions Pvt. 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2133	https://www.youtube.com/watch?v=yPf1m4n4z7Q	Algebra 2: 2.6 The graph of y=k/x and y=k/x^2 GRCHSFalk 239 subscribers Description 2254 views Posted: 6 Aug 2015 The notes introduce hyperbolas and inverse square functions. The impact of the constant (k) is explored as are asymptotes and discontinuous graphs. Transcript: Intro So for today's notes, we're doing section six of chapter 2. It's the graphs of y= kx^2 and y= kx. Um for those uh first thing I want to mention is the names. Although these also show up on the sheet, but uh this here is a hyperola. That's y= kx and y= kx^2. Technically that's an inverse square. And that should make sense if you stop and think about it for a minute. Uh, one, it's going to be inverse variation. We have the constant K / the variable squared. So when X goes up, Y is going to go down. It's inversely related. X and Y. The other thing is X is squared. So inverse square. Uh, another name that I've kind of used often is tornado. That's kind of how it was taught to me. and we'll get into Hyperbola so flipping over here to the chart. Uh we're going to go I'm going to go through the hyperola part first. And hyperola then the general equation is y= kx. And uh that's inverse inversely related because you have your constant k divided by the variable which means that if x goes up y will go down. If x goes down y will go up like normal inverse variation. The reason it's discontinuous is if you go back to chapter 1 when we have y= kx and you look at what's the domain for x. Well, the domain cannot include zero because we can't divide by zero. That gives us an interesting situation on the graph. And by that we have our yaxis here. Uh the graph can't have a y intercept. There is no possible y value um when x equals zero. Uh so that means then we're we'll have a graph over here on one side of the y- axis. I have it in quadrant one up here, but then also in quadrant three on the other side, but it has to try to jump across the y ais since x can't be zero. So that's why we call this graph discontinuous. You couldn't draw it all at once because it has to hop over x equals z. As far as the quadrants go, I'm going to come back to the asmtopes. If you think back to y= kx, if k is positive, um then if we're talking about over here on the right side of the y ais, those are going to be my positive x values. So positive / positive always gives you a positive, which means uh if you k is positive, one option is to be in quadrant 1. If our x values though are negative over here, so we're talking the left side of the y- axis, now we have a positive divided by negative, which will always give you a negative. Uh, so then when k is positive, your two options are either quadrant 1 or quadrant 3. If we have a negative k value, so if I change my top to a negative, and we have now we're looking over again at the right side of the y- axis, we're going to be down here in quadrant 3 because a negative divided by a positive will always give us a negative. Therefore, y is going to be negative uh over here. So that puts us in quadrant 4. And our other option then if our x's are negative so we're talking the left side of the y- ais again negative divided by a negative gives us a positive. So therefore then that kind of rounds out all the options. Uh k is negative. Definition All right. So to define an asmtope an asmtope t o t e asmtope the official definition is it's a line the graph gets increas in increasingly close to or closer and closer to but never touches. So asmtope you may want to try squeezing this on your chart or you could write it in some notes. Um it's asmtope a line graph gets increasingly close to but never touches. First time trying to type while taking a video. Another thing you could say though instead of that increasingly close be closer and closer to um if I flip back then over here enter. Negative side Okay. So if I flip back to my chart the asmtopes then there's going to be two the y- axis and the x axis. One would be the y ais because x can't be zero. And but x could be 0.1 or 01 or 0001 and the same on the negative side that the graph can get closer and closer to the y- axis but it never will reach it since x cannot be zero. Same thing is true for the x-axis that as you get a larger and larger value for x k say it's five divided by 50 is a small number. 5 divided by 5 million is even smaller. So as your k your x's get larger and larger, your y's are going to get increasingly close to zero. But it is impossible for it to actually get to zero. Graphing I want to show the asintope thing with the graphing calculator. So I pulled up my calculator program here and we'll just go with the equation y= 5 / x. So get over here. So go to your y equals menu and you can get 5 / your x value and if I were to choose to graph this thing um you get a hyperola and it shows me the hyperola here in uh quadrant 1 and you see that it gets closer and closer and closer to the x-axis. Um if you want to actually check it more and this is when I used to teach this in class. You get some people that spend the next 20 minutes just holding the right arrow. But if I press my trace key, I can see that my x value is zero and my y-val is undefined, which makes sense because there is no y intercept. Soon as I press my right arrow once, my x value.32, my y-value 15.5, and as I keep pressing my my right arrow, I'll just hold it down. Uh you should see that the y values keep getting smaller and smaller and smaller. So, right now I'm down to 0.19. X is 25. Um, if I keep holding this thing down, gets smaller and smaller. Eventually, it's going to drop into scientific notation. And you could feasibly hold this down for forever and have it never actually get to zero. But you get to a limit of the calculator at some point. That's enough for the sake of this video. Tornado The last part of the video, we just need to hit this uh last column uh which is inverse square or tornado. Um with this, you'll notice really it looks real similar to the hyperola if it were drawn. Well, uh the graph would actually drop a little bit quicker to the x-axis and then kind of even out. Uh but looks pretty similar. Uh difference in the equation is now the variables squared in the denominator. y = kx^2. Still inverse variation because our x is in the denominator of the fraction and x and y will do opposite things. When x gets larger, y will get smaller and vice versa. Discontinuous because we still can't divide by 0. 0 squ is zero. Um asmtopes, same sort of thing. Uh still can't have a y intercept. So the y- axis will be an asmtopte. uh no matter what you divide K by you're never actually going to get down to zero. Uh so this should have an uh x asmtope as well. The x axis is an asmtope. The difference here between the two is the quadrants that they're in. Now when we have this if we're looking kind of like before you're always going to have a positive denominator uh no matter what. If x is negative you square a negative you get a positive. x is positive, you square it positive, you get a positive. So I'm looking at these, the denominator is always going to be a positive number. The numerator then is kind of what makes the difference. If you have a positive k, uh like I have here in this line, uh it'll be in quadrants 1 and two. And the reason is positive number divided by a positive number is always positive. So your y value is always going to be positive if you have a positive k. If we change it then get rid of my positive here and make that into a negative instead. Uh a negative / x^2 will always be positive. It's going to put you down here in quadrants three and four. One way to possibly remember the difference and I don't know if this is where the name tornado came from and the person that taught me uh but you could remember that the downward the dashed ones in the graph there kind of look like a funnel cloud like a tornado. They're kind of skinnier at the bottom, getting increasingly large. Um, so that's one way perhaps remember that a negative K gets you down below if you don't want to try to reason through it. And that concludes the notes from section six.
2134	https://helpingwithmath.com/irrational-numbers/	Skip to content Home » Math Theory » Numbers » Irrational Numbers Irrational Numbers What are Irrational numbers? We have learnt about decimal numbers. We also know that decimal numbers can be terminating or non-terminating. But are there any other decimal numbers other than these two categories? Let us find out. Suppose we have a number 0.101001000100001……. If we observe this number, we will find out in this decimal expression, on the right of the decimal point there are either 1’s or 0’s and that the 1’s are separated respectively by one zero, two zeros, three zeros and so on. Thus the number of zeros separating two successive 1’s goes on increasing by one. It can be seen that we can go on writing this number in an endless manner. This shows that the expansion of this decimal is non-terminating as well as non-repeating. Hence, it cannot be represented as a rational number as a rational numbr is either terminating or non-terminating and non-recurring. So, what do we call such numbers? There are infinitely many decimal numbers of the type that has been discussed above such as 0.121121112…….., 0.020020002…….m 0.3000300003……… which are non-terminating and non-recurring decimals. We say that the decimal expansion which is non-terminating and non-repeating, by definition, represents an “irrational number”. Thus, a number is irrational if and only if its decimal representation is non-terminating and non-repeating. Do you know that the numbers such as , $\sqrt{2}, \,\sqrt{3}$ etc. are all irrational numbers. For example, $\sqrt{2}$ = 1.4142135…….., $\sqrt{3}$ = 1.7320508……. An irrational number is a non-terminating and non-recurring decimal, that is, it cannot be written in the form of $\frac{p}{q}$, where p and q are both integers and q ≠ 0. Now, let us find the square root of 2. We can see that $\sqrt{2}$ has non-terminating and non0recirring decimal expansion and it is given by 1.4142135……. Now let us find the square root of 3. Similarly, $\sqrt{5}, \,\sqrt{6}$ and other such square roots can also be expressed as non-terminating and non-repeating decimals and therefore they are all irrational numbers. It is important to note here that an irrational number can be approximated as closely as we like by rational numbers. The process of finding the square root of 2 by the division method can give us the decimal representation of $\sqrt{2}$. This can also be done using the following elementary method: We know that 12 = 1 < 2 < 4 = 22 Taking positive square roots, we get 1 < $\sqrt{2}$ < 2 Next, (1.4)2 = 1.96 < 2 < 2.25 = (1.5)2 Taking positive square roots again we get 1.4 < $\sqrt{2}$ < 1.5 Further, (1.41)2 = 1.9881 < 2 < 2.0.164 = (1.42)2 Again, taking positive square roots we get 1.41 < $\sqrt{2}$ < 1.42 By continuing in this manner, the next step leads us to the following inequalities: 1.414 < $\sqrt{2}$ < 1.415 If we proceed in this manner, every step will give us a closer decimal approximation of $\sqrt{2}$ than in the previous step. The eighth step will give us the following inequalities: (1.4142315)2 = 1.99999982358225 < 2 < 2.00000010642496 = (1.4142316)2 This is a very close approximation of $\sqrt{2}$. Since V is not a rational number, this process will not terminate and will lead to decimal expansion, which will be non-terminating and non-repeating. Hence, the non-repeating decimal expansion of $\sqrt{2}$ will be given by $\sqrt{2}$ = 1.4142135…. where the dots indicate that this decimal representation will not terminate. Hence, $\sqrt{2}$ = 1.4142135…… Similarly, we can find the value of other square roots which are irrational numbers. Notation for Irrational Numbers The common number sets such as the natural numbers, whole numbers, real numbers etc. all have standard symbols that are used to denote them. For instance, we use N for natural numbers, R for real numbers, W for whole numbers, Z for integers and so on. Do we have such a notation of irrational numbers? There is no standard notation for the set of irrational numbers, but the notations where the bar, minus sign, or backslash indicates the set complement of the rational numbers over the reals R could all be used such as (R – Q). Surds When we can’t simplify a number to remove a square root (or cube root etc,) then it is a surd. For example, √2 (square root of 2) can’t be simplified further so it is a surd. Properties of Irrational Numbers Irrational numbers are known to have the following properties: Negative numbers of an irrational number is an irrational number. If r is a rational number and s is an irrational number, then r + s and r – s are irrational numbers and rs and r/s are also irrational numbers. This means that any operation between a rational and an irrational number, be it addition, subtraction, multiplication or divisions will always result in an irrational number only. If r is one irrational number and s is another irrational number, then r + s and r – s may or may not be irrational numbers and rs and r/s are may or may not be irrational numbers. This means that any operation between two irrational numbers, be it addition, subtraction, multiplication or divisions will not always result in an irrational number. If a and b are two distinct positive rational numbers such that ab is not a perfect square of a rational number, then $\sqrt{ab}$ is an irrational number lying between a and b. If a and b are two distinct positive irrational numbers, then $\sqrt{ab}$ is an irrational number lying between a and b. Irrational numbers are always real numbers. Irrational numbers consist of non-terminating and non-recurring decimals. For any two irrational numbers, their least common multiple (LCM) may or may not exist. Famous Irrational Numbers Some irrational numbers are quite famous as they are used in everyday calculations and in basic mathematics as well. Let us learn about these famous irrational numbers. The first such famous irrational number is Pi (π). People have calculated Pi to over a quadrillion decimal places and still, there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795. It is defined in Euclidean geometry as the ratio of a circle’s circumference to its diameter, and also has various equivalent definitions. The number appears in many formulas in all areas of mathematics and physics. The earliest known use of the Greek letter π to represent the ratio of a circle’s circumference to its diameter was by Welsh mathematician William Jones in 1706. It is also referred to as Archimedes’s constant. Another famous irrational number is e (Euler’s number). The first few digits of e are 2.7182818284590452353602874713527. Euler’s number is the base of a natural logarithm. This number was first introduced in 1731 by the Swiss mathematician Leonhard Euler. It is also known as Napier Number which is frequently used in trigonometry as well as logarithms. The Golden Ratio number is another famous irrational number. The symbol for the golden ratio number is the Greek letter “phi” Φ. It is a special number approximately equal to 1.618. $\sqrt{2}$ that is so commonly sued in everyday calculations is another famous irrational number. It is also known as Pythagoras’s constant. Commonly used Irrational Numbers Below we have a list of commonly sued irrational numbers and their approximate values that prove helpful for mathematical calculations. \| \| \| --- \| \| Irrational Number \| Approximate Value \| \| $\sqrt{2}$ \| 1.414 \| \| $\sqrt{3}$ \| 1.732 \| \| $\sqrt{5}$ \| 2.236 \| \| π \| 3.14 \| \| e \| 2.718 \| \| $\sqrt{7}$ \| 2.645 \| Difference between Rational and Irrational Numbers Now that we know about both rational well as irrational numbers, we can clearly explain the difference between them. Some of the major differences between rational and irrational numbers include – \| \| \| --- \| \| Rational Numbers \| Irrational Numbers \| \| Rational numbers can be expressed in the form of a fraction or ratio i.e. p/q, where q ≠ 0. \| Irrational numbers cannot be expressed in the form of a fraction or ratio. \| \| Rational numbers refer to a number that can be expressed in a ratio of two integers. \| An irrational number is one that can’t be written as a ratio of two integers. \| \| Rational numbers include perfect squares \| Irrational numbers include Surds \| \| Rational numbers are finite or recurring decimals. This means that The decimal expansion is terminating or non-terminating recurring (repeating) \| Irrational numbers are non-finite or non-recurring decimals. This means that The decimal expansion is non-terminating and non-recurring at any point. \| \| Example – 5/8, 0.65 \| Example $-\sqrt{2}, \sqrt{3}$, \| \| In rational numbers, both numerator and denominator are whole numbers, where the denominator is not equal to zero. \| An irrational number cannot be written in a fraction. \| Fun Facts about Irrational Numbers It was Hippasus who was one of the students of Pythagoras who discovered irrational numbers. He discovered it when he was trying to write the square root of 2 as a fraction which is believed that he was doing it using geometry. While doing it, instead he proved the square root of 2 could not be written as a fraction, and therefore should be termed as irrational. However, the other followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods! Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. However, Hippasus was not credited or appreciated for his explanation. Theodorus of Cyrene did prove the irrational nature of the surds but it could do so only up to the whole number 17. Rationalisation Now we understand both rational and irrational numbers. We are aware of the techniques to solve compute values using rational numbers. If irrational numbers are present in the numerator of a fraction, we can do the calculations. But when these irrational numbers exist in denominators of fractions, they make calculations more difficult and complicated. To avoid such complications in the numeric calculations, we use the method of rationalization. Hence, rationalization can be defined as the process by which we eliminate radicals present in the denominators of fractions. So, how do calculate when we have irrational numbers in the denominator? It is important to note here that If a surd or surd with rational numbers is present in the denominator of an equation, to simplify it or to omit the surds from the denominator, rationalization of surds is used. Surds are irrational numbers but if multiply surd with a suitable factor, the result of multiplication will be a rational number. This is the basic principle involved in the rationalization of surds. In other words, the process of reducing a given surd to a rational form after multiplying it by a suitable surd is known as rationalization. So, what are the steps involved in rationalisation? Let us find out. Before understanding the steps, it is important to understand the meaning of the term “conjugate”. A conjugate is a similar surd but with a different sign. For example, the conjugate of $\sqrt{2}+\sqrt{3}$ would be $\sqrt{2}-\sqrt{3}$. Similarly, The conjugate of (5 + 7–√7) is (5 – 7–√7) The conjugate of (5 – 7–√7) is (5 + 7–√7) The conjugate of (10 + 3–√3) is (10 – 3–√3) The conjugate of (10 – 3–√3) is (10 + 3–√3) We now understand what conjugates are, let us check how we are supposed to rationalise an irrational number. There are two methods of rationalisation of irrational numbers. These two methods are: Conjugate Method Long Division Method Let us now understand these two methods in detail. Conjugate method This method comes into place when we have an irrational number in the denominator. The following steps are followed for rationalising in this case: Our first step is to remove the radicals from the denominator. To do this, we multiply the numerator and the denominator by a suitable conjugate. Next, we need to make sure that all the surds in the given fraction are in their simplified form. The last step of rationalisation is to simplify the fraction if required. Let us understand this through an example Example Suppose we want to rationalise $\frac{1}{4+\sqrt{2}}$. Solution Let us perform the steps that have been listed above for rationalisation. First, we need to find the conjugate of the denominator. The denominator is $4+\sqrt{2}$. Its conjugate will be $4-\sqrt{2}$. Therefore, we will multiply both the numerator and the denominator by $4-\sqrt{2}$. We will get: $\frac{1}{4+\sqrt{2}}\times \frac{4-\sqrt{2}}{4-\sqrt{2}}=\frac{4-\sqrt{2}}{4^{2}-\sqrt{2}^{2}}$ $=\frac{4-\sqrt{2}}{16-2}$ $=\frac{4-\sqrt{2}}{14}$ Therefore, the required rationalized number is $\frac{4-\sqrt{2}}{14}$which is in its simplest form. Long Division Method The Long Division method is the same method we use to find the square root of a number, whether rational or irrational. A limitation of this method is that it is only checking an irrational number that has the power of $\frac{1}{2}$. So, we recall the long division we have just done for checking whether $\sqrt{3}$ is a rational or an irrational number. Solved Examples Example 1 Prove that √5 is an irrational number. Solution Let us prove this by the method of contradiction- Say, √5 is a rational number. It can be expressed in the form p/q where p,q are co-prime integers. ⇒√5=p/q ⇒5=p²/q² {Squaring both the sides} ⇒5q²=p² (1) ⇒p² is a multiple of 5. {Euclid’s Division Lemma} ⇒p is also a multiple of 5. {Fundamental Theorem of arithmetic} ⇒p=5m ⇒p²=25m² (2) From equations (1) and (2), we get, 5q²=25m² ⇒q²=5m² ⇒q² is a multiple of 5. {Euclid’s Division Lemma} ⇒q is a multiple of 5. {Fundamental Theorem of Arithmetic} Hence, p,q have a common factor 5. which contradicts that they are co-primes. Therefore, p/q is not a rational number. This proves that √5 is an irrational number.Example 2 Find an irrational number between $\sqrt{2}$ and $\sqrt{3}$. Solution An irrational number between $\sqrt{2}$ and $\sqrt{3}$ is $\sqrt{\sqrt{2\times \sqrt{3}}}=\sqrt{\sqrt{6}}=6^{\frac{1}{4}}$ Example 3 Rationalize $\frac{1}{\sqrt{5}}$. Solution Since √5 is an irrational number and is present in the denominator of the fraction. So, we first need to rationalize it. This can be done by multiplying both numerator and denominator by √5. We will get $\frac{1}{\sqrt{5}}\times \frac{sqrt{5}}{sqrt{5}}$ $\frac{sqrt{5}}{5}$ Hence, the rationalised form of $\frac{1}{\sqrt{5}}$is $\frac{sqrt{5}}{5}$. Remember A number is irrational if and only if its decimal representation is non-terminating and non-repeating. There is no standard notation for the set of irrational numbers, but the notations where the bar, minus sign, or backslash indicates the set complement of the rational numbers over the reals R could all be used such as (R – Q). A conjugate is a similar surd but with a different sign. The conjugate or the rationalizing factor of (√a +√b) is (√a -√b). Each irrational number can be expressed as an infinite decimal expansion with no regularly repeating digit or group of digits. Together with the rational numbers, they form the real numbers. Irrational numbers are always real numbers. Irrational numbers consist of non-terminating and non-recurring decimals. For any two irrational numbers, their least common multiple (LCM) may or may not exist. Recommended Worksheets Understanding Irrational Numbers 8th Grade Math Worksheets Rounding Whole Numbers and Decimals (Basketball Themed) Worksheets Solving Word Problems Involving Rational Numbers 7th Grade Math Worksheets Browse All Worksheets Link/Reference Us We spend a lot of time researching and compiling the information on this site. If you find this useful in your research, please use the tool below to properly link to or reference Helping with Math as the source. We appreciate your support! Irrational Numbers "Irrational Numbers". Helping with Math. Accessed on August 31, 2025. "Irrational Numbers". Helping with Math, Accessed 31 August, 2025. Irrational Numbers. Helping with Math. Retrieved from Additional Numbers Theory: Multiplication Property – An Essential Building Block in Mathematics Composing Numbers Comparing and Ordering Integers Gratuity Numbers Up To Three Digits Prime and Composite Numbers Laws of Integral Exponent Appreciation and Depreciation Integers Place Value of Tenths, Hundredths, and Thousandths Latest Worksheets The worksheets below are the mostly recently added to the site. Rational Functions Math Activities Piecewise Functions Math Activities Evaluating Functions Math Activities Relative Frequencies Math Activities Normal Distribution Math Activities Margin of Error Math Activities Exponential Functions Advanced Math Activities Linear Functions Math Activities Inverse Functions Math Activities Distance Formula Math Activities
2135	https://www.startups.com/articles/founder-vs-ceo	Sitemaps Home Finding Investors How Funding Works Idea Validation Bootcamp Pitch Deck Bootcamp Pitching Investors Product MVP Articles Founder Interviews Podcast Videos Playbooks Will the Payout be Worth the Sacrifice? How Founders Blow Their Fortunes When Should We STOP Asking for "More"? Can Entrepreneurship Help Alleviate Poverty? Should Employees Really be Our Friends? Can Founders be Replaced by AI? The Best Startup Fractional CMOs and Growth Leaders The Best Growth Marketing Agencies: How to Choose Them and Who They Are What if We Run Out of Goals? Reinvention is Our Only Constant We're So Connected — And Totally Lonely The 10 Best Growth Agencies for Startups My Roadmap to Becoming a Confident Leader Is College a Waste of Time for Founders? Are We Preventing Our Startup From Evolving? Building a Startup That Loves You Back When Our Friends Resent Our Success When Our Startup Outgrows Us When Being in the 1% Feels like Failure We Need Outside Interests that Consume Us If You're Not Terrified, You're Doing it Wrong Why do Founders Suck at Asking for Help? The Ideal Client Profile Is Your Startup’s North Star (Stop Ignoring It) Are We Growing or Just Getting Fat? Let's Get Back to Our Why Does Startup Success Validate Us Personally? How We Secretly Lose Control of Our Startups Should Kids Follow in Our Founder Footsteps? The Evolution of Entry Level Workers Assume Everyone Will Leave in Year One Stop Listening to Investors Was Mortgaging My Life Worth it? What's My Startup Worth in an Acquisition? When Our Ambition is Our Enemy Are Startups in a "Silent Recession"? The 5 Types of Startup Funding What Is Startup Funding? Do Founders Deserve Their Profit? 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The Hidden Treasure of Failed Startups If It Makes Money, It Makes Sense Why do VCs Keep Giving Failed Founders Money? $10K Per Month isn't Just Revenue — It's Life Support The Ridiculous Spectrum of Investor Feedback Startup CEOs Aren't Really CEOs Series A, B, C, D, and E Funding: How It Works Best Pitch Decks Ever: The Most Successful Fundraising Pitches You Need to Know When to Raise Funds Why Aren't Investors Responding to Me? Should I Regret Not Raising Capital? Unemployment Cases — Why I LOOOOOVE To Win Them So Much. How Much to Pay Yourself Heat-Seeking Missile: WePay’s Journey to Product-Market Fit — Interview with Rich Aberman, Co-Founder of Wepay The R&D technique for startups: Rip off & Duplicate Why Some Startups Win. Chapter #1: First Steps To Validate Your Business Idea Product Users, Not Ideas, Will Determine Your Startup’s Fate Drop Your Free Tier Your Advisors Are Probably Wrong Growth Isn't Always Good How to Shut Down Gracefully How Does My Startup Get Acquired? Can Entrepreneurship Be Taught? How to Pick the Wrong Co-Founder Staying Small While Going Big Investors are NOT on Our Side of the Table Who am I Really Competing Against? Why Can't Founders Replace Themselves? Actually, We Have Plenty of Time Quitting vs Letting Go How Startups Actually Get Bought What if I'm Building the Wrong Product? Are Founders Driven by Fear or Greed? Why I'm Either Working or Feeling Guilty Startup Financial Assumptions Why Every Kid Should be a Startup Founder We Only Have to be Right Once If a Startup Sinks, Founders Go Down With it Founder vs CEO: What's the Difference? SH Sarah Humphreys CEO vs Founder; What is the best title? “What ARE we?” Ah, the age-old question — defining the relationship (or in this case, the title) of Founder and CEO, or Founder or CEO. While both titles sound impressive in conversation, does it really matter what the title is, or is that just something for teenagers trying to land their first relationship, or what to write on their business card? I have a stack of business cards near me, and they are spattered with a variety of business owner titles, and founder synonyms. So, should we be worried about the title, keep searching for related words, or another word for founder? Well, we’re about to have “the talk” right now. For leaders of companies who don’t know whether they should introduce themselves as the Founder vs CEO (or Founder CEOs) — here’s some information about what those titles mean, with insight from our amazing Startups.com Founders community on which title (if either) is the best fit. Do business owner titles matter in startups? Technically, yes, but there isn’t really a definite answer to this question. Some people claim that job titles don't matter, particularly at startups, but they do, and there is plenty of research to back it up. Job titles define employee roles, which extend even up to the Founder and CEO level, and according to various studies, they do a lot more than that. Titles can help you attract talent, get introductions, and network more strategically. Everyone wants to be part of something exciting and new — and everyone knows that titles are a shortcut to getting people interested in your business. When it comes to the roles of Founder and CEO, the weight of responsibility is a lot heavier than say the title of “office clown." Arguably, the amount of juggling is the same. But executive titles can also be a distraction from the real work at hand (“I don't care what your title is, get me those numbers!”) within the startup. At the same time, they can set expectations for roles and responsibilities within an organization, which is useful for the business owner, employees, potential investors and other stakeholders. What if I'm the only member of the executive team? At the early stages of a new company, you'll find many Founder / CEOs simply by virtue of being the only member of the team. Not only the top position, but all the executive positions in the company are filled by the Founder. As the business grows the Founder / CEO may shed some of the other titles Seriously though, does it matter if I'm a Founder / CEO? So do titles matter when it comes to the Founder or CEO of a company? I'm still the business owner, right? It depends on how you use them and the role of responsibilities you are performing in the organization. Community member, Nigel Hibbert, chimes in to share thoughts on this in a little more detail: “Founder” would be the term if you started the company from the start. Apple would have been started by Steve Jobs and Steve Wozniak, Microsoft (was) founded by Bill Gates and Paul Allen, and Walt Disney would have founded his company in his own name. As a startup head, the title of Founder can help you find investors and funding. The main drawback is that if you list yourself as a Founder, it will show that you are in the early stages of your business and therefore can be perceived as being a risky or potential high-risk prospect. Tim Cook of Apple or Steve Ballmer of Microsoft, Robert Iger, and Michael Eisner of Disney are examples of the role of the CEO and they have a much greater responsibility to the whole business operation. You would have to demonstrate that you have the necessary knowledge and experience to be holding the position of a CEO within the company and with all the necessary capability of what that means.” Founder vs CEO It’s important to know the definition of Founder and Chief Executive Officer and what those titles mean and the different skills they require. Founder: The person who started the company. It is someone who has an idea and creates a business around that idea. They are the “Founding Father” or "Founding Mother" of the company, as the company would have never existed without them creating it. They are often focused on vision and big picture of the start up. They are generally the business owner, or at least one of them. CEO: The head of the company, responsible for overseeing all aspects of it and making sure everything runs smoothly. The Chief Executive Officer runs it as a business, sets the long term plans and drives towards success. They also communicate directly to the board of directors. A Professional CEO will have the distinction of having risen through the ranks, and brings a different set of experiences and skills than a new CEO. Experience in how a CEO works at large companies may not translate directly to how a CEO will work in newer, smaller businesses. However, professional CEOs will have better understanding and experience in communicating with a board, setting success metrics, and providing an example to the rest of the team. Head of a company or managing director? The difference between these roles is good to understand in further detail. A head of the company (e.g., chairperson or chairman) might be responsible for administrative issues such as the finances, whereas a managing director (MD) will typically be involved in day-to-day operations and decision-making. But don’t just take it from us, here is feedback from a couple of our community members sharing their take on Founder vs CEO position titles: “The most important message to convey via a practical title is: a) did you start the company - ex: Founder, Co-Founder b) what is your key role (whom you make most of the decisions as) - ex: CEO, COO, Business Development, Product, Marketing.” — Heislyc Loh “The term "founder" describes your relationship to the history of the business. Page and Brin will always be Google's founders. The term "CEO" is about your position in the current organization's hierarchy. Some founders will be CEOs, at least for a while.” — William Pietri If you're a founder without another title, you may have to work extra hard to earn respect in some cases. This is especially true for female founders. Establishing authority by using the right title can play a part in getting funding and customers. However, professional CEOs bring skills and experience in hiring employees and recruiting advisors, talking to venture capitalists, communicating with the board members, partnering with other companies and distinguishing your specific role within the executive team. A Title Doesn’t Define Your Role. It’s What You Do That Matters. While titles are a great way to understand the roles on your team, they aren't the same thing as roles. A title doesn't define someone's role in a new business (e.g. "CEO" doesn't mean you're responsible for everything). In fact, thinking of titles as defining roles can be a bit misleading because it can make you feel like titles are something that's set in stone — they're not! Titles can change over time. At the end of the day, having the right title will not be what determines if you build a great company or not. Titles are also just one part understanding what each person does on your team — they also indicate what experience someone had before coming into their current position and provide clarity to people outside of your company. Founder / CEOs Don't forget the option of Founder / CEOs. From Steve Jobs, to Jeff Bezos, Mark Zuckerberg to Bill Gates, Founder CEOs are a common occurrence. It is also useful when we have a co-founder to distinguish between their respective roles of the co-founders. Understanding who is the most senior executive when there is more than one founder is useful. This is true from a small business to one seeking funding from venture capital firms. This is even more important when we have many founders. Obviously, in the case of multiple co-founders, not every founder can be a Founder CEO. Founder CEOs may also find that they hold this title temporarily - eventually being replaced by another CEO. For example, Google Founder and CEO Larry Page resigned from his role as CEO in favor of Sindar Pichai some 20 years after Google's creation. This is common among Founder CEOs. Can there be a different Founder and a CEO? It is certainly the case that companies exist with a Founder AND a CEO. Some Founder CEOs step back from the CEO role, retaining the Founder title, possibly switching out a board of directors seat. In these cases either a co-founder, one of the other employees, or an external ceo / professional CEO can fill the role within the company. This often comes with very different cash compensation than the Founder / CEO enjoyed during their tenure, and is often accompanied by equity grants. Creative alternatives for the heads of companies While Founder / CEO are definitely the most common titles, if you're the head of a company and want a more creative title than Founder / CEO here are some suggestions. Additionally, some companies have a business model that might dictate different roles to suit organizational operations. President This position is ideal for someone who loves to lead and make decisions in a fast-paced environment. They manage the staff and budget of a company, guide their direction, and make sure it's compliant with their mission statement. Presidents will also make presentations to other executives, reviewing strategy, and plans for short-term and long-term projects. Principal The principal of a company is responsible for the management of daily business operations. They work to develop their company's mission and think of strategies to achieve the organization's goals. Director of X A director of a company is responsible for the organization's development and implementation of policies, procedures, and operational reporting/metrics. Directors develop and implement consistent inventory and cost accounting policies, procedures, and operational reporting/metrics. Managing Partner. A managing partner is a challenging role that requires considerable expertise, experience, and leadership skills. The person in this role should be equipped to handle tasks from supervising daily operations to achieve organizational goals and objectives. Entrepreneur If you're not yet at a stage where you feel justified in referring to yourself as a Founder, a CEO, a Founder / CEO or similar, but want people to understand that you're working on an original idea - Entrepreneur can be a good bridge title until you feel like you fit in with the other Founder / CEOs. Summary/Conclusion Titles are good for business cards and social media profiles, and they can help bring you a sense of authority or confidence. But when it comes to running the company, what you do matters more than your title. A leader is there to inspire the team and make sure everyone is on the same page in terms of what's important to the organization as a company. So, which title is best for the head of a startup, Founder vs CEO? It all comes down to what the head does within the company. Are you an active manager or an absentee business owner? Use the tips and definitions above to make sure it aligns with the description of the work you are doing and you will have the best title for your role. Don't worry about keeping up with all the other Founder CEOs, Founders, CEOs, and Founder / CEOs :D. Find this article helpful? This is just a small sample! Register to unlock our in-depth courses, hundreds of video courses, and a library of playbooks and articles to grow your startup fast. Let us Let us show you! Submission confirms agreement to our Terms of Service and Privacy Policy. No comments yet. Start a Membership to join the discussion. Already a member? Login Create Free Account
2136	https://www.red-gate.com/simple-talk/databases/sql-server/t-sql-programming-sql-server/the-sql-of-membership-equivalence-classes-cliques/	The SQL of Membership: Equivalence Classes & Cliques - Simple Talk Articles Podcasts Contributors About About Simple Talk Author AI Usage Policy FAQ Contact Us Resources Books RegisterLog in Register for Simple Talk Get the latest Opinion, Podcasts, Webinars, and Events, delivered straight to your inbox. RegisterLog in Articles Podcasts Contributors About About Simple Talk Author AI Usage Policy FAQ Contact Us Resources Books Articles /Databases /SQL Server /T-SQL Programming The SQL of Membership: Equivalence Classes & Cliques Joe Celko28th July 2014 Article Recommended About the author Comments 3Share Share to social media Equivalence Classes & Cliques We keep harping that SQL is based on sets, but how many of us ever go back and re-read any old text books on set theory? In particular, one concept gets forgotten is equivalence relations on sets, which create equivalence classes. These classes are disjoint and we can put an element from a set into one of them with some kind of rule. Let’s use ~ as the notation for an equivalence relation. The definition is that it has three properties: The relation is reflexive: A ~ A. This means the relation applies to itself. The relation is symmetric: : if (A ~ B) ⇔ (B ~ A). This means when the relation applies to two different elements in the set, it applies both ways. The relation is transitive: (A ~ B) Λ (B ~ C) ⇒ (A ~ C). This means we can deduce all elements in a class by applying the relation to any known members of the class. This is pretty important for programming, as we will see. So, if this is such a basic set operation, why isn’t it in SQL? The problem is that it produces classes, not a set! A class is a set of sets, but a table is just a set. The notation for each class is a pair of square bracket that contain some representative of each set. Assume we have a ∈ A b ∈ B These expressions are all the same: A ~ B [a] = [b] [a] ∩ [b] ≠ Φ A common example is the set Z of integers and anyMOD()operator will give you equivalence classes. The MOD(n, 2) operation gives you one class consisting of all even numbers, and the other consisting of all odd numbers. The nice part is that you can compute the MOD() function with arithmetic. An equivalence classes can also be defined by a set of sets. This is an important concept in graph theory and graph database. If you have not seen it, Google “Six Degrees of Kevin Bacon“. The idea is that any individual involved in the Hollywood film industry can be linked through his or her film roles to Kevin Bacon within six steps. Let’s steal a sample set from a SQL Forum posting of friends, where ~ now means “is a friend of” and that we can use “foaf” — “Friend of a friend” — to build cliches. {Fred, Jessica, John, Mary} {Albert, Nancy, Peter} {Abby} {Frank, Joe} A complete graph means that each node is connected to every other node by one edge. If a subgraph is complete, it is actually called a Clique in graph theory! The complete graph Kn of order n is a simple graph with n vertices in which every vertex is adjacent to every other. The complete graph on n vertices has n(n-1)/2 edges, which corresponding to all possible choices of pairs of vertices. But this is not an equivalence relation because it does not include the reference of each node to itself. Remember A~A? Think about Abby in this set, assuming that she likes herself, in spite of her lack of social skills. Obviously, if you have a clique with (k) nodes in it, you can a complete subgraph of any size (j <k). A maximal clique is a clique that is not a subset of any other clique (some authors reserve the term clique for maximal clique. SQL seems to have two equivalence relations that are major parts of the language. The first is plain old vanilla equal (=) for scalar values in the base data types. SQL is just like any other programming language, but we also have NULLs to consider. Oops! We all know that (NULL = NULL) is not true. So we have to exclude NULLs or work around them. The other “almost” equivalence relation is GROUP BY in which the class is the grouping columns. A quick example would be “SELECT city_name, state_code FROM Customers GROUP BY state_code;” and this is still not quite right because we have to do some kind of aggregation on the non-grouping columns in the query. Graphs in SQL The idiom for graphs in SQL is to use a table with the nodes involved and a second table with pairs of nodes (a, b) to model the edges, which model our ~ relation.. This is classic RDBMS; the first table is entities (e.g. cities on a map, electronic components in a device, etc) and the second table is a relationship (e.g. roads between cities, wiring between components, etc). The bad news is that when you use it for equivalence relations, it can get big. A class with one member is has one row to show the edge back to itself. A class with two members {a, b} has{(a, a), (b, b), (a, b), (b, a)}to show the ~ properties as edges. Three members gives us six rows;{ (a, a), (b, b), (c, c), (a, b), (a, c), (b, a), (b, c), (c, a), (c, b)}. The general formula is(n (n-1)) + n, which is pretty close to (n!). First, load the table a few facts we might already know: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 CREATE TABLE Friends (lft_member VARCHAR(20) NOT NULL, rgt_member VARCHAR(20) NOT NULL, PRIMARY KEY (lft_member, rgt_member)) INSERT INTO Friends (lft_member, rgt_member) VALUES ('John', 'Mary'), ('Mary', 'Jessica'), ('Peter', 'Albert'), ('Peter', 'Nancy'), ('Abby', 'Abby'), ('Jessica', 'Fred'), ('Joe', 'Frank'); Reflexive Rows The first thing we noticed is that only Abby has a reflexive property row. We need to add those rows and can do this with basic set operators. But it is not that easy, as you can see with this insertion statement: 1 2 3 4 5 6 7 8 9 INSERT INTO Friends SELECT X.lft_member, X.rgt_member FROM ((SELECT lft_member, lft_member FROM Friends AS F1 UNION SELECT rgt_member, rgt_member FROM Friends AS F2) EXCEPT SELECT lft_member, rgt_member FROM Friends AS F3) AS X (lft_member, rgt_member); The use of table aliases is tricky. You have to be sure that the SQL engine will construct a table in such a way that you do not get scoping problems. The UNIONand EXCEPToperators are used to assure that we do not have primary key violations. Abby Abby Albert Albert Frank Frank Fred Fred Jessica Jessica Jessica Fred Joe Joe Joe Frank John Mary John John Mary Mary Mary Jessica Nancy Nancy Peter Peter Peter Nancy Peter Albert Symmetric Rows Look at the update table and there are no{ (a, b), (b, a)} pairs. We need to add this second property to the relation table. This follows the pattern of set operators we just used: 1 2 3 4 5 6 7 8 9 10 INSERT INTO Friends SELECT X.lft_member, X.rgt_member FROM ( (SELECT F1.rgt_member, F1.lft_member FROM Friends AS F1 WHERE F1.lft_member <> F1.rgt_member) EXCEPT (SELECT F2.lft_member, F2.rgt_member FROM Friends AS F2) ) AS X (lft_member, rgt_member); This will give us: Abby Abby Albert Peter Albert Albert Frank Joe Frank Frank Fred Jessica Fred Fred Jessica Mary Jessica Jessica Jessica Fred Joe Joe Joe Frank John Mary John John Mary Mary Mary John Mary Jessica Nancy Peter Nancy Nancy Peter Peter Peter Nancy Peter Albert If you do quick GROUP BY query, you see that Abby is seriously anti-social with a count of 1 , but Mary and Jessica look very social with a count of 3. This is not quite true because the property is the dreaded “Rock-paper-scissors-lizard-Spock” relationship.” Transitive Rows Transitivity goes on forever. Well, until we have what mathematicians call a closure. This means we have gotten a set of elements that have all of the valid relationships. In some cases, these sets are infinite. But in database, we can only have insanely large tables. Let’s pull a subset that is part oi a clique of 4 friends. Fred Jessica Fred Fred Jessica Mary Jessica Jessica Jessica Fred John Mary John John Mary Mary Mary John Mary Jessica Now apply the transitive relation to get some of the missing edges of a graph: 1 2 3 4 5 6 7 8 9 INSERT INTO Friends SELECT X.lft_member,X.rgt_member FROM ((SELECT F1.lft_member,F2.rgt_member FROM Friends AS F1,Friends AS F2 WHERE F1.rgt_member=F2.lft_member) EXCEPT (SELECT F3.lft_member,F3.rgt_member FROM Friends AS F3) )AS X (lft_member,rgt_member); This is a simple statement using the definition of transitivity, without a universal quantifier or loop on it. It will add these rows to this subset: Fred Mary Jessica John John Jessica Mary Fred When you look at Mary and Jessica, you have all of their friends. However, Fred and John do not know that they are friends. So we invoke the statement again and get those two rows. If you try it a third time, there are zero rows added. The first thought is this sounds like a job for a recursive statement. But it is not that easy. If the original graph has a cycle in it, you can hang in an infinite loop if you try to use a recursive CTE. The assumption is that each clique has a spanning graph in the pairs in. Oops! New term: a spanning graph is a sub-graph that includes the nodes and some or all of the edges of the graph. A complete graph is the spanning graph has all the possible edges, so each node is directly connected to any other node. Cliques Now change your mindset from graphs to sets. Let’s look at the size of the cliques: 1 2 3 4 WITH X (member,clique_size) AS (SELECT lft_member,COUNT()FROM Friends GROUP BY lft_member) SELECTFROM X ORDER BY clique_size; Abby 1 Frank 2 Joe 2 Nancy 3 Peter 3 Albert 3 Fred 4 Jessica 4 John 4 Mary 4 What we want to do is to assign a number to each clique. This sample data is biased by the fact that the cliques are all different sizes. You cannot simply use the size to assign a clique_nbr. Create this table load it with the names. 1 2 3 CREATE TABLE Cliques (clique_member VARCHAR(20)NOT NULL PRIMARY KEY, clique_nbr INTEGER); We can start by assigning everyone their own clique, using whatever your favorite method is: 1 2 3 4 5 6 7 8 9 10 11 12 INSERT INTO Cliques VALUES ('Abby',1), ('Frank',2), ('Joe',3), ('Nancy',4), ('Peter',5), ('Albert',6), ('Fred',7), ('Jessica',8), ('John',9), ('Mary',10); Everyone is equally a member of their clique in this model. That means we could start with anyone to get the rest of their clique. The update is very straight forward. Take any clique number, find the member to whom it belongs and use that name to build that clique. But which number to we use? We could use the MIN, MAX or a random number in the clique; I will use the MAX for no particular reason I keep thinking that there is a recursive update statement that will do this in one statement. But I know it will not port (Standard SQL has no recursive update statement right now. We would do it with SQL/PSM or a host language) and I think it would be expensive. Recursion will happen for each member of the whole set, but if we consolidate a clique for one person, we have removed his friends from consideration. The worst situation would be a bunch of hermits, so the number of clique would be the cardinality of set. That is not likely or useful in the real world. Let’s put the update in the body of a loop. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 BEGIN DECLARE@clique_loop INTEGER; DECLARE@mem CHAR(10); SET@clique_loop=(SELECT MAX(clique_nbr)FROM Cliques); WHILE@clique_loop>0 BEGIN SET@mem=(SELECT clique_member FROM Cliques WHERE clique_nbr=@clique_loop); UPDATE Cliques SET clique_nbr =(SELECT clique_nbr FROM Cliques WHERE clique_member=@mem) WHERE clique_member IN(SELECT rgt_member FROM Friends WHERE lft_member=@mem); SET@clique_loop=@clique_loop-1; SELECTFROM Cliques; END; END; As a programming assignment, replace the simple counting loop with one that does not do updates with clique_loop values that were removed in the prior iteration. This can save a lot of work in a larger social network than the sample data used here. Adding Members Now that we have a Cliques table, we can do a simple insertion if the new guy belongs to one clique. However, we can have someone who knows a people in different cliques. This will merge the two cliques into one. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 CREATE PROCEDURE Clique_Merge @clique_member_1 CHAR(10), @clique_member_2 CHAR(10) AS UPDATE Cliques SET clique_nbr =(SELECT MAX(clique_nbr) FROM Cliques WHERE clique_member IN(@clique_member_1,@clique_member_2)) WHERE clique_nbr =(SELECT MIN (clique_nbr) FROM Cliques WHERE clique_member IN(@clique_member_1,@clique_member_2)); Deleting a member or moving him to another clique is trivial. Conclusion This article is full of handy tricks to use with SQL. But cliques and equivalence classes are better handled with a graph database than with SQL. This is no surprise; that is what they were meant to do! The graph database can also handle relationships that do not have all three of the mathematical properties needed for an equivalence relationship. If any readers want to add to this set of tricks, please do. Spotted an error? If you've seen something in this article that needs changing, whether it's a technical inaccuracy or a typo, please let us know by reaching out to the team. Get in touch ### Time is running out to sponsor PASS Summit 2025! Showcase your brand at one of the largest data gatherings in North America. Sponsorship applications close October 9th. Request a prospectus ### Chance to win a $250 Amazon voucher Redgate’s latest State of the Database Landscape survey is now open – share your experience for a chance to win. Take part On this page Equivalence Classes & Cliques Graphs in SQL Reflexive Rows Symmetric Rows Transitive Rows Cliques Adding Members Conclusion Recommended #### Introduction to artificial intelligence Artificial intelligence applications are all around us, but what does it really mean? In this article, Kumar Abhishek explains the history and progress of artificial intelligence. Read more #### Why Test-Driven Development? (Part 1) Software development is a very tough discipline. Robert Martin (better known as Uncle Bob) holds that software development is the toughest discipline Read more #### Text Mining and Sentiment Analysis: Analysis with In the third article of this series, Sanil Mhatre demonstrates how to perform a sentiment analysis using R including generating a word cloud, word associations, sentiment scores, and emotion classification. Read more Subscribe for more Get selected articles, event information, podcasts and other industry content delivered straight to your inbox. Subscribe About the author Joe Celko See Profile Joe Celko is one of the most widely read of all writers about SQL, and was the winner of the DBMS Magazine Reader's Choice Award four consecutive years. He is an independent consultant living in Austin, TX. He has taught SQL in the US, UK, the Nordic countries, South America and Africa. 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2137	https://www.reddit.com/r/askmath/comments/ax8oib/question_is_zero_a_rational_number/?tl=zh-hans	[问题] 零是有理数吗？ : r/askmath Skip to main content[问题] 零是有理数吗？ : r/askmath Open menu Open navigationGo to Reddit Home r/askmath A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to askmath r/askmath r/askmath 这个子版块是用来提问数学问题的。请在发帖前阅读下面的子版块规则。 209K Members Online •7 yr. ago NEXT_VICTIM [问题] 零是有理数吗？我正在看一些关于各种数字的东西（感谢 YouTube 上的 numberphile），他们展示了一个数字分类图。我试图弄清楚 0 应该放在哪里。额外问题！ Null ( ø ) 也会有一个分类吗？会和 0 不同吗？ Read more Archived post. New comments cannot be posted and votes cannot be cast. Share New to Reddit? Create your account and connect with a world of communities. Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community 0 0 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
2138	https://www.ams.org/journals/bull/1942-48-08/S0002-9904-1942-07740-8/S0002-9904-1942-07740-8.pdf	ON THE THEORY OF THE TETRAHEDRON N. A. COURT I. DEFINITION. We associate with the general tetrahedron (T) — ABCD a sphere (Q) whose center is the Monge point M of (T) and the square of whose radius is (a) q2 = (MO2 - R2)/3, where O and R are the center and the radius of the circumsphere (0) of (T). In what follows, a number of propositions regarding the sphere (<2) will be established and it will be shown that from the properties of (Q) may be derived, as special cases, properties of the polar sphere (H) of the orthocentric tetrahedron (Th). For want of a better name we shall refer to (Q) as the "quasi-polar" sphere of the general tetrahedron (T). The expression M02 — R2 is the power of the Monge point M of (T) for the sphere (O). THEOREM 1. The square of the radius of the quasi-polar sphere of the general tetrahedron is equal to one-third of the power of the Monge point of the tetrahedron f or its circumsphere. The sphere (Q) is real, a point sphere, or imaginary according as MO is greater than, equal to, or smaller than R. Moreover, we have MO<2R, for the mid-point of MO is the centroid G of (T), and G necessarily lies within the sphere (0). COROLLARY. In an orthocentric tetrahedron (Th) the Monge point coincides with the orthocenter H, and the above properties of (Q) are valid for the polar sphere (H) of (Th).1 The Monge point M of (T) is a center of similitude of the circum sphere (0) and the twelve point sphere (L) of (T)2 hence M is the center of a sphere of antisimilitude of (0) and (L), that is, a sphere with respect to which the spheres (O) and (L) are inverse of one an-other. Presented to the Society, December 31, 1941; received by the editors November 22, 1941. 1 Nathan Altshiller-Court, Modern Pure Solid Geometry, New York, 1935, p. 265, §813. This book will be referred to as MPSG. 2 MPSG, p. 251, §764. 583 584 N. A. COURT [August The line of centers LO of the spheres (L), (0) meets (0) in the points whose distances from M are MO — R and MO+R. Again, the line LO meets (L) in two points whose distances from M are ML — R/3 and ML+R/3, for the radius of (L) is equal to R/3. Now the points MO+R, ML —R/3 correspond to each other in the inversion con-sidered (and so do the points MO — R, ML + R/3), hence if x2 is the constant of inversion, we have x2 = (MO + R)(ML - R/3) = (MO + R)(3ML - R)/3 = (MO2 - R2)/3 = q2. But the constant of inversion is the square of the radius of the sphere of inversion, and the sphere of inversion is coaxial with the two inverse spheres. THEOREM 2. The quasi-polar sphere of the general tetrahedron is co-axial with the circumsphere and the general twelve point sphere of the tetrahedron. COROLLARY. TO the square (L) of (T) corresponds in the ortho-centric tetrahedron (T) the second twelve point sphere, and from the above property of (Q) we obtain the known property of the polar sphere THEOREM 3. The sum of the powers of the vertices of a general tetra-hedron with respect to the quasi-polar sphere of that tetrahedron is equal to one-third of the sum of the squares of the edges of the tetrahedron. The power Dq of the vertex D of (7") for the sphere (Q) is Dq = MB2 - q2 = MD2 - (MO2 - R2)/3, and from the triangle DMO we have DM2 + DO2 = 2DG2 + M02/2 = WG2 + 20G2, hence Dq = 2DG2 + 2(OG2 - R2)/3. Adding to Dq the analogous formulas relative to the vertices A, B, C of (T) we obtain Aq + Bq + Cq + Dq = 2(AG2 + BG2 + CG2 + DG2) + 8(OG2 - R2)/3. Now if we denote by k2 the sum of the squares of the edges of (T), we have s MPSG, p. 264, §805. i942] THEORY OF THE TETRAHEDRON 585 AG2 + EG2 + CG2 + DG2 = &2/4,4 (b) OG2 - R2 = - &2/16,6 hence Aq + Bq + Cq + Dq = k2/2 - k2/6 = k2/3. COROLLARY. In an orthocentric tetrahedron the sum of the squares of a pair of opposite edges is constant, hence the sum of the powers of the vertices of the tetrahedron for the polar sphere is equal to the sum of the squares of a pair of opposite edges THEOREM 4. The sum of the squares of the distances of the Monge point of a tetrahedron from the mid-points of the edges is equal to three times the sum of the squares of the radii of the circumsphere and the quasi-polar sphere of the tetrahedron. Let £, F be the mid-points of the edges DA, BC of (T). The mid-point of the bimedian EF — ma of (T) is the centroid G and therefore we have, from the triangle MEF, ME2 + MF2 = 2MG2 + EF2/2 = 20d + ml/2. We have two analogous formulas relative to the other two pairs of opposite edges of (T). Adding the three relations we have £ ME2 = 60G2 + (ml + ml + mc)/2. Now the expression in the parenthesis is equal to7 k2/4, hence, making use of the formula (b), we have X) ME2 = 6(R2 - &2/16) + k2/S = 3R2 + 3(R2 - k2/\2). But, eliminating OG between the formulas (a) and (b) we have (c) R2 - k2/\2 = q2, hence 2 ME2 = 3(R2 + q2). COROLLARY. The proposition is valid for the orthocentric tetrahedron.8 THEOREM 5. The sum of the powers of the Monge point of a tetra-4 National Mathematics Magazine, vol. 15 (1941), p. 273, §3. 0 V. Thebault, Nouvelles Annales de Mathématiques, (4), vol. 19 (1919), p. 425. 6 Mathesis, vol. 42 (1928), p. 338, §2. 7 MPSG, p. 56, §186. 8 MPSG, p. 275, Example 4. 586 N. A. COURT [August hedron with respect to the six spheres having for diameters the edges of the tetrahedron is equal to six times the square of the radius of the quasi-polar sphere of the tetrahedron. The power of the Monge point M for the sphere (BC) having for diameter the edge BC = a of (T) is equal to MF2 — a2/^. We have five analogous expressions relative to the other five edges of (T). Adding the six expressions we have £ MF2 - &2/4, or, using the value oî^MF2 from the preceding article, 6(R2 - &2/16) + k2/S - &2/4 = 6q\ In an orthocentric tetrahedron the spheres having for diameters the edges of the tetrahedron are orthogonal to the polar sphere. COROLLARY. The power of the orthocenterfor the six spheres is equal to the square of the radius of the polar sphere. The powers of the Monge point of (T) for the two spheres having for diameters a pair of opposite edges of (T) are equal.9 THEOREM 6. The sum of the powers of the Monge point of a tetra-hedron with respect to three spheres having f or diameters three concurrent (or coplanar) edges of the tetrahedron is equal to three times the square of the radius of the quasi-polar sphere. The centroid G of (T) bisects the segment AA ' joining A to the corresponding vertex A' of the tetrahedron (T') twin to (T), hence A MA 'O is a parallelogram. Let Ai be the diametric opposite of A on the circumsphere (0) of (T). The quadrilateral MArA\0 is a parallelogram, for OA\ and MA' are equal and parallel. Hence the diagonal MA\ bisects the diagonal OA', and therefore meets the median A'G of the triangle AfOM in the centroid Ga of that triangle. Thus GGa = A'Ga/2 = AG/3, and Ga is therefore the centroid of the face BCD of (T). Moreover MGa = MAi/3. Let A2 be the second point of intersection, besides A, of the line AM with the sphere (0). The line AiA2 is perpendicular to the line AMA2, hence if Fis the foot of the perpendicular from Ga upon AMA2 we have 9 G. Gallucci, Nouvelles Annales de Mathématiques, (3), vol. 16 (1897), p. 17, §5. 1942 J THEORY OF THE TETRAHEDRON 587 MF:MA2 = MGa'.MÂ! = 1:3, and therefore MA-MA2 = 3MA-MF. But MA • MA2 is the power of M for the sphere (0), hence by Theo-rem 1, MA • M F is equal to the square of the radius of the quasi-polar sphere (Ç), that is, the points A, F are inverse with respect to (Q). The median AGa of (T) subtends a right angle at F, hence the sphere (AGa) having AGa for diameter is orthogonal to (Q), simi-larly for the three analogous spheres (5GT>), (CGC), (DGd). Thus this theorem follows : THEOREM 7. The quasi-polar sphere of a tetrahedron is orthogonal to the four spheres having for diameters the four medians of the tetrahedron.10 Since the points A, F are inverse for (Q) (Theorem 7), the polar plane a of A for (Q) is perpendicular to A M at F\ hence the line FGa is the trace of a in the plane A MGO. THEOREM 8. The polar reciprocal tetrahedron of a given tetrahedron (T) with respect to the quasi-polar sphere of (T) is circumscribed about the medial tetrahedron of (T). THEOREM 9. The faces of the polar reciprocal tetrahedron of a given tetrahedron (T) with respect to the quasi-polar sphere of (T) cut the spheres having for diameters the corresponding medians of (T) along four circles lying on the same sphere, namely, the twelve point sphere of(T). The point Ga lies on the twelve point sphere (L) of (T), and the diametric opposite of Ga on (L) lies on the line AM',11 hence the point F also belongs to the sphere (L). Thus the plane a (Theorem 8) cuts the two spheres (AGa) and (L) along the same circle, and FGa is a diameter of that circle. THEOREM 10. The faces of the polar reciprocal tetrahedron of the medial tetrahedron of a given tetrahedron (T) with respect to the quasi-polar sphere of (T) cuts the spheres having f or diameters the respective medians of (T) along f our circles lying on the same sphere, namely the circumsphere of (T). The polar plane of Ga with respect to the quasi-polar sphere (Q) passes through A and is perpendicular to MGa, hence the trace K 10 American Mathematical Monthly, vol. 39 (1932), pp. 196, 197, §§11, 13. 11 MPSG, p. 251. 588 N. A. COURT [August of this plane on MGa lies on the sphere (AGa) having for diameter the median AGa of (T). On the other hand, the line MGa passes through the diametric opposite A\ of A on the circumsphere (0) of (T), hence K lies on (0). Thus the line AK is a common diameter of the two circles along which the polar plane considered cuts the two spheres (AGa) and (0). II. DEFINITIONS. We associate with the general tetrahedron (T) the sphere having for center the centroid G of (T) and for the square of its diameter one-third of the sum of the squares of the bimedians of (T). The sphere will be denoted by (G) and referred to as the G-sphere of (T). The sphere (G) is concentric with the three spheres having for diameters the bimedians of (T). In an orthocentric tetrahedron the bimedians are equal, so that the three spheres having these bimedians for diameters coincide, and the G-sphere coincides with them, in the first twelve point sphere of the orthocentric tetrahedron, the three bimedians being diameters of that sphere. THEOREM 1. The quasi-polar sphere and the (G)-sphere of the general tetrahedron are orthogonal. The square of the radius, g2, of the sphere (G) is equal to7 (§1, Theorem 4) (ma + mh + mc)/3:4: = k /48, hence g2 + g2 = (R2 __ £2/12) + £2/ 4 8 = R2 _ £2/16 = QQ2 = MQ2} that is, the square of the line of centers of the spheres (<2), (G) is equal to the sum of the squares of their radii. COROLLARY. The polar sphere and the first twelve point sphere of the orthocentric tetrahedron are orthogonal.12 THEOREM 2. The sphere (G) belongs to the coaxial pencil formed by the circumsphere, the twelve point sphere, and the quasi-polar sphere of the tetrahedron. The centroid G is the second center of similitude, besides the point M, of the circumsphere (0) and the twelve point sphere (L) of (T). Now the quasi-polar sphere (Q) being a sphere of antisimilitude of « MPSG, p. 262, §799. 194^] TERNARY CUBIC 589 (O) and (L) (§1, Theorem 2), the sphere (G) having G for center and orthogonal to (Q) (§11, Theorem 1) is the second sphere of anti-similitude of (0) and (L) ; hence (G) is coaxial with these spheres. THEOREM 3. The four spheres having for centers the vertices of a tetrahedron and orthogonal to the quasi-polar sphere cut the spheres having for diameters the respective medians of the tetrahedron along four circles belonging to the same sphere, namely, the (G)-sphere of the tetrahedron. The sphere (A) having A for center and orthogonal to the sphere (<2) is coaxial with the spheres (G) and {AGa), for the centers of these three spheres are collinear and all three are orthogonal to (Q). Simi-larly for the vertices B, C, D of (T). UNIVERSITY OF OKLAHOMA EUCLIDEAN CONCOMITANTS OF THE TERNARY CUBIC T. L. WADE 1. Introduction; construction of concomitants. In this paper we use the results of Cramlet [l] and the writer to study the euclidean concomitants of the ternary cubic curve TahcXaXbXc = 0, where a, b, c — \, 2, 3 and Tabc is symmetric. With tensor algebra as the medium of investigation all types of concomitants are readily con-structed, and their geometric interpretations are also readily made in most cases. As is conventional in classical invariant theory, the word concomitant will be used as meaning rational integral concomitant unless stated to the contrary. As a consequence of Theorem 3 in , we have the following theo-rem. THEOREM I. Every euclidean concomitant of the ground form TabcXaXbXc (a, b, c = l, 2, 3) is expressible by composition as a tensor of order zero with the use of the coefficient tensor Tabc, the variable coordi-nate tensors Xa and Ua, and the numerical tensors eabc, La, and Eab. Presented to the Society, September 5, 1941 ; received by the editors of the Trans-sctions of this Society, September 25, 1941 ; accepted by them and later transferred to this Bulletin.
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Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Screening (multi)collinearity in a regression model Ask Question Asked 15 years, 3 months ago Modified11 years, 2 months ago Viewed 38k times Part of R Language Collective This question shows research effort; it is useful and clear 65 Save this question. Show activity on this post. I hope that this one is not going to be "ask-and-answer" question... here goes: (multi)collinearity refers to extremely high correlations between predictors in the regression model. How to cure them... well, sometimes you don't need to "cure" collinearity, since it doesn't affect regression model itself, but interpretation of an effect of individual predictors. One way to spot collinearity is to put each predictor as a dependent variable, and other predictors as independent variables, determine R 2, and if it's larger than .9 (or .95), we can consider predictor redundant. This is one "method"... what about other approaches? Some of them are time consuming, like excluding predictors from model and watching for b-coefficient changes - they should be noticeably different. Of course, we must always bear in mind the specific context/goal of the analysis... Sometimes, only remedy is to repeat a research, but right now, I'm interested in various ways of screening redundant predictors when (multi)collinearity occurs in a regression model. R Language Collective r statistics regression Share Share a link to this question Copy linkCC BY-SA 3.0 Improve this question Follow Follow this question to receive notifications edited Dec 31, 2013 at 22:42 abcd 10.8k 16 16 gold badges 56 56 silver badges 91 91 bronze badges asked Jun 15, 2010 at 2:10 aL3xaaL3xa 36.2k 18 18 gold badges 81 81 silver badges 112 112 bronze badges 13 5 I'm very pleased that no one marked this as not "programmy" enough and many people up voted this. This is a very good question that many of us who "program with data" struggle with.JD Long –JD Long 2010-06-15 17:04:53 +00:00 Commented Jun 15, 2010 at 17:04 Great question, and wonderful answers. A very rewarding read - thank you.Tal Galili –Tal Galili 2010-06-15 17:12:44 +00:00 Commented Jun 15, 2010 at 17:12 Credits should go to friend of mine... she asked me about collinearity, and after searching topics on SO, I figured out that there are no questions about it... which was quite odd, since the collinearity problem is mundane in statistical analysis. Thank you lads for these great answers!aL3xa –aL3xa 2010-06-15 21:07:12 +00:00 Commented Jun 15, 2010 at 21:07 Great stuff guys, I really appreciate seeing an R community spring up here at SO.Pierreten –Pierreten 2010-06-16 17:07:54 +00:00 Commented Jun 16, 2010 at 17:07 1 This question appears to be off-topic because it is about statistical practice. It should be migrated to CrossValidated (which didn't exist when the question was originally asked ...)Ben Bolker –Ben Bolker 2013-12-31 23:08:58 +00:00 Commented Dec 31, 2013 at 23:08 \|Show 8 more comments 5 Answers 5 Sorted by: Reset to default This answer is useful 40 Save this answer. Show activity on this post. The kappa() function can help. Here is a simulated example: ```r set.seed(42) x1 <- rnorm(100) x2 <- rnorm(100) x3 <- x1 + 2x2 + rnorm(100)0.0001 # so x3 approx a linear comb. of x1+x2 mm12 <- model.matrix(~ x1 + x2) # normal model, two indep. regressors mm123 <- model.matrix(~ x1 + x2 + x3) # bad model with near collinearity kappa(mm12) # a 'low' kappa is good 1.166029 kappa(mm123) # a 'high' kappa indicates trouble 121530.7 ``` and we go further by making the third regressor more and more collinear: ```r x4 <- x1 + 2x2 + rnorm(100)0.000001 # even more collinear mm124 <- model.matrix(~ x1 + x2 + x4) kappa(mm124) 13955982 x5 <- x1 + 2x2 # now x5 is linear comb of x1,x2 mm125 <- model.matrix(~ x1 + x2 + x5) kappa(mm125) 1.067568e+16 ``` This used approximations, see help(kappa) for details. Share Share a link to this answer Copy linkCC BY-SA 2.5 Improve this answer Follow Follow this answer to receive notifications edited Jun 15, 2010 at 3:06 answered Jun 15, 2010 at 2:58 Dirk is no longer hereDirk is no longer here 370k 60 60 gold badges 668 668 silver badges 742 742 bronze badges Comments Add a comment This answer is useful 35 Save this answer. Show activity on this post. Just to add to what Dirk said about the Condition Number method, a rule of thumb is that values of CN > 30 indicate severe collinearity. Other methods, apart from condition number, include: 1) the determinant of the covariance matrix which ranges from 0 (Perfect Collinearity) to 1 (No Collinearity) ```r using Dirk's example det(cov(mm12[,-1])) 0.8856818 det(cov(mm123[,-1])) 8.916092e-09 ``` 2) Using the fact that the determinant of a diagonal matrix is the product of the eigenvalues => The presence of one or more small eigenvalues indicates collinearity ```r eigen(cov(mm12[,-1]))$values 1.0876357 0.8143184 eigen(cov(mm123[,-1]))$values 5.388022e+00 9.862794e-01 1.677819e-09 ``` 3) The value of the Variance Inflation Factor (VIF). The VIF for predictor i is 1/(1-R_i^2), where R_i^2 is the R^2 from a regression of predictor i against the remaining predictors. Collinearity is present when VIF for at least one independent variable is large. Rule of Thumb: VIF > 10 is of concern. For an implementation in R see here. I would also like to comment that the use of R^2 for determining collinearity should go hand in hand with visual examination of the scatterplots because a single outlier can "cause" collinearity where it doesn't exist, or can HIDE collinearity where it exists. Share Share a link to this answer Copy linkCC BY-SA 2.5 Improve this answer Follow Follow this answer to receive notifications edited Jun 16, 2010 at 9:14 answered Jun 15, 2010 at 8:23 YorgosYorgos 30.6k 20 20 gold badges 113 113 silver badges 152 152 bronze badges 3 Comments Add a comment aL3xa aL3xaOver a year ago Thanks Γιώργος, +2 for this one! Great answer! 2010-06-15T16:50:13.597Z+00:00 1 Reply Copy link kevinykuo kevinykuoOver a year ago why would the determinant of the covariance matrix be capped at 1?? 2014-02-24T20:24:20.51Z+00:00 1 Reply Copy link Yorgos YorgosOver a year ago blog.exploratory.io/… 2019-11-24T06:07:16.63Z+00:00 0 Reply Copy link Add a comment This answer is useful 19 Save this answer. Show activity on this post. You might like Vito Ricci's Reference Card "R Functions For Regression Analysis" It succinctly lists many useful regression related functions in R including diagnostic functions. In particular, it lists the vif function from the car package which can assess multicollinearity. Consideration of multicollinearity often goes hand in hand with issues of assessing variable importance. If this applies to you, perhaps check out the relaimpo package: Share Share a link to this answer Copy linkCC BY-SA 2.5 Improve this answer Follow Follow this answer to receive notifications answered Jun 15, 2010 at 9:08 Jeromy AnglimJeromy Anglim 35.2k 31 31 gold badges 122 122 silver badges 176 176 bronze badges 1 Comment Add a comment aL3xa aL3xaOver a year ago Technically, and arithmetically, VIF = 1(1 - R^2), where R^2 refers to example I stated in my question. I forgot to mention VIF, so thanks for helping on this one! relaimpo is a great find! 2010-06-15T11:41:07.88Z+00:00 1 Reply Copy link This answer is useful 9 Save this answer. Show activity on this post. See also Section 9.4 in this Book: Practical Regression and Anova using R [Faraway 2002]. Collinearity can be detected in several ways: Examination of the correlation matrix of the predictors will reveal large pairwise collinearities. A regression of x_i on all other predictors gives R^2_i. Repeat for all predictors. R^2_i close to one indicates a problem — the offending linear combination may be found. Examine the eigenvalues of t(X) %% X, where X denotes the model matrix; Small eigenvalues indicate a problem. The 2-norm condition number can be shown to be the ratio of the largest to the smallest non-zero singular value of the matrix ($\kappa = \sqrt{\lambda_1/\lambda_p}$; see ?kappa); \kappa >= 30 is considered large. Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications edited Jul 26, 2014 at 1:44 answered Jun 15, 2010 at 7:50 rcsrcs 69.1k 24 24 gold badges 177 177 silver badges 157 157 bronze badges 1 Comment Add a comment Ben Bolker Ben BolkerOver a year ago link-only answers are deprecated on SO 2014-07-25T20:54:38.633Z+00:00 1 Reply Copy link This answer is useful 8 Save this answer. Show activity on this post. Since there is no mention of VIF so far, I will add my answer. Variance Inflation Factor>10 usually indicates serious redundancy between predictor variables. VIF indicates the factor by which variance of the co-efficient of a variable would increase if it was not highly correlated with other variables. vif() is available in package cars and applied to an object of class(lm). It returns the vif of x1, x2 . . . xn in object lm(). It is a good idea to exclude variables with vif >10 or introduce transformations to the variables with vif>10. Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications answered Jul 25, 2014 at 20:50 vagabondvagabond 3,594 7 7 gold badges 46 46 silver badges 76 76 bronze badges Comments Add a comment Your Answer Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. To learn more, see our tips on writing great answers. 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2140	https://www.msdmanuals.com/professional/multimedia/video/overview-of-phobias	Video: Overview of Phobias-MSD Manual Professional Edition honeypot link skip to main content Professional Consumer Professional edition active ENGLISH MSD Manual Professional Version MEDICAL TOPICSRESOURCESCOMMENTARYPROCEDURESQUIZZESABOUT US MEDICAL TOPICSRESOURCESCOMMENTARYPROCEDURESQUIZZES Professional/ Videos/ Overview of Phobias/ Overview of Phobias Overview of Phobias Phobias ( by Osmosis ( is licensed under CC-BY-SA 4.0 ( In these topics Specific Phobias> Social Anxiety Disorder> Agoraphobia> Brought to you by Merck & Co, Inc., Rahway, NJ, USA (known as MSD outside the US and Canada) —dedicated to using leading-edge science to save and improve lives around the world. Learn more about the MSD Manuals and our commitment toGlobal Medical Knowledge. About us Disclaimer Permissions Privacy Cookie Preferences Terms of use Partnerships Contact Us Global Medical Knowledge Veterinary Manual Mobile App Copyright© 2025 Merck & Co., Inc., Rahway, NJ, USA and its affiliates. All rights reserved. Find In Topic This Site Uses Cookies and Your Privacy Choice Is Important to Us We suggest you choose Customize my Settings to make your individualized choices. Accept Cookies means that you are choosing to accept third-party Cookies and that you understand this choice. See our Privacy Policy Customize my Settings Reject Cookies Accept Cookies
2141	https://www.nationalgeographic.com/science/article/chimps-humans-96-percent-the-same-gene-study-finds	Chimps, Humans 96 Percent the Same, Gene Study Finds This breakthrough will aid scientists in their mission to learn what sets us apart from other animals. ByStefan Lovgren August 31, 2005 •4 min read Scientists have sequenced the genome of the chimpanzee and found that humans are 96 percent similar to the great ape species. "Darwin wasn't just provocative in saying that we descend from the apes—he didn't go far enough," said Frans de Waal, a primate scientist at Emory University in Atlanta, Georgia. "We are apes in every way, from our long arms and tailless bodies to our habits and temperament." Because chimpanzees are our closest living relatives, the chimp genome is the most useful key to understanding human biology and evolution, next to the human genome itself. The breakthrough will aid scientists in their mission to learn what sets us apart from other animals. By comparing human and chimpanzee genomes, the researchers have identified several sequences of genetic code that differ between human and chimp. These sequences may hold the most promise for determining what creates human-specific traits such as speech. "If people are asking what makes us human, they're not going to find a smoking gun [in this study]," said Evan Eichler, a genome scientist at the University of Washington in Seattle who was part of the research team. "But they're going to find suggestions for where to look." The project was conducted by an international group of scientists called the Chimp Sequencing and Analysis Consortium. Sixty-seven researchers co-authored the study, which is detailed tomorrow in the journal Nature. Genetic Blueprints To map the chimp genome, researchers used DNA from the blood of a male common chimpanzee (Pan troglodytes) named Clint, who lived at the Yerkes National Primate Research Center in Atlanta. Clint died last year from heart failure at the relatively young age of 24. A comparison of Clint's genetic blueprints with that of the human genome shows that our closest living relatives share 96 percent of our DNA. The number of genetic differences between humans and chimps is ten times smaller than that between mice and rats. Scientists also discovered that some classes of genes are changing unusually quickly in both humans and chimpanzees, as compared with other mammals. These classes include genes involved in the perception of sound, transmission of nerve signals, and the production of sperm. Despite the similarities in human and chimp genomes, the scientists identified some 40 million differences among the three billion DNA molecules, or nucleotides, in each genome. The vast majority of those differences are not biologically significant, but researchers were able to identify a couple thousand differences that are potentially important to the evolution of the human lineage. "The goal is to answer the basic question: What makes us humans?" said Eichler. Eichler and his colleagues found that the human and chimp sequences differ by only 1.2 percent in terms of single-nucleotide changes to the genetic code. But 2.7 percent of the genetic difference between humans and chimps are duplications, in which segments of genetic code are copied many times in the genome. "If genetic code is a book, what we found is that entire pages of the book duplicated in one species but not the other," said Eichler. "This gives us some insight into the genetic diversity that's going on between chimp and human and identifies regions that contain genes that have undergone very rapid genomic changes." Mutations Humans and chimps originate from a common ancestor, and scientists believe they diverged some six million years ago. Given this relatively short time since the split, it's likely that a few important mutations are responsible for the differences between the two species, according to Wen-Hsiung Li, a molecular evolutionist at the University of Chicago in Illinois. "If you look at two species of frogs over 10 million years, you probably won't see a lot of the morphological or behavioral differences that you see between humans and chimps," said Li, who wrote an accompanying commentary on the chimp genome sequencing for Nature. There are several hypotheses that account for the evolution of human traits. Li believes these traits come from changes in the parts of the genome that regulate other gene activity. Scientists agree that many questions remain unanswered but the chimp genome provides important clues to understanding what makes us human. "We're in a very nice intermediate stage of understanding human-chimp differences," said Eichler. 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2142	https://web.hypothes.is/blog/back-to-school-with-annotation-10-ways-to-annotate-with-students/	We value your privacy We use cookies to enhance your browsing experience, serve personalised ads or content, and analyse our traffic. By clicking "Accept All", you consent to our use of cookies. Skip to main content Education Back to School With Annotation: 10 Ways to Annotate With Students By jeremydean \| 25 August, 2015 It’s back-to-school season and I find myself once again encouraging teachers to discuss course readings with their students using collaborative web annotation technologies like Hypothesis. Though relatively new to Hypothesis, I’ve been making this pitch for a few years now, but in conversations with educators of late I’ve come to realize that we often mean different things by the word “annotate.” Annotation connotes something distinct in specific subject areas, at different grade and skill levels, and within certain teaching philosophies. This will be the first semester during which Hypothesis has an active education department and so in the spirit these first days of the school year, I thought it might be worth exploring what we really mean when we say, “annotate.” Annotation is typically perceived as a means to an end. As marginal note-taking it often is the basis for questions asked in class discussion or points made in a final paper. But annotation can also be a kind of end in itself, or at least more than a rest-stop on the way to intellectual discovery. This becomes especially true when annotation is brought into the relatively public and collaborative space of social reading online. Digital marginalia as such requires a redefinition or at least expanded understanding of what is traditionally meant by the act of “annotation.” Billy Collins’ poem “Marginalia” outlines various ways that people have annotated throughout history, including in formal education contexts. But even within pre-digital student marginalia there can be a wide range of types of annotation from defining terms and explaining allusions to analytic commentary to more creative responses to the text at hand. As annotation becomes social and media-rich as it does using Hypothesis and other web annotation technologies, these species of marginalia only further proliferate. For those curious about integrating annotation exercises into an assignment or a course, below I outline ten practical ways that one might annotate with a class. This list is by no means exhaustive–the larger point is that there are a lot of different ways for students and teachers to annotate. I’d love to hear about your experiences with annotation in the classroom in notes and comments here or even in your own blog posts on Hypothesis. My hope is that over the course of the coming semesters, Hypothesis will develop as a community of educators sharing their ideas, assignments, successes, and failures. As always, feel free to contact me at jeremydean@hypothes.is to chat further about collaborative annotation! (For a more technical guide to using Hypothesis, see our tutorials on getting started here.) 1. Teacher Annotations Pre-populate a text with questions for students to reply to in annotations or notes elucidating important points as they read. One of the amazing aspects of social reading is that you can be inside the text with your students while they are reading, facilitating their comprehension, inspiring their analysis, and observing their confusion and insight. It’s about as close as you can get to the intimacy of in-class interaction online. You can guide students through the reading with your annotations, offering context and possible interpretations. This allows you to be the Norton editor of your course readings, but attentive to the particular themes of your course or local contexts of your classroom community. Or you can provoke student responses to the text through annotating with questions to be answered in replies to your annotations. Looking at responses to a question posed in the margin of a text is a great starting point for class discussion in a blended course. In the classroom, students can be prompted to expand on points begun as annotations to jumpstart the conversation. And when there is no physical classroom space, as in online courses, annotation can be a means for the instructor to have a similar guiding presence and to create an engaging and engaged community of readers. The real pedagogical innovation of collaborative annotation, however, is that students are empowered as knowledge producers in their own right, so most of my suggested classroom annotation practices revolve around a variety of student-centered annotation activities in which they are the ones posing the questions and teachers are co-learners in the reading process. There are also other use cases, however, for teacher annotation, such as using web annotation to comment on student writing published online. 2. Annotation as Gloss Have students look up difficult words or unknown allusions in a text and share their research as annotations. Practical across a wide range of skill levels, this exercise can span from simply creating a list of vocabulary words from a text for study to presenting, as a class or individually, a text annotated like a scholarly volume. We’ve seen this kind of exercise completed on great works of literature as well as scientific research papers. Think of the activity as creating a kind of inline Wikipedia on top of your course reading. For difficult texts, sharing the burden of the research necessary for comprehension can help students better understand their reading. And there is something incredibly powerful about students beginning to imagine themselves as scholars, responsible for guiding a real audience through a text, whether their own peers or a broader intellectual community. Students can be encouraged to practice skills like rephrasing research material appropriately and citing sources using different formatting styles. And, of course, glossing can be combined with more insightful annotation as well. Protip: If you plan to annotate across multiple texts with a class, have students use a course tag (like “Eng101DrDeanFall15”) in all of their annotations. Tagging in this way allows both teacher and students to follow the group’s work on a class stream of activity. Here’s an example of what such a class tag stream looks like from one of our most active educators, Greg McVerry. More on the pedagogy of tags in this tutorial. Note: very soon (in a matter of weeks) we will be launching a private group feature that will streamline this workflow–annotations will be publishable to a specific group and that group will have a stream that can be followed. 3. Annotation as Question Have students highlight, tag, and annotate words or passages that are confusing to them in their readings. An annotation need not be, and often is not, an answer. A simple question mark in the margin of a book can flag a word or passage for discussion. And such discussions can be generative of important explication and analysis. Directing students to annotate in this way creates a sort of heat map for the instructor that can be used to zero in on troubling sections and subjects or spark class discussion. Tags categorizing the particular problem could be used as a simple way to prompt clarification (vocab, plts, research methods, etc.). While the teacher can respond to such student annotations, a possible follow up exercise could have students respond to each other’s questions instead. 4. Annotation as Close Reading Have students identify formal textual elements and broader social and historical contexts at work in specific passages. Online annotation powerfully enacts the careful selection of text for in-depth analysis that is the hallmark of much high school and college English and language arts curriculum. Using web annotation, students are required to literally select small pieces of meaningful evidence from a document for specific analysis. Teachers can direct students to identify textual features (word choice, repetition, imagery, metaphor, etc.) or relevant broader contexts (historical, biographical, cultural, etc.) for passages of a text, and then prompt them to develop a short argument based on such evidence. Collaborative close reading can be especially effective in that multiple students can build off each other’s interpretations to demonstrate how deep textual analysis can go. Teachers implementing the Common Core State Standards for reading might pay special attention to this use case for annotation in the classroom. Some teachers will use web annotation as a tool throughout the semester for this purpose. Students thus gain regular practice in close reading and build ideas towards more substantive, summative assignments. Such assignments can also begin as collaborative exercises done by the entire class and culminate with individual or small group annotation projects. 5. Annotation as Rhetorical Analysis Have students mark and explain the use of rhetorical strategies in online articles or essays. Whether analyzed as a class or individually, a clearly argumentative piece should be chosen for this assignment, perhaps from an op-ed page in a newspaper or magazine. Students might be asked first to simply identify rhetorical strategies (like ethos, pathos, and logos) using the tag feature in annotations created with Hypothesis. On a second pass, they can be asked to elaborate on how and why a certain strategy is being used by the author. Identification of rhetorical fallacies could be built into this or a related assignment as well. Note: in order to make such an exercise more streamlined, we plan in the near to future allow users to pre-populate a set of controlled terms with which a group can tag their annotations. Combined with exercises six and nine, annotation as rhetorical analysis could be part of a composition course that also has students map arguments in a controversy using annotation and then begin their own advocacy through annotation of primary sources mapped and analyzed. (This is how the rhetoric department at UT-Austin, where I taught while getting my PhD., structures their freshman composition courses.) A twist on this assignment could ask students to analyze their own persuasive prose in this way–discussion of such self-reflexive annotation on one’s own writing is a whole other category of annotation, probably deserving of a blog post in itself. 6. Annotation as Opinion Have students share their personal opinions on a controversial topic as discussed by an article. A lot of how people are interacting with content online today—commenting on web articles, Tweeting about them with brief notes–is a kind of annotation. At Hypothesis we might think of web annotation as a more rigorous form of such engagement with language and ideas on the Internet. Framing one’s opinions as annotations of specific statements or facts is a reminder that our arguments should be grounded in actual evidence. In any case, allowing students to express their opinions in the margins of the Web, and helping them become responsible and thoughtful in those expressions, is a huge part of what it means to be literate both on the Web and in democratic society more generally. Students could be asked simply to respond to the reading with their thoughts, as in a dialectical reading journal, or employ specific cultural or persuasive strategies in their rhetorical intervention. Again, this advocacy exercise could be a summative assignment within a unit that uses Hypothesis to complete annotation activities like those described in ways five and nine here. 7. Annotation as Multimedia Writing Have students annotate with images and video or integrate images and video into other types of annotations. One of the unique aspects of online annotation (and online writing in general) is the ability to include multimedia elements in the composition process. We’ve found many teachers and students excited to make use of animated GIFs in annotation. The use of images can simply be representative (this is a reference to Lincoln with a photo of the 16th president), but more advanced students can be taught to think about how images themselves make arguments and serve other rhetorical purposes. It is advisable to spend a lesson introducing the idea of digital writing to students with particular attention to the use of images, covering everything from search to use policies and attribution. More traditional teachers may be less accustomed to assessing such multimedia compositions and should spend some time thinking about and explaining to their students a grading rubric. 8. Annotation as Independent Study Have students explore the Internet on their own with some limited direction (find an article from a respectable source on a topic important to you personally), exercising traditional literacy skills (define difficult words, identify persuasive strategies, etc.). Many of the above exercises presume that students are annotating for the most part together on a shared course text. But the nature of web annotation is that we can see the notes of others even if we are not reading the same text. In this way, we can attend to annotations as texts themselves. Like a friend’s Facebook page or Twitter feed, seeing someone else navigate the world can be interesting. And through web annotation students can be taught to navigate the digital world responsibly and thoughtfully. Protip: because each Hypothesis user’s annotations are streamed on their public “My Annotations” page, teachers can monitor and assess student work there rather than on individual texts if so desired. (You can click on a username attached to an annotation or search the Hypothesis stream for a username to locate this page. Here’s mine.) Whether or not one goes so far as to let students roam free on the open Web applying their classroom learning, we have found it to be valuable to have a unit develop from collaborative work to independent or small group work. Students might start off annotating together on a few select texts, getting a sense of what annotation means and how a particular platform like Hypothesis works, but by the end of a term become responsible for glossing or analyzing a single text or set of texts themselves. 9. Annotation as Annotated Bibliography Have students research a topic or theme and tag and annotate relevant texts across the Internet. This is a different kind of annotation than largely discussed above. Here we are annotating on the level of the document rather than on a particular selection of text. (Users can create unanchored annotations for this purpose using the annotation icon on the sidebar without selecting any particular text within a document.) But this annotation exercise practices solid research skills and can be used as preparation for research assignments. In fact, using annotation as a annotated bibliography assignment is an excellent way for teachers to follow and guide student research during the process itself. The result of this assignment will be something useful for a paper such as a summary of the major stakeholders of a particular issue and how they articulate their positions. Of course this kind of annotation as page-level commentary can be combined with more fine-grained attention through annotation to the texts of these tagged documents. In addition to outlining sources needed for a paper, the student can begin to break down those sources for close reading within an essay. 10. Annotation as Creative Act Have students respond creatively to their reading with their own poetry or prose or visual art as annotations. Annotation need not be overtly analytical. Whether in writing or using other media, students can respond creatively to texts under study through annotation as well, inserting themselves within the intertextual web that is the history of literature and culture. One creative writing exercise might be to have students annotate in the voices of a characters from a novel being read. Or to have them re-imagine passages written as newspaper stories. Nathan Blom’s Annotated Lear Project at LaGuardia Arts High School is a great example of students creatively responding to a text through annotation. Students can also use their imaginations to annotate texts with their own drawings, photographs, or videos inline with the relevant sources of textual inspiration. Whether completed individually or collaboratively this exercise can result in some wonderful, illustrated editions of course texts. Share this article
2143	https://www.math.purdue.edu/~eremenko/Springer/levinUMZ.pdf	v/:~ E WL.o (A;,) : I~x E w~ (H) c ~o. By choosing a suitable sequence of random elements {Bnx; n -> i}, we see that Lemma 3 holds. Proof of the Assertions in Example 2. Using the random variable ~ we can construct a sequence of random variables {~n =~ n (~), ~n 9 cl'b', n > _ I} such that i) ~n------1 for ~7>--I , ' n~1; (% n 3) ~EW ~, tJD~IIEL~(.~ By considering the sequence {Bn~X; n -> t} we have x 9 ~. In addition for each n > 1 integrating by parts we obtain (13~x; ~> -- [3 n (~x; ~> -b (~.x; D[3n). On the set (%>n Eg:D~=--7. D(%. Consequently, 1 1 rl..+ ~ l, 2. 3. 4. LITERATURE CITED A. V. Skorokhod, "On a generalization of stochastic integrals," Teor. Veroyatn. Primen., 20, No. 2, 223-237 (1975). Yu. L. Daletskii and S. V. Fomin, Measures and Differential Equations in Infinite Dimen- sional Spaces [in Russian], Nauka, Moscow (1983). T. Sekiguchi and Y. Shiola, "L2-theory of noncausal stochastic integrals," Math. Reports Toyama Univ., 8, 119-195 (1985). E. Pardoux and P. Protter, "A two-sided stochastic integral and its calculus," Prob. Theory and Rel. Fields, 78, 535-581 (1988). PERIODIC POINTS OF POLYNOMIALS A. ~. Eremenko and G. M. Levin UDC 517.53 We recall the most important facts of the Julia-Fatou theory as related to the iteration of polynomials (cf. [1-3]). Let P be a polynomial of degree m e 2, and pn its n-th iteration. A point z is called periodic if Pnz = z for some n 9 ~ The set {pkz}k=in is then called a cycle, and its cardinality is called the order of the cycle. The number % = (Pn)'(z), where z ~ ~, is called a multiplicator of a cycle of order n. A cycle is called repulsive if l~I > i. Let D~ = {z 9 @ : Pnz ~ ~, n ~ ~}. It is easy to see that D~ is a region and that 9 D~. The boundary of this region is called the Julia set J = J(P). An equivalent defi- nition is as follows. Let N(P) be the largest open set in 9 on which the family {pn} is normal. Then J(P) =9 The Julia set is perfect and fully invariant, i.e., p-l(j) = j. Furthermore, j(pn) = j, n 9 ~. The polynomial P has no more than m - i nonrepulsive cycles . On the other hand, the number of repulsive cycles is infinite; their union is a dense subset of J. Let D be a region, and z0 9 8D. A point z 0 is called attainable (from D) if there exists a curve F c D which ends on z0. THEOREM i. Repulsive periodic points of the polynomial P are attainable from D=. Physical and Technical Institute for Low Temperatures, Academy of Sciences of the Ukrainian SSR, Khar'kov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. Ii, pp. 1467-1471, November,. 1989. Original article submitted November 23, 1987. 1258 0041-5995/89/4111-1258512.50 9 1990 Plenum Publishing Corporation In the case where the set J(P) is connected, Theorem 1 was announced by Douady . As far as we know, the proof was not published, Denote by T m a polynomial defined by the functional equation cos mm = Tm(cOs ~). Julia set J(T m) is the set [-i, i]. If Rm(z) = z m, then J(R m) is the unit circle. nomials T m and R m play an extremely important role in iteration theory [i, 3]. THEOREM 2. Suppose the set J(P) is connected. Then the equation m , m = deg P The The poly- (1) for the multiplicator ~ holds for any cycle of order n. Equality is achieved in Eq. (i) if and only if P is conjugate to T m by a linear transformation, and ~ is the multiplicator of the endpoint of the interval J(P) (which is a fixed point). Note that if Theorem i is proved, then Eq. (I) (without the case of equality) follows from Theorem 3 of Pommerenke's paper . THEOREM 3. For any polynomial P of degree m, one of the following is true: i) there exists a cycle of order n with multiplicator ~, such that IX[ > mn; 2) P is conjugate to R m by a linear transformation. Note that it is enough to prove Theorems 1 and 2 for fixed points, i.e., cycles of order i. The proof of Theorems 1 and 2 is cased on the study of the entire function introduced by Poincare. Suppose P(z 0) = z0, P'(z0 = X, IiI> i. From Poincare's theorem [i], the functional equation [ (}~z) ---- P ([ (z)) ( 2 ) has an entire solution f; moreover, this solution is uniquely determined by the conditions [(O)=zo, f'(O)=l- (3) [The simplest proof of these facts (it belongs to Poincare) goes as follows: we first deter- mine the formal power series f(z) = z 0 + z + ciz 2 + ... satisfying Eq. (2), and then show, by a direct analysis of the coefficients, that the series converges in some neighborhood of zero; finally, we extend the function f into r with the help of Eq. (2), taking into account that ]~] > i.] Denote by I the set of points on which the family {f(lnz) : n e ~ } is not normal. It is clear that I = f-1(j). Set D = f-l(D~). It follows from Eq. (2) and the full invariance of J and D~ that LI=I, kD=D. (4) Let G be the Green's function for the region D~ with a pole in ~ which is continued by the zero function on 9 The function G is continuous and subharmonic in C and obeys the functional equation found in : G (p (z)~ = mG (z), z E r (5) [This property follows immediately from the evident relation G(z)=lim m-nln [ pn (z) i" ] The function u(z) = G(f(z)) is continuous and subharmonic in ~. It follows from Eqs. (2) and (5) that u(~z) = mu(z~, zE~. (6) Subharmonic functions satisfying Eq. (6) play an important role in the theory of entire func- tions (see, for example, ). The order of a subharmonic function u in ~ is determined by the formula 9= lim InB(r,u)/ inr, where B(r, u) = max{u(z) : Izl = r}. It follows from (6) that p=lnm/lnlLI 9 (7) 1259 According to the "subharmonic version of the Denjoy-Carleman-Ahlfors theorem" [6, Theorem 4.16], the number of connected components of the set {z : u(z) > 0} does not exceed max{2p, i}. We denote these components by D i ..... Dp. From Eq. (4) there exists an N such that INDi = D i. Choose a point w0 e Di and connect it by a curve 10 c D i with the point %-Nw 0 e D i. Then F =U)~-N'zF0 ~ D i is a curve which goes to zero. Its image f(F) c D~ is a curve which t2~O goes to z 0 by virtue of Eq. (3). This proves Theorem i. To prove Theorem 2, we note that if the Julia set is connected, then the set I = f-i(j) contains the continuum K connecting 0 and =. Indeed, if this is not true, then there exists a closed Jordan curve y which separates 0 and ~; moreover, X D I = r Let V be a neighborhood o~ zero on which f is bijective [this exists by virtue of Eq. (3)]. Choose V small enough so that J is not contained in f(V). Let M be a number large enough so that l-M~ c V. Taking Eq. (4) into account, we find h_My q I = r Thus f(l-My) f l J = r and the curve f(h-MT) sepa- rates z 0 and ~. This contradicts the fact that the set J is connected. A classical theorem of Wiman (see, for example, ) states that for a harmonic function v of order p < i, the inequality lim A (r, v)/B (r, v) .~ cos ap, holds, where A(r, v) = inf {v(z) : ]z[ = r}. Since u(z) = 0, z e K, we have that A(r, u) ~ 0, and hence O e 0.5. The inequality (i) (with n = i) now follows from Eq. (7). Suppose now that equality holds in Eq. (i). Then p = 0.5. We show that the subharmonic function u e 0 of order 0.5 satisfying conditions (6) and A(r, v) ~ 0 necessarily takes the form I (0--00), 10]~<~, (8) u (re ~~ = cr i %os -~ where c > 0 and @0 ~ [-~, ~] are some constants. This result may be derived from [7, 8]; nonetheless, we present an independent simple proof. The function u can be represented as u(z)=,(ln 1-- dF;, c where ~ is some Borel measure. Let n(t) = ~{~:]~} ~ t}. Then O = lira !n n (t)/ln t. Let u (z) =- ,, 1 -- -T dn (t). The subharmonic function u is of order p. We show that the measure ~ is concentrated on the ray s = {~ : arg~ = 80}. Suppose this is not so. Taking into account the fact that for fixed r > 0 the quantity in ]i - reie I has a strict minimum for % = 0, we obtain u(r)=A(r,l~)0. Then it follows from Eq. (6) that u([i[z) = mu(z), and hence lim A(r,u)/B(r,u) 0 holds in C \s Moreover, u = 0 on ~, since A(r, u) ~ 0. Thus, u has the form of Eq. (8). From this it follows that I is a ray. Since I = f-i(j), there exists a circle V such that J N V is an analytic curve. Then it follows from Fatou's theorem [1, p. 225] that the 1260 polynomial P is conjugate to either T m or R m. The latter case is eliminated by direct check- ing. Theorem 2 is proved. Remark. Let G be a region, and let z 0 e 8G be an attainable boundary point. Two curves FI, F 2 c G ending on the point z 0 are called equivalent if there exists a sequence of curves ~n c G, 7n ~ z0 connecting F i and F~. From the results of Douady [2, Sec. 6, Lemma i], it follows that if the Julia set is connected, then there exists a finite number of classes of equivalent curves in D~ which end on the periodic point z0. It is possible to show that the number p of these classes is equal to the number of connected components of the set D = {z : u(z) > 0}. Applying the Denjoy-Carleman-Ahlfors theorem, we find p ~ 2p. From Eq. (7) we find that IXl ~ m21P for a fixed point, or IXl ~ m2n/p for a cycle of order n. More delicate arguments show that the equality in these estimates is possible in only two cases: I) p = I; P is conjugate to T m and z0 is the endpaint of the interval J(P); 2) p = 2; P is conjugate to T m and z 0 is an interior point of the interval J(P). We now go to the proof of Theorem 3. Without loss of generality, we can assume that the leading coefficient of the polynomial P is equal to i. This is always possible to achieve by conjugating with a linear function which does not change the multiplicators. We will need the following lemmas. LEMMA i. Let c=c(A)= V p, ,~ (z). P(zI~A Then c does not depend on A. Moreover, ~q (P~)'(z)=c ~, nell, Pn(z)=A (P~)' (z)=mn(m '~- l)--[-c ~, n6~, ,.,.,.d Pn(z}=z m = deg P. (9) (io) Proof. From the residue theorem 9 S { P--A Iz[=r (~,)2 P--B }dz, where r is sufficiently large. The expression in the integral is O(z-2), z ~ ~; hence c(A) = c(B). We prove Eq. (9) by induction: V (P~+')'(z)= V (P~)'(o) V P'(z)=c ~ (pk),(o). pk+I (z)=A Pk(o})mA P( z)~o~ pk ( o~}=A In view of Eq. (9), it is enough to prove Eq. (i0) for n = i. Then Eq. (i0) follows from the fact that the residue of the function P' (z) (P (z) -- z)' (p,)2 (z) P (z) -- z P (z) in the point ~ is equal to m(m- 1). The lemma is proved. LEMMA 2. Suppose all the fixed points of the polynomial P, wit:h the possible exception of one, have multiplicators equal to m = deg P. Then P is conjugate to R m. Proof. By conjugating with the function z + a, we make the exceptional point be equal to 0. From the hypotheses of the len~na, it follows that P(z) - z = z/m(P' (z) - m). Solving this differential equation with the boundary condition P(0) = 0, we find that P = R m. LEMMA 3. Let c, I e ~. If Re(X n) > Re(c n) for all n e ~, then i > 0. Proof. The cases cl = 0, argc = -+z/2, argk = -+~/2 are easily eliminated. There exists a 6 > 0 such that infinitely many points c n lie within the angle {z : largzl <- v/2 -- 6}. 1261 This implies that IX[ ~ [c[. If arg ~ ~ 0, then infinitely many points A n lie within some angle of the form {z : Iargz - zl ~ ~/2 - 6}. Thus 141 = Icl. Setting 81 = argk, 82 = arg c, we obtain cosnOl>cosnO~, n C~. (ii) In particular, cos 2n0~ > cos 2nO 2, which implies cos2nO1>cos2nOv nE~. (12) It follows from Eqs. (11) and (12) that cos nO 1 > O, n e ~. Thus 0 z = O, which is shat was needed. We now finish the proof of Theorem 3. Suppose that the moduli of the multiplicators of all cycles do not exceed m n, where n is the order of the cycle. Let ~ be the multiplicator of any fixed point. Then, by assumption, Re E (P")'(z)~ Re(cn), n e ~. From Lemma 3, ~ > 0. This holds for all fixed points. If all their multiplicators are equal to m, we apply Lenm~a 2 again. Suppose that the equa- tion h i < m - e, i =1,2, e > 0 holds for two multiplicators. Choose a sequence nk such that Re (c nk) ~ 0, k e~ is satisfied. By virtue of Eq. (i0), we find that either m~k(mnk- 1 <. E Re(Pnk)'(z)~mnk(m"k--2) +2(m-~)"~~ 2(m -- e) nk, which is impossible. Theorem Pnk(z)=z 3 is proved. Remark. Suppose P(z) is a polynomial of degree m ~ 2 whose Julia set is connected. Denote by K(P) the lower bound of those x > 0, for which the inequality Ix I ~ m nx is satis- fied for multiplicators % of all cycles of order n e ~. We have proved that 1 ~ K(P) ~ 2. By the method of extremal lengths it is possible to prove the strict inequality K(P) < 2 for the case in which the mapping P : J(P) ~-J(P) is hyperbolic . i. 2. 3. 4. 5. 6. 7. 8. 9. LITERATURE CITED P. Fatou, "Memoire sur les equations fonctionnelles," Bull. Soc~ Math. Fr., 47, 161- 271; 48, 33-94, 208-314 (1919). A. Douady, "Systemes dynamiques holomorphes," Seminaire Bourbaki, 35e annee, 599, 1-25 (1982/83). M. Yu. Lyubich, "Dynamics of rational transformations: the topological picture," Usp. Mat. Nauk, 417 No. 4, 35-95 (1986). C. Pommerenke, "On conformal mappings and iteration of rational functions," Complex Variables, ~, Nos. 2-4, 117-126 (1986). B. Kjellberg, "On certain integral and harmonic functions," Dissertation, Uppsala (1948). W. K. Hayman and P. B. Kennedy, Subharmonic Functions, Academic Press, New York (1976). A. Edrei, "Extremal problems of the cosT0 type," J. D'anal. Math., 29, 19-66 (1976). D. Drasin and D. Shea, "Convolution inequalities, regular variation, and exceptional sets," J. D'anal. Math., 29, 232-292 (1976). V. S. Azarin, "Concerning the asymptotic behavior of subharmonic functions of finite order," Mat. Sb., 108, No. 2, 147-167 (1979). 1262
2144	https://researchsafety.gwu.edu/pathogen-data-sheets/streptococcus-pneumoniae	Office of Research Safety Office of the Vice Provost for Research Main Bootstrap Navigation Our Office Meet Our Team Policies & Regulations Emergency Procedures Biological Spill Emergency Contacts Incident Reporting Biohazard Spill Kit Biological Exposure Research Safety Programs Biosafety Biosafety Manual Institutional Biosafety Committee Charter Recombinant DNA FAQs Submitting Protocols for Review IBC Meeting Dates Biosafety Training Biosafety Cabinets Biological Safety Cabinet Placement Considerations Biosafety Cabinet Maintenance Biosafety Cabinet Usage Guidelines Biosafety Cabinets: Repair and Movement Setting up your Biosafety Cabinet BSL2 Manual Template Biosafety Level 2 with Enhanced Practices (BSL-2+) guidelines Lentiviral Vector Safety Manual Template Nanotechnology Safety Nanomaterials Registration Form CDC/NIOSH General Safe Practices for Working with Engineered Nanomaterials in Research Laboratories Shipping Biologicals CDC Import Permit USDA Import Permit Biohazard Labeling Guidelines Biosafety FAQS Radiation Safety Radiation Safety Committee Radiation Safety Training Manual Uranyl Acetate & Uranyl Nitrate Laser Safety Standards and Regulations Laser Safety Training Manual IACUC IACUC Forms and Policies IACUC Deadlines and Meetings Protocol Submission and Review Process Training and Health Requirements Post-Approval Monitoring GW iRIS Help Training Mandatory Training Specialized Training Training Videos Safety Guidance I Need To... Field Research Policies General Safety Guidelines Lab Glove Usage Glove Selection Glove Removal Preventing Aerosol Production Sharps Safety Sterilization, Disinfection, and Decontamination PPE: Donning & Doffing Lab-Specific Safety Guidance Animal Biosafety Autoclave Safety Biological Safety Cabinet Safety Considerations Centrifuge safety and spill clean-up Caring for your centrifuge Cold & Warm Room Safety Cold Room Guidelines General Precautions for Handling Compressed Gases Liquid Nitrogen Storage of Cryovials Safety Recombinant DNA Safety Transporting Infectious Substances by Ground Lab Signage Eyewash Checklist Biohazard Warning Sign Hazard Communication Sign Pathogen Data Sheets COVID-19 Planning SARS-CoV-2 Guidelines & Checklists SARS-CoV-2 Infographics Institutional Biosafety Committee (IBC) Interim Lab Biosafety Guidance for Research with SARS-CoV-2 (COVID-19) Updated COVID-19 Laboratory Resources COVID-19 Laboratory Resources Infographic from ABSA International on the Differences between Face Coverings and Masks Contact Us Related Departments Streptococcus pneumoniae \| Characteristics \| \| Morphology \| S. pneumoniae is of the Streptococcaceae family. The gram-positive, oval/lancet-shaped cocci are often arranged in pairs, known as a diplococcus, or can be present in short chains. There are around 90 serotypes and its surface capsule, which is the distinguishing trait of the pneumococcus and is the major virulent factor, holds a complex mosaic of monosaccharides, oligosaccharides, and other polymer components that are of high-molecular weight \| \| GrowthConditions \| Cultures can be placed in various growth media and incubated at 37°C. When grown on blood agar, it forms round facultative anaerobic colonies surrounded by α-hemolysis. \| \| Health Hazards \| \| Host Range \| Humans, mice, rats, guinea pigs, chimpanzees, rhesus monkeys, and other mammals associated with humans \| \| Modes of Transmission \| Infectious cells can be disseminated via microaerosol droplets created by coughing or sneezing, or person-person oral contact. Transmission is common, but infection is infrequent as healthy individuals carry S. pneumoniae in the nasopharyngeal region without any presence of infection. Pneumococcus bacteria can cause infections in many parts of the body including: lungs, ears, sinuses, brain, and spinal tissue \| \| Signs and Symptoms \| Symptoms of pneumococcal infection depend on the part of the body affected. Symptoms can include fever, cough, shortness of breath, chest pain, stiff neck, confusion, increased sensitivity to light, joint pain, chills, ear pain, sleeplessness, and irritability. In severe cases, pneumococcal disease can cause hearing loss, brain damage, sepsis, and death. \| \| Infectious Dose \| Unknown \| \| Incubation Period \| About 1 – 3 days upon breaching primary immune barriers \| \| Medical Precautions/Treatment \| \| Prophylaxis \| Antibiotic prophylaxis is available \| \| Vaccines \| Vaccines are available, Pneumococcal conjugate vaccine (PCV13) and pneumococcal polysaccharide vaccine (PPV23) \| \| Treatment \| Administer appropriate drug treatment. Inflammation caused by pneumococcal lysis makes the treatment of pneumococcal diseases less effective with antibiotics alone, and even a highly effective bactericidal such as β-lactam may actually enhance the harmful effects of the disease in some cases \| \| Surveillance \| Monitor for symptoms. Gram-negative stains or bacteriology studies of direct smears can be used to detect symptomatic or asymptomatic infections. ELISA, PCR fingerprinting analysis, and radiography techniques are also useful for diagnosis. \| \| GWU Requirements \| Report all incidents to the Office of Risk Management as well as the Office of Research Safety (ORS) IMMEDIATELY FOLLOWING THE OCCURRENCE. \| \| Containment \| \| BSL-2/ABSL-2 \| Risk Group 2 classification is applied VC. BSL-2 practices, containment equipment, and facilities are recommended for activities with cultures or potentially infectious clinical materials. ABSL-2 practices, containment equipment, and facilities are recommended for activities with naturally or experimentally infected animals. \| \| Spill Procedures \| \| Small Spills (<1 liter) \| If the spill occurred inside a biological safety cabinet, close the sash and allow the cabinet to operate for 15 minutes before continuing with the spill cleanup. S. pneumoniae is not commonly transmitted via aerosols, but caution should be taken during a spill outside of a BSC. Leave the room immediately and allow the aerosols to dissipate for 15 minutes. Notify others working in the lab. Don appropriate PPE. Cover area of the spill with paper towels or any absorbent material and apply an EPA registered disinfectant effective against the bacteria (bleach, ethanol, glutaraldehyde), working from the perimeter towards the center. Allow 30 minutes of contact time before disposal and cleanup of spill materials. \| \| Large Spills \| Alert lab personnel in the laboratory to the spill and keep people out of the area to prevent spread of the contamination. Check if you have been contaminated or if any of your PPE has been breached. If so follow exposure procedures. Remove any contaminated clothing and place it the biohazard waste. Wash your hands and post a sign on the door. Notify your supervisor of the incident and call ORS (4-8258) for assistance. If the situation involves an imminently life-threatening injury or has catastrophic potential, call 911. \| \| Exposure Procedures \| \| Mucus Membrane \| Flush eyes, mouth or nose for 15 minutes at eyewash station. \| \| Other Exposure \| For an area not protected by skin, wash with soap and water for 15 minutes (open wounds, sores, etc.) \| \| Reporting \| Report ALL injuries to the PI immediately and reported to the Office of Risk Management at riskgwu [dot] edu (risk[at]gwu[dot]edu) IMMEDIATELY FOLLOWING THE OCCURRENCE. Exposures that involve a bloodborne-pathogen or recombinant DNA also need to be reported to the Office of Research Safety at labsafetygwu [dot] edu. If the injury requires immediate medical attention, call GWPD at 202-994-6111 or call 911. \| \| Medical Monitoring \| Seek immediate medical evaluation, treatment, and post exposure follow-up at the Employee Health Office at GWU Hospital (900 23rd St., NW, Suite G-1090, Phone: 202-715-4275). Students should go to the Students Health Office at Marvin Center. After hours treatment can be received at the GWU hospital emergency room. \| \| Stability \| \| Disinfection \| Susceptible to 10% bleach, 0.5% glutaraldehyde, 70% ethanol \| \| Inactivation \| Cells can be inactivated by heat suspension in a water bath at 56˚C for 30 minutes \| \| Survival Outside Host \| Can survive in dental plaque and sputum for up to 7 days, in dust for up to 20 days, on glass for 1 – 11 days, up to 180 days in frozen fish, and on mouse carcasses for up to 180 to 270 days. \| \| Personal Protective Equipment (PPE) \| \| Minimum PPE Requirements \| At minimum, personnel are required to don gloves, closed toed shoes, lab coat, and appropriate face and eye protection prior to working with S. pneumoniae. Additional PPE may be required depending on lab specific SOPs. \| \| Additional Precautions \| Additional protection may be worn over laboratory clothing when infectious materials are directly handled, such as solid-front gowns with tight fitting wrists, gloves, and respiratory protection. Wash hands with soap and water after removing gloves. \| \| References \| \| CDC ( BMBL 6th Edition ( Canadian PHAC ( \| Print this page
2145	https://www.vcalc.com/wiki/cylinder-side-area	The Lateral Surface Area of a Cylinder calculator computes the lateral (sides only) surface area of a cylinder. INSTRUCTIONS: Choose units and enter the following: Surface Area (A): The calculator returns the surface area in square meters. However, this can be automatically converted to other area units via the pull-down menu. The Math The formula for the lateral surface area of a cylinder is: A = 2π•r•h. where: Many manufactured objects are in a cylindrical form. The surface area of a cylinder formula can be used to compute: --- Cylinder Calculators 1. Cylinder Side Area: Computes the surface area of the sides of a cylinder based on height and radius. 2. Cylinder Surface Area: Computes the total surface area of a cylinder including sides, top and bottom. 3. Cylinder Volume: Computes the volume of a cylinder based on cylinder height and radius. 4. Cylinder Volume from Height and Circumference:Computes the volume of a cylinder based on the height and circumference. It also returns the diameter of the cylinder. 5. Cylinder Height: Computes the height of a cylinder based on the volume and radius. 6. Cylinder Radius: Computes the radius and diameter of a cylinder based on the volume and height. 7. Cylinder Weight: Computes the weight, mass and volume of a cylinder based on the height, radius and density. 8. Cylinder Mass Computes the mass of a cylinder based on the radius and height and mean density of the cylinder. 9. Volume in Horizontal Cylinder: Computes partial volume of a horizontal circular cylinder based on the radius and length of the cylinder and the vertical height of the contents in the cylinder (see diagram). 10. Cylinder Density: Computes the mean density of a cylinder based on the mass and computed volume from the radius and height. 11. Slanted Cylinder Surface Area: Computes the surface area of a slanted cylinder based on the radius and side length (L). 12. Slanted Cylinder Area from Angle: Computes the lateral area (surface area of the sides) of a slanted cylinder based on its radius, height and slant angle (θ). 13. Slanted Cylinder Volume: Computes the volume of a slanted cylinder based on the radius of the base, side length and slant angle. 14. Slanted Cylinder Weight:Computes the weight or mass of a slanted cylinder based on the radius, side length, slant angle and density. 15. Cylinder MoI about Central Axis:Computes the moment of inertia of a cylinder about its central axis based on the mass and radius. 16. Cylinder MoI about the End:Computes the moment of inertia of a solid circular cylinder of uniform density about an axis along the diameter of the cylinder at one end. 17. Cylinder MoI Perpendicular to Axis: Computes the moment of inertia of a solid circular cylinder of uniform density about an axis perpendicular to the axis of the cylinder and through the center of mass. Cylinder Calculators Enhance your vCalc experience with a free account Sorry, JavaScript must be enabled. Change your browser options, then try again.
2146	https://puzzling.stackexchange.com/questions/116708/cover-an-n-times-n-grid-with-non-diagonal-non-intersecting-n-1-shortest-paths	mathematics - Cover an n times n grid with non-diagonal non-intersecting n-1 shortest paths - Puzzling Stack Exchange Join Puzzling By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Cover an n times n grid with non-diagonal non-intersecting n-1 shortest paths Ask Question Asked 3 years, 3 months ago Modified3 years, 3 months ago Viewed 495 times This question shows research effort; it is useful and clear 6 Save this question. Show activity on this post. This puzzle was given to me by PhD student colleagues. Suppose that you have a n×n grid. Is it possible, for a given n to cover all its n 2 nodes with n−1 non-diagonal and non-intersecting shortest paths? If it is not possible, can you prove it for all n? With n non-diagonal shortest path, it is possible for every n. Here is an example with n=4. The following example aims to show you what is a diagonal shortest path which are not allowed: Two problems with this latter example: There is a diagonal shortest path Three nodes are not covered A "shortest path" is a shortest path between any two nodes of the grid. It is not necessarily of length 2 if you chose two far away nodes. The aim is try having n−1 of those paths covering the n 2 nodes. The length of each paths does not matter as long as it is shortest. The non-intersecting property means that no two paths shall share an edge nor a node. mathematics graph-theory Share Share a link to this question Copy linkCC BY-SA 4.0 Improve this question Follow Follow this question to receive notifications edited Jun 22, 2022 at 3:29 bobble 11.8k 4 4 gold badges 38 38 silver badges 96 96 bronze badges asked Jun 20, 2022 at 11:37 JKHAJKHA 7,948 16 16 silver badges 50 50 bronze badges 5 Risking to sound like an idiot, what is a shortest path, and do paths need to touch all nodes (and why are there three paths in the 3x3 grid when there should only be two)Auribouros –Auribouros 2022-06-20 11:47:18 +00:00 Commented Jun 20, 2022 at 11:47 @Auribouros, there are three shortest path because I was showing that for n non diagonal shortest path, it is possible. The difficulties lie when you ask for n−1. Yes, path must touch all nodes, they must cover the nodes.JKHA –JKHA 2022-06-20 12:00:06 +00:00 Commented Jun 20, 2022 at 12:00 Going back to my initial "What is a shortest path" question, on the first grid, you mention there are 3 shortest paths, which, by the definition of "shortest", doesn't make sense to me. I see a path of length 2, one of length 4, and one of length 6. The shortest path should be the one of length 2 solely, right? And if those are just the "walls" for the paths, it still doesn't fit, there is a length 6 path in the first grid amongst the length 5 ones.Auribouros –Auribouros 2022-06-20 12:08:21 +00:00 Commented Jun 20, 2022 at 12:08 Me again, do the "edge nodes" count as needing to be passed, or just the center nodes?Auribouros –Auribouros 2022-06-20 12:24:47 +00:00 Commented Jun 20, 2022 at 12:24 @Auribouros, all the nodes, all the n 2 nodes JKHA –JKHA 2022-06-20 12:26:16 +00:00 Commented Jun 20, 2022 at 12:26 Add a comment\| 4 Answers 4 Sorted by: Reset to default This answer is useful 6 Save this answer. Show activity on this post. Terminology: There are two general "orientations" a path can have: bottom left to top right; let's call that "slashy" top left to bottom right; let's call that "backslashy" These are not mutually exclusive: We'll use the convention that purely horizontal or purely vertical paths belong to both classes. We'll now prove the claim that given n>0 a covering of an n x n grid by n-1 shortest on-grid paths does not exist. Otherwise there must be a smallest counter example X. Let n be its size. We'll show that we can construct from this a counter example Y of size n-1 contradicting minimality of X. Lemma: If there is a path p in X that contains two distinct points touching two distinct boundaries then Y as above exists. Proof: Wlog p is slashy. Extend p to the bottom left and top right corners (erasing the bits of other paths that are in the way). As this extension hugs the boundaries it will not break up any other path. The resulting configuration X' therefore still has at most n-1 paths. To obtain Y, simply remove (the extended) p and push the resulting two leftovers together. To conclude the proof of the main claim we will show that a similar extension can always be made even if p only touches one boundary. (Such a p always exists.) Still wlog p is slashy and starts at the left or bottom boundary. Let y,x be the coordinates where it ends. We now show that we can extend p (and keep doing so until we reach the top or right boundary). If either y+1,x or y,x+1, i.e. the points just above or to the right is part of a slashy path q in X we can extend p along q in the top right direction. If we keep going until the end of q this will not change the total number of paths (at least not upwards, it could in theory reduce it as @Jaap Scherphuis points out). Otherwise both y+1,x or y,x+1 are in backslashy paths. If either is an endpoint we can add it to p. This only leaves the case that y+1,x and y,x+1 are part of the same backslashy path q and connected at y+1,x+1. But then we can simply extend p to y+1,x+1 (which way does not matter). This will temporarily create an additional path because q is cut in two. But q will be rejoined at the final "push-the-two-leftovers-together" stage. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications edited Jun 23, 2022 at 1:42 answered Jun 22, 2022 at 20:07 loopy waltloopy walt 21.6k 1 1 gold badge 36 36 silver badges 97 97 bronze badges 4 Very nice! I had been trying something similar, but got fixed on the idea that p had to fully lie on the boundary to reduce to the n-1 case. Then the any solution with minimal n must have path endpoints at all four corners (any corner that is not an endpoint must lie on a path that goes along the boundary). And that is where I got stuck.Jaap Scherphuis –Jaap Scherphuis 2022-06-22 20:46:48 +00:00 Commented Jun 22, 2022 at 20:46 1 One minor correction/clarification: "this will not change the total number of paths". Actually, it may theoretically reduce the number of paths: If it first hits q at an end point, then extending p fully absorbs q, decrementing the number of paths. This would seem unlikely to occur in a minimal solution, though I can't immediately rule it out. Of course, this does not matter at all to the argument of the proof.Jaap Scherphuis –Jaap Scherphuis 2022-06-22 20:58:39 +00:00 Commented Jun 22, 2022 at 20:58 @JaapScherphuis very astute, Fixed. One can probably rule it out in a minimal solution, But at a later iteration when we already have some temporally cut in two q's lying around I don't see how to rule it out.loopy walt –loopy walt 2022-06-22 21:13:47 +00:00 Commented Jun 22, 2022 at 21:13 Hi, nice solution, this is related to a research question. Would you like to contact me? (contact details on my profile)Florent Foucaud –Florent Foucaud 2022-09-03 08:21:01 +00:00 Commented Sep 3, 2022 at 8:21 Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. Impossible. For any set of nodes that make up a shortest path from the bottom-left corner to the top-right corner (or similarly from the bottom-right to the top-left), if we remove these nodes, the remaining nodes can be slid together together to form a n−1 by n−1 grid. It may help to think of shifting the nodes below the removed path up and left by one unit. In this process, any path that does not involve removed nodes is preserved. Any path that only involves removed nodes is removed. A path the crosses from one side to the other might not remain a shortest path (and may not even remain connected). To make this precise, the transformed path after removal of a set of nodes will contain all the non-removed nodes that were part of the original path, and after transforming the grid, any newly adjacent pair of nodes that are both part of the path will have a new edge added. One case where a crossing path remains a shortest path is when exactly two nodes are removed: a corner (meaning one connection at that node is vertical and one is horizontal) and one of its neighbors can be safely removed if the slope of the path is negative (meaning it's left endpoint is higher than its right endpoint). This is because shifting the cells up and left by one will put one of the corner's neighbors on top of the other one. It is also possible for a path to intersect the removed nodes in the middle, but not cross from one side to the other. This case won't be of any concern. Given a covering of the grid with shortest paths, we will remove a set of nodes that make up a shortest path from one corner to the opposite corner. This path might not be one of the paths in our covering, but we will choose it so that it completely contains at least one path from the covering and also so that all of paths of the covering that aren't completely removed remain shortest paths in the smaller grid. Start with the node in the lower-left corner. If this is a corner of a path in the covering, then we will just remove the bottom row and left column. This will remove the path that contains this corner node entirely and clearly won't cause problems for other paths. If it is not a corner, it is an endpoint and we will follow the path until its end. Then (if we are not in the upper-right corner yet), we must choose to step either up or right. If we can step onto a horizontal, vertical, or positively sloped path from the covering, we will do so and follow it to its end. Thus, this path will just be truncated (or maybe completely removed) by removal of the chosen path. Otherwise, if we can step onto the endpoint of a path from the covering, we will do so and follow it as long as we can without stepping down or left. Thus, this path will also just be truncated; and since it is negatively sloped, after leaving it, we cannot meet if again by traveling right and up. The last case is the node above and the node to the right are both part of the same negatively sloped path from the covering. In this case, we can step up by one and right by one (in either order) which will remove a corner and one adjacent node from a negatively sloped path which, as noted above, will preserve that path. Since this path is negatively sloped, we won't meet it again. This process is repeated until reaching the top-right corner. Then the chosen path is removed. This reduces the grid to n−1 by n−1 and completely removes at least one path (the path that covered the original bottom-left node). The remaining grid is still covered by shortest paths. Thus, if it was possible to cover an n by n grid with n−1 paths, we could repeat this removal process n−1 times to reduce it to a 1 by 1 grid covered by at most 0 paths. This is clearly impossible, so it is impossible to cover an n by n grid with n−1 paths. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications answered Jun 22, 2022 at 20:06 tehtmitehtmi 4,272 20 20 silver badges 27 27 bronze badges 4 Did you just beat me by 30 seconds?loopy walt –loopy walt 2022-06-22 20:08:09 +00:00 Commented Jun 22, 2022 at 20:08 And our proofs are more or less identical!loopy walt –loopy walt 2022-06-22 20:14:07 +00:00 Commented Jun 22, 2022 at 20:14 @loopywalt: Indeed. Great minds think alike, I guess. :)tehtmi –tehtmi 2022-06-22 20:14:40 +00:00 Commented Jun 22, 2022 at 20:14 Hi, nice solution, this is related to a research question. Would you like to contact me? (contact details on my profile)Florent Foucaud –Florent Foucaud 2022-09-03 08:21:08 +00:00 Commented Sep 3, 2022 at 8:21 Add a comment\| This answer is useful -1 Save this answer. Show activity on this post. The minimum amount of paths you need to fill an n∗n grid is: n Proof: Greedy version: An n∗n grid contains a total of n 2 nodes, in the example shown above, 16. The maximum amount of nodes a path can pass depends on: The number of paths put down previously The shape of the paths (An L shape or a staircase occupy the most space, a U would break the rules and make it a path longer than it needs to be) The size of the grid The maximum number of nodes passed for any path p x in an n∗n size grid is: N x=2(n−x−1)−1 starting with N 1 A path can clear a full row, along with a full column, the however they will have a node in common. So that's 2 n−1 nodes cleared. The path after that will see its field reduced to an (n−1)∗(n−1) square, and will continue on. In the example given above, that's 7, then 5, then 3, then 1. Which shows we can only use a minimum of n paths to clear all nodes. Analysis of the square and paths: Each path can only have two directions, up or down, and right or left. After choosing those directions, the path can at most pass n nodes in both axis (n nodes horizontally, and n nodes vertically), but by removing a full row and a full column of nodes, there will inevitably be an intersection point, robbing us of one node pass. We need to clear n 2 nodes in (n−1) steps, taking an example with n=3 that's 9 nodes in 2 steps. The only way to clear those 9 nodes would require us to clear 5 nodes (as we figured in the greedy proof, we cannot clear anymore than 5 nodes in one go) and 4 nodes with the next path. To do so, we need to have as much space possible after our first clear, which is sadly also equal to our greedy algorithm. And despite our best efforts, we will only be able to clear 3 nodes on the second go, making it impossible to reach the 9th node in only 2 steps. Now for any n, the issue is going to be similar, for n=4, 16 can be deconstructed in chunks: 16=7+5+4 (greedy),6+6+4,5+5+6 Every deconstruction of n 2 has the square of a smaller n inside it (5 + 5 + 6 doesn't, but we'll see that later), and those smaller squares do the same, they can be deconstructed until we reach the fatal square of side 1, the single node that needs to be cleared without any paths. For those deconstructions, it's impossible to clear all nodes of a square of side n with n−1 paths. For the other deconstructions, the problem comes in the form of a crossroads, we can use our smaller length paths to fill in the blanks, but as the area inevitably shrinks, so does the maximum length of a path when a square is present on the shape, and as soon as a square of size n is formed in one of the resulting shape, we fall into the same issue as we did before, deconstructing the resulting shape, sooner or later, gives us a smaller square, and with it comes the impossible resulting 1 side square. And if n paths are needed, we can do simply make stripes, and that fulfills every requirement. That is, unless circular paths are allowed.. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications edited Jun 20, 2022 at 14:23 answered Jun 20, 2022 at 12:55 AuribourosAuribouros 6,661 13 13 silver badges 50 50 bronze badges 3 I don't think that is a complete proof. It basically shows that the greedy algorithm for choosing the paths does not work. But maybe some other strategy exists that works?Jaap Scherphuis –Jaap Scherphuis 2022-06-20 13:20:10 +00:00 Commented Jun 20, 2022 at 13:20 You right, more in-depth proof incoming Auribouros –Auribouros 2022-06-20 13:41:08 +00:00 Commented Jun 20, 2022 at 13:41 2 Looking forward to it. I'm stuck so far.Jaap Scherphuis –Jaap Scherphuis 2022-06-20 13:41:58 +00:00 Commented Jun 20, 2022 at 13:41 Add a comment\| This answer is useful -1 Save this answer. Show activity on this post. If we assume that the minimal length of a path is independent of the rest of the grid, then Auriboros' result is optimal. However, if the minimal length of a path is defined with regards to the other endpoints, then achieving n-1 is easy: It is straightforward to see that each path cannot get any shorter even if you allow paths to cross. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications answered Jun 20, 2022 at 14:28 AxiomaticSystemAxiomaticSystem 18.2k 1 1 gold badge 36 36 silver badges 72 72 bronze badges 1 A U-shaped path is not a "shortest path". (Oh, okay... it's an answer to a modified problem.)Rebecca J. Stones –Rebecca J. Stones 2022-06-23 04:35:55 +00:00 Commented Jun 23, 2022 at 4:35 Add a comment\| Your Answer Thanks for contributing an answer to Puzzling Stack Exchange! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. 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2147	https://allen.in/dn/qna/118619596	If P be any point in the plane of square ABCD, prove that PA^(2)+PC^(2)=PB^(2)+PD^(2) Courses NEET Class 11th Class 12th Class 12th Plus JEE Class 11th Class 12th Class 12th Plus Class 6-10 Class 6th Class 7th Class 8th Class 9th Class 10th View All Options Online Courses Distance Learning Hindi Medium Courses International Olympiad Test Series NEET Class 11th Class 12th Class 12th Plus JEE (Main+Advanced) Class 11th Class 12th Class 12th Plus JEE Main Class 11th Class 12th Class 12th Plus Classroom Results NEET 2025 2024 2023 2022 JEE 2025 2024 2023 2022 Class 6-10 Scholarships NEW TALLENTEX AOSAT ALLEN E-Store More ALLEN for Schools About ALLEN Blogs News Careers Request a call back Book home demo Login HomeClass 12MATHSIf P be any point in the plane of square... If P be any point in the plane of square ABCD, prove that P A 2+P C 2=P B 2+P D 2 To view this video, please enable JavaScript and consider upgrading to a web browser thatsupports HTML5 video Video Player is loading. Play Video Play Skip Backward Skip Forward Mute Current Time 0:00 / Duration-:- Loaded: 0% Stream Type LIVE Seek to live, currently behind live LIVE Remaining Time-0:00 1x Playback Rate 2x 1.5x 1x, selected 0.5x 0.25x Chapters Chapters Descriptions descriptions off, selected Captions captions settings, opens captions settings dialog captions off, selected Audio Track Picture-in-Picture Fullscreen This is a modal window. Beginning of dialog window. Escape will cancel and close the window. Text Color Opacity Text Background Color Opacity Caption Area Background Color Opacity Font Size Text Edge Style Font Family Reset Done Close Modal Dialog End of dialog window. Text Solution AI Generated Solution \| Share Topper's Solved these Questions COORDINATE SYSTEM AND COORDINATES ARIHANT MATHS\|Exercise Exercise For Session 3\|19 Videos View Playlist COORDINATE SYSTEM AND COORDINATES ARIHANT MATHS\|Exercise Exercise For Session 4\|17 Videos View Playlist COORDINATE SYSTEM AND COORDINATES ARIHANT MATHS\|Exercise Exercise For Session 1\|10 Videos View Playlist CONTINUITY AND DIFFERENTIABILITY ARIHANT MATHS\|Exercise Exercise (Questions Asked In Previous 13 Years Exam)\|20 Videos View Playlist DEFINITE INTEGRAL ARIHANT MATHS\|Exercise Exercise (Questions Asked In Previous 13 Years Exam)\|38 Videos View Playlist Similar Questions Explore conceptually related problems If G be the centroid of a triangle ABC and P be any other point in the plane prove that PA^(2)+PB^(2)+PC^(2)=GA^(2)+GB^(2)+GC^(2)+3GP^(2) Watch solution Let ABCD be a rectangle and P be any point in its plane.Show that AP^(2)+PC^(2)=PB^(2)+PD^(2) Watch solution Knowledge Check ABC is an equilateral triangle where AB = a and P is any point in its plane such that PA = PB + PC. Then (PA^(2)+PB^(2)+PC^(2))/(a^(2)) is A 3 B 3 4 C 3 4 D 2 Submit Let G be the centroid of the DeltaABC , whose sides are of lengths a,b,c. If P be a point in the plane of triangleABC , such that PA=1,PB=3, PC=4 and PG=2 , then the value of a^(2)+b^(2)+c^(2) is A 42 B 40 C 36 D 28 Submit If P is a point in th e plane of D ABCD such that bar(AP) + bar(PB) + bar(CP) + bar(PD) = bar0 , then the quadrilateral is in general, a A kite B parallelogram C trapezoid D rectangle Submit Similar Questions Explore conceptually related problems P is an interior point of square ABCD. Prove that PA+PB+PC+PD gt 2 AB . Watch solution O is any point inside a rectangle ABCD.Prove that OB^(2)+OD^(2)=OA^(2)+OC^(2) Watch solution An equilateral triangle ABC is inscribed in a circle of radius r if P be any point on the circle then find the value of PA^(2)+PB^(2)+PC^(2) Watch solution If P is a point inside a convex quadrilateral ABCD such that PA^2+PB^2+PC^2+PD^2 is twice the area of the quadrilateral , then the correct statements is/ are Watch solution Let ABCD be a square with sides of unit lenght. Points E and F are taken on sides AB and AD, respectively,so that AE=AF . Let P be a point inside the squre ABCD. The value of (PA)^2-(PB)^2+(PC)^2-(PD)^2 is equal to Watch solution ARIHANT MATHS-COORDINATE SYSTEM AND COORDINATES -Exercise For Session 2 If the distance between the points (a, 2) and (3, 4) be 8, then a equa...01:51 \| Play The three points (-2, 2), (8, -2) and (-4, -3) are the vertices of 02:15 \| Play The distance between the points (3,pi/4) and (7,(5pi)/4)02:30 \| Play Let A(6, -1), B (1, 3) and C (x, 8) be three points such that AB = BC ...01:48 \| Play The points (a+1,1), (2a+1,3) and (2a+2,2a) are collinear if 02:31 \| Play Let A=(3,4) and B is a variable point on the lines \|x\| =6. IF A Blt...02:07 \| Play The number of points on X-axis which are at a distance c units (c lt 3...02:37 \| Play The point on the axis of y which its equidistant from (-1, 2) and (3, ...01:55 \| Play Find the distance between the points (at(1)^(2), 2 at(1)) and (at(2)^(...05:36 \| Play If P and Q are two points whose coordinates are (a t^2,2a t)a n d(a/(t...05:57 \| Play Show that the points (3, 4), (8, -6) and (13, 9) are the vertices of a...02:55 \| Play Prove that the points (0, -1), (6, 7), (-2, 3) and (8, 3) are the vert...04:09 \| Play Find the circumcentre and circumradius of the triangle whose vertices ...06:51 \| Play The vertices of a triangle are A(1,1),\ B(4,5)a n d\ C(6, 13)dot Find ...02:49 \| Play The opposite vertices of a square are (2, 6) and (0, -2). Find the coo...07:20 \| Play If the point (x , y) is equidistant from the points (ab , b-a) and (a-...03:19 \| Play if a and bbetween 0 and 1 such that the points (a, 1). 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2148	https://www.quora.com/How-many-ways-there-exist-such-that-an-integer-natural-number-N-can-be-written-as-sum-of-consecutive-integers	How many ways there exist such that an integer (natural number) N can be written as sum of consecutive integers? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Consecutive Odd Integers Calculate with Integers Natural Numbers Integer Sequences Math Arithmetic Number Theory Consecutive Multiples Integer Division 5 How many ways there exist such that an integer (natural number) N can be written as sum of consecutive integers? All related (36) Sort Recommended Nick Shales Combinatorics enthusiast · Upvoted by Parag Kalita , Interested in numbers. · Author has 419 answers and 1.5M answer views ·Updated 4y Related How many ways can a positive integer N be written as the sum of consecutive positive integers? Instead of concerning ourselves immediately with the problem of some particular integer n n let's go through all the possible consecutive positive integer sums which I will call consecutive part partitions (or cpp for short). It would be handy if we could come up with an organised way of listing them. Well, the first thing that comes to mind is to start small and work up. Let's begin 1,1+2,1+2+3,1+2+3+4,1+2+3+4+5,…1,1+2,1+2+3,1+2+3+4,1+2+3+4+5,… okay we've got all of those beginning with 1, now let's start with 2 2,2+3,2+3+4,2+3+4+5,2+3+4+5+6,…2,2+3,2+3+4,2+3+4+5,2+3+4+5+6,… beginning with 3 3,3+4,3+4+5,3+3,3+4,3+4+5,3+ Continue Reading Instead of concerning ourselves immediately with the problem of some particular integer n n let's go through all the possible consecutive positive integer sums which I will call consecutive part partitions (or cpp for short). It would be handy if we could come up with an organised way of listing them. Well, the first thing that comes to mind is to start small and work up. Let's begin 1,1+2,1+2+3,1+2+3+4,1+2+3+4+5,…1,1+2,1+2+3,1+2+3+4,1+2+3+4+5,… okay we've got all of those beginning with 1, now let's start with 2 2,2+3,2+3+4,2+3+4+5,2+3+4+5+6,…2,2+3,2+3+4,2+3+4+5,2+3+4+5+6,… beginning with 3 3,3+4,3+4+5,3+4+5+6,3+4+5+6+7,….3,3+4,3+4+5,3+4+5+6,3+4+5+6+7,…. Well, we can see where this is going. This list of lists will continue ad infinitum, but we can express it more succinctly if we write these possibilities in a clever way: as powers of x x in an infinite series. Our three example lists above now look like this x 1+x 1+2+x 1+2+3+x 1+2+3+4+⋯+x 1+2+3+4+5+⋯+k+⋯x 2+x 2+3+x 2+3+4+x 2+3+4+5+⋯+x 2+3+4+5+6+⋯+(k+1)+⋯x 3+x 3+4+x 3+4+5+x 3+4+5+6+⋯+x 3+4+5+6+7+⋯+(k+2)+⋯.x 1+x 1+2+x 1+2+3+x 1+2+3+4+⋯+x 1+2+3+4+5+⋯+k+⋯x 2+x 2+3+x 2+3+4+x 2+3+4+5+⋯+x 2+3+4+5+6+⋯+(k+1)+⋯x 3+x 3+4+x 3+4+5+x 3+4+5+6+⋯+x 3+4+5+6+7+⋯+(k+2)+⋯. We will have a sum of all these lists, let's call it f(x)f(x) then such a sum may be written f(x)=∑r≥1(x r+x r+(r+1)+x r+(r+1)+(r+2)+x r+(r+1)+(r+2)+(r+3)+⋯+x r+(r+1)+(r+2)+(r+3)+⋯+(k+r−1)+⋯).(1)(1)f(x)=∑r≥1(x r+x r+(r+1)+x r+(r+1)+(r+2)+x r+(r+1)+(r+2)+(r+3)+⋯+x r+(r+1)+(r+2)+(r+3)+⋯+(k+r−1)+⋯). Now here comes the key observation: The number of times x n x n occurs in this sum is the number of cpp’s of n n. In other words, the coefficient of x n x n in f(x)f(x) is our desired answer. We denote this coefficient of x n x n in f(x)f(x) as [x n]f(x)[x n]f(x). [x n][x n] is an operator that finds the x n x n coefficient of a series. We may simplify (1)(1) somewhat since the general sum of the finite arithmetic series is r+(r+1)+(r+2)+⋯+(r+k−1)=1 2 k(2 r+(k−1))r+(r+1)+(r+2)+⋯+(r+k−1)=1 2 k(2 r+(k−1)) we then get f(x)=∑r≥1(x 1(2 r+0)/2+x 2(2 r+1)/2+x 3(2 r+2)/2+⋯+x k(2 r+(k−1))/2+⋯),f(x)=∑r≥1(x 1(2 r+0)/2+x 2(2 r+1)/2+x 3(2 r+2)/2+⋯+x k(2 r+(k−1))/2+⋯), or f(x)=x 0∑r≥1 x 1 r+x 1∑r≥1 x 2 r+x 3∑r≥1 x 3 r+⋯+x k(k−1)/2∑r≥1 x k r+⋯.f(x)=x 0∑r≥1 x 1 r+x 1∑r≥1 x 2 r+x 3∑r≥1 x 3 r+⋯+x k(k−1)/2∑r≥1 x k r+⋯. Each summation here can be written as a series f(x)=x 0 x 1 1−x 1+x 1 x 2 1−x 2+x 3 x 3 1−x 3+⋯+x k(k−1)/2 x k 1−x k+⋯,f(x)=x 0 x 1 1−x 1+x 1 x 2 1−x 2+x 3 x 3 1−x 3+⋯+x k(k−1)/2 x k 1−x k+⋯, again simplifying by writing in summation notation f(x)=∑k≥1 x k(k−1)/2 x k 1−x k=∑k≥1 x k(k+1)/2 1−x k.(2)f(x)=∑k≥1 x k(k−1)/2 x k 1−x k(2)=∑k≥1 x k(k+1)/2 1−x k. We could leave it here if we wished. If we take [x n]f(x)[x n]f(x) then this will give our answer but does this admit an easy description of the answer? Well noting that [x n][x n] is distributive over addition and [x n]x t=[x n−t][x n]x t=[x n−t] we get [x n]f(x)=∑k≥1[x n−k(k+1)/2]1 1−x k[x n]f(x)=∑k≥1[x n−k(k+1)/2]1 1−x k remembering that 1 1−x k=1+x k+x 2 k+x 3 k+⋯1 1−x k=1+x k+x 2 k+x 3 k+⋯ we see that [x n−k(k+1)/2]1 1−x k[x n−k(k+1)/2]1 1−x k is 1 1 if n−k(k+1)/2 n−k(k+1)/2 is some positive multiple of k k and 0 0 otherwise. So, in effect we want to find out for each k k if n−k(k+1)/2 n−k(k+1)/2 is divisible by k k. In other words we want to answer the question: When is n−k(k+1)/2 k=n k−(k+1)2 n−k(k+1)/2 k=n k−(k+1)2 a positive integer? We can make progress with this description and it allows us to do some actual number crunching but there is a cleaner description available. Okay, let’s write out in the following array, this will appear unmotivated at first but bear with me. c∖r 1 2 3 4⋯1 x x 2 x 3 x 4⋯2 x 3 x 6 x 9 x 12⋯3 x 5 x 10 x 15 x 20⋯4 x 7 x 14 x 21 x 28⋯⋮⋮⋮⋮⋮⋱c∖r 1 2 3 4⋯1 x x 2 x 3 x 4⋯2 x 3 x 6 x 9 x 12⋯3 x 5 x 10 x 15 x 20⋯4 x 7 x 14 x 21 x 28⋯⋮⋮⋮⋮⋮⋱ This table is constructed so that each successive row has a common ratio x 2 c x 2 c where c c is the column number and each successive column had common ratio x 2 r−1 x 2 r−1 where r r is the row number, finally the top left cell is occupied by x x. Clearly the staircase of cells ((c,r))=((1,1),(2,1),(2,2),(3,2)…)((c,r))=((1,1),(2,1),(2,2),(3,2)…) will start with x x and have ratios (x 2,x 3,x 4,x 5,…)(x 2,x 3,x 4,x 5,…). Thus row 1 starting at (1,1)(1,1) is the series x 1 1−x x 1 1−x, column 1 starting at (2,1)(2,1) is the series x 1+2 1−x 2 x 1+2 1−x 2, row 2 starting at (2,2)(2,2) is the series x 1+2+3 1−x 3 x 1+2+3 1−x 3 and so on. Adding these up we find that the sum of this entire table is ∑k≥1 x k(k+1)/2 1−x k=f(x),∑k≥1 x k(k+1)/2 1−x k=f(x), fabulous! So we have reorganised the terms in our sum f(x)f(x). Right, now if we decide to sum rows then we see that the k th k th row is the series x 2 k−1 1−x 2 k−1 x 2 k−1 1−x 2 k−1, summing over all rows gives f(x)=∑k≥1 x 2 k−1 1−x 2 k−1.f(x)=∑k≥1 x 2 k−1 1−x 2 k−1. Taking the x n x n coefficient we find [x n]f(x)=∑k≥1[x n]x 2 k−1 1−x 2 k−1[x n]f(x)=∑k≥1[x n]x 2 k−1 1−x 2 k−1 and [x n]x 2 k−1 1−x 2 k−1[x n]x 2 k−1 1−x 2 k−1 is 1 1 if n n is a positive multiple of 2 k−1 2 k−1 and 0 0 otherwise. Thus the sum therefore represents the total count of odd factors of n n. I found this clever reorganisation of terms in MacMahon’s 1915 text ‘Combinatory Analysis’ [R1] volume 2 page 27. The number of cpp’s for n n is therfore given by: [x n]f(x)=#(odd factors of n).(Answer)(Answer)[x n]f(x)=#(odd factors of n). This answer is verified by the oeis sequence A001227 as described by the second paragraph of the comments. Actually calculating this may require a prime decomposition of n n so that if n=2 n 2 3 n 3 5 n 5 7 n 7⋯n=2 n 2 3 n 3 5 n 5 7 n 7⋯ then [x n]f(x)=(n 3+1)(n 5+1)(n 7+1)⋯[x n]f(x)=(n 3+1)(n 5+1)(n 7+1)⋯ Reference [R1] Combinatory Analysis volumes 1 & 2, P. A. MacMahon, Dover republication, originally published 1915. Footnotes A001227 - OEIS Upvote · 99 15 9 9 Promoted by Coverage.com Johnny M Master's Degree from Harvard University (Graduated 2011) ·Updated Sep 9 Does switching car insurance really save you money, or is that just marketing hype? This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. Continue Reading This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take. But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious. They claimed I could save over $1,000 per year. I ended up exceeding that number and I cut my monthly premium by over $100. That’s over $1200 a year. For the exact same coverage. No phone tag. No junk emails. Just a better deal in less time than it takes to make coffee. Here’s the link to two comparison sites - the one I used and an alternative that I also tested. If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying. Upvote · 999 485 999 103 99 17 Related questions More answers below The sum of three consecutive integers is 258. How do you find the three integers? How many ways can a positive integer N be written as the sum of consecutive positive integers? In how many ways can 45 be written as the sum of consecutive positive integers? The sum of two consecutive integers is 41. What are the integers? How is the sum of two consecutive integers odd? Doug Dillon Ph.D. Mathematics · Author has 12.4K answers and 11.4M answer views ·4y I will treat this as an investigation involving experimentation and inductive reasoning. When we can offer an answer, we may or may not prove it. You have set the direction for the question by equating integers with natural numbers. Consider a number of natural numbers, chosen in an organized way. Group 1: 3, 5, 7. These can be written as the sums of consecutive integers in only one way each (1+2, 2+ Continue Reading I will treat this as an investigation involving experimentation and inductive reasoning. When we can offer an answer, we may or may not prove it. You have set the direction for the question by equating integers with natural numbers. Consider a number of natural numbers, chosen in an organized way. Group 1: 3, 5, 7. These can be written as the sums of consecutive integers in only one way each (1+2, 2+3 and 3+4). If we wish to include singletons, then 2 ways each. Henceforth, I will include singletons for the ease of counting. At this point, our intuition might flare up and suggest that for all odd numbers, there are two ways off writing a the sum of consecutive integers. But maybe only certain odds. For instance, try 9. 9=4+5=2+3+4=9. So our intuition got a little excited. So, given that 11 and 13 “behave” the next to try is 15. Well, 15=7+8=4+5+6=15. To summarize, so far, prime numbers can be written as the sum of consecutive integers in 2 ways. Group 2: 2, 4, 8 These can only be written as the sums of consecutive in 1 way each Perhaps that is true for all powers of 2. Group 3: 6, 10, 14 This group have a common trait: each is the product o... Upvote · 9 1 Alon Amit PhD in Mathematics; Mathcircler. · Upvoted by Horst H. von Brand , PhD Computer Science & Mathematics, Louisiana State University (1987) and Nathan Hannon , Ph. D. Mathematics, University of California, Davis (2021) · Author has 8.8K answers and 173.8M answer views ·9y Related Which positive integers can be represented as a sum of two or more consecutive positive integers? As Paulina Jonušaitė showed, the answer is all positive integers except for powers of 2. Here's a simple proof of that fact which, I hope, is a little clearer and better motivated. It's not short, because I'm not going for brevity; I'm going for simplicity. But this proof can be rewritten as a very short argument. Which numbers are a sum of exactly two consecutive positive integers? That's easy: 1+2=3 1+2=3, 2+3=5 2+3=5, 3+4=7 3+4=7, 4+5=9 4+5=9 and so on. Obviously we get all odd numbers greater than 1. Now, what happens if we take such a simple sum such as 10+11=21 10+11=21, and extend it by including the previous number a Continue Reading As Paulina Jonušaitė showed, the answer is all positive integers except for powers of 2. Here's a simple proof of that fact which, I hope, is a little clearer and better motivated. It's not short, because I'm not going for brevity; I'm going for simplicity. But this proof can be rewritten as a very short argument. Which numbers are a sum of exactly two consecutive positive integers? That's easy: 1+2=3 1+2=3, 2+3=5 2+3=5, 3+4=7 3+4=7, 4+5=9 4+5=9 and so on. Obviously we get all odd numbers greater than 1. Now, what happens if we take such a simple sum such as 10+11=21 10+11=21, and extend it by including the previous number and the next one? We get 9+10+11+12=42 9+10+11+12=42. Double what we had before. Why? Well clearly, 10+11 10+11 and 9+12 9+12 are the same thing, so by adding them up we get twice the original sum. We can continue this extension trick to get 8+9+10+11+12+13=63 8+9+10+11+12+13=63, three time the original, because 8+13 8+13 and 9+12 9+12 and 10+11 10+11 are all the same thing: 21 21. And then 7+8+…+14=84 7+8+…+14=84 is four times the original, and so on. You may object to this "and so on": we're making the starting number smaller and smaller; what happens when it hits 0 0 and then goes even further into negative territory? We're supposed to only look at sums of positive integers. So does the process break at some point? Oh, but it doesn't. Let's say we started with 1+2=3 1+2=3. Then we extended it to 0+1+2+3=6 0+1+2+3=6. The 0 0 is not legitimate, so just ignore it – of course the sum doesn't change! The number 6 6 is still a sum of consecutive positive integers. In the next step we get (−1)+0+1+2+3+4=9(−1)+0+1+2+3+4=9. That looks illegal, but no worries, just ignore the 0 0 and also ignore the (−1)+1(−1)+1. The sum 9 9 is still a legitimate 2+3+4 2+3+4. And so on. So in fact, allowing negative numbers doesn't alter the result at all. Ok. Now, what numbers can we get by using this method repeatedly? We had all odd numbers (except 1) to begin with. Then we doubled all of these, which gives all numbers that are twice an odd number (except 2×1 2×1). Then we tripled them, but that's uninteresting since tripling an odd number is still odd. Then we quadrupled them, giving us all numbers which are four times an odd number (except 4×1 4×1). And so on and so forth. What are those numbers? Well of course, any number is an odd number times some power of 2 2 (possibly the "trivial" power 2 0 2 0). Why? Because starting with any number, just divide it by 2 2 and divide the result by 2 2 again and keep going until you get an odd number. For example 100=25×2 2 100=25×2 2 and 1728=27×2 6 1728=27×2 6. Odd numbers are just like 7=7×2 0 7=7×2 0. So did we just prove that all positive integers are sums of consecutive integers? No, because remember that the odd part isn't allowed to be 1 1. Meaning, we get all numbers that have some genuine odd divisor; in other words, everything except powers of 2 2. Any number that's not a power of 2 2 can be written as 2 k b 2 k b where b b is an odd integer greater than 1 1, and we just proved using the extension trick that all such numbers are representable as a sum of consecutive integers. Done? Not quite. We showed how to get all non-powers-of-2 2, but we still need to show that the powers of 2 2 are really not achievable. Here's an elementary way of seeing why, using the same kind of reasoning: When you add up an even number of consecutive integers, you can pair them up (first, last), (second, next-to-last) and so on, and each such pair has an odd sum. So the whole thing adds up to (something odd) times whatever, and it cannot be a power of 2 2. (Observe how the paired up numbers must have opposite parity, because the distance between them is odd). When you add up an odd number of consecutive integers, there's some number in the middle. The pair around it (right before and right after) add up to twice that number, and the next pair outwards also ads up to twice that number, and so on; so overall we get that middle number times something odd. Again, this can't be a power 2 2, and we're done. Upvote · 999 173 99 20 9 4 David Vanderschel PhD in Mathematics&Physics, Rice (Houston neighborhood) (Graduated 1970) · Author has 37.6K answers and 50.1M answer views ·4y Let the f f be the first term of the sequence and let m m be the number of terms. We seek solutions of m(f+(f+m−1))/2=N m(f+(f+m−1))/2=N or m(2 f+m−1)=2 N.m(2 f+m−1)=2 N. A divisor m m of 2 N 2 N will work if f=N/m−(m−1)/2 f=N/m−(m−1)/2 is an integer and that holds when m m is any odd divisor of N.N. If m m is even, it will work only if N N is odd. So let k k be the number of odd divisors of N.N. If N N is even, there are k k such series. If N N is odd, there are 2 k 2 k such series. Upvote · Promoted by The Hartford The Hartford We help protect over 1 million small businesses ·Updated Sep 19 What is small business insurance? Small business insurance is a comprehensive type of coverage designed to help protect small businesses from various risks and liabilities. It encompasses a range of policies based on the different aspects of a business’s operations, allowing owners to focus on growth and success. The primary purpose of small business insurance is to help safeguard a business’s financial health. It acts as a safety net, helping to mitigate financial losses that could arise from the unexpected, such as property damage, lawsuits, or employee injuries. For small business owners, it’s important for recovering quickl Continue Reading Small business insurance is a comprehensive type of coverage designed to help protect small businesses from various risks and liabilities. It encompasses a range of policies based on the different aspects of a business’s operations, allowing owners to focus on growth and success. The primary purpose of small business insurance is to help safeguard a business’s financial health. It acts as a safety net, helping to mitigate financial losses that could arise from the unexpected, such as property damage, lawsuits, or employee injuries. For small business owners, it’s important for recovering quickly and maintaining operations. Choosing the right insurance for your small business involves assessing your unique needs and consulting with an advisor to pick from comprehensive policy options. With over 200 years of experience and more than 1 million small business owners served, The Hartford is dedicated to providing personalized solutions that help you focus on growth and success. Learn about our coverage options! Upvote · 999 557 9 1 9 3 Related questions More answers below Are there as many integers as even integers? The sum of two consecutive even integers is 126. What are the integers? What 3 consecutive even integers have a sum of -36? What is the smallest possible sum of four consecutive integers? How do you prove that every integer is a rational number? David Vanderschel PhD in Mathematics&Physics, Rice (Houston neighborhood) (Graduated 1970) · Author has 37.6K answers and 50.1M answer views ·Updated 6y Related How many ways can a positive integer N be written as the sum of consecutive positive integers? Any number is the sum of a sequence of 1 1 consecutive number(s), namely itself. So let us forget about this trivial solution which can also be discounted on the grounds that we don’t have integer s plural. However, it does provide a solution for N=1 N=1 which otherwise would not have one. In general, a sum of consecutive integers must always be of the form n(f+(f+n−1))/2 n(f+(f+n−1))/2 where n n is the number of terms, f f is the first, and f+n−1 f+n−1 is the last. We have already discarded the case for n=1 n=1 and f=N.f=N. We have 2 N=n(2 f+n−1).2 N=n(2 f+n−1).f=1 2(2 N n−n+1)f=1 2(2 N n−n+1) must be an integer. That can be achieved if n n is an od Continue Reading Any number is the sum of a sequence of 1 1 consecutive number(s), namely itself. So let us forget about this trivial solution which can also be discounted on the grounds that we don’t have integer s plural. However, it does provide a solution for N=1 N=1 which otherwise would not have one. In general, a sum of consecutive integers must always be of the form n(f+(f+n−1))/2 n(f+(f+n−1))/2 where n n is the number of terms, f f is the first, and f+n−1 f+n−1 is the last. We have already discarded the case for n=1 n=1 and f=N.f=N. We have 2 N=n(2 f+n−1).2 N=n(2 f+n−1).f=1 2(2 N n−n+1)f=1 2(2 N n−n+1) must be an integer. That can be achieved if n n is an odd divisor of N N or n n is an even divisor of 2 N 2 N for which the quotient is odd. For an n n which satisfies, N N is the sum of the n n consecutive integers starting at f.f. To achieve f>0,f>0, we must have 2 N>n(n−1).2 N>n(n−1). So it comes down to the size of the following set: {n\|n>1;2∤n{n\|n>1;2∤n⟹n∣N;2∣n⟹n∣N;2∣n⟹n∣2 N⟹n∣2 N and 2∤2 N n;n 2−n<2 N}2∤2 N n;n 2−n<2 N} If the prime factorization of N N has no odd factors then that set is empty. I.e., no power of 2 2 can be expressed as the sum of such a sequence. Note that, if N N is odd, then we have a solution with n=2 n=2 and f=(N−1)/2.f=(N−1)/2. If N N is also prime, that is the only solution. (If N N has q q distinct odd prime factors, let m i m i for i∈[1,q]i∈[1,q] be their multiplicities. Then the number of distinct odd divisors of N N is (∏q i=1(m i+1))−1.(∏i=1 q(m i+1))−1. (You can take any power in [0,m i][0,m i] for the i th prime, but you can’t take 0 for all of them.) Alas, I do not know a good way to express the number of those satisfying n(n−1)<2 N.n(n−1)<2 N. (I only know how to brute force it for a given prime factorization.) Similarly for even n,n,n n must have one more than all the powers of 2 2 in N N plus any proper subset of the odd prime factors. (Any power on the i th odd prime except not m i m i for all.) So it’s the same number for this sort of n, though the size limit is going to weed out more of them.) UPDATE: Privately I asked Nick Shales if he had any insight on my impasse described in the preceding parenthetical paragraph. He did. Indeed, if one allows the trivial length 1 solutions, the solutions to this problem are a sequence which arises in a number of contexts and it is well documented here: A001227 - OEIS. The second additional description there is “Also number of partitions of n into consecutive positive integers including the trivial partition of length 1 …”. However, the primary description is “Number of odd divisors of N.”N.” So I was on the right path but was missing the fact that the odd divisors of N N for which the resulting sequence is not all positive correspond 1-to-1 with the qualifying even divisors of 2 N.2 N. So I will add that clarification here. Let o o be an odd divisor of N,N, and let e=2 N/o.e=2 N/o.e e is even and 2 N/e 2 N/e is odd, so there are sequences of length e e which sum to N.N.o≠e,o≠e, so let n=min(o,e)n=min(o,e) and x=max(o,e).x=max(o,e).x(x−1)≥x n=2 N,x(x−1)≥x n=2 N, so the sequence of length x x does not start positive. n(n−1)<x n=2 N,n(n−1)<x n=2 N, so that with length n n does. Thus there is a 1-to-1 correspondence between odd divisors of N N and sequences of consecutive positive integers which sum to N.N. However, not all those divisors are the lengths such sequences. When not, 2 N 2 N divided by that divisor is. Suppose that the prime factorization of N N has q q distinct odd prime factors and let their multiplicities be m i m i for i i in [1,q].[1,q]. Then the number of distinct odd factors of N N is ∏q i=1(m i+1)∏i=1 q(m i+1) and that is an answer to the question (including the trivial length 1 sequence). Nick has a way of writing an expression for that number based on the generating function corresponding to the sequence which answers the question. I will leave it to him to present that expression. UPDATE2: I was wrong about Nick having an expression for the number of odd divisors. Indeed, he suggests evaluating it the same way I did. He just had a different way of showing that that is the answer to the question. Upvote · 9 3 9 1 Lucas Curtis Science teacher (2001–present) · Author has 7.4K answers and 22.1M answer views ·Updated 4y Related In how many ways can 45 be written as the sum of consecutive positive integers? Several people have given you the correct answer: there are five ways to express 45 as the sum of consecutive positive integers. I would like to probe for a more general solution; one that can be used for any starting whole number. WARNING: THIS ANSWER WILL BE LONG! If you’re not interested in the mechanics of this problem, just keep scrolling. No hard feelings. Let’s say you have a number N N and you want to find a sequence of consecutive positive integers (hereafter, CPIs), starting with a a, such that: N=a+(a+1)+(a+2)+…+(a+(k−1))+(a+k)N=a+(a+1)+(a+2)+…+(a+(k−1))+(a+k) Rearranging we get: N=(k+1)a+(1+2+3+…+(N=(k+1)a+(1+2+3+…+( Continue Reading Several people have given you the correct answer: there are five ways to express 45 as the sum of consecutive positive integers. I would like to probe for a more general solution; one that can be used for any starting whole number. WARNING: THIS ANSWER WILL BE LONG! If you’re not interested in the mechanics of this problem, just keep scrolling. No hard feelings. Let’s say you have a number N N and you want to find a sequence of consecutive positive integers (hereafter, CPIs), starting with a a, such that: N=a+(a+1)+(a+2)+…+(a+(k−1))+(a+k)N=a+(a+1)+(a+2)+…+(a+(k−1))+(a+k) Rearranging we get: N=(k+1)a+(1+2+3+…+(k−1)+k)N=(k+1)a+(1+2+3+…+(k−1)+k) Let’s focus on this part: 1+2+3+…+(k−1)+k.1+2+3+…+(k−1)+k. If we combine the first and last term, we get k+1 k+1. If we combine the second and second-to-last term, we also get k+1 k+1. Because we’re pairing terms, the sum of k+1 k+1 will turn up k 2 k 2 times. Pulling it all together: N=(k+1)a+(k+1)(k 2)N=(k+1)a+(k+1)(k 2) N=(k+1)a+k(k+1)2 N=(k+1)a+k(k+1)2 Rearranging some more: N=2 a(k+1)+k(k+1)2 N=2 a(k+1)+k(k+1)2 N=(2 a+k)(k+1)2 N=(2 a+k)(k+1)2 Now we’re set. We can quickly check for viable solutions using this equation. Here’s what you do. Pick a whole number N N. In this case, N=45 N=45. Pick a starting number for k k. I recommend starting with k=1 k=1. Solve for a a. If you get a whole number for a a, then congratulations! That’s the first number in a sequence of CPIs that will add up to N N. (The last number in the sequence will be a+k a+k.) If you don’t get a whole number for a a, then increase k k by 1 and try again. You can stop searching when the value of a a drops below 1. So let’s take our algorithm for a spin with N=45 N=45 and see if it yields the correct results: Starting with k=1 k=1: 45=(2 a+(1))((1)+1)2 45=(2 a+(1))((1)+1)2 90=(2 a+1)(2)90=(2 a+1)(2) 45=2 a+1 45=2 a+1 44=2 a 44=2 a 22=a 22=a Because 22 22 is a whole number, it is the first number in a sequence of CPIs that will add up to 45 45. The last number is a+k=22+1=23 a+k=22+1=23. Starting with k=2 k=2: 45=(2 a+(2))((2)+1)2 45=(2 a+(2))((2)+1)2 90=(2 a+2)(3)90=(2 a+2)(3) 30=2 a+2 30=2 a+2 28=2 a 28=2 a 14=a 14=a Here we have another number that can serve as the starting point of a sequence of CPIs whose sum is 45 45. The last number is a+k=14+2=16 a+k=14+2=16. Starting with k=3 k=3: 45=(2 a+(3))((3)+1)2 45=(2 a+(3))((3)+1)2 90=(2 a+3)(4)90=(2 a+3)(4) 22.5=2 a+3 22.5=2 a+3 19.5=2 a 19.5=2 a a=9.75 a=9.75 No solution here! Let’s keep on searching! Starting with k=4 k=4: 45=(2 a+(4))((4)+1)2 45=(2 a+(4))((4)+1)2 90=(2 a+4)(5)90=(2 a+4)(5) 18=2 a+4 18=2 a+4 14=2 a 14=2 a 7=a 7=a So 7 7 is the first number of a sequence of CPIs whose sum is 45 45. The last number in this sequence is a+k=7+4=11 a+k=7+4=11. You can see that the values of a a are getting smaller as the value of k k increases. Eventually the value of a a will be lower than 1 1, and at that point we can stop searching for solutions. If you want to speed up the process of searching for solutions, here’s a PRO TIP: You can avoid any values of k k if k+1 k+1 is not a factor of 2 N 2 N. In the case of k=3 k=3 and N=45 N=45, we could see that k+1 k+1 (or 4) is not a factor of 2 N 2 N (or 90), so we would know in advance that k=3 k=3 would not yield a valid starting point. Similarly, we can save time by not checking k=6 k=6 and k=7 k=7, since neither 6+1 6+1 nor 7+1 7+1 is a factor of 2 N 2 N (90). Once we get to k=9 k=9, the value of a is less than 1 1: Starting with k=9 k=9: 45=(2 a+(9))((9)+1)2 45=(2 a+(9))((9)+1)2 90=(2 a+9)(10)90=(2 a+9)(10) 9=2 a+9 9=2 a+9 0=2 a 0=2 a 0=a 0=a Since we’re only looking for positive integer sequences, we cannot use a sequence that starts with zero or lower. By this point, we will have found any solutions that exist to be found, so we can stop searching and go do other things! Upvote · 99 21 9 1 Promoted by US Auto Insurance Now US Auto Insurance Now Helping Drivers Find Great Car Insurance Deals ·Sep 23 What are some of the most effective ways to save money? Making smart financial decisions doesn't have to be complicated. In 2025, there are several simple yet highly effective money hacks that can make a huge difference in your financial health. These aren't complicated investment strategies; they are practical, everyday habitsthat help you keep more of your hard-earned money. Here are 5 easy ways to boost your savings and make your income work for you: Automate Your Savings and Investments Set up an automatic transfer to your savings account or investment portfolio the same day your paycheck hits. Even if it's a small amount like $25 a week, it Continue Reading Making smart financial decisions doesn't have to be complicated. In 2025, there are several simple yet highly effective money hacks that can make a huge difference in your financial health. These aren't complicated investment strategies; they are practical, everyday habitsthat help you keep more of your hard-earned money. Here are 5 easy ways to boost your savings and make your income work for you: Automate Your Savings and Investments Set up an automatic transfer to your savings account or investment portfolio the same day your paycheck hits. Even if it's a small amount like $25 a week, it adds up quickly and builds wealth with zero effort. The key is "out of sight, out of mind"—you won't be tempted to spend what you never see. Leverage High-Yield Savings Accounts Don't let your emergency fund or short-term savings sit idle in a traditional bank account earning a fraction of a percent. Look for a high-yield savings account (HYSA). These accounts often have no fees and offer significantly higher interest rates, allowing your money to grow passively and effortlessly. Cancel Unused Subscriptions Take an hour to review your bank and credit card statements from the last few months. You'll likely find subscriptions for streaming services, apps, or gym memberships you rarely use. Canceling these small, recurring expenses can free up dozens of dollars each month that can be redirected toward savings or debt payment. Compare Car and Home Insurance Rates with a Free Tool Did you know you could be overpaying on your car or home insurance by hundreds of dollars each year without even knowing it? Insurance rates can change dramatically based on market conditions, your driving history, and even your credit score. The best part is that instead of calling dozens of companies, there is a tool that does the hard work for you. It lets you compare real-time quotes from a network of over 1000 insurers in minutes. You simply enter your information once and see all your options side-by-side. Current users save, on average, up to $600 per year by finding a better rate. 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Upvote · 99 26 9 1 Akash Sivaraman Integrated M.sc. in Data Science&Mathematics, Student at Amrita Vishwa Vidhyapeetham,Coimbatore (Graduated 2023) · Author has 51 answers and 112.7K answer views ·5y Originally Answered: In how many ways can a number be expressed as a sum of consecutive numbers? · Only one way It is impossible to express a even number as a sum of two consecutive numbers as the sum of two consecutive numbers is always odd For an odd number of the from 2n + 1,it can be expressed as the sum of two consecutive numbers which are n and n+1 2n+1 = n + (n+1) Upvote · Larry Stark Former CEO at PROFITsystems (company) (1980–2007) · Author has 468 answers and 369K answer views ·5y Originally Answered: In how many ways can a number be expressed as a sum of consecutive numbers? · If you mean 3 consecutive numbers, the largest of which is the sum of the other two, then there is only one way: 1+2=3. Of this is not the question, then I think the question needs clarification. Upvote · Sponsored by StealthGPT All your schoolwork, one app. Trusted by millions of students, writing countless essays and answering any question. Learn More 99 83 Paulina Jonušaitė is an MSc in Applied Mathematics. · Author has 2.2K answers and 14.2M answer views ·9y Related Which positive integers can be represented as a sum of two or more consecutive positive integers? Assume n n is the number we start adding from. n+(n+1)=2 n+1 n+(n+1)=2 n+1 n+(n+1)+(n+2)=3 n+3 n+(n+1)+(n+2)=3 n+3 n+(n+1)+(n+2)+(n+3)=4 n+6 n+(n+1)+(n+2)+(n+3)=4 n+6 Notice that n n is multiplied by the number of consecutive numbers, which we will denote k k, and the other number we add is a (k−1)(k−1)-th triangular number. Therefore, n+(n+1)+…+(n+k−1)=k n+k(k−1)2 n+(n+1)+…+(n+k−1)=k n+k(k−1)2 So, for a number N N to be represented as a sum of consecutive natural numbers, this equation has to have at least one solution (n,k)(n,k) in natural numbers, where k≠1 k≠1: N=k n+k(k−1)2 N=k n+k(k−1)2 In particular, all odd numbers except 1 1 can b Continue Reading Assume n n is the number we start adding from. n+(n+1)=2 n+1 n+(n+1)=2 n+1 n+(n+1)+(n+2)=3 n+3 n+(n+1)+(n+2)=3 n+3 n+(n+1)+(n+2)+(n+3)=4 n+6 n+(n+1)+(n+2)+(n+3)=4 n+6 Notice that n n is multiplied by the number of consecutive numbers, which we will denote k k, and the other number we add is a (k−1)(k−1)-th triangular number. Therefore, n+(n+1)+…+(n+k−1)=k n+k(k−1)2 n+(n+1)+…+(n+k−1)=k n+k(k−1)2 So, for a number N N to be represented as a sum of consecutive natural numbers, this equation has to have at least one solution (n,k)(n,k) in natural numbers, where k≠1 k≠1: N=k n+k(k−1)2 N=k n+k(k−1)2 In particular, all odd numbers except 1 1 can be expressed very easily with two consecutive natural numbers (divide by 2, take the closest smaller and larger integers, add them together), so the interesting question now becomes, which even natural numbers are not representable? Any even multiple of an odd number k k can be represented by a sum of k k consecutive numbers. Indeed, assuming N=2 m k N=2 m k, this follows: 2 m k=k n+k(k−1)2 2 m k=k n+k(k−1)2 2 m=n+k−1 2 2 m=n+k−1 2 n=2 m−k−1 2 n=2 m−k−1 2 2 m 2 m is obviously a natural number, and k−1 k−1 is an even number, so the result is definitely an integer. However, it is a natural number only if 4 m>k−1 4 m>k−1 or alternatively, 2 N>k(k−1)2 N>k(k−1). This, however, is not an issue: the opposite integers will simplify, and zero can be removed, leaving a shorter sum that includes only natural numbers. Reverting back to N N, we can express n n as follows. n=N k−k−1 2 n=N k−k−1 2 For example, 42 42 has a prime factorization 2×3×7 2×3×7, so it can be expressed as a sum of 3 3 or 7 7 consecutive natural numbers: n 3=42 3−3−1 2=14−1=13 n 3=42 3−3−1 2=14−1=13 13+14+15=42 13+14+15=42 n 7=42 7−7−1 2=6−3=3 n 7=42 7−7−1 2=6−3=3 3+4+5+6+7+8+9=42 3+4+5+6+7+8+9=42 It cannot be expressed as a sum of 21 21 natural numbers, because 2×42=84<420=21×20 2×42=84<420=21×20, but we can use this result to obtain a sum that does qualify: n 21=42 21−21−1 2=2−10=−8 n 21=42 21−21−1 2=2−10=−8 (−8)+(−7)+…+7+8+9+10+11+12=9+10+11+12=42(−8)+(−7)+…+7+8+9+10+11+12=9+10+11+12=42 Having eliminated all odd numbers except 1, and all even multiples of odd numbers, we have only one class of numbers remaining – the powers of two. Assuming N=2 m N=2 m, n=2 m k−k−1 2 n=2 m k−k−1 2 If k k is odd, it is obvious, that that the first fraction is not a natural number, and the second one is. If k k is even, i.e., k=2 s p k=2 s p, where s s is a non-negative integer, and p p is an odd number, then n=2 m 2 s p−2 s p−1 2=2 m−s p−2 s−1 p+1 2 n=2 m 2 s p−2 s p−1 2=2 m−s p−2 s−1 p+1 2 Combining fractions into one, we obtain n=2 m−s+1+p 2 p−2 s−1 p n=2 m−s+1+p 2 p−2 s−1 p Since we assumed p p is odd, the numerator is odd, and the denominator is even, unless s=m+1 s=m+1, as 2 0=1 2 0=1 is the only odd power of two. Therefore, n=1+p 2 p−2 m p=1+p 2 p−N p n=1+p 2 p−2 m p=1+p 2 p−N p Remember that p p is odd, i.e., we can write it down like p=2 q+1 p=2 q+1? n=1+2 q+1 2(2 q+1)−N(2 q+1)=q+1 2 q+1−N(2 q+1)n=1+2 q+1 2(2 q+1)−N(2 q+1)=q+1 2 q+1−N(2 q+1) The fraction is a natural number if and only if q=0 q=0, otherwise the numerator is strictly smaller than denominator, therefore, p=1 p=1, and k=2 m+1 k=2 m+1. Finally! We can plug this value of k k into our formula for n n and reap the rewards: n=2 m 2 m+1−2 m+1−1 2=1 2−2 m+1 2=−2 m+1 n=2 m 2 m+1−2 m+1−1 2=1 2−2 m+1 2=−2 m+1 However, if we now construct our sum with the intention of simplifying opposite integers and zero out, we can see the result: N=(−2 m+1)+(−2 m+2)+…+(2 m−2)+(2 m−1)+2 m=2 m N=(−2 m+1)+(−2 m+2)+…+(2 m−2)+(2 m−1)+2 m=2 m So now we see that the only way to express a power of two is itself. Too Long Didn't Read: you can express any odd natural number except 1 as a sum of two consecutive natural numbers; you can express any even multiple of an odd natural number k k as a sum of k k consecutive integers, which can then be simplified by eliminating zero and opposite integers; you cannot express any integer powers of 2 2 as a sum of two or more consecutive natural numbers. Still Too Long: any number that is not an integer power of 2 2. Upvote · 99 31 Phil Albert I can count from 2 to P · Author has 6.6K answers and 4.9M answer views ·6y Related How many ways can a positive integer N be written as the sum of consecutive positive integers? Let N be a positive integer that is the sum of k consecutive integers for some positive integer k, and let r denote the smallest of the k consecutive integers. For now, do not assume that r is positive. For a positive integer x, there is a function, d(x), that is the number of positive divisors of x. This can be derived from the powers of the prime factors of x in a straightforward manner. This will be relevant later. There is some odd M and some integer b greater than or equal to zero for which N=2 b M N=2 b M. Since 2 and M are relatively prime, d(N)=(b+1)d(M). This will also be relevant later. If N is Continue Reading Let N be a positive integer that is the sum of k consecutive integers for some positive integer k, and let r denote the smallest of the k consecutive integers. For now, do not assume that r is positive. For a positive integer x, there is a function, d(x), that is the number of positive divisors of x. This can be derived from the powers of the prime factors of x in a straightforward manner. This will be relevant later. There is some odd M and some integer b greater than or equal to zero for which N=2 b M N=2 b M. Since 2 and M are relatively prime, d(N)=(b+1)d(M). This will also be relevant later. If N is the sum of the k consecutive integers from r to r+(k-1), then: N=k r+k(k−1)/2 N=k r+k(k−1)/2 (1) For a given N and k, if r is an integer, then (k, r) defines one of the ways to write N as a sum of k consecutive integers, namely the sum of r to r+(k-1). Define the functions S(N), S o(N), and S e(N), where S o(N) is the number of different odd values of k for which (2) yields an integer value of r, S e(N) is the number of different even values of k for which (2) yields an integer value of r, and S(N) is S o(N)+S e(N). Rearranging terms of (1), we get: 2 r=(2 N/k)−k+1 2 r=(2 N/k)−k+1 (2) Considering (2) for odd k, it is easy to see that r is an integer when 2N/k is an integer, which is if and only if k divides N (k only divides 2 N when k divides N, since k is odd). That means that S o(N) is simply the total number of odd positive divisors of N, which is equal to d(M) because all of the other bd(M) divisors of N are even. For even k, there is an integer c such that 2 c=k, so from (2), we get: 2 r=(2 N/2 c)−2 c+1 2 r=(2 N/2 c)−2 c+1 (3) 2(r+c)=(2 N/2 c)+1 2(r+c)=(2 N/2 c)+1 (4) 2(r+c)=(N/c)+1 2(r+c)=(N/c)+1 (5) From (5) it is clear that r will be an integer if and only if N/c is an odd integer. For N/c to be an integer, c must divide N. For N/c to be an odd integer, c must also be a multiple of 2 b 2 b, so c must be a divisor of M. Therefore, the number of unique values of c, and thus the number of unique values of k, for which r is an integer, is equal to d(M). So S(N)=S o(N)+S e(N)=2d(M). Now, that is if we did not limit the values of r to be positive integers. For example, in the case of N=12, k=8, we determine that b=2 and M=3, so there are 2d(3)=4 solutions for N=12. Indeed there are: (k=1, r=12), (k=3, r=3), (k=8, r=-2), and (k=24, r=-11). The first one is a trivial solution, (k=1, r=N), and the last one is also a trivial solution, (k=2 N, r=1-N). To check the nontrivial solutions: 3+4+5=12 and -2-1+0+1+2+3+4+5=12. If we add the limitation of r being a positive integer, then from (2), we have the additional constraint that k<√2 N k<2 N. I don’t know that S(N) is easily calculated without just running through all the divisors of N. Upvote · Dana F Anderson Just interested in Math, particularly PRIME NUMBERS · Author has 10.5K answers and 8.7M answer views ·4y There are an INFINITE number of ways. 1 + 2 = 3. 2 + 3 = 5. 3 + 4 = 7. ………………………………………………………. 1234567899 + 12345678900 = 2,469,135,799……………………………………………and so on FOREVER! Upvote · Raziman T.V. Recreational mathematics addict · Author has 2.6K answers and 28.1M answer views ·7y Related How many ways can 1,000 be written as a sum of two or more consecutive natural numbers? Three. Suppose that 1000 is the sum of natural numbers from x x to y y. Then y(y+1)2−x(x−1)2=1000 y(y+1)2−x(x−1)2=1000 ⟹(y+x)(y−x+1)=2000⟹(y+x)(y−x+1)=2000 Both terms on the LHS should be integers with their product equal to 2000 2000. Also, note that they should have opposite parity and the first number should be greater than the second. 2000=2 4 5 3 2000=2 4 5 3. Hence the combinations which work are y+x=5 3,y−x+1=2 4⟹x=55,y=70 y+x=5 3,y−x+1=2 4⟹x=55,y=70 y+x=2 4 5,y−x+1=5 2⟹x=28,y=52 y+x=2 4 5,y−x+1=5 2⟹x=28,y=52 y+x=2 4 5 2,y−x+1=5⟹x=198,y=202 y+x=2 4 5 2,y−x+1=5⟹x=198,y=202 y+x=2 4 5 3,y−x+1=1⟹x=1000,y=1000 y+x=2 4 5 3,y−x+1=1⟹x=1000,y=1000 should not be consider Continue Reading Three. Suppose that 1000 is the sum of natural numbers from x x to y y. Then y(y+1)2−x(x−1)2=1000 y(y+1)2−x(x−1)2=1000 ⟹(y+x)(y−x+1)=2000⟹(y+x)(y−x+1)=2000 Both terms on the LHS should be integers with their product equal to 2000 2000. Also, note that they should have opposite parity and the first number should be greater than the second. 2000=2 4 5 3 2000=2 4 5 3. Hence the combinations which work are y+x=5 3,y−x+1=2 4⟹x=55,y=70 y+x=5 3,y−x+1=2 4⟹x=55,y=70 y+x=2 4 5,y−x+1=5 2⟹x=28,y=52 y+x=2 4 5,y−x+1=5 2⟹x=28,y=52 y+x=2 4 5 2,y−x+1=5⟹x=198,y=202 y+x=2 4 5 2,y−x+1=5⟹x=198,y=202 y+x=2 4 5 3,y−x+1=1⟹x=1000,y=1000 y+x=2 4 5 3,y−x+1=1⟹x=1000,y=1000 should not be considered Thus there are only three solutions. More generally, if N=2 a∏i p q i i N=2 a∏i p i q i where p i p i are distinct odd primes, the answer is ∏i(q i+1)−1∏i(q i+1)−1. Upvote · 99 18 9 2 Kazi Istiak Toriqe Knows Bengali ·Updated 4y Let us take a natural number x<N x<N and We can conclude that x+x+1+x+2+……+x+k−1=N x+x+1+x+2+……+x+k−1=N That means : k x+1+2+3+….+k−1=N k x+1+2+3+….+k−1=N k x+k(k−1)2=N k x+k(k−1)2=N 2 k x+k 2−k=2 N 2 k x+k 2−k=2 N 2 x+k−1=2 N k 2 x+k−1=2 N k x=N k+1 2−k 2....(i)x=N k+1 2−k 2....(i) In the right side of equation k must be odd and divisor of N N and k>1 k>1 so that , (1-k)/2 is even and (N/k) is an integer. So number of ways = Odd divisors of (N)−1(N)−1 Upvote · 9 1 9 4 Related questions The sum of three consecutive integers is 258. How do you find the three integers? How many ways can a positive integer N be written as the sum of consecutive positive integers? In how many ways can 45 be written as the sum of consecutive positive integers? The sum of two consecutive integers is 41. What are the integers? How is the sum of two consecutive integers odd? Are there as many integers as even integers? The sum of two consecutive even integers is 126. What are the integers? What 3 consecutive even integers have a sum of -36? What is the smallest possible sum of four consecutive integers? How do you prove that every integer is a rational number? How do you prove that n can be written as a sum of (at least two) consecutive integers (number theory, integers, math)? What are all possible integers less than 30? Can you find the two integers, the sum of two consecutive even integers is 78? Can any number be written as a sum of positive integers? What is the number of ways to write down all natural numbers as sums of consecutive integers? Related questions The sum of three consecutive integers is 258. How do you find the three integers? How many ways can a positive integer N be written as the sum of consecutive positive integers? In how many ways can 45 be written as the sum of consecutive positive integers? The sum of two consecutive integers is 41. What are the integers? How is the sum of two consecutive integers odd? Are there as many integers as even integers? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
2149	https://nursehub.com/courses/hesi-a2-math-review-3/lessons/preliminary-skills/topics/translating-word-problems-into-mathematical-expressions-lesson/	Back to Course HESI A2 Math Course 0% Complete 0/0 Steps Introduction HESI A2 Math Diagnostic Test 1 Quiz HESI A2 Math Diagnostic Test (50 Questions) Topic Quizzes Preliminary Skills 5 Lessons \| 5 Quizzes Place Value Place Value Quiz (20 Questions) Positive and Negative Integers Positive and Negative Numbers Quiz (20 Questions) Translating Word Problems into Mathematical Expressions Translating Word Problems into Mathematical Expressions Quiz (20 Questions) Rounding Rounding Quiz (20 Questions) Order of Operations Order of Operations Quiz (20 Questions) Whole Numbers 3 Lessons \| 3 Quizzes Adding and Subtracting Whole Numbers Adding and Subtracting Whole Numbers Quiz (20 Questions) Multiplying & Dividing Whole Numbers Multiplying & Dividing Whole Numbers Quiz (20 Questions) Solving Word Problems with Whole Numbers Solving Word Problems with Whole Numbers Quiz (20 Questions) Decimals 3 Lessons \| 3 Quizzes Adding and Subtracting Decimals Adding and Subtracting Decimals Quiz (20 Questions) Multiplying & Dividing Decimals Multiplying & Dividing Decimals Quiz (20 Questions) Solving Word Problems with Decimals Solving Word Problems with Decimals Quiz (20 Questions) Fractions 10 Lessons \| 10 Quizzes Simplifying Fractions Simplifying Fractions Quiz (20 Questions) Expanding Fractions Expanding Fractions Quiz (20 Questions) Converting Between Mixed Numbers & Improper Fractions Converting Between Mixed Numbers & Improper Fractions Quiz (20 Questions) Adding and Subtracting Fractions Adding and Subtracting Fractions Quiz (20 Questions) Adding and Subtracting Mixed Numbers Adding and Subtracting Mixed Numbers Quiz (20 Questions) Adding and Subtracting Fractions and Mixed Numbers: Word Problems Adding & Subtracting Fractions & Mixed Numbers: Word Problems Quiz (20 Questions) Multiplying Fractions & Mixed Numbers Multiplying Fractions & Mixed Numbers Quiz (20 Questions) Dividing Fractions and Mixed Numbers Dividing Fractions & Mixed Numbers Quiz (20 Questions) Multiplying and Dividing Fractions and Mixed Numbers: Word Problems Multiplying & Dividing Fractions and Mixed Numbers: Word Problems Quiz (20 Questions) Solving Word Problems with Fractions and Mixed Numbers Solving Word Problems with Fraction & Mixed Numbers Quiz (20 Questions) Converting Between Fractions and Decimals 3 Lessons \| 3 Quizzes Converting Decimals to Fractions Converting Decimals to Fractions Quiz (20 Questions) Converting Fractions to Decimals Converting Fractions to Decimals Quiz (20 Questions) Converting Mixed Numbers to Decimals Converting Mixed Numbers to Decimals Quiz (20 Questions) Ratios and Proportions 3 Lessons \| 3 Quizzes Ratios & Proportions: Find the Missing Value Ratios & Proportions: Finding the Missing Value Quiz (20 Questions) Unit Rates Unit Rates Quiz (20 Questions) Ratios, Rates & Proportions: Word Problems Ratios, Rates, & Proportions: Word Problems Quiz (20 Questions) Percentages 5 Lessons \| 5 Quizzes Converting Decimals to Percents Converting Decimals to Percents Quiz (20 Questions) Finding Percent of a Whole Finding Percent of a Whole Quiz (20 Questions) Finding the Whole, Given the Percent Finding the Whole, Given the Percent Quiz (20 Questions) Finding Percentage Finding Percentage Quiz (20 Questions) Solving Word Problems with Percents Solving Word Problems with Percents Quiz (20 Questions) Conversions 10 Lessons \| 10 Quizzes Metric Conversions Metric Conversions Quiz (20 Questions) Standard Conversions: Distance Standard Conversions: Distance Quiz (20 Questions) Standard Conversions: Liquid Standard Conversions: Liquid Quiz (20 Questions) Standard Conversions: Weight/Mass Standard Conversions: Weight/Mass Quiz (20 Questions) Metric & Standard Conversion: Distance Metric & Standard Conversions: Distance Quiz (20 Questions) Metric & Standard Conversion: Liquid Metric & Standard Conversions: Liquid Quiz (20 Questions) Metric & Standard Conversions: Weight/Mass Metric & Standard Conversions: Weight/Mass Quiz (20 Questions) Celsius and Fahrenheit Conversions Converting Between Celsius & Fahrenheit Quiz (20 Questions) Roman Numerals Roman Numerals Quiz (20 Questions) Standard and Military Time Converting Between Standard & Military Time Quiz (20 Questions) Elementary Algebra 4 Lessons \| 4 Quizzes Evaluating Expressions: One Variable Evaluating Expressions: One Variable Quiz (20 Questions) Evaluating Expressions: Two or More Variables Evaluating Expressions: Two or More Variables Quiz (20 Questions) Solving Equations with One Variable: The Basics Solving Equations with One Variable: The Basics Quiz (20 Questions) Solving Equations with One Variable: Multi-Step Solving Equations with One Variable: Multi-Step Quiz (20 Questions) Timed Practice Test Simulations HESI A2 Math Practice Tests 4h 10m\| 5 Quizzes HESI A2 Math Test 1 (50 Questions, 50 Minutes) HESI A2 Math Test 2 (50 Questions, 50 Minutes) HESI A2 Math Test 3 (50 Questions, 50 Minutes) HESI A2 Math Test 4 (50 Questions, 50 Minutes) HESI A2 Math Test 5 (50 Questions, 50 Minutes) HESI A2 Math Course Preliminary Skills Translating Word Problems into Mathematical Expressions Topic 2, Lesson 3 In Progress ← Previous Next→ Translating Word Problems into Mathematical Expressions Topic Progress 0% Complete This content is only accessible to NurseHub Premium members. 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2150	https://tex.stackexchange.com/questions/285901/span-figure-across-margin-and-textbody-in-twoside-assigning-savebox-locally	grouping - Span figure across margin and textbody in twoside: Assigning savebox locally - TeX - LaTeX Stack Exchange Join TeX - LaTeX By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community TeX - LaTeX helpchat TeX - LaTeX Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Companies Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Span figure across margin and textbody in twoside: Assigning savebox locally Ask Question Asked 9 years, 9 months ago Modified9 years, 9 months ago Viewed 157 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. For twosidedocuments, I'd like my wide images and tables to span both the margin and the text. In my approach, I have created a command, which takes the input and stores in in a save box, measures its total height, and places in in a minipagein the margin. The minipage spans the width of the margin and the text body. The reason I had to use a saveboxwas that, since I am placing a minipagein the margin, it takes 0 vertical space in the text body, so I need to measure the height of it and place a vspace in the text to shift the material that follows down accordingly to that height. Now this works great for the first run, but as soon as I call the command again to place another wide image, I get an error. I understand why I get this error; I am trying to overwrite an existing savebox. I thought when writing the code that it would be saved only locally, not globally. Is there any way I could do that? Assign it locally, that is, or perhaps just delete it afterwards. In the following MWE, commenting out the line will give the error. latex \documentclass[11pt, twoside]{article} \usepackage{graphicx, showframe} \newlength\marginandtext \addtolength{\marginandtext}{\textwidth} \addtolength{\marginandtext}{\marginparwidth} \addtolength{\marginandtext}{\marginparsep} \newcommand{\leftsidefullwidth}{% \newsavebox\mybox \newlength\myheight \newlength\mydepth \savebox\mybox{\noindent#1} \settoheight\myheight{\usebox\mybox} \settodepth\mydepth{\usebox\mybox} \addtolength{\myheight}{\mydepth} \marginpar{\begin{minipage}{\marginandtext} \usebox\mybox \end{minipage}} \vspace{\myheight} } \begin{document} This page is left empty for this example, since it needs to be shown on left pages. \clearpage \leftsidefullwidth{\includegraphics[width=\marginandtext,height=6cm]{example-image-a}} foo %\leftsidefullwidth{\includegraphics[width=\marginandtext,height=3cm]{example-image-b}} foo \end{document} grouping Share Share a link to this question Copy linkCC BY-SA 3.0 Improve this question Follow Follow this question to receive notifications asked Jan 4, 2016 at 10:47 RunarRunar 6,232 1 1 gold badge 18 18 silver badges 40 40 bronze badges 3 Since you are taking up space in teh text flow it would be more natural to use a normal float than a marginpar and just make the image as wide as you need and shift to the right or left so that it sticks into the margin space.David Carlisle –David Carlisle 2016-01-04 10:55:15 +00:00 Commented Jan 4, 2016 at 10:55 as for the actual question always have the \newxxx command at the top level, you do not need to allocate a new box each time, just allocate a box for your package and re-use that box each time. Also you need% the ends of all those lines or your command will generate spurious white space in the text.David Carlisle –David Carlisle 2016-01-04 10:56:33 +00:00 Commented Jan 4, 2016 at 10:56 really? I thought I checked that a while ago. Guess I was completely wrong here. Okey, I guess this was a really stupid question on my part.Runar –Runar 2016-01-04 11:04:29 +00:00 Commented Jan 4, 2016 at 11:04 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 3 Save this answer. Show activity on this post. you do not need a box or measurement at all here as latex \savebox\mybox{\noindent#1} \savebox is a horizontal construct (like \mbox) so \noindent is a no-op and the box is the size of its contents. You know that size in advance as you have latex [width=\marginandtext,height=6cm] You should use a normal float not a marginpar, as the float takes space in the main text flow, then use \makebox[\textwidth][l]{begin{minipage}{\marginandtext}} on one side and [r] on the other. Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications edited Jan 4, 2016 at 11:08 answered Jan 4, 2016 at 11:03 David CarlisleDavid Carlisle 823k 74 74 gold badges 1.7k 1.7k silver badges 2.6k 2.6k bronze badges 6 Just to be clear, I wouldn't always know the size of the box (for instance a table), that was just for example images.Runar –Runar 2016-01-04 11:09:45 +00:00 Commented Jan 4, 2016 at 11:09 @runartrollet you could just use a scratch box (such as 0) for local measurement, but actually you know the width in advance (\marginandtext) and you only need the height to correct for the misuse of \marginpar, if you used a normal float you wouldn't need that vspace correction.David Carlisle –David Carlisle 2016-01-04 11:12:07 +00:00 Commented Jan 4, 2016 at 11:12 Any way to dynamically set [r]and [l]in \makebox[\textwidth][l]{begin{minipage}{\marginandtext}}depending on odd/even pages?Runar –Runar 2016-01-04 11:45:06 +00:00 Commented Jan 4, 2016 at 11:45 @runartrollet I have an answer on site somewhere about page dependent formatting in floats I'll see if I can find....David Carlisle –David Carlisle 2016-01-04 11:49:00 +00:00 Commented Jan 4, 2016 at 11:49 @runartrollet this is probably more general than you need for just odd/even but... tex.stackexchange.com/questions/118001/…David Carlisle –David Carlisle 2016-01-04 11:51:00 +00:00 Commented Jan 4, 2016 at 11:51 \|Show 1 more comment You must log in to answer this question. 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2151	https://math.libretexts.org/Bookshelves/Applied_Mathematics/Quantway_College_with_Corequisite_(Carnegie_Math_Pathways)/04%3A_Corequisite_Module_N/4.02%3A_Corequisite-_Conversion_Estimation_Percentages_and_Ratios_Scientific_Notation	4.2: Corequisite- Conversion, Estimation, Percentages and Ratios, Scientific Notation - Mathematics LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 4: Corequisite Module N Quantway College with Corequisite (Carnegie Math Pathways) { "4.2.01:_Practice_Problems_Corequisite_N.2" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } { "4.01:_Corequisite-_Estimation_Vs_Calculation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.02:_Corequisite-_Conversion_Estimation_Percentages_and_Ratios_Scientific_Notation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.03:_Corequisite-_Interest_Doubling_Time" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.04:_Corequisite-_Percentages_and_Ratios_Absolute_and_Relative_Change_Rates" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.05:_Corequisite-_Percentages_and_Taxes_Ratios_Proportions_and_Dimensional_Analysis" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.06:_Corequisite-_Dimensional_Analysis_Unit_Conversion" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.07:_Corequisite-_Buying_Power_Consumer_Price_Index_Absolute_and_Relative_Change_Reading_and_Interpreting_Graphs" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.08:_Corequisite-_Percentages_Division_Extracting_Relevant_Information" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } { "00:_Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_College_Module_N" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_College_Module_M" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_College_Module_S" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Corequisite_Module_N" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Corequisite_Module_M" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Corequisite_Module_S" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } Sat, 13 Apr 2024 01:28:39 GMT 4.2: Corequisite- Conversion, Estimation, Percentages and Ratios, Scientific Notation 148600 148600 Delmar Larsen { } Anonymous Anonymous 2 false false [ "article:topic", "showtoc:no", "license:ccbync", "licenseversion:40", "authorname:wested", "source@ ] [ "article:topic", "showtoc:no", "license:ccbync", "licenseversion:40", "authorname:wested", "source@ ] Search site Search Search Go back to previous article Sign in Username Password Sign in Sign in Sign in Forgot password Contents 1. Home 2. Bookshelves 3. Applied Mathematics 4. Quantway College with Corequisite (Carnegie Math Pathways) 5. 4: Corequisite Module N 6. 4.2: Corequisite- Conversion, Estimation, Percentages and Ratios, Scientific Notation Expand/collapse global location Quantway College with Corequisite (Carnegie Math Pathways) Front Matter 1: College Module N 2: College Module M 3: College Module S 4: Corequisite Module N 5: Corequisite Module M 6: Corequisite Module S Back Matter 4.2: Corequisite- Conversion, Estimation, Percentages and Ratios, Scientific Notation Last updated Apr 13, 2024 Save as PDF 4.1: Corequisite- Estimation Vs Calculation 4.2.1: Practice Problems Corequisite N.2 Page ID 148600 Carnegie Math Pathways WestEd ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. SPECIFIC OBJECTIVES 2. PROBLEM SITUATION: COMPARING POPULATIONS SPECIFIC OBJECTIVES By the end of this lesson, you should understand that the magnitude of large numbers is seen in place value and in scientific notation. proportions are one way to compare numbers of varying magnitudes. different comparisons may be needed to accurately compare two or more quantities. estimation is a valuable skill. standard benchmarks can be used in estimation. there are many strategies for estimating. percentages are an important quantitative concept. By the end of this lesson, you should be able to express numbers in scientific notation. use multiple computations to compare quantities. convert between millions, billions, and trillions. compute basic percentages and ratios. express one number as a percentage of another. PROBLEM SITUATION: COMPARING POPULATIONS In this problem situation, you are going to compare the populations of China and the United States, and you will compare the population of each country to the population of the world. Since the population of each of these countries is very large, you will use scientific notation to write your answers. Recall that scientific notationis a special way of writing very large or very small numbers. A number in scientific notation is written in the form: a × 10 n where 1 ≤ a < 10; and n is an integer. An integer, commonly known as a whole number, is a number that can be written without a fractional component. Below are two examples of very large numbers written in scientific notation. Examples: In 2020, the total population of Asia was approximately 4.7 billion people.2 You can write this as 4,700,000,000 or you can use scientific notation to write it as 4.7 × 109 people. 28,930,000 can be written in scientific notation as 2.893 × 107. (1) In late-2022, the population of the United States was 333,000,000.3 In late-2022, Earth’s population was about 7.9 billion. Write these numbers in scientific notation. (a) U.S. population in scientific notation: (b) World population in scientific notation: (2) What are some other ways you could compare the population of the United States to the population of the earth besides writing the two numbers in scientific notation? Think about forms of comparisons using both estimation and calculation. (3) A ratiois an expression that compares how much there is of one thing to another. Ratios are often written as fractions. In mid-2022, the population of China was 1.412 billion.4 Use a ratio to compare the population of China in 2022 to the population of Earth in 2022 (7.9 billion). Write your answer as a fraction. (4) Compare China’s population with the population of the United States using a ratio (with the U.S. population as the reference value). Write a sentence that interprets this ratio in the given context. 2 3 4 This page titled 4.2: Corequisite- Conversion, Estimation, Percentages and Ratios, Scientific Notation is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Carnegie Math Pathways (WestEd) via source content that was edited to the style and standards of the LibreTexts platform. Back to top 4.1: Corequisite- Estimation Vs Calculation 4.2.1: Practice Problems Corequisite N.2 Was this article helpful? Yes No Recommended articles 4.2.1: Practice Problems Corequisite N.2 4.1: Corequisite- Estimation Vs Calculation 4.3: Corequisite- Interest, Doubling Time 4.4: Corequisite- Percentages and Ratios, Absolute and Relative Change, Rates 4.5: Corequisite- Percentages and Taxes, Ratios, Proportions and Dimensional Analysis Article typeSection or PageAuthorCarnegie Math Pathways/WestEdLicenseCC BY-NCLicense Version4.0Show Page TOCno Tags source@ © Copyright 2025 Mathematics LibreTexts Powered by CXone Expert ® ? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. 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2152	https://www.thoughtco.com/collocations-with-money-1212318	Collocations With Money for ESL Learners Skip to content Menu Home Science, Tech, Math Science Math Social Sciences Computer Science Animals & Nature Humanities History & Culture Visual Arts Literature English Geography Philosophy Issues Languages English as a Second Language Spanish French German Italian Japanese Mandarin Russian Resources For Students & Parents For Educators For Adult Learners About Us Search Close Search the site GO Science, Tech, Math Science Math Social Sciences Computer Science Animals & Nature Humanities History & Culture Visual Arts Literature English Geography Philosophy Issues Languages English as a Second Language Spanish French German Italian Japanese Mandarin Russian Resources For Students & Parents For Educators For Adult Learners About Us Contact Us Editorial Guidelines Privacy Policy Languages› English as a Second Language› Vocabulary› Collocations With Money Print Money.Adam Gault / Getty Images English as a Second Language Vocabulary Basic Conversations for English Language Learners Pronunciation & Conversation Writing Skills Reading Comprehension Grammar Business English Resources for Teachers By Kenneth Beare Kenneth Beare English as a Second Language (ESL) Expert TESOL Diploma, Trinity College London M.A., Music Performance, Cologne University of Music B.A., Vocal Performance, Eastman School of Music Kenneth Beare is an English as a Second Language (ESL) teacher and course developer with over three decades of teaching experience. Learn about ourEditorial Process Updated on January 10, 2020 Close An important step to improving your Englishvocabulary is to not only learn the appropriate terms but to learn the words that commonly go together with those terms. These word combinations are often, adjective + noun, verb + noun, and noun + verb pairs. Each of these collocation sheets provide collocations for commonly used words arranged into categories. Each collocation is illustrated with an example sentence. Adjectives + "Money" The following list includes adjectives that are commonly used with the noun 'money'. Adjectives that are similar in meaning are grouped together. Each adjective or adjective group has an example sentence to illustrate usage. easy He thinks working in marketing is easy money. I think he'll find it's quite a different story. bonus, extraIf you complete the project before next Tuesday, there'll be some bonus money. hard-earnedThe best way to feel good about any purchase is if it's been made with hard-earned money. government, public, taxpayers'It's not right to waste taxpayers' money on projects that benefit those who are already wealthy. pocket, spendingWould you like a little extra pocket money this weekend? gas, lunch, petrol, rent, etcCould you lend me some lunch money today? prize, grant, scholarshipThey won a lot of grant money for their research into DNA. stolen, dirty, bribe, ransomI don't want your dirty money! hush, protectionThat gang is demanding protection money from every store on the street. It's scandalous! pension, retirementWe plan to move to Hawaii with our retirement money. counterfeit, fakeThe police discovered more than $2 million in fake money. Verb + "Money" The following list includes verbs that are commonly followed by the noun 'money' or an amount of a particular type of money or currency. Verbs that are similar in meaning are grouped together. Each verb or verb group has an example sentence to illustrate usage. coin, printThe government printed a lot of money in 2001. count Let's count your money and see if you have enough to buy that. bring in, earn, make,The company brought in more than $4 million. borrow Could I borrow some money for this weekend? lend I'll lend you some money until next month. bank, deposit, pay in, pay into the bank, put in the bankI deposited a large amount of money last Friday. draw out, get out, take out, withdrawShe took $500 out of our account. pay out, shell out, spendThey paid out more than $300 dollars for that lamp. fritter away, squander, throw awayI hate it when you squander our savings! hoard, save, set aside, stash awayThey set aside $200 each week for savings. contribute, donate, giveThey donated more than $200,000 to charity last year. give back, pay back, refund, repayI'll pay you back the money by the end of next week. owe She owes Thomas a lot of money. share Let's share the money we've found! accept, takeI'm afraid I can't accept your money. be worthThat painting is worth a lot of money. change, exchangeI'd like to change twenty dollars, please. Could you give me four five dollar bills? allocate, earmarkThe committee decided to allocate $50,000 for the project. channel, direct, funnelThe program directs more than $5 billion to help the homeless. embezzle, extort, siphon off, stealHe was charged with embezzling money from the company. launder They used the internet to launder the stolen money. "Money" + Verb The following list includes verbs that commonly follow the noun 'money'. Verbs that are similar in meaning are grouped together. Each verb or verb group has an example sentence to illustrate usage. come from somethingMoney for the exhibit comes from donations to the museum. go to somethingThe money goes to research. come in, flow in, pour inThe money just kept pouring in! It was amazing! buy somethingWho says that money can't buy happiness? "Money" + Noun The following list includes nouns that commonly follow the noun 'money'. Nouns that are similar in meaning are grouped together. Each noun or noun group has an example sentence to illustrate usage. management, managerI think you should hire a money manager for your savings. supply The money supply is very tight at the moment. order You can pay by money order. Phrases With "Money" The following list includes phrases made with the noun 'money'. Each phrase has an example sentence to illustrate usage. bet money on somethingLet's bet $400 dollars on the race. get money off somethingAsk if you can get some money off the display model. get your money's worthMake sure to spend the whole day at the park to get your money's worth. on the moneyYour prediction was on the money! the smart money is onThe smart money is on Tom for the director's position. throw money at somethingDon't just throw money at the project. Make sure you demand results. throw your money aroundPeter throws his money around like it meant nothing. Cite this Article Format mlaapachicago Your Citation Beare, Kenneth. "Collocations With Money." ThoughtCo, Apr. 5, 2023, thoughtco.com/collocations-with-money-1212318.Beare, Kenneth. (2023, April 5). Collocations With Money. Retrieved from Beare, Kenneth. "Collocations With Money." ThoughtCo. (accessed September 28, 2025). copy citation Sponsored Stories North Charleston New Policy for Senior Drivers thequotegeneral.com Jeff Bezos Claims the 1-Hour Rule Makes Him Smarter. I Gave It A Go.Blinkist Magazine Heart Surgeon Begs: Throw Out Your Olive Oil Now (Here’s Why)enrichyourfood.com Drivers Age 30–65 Could Save Big on Auto Insurance [Check Now]carinsurance4less.co How to Improve Your Vocabulary Collocation Examples For English Learners Top Vocabulary Building Books 5 Top English Learner Dictionaries Vocabulary Building With Collocation Lists of Sports Phrasal Verbs Reference Advanced Word Formation - B, C Advanced Word Formation - A Sponsored Stories Doctor’s Plea: Avoid Blueberries Until You Learn This thehealthyfat.com Ask A Pro: "I'm 70 with $1.4M in IRAs. 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2153	https://math.stackexchange.com/questions/4982786/filling-a-triangular-trough-from-a-large-aperture	ordinary differential equations - Filling a triangular trough from a large aperture - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Filling a triangular trough from a large aperture Ask Question Asked 11 months ago Modified11 months ago Viewed 80 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. A triangular trough of width W, length L, and vertical depth D fills with water from an aperture of width w and height h. The supply height of water exactly matches the aperture. Water flows at a constant rate Q according to Torricelli's Law. In terms of w and h, what is y the depth of the water in the trough at a given time? Here's what I've got: Q=2 3 w h 3 2 2 g−−√ by integrating vertically, according to an AI assistant but I don't think the dimensional analysis checks out partly filled, the depth is y with surface width x=W D y water volume is V(t)=L x y=L W 2 D y 2=Q t y=2 Q D t L W−−−−√ I have a hunch this is wrong. Can you help me finish please? ordinary-differential-equations Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications edited Oct 16, 2024 at 11:43 spraffspraff asked Oct 10, 2024 at 16:19 spraffspraff 1,275 11 11 silver badges 23 23 bronze badges 6 Why the AI assistance?Cesareo –Cesareo 2024-10-15 10:43:54 +00:00 Commented Oct 15, 2024 at 10:43 There seems to be some missing data here. How far, H, does water extend above the aperture? You need this to apply Torricelli's law, in which case the rate law you have is almost correct. It would be 2 3 w((h+H)3/2−H 3/2)2 g−−√. The reason why it looked like the dimensional analysis was wrong was because you inputted the Newtonian gravitational constant (m 3⋅kg−1⋅s−1), rather than the gravitational acceleration (m⋅s−2).Jacob –Jacob 2024-10-15 19:22:15 +00:00 Commented Oct 15, 2024 at 19:22 The incoming water exactly fills the aperture.spraff –spraff 2024-10-16 11:41:34 +00:00 Commented Oct 16, 2024 at 11:41 So if H=0 then the formula is correct? Dimensions of Q would then be L.L 3 2 L.T−2−−−−−−√=L 1+3 2+−1 2 T−1=L 2.T−1 but it should be L 3.T−1, right?spraff –spraff 2024-10-16 11:49:50 +00:00 Commented Oct 16, 2024 at 11:49 @spraff You subtracted 1 2 from the exponent of L when you should have added. Correcting that gives the L 3 term you need.Jade Vanadium –Jade Vanadium 2024-10-16 15:05:21 +00:00 Commented Oct 16, 2024 at 15:05 \|Show 1 more comment 0 Sorted by: Reset to default You must log in to answer this question. 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2154	https://nsidc.org/learn/ask-scientist/how-ice-shelf-disintegration-different-calving	How is ice shelf disintegration different from calving? \| National Snow and Ice Data Center National Snow and Ice Data Center NSIDC a part of CIRES at the University of Colorado BoulderSkip to main content Search Close menu Menu News & AnalysesToggle submenu News & Stories Sea Ice Today Ice Sheets Today Snow Today About our Analyses Multimedia DataToggle submenu Explore Data Visualize Data Submit DataToggle submenu Submit NASA Data to NSIDC DAAC Submit Data to Other NSIDC Programs User ResourcesToggle submenu Get Started with Data Data Announcements Help Center Data Tools Documents Levels of Service NASA Earthdata Forum Data ProgramsToggle submenu About our Programs NASA National Snow and Ice Data Center Distributed Active Archive Center (NSIDC DAAC) NOAA at NSIDC Exchange for Observations and Local Knowledge of the Arctic (ELOKA) Data Policies Our Research LearnToggle submenu What is the Cryosphere? 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Parts of the Cryosphere Arctic Weather & Climate Frozen Ground & Permafrost Glaciers Ice Sheets Ice Shelves Sea Ice Snow Ask a Scientist Cryosphere glossary About About NSIDC What We Do Our People Published Research Our History Careers For the Media Contact Us Citation Policies Protecting Your Privacy Land Acknowledgement Search Share News & Analyses News & Stories Sea Ice Today Ice Sheets Today Snow Today About our Analyses Multimedia Data Explore Data Visualize Data Submit Data Submit NASA Data to NSIDC DAAC Submit Data to Other NSIDC Programs User Resources Get Started with Data Data Announcements Help Center Data Tools Documents Levels of Service NASA Earthdata Forum Data Programs About our Programs NASA National Snow and Ice Data Center Distributed Active Archive Center (NSIDC DAAC) NOAA at NSIDC Exchange for Observations and Local Knowledge of the Arctic (ELOKA) Data Policies Our Research Learn What is the Cryosphere? Parts of the Cryosphere Arctic Weather & Climate Frozen Ground & Permafrost Glaciers Ice Sheets Ice Shelves Sea Ice Snow Ask a Scientist Cryosphere glossary About About NSIDC What We Do Our People Published Research Our History Careers For the Media Contact Us Citation Policies Protecting Your Privacy Land Acknowledgement Home Learn Ask a Scientist How is ice shelf disintegration different from calving? < Previous article Next article > In this section Calving: a natural process Disintegration: new and disconcerting Calving: a natural process Disintegration: new and disconcerting Ask a Scientist How is ice shelf disintegration different from calving? Last Updated: THURSDAY, SEPTEMBER 2, 2021 Ice shelves are thick slabs of ice that cling to cold-region coastlines. Although some ice shelves form through the accumulation of sea ice, the world’s largest ice shelves are fed by glaciers. When a glacier reaches the coast, the ice floats on near-freezing water and continues to spread outward. Two kinds of events have been observed in ice shelves: iceberg calving and disintegration. Calving is a natural phenomenon, but disintegration is a new phenomenon, likely linked to climate change. Calving: a natural process The breaking off of icebergs, known as calving, is a normal part of the life of a glacier-fed ice shelf, even when the icebergs calved are massive. When a large iceberg calves off an ice shelf in a stable glacier-fed ice shelf system, the lost ice is replaced over the course of years to decades by continuing outflow. Only when the pace of glacier calving accelerates significantly over an extended period of time is the system considered to be in retreat. Operation IceBridge snapped this photo of two tabular icebergs off the Antarctic Peninsula on October 16, 2021.— Credit:NASA Disintegration: new and disconcerting In contrast to iceberg-calving events, disintegration is a newly observed phenomenon, only documented since the 1990s. Since then, studies have indicated that ice shelf disintegration results from warming conditions, which stress ice shelves in multiple ways. One way that warmer conditions destabilize ice shelves is through higher air temperatures. Ice shelves are typically blanketed by snow, which contributes to the ice shelf’s mass over time. Warmer summer air temperatures cause snow cover to melt, changing the permeable snow surface into an impermeable ice surface—the same way a few warm days can turn snow in a sidewalk gutter into ice. In subsequent summers, meltwater on the ice shelf does not sink into the snow. Instead, it pools on the shelf surface. The water can find its way into pre existing cracks in the ice and, because water is denser than ice, slice deeper into the ice shelf. Eventually, the water can cut completely through the ice. Another way that warming conditions destabilize ice shelves is through ocean warming. As warm ocean water slithers under the ice shelf, it melts the ice from below, thinning the ice shelf. Warmer ocean water can also contribute to sea ice decline. As sea ice diminishes, wave action in the region increases. Now unfettered by sea ice, waves can flex and bend the shelf, stressing it further. One or more of these pressures can lead to ice shelf disintegration. Once disintegration starts, it may progress quickly. Some ice shelves change so dramatically that the changes are visible from space in a matter or days or even hours. Situated on the west coast of the Antarctic Peninsula, the Wilkins Ice Shelf underwent a rapid disintegration in early 2008, collapsing in a matter of hours. Moderate Resolution Imaging Spectroradiometer (MODIS) sensors on NASA’s Terra and Aqua satellites captured these images of the disintegration on February 28, 2008 at 14:00 UTC (left); February 28, 2008 at 21:00 UTC (center); and February 29, 2008 at 13:05 UTC (right).— Credit:NASA Ice shelf disintegration not only has the potential to happen quickly, it also has the potential to destabilize the glacier(s) feeding the ice shelf. In a glacier-fed ice shelf system, the point where glacier ice begins to float is known as the grounding line, and it is an important part of ice shelf stability. If an ice shelf retreats past the grounding line, the tributary glacier(s) begin flowing faster. This process can eventually contribute to sea level rise as glaciers slide from land into the water. For an example of how this process develops, read about the aftermath of the Larsen B Ice Shelf disintegration. In short, glacier calving is a normal, natural process. Ice shelf disintegration is not. 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2155	https://whzecomjm.com/p/2018/10/trigonometric-identities/	三角恒等式的几何表达 🍥 Transcendence whzecomjm's blog Home About Archives Links Search Dark Mode 组合几何三角恒等式的几何表达 trigonometric identities Oct 07, 2018 1 minute read 高中数学记忆三角恒等式一直是很多人苦于去做的，虽然有 sinco cosin coco sinsin 之类的记忆“口诀”，但是过了一段时间不复习中间的细节比如两倍、正负号还是容易忘记的。下面我们来梳理一下这些恒等式的几何化表达。角和差三角恒等式角和差三角恒等式是所有公式的基础，容易记忆、也容易得到几何化的表达。正余弦的角和公式 sin⁡(α+β)=sin⁡α cos⁡β+cos⁡α sin⁡β cos⁡(α+β)=cos⁡α cos⁡β−sin⁡α sin⁡β \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\ \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta sin(α+β)=sin α cos β+cos α sin β cos(α+β)=cos α cos β−sin α sin β 上述几何图示是 α+β<π 2\alpha+\beta<\frac{\pi}{2}α+β<2 π 的情况, 其余情况也能画出类似的图示, 只不过要注意正负号. 例如对于 α+β>π 2\alpha+\beta>\frac{\pi}{2}α+β>2 π, 且都为锐角情况, 我们有如下图 (图好像画歪了 XD): 正切的角和公式和差化积恒等式下面是和差化积的一个几何表达，角BAE为 α\alpha α，角 C A E CAE C A E 为 β\beta β，A C=A B=1 AC=AB=1 A C=A B=1，D D D 是等腰三角形底边中点。上图可以很好地表达和差化积公式，比如 sin⁡α+sin⁡β=2 sin⁡α+β 2 cos⁡α−β 2.\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}.sin α+sin β=2 sin 2 α+βcos 2 α−β. 上式左边两个sin是B、C两点到边 l l l 的距离，乘以 ∣l∣\|l\|∣l∣ 则是三角形ABC面积的两倍；等式右边等于 2∣C D∣×∣h∣∣l∣, 2\|CD\|\times\frac{\|h\|}{\|l\|}, 2∣C D∣×∣l∣∣h∣, 是三角形ABC面积的两倍除以 ∣l∣\|l\|∣l∣。故两边相等。半角公式此图一目了然。参考文献维基百科三角恒等式半角公式几何化来自微博进一步的内容: 数学无字证明——三角函数. 三角恒等式Licensed under CC BY-NC-SA 4.0 Related content 每周定理 ---- 向量法求根式函数值域 ---------- 幂平均不等式和对称法 ---------- 拉姆齐数一个下界的概率证明 ------------- 射影平面的直观理解 --------- © 2014 - 2023 Transcendence Built with Hugo Theme Stack designed by Jimmy Table of contents 角和差三角恒等式正余弦的角和公式正切的角和公式和差化积恒等式半角公式参考文献
2156	https://www.verywellmind.com/what-is-episodic-memory-2795173	What Is Episodic Memory? Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book." Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk, "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time. An episodic memory is a memory of a specific event. Because each person has a different perspective and experience of an event, episodic memories of that event are unique to each person. Closely related to this is what researchers refer to as autobiographical memory—memories of your own life history. As you can imagine, episodic and autobiographical memories play an important role in your self-identity. At a Glance The memories of specific events and experiences, such as your high school graduation or wedding day, are examples of episodic memory. It includes your memories of specific events, personal facts, general events, and snapshots of particular moments in time. Such memories play an important role in shaping who you are but can be affected by certain types of damage or disease. How Episodic Memory Works Imagine that you get a phone call from an old college friend. You spend an evening reminiscing about amusing moments from your days at school. Your memories of those specific events and experiences are examples of episodic memory. Episodic memories are important for shaping your personal sense of identity. These memories provide you with a sense of personal history and a shared history with others in your life. Episodic memories are important because they allow you to recall personal experiences that helped shape your life and perceptions. Areas of the Brain Involved in Episodic Memory Several regions of the brain have been implicated in episodic memory. These regions of the brain are part of what is known as the default mode network. These areas include: The hippocampus and parts of the brain found in the temporal lobe are also important in encoding, consolidating, and retrieving episodic memories. Several parts of the brain work in concert to help you store and retrieve episodic memories: the temporal, parietal, and frontal cortices, diencephalon, and cerebellum. Practice, genetics, and experience all influence your ability to recall episodic memories. Episodic vs. Semantic Memory The term episodic memory was first introduced by Endel Tulving in 1972 to distinguish between remembering events from the past (episodic memory) and knowing factual information, known as semantic memory. Semantic memory is focused on general knowledge about the world and includes facts, concepts, and ideas. Episodic memory and semantic memory are part of the division of memory known as explicit or declarative memory. Semantic and Episodic Memory Are Interdependent Researchers have found that episodic memory can also be interdependent with semantic memory. On learning tasks, participants performed better when new information was aligned with prior knowledge, suggesting that semantic knowledge of a task provides a sort of framework for new episodic learning. Participants were asked to remember the prices of grocery items. Those in the control group were better able to remember these prices when the new information was congruent with their existing episodic memories of grocery prices. Amnesiac participants in the experimental group, however, performed much worse at remembering new information because they did not have access to episodic information from their past. Researchers have also found that episodic memories also play a role in the retrieval of semantic memories. In experiments where participants were asked to generate lists of items in particular categories, those who were able to rely on episodic memories performed better than amnesiac participants who did not have access to episodic memories. Types of Episodic Memories Episodic memories fall into a few main categories. Specific Events These involve memories of particular moments from personal history. Your first kiss, first day of school, a friend's birthday party, and your brother's graduation are all examples of episodic memories. In addition to your overall recall of the event itself, the episodic memory include the locations and times of the events. Personal Facts Knowing who was president the year that you got married, the make and model of your first car, and the name of your first boss are all examples of personal fact episodic memories. General Events Remembering what a kiss feels like is an example of this general type of memory. You do not remember each and every kiss you've ever shared, but you can recall what it feels like based on your personal experiences. Flashbulb Memories Flashbulb memories are vivid and detailed "snapshots" related to finding out particularly important news. Sometimes these moments might be highly personal, like the moment you found out that your grandmother had died. In other cases, these memories might be shared by many people in a social group. The moments you found out about the 9/11 attacks or the Paris concert theater attacks are examples of shared flashbulb memories. Examples of Episodic Memories Some examples of episodic memories might include: Remember, each person's episodic memory of an event is entirely unique. Even other people who shared the same experience may have different recollections of what happened. Episodic memories can be of important details or events from your life (like the day you met your partner or your social security number). They also include much more mundane information. Your memory of a particular the year you attended your first concert or of what you had for breakfast yesterday morning are also examples of episodic memory. Nine Properties of Episodic Memory Research indicates that episodic memory carries nine attributes. Other types of memory feature some of these, too—but only episodic memories comprise all of them. Episodic memories: Studies also suggest that there are sex differences in episodic memory. Research has found, for example, that women tend to outperform men on tests of episodic memory function, particularly on verbal-based episodic memory. Studies also show that women can access these memories faster and date them more accurately than men. Impact of Episodic Memory Episodic memory has a profound effect on your life. Tulving described it as a "mental time machine." Anytime you revisit a moment from your past, whether you're remembering your first day of kindergarten, what you wore to work yesterday, or some other personal detail, you're drawing on your episodic memory. Such memories help create a cohesive experience. They allow you to form a personal history that plays a role in your identity and sense of self. How Damage and Disease Can Affect Episodic Memory Aging and neurodegenerative diseases take an extreme toll on episodic memory. For example, a decline in the ability to retrieve this kind of memory is among the first signs of Alzheimer’s disease. Other psychiatric conditions that can cause deficits in episodic memory: However, such damage is difficult to assess because memories are difficult to induce and measure. Likewise, traumatic brain injury, such as concussion, tends to impede episodic memory. Semantic memory appears to be less susceptible to this kind of damage. What This Means For You Episodic memory is important because it creates the fabric of your life history. All of the tidbits of knowledge, memories of things that have happened, and details about your personal experiences are also stored in your episodic memory. It's a bit like a mental history that allows you to revisit your paster and create a cohesive narrative about who you are and the life you've lived so far. If you are experiencing problems with episodic memory, it is important to talk to your doctor for further evaluation and treatment. Brandman T, Malach R, Simony E. The surprising role of the default mode network in naturalistic perception. Commun Biol. 2021;4(1):79. doi:10.1038/s42003-020-01602-z Byrne JH. Learning and Memory: A Comprehensive Reference. Elsevier Science. Renoult L, Rugg MD. An historical perspective on Endel Tulving's episodic-semantic distinction. Neuropsychologia. 2020;139:107366. doi:10.1016/j.neuropsychologia.2020.107366 Kan IP, Alexander MP, Verfaillie M. Contribution of prior semantic knowledge to new episodic learning in amnesia. J Cogn Neurosci. 2009;21:938-944. doi:10.1162/jocn.2009.21066 Tulving E. Episodic memory: from mind to brain. Annu Rev Psychol. 2002;53(1):1-25. doi:10.1146/annurev.psych.53.100901.135114 Conway MA. Episodic memories. Neuropsychologia. 2009;47(11):2305-2313. doi:10.1016/j.neuropsychologia.2009.02.003 Lundervold AJ, Wollschlager D, Wehling E. Age and sex related changes in episodic memory function in middle aged and older adults. Scand J Psychol. 2014;225-32. doi: 10.1111/sjop.12114 Loprinzi PD, Frith E. The role of sex in memory function: Considerations and recommendations in the context of exercise. J Clin Med. 2018;7(6):132. doi:10.3390/jcm7060132 Matthews BR. Memory dysfunction. Continuum (Minneap Minn). 2015;21(3 Behavioral Neurology and Neuropsychiatry):613-626. doi:10.1212/01.CON.0000466656.59413.29 Pause BM, Zlomuzica A, Kinugawa K, Mariani J, Pietrowsky R, Dere E. Perspectives on episodic-like and episodic memory. Front Behav Neurosci. 2013;7. doi:10.3389/fnbeh.2013.00033 Wammes JD, Good TJ, Fernandes MA. Autobiographical and episodic memory deficits in mild traumatic brain injury. Brain Cogn. 2017;111:112-126. doi:10.1016/j.bandc.2016.11.004 Greenberg DL, Keane MM, Verfaellie M. Impaired category fluency in medial temporal lobe amnesia: The role of episodic memory. J Neurosci. 2009;29(35):10900-10908. doi:10.1523/JNEUROSCI.1202-09.2009 By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book." Thank you, {{form.email}}, for signing up. There was an error. Please try again.
2157	https://www.mathros.net.ua/en/isosceles-trapezoid.html	Mastering the Isosceles Trapezoid: A Step-by-Step Guide Skip to content www.mathros.net.ua Site for students majoring in computer science ua Menu Home Sitemap Online Tools Isosceles Trapezoid: Definition, Properties, and Examples 15/02/2025Rostyslav VereshchakPolygons An isosceles trapezoid is a special kind of trapezoid whose legs (the non-parallel sides) are equal in length. It has exactly one pair of parallel sides, known as the bases. Thanks to its symmetry and unique geometric features, the isosceles trapezoid frequently appears in geometry problems. But how can you recognize this shape, and what are its most important characteristics? Let’s explore these questions step by step. Isosceles Trapezoid: What It Is and How It Looks An isosceles trapezoid is a quadrilateral with: One pair of parallel sides (called the bases). Two legs that are equal in length. The equal legs give this trapezoid its special symmetry. There is a line of symmetry that runs through the midpoints of both bases and is perpendicular to them. This symmetry sets the isosceles trapezoid apart from the general trapezoid, in which the non-parallel sides can be any length. A helpful way to visualize this shape is to see it as part of an isosceles triangle. If you draw a line parallel to the base of that triangle, the enclosed portion is an isosceles trapezoid. Geometric Properties of the Isosceles Trapezoid: Proofs and Explanations An isosceles trapezoid has several important properties that make it especially interesting. It’s symmetrical, and its angles and diagonals have unique relationships. Let’s look at these properties in more detail. Equal Angles at Each Base One defining feature of an isosceles trapezoid is that the angles adjacent to each base are equal. This means: The angles at the longer base are equal. The angles at the shorter base are also equal. Why is this True? Consider an isosceles trapezoid ABCD, where AD and BC are the bases, and AB and CD are the equal legs. Now, draw a line from C parallel to AB, intersecting AD at point K. Since CK is parallel to AB, the quadrilateral ABCK forms a parallelogram. In a parallelogram: Opposite sides are equal, so CK=AB. By the isosceles trapezoid condition, AB=CD, which means CK=CD. This shows that triangle CKD is isosceles, and its base angles are equal. Thus, ∠A=∠D. Similarly, the sum of angles on the same leg must be 180°, proving that ∠B=∠C. Equal Diagonals Another special property of an isosceles trapezoid is that its diagonals are always equal. Proof Sketch: In trapezoid ABCD, we need to prove that diagonal AC equal to diagonal BD. If we look at triangles ABD and ACD, we can see that: AD is a common side. AB=CD (equal legs). ∠BAD=∠ADC (from the previous property about equal angles). Since these two triangles have two equal sides and the included angle is the same, they are congruent by the SAS (Side-Angle-Side) criterion. This means their corresponding sides are also equal, so AC=BD. Key takeaway: If the diagonals of a trapezoid are equal, then it must be isosceles! Isosceles Trapezoid as Part of an Isosceles Triangle A cool way to look at an isosceles trapezoid is by considering it as part of a larger isosceles triangle. If you extend the equal legs, they will meet at a point, forming an isosceles triangle with the longer base of the trapezoid. Why is this Interesting? If you extend AB and CD, they intersect at point K. The triangle AKD is isosceles because AB=CD. If you draw a perpendicular KL from K to AD, it acts as a line of symmetry for both the trapezoid and the triangle. This explains why an isosceles trapezoid maintains symmetry and is closely related to an isosceles triangle. Practical Problems: Applying Isosceles Trapezoid Properties To gain a better grasp of the isosceles trapezoid, it’s helpful to apply these properties in practice. Here are a few examples to help you solidify your understanding and build problem-solving skills. Example 1: In an Isosceles Trapezoid ABCD, the Heights From Vertices B and C Onto the Base AD Intersect It at Points K and L, Respectively. Find the Lengths of AK and LD If AD=15 cm and BC=5 cm Since the trapezoid is isosceles, the heights BK and CL divide it into three figures: two congruent right triangles ABK and CLD, plus a rectangle KBCL. In a rectangle, opposite sides are equal, so KL=BC=5 cm. To find segments AK and LD: So, AK=LD=5 cm. Example 2: In an Isosceles Trapezoid ABCD, ∠ABC is Three Times Larger than ∠BAD. Find All the Angles of the Trapezoid The sum of the angles adjacent to the same leg in an isosceles trapezoid is 180°. Let ∠BAD=x. Then, according to the problem, ∠ABC=3⋅x. We have: Since the trapezoid is isosceles, angles at the same base are equal: Hence, the angles of the trapezoid are 45°, 135°, 45°, 135°. Example 3: The Height of an Isosceles Trapezoid Is 10 cm, and Its Diagonals Intersect at a 60° Angle. Find the Length of the Trapezoid Diagonals Let O be the intersection point of the diagonals. In an isosceles trapezoid, triangle AOD is isosceles, meaning AO=OD. The angle ∠COD=60° is an external angle to triangle AOD. This makes the angles ∠OAD and ∠ODA each 30°. Now, consider the right triangle BKD, where BK is the trapezoid’s height. The angle ∠BDK=30°. Using the relationships in a 30°-60°-90° right triangle, the hypotenuse BD is twice the shorter leg BK: Thus, BD=20 cm. And since an isosceles trapezoid has equal diagonals, AC=BD=20 cm. Keep Learning: More Resources to Boost Your Geometry Skills Mastering the isosceles trapezoid is just the beginning! If you want to expand your knowledge and build even more confidence in solving geometry problems, here are a few key topics worth exploring: Midsegment (Median) of a Trapezoid – Learn how to calculate the midsegment (sometimes called the median) of a trapezoid and see how to apply this in practice. Perimeter of a Trapezoid – Discover quick methods for finding the perimeter, which is often crucial for solving more complex geometry questions. Area of a Trapezoid – Explore various ways to calculate the area and find practical examples demonstrating each method. Diving into these topics will strengthen your understanding, sharpen your problem-solving skills, and prepare you for even more challenging geometry concepts. Keep practicing, and soon, you’ll be tackling any trapezoid problem with confidence! Isosceles Trapezoid in Programming: How to Check? Modern technology makes it easier than ever to explore geometry through programming. If you enjoy both math and coding, why not combine them by writing a simple program to identify an isosceles trapezoid? By using logical conditions and geometric properties, you can create an algorithm that quickly determines whether a given quadrilateral meets the criteria for being an isosceles trapezoid. The flowchart above provides a great starting point for your program. Your code should check whether exactly one pair of opposite sides is parallel and whether the other two sides are equal. Based on these conditions, the program should output either “Yes, this is an isosceles trapezoid” or “No, this is not an isosceles trapezoid”. Try implementing this logic in your favorite programming language and see how quickly your algorithm can classify different quadrilaterals! geometry basicsIsosceles TrapezoidTrapezoid GeometryTrapezoid PropertiesTrapezoid Types Post navigation Previous Post:Midsegment of a Trapezoid: Definition, Properties, and Examples Next Post:What is Python? 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2158	https://www.echemi.com/community/the-kb-for-hydrazine-n2h4-g-a-rocket-fuel-is-1-7x10-6-what-is-the-ka-of-its-conjugate-acid-n2h5_mjart2204108625_157.html	The Kb for hydrazine, N2H4(g) a rocket fuel, is 1.7x10—6. What is the Ka of its conjugate acid, N2H5 +? - ECHEMI  Community  Sign in \| Join free Home>Community>The Kb for hydrazine, N2H4(g) a rocket fuel, is 1.7x10—6. What is the Ka of its conjugate acid, N2H5 +? Upvote 17 Downvote Acids Chemistry Posted by Michael Bottenfield The Kb for hydrazine, N2H4(g) a rocket fuel, is 1.7x10—6. What is the Ka of its conjugate acid, N2H5 +? C6H5NH3+ Doctor LeeFollowFollowing C6H5NH3+ More Upvote VOTE Downvote [math]pK_{a}+pK_{b}=14[/math], you have [math]K_{a}[/math], do we need to go on? Antonio MoroniFollowFollowing [math]pK_{a}+pK_{b}=14[/math], you have [math]K_{a}[/math], do we need to go on? More Upvote VOTE Downvote Related Posts How many ml are in 1 glass of water? 26 Upvotes · 2 Comments What is the best solution for cleaning a meth pipe? 15 Upvotes · 3 Comments What is dicyanin, and what is it used for? 14 Upvotes · 3 Comments How do you prepare 0.5M of an HCl solution? 14 Upvotes · 8 Comments Is it safe to use expired salt packets from neti pots for nasal irrigation? 19 Upvotes · 2 Comments How do I prepare a 1M KOH solution? 16 Upvotes · 8 Comments How do I make a 1N H2SO4 solution? 15 Upvotes · 10 Comments Wholesale About ECHEMI Encyclopedia Contact Us Local Mall Partners Market Price & Insight Marketing Kit Trade Data ECHEMI Group Join Us News Sitemap For Buyers For Suppliers Copyright@Qingdao ECHEMI Digital Technology Co., Ltd. TOP
2159	https://kknews.cc/education/gzr98y9.html	初中數學：五道常見一元一次不等式考試題型解題技巧 - 每日頭條每日頭條 ==== 首頁健康娛樂時尚遊戲 3C 親子文化歷史動漫星座健身家居情感科技寵物初中數學：五道常見一元一次不等式考試題型解題技巧 2017-12-02由方老師數學課堂發表于教育方老師感謝大家來到並主動關注方老師數學課堂。一元一次不等式的考試題目一般都是很求取值範圍有關。熟練掌握和運用不等式的基本性質，是解題的基礎關鍵點。 ... close Advertisements custom_chain_chinese_zodiac[webstory]-new-20250922-22:01 CANCEL NEXT VIDEO Pause Play % buffered 00:00 00:00 00:00 Unmute Mute Play Powered by GliaStudios 例1，解得二元一次方程組分別到x和y的，用含K的代數式表示。然後因為x大於1，y小於或等於1,得到一個元一次方程組就，解得K的取值範圍。然後K是整取，在這個範圍內取整數值就可以了。 ... 例題2，這是考查等式基本性質的常見題型。等式兩天同時乘以一個數或者式子，不等號的開口方向改變了。那麼這個數字或者式子肯定是個負數。此題得解。 ... 例題3，先解得原一元一次不等式，得到x大於-3，然後得到x的最小整數解釋x=-2。然後帶入第二個方程，解得a的值。深入瞭解星座運勢諮詢親子教育課程購買中國排球聯賽門票排球裝備親子旅遊文化講座科技產品媒體公關服務阿根廷探戈課程馬可波羅展覽門票 ... 例題4，這個題和例題1類似，先解得一元一次不等式組，得到解集。然後套用解集的數值。分別求出m和n的值。把m和n的值帶入，求值結果。 ... 例題5，這個題目字數不多，但是初一看，也挺不容易的。其實，仔細看看，就會發現x的係數肯定是一個負數啊，那麼不等式的兩邊同時除以一個負數，不等號的開口方向改變就好了。總結起來，關於不等式的題目不外乎就是不等式的基本性質的知識點。特別要留意兩邊同時乘以或者除以一個負數的時候，不等號的開口方向要改變。點我分享到Facebook 深入瞭解馬可波羅展覽門票明星八卦雜誌親子教育課程購買寵物用品線上商店媒體公關服務媒體會員家居裝飾設計服務科技產品 3C產品澳洲旅遊保險購買相關文章人教版中考數學不等式及不等式組知識點總結（3大知識點+例題） 2017-12-21 不等式及不等式組知識點：一、不等式與不等式的性質 1、不等式：表示不等關係的式子。（表示不等關係的常用符號：≠，＜，＞）。初中數學：不等式必須要掌握的基礎知識點（考試必備） 2017-07-27 同學們，大家好，今天笑老師給大家講解的是初中數學不等式的相關知識點首先要了解一下是用大於號或小於號連接起來表示大小關係的式子，這樣的式子叫做不等式，例如x+1＞6，3x+8≤7。那什麼叫做不等式的解呢，使不等式成立的未知數的值，叫做不等式的解。代數複習——不等式及不等式組 2018-01-28 同一個數，不等號方向不改變，如a＞b，c為實數，則a＋c＞b＋c。同一個負數，不等號方向改變，如a＞b，c＜0，則ac＜bc.。乾貨來襲數學加帶你巧妙解開不等式（初一）上篇 2016-09-07 前幾天，數學加講解了七年級的重點知識點有理數，今天帶大家走進「不等式與不等式組」的花樣世界裡吧！在這裡你可以學習到：不等式的概念、不等式的基本性質、一元一次不等式、一元一次不等式組、含字母係數的不等式、不等式的應用、易錯點分析以及相關的練習等多方面的內容學習，以讓你牢牢掌握好它！一元一次不等式/不等式組的解法 2016-06-23 文章來源：呼和浩特中小學數學（hhhtshuxue），謝謝合作！一、知識梳理1知識結構圖二、知識點回顧1不等式用不等號連接起來的式子叫做不等式。「七年級」含字母已知數的一元一次不等式（組） 2018-07-15 如果不等式x＞a+1的解集為x＜1，則a的取值範圍是（）A、a＜0B、a＜1C、a＜-1D、a＞1我們知道，當不等式的兩邊同時除以一個負數時，不等號的方向要改變；反過來呢。一元一次不等式基礎知識上的誤區 2018-05-16 為了更好的幫助大家，整理了有關基礎知識容易犯錯的方面。不等式的定義像x>3，5<6，這樣用符號">"、"<"表示大小關係的式子，叫不等式。初中數學：一元一次不等式有如此特別之處 2017-10-14 一般的，用符號「<」（或「≤」），「>」（或「≥」）,「≠」連接的式子叫做不等式。一元一次不等式（組）：已知解集求參數（字母）題型（初一數學） 2017-03-24 在不等式中經常會遇到這類題：已經知道不等式（組）的解集或解的情況，來求不等式（組）中含有的某個未知字母的範圍！那這類題我們可以怎麼考慮？其實,萬變不離其宗，這類題我們把參數作為「已知的數字」，仍可先得到不等式（組）的解集，藉助於數軸來考慮。極客數學幫講初中方程的解知識點，開學手冊拿好了 2017-08-31 方程的解是使方程兩邊左右相等的的一個未知數，方程的解不唯一，解方程是要注意很多問題。今天極客數學幫就來總結歸納一下關於方程的解中有哪些知識點。同學們都來看看吧。七下不等式與不等式組知識點及精選計算 2018-02-06 知識點1：不等式的概念像2x＜3，x＞50這樣用符號「＜」或「＞」表示大小關係的式子，叫不等式.像a+2≠a-2這樣用「≠」表示不等關係的式子也是不等式.知識點2：不等式的解和解集我們曾經學過使方程兩邊相等的未知數的值就是方程的解，我們也可以把使不等式成立的未知數的值叫做不等式的初中數學一元一次不等式 2018-07-21 1．不等式的基本性質通過對比不等式和方程的性質，使學生學會用類比的方法看問題。性質1：不等式的兩邊同時加上同一個數或同一個整式，不等號方向不改變。初中數學：不等式的最常見易錯點是它 2017-07-28 今天的笑老師跟大家分享一下不等式的性質的話題。人教版七年級數學下冊不等式與不等式組複習提綱 2018-06-03 9.1不等式 1、用小於號或大於號表示大小關係的式子，叫做不等式。 3、能使不等式成立的x的取值範圍，叫做不等式的解的集合，簡稱解集。七年級（數學）下重要內容詳解五！ 2018-04-12 ②把二元一次方程組中一個方程的一個未知數用含另一個未知數的式子表示出來，再代入另一個方程，實現消元，進而求得這個二元一次方程組的解，這種方法叫做代入消元法，簡稱代入法。初中數學一元一次不等式和一元一次不等式組知識點總結及練習題解 2017-11-23 一元一次不等式和一元一次不等式組知識點總結：一. 不等關係1. 一般地,用符號「<」(或「≤」), 「>」(或「≥」)連接的式子叫做不等式.2. 區別方程與不等式：方程表示是相等的關係，不等式表示是不相等的關係。3. 準確「翻譯」不等式,正確理解「非負數」、「不小於」等數學術語. 初中數學：不等式的性質，很容易考的地方在哪裡 2017-10-13 不等式性質這樣講要達到哪些目標呢？1. 了解不等式、不等式的解集的概念，會在數軸上表示不等式的解集。2. 掌握不等式的三條基本性質，並會用它們解一元一次不等式。這一講的重點和難度在哪裡？1. 重點：不等式的基本性質和一元一次不等式的解法。2. 2018中考數學 —— 一元一次不等式（組） 2018-03-15 基本性質1、不等式兩邊都加上同一個，即：若a<b,則a+c<b+c基本性質2：不等式兩邊都乘以同一個，即：若a0則ac<bc基本性質3、不等式兩邊都乘以同一個，即：若a<b，c<0則ac>bc【名師提醒：運用不等式的基本性質解題時要主要與等式基本性質的區別與聯繫，特別強「等式的性質」與「不等式的性質」的區別和聯繫，你掌握了嗎？ 2018-06-12 大家好，學完不等式的性質，夥伴們是不是自然而然地就想起了等式的性質；是的，今天，小隴哥就帶大家重溫等式的性質與不等式的性質的異同點。「中考專題」不等式與不等式組（04） 2017-10-25 之前三篇我們學了實數，代數，解方程。今天我們說下不等式和不等式組，我先大概說下比較重要的幾點1，我們要會解簡單的不等式，這裡指單個不等式2，單個不等式的解集要會表現在數軸上3，要會解不等式組，並把解集表示出來4，對於應用題，要學列不等式。求出最大利潤，最佳方案等等。初中數學二次根式計算難點題型：無理數整數部分和小數部分再求值 2017-11-30 這個題目是一個初三同學發給我的題目，說方老師這個題目怎麼做啊？這個題目考察了兩個知識點，首先要對這個無理數進行估值，然後得出它的整數部分，小數部分就是用它本身去減整數部分就可以了。衝刺2018年高考數學, 典型例題分析21：絕對值相關的綜合題型 2018-04-01 f的圖象可以圍成一個三角形，由直線過可得k=3/5，由直線過可得k=3/8。∴3/8＜k≤3/5．考點分析：絕對值不等式的解法．利用不等式性質可以求某些代數式的取值範圍，但應注意兩點：一是必須嚴格運用不等式的性質。高分必備 \| 數學易錯知識點匯總（一） 2018-04-02 一、數與式易錯點1有理數、無理數以及實數的有關概念理解錯誤，相反數、倒數、絕對值的意義概念混淆。靈活地運用各種運算律，關鍵是把好符號關；在較複雜的運算中，不注意運算順序或者不合理使用運算律，從而使運算出現錯誤。人教版初中一年級數學下學期第五章：不等式不等式組，習題與詳析 2018-03-04 要注意的是「三定」：一是定邊界點，二是定方向，三是定空實.3.應用：列不等式解應用題的基本步驟與列方程解應用題的步驟相類似，即：設：設出適當的未知數；列。一元一次不等式與一次函數 2018-04-16 一、基本知識點考點：1、一次函數y=kx+b與一元一次不等式的關係：y＞0,則kx+b>0；y﹤0,則kx+b<0。中考數學備考專題-不等式組的解法 2018-02-13 一元一次不等式及不等式組是全國各地中考數學的又一大熱點和考點，且它的考查範圍更廣泛。從不等式及其解法到應用，可以遍布到全卷的選擇題、填空題和解答題中。衝刺2018年高考數學, 典型例題分析11：不等式綜合問題 2018-03-22 ∵關於x的不等式f有解，∴a＞4，即實數a的取值範圍是．=5，則g=-7/2+a+2=5，解得a=13/2，∴當x＜2時，g=x+9/2=4，得x=-1/2∈。深入瞭解健身器材房地產投資諮詢娛樂新聞訂閱運動賽事轉播歷史文化書籍演藝圈新聞動漫周邊時尚品牌服飾寵物用品科技產品評論 Copyright ©2025/服務條款/DMCA/聯絡我們
2160	https://senecalearning.com/en-GB/revision-notes/gcse/maths/edexcel/higher/6-2-14-scatter-graphs-predictions	Maths: Edexcel GCSE Higher [x] GCSE GCSE Biology RevisionGCSE Chemistry RevisionGCSE Physics RevisionGCSE Geography RevisionGCSE English Language RevisionGCSE Computer Science Revision A Level A Level Biology RevisionA Level Chemistry RevisionA Level Physics RevisionA Level Psychology RevisionA Level Business RevisionA Level Sociology RevisionA Level Political Studies Revision KS3 KS3 Science RevisionKS3 Maths RevisionKS3 Geography RevisionKS3 History Revision GCSE GCSE Biology Revision GCSE Chemistry Revision GCSE Physics Revision GCSE Geography Revision GCSE English Language Revision GCSE Computer Science Revision A Level A Level Biology Revision A Level Chemistry Revision A Level Physics Revision A Level Psychology Revision A Level Business Revision A Level Sociology Revision A Level Political Studies Revision KS3 KS3 Science Revision KS3 Maths Revision KS3 Geography Revision KS3 History Revision Sign up GCSEMathsEdexcelMaths: Edexcel GCSE Higher6.2.14 6.2.14 Scatter Graphs - Predictions Test yourself Scatter Graphs - Making Predictions The line of best fit can be used to make predictions. Interpolation Making predictions based within the range of the original data is called interpolation. It is more reliable than extrapolation because we know the pattern of data in this range. Interpolation - predictions within the range of data Extrapolation You can make predictions outside the range of data by extending the line of best fit. This is called extrapolation. Extrapolation is less reliable than interpolation because we cannot be sure the same pattern occurs in data outside our range. Extrapolation - predictions outside the range of data 1 Number 1.1 Using Numbers 1.1.1 Equal Symbols1.1.2 Negative Numbers1.1.3 BIDMAS1.1.4 Inverse Operations1.1.5 End of Topic Test- Ordering1.1.6 Factors1.1.7 Multiples1.1.8 Highest Common Factor1.1.9 Lowest Common Multiple1.1.10 Number of Ways to Do a Task1.1.11 Number of Combinations1.1.12 Number of Ways To Arrange Objects1.1.13 End of Topic Test - Factors, Multiples & Combos1.1.14 Standard Form1.1.15 Ordinary Numbers to Standard Form1.1.16 Addition in Standard Form1.1.17 Subtraction in Standard Form1.1.18 Multiplication in Standard Form1.1.19 Division in Standard Form1.1.20 Exam-Style Questions - Standard Form & Primes1.1.21 Long Division1.1.22 Prime Numbers1.1.23 End of Topic Test - Standard Form & Long Division1.1.24 Diagnostic Misconceptions - 1 NOT a Prime Number1.1.25 Diagnostic Misconceptions - BIDMAS1.1.26 Diagnostic Misconceptions - Adding Negatives 1.2 Fractions, Decimals & Percentages 1.2.1 Adding Fractions1.2.2 Subtracting Fractions1.2.3 Multiplying Fractions1.2.4 Dividing Fractions1.2.5 Mixed Fractions1.2.6 Calculating Percentages of Numbers1.2.7 Calculating Fractions of Numbers1.2.8 Multiply Decimals1.2.9 Place Value1.2.10 Recurring Decimals1.2.11 Exam-Style Questions - Recurring Decimals1.2.12 Convert Fractions & Terminating Decimals1.2.13 Convert Recurring Decimals1.2.14 Convert Decimals & Percentages1.2.15 Compare Decimals1.2.16 Compare Fractions1.2.17 Exam-Style Questions - Percentages1.2.18 End of Topic Test - Fractions & Decimals1.2.19 Diagnostic Misconceptions - Adding Fractions 1.3 Powers & Roots 1.3.1 Powers1.3.2 Roots1.3.3 Indices Advanced1.3.4 Estimating Powers1.3.5 Estimating Roots1.3.6 Multiplying Powers1.3.7 Dividing Powers1.3.8 Power Raised to Power1.3.9 Fractions Raised to a Power1.3.10 Numbers Raised to Negative Powers1.3.11 Roots of Numbers1.3.12 Numbers to the Power of 1 & 01.3.13 Surds1.3.14 Simplify Surds1.3.15 Rationalise Denominator1.3.16 Exam-Style Questions - Indices1.3.17 End of Topic Test - Powers & Roots1.3.18 Diagnostic Misconceptions - Squaring1.3.19 Diagnostic Misconceptions - Simplifying Surds 1.4 Accuracy 1.4.1 Decimal Rounding1.4.2 Significant Figure Rounding1.4.3 Estimation1.4.4 Truncation1.4.5 Limits of Accuracy1.4.6 Bounds Addition1.4.7 Bounds Subtraction1.4.8 Bounds Multiplication1.4.9 Bounds Division1.4.10 End of Topic Test - Rounding & Estimation1.4.11 Diagnostic Misconceptions - Upper Bounds 2 Algebra 2.1 Introduction to Algebra 2.1.1 Algebra Basics2.1.2 Substituting Values2.1.3 Algebraic Vocabulary2.1.4 Collecting Terms2.1.5 End of Topic Test - Algebra Basics & Power Rules2.1.6 Expanding Brackets I2.1.7 Common Factors2.1.8 Expanding Brackets II2.1.9 Rearrange For x2.1.10 Rearrange with Powers2.1.11 Exam-Style Questions - Basics of Algebra2.1.12 End of Topic Test - Expanding & Rearranging 2.2 Manipulating Algebra 2.2.1 How to Factorise2.2.2 Factorising Quadratics2.2.3 Algebraic Fractions - Addition & Subtraction2.2.4 Algebraic Fractions - Multiplication & Division2.2.5 Diagnostic Misconceptions - Common Denominators 2.3 Proofs & Functions 2.3.1 Identities & Proofs2.3.2 Exam-Style Questions - Algebraic Proof2.3.3 End of Topic Test - Algebra & Proofs2.3.4 Functions2.3.5 Combine Functions2.3.6 Composite Functions2.3.7 Inverse Functions 2.4 Straight Line Graphs 2.4.1 Straight Line Graphs2.4.2 y = mx + c2.4.3 Grad + YIntercept2.4.4 Exam-Style Questions - Straight Line Graphs2.4.5 End of Topic Test - Functions & Straight Lines2.4.6 Diagnostic Misconceptions - The x Coefficient 2.5 Common Graphs 2.5.1 Shape of Quadratic Graphs2.5.2 Roots of a Quadratic Graph2.5.3 Completing Squares2.5.4 Turning Points2.5.5 Common Functions 12.5.6 Common Functions 22.5.7 Trigonometric functions2.5.8 End of Topic Test - Common Graphs 2.6 Transformations & Tangents 2.6.1 Transformations2.6.2 Tangent on a Circle 2.7 Properties of Graphs 2.7.1 Estimate Gradient2.7.2 Estimate Area2.7.3 Represent Gradient & Area2.7.4 Distance-Time Graph2.7.5 Velocity-Time Graph2.7.6 End of Topic Test - Transformations & Graphs 2.8 Solving Equations 2.8.1 Solve Linear Equations2.8.2 Linear Equations with Fractions2.8.3 Solve Linear Equations Graphically2.8.4 Quadratic ±2.8.5 Factorise Quadratics2.8.6 Quadratic Formula2.8.7 Complete the Square2.8.8 Solve Quadratics Graphically2.8.9 End of Topic Test - Solving Equations2.8.10 Simultaneous Equations2.8.11 Quadratic Simultaneous Equations2.8.12 Simultaneous Equations - Graphs2.8.13 Simultaneous Equations - Word Problems2.8.14 Approximate Solutions - Number of Iterations2.8.15 Approximate Solutions - Iterations to 1 DP2.8.16 Approximate Solutions - Iteration Formula2.8.17 Exam-Style Questions - Simultaneous Equations2.8.18 End of Topic Test - Simultaneous Equations2.8.19 Diagnostic Misconceptions - Solving Quadratics 2.9 Inequalities 2.9.1 Solving Inequalities2.9.2 Solving Double Inequalities2.9.3 Solving Quadratic Inequalities2.9.4 Representing Inequalities2.9.5 Exam-Style Questions - Inequalities2.9.6 End of Topic Test - Inequalities 2.10 Sequences 2.10.1 Nth Term Rules2.10.2 Recurrence Relationships2.10.3 Find Nth Term in Linear Sequence2.10.4 Check If a Number is In a Sequence2.10.5 Exam-Style Questions - Sequences2.10.6 Arithmetic & Geometric Progressions2.10.7 Key Sequences2.10.8 Quadratic Sequences2.10.9 Find Nth Term in Quadratic Sequence2.10.10 Exam-Style Questions - Geometric Series2.10.11 End of Topic Test - Sequences 3 Ratio 3.1 Ratios in Practice 3.1.1 Converting Between Units3.1.2 Formula Triangles3.1.3 Scale Factors3.1.4 Ratios in Shapes3.1.5 Diagnostic Misconceptions - Writing Ratios 3.2 Manipulating Ratios 3.2.1 Simplifying Ratios3.2.2 Turn Ratios into Equations3.2.3 Ratio Notation & Reduction3.2.4 End of Topic Test - Units & Ratios3.2.5 Part : Part Ratio3.2.6 Part : Whole Ratio3.2.7 Two Ratios3.2.8 Differences in Ratios3.2.9 Fractions in Ratios3.2.10 Numbers as Fractions3.2.11 Ratios to Fractions3.2.12 Exam-Style Questions - Ratios3.2.13 End of Topic Test - Fractions & Ratios3.2.14 Diagnostic Misconceptions - Combining Ratios 3.3 Percentage & Interest 3.3.1 Convert Percentages to Decimals3.3.2 Convert Percentages to Fractions3.3.3 Find the Percentage3.3.4 More/Less Percentage3.3.5 Price from Percentage Change3.3.6 Calculate Percentage Change from Price3.3.7 End of Topic Test - Percentages3.3.8 Profit & Interest & Discount3.3.9 Original Price after Discounts3.3.10 Simple Interest3.3.11 Compound Growth & Decay3.3.12 Compound Interest3.3.13 Compound Interest - After How Many Days?3.3.14 End of Topic Test - Percentages in Finance 3.4 Proportion 3.4.1 Proportion3.4.2 Direct Proportion3.4.3 Inverse Proportion3.4.4 Exam-Style Questions - Proportionality 3.5 Gradient 3.5.1 Gradient on a Straight Line3.5.2 Direct & Inverse Proportions as Graphs3.5.3 Gradient of a Curve3.5.4 Positive & Negative Gradients3.5.5 End of Topic Test - Proportion & Gradient 4 Geometry 4.1 Introduction to Geometry 4.1.1 Geometry Basics4.1.2 Geometry Basics 24.1.3 Regular Polygons4.1.4 Triangles & Angles4.1.5 Midpoint4.1.6 Bisectors & Perpendiculars4.1.7 Locus4.1.8 Angle rules4.1.9 Parallel Angle Rules4.1.10 Angles in a Polygon4.1.11 Angles in a Triangle4.1.12 Exam-Style Questions - Angles in Shapes4.1.13 End of Topic Test - Geometry 4.2 Triangles & Quadrilaterals 4.2.1 Types of Triangle4.2.2 Angles & Triangles4.2.3 Congruence4.2.4 Similarity4.2.5 Quadrilaterals 4.3 Transformations 4.3.1 Types of Transformations4.3.2 Invariant Points4.3.3 Effects of Transformations4.3.4 Exam-Style Questions - Reflection 4.4 Circle Basics 4.4.1 Properties of a Circle 14.4.2 Chords4.4.3 Properties of a Circle 24.4.4 Area of a Circle4.4.5 Circumference of a Circle4.4.6 Fractions of Circles4.4.7 Exam-Style Questions - Circle Formulae4.4.8 End of Topic Test - Triangles & Circles 4.5 Circle Theorems 4.5.1 Perpendicular Bisector of a Chord4.5.2 Angle at the Centre4.5.3 Angle of a Semicircle4.5.4 Opposite Angles in Cyclic Quadrilaterals4.5.5 Tangent Lengths4.5.6 Angle in the Same Segment4.5.7 Alternate Segment Theorem4.5.8 Exam-Style Questions - Circle Theorems4.5.9 Diagnostic Misconceptions - Circumference & Radius 4.6 Measurements & Units 4.6.1 Length4.6.2 Area4.6.3 Volume4.6.4 Capacity4.6.5 Time4.6.6 Money4.6.7 Mass4.6.8 Bearings4.6.9 Scale Factor & Enlargement4.6.10 Exam-Style Questions - Enlargements4.6.11 Diagnostic Misconceptions - Hours 4.7 Calculating Area 4.7.1 Area of 2D Shapes4.7.2 Exam-Style Questions - Area4.7.3 End of Topic Test - Circle Theorems, Units & Area 4.8 Triangle Formulae 4.8.1 Pythagoras' Theorem4.8.2 Pythagoras in 3D4.8.3 Trigonometry4.8.4 Exact Values4.8.5 Sine Rule4.8.6 Cosine Rule4.8.7 Area of a Triangle4.8.8 Exam-Style Questions - Trigonometry & Area4.8.9 Diagnostic Misconceptions - Bearings from Angles4.8.10 Diagnostic Misconceptions - Pythagoras' Theorem 4.9 3D Shapes 4.9.1 Types of 3D Shape4.9.2 Plans & Elevations4.9.3 Nets4.9.4 Volume of 3D Shapes4.9.5 Volume of a Cuboid4.9.6 Volume of a Cylinder4.9.7 Volume of a Sphere4.9.8 Volume of a Hemisphere4.9.9 Volume of a Cone4.9.10 Volume of a Pyramid4.9.11 Surface Area & Composite Shapes4.9.12 Surface Area of a Cuboid4.9.13 Surface Area of a Cylinder4.9.14 Surface Area of a Sphere/Hemisphere4.9.15 Surface Area of a Cone4.9.16 Exam-Style Questions - Volume 4.10 Vectors 4.10.1 Vectors Introduction4.10.2 Vectors4.10.3 Translation4.10.4 Scalar4.10.5 Exam-Style Questions - Vectors4.10.6 End of Topic Test - Trigonometry, Shapes & Vectors4.10.7 Diagnostic Misconceptions - Column Vectors 5 Probability 5.1 Introduction to Probability 5.1.1 Randomness & Fairness5.1.2 Sum of Probabilities5.1.3 Relative & Expected Frequency5.1.4 Frequency Tables5.1.5 Using Frequency Trees5.1.6 Complete Frequency Trees 5.2 Representing Probabilities 5.2.1 Probability5.2.2 Probability Scale5.2.3 Sample Space Diagrams5.2.4 Sets5.2.5 Venn Diagrams5.2.6 Probabilities in Venn Diagrams5.2.7 Exam-Style Questions - Venn Diagrams 5.3 Probability Calculations 5.3.1 Theoretical Probabilities5.3.2 Multiple Events5.3.3 Probability Calculations5.3.4 Dependent Events5.3.5 Independent Events5.3.6 Probability Tree Diagram5.3.7 Complex Tree Diagrams5.3.8 Conditional Probability5.3.9 Conditional Probability From a Two Way Table5.3.10 Conditional Probability Venn Diagrams5.3.11 Conditional Probability Tree Diagrams5.3.12 Exam-Style Questions - Probability5.3.13 End of Topic Test - Probability5.3.14 Diagnostic Misconceptions - Denominator 6 Statistics 6.1 Introduction to Statistics 6.1.1 Statistics Basics6.1.2 Statistics Key Terms6.1.3 Frequency Tables6.1.4 Exam-Style Questions - Statistics & Probability6.1.5 Diagnostic Misconceptions - Median6.1.6 Diagnostic Misconceptions - Mode6.1.7 Diagnostic Misconceptions - Mean 6.2 Charts & Diagrams 6.2.1 Bar Charts6.2.2 Pie Charts6.2.3 Stem & Leaf Diagrams6.2.4 Back to Back Stem & Leaf Diagrams6.2.5 End of Topic Test - Statistics6.2.6 Pictograms6.2.7 Time Series Graphs6.2.8 Histograms6.2.9 Histograms 26.2.10 Frequency Polygons6.2.11 Cumulative Frequency6.2.12 Cumulative Frequency 26.2.13 Scatter Graphs6.2.14 Scatter Graphs - Predictions6.2.15 Correlation & Causation6.2.16 Exam-Style Questions - Cumulative Frequency 6.3 Comparing Datasets 6.3.1 Comparing Datasets6.3.2 End of Topic - Datasets Jump to other topics 1 Number 1.1 Using Numbers 1.1.1 Equal Symbols1.1.2 Negative Numbers1.1.3 BIDMAS1.1.4 Inverse Operations1.1.5 End of Topic Test- Ordering1.1.6 Factors1.1.7 Multiples1.1.8 Highest Common Factor1.1.9 Lowest Common Multiple1.1.10 Number of Ways to Do a Task1.1.11 Number of Combinations1.1.12 Number of Ways To Arrange Objects1.1.13 End of Topic Test - Factors, Multiples & Combos1.1.14 Standard Form1.1.15 Ordinary Numbers to Standard Form1.1.16 Addition in Standard Form1.1.17 Subtraction in Standard Form1.1.18 Multiplication in Standard Form1.1.19 Division in Standard Form1.1.20 Exam-Style Questions - Standard Form & Primes1.1.21 Long Division1.1.22 Prime Numbers1.1.23 End of Topic Test - Standard Form & Long Division1.1.24 Diagnostic Misconceptions - 1 NOT a Prime Number1.1.25 Diagnostic Misconceptions - BIDMAS1.1.26 Diagnostic Misconceptions - Adding Negatives 1.2 Fractions, Decimals & Percentages 1.2.1 Adding Fractions1.2.2 Subtracting Fractions1.2.3 Multiplying Fractions1.2.4 Dividing Fractions1.2.5 Mixed Fractions1.2.6 Calculating Percentages of Numbers1.2.7 Calculating Fractions of Numbers1.2.8 Multiply Decimals1.2.9 Place Value1.2.10 Recurring Decimals1.2.11 Exam-Style Questions - Recurring Decimals1.2.12 Convert Fractions & Terminating Decimals1.2.13 Convert Recurring Decimals1.2.14 Convert Decimals & Percentages1.2.15 Compare Decimals1.2.16 Compare Fractions1.2.17 Exam-Style Questions - Percentages1.2.18 End of Topic Test - Fractions & Decimals1.2.19 Diagnostic Misconceptions - Adding Fractions 1.3 Powers & Roots 1.3.1 Powers1.3.2 Roots1.3.3 Indices Advanced1.3.4 Estimating Powers1.3.5 Estimating Roots1.3.6 Multiplying Powers1.3.7 Dividing Powers1.3.8 Power Raised to Power1.3.9 Fractions Raised to a Power1.3.10 Numbers Raised to Negative Powers1.3.11 Roots of Numbers1.3.12 Numbers to the Power of 1 & 01.3.13 Surds1.3.14 Simplify Surds1.3.15 Rationalise Denominator1.3.16 Exam-Style Questions - Indices1.3.17 End of Topic Test - Powers & Roots1.3.18 Diagnostic Misconceptions - Squaring1.3.19 Diagnostic Misconceptions - Simplifying Surds 1.4 Accuracy 1.4.1 Decimal Rounding1.4.2 Significant Figure Rounding1.4.3 Estimation1.4.4 Truncation1.4.5 Limits of Accuracy1.4.6 Bounds Addition1.4.7 Bounds Subtraction1.4.8 Bounds Multiplication1.4.9 Bounds Division1.4.10 End of Topic Test - Rounding & Estimation1.4.11 Diagnostic Misconceptions - Upper Bounds 2 Algebra 2.1 Introduction to Algebra 2.1.1 Algebra Basics2.1.2 Substituting Values2.1.3 Algebraic Vocabulary2.1.4 Collecting Terms2.1.5 End of Topic Test - Algebra Basics & Power Rules2.1.6 Expanding Brackets I2.1.7 Common Factors2.1.8 Expanding Brackets II2.1.9 Rearrange For x2.1.10 Rearrange with Powers2.1.11 Exam-Style Questions - Basics of Algebra2.1.12 End of Topic Test - Expanding & Rearranging 2.2 Manipulating Algebra 2.2.1 How to Factorise2.2.2 Factorising Quadratics2.2.3 Algebraic Fractions - Addition & Subtraction2.2.4 Algebraic Fractions - Multiplication & Division2.2.5 Diagnostic Misconceptions - Common Denominators 2.3 Proofs & Functions 2.3.1 Identities & Proofs2.3.2 Exam-Style Questions - Algebraic Proof2.3.3 End of Topic Test - Algebra & Proofs2.3.4 Functions2.3.5 Combine Functions2.3.6 Composite Functions2.3.7 Inverse Functions 2.4 Straight Line Graphs 2.4.1 Straight Line Graphs2.4.2 y = mx + c2.4.3 Grad + YIntercept2.4.4 Exam-Style Questions - Straight Line Graphs2.4.5 End of Topic Test - Functions & Straight Lines2.4.6 Diagnostic Misconceptions - The x Coefficient 2.5 Common Graphs 2.5.1 Shape of Quadratic Graphs2.5.2 Roots of a Quadratic Graph2.5.3 Completing Squares2.5.4 Turning Points2.5.5 Common Functions 12.5.6 Common Functions 22.5.7 Trigonometric functions2.5.8 End of Topic Test - Common Graphs 2.6 Transformations & Tangents 2.6.1 Transformations2.6.2 Tangent on a Circle 2.7 Properties of Graphs 2.7.1 Estimate Gradient2.7.2 Estimate Area2.7.3 Represent Gradient & Area2.7.4 Distance-Time Graph2.7.5 Velocity-Time Graph2.7.6 End of Topic Test - Transformations & Graphs 2.8 Solving Equations 2.8.1 Solve Linear Equations2.8.2 Linear Equations with Fractions2.8.3 Solve Linear Equations Graphically2.8.4 Quadratic ±2.8.5 Factorise Quadratics2.8.6 Quadratic Formula2.8.7 Complete the Square2.8.8 Solve Quadratics Graphically2.8.9 End of Topic Test - Solving Equations2.8.10 Simultaneous Equations2.8.11 Quadratic Simultaneous Equations2.8.12 Simultaneous Equations - Graphs2.8.13 Simultaneous Equations - Word Problems2.8.14 Approximate Solutions - Number of Iterations2.8.15 Approximate Solutions - Iterations to 1 DP2.8.16 Approximate Solutions - Iteration Formula2.8.17 Exam-Style Questions - Simultaneous Equations2.8.18 End of Topic Test - Simultaneous Equations2.8.19 Diagnostic Misconceptions - Solving Quadratics 2.9 Inequalities 2.9.1 Solving Inequalities2.9.2 Solving Double Inequalities2.9.3 Solving Quadratic Inequalities2.9.4 Representing Inequalities2.9.5 Exam-Style Questions - Inequalities2.9.6 End of Topic Test - Inequalities 2.10 Sequences 2.10.1 Nth Term Rules2.10.2 Recurrence Relationships2.10.3 Find Nth Term in Linear Sequence2.10.4 Check If a Number is In a Sequence2.10.5 Exam-Style Questions - Sequences2.10.6 Arithmetic & Geometric Progressions2.10.7 Key Sequences2.10.8 Quadratic Sequences2.10.9 Find Nth Term in Quadratic Sequence2.10.10 Exam-Style Questions - Geometric Series2.10.11 End of Topic Test - Sequences 3 Ratio 3.1 Ratios in Practice 3.1.1 Converting Between Units3.1.2 Formula Triangles3.1.3 Scale Factors3.1.4 Ratios in Shapes3.1.5 Diagnostic Misconceptions - Writing Ratios 3.2 Manipulating Ratios 3.2.1 Simplifying Ratios3.2.2 Turn Ratios into Equations3.2.3 Ratio Notation & Reduction3.2.4 End of Topic Test - Units & Ratios3.2.5 Part : Part Ratio3.2.6 Part : Whole Ratio3.2.7 Two Ratios3.2.8 Differences in Ratios3.2.9 Fractions in Ratios3.2.10 Numbers as Fractions3.2.11 Ratios to Fractions3.2.12 Exam-Style Questions - Ratios3.2.13 End of Topic Test - Fractions & Ratios3.2.14 Diagnostic Misconceptions - Combining Ratios 3.3 Percentage & Interest 3.3.1 Convert Percentages to Decimals3.3.2 Convert Percentages to Fractions3.3.3 Find the Percentage3.3.4 More/Less Percentage3.3.5 Price from Percentage Change3.3.6 Calculate Percentage Change from Price3.3.7 End of Topic Test - Percentages3.3.8 Profit & Interest & Discount3.3.9 Original Price after Discounts3.3.10 Simple Interest3.3.11 Compound Growth & Decay3.3.12 Compound Interest3.3.13 Compound Interest - After How Many Days?3.3.14 End of Topic Test - Percentages in Finance 3.4 Proportion 3.4.1 Proportion3.4.2 Direct Proportion3.4.3 Inverse Proportion3.4.4 Exam-Style Questions - Proportionality 3.5 Gradient 3.5.1 Gradient on a Straight Line3.5.2 Direct & Inverse Proportions as Graphs3.5.3 Gradient of a Curve3.5.4 Positive & Negative Gradients3.5.5 End of Topic Test - Proportion & Gradient 4 Geometry 4.1 Introduction to Geometry 4.1.1 Geometry Basics4.1.2 Geometry Basics 24.1.3 Regular Polygons4.1.4 Triangles & Angles4.1.5 Midpoint4.1.6 Bisectors & Perpendiculars4.1.7 Locus4.1.8 Angle rules4.1.9 Parallel Angle Rules4.1.10 Angles in a Polygon4.1.11 Angles in a Triangle4.1.12 Exam-Style Questions - Angles in Shapes4.1.13 End of Topic Test - Geometry 4.2 Triangles & Quadrilaterals 4.2.1 Types of Triangle4.2.2 Angles & Triangles4.2.3 Congruence4.2.4 Similarity4.2.5 Quadrilaterals 4.3 Transformations 4.3.1 Types of Transformations4.3.2 Invariant Points4.3.3 Effects of Transformations4.3.4 Exam-Style Questions - Reflection 4.4 Circle Basics 4.4.1 Properties of a Circle 14.4.2 Chords4.4.3 Properties of a Circle 24.4.4 Area of a Circle4.4.5 Circumference of a Circle4.4.6 Fractions of Circles4.4.7 Exam-Style Questions - Circle Formulae4.4.8 End of Topic Test - Triangles & Circles 4.5 Circle Theorems 4.5.1 Perpendicular Bisector of a Chord4.5.2 Angle at the Centre4.5.3 Angle of a Semicircle4.5.4 Opposite Angles in Cyclic Quadrilaterals4.5.5 Tangent Lengths4.5.6 Angle in the Same Segment4.5.7 Alternate Segment Theorem4.5.8 Exam-Style Questions - Circle Theorems4.5.9 Diagnostic Misconceptions - Circumference & Radius 4.6 Measurements & Units 4.6.1 Length4.6.2 Area4.6.3 Volume4.6.4 Capacity4.6.5 Time4.6.6 Money4.6.7 Mass4.6.8 Bearings4.6.9 Scale Factor & Enlargement4.6.10 Exam-Style Questions - Enlargements4.6.11 Diagnostic Misconceptions - Hours 4.7 Calculating Area 4.7.1 Area of 2D Shapes4.7.2 Exam-Style Questions - Area4.7.3 End of Topic Test - Circle Theorems, Units & Area 4.8 Triangle Formulae 4.8.1 Pythagoras' Theorem4.8.2 Pythagoras in 3D4.8.3 Trigonometry4.8.4 Exact Values4.8.5 Sine Rule4.8.6 Cosine Rule4.8.7 Area of a Triangle4.8.8 Exam-Style Questions - Trigonometry & Area4.8.9 Diagnostic Misconceptions - Bearings from Angles4.8.10 Diagnostic Misconceptions - Pythagoras' Theorem 4.9 3D Shapes 4.9.1 Types of 3D Shape4.9.2 Plans & Elevations4.9.3 Nets4.9.4 Volume of 3D Shapes4.9.5 Volume of a Cuboid4.9.6 Volume of a Cylinder4.9.7 Volume of a Sphere4.9.8 Volume of a Hemisphere4.9.9 Volume of a Cone4.9.10 Volume of a Pyramid4.9.11 Surface Area & Composite Shapes4.9.12 Surface Area of a Cuboid4.9.13 Surface Area of a Cylinder4.9.14 Surface Area of a Sphere/Hemisphere4.9.15 Surface Area of a Cone4.9.16 Exam-Style Questions - Volume 4.10 Vectors 4.10.1 Vectors Introduction4.10.2 Vectors4.10.3 Translation4.10.4 Scalar4.10.5 Exam-Style Questions - Vectors4.10.6 End of Topic Test - Trigonometry, Shapes & Vectors4.10.7 Diagnostic Misconceptions - Column Vectors 5 Probability 5.1 Introduction to Probability 5.1.1 Randomness & Fairness5.1.2 Sum of Probabilities5.1.3 Relative & Expected Frequency5.1.4 Frequency Tables5.1.5 Using Frequency Trees5.1.6 Complete Frequency Trees 5.2 Representing Probabilities 5.2.1 Probability5.2.2 Probability Scale5.2.3 Sample Space Diagrams5.2.4 Sets5.2.5 Venn Diagrams5.2.6 Probabilities in Venn Diagrams5.2.7 Exam-Style Questions - Venn Diagrams 5.3 Probability Calculations 5.3.1 Theoretical Probabilities5.3.2 Multiple Events5.3.3 Probability Calculations5.3.4 Dependent Events5.3.5 Independent Events5.3.6 Probability Tree Diagram5.3.7 Complex Tree Diagrams5.3.8 Conditional Probability5.3.9 Conditional Probability From a Two Way Table5.3.10 Conditional Probability Venn Diagrams5.3.11 Conditional Probability Tree Diagrams5.3.12 Exam-Style Questions - Probability5.3.13 End of Topic Test - Probability5.3.14 Diagnostic Misconceptions - Denominator 6 Statistics 6.1 Introduction to Statistics 6.1.1 Statistics Basics6.1.2 Statistics Key Terms6.1.3 Frequency Tables6.1.4 Exam-Style Questions - Statistics & Probability6.1.5 Diagnostic Misconceptions - Median6.1.6 Diagnostic Misconceptions - Mode6.1.7 Diagnostic Misconceptions - Mean 6.2 Charts & Diagrams 6.2.1 Bar Charts6.2.2 Pie Charts6.2.3 Stem & Leaf Diagrams6.2.4 Back to Back Stem & Leaf Diagrams6.2.5 End of Topic Test - Statistics6.2.6 Pictograms6.2.7 Time Series Graphs6.2.8 Histograms6.2.9 Histograms 26.2.10 Frequency Polygons6.2.11 Cumulative Frequency6.2.12 Cumulative Frequency 26.2.13 Scatter Graphs6.2.14 Scatter Graphs - Predictions6.2.15 Correlation & Causation6.2.16 Exam-Style Questions - Cumulative Frequency 6.3 Comparing Datasets 6.3.1 Comparing Datasets6.3.2 End of Topic - Datasets Unlock your full potential with Seneca Premium Unlimited access to 10,000+ open-ended exam questions Mini-mock exams based on your study history Unlock 800+ premium courses & e-books Get started with Seneca Premium Scatter Graphs Correlation & Causation GCSEGCSE Biology RevisionGCSE Chemistry RevisionGCSE Physics RevisionGCSE Geography RevisionGCSE English Language RevisionGCSE Computer Science Revision A LevelA Level Biology RevisionA Level Chemistry RevisionA Level Physics RevisionA Level Psychology RevisionA Level Business RevisionA Level Sociology RevisionA Level Political Studies Revision KS3KS3 Science RevisionKS3 Maths RevisionKS3 Geography RevisionKS3 History Revision FAQs learnmore@seneca.ioHelp centreFind a tutorExam stress & wellbeingSeneca premiumPress kit
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General Biology 2e (OpenStax) 5. 6: Unit VI- Plant Structure and Function 6. 6.1: Plant Form and Physiology Expand/collapse global location General Biology 2e (OpenStax) Front Matter 1: Unit I- The Chemistry of Life 2: Unit II- The Cell 3: Unit III- Genetics 4: Unit IV- Evolutionary Processes 5: Unit V- Biological Diversity 6: Unit VI- Plant Structure and Function 7: Unit VII- Animal Structure and Function 8: Unit VIII- Ecology 9: Appendix Back Matter 6.1: Plant Form and Physiology Last updated Dec 25, 2022 Save as PDF 6: Unit VI- Plant Structure and Function 6.1.1: Introduction picture_as_pdf Full Book Page Downloads Full PDF Import into LMS Individual ZIP Buy Print Copy Print Book Files Buy Print CopyReview / Adopt Submit Adoption Report Submit a Peer Review HomeworkView on CommonsDonate Page ID 97183 OpenStax OpenStax ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents No headers Like animals, plants contain cells with organelles in which specific metabolic activities take place. Unlike animals, however, plants use energy from sunlight to form sugars during photosynthesis. In addition, plant cells have cell walls, plastids, and a large central vacuole: structures that are not found in animal cells. Each of these cellular structures plays a specific role in plant structure and function. 6.1.1: IntroductionWhile individual plant species are unique, all share a common structure: a plant body consisting of stems, roots, and leaves. They all transport water, minerals, and sugars produced through photosynthesis through the plant body in a similar manner. All plant species also respond to environmental factors, such as light, gravity, competition, temperature, and predation. 6.1.2: The Plant BodyLike animals, plants contain cells with organelles in which specific metabolic activities take place. Unlike animals, however, plants use energy from sunlight to form sugars during photosynthesis. In addition, plant cells have cell walls, plastids, and a large central vacuole: structures that are not found in animal cells. Each of these cellular structures plays a specific role in plant structure and function. 6.1.3: StemsPlant stems, whether above or below ground, are characterized by the presence of nodes and internodes. Nodes are points of attachment for leaves, aerial roots, and flowers. The stem region between two nodes is called an internode. The stalk that extends from the stem to the base of the leaf is the petiole. An axillary bud is usually found in the axil—the area between the base of a leaf and the stem—where it can give rise to a branch or a flower. 6.1.4: RootsThe roots of seed plants have three major functions: anchoring the plant to the soil, absorbing water and minerals and transporting them upwards, and storing the products of photosynthesis. Some roots are modified to absorb moisture and exchange gases. Most roots are underground. Some plants, however, also have adventitious roots, which emerge above the ground from the shoot. 6.1.5: LeavesLeaves are the main sites for photosynthesis: the process by which plants synthesize food. Most leaves are usually green, due to the presence of chlorophyll in the leaf cells. However, some leaves may have different colors, caused by other plant pigments that mask the green chlorophyll. The thickness, shape, and size of leaves are adapted to the environment. Each variation helps a plant species maximize its chances of survival in a particular habitat. 6.1.6: Transport of Water and Solutes in PlantsThe structure of plant roots, stems, and leaves facilitates the transport of water, nutrients, and photosynthates throughout the plant. The phloem and xylem are the main tissues responsible for this movement. Water potential, evapotranspiration, and stomatal regulation influence how water and nutrients are transported in plants. To understand how these processes work, we must first understand the energetics of water potential. 6.1.7: Plant Sensory Systems and ResponsesAnimals can respond to environmental factors by moving to a new location. Plants, however, are rooted in place and must respond to the surrounding environmental factors. Plants have sophisticated systems to detect and respond to light, gravity, temperature, and physical touch. Receptors sense environmental factors and relay the information to effector systems—often through intermediate chemical messengers—to bring about plant responses. 6.1.8: Key Terms 6.1.9: Chapter Summary 6.1.10: Visual Connection Questions 6.1.11: Review Questions 6.1.12: Critical Thinking Questions Thumbnail: Rose thorns. (CC BY 2.0; macrophile via Flickr). This page titled 6.1: Plant Form and Physiology is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform. 30: Plant Form and Physiology by OpenStax is licensed CC BY 4.0. Toggle block-level attributions Back to top 6: Unit VI- Plant Structure and Function 6.1.1: Introduction Was this article helpful? Yes No Recommended articles 7: Plant Form and FunctionLike animals, plants contain cells with organelles in which specific metabolic activities take place. Unlike animals, however, plants use energy from ... 7: Plant Form and PhysiologyLike animals, plants contain cells with organelles in which specific metabolic activities take place. Unlike animals, however, plants use energy from ... 6.1.1: IntroductionWhile individual plant species are unique, all share a common structure: a plant body consisting of stems, roots, and leaves. They all transport water... 6.1.2: The Plant BodyLike animals, plants contain cells with organelles in which specific metabolic activities take place. Unlike animals, however, plants use energy from ... 6.1.3: StemsPlant stems, whether above or below ground, are characterized by the presence of nodes and internodes. Nodes are points of attachment for leaves, aeri... Article typeSection or PageAuthorOpenStaxPrint CSSDenseLicenseCC BYLicense Version4.0OER program or PublisherOpenStaxShow TOCno Tags source-bio-1819 source@ © Copyright 2025 Biology LibreTexts Powered by CXone Expert ® ? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. 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2163	https://pmc.ncbi.nlm.nih.gov/articles/PMC9611905/	Cork Development: What Lies Within - 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. Service Alert: Planned Maintenance beginning July 25th Most services will be unavailable for 24+ hours starting 9 PM EDT. Learn more about the maintenance. 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Learn more: PMC Disclaimer \| PMC Copyright Notice Plants (Basel) . 2022 Oct 11;11(20):2671. doi: 10.3390/plants11202671 Search in PMC Search in PubMed View in NLM Catalog Add to search Cork Development: What Lies Within Rita Teresa Teixeira Rita Teresa Teixeira 1 BioISI—Biosystems & Integrative Sciences Institute, Faculty of Sciences, University of Lisboa, 1749-016 Lisboa, Portugal; rtteixeira@fc.ul.pt Find articles by Rita Teresa Teixeira 1 Editor: Marcello Salvatore Lenucci 1 Author information Article notes Copyright and License information 1 BioISI—Biosystems & Integrative Sciences Institute, Faculty of Sciences, University of Lisboa, 1749-016 Lisboa, Portugal; rtteixeira@fc.ul.pt Roles Marcello Salvatore Lenucci: Academic Editor Received 2022 Aug 12; Accepted 2022 Oct 6; Collection date 2022 Oct. © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( PMC Copyright notice PMCID: PMC9611905 PMID: 36297695 Abstract The cork layer present in all dicotyledonous plant species with radial growth is the result of the phellogen activity, a secondary meristem that produces phellem (cork) to the outside and phelloderm inwards. These three different tissues form the periderm, an efficient protective tissue working as a barrier against external factors such as environmental aggressions and pathogen attacks. The protective function offered by cork cells is mainly due to the abundance of suberin in their cell walls. Chemically, suberin is a complex aliphatic network of long chain fatty acids and alcohols with glycerol together with aromatic units. In most woody species growing in temperate climates, the first periderm is replaced by a new functional periderm upon a few years after being formed. One exception to this bark development can be found in cork oak (Quercus suber) which display a single periderm that grows continuously. Quercus suber stands by its thick cork layer development with continuous seasonal growth. Cork raw material has been exploited by man for centuries, especially in Portugal and Spain. Nowadays, its applications have widened vastly, from the most known product, stoppers, to purses or insulating materials used in so many industries, such as construction and car production. Research on how cork develops, and the effect environmental factors on cork oak trees is extremely important to maintain production of good-quality cork, and, by maintaining cork oak stands wealthy, we are preserving a very important ecosystem both by its biodiversity and its vital social and economic role in areas already showing a population declination. Keywords: cork, exploitation, phellem, phellogen, stress, suberin 1. Growth and Uses The cork oak Quercus suber L. (Fagaceae) is an endemic tree of the Mediterranean basin growing mainly in Portugal, Spain, southern France and Italy, and northern Morocco. Cork oak stands, known as “montados” in Portugal and “dehesas” in Spain, are complex agro-silvo-pastoral systems managed by human labor to provide a habitat for a diverse wildlife and cultural ecosystems services. Management and exploitation of cork oak stands dates back to roman times and, throughout the centuries, Portugal alone has become the world leader of cork production (55% of world’s production) and cork transformation [1,2,3]. Cork possesses a set of properties, i.e., a low material density and an extremely low permeability to liquids and gases, it is biologically and chemically inert and mechanically elastic, and the conferring cork material provide high insulation and damping capacities . Due to all properties of cork, it is used in a plethora of products such as sealants, agglomerates and composites that can be transformed into bottle stoppers, insulation and surfacing panels for construction and aeronautics, for pollutants absorbers, for clothing, fashion items and decorative furniture pieces [4,5]. Besides the specific uses for cork singled out in this review, other compounds can also be extracted from bark, which are later used in medicine, construction, chemistry, clothing, energy and biofuels . Cork has also been found to work as biosorbents for heavy metals [7,8], oils and aromatic hydrocarbons . Cork can be extracted from the tree as a single plank because the phellogen (the meristematic layer producing cork outwardly) forms a continuous cylinder layer all around the trunk. Cork oak debarking for cork exploitation can only occur during a short time window from June till July (highly dependent on the environmental conditions), when phellogen cells are fully active and the new cells produced still display thin and fragile cell walls. As far as industrial requirements are concerned, for cork quality, the first cork produced by the tree and harvested, also known as virgin cork (Figure 1a), cannot be used for high end products because it is a hard-rough cork, very irregular in cell/material density and thickness. After the first cork, each debarking occurs every 9-12 years allowing the tree, the time necessary to regrow the tree’s outer periderm. This cork, used by industry, is called reproduction cork and displays enough thickness to allow punching natural cork stoppers (planks must be at least 27 mm thick) (Figure 1b). Regardless, all cork removed from the tree can be exploited commercially as raw material for final products as agglomerates. Only second reproduction cork and following harvests named “amadia” cork meets the quality standards necessary for stopper production demanded by wine companies (Figure 1b). Stopper bottles are the product with the higher industrial revenue; however, the amount of available good-quality cork needed is becoming scarcer in the field, compromising cork manufacturing goals. For several years now, the transformation industry is developing new products made from lower quality cork and leftovers from stopper production, reducing the dependency on the wine sector [2,12]. Furthermore, in cork exploitation, there is a growing interest in understanding periderm development to improve plant resilience and as sinks for CO 2 sequestration . Figure 1. Open in a new tab The cork layer of Quercus suber. (a) Virgin cork. This cork oak has never been debarked is still displaying its first cork layer. (b,c) Different quality types of amadia cork. (b) Cork plank thick enough and with very few discontinuities allowing stopper punching (red dots circle). It is possible to distinguish the growth rings (black arrow). (c) Example of a reduced-quality cork plank, thin in thickness and intersected by numerous lenticular channels (black arrowhead). White arrows indicate the growth direction from the surface in contact with the trunk and where the phellogen is (also known as cork belly) towards the surface in contact with the environment. In—cork plant inside; out—outermost cork plank surface. Albeit, the importance of cork products in countries like Portugal, which accounts for 2% of all exported goods, cork production has shown a reduction of 2.7% over the last decade [11,12] reflecting a cork oak decline associated with abiotic stresses phenomena such as prolonged drought events and ever more frequent and long-lasting heat waves . These authors came to the conclusion that besides environment, tree genetics have a strong contribution to the cork chemical composition. 2. Biology and Adaptation Several herbaceous species display a limited secondary growth, produced by interfascicular and fascicular meristems, therefore lacking phellogen and a periderm. Most herbaceous or woody plants display secondary growth of plant organs, which confers widening and is accomplished through the activation of two secondary meristems; the vascular cambium and the phellogen (cork cambium). The increase in diameter reflects the periclinal divisions on these two cambiums, producing derivatives towards both ends with further distinct specifications. The vascular cambium produces xylem inwardly and phloem outwardly resulting in stem and root thickening. This vascular growth forces the primary epidermis to break down with concomitant development of a complete periderm, a protective tissue of mature organs . The periderm is composed of the phellogen, the phelloderm (produced by the phellogen inwards) and the phellem (cork) present outward of the phellogen (Figure 2j,k). In most woody species growing in temperate climates, the first periderm is replaced by a new functional periderm a few years after being formed. This way, bark begins to accumulate on the outside comprising layers of dead periderms and remaining non-functional phloem tissue between them. This structure is called rhytidome . One exception to this bark development can be found in Quercus suber which displays a single periderm that grows continuously. Such continuous growth of the phellem gives rise to cork rings that can be clearly distinguished because spring forming cells have thinner walls and a larger diameter compared to the cells formed later in the season (Figure 1b). Figure 2. Open in a new tab Cork develops as early as one-year stems in Quercus suber. (a–i) Cross-sectional stereomicroscopic images of one-, two- and three-year-old stems. (Scale bars: (a,b) 100 µm, c 500 µm). (d–f) Details of the xylem with empty tracheids (). Moving outwardly, one finds the phloem (dark layer) surrounded by the periderm comprising the phelloderm (orange arrow -phd), a thin layer of phellogenic cells (white arrowhead) and several rows of suber cells. The most external layer is the remaining epidermis. (Scale bars: (d,e) 100 µm, f 500 µm). (g–i) Cork cells exhibit autofluorescence when excited with UV light . The stems observed in d-f were excited with UV using a stereoscope filter for “Lumar01” (BP 365/12; LP397) after which, the suberin layer is clearly distinguishable (white arrow). In older stems, as xylem cells mature, autofluorescence can be observed since lignin also possesses some autofluorescence properties under UV light. (j,k) Light microscopy of one-year stem cross sections stained with toluidine blue. (Scale bars: (g,h) 100 µm, i 500 µm). (j) Some remaining epidermis (ep) is still present and right below, six to seven cell cork layers with cells displaying a hyaline cell wall appearance and filled with electrodense material (phenolic compounds) (orange line) can be observed. Inwardly and adjacent to suberin cells, is the phellogen composed of only one to two cells (red line). Cells are filled with cytoplasm and cell walls do not present a secondary thickening. Right below is the phelloderm with the characteristic round-shaped parenchymatous cells. (Scale bar: 10 µm). (k) Cross section showing a lenticel forming in a one-year stem (black arrow). The disorganized division of the meristematic cells inside the lenticel structure is starting to push the suberin layer outwards leading to a complete rupture of this layer to form an aperture that allows gas exchanges. (Scale bar: 40 µm). (l) Transmission electron microscopy image of amadia cork cell wall clearly showing a plasmodesmata (black arrow) crossing both suberized cell walls. It is possible to observe cytoplasmic deposition in the inner part of the cell, with thickening appearing on both sides of the plasmodesmata channel (amp. 20,000×). ep—epidermis; ph—phloem; phd—phelloderm; phl—phellogen; s—suberin; xy—xylem. The activity of the phellogen begins in April and continues to be active till the end of October . During these months, growth is not uniform. In early season (April–July), cells are formed faster, thus display thinner walls and larger lumina diameter than cells growing later in the season (August–October). Such difference in cell anatomy permits distinguishing cork rings in a similar fashion to tree-rings (Figure 1b). Weather plays a role in cork-ring growth with a higher growth resisted in rainy years with moderate-low temperatures . In fact, even outside of growing season (November–December), precipitation positively influences cork growth as well as moderate temperatures during the summer . However, it has been verified that high summer temperatures negatively influenced cork production . The constrains that weather pressure exert on cork development is of great worry for cork producers and the cork industry. Costa et al. (2022) reported that under water stress, smaller cork cells with reduced lumina and thicker cell-wall were produced resulting in denser cork rings. With climate changes hitting the Mediterranean area stronger than in many other zones of the globe, with the forecasted increased summer temperatures and longer, more extreme drought periods, it is very important deepen our understanding on the mechanisms governing cork development so that tree improvement programs can be put in place. We ought to keep in mind that the periderm is first and foremost a protective tissue that works as the first line of defense of trees against abiotic and biotic stresses. The periderm confers resistance to pathogen invasion due to suberin deposits on cell walls and the presence of metabolites in cork tissues . Since suberized cells are impervious, water vapor and the gases CO 2 and O 2 are regulated through lenticels. Lenticels are aerenchymatous structures composed mainly of active meristematic cells interspersing the phellem layer allowing gaseous exchange between the tree and the environment (Figure 2k). Lenticels, unlike the neighboring suberized cells lack suberin but are essentially composed of lignin . Lenticels arise from previous structures involved in gases exchange, the stomata, which are present in the epidermis and are genetically determined. For the cork industry, the presence of a high number of lenticels downgrades the cork quality, thus bringing its price down for producers (Figure 1c). Therefore, understanding the ontogeny of lenticels and the reason why some trees display a much higher number of these structures is of great importance to cork production. 3. Chemical Composition The characteristic cork properties are a direct reflection of the cell wall’s chemical composition and structure. Cork’s chemical nature dictates its function as a protective layer of the internal tree’s tissues against environmental aggression, which is achieved by its main cell wall component, suberin. Other important cork cell wall components are lignin, the polysaccharide cellulose and the hemicelluloses and polar and non-polar extractives . Cork cells present a wide chemical variation concerning suberin contents (23.1% to 54.2%) and lignin (17.1–36.4%) with the ratio suberin-to-lignin playing a pivotal role in physical cork properties such as compression [25,26]. Comparing the chemical composition of Q. suber’s cork, phloem and wood tissues, cork cells are the only ones with suberin on its cell walls (comprising 42.3%). Phloem on the other hand, shows a higher degree of lignification in comparison to wood and have less polysaccharides . Suberin is made up of two domains: an aliphatic zone and an aromatic zone that includes ferulic acid . One of the most recent accounts of suberin’s ferulic acid content was of 2.7% . The polymeric aliphatic macromolecule is composed of two types of monomers: glycerol and long chain fatty acids and alcohols, whose hydroxyl and carboxylic groups are linked by ester bonds . The monomer glycerol represents 40.8% of suberin and the long-chain monomers comprising mainly α-ω-diacides at 36.4% . The fatty acids present chain lengths that can vary from C16 to C30, esterified to glycerol and cross-esterified . At suberin’s macromolecular structural level, the polyester aliphatic structure is extensively linked to aromatic moieties conferring the secondary cell wall, the appearance of lamellate structure as seen by transmission electron microscopy (TEM) . These lamellae show alternate opaque and translucent contrasting structures, with translucent lamellae showing a regular thickness of ca. 30 Å and the opaque lamellae varying in thickness of ca. 70 to 100 Å. Between 30 and 60 lamellae have been counted in the suberized cell walls of cork of different species . Recent work comparing the ultrastructural analysis of secondary cell walls in Quercus suber, Quercus cerris, Calotropis procera and Solanum tuberosum found that the lamellar structures with alternating dark and light bands were present in suberized cells of potato tuber periderm and Calotropis bark, whereas the cork cells of Q. cerris and Q. suber lacked defined lamellae . Such findings led to the hypothesis that the chemical composition of suberin, which differs between species, may play a role in the cell wall topochemistry due to different spatial development of the suberin macromolecule . The second most important cork cell wall component is lignin which confers strength to the cell. The remaining cell wall polysaccharides are cellulose and hemicellulose accounting for 20% of the cork’s cell wall structural components [36,37]. Lignin is present in most plants’ secondary tissues. It is a polymer comprising three different types of phenyl propane monomers, conferring an aromatic nature to lignin. The monomers ρ-coumaryl, coniferyl and sinapyl alcohols are linked through enzymatic phenoxy radical formation [1,30]. Lignin is also linked to ferulic acid by ester bonds accounting for 3% . Non-structural components, soluble in different solvents, are also present in cork, such as lipophilic extractives that include fatty acids and alcohols, sterols and terpenes and phenolic compounds [6,38]. Ash is an inorganic material found in percentages ranging from 1% to 2% and is the end result upon total combustion. Amongst all ash components, calcium, phosphorous, sodium, potassium and magnesium are the minerals present in the highest concentrations [37,39]. In sum, in cork cells, the composition of the secondary cell wall depositions that takes place in the interior of the cell is of about 42% suberin, 22% of lignin, 19% polysaccharides and 16% of extractives . The extractive portion is a myriad of low to medium molecular weight molecules and includes: n-alkanes, n-alkanols, waxes, triterpenes, fatty acids, glycerides, sterols, phenols and polyphenols that can be sorted in two main groups; aliphatic and phenolic . Suberin, requires the presence of a lignin-like polymerized aromatic domain to adhere to the cell wall , whereas waxes are aliphatic compounds that do not covalently link to the cell wall . Suberization is a fast process which can be visualized as soon as phellogen cells grow giving rise to a differentiated phellem (Figure 2j) . Phellem development begins as soon as one-year stems grow (Figure 2a,d,g) and become thicker as the tree ages (Figure 2b,c,e,f,h,i). This can be detected on the roots and the aerial parts of the tree . Polyphenolics are another group of extractives found in cork cells that are much less studied, with more heterogenous composition among phellems, which include simple phenols and/or polymeric phenols such as tannins. Out of the few studies on polyphenols in cork, Pinheiro et al. (2019) were able to establish a comparison of phenolic compositions between cork samples of higher and lower quality findings that in cork of higher quality, aromatic phenylpropanoid components were incorporated into the cell wall in larger amounts than hydrolysable tannins. This work was carried out in parallel with a transcriptomic analysis using the same cork samples. Here, it was possible to see that during the development of cork of superior and inferior quality, and in the former, the shikimate pathway shifts towards the synthesis of cell wall-bound phenolics. In the latter, phellogenic cells invested more into the biosynthesis of soluble phenolics, especially hydrolysable tannins, displaying a higher reducing capacity (40% more than in cork of higher quality) [44,45]. 4. Molecular Basis of Cork Development From a transcriptomic point of view, at the beginning of the season, cell division and differentiation genes such as cyclins, are enriched, while later in the season, wall biogenesis and factors involved in secondary metabolites are elevated . Genetics plays a decisive role in cork growth and its chemical composition [46,47]. High-throughput sequencing of cork oak phellogen samples isolated from thick and thin cork planks (examples in Figure 1b,c) collected on the same day, showed differences in the gene expression [45,48]. Trees producing a thinner cork layer and with higher number of lenticular channels (Figure 1c) exhibited a higher number of transcripts involved in DNA synthesis, RNA processing, proteolysis, and especially transcription factors associated with the abiotic stress response and stomatal/lenticular-associated genes. On the other hand, trees producing more uniform and thicker cork planks expressed genes encoding for heat-shock proteins were up-regulated . When a transcriptomic analysis of phellogen cells isolated by laser microdissection was performed on samples collected from trees producing cork of good- and bad-quality, stress-related genes were enriched in samples from bad-quality cork which also showed an up-regulation of genes belonging to the flavonoid pathway . These authors demonstrated that cork of lower quality displayed a higher content of free phenolics and, when analyzing the phenylpropanoid pathway, verified that in lower quality cork there was an up-regulation of genes involved in the biosynthesis of free phenolic compounds; while in good quality cork, the synthesis pathways for lignin and suberin was promoted . Both these works showed that besides other environmental and genetic factors, specific gene expression patterns dictate which phenylpropanoid pathway branch is promoted (suberin synthesis or free phenolic production) resulting in differences in cork layer thickness. However, only part of the mechanism governing cork development can be explained purely by gene expression. For example, distinct epigenetic patterns and single-methylation events associated with climate variations influence cork quality . Epigenetic gene regulation is done through the action of histones and DNA modifications which include methylation and acetylation. Basically, DNA and histone modification enzymes have the capacity to alter both euchromatin and heterochromatin conformations creating regions of condensed chromatin which physically prevents a given gene from being transcribed at a particular time point. Such regulation can be rapidly reversed every time a tree’s growth and/or environment conditions change, requiring differential gene expression. Epigenetic gene regulation processes are essential for plant development but have a special role when it comes to environmental adaptation, hence the abiotic/biotic stress responses. Recently, Silva et al. (2020) identified in Q. suber, DNA methyltransferases (DNAMtases) and DNA demethylases (DDMEs) involved in histones methylation, demethylation, acetylation and deacetylation. In Q. suber phellogen cells, epigenetic-related genes were investigated to establish a relationship between epigenetic regulation and cork quality . The research group found that all classes of DNA methyltransferases were present in Q. suber and that QsDRM2 was the methyltransferase most active in the phellogen. The specific DNA methylation patterns are pivotal for proper tissue establishment and differentiation [52,53]. Interestingly, expression of QsDMAP1 (DNA Methyltransferase 1-associated Protein 1), involved in transcription repression and activation and in genomic instability , was higher in cork of low quality , suggesting that cork inclusion could be a mechanism for the tree to cope with the environment. During heat-stress experiments, Correia et al. (2013) showed that DNA methylation and histone H3 acetylation acted contrary to each other, whereas DNA methylation increased in Q. suber plants as temperatures rose from 35 °C to 55 °C under a controlled environment. On the other hand, acetylated histone H3 (AcH3) levels decreased during the same temperature range experiment. Besides adaptation to stress conditions, epigenetic gene regulation also plays an important role on well-programmed seasonally regulated growth conditions, such as dormancy. Trees growing in temperate and boreal forests have developed an elaborate mechanism to survive the cold temperatures during winter and the capacity to resume growth once spring conditions arrive . As the photoperiod diminish, a set of gene responsible for dormancy regulation mechanisms are initiated which are highly dependent on epigenetic gene silencing . The epigenetic regulation of DORMANCY-ASSOCIATED MADS-box (DAM)-related genes and FLOWERING LOCUS C gene are two examples of genome-wide epigenetic modifications associated with dormancy events [58,59]. Upon transcriptomic analysis of cork oak phellogen samples collected in April (cork active growing season) it was possible to observe that meristem-associated genes, such as WUSCHEL (WUS), SHOOT MERISTEMLESS (STM), APETALA2 (AP2), PHABULOSA (PHB), AINTEGUMENTA (ANT) and AINTEGUMENTA-LIKE6 (AIL6) were up-regulated. Later on, during the active growth season (June till July) cell wall genes linked to lignin, suberin and secondary cell wall deposition were highly expressed . The transition of gene expression from meristem identity/maintenance genes toward genes encoding elements necessary for secondary wall establishment reflects the developmental nature of cork layer from the phellogen. More recent works have contributed to elucidate gene activation involved in the phellogen, e.g., chromatin-remodeling genes and other genes already known to be involved in meristem regulation, such as FLOWERING LOCUS C (FLC) or stem-cell maintenance-related transcription factors [47,60]. The transcriptome of potato phellogen showed that genes associated with cell niche and radial patterning, such as the transcription factor SHORT-ROOT 2B (TPR-2), NO APICAL MWEISTEM (NAM), HD2B, PHAVOLUTA-Like HD-ZIPIII, and PHLOEM INTERCALATED WITH XYLEM (PXY) were present and highly expressed . This study on potato phellogen showed an up-regulation of genes related to ribosomal proteins as well as stress-associated genes, including heat-shock proteins, a feature also observed in cork oak phellogen . When it comes to cork phellogen regulation, phytohormones, auxins and brassinosteroids, seem to play an important role. It was shown that increased auxin levels proceeded ethylene production which, in turn, worked as a principal activator of genes involved in the phellogen’s initial development . A putative PIN3 orthologue, an auxin transporter, and DWF1, encoding an enzyme involved in the synthesis of brassinosteroid, are up regulated in cork phellogenic cells . ABA responsive element-binding protein/ABA binding factor (AREB/ABF) and MYC/MYB transcription factors together with ABA-related genes displayed a higher number of elements up regulated in trees with a more pronounced cork layer development . Additionally, in these cork samples, genes encoding ethylene-responsive transcription factors; EIN3 (Ethylene insensitive 3) and ETR1/ETR2 and EIN4 (ethylene receptors) were also highly expressed . These genes are membrane-associated receptors with the capacity to bind ethylene further triggering the downstream ethylene response cascade . Lopes et al. (2019) also detected an upregulation of the AP2/ERF ethylene responsive transcription factor in cork phellogen samples. Besides ethylene response-associated genes important in developmental mechanisms , jasmonate-associated genes, important in plant defense , were detected to be up regulated in cork oak . Other jasmonate signaling-associated genes like jasmonate-ZIM domain-containing proteins and the jasmonate signaling mediator MYC transcription factor, were up regulated in thick cork samples suggesting the participation of jasmonates in stress responses during proper cork development . Transcription factors are also regulated by abiotic stresses in cork oak phellogen. Consistently, the Arabidopsis ortholog MYB84, QsMybB1 is down regulated in response to drought and heat [67,68] (Almeida et al., 2013a,b) and was shown to accumulate in poplar phellem . Cork oak phellogen has the capacity to produce thick suber layers at a rate of 1–3 mm/year with pronounced suberin secondary deposition on the cell wall. One of the genes responsible for the suberization process is FHT (fatty ω-hydroxyacid/fatty alcohol hydroxycinnamayl transferase) which is specifically expressed in cells undergoing suberization regulated by ABA and salicylic acid (SA) . Its activation is detected in phellogen cells right before suberin deposition is initiated . FHT is also involved in wax biosynthesis, essential for periderm integrity . The number of differently expressed genes involved in the phenylpropanoid pathway during cork development is elevated. Especially in suberin synthesis and assembling, several transcripts associated with fatty acid elongation are consistently found to be expressed, i.e., 3-ketoacyl-CoA synthesis (KCSs); long-chain acyl-CoA synthase (LACSs), fatty acyl-CoA reductases (FAR) and glycerol-3-phosphate acyltransferases (GTPAs) [45,64]. Besides the detection of genes related to wax biosynthesis in cork oak phellogen, the suberin biosynthesis genes GPAT5, CYP86A1, CYP86B1 and AtHHT/ASFT, CYP86A33 and FHT were also found to be up regulated compared to holm oak Quercus ilex L. (not producing a cork layer but a rhytidome) . All genetic analyses and tissues producing suberin have shown expression of cytochrome P450-family members involved in the catalysis of ω-hydroxylation of fatty acids necessary for suberin monomer production . In fact, the cytochrome P450 transcripts CYP86A1 and CYP86B1, CYP72A, CYP81E, CYP82A and CYP87A were identified in cork tissues [45,48,72,73]. Recently, Lopes et al., (2019) identified CYP82D, CYP76A and CYP705A in phellogen samples. When Arabidopsis root phellem was studied for suberin synthesis, it was possible to detect the expression of suberin synthesis components; CYP86A1, CYP86B1, FAR1/4/5, GTPAT5/7, FACT, and ASFT indicating a conservative mechanism for suberin biosynthesis. Albeit suberin synthesis confers singularity to suberin tissues, the most transcripts enriched category in cork oak phellogenic cell was “phenylpropanoid metabolic processes”, belonging to the phenylpropanoid pathway . Genes encoding key enzymes of this pathway such as phenylalanine ammonia-lyase (PAL), cinnamate-4-hydroxylase (C4H) and 4-coumarate-CoA ligase (4CL) were up-regulated in cork samples. In potato leaves that have been mechanically injured, PAL expression peaked 2 h after wounding then returned to a basal level 6 h later . PAL seems to require the action of ABA since low levels of the phytohormone leads to a reduction of suberin aromatic components by affecting PAL activation . Further down the phenylpropanoid pathway acting in the early steps of the lignin branch, the enzyme hydroxycinnamoyltransferase (HCT) whose corresponding gene, when down-regulated, leads to a strong decrease of suberin-linked ferulic acid levels . Proanthocyanidins (PA), the end product of the flavonoid pathway, a phenylpropanoid pathway branch , accumulates in the vacuoles of cells of the traumatic periderms and young cork cells, and associate with the cell walls of suberized empty cells . Anatomy studies have already shown us that the first cork cells formed can be distinguished by their electrodense fillings and brown inclusion of tannins , a feature also observed in cork oak’s one-year stems (Figure 2j). Tannins are part of proanthocyanidin (PA) secondary metabolites . Both PA and hydrolysable tannins accumulate inside the vacuoles and have been found in Q. suber [44,81] contributing to the plant’s defense mechanism . The biosynthesis of PA components is achieved by the action of three enzymes: leucoanthocyanidin reductase (LAR), anthocyanidin reductase (BAN/ANR), two NADPH-dependent enzymes [82,83] and leucoanthocyanidin dioxygenase/anthocyanidin synthase (LDOX/ANS) and found to be up-regulated on cork samples with higher amounts of free phenolics . Suberin, lignin and polyphenol production are amongst the most enriched categories during cork tissue development, but autophagy and programmed cell death (PCD) are also categories with high number of up-regulated genes . Besides the high suberin content of periderm cork cells, another characteristic of these cells is the fact they are dead by the time functional maturity is reached, implying they undergo PCD . In general, plant PCD is carried out by caspase-like enzymes and proteasomes degrading ubiquitination proteins . Particularly during periderm development, PCD has been regarded as a crucial event for cell maturation and a set of PCD marker genes are exclusively expressed in the endodermis and dying phellem cells in Arabidopsis during periderm formation . Focusing on the periderm, PCD begins simultaneously as soon as fast cell wall suberization begins and phellem cells differentiate [18,88]. Programmed cell death occurring in cork oak has a developmental (dPCD) nature rather than being an effect of environmental factors . Regardless lenticels support the exchange of vital gases, such as CO 2, O 2 and water vapor between the inside and outside of the stem, the presence of lenticel channels affects the quality and price of natural cork stoppers since cork planks quality, is defined according to the external surface porosity promoted by these lenticels (Figure 1b,c). A higher number of lenticels is indicative of a lower quality of the cork (Figure 1c) . Lenticular phellogen differentiation leads to the development of unsuberized cells, which are mainly composed of lignin . Their development has a genetic background comprising at least three basic helix–loop–helix (bHLH) transcription factors forming a single clade [89,90], i.e., SPEECHLESS (SPCH), MUTE, and FAMA, identified in Arabidopsis and essential for stomatal formation [91,92]. When a comparative transcriptomic analysis was carried out on phellogen from good- and low-quality cork samples, it was possible to observe an up regulation of two orthologues included in the same bHLH clade in low-quality cork, samples characterized by the present of a much higher number of lenticular channels (Figure 1c). More and more attention has been directed to cell proteomes but when it comes to cork oak suberin development, the amount of work reported is practically non-existent. One of the first reports was conducted by Ricardo et al. (2011) where it was found that the most significant proteins associated with cork development were “carbohydrate metabolism” (28%), “defense” (22%), “protein folding, stability and degradation” (19%), “regulation/signaling” (11%), “secondary metabolism” (9%), “energy metabolism” (6%), and “membrane transport” (2%). Recently, the work performed on phellogenic cells isolated from fresh debarked planks showed high amounts of oxidoreductases (26%) and metal ion binding proteins (16%) . Examples include, dehydrogenases and enolases (involved in glycolysis and respiration) and signaling-related proteins, such as annexins and other Ca 2+-binding proteins . In accordance with what has been observed in numerous transcriptomic analyses during cork development, the proteomic study carried out by Ricardo et al. (2011) indicated a high involvement of defense proteins (thioredoxin-dependent peroxidase, glutathione-S-transferase, SGT1 protein, cystatin, and chitinases) in phellem differentiation, indicating that protection of tree’s inner tissues is the main function of the cork layer. 5. Conclusions Knowledge on periderm development has broadened in these last years due to its protective role especially when coping with a climate emergency. It has now been regarded as not only as a physical shield of the plant’s internal tissues, but also as a tissue that accumulates secondary metabolites which can be used in industry for various proposes. This review focused on Quercus suber for displaying the formation of a continuous subereous layer that has been exploited by man for thousands of years. Nowadays, everyone is familiar with cork stoppers, the cork product that still carries out the most revenue for the transformation industry. Cork oak allows the exploitation of its raw cork layer while keeping the tree alive permitting silviculture practices on cork oak stands to deliver a win-win situation at all levels. Ecologically because cork oak stands harbor a rich ecosystem in terms of biodiversity and also works as powerful CO 2 sequestering sinkers. Additionally, it’s an important social player since cork exploitation enables economically profitable activities in regions where human population is declining fast. To some extent, keeping cork oak trees healthy works as a buffer against putative soil and human desertification. After so many centuries protecting and exploiting cork oak trees empirically, it is time scientific knowledge to step in and contribute toward an even better sustainable silviculture practices using molecular markers for improved clone selection which can growth with a better successful rate under stressful conditions and continue to provide cork of good quality which is demanded by industry. Acknowledgments I would like to thank Luis Marques from “Faculty of Sciences of the University of Lisbon’s Microscopy Facility is a node of the Portuguese Platform of BioImaging, reference PPBI-POCI-01-0145-FEDER-022122” for his precious help with the pictures. Data Availability Statement Not applicable. Conflicts of Interest The authors declare no conflict of interest. Funding Statement This research received no external funding. 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Chemical Composition 4. Molecular Basis of Cork Development 5. Conclusions Acknowledgments Data Availability Statement Conflicts of Interest Funding Statement Footnotes References Associated Data 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
2164	https://ntrs.nasa.gov/api/citations/20020092087/downloads/20020092087.pdf	6.9 REFERENCE AND STANDARD ATMOSPHERE MODELS Dale L. Johnson and Barry C. Roberts NASA Marshall Space Flight Center, Huntsville, Alabama 35812 William W. Vaughan University of Alabama in Huntsville, Huntsville, Alabama 35899 1. INTRODUCTION This paper describes the development of standard and reference atmosphere models along with the history of their origin and use since the mid 19_h century. The first "Standard Atmospheres" were established by international agreement in the 1920's. Later some countries, notably the United States, also developed and published "Standard Atmospheres". The term "Reference Atmospheres" is used to identify atmosphere models for specific geographical locations. Range Reference Atmosphere Models developed first during the 1960's are examples of these descriptions of the atmosphere. This paper discusses the various models, scopes, applications and limitations relative to use in aerospace industry activities. 2. DISCUSSION A "Standard Atmosphere" is defined as a vertical distribution of atmospheric temperature, pressure, and density which by international agreement is taken to be representative of the Earth's atmosphere. The first "Standard Atmospheres" established by international agreement were developed in the 1920's primarily for the purposes of pressure altimeter calibrations, aircraft performance calculations, aircraft and rocket design, ballistic tables, etc.. Later some countries, notably the United States, also developed and published "Standard Atmospheres". The term "Reference Atmosphere" is used to identify vertical descriptions of the atmosphere for specific geographical locations or globally. These were developed by organizations for specific applications, especially as the aerospace industry began to mature after WWll. The term "Standard Atmosphere" has in recent years also been used by national and international organizations to describe vertical descriptions of atmospheric trace constituents, the ionosphere, atomic oxygen, aerosols, ozone, winds, water vapor, planetary atmospheres, etc. A standard unit of atmospheric pressure is defined as that pressure exerted by a 760 millimeter column of mercury at standard gravity (980.665 cms 2) at 45.5425 ° N latitude and sea level at a temperature of 273.15 ° K (0° C). The recommended unit for Corresponding author address: Dale Johnson, NASA, Marshall Space Flight Center, Mail Code ED44, Huntsville, AL 35812; e-mail: dale.johnson @msfc.nasa.gov meteorological use is 1013.25 hectopascals (millibars). Standard temperature is used in physics to indicate a temperature of 0° C, the ice point, and a pressure of one standard atmosphere (1013.25 hectopascals). In meteorology, the term standard temperature has no generally accepted meaning, except that it may refer to the temperature at zero pressure-altitude in the standard atmosphere (15 ° C) with a density of 1225.00 gm 3. The standard sea-level values of temperature, pressure, and density that have been used for decades are: temperature of 288.15 ° K, or 15° C; pressure of 1013.25 millibars, or 760 millimeters of Hg; and density of 1225.00 gm3. As early as the middle of the 19th century, a Standard Atmosphere was needed as a basis for calibrating aneroid barometers used in measuring altitudes. These instruments provided the means of obtaining a rough measure of the height of mountains and other land areas. They were later used for altitude determination in manned balloon flights. Similar atmospheres in England as well as in the United States were computed on the basis of a constant temperature independent of altitude. Shortly after the turn of the century, several atmospheres were developed on the basis of observed or assumed temperature-altitude profiles, in which the temperature decreased with increasing altitude. These atmospheres were adopted by France, Italy, and Germany. The development of the airplane, plus the desire to improve the direct reading accuracy of barometer altimeters, stimulated the measurement of atmospheric temperature to the greatest possible altitude at various locations, particularly in England, France, Germany, and Italy. With the more general use of airplanes during World War I from 1914 to 1917, the need for one standard atmosphere to serve as the basis for comparison of aircraft performance became evident. The general international desire for unity of national atmospheres following the end of World War I, as well as the unreality and complexity of several of the existing aeronautical atmospheres, prompted the study of the problem with the aim of recommending a simple compromise model. The result of this study was the adoption of an atmosphere model for France in 1920 as the official standard atmosphere in aircraft performance tests. Italy also adopted this atmosphere model in 1920 and England in 1921. It was not until 1925, however, that this atmosphere model was adopted in England as the basis for altimeter calibration. In 1924 the International Commission for Aerial Navigation (ICAN) adopted the model as the basis for an international standard known as the ICAN Standard Atmosphere. Though never adopted bytheUnited States (U.S.), this standard served much of theworlduntil1952 when slight differences were reconciled andit wasmodified slightly under the International Civil Aviation Organization (ICAO), which included theUnited States. Thisstandard atmosphere formed the basisof the tablesgivenin National Advisory committee for Aeronautics (NACA) Report 1235. In 1922the United States NACA Standard Atmosphere (orfirstU.S.Standard Atmosphere) was published. It wasofficially approved on2 December 1924by the Executive committee of NACAas described in NACA TR-218. TheWar and Navy Departments, theWeather Bureau andtheBureau of Standards adoptedit for use in aeronautical calculations. Table1 givesa timehistory of the documented technical reports dealing with theupdates to this U.S.Standard Atmosphere. In 1952the International Civil Aeronautical Organization (ICAO) produced theICAO Standard Atmosphere andin1964 anextension to32km.Subsequent tothistimethere have beena succession of Standard andReference Atmospheres, some extending toaltitudes above 1000 km,produced bytheU.S. Committee onExtension to the Standard Atmosphere (COESA), Committee on Space Research (COSPAR), ComitetStandartov (USSR), International Standardization Organization (ISO), U. S. Air ForceResearch andDevelopment Command (ARDC), U.S.Range Commanders Council (RCC), andU. S. National Aeronautics andSpace Administration (NASA) plusothers. COESA was established in 1953 andledto thepublication of the 1958,1962,1966and 1976versions of the U.S. Standard Atmosphere. In 1975 the International Standards Organization published a Standard Atmosphere for altitudes from -2 to50kmthat isidentical totheICAO Standard Atmosphere from -2 to32kmoSubsequently theISOpublished in 1982 a family offiveReference Atmospheres for Aerospace Use for altitudes up to 80 km and latitudes of 15, 30, 45, 60, and 80° N. The portion of the U. S. Standard Atmosphere up to 32 km is identical with the ICAO Standard Atmosphere, 1964; and identical below 50 km with the ISO Standard Atmosphere, 1973. For this reason, in addition to providing an excellent description of the atmosphere model development extending beyond conventional aircraft operations, the U.S. Standard Atmosphere, 1976 is used here to illustrate the vertical distribution of atmospheric temperature. Figure 1 provides an illustration of the temperature-height profiles to 100 km of the COESA U. S. Standard Atmosphere, 1976, and the lowest and highest mean monthly temperatures obtained for any location between the Equator and pole. For altitudes above approximately 100 km, significant variations in the temperature, and thus density, occur due to solar and geomagnetic activity over the period of a solar cycle. Variations in the temperature-height profiles for various degrees of solar and geomagnetic activity are presented in Figure 2. Profile (A) gives the lowest temperature expected at solar cycle minimum; profile (B) represents average conditions at solar cycle minimum; (C) represents average conditions at a typical solar cycle maximum; and (D) gives the highest temperatures to be expected during a period of exceptionally high solar and geomagnetic activity. In the early 1970's, during the initial development of the Space Shuttle vehicle, it was determined that the various reference atmosphere models developed up to that time frame might not be sufficient to use for a vehicle which could land at any location on the Globe. This prompted the development of the NASNMSFC Global Reference Atmosphere Model (GRAM), and its many revisions. GRAM gives the engineer a monthly, average atmospheric profile (thermodynamic parameters and wind), with their variability, either at any given lat/Iong location or along any inputted trajectory. In 1996 the American Institute of Aeronautics and Astronautics (AIAA) published a Guide to Reference and Standard Atmosphere Models. This document provides information on the principal features for a number of global, regional, middle atmosphere, thermosphere, test ranges, and planetary atmosphere models. Summary information on these reference and standard atmosphere models is given relative to geographic region, altitude range, parameters, species, temporal variation, output data, and principal application. Currently some of the most commonly used Standard and Reference Atmospheres include the ICAO Standard Atmosphere, 1952/1964, the ISO Standard Atmosphere, 1975, the U. S. Standard Atmosphere, 1976, the COSPAR International Reference Atmosphere (CIRA), 1986 , the NASA/MSFC Global Reference Atmosphere Model (GRAM), 1999 and the RCC/MG Range Reference Atmospheres (see Table 2 for a complete Range listing). 3. CONCLUSION The intent of this paper is to present a summary historical account regarding the establishment of International and domestic Standard and Reference Atmospheres. These atmospheres were developed to provide a standard type of atmospheric input for the various aeronautical and space vehicle design, development, and operational applications. previously issuedas CIRA 1961, CIRA 1965 and CIRA 1972. previously issued as GRAM-86, GRAM-88, GRAM-90 and GRAM-95. Some site specific annual reference atmospheres (and Hot and Cold atmospheres) have been created by NASNMarshall Space Flight Center for NASA sites of interest, as for Patrick AFB/NASA Kennedy Space Center, Vandenberg AFB and Edwards AFB. 4. BIBLIOGRAPHY OF DOCUMENTS, INCLUDING REFERENCES THEREIN, ON WHICH THE CONTENTS OF THIS ARTICLE IS BASED Champion, K.S.W., 1995: Early Years of Air Force Geophysics Research Contributions to Internationally Recognized Standard and Reference Atmospheres. Technical Report PL-TR-95-2164, Air Force Phillips Laboratory, Hanscom AFB, MA, USA. Sissenwine, N., M. Dubin, and S. Teweles, COESA Co-Chairmen, 1976: U. S. Standard Atmosphere, 1976. Stock No. 003-017-00323-0, U. S. Government Printing Office, Washington, D. C. USA. Vaughan, W.W., D.L. Johnson, C.G. Justus, and S. Pavelitz, 1996: Guide to Reference and Standard Atmosphere Models. Document ANSl/AIAA G-003A-1996, American Institute of Aeronautics and Astronautics, Reston, VA, USA. Table 1. Timeline History of U.S. Standard Atmosphere Publications. Date 1922/6 1922 1925 1926 1932 1936 1947/1 1954/5 1955 1956/12 Title Notes on the Standard Atmosphere Standard Atmosphere (SA) Standard Atmosphere Tables & Data Tables for Calibrating Altimeters etc Some Approximate Equations for the Std Atmos ,ltitude-pressure Tables Based on US Std Atm Tentative Tables for the Prop of Upper Atmosphere Manual of ICAO Std Atm Calc by NACA Standard Atmosphere-Tables to 65800' The ARDC Model Atmosphem, 1956 Repod No. NACA TN-99 NACA Rpt-147 NACA Rpt-218 NACA Rpt-246 NACA Rpt-376 NACA Rpt-538 NACA TN-1200 NACA TN-3182 Author Diehl Gregg Diehl Brombacher Diehl Brombacher Warfield Anon NACA Rpt-Anon 1235 AFSG Minzner TN56-204 #86 1958 1959/8 1962/12 1966 1976 1976 US Extension to ICAO Std Atm - to 300 km The ARDC Model Atmosphem, 1959 US Standard Atmosphere, 1962 US Standard Atmosphere Supplements, 1966 The 1976 Standa Atmos. Above 86-km Altitude US Standard Atmosphere, 1976 USGovPO AFCRC TR59-267 #115 USGovPO USGovPO NASA SP-398 USGovPO Minzner Minzner Anon Anon Minzner Anon Table 2. Listing of Published IRIG & RCC Range Reference Atmospheres. 1, 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. Argentia, New Foundland Ascension Island, South Atlantic Barking Sands, Hawaii Cape Canaveral, Florida China Lake, California Dugway Proving Ground, Utah Edwards AFB, California Eglin AFB, Florida Eniwetok, Marshall Islands, Pacific Fairbanks, Alaska Fort Churchill, Canada Fort Greeley, Alaska Fort Huachuca, Arizona Johnston Island, Pacific Kodiak, Alaska Kwajalein, Marshall Islands, Pacific Lihue Kauai, Hawaii Nellis AFB, Nevada Point Arguello, California Point Mugu, California Roosevelt Roads, Puerto Rico Shemya, Alaska Taguac, Guam, Pacific Thule, Greenland Vandenberg AFB, California Wake Island Pacific Wallops Island, Virginia White Sands, New Mexico Yuma PG, Arizona 3 9O 80 ¸ 7O E 6O O p-I-5O .1 ¢J E 40 I-1,4,1 o 30 UJ L0 2O 10 --I I I I I 1 I i ' O _ _o _ ° -/ "\ \ • 1%EXTREMES / \ " / . " ,. ) / / • = . . ,o --/I J / i/./ J ] = =." __ 2,./ ...-/ ., I l I I j_o j __-4..1,.---20 140 160 180 200 220 240 260 280 300 32C TEMPERATURE, K 50O 400 E =u" I---300 .< L.) o= L9 2O0 A) (s) lOO .500 I (C) ) (O) 0 +500 TEMPERATURE DIFFERENCE, K +1000 FIGURE 1. RANGE OF SYSTEMATIC VARIABILITY OF TEMPERATURE AROUND THE U. S. STANDARD ATMOSPHERE, 1976 (Source: U. S. Standard Atmosphere, 1976) FIGURE 2. DEPARTURES OF THE TEMPERATURE-ALTITUDE PROFILES FROM THAT OF THE U.S. STANDARD ATMOSPHERE, 1976, FOR VARIOUS DEGREES OF SOLAR ACTIVITY (Source: U. S. Standard Atmosphere, 1976) 4
2165	https://www.forecastr.co/blog/binomial-option-pricing-model-explained	3 min read Binomial option pricing model explained simply Evan Diaz de Arce January 29, 2025 Financial modeling Do you know how investors predict the price of options with a good degree of accuracy? Enter the stage - the binomial option pricing model. This technique might sound like something straight out of a finance textbook, but trust me, understanding its basics can be quite straightforward. Imagine having a magic crystal ball that lets you peek into the future of stock prices, all without having to swing by a fortune teller's booth. The secret sauce lies in breaking down complex predictions into simpler binary outcomes – think flipping coins but much more sophisticated. At its core, this model offers a pragmatic approach to valuing options by envisioning multiple possible paths an asset’s price could take over time. If numbers aren't your thing, don’t worry; we’ve got you covered with an easy-to-follow breakdown. To explore more about how this and other types of financial models work, check out our detailed guide. Key takeaways: The binomial option pricing model is a simple but powerful tool for pricing options based on a no-arbitrage assumption. The model uses a binomial tree to show possible stock price paths, with the option price calculated at each node. Key inputs include current stock price, strike price, time to expiration, risk-free rate, and up/down factors. The model is advantageous for its simplicity and multi-period view but assumes constant volatility. It differs from Black-Scholes by using discrete time steps, while both models share theoretical foundations. What is the binomial option pricing model? The binomial option pricing model is a mathematically simple but powerful way to price options. It's based on a no-arbitrage assumption, meaning markets are efficient and investments earn the risk-free rate of return. This option pricing model uses an iterative procedure, allowing you to specify nodes (points in time) between the valuation date and the option's expiration date. Unlike the Black-Scholes model which gives a numerical result, the binomial model lets you calculate the asset and option price for multiple periods, showing the range of possible results each period. The binomial option pricing model is used frequently in practice. It's intuitive and relatively simple to implement in Excel, especially compared to complex stochastic differential equations. Basics of the binomial option pricing model At its core, the binomial option pricing model has two possible outcomes in each iteration - the underlying asset price can move up or down, following a binomial tree. Key inputs include the current stock price, strike price, time until the option expires, and risk-free interest rate. The model assumes investors are risk-neutral. Understanding the binomial tree The binomial tree shows the different paths the stock price could take over the life of the option. At expiration, the option price is simply its intrinsic value (underlying stock price minus strike price). Working backward from expiration, you can calculate the option's price at each node of the tree. Calculating price with the binomial model To calculate the option's price, start with the current stock price. Then estimate the stock price at expiration, with two possible outcomes - up or down. The model needs the following inputs: Current stock price Strike price Time to expiration Risk-free interest rate Up and down factors for the binomial tree Plug these into the binomial option pricing formula to calculate the option price at each node, working backward from expiration to the current price point. This gives a multi-period view of how the option's price could evolve. Advantages and disadvantages of the binomial option pricing model The biggest advantage is that the binomial option pricing model is mathematically simple. It provides a multi-period view of the option price path until expiration. The disadvantage is that a key assumption is that volatility remains constant over the option's life, which may not reflect reality. The model can also get complex with many time steps. Real-world example of binomial option pricing model The binomial model is often used to price American options, which can be exercised anytime before expiration. It also works well for European options, options embedded in other securities, and even currency options. For example, let's say you're pricing an American call option with a $50 strike price. The current stock price is $50, and there are 3 months until expiration. The risk-free rate is 2%. You believe the stock has a 50/50 chance of going up or down 10% each month. Plugging this into a binomial option pricing model in Excel would give you the theoretical price of the option and show how that price might change over the next 3 months. What is the difference between black-scholes and binomial option pricing models? The key difference is that the Black-Scholes model solves a stochastic differential equation to arrive at a single option price. The binomial option pricing model, developed by Cox, Ross and Rubinstein, uses a tree diagram with discrete time steps to show a range of possible option prices. Black-Scholes is more mathematically complex, while the binomial model is relatively simple to implement in Excel. However, both models share the same theoretical foundations, like assuming markets are efficient with no arbitrage opportunities. Simplifying the binomial option pricing model So there we have it - navigating through the intricacies of the binomial option pricing model isn't as daunting as one might think at first glance. By peeling back layers of financial jargon and mathematical formulas, what emerges is a highly practical tool designed not just for high-flying traders but also for anyone keen on making informed decisions about their investments. This approach shows us that blending simple techniques with clever math can lead to powerful ways of predicting market movements and making your money work harder for you. The beauty here doesn’t lie solely in predicting future stock prices accurately every single time; rather it’s about reducing vast uncertainties into manageable probabilities. For more information, contact Forecastr today! New video Subscribe on YouTube to get notified when we post new content Trending New videos Raising capital CFO services Financial hygiene Know your numbers All content Join our community Get notified about new events, free resources, and fresh content Popular downloads \| \| \| PLAYBOOK 10 Steps to Fundraising Success \| \| VIDEO SERIES Execute the Perfect Fundraise \| \| REPORT Fundraising Benchmarks Report \| Related content Financial Modeling for Operations: Part 3 – Capitalization in Finance Logan Burchett If you haven’t already, check out the first and second installments of of this series where I shared the first three key initiatives in optimizing... Financial modeling Know your numbers 8 Best Financial Model Software Tools Compared (2024) Jeff Erickson A financial model can help you build trust with investors, improve the efficiency of your operations, and unlock valuable insights about your... Financial modeling Know your numbers Enhancing predictive analytics: The integration of forecasting tools with SaaS platforms Guest Accurate forecasts are the holy grail of business planning. The ability to reliably predict future outcomes allows companies to strategize more... Financial modeling Know your numbers
2166	https://pubmed.ncbi.nlm.nih.gov/38356659/	Prevalence of pulmonary hypertension in children with obstructive sleep apnea living at high altitude - 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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Prevalence of pulmonary hypertension in children with obstructive sleep apnea living at high altitude Elida Duenas-Meza1,Diego Fernando Severiche-Bueno1,Carolina Santos Quintero2,Jenny Talani Ochoa2,Miguel Ronderos Dummit3,Claudia Stapper3,Carlos Granados G2 Affiliations Expand Affiliations 1 Fundación Neumológica Colombiana, Bogotá, Colombia. 2 Departamento de Pediatría, Universidad de La Sabana, Chía, Cundinamarca, Colombia. 3 Fundación CardioInfantil, Bogotá, Colombia. PMID: 38356659 PMCID: PMC10864626 DOI: 10.1016/j.sleepx.2024.100106 Item in Clipboard Prevalence of pulmonary hypertension in children with obstructive sleep apnea living at high altitude Elida Duenas-Meza et al. Sleep Med X.2024. Show details Display options Display options Format Sleep Med X Actions Search in PubMed Search in NLM Catalog Add to Search . 2024 Feb 2:7:100106. doi: 10.1016/j.sleepx.2024.100106. eCollection 2024 Dec. Authors Elida Duenas-Meza1,Diego Fernando Severiche-Bueno1,Carolina Santos Quintero2,Jenny Talani Ochoa2,Miguel Ronderos Dummit3,Claudia Stapper3,Carlos Granados G2 Affiliations 1 Fundación Neumológica Colombiana, Bogotá, Colombia. 2 Departamento de Pediatría, Universidad de La Sabana, Chía, Cundinamarca, Colombia. 3 Fundación CardioInfantil, Bogotá, Colombia. PMID: 38356659 PMCID: PMC10864626 DOI: 10.1016/j.sleepx.2024.100106 Item in Clipboard Cite Display options Display options Format Abstract Introduction: The prevalence of obstructive sleep apnea (OSA) is 1-4 %. Some reports describe its association with pulmonary hypertension (PH), but its prevalence is unknown. No studies at high altitude have determined the relationship between OSA and PH. The aim of this study was to establish the prevalence of PH in children diagnosed with OSA living in a high-altitude city at 2640 m above sea level. Methods: Children between 2 and 16 years of age referred to the Sleep Laboratory of the Fundación Neumológica Colombiana in Bogotá with a positive polysomnogram for OSA were included, and a two-dimensional transthoracic echocardiogram (TTE) was performed to evaluate PH. Statistical analysis was performed using median, interquartile range, chi-squared test, and Kruskall-Wallis test. Results: Of the 55 patients (n: 55), 63.6 % were male, with a median age of 6 years, 14 children (25.5 %) were overweight; 12 children (21.8 %) had mild OSA, 12 (21.8 %) had moderate OSA and 31 (56.4 %) severe OSA. In patients with severe OSA, the minimum saturation during events was 78 % with a desaturation index (DI) of 33.8/hour (p < 0.01). T90 and T85 increased proportionally with OSA severity (p < 0.05). Of the 55 patients with OSA, none had PH according to echocardiography; 4 patients (7.2 %) had pulmonary artery systolic pressure (PASP) at the upper limit of normal (ULN), and it was not related to a higher body mass index (BMI). Conclusions: We found no association between OSA and PH in children with OSA at high altitude. Keywords: Children; High altitude; Obstructive sleep apnea; Prevalence; Pulmonary hypertension. © 2024 The Authors. PubMed Disclaimer Conflict of interest statement The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Figures Fig. 1 Patient recruitment flowchart. Fig. 1 Patient recruitment flowchart. Fig. 1 Patient recruitment flowchart. See this image and copyright information in PMC Similar articles Prevalence of pulmonary hypertension on echocardiogram in children with severe obstructive sleep apnea.Maloney MA, Ward SLD, Su JA, Durazo-Arvizu RA, Breunig JM, Okpara DU, Gillett ES.Maloney MA, et al.J Clin Sleep Med. 2022 Jun 1;18(6):1629-1637. doi: 10.5664/jcsm.9944.J Clin Sleep Med. 2022.PMID: 35212261 Free PMC article. Sleep apnea and oxygen saturation in adults at 2640 m above sea level.Bazurto Zapata MA, Dueñas Meza E, Jaramillo C, Maldonado Gomez D, Torres Duque C.Bazurto Zapata MA, et al.Sleep Sci. 2014 Jun;7(2):103-6. doi: 10.1016/j.slsci.2014.09.003. Epub 2014 Sep 16.Sleep Sci. 2014.PMID: 26483911 Free PMC article. Prevalence of central sleep apnea during continous positive airway pressure (CPAP) titration in subjects with obstructive sleep apnea syndrome at an altitude of 2640 m.Bazurto Zapata MA, Martinez-Guzman W, Vargas-Ramirez L, Herrera K, Gonzalez-Garcia M.Bazurto Zapata MA, et al.Sleep Med. 2015 Mar;16(3):343-6. doi: 10.1016/j.sleep.2014.09.022. Epub 2015 Jan 28.Sleep Med. 2015.PMID: 25683561 Anti-inflammatory medications for obstructive sleep apnoea in children.Kuhle S, Hoffmann DU, Mitra S, Urschitz MS.Kuhle S, et al.Cochrane Database Syst Rev. 2020 Jan 17;1(1):CD007074. doi: 10.1002/14651858.CD007074.pub3.Cochrane Database Syst Rev. 2020.PMID: 31978261 Free PMC article. Association of obstructive sleep apnea and nocturnal hypoxemia with all-cancer incidence and mortality: a systematic review and meta-analysis.Tan BKJ, Teo YH, Tan NKW, Yap DWT, Sundar R, Lee CH, See A, Toh ST.Tan BKJ, et al.J Clin Sleep Med. 2022 May 1;18(5):1427-1440. doi: 10.5664/jcsm.9772.J Clin Sleep Med. 2022.PMID: 34755597 Free PMC article. See all similar articles Cited by Breathing patterns during sleep and their relationship with FEV1 in pediatric patients with cystic fibrosis residing at high altitude.Duenas-Meza E, Proaños-Jurado NJ, Pulido-Fentanes S, Severiche-Bueno DF, Escamilla-Gil MI, Bazurto-Zapata MA, Jurado JL, Suarez MR, Giraldo-Cadavid LF.Duenas-Meza E, et al.Front Pediatr. 2024 Aug 15;12:1360227. doi: 10.3389/fped.2024.1360227. eCollection 2024.Front Pediatr. 2024.PMID: 39210986 Free PMC article. References Marcus C.L., Brooks L.J., Draper K.A., Gozal D., Halbower A.C., Jones J., et al. Diagnosis and management of childhood obstructive sleep apnea syndrome. Pediatrics. 2012;130(3):e714–e755. - PubMed Blechner M., Williamson A.A. Consequences of obstructive sleep apnea in children. Curr Probl Pediatr Adolesc Health Care. 2016;46(1):19–26. - PubMed Parish J.M., Somers V.K. Obstructive sleep apnea and cardiovascular disease. Mayo Clin Proc. 2004;79(8):1036–1046. - PubMed Dyugovskaya L., Lavie P., Hirsh M., Lavie L. Activated CD8+ T-lymphocytes in obstructive sleep apnoea. Eur Respir J. 2005;25(5):820–828. - PubMed Phillips B.G., Narkiewicz K., Pesek C.A., Haynes W.G., Dyken M.E., Somers V.K. Effects of obstructive sleep apnea on endothelin-1 and blood pressure. J Hypertens. 1999;17(1):61–66. - PubMed Show all 58 references Related information MedGen [x] Cite Copy Download .nbib.nbib Format: Send To Clipboard Email Save My Bibliography Collections Citation Manager [x] NCBI Literature Resources MeSHPMCBookshelfDisclaimer The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). 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2167	https://spirometry.com/en/news/understanding-fvc-normal-range-in-spirometry/	03 Mar 25 Understanding FVC normal range in spirometry Spirometry is an insightful tool for assessing lung function and supporting the diagnosis and management of several respiratory diseases. One of its key measurements is Forced Vital Capacity, or FVC, an essential parameter for identifying potential lung conditions. In this article, we will explain how to interpret FVC results and compare them to normal values for a better understanding of spirometry. What is FVC? The acronym FVC stands for Forced Vital Capacity, referring to the total amount of air a person can forcefully exhale after taking a deep breath. FVC therefore provides insights into a patient’s lung capacity. In relation to other parameters, FVC is decisive in identifying obstructive and/or restrictive patterns, which may be caused by conditions like COPD (Chronic Obstructive Pulmonary Disease) or fibrosis. Is it correct to talk about normal FVC values? Although the expression FVC normal values is commonly used in popular language, a more precise form would be Reference range for FVC or Predicted FVC range. The expression normal range refers to the expected values based on factors such as age, sex, height, and ethnicity. Before a spirometry test, spirometers are usually calibrated on the basis of average values from healthy patients sharing the same anthropometric data. In clinical settings, such values are called predicted values, and the measurements related to the single parameters are often expressed as a percentage (% predicted) of the predicted values. A value of 80% or higher of the predicted is considered within the normal range. How FVC normal range may vary based on anthropometric data Let’s examine how the so-called normal FVC range is affected by different factors: Age. Lung capacity tends to decline with older age, as the lung tissue loses elasticity and the diaphragm gets weaker, especially in patients who don’t exercise regularly Gender. Males typically have higher FVC values than females, because women’s lungs are proportionally smaller Height. As surprising as it may sound, height plays a role in the definition of lung capacity, as taller individuals usually have greater lung volumes Ethnicity. Reference values can differ among ethnic groups, with Caucasians generally having higher volumes than Asians, because of an increased number of alveoli and larger chest cavities As a general reference, an acceptable FVC range may fall between 4.8 and 6.0 liters in adult males, and between 3.2 and 4.5 liters in adult females. How to interpret FVC results When it comes to spirometry results, FVC values are generally interpreted as follows: 80% predicted values or higher: no significant impairment 70-79% predicted values: mild lung restriction 60-69% predicted values: moderate restriction 50-59% predicted values: moderately severe restriction <50% predicted values: severe restriction FVC values in comparison to other spirometric parameters Besides being a means for measuring lung capacity, comparing FVC values to other measurements provides a more comprehensive frame of a patient’s lung functionality. Normal FVC and normal FEV1/FVC Ratio are generally associated with healthy lung function. Low FVC and normal or high FEV1/FVC Ratio may be indicators of restrictive patterns, often due to underlying conditions like pulmonary fibrosis, obesity, or neuromuscular diseases. Low FVC and low FEV1/FVC Ratio suggest the presence of airway obstructions, often caused by lung diseases like COPD or asthma. In conclusion… Forced Vital Capacity is one of the main measurements in spirometry, helping diagnose and monitor the progression of respiratory diseases. If you wish to learn more about spirometry results and how to interpret them, we explored the topic in this article. Share: How spirometry can help manage chronic bronchitis Respiratory therapy for improving quality of life in respiratory disorders Events ### ERS Congress 2025 27/09/2025 - 01/10/2025 Amsterdam, Netherlands ### Medbaltica 2025 25/09/2025 - 26/09/2025 Riga, Latvia ### SIMRI 2025 11/09/2025 - 13/09/2025 Verona, Italy View all Where to buy, how to order? Want to become a dealer? Learn more with our developer program Want to join the next webinar? Get in touch now mir@spirometry.com Facebook Instagram YouTube LinkedIn Scroll to top
2168	https://demonstrations.wolfram.com/IntersectingSecantsTheorem	Intersecting Secants Theorem \| Wolfram Demonstrations Project 13,000+ Open Interactive Demonstrations From the makers of Wolfram Language and Mathematica Wolfram Demonstrations Project Topics Latest Random Authoring Notebook Intersecting Secants Theorem P B×P A 7.0 0 P D×P C 7.0 0 Initializing live version Open Notebook in CloudCopy Manipulate to Clipboard Source Code Contributed by: Jay Warendorff(2011) Open content licensed under CC BY-NC-SA Snapshots External Links Chord (Wolfram MathWorld)» Secant Line (Wolfram MathWorld)» Permanent Citation Cite this as Jay Warendorff (2011), "Intersecting Secants Theorem" Wolfram Demonstrations Project. demonstrations.wolfram.com/IntersectingSecantsTheorem/ Related Demonstrations Intersecting Chords Theorem Jay Warendorff A Theorem about Two Tangent Circles Jay Warendorff Circle-Point Midpoint Theorem Jay Warendorff Miquel's Theorem Jay Warendorff Secant Intersection with Two Internally Tangent Circles Jay Warendorff The Chordal Theorem Jay Warendorff Carnot's Theorem Chris Boucher Thales' Theorem Michael Schreiber Angle Bisector Theorem Jay Warendorff Bottema's Theorem Jay Warendorff Concyclic Circumcenters Jay Warendorff Crossed Ladders Theorem Jay Warendorff Cross's Theorem Jay Warendorff Equally Distant Chords Jay Warendorff Four Concyclic Points Jay Warendorff, D. Grinberg Hjelmslev's Theorem Jay Warendorff Inscribed and Central Angles in a Circle Jay Warendorff Inscribed Angles That Intercept the Same Arc Jay Warendorff Inscribing Four Circles in a Triangle Jay Warendorff Related Topics High School Geometry Plane Geometry Browse all topics Wolfram LanguageMathematicaWolfram UCommunityMore» ©2025 Wolfram Demonstrations Project &Contributors \| Terms of Use \| Privacy Policy \| Give feedback
2169	https://www.investopedia.com/terms/r/retail-inventory-method.asp	Skip to content Top Stories Will Taylor Swift and Travis Kelce Get a Prenup? This New Senior Tax Break Could Lower Your Taxes Think Tiny Homes Save Big Money? Here’s the Real Cost 6 Important Money Conversations to Have with Your Partner Right Now Retail Inventory Method: Definition, Calculation, and Example By Will Kenton Full Bio Will Kenton is an expert on the economy and investing laws and regulations. He previously held senior editorial roles at Investopedia and Kapitall Wire and holds a MA in Economics from The New School for Social Research and Doctor of Philosophy in English literature from NYU. Learn about our editorial policies Updated August 15, 2024 Reviewed by Charlene Rhinehart Reviewed by Charlene Rhinehart Full Bio Charlene Rhinehart is a CPA , CFE, chair of an Illinois CPA Society committee, and has a degree in accounting and finance from DePaul University. Learn about our Financial Review Board What Is the Retail Inventory Method? The retail inventory method is an accounting method used to estimate the value of a store's merchandise. The retail method provides the ending inventory balance for a store by measuring the cost of inventory relative to the price of the merchandise. Along with sales and inventory for a period, the retail inventory method uses the cost-to-retail ratio. Key Takeaways The retail inventory method is an accounting method used to estimate the value of a store's merchandise. The retail method provides the ending inventory balance for a store by measuring the cost of inventory relative to the price of the goods. Along with sales and inventory for a period, the retail inventory method uses the cost-to-retail ratio. The retail method of valuing inventory only provides an approximation of inventory value since some items in a retail store will most likely have been shoplifted, broken, or misplaced. The retail inventory method is only an estimate and should always be supported by period physical inventory counts. Understanding the Retail Inventory Method Having a handle on your inventory is an important step in managing a successful business. It allows you to understand your sales, when to order more inventory, how to manage the cost of your inventory, as well as how much of your inventory is making it into the hands of consumers, as opposed to being stolen or broken. The retail inventory method should only be used when there is a clear relationship between the price at which merchandise is purchased from a wholesaler and the price at which it is sold to customers. For example, if a clothing store marks up every item it sells by 100% of the wholesale price, it could accurately use the retail inventory method, but if it marks up some items by 20%, some by 35%, and some by 67%, it can be difficult to apply this method with accuracy. The retail method of valuing inventory only provides an approximation of inventory value since some items in a retail store will most likely have been shoplifted, broken, or misplaced. It's important for retail stores to perform a physical inventory valuation periodically to ensure the accuracy of inventory estimates as a way to support the retail method of valuing inventory. Calculating Ending Retail Inventory The retail inventory method calculates the ending inventory value by totaling the value of goods that are available for sale, which includes beginning inventory and any new purchases of inventory. Total sales for the period are subtracted from goods available for sale. The difference is multiplied by the cost-to-retail ratio (or the percentage by which goods are marked up from their wholesale purchase price to their retail sales price). The cost-to-retail ratio, also called the cost-to-retail percentage, provides how much a good's retail price is made up of costs. If, for example, an iPhone costs $300 to manufacture and it sells for $500 each, the cost-to-retail ratio is 60% (or $300/$500) 100 to move the decimal. Disadvantages of the Retail Inventory Method The retail inventory method's primary advantage is the ease of calculation, but some of the drawbacks include: The retail inventory method is only an estimate. Results can never compete with a physical inventory count. The retail inventory method only works if you have a consistent markup across all products sold. The method assumes that the historical basis for the markup percentage continues into the current period. If the markup was different (as may be caused by an after-holiday sale), then the results of the calculation will be inaccurate. The method does not work if an acquisition has been made, and the acquiree holds large amounts of inventory at a significantly different markup percentage from the rate used by the acquirer. Example of the Retail Inventory Method Using our earlier example, the iPhone costs $300 to manufacture and it sells for $500. The cost-to-retail ratio is 60% ($300/$500 100). Let's say that the iPhone had total sales of $1,800,000 for the period. Beginning inventory: $1,000,000 New Purchases: $500,000 Total goods available for sale: $1,500,000 Sales: $1,080,000 (Sales of $1,800,000 x 60% cost-to-retail ratio) Ending inventory: $420,000 ($1,500,000 - $1,080,000) The offers that appear in this table are from partnerships from which Investopedia receives compensation. This compensation may impact how and where listings appear. Investopedia does not include all offers available in the marketplace. 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2170	https://www2.chemistry.msu.edu/courses/cem151/chap24lect_2019.pdf	Chapter 23 Chemistry of Coordination Compounds Coordination Compounds • How do we think about transition metals binding to other atoms? • What do those d orbitals do? • We call them, coordination compounds. Complexes • A central metal atom can bond to a group of molecules or ions: metal complex. • If it’s charged: complex ion. • Compounds containing complexes are coordination compounds. Question, is this an “ionic compound” Does it dissociate in water? Complexes • The molecules or ions coordinating to the metal are the ligands. • They are usually anions or polar molecules. • They must have lone pairs to interact with metal Ligands Complexes • Examples of some common ligands • Note, all have lone pairs • Some are charged, others are not. A chemical mystery: Same metal, same ligands, but different behavior • Same ligands, but different stochiometry • Same ligands, different ionic species when dissolved in water. • Many coordination compounds are brightly colored, but again, same metal, same ligands, different colors. Alfred Werner Father of coordination chemistry First inorganic chemist to win Nobel Prize • suggested in 1893 that metal ions have primary and secondary valences. ØPrimary valence equals the metal’s oxidation number ØSecondary valence is the number of atoms directly bonded to the metal (coordination number) Co(III) oxidation state Coordination # is 6 Cl-Werner’s Theory • The central metal and the ligands directly bonded to it make up the coordination sphere of the complex. • In CoCl3 ∙6 NH3, all six of the ligands are NH3 and the 3 chloride ions are outside the coordination sphere. Werner’s Theory In CoCl3 ∙5 NH3 five NH3 groups and one chlorine are bonded to the cobalt, the other two chloride ions are outside the sphere. Werner’s Theory Given this insight: Ligands inside brackets, part of coordination sphere, bonded directly to the metal. Those outside, float away in water, counterions Werner’s Theory • A good theory must predict. • Werner correctly predicts that there are two forms of CoCl3 ∙ 4 NH3. ØCorrect formula: Ø [Co(NH3)4Cl2]Cl. ØTwo stereoisomers ØTop, Cl’s next to each other ØBottom, Cl’s across from each other.. Oxidation Numbers Key issue in metal complexes Easily found using: Charge of the complex ion Charge of each of the ligands. Oxidation Numbers Or, knowing the oxidation number on the metal and the charges on the ligands, one can calculate the charge on the complex ion. Example: Cr(III)(H2O)4Cl2 + Electron configuration of metals in complexes. • Generally a transition metal has an electron configuration like this: • [n noble gas]ns2(n-1)dm • Example: • V: [Ar]4s23d3 • Pt: [Xe]6s24f145d8 Electron configuration of metals in complexes. • But, when transition metals lose electrons to become cations. They lose the s electrons first: • Example: ØV: [Ar]4s23d3 • V2+: [Ar]3d3 ØPt: [Xe]6s24f145d8 (we can drop the f’s, they don’t do anything) • Pt2+: [Xe]5d8 • Pt3+: [Xe]5d7 What is Coordination? • When an orbital from a ligand with lone pairs in it overlaps with an empty orbital from a metal So ligands must have lone pairs of electrons. Because metals don’t share no electrons! Why? Because they are cations. M L Sometimes called a coordinate covalent bond Metal d orbital The concept of a Lewis Acid • This bond is formed between a Lewis acid and a Lewis base. ØLewis base: electron donor. ØLewis acid: electron acceptor. OH H + H2O Lewis acid Lewis Base electron electron acceptor donor Metal-Ligand Bond • This bond is formed between a Lewis acid and a Lewis base. ØThe ligands (Lewis bases) have nonbonding electrons. ØThe metal (Lewis acid) has empty orbitals. ØThe concept of a Lewis acid/base is very important! What is Coordination? • There are 3 ways of looking at bonding in coordination compounds: ØValence bond theory ØLigand Field Theory (adaptation of MO theory) ØCrystal Field Theory (theory of pure electrostatic interactions So ligands must have lone pairs of electrons. Valence Bond theory • Just like we learned before. ØWe mix the atomic orbitals on the metal before we bond the ligands: • For Transition metals we have 9-14 valence orbitals Ø1 ns Ø5 (n-1)d Ø3 np Ø5 nd Valence Bond theory • Just like we learned before. ØWe mix the atomic orbitals on the metal before we bond the ligands: • For Transition metals we have 14 valence orbitals Ø1 ns Ø5 (n-1)d Ø3 np Ø5 nd if needed. Valence Bond theory • Example • Co(NH3)6 3+ • Co electron configuration: 4s23d7 • Co3+ electron configuration: 4s03d6 • Need six orbitals for six ligands so: ØHybridize 1 4s, 3 4p and 2 3d to give: Øsp3d2 orbitals. The 6 valence electrons of Co+3 sit in the other 3 d orbitals. Valence Bond theory • Example • Ni(NH3)6 2+ • Ni2+ electron configuration (8 valence electrons): • 4s23d6 ----à 4s03d8 • Need six orbitals for six ligands but: Ø4 3d orbitals are full, only 1 3d orbital left ØMust hybridize 1 4s, 3 4p and 2 4d to give: Øsp3d2 orbitals. The 8 electrons of Ni2+ sit in five 3d orbitals. Ligand Field theory (MO theory for coordination compounds) 3d 4s 4p Ti3+ 6NH3 Example: Ti(NH3)3+ Metal-Ligand Bond The metal’s ligands and geometry greatly alter its properties, such as color, ease of oxidation. Magnetic properties Etc. Coordination Number • The atom that supplies the lone pairs of electrons for the metal-ligand bond is the donor atom. • The number of these atoms is the coordination number. Coordination Number • Some metals, such as chromium(III) and cobalt(III), consistently have the same coordination number (6 in the case of these two metals). • The most commonly encountered numbers are 4 and 6. Geometries • Metal ions with d8 configuration are often 4 coordinate • There are two common geometries for metals with a coordination number of four: ØTetrahedral ØSquare planar Tetrahedral Square planar Why square planar? We’ll get to that Geometries By far the most-encountered geometry, when the coordination number is six, is octahedral. Polydentate Ligands • Some ligands have two or more donor atoms. • These are called polydentate ligands or chelating agents. • In ethylenediamine, NH2CH2CH2NH2, represented here as en, each N is a donor atom. • Therefore, en is bidentate. • 3 en’s give coordination number of six. Polydentate Ligands Ethylenediaminetetraacetate, mercifully abbreviated EDTA, has six donor atoms. Wraps around the central atom like an octopus Polydentate Ligands Chelating agents generally form more stable complexes than do monodentate ligands. Chelating Agents • Bind to metal ions removing them from solution. • Phosphates are used to tie up Ca2+ and Mg2+ in hard water to prevent them from interfering with detergents. 5------.. .. .. .. : : : : : : : : .. .. .. Chelating Agents • Porphyrins are complexes containing a form of the porphine molecule shown at right. • Important biomolecules like heme and chlorophyll are porphyrins. Chelating Agents Porphines (like chlorophyll a) are tetradentate ligands. Porphyrin Heme binds the oxygen in your blood Part of Hemoglobin molecule Hemoglobin tetramer has 4 hemes Oxygen binding causes conformational change Makes the other sites bind oxygen better Nomenclature of Coordination Compounds • coordination complex nomenclature: Øname the ligands as prefixes before the metal name. Nomenclature of Coordination Compounds • Cation appears first (as always) • Anion is named last. • Ligands are listed alphabetically before the metal. Prefixes ignored when alphabetizing. Nomenclature of Coordination Compounds • Anionic ligands end in “o”; • neutral ligands are not changed. • Prefixes = number of each ligand. • If the name of the ligand itself has such a prefix, alternatives like bis-, tris-, etc., are used. • If complex is anion, its ending is changed to -ate. • The oxidation number of the metal is given by a Roman numeral in parentheses after the metal. Nomenclature of Coordination Compounds Isomers Isomers have the same molecular formula, but either: Their bonding is different (structural isomers) or Their spatial arrangement is different (stereoisomers). Structural Isomers If a ligand (like the NO2 group at the bottom of the complex) has more than one donor atom (atom with lone pairs) as the donor atom, linkage isomers are formed. Is this a structural or geometric isomer? Structural Isomers If a ligand (like the NO2 group at the bottom of the complex) can bind to the metal with one or another atom as the donor atom, linkage isomers are formed. Is this a structural or geometric isomer? Structural, bonding different Structural Isomers • Some isomers differ in what ligands are bonded to the metal (coordination sphere) and which are not. • these are coordination-sphere isomers. • Example: • Three isomers of CrCl3(H2O)6 are ØThe violet [Cr(H2O)6]Cl3, ØThe green [Cr(H2O)5Cl]Cl2 ∙ H2O, and ØThe (also) green [Cr(H2O)4Cl2]Cl ∙ 2 H2O. Geometric isomers • Pt(NH3)2Cl2 • Has two geometric isomers, two chlorines and two NH3 groups are bonded to the platinum metal, but are clearly different. Øcis-Isomers have like groups on the same side. Øtrans-Isomers have like groups on opposite sides. of each atom the same Bonding the same Arrangement in space different Geometric isomers This compound binds DNA Kills rapidly dividing cancer cells This one doesn’t These include testicular cancer, ovarian cancer, cervical cancer, breast cancer, bladder cancer, head and neck cancer, esophageal cancer, lung cancer, mesothelioma, brain tumors and neuroblastoma. Stereoisomers • Other stereoisomers, called optical isomers or enantiomers, are mirror images of each other. • Just as a right hand will not fit into a left glove, two enantiomers cannot be superimposed on each other. Enantiomers A molecule or ion that exists as a pair of enantiomers is said to be chiral. Enantiomers • Physical properties of chiral molecules are identical (boiling point, freezing point, density, etc.) • One exception: Øinteraction of a chiral molecule with plane-polarized light. Enantiomers • A chiral compound will rotate plane polarized light. • If one enantiomer rotates the light 32° to the right, the other will rotate it 32° to the left. • Generally, only when 2 chiral things interact is there a difference in properties. Explaining the properties of transition metal coordination complexes 1. Magnetism 2. color Metal complexes and color The ligands of a metal complex effect its color Addition of NH3 ligand to Cu(H2O)4 changes its color Why does anything have color? Light of different frequencies give different colors We learned that elements can emit light of different frequency or color. But these coordination complexes are not emitting light They absorb light. How does that give color? Light can bounce off an object or get absorbed by an object No light absorbed, all reflected get white color All light absorbed, none reflected get black color What if only one color is absorbed? Complimentary color wheel If one color absorbed, the color opposite is perceived. Absorb Orange See Blue Absorb Red See Green [Ti(H2O)6]3+ Absorbs in green yellow. Looks purple. How is an absorption spectrum of a Compound measured? A spectrophotometer. So color comes from: (metal complexes) (element line spectra) How is light absorbed in a metal complex? Ligand Field theory: 2 possibilities 3d 4s 4p Ti(NH3)6 3+ 2. Electronic transition From ligand to metal orbital (“charge transfer”) 1. Metal d electron transition Ti 6NH3 Metal complexes and color But why do different ligands on same metal give Different colors? Why do different ligands change absorption? Addition of NH3 ligand to Cu(H2O)4 changes its color Model of ligand/metal bonding. Electron pair comes from ligand Bond very polarized. Assumption: interaction pure electrostatic. Crystal Field Theory Now, think of point charges being attracted to the positively Charged metal. What happens to the d orbitals? Ligand negative charge Is repelled by d electrons, d orbital energy goes up Ligands will interact with some d orbitals more than others Depends on relative orientation of orbital and ligand Ligands point right at lobes Orbitals go In these orbitals, the ligands are between the lobes Interact less strongly, Splitting due to ligand/orbirtal orientation. = 495 nm Absorption of light promotes an electron to a higher in E d orbital. D is E of the photon that can be absorbed. Different ligands interact more or less, Change energy spacing of d orbitals. Spectrochemical series (strength of ligand interaction) Cl- < F- < H2O < NH3 < en < NO2 - < CN-Increasing D Increasing D High field Low field Electron configurations of some octahedral complexes As Energy difference increases, electron configuration Changes. Hund’s rule breaks down because d orbitals are not degenerate “High spin” “Low spin” Co(III) is d6 The 2 choices for a d5 metal, high spin (more unpaired electrons) or low spin (more paired electrons) In tetrahedral complexes, orbitals are inverted. Again because of orientation of orbitals and ligands. D is always small, always high spin (less ligands) Tetrahedral Complexes Square planar complexes are different still Tetrahedral Square Planar D8 complexes like square planar KMnO4 K2CrO4 KClO4 Intense color can come from “charge transfer” Ligand electrons jump to empty metal orbitals No d orbitals in Cl, orbitals higher In energy Exam 4 Topics 1. Valence bond theory 2. Molecular orbital theory 3. Chapter 24, coordination chemistry 4. Chapter 25, Organic (a little) Valence bond theory: 1.Hybridization (mostly covered in last exam) 2.Double bonds due to overlap of atomic p orbitals (pi bonds) 3.Concept of delocalization what orbitals are overlaping in a delocalized system? Exam 4, MO theory and coordination compounds Chapter 9, end and Chapter 24. MO theory: Rules: • 1. The number of MO’s equals the # of Atomic orbitals • 2. The overlap of two atomic orbitals gives two molecular orbitals, 1 bonding, one antibonding • 3. Atomic orbitals combine with other atomic orbitals of similar energy. • 4. Degree of overlap matters. More overlap means bonding orbital goes lower in E, antibonding orbital goes higher in E. • 5. Each MO gets two electrons • 6. Orbitals of the same energy get filled 1 electron at a time until they are filled. Difference between pi and sigma orbitals End on Side to side. A typical MO diagram, like the one below. For 2p and 2s atomic orbital mixing. Oxygen O2 is Paramagnetic, why? Show me why. Exam 4 Chapter 23. Concentrate on the homeworks and the quiz! Terms: 1. Coordination sphere 2. Ligand 3. Coordination compound 4. Metal complex 5. Complex ion 6. Coordination 7. Coordination number Same ligands different properties? Figuring oxidation number on metal Polydentate ligands (what are they)? Only ethylene diamine will be used (en) NH2-CH2-CH2NH2 Isomers. structural isomers (formula same, bonds differ) geometric isomers (formula AND bonds same, structure differs) Stereoisomers: Chirality, handedness, Stereoisomers Explaining the properties of metal complexes Magnetism and color How does seeing color work? Absorb Orange See Blue Absorb Red See Green Addition of NH3 ligand to Cu(H2O)4 changes its color Different ligands on same metal give different colors dxy dyz dxz dz 2 dx 2 -y 2 Splitting of d orbitals in an octahedral ligand field Spectrochemical series (strength of ligand interaction) Cl- < F- < H2O < NH3 < en < NO2 - < CN-Increasing D Increasing D Know low spin versus high spin There is also splitting from tetrahedral And square planar. Know they are different, don’t remember exactly what square planer looks like.
2171	https://www.ime.usp.br/~lucchesi/publications/clm13.pdf	On the Number of Perfect Matchings in a Bipartite Graph∗ Marcelo H. de Carvalho Cl´ audio L. Lucchesi facom – ufms, Campo Grande, Brasil U. S. R. Murty University of Waterloo, Waterloo, Canada February, 2012 Abstract In this paper we show that, with 11 exceptions, any matching cov-ered bipartite graph on n vertices, with minimum degree greater than two, has at least 2n −4 perfect matchings. Using this bound, which is best possible, and McCuaig’s Theorem on brace generation, we show that any brace on n vertices has at least (n −2)2/8 perfect matchings. A bi-wheel on n vertices has (n−2)2/4 perfect matchings. We conjecture that there exists an integer N such that every brace on n ≥N vertices has at least (n −2)2/4 perfect matchings. Contents 1 Matching Covered Graphs 2 1.1 Function f(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Tight cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Bi-contractions and retracts . . . . . . . . . . . . . . . . . . . 5 ∗Supported by cnpq 2 Braces 6 2.1 McCuaig braces . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Removable edges and their indices . . . . . . . . . . . . . . . . 9 2.3 McCuaig’s Theorem on brace generation . . . . . . . . . . . . 10 2.4 Braces of orders six to twelve . . . . . . . . . . . . . . . . . . 12 3 A Linear Lower Bound for f(n) 15 3.1 The list E of exceptions . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Minimal graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Solitary edges . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 A generalization and its proof . . . . . . . . . . . . . . . . . . 18 4 A Quadratic Lower Bound for b(n) 23 1 Matching Covered Graphs For graph theoretical notation and terminology, we essentially follow Bondy and Murty . The order of a graph is the number of its vertices, and its size is the number of its edges. We denote the number of perfect matchings in a graph G by Φ(G), and the minimum degree of vertices of G by δ(G). A graph is matching covered if it is connected, has at least two vertices, and each of its edges lies in a perfect matching. In studying questions related to the number of perfect matchings in a graph one may clearly restrict one-self to matching covered graphs. Thus, all the graphs we consider here are matching covered graphs. The treatise by Lov´ asz and Plummer , and the seminal work by Lov´ asz on the matching lattice, contain the basic theory of matching covered graphs. Let G be a bipartite matching covered graph on n vertices. Voorhoeve showed that if G is 3-regular, then Φ(G) ≥( 4 3)n/2. (Establishing a more general conjecture of Lov´ asz and Plummer, Esperet et al have shown that any 2-connected cubic graph on n vertices has at least 2n/3656 perfect matchings.) As mentioned in the abstract, we shall show here that if δ(G) ≥ 3, and n is large enough, then Φ(G) ≥2n −4. We shall also show that if G is a brace, then Φ(G) ≥(n−2)2 8 . (For the deﬁnition of a brace, see Section 2.) For the convenience of the reader, we shall brieﬂy review here the termi-nology and results which are pertinent to this article. 2 1.1 Function f(n) We denote the class of all bipartite matching covered graphs with minimum degree at least three by F, and the subclass of those graphs on n vertices in F by Fn. For each even integer n ≥2, we deﬁne the function f(n) as min{Φ(G)}, where the minimum is taken over all graphs in Fn. It can be veriﬁed that f(2) = 3, f(4) = 5, f(6) = 6, and f(8) = 9. A graph G∗in Fn is extremal if Φ(G∗) = f(n). Figure 1 shows extremal graphs on two, four, six and eight vertices. (a) (b) (c) (d) Figure 1: (a) The Theta graph (b) P4 (c) K3,3 (d) B8 = P8 For each even integer n ≥6, let An be the graph obtained from a matching {u1u2, u3u4, . . . , un−3un−2}, by adjoining two vertices, h1 and h2, and then joining h1 to each of u1, u3, . . . , un−3 by a pair of multiple edges, and h2 to each of u2, u4, . . . , un−2 by a pair of multiple edges. Figure 2 shows a drawing of A10. (Compare the deﬁnition of An with the deﬁnition of bi-wheel Bn given in Section 2.1.) We note that An is a matching covered graph with minimum degree three, that is, for n ≥6, the graph An belongs to Fn. In the graph An, vertex h1 has degree n−2, and each edge incident with h1 is in two perfect matchings. Therefore, Φ(An) = 2n −4. Thus: f(n) ≤2n −4, for n ≥6 (1) This upper bound is not attained for n < 14. But we shall see later on that f(n) = 2n −4, for all n ≥14. 1.2 Tight cuts Let G be a matching covered graph. We shall refer to a subset of edges of G of the form ∂(X), where X is a nonempty proper subset of V , as a cut of G 3 u1 u2 u3 u4 u5 u6 u7 u8 h1 h2 A10 Figure 2: The graph A10 with X and X = V \ X as its shores. A cut is trivial if one of its shores consists of exactly one vertex. Given any cut C := ∂(X) of G, one may obtain two graphs G/X and G/X by shrinking the shores of C to single vertices; they are called the C-contractions of G. A cut C := ∂(X) of a matching covered graph G is tight if every perfect matching of G has exactly one edge in common with C. Trivial cuts are examples of tight cuts. As we shall see below, tight cuts in bipartite matching covered graphs may be described in a simple manner. We denote a bipartite graph G with bipartition (A, B) by G[A, B]. Let G := G[A, B] be a bipartite matching covered graph. If X is an odd subset of V then the larger of the two sets \|X ∩A\| and \|X ∩B\| is called the majority part of X and is denoted by X+, and the other is called the minority part and is denoted by X−. If X is a subset of V such that \|X+\| = \|X−\| + 1 and all edges of ∂(X) have one end in X+ and one end in X+, then ∂(X) is a tight cut of G. It is not diﬃcult to see that every tight cut of a bipartite matching covered graph is of this form. Let F and H be two given matching covered graphs. A graph G is called a splicing of F and H, and is denoted by F ⊙H, if there is a tight cut ∂(X) of G such that F ∼ = G/X and H ∼ = G/X. In general, there may be many graphs G with this property. But in cases of interest to us here, F ⊙H is unique up to isomorphism. The two graphs in the top row of Figure 3 are P4 ⊙K3,3 and P4 ⊙B8, where P4 is the graph shown in Figure 1(b), and B8 is the cube, shown in Figure 1(d). The two graphs in the bottom row are 4 X X X X Figure 3: Top: P4 ⊙K3,3 and P4 ⊙B8; bottom: K3,3 ⊙K3,3 and K3,3 ⊙B8 K3,3 ⊙K3,3 and K3,3 ⊙B8. Observation 1.1 Since Φ(P4 ⊙K3,3) = 10 < 2 × 8 −4 = 12, Φ(P4 ⊙B8) = 15 < 2 × 10 −4 = 16, Φ(K3,3 ⊙K3,3) = 12 < 2 × 10 −4 = 16, and Φ(K3,3 ⊙B8) = 18 < 2 × 12 −4 = 20, we note that the four graphs in Figure 3 will have to be in the list of exceptions mentioned in the abstract. 1.3 Bi-contractions and retracts Vertices of degree two in a matching covered graph give rise to a very simple type of tight cuts. Let v be a vertex of degree two in a matching covered graph G, and let X := {v, v1, v2}, where v1 and v2 are the two neighbours of v. Then ∂(X) is a tight cut of G, and the the graph G/X is said to be obtained from G by the bi-contraction of v. Lemma 1.2 Let G be a graph on four or more vertices, let v be a vertex of degree two in G, and let H be the graph obtained from G by bi-contracting v. Then, Φ(G) = Φ(H). Proof: Every perfect matching of H has a unique extension to a perfect matching of G. Conversely, the restriction of any perfect matching of G is a perfect matching of H. ✷ If a graph G has at least three vertices and is not a cycle, then one can obtain a graph of minimum degree greater than two from G by means of 5 bi-contractions. Up to isomorphism, the graph so obtained does not depend on the sequence of bi-contractions performed (see ). We refer to it as the retract of G, and denote it by b G. Corollary 1.3 For any graph G, Φ(G) = Φ( b G). ✷ We denote the number of perfect matchings of a graph G containing a given edge e by Φe(G). If u and v are the two ends of e, then clearly Φe(G) = Φ(G −{u, v}). The following simple lemma plays a useful role in the proofs of many results. Lemma 1.4 (Recursion Lemma) Let G be a graph and let e = uv be any edge of G. Then Φ(G) = Φ(\ G −e) + Φ(G −{u, v}) (2) Proof: The number of perfect matchings not containing the edge e is equal to Φ(G −e), but Φ(G −e) = Φ(\ G −e). On the other hand, the number Φe(G) of perfect matchings containing e is Φ(G −{u, v}). Hence the identity. ✷ Lemma 1.5 Let G be a graph in F, let v be a vertex of G, let d denote the degree of v. Then, Φ(G) = 1 d −1 X e∈∂(v) \ Φ(G −e). Proof: The following equalities hold: X e∈∂(v) \ Φ(G −e) = (d −1) X e∈∂(v) Φe(G) = (d −1)Φ(G). ✷ 2 Braces A matching covered graph without nontrivial tight cuts is called a brace if it is bipartite, and a brick if it is nonbipartite. The theorem below provides a characterization of braces: 6 Theorem 2.1 (See , ) Let G[A, B] be a bipartite matching covered graph. The following statements are equivalent: (a) G is a brace; (b) G −a1 −a2 −b1 −b2 has a perfect matching, for any two vertices a1 and a2 in A and any two vertices b1 and b2 in B; (c) for every subset X of A such that 0 < \|X\| < \|A\|−1, we have \|N(X)\| > \|X\| + 1. ✷ A graph G is said to be 2-extendable if it has two non-adjacent edges and for any two non-adjacent edges e and f graph G has a perfect matching which includes {e, f}. Theorem 2.1 implies that every brace of order four or more is 2-extendable. It also implies that every vertex of a brace of order six or more has at least three distinct neighbors. The following observation will be found useful later on. The only simple cubic bipartite graphs on six and eight vertices are, respectively, K3,3 and the cube (see Figure 1(d)). Both of them happen to be braces. However, this not the case for larger orders. The M¨ obius ladder M10, introduced in the next subsection, is a cubic brace of order 10, and there is a simple cubic bipartite graph of order 10 which is diﬀerent from it. Proposition 2.2 The only simple cubic bipartite graph of order 10 that is not a brace is K3,3 ⊙K3,3. Proof: Let G be a simple cubic bipartite graph, and let ∂(X) be a nontrivial tight cut of G. Since each vertex has three distinct neighbours, and the vertices of X−are joined only to those in X+, it follows that \|X\| ≥5. Similarly, \|X\| ≥5. Thus, \|X\| = 5 = \|X\|. It is now easy to deduce that G is K3,3 ⊙K3,3. ✷ 2.1 McCuaig braces There are three families of braces of special importance, namely prisms, M¨ obius ladders and bi-wheels. McCuaig showed that all braces of or-der at least six may be generated from these special braces by using three 7 elementary operations. (We shall brieﬂy review McCuaig’s result in Sec-tion 2.3.) Prisms: The prism P2r of order 2r, r ≥2, is the cartesian product of the r-cycle Cr and the complete graph K2. For each even integer r ≥2, the prism P2r is a brace. Figures 1(b) and (d) depict the prisms P4 and P8. Figure 4(a) depicts the prism P12. Consider the sequence F1 = 1, F2 = 1, F3 = 2, F4 = 3, F5 = 5, F6 = 8, F7 = 13, F8 = 21, . . ., of Fibonacci numbers deﬁned by F1 = F2 = 1 and by the recursion Fi = Fi−1 + Fi−2, for i ≥3. Using simple inductive arguments, it is easy to see that the number of perfect matchings in the prism P2r may be expressed in terms of Fibonacci numbers as follows: Φ(P2r) = Fr+1 + Fr−1 + 2 (3) In particular, Φ(P12) = F7 + F5 + 2 = 13 + 5 + 2 = 20. (a) (b) (c) Figure 4: (a) P12 (b) M10 (c) B10 M¨ obius ladders: The M¨ obius ladder M2r, r ≥3, is the cubic graph obtained from the 2r-cycle by joining each vertex of the cycle to the vertex that is antipodal to it. For each odd integer r ≥3, the M¨ obius ladder M2r is a brace. Furthermore, Φ(M2r) = Fr+1 + Fr−1 + 2 (4) The M¨ obius ladder M6 is isomorphic to K3,3. Figure 4(b) depicts the M¨ obius ladder M10. It follows from (4) that Φ(M10) = F6 +F4 +2 = 8+3+2 = 13. Bi-wheels: For r ≥4, the bi-wheel B2r of order 2r is the graph obtained from a (2r −2)-cycle (u1, u2, . . . , u2r−2, u1), called the rim, by adjoining 8 two vertices, h1 and h2, called the hubs, and then joining h1 to each of u1, u3, . . . , u2r−3, and h2 to each of u2, u4, . . . , u2r−2. (Figure 4(c) depicts the bi-wheel B10.) It is easy to verify that B2r is a brace and that any edge incident with h1 and any edge incident with h2 are contained together in a unique perfect matching of B2r. Consequently, Φ(B2r) = (r −1)2 (5) The cube, which we noted is a prism, is also a bi-wheel, the smallest bi-wheel B8. Using the above formula, we have Φ(B8) = (4 −1)2 = 9. Extended bi-wheels: The brace obtained by adding an edge to B2r joining its two hubs is denoted by B+ 2r and is called an extended bi-wheel. This auxiliary family of braces also plays an important role in McCuaig’s work . Figure 5 shows drawings of B+ 8 and B+ 10. It is easy to see that Φ(B+ 2r) = (r −1)2 + 2. In particular, Φ(B+ 8 ) = 9 + 2 = 11. h1 h1 h2 h2 Figure 5: The extended bi-wheels B+ 8 and B+ 10 2.2 Removable edges and their indices An edge e of a matching covered graph G is removable if G−e is also matching covered. Suppose that the order of G is at least four. In this case, if e is removable, then both ends of e have degree at least three in G. The index of a removable edge is the number of its ends which have degree exactly three. Thus, the index of e is zero if both ends of e have degree greater then three, one if exactly one end of e has degree three, and two if both ends of e have degree three. The following statement concerning braces may be deduced from Theo-rem 2.1. 9 Corollary 2.3 Every edge in a brace of order six or more is removable. ✷ If G is a bipartite matching covered graph, then it is known (see ) that the dimension of the linear space generated by the incidence vectors of perfect matchings of G is \|E(G)\| −\|V (G)\| + 2, implying that: Φ(G) ≥\|E(G)\| −\|V (G)\| + 2 (6) The proof of the next lemma uses this fact. However, the above lower bound is rarely attained. A characterization of bipartite graphs for which equality holds in (6) is given in . Lemma 2.4 Let G = G[A, B] be a brace of order 2r ≥6, and let e = uv be a any edge of positive index of G, with u ∈A and v ∈B. Then H := G −{u, v} is matching covered, and Φ(H) ≥r −1. Proof: The fact that H is matching covered follows from the fact that G is 2-extendable. Since G has positive index, at least one of the ends of e has degree two in G −e. Without loss of generality, suppose that v has degree two in G −e, and let u1 and u2 denote the neighbours of v in A −u. Then dH(u1) = dG(u1) −1 ≥2, dH(u2) = dG(u2) −1 ≥2, and for u′ ∈A −u −u1 −u2, dH(u′) = dG(u′) ≥3. We therefore have: \|E(H)\| ≥2 + 2 + 3(r −3) = 3r −5 Clearly, \|V (H)\| = 2r −2. Thus, using (6), we have Φ(H) ≥(3r −5) −(2r − 2) + 2 = r −1. ✷ We note that the assumption that e has positive index is necessary. For example, if G is the extended bi-wheel B+ 2r, and e = h1h2 is the edge joining the two hubs, then Φ(B+ 2r −{h1, h2}) = 2, for all r. 2.3 McCuaig’s Theorem on brace generation Let G be a brace on six or more vertices, let e be an edge of G. We note that the number of bi-contractions required to obtain \ G −e from G −e is equal to the index of e. Edge e is thin if \ G −e is a brace, and is strictly thin 10 if \ G −e is a simple brace. The three edges e0, e1 and e2 in Figure 6 are, respectively, thin edges of index zero, one and two in that brace. Edges e0 and e1 are strictly thin, but e2 is not. e0 e1 e2 Figure 6: Edges e0 and e1 are strictly thin, but e2 is not Prisms, M¨ obius ladders, and bi-wheels do not have any strictly thin edges. We shall refer to braces in these families as McCuaig braces. In McCuaig proved the following fundamental theorem. (In fact, he proved a stronger (splitter) version of the statement given here.) Theorem 2.5 (The strictly thin edge theorem) Every simple brace of order six or more which is not a McCuaig brace has a strictly thin edge. A brace G is called an extension of index i of another brace H if G has a thin edge e of index i such that H = \ G −e. As an immediate consequence of the above theorem, we have: Theorem 2.6 (Brace generation theorem) Given any simple brace G of order at least six, there exists a sequence (G1, G2, . . . , Gr) of simple braces such that: (i) G1 is a McCuaig brace, (ii) Gr = G, (iii) for 1 ≤i ≤r −1, Gi+1 is an extension of Gi. 11 McCuaig braces do not have strictly thin edges, but they do have thin edges. Thus we may conclude from McCuaig’s Theorem the following: Theorem 2.7 (The thin edge theorem) Every brace of order six or more has a thin edge. We gave a direct proof of the above theorem in (unpublished) and showed how Theorem 2.5 may be deduced from it. (Our paper contains the proof of an analogous theorem for bricks. An extension of that result has been published by Norine and Thomas ). 2.4 Braces of orders six to twelve Braces of order six: The only simple brace of order six is K3,3, and it has six perfect matchings. Observation 2.8 Since Φ(K3,3) = 6 < 2×6−4 = 8, we note that K3,3 will have to be in the list of exceptions mentioned in the abstract. Braces of order eight: Every simple brace of order eight is obtained by deleting the edges of a matching from K4,4, and thus the sizes (numbers of edges) of such braces range from 12 to 16. The one of size twelve is the bi-wheel B8 (which is the cube), the one of size thirteen is the extended bi-wheel B+ 8 . We have seen that Φ(B8) = 9, and Φ(B+ 8 ) = 11. The following proposition is easily veriﬁed. Proposition 2.9 Every simple brace of order eight, other than B8, and B+ 8 , has at least 14 perfect matchings. Observation 2.10 Since 2 × 8 −4 = 12, we note that the bi-wheel B8, with 9 perfect matchings, and the extended bi-wheel B+ 8 , with 11 perfect matchings, will have to be in the list of exceptions. McCuaig’s Stricly This Edge Theorem 2.5 plays a crucial role in the study of braces of order more than eight. Braces of order 10: Lemma 2.11 In a brace of order 10, no thin edge of index two is strictly thin. 12 Proof: Let G be a brace of order 10, and let e be a thin edge in G of index two. Then both ends of e have degree three in G, and \|V (\ G −e)\| = \|V (G)\|−4 = 6 and \|E(\ G −e)\| = \|E(G)\| −5 ≥10. A simple bipartite graph on six vertices has at most nine edges. Hence, the brace \ G −e is not simple, and e is not strictly thin. ✷ Corollary 2.12 The M¨ obius ladder M10 is the only cubic brace on 10 vertices. Proof: Let G be a cubic brace on 10 vertices. Since all degrees of G are three, G cannot have any strictly thin edges of index zero or of index one. By Lemma 2.11, G cannot have any strictly thin edges of index two either. Thus, by the Strictly Thin Edge Theorem, G is either M10 or B10. But B10 is not cubic. ✷ Lemma 2.13 The only simple braces of order 10 and size 16 are M + 10 and (K3,3 ⊙K3,3)+ (shown in Figure 7), and B10. u u v v (a) (b) Figure 7: (a) M + 10, (b) (K3,3 ⊙K3,3)+ Proof: Let G be a simple brace of order 10 and size 16. Then, clearly, G has two vertices u and v of degree four, one in each part of the bipartition of G, and all the remaining vertices have degree three. Suppose ﬁrst that u and v are adjacent. If G −uv is a brace then, by Corollary 2.12, G is isomorphic to the graph M + 10. If G −uv is not a 13 brace then, by Proposition 2.2, G −e is = K3,3 ⊙K3,3, and G is the graph (K3,3 ⊙K3,3)+ obtained by adding an edge to K3,3 ⊙K3,3. Now suppose that u and v are not adjacent. It follows from Theo-rem 2.1(b) that graph G −u −v is matching covered. As every vertex of G −u −v has degree two, G −u −v is a cycle on eight vertices. Thus G is the bi-wheel B10 with u and v as its hubs. ✷ Observation 2.14 Note that 2 × 10 −4 = 16. Since Φ(M10) = 13, and Φ((K3,3 ⊙K3,3)+) = 15, the two braces M10 and (K3,3 ⊙K3,3)+ will have to be included in the list of exceptions. Braces of order twelve: Lemma 2.15 There are three cubic braces of order twelve, namely the prism P12, and the two braces G1 12 and G2 12 shown in Figure 8. (a) (b) Figure 8: (a) G1 12 (b) G1 12 Proof: Let G be a cubic brace of order twelve. Clearly G cannot have any strictly thin edges of index zero or one. If it has no strictly thin edges of index two either, then by the Strictly Thin Edge Theorem, G is P12. So, suppose that G has a strictly thin edge e of index two. Then \|V (\ G −e)\| = 12−4 = 8, and \|E(\ G −e)\| = 18−5 = 13. It follows that \ G −e is the extended bi-wheel B+ 8 . Thus G is an extension of index two of B+ 8 . It can be veriﬁed that G1 12 and G2 12 are the only such extensions. ✷ 14 Observation 2.16 The numbers of perfect matchings of G1 12 and G2 12 are, respectively, 17 and 18. Since 2 × 12 −4 = 20, it follows that G1 12 and G2 12 will have to be included in the list of exceptions. 3 A Linear Lower Bound for f(n) 3.1 The list E of exceptions Recall that F is the class of all bipartite matching covered graphs with min-imum degree at least three. We deﬁne E to be the family of 11 graphs in F consisting of: • the brace K3,3 of order six (Figure 1(c)); • the graph P4⊙K3,3 (Figure 3, top left), and the braces B8 (Figure 1(d)) and B+ 8 (Figure 5, left) of order eight; • the graphs P4 ⊙B8 (Figure 3, top right) and K3,3 ⊙K3,3 (Figure 3, bottom left), and the braces M10 (Figure 4(b)) and (K3,3 ⊙K3,3)+ (Figure 7(b)) of order ten; and • the graph K3,3 ⊙B8 (Figure 3, bottom right), and the the braces G1 12 and G2 12 (Figure 8) of order twelve. It would be useful to keep in mind that all members of E, except B+ 8 and (K3,3 ⊙K3,3)+, are cubic. We have observed that if G ∈E, then Φ(G) < 2\|V (G)\| −4. We wish to prove the following theorem: Theorem 3.1 Any graph G in F \ E has at least 2\|V (G)\| −4 perfect matchings. In order to prove Theorem 3.1 by induction, it turns out to be convenient to prove a more general assertion. The notions of minimal graph and solitary edge play an important role in this more general setting. 3.2 Minimal graphs A graph G in F is minimal if there is no edge e of G such that G−e belongs to F. As an immediate consequence of the deﬁnition we have: 15 Lemma 3.2 If G is a minimal graph in F then every removable edge of G is incident with a vertex of degree three. ✷ Corollary 3.3 If G is a minimal graph in F and e and f are multiple edges of G then at least one end of e has degree three in G. ✷ Lemma 3.4 Let G be a graph in F. If G is minimal and simple then it has a vertex v of degree three such that each edge in ∂(v) is removable in G. Proof: The minimality of G implies that every removable edge of G is incident with a vertex of degree three. Consider ﬁrst the case in which G is a brace. In that case, every edge of G is removable and the assertion holds. Assume thus that G is not a brace. Let X be a minimal set of vertices of G such that C := ∂(X) is a non-trivial tight cut of G. Then, H := G/X is a brace. Let X+ and X−denote, respectively, the majority and the minority parts of X. Every edge of G having both ends in X is removable in H, hence removable in G. To complete the proof, we now show that X−has a vertex of degree three. Consider ﬁrst the case in which X+ has a vertex w of degree four or more. Every vertex of X+ is adjacent to some vertex of X−. Every edge having both ends in X is removable. Every vertex of X−adjacent to w has degree three. The assertion holds in this case. Alternatively, assume that every vertex of X+ has degree three. Then, \|∂(X−)\| = \|∂(X+)\| −\|C\| ≤3\|X+\| −2 = 3\|X−\| + 1. The simplicity of G implies that \|X+\| ≥3. Thus, \|X−\| ≥2. Then, the average degree of vertices in X−is 3.5 or less. Consequently, at least one vertex of X−has degree three. ✷ Lemma 3.5 Let G be a minimal brace in F of order 2r ≥6. If G has a vertex of degree at least four then Φ(G) ≥2\|V (G)\| −4. 16 Proof: Let u be a vertex of degree at least four in G. By Corollary 2.3, every edge of G is removable. As G is minimal, every neighbor of u has degree three. Thus, every edge incident with u has index one. By Lemma 2.4, every edge incident with u lies in at least r −1 perfect matchings. Therefore, Φ(G) ≥4(r −1) = 2(2r) −4 = 2\|V (G)\| −4. ✷ 3.3 Solitary edges For an edge e of a graph G, recall that we denote the number of perfect matchings of G containing e by Φe(G). We say that an edge e of G is solitary if Φe(G) = 1. A brace of order six or more cannot have any solitary edges, but non-braces in F may have any number of them. Consider, for example, the graph A(k) n obtained from An (described in Section 1) by joining the two vertices h1 and h2 by k multiple edges. Then each of those k edges is a solitary edge. The following property of solitary edges will be found to be very useful. Lemma 3.6 Let G be a graph in F and let e = uv be a solitary edge of G. If G has four or more vertices then there exists a vertex w in V (G)−u−v that has degree one in G −u −v, and is joined to v by two or more parallel edges in G. Proof: Since G has at least four vertices, H := G −u −v has at least two vertices. By hypothesis e is solitary, hence H has precisely one perfect matching. Let M be the perfect matching of H, and let P be a maximal M-alternating path in H. Path P has odd length, its ends lie in distinct color classes of G. Moreover, both ends of P have degree one in H. Let w be the end of P that lies in the same colour class of u in G. Then, w is joined to v by two or more edges. ✷ Using the above lemma, we deduce that if G is a graph in F having solitary edges then G has two non-adjacent sets of parallel edges. One may then easily verify the following corollary: 17 Corollary 3.7 Let H be a graph in the family E. Then both H and H + e, where H + e is any graph obtained by adding an edge e joining two vertices of H, are free of solitary edges. ✷ For a graph G, we denote by µ(G) the maximum multiplicity of solitary edges; in other words, µ(G) is the maximum number of parallel solitary edges joining any pair of vertices of G. If G has no solitary edges, then µ(G) = 0. In the graph A(k) n described in Section 3.3, the k parallel edges joining h1 to h2 are solitary, but no other edge is solitary. Therefore, µ(A(k) n ) = k. Note that Φ(A(k) n ) = 2n −4 + µ(A(k) n ) . 3.4 A generalization and its proof We are now ready to state and prove an assertion that is more general than Theorem 3.1. Theorem 3.8 Any graph G in F \ E has at least 2\|V (G)\| −4 + µ(G) perfect matchings. Proof: By induction on \|E(G)\|. Let G be a graph of order n in F\E. Assume inductively that if H is any graph in F \ E with \|E(H)\| < \|E(G)\|, then Φ(H) ≥2\|V (H)\| −4 + µ(H). We shall deduce that the asserted inequality holds for G by analyzing various cases. Case 1 Graph G is not minimal. By the deﬁnition of minimality, there exists an edge e of G such that H := G −e ∈F. Case 1.1 Suppose ﬁrst that H is not in E. Then, by induction hypothesis, Φ(H) ≥2n −4 + µ(H). Every edge of G distinct from e that is solitary in G is also solitary in H. Thus, µ(H) ≥ µ(G) −1. Therefore, Φ(G) ≥Φ(H) + 1 ≥2n −4 + µ(H) + 1 ≥2n −4 + µ(G), showing that the assertion holds in this case. Case 1.2 Now suppose that H is in E. 18 If H is a graph in E other than B8, K3,3 ⊙K3,3, M10 and G1 12 then, as we have seen, the diﬀerence between Φ(H) and 2\|V (H)\| −4 is at most two. By Corollary 3.7, µ(G) = 0, that is, e is not solitary in G. Thus, Φ(G) ≥ 2\|V (G)\| −4 in this case. If H = B8 then either G = B+ 8 and belongs to E or e is a parallel edge and Φ(G) = 12 = 2\|V (G)\| −4. If H is M10 or G1 12 then the diﬀerence between Φ(H) and 2\|V (H)\| −4 is three, and if H is K3,3 ⊙K3,3 then the diﬀerence between Φ(H) and 2\|V (H)\| −4 is four. One can now verify that if H is M10 or G1 12 then e lies in at least three perfect matchings, and if H is K3,3 ⊙K3,3 then G is (K3,3 ⊙K3,3)+, a member of E, or e lies in at least four perfect matchings. Case 2 Graph G is minimal but has multiple edges. Let us ﬁrst suppose that µ(G) > 1. In this case, we assert that G is the theta graph (shown in Figure 1(a)). Let e′ and e′′ be two parallel edges that are solitary. By the minimality of G, at least one end of e′ has degree three; otherwise G −e′ ∈F, in contradiction to the minimality of G. By Lemma 3.6, G has only two vertices. We deduce that G is the theta graph in this case. Then, Φ(G) = µ(G) = 3 and the assertion holds. We may thus assume that µ(G) ≤1. Then, G has more than two vertices. By the hypothesis of the case, G has multiple edges. Let e′ and e′′ be two parallel edges of G. If possible, choose e′ and e′′ so that they are adjacent to some solitary edge of G. Let u1 and v1 denote the ends of e′ and e′′. The minimality of G implies that at least one of u1 and v1 has degree three. Adjust notation so that u1 has degree three. Then, e′ and e′′ are the only edges that join u1 and v1. Let u1v2 be the edge of G incident with u1 but not with v1 (Figure 9). Let X := {u1, v1, v2}. Clearly, C := ∂(X) is a tight cut of G. For i = 1, 2, let Ci denote the set of edges of C that are incident with vertex vi. Let H := G/X. Clearly, C1 is non-empty and C2 has two or more edges. Thus H ∈F. Case 2.1 Graph H does not lie in E. For i = 1, 2, let si denote the number of edges in Ci that are solitary in H. Then, 19 e′ e′′ X C1 C2 u1 v1 v2 Figure 9: The case in which G has multiple edges Φ(G) = Φ(H) + X f∈C2 Φf(H), X f∈C2 Φf(H) ≥2\|C2\| −s2 ≥4 −s2, Φ(H) ≥2(n −2) −4 + s1 + s2 = 2n −8 + s1 + s2, where the last inequality follows by induction. Addition and simpliﬁcation yields Φ(G) ≥2n −4 + s1. To complete the analysis of the case, we prove that s1 ≥µ(G). This inequality certainly holds if µ(G) = 0. We have assumed that µ(G) ≤1. We may thus assume that µ(G) = 1. By Lemma 3.6 and the criterion used for choosing e′ and e′′, it follows that a solitary edge of G is incident with one of u1 and v1. If u1v2 is solitary in G then v1 also has degree three and the edge of C1 is solitary. Alternatively, if u1v2 is not solitary then, by the choice of e′ and e′′, some edge of C1 is solitary in G. In both alternatives, we may assume that some edge of C1 is solitary in G. Every perfect matching of H is extendable to a perfect matching of G. Thus, every solitary edge of G that lies in E(H) is solitary in H. We conclude that some edge of C1 is solitary in H. In other words, s1 > 0. The assertion holds in this case. Case 2.2 Graph H is in E. 20 By Corollary 3.7, H is free of solitary edges. Every edge of H that is solitary in G is also solitary in H. Edges e′ and e′′ are not solitary in G. Edge u1v2 is not solitary, otherwise C1 would consist of a single edge, an edge of H solitary in G. We conclude that G is free of solitary edges. We must thus prove that Φ(G) ≥2\|V (G)\| −4. Deﬁne the parameter φ2(H) to be the minimum of Φe(H)+Φf(H), where the minimum is taken over all pairs {e, f} of adjacent edges of H. Clearly, Φ(G) ≥Φ(H) + φ2(H) (7) If H is either K3,3 or B8, then G also belongs to E. So, we need only examine the other nine graphs in E. For each of those graphs, the parameter φ2 can be computed, and it can be veriﬁed that Φ(G) ≥2\|V (G)\| −4. The details are included in Table 1. Graph H Φ(H) φ2(H) 2\|V (G)\| −4 P4 ⊙K3,3 10 6 16 B+ 8 11 5 16 P4 ⊙B8 15 9 20 K3,3 ⊙K3,3 12 8 20 (K3,3 ⊙K3,3)+ 15 7 20 M10 13 8 20 K3,3 ⊙B8 18 12 24 G1 12 17 11 24 G2 12 18 12 24 Table 1: Values of Φ(H), φ2(H) and 2\|V (G)\| −4 Case 3 Graph G is minimal and free of multiple edges. We remark that as G is free of multiple edges it is also free of solitary edges, by Lemma 3.6. We must thus prove that Φ(G) ≥2n −4. Every graph in F having fewer than six vertices has multiple edges. Thus, G has six or more vertices. The brace K3,3 is the only simple graph on six vertices in F. But K3,3 is a member of E. Thus, G has eight or more vertices. Every graph in F on eight vertices and free of multiple edges is a brace. Every brace on eight vertices includes the cube B8 as a subgraph. The minimality of G then 21 implies that G is B8. But B8 is a member of E. We deduce that G has more than eight vertices. Case 3.1 n = 10. Consider ﬁrst the case in which G has a non-trivial tight cut C := ∂(X). The absence of multiple edges in G implies that H := G/X is K3,3, up to multiple edges in C. Every edge in C lies in two perfect matchings of H. Likewise, H′ := G/X is K3,3 up to multiple edges in C and each edge in C lies in two perfect matchings of H′. We deduce that every edge of C lies in four perfect matchings of G. As C is tight in G it follows that Φ(G) = 4\|C\|. But 2n−4 = 16. Thus, the asserted inequality holds, unless \|C\| = 3. In that case, G is K3,3 ⊙K3,3, which is a member of E. We consider now the case in which G is a brace. If G is cubic, then by Corollary 2.12, G is M10. But M10 belongs to the list E. If G is not cubic then by Lemma 3.5, Φ(G) ≥2n −4. Case 3.2 n = 12. Consider ﬁrst the case in which G is not a brace. The simplicity of G implies that it has a non-trivial tight cut C := ∂(X), where H := G/X is K3,3, up to multiple edges of C. Let H′ := G/X. • If H′ ̸∈E then, by induction hypothesis, Φ(H′) ≥12, and Φ(G) = 2Φ(H′) ≥24 > 2n −4; • If H′ = P4 ⊙K3,3 then Φ(H′) = 10 and Φ(G) = 2Φ(H′) = 20 = 2n −4; • If H′ = B8 then G is the graph K3,3 ⊙B8 which is a member of E; • If H′ = B+ 8 , then Φ(H′) = 11, and Φ(G) ≥2Φ(H′) = 22 > 20 = 2n−4. Hence the assertion holds for graphs of order 12 that are not braces. We now consider the case in which G is a brace. If G is not cubic then, by Lemma 3.5, Φ(G) ≥2n−4. Assume thus G to be cubic. By Lemma 2.15, G is either P12, or one of the two braces G1 12 and G2 12 shown in Figure 8. The number of perfect matchings of P12 is equal to the required lower bound 20 = 2n −4. On the other hand, the two braces G1 12 and G2 12 are members of E. Hence the assertion holds for braces of order 12. Case 3.3 n ≥14. 22 By Lemma 3.4, G has a vertex v of degree three such that every edge in ∂(v) is removable in G. Let e be any edge in ∂(v). The retract \ G −e of G −e is not cubic and has n −4 or more vertices. Thus, either \ G −e does not lie in E or it is (K3,3 ⊙K3,3)+. By induction hypothesis, Φ(\ G −e) ≥2(n −4) −5 = 2n −13, with equality only if (i) n = 14, (ii) edge e has index two and (iii) \ G −e = (K3,3 ⊙K3,3)+. Assume that equality does not hold, for any edge e in ∂(v). In that case, by Lemma 1.5, Φ(G) ≥3 · (2n −12) 2 = 3n −18 ≥2n −4. To complete the analysis of the case, we must consider the situation in which n = 14 and ∂(v) has an edge e of index two such that the retract of G −e is (K3,3 ⊙K3,3)+. Up to isomorphism, there are two graphs of order fourteen, denoted by G1 14 and G2 14, which are expansions of index two of (K3,3 ⊙K3,3)+. They are shown in Figure 10. The ﬁrst graph G1 14 has 25 perfect matchings, the second graph G2 14 has 24. We may now conclude that the assertion holds, by induction, for every minimal simple graph. ✷ We may deduce from the proof of the above theorem that f(6) = 6, f(8) = 9, f(10) = 12, and f(12) = 17, and that the unique extremal graphs of orders six, eight, 10, and twelve are, respectively, K3,3, B8, K3,3 ⊙K3,3, and G1 12. The value of f(14) is 24, and A14, G1 14 are extremal graphs of order fourteen. The well-known Heawood graph, which is an expansion of B10, is also an extremal graph of order fourteen. 4 A Quadratic Lower Bound for b(n) We denote the class of all braces of order n by Bn. Analogous to f(n), we deﬁne b(n) to be min{Φ(G) : G ∈Bn}. Clearly, b(4) = 2. Moreover, Bn ⊆Fn for n ≥6. Thus: Proposition 4.1 For all n ≥6, f(n) ≤b(n). 23 e e (a) (b) (c) x y Figure 10: (a) (K3,3 ⊙K3,3)+, (b) G1 14 and (c) G2 14 It so happens that, for n = 6, 8, the values of f(n) and b(n) coincide. But the brace of order 10 with the fewest number of perfect matchings is M10, and it has thirteen perfect matchings, that is, b(10) = 13, whereas f(10) = 12. A brace B∗of order n is extremal if Φ(B∗) = b(n). By Proposition 4.1, if an extremal graph of order n happens to be a brace, then it is also an extremal brace of order n. Thus, b(6) = 6, b(8) = 9, b(10) = 13, and b(12) = 17. Since the bi-wheel of order n is a brace and has (n−2)2/4 perfect match-ings, it follows that, for n ≥8, b(n) ≤(n −2)2/4 (8) As mentioned in the abstract, we shall show that, for all n ≥2, b(n) ≥ (n −2)2/8. We ﬁnd it compelling to believe that, for large enough n, b(n) = (n −2)2/4. 24 Theorem 4.2 A brace of order n, where n ≥4, has at least ℓ(n) := (n−2)2 8 perfect matchings. Proof: Let us compare ℓ(n) and b(n) for small values of n. The following table shows the values of ⌈ℓ(n)⌉and b(n) for 4 ≤n ≤12. n ⌈ℓ(n)⌉ b(n) 4 1 2 6 2 6 8 5 9 10 8 13 12 13 17 Thus ℓ(n) ≤b(n), for 4 ≤n ≤12. This inequality also holds for 14 ≤n ≤18. To see this, ﬁrst observe that, in this range, ℓ(n) ≤2n −4. By Theorem 3.1, 2n −4 ≤f(n), for n ≥14. On the other hand, by Propo-sition 4.1, f(n) ≤b(n). Thus, for 14 ≤n ≤18, we have ℓ(n) ≤b(n). We shall prove the validity of the inequality for n ≥20 by induction on the number of edges. Consider any brace G of order n, where n ≥20. By Theorem 2.7, G has a thin edge, say, e = uv. Case 0 Index of e is zero. In this case, \ G −e = G −e is a brace on n vertices. By the induction hypothesis, Φ(G −e) ≥ℓ(n), implying that Φ(G) > ℓ(n). Case 1 Index of e is one. In this case, \ G −e has n −2 vertices, and by induction hypothesis, Φ(\ G −e) ≥ℓ(n −2) = ℓ(n) −4n −12 8 (9) On the other hand, by Lemma 2.4, Φ(G −{u, v}) ≥n −2 2 = 4n −8 8 (10) Adding inequalities (9) and (10), and using Lemma 1.4 we deduce that Φ(G) > ℓ(n). Case 2 Index of e is two. 25 In this case, \ G −e has n −4 vertices. By the induction hypothesis, Φ(\ G −e) ≥ℓ(n −4) = ℓ(n) −n + 4 (11) The graph G −{u, v} is a matching covered graph on n −2 ≥18 vertices and at least 25 edges. It has at most four vertices of degree two. Let H denote the retract of G −{u, v}. Each bi-contraction decreases the number of vertices and edges by two. Since H is obtained from G−{u, v} by at most four bi-contractions, the number of vertices of H is at least n −10 ≥n 2. If at most two bi-contractions were required to obtain H from G−{u, v}, then \|V (H)\| ≥14. If three bi-contractions were required to obtain H then H would have 12 vertices and 19 edges. The members of E on 12 vertices (graphs G1 12 and G2 12) have 18 edges. If four bi-contractions were required to obtain H then H would have 10 vertices and 17 edges. The members of E on 10 vertices have at most 16 edges. Hence H belongs to F, but is not a member of E. Consequently, by Theorem 3.1, Φ(G −{u, v}) = Φ(H) ≥2 n 2 −4 = n −4 (12) Now, adding the two inequalities (11), and (12), and using Lemma 1.4 we deduce that Φ(G) ≥ℓ(n). ✷ References J. A. Bondy and U. S. R. Murty. Graph Theory. Springer, 2008. M. H. Carvalho, C. L. Lucchesi, and U. S. R. Murty. Graphs with independent perfect matchings. J. Graph Theory, 48:19–50, 2005. M. H. Carvalho, C. L. Lucchesi, and U. S. R. Murty. How to build a brick. Discrete Math., 306:2383–2410, 2006. M. H. Carvalho, C. L. Lucchesi, and U. S. R. Murty. Generating simple bricks and braces. Technical Report IC-08-16, Institute of Computing, University of Campinas, July 2008. L. Esperet, F. Kardoˇ s, A. D. King, D. Kr´ al’, and S. Norine. Exponen-tially many perfect matchings in cubic graphs. Advances in Mathemat-ics, 227:1646–1664, 2011. 26 L. Lov´ asz. Matching structure and the matching lattice. J. Com-bin. Theory Ser. B, 43:187–222, 1987. L. Lov´ asz and M. D. Plummer. Matching Theory. Number 29 in Annals of Discrete Mathematics. Elsevier Science, 1986. W. McCuaig. Brace generation. J. Graph Theory, 38:124–169, 2001. S. Norine and R. Thomas. Generating bricks. J. Combin. Theory Ser. B, 97:769–817, 2007. M. Voorhoeve. A lower bound for the permanent of certain (0,1)-matrices. Indag. Math., 41:83–86, 1979. 27
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Test New Past Papers Test Builder New Learning hub Glossaries Ambassadors About us Help and support Revision NotesExam QuestionsTarget TestPast Papers Revision Notes 1. Structure, Bonding & Introduction to Organic Chemistry 10 Topics · 44 Revision Notes 1. Formulae & Equations 1. ##### Formulae & Mass 2. ##### Avogadro & the Mole 3. ##### Full & Ionic Equations 4. ##### Calculating Formulae 2. Amount of Substance 1. ##### Concentration Calculations 2. ##### Reacting Mass Calculations 3. ##### Reacting Volume Calculations 4. ##### Calculations of Product 3. Atomic Structure 1. ##### Sub-Atomic Particles 2. ##### Isotopes & Mass Spectra 4. Electrons & Ions 1. ##### Endothermic Ionisation Energy 2. ##### Electronic Structures 3. ##### Shapes of Orbitals 4. ##### Electronic Configurations & Chemical Properties 5. The Periodic Table 1. ##### Periodicity - Ionisation Energy 2. ##### Periodicity - Thermal Trends 6. Ionic & Metallic Bonding & Structure 1. ##### Evidence of Ions 2. ##### Ionic Bonding & Structures 3. ##### Ionic Bond Strength 4. ##### Polarisation 5. ##### Metallic Bonding & Lattices 7. Covalent Bonding & Structure 1. ##### Evidence of Covalent Bonding 2. ##### Covalent Bonding 3. ##### Carbon Allotropes 4. ##### Electronegativity 5. ##### VSEPR Theory 6. ##### Predicting Shapes & Angles 8. Introductory Organic Chemistry 1. ##### Hazards & Risks 2. ##### Functional Groups & Homologous Series 3. ##### Nomenclature & Classification 4. ##### Isomers - Structural 9. Alkanes 1. ##### Alkanes - Introduction 2. ##### Alkanes as Fuels 3. ##### Combustion of Alkanes 4. ##### Alternative Fuels 5. ##### Free Radicals & Fission 6. ##### The Free Radical Substitution Mechanism 10. Alkenes 1. ##### Alkenes - Introduction 2. ##### Isomers - Geometric 3. ##### Reactions of Alkenes 4. ##### Saturation Test 5. ##### Electrophilic Addition - Mechanism 6. ##### Addition Polymerisation 7. ##### Polymer Disposal 2. Energetics, Group Chemistry, Halogenoalkanes & Alcohols 10 Topics · 45 Revision Notes 1. Energetics 1. ##### Enthalpy Level Diagrams 2. ##### Enthalpy Change - Definitions 3. ##### Using Calorimetry 4. ##### Using Hess Cycles 5. ##### Bond Enthalpies 2. Intermolecular Forces 1. ##### Intermolecular Forces - Introduction 2. ##### Hydrogen Bonding 3. ##### Physical Properties 4. ##### Solvent Choice 3. Redox Chemistry & Acid-Base Titrations 1. ##### Oxidation Numbers - Introduction 2. ##### Oxidation & Reduction 3. ##### Disproportionation Reactions 4. ##### Ionic Equations 5. ##### Acid-Base Titrations with Indicators 4. Groups 1 & 2 1. ##### Ionisation Energy - Groups 1 & 2 2. ##### Reactivity - Groups 1 & 2 3. ##### Oxides & Hydroxides 4. ##### Group 2 Hydroxides & Sulfates 5. ##### Nitrates & Carbonates 6. ##### Flame Tests 7. ##### Qualitative Tests 5. Group 7 1. ##### Trends of the Group 7 Elements 2. ##### Halogen Displacement Reactions 3. ##### Redox Reactions of the Halogens 4. ##### Other Reactions of the Halogens 6. Introduction to Kinetics & Equilibria 1. ##### Collision Theory 2. ##### Rates of Reaction 3. ##### Maxwell-Boltzmann Distributions 4. ##### Catalysts & Energy 5. ##### Dynamic Equilibrium in Reversible Reactions 6. ##### Le Chatelier's Principle 7. ##### Industrial Compromises 7. Organic Chemistry: Introduction 1. ##### Types of Reaction 2. ##### Reaction Mechanisms: Introduction 8. Organic Chemistry: Halogenoalkanes 1. ##### Halogenoalkanes - Introduction 2. ##### Reactions of Halogenoalkanes 3. ##### The Nucleophilic Substitution Mechanism 4. ##### Hydrolysis of Halogenoalkanes 5. ##### Trends in Halogenoalkanes 9. Organic Chemistry: Alcohols 1. ##### Alcohols - Introduction 2. ##### Reactions of Alcohols 3. ##### Oxidation of Alcohols 10. Organic Chemistry: Techniques & Spectra 1. ##### Organic Chemistry Techniques 2. ##### Interpreting Mass Spectra 3. ##### Using Infrared Spectrometry 3. Practical Skills in Chemistry I 2 Topics · 8 Revision Notes 1. Physical Chemistry Core Practicals 1. ##### Molar Volume of a Gas 2. ##### Determining Enthalpy Change of Reaction 3. ##### Determining Concentrations 4. ##### Standard Solutions 2. Inorganic & Organic Chemistry Core Practicals 1. ##### Hydrolysis of Halogenoalkanes 2. ##### Chlorination of 2-Methylpropan-2-ol 3. ##### Propan-1-ol Oxidation 4. ##### AS Qualitative Analysis 4: Rates, Equilibria & Further Organic Chemistry 9 Topics · 51 Revision Notes 1. Kinetics 1. ##### Kinetic Rates 2. ##### Rate Equations & Orders 3. ##### Reaction Orders - Graphs 4. ##### Obtaining Rate Data 5. ##### Initial-Rate Method 6. ##### Continuous Monitoring Method 7. ##### Rate-Determining Steps from Equations 8. ##### Reaction Mechanisms 9. ##### Acid-Catalysed Iodination of Propanone 10. ##### Activation Energy & Arrhenius 2. Entropy 1. ##### Entropy - Introduction 2. ##### Entropy Changes 3. ##### Total Entropy 4. ##### Reaction Feasibility 3. Energetics 1. ##### Born-Haber Cycles - Key Terms 2. ##### Born-Haber Cycles 3. ##### Polarisation 4. ##### Enthalpy of Solution - Key Terms 5. ##### Enthalpy of Solution - Calculations 6. ##### Enthalpy of Solution - Ionic Charge & Radius 4. Chemical Equilibria 1. ##### Equilibrium Constant, Kc 2. ##### Equilibrium Constant, Kp 3. ##### Equilibrium Constant & Conditions 4. ##### Equilibrium Constant & Temperature 5. ##### Equilibrium Constant & Entropy 5. Acid-base Equilibria 1. ##### Brønsted–Lowry Acid & Base Theory 2. ##### pH 3. ##### Acid Strength 4. ##### pH Calculations of Acids 5. ##### Ionic Product of Water 6. ##### Analysing pH Data 7. ##### Titration Curves 8. ##### Buffers 9. ##### Buffer Calculations 6. Organic Chemistry: Chirality 1. ##### Chirality in Optical Isomers 2. ##### Racemates & Optical Activity 3. ##### Mechanisms & Optical Activity 7. Organic Chemistry: Carbonyls 1. ##### Carbonyls 2. ##### Reactions of Carbonyls 8. Organic Chemistry: Carboxylic Acids 1. ##### Carboxylic Acids 2. ##### Preparation of Carboxylic Acids 3. ##### Reactions of Carboxylic Acids 4. ##### Acyl Chlorides & Esters 5. ##### Acid & Alkaline Hydrolysis of Esters 6. ##### Forming Polyesters 9. Organic Analytical Techniques 1. ##### High Resolution Mass Spectrometry 2. ##### Carbon-13 NMR Spectroscopy 3. ##### Low Resolution Proton NMR 4. ##### High Resolution Proton NMR 5. ##### Chromatography - TLC 6. ##### Chromatography - HPLC & GC 5. Transition Metals & Organic Nitrogen Chemistry 6 Topics · 38 Revision Notes 1. Redox Equilibria 1. ##### Reduction & Oxidation 2. ##### Standard Electrode Potential 3. ##### Measuring Standard Electrode Potential 4. ##### Conventional Cell Representation 5. ##### Thermodynamics & Electrode Potential 6. ##### Fuel Cells 7. ##### Redox Titration Calculations 2. Transition Metals & Complexes 1. ##### Transition Metals 2. ##### Transition Metal Complexes 3. ##### Colour in Aqueous ions 4. ##### Ligands 5. ##### Ligand Exchange 6. ##### Octahedral & Tetrahedral Complexes 7. ##### Square Planar Complexes 8. ##### Haemoglobin & Ligand Exchange 3. Transition Metal Reactions 1. ##### Vanadium Chemistry 2. ##### Chromium Chemistry 3. ##### Ions in Aqueous Solution 4. ##### Catalysts 5. ##### Homogeneous Catalysis 6. ##### Autocatalysis 4. Organic Chemistry: Arenes 1. ##### Benzene - Structure & Stability 2. ##### Benzene Reactions 3. ##### Benzene - Electrophilic Substitution 4. ##### Phenol Bromination 5. Organic Chemistry: Nitrogen Compounds 1. ##### Amines, Amides & Amino Acids - Introduction 2. ##### Primary Aliphatic Amines 3. ##### Aromatic Amines 4. ##### Amides 5. ##### Polymerisation 6. ##### Polyamides & Poly(ethenol) 7. ##### Characteristic Behaviour of Amino Acids 6. Organic Synthesis 1. ##### Deducing Organic Structures 2. ##### Planning Reaction Schemes 3. ##### Increasing the Carbon Chain Length 4. ##### Organic Techniques - Preparation 5. ##### Organic Techniques - Purification 6. ##### Organic Techniques - Purity 6. Practical Skills in Chemistry II 2 Topics · 10 Revision Notes 1. Advanced Physical Chemistry Core Practicals 1. ##### Rates of Reaction - Titrimetric Method 2. ##### Rates of Reaction - Clock Reaction 3. ##### Activation Energy 4. ##### Finding Ka Values 5. ##### Electrochemical Cells 2. Advanced Inorganic & Organic Chemistry Core Practicals 1. ##### Redox Titration - Iron(II) & Manganate(VII) 2. ##### Redox Titration -Thiosulfate & Iodine 3. ##### Transition Metal Complexes 4. ##### A-Level Qualitative Analysis 5. ##### Aspirin Preparation International A Level (IAL)ChemistryEdexcelRevision Notes2. Energetics, Group Chemistry, Halogenoalkanes & Alcohols Energetics Using Calorimetry Using Calorimetry(Edexcel International A Level (IAL) Chemistry):Revision Note Exam code:YCH11 Download PDF Author Richard Boole Last updated 8 January 2025 Using Calorimetry Calorimetry is the measurement enthalpy changes in chemical reactions There are two types of calorimetry experiments you need to know: Enthalpy changes of reactions in solution, e.g. displacement and neutralisation Enthalpy changes of combustion In both cases, you should be able to give an outline of the experiment and be able to process experimental data using calculations or graphical methods Enthalpy changes for reactions in solution The principle of these calorimetry experiments is to carry out the reaction with an excess of one reagent and measure the temperature change over the course of a few minutes The apparatus needed to carry out an enthalpy of reaction in solution calorimetry experiment is shown below Acalorimetercan be made up of apolystyrene drinking cup, avacuum flaskormetal can A polystyrene cup can act as a calorimeter to find enthalpy changes in a chemical reaction The energy needed to raise the temperature of 1 g of a substance by 1 K is called thespecific heat capacity(c) of the liquid Thespecific heat capacityof water is 4.18 J g-1 K-1 The energy transferred as heat can be calculated by: Equation for calculating energy transferred in a calorimeter Sample method for a displacement reaction Using a measuring cylinder place 25 cm 3 of the 1.0 mol dm-3 copper(II) sulphate solution into the polystyrene cup Weigh about 6 g of zinc powder - as this is an excess there is no need to be very accurate Draw a table to record the initial temperature and then the temperature and time every half minute up to 9.5 minutes Put a thermometer or temperature probe in the cup, stir, and record the temperature every half minute for 2.5 minutes At precisely 3 minutes, add the zinc powder to the cup (DO NOT RECORD THE TEMPERATURE AT 3 MINUTES) Continue stirring and record the temperature for an additional 6 minutes For the purposes of the calculations, some assumptions are made about the experiment: That the specific heat capacity of the solution is the same as pure water, i.e.4.18 J g-1K-1 That the density of the solution is the same as pure water, i.e.1 g cm-3 The specific heat capacity of the container is ignored The reaction is complete There are negligible heat losses Examiner Tips and Tricks As well as any assumptions made for the experiment, you can be expected to comment on: Sources of error, e.g. incorrect masses or volumes of chemicals, heat loss to the surroundings, etc. Uncertainty, e.g. measurements for mass, volume and temperature Be aware that you will have to double the uncertainty reading for temperature as the calculation requires an initial and final temperature to determine the temperature change Temperature correction graphs For reactions which are not instantaneous there may be a delay before the maximum temperature is reached During that delay the substances themselves may be losing heat to the surroundings, so that the true maximum temperature is never actually reached To overcome this problem we can use graphical analysis to determine the maximum enthalpy change A temperature correction graph for a metal displacement reaction between zinc and copper sulfate solution. The zinc is added after 4 minutes The steps to make a temperature correction graph are: Take a temperature reading before adding the reactants for a few minutes to get a steady value Add the second reactant and continue recording the temperature and time Plot the graph andextrapolatethe cooling part of the graph until you intersect the time at which the second reactant was added Analysis Use both extrapolated lines to calculate Δ T as shown on the graph Use the equation q =mc Δ T to calculate the energy transferred q = energy transferred m = mass - this will be the mass of the 25 cm 3 solution which will be 25 g (assuming a density of 1 g cm-3) c = specific heat capacity - this will be assumed to be 4.18 J g-1 K-1, which is the specific heat capacity of water Δ T = the temperature change from the graph Convert your value for energy transferred from J into kJ Then use the equation Δ H = to calculate the enthalpy change for the reaction q = energy transferred n= number of moles - this would be the number of moles of the limiting reagent, which means that you will have an extra calculation to do to determine whether this is the zinc or the copper sulfate Remember that in the example above, the temperature of the reaction mixture increased which means that the reaction is exothermic and should, therefore, have a negative value Worked Example Enthalpy of neutralisation calculation The initial temperature of 25.0 cm 3 of sodium hydroxide and 50.0 cm 3 of sulfuric acid was measured every 30 seconds for 3 minutes. The two liquids were then mixed and the temperature of the resulting solution was taken every 30 seconds for 5 minutes.A temperature correction graph of the results was plotted and the temperature change was determined to be 6.5 o C. Calculate the energy transferred during this reaction. Answer q = m x c x Δ T m (of resulting solution) = 25.0 + 50.0 = 75.0 g c (of water) = 4.18 J g-1 o C-1 Δ T (of water) = 6.5 o C q = 75.0 x 4.18 x 6.5 = 2038 J Examiner Tips and Tricks It is more likely that you would be asked to calculate the enthalpy change for this neutralisation reaction Calculate the number of moles of NaOH and H 2 SO 4 to determine which reactant is the limiting reagent Calculate the enthalpy change by dividing the energy transferred by the number of moles of the limiting reagent Another twist to this calculation could be for the NaOH and H 2 SO 4 to have different initial temperatures In this case, you work out the energy transferred for the NaOH and for the H 2 SO 4 separately and then add them together After that you follow steps 1 and 2 above to calculate the enthalpy change Enthalpy of combustion The principle here is to use the heat released by a combustion reaction to increase the heat content of water A typical simple calorimeter is used to measure the temperature changes to the water A simple combustion calorimeter To complete this experiment, the following steps will need to be completed: It is important that you record the starting temperature, and the final temperature in order to complete the calculations You must also record the starting mass of the spirit burner and the final mass of the spirit burner, so that you can work out the mass of the fuel burned during the reaction This will then be used to calculate the moles, which will be used to convert Q to an enthalpy change in your calculations Key points to consider Not all the heat produced by the combustion reaction is transferred to the water Some heat is lost to the surroundings Some heat is absorbed by the calorimeter To minimise the heat losses the copper calorimeter should not be placed too far above the flame and a lid placed over the calorimeter Shielding can be used to reduce draughts In this experiment the main sources of error are Heat losses Incomplete combustion Worked Example Enthalpy of combustion In a calorimetry experiment 2.50 g of methane is burnt in excess oxygen. 30% of the energy released during the combustion is absorbed by 500 g of water, the temperature of which rises from 25 °C to 68 °C. The specific heat capacity of water is 4.18 J g-1 K−1 What is the total energy released per gram of methane burnt? Answer Step 1: q = m x c x Δ T m (of water) = 500 g c (of water) = 4.18 J g-1 o C-1 Δ T (of water) = 68 o C - 25 o C = 43 o C Step 2: q= 500 x 4.18 x 43 = 89 870 J Step 3: This is only 30% of the total energy released by methane Total energy x 0.3 = 89 870 J Total energy = 299 567 J Step 4: This is released by 2.50 g of methane Energy released by 1.00 g of methane = 299 567 ÷ 2.50 = 120 000 J g-1 (to 3 s.f.) or 120 kJ g-1 Calculating Enthalpy Changes Aqueous solutions of acid, alkalis and salts are assumed to be largely water so you can just use the m and c values of water when calculating the energy transferred. To calculate any changes in enthalpy per mole of a reactant or product the following relationship can be used: Δ H = When there is a rise in temperature, the value for Δ H becomes negative suggesting that the reaction is exothermic This means that your value should be negative for an exothermic reaction, e.g. combustion When the temperature falls, the value for Δ H becomes positive suggesting that the reaction is endothermic This means that your value should be positive for an endothermic reaction, e.g. ammonium salts dissolving in water Worked Example 1.50 g of an organic liquid (M r = 58.0) underwent complete combustion. The heat formed raised the temperature of 100 g of water from 20 o C to 75 o C. Calculate the enthalpy of combustion for the organic liquid Answer Step 1:Calculate the energy released by the organic liquid Q=mc Δ T Q= 100 x 4.18 x (75 - 20) Q= 22990 J Q= 22.99 kJ Step 2: Calculate the number of moles of the organic liquid Number of moles =0.0259 moles (to 3s.f.) Step 3: Calculate the enthalpy change of combustion Δ c H θ-887 kJ mol-1 (to 3s.f.) Remember, combustion is an exothermic process and will, therefore, be a negative enthalpy change value Unlock more, it's free! Join the 100,000+ Students that ❤️ Save My Exams the (exam) results speak for themselves: I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top grades in science and maths. You really did save my exams! Thank you. Beth IGCSE Student > This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS! Fathima A Level Student > Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! 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Yes No Previous:Enthalpy Change - DefinitionsNext:Using Hess Cycles More Exam Questions you might like Energetics Intermolecular Forces Redox Chemistry & Acid-Base Titrations Groups 1 & 2 Group 7 Introduction to Kinetics & Equilibria Organic Chemistry: Halogenoalkanes Organic Chemistry: Alcohols Organic Chemistry: Techniques & Spectra Physical Chemistry Core Practicals Author:Richard Boole Expertise:Chemistry Content Creator Richard has taught Chemistry for over 15 years as well as working as a science tutor, examiner, content creator and author. He wasn’t the greatest at exams and only discovered how to revise in his final year at university. That knowledge made him want to help students learn how to revise, challenge them to think about what they actually know and hopefully succeed; so here he is, happily, at SME. 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2173	https://en.grob-antriebstechnik.de/learning-center/selection-criteria/dimensioning-screw-jack	DeutschFrançais Got any questions? We're happy to advise you CallCallbackMessage Don’t waste any time - call us now! +49 7261 9263-63 📞 Request a callback! ✉️ Send us a message! Learning Center Selection Criteria Dimensioning Screw Jacks Dimensioning Screw Jacks Selection Criteria Contents 1. Introduction Proper dimensioning of a Screw Jack is essential for safe and efficient operation. This learning article explains the necessary calculations – from determining the required drive power to checking for buckling and critical speeds. 2. Motor power Motor power is crucial to ensure that the lifting element can lift or move the required load. The motor's power should be sufficient to achieve the desired torque and speed, depending on the size and weight of the load to be moved. 2.1 Determining motor power Dynamic lifting force Fdyn = m g in kN (g = 9.81 m/s² where 10 m/s² is assumed) Required drive speed Drive torque of the system From the drive torque and the required drive speed, the formula for determining the motor power PM is derived. Efficiency of a Multi-Jack System 3. What is efficiency (%) and how is it calculated? Imagine you want to lift a load using a screw. You turn the screw with force – that’s the input energy. The screw moves the load – that’s the useful (output) energy. The efficiency tells you how much of your input energy is actually used to move the load – in other words, how efficient the system is. Simply put: 100% efficiency = no energy is lost 50% efficiency = half of the energy is lost, e.g., due to friction 10% efficiency = only a small part of the energy actually moves the load So, efficiency helps you understand how “well” a drive system works. 3.1 Calculation Efficiency is measured by comparing how much energy you put in and how much of it actually reaches the output. In other words: Specifically for a Screw Drive Measure the input power: You measure the torque (how strongly you turn) and the rotational speed (how fast you turn), like this: Measure the useful (output) power: You measure how much force actually acts on the load and how fast it moves: Then compare the two values: Example: You input 100 W (e.g., motor power) At the end, 85 W actually move the load → The efficiency is 85% Important to know: Some energy is always lost – for example, through friction, heat, or deformation. A high efficiency means that only a small amount of energy is lost. 4. Speed The speed of a Screw Jack refers to the rateat which the linear motion of the Screw Jack occurs. It's important to understand how the speed works in your Screw Jack, as it has a significant impact on the performance and application. The allowable spindle speed is then determined by: The allowable speed is typically calculated without the nut. As a result, the unsupported spindle length changes permanently, and a higher speed may be permitted under certain circumstances. 5. Buckling Buckling refers to the deformation or bending of lifting system components under the influence of compressive forces or loads. Buckling can occur when an elongated component, such as a spindle or a push rod, is subjected to a compressive load that exceeds a critical threshold. Euler buckling cases Buckling can be avoided by selecting components with sufficient stiffness and limiting the load to values below the buckling load. For a rough preliminary dimensioning, the following formula is sufficient: 5.1 Euler 1 5.2 Euler 2 5.3 Euler 3 To ensure that the components of your Screw Jack withstand the loads, buckling can be analyzed through more detailed calculations, which are shown below: The limiting slenderness ratio indicates how slender or relatively thin a spindle is in relation to its length. A lower value means the spindle is relatively thick compared to its length, while a higher value indicates that the spindle is more slender in relation to its length. A too high limiting slenderness ratio can lead to instability, causing vibrations or oscillations in the spindle. Conversely, a low limiting slenderness ratio may mean that the spindle is unnecessarily large and heavy. If the slenderness ratio λ of the spindle is < λ0, the buckling calculation is performed according to Tetmajer (inelastic). The slenderness ratio can be calculated using the following formula: 5.4 Determination of buckling stress according to Tetmajer 5.5 Determination of buckling stress according to Euler 6. Bending critical speed Long, rapidly rotating spindles can begin to oscillateunder both compressive and tensile loads. This calculation is considered in the critical speed. The critical speed must be taken into account only for the traveling nut version, as spindle rotation occurs only in this case. Factors to consider include the diameter and length of the spindle, as well as its bearing. 7. Lead angle The lead angle is an important parameter in the dimensioning of Screw Jacks. It refers to the angle at which the threaded spindle or worm shaft of your Screw Jack is inclined. The lead angle affects how your Screw Jack converts linear motion into rotational movement and vice versa. A larger lead angle results in more linear movement per revolution of the spindle or worm. A smaller lead angle leads to higher speed but lower force transmission, whereas a larger lead angle results in lower speed but higher force transmission. Here is how to calculate the lead angle: When dimensioning lifting platforms with threaded spindles as the drive mechanism, the following rules apply for the lead angle of the screw as well as any potential self-locking of the thread: How did you like the article? Let us know your opinion. Want to know more? Accuracy and precision In electromechanical drive technology, accuracy determines the success or failure of your applications. But what does precision really mean, and why is it so crucial? Our expert article explores what truly matters in planning and implementation. Read more Want to save precious time? ...Get CAD data and configure your product yourself with our product configurator 📐 Configure now
2174	https://www.yumpu.com/en/document/view/11335895/table-of-acids-with-ka-and-pka-values-clas	Table of Acids with Ka and pKa Values CLAS EN EnglishDeutschFrançaisEspañolPortuguêsItalianoRomânNederlandsLatinaDanskSvenskaNorskMagyarBahasa IndonesiaTürkçeSuomiLatvianLithuaniančeskýрусскийбългарскиالعربيةUnknown Login to YUMPU NewsLogin to YUMPU Publishing TRY ADFREE Discover productsNewsPublishing Magazines Create ePaper Login to YUMPU NewsLogin to YUMPU Publishing × Attention! Your ePaper is waiting for publication! By publishing your document, the content will be optimally indexed by Google via AI and sorted into the right category for over 500 million ePaper readers on YUMPU. This will ensure high visibility and many readers! PUBLISH DOCUMENT No, I renounce more range. Your ePaper is now published and live on YUMPU! You can find your publication here: view Share your interactive ePaper on all platforms and on your website with our embed function share: Design embed now ⬤⬤ 21.03.2013 • Views ShareEmbedReport Table of Acids with Ka and pKa Values CLAS Table of Acids with Ka and pKa Values CLAS Table of Acids with Ka and pKa Values CLAS SHOW MORE SHOW LESS ePAPER READ DOWNLOAD ePAPER TAGS acids appendix clas phosphoric arsenic citric conjugate bases sulfuric oxalic clas.sa.ucsb.edu clas.sa.ucsb.edu Transform your PDFs into Flipbooks and boost your revenue! Leverage SEO-optimized Flipbooks, powerful backlinks, and multimedia content to professionally showcase your products and significantly increase your reach. Start now More Magazines Recommendations Info TableofAcidswithKaand pKaValuesCLAS Acid HA A - Hydroiodic HI I - Hydrobromic HBr Br - Perchloric HClO4 ClO4 - Hydrochloric HCl Cl - Chloric HClO3 ClO3 - Sulfuric (1) H2SO4 HSO4 - Nitric HNO3 NO3 - Hydronium ion H3O + Iodic HIO3 IO3 - Oxalic (1) H2C2O4 HC2O4 - Sulfurous (1) H2SO3 HSO3 - Sulfuric (2) HSO4 - SO4 2- Chlorous HClO2 ClO2 - Phosphoric (1) H3PO4 H2PO4 - Arsenic (1) H3AsO4 H2AsO4 - Chloroacetic CH2ClCOOH CH2ClCOO - Citric (1) H3C6H5O7 H2C6H5O7 - Hydrofluoric HF F - Nitrous HNO2 NO2 - Formic HCOOH HCOO - Lactic HCH3H5O3 CH3H5O3 - Ascorbic (1) H2C6H6O6 HC6H6O6 - Benzoic C6H5COOH C6H5COO - Oxalic (2) HC2O4 - C2O4 2- Hydrazoic HN3 N3 - Citric (2) H2C6H5O7 - HC6H5O7 2- Acetic CH3COOH CH3COO - Propionic CH3CH2COOH CH3CH2COO - Pyridinium ion C5H4NH + Citric (3) HC6H5O7 2- C6H5O7 3- Carbonic (1) H2CO3 HCO3 - Sulfurous (2) HSO4 - SO4 2- Arsenic (2) H2AsO4 - HAsO4 2- Ka pKa Acid Strength Strong acids completely dissociate in aq solution (Ka> 1, pKa< 1). Conjugate bases of strong acids are ineffective bases. H2O 1 0.0 1.6 x 10 -1 0.80 5.9 x 10 -2 1.23 1.54 x 10 -2 1.81 1.2 x 10 -2 1.92 1.1 x 10 -2 1.96 7.52 x 10 -3 2.12 5.0 x 10 -3 2.30 1.4 x 10 -3 2.85 8.4 x 10 -4 3.08 7.2 x 10 -4 3.14 4.0 x 10 -4 3.39 1.77 x 10 -4 3.75 1.38 x 10 -4 3.86 7.9 x 10 -5 4.10 6.46 x 10 -5 4.19 6.4 x 10 -5 4.19 1.9 x 10 -5 4.72 1.8 x 10 -5 4.74 1.76 x 10 -5 4.75 1.34 x 10 -5 4.87 C5H4N 5.6 x 10 -6 5.25 4.0 x 10 -6 5.40 4.3 x 10 -7 6.37 1.02 x 10 -7 6.91 8/9.3 x 10 -8 7.10/7.03 Hydrosulfuric H2S HS - 1.0 x 10 -7 /9.1 x 10 - 7/7.04 Phosphoric (2) H2PO4 - HPO4 2- Hypochlorous HClO ClO - Hypobromous HBrO BrO - Hydrocyanic HCN CN - Boric (1) H3BO3 H2BO3 - Ammonium ion NH4 + Phenol C6H5OH NH3 C6H5O - Carbonic (2) HCO3 - CO3 2- Hypoiodous HIO IO - Arsenic (3) HAsO4 2- AsO4 3- Hydrogen peroxide H2O2 HO2 - Ascorbic (2) HC6H6O6 - Phosphoric (3) HPO4 2- PO4 3- Water H2O OH - Group I metal hydroxides (LiOH, NaOH, etc.) C6H6O6 2- Group II metal hydroxides (Mg(OH)2, Ba(OH)2, etc.) 8 6.23 x 10 -8 3.5/3.0 x 10 -8 2 x 10 -9 6.17 x 10 -10 5.8 x 10 -10 5.6 x 10 -10 1.6 x 10 -10 4.8 x 10 -11 2 x 10 -11 6.0 x 10 -10 /3.0 x 10 -12 2.4 x 10 -12 7.21 7.46/7.53 8.70 9.21 9.23 9.25 9.80 10.32 10.70 9.22/11.53 11.62 1.6 x 10 -12 11.80 4.8/2.2 x 10 -13 12.32/12.66 1.0 x 10 -14 14.0 Strong bases completely dissociate in aq solution (Kb > 1, pKb < 1). Conjugate acids (cations) of strong bases are ineffective bases. Conjugate Base Strength Compiled from Appendix 5 Chem 1A, B, C Lab Manual and Zumdahl 6 th Ed. The pKa values for organic acids can be found in Appendix II of Bruice 5 th Ed. Page 2: Table of Acids with Ka and pKa Valu × Share or Link Short-link Embed Copy Copy Copy Extended embed settings × Inappropriate Loading... Report as Inappropriate Cancel × Inappropriate You have already flagged this document. Thank you, for helping us keep this platform clean. The editors will have a look at it as soon as possible. × Embed Loading... Cancel Delete template? Are you sure you want to delete your template? Cancel Delete × DOWNLOAD ePAPER This ePaper is currently not available for download. You can find similar magazines on this topic below under ‘Recommendations’. Cancel × Save as template ? 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2175	https://math.stackexchange.com/questions/2565157/non-negative-real-values-of-a-b-c	algebra precalculus - Non negative real values of $a,b,c$ - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Non negative real values of a,b,c a,b,c Ask Question Asked 7 years, 9 months ago Modified7 years, 9 months ago Viewed 596 times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. The question is to find out the minimum value of non negative real values of a,b,c a,b,c given that the equation x 4+a x 3+b x 2+c x+1=0 x 4+a x 3+b x 2+c x+1=0 has real roots. I tried dividing both sides by x 2 x 2 the resultant equation becomes x 2+a x+b+c x+1 x 2=0 x 2+a x+b+c x+1 x 2=0 I couldn't transfer this to a quadratic equation.If I take a=c then a quadratic equation may be formed by substituting x+1/x=t x+1/x=t with this assumption the equation reduces to t 2+a t+b−2=0 t 2+a t+b−2=0 Now we know that t≥2 t≥2 when x x is positive and t≤−2 t≤−2 when x x is negative. So we can check that if the roots of t t be in the interval (−2,2)(−2,2) the corresponding value of x x will be imaginary.For this we have three conditions f(−2)>0 f(−2)>0,f(2)>0 f(2)>0 and −a 2∈(−2,2)−a 2∈(−2,2) [This comes from the fact that the minimum must be in the interval of the roots] and the discriminant must be greater than or equal to zero.Using these conditions I get that (4,6,4) are the values of (a,b,c) However I am not sure how to proceed if a≠c a≠c .Any ideas? algebra-precalculus Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Dec 13, 2017 at 17:58 user471651user471651 asked Dec 13, 2017 at 17:34 user471651user471651 1,204 10 10 silver badges 24 24 bronze badges 9 Try completing the square twice to eliminate the x 3,x x 3,x terms.vadim123 –vadim123 2017-12-13 17:35:54 +00:00 Commented Dec 13, 2017 at 17:35 1 Just to clarify: a,b,c are nonnegative reals? The equation must have a real root or only real roots?Andreas –Andreas 2017-12-13 17:44:40 +00:00 Commented Dec 13, 2017 at 17:44 1 Have you checked this?: en.wikipedia.org/wiki/…Andreas –Andreas 2017-12-13 17:49:25 +00:00 Commented Dec 13, 2017 at 17:49 @Andreas they are non negative real.sorry for not mentioning that user471651 –user471651 2017-12-13 17:57:49 +00:00 Commented Dec 13, 2017 at 17:57 An already asked question: The equation must have at least one real root or only real roots?user261263 –user261263 2017-12-13 18:01:20 +00:00 Commented Dec 13, 2017 at 18:01 \|Show 4 more comments 1 Answer 1 Sorted by: Reset to default This answer is useful 3 Save this answer. Show activity on this post. The idea followed here is, for a≠c a≠c, to apply a shift to x x in order to transform the equation into a form for which the OP has already given a solution. From that, we can infer back to the values of a,b,c a,b,c. Applying a shift x=y+q x=y+q gives 1+c q+b q 2+a q 3+q 4+(c+2 b q+3 a q 2+4 q 3)y+(b+3 a q+6 q 2)y 2+(a+4 q)y 3+y 4=0 1+c q+b q 2+a q 3+q 4+(c+2 b q+3 a q 2+4 q 3)y+(b+3 a q+6 q 2)y 2+(a+4 q)y 3+y 4=0 Now you can select the q=q(a,b,c)q=q(a,b,c) which solves c+2 b q+3 a q 2+4 q 3=a′r a+4 q=a′1+c q+b q 2+a q 3+q 4=r 2 c+2 b q+3 a q 2+4 q 3=a′r a+4 q=a′1+c q+b q 2+a q 3+q 4=r 2 (coming back to that later). Further, identify b′=b+3 a q+6 q 2 b′=b+3 a q+6 q 2. This gives 0=r 2+a′r y+b′y 2+a′y 3+y 4=y 2(r 2/y 2+a′r/y+b′+a′y+y 2)0=r 2+a′r y+b′y 2+a′y 3+y 4=y 2(r 2/y 2+a′r/y+b′+a′y+y 2) Setting z=y/r√z=y/r gives 0=1/z 2+(a′/r√)/z+b′/r+(a′/r√)z+z 2 0=1/z 2+(a′/r)/z+b′/r+(a′/r)z+z 2 Now we can apply the OP's solution which gives a′/r√=4 a′/r=4 and b′/r=6 b′/r=6. Plugging that back into the equations above gives c+2 b q+3 a q 2+4 q 3=4 r 3−−√a+4 q=4 r√1+c q+b q 2+a q 3+q 4=r 2 b+3 a q+6 q 2=6 r c+2 b q+3 a q 2+4 q 3=4 r 3 a+4 q=4 r 1+c q+b q 2+a q 3+q 4=r 2 b+3 a q+6 q 2=6 r [One can check that for a=c a=c, indeed r=1 r=1 and q=0 q=0 solve this system.] Now here again fourth-order equations are present so it won't be easy. We need to identify a,b,c a,b,c from these four equations by eliminating q q and r r. (is there a fifth equation hidden somewhere? This would generally be necessary for eliminating r r and q q, however: wait for what follows.) We get r=(q+a/4)2 r=(q+a/4)2 and hence b/6+a q/2+q 2=q 2+a q/2+a 2/16 b/6+a q/2+q 2=q 2+a q/2+a 2/16 or b=3 a 2/8 b=3 a 2/8. Putting the results for r r and b b into the first system equation gives c+6 a 2 q/8+3 a q 2+4 q 3=a 3/16+(3 a 2 q)/4+3 a q 2+4 q 3 c+6 a 2 q/8+3 a q 2+4 q 3=a 3/16+(3 a 2 q)/4+3 a q 2+4 q 3, so c=a 3/16 c=a 3/16 Further, r 2=(q+a/4)4=a 4/256+(a 3 q)/16+(3 a 2 q 2)/8+a q 3+q 4=1+c q+b q 2+a q 3+q 4 r 2=(q+a/4)4=a 4/256+(a 3 q)/16+(3 a 2 q 2)/8+a q 3+q 4=1+c q+b q 2+a q 3+q 4. Plugging in b b and c c we have a 4/256=1 a 4/256=1 which gives a=4 a=4no matter which shift q q. So collecting we have (a,b,c)=(4,6,4)(a,b,c)=(4,6,4) as the smallest values even if we started without the condition a=c a=c. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Dec 13, 2017 at 22:25 answered Dec 13, 2017 at 22:09 AndreasAndreas 16.7k 28 28 silver badges 63 63 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions algebra-precalculus See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Linked 2Find the GCD of the minimum non-negative values of a a, b b and c c, given that the equation x 4+a x 3+b x 2+c x+1=0 x 4+a x 3+b x 2+c x+1=0 has real roots only. Related 1distinct possible real roots of the equation >(3 x 2+k x+3)(x 2+k x−1)=0.(3 x 2+k x+3)(x 2+k x−1)=0. 1Finding equation of parabola when given point and tangent line 3Show that the roots of a x 2+(a+b)x+b=0 a x 2+(a+b)x+b=0 are real for all values of a and b. 2Condititon for roots of quartic to be real and two be coincident 3Finding the minimum value. 4Intermediate Algebra Quadratic Inequalities 1Discriminant of Quadratic Equation Hot Network Questions What meal can come next? 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2176	https://www.distancecalculus.com/compare/opencourseware-mit/	Distance Calculus @ Roger Williams University Enrollment Application Celebrating Our28th Year Teaching Online OpenCourseWare MIT versus Distance Calculus Comparison We are HUGE fans of the OpenCourseware Project at MIT, located here. Specifically, the Linear Algebra courses by Gilbert Strang, longtime master teacher at MIT, are fabulous! If you look through these course offerings, you will see that all of them are "snapshots" of previous courses over the past 10-15 years. They include fantastic lectures, awesome course materials, and challenging problem sets. Keep in mind, however, that these OCW @ MIT courses do not grant academic credit. They are presented more as a resource for learning, and certainly an awesome resource! But if you "enroll" in one of these courses and start engaging the course materials and lectures, you will not be on a path towards earning academic credits for these courses. Perhaps someday OCW courses will grant academic credit, but for now, they do not. If you are looking for a non-credit alternative to just learn the course materials, OCW @ MIT is an excellent resource. You really feel like you are in a lecture hall at MIT taking one of these courses. But if your goal is to earn academic credits for your Calculus course effort, you need to seek out an academic credit-granting program, like Distance Calculus @ Roger Williams University. Comparison of OpenCourseWare @ MIT; vs Distance Calculus \| Topic \| MyMathLab® \| Distance Calculus \| --- \| Multiple Choice \| No \| No \| \| Use of Computer Algebra System \| No \| Yes, Extensive \| \| Tied to Traditional Textbook \| Sometimes \| No, Reform e-Curriculum \| \| Recursive Grading \| No \| Yes \| \| Comprehensive Course System \| No, just access to lectures and homework problems \| Yes \| \| Drill & Practice \| No \| No \| Here are a few videos about Distance Calculus @ Roger Williams University: Is Distance Calculus for You? Success Indicators for Distance Calculus Distance Calculus - Student Reviews Date Posted: Aug 16, 2020Review by: Jennifer S. Courses Completed: Calculus IReview: The course was intense and required a lot of hard work. Professors ready available to assist when needed. Professors presented and explained materials/course work in detail and provided explanations and resources. Transferred Credits to: University of New Haven, West Haven, CT Date Posted: Jan 19, 2020Review by: William WilliamsStudent Email: wf.williamster@gmail.comCourses Completed: Linear Algebra, Probability TheoryReview: I have difficulty learning calculus based math, akin to dyslexia when examining the symbolic forms, equations, definitions, and problems. Mathematica based calculus courses allowed me to continue with my studies because of the option of seeing the math expressed as a programming language for which I have no difficulty in interpreting visually and the immediate feedback of graphical representations of functions, equations, or data makes a huge impact on understanding. Mathematica based calculus courses should be the default method of teaching Calculus everywhere.Transferred Credits to: Thomas Edison State College Date Posted: Apr 30, 2020Review by: Hannah J. Courses Completed: Probability TheoryReview: Probability Theory was a great course. Very very thorough. I thought it would never end :). I was very prepared for my coursework in economics. Excellent refereshher of derivatives and integrals - really forced me to remember that stuff from freshman cal. Transferred Credits to: Boston University Distance Calculus - Curriculum Exploration Systems of Equations P10: Systems of Equations: P10.1: Systems of Linear 2D Equations P10.1.a: Graphing P10.1.b: Solving by Substitution P10.1.c: Using Matrix Notation P10.1.d: Homework Problems P10.2: Matrices P10.2.a: Matrix Addition P10.2.b: Matrix Multiplication P10.2.c: Determinants P10.2.d: Matrix Inverse P10.2.e: Solving Linear Matrix Equation P10.2.f: Homework Problems P10.3: Systems of Linear 3D Equations P10.3.a: Graphing P10.3.b: Solve via Substitution P10.3.c: Solve Using Matrix Methods P10.3.d: Different Solution Cases in 3D P10.3.e: Homework Problems P10.4: Non-Linear Systems P10.4.a: Non-Linear 2D Systems P10.4.b: Non-Linear 3D Systems P10.4.c: Homework Problems Course Information How Distance Calculus Courses Work Accreditation About Mastery Learning Flexible Asynchronous Course Maximum Course Time - 1 Year Computer Requirements Completion Time Estimates Course Prerequisites All Course Syllabi Who Can Take These Courses? Student Reviews Helpful Videos HONORS Courses Frequent Questions When Can I Enroll? Term Dates? Summer? Winter? When Can I Start Course? Costs - Tuition/Software/E-Text Will This Course TRANSFER? How Grades Work: A,B,C,D,F Other Frequently Asked Questions Same as AP Calculus? Letters of Recommendation Are These Computer-Based Courses? Are These ONLINE Courses? Freshman Mathematics Courses SemesterCreditHours Applied Calculus for Business 33 Applied Calculus for Life Science 33 STEM Calculus I44 STEM Calculus II44 Honors STEM Calculus I 55 Honors STEM Calculus II 55 Honors Calculus I+II for Data Science 55 Sophomore Mathematics Courses SemesterCreditHours Multivariable/Vector Calculus 44 Differential Equations 33 Linear Algebra 44 Probability Theory 33 Honors Multivariable/Vector Calculus 55 Honors Differential Equations 44 Honors Linear Algebra 55 Honors Linear Algebra for Data Science 55 General Mathematics Courses SemesterCreditHours Introductory Statistics 44 Precalculus with Trigonometry 44 Finite Mathematics 44 Discrete Mathematics 44 Mathematics Major Courses SemesterCreditHours Computational Abstract Algebra 44 Computational Differential Geometry 44 Distance Calculus - Student Reviews Douglas Z.★★★★★ Posted: Jun 6, 2020 Courses Completed: Multivariable Calculus, Differential Equations, Linear Algebra, Probability Theory I loved these courses. So in depth and comprehensive. The mix of software and math curriculum was tremendously helpful to my future studies and career in engineering. I highly recommend these courses if you are bored of textbook courses. Transferred Credits To: University of Massachusetts, Amherst Howard B.★★★★★ Posted: May 17, 2025 Courses Completed: Applied Calculus I truly loved this class—it's one of the most enjoyable math courses I’ve ever taken.Pros:-- Exceptional Instruction and Support: Dr. Curtis was incredibly responsive and helpful whenever I had questions. The TA was also very supportive, and thanks to their guidance, I was proud to earn a 100% in the course—even without having taken pre-calculus beforehand.-- Innovative Software Platform: The custom software used in the course made a huge difference for me. I found it intuitive and engaging, and it helped reinforce the concepts in a way traditional textbooks never did.-- Thorough, Rigorous Curriculum: The structure of the course really pushed me to stay organized and plan ahead. I felt like I was being challenged in all the right ways.Potential Considerations for Others:-- Requires Strong Time Management: If you haven’t taken pre-calc, like me, you’ll need to be extra proactive. The course can move quickly if you need, and pacing yourself is essential.-- Software Learning Curve: While I personally loved the software, students who aren’t comfortable adapting to new digital tools might need a bit of extra time upfront to get used to it.-- Helpful to Have Supplementary Resources: One improvement might be to offer a short list of "starter resources" (videos, concept overviews, etc.) for students who need a broader intro to calculus before diving in.Overall, I highly recommend this course to motivated students, especially those comfortable with self-paced learning and open to using new tools. Dr. Curtis is a fantastic instructor, and the course setup really works. Transferred Credits To: MIT Genevieve P.★★★★★ Posted: Sep 20, 2020 Courses Completed: Applied Calculus I found out from my grad school after being accepted that I needed a Calculus course before starting their MBA program. I had less than 6 weeks to do it (and as a non-STEM undergrad no less). The video lectures were informative, the pre-calc refresher was great to get re-conditioned, and the asynchronous format worked so well as I did this at night/weekends after work. I completed it in 4 weeks. Professor Curtis was extremely responsive, graded assignments quickly, and a supportive guide providing constructive feedback to me to excel at the assignments. I highly recommend this course for those who need a pre-req in a hurry or like learning on their own schedule. Thanks, Distance Calculus and Professor Curtis! Transferred Credits To: Massachusetts Institute of Technology (MIT) Trevor★★★★★ Posted: Jun 19, 2025 Courses Completed: Calculus I POSITIVES:One of the best math classes I have ever taken. The lessons made the failures of my previous professors very apparent. In a few short minutes, things that I used to struggle with just clicked. This professor is top notch and really wants you to understand how to use the material.NEGATIVES:The SOFTWARE is extremely frustrating. Even after taking the time to learn, there are countless glitches. You learn to work around them, and overall, the software makes the math convenient, but its failures are sorely felt throughout the course. Make sure you save often as it crashes regularly, especially with graphs.The assignments are easy enough but some of them don't line up with the taught material. Be prepared to do some of your own independent research to get a deeper understanding of why things are the way they are. Transferred Credits To: US Army Taylor C.★★★★★ Posted: May 27, 2025 Courses Completed: Multivariable Calculus Im a visual learner and had a difficult time understanding multivariable calculus my first go around, and found this course offered by Roger Williams University, which uses real-time 3d graphing of the equations. I would see my work be manipulating the models, which gave a robust understanding of what each variable did. absolutely loved the course and class recordings. Transferred Credits To: Califorina Baptist University Emmy★★★★★ Posted: May 26, 2025 Courses Completed: Applied Calculus Distance Calculus was the perfect answer to getting ahead in math over the summer of my junior year in HS. I was able to complete the entire course over the summer. The teacher was responsive and the course was understandable. Highly recommend. Transferred Credits To: Syracuse University Video Player Document Viewer
2177	https://harvesttotable.com/trowel-buyers-guide/	Trowel Buyer's Guide - Harvest to Table Skip to content Join our gardening family to receive the latest tips. 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Facebook Twitter Save A trowel is a small, narrow, hand-held metal scoop used for small digging jobs, such as digging holes for flower and vegetrable transplants. Trowels are used while you kneel or sit. Trowels are commonly made from one piece of cast metal or a metal blade with a plastic or wooden handle. Trowels have various blade and handle sizes that are suited for a variety of tasks. Transplanting trowels, for example, have extra-narrow blade while multi-purpose towels have wide-blades. Table of Contents Toggle Trowel uses Trowel construction Trowel types Standard trowel One-piece trowel Planting or bulb trowel Trowel use tips Trowel maintenance Trowel buying tips Trowel uses Trowels are excellent for digging small planting holds and for removing weeds. Use a trowel for digging up and dividing a small perennial. Use a trowel to break up a bale of peat moss or compost. Use a trowel or to scoop potting mix into containers and for planting bulbs. Use a wide blade trowel to move soil. Use a long narrow blade trowel are for digging a hole in compacted clay soil. Some trowels have measurements scribed onto the blade to help dig to a desired planting depth. Trowel construction Trowel blades are can be made of tough fordged steel. Some blades are made of light weight and rust-resistant aluminum. Stainless steel trowel blades offer durability. Trowel handles can be made of wood, fiberglras or metal. The handles are sometimes wrapped in rubber or plastic for a cushioned grip. Some trowels have larger-diameter handles for gardeners who have a weak grip or suffer from arthritis. Som trowels have curved and foam-padded handles. Some manufacturers offer long-handled trowels with 2- or 3-feet long lightweight handles for use by those gardeners who have trouble bending over. Houseplant trowels have small, nearly minature blades for use in container gardening. Trowel with scribed measurements on blade Trowel types Standard trowel A standard trowel is made from two pieces: a wooden, metal ,plastic, or fiberglass handle and metal blade with a sharp edge. One-piece trowel A one-piece trowel is made from cast lighweight aluminium or other metal alloy or forged heavy-gauge steel. Usually the handle is covered with a colorfull platic grip. The blade on a one-piece steel trowel is often V-shaped rather than curved (adding strength). Planting or bulb trowel A bulb trowel has a narrower blade, ususally 2 to 3 inches at the top, tapering down to about 1 inch at the tip, and as much as 6 inches long. The smaller blade can be used to dig a small, narrow hole suitable for planting bulbs, bedding plants, or seedlings. Most bulb trowels have measuring lines stamped into the blades for determining how deep to plant each bulb. Trowel use tips Hold a trowel with your knuckles wraped around the back of the handle and your thumb pointed up, as if if where an icepick; stab the blade into the soil. When transplanting, insert the trowel into the soil and open a hole by moving the trowel from side to side; the curve of the blade will form the hole. Use a narrow and V-shped blades for digging out weeds and planting bulbs and for piercing heavy soil. Use a wider and flatter blade for moving soil and digging large holes in loose soil. Use a long handle trowel if you have trouble bending over or kneeling. Trowel maintenance Metal trowel blades should be cleaned with an oiled cloth after use and stored in a dry place to avoid rusting. Metal-blade trowel with wooden handle Trowel buying tips Choose a trowel that feels comfortable in your hand. Choose a trowel with a bright handle that is easy to find if you lay it down while you are working. Try various grips for comfort. Usually a larger grip will be eaiser to use. A one-piece aluminum or steel trowel with a plastic or rubber wrapped-handle will be most durable an comfortable. A tang-and-ferrule trowel blade that fits into a separate handle may eventually detach. V-shaped blades are strong and less likely to bend. Aluminium trowels are lightweight, a consideration for people with arthritis or weak grip but the blade will not be as sharp as a steel blade and cannot be sharpened. You may want to buy several sizes of trowels for use in different jobs. Trowels we like Stephen Albert Stephen Albert is a horticulturist, master gardener, and certified nurseryman who has taught at the University of California for more than 25 years. He holds graduate degrees from the University of California and the University of Iowa. His books include Vegetable Garden Grower’s Guide, Vegetable Garden Almanac & Planner, Tomato Grower’s Answer Book, and Kitchen Garden Grower’s Guide. His Vegetable Garden Grower’s Masterclass is available online. Harvesttotable.com has more than 10 million visitors each year. Similar Posts ### Peat Pellet Buyer’s Guide ByStephen Albert Peat pellets are used for seed starting. Peat pellets are compressed disks of dehydrated peat moss and other organic materials. They are used to germinate seeds indoors before transplanting seedlings into the garden. A peat pellet is a self-contained pot and planting medium all in one. When it comes to gardening and horticulture, peat pellets… Read More Peat Pellet Buyer’s Guide ### Planting Pot and Container Guide ByStephen Albert Growing plants in pots and containers is not difficult. One of the first decisions to make when growing plants in pots and containers is the type of container that will work best for the plant and the place you want to grow it. 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2178	https://www.sciencing.com/real-life-uses-pythagorean-theorem-8247514/	Math Geometry Real Life Uses Of The Pythagorean Theorem By Jon Zamboni Updated BackyardProduction/iStock/GettyImages The Pythagorean Theorem is a statement in geometry that shows the relationship between the lengths of the sides of a right triangle – a triangle with one 90-degree angle. The right triangle equation is a2 + b2 = c2. Being able to find the length of a side, given the lengths of the two other sides makes the Pythagorean Theorem a useful technique for construction and navigation. Architecture and Construction Architecture and Construction Given two straight lines, the Pythagorean Theorem allows you to calculate the length of the diagonal connecting them. This application is frequently used in architecture, woodworking, or other physical construction projects. For instance, say you are building a sloped roof. If you know the height of the roof and the length for it to cover, you can use the Pythagorean Theorem to find the diagonal length of the roof's slope. You can use this information to cut properly sized beams to support the roof, or calculate the area of the roof that you would need to shingle. Laying Out Square Angles Laying Out Square Angles The Pythagorean Theorem is also used in construction to make sure buildings are square. A triangle whose side lengths correspond with the Pythagorean Theorem – such as a 3 foot by 4 foot by 5 foot triangle – will always be a right triangle. When laying out a foundation, or constructing a square corner between two walls, construction workers will set out a triangle from three strings that correspond with these lengths. If the string lengths were measured correctly, the corner opposite the triangle's hypotenuse will be a right angle, so the builders will know they are constructing their walls or foundations on the right lines. Navigation Navigation The Pythagorean Theorem is useful for two-dimensional navigation. You can use it and two lengths to find the shortest distance. For instance, if you are at sea and navigating to a point that is 300 miles north and 400 miles west, you can use the theorem to find the distance from your ship to that point and calculate how many degrees to the west of north you would need to follow to reach that point. The distances north and west will be the two legs of the triangle, and the shortest line connecting them will be the diagonal. The same principles can be used for air navigation. For instance, a plane can use its height above the ground and its distance from the destination airport to find the correct place to begin a descent to that airport. Surveying Surveying Surveying is the process by which cartographers calculate the numerical distances and heights between different points before creating a map. Because terrain is often uneven, surveyors must find ways to take measurements of distance in a systematic way. The Pythagorean Theorem is used to calculate the steepness of slopes of hills or mountains. A surveyor looks through a telescope toward a measuring stick a fixed distance away, so that the telescope's line of sight and the measuring stick form a right angle. Since the surveyor knows both the height of the measuring stick and the horizontal distance of the stick from the telescope, he can then use the theorem to find the length of the slope that covers that distance, and from that length, determine how steep it is. References NASA: Pythagorean Theorem Math Central: Navigation NOVA Teachers: War Machines of Tomorrow American Mathematical Society: The Mathematics of Surveying University of Illinois: Using the Pythagorean Theorem in Construction Cite This Article MLA Zamboni, Jon. "Real Life Uses Of The Pythagorean Theorem" sciencing.com, 13 March 2018. APA Zamboni, Jon. (2018, March 13). Real Life Uses Of The Pythagorean Theorem. sciencing.com. Retrieved from Chicago Zamboni, Jon. Real Life Uses Of The Pythagorean Theorem last modified March 24, 2022.
2179	https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Statistics_with_Technology_2e_(Kozak)/07%3A_One-Sample_Inference/7.02%3A_One-Sample_Proportion_Test	Skip to main content 7.2: One-Sample Proportion Test Last updated : Jan 11, 2021 Save as PDF 7.1: Basics of Hypothesis Testing 7.3: One-Sample Test for the Mean Page ID : 5200 Kathryn Kozak Coconino Community College ( \newcommand{\kernel}{\mathrm{null}\,}) There are many different parameters that you can test. There is a test for the mean, such as was introduced with the z-test. There is also a test for the population proportion, p. This is where you might be curious if the proportion of students who smoke at your school is lower than the proportion in your area. Or you could question if the proportion of accidents caused by teenage drivers who do not have a drivers’ education class is more than the national proportion. To test a population proportion, there are a few things that need to be defined first. Usually, Greek letters are used for parameters and Latin letters for statistics. When talking about proportions, it makes sense to use p for proportion. The Greek letter for p is , but that is too confusing to use. Instead, it is best to use p for the population proportion. That means that a different symbol is needed for the sample proportion. The convention is to use, , known as p-hat. This way you know that p is the population proportion, and that is the sample proportion related to it. Now proportion tests are about looking for the percentage of individuals who have a particular attribute. You are really looking for the number of successes that happen. Thus, a proportion test involves a binomial distribution. Hypothesis Test for One Population Proportion (1-Prop Test) State the random variable and the parameter in words. x = number of successes I = proportion of successes State the null and alternative hypotheses and the level of significance , where is the known proportion , use the appropriate one for your problem Also, state your level here. 3. State and check the assumptions for a hypothesis test 1. A simple random sample of size n is taken. 2. The conditions for the binomial distribution are satisfied 3. To determine the sampling distribution of , you need to show that and , where . If this requirement is true, then the sampling distribution of is well approximated by a normal curve. 4. Find the sample statistic, test statistic, and p-value Sample Proportion: Test Statistic: p-value: TI-83/84: Use normalcdf(lower limit, upper limit, 0, 1) Note if , then lower limit is and upper limit is your test statistic. If , then lower limit is your test statistic and the upper limit is . If , then find the p-value for , and multiply by 2. R: Use pnorm(z, 0, 1) Note If , then you can use pnorm. If , then you have to find pnorm and then subtract from 1. If , then find the p-value for , and multiply by 2. 5. Conclusion This is where you write reject or fail to reject . The rule is: if the p-value < , then reject . If the p-value , then fail to reject . 6. Interpretation This is where you interpret in real world terms the conclusion to the test. The conclusion for a hypothesis test is that you either have enough evidence to show is true, or you do not have enough evidence to show is true. Example hypothesis test for one proportion using formula A concern was raised in Australia that the percentage of deaths of Aboriginal prisoners was higher than the percent of deaths of non-Aboriginal prisoners, which is 0.27%. A sample of six years (1990-1995) of data was collected, and it was found that out of 14,495 Aboriginal prisoners, 51 died ("Indigenous deaths in," 1996). Do the data provide enough evidence to show that the proportion of deaths of Aboriginal prisoners is more than 0.27%? State the random variable and the parameter in words. State the null and alternative hypotheses and the level of significance. State and check the assumptions for a hypothesis test. Find the sample statistic, test statistic, and p-value. Conclusion Interpretation Solution x = number of Aboriginal prisoners who die p = proportion of Aboriginal prisoners who die 2. Example b argued that the . 3. A simple random sample of 14,495 Aboriginal prisoners was taken. However, the sample was not a random sample, since it was data from six years. It is the numbers for all prisoners in these six years, but the six years were not picked at random. Unless there was something special about the six years that were chosen, the sample is probably a representative sample. This assumption is probably met. There are 14,495 prisoners in this case. The prisoners are all Aboriginals, so you are not mixing Aboriginal with non-Aboriginal prisoners. There are only two outcomes, either the prisoner dies or doesn’t. The chance that one prisoner dies over another may not be constant, but if you consider all prisoners the same, then it may be close to the same probability. Thus the conditions for the binomial distribution are satisfied In this case p = 0.0027 and n = 14,495. and . So, the sampling distribution for is a normal distribution. Sample Proportion: x = 51 n = 14495 Test Statistic: p-value: TI-83/84: p-value = R: p-value = Since the p-value < 0.05, then reject . There is enough evidence to show that the proportion of deaths of Aboriginal prisoners is more than for non-Aboriginal prisoners. Example hypothesis test for one proportion using technology A researcher who is studying the effects of income levels on breastfeeding of infants hypothesizes that countries where the income level is lower have a higher rate of infant breastfeeding than higher income countries. It is known that in Germany, considered a high-income country by the World Bank, 22% of all babies are breastfeed. In Tajikistan, considered a low-income country by the World Bank, researchers found that in a random sample of 500 new mothers that 125 were breastfeeding their infant. At the 5% level of significance, does this show that low-income countries have a higher incident of breastfeeding? State you random variable and the parameter in words. State the null and alternative hypotheses and the level of significance. State and check the assumptions for a hypothesis test. Find the sample statistic, test statistic, and p-value. Conclusion Interpretation Solution x = number of woman who breastfeed in a low-income country p = proportion of woman who breastfeed in a low-income country 2. 3. A simple random sample of 500 breastfeeding habits of woman in a low-income country was taken as was stated in the problem. There were 500 women in the study. The women are considered identical, though they probably have some differences. There are only two outcomes, either the woman breastfeeds or she doesn’t. The probability of a woman breastfeeding is probably not the same for each woman, but it is probably not very different for each woman. The conditions for the binomial distribution are satisfied In this case, n = 500 and p = 0.22. and , so the sampling distribution of is well approximated by a normal curve. This time, all calculations will be done with technology. On the TI-83/84 calculator. Go into the STAT menu, then arrow over to TESTS. This test is a 1-propZTest. Then type in the information just as shown in Figure . Once you press Calculate, you will see the results as in Figure . The z in the results is the test statistic. The p = 0.052683219 is the p-value, and the is the sample proportion. The p-value is approximately 0.053. On R, the command is prop.test(x, n, po, alternative = "less" or "greater"), where po is what says p equals, and you use less if your is less and greater if your is greater. If your is not equal to, then leave off the alternative statement. So for this example, the command would be prop.test(125, 500, .22, alternative = "greater") 1-sample proportions test with continuity correction data: 125 out of 500, null probability 0.22 X-squared = 2.4505, df = 1, p-value = 0.05874 alternative hypothesis: true p is greater than 0.22 95 percent confidence interval: 0.218598 1.000000 sample estimates: p 0.25 Note R does a continuity correction that the formula and the TI-83/84 calculator do not do. You can put in a command that says not to use the continuity correction, but it is correct to use it. Also, R doesn’t give the z test statistic, so you don’t need to worry about this. It does give a p-value that is slightly off from the formula and the calculator due to the continuity correction. p-value = 0.05874 Since the p-value is more than 0.05, you fail to reject . There is not enough evidence to show that the proportion of women who breastfeed in low-income countries is more than in high-income countries. Notice, the conclusion is that there wasn't enough evidence to show what said. The conclusion was not that you proved true. There are many reasons why you can’t say that is true. It could be that the countries you chose were not very representative of what truly happens. If you instead looked at all high-income countries and compared them to low-income countries, you might have different results. It could also be that the sample you collected in the low-income country was not representative. It could also be that income level is not an indication of breastfeeding habits. There could be other factors involved. This is why you can’t say that you have proven is true. There are too many other factors that could be the reason that you failed to reject . Homework Exercise In each problem show all steps of the hypothesis test. If some of the assumptions are not met, note that the results of the test may not be correct and then continue the process of the hypothesis test. Eyeglassomatic manufactures eyeglasses for different retailers. They test to see how many defective lenses they made in a given time period and found that 11% of all lenses had defects of some type. Looking at the type of defects, they found in a three-month time period that out of 34,641 defective lenses, 5865 were due to scratches. Are there more defects from scratches than from all other causes? Use a 1% level of significance. In July of 1997, Australians were asked if they thought unemployment would increase, and 47% thought that it would increase. In November of 1997, they were asked again. At that time 284 out of 631 said that they thought unemployment would increase ("Morgan gallup poll," 2013). At the 5% level, is there enough evidence to show that the proportion of Australians in November 1997 who believe unemployment would increase is less than the proportion who felt it would increase in July 1997? According to the February 2008 Federal Trade Commission report on consumer fraud and identity theft, 23% of all complaints in 2007 were for identity theft. In that year, Arkansas had 1,601 complaints of identity theft out of 3,482 consumer complaints ("Consumer fraud and," 2008). Does this data provide enough evidence to show that Arkansas had a higher proportion of identity theft than 23%? Test at the 5% level. According to the February 2008 Federal Trade Commission report on consumer fraud and identity theft, 23% of all complaints in 2007 were for identity theft. In that year, Alaska had 321 complaints of identity theft out of 1,432 consumer complaints ("Consumer fraud and," 2008). Does this data provide enough evidence to show that Alaska had a lower proportion of identity theft than 23%? Test at the 5% level. In 2001, the Gallup poll found that 81% of American adults believed that there was a conspiracy in the death of President Kennedy. In 2013, the Gallup poll asked 1,039 American adults if they believe there was a conspiracy in the assassination, and found that 634 believe there was a conspiracy ("Gallup news service," 2013). Do the data show that the proportion of Americans who believe in this conspiracy has decreased? Test at the 1% level. In 2008, there were 507 children in Arizona out of 32,601 who were diagnosed with Autism Spectrum Disorder (ASD) ("Autism and developmental," 2008). Nationally 1 in 88 children are diagnosed with ASD ("CDC features -," 2013). Is there sufficient data to show that the incident of ASD is more in Arizona than nationally? Test at the 1% level. Answer : For all hypothesis tests, just the conclusion is given. See solutions for the entire answer. 1. Reject Ho. 3. Reject Ho. 5. Reject Ho. 7.1: Basics of Hypothesis Testing 7.3: One-Sample Test for the Mean
2180	https://www.youtube.com/watch?v=gl79sSw5aFI	Geometry Meets Algebra: Find the Missing Angles! TabletClass Math 880000 subscribers 156 likes Description 4209 views Posted: 16 Apr 2025 In this video, we blend geometry and algebra to solve a classic problem: How do you find the missing angles of a quadrilateral? 👉 Need Help with Math? Get full lessons, practice problems, and expert teacher instruction at TabletClass Math Academy: 🔗 Perfect for: ✅ Homeschool Students (Grades 6–12) ✅ High School & College Math Review ✅ Test Prep: SAT, ACT, GED, ASVAB & More ✅ Adults Brushing Up on Geometry & Algebra Popular Math Courses: Math Foundations Math Skills Rebuilder Course: Pre-Algebra Algebra Geometry Algebra 2 Pre-Calculus Math Notes: If you’re looking for a math course for any of the following, check out my full Course Catalog at: • MIDDLE & HIGH SCHOOL MATH • HOMESCHOOL MATH • COLLEGE MATH • TEST PREP MATH • TEACHER CERTIFICATION TEST MATH 23 comments Transcript: So let's take a look at a type of problem that you will definitely see in all geometry and algebra courses and that is finding the angles for a particular shape or figure. All right. So the question here is find the angles of this quadilateral and a quadilateral is a foursided polygon. But the actual uh question is we have this angle here being x° this angle right here being x°. this angle 130° and this angle right here is x + 20°. We want to find the actual angles in degrees. Okay, so if you know how to do this problem and feel free to use a calculator, go ahead and put your answer into the comment section and of course I'll walk through exactly how to solve this problem. This is not that difficult and this is something that you absolutely need to understand if you are taking any sort of geometry or algebra course. All right, but uh before we get started, let me quickly introduce myself. My name is John and I have been teaching middle and high school math for decades. And if you need help learning math, well, check out my math help program at tcmathacademy.com. You can find a link to that in the description below. And if this video helps you out or if you just enjoy this content, make sure to like and subscribe as that definitely helps me out. So let's go and get into the solution right now. Okay, so as I indicated, uh these type of problems are all over uh the place in geometry and algebra. Okay, so you might be saying to yourself, well, you know, when you study algebra, you don't really do geometry. No, that's not the case. when you study algebra whether it's basic algebra like pre-alggebra you do a lot of geometry problems and then of course in a full geometry course you uh definitely need to know algebra as that is a typical prerequisite to take geometry all right but the fact here that we need to understand is that the sum of the angles in a quadilateral is 360° so this is very much along the line of the sum of the angles of a triangle is 180° you need to know these basic facts facts because if you understand these basic facts then we can just use some basic algebra to solve this problem. Because if the sum of the angles of this quadrilateral is 360°, uh what we're saying is that this angle x plus this angle over here x plus this angle x + 20° plus this angle 360° has to add up to uh I'm sorry 130 degrees has to add up to 360 degrees. So we can simply build a lovely algebraic equation. Solve for x and then go ahead and uh find the actual angles. Okay. So let's go and take a look at the equation. Hopefully you were uh kind of figuring this out as I was speaking about this. Now some of you might be like, "Oh, this problem is not that difficult. I just uh forgot the sum of the angles of a quadilateral." And that's okay. But uh here's the thing. If there's two figures that you want to know a lot about in geometry, those two things are triangles and quadilaterals. There's all different sorts of quadilaterals that you need to understand. Things like the trapezoid, rectangle, square, a rhombus. Uh quadrilaterals are big topic in geometry. So at a minimum, you just want to know that uh the sum of the angles of a quadrilateral is 360°. Okay. So let's go and build this uh algebra equation. You can see here that I already did this. So, we're going to have this angle uh x + 20. So, I'll write that here in parenthesis plus 130°. That's that angle plus x + x. The order doesn't really make a difference here. You could have uh start at x plus x + 20 + 130 uh plus x doesn't make a difference because the sum total of all these angles is 360°. Okay. So, now that we have our equation, uh really it's uh up to you uh and your algebra skills to solve for x. So, we got to remember when we do solve this equation for X, that's going to be these two angles right here. And then to get this angle, we're going to have to take X and add it to 20. Okay, so let's go and do this basic algebra. This is not that difficult. And uh if there's anything in this little equation that you don't understand, well, that tells me you need to review some algebra. And I'll give you some suggestions on how you can review algebra and or geometry in a second. But let's go ahead and walk through the solution here and uh take a look at how many x's we have because we want to combine like terms. So we have x plus x plus x that gives us 3x. Then I have 20 and 130. That's going to give me one 50. So what I want to do here like any uh linear equation is I want to get all the variables on one side. Typically the left hand side of the equation all my numbers to the right side. So we have 3x + 150 is equal to uh 360. So my next move is to subtract 150 from both sides of the equation. So that's going to give us 3x is equal to 210, right? 360 minus uh 150 is 210. Now to solve for x, all I have to do is divide both sides of the equation by 3. So 3id 3 is 1 or 1 x or x. And then of course uh 210 / 3 is 70. Okay, so x is equal to 70. Pretty straightforward. So now before we continue on, it would really mean a lot to me if you hit that subscribe button. Now the reason I want more subscribers is basically I look at everybody that subscribes to my channel as a new student and as a math teacher that makes me very happy. So uh the best way to support this channel and what I do is to simply hit that subscribe button. And if you're going to do that, hit that bell notification as well, so you can get my latest videos. Now, remember, uh, irrespective of whether you're a math student or not. If you want to relearn math, for example, and you've been out of school for many, many years, I have two great courses, my math foundation and my math skills rebuilder course. You can find links to all of this in the description of this video. But if you happen to be a student, make sure to check out my full uh course library. Again, you can find the links to all of this in the description below. Okay, so let's go ahead and uh finish this problem up because now that we know that x is equal to 70, we are not done. Okay, we're going to have to go back and actually find the angles. So, we know these two uh these two angles down here in this quadrilateral are x degrees and x degrees. And by the way, this is a part of the problem that a lot of students um forget. They'll solve the equation, but they won't answer the uh the actual question. Okay, the question is, what is the angles of the quadrilaterals? So, we're going to have to go back and make sure we answer the question 100%. So, x degrees and x degrees, x is equal to 70. So, these two angles here um are 70°. And then this angle right here is x + 20, which is going to be 70 + 20, which of course is 90°. And then, of course, we can just uh check this, right? 90 + 130 + 70 + 70 will give us a sum total of 360°. Okay, so as I indicated in the beginning of this video, you are guaranteed to see this type of problem in all geometry and algebra courses. All right, you got to be really good at solving these type of um you know find angle problems, but they work the same way. Okay, so again, two figures that you want to know a lot about, quadrilaterals and triangles. But hopefully this little video helps you out. And if that's the case, don't forget to like and subscribe. And with that being said, I definitely wish you all the best in your math adventures. Thank you for your time and have a great day.
2181	https://www.etd.ceu.edu/2021/salia_nika.pdf	Extremal problems for paths and cycles Nika Salia Supervisor: Prof. Ervin Gy˝ ori Department of Mathematics and its Applications Central European University This dissertation is submitted for the degree of Doctor of Philosophy May 2021 CEU eTD Collection CEU eTD Collection For my mother CEU eTD Collection CEU eTD Collection It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us. David Hilbert, Mathematical Problems, International Congress of Mathematicians, Paris, 1900. CEU eTD Collection CEU eTD Collection Acknowledgements I am grateful to my supervisor, Prof. Ervin Gy˝ ori. Without his guidance, critiques, and unlimited hours of conversations, this work would never have been written. More importantly, Ervin has supported me in my personal and professional life outside this work. That support was essential for my well-being. I am deeply grateful for the time, energy, and emotions he invested in me. I would like to express my gratitude to researchers from Alfréd Rényi Institute of Mathematics. Special thanks to Prof. Gyula O.H. Katona for his constant support and inspiring conversations, I am deeply indebted to him. I would like to thank my colleague Casey Tompkins, who has constantly provided me with new problems and ideas, and also with his passion and drive to pursue the unknown. I want to thank prof. Rob Morris for his hospitality at IMPA. I also want to thank my collaborators- Beka Ergemlidze, Grahame Erskine, Dániel Gerbner, Oliver Janzer, Nathan Lemons, Ryan R. Martin, Abhishek Methuku, Dániel T. Nagy, Balázs Patkós, Addisu Paulos, Christoph Spiegel, James Tuite, Máté Vizer, Zhiyu Wang, Chuanqi Xiao, and Oscar Zamora Luna. I am thankful for productive conversations with my colleagues- Ferenc Bencs, Lucas Colucci, Gábor Mészáros, Tamás Róbert Mezei, and Dávid Tóth, I am grateful to Nino Latsabidze, Gela Kvirikashvili, Vano Kvitashvili, and Lasha Ephremidze. They have been influential in my life and have played an important role in empowering me to pursue this path. And I thank Csilla, who has enriched my life. Without her love, patience, and encourage-ment, I may never have finished this project. CEU eTD Collection CEU eTD Collection Abstract A classical result of Erd˝ os and Gallai determines the maximum number of edges in a simple n vertex graph without a path of given length as a subgraph, i.e. they determined Turán number of paths. They also determined Turán number of a class of long cycles. In this dissertation, we extend those results for Hypergraphs. We follow one of the most general definitions of paths and cycles in hypergraphs. A Berge-path of length k in a hypergraph H is a sequence v1,e1,v2,e2,...,vk,ek,vk+1 of distinct vertices and hyperedges with vi+1 ∈ei,ei+1 for all i ∈[k]. Berge-cycles are defined similarly. We study several generalizations of Erd˝ os-Gallai theorem for hypergraphs forbidding Berge families of paths and cycles and some related problems. CEU eTD Collection CEU eTD Collection Table of contents List of figures xiii Notations, symbols xv 1 Introduction 1 1.1 The Turán number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Turán number of degenerate (bipartite) graphs . . . . . . . . . . . . . 5 1.3 Erd˝ os-Gallai Theorem, paths and cycles . . . . . . . . . . . . . . . . . . . 7 1.4 Erd˝ os-Gallai Theorem, for vertex colored graphs . . . . . . . . . . . . . . 11 1.5 The generalized Turán number . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 Berge-hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7 The Turán number of Berge-hypergraphs . . . . . . . . . . . . . . . . . . . 17 1.8 Extensions of Erd˝ os-Gallai Theorem . . . . . . . . . . . . . . . . . . . . . 20 1.8.1 Hypergraphs without long Berge-paths . . . . . . . . . . . . . . . 20 1.8.2 Hypergraphs without long Berge-cycles . . . . . . . . . . . . . . . 24 1.8.3 Hypergraphs without Berge-trees . . . . . . . . . . . . . . . . . . 27 1.9 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 Erd˝ os-Gallai theorem, for vertex colored graphs 33 2.1 Forbidden paths in vertex colored graphs . . . . . . . . . . . . . . . . . . . 34 2.2 Forbidden trees in vertex colored graphs . . . . . . . . . . . . . . . . . . . 36 3 Hypergraph girth problem 39 3.1 Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Pentagons vs. triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 3-uniform BC4-free hypergraphs . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 CEU eTD Collection xii Table of contents 4 The Structure of Hypergraphs avoiding long Berge-cycles 49 4.1 Avoiding long Berge-cycles, cases k = r +1 and k = r +2 . . . . . . . . . 49 4.1.1 Basic Lemmas, used in Subsection 4.1.2 and 4.1.3 . . . . . . . . . 50 4.1.2 Extremal hypergraphs without BC≥r+1 . . . . . . . . . . . . . . . . 51 4.1.3 Extremal hypergraphs without BC≥r+2 . . . . . . . . . . . . . . . . 59 4.1.4 A corollary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Hypergraphs with a circumference at most uniformity . . . . . . . . . . . . 62 4.2.1 Proof of Theorem 1.8.16, 1.8.18, 1.8.20 . . . . . . . . . . . . . . . 64 4.2.2 Proof of Lemma 4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2.3 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5 Connected Hypergraphs without long Berge-paths 73 5.1 Proof of asymptotic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2 Proof of stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.1 Preliminary technical claims . . . . . . . . . . . . . . . . . . . . . 83 5.2.2 Possible embeddings of Hypergraph . . . . . . . . . . . . . . . . . 88 5.2.3 Case-by-case analysis . . . . . . . . . . . . . . . . . . . . . . . . 90 References 101 Index 109 CEU eTD Collection List of figures 1.1 The extremal graph of Theorem 1.3.1, ex(n,P ℓ). . . . . . . . . . . . . . . 8 1.2 The extremal graph of Theorem 1.3.2, ex(n,C≥ℓ). . . . . . . . . . . . . . . 9 1.3 The graph Gn,k,s, Definition 1.3.6. . . . . . . . . . . . . . . . . . . . . . . 11 1.4 A block tree and S(r) n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 The hypergraph Hn,a,b1,b2,...,bt. . . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 The hypergraphs Hn,⌊k−1 2 ⌋and Hn,⌊k−1 2 ⌋,2 . . . . . . . . . . . . . . . . . . 23 1.7 An extremal graph, Theorem 1.8.28. . . . . . . . . . . . . . . . . . . . . . 29 2.1 Sketch of the proof of Theorem 1.4.2. The picture on the left is for Case 1, and the other pictures show Case 4. . . . . . . . . . . . . . . . . . . . . . . 36 2.2 The graph Sa,b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1 An example of a crown-block and a K4-block. . . . . . . . . . . . . . . . . 40 5.1 The hypergraphs Hn,⌊k−1 2 ⌋and Hn,⌊k−1 2 ⌋,2. . . . . . . . . . . . . . . . . . . 74 5.2 The hypergraph Hn,a,b1,b2,...,bt. . . . . . . . . . . . . . . . . . . . . . . . . 77 5.3 Sketch of the proof of Claim 5.2.6. . . . . . . . . . . . . . . . . . . . . . . 83 5.4 Sketch of the proof of Claim 5.2.7. . . . . . . . . . . . . . . . . . . . . . . 84 CEU eTD Collection CEU eTD Collection Notations, symbols Functions χ(G) Chromatic number of G. ex(n,H) The maximum number of edges in an n-vertex H-free graph. ex(n,H,F) max{N(G,F) : G ⊆Kn, H ̸⊆G} exc(n,T) Extremal function for vertex colored graphs. exconn(n,H) The maximum number of edges in an n-vertex H-free connected graph. N(G,F) The number of sub-graphs of G isomorphic to F ∂k(H) k-shadow of a hypergraph H, i.e. all k-sets contained in a hyperedge of H. E(G) The edge set of graph / hypergraph G. e(G) The number of edges of graph / hypergraph G. fs(n,k,a) k−a s +(n−k +a) a s−1 . G1 +G2 A graph with vertex set V(G1)∪V(G2) and edge set E(G1)∪E(G2)∪{(v,u) : v ∈ V(G1),u ∈V(G2)}. G1 ∪G2 Disjoint union of graphs, i.e. it is a graph with vertex set V(G1)∪V(G2) and edge set E(G1)∪E(G2). N(G,F) The number of sub-graphs of G isomorphic to F. tr(n) The number of edges of Turán graph Tr(n) i.e. tr(n) = e(Tr(n)) V(G) The vertex set of graph / hypergraph G. v(G) The number of vertices of graph / hypergraph G. CEU eTD Collection xvi Notations, symbols General [n] {1,2,3,··· ,n}. 1A(·) Indicator function, 1A(x) = 1 if x ∈A and 1A(x) = 0 otherwise. Graphs C The family of all cycles, {C3,C4,...}. C≥k The family of cycles of length at least k, {Ck,Ck+1,...}. Codd The set of all cycles of odd length, i.e. {C3,C5,...}. S(r) n The n-vertex r-uniform hypergraph with n−r +1 hyperedges all intersecting in the same r −1 set. Cn A cycle with n vertices. Gn,k,s Kk−2s ∪Kn−k+1 +Ks. Kn The complete graph with n vertices. Kn,m Complete bipartite graph, with partite sets of size n and m. Kr n The complete r-uniform hypergraph on n vertices. P n A path of length n, i.e. n+1-vertex path. Tr(n) Complete r-partite Turán graph with n-vertices. CEU eTD Collection Chapter 1 Introduction Extremal graph theory, in its strictest sense, is a branch of graph theory de-veloped and loved by Hungarians Extremal Graph Theory , Béla Bollobás Extremal combinatorics is a branch of discrete mathematics that studies the maximum or the minimum size of discrete structures under given constraints. For example, a classical question studied by Mantel 1907 asks, ‘What is the maximum number of edges that a triangle-free graph can have?’, see Theorem 1.1.4. Thus, extremal combinatorics can be thought of as solving certain optimization problems, and as such has many real-world applications. Extremal combinatorics, as one might guess, is about extreme behaviors therefore it fascinates a wide range of audiences. In particular, since we are all naturally fascinated by everyday objects that are extreme for various properties like speed, color, size, or time in much the same way extremal objects in mathematics are inherently fascinating. In addition to the natural beauty, extremal combinatorics is a useful tool for other fields of mathematics, even more, extremal combinatorial problems encourage elegant mathematics that uses a variety of techniques from different fields of mathematics. In some sense, the first extremal result in graph theory was by Euler in 1758, when he showed that the maximum number of edges in a planar graph is at most 3n−6. After a couple of centuries in 1907 Mantel determined the maximum number of edges in a triangle-free graph. While working on a problem of number theory, Erd˝ os maximized the number of edges in an n-vertex graph without 4-cycles in 1938 . Surprisingly, after obtaining this result Erd˝ os did not discover extremal graph theory as a research subject as he said. Extremal combinatorics as a research subject was not discovered until 1941 when Turán determined CEU eTD Collection 2 Introduction the maximum number of edges in an n-vertex graph without a complete graph of a given size, Theorem 1.1.5. This result is counted as the birth of extremal combinatorics, naturally, we refer to the extremal number of a graph as the Turán number of that graph. Soon after Turán’s Theorem 1.1.5, Erd˝ os and Stone in 1946, later Erd˝ os and Simonovits in 1966 determined the asymptotic behavior of Turán function for all non-bipartite graphs. On one hand, it seems they have settled the majority of the problems but in the case of a bipartite graphs Turán function is hard to determine and in some cases, we do not even know the order of magnitude. This phenomenon having multiple results without discovering the subject repeats with so-called Generalized Turán numbers also. If the Turán number of a graph counts the maximum number of edges under some constraints, the generalized Turán number counts the maximum number of given substructures under the same constraints. In particular, in 1949 Zykov determined the maximum number of cliques of a given size in an n vertex graph without cliques of larger size, since this result was about linear complexes. The same result was done later by Erd˝ os , independently. While Erd˝ os was measuring how far are the triangle-free graphs from bipartite graphs he naturally asked a question ‘What is the maximum number of pentagons in a triangle-free graph’ . This question was settled half a century later by two groups of mathematicians using flag algebras. In 1991, Gy˝ ori, Pach, Simonovits , defined the generalized Turán number and obtained some results. In particular, they maximized copies of a bipartite graph with 1-factor in triangle-free graphs. While investigating pentagon-free 3-uniform hypergraphs Bollobás-Gy˝ ori initiated the study of the converse of the problem of Erd˝ os. They asked the following question ‘What is the maximum number of triangles in pentagon-free graphs’. Surprisingly this question is still open but we show a simple proof of an upper-bound in Chapter 3. After a decade Alon, Shikhelman defined the function ex(n,H,F), and obtained some general results after which the generalized Turán number gained a wide range of interest. Extremal questions for hypergraphs are even harder. Lately, numerous mathematicians are investigating the Turán number for hypergraphs. In here there are lots of questions and lots of open problems the majority of which are hard. The underlying theme of this work is investigating variants of the Turán extremal problem for Berge-hypergraphs. In this work, we determine extremal numbers of various kinds of paths and cycles in hypergraphs in the spirit of the Erd˝ os-Gallai theorem. In some sections, we prove some results for graphs which are useful tools for getting results for hypergraphs. CEU eTD Collection 1.1 The Turán number 3 1.1 The Turán number Extremal Graph theory is a branch of discrete mathematics that studies relations between graph invariants. In particular, how global parameters, such as the number of edges, can influence local substructures. One of the most studied problems of this field is to determine the Turán number of a graph. Definition 1.1.1. The Turán number of a graph H, ex(n,H), is the maximum number of edges in a simple graph on n vertices which does not have H as a sub-graph. In particular, ex(n,H) = max e(G) : G ⊆Kn and H ⊈G . For a given graph H, all n-vertex H-free graphs with ex(n,H) edges are extremal graphs. Let us denote a path of length n by P n, and cycle of length n by Cn. Note that number of vertices of P n is n+1. Example 1.1.2. It is easy to note that ex(n,P 2) = jn 2 k . even more, for a fixed n the only extremal graph is a perfect matching if n is even and almost perfect matching otherwise. This function naturally generalizes to a setting where rather than forbidding just a graph H but a class of graphs H. In particular, ex(n,H) = max e(G) : G ⊆Kn and H ⊈G for all H ∈H . Let us denote class of all cycles by C, i.e. C = {C3,C4,...}. We give you another example from classical graph theory. Example 1.1.3. We know the maximum number of edges in an n-vertex graph without a cycle is n−1. Therefore we have another Turán-type result ex(n,C) = n−1. In this case, extremal graphs are all n-vertex trees. The basic theorem from extremal graph theory is Mantel’s theorem from 1907.1 1For the reader, not familiar with extremal graph theory we recommend to check different proofs of Mantel’s theorem, see . CEU eTD Collection 4 Introduction Theorem 1.1.4 (Mantel ). If a graph G is triangle-free, then it contains at most j n2 4 k edges. Equality holds if and only if G is a complete balanced bipartite graph K⌊n 2⌋,⌈n 2⌉. Mantel’s theorem in extremal graph theory language is ex(n,K3) = j n2 4 k . The only extremal graph for fixed integer n, is a balanced bipartite graph K⌊n 2⌋,⌈n 2⌉. Later Mantel’s theorem was generalized for all complete graphs by Turán in 1941. This result is counted as the birth of extremal combinatorics. Theorem 1.1.5 (Turán ). For fixed integers n and r, let Turán graph T(n,r) be a balanced, n-vertex r-partite graph. Let us denote the number of edges in T(n,r) by t(n,r). Then we have ex(n,Kr+1) = t(n,r) ≤ 1−1 r n 2 . The only extremal graph is T(n,r). Since then researchers studied the Turán number of various graphs. Note that there are different stages while searching for ex(n,·). In particular, for a fixed graph it may be difficult to find the exact Turán number, hence there are different relaxations of the problem. At first, we try to find the order of magnitude of ex(n,·) as a function of n, then asymptotic, only after the exact value. Which is followed by characterizing all extremal graphs. Even more, the research may be continued afterward by asking saturation and stability-type questions, see Section 1.9. Note that there is a spectrum of other steps in between those steps. Like sometimes it is hard to get the exact result for every n but it is possible to get the exact result for infinitely many n. This spectrum of different problems, excited mathematicians of different tastes which made this subject widely popular. Soon after Turán’s Theorem 1.1.5, Erd˝ os and Stone in 1946 proved a theorem, this was later strengthened by Erd˝ os and Simonovits in 1966, Theorem 1.1.6. This theorem at first glance suggests that the majority of Turán problems are settled asymptotically. Theorem 1.1.6 (Erd˝ os, Stone, Simonovits [28, 33]). ex(n,H) = 1− 1 χ(H)−1 +o(1) n 2 . Note that if H is not bipartite i.e. χ(H) > 2 the this theorem determines asymptotic of ex(n,H). This was a motivation for Zarankiewicz to generalize Turán’s problem in 1951 . Before we state Zarankiewicz problem let us make a cosmetic change in the definition of the function ex(·,·). As we have seen ex(n,H) denotes the maximum number of edges in an n-vertex graph not containing H as a sub-graph. Therefore we are searching CEU eTD Collection 1.2 The Turán number of degenerate (bipartite) graphs 5 between all sub-graphs of Kn one with the maximum number of edges and no H. Naturally, one may generalize Turán function. ex(F,H) = max e(G) : G ⊆F and H ⊈G . or even more, let Fn be a family of n-vertex graphs. Then we may ask the following variant of the Turán number ex(Fn,H) = max e(G) : G ∈Fn and H ⊈G . For the reader, we specify F can be all n-vertex planar graphs, or all n-vertex connected graphs,2 or all n-vertex bipartite graphs. In this settings ex(n,H) is the same as ex(Kn,H). Zarankiewicz proposed to study ex(Kn,n,·) in 1951 . 1.2 The Turán number of degenerate (bipartite) graphs As we have seen Erd˝ os-Stone-Simonovits Theorem 1.1.6 settling Turán problem for all but non-bipartite graphs. Naturally, mathematicians started investigating Turán functen for bipartite graphs. Füredi and Simonovits devote a one-hundred-page survey to this topic, therefore we refer to this survey all the readers who are deeply interested in this topic. In this section, we try to present some central theorems from this area. Some of those results will be applied later. Let Codd be the set of all cycles of odd length. Theorem 1.2.1 (K˝ ovári, Sós, Turán ). ex(n,C4 ∪{Codd}) = 1 2 √ 2 n 3 2 +o(n 3 2). Theorem 1.2.2 (K˝ ovári, Sós, Turán ). For all positive integers a,b, a ≤b, we have ex(n,Ka,b) ≤1 2 a √ b−1n2−1 a + a−1 2 n. The proof of Theorem 1.2.2 is an intuitive double counting idea. On one hand they count the number of a-stars,3 which is ∑v∈V(G) d(v) a . If G is Ka,b-free then the number of a-stars is at most (b−1) V(G) a . Finally they apply Jansen’s Inequality to get the desired result. 2A graph is connected if there is a path between any two vertices of it. 3K1,a. CEU eTD Collection 6 Introduction Conjecture 1.2.3 (K˝ ovári, Sós, Turán ). The order of the upper bound in Theorem 1.2.2 is sharp. This conjecture holds for some special cases. For example we have Theorem 1.2.4 (Erd˝ os, Rényi, Sós , Brown ). ex(n,C4) = 1 2n3/2 +O(n 3 2 −c). The upper bound of this theorem comes from cherry4 counting argument as before. A corresponding lower bound comes from finite geometry constructions. Example 1.2.5 ([19, 27], see also [16, 52]). Let q be a prime power. The vertices of our graph G are the equivalence classes of the non-zero triples (a,b,c) ∈GF(q)3, Where GF(q) is a finite field. Two triples (a,b,c) and (x,y,z) are considered in the same equivalence class if (a,b,c) = (λx,λy,λz) for some non-zero element λ ∈GF(q). We have v(G) = q3−1 q−1 = q2 +q+1. The edge set of graph G is E(G) = {((a,b,c),(x,y,z)) : ax+by+cz = 0 and (a,b,c) ̸= (x,y,z)} One can easily see that the graph G from Example 1.2.5 is C4-free. Theorem 1.2.6 (Füredi ). Let n = q2 + q + 1 where q is an integer such that q ̸= 1,7,9,11,13, then ex(n,C4) ≤1 2q(q+1)2. Moreover, if q is a power of a prime, then ex(n,C4) = 1 2q(q+1)2. Here we would like to highlight our favorite conjecture of Erd˝ os. Conjecture 1.2.7 (Erd˝ os). ex(n,{C3,C4}) = 1 2 √ 2 n 3 2 +o(n 3 2). On the other hand, there is a counter-conjecture from Allen, Keevash, Sudakov, and Verstraëte. 4Cherry is a bipartite graph K1,2 CEU eTD Collection 1.3 Erd˝ os-Gallai Theorem, paths and cycles 7 Conjecture 1.2.8 (Allen, Keevash, Sudakov, Verstraëte ). limsup n→∞ ex(n,{C3,C4}) ex(n,Codd ∪{C4}) > 1. Erd˝ os, Simonovits determined extremal number of {C4,C5}. Theorem 1.2.9 (Erd˝ os, Simonovits ). ex(n,{C5,C4}) = 1 2 √ 2 n 3 2 +o(n 3 2). Allen, Keevash, Sudakov, Verstraëte generalized Erd˝ os, Simonovits theorem. Theorem 1.2.10 (Allen, Keevash, Sudakov, Verstraëte ). Let ℓand t be integers, such that ℓ≥2. Then we have limsup n→∞ ex(n,{K2,t,C2ℓ+1}) ex(n,{K2,t,Codd})). Here comes another mysterious open problems for cycles. A well-known result of Bondy-Simonovits asserts that for all ℓ≥2 we have ex(n,C2ℓ) = O(n1+1/ℓ), however the order of magnitude is only known to be sharp in the cases ℓ= 2,3,5. In the following section, we concentrate on paths and long cycles. 1.3 Erd˝ os-Gallai Theorem, paths and cycles As we have seen Erd˝ os-Stone-Simonovits Theorem 1.1.6 settling Turán problem for all but non-bipartite graphs. Erd˝ os-Gallai determined the Turán number of paths and a class of long cycles in 1959. Theorem 1.3.1 (Erd˝ os, Gallai ). For two integers n and ℓ, ex(n,P ℓ) ≤(ℓ−1)n 2 . The equality holds if and only if ℓ\|n and extremal graph is the disjoint union of n ℓcliques of size ℓ, see Figure 1.1. Strictly speaking, this theorem was a corollary of the following more general theorem. Let us denote a family of cycles of length at least ℓby C≥ℓ. Theorem 1.3.2 (Erd˝ os, Gallai ). For two integers n and ℓ, ex(n,C≥ℓ) ≤(ℓ−1)(n−1) 2 . CEU eTD Collection 8 Introduction Kℓ Kℓ ··· Kℓ Fig. 1.1 The extremal graph of Theorem 1.3.1, ex(n,P ℓ). The equality holds if and only if ℓ−2\|n−1 and G is the union of n−1 ℓ−2 disjoint cliques of size ℓ−1 sharing a vertex in a tree-like structure 5, see Figure 1.2. Let G be an n vertex P ℓ-free graph. We construct an auxiliary graph G′ from G, by adding a vertex v to G which is joined to all vertices of G. In particular, we have V(G′) =V(V)∪{v} and E(G′) = E(G)∪{(v,u) : u ∈V(G)}. Note that since G is P ℓ-free, G′ is C≥ℓ+2-free. Hence from Theorem 1.3.2 we have the desired inequality e(G) = e(G′)−n ≤(ℓ+2−1)(n+1−1) 2 −n = (ℓ−1)n 2 . Since we find the following trick is a useful tool we would like to show the proof of Theorem 1.3.1 using Theorem 1.3.2 as in . Let G be an n vertex P ℓ-free graph. We construct an auxiliary graph G′ from G, by adding a vertex v to G which is joined to all vertices of G. In particular, we have V(G′) =V(V)∪{v} and E(G′) = E(G)∪{(v,u) : u ∈V(G)}. Note that since G is P ℓ-free, G′ is C≥ℓ+2-free. Hence from Theorem 1.3.2 we have the desired inequality e(G) = e(G′)−n ≤(ℓ+2−1)(n+1−1) 2 −n = (ℓ−1)n 2 . After finding the Turán number of P ℓ, one may attempt to determine the Turán number of ℓ+ 1-vertex trees. Surprisingly this problem remains still open despite of the interest around it. 5Every vertex-maximal two-connected component of G is isomorphic to Kℓ−1. A graph is two-connected if there are two internally disjoint paths between any two vertices. CEU eTD Collection 1.3 Erd˝ os-Gallai Theorem, paths and cycles 9 Fig. 1.2 The extremal graph of Theorem 1.3.2, ex(n,C≥ℓ). Conjecture 1.3.3 (Erd˝ os, Sós ). Let Tℓ+1 be an arbitrary ℓ+1 vertex tree, then ex(n,Tℓ+1) ≤(ℓ−1)n 2 . A corresponding lower-bound comes from the same construction as in Theorem 1.3.1, see Figure 1.1. It is believed that this conjecture is true since there are some specific cases done. For the reader, interested in this subject we refer to the sixth chapter of the Survey . Mordechai Lewin extended Erd˝ os-Gallai Theorem 1.3.2 for directed cycles and obtained sharp results for all n. Woodall extended Theorem 1.3.2, they obtained sharp results for every n. Theorem 1.3.4 (Woodall ). Let 0 ≤t and 0 ≤r < ℓ−2 ex(t(ℓ−2)+r +1,C≥ℓ) = t l −1 2 + r +1 2 A corresponding lower-bound comes from a graph containing t copies of Kℓ−1 and a Kr+1 sharing a vertex. Similarly, as before one may get a corollary for the paths, this was already obtained by Faudree and Schelp including the classification of extremal graphs. Theorem 1.3.5 (Faudree, Schelp ). Let 0 ≤t and 0 ≤r < ℓ ex(tℓ+r,P ℓ) = t ℓ 2 + r 2 . CEU eTD Collection 10 Introduction Extremal graphs are G1 and G2, j. Where G1 := t [ i=1 Kℓ ! [ Kr. When ℓis odd, t > 0 and r = n±1 2 then G2, j := j [ i=1 Kℓ [ Kℓ−1 2 +K ℓ+1 2 +(t−j−1)ℓ+r for every j, 0 ≤j ≤k −1. Jackson , Gyárfás obtained extremal numbers of paths in bipartite graphs. Another extension of Theorem 1.3.2 was suggested by Woodall . They considered 2-connected graphs without long cycles and obtained partial results. Later in 1977,Kopylov settled the proposed conjecture in . They determined extremal number for 2-connected n-vertex graphs without a cycle of length at least ℓ. This result with the already mentioned trick was implying result for paths too. Which was later also proved by Balister, Gy˝ ori, Lehel, Schelp in , including Extremal graphs. Let us denote the maximum number of edges in an n-vertex H-free connected graph by exconn(n,H). Definition 1.3.6. For n ≥k ≥2s let Gn,k,s := Kk−2s ∪Kn−k+1 +Ks. See figure 1.3. Theorem 1.3.7 (Kopylov , Balister, Gy˝ ori, Lehel, Schelp ). Let n > ℓ≥3 exconn(n,P ℓ) = max n e(Gn,ℓ,1),e(Gn,ℓ,⌊ℓ−1 2 ⌋) o . Extremal graphs are Gn,ℓ,1 or Gn,ℓ,⌊ℓ−1 2 ⌋, see Definition 1.3.6. Another variant of Erd˝ os-Gallai problem is to determine the Turán number of paths for Erd˝ os-Rényi random graphs. In particular, the random variable ex(Gn,p,H), where Gn,p is the Erd˝ os–Rényi random graph, was introduced by Babai, Simonovits, Spencer , and by Frankl, Rödl . For recent developments in this direction see [93, 12]. CEU eTD Collection 1.4 Erd˝ os-Gallai Theorem, for vertex colored graphs 11 Kk−2s Ks Kn−k+s Fig. 1.3 The graph Gn,k,s, Definition 1.3.6. 1.4 Erd˝ os-Gallai Theorem, for vertex colored graphs In a relatively recent paper, Gy˝ ori, Lemons investigated the extremal number of hyper-graphs avoiding Berge-cycles. To this end, they introduced a generalization of the theorem of Erd˝ os-Gallai about paths. By a proper vertex coloring of a graph G, we mean a coloring of the vertices of G such that no two adjacent vertices are the same color. Gy˝ ori, Lemons proved the following lemma. Lemma 1.4.1 (Gy˝ ori-Lemons ). Let k be a positive integer and G be an n-vertex graph with a proper vertex coloring such that G contains no P 2k+1 with endpoints of different colors, then \|E(G)\| ≤2kn. We show that the factor of 2 in Theorem 1.4.1 is not needed and, thus, recover the original upper bound from the Erd˝ os-Gallai theorem. We also determine which graphs achieve this upper bound. Theorem 1.4.2 (Salia, Tompkins, Zamora ). Let k ≥0 and G be an n-vertex graph with a proper vertex coloring such that G contains no P 2k+1 with endpoints of different colors, then \|E(G)\| ≤kn, and equality holds if and only if 2k +1 divides n and G is the union of n 2k+1 disjoint cliques of size 2k +1. We believe that an analog of Theorem 1.4.1 should hold in the setting of trees. Recall that the extremal number ex(n,H) of a graph H is defined to be the largest number of edges an n-vertex graph may have if it does not contain H as a sub-graph. We introduce a new variation of the extremal function ex(n,T) in the case of trees. CEU eTD Collection 12 Introduction Definition 1.4.3. Let exc(n,T) denote the maximum number of edges possible in an n-vertex graph G with a proper vertex coloring (using any number of colors), such that in every copy of T in G the leaves of T are all the same color. As for what we experienced for paths we have two different cases for trees. The first when the tree T has leaves from different color classes. Theorem 1.4.4 (Salia, Tompkins, Zamora ). Let T be a tree with k edges such that in the (unique) proper vertex 2-coloring of T all leaves are not in the same color, then exc(n,T) ≤(k −1)n. The second when all leaves are from the same color class. Theorem 1.4.5 (Salia, Tompkins, Zamora ). Let T be a tree with k edges such that in the proper vertex 2-coloring of T all leaves are the same color, then exc(n,T) = j n2 4 k , provided n is sufficiently large. We believe that a strengthening of Conjecture 1.3.3 should hold for trees whose 2-coloring yields two leaves of different colors. Conjecture 1.4.6 (Salia, Tompkins, Zamora ). Let T be a tree with k ≥1 edges such that in the proper vertex 2-coloring of T all leaves are not the same color, then exc(n,T) ≤(k−1)n 2 . One would hope that Conjecture 1.4.6 could be deduced directly from Conjecture 1.3.3, but unfortunately, this does not seem to be the case. We take the first step towards Conjec-ture 1.4.6 by proving it in the case of double stars. Theorem 1.4.7 (Salia, Tompkins, Zamora ). For positive integers a and b, let Sa,b denote the tree on a + b + 2 vertices consisting of an edge {u,v} where \|N(u)\v\| = a, \|N(v)\u\| = b and N(u)∩N(v) = ∅(See Figure 2.2, left). We have exc(n,Sa,b) ≤a+b 2 n. Proofs of these theorems are in Chapter 2. 1.5 The generalized Turán number Recall, for a fixed graph F, the classical Turán number ex(n,F) is defined to be the maximum number of edges possible in an n-vertex graph not containing F as a sub-graph. This function naturally generalizes to a setting where, rather than edges, we maximize the number of copies of a given graph H in an n-vertex F-free graph. Following Alon, Shikhelman (see also ), we denote this more general function by ex(n,H,F). In particular, we have ex(n,F,H) = max{N(G,F) : G ⊆Kn, H ̸⊆G}. CEU eTD Collection 1.5 The generalized Turán number 13 Where N(G,F) denotes the number of sub-graphs of G (not necessarily induced) isomorphic to F. Problems of this type have a long history beginning with a result of Zykov (and later independently Erd˝ os ) who determined the value of ex(n,Kr,Kt) for any pair of cliques. While Erd˝ os was measuring how far are the triangle-free graphs from bipartite graphs he naturally asked a question ‘What is the maximum number of pentagons in a triangle-free graph’ . This question was settled half a century later by Hatami, Hladký, Král, Norine, Razborov and independently by Grzesik , using flag algebras. In 1991, Gy˝ ori, Pach, Simonovits , defined the generalized Turán number and obtained some results. In particular, they maximized copies of a bipartite graph with 1-factor in triangle-free graphs. While investigating pentagon-free 3-uniform hypergraphs Bollobás-Gy˝ ori initiated the study of the converse of the problem of Erd˝ os. They asked the following question ‘What is the maximum number of triangles in a pentagon-free graph’. Surprisingly this question is still open but we show a simple proof of an upper-bound in Chapter 3. After a decade Alon, Shikhelman defined the function ex(n,H,F), and obtained some general results after which the generalized Turán number gained a wide range of interest. Bollobás, Gy˝ ori showed that Theorem 1.5.1 (Bollobás, Gy˝ ori ). (n) 3 2 3 √ 3(1+o(1)) ≤ex(n,C3,C5) ≤n3/2(5 4 +o(1)). The lower-bound of Theorem 1.5.1 comes from the following construction. Example 1.5.2. There exists a balanced bipartite C4-free extremal graph G′ with 2n 3 vertices and 1 3 √ 3n 3 2 edges, from Theorem 1.2.1. Let us denote partite sets of G′ by A = {a1,a2,...,a n 3} and B = {b1,b2,...,b n 3}. Let A′ be a disjoint copy of A, A′ = {a′ 1,a′ 2,...,a′ n 3}. Let G be a graph on n vertices, such that 3\|n. The vertex set of G is A∪A′ ∪B. The edge set of G is E(G) = {(ai,a′ i) : i ∈[n 3]}∪E(G′)∪{(a′ i,b j) : (ai,b j) ∈E(G′), i, j ∈[n 3]}. Theorem 1.5.1 was improved since then but it still stays open [3, 36, 39]. In , Ergemlidze and Methuku prove the best known upper-bound. Theorem 1.5.3 (Ergemlidze, Gy˝ ori, Methuku, Salia ). ex(n,C3,C5) ≤ 1 2 √ 2 (1+o(1))n3/2. CEU eTD Collection 14 Introduction See proof of this theorem in Section 3.2. Luo determined the maximum number of cliques in a graph without long paths. Before we need to introduce a function fs(n,k,a). fs(n,k,a) = k −a s +(n−k +a) a s−1 . Theorem 1.5.4 (Luo ). Let n−1 ≥k ≥4. Let G be a connected n-vertex graph with no P k, then the number of s-cliques in G is at most max{ fs(n,k,⌊(k −1)/2⌋), fs(n,k,1)}. As a corollary, she also showed Corollary 1.5.5 (Luo). Let n ≥k ≥3. Assume that G is an n-vertex graph with no cycle of length k or more, then Ns(G) ≤n−1 k −2 k −1 s . Nowadays the generalized Turán number is in the center of attention of extremal combi-natorialists, therefore there are many fresh results obtained in this subject [74, 77, 113, 59, 95, 75, 54, 95, 60, 79]. 1.6 Berge-hypergraphs A hypergraph H is a pairV(H) and E(H) whereV(H) is a vertex set and E(H) is a hyperedge set. Similarly, as for graphs v(H) := \|V(H)\|, e(H) := \|E(H)\|. Where E(H) ⊆2V(H). For a fixed set of positive integers R, a hypergraph H is R-uniform if the cardinality of each edge belongs to R, i.e. ∀h ∈H we have \|h\| ∈R. If R = {r}, then an R-uniform hypergraph is simply an r-uniform hypergraph. We say a hypergraph is F-free if it does not contain a copy of any hypergraph from the family F as a sub-hypergraph. Let H be a hypergraph. Then its k-shadow, denoted by ∂kH, is a hypergraph on the same set of vertices and the hyperedge set is the collection of all k-sets that lie in some hyperedge of H. In particular, E(∂kH) = {e : \|e\| = k and e ⊆h ∈H} A hypergraph H is connected if ∂2(H) is a connected graph. Incidence bipartite graph G of a hypergraph H is a bipartite graph with partite sets V(H) and E(H), i.e. V(G) = V(H)∪E(H). Two vertices of G, v and h, v ∈V(H) and h ∈E(H), are joined by an edge in G if v ∈h. CEU eTD Collection 1.6 Berge-hypergraphs 15 Fig. 1.4 A block tree and S(r) n . Let n,k,r be integers such that k ≤r. Fix s, s ∈{r,r + 1}. An r-uniform hypergraph H is called an (s,k −1)-block tree if ∂2(H) is connected and every 2-connected block of ∂2(H) consists of s vertices which induce k −1 hyperedges in H. An (s,k −1)-block tree contains no Berge-cycle of length at least k, because each of its blocks contain fewer than k hyperedges, see Figure 1.4. We define the r-star, S(r) n , as the n-vertex r-uniform hypergraph with vertex set V(S(r) n ) = {v1,v2,...,vn} and edge set E(S(r) n ) = {{v1,v2,...,vr−1,vi} : r ≤i ≤n}, the set {v1,v2,...,vr−1} is called the center of the star. Since S(r) n has r−1 vertices of degree larger than 1, S(r) n contains no Berge-cycle of length at least r. Definition 1.6.1. For a set S ⊆V, the hyperedge neighborhood of S in an r-uniform hyper-graph H is the set NH(S) := {h ∈E(H)\|h∩S ̸= / 0} of hyperedges that are incident with at least one vertex of S. There are no natural ways to generalize graph paths and cycles for hypergraphs. Hence there are numerous definitions for them. In this work, we follow the definition of Berge . Berge-paths and Berge-cycles are some of the most general definitions of paths and cycles in hypergraphs. For example, there are linear-paths/cycles, loose-paths/cycles, and tight-paths/cycles all of which are specific examples of a Berge-path/cycles. All such paths/cycles coincide with the classical definition of path/cycle for graphs. Definition 1.6.2. A hypergraph BCℓis a Berge-cycle of length ℓif • V(BCℓ) ⊇{v1,v2,...,vℓ}; • E(BCℓ) = {e1,e2,...,eℓ}; CEU eTD Collection 16 Introduction • vi,vi+1 ∈ei for all i ∈[ℓ].6 The vertices v1,v2,...,vℓare called defining vertices and hyperedges e1,e2,...,eℓdefin-ing hyperedges of the Berge-cycle. The class of all Berge-cycles of length ℓis denoted by BCℓand the class of all Berge-cycles of length at least ℓby BC≥ℓ. Remark 1.6.3. We have defined that a hypergraph H is connected if ∂2(H) is a connected graph. We can give an equivalent definition to this one. A hypergraph H is connected if for all pairs of vertices v and u there is a Berge-path from v to u. For the completeness of this work we define Berge-paths. Definition 1.6.4. A hypergraph BP ℓis a Berge-path of length ℓif • V(BP ℓ) ⊇{v1,v2,...,vℓ+1}; • E(BP ℓ) = {e1,e2,...,eℓ}; • vi,vi+1 ∈ei for all i ∈[ℓ]. The vertices v1,v2,...,vℓ+1 are called defining vertices and hyperedges e1,e2,...,eℓ defining hyperedges of the Berge-path. The class of all Berge-paths of length ℓis denoted by BPℓ Note that the number of vertices in BPℓis at least ℓ+1 but it can be larger. The index of path classically denotes the number of vertices in a path, but as we have seen it is not a case for hypergraphs therefore we decided to use the index to denote the length of the path. Those definitions of Berge-paths and cycles naturally generalize to other graphs. Definition 1.6.5. A hypergraph BG is a Berge-G, for some fixed graph G if • There is an injective function f1 : V(G) →V(BG); • There is a bijective function f2 : E(G) →E(BG); • If {v1,v2} ∈E(G), then { f1(v1), f1(v2)} ⊆f2({v1,v2}) The vertices from f1(V(G)) are called defining vertices and hyperedges from f2(E(G)) defining hyperedges of the BG. The class of all Berge-G is denoted by BG. 6indices are taken modulo ℓ CEU eTD Collection 1.7 The Turán number of Berge-hypergraphs 17 Recently there has been extensive study of various parameters of Berge-hypergraphs. For example, it is a popular subject to determine Ramsey number for Berge-hypergraphs. Ramsey theory is among the oldest and most intensely investigated topics in combinatorics. It began with the seminal result of Ramsey from 1930. Theorem 1.6.6 (Ramsey ). Let r,t and k be positive integers. Then there exists an integer N such that any coloring of the N-vertex r-uniform complete hypergraph with k colors contains a monochromatic copy of the t-vertex r-uniform complete hypergraph. Estimating the smallest value of such an integer N (the so-called Ramsey number) is a notoriously difficult problem and only weak bounds are known. Given the difficulty of this problem, many people began investigating variations of this problem where graphs other than the complete graphs are considered. An example of an early result in this direction due to Chvátal asserts that the Ramsey number of a t-clique versus any m-vertex tree is precisely N = 1+(m−1)(t −1). That is any red-blue coloring of the complete graph KN yields a red Kt or a blue copy of a given m-vertex tree. Ramsey problems for a variety of hypergraphs and classes of hypergraphs have been considered (for a recent survey of such problems see ). The Ramsey problem for Berge-paths and cycles has received much attention. Of particular interest is a result of Gyárfás and Sárközy showing that the 3-color Ramsey number of a 3-uniform Berge-cycle of length n is asymptotic to 5n 4 (the 2-color case was settled exactly in ). Since this is not a subject of this work we refer the reader to the following manuscripts [7, 105, 55, 96, 53, 98]. 1.7 The Turán number of Berge-hypergraphs The Turán number naturally generalizes for hypergraphs. Definition 1.7.1. The Turán number of a family of R-uniform hypergraphs F, denoted exR(n,F), is the maximum number of hyperedges in an n-vertex, R-uniform, simple-hypergraph which does not contain an isomorphic copy of H as a sub-hypergraph, for all F ∈F. The same question may be asked for multi-hypergraphs, we denote the Turán number for multi-hypergraphs by exmulti R (n,F). But this question is not always interesting. Remark 1.7.2. If every hypergraph in F has at least r + 1 vertices, then exmulti r (n,F) is infinite, since a hypergraph on r vertices and multiple copies of the same hyperedge is F-free. Gerbner, Palmer obtained general bounds for the Turán number of r-uniform Berge-hypergraphs using the classical Turán number. CEU eTD Collection 18 Introduction Theorem 1.7.3 (Gerbner, Palmer ). For all integer r ≥2 and a graph H we have ex(n,F) ≤exr(n,BF) ≤ex(n,Kr,F)+ex(n,F). In 2004 Gy˝ ori extended mantels theorem for 3 and 4 uniform hypergraphs . Theorem 1.7.4 (Gy˝ ori ). Let H be a BC3-free, n-vertex hypergraph. Then ∑ h∈E(H) (\|h\|−2) ≤n2 8 . for all n, n ≥100. Example 1.7.5 (Gy˝ ori ). For an integer n multiple of 4, let H4 be a 4-uniform hypergraph. The vertex set of H4 is partitioned to four k-sets A, A′, B, and B′. The edge set of H4 is E(H4) = n {ai,a′ i,bj,b′ j} : 1 ≤i, j ≤n 4 o Clearly H4 is BC3-free and e(H4) = n2 16. Hence we have equality in Theorem 1.7.4 ∑h∈E(H4) (\|h\|−2) = 2e(H4) = n2 8 . Example 1.7.6 (Gy˝ ori ). For an integer n multiple of 4, let H3 be a 3-uniform hypergraph. The vertex set of H3 is partitioned to three sets A, A′, and B of sizes n 4, n 4, and n 2 accordingly. The edge set of H3 is E(H3) = {ai,a′ i,bj} : 1 ≤i, j ≤n 4 Clearly H3 is BC3-free and e(H3) = n2 8 . Hence we have equality in Theorem 1.7.4 ∑h∈E(H3) (\|h\|−2) = 2e(H4) = n2 8 . Putting together Theorem 1.7.4 and Examples 1.7.5, 1.7.6 we have a corollary. Corollary 1.7.7 (Gy˝ ori ). For all n ≥100, such that 4\|n, we have ex3(n,BC3) = n2 8 , and ex4(n,BC3) = n2 16. Later Gy˝ ori, Lemons gave an upper bound for the size of 3-uniform hypergraphs avoiding BC2k+1. In particular, they show ex3(n,BC2k+1) ≤4k4n1+ 1 k +O(n). They provide constructions showing that these bounds are best possible for k ∈{1,2,3,5} up to the constant factor. Gy˝ ori, Lemons extended their results for r-uniform hypergraphs and non-uniform hypergraphs as well. CEU eTD Collection 1.7 The Turán number of Berge-hypergraphs 19 The Turán number of C4 was determined in Theorem 1.2.4. ex(n,C4) = n3/2 2 +o(n3/2). Füredi, Özkahya proved ex3(n,BC4) ≤(1+o(1))2 3n3/2. Recently we improved this result. Theorem 1.7.8 (Ergemlidze, Gy˝ ori, Methuku, Salia, Tompkins ). ex3(n,BC4) ≤(1+o(1)) n3/2 √ 10. See proof in Section 3.3. On the other hand, we have a lower-bound. Theorem 1.7.9 (Bollobás, Gy˝ ori , see also ). (1−o(1)) n3/2 3 √ 3 ≤ex3(n,BC4). This lower bound comes from the following example. Example 1.7.10 ( Bollobás, Gy˝ ori ). There exists a balanced bipartite C4-free extremal graph G with 2n 3 vertices and 1 3 √ 3n 3 2 edges, 1.2.1. Let us denote partite sets of G by A = {a1,a2,...,a n 3} and B = {b1,b2,...,b n 3}. Let A′ be a disjoint copy of A, A′ = {a′ 1,a′ 2,...,a′ n 3}. Let H be a 3-uniform hyper-graph on n vertices, such that 3\|n. The vertex set of H is E(H) = A∪A′ ∪B. The edge set of H is E(H) = {(ai,a′ i,b j) : (ai,b j) ∈E(G), i, j ∈[n 3]}. Interestingly the example of Bollobás, Gy˝ ori initially was for pentagon-free hypergraphs but it happened to be C4-free too. Which is an interesting phenomenon. We know even and odd cycles behave extremely differently in the case of graphs. Here I would suggest an exciting conjecture from Gy˝ ori. Conjecture 1.7.11 (Gy˝ ori ). For all integer k ≥2, we have lim n→∞ ex3(n,BC2k) ex3(n,BC2k+1) = 1. CEU eTD Collection 20 Introduction This version of the conjecture is a modest version of the one we believe. The Turán problem for Berge-cliques has been investigated heavily by several authors in [67, 99, 63]. For further results see [6, 103, 61] also. 1.8 Extensions of Erd˝ os-Gallai Theorem In the last decade, there has been extensive study of hypergraphs without long Berge-paths and cycles. But in contrast to graphs, the pattern shows that there are two distinct cases for hypergraphs. The first when the forbidden structure has more vertices than uniformity and the second when uniformity is ’larger’ than the number of defining vertices in a forbidden structure. But in some cases, there are different constructions and situations when the number of defining vertices of a forbidden structure is close to uniformity. 1.8.1 Hypergraphs without long Berge-paths We open this section with a theorem of Gy˝ ori, Katona, Lemons. Theorem 1.8.1 (Gy˝ ori, Katona, Lemons ). Let r ≥k ≥3, then ex(n,BPk) ≤(k −1)n r +1 . Theorem 1.8.2 (Gy˝ ori, Katona, Lemons ). Let k > r +1 > 3, then ex(n,BPk)n k k r . In the remaining case when k = r + 1 was solved later by Davoodi, Gy˝ ori, Methuku, Tompkins , the extremal number matches the upper bound of Theorem 1.8.2. Theorem 1.8.3 (Davoodi, Gy˝ ori, Methuku, Tompkins ). Let r > 2, then ex(n,BPr+1) ≤ n r +1 k r . Similarly, as in the graph case in Theorem 1.3.7, it makes sense to ask the same question requiring connectivity. We obtained the first partial result in this direction. Theorem 1.8.4 (Gy˝ ori, Methuku, Salia, Tompkins, Vizer ). Let Hn,k be a largest r-uniform connected n-vertex hypergraph with no Berge-path of length k, then lim k→∞lim n→∞ E(Hn,k) kr−1n = 1 2r−1(r −1)!. CEU eTD Collection 1.8 Extensions of Erd˝ os-Gallai Theorem 21 See proof of this theorem in Chapter 5. In the recent work of Füredi, Kostochka, Luo , they investigate 2-connected hyper-graphs7 and obtain many interesting results. To present their results, we start by introducing the following two functions, f(n,k,r,a) := k −a min{r,⌊k−a 2 ⌋} +(n−k +a) a min{r −1,⌊a/2⌋} , and f ∗(n,k,r,a) := k −a r +(n−k +a) a r −1 . Let us introduce another notion, a family of sets F is called a Sperner family, if for all f ⊂F, there is no f ′ ⊂F such that f ′ ̸⊂f. Füredi, Kostochka, Luo obtained a sharp upper bound for 2-connected Sperner families without a long Berge-cycle. Theorem 1.8.5 (Füredi, Kostochka, Luo ). Let n ≥k ≥r ≥3. If H is an n-vertex Sperner 2-connected r-uniform hypergraph with no Berge-cycle of length at least k, then e(H) ≤max{ f(n,k,r,⌊(k −1)/2⌋), f(n,k,r,2)}. They also obtained a sharp upper-bound for 2-connected r-uniform hypergraphs without a long Berge-cycle. Theorem 1.8.6 (Füredi, Kostochka, Luo ). Let n ≥nk,r ≥k ≥4r ≥12. If H is an n-vertex 2-connected r-uniform hypergraph with no Berge-cycle of length k or longer, then e(H) ≤f(n,k,r,⌊(k −1)/2⌋) = f ∗(n,k,r,⌊(k −1)/2⌋). The following two results are for 2-connected hypergraphs, the first when the hypergraph is a Sperner family and the second when the hypergraph is uniform. Theorem 1.8.7 (Füredi, Kostochka, Luo ). Let n ≥k ≥r ≥3. If H is an n-vertex Sperner connected ≤r-uniform hypergraph with no Berge-path of length k, then e(H) ≤max{ f(n,k,r,⌊(k −1)/2⌋), f(n,k,r,1)}. 7A hypergraph is k-connected if it’s incidence bipartite graph is k-connected. CEU eTD Collection 22 Introduction Theorem 1.8.8 (Füredi, Kostochka, Luo ). Let n ≥n′ k,r ≥k ≥4r ≥12. If H is an n-vertex connected r-uniform hypergraph with no Berge-path of length k, then e(H) ≤f(n,k,r,⌊(k −1)/2⌋) = f ∗(n,k,r,⌊(k −1)/2⌋). Theorem 1.8.8 has the sharp upper bound. We extended Theorem for k ≥2r + 13 as well as we prove that there is only one hypergraph with the extremal number of hyperedges. To describe the extremal result and to introduce our contributions, we need the following definition that can be considered as an analog of Definition 1.3.6 for higher uniformity. Definition 1.8.9. For integers n,a ≥1 and b1,...,bt ≥2 with n ≥2a+∑t i=1 bi let us denote by Hn,a,b1,b2,...,bt the following r-uniform hypergraph. • Let the vertex set of Hn,a,b1,b2,...,bt be A∪L∪St i=1 Bi, where A,B1,B2,...,Bt and L are pairwise disjoint sets of sizes \|A\| = a, \|Bi\| = bi (i = 1,2,...,t) and \|L\| = n−a−∑t i=1 bi. • Let the hyperedges of Hn,a,b1,b2,...,bt be A r ∪ t [ i=1 A∪Bi r ∪ {c}∪A′ : c ∈L,A′ ∈ A r −1 . Fig. 1.5 The hypergraph Hn,a,b1,b2,...,bt. Observe that the number of hyperedges in Hn,a,b1,b2,...,bt is n−a− t ∑ i=1 bi ! a r −1 + t ∑ i=1 a+bi r −(t −1) a r . Note that, if a ≤a′ and bi ≤b′ i for all i = 1,2,...,t, then Hn,a,b1,b2,...,bt is a sub-hypergraph of Hn,a′,b′ 1,b′ 2,...,b′ t. Finally, the length of the longest path in Hn,a,b1,b2,...,bt is 2a−t +∑t i=1 bi if t ≤a+1, and a−1+∑a+1 i=1 bi if t > a+1 and the bi’s are in non-increasing order. CEU eTD Collection 1.8 Extensions of Erd˝ os-Gallai Theorem 23 If k is odd. If k is even. Fig. 1.6 The hypergraphs Hn,⌊k−1 2 ⌋and Hn,⌊k−1 2 ⌋,2 . With a slight abuse of notation, we define H+ n,a to be a hypergraph obtained from Hn,a by adding an arbitrary hyperedge. Hyperedges containing at least r −1 vertices from A are already in Hn,a, therefore there are r −1 pairwise different hypergraphs that we denote by H+ n,a depending on the number of vertices from A in the extra hyperedge. Observe that the length of the longest path in H+ n,a is one larger than in Hn,a, in particular, if k is even, then H+ n,⌊k−1 2 ⌋does not contain a Berge-path of length k. Let us denote the maximum number of edges in an n-vertex H-free connected r-uniform hypergraph by exconn r (n,H). Theorem 1.8.10 (Gy˝ ori, Salia , Zamora ). For all integers k,r with k ≥2r + 13 ≥18 there exists nk,r such that if n > nk,r, then we have • exconn r (n,BPk) = \|Hn,⌊k−1 2 ⌋\|, if k is odd, and • exconn r (n,BPk) = \|Hn,⌊k−1 2 ⌋,2\|, if k is even. Depending on the parity of k, the unique extremal hypergraph is Hn,⌊k−1 2 ⌋or Hn,⌊k−1 2 ⌋,2, (see Figure 1.6). Later we extended this result to a stability result, since the proof is similar we will skip the proof of this theorem in this work. Instead, we show the idea of it here. At first let us study the extremal hypergraphs Hn,⌊k−1 2 ⌋and Hn,⌊k−1 2 ⌋,2, Figure 1.6. For fixed parity of k the extremal hypergraph is unique, let us describe the case when k is odd.8 The hypergraph Hn,⌊k−1 2 ⌋has two classes of vertices, one with degree Θ(n) and the others with constant.9 It is easy to note that two vertices from the sparse family can not be neighbors in a Berge-path, since there is no hyperedge incident with both. Therefore the longest Berge-path and cycle of Hn,⌊k−1 2 ⌋has length k −1. This special structure was an inspiration 8This case is technically easier than the other. 9As function of n. CEU eTD Collection 24 Introduction to use the following argument for the desired result. Proof is by contradiction, let H be a connected r-uniform BPk-free hypergraph with e(H) ≥e(Hn,⌊k−1 2 ⌋). At first we found a sub-hypergraph with a large minimum degree, by a classical argument, removing low degree vertices. The second important step was to show that the hypergraph we got was still connected. Then by classical Dirac argument , we show that the hypergraph contains a Berge-cycle of length k −1 as the unique extremal hypergraph Hn,⌊k−1 2 ⌋. The rest is to show that there is a partitioning of vertices in two sets with the desired properties, which is deduced from the Berge-cycle of length k −1. This result was followed by a stability result which we show in Section 1.9, and prove in Chapter 5. 1.8.2 Hypergraphs without long Berge-cycles Similarly, as for the forbidden path case, the extremal hypergraphs when Berge-cycles of length at least k are forbidden are different in the cases when k ≥r +2 and k ≤r +1 with an exceptional third case when k = r case. The latter has a surprisingly different extremal hypergraph. Füredi, Kostochka, Luo provide sharp bounds and extremal constructions for infinitely many n, for k ≥r +3 ≥6. Later in they also determined the exact bounds and extremal constructions for all n, for the case k ≥r +4. Kostochka, Luo determine a bound for k ≤r −1 which is sharp for infinitely many n. Ergemlidze, Gy˝ ori, Methuku, Salia, Tompkins, Zamora determine a bound in the cases where k ∈{r +1,r +2}. The case when k = r remained open. Both papers [90, 38] conjectured the maximum number of edges to be bounded by max n (n−1)(r−1) r ,n−(r −1) o (See Figure 1.4, the hypergraph S(r) n is on the right). Theorem 1.8.11 (Füredi, Kostochka, Luo [48, 47]). Let r ≥3 and k ≥r +3, then ex(n,BC≥k) ≤n−1 k −2 k −1 r . Moreover, equality is achieved if and only if ∂2(H) is connected and for every block D of ∂2(H), D = Kk−1 and H[D] = Kr k−1, (see Figure 1.4). Theorem 1.8.12 ( Gy˝ ori, Methuku, Salia, Tompkins, Zamora ). If r ≥3 then ex(n,BC≥r+2) ≤(n−1)(r +1) r . Moreover, equality is achieved if and only if ∂2(H) is connected and for every block D of ∂2(H), D = Kr+1 and H[D] = Kr r+1, (see Figure 1.4). CEU eTD Collection 1.8 Extensions of Erd˝ os-Gallai Theorem 25 Theorem 1.8.13 (Kostochka, Luo ). Let k ≥4,r ≥k +1 and ex(n,BC≥k) ≤(k −1)(n−1) r . Moreover, equality is achieved if and only if ∂2(H) is connected and for every block D of ∂2(H), D = Kr+1 and H[D] consists of k −1 hyperedges, (see Figure 1.4). Theorem 1.8.14 ( Gy˝ ori, Methuku, Salia, Tompkins, Zamora ). If r ≥3 then ex(n,BC≥r+1) ≤n−1. Moreover, equality is achieved if and only if ∂2(H) is connected and for every block D of ∂2(H), D = Kr+1 and H[D] consists of r hyperedges, (see Figure 1.4). Let us note that our proof method is very different from the proof of Füredi, Kostochka, Luo . One of our main ideas in proving the above two theorems is an unusual application of Hall’s theorem to vertices of the hypergraph (instead of applying it to edges in the shadow of the hypergraph). This allows us to assign a distinct hyperedge to each vertex which is then very helpful in finding Berge-cycles. The notion of connectivity (i.e., the notion of “cut hyperedges") in Berge-hypergraphs has also been very important. These ideas may have further applications (see for e.g. ). Note that Theorem 1.8.14 implies Theorem 1.8.3. In fact, it gives the following stronger form. We prove this implication in Chapter 4. Theorem 1.8.15 ( Gy˝ ori, Methuku, Salia, Tompkins, Zamora ). Fix k = r+1 > 2 and let H be an r-uniform hypergraph containing no Berge-path of length k. Then, e(H) ≤n k k r = n. Moreover, equality holds if and only if each connected component D of ∂2(H) is Kr+1, and H[D] = Kr r+1. Theorem 1.8.16 (Gy˝ ori, Lemons, Salia, Zamora ). Let k,n and r be positive integers such that 3 ≤k < r. Then exr(n,BC≥k) = n−1 r (k −1)+1rN∗(n). If r\|(n−1), then the only extremal n-vertex r-graphs are the (r +1,k −1)-block trees, (see Figure 1.4). We note that as a corollary of Theorem 1.8.16, we obtain a slightly stronger version of Theorem 1.8.1. CEU eTD Collection 26 Introduction Corollary 1.8.17 (Gy˝ ori, Lemons, Salia, Zamora ). Let k,n and r be positive integers with 3 ≤k ≤r. Then exr(n,BPk) = n r +1 (k −1)+1(r+1)N∗(n+1). Theorem 1.8.18 (Gy˝ ori, Lemons, Salia, Zamora ). Let r > 2 and n be positive integers. Then exr(n,BC≥r) = max n−1 r (r −1),n−r +1 . When n−r +1 > n−1 r (r −1) the only extremal graph is S(r) n . When n−1 r (r −1) > n−r +1 and r\|(n−1) the only extremal graphs are the (r +1,k −1)-block trees, (see Figure 1.4). Remark 1.8.19 (Gy˝ ori, Lemons, Salia, Zamora ). In particular, when n ≥r(r −2)+2, we have that exr(n,BC≥r) = n−r +1 and S(r) n is the only extremal hypergraph. Theorem 1.8.20 (Gy˝ ori, Lemons, Salia, Zamora ). Let k,n and r be positive integers such that 2 ≤k ≤r. Then exmulti r (n,BC≥k) = n−1 r −1 (k −1). If r −1\|(n−1) then the only extremal graphs with n vertices are the (r,k −1)-block trees. As a corollary of Theorem 1.8.20 we obtain a version of Theorem 1.8.1 with multiple hyperedges. Corollary 1.8.21 (Gy˝ ori, Lemons, Salia, Zamora ). Let k,n and r be positive integers with 2 ≤k ≤r then exmulti r (n,BPk) = jn r k (k −1). Kostochka and Luo obtain Theorem 1.8.13 from the incidence bipartite graph by investi-gating the structure of 2-connected bipartite graphs. Similarly, Jackson gives an upper bound on the number of edges of a multi r-uniform hypergraph with no Berge-cycle of length at least r. Theorem 1.8.22 (Jackson ). Let G be a bipartite graph with bipartition A and B such that \|A\| = n and every vertex in B has degree at least r. If \|B\| > n−1 r−1 (r −1), then G contains a cycle of length at least 2r. We study the structure of r-uniform hypergraphs containing no Berge-cycles of length at least k, for all 3 ≤k ≤r. By exploring the structure of the hypergraphs, instead of bipartite CEU eTD Collection 1.8 Extensions of Erd˝ os-Gallai Theorem 27 graphs, we can find the extremal number in the case when k = r, which also gives us a simple proof for Theorem 1.8.13. Furthermore, our method lets us determine the extremal number for every value of n in both simple r-uniform hypergraphs and multi r-uniform hypergraphs. At the end of this section, we want to present one of our favorite conjectures. Super-cyclic bipartite graphs Kostochka, Lavrov, Luo, Zirlin [89, 91, 88] investigated pan-cyclic and super-pan-cyclic graphs and hypergraphs. Without getting into the details we would like to state a version of their conjecture here. Conjecture 1.8.23 (Kostochka, Lavrov, Luo, Zirlin ). Let G be a bipartite graph, with partite sets A and B. Let for every subset A′ of A, \|A′\| ≥2, the number of vertices incident with at least two vertices of A′ is at least \|A′\| in G. Then for every subset A′ of A, \|A′\| ≥2, there is a cycle CA′ in G such that V(CA′)∩A = A′. Note that this conjecture has an equivalent variant for non-uniform Hypergraphs. 1.8.3 Hypergraphs without Berge-trees The Turán number of certain kinds of trees in r-uniform hypergraphs has long been a major topic of research. For example, there is a notoriously difficult conjecture of Kalai which is more general than the Erd˝ os-Sós conjecture (see Conjecture 1.3.3). The trees which Kalai considers are generalizations of the notion of tight paths in hypergraphs. In another direction, Füredi investigated linear trees, constructed by adding r −2 new vertices to every edge in a (graph) tree. In this setting, he proved asymptotic results for all uniformities at least 4. Whereas, the articles above considered classes of trees containing tight and linear paths, respectively, we will consider the setting of Berge-trees. In the range when k > r, a number of results on forbidding Berge-trees were obtained by Gerbner, Methuku, Palmer . In particular, they proved that if we assume the Erd˝ os-Sós Conjecture 1.3.3 holds for a tree T with k edges and all of its sub-trees and also that k > r+1, we have exr(n,BT) ≤n k k r (a construction matching this bound when k divides n is given by n/k disjoint copies of the complete r-uniform hypergraph on k vertices). In the present paper, we will consider the range r > k, where we prove some exact results. Considering multi-hypergraphs, we prove the following. CEU eTD Collection 28 Introduction Theorem 1.8.24 (Gy˝ ori, Salia, Tompkins, Zamora ). Let n,k,r be positive integers and let T be a k-edge tree, then for all r ≥(k −1)(k −2), exmulti r (n,BT) ≤n(k −1) r . If r > (k −1)(k −2) and T is not a star, equality holds if and only if r divides n and the extremal multi-hypergraph is n r disjoint hyperedges, each with multiplicity k −1. If T is a star equality holds only for all (k −1)-regular multi-hypergraphs. We conjecture that Theorem 1.8.24 holds for the following wider set of parameters. Conjecture 1.8.25 (Gy˝ ori, Salia, Tompkins, Zamora ). Let n,k,r be positive integers and let T be a k-edge tree, then for all r ≥k +1, exmulti r (n,BT) ≤n(k −1) r . For all trees T, where T is not a star, equality holds if and only if r divides n and the extremal multi-hypergraph is n r disjoint hyperedges each with multiplicity k −1. The special case of Conjecture 1.8.25, when the forbidden tree is a path, was settled by Gy˝ ori, Lemons, Salia, Zamora (see the first corollary). We now define a class of hypergraphs which we will need when we classify the extremal examples in our main result about simple hypergraphs, Theorem 1.8.28. Definition 1.8.26. An r-uniform hypergraph H is two-sided if V(H) can be partitioned into a set X and pairwise disjoint sets Ai, i = 1,2,...,t (also disjoint from X) of size r −1, such that every hyperedge is of the form {x}∪Ai for some x ∈X. We say that a two-sided r-uniform hypergraph is (a,b)-regular if every vertex of X has degree a and every vertex of t [ i=1 Ai has degree b. Remark 1.8.27. A two-sided r-uniform hypergraph can also be viewed as a graph obtained by taking a bipartite graph G with bipartite classes X and Y, and “blowing up" each vertex of Y to a set of size r −1, and replacing each edge {x,y} by the r-hyperedge containing x together with the blown-up set for y. Theorem 1.8.28 (Gy˝ ori, Salia, Tompkins, Zamora ). Let n,k,r be positive integers and let T be a k-edge tree which is not a star, then for all r ≥k(k −2), exr(n,BT) ≤n(k −1) r +1 . CEU eTD Collection 1.9 Stability 29 r +1 vertices k −1 hyperedges r +1 vertices k −1 hyperedges r +1 vertices k −1 hyperedges \|Ai\| = r −1, d(Ai) = k−1 2 d(x) = k −1 Fig. 1.7 An extremal graph, Theorem 1.8.28. Equality holds if and only if r + 1 divides n, and the extremal hypergraph is obtained from n r+1 disjoint sets of size r +1, each containing k −1 hyperedges. Unless k is odd, and T is the balanced double star, where the balanced double star is the tree obtain from an edge by adding k−1 2 incident edges to each of the ends of the edge, in which case equality holds if and only if r +1 divides n and H is obtained from the disjoint union of sets of size r +1 containing k −1 hyperedges each and possibly a (k −1, k−1 2 )-regular two-sided r-uniform hypergraph (see Figure 1.7). 1.9 Stability As we have seen, the Turán number depicts how a global parameter number of edges influence local substructures. In particular, for a given substructure, we determine the maximum number of edges. The graphs with the maximum number of edges without a given substructure are called Extremal graphs. Often such graphs are finite and unique. In other words, if we have ’many’ edges and no given substructure, our graph must be one from the given set of Extremal graphs. Naturally, arises another question are these properties stable? in particular, if we have ’nearly as many’ edges as in an extremal graph and no substructure are we ’close to’ a graph from the extremal family? One needs to define what do we mean in ’nearly as many’ and ’close to’. Those concepts are always dependent on the settings of the problem. CEU eTD Collection 30 Introduction We open this section with a famous stability Theorem of Erd˝ os, Simonovits [25, 108], which has many useful applications [2, 10, 107, 112]. Theorem 1.9.1 (Erd˝ os, Simonovits [25, 108]). Let G be an n vertex Kr+1 graph with t(n,r)+ o(n2) edges, then by changing o(n2) edges we can get T(n,r) graph from G. While studying almost extremal graphs without Kr+1, Simonovits observed that by replacing a vertex from one color class to another in a Turán graph T(n,r) we lose a constant number of edges but the distance to the Turán graph is θ(n). Therefore one may ask a modified question- what is the minimum number of edges for a Kr+1-free graph to be an r-partite graph. This type of questions were investigated in a series of papers [5, 87, 11, 18, 82, 32]. Let us recall Theorem 1.3.7, exconn(n,P ℓ) = max n e(Gn,ℓ,1),e(Gn,ℓ,⌊ℓ−1 2 ⌋) o . Extremal graphs for this problem were Gn,ℓ,1 or Gn,ℓ,⌊ℓ−1 2 ⌋depending how large was n. The stability version of these results was proved by Füredi, Kostochka, Verstraëte . Theorem 1.9.2 (Füredi, Kostochka, Verstraëte ). Let t ≥2, n ≥3t −1 and k ∈{2t,2t +1}. Suppose we have a n-vertex connected P k-free graph G with more edges than Gn+1,k+1,t−1 − n, where the graph Gn+1,k+1,t−1 is described in Definition 1.3.6. Then we have either • k = 2t, k ̸= 6 and G is a sub-graph of Gn,k,t−1, or • k = 2t +1 or k = 6, and G\A is a star forest for A ⊆V(G) of size at most t −1. Our main result provides a stability version (and thus a strengthening) of Theorem 1.8.10 and also an extension of Theorem 1.9.2 for uniformity at least 3. First we state it for hypergraphs with minimum degree at least 2, and then in full generality. In the proof, the hypergraphs Hn, k−3 2 ,3 and Hn, k−3 2 ,2,2 will play a crucial role in case k is odd, while if k is even, then the hypergraphs Hn,⌊k−3 2 ⌋,4, Hn,⌊k−3 2 ⌋,3,2 and Hn,⌊k−3 2 ⌋,2,2,2 will be of importance (Definition 1.8.9), note that all of them are n-vertex, maximal, BPk-free hypergraphs. In both cases, the hypergraph listed first contains the largest number of hyperedges. This number gives the lower bound in the following theorem. Theorem 1.9.3 (Gerbner, Nagy, Patkós, Salia, Vizer ,). For any ε > 0 there exist integers q = qε and nk,r such that if r ≥3, k ≥(2+ε)r +q, n ≥nk,r and H is a connected n-vertex, r-uniform hypergraph with minimum degree at least 2, without a Berge-path of length k, then we have the following. • If k is odd and \|H\| > \|Hn, k−3 2 ,3\| = (n−k+3 2 ) k−3 2 r−1 + k+3 2 r , then H is a sub-hypergraph of Hn, k−1 2 . CEU eTD Collection 1.9 Stability 31 • If k is even and \|H\| > \|Hn,⌊k−3 2 ⌋,4\| = (n −⌊k+5 2 ⌋) ⌊k−3 2 ⌋ r−1 + ⌊k+5 2 ⌋ r , then H is a sub-hypergraph of Hn,⌊k−1 2 ⌋,2 or H+ n,⌊k−1 2 ⌋. Let H′ n′,a,b1,b2,...,bt be the class of hypergraphs that can be obtained from Hn,a,b1,b2,...,bt for some n ≤n′ by adding hyperedges of the form A′ j ∪D j, where the D j’s partition [n′][n], all D j’s are of size at least 2 and A′ j ⊆A for all j. Let us define H+ n′,⌊k−1 2 ⌋analogously. Theorem 1.9.4 (Gerbner, Nagy, Patkós, Salia, Vizer ,). For any ε > 0 there exist integers q = qε and nk,r such that if r ≥3, k ≥(2+ε)r +q, n ≥nk,r and H is a connected n-vertex, r-uniform hypergraph without a Berge-path of length k, then we have the following. • If k is odd and \|H\| > \|Hn, k−3 2 ,3\|, then H is a sub-hypergraph of some H′ ∈H′ n, k−1 2 . • If k is even and \|H\| > \|Hn,⌊k−3 2 ⌋,4\|, then H is a sub-hypergraph of some H′ ∈H′ n,⌊k−1 2 ⌋,2 or H+ n,⌊k−1 2 ⌋. We prove Theorem 1.9.3 and 1.9.4 in Capter 5. CEU eTD Collection CEU eTD Collection Chapter 2 Erd˝ os-Gallai theorem, for vertex colored graphs Working with Paul Erdos was like taking a walk in the hills. Every time when I thought that we had achieved our goal and deserved a rest, Paul pointed to the top of another hill and off we would go. Fan Chung In a relatively recent paper, Gy˝ ori, Lemons investigated the extremal number of hypergraphs avoiding Berge-cycles. To this end, they introduced a generalization of the theorem of Erd˝ os-Gallai about paths. Gy˝ ori, Lemons proved the following Lemma 1.4.1. Lemma (Gy˝ ori-Lemons ). Let k be a positive integer and G be an n-vertex graph with a proper vertex coloring such that G contains no P 2k+1 with endpoints of different colors, then \|E(G)\| ≤2kn. This was a useful tool for determining the Turán number of Hypergraphs avoiding Berge-cycles. One can see this lemma as a version of Theta-lemma of Bondy-Simonovits , which has many useful applications like in . In this Chapter we show proofs of theorems raised in Section 1.4. In the following section at first we prove Theorem 1.4.2 for paths and in Section 2.2 we prove theorems for trees-Theorem 1.4.4, Theorem 1.4.5, and Theorem 1.4.7. CEU eTD Collection 34 Erd˝ os-Gallai theorem, for vertex colored graphs 2.1 Forbidden paths in vertex colored graphs We start with a proof of Theorem 1.4.2, which eliminates the factor of 2 in Theorem 1.4.1. Thus, we recover the original upper bound from the Erd˝ os-Gallai theorem 1.3.1. We also determine the family of extremal graphs. Theorem (Salia, Tompkins, Zamora ). Let k ≥0 and G be an n-vertex graph with a proper vertex coloring such that G contains no P 2k+1 with endpoints of different colors, then \|E(G)\| ≤kn, and equality holds if and only if 2k +1 divides n and G is the union of n 2k+1 disjoint cliques of size 2k +1. Proof. By induction on the number of vertices, we may assume that G is connected and has minimum degree δ(G) ≥k. Indeed, if δ(v) < k then e(G) = e(G−v)+δ(v) ≤k(n−1)+k −1 < kn. If G is Cℓ-free for all ℓ≥2k +1, then by Theorem 1.3.2 we have \|E(G)\| ≤(n−1)2k 2 < kn. Thus, assume there is a cycle of length at least 2k +1, and let C be the smallest such cycle with length ℓ. Let the vertices of C be v0,v1,v2,...,vℓ−1,v0, consecutively. Addition and subtraction in subscripts will always be taken modulo ℓ. We say that an edge e is outgoing if it has one vertex in V(C) and the other in V(G)\V(C). We say a vertex v ∈V(C) is outgoing if it is contained in an outgoing edge. We will consider cases based on the value of ℓ. Observe that ℓ= 2k +2 is impossible since v0,v1,...,v2k+1 is a path of length 2k +1 but v0 and v2k+1 are adjacent, contradiction. Case 1. Suppose ℓ≥2k + 4. Since we have chosen ℓto be the length of the smallest Cℓwith ℓ≥2k + 1, we have v0 cannot be adjacent to any of v2,v3,...,vℓ−2k nor any of v2k,v2k+1,...,vℓ−2, for otherwise we would have a shorter cycle of length at least 2k + 1. Also note that v0 is adjacent to v1 and vℓ−1. Observe that v0 cannot have two consecutive neighbors in the ℓ-cycle. Indeed, if vi and vi+1 are neighbors of v0, then we have the following (2k + 1)-paths starting at v1: v1,v2,...,v2k+1,v2k+2 and v1,v2,...,vi,v0,vi+1,vi+2,...,v2k,v2k+1. Thus, v2k+1 and v2k+2 would have to be colored the same, but this is impossible since they are neighbors. CEU eTD Collection 2.1 Forbidden paths in vertex colored graphs 35 If v0 has a neighbor outside of C, say u0, then we have two paths of length 2k + 1: u0,v0,v1,...,v2k and v2k,v2k−1,...,v0,vℓ−1. It follows that u0 and vℓ−1 have the same color. Similarly, u0 and v1 have the same color. Thus, vℓ−1 and v1 also have the same color, and similarly, for every i such that vi is outgoing, we can conclude vi−1 and vi+1 have the same color. If ℓ= 2k+4 and there is an outgoing vertex, say v0, then v1 and v2k+3 have the same color (from above), a contradiction since v1 and v2k+2 also have the same color (they are endpoints of a length 2k + 1 path along the cycle C). If there is no outgoing vertex in V(C), then C uses all vertices of the graph. Since no vertex of the cycle has two consecutive neighbors, it follows that each degree is bounded by 2+⌈2k−5 2 ⌉≤k and so the number of edges is at most (2k+4)k 2 = nk 2 < nk. If ℓ≥2k +5, we will show that v0 has an outgoing edge from the ℓ-cycle C. Suppose not, then since v0 does not have consecutive neighbors, it follows that v0 has at most 2+ 2k −(ℓ−2k +1) 2 ≤k −1 neighbors, a contradiction. Thus, v0 and similarly every other vi has an outgoing neighbor, and it follows that for every i, the vertices vi and vi+2 have the same color. Hence v0 and v2k have the same color, contradicting that v0 and v2k+1 have the same color, since they are endpoints of a P 2k+1. Case 2. Suppose ℓ= 2k +3. For all 0 ≤i ≤ℓ−1, vi+2,vi+1,...,vℓ−1,v0,...,vi is a path of length 2ℓ+1, and so vi and vi+2 have the same color. Thus, v0 and v2k+2 have the same color, but they are adjacent, contradiction. Case 3. Finally, suppose ℓ= 2k +1. If no edge is outgoing, then we are done, since by connectivity the total number of edges in the graph is at most 2k+1 2 = kn. If indeed the total number of edges is kn, then G is a clique. This is the only case when equality holds. From here on, we will assume there is an outgoing edge. Observe that if u is not a vertex of C, then u cannot have two consecutive neighbors in C, for otherwise we would have a cycle of length 2k +2. Moreover, u cannot be connected to vi and vi+3, since there would be paths of length 2k +1 from u to vi+1 and vi+2. It follows that u can have at most k −1 neighbors in C and, thus, must have a neighbor outside C. If there are some two consecutive non outgoing vertices in C, then we may take two such vertices vi and vi+1, for some index i, so that the next vertex vi+2 is outgoing. Suppose {vi+2,u} is an outgoing edge. By the previous observation, there is an edge {u,w} where w ̸∈C. So we have a 2k +1 length path from vi to w, then vi+1 cannot have two consecutive neighbors from C, since that would also imply that there is also 2k +1 length path from w to CEU eTD Collection 36 Erd˝ os-Gallai theorem, for vertex colored graphs v0 vivi+1 v1 v2k+2 v2k+1 v0 u0 v1 vℓ−1 v2k vi+2 ui+2 wi+2 vi+1 vi vjvj+1 Fig. 2.1 Sketch of the proof of Theorem 1.4.2. The picture on the left is for Case 1, and the other pictures show Case 4. vi−1, similarly vi cannot have two consecutive neighbors in C, hence vi and vi+1 have degree at most k. By removing these two vertices, we remove 2k −1 edges, and by the induction hypothesis the resulting graph has at most k(n−2) edges. So e(G) < kn. For every i, either vi+1 or vi+2 is an outgoing vertex. Hence the vertex vi has either the same color as vi+2, if vi+1 is an outgoing vertex, or the same color as vi+4, if vi+2 is an outgoing vertex. Hence by repeatedly applying this argument we obtain that v0 has the same color as v2k or v1, contradiction. 2.2 Forbidden trees in vertex colored graphs Let us recall Theorem 1.4.4. Theorem. Let T be a tree with k edges such that in the (unique) proper vertex 2-coloring of T all leaves are not the same color, then exc(n,T) ≤(k −1)n. Proof. There is a path of odd length in T with endpoints which are leaves. Let G be an n-vertex graph with more than (k −1)n edges with a proper vertex coloring. We may find a subgraph G′ of G with the average degree at least that of G and minimum degree greater than k −1. The proper coloring of G induces a proper coloring of G′ and so applying Theorem 1.4.1 we may find a copy of P 2ℓ+1 in G′ with endpoints of distinct colors. We may now build up the rest of the tree with a greedy argument, as every degree in G′ is at least k and T has k +1 vertices. Thus, we have found a copy of T in the graph G with leaves of at least two colors. We recall Theorem 1.4.5. CEU eTD Collection 2.2 Forbidden trees in vertex colored graphs 37 u v . . . A . . . B u v . . . B w . . . C u x . . . B . . . C y Fig. 2.2 The graph Sa,b. Theorem. Let T be a tree with k edges such that in the proper vertex 2-coloring of T all leaves are the same color, then exc(n,T) = j n2 4 k , provided n is sufficiently large. Proof. The fact that all leaves are colored the same by a 2-coloring implies that all paths between a pair of leaves have even length. We add an edge e to T connecting an arbitrary pair of leaves and let G be the resulting graph. Since G has an odd cycle, its chromatic number is 3, and the deletion of e yields a 2-chromatic graph. It follows from a theorem of Simonovits that if n is sufficiently large, the extremal number of G is precisely ex(n,G) = j n2 4 k . Thus, in any n-vertex graph with more than j n2 4 k edges we have a copy T with two adjacent leaves, and so in any proper coloring of this graph we have a copy of T with leaves of at least 2 colors. It follows that exc(n,T) ≤ j n2 4 k , and this bound is realized by the complete bipartite graph K⌊n 2⌋,⌈n 2⌉. Remark 2.2.1. The paths of even length P 2k are a special case of Theorem 1.4.5. Better bounds on n are known to exist. For example, the result of Füredi on the extremal number of odd cycles implies that n ≥4k is sufficient. We believe that a strengthening of Conjecture 1.4.6 should hold for trees whose 2-coloring yields two leaves of different colors. Let us recall and prove a special case of conjecture 1.4.6, Theorem 1.4.7. Theorem. For positive integers a and b, let Sa,b denote the tree on a + b + 2 vertices consisting of an edge {u,v} where \|N(u)\v\| = a, \|N(v)\u\| = b and N(u)∩N(v) = ∅(See Figure 2.2, left). We have exc(n,Sa,b) ≤a+b 2 n. Proof. Let G be a vertex colored graph with \|E(G)\| > \|V(G)\| a+b 2 . Without loss of generality, suppose a ≤b. We may assume by induction that δ(G) > a+b 2 ≥a. Since ex(m,Sa,b) = ma+b 2 (see, for example ), it follows that G contains a copy of Sa,b. Suppose this copy is defined by the edge {u,v} together with the disjoint sets A ⊆N(u), B ⊆N(v) with \|A\| = a,\|B\| = b. Now, if there is more than one color in A∪B, then we are done. So suppose the color of all vertices in A∪B is the same. Hence A∪B is an independent set. CEU eTD Collection 38 Erd˝ os-Gallai theorem, for vertex colored graphs If u is not adjacent to some w ∈B (See Figure 2.2, middle), since \|N(w)\| ≥a+1, we can pick C ⊆N(w){u,v} of size a. So the edge {v,w} together with the sets B′ = (B∪{u}){w} and C define a Sa,b, where the colors of all vertices in C are different from the colors of B′{u}. If u is adjacent to all w ∈B, then fix x ∈B (See Figure 2.2, right). Since \|N(x)\| ≥a+1, we can pick C ⊆N(x){u} of size a. Let y ∈A and define B′ = (B∪{y}){x}. Observe that B′ ⊆N(u), and the edge {u,x} together with the sets B′ and C defines a Sa,b, where again the color of the vertices in C is different from the color of vertices in B′. CEU eTD Collection Chapter 3 Hypergraph girth problem It’s like asking why is Ludwig van Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is. Paul Erd˝ os 3.1 Connection In this chapter, we prove an upper bound on the number of triangles in pentagon-free graphs Theorem 1.5.3 and an upper bound on the number of hyperedges in three uniform hypergraphs without Berge-cycle of length 4, Theorem 1.7.8. At first one may think these two problems are non-related, and it may seem an odd choice to put them together in this chapter. We try to justify our choice at the end of this chapter, in Section 3.4. 3.2 Pentagons vs. triangles We open this section by recalling Theorem 1.5.3. Theorem (Ergemlidze, Gy˝ ori, Methuku, Salia ). ex(n,C3,C5) ≤ 1 2 √ 2 (1+o(1))n3/2. CEU eTD Collection 40 Hypergraph girth problem Fig. 3.1 An example of a crown-block and a K4-block. Proof. Let G be a C5-free graph with the maximum possible number of triangles. We may assume that each edge of G is contained in a triangle because otherwise, we can delete it without changing the number of triangles. Two triangles T,T ′ are said to be in the same block if they either share an edge or if there is a sequence of triangles T,T1,T2,...,Ts,T ′ where each triangle of this sequence shares an edge with the previous one (except the first one of course). It is easy to see that all the triangles in G are partitioned uniquely into blocks. Below we will characterize the blocks of G. A block of the form {abc1,abc2,...,abck} where k ≥1, is called a crown-block (i.e., a collection of triangles containing the same edge) and a block consisting of all triangles contained in the complete graph K4 is called a K4-block. See Figure 3.1. Claim 3.2.1. Every block of G is either a crown-block or a K4-block. Proof. If a block contains only one or two triangles, then it is easy to see that it is a crown-block. So we may assume that a block of G contains at least three triangles and let abc1,abc2 be two of them. We claim that if bc1x or ac1x is a triangle in G which is different from abc1, then x = c2. Indeed, if x ̸= c2, then the vertices a,x,c1,b,c2 contain a C5, a contradiction. Similarly, if bc2x or ac2x is a triangle in G which is different from abc2, then x = c1. Therefore, if aci or bci (for i = 1,2) is contained in two triangles, then abc1c2 forms a K4. However, then there is no triangle in G which shares an edge with this K4 but is not contained in it, because otherwise, it is easy to find a C5 in G, a contradiction. So in this case, the whole block consists only of a K4, and we are done. So we can assume that whenever abc1,abc2 are two triangles then the edges ac1, bc1, ac2, bc2 are each contained in exactly one triangle. Therefore, any other triangle which shares an edge with either abc1 or abc2 must contain ab. Let abc3 be such a triangle. Then applying the same argument as before for the triangles abc1,abc3 one can conclude that the edges ac3,bc3 are contained in exactly one triangle and so, any other triangle of G which shares an edge with one of the triangles abc1,abc2,abc3 must contain ab again. So by induction, CEU eTD Collection 3.3 3-uniform BC4-free hypergraphs 41 it is easy to see that all of the triangles in this block must contain ab. Therefore, it is a crown-block, as needed. Notice that, by the maximality, blocks of G are edge-disjoint. We claim that there is no C4 in G whose edges lie in different blocks of G. Claim 3.2.2. The edge set of every C4 is contained in some block of G. Proof. Let xyzw be a 4-cycle in G. Every edge of G is contained in a triangle. So in particular, let xyu be a triangle containing the edge xy. If u ̸∈{x,y,z,w} then uxwzy is a C5, a contradiction. Therefore, u = z or u = w. So either xyz and yzw or xyw and ywz are triangles of G. In both cases, the two triangles share an edge, so they belong to the same block. Hence, all four edges of xyzw lie in the same block. We are now ready to prove the theorem. We want to select a C4-free subgraph G0 of G such that the number of edges in G0 is the same as the number of triangles in G. By Claim 3.2.1 the edge set of every C4 is contained in some block of G. To make sure the selected subgraph G0 is C4-free, it suffices to make sure the edges selected from each block of G do not contain a C4, which is done as follows: From each crown-block {abc1,abc2,...,abck}, we select the edges ac1,ac2,...,ack to be in G0. From each K4-block abcd we select the edges ab,bc,ac,ad to be in G0 (since every block is either a crown-block or a K4-block by Claim 3.2.1, we have dealt with all blocks of G). Finally, notice that the number of selected edges in each block is exactly the number of triangles in that block. Moreover, since blocks are edge-disjoint, we never select the same edge twice. Therefore, since every triangle of G is contained in some block, the total number of triangles in G is the same as the number of edges in G0. On the other hand, as G0 is C4-free and also C5-free (as it is a subgraph of G), we can use Theorem 1.2.9, to show that the number of edges in it is at most 1 2 √ 2(1+o(1))n3/2, completing the proof of Theorem 1.5.3. 3.3 3-uniform BC4-free hypergraphs In this section we prove Theorem 1.7.8. Theorem (Ergemlidze, Gy˝ ori, Methuku, Salia, Tompkins ). ex3(n,C4) ≤(1+o(1)) n3/2 √ 10. CEU eTD Collection 42 Hypergraph girth problem Proof. Let H be a 3-uniform hypergraph with no BC4 and no isolated vertices. A block B of a hypergraph H is defined to be a maximal subparagraph of H with the property that for any two edges e, f ∈E(B), there is a sequence of edges of H, e = e1,e2,...,et = f, such that \|ei ∩ei+1\| = 2 for all 1 ≤i ≤t −1 and V(B) = ∪h∈E(B)h. It is easy to see that the blocks of H define a unique partition of E(H). For a block B and an edge h ∈E(B), we say h is a leaf if there exists x ∈h such that the only edge of B incident to x is h. It is simple to observe that the set of non-leaf edges of a block B is either the empty set, a single edge, or the edges of a complete hypergraph on 4-vertices minus an edge, K(3)− 4 . Even more, if the set of non-leaf edges of B is E(K(3)− 4 ), then B = K(3)− 4 . This implies that the set B(H) = {B \| B is a block in H} of all blocks of H, can be partitioned into the following types of blocks: 1. We say B ∈B(H) is type 1 if there exists an edge e ∈E(B) such that for all distinct f1, f2 ∈E(B), f1, f2 ̸= e, we have \|e∩fi\| = 2, for i = 1,2 and f1 ∩f2 ⊆e. 2. We say B ∈B(H) is type 2 if B = K(3)− 4 . Define the 2-shadow of a hypergraph to be the graph on the same set of vertices whose edges are all pairs of vertices {x,y} for which there exists an edge e ∈E(H) such that {x,y} ⊂e. We denote the 2-shadow of a hypergraph H by ∂H. The proof of Theorem 1.7.9 will proceed by estimating the number of 3-paths (3-vertex paths) in the 2-shadow of a Berge-C4-free hypergraph in two different ways. Given a vertex v in a hypergraph H, d(v) denotes the classical hypergraph degree of v. In particular, d(v) = \|{h ∈E(H) : v ∈h}\|. Let ds(v) be the (graph) degree of v in the 2-shadow of the hypergraph. In particular, ds(v) = \|{e ∈E(∂H) : v ∈e}\|. Then, we define the excess degree of the vertex v to be dex(v) = ds(v) −d(v). Finally, we define the block degree db(v) to be the total number of blocks containing an edge which contains v. Notice that for every 4-cycle x1,x2,x3,x4,x1 of ∂H, there exists three distinct integers 1 ≤i < j < k ≤4 such that {xi,xj,xk} ∈E(H), otherwise, H contains a copy of Berge-C4. We call this edge a representative edge of this 4-cycle. Note that each 4-cycle of ∂H has either 1, 2 or 3 representative edges. Two edges of H sharing two vertices yield a C4 in ∂H. However these are not the only types of C4’s in ∂H. We call a 4-cycle of ∂H rare if the induced subhypergraph of H on the vertices of cycle does not contain two edges sharing a diagonal pair of vertices of the 4-cycle. In the following claim, we show that the number of such cycles is small. We define a particular type of 3-path of ∂H. A 3-path, x1,x2,x3, is called good if {x1,x2,x3} / ∈E(H) and there is no x ∈V(H) such that x,x1,x2,x3,x is a rare cycle of ∂H. CEU eTD Collection 3.3 3-uniform BC4-free hypergraphs 43 Claim 3.3.1. For any a,b ∈V(H), there are at most two good 3-paths in ∂H with end points a and b. Proof. Suppose, by contradiction, that there are three distinct vertices v1,v2,v3 different from a and b such that a,vi,b forms a good 3-path of ∂H for all integer 1 ≤i ≤3. It follows that there are three Berge-paths a,ei,vi, fi,b, for all integer 1 ≤i ≤3 in H. Note that those edges are not necessarily distinct. But we have ei ̸= fi and ei ̸= f j, i ̸= j, since {a,vi} ⊂ei and {b,vj} ⊂f j and H is 3-uniform. Note that if e2 = e3, then e2 = {a,v2,v3}, hence e1 ̸= e2. Similarly, we have either f1 ̸= f2 or f1 ̸= f3. We may assume, without loss of generality, that e1 ̸= e2,e3. It follows that either a,e1,v1, f1,b, f2,v2,e2,a or a,e1,v1, f1,b, f3,v3,e3,a is a Berge-C4, a contradiction. Claim 3.3.2. There are at most 6\|E(H)\| rare 4-cycles in ∂H. Proof. We fix an edge {a,b,c} ∈E(H). It suffices to show that the edge {a,b,c} is repre-sentative of at most 6 rare 4-cycles (that is, {a,b,c} is contained in the vertex set of at most 6 rare 4-cycles). Suppose by contradiction that this is not true. Observe that there are three possible positions for a fixed vertex v among the vertices of a 4-cycle in ∂H with {a,b,c}. By the pigeonhole principle there are 3 distinct vertices v1,v2,v3 different from a, b or c with the same position in the 4-cycle. Without loss of generality, we may assume they form a 4-cycle in the order vi,a,c,b,vi. Therefore from the definition of a rare 4-cycle, there are at least three good 3-paths in ∂H from a to b, a contradiction to Claim 3.3.1. Using Claim 3.3.2, it is easy to see that the number of 3-paths in ∂H which are not good is at most 3\|E(H)\|+3·6\|E(H)\| = 21\|E(H)\|. Here we use the fact that each rare 4-cycle induces an edge of H. By conditioning on the middle vertex of the 3-path, we have the following estimate on the number of 3-paths in ∂H: #(3-paths in ∂H) = ∑ v∈V(H) ds(v) 2 = ∑ v∈V(H) d(v)+dex(v) 2 . The following claim provides an upper bound on the number of good 3-paths in ∂H. Claim 3.3.3. (good 3-paths in ∂H) ≤2 n 2 −4 ∑ v∈V(H) db(v) 2 . Proof. Fix a vertex v and consider two adjacent edges {v,x1,x2} and {v,y1,y2} such that they belong to the different blocks; clearly the vertices v,x1,x2,y1,y2 are all distinct. We CEU eTD Collection 44 Hypergraph girth problem claim that there is at most one good 3-path, namely xi,v,yj, between xi and yj, for each i, j ∈{1,2}. Suppose this is not the case, then without loss of generality, there exists u ̸= v such that x1,u,y1 is a good 3-path. By the definition of a good 3-path, there are two distinct edges hx,hy ∈H such that x1,u ∈hx and y1,u ∈hy. If {v,x1,x2}, {v,y1,y2}, hx and hy are all different edges, then clearly there is a Berge-4-cycle. Therefore either {v,x1,x2} = hx or {v,y1,y2} = hy. Hence we have u ∈{x2,y2}, without loss of generality we may assume u = x2. Observe that the 4-cycle x1,x2,y1,v of ∂H contains a good 3-path and so by definition the 4-cycle x1,x2,y1,v is not a rare 4-cycle. Hence we have a contradiction to the statement that edges {v,x1,x2} and {v,y1,y2} belong to the different blocks. Concluding that there is at most one good path between xi and yj. So there are at least 4∑v∈V(H) db(v) 2 pairs of vertices which have at most one good 3-path between them. From Claim 3.3.1, for each pair of vertices there are at most two good 3-paths in ∂H. These observations complete the proof of Claim 3.3.3. Thus, since the number of 3-paths which are not good is at most 21\|E(H)\|, we have ∑ v∈V(H) d(v)+dex(v) 2 = #(3-paths in ∂H) ≤2 n 2 −4 ∑ v∈V(H) db(v) 2 +21\|E(H)\|. (3.1) Now, we will obtain estimates for ∑v∈V(H) dex(v) and ∑v∈V(H) db(v). For each block B and v ∈V(B), let dB ex(v) denote an excess degree of v inside the hypergraph B. If B is type 1, then every vertex v ∈V(B) has dB ex(v) ≥1, so for type 1 blocks, ∑v∈V(B) dB ex(v) ≥\|V(B)\|. It is easy to see that for every block B we have \|V(B)\| > \|E(B)\|, so ∑v∈V(B) dB ex(v) > \|E(B)\|, for every type 1 block B. If B is a type 2 block, then ∑v∈V(B) dB ex(v) = 3 = \|E(B)\|. Therefore, ∑ v∈V(B) dB ex(v) ≥\|E(B)\| for every block B in B(H). This together with the fact that the blocks define a partition of the edges E(H) implies ∑ v∈V(H) dex(v) = ∑ B∈B(H) ∑ v∈V(B) dB ex(v) ≥∑ B∈B(H) \|E(B)\| = \|E(H)\|. (3.2) On the other hand, a simple double counting argument yields ∑ v∈V(H) db(v) = ∑ B∈B(H) \|V(B)\|. Therefore, CEU eTD Collection 3.4 Concluding remarks 45 ∑ v∈V(H) db(v) = ∑ B∈B(H) \|V(B)\| ≥∑ B∈B(H) \|B\| = \|E(H)\|. (3.3) Now we will use the inequalities derived so far to get desired upper bound on \|E(H)\|. By (3.2), 4\|E(H)\| = 3\|E(H)\|+\|E(H)\| ≤∑ v∈V(H) (d(v)+dex(v)). Since x 2 is a convex function, by Jensen’s inequality we have 1 n ∑v∈V(H)(d(v)+dex(v)) 2 ≤1 n ∑ v∈V(H) d(v)+dex(v) 2 . Combining the above two inequalities we get n 4\|E(H)\| n 2 ≤∑ v∈V(H) d(v)+dex(v) 2 . (3.4) Similarly, by (3.3) and Jensen’s inequality, we have n \|E(H)\| n 2 ≤∑ v∈V(H) db(v) 2 . (3.5) Combining (3.1), (3.4) and (3.5) we obtain n 4\|E(H)\| n 2 +4n \|E(H)\| n 2 ≤2 n 2 +21\|E(H)\|. (3.6) Rearranging (3.6) yields the desired bound, \|E(H)\| ≤(1+o(1)) n3/2 √ 10. 3.4 Concluding remarks We start with Lazebnik-Verstraëte theorem . Theorem 3.4.1 (Lazebnik, Verstraëte ). ex3(n,{BC2,BC3,BC4}) = 1 6n3/2 +o(n3/2). CEU eTD Collection 46 Hypergraph girth problem Surprisingly it is not known if there is a similar theorem for 4-uniform hypergraph of girth six1. Let us state a bold conjecture. Conjecture 3.4.2. ex4(n,{BC2,BC3,BC4,BC5}) = o(n3/2). Even more, if one believes Conjecture 3.4.2 is false, one may try to confirm the following conjecture for 3-uniform hypergraphs. Conjecture 3.4.3. ex3(n,{BC2,BC3,BC4,BC5}) = Θ(n3/2). Conjecture 3.4.3 is a strengthening of Theorem 3.4.1. Note that construction Lazebnik-Verstraëte in Theorem is full of BC5’s. Example 3.4.4 (Lazebnik, Verstraëte ). Take Füredi graph G from Theorem 1.2.6. In particular, G is a C4-free graph with a property that for every pair of vertices there is the unique vertex connected with both of them. Therefore every edge is in a unique triangle. Let H be a 3-uniform hypergraph on the vertex set V(G) and a triple is a hyperedge if and only if it induces a triangle in G. From the unique property of G, that every edge is in a unique triangle, H is linear. Even more, since a Berge-triangle in H is a triangle in the shadow G, the same property implies that H is Berge-triangle-free. Similarly, H is BC4-free too. The graph G is full of C5’s. Since G is C4-free then every C5 is an induced C5 in G. Finally, since every edge is in a triangle, we have for every C5 in G there is a BC5 in H. All of these problems are closely related with the problem to determine ex3(n,BC4). Since we conjectured the lower bound is the correct asymptotic see Theorem 1.7.9, and the hypergraph in Example 1.7.10 is not only BC4-free but BC3 and BC5-free, we have the following relaxed conjecture. Conjecture 3.4.5. ex3(n,{BC3,BC4,BC5}) = (1+o(1)) n3/2 3 √ 3. Theorem 3.4.6 (Ergemlidze, Gy˝ ori, Methuku ). ex3(n,{BC2,BC3,BC5}) = ex3(n,{BC2,BC5)} = 1 3 √ 3n3/2 +O(n). 1{BC2,BC3,BC4,BC5}-free CEU eTD Collection 3.4 Concluding remarks 47 Ergemlidze, Gy˝ ori, Methuku provided the construction which has a beautiful geometric interpretation. This geometric construction was found by Mubayi-Solymosi. Example 3.4.7 (Mubayi, Solymosi ). For a fixed integer n, take a cube [n]3. The axes parallel lines are {(a1,b1,x) : x ∈[n]} for all a1,b1 ∈[n], {(a2,x,c2) : x ∈[n]} for all a2,c2 ∈[n], and {x,b3,c3) : x ∈[n]} for all b3,c3 ∈[n]. The vertices of the hypergraph H are those lines, hence e(H) = 3n2. Three vertices form a hyperedge of size three if and only if they meet in exactly one point-E(H) = {((a,b,·),(a,·,c),(·,b,c)) : a,b,c ∈[n]}. In other words, all vertices (a,b,c) of the cube are hyperedges. It is easy to verify that, this hypergraph is linear, BC3 and BC5-free but full of BC4. Finally, we give you reasoning why the problem of maximizing triangles in a C5-free graph is closely related to these problems. Let G be a C5-free graph. As we have seen in the proof of Theorem 1.5.3, triangles are distributed in G as crown blocks or K4-blocks, see Figure 3.1. It is easy to see that if one constructs a 4-uniform hypergraph on the same set of vertices, taking 4-edges on vertices inducing K4 in G we would get a 4-uniform girth-6 graph. Even more, by the conjectured lower-bound construction, Example 1.5.2, does not contain any copy of K4. Therefore proving Conjecture 3.4.2 may be an important step towards proving conjectured upper bound of ex(n,C3,C5). CEU eTD Collection CEU eTD Collection Chapter 4 The Structure of Hypergraphs avoiding long Berge-cycles Simplicity is the final achievement. Af-ter one has played a vast quantity of notes and more notes, it is simplicity that emerges as the crowning reward of art. Frederic Chopin In this Chapter we consider a problem ex(n,BC≥k). Naturally, this problem is divided in three parts with different extremal values and constructions. 1. k ≥r +2; 2. k = r; 3. k < r and k = r +1; Recently, Füredi, Kostochka, Luo [48, 47] proved exact bounds for k > r +2. Kostochka, Luo settled k < r case. In the following section we prove cases k = r +1 and k = r +2, fully settling 1 and 3. After we prove k = r case with a lemma which also slightly improves Kostochka, Luo result in the case k < r. 4.1 Avoiding long Berge-cycles, cases k = r+1 and k = r+2 In this section we prove Theorem 1.8.14 and theorem 1.8.12. Let us recall those theorems. CEU eTD Collection 50 The Structure of Hypergraphs avoiding long Berge-cycles Theorem ( Gy˝ ori, Methuku, Salia, Tompkins, Zamora ). If r ≥3 then ex(n,BC≥r+1) ≤n−1. Moreover, equality is achieved if and only if ∂2(H) is connected and for every block D of ∂2(H), D = Kr+1 and H[D] consists of r hyperedges, (see Figure 1.4). Theorem ( Gy˝ ori, Methuku, Salia, Tompkins, Zamora ). If r ≥3 then ex(n,BC≥r+2) ≤(n−1)(r +1) r . Moreover, equality is achieved if and only if ∂2(H) is connected and for every block D of ∂2(H), D = Kr+1 and H[D] = Kr r+1, (see Figure 1.4). 4.1.1 Basic Lemmas, used in Subsection 4.1.2 and 4.1.3 Lemma 4.1.1. For any r ≥3, if a set S of size r + 1 contains r hyperedges of size r, then between any two vertices u,v ∈S, there is a Berge-path of length r consisting of these hyperedges. Proof. Let H be the hypergraph consisting of r hyperedges on r +1 vertices. First notice that for any pair of vertices x,y ∈S, the number of hyperedges h ⊂S such that {x,y} ̸⊂h is at most 2. (Indeed, there is at most one hyperedge that does not contain x and at most one hyperedge that does not contain y.) This means that every pair x,y ∈S is contained in some hyperedge, as there are at least 3 hyperedges contained in S. In other words, ∂2(H) = Kr+1. Consider an arbitrary path x1x2,...,xr+1 of length r in the ∂2(H) connecting u = x1 and v = xr+1. We want to show that there are distinct hyperedges containing the pairs xixi+1 for each 1 ≤i ≤r. To this end, we consider an auxiliary bipartite graph with pairs {x1x2,x2x3,...,xrxr+1} in one class and the r hyperedges h ⊂S in the other class, and a pair is connected to a hyperedge if it is contained in the hyperedge. We will show that Hall’s condition holds . As noted before, every pair is contained in a hyperedge. Given any two distinct pairs xixi+1 and xjxj+1, there is at most one hyperedge that does not contain either of them; i.e., at least r −1 hyperedges contain one of them. Thus we need 2 ≤r −1 for Hall’s condition to hold, but this is true as we assumed r ≥3. Moreover, if we take any 3 ≤j ≤r distinct pairs, then every hyperedge contains one of them. Therefore, we need j ≤r, but this is true by assumption. This finishes the proof of the lemma. CEU eTD Collection 4.1 Avoiding long Berge-cycles, cases k = r +1 and k = r +2 51 Lemma 4.1.2. For any r ≥4, if a set S of size r +1 contains r −1 hyperedges of size r, then between any two vertices u,v ∈S, there is a Berge-path of length r −1 consisting of these hyperedges. Proof. The proof is similar to that of Lemma 4.1.1. Let H be the hypergraph consisting of r −1 hyperedges on r +1 vertices. First notice that for any pair of vertices x,y ∈S, the number of hyperedges h ⊂S such that {x,y} ̸⊂h is at most 2. This means that every pair x,y ∈S is contained in some hyperedge, as there are at least r −1 ≥3 hyperedges contained in S. In other words, ∂2(H) = Kr+1. Consider an arbitrary path x1x2 ...xr of length r −1 in the ∂2(H) connecting u = x1 and v = xr. We want to show that there are distinct hyperedges containing the pairs xixi+1 for each 1 ≤i ≤r −1. To this end, we consider an auxiliary bipartite graph with pairs {x1x2,x2x3,...,xr−1xr} in one class and the r −1 hyperedges h ⊂S in the other class, and a pair is connected to a hyperedge if it is contained in the hyperedge. We show that Hall’s condition holds: As noted before, every pair is contained in a hyperedge. Given any two distinct pairs xixi+1 and xjxj+1, there is at most one hyperedge that does not contain either of them; i.e., at least r −2 hyperedges contain one of them. Thus we need 2 ≤r −2 for Hall’s condition to hold, but this is true as we assumed r ≥4. Moreover, if we take any 3 ≤j ≤r −1 distinct pairs, then every hyperedge contains one of them. Therefore, we need j ≤r −1 for Hall’s condition to hold, and this is true by assumption. This finishes the proof of the lemma. 4.1.2 Extremal hypergraphs without BC≥r+1 We recall Theorem 1.8.14. Theorem ( Gy˝ ori, Methuku, Salia, Tompkins, Zamora ). If r ≥3 then ex(n,BC≥r+1) ≤n−1. Moreover, equality is achieved if and only if ∂2(H) is connected and for every block D of ∂2(H), D = Kr+1 and H[D] consists of r hyperedges, (see Figure 1.4). Proof. We use induction on n. For the base cases, notice that the statement of the theorem is trivially true if 1 ≤n ≤r. Moreover, if n = r + 1, then e(H) ≤r because otherwise, H = Kr r+1 and then it is easy to see that there is a (Hamiltonian) Berge-cycle of length r +1 in H, a contradiction. Therefore, e(H) ≤r = n−1. Moreover, equality holds if and only if ∂2(H) = Kr+1 and H consists of r hyperedges. CEU eTD Collection 52 The Structure of Hypergraphs avoiding long Berge-cycles We will show the statement is true for n assuming it is true for all smaller values. Let H be an r-uniform hypergraph on n vertices having no Berge-cycle of length r +1 or longer. We show that we may assume the following two properties hold for H. (1) For any set S ⊆V(H) with \|S\| ≤\|V(H)\|−1 = n−1, the number of hyperedges of H incident to the vertices of S is at least \|S\|. Indeed, suppose there is a set S ⊂V(H) (i.e., \|S\| ≤\|V(H)\| −1) with fewer than \|S\| hyperedges incident to the vertices of S. We delete the vertices of S from H to obtain a new hypergraph H′ on n−\|S\| vertices. By induction, H′ contains at most (n−\|S\|−1) hyperedges, so H contains less than (n −1 −\|S\|) + \|S\| = (n −1) hyperedges, as required. (2) There is no cut-hyperedge in H. Indeed, if h ∈E(H) is a cut-hyperedge, then ∂2(H \ {h}) is not a connected graph, so there are disjoint non-empty sets V1 and V2 such that V(H) = V1 ∪V2 and there are no edges of ∂2(H{h}) between V1 and V2. So the hypergraphs H[V1] and H[V2] do not contain a Berge-cycle of length r + 1 or longer. Therefore, by induction, e(H[V1]) ≤\|V1\|−1 and e(H[V2]) ≤\|V2\|−1. In total, e(H) = e(H[V1])+e(H[V2])+ 1 ≤(\|V1\|+\|V2\|−2)+1 = \|V(H)\|−1, as desired. Moreover, we claim that the equality e(H) = \|V(H)\|−1 cannot hold in this case (i.e., if there is a cut-hyperedge). Indeed, if equality holds, then we must have e(H[V1]) = \|V1\| −1 and e(H[V2]) = \|V2\| −1. Notice that since r ≥3, the hyperedge h either contains at least two vertices x,y ∈V1 or two vertices x,y ∈V2. Without loss of generality, assume the former is true. By induction, ∂2(H[V1]) is connected and for every block D of ∂2(H[V1]), we have D = Kr+1 and the subhypergraph induced by D consists of r hyperedges. So by Lemma 4.1.1, there is a Berge-path of length r (consisting of the r hyperedges induced by D) between any two vertices of a block D. Then it is easy to see that since ∂2(H[V1]) is connected, there is a Berge-path P of length at least r between any two vertices of V1. In particular, between x and y. Then P together with h forms a Berge-cycle of length r +1 in H, a contradiction. Consider an auxiliary bipartite graph B consisting of vertices of H in one class and hyperedges of H on the other class. Then property (1) shows that Hall’s condition holds for all subsets of V(H) of size up to \|V(H)\|−1. Therefore, there is a matching in B that matches all the vertices in V(H), except at most one vertex, say x. In other words, there exists an injection f :V(H){x} →E(H) such that for every v ∈V(H){x}, we have v ∈f(v). Given an injection f : V(H) \ {x} →E(H) with v ∈f(v), let P f be a longest Berge-path of the CEU eTD Collection 4.1 Avoiding long Berge-cycles, cases k = r +1 and k = r +2 53 form v1 f(v1)v2 f(v2)...vl−1 f(vl−1)vl where for each 1 ≤i ≤l −1, vi+1 ∈f(vi). Moreover, among all injections f : V(H){x} →E(H) with v ∈f(v), suppose φ : V(H){x} →E(H) is an injection for which the path Pφ = v1φ(v1)v2φ(v2)...vl−1φ(vl−1)vl is a longest path. Because of the way Pφ was constructed, it is also clear that x ̸∈{v1,v2,...,vl−1}. We consider two cases depending on whether vl is equal to x or not. Case 1: vl ̸= x. Our aim is to get a contradiction, and show that this case is impossible. Claim 4.1.3. If vl ̸= x, then φ(vl) = {vl−r+1,vl−r+2,...,vl}. Proof. If vl ̸= x, then we claim φ(vl) = {vl−r+1,vl−r+2,...,vl}. Indeed, if φ(vl) contains a vertex vi ∈{v1,v2,...,vl−r}, then the Berge-cycle viφ(vi)vi+1φ(vi+1)...vlφ(vl)vi is of length r +1 or longer, a contradiction. Moreover, if φ(vl) contains a vertex v ̸∈{v1,v2,...,vl}, then P φ can be extended to a longer path v1φ(v1)v2φ(v2),...,vl−1φ(vl−1)vlφ(vl)v, a contradiction again, proving that φ(vl) = {vl−r+1,vl−r+2,...,vl}. Fix some i ∈{l −r + 1,l −r + 2,...,l −1}. Let us define a new injection ψ : V(H) \ {x} →E(H) as follows: ψ(v) = φ(v) for every v ̸∈{x,v1,v2,...,vl}, and for every v ∈ {v1,v2,...,vi−1}. Moreover, let ψ(vi) = φ(vl) and ψ(vk) = φ(vk−1) for each l ≥k ≥i + 1. Now consider the Berge-path v1φ(v1)v2φ(v2) ... viφ(vl)vlφ(vl−1)...vi+2φ(vi+1)vi+1 = v1ψ(v1)v2ψ(v2) ... viψ(vi)vlψ(vl) ...vi+2ψ(vi+2)vi+1. This path has the same length as Pφ, so it is also a longest path. Moreover, vi+1 ̸= x, so we can apply Claim 4.1.3 to conclude that ψ(vi+1) = {vl−r+1,vl−r+2,...,vl} = φ(vi). But then φ(vi) = φ(vl), a contradiction to the fact that φ was an injection. Case 2: vl = x. Claim 4.1.4. φ(vl−1) ⊂{vl−r,vl−r+1,...,vl}. Proof. If φ(vl−1) contains a vertex v ̸∈{v1,v2,...,vl}, then we consider the Berge-path v1φ(v1)v2 φ(v2) ,..., vl−1φ(vl−1)v. Since v ̸= x, we get a contradiction by Case 1. More-over, if φ(vl−1) contains a vertex vi with i ∈{1,2,...,l −r −1}, then the Berge-cycle viφ(vi)vi+1φ(vi+1)...vl−1φ(vl−1)vi is of length r + 1 or longer, a contradiction. This fin-ishes the proof of the claim. By Claim 4.1.4, we know that φ(vl−1) = {vl−r,vl−r+1,...,vl−1,vl} \ {vj} for some j with l −r ≤j ≤l −2. In the rest of the proof we fix this j. Claim 4.1.5. For any i ∈{l −r,l −r +1,...,l −1}{j}, we have φ(vi) ⊂{vl−r,vl−r+1,...,vl−1,vl}. CEU eTD Collection 54 The Structure of Hypergraphs avoiding long Berge-cycles Proof. When i = l −1, we know the statement is true by Claim 4.1.4. Suppose i ∈{l −r,l −r +1,...,l −2}{j}. Let us define a new injection ψ : V(H)\ {x} →E(H) as follows: ψ(v) = φ(v) for every v ̸∈{v1,v2,...,vl}, and for every v ∈ {v1,v2,...,vi−1}. Moreover, let ψ(vi) = φ(vl−1) and ψ(vk) = φ(vk−1) for each l −1 ≥ k ≥i + 1. Now consider the Berge-path v1φ(v1)v2φ(v2) ... viφ(vl−1)vl−1φ(vl−2)...vi+1 = v1ψ(v1)v2ψ(v2) ... viψ(vi)vl−1ψ(vl−1)...vi+1. (Note that when i = l −2, the Berge-path is simply v1φ(v1)v2φ(v2) ... viφ(vl−1)vl−1 = v1ψ(v1)v2 ψ(v2)...viψ(vi)vl−1.) If ψ(vi+1) contains a vertex v ̸∈{v1,v2,...,vl}, then the Berge-path v1ψ(v1)v2ψ(v2) ... viψ(vi) vl−1ψ(vl−1)...vi+2ψ(vi+2)vi+1ψ(vi+1)v has the same length as Pφ, so it is also a longest path. Moreover, since v ̸= x, we get a contradiction by Case 1. If ψ(vi+1) contains a vertex vk ∈{v1,v2,...,vl−r−1} then one can see that the Berge-cycle vkψ(vk)vk+1ψ(vk+1) ... vl−1ψ(vl−1)vk is of length r +1 or longer, a contradiction. There-fore, we have ψ(vi+1) ⊂{vl−r,vl−r+1,...,vl}. But we defined ψ(vi+1) = φ(vi), proving the claim. Note that Claim 4.1.5 shows that r −1 hyperedges of H are contained in a set S := {vl−r,vl−r+1,..., vl−1,vl} of size r +1. The following claim shows that if we can find one more hyperedge of H contained in S, then S must induce a block of ∂2(H). Claim 4.1.6. Suppose r ≥3. If a set S of size r+1 contains r hyperedges of H then it induces a block of ∂2(H). Proof. Since the set S contains at least 3 hyperedges every pair x,y ∈S is contained in some hyperedge. Thus ∂2(H[S]) = Kr+1. Consider a (maximal) block D of ∂2(H) containing S. Suppose D contains a vertex t ̸∈S. Then since D is 2-connected, there are two paths P 1,P 2 in ∂2(H) between t and S, which are vertex-disjoint besides t. Let V(P 1) ∩S = {u} and V(P 2)∩S = {v}. For each edge xy ∈E(P 1)∪E(P 2), fix an arbitrary hyperedge hxy of H containing xy. It is easy to see that a subset of the hyperedges {hxy \| xy ∈E(P 1) ∪E(P 2)} forms a Berge-path P between u and v. On the other hand, by Lemma 4.1.1, there is a Berge-path P′ of length r between u and v consisting of the r hyperedges contained in S. Note that P and P′ do not share any hyperedges (indeed, each hyperedge of P contains a vertex not in S, while hyperedges of P′ are contained in S). Therefore, P together with P′ forms a Berge-cycle of length r +1 or longer, a contradiction. Therefore, D contains no vertex outside S; thus S induces a block of ∂2(H), as required. We will use the above claim several times later. At this point, we need to distinguish the cases r = 3 and r ≥4 since Lemma 4.1.2 only applies in the latter case. CEU eTD Collection 4.1 Avoiding long Berge-cycles, cases k = r +1 and k = r +2 55 The case r ≥4 Since r ≥4, by Claim 4.1.5 and Lemma 4.1.2 there is a Berge-path of length r −1 between any two vertices of S = {vl−r,vl−r+1,...,vl−1,vl}. This will allow us to show the following. (Recall that j is defined just before Claim 4.1.5.) Claim 4.1.7. φ(vj) ⊂{vl−r,vl−r+1,...,vl−1,vl} = S. Proof. Suppose for a contradiction that φ(vj) contains a vertex v ̸∈S. The hyperedge φ(vj) contains at least two vertices from S, namely vj and vj+1. By property (2), φ(vj) is not a cut-hyperedge of H. So after deleting φ(vj) from H, the hypergraph H \ {φ(vj)} is still connected – so there is a (shortest) Berge-path Q in H{φ(vj)} between v and a vertex s ∈S (note that the hyperedges of Q are not contained in S). The vertex s is different from either vj or vj+1, say s ̸= vj, without loss of generality. By Lemma 4.1.2, there is a Berge-path Q′ of length r −1 between s and vj (consisting of the hyperedges contained in S). Then Q,Q′ and φ(vj) form a Berge-cycle of length at least r +1 in H, a contradiction. Claim 4.1.5 and Claim 4.1.7 together show that there are at least r hyperedges of H contained in S. If all r +1 subsets of S of size r are hyperedges of H, then S induces Kr r+1 and it is easy to show that it contains a Berge-cycle of length r + 1, a contradiction. This means S contains exactly r hyperedges of H. Then by Claim 4.1.6, we know that S induces a block of ∂2(H). Let D1,D2,...,Dp be the unique decomposition of ∂2(H) into 2-connected blocks. Claim 4.1.6 shows that one of these blocks, say D1, is induced by S. Let us contract the vertices of S to a single vertex, to produce a new hypergraph H′. Then it is clear that the block decomposition of ∂2(H′) consists of the blocks D2,...,Dp. So H′ does not contain any Berge-cycle of length r +1 or longer, also; moreover, \|V(H′)\| = \|V(H)\|−r and e(H′) = e(H)−r. By induction, we have e(H′) ≤\|V(H′)\|−1. Therefore, e(H) = e(H′)+r ≤(\|V(H′)\|−1)+r = (\|V(H)\|−r −1)+r = \|V(H)\|−1. If e(H) = \|V(H)\|−1, then we must have e(H′) = \|V(H′)\|−1 and S must contain exactly r hyperedges. Moreover, since equality holds for H′, by induction, ∂2(H′) is connected and for each block Di (with 2 ≤i ≤p) of ∂2(H′), Di = Kr+1 and H′[Di] contains exactly r hyperedges. This means that for every block D of ∂2(H), we have D = Kr+1 and H[D] contains exactly r hyperedges, completing the proof in the case r ≥4. CEU eTD Collection 56 The Structure of Hypergraphs avoiding long Berge-cycles The case r = 3 Recall that using Claim 4.1.5 we can find a set S of size 4 which contains 2 hyperedges of H. Let S = {x,y,a,b} and the two hyperedges be xab and yab. By property (2), xab is not a cut-hyperedge of H. So after deleting xab from H, the hypergraph H {xab} is still connected – so there is a (shortest) Berge-path Q between x and {y,a,b}. If Q is of length at least 2, then it is easy to see that Q together with yab and xab form a Berge-cycle of length at least 4, a contradiction. So Q consists of only one hyperedge, say h. Our goal is to find a set of vertices that induces a block of ∂2(H) so that we can apply induction. If \|h∩{y,a,b}\| = 2 then h,xab,yab are 3 hyperedges of H contained in S, so by Claim 4.1.6, we can conclude that S induces a block of ∂2(H). (Notice that S contains exactly \|S\|−1 = 3 hyperedges of H, otherwise it is easy to find a Berge-cycle of length 4; this will be useful later.) So we can suppose \|h∩{y,a,b}\| = 1. We consider two cases depending on whether h ∈{xat,xbt}, or whether h = xyt for some t ̸∈S. Case 1. First suppose without loss of generality that h = xat for some t ̸∈S. Consider the set D of all hyperedges of H containing the pairs xa, ab or xb and let D be the set of vertices spanned by them. For each pair of vertices i, j ∈{x,a,b}, let Vi j = {v \| ijv ∈H}{x,a,b}. We claim that the sets Vxa,Vab,Vxb are pairwise disjoint. Suppose for the sake of contradiction that t′ ∈Vxa ∩Vab. Then the hyperedges xat′,abt′,xab are contained in a set of 4 vertices {x,a,b,t′}. Thus by Claim 4.1.6, this set induces a block of ∂2(H) and we are done. Thus we can suppose Vxa ∩Vab = / 0. Similarly, Vab ∩Vxb = / 0 and Vxa ∩Vxb = / 0. This shows that \|D\| = 3 + \|Vxa\| + \|Vxb\| + \|Vab\|. On the other hand, D consists of 1 + \|Vxa\| + \|Vxb\| + \|Vab\| hyperedges, so \|D\| = \|D\|−2. We will now show that D induces a block of ∂2(H). Let D′ be a (maximal) block of ∂2(H) containing D and suppose for the sake of a contradiction that it contains a vertex p ̸∈D. Then since D′ is 2-connected, there are two paths P 1,P 2 in ∂2(H) between p and D, which are vertex-disjoint besides p. Let V(P 1)∩D = {u} and V(P 2)∩D = {v}. For each edge xy ∈E(P 1)∪E(P 2), fix an arbitrary hyperedge hxy of H containing xy. It is easy to see that a subset of the hyperedges {hxy \| xy ∈E(P 1)∪E(P 2)} forms a Berge-path P between u and v. If uv ̸∈{xa,ab,xb}, then it is easy to see that there is a path P′ of length 3 between u and v consisting of the hyperedges of D. Then P together with P′ forms a Berge-cycle of length at least 4 in H, a contradiction. On the other hand, if uv ∈{xa,ab,xb}, then P must contain at least two hyperedges of H because otherwise P = {puv} but then puv should have been in D (since by definition D must contain all the hyperedges of H containing the pair uv); moreover, it is easy to check that between u and CEU eTD Collection 4.1 Avoiding long Berge-cycles, cases k = r +1 and k = r +2 57 v there is a Berge-path P′ of length 2 consisting of the hyperedges of D. Then again, P together with P′ forms a Berge-cycle of length at least 4 in H, a contradiction. Therefore, D′ contains no vertex outside D; so D induces a block of ∂2(H) (which contains \|D\| −2 hyperedges of H), as desired. Case 2. Finally suppose h = xyt for some t ̸∈S. Let D be the set of all hyperedges of H containing the pair xy plus the hyperedges xab and yab, and let D be the set of vertices spanned by the hyperedges of D. Let Vxy = {v \| xyv ∈H}. We claim that a ̸∈Vxy and b ̸∈Vxy. Indeed suppose for the sake of a contradiction that a ∈Vxy. Then the hyperedges xab,yab,xya are contained in a set of 4 vertices {x,y,a,b}. So by Claim 4.1.6, this set induces a block of ∂2(H), and we are done. So a ̸∈Vxy. Similarly, we can conclude b ̸∈Vxy. Therefore, \|D\| = Vxy +4. On the other hand, \|D\| = Vxy +2, so \|D\| = \|D\|−2. We claim that D induces a block of ∂2(H). The proof is very similar to that of Case 1, but we still give it for completeness. Let D′ be a (maximal) block of ∂2(H) containing D and suppose for the sake of a contradiction that it contains a vertex p ̸∈D. Then since D′ is 2-connected, there are two paths P 1,P 2 in ∂2(H) between p and D, which are vertex-disjoint besides p. Let V(P 1)∩D = {u} and V(P 2)∩D = {v}. For each edge xy ∈E(P 1)∪E(P 2), fix an arbitrary hyperedge hxy of H containing xy. It is easy to see that a subset of the hyperedges {hxy \| xy ∈E(P 1)∪E(P 2)} forms a Berge-path P between u and v. If uv ̸= xy, then it is easy to see that there is a path P′ of length 3 or 4 between u and v consisting of the hyperedges of D. (Indeed if u,v ∈Vxy, then P′ is of length 4, otherwise it is of length 3.) Then P together with P′ forms a Berge-cycle of length at least 4 in H, a contradiction. On the other hand, if uv = xy, then P must contain at least two hyperedges of H because otherwise P = {puv} but then puv should have been in D (since by definition D must contain all the hyperedges of H containing the pair uv); moreover, it is easy to check that between u and v there is a Berge-path P′ of length 2 consisting of the hyperedges of D. Then again, P together with P′ forms a Berge-cycle of length at least 4 in H, a contradiction. Therefore, D′ contains no vertex outside D; so D induces a block of ∂2(H) (and contains \|D\|−2 hyperedges of H), as desired. Let D1,D2,...,Dp be the unique decomposition of ∂2(H) into 2-connected blocks. In Case 1 and Case 2 we showed that one of these blocks, (say) D1 = D is such that H[D1] contains \|D1\| −2 hyperedges of H, otherwise, D1 is a set of 4 vertices such that H[D1] contains exactly \|D1\| −1 = 3 hyperedges of H. In all these cases, note that e(H[D1]) ≤ \|D1\|−1. Let us contract the vertices of D1 to a single vertex, to produce a new hypergraph H′. Then it is clear that the block decomposition of ∂2(H′) consists of the blocks D2,...,Dp. So H′ does not contain any Berge-cycle of length 4 or longer, also; moreover, \|V(H′)\| = CEU eTD Collection 58 The Structure of Hypergraphs avoiding long Berge-cycles \|V(H)\|−\|D1\|+1 and e(H′) = e(H)−e(H[D1]). By induction, we have e(H′) ≤\|V(H′)\|−1. Therefore, e(H) = e(H′)+e(H[D1]) ≤\|V(H′)\|−1+\|D1\|−1 = (\|V(H)\|−\|D1\|+1)−1+\|D1\|−1 = \|V(H)\|−1. If e(H) = \|V(H)\|−1, then we must have e(H′) = \|V(H′)\|−1 and H[D1] must contain exactly \|D1\|−1 hyperedges. As noted before, this is only possible if D1 has 4 vertices and induces exactly 3 hyperedges of H. Moreover, since equality holds for H′, by induction, ∂2(H′) is connected and for each block Di (with 2 ≤i ≤p) of ∂2(H′), Di = K4 and H′[Di] contains exactly 3 hyperedges. This means for every block D of ∂2(H), we have D = K4 and H[D] contains exactly 3 hyperedges of H, completing the proof in the case r = 3. Note that Theorem 1.8.14 easily implies Theorem 1.8.3. In fact, it gives the following stronger form. Theorem 4.1.8. Fix k = r + 1 > 2 and let H be an r-uniform hypergraph containing no Berge-path of length k. Then, e(H) ≤n k k r = n. Moreover, equality holds if and only if each connected component D of ∂2(H) is Kr+1, and H[D] = Kr r+1. Proof. We proceed by induction on n. The base cases n ≤r +1 are easy to check. Let H be an r-uniform hypergraph containing no Berge-path of length k = r +1 such that e(H) ≥n. Then by Theorem 1.8.14, H contains a Berge-cycle C of length r + 1 or longer. C must be of length exactly r + 1, otherwise it would contain a Berge-path of length r + 1. Let v1,...,vr+1 and e1,...,er+1 be the vertices and edges of C where {vi,vi+1} ⊆ei (indices are taken modulo r +1). For any i with 1 ≤i ≤r +1, if ei contains a vertex v ̸∈{v1,...,vr+1}, then vi+1ei+1vi+2ei+2 ...ei−1vieiv forms a Berge-path of length r +1 in H, a contradiction. Therefore, all of the edges ei (for 1 ≤i ≤r +1) are contained in the set S := {v1,...,vr+1}. That is, H[S] = Kr r+1. It is easy to see that S forms a connected component in ∂2(H) because if any hyperedge h of H (with h ̸∈C) contains a vertex of C, then C can be extended to form a Berge-path of length r +1. Let S1,S2,...,St be the vertex sets of connected components of ∂2(H). As noted before, one of them, say S1, is equal to S. We delete the vertices of S1 from H to form a new hypergraph H′; note that \|V(H′)\| = \|V(H)\| −(r + 1) and \|E(H′)\| = \|E(H)\| −(r + 1) and the connected components of ∂2(H′) are S2,...,St. By induction \|E(H′)\| ≤\|V(H′)\|. Thus \|E(H)\| = \|E(H′)\|+(r +1) ≤\|V(H′)\|+(r +1) = \|V(H)\|. Moreover, if \|E(H)\| = \|V(H)\|, then \|E(H′)\| = \|V(H′)\|, so by the induction hypothesis each connected component Si (i ≥2) of ∂2(H′) is Kr+1, and H′[Si] = Kr r+1, proving the theorem. CEU eTD Collection 4.1 Avoiding long Berge-cycles, cases k = r +1 and k = r +2 59 4.1.3 Extremal hypergraphs without BC≥r+2 We will prove the theorem by induction on n. For the base cases, note that if 1 ≤n ≤r then the statement of the theorem is trivially true. If n = r +1, the statement is true since there are at most r +1 hyperedges of size r on r +1 vertices. Moreover, equality holds if and only if H = Kr r+1. We will show the statement is true for n ≥r +2 assuming it is true for all smaller values. Let H be an r-uniform hypergraph on n vertices having no Berge-cycle of length r +2 or longer. We show that we may assume the following two properties hold for H. (1) For any set S ⊆V(H) of vertices, the number of hyperedges of H incident to the vertices of S is at least \|S\|. Indeed, suppose there is a set S ⊆V(H) with fewer than \|S\| hyperedges incident to the vertices of S. If \|S\| = n we immediately have the required bound on e(H), so assume n > \|S\|. We can delete the vertices of S from H to obtain a new hypergraph H′ on n−\|S\| vertices. By induction, H′ contains at most r+1 r (n−\|S\|−1) hyperedges, so H contains less than r+1 r (n−1−\|S\|)+\|S\| < r+1 r (n−1) hyperedges, as desired. (2) There is no cut-hyperedge in H. Indeed, if h ∈E(H) is a cut-hyperedge, then ∂2(H{h}) is not a connected graph, so there are non-empty disjoint sets V1 and V2 such that V(H) = V1 ∪V2, and there are no edges of ∂2(H{h}) between V1 and V2. So both hypergraphs H[V1] and H[V2] do not contain a Berge-cycle of length r+2 or longer. By induction, e(H[V1]) ≤r+1 r (\|V1\|−1) and e(H[V2]) ≤r+1 r (\|V2\|−1). In total, e(H) = e(H[V1])+e(H[V2])+1 ≤r+1 r (\|V1\|+ \|V2\|−2)+1 < r+1 r (\|V(H)\|−1), as desired. Consider an auxiliary bipartite graph B consisting of vertices of H in one class and hyperedges of H in the other class, and the edges of B are defined as follows: xh ∈E(B) if and only if the vertex x is contained in the hyperedge h. Then property (1) shows that Hall’s condition holds in B. Therefore, there is a perfect matching in B. In other words, there exists an injection f : V(H) →E(H) such that v ∈f(v). Given an injection f : V(H) →E(H) with v ∈f(v), let P f be a longest Berge-path of the form v1 f(v1)v2 f(v2)...vl−1 f(vl−1)vl where for each 1 ≤i ≤l −1, vi+1 ∈f(vi). Moreover, among all injections f : V(H) →E(H) with v ∈f(v), suppose φ : V(H) →E(H) is an injection for which the path Pφ = v1φ(v1)v2φ(v2)...vl−1φ(vl−1)vl is a longest path. Claim 4.1.9. φ(vl) ⊂{vl−r,vl−r+1,...,vl−1,vl}. CEU eTD Collection 60 The Structure of Hypergraphs avoiding long Berge-cycles Proof. First notice that if φ(vl) contains a vertex vi ∈{v1,v2,...,vl−r−1}, then the Berge-cycle viφ(vi)vi+1φ(vi+1)...vlφ(vl)vi is of length r +2 or longer, a contradiction. Moreover, if φ(vl) contains a vertex v ̸∈{v1,v2,...,vl}, then Pφ can be extended to a longer path v1φ(v1)v2φ(v2) ...vl−1φ(vl−1)vlφ(vl)v, a contradiction. This completes the proof of the claim. By Claim 4.1.9, we know that φ(vl) = {vl−r,vl−r+1,...,vl−1,vl}{vj} for some l −r ≤ j ≤l −1. Claim 4.1.10. For any i ∈{l −r,l −r +1,...,l}{j}, we have φ(vi) ⊂{vl−r,vl−r+1,...,vl−1,vl}. Proof. When i = l, we know the statement is true. Suppose i ∈{l −r,l −r +1,...,l −1}\ { j}. Let us define a new injection ψ : V(H) →E(H) as follows: ψ(v) = φ(v) for every v ̸∈{v1,v2,...,vl}, and for every v ∈{v1,v2,...,vi−1}. Moreover, let ψ(vi) = φ(vl) and ψ(vk) = φ(vk−1) for each l ≥k ≥i+1. Now consider the Berge-path v1φ(v1)v2φ(v2)...viφ(vl)vlφ(vl−1)...vi+2φ(vi+1)vi+1, equivalently v1ψ(v1)v2ψ(v2)...viψ(vi)vlψ(vl)...vi+2ψ(vi+2)vi+1. This path has the same length as Pφ, so it is also a longest path. Moreover, notice that the sets of last r + 1 vertices of both paths are the same. Thus we can apply Claim 4.1.9 to conclude that φ(vi) = ψ(vi+1) ⊂{vl−r,vl−r+1,...,vl−1,vl}, as desired. Claim 4.1.10 shows that there are r hyperedges (each of size r) contained in the set S := {vl−r,vl−r+1 ,..., vl−1,vl} of size r +1. We will apply Lemma 4.1.1 to S. Claim 4.1.11. The set S = {vl−r,vl−r+1,...,vl−1,vl} induces a block of ∂2(H). Proof. Since the set S = {vl−r,vl−r+1,...,vl−1,vl} contains r ≥3 hyperedges every pair x,y ∈S is contained in some hyperedge. Thus ∂2(H[S]) = Kr+1. Consider a (maximal) block D of ∂2(H) containing S. Suppose D contains a vertex t ̸∈S. Then since D is 2-connected, there are two paths P 1,P 2 in ∂2(H) between t and S, which are vertex-disjoint besides t. Let V(P 1) ∩S = {u} and V(P 2)∩S = {v}. For each edge xy ∈E(P 1)∪E(P 2), fix an arbitrary hyperedge hxy of H containing xy. It is easy to see that a subset of the hyperedges {hxy \| xy ∈E(P 1) ∪E(P 2)} forms a Berge-path P between u and v. CEU eTD Collection 4.1 Avoiding long Berge-cycles, cases k = r +1 and k = r +2 61 On the other hand, by Lemma 4.1.1, there is a Berge-path P′ of length r between u and v consisting of the r hyperedges contained in S. Note that P and P′ do not share any hyperedges (indeed, each hyperedge of P contains a vertex not in S, while hyperedges of P′ are contained in S). Therefore, P ∪P′ forms a Berge-cycle of length r +2 or longer unless P consists of only one hyperedge, say h. Note that h contains a vertex x ̸∈S and u,v ∈h; moreover by property (2), h is not a cut-hyperedge of H. So after deleting h from H, the hypergraph H \ {h} is still connected – so there is a (shortest) Berge-path Q in H \ {h} between x and a vertex s ∈S. (Note that the hyperedges of Q are not contained in S, and as it is a shortest Berge-path, both u and v do not appear among the “defining" vertices of Q.) The vertex s is different from either u or v, say s ̸= u without loss of generality. By Lemma 4.1.1, there is a Berge-path Q′ of length r between s and u (consisting of hyperedges contained in S). Then, Q,Q′ and h form a Berge-cycle of length at least r +2, a contradiction. Therefore, D contains no vertex outside S; thus S induces a block of ∂2(H), as required. Let D1,D2,...,Dp be the unique decomposition of ∂2(H) into 2-connected blocks. Claim 4.1.11 shows that one of these blocks, say D1, is induced by S. Let us contract the vertices of S to a single vertex, to produce a new hypergraph H′. Then it is clear that the block decomposition of ∂2(H′) consists of the blocks D2,...,Dp. So H′ does not contain any Berge-cycle of length r + 2 or longer, also; moreover \|V(H′)\| = \|V(H)\| −r. Thus, by induction, we have e(H′) ≤r+1 r (\|V(H′)\|−1). Therefore, e(H) ≤r +1 r (\|V(H′)\|−1)+(r+1) = r +1 r (\|V(H)\|−r−1)+(r+1) = r +1 r (\|V(H)\|−1). Now if e(H) = r+1 r (\|V(H)\| −1), then we must have e(H′) = r+1 r (\|V(H′)\| −1) and S must contain all r+1 subsets of size r (i.e., H[S] = H[D1] = Kr r+1). Moreover, since equality holds for H′, by induction, ∂2(H′) is connected and for each block Di (with 2 ≤i ≤p) of ∂2(H′), Di = Kr+1 and H′[Di] = Kr r+1. This means that for every block D of ∂2(H), we have D = Kr+1 and H[D] = Kr r+1, completing the proof. 4.1.4 A corollary We note that Theorem 1.8.14 implies Theorem 1.8.3. In fact, it gives the following stronger form. Here we prove this implication. Theorem 4.1.12. Fix k = r + 1 > 2 and let H be an r-uniform hypergraph containing no Berge-path of length k. Then, e(H) ≤n k k r = n. Moreover, equality holds if and only if each connected component D of ∂2(H) is Kr+1, and H[D] = Kr r+1. CEU eTD Collection 62 The Structure of Hypergraphs avoiding long Berge-cycles Proof. We proceed by induction on n. The base cases n ≤r +1 are easy to check. Let H be an r-uniform hypergraph containing no Berge-path of length k = r +1 such that e(H) ≥n. Then by Theorem 1.8.14, H contains a Berge-cycle C of length r + 1 or longer. C must be of length exactly r + 1, otherwise it would contain a Berge-path of length r + 1. Let v1,...,vr+1 and e1,...,er+1 be the vertices and edges of C where {vi,vi+1} ⊆ei (indices are taken modulo r +1). For any i with 1 ≤i ≤r +1, if ei contains a vertex v ̸∈{v1,...,vr+1}, then vi+1ei+1vi+2ei+2 ...ei−1vieiv forms a Berge-path of length r +1 in H, a contradiction. Therefore, all of the edges ei (for 1 ≤i ≤r +1) are contained in the set S := {v1,...,vr+1}. That is, H[S] = Kr r+1. It is easy to see that S forms a connected component in ∂2(H) because if any hyperedge h of H (with h ̸∈C) contains a vertex of C, then C can be extended to form a Berge-path of length r +1. Let S1,S2,...,St be the vertex sets of connected components of ∂2(H). As noted before, one of them, say S1, is equal to S. We delete the vertices of S1 from H to form a new hypergraph H′; note that \|V(H′)\| = \|V(H)\| −(r + 1) and \|E(H′)\| = \|E(H)\| −(r + 1) and the connected components of ∂2(H′) are S2,...,St. By induction \|E(H′)\| ≤\|V(H′)\|. Thus \|E(H)\| = \|E(H′)\|+(r +1) ≤\|V(H′)\|+(r +1) = \|V(H)\|. Moreover, if \|E(H)\| = \|V(H)\|, then \|E(H′)\| = \|V(H′)\|, so by the induction hypothesis each connected component Si (i ≥2) of ∂2(H′) is Kr+1, and H′[Si] = Kr r+1, proving the theorem. 4.2 Hypergraphs with a circumference at most uniformity In this Section we prove Theorem 1.8.16, Theorem 1.8.18 and Theorem 1.8.20 using a powerful Lemma 4.2.2. We recall Theorem 1.8.16. Theorem. Let k,n and r be positive integers such that 3 ≤k < r. Then exr(n,BC≥k) = n−1 r (k −1)+1rN∗(n) If r\|(n−1) the only extremal n-vertex r-graphs are the (r +1,k −1)-block trees. We recall Theorem 1.8.18. Theorem. Let r > 2 and n be positive integers. Then exr(n,BC≥r) = max n−1 r (r −1),n−r +1 CEU eTD Collection 4.2 Hypergraphs with a circumference at most uniformity 63 When n−r +1 > n−1 r (r −1) the only extremal graph is S(r) n . When n−1 r (r −1) > n−r +1 and r\|(n−1) the only extremal graphs are the (r +1,k −1)-block trees. Remark 4.2.1. In particular when n ≥r(r −2)+2, we have that exr(n,BC≥r) = n−r +1 and S(r) n is the only extremal hypergraph. We recall Theorem 1.8.20. Theorem. Let k,n and r be positive integers such that 2 ≤k ≤r. Then exmulti r (n,BC≥k) = n−1 r −1 (k −1) If r −1\|(n−1) the only extremal graphs with n vertices are the (r,k −1)-block trees. All three theorems have essentially the same proof since, these results follow from our Lemma 4.2.2, which to some extent lets us understand the structure of long Berge-cycle free hypergraphs. Lemma 4.2.2. Let r,k,n and m be positive integers, with r ≥k ≥3, and let H be an n-vertex r-graph which is BC≥k-free such that every hyperedge has multiplicity at most m. Then at least one of the following holds. i) There exists S ⊆V of size r −1 such that \|Nh(S)\| ≤m. Moreover, if m < k −1 there exists a set S of size r −1 such that Nh(S) is d ≤m copies of a hyperedge h and S ⊂h. ii) There exists S ⊆V of size r such that \|Nh(S)\| ≤k −1. iii) k = r, m < k −1, and there exists e ∈E(H) such that after removing e from H the resulting r-graph can be decomposed in two r-graphs, S and K sharing one vertex, such that S is an r-star with at least r −1 edges, the shared vertex is in the center of S, e∩V(S) is a subset of the center of S and v(K) ≥2. In particular, since no hyperedge can have multiplicity larger than k −1, by setting m = k −1 we have that there exists a set S of size r −1 incident with at most k −1 edges. In Subsection 4.2.1 we deduce Theorems 1.8.16, 1.8.18 and 1.8.20 from Lemma 4.2.2. We deduce corollaries of those theorems in the end of this chapter Subsection 4.2.3. \| We give the proof of Lemma 4.2.2 in Subsection 4.2.2. CEU eTD Collection 64 The Structure of Hypergraphs avoiding long Berge-cycles 4.2.1 Proof of Theorem 1.8.16, 1.8.18, 1.8.20 To obtain the extremal constructions in Theorem 1.8.16, first we are going to show that in an (r +1,k −1)-block tree, for every pair of vertices there exists a Berge-path of length k −1 joining them. For this, we prove the following statement by induction. Claim 4.2.3. Let r ≥4 and H1 a multi (not necessarily uniform) hypergraph such that v(H1) = r +1 > e(H1) ≥2, and every hyperedge h ∈E(H1), h ̸= V(H1), has size at least r and multiplicity at most one. Then every pair of vertices of H1 are joined by a Berge-path of length e(H1). Proof. The proof is by induction on r. The case where r = 4 is simple to check, as well as the case when e(H1) = 2, since every edge contains all but at most one vertex. So suppose r > 4 and e(H1) > 2. Let v,u be two distinct vertices, take any hyperedge h containing v, then choose w ∈H{v,u}, consider H2 obtain by removing v and h from H1 and by deleting v from the remaining hyperedges, then H2 satisfy the conditions of the claim, hence there exists a Berge-path of length e(H2) = e(H1)−1 joining w and u, we can extend this path with h to be a Berge-path of length e(H1) joining v and u. Therefore we proved that, when r ≥4, in an (r +1,k −1)-block tree for every pair of vertices from the same block there exists a Berge-path of length k −1 joining them, hence the statement trivially holds for every pair of vertices too, since an (r +1,k −1)-block tree is a connected hypergraph. The same is true for (r,k −1)-block trees. Proof of Theorem 1.8.16. For the lower bound we can observe that an (r +1,k −1)-block tree on ar + 1 vertices is a BC≥k-free graph with a(k −1) edges, for n ∈{ar + 1,ar + 2,...,(a+1)r −1} this proves the lower bound, if n = (a+1)r add an extra edge containing r −1 new vertices to this construction and we will get a desired lower bound. For the upper bound, let H is an r-uniform, n-vertex, hypergraph, without a Berge-cycle of length at least k. The proof is by induction on the number of vertices. The theorem trivially holds for n ≤r. So suppose n > r and that the theorem holds for any graph with less than n vertices, by Lemma 4.2.2 there exists a set S ⊆V such that either \|S\| = r−1 and \|NH(S)\| = 1 or \|S\| = r and \|NH(S)\| = k −1. Let H′ be the graph induced by V ′ = V\S. Then either e(H) ≤1+e(H′) ≤1+ n−r r (k−1)+1rN∗(n−r+1) ≤ n−1 r (k−1)+·1rN∗(n), or e(H) ≤(k −1)+e(H′) ≤(k −1)+ n−(r −1)−1 r (k −1)+1rN∗(n−r) = n−1 r (k −1)+1rN∗(n). CEU eTD Collection 4.2 Hypergraphs with a circumference at most uniformity 65 From the above calculations equality holds, only when \|S\| = r, \|NH(S)\| = k −1 and e(H′) = n−r−1 r (k −1) or r\|n, \|S\| = r −1, \|NH(S)\| = 1 and e(H′) = n−r−1 r (k −1) + 1. If r\|n −1, we will prove that the only extremal hypergraph is an (r + 1,k −1)-block tree. We have \|S\| = r and by induction H′ is an (r + 1,k −1)-block tree. For any hyperedge h incident with S we have that \|h∩V ′\| ≤1, otherwise, since any two vertices of V ′ are joined by a Berge-path of length at least k −1 in H′, we have a Berge-cycle of length at least k in H, a contradiction. If there exist two hyperedges h, h′ incident with S such that, there exists two distinct vertices v, v′ ∈V ′, such that v ∈h and v′ ∈h′ then both h{v} and h′{v′} have r −1 elements in S, then these hyperedges must intersect in a vertex x, x ∈S. So v,h,x,h′,v′ together with a Berge-path of length at least k −1 joining v to v′ in H′ is a Berge-cycle of length at least k +1. Therefore every edge in NH(S) is either S or intersects the same vertex v of V ′, hence H is an (r +1,k −1)-block tree. Remark 4.2.4. If H is an n-vertex multi r-graph in which each edge has multiplicity at most m ≤k −1 and contains no Berge-cycle of length at least k, then Lemma 4.2.2 implies e(H) ≤max{a(k −1)+bm : ar +b(r −1) < n}, this holds for all k ≥2. Proof of Theorem 1.8.20. This theorem follows by induction in the same way as Theo-rem 1.8.16 since we can always find a set S of size r −1 incident with at most k −1 edges. Proof of Theorem 1.8.18. The Theorem trivially holds for n ≤r. So suppose n ≥r +1 We will assume by induction that Theorem 1.8.18 holds for n′ < n. Note that for n′ > 2, we have that any n′ vertex BC≥r-free r-graph have at most n′ −2 hyperedges, more over the only such r-graphs with precisely n′ −2 hyperedges are S(3) n′ , when r = 3, or an (r + 1,r −1)-block when n′ = r +1, both of which are connected. Let H be an n-vertex BC≥k-free r-graph, with maximum number of hyperedges. Applying Lemma 4.2.2, one of i), ii) or iii) must hold. Suppose iii) holds in Lemma 4.2.2, and let S and K be the given decomposition of H after removing the hyperedge e. Let v be the only vertex in V(S)∩V(K). If e∩V(S) = {v}, then we can in fact decompose H into S and K′, where K′ is obtain from K by adding e. By induction we have that e(H) ≤e(S)+e(K′) ≤v(S)−(r −1)+v(K)−2 < n−(r −1), a contradiction. Hence we have that e∩V(S) ̸= {v}. Let u ∈e∩V(S) and w ∈e∩V(K), with both u,w different from v. If v(K) ≥3, then e(H) = e(S) + e(K) + 1 ≤v(S) −(r − 1)+v(K)−2+1 = n−(r −1), but equality is not possible, since by connectivity of K there is a Berge-path from w to v in K and we have a Berge-path of length r −2 in S from v to u, finally we can use the hyperedge e to connect u to w, we get a Berge-path of length at least k, a contradiction. So H has n−(r −1) edges only if v(K) = 2. In this case e(K) = 0, CEU eTD Collection 66 The Structure of Hypergraphs avoiding long Berge-cycles and therefore e contains the center of S and the only vertex of V(K){v}, hence H = S(r) n . Finally we have e(H) ≤n−(r −1) and equality holds when H = S(r) n . If n−1 r (r−1) ≥n−r+1 then either iii) in Lemma 4.2.2 holds and e(H) ≤n−(r−1), or the proof of extremal number follows by induction in the similar way as Theorem 1.8.16. Suppose n −r + 1 > n−1 r (r −1). If i) holds in Lemma 4.2.2 then we have e(H) < n−(r−1) since n′−r+1 ≥ j n′−1 r k (r−1) for n′ = n−(r−1), a contradiction. If ii) holds in Lemma 4.2.2 then we have e(H) < n−(r−1) since n′ −r+1 ≥ j n′−1 r k (r−1) for n′ = n−r, which is also a contradiction. Therefore iii) holds in Lemma 4.2.2, hence H = S(r) n . 4.2.2 Proof of Lemma 4.2.2 Definition 4.2.5. A semi-path of length t in a hypergraph, is an alternating sequence of distinct hyperedges and vertices, e1,v1,e2,v2,...,et,vt (starting with a hyperedge and ending in a vertex) such that, v1 ∈e1 and vi−1,vi ∈ei, for i = 2,3,...t. Let r ≥k ≥3 be fixed integers and let H be a BC≥k-free multi r-graph, consider a semi-path P = e1,v1,e2,v2,...,et,vt of maximal length. Consider P′ the semi-path e1,v1,e2,v2,..., eℓ,vℓobtained from the first ℓvertices and hyperedges of P, where ℓ= min{k −1,t}, let F = {e1,e2,...,eℓ} and U = {v1,v2,...,vℓ}, the defining vertices and hyperedges of this path. Note that \|e1 ∩U\| ≤k −1 < r, so e1\U ̸= / 0. First, we will show that any vertex from e1 ∩U is the only incident with the defining hyperedges from F. Lemma 4.2.6. Suppose w ∈e1\U, then NH(w) ⊆F. Hence NH(e1\U) ⊆F. Proof. If w is incident with a hyperedge of P not in F, then t ≥k. Let j ≥k be the smallest index such that w is incident with ej, then v1,e2,v2 ...,vj−1,ej,w,e1,v1 is a Berge-cycle of length at least k, a contradiction. If w is incident with an edge e not in the semi-path P, then e,w,P is a longer semi-path, a contradiction to the maximality of P. For simplicity Lemma 4.2.6 was stated and proved for the maximal semi-path P, but similarly, it holds for every maximal semi-path. Hence we may apply Lemma 4.2.6 for other maximal semi-paths. For each defining vertex vi, vi ∈e1∩U, we find another maximal semi-path by rearranging P, starting at ei, without changing the set of the first ℓvertices and hyperedges. Lemma 4.2.7. If for some i we have that vi ∈e1 ∩U, then NH(ei\U) ⊆F. CEU eTD Collection 4.2 Hypergraphs with a circumference at most uniformity 67 e1 e2 ej ei−1 ei et v1 v2 vj vi vt vj u u e1 ei Fig. 4.1 Semi-path P 1 in the proof of Lemma 4.2.8 Proof. Consider the semi-path ei,vi−1,ei−1,vi−2,...,e2,v1,e1,vi,ei+1,vi+1,...,et,vt, this semi-path has length t, so it is maximal, then NH(ei\U) ⊆F follows from Lemma 4.2.6 for this path. Lemma 4.2.8. If there are two vertices vi,vj ∈e1 ∩U, with i > j such that (ei ∩ej)\U ̸= / 0, then NH(vi−1) ⊆F and NH(vj) ⊆F. Proof. Fix a vertex u ∈(ei ∩ej)\U and consider maximal length semi-paths P 1 and P 2, (see Figure 4.1). P 1 = ei−1,vi−2,ei−2,vi−3,...,ej+1,vj,e1v1,e2,v2,...,vj−1,ej,u,ei,vi,ei+1,vi+1,...,et,vt, P 2 = ej+1,vj+1,ej+2,vj+2,...,vi−1,ei,u,ejvj−1,ej−1,vj−2,...,e2,v1,e1,vi,ei+1,...,et,vt. Applying Lemma 4.2.6 for P 1 and P 2, we get NH(vi−1) ⊆F and NH(vj+1) ⊆F. Let d ≤m be an integer such that V(e1) = V(e2) = ··· = V(ed) ̸= V(ed+1). Claim 4.2.9. If e1 ∩U = {v1,v2,v3,...,vd} then either e1{vd} is incident with d, d ≤m, hyperedges or there exists a set S of size r such that NH(S) ⊆F. In particular if e1 ∩U = {v1,v2,v3,...,vd} then Lemma 4.2.2 holds. Proof. First note that the vertices v1,v2,...,vd−1 can be exchanged with the vertices of e1\U ̸= / 0, hence from the Lemma 4.2.6, we have Nh(e1{vd}) ⊆F. Suppose w ∈e1{vd} is incident with a hyperedge ej, ℓ≥j > d, we may assume w ∈e1\U, then the semi-path P 1 = ej−1,vj−2,ej−2,vj−3,... ...,e2,v1,e1,w,ej,vj,...,et,vt has maximal length. Since vj−1 is a non-defining vertex in the first hyperedge P 1, applying Lemma 4.2.6 to P 1, we have that NH(vj−1) ⊆F, therefore the set (v1{vd})∪{vj−1} is a set of r vertices incident with at most k−1 hyperedges from F. Otherwise, if there is no such w then we have a set of r −1 vertices, e1{vd} incident with at most m hyperedges. CEU eTD Collection 68 The Structure of Hypergraphs avoiding long Berge-cycles From here we may assume that \|e1 ∩U\| > d. Let e1 ∩U = {vi0,vi1,vi2,...,vis}, where 1 = i0 < i1 < i2 < ··· < is, define recursively the sets A1 := e1\U and for j = 1,...,s, if (ei j\U)∩Aj = / 0, take Aj+1 := A j ∪(ei j\U), otherwise take A j+1 := A j ∪(ei j\U)∪{vij−1}. Note that the only possible defining vertices in A j are vi1−1,vi2−1,...,vi j−1−1, therefore vij−1 is not contained in A j. Let us denote A := As+1. We have that A j < A j+1 , for all j ∈{1,2,...,s}, so \|A\| ≥\|A1\|+s ≥r −1, by Lemmas 4.2.6, 4.2.7 and 4.2.8, we have that NH(A) ⊆F. If m ≥k −1 then A is a set of at least r −1 vertices incident with at most m hyperedges, hence Lemma 4.2.2 holds. If \|A\| ≥r then A is a set of at least r vertices incident with at most k −1 hyperedges, hence Lemma 4.2.2 holds. From here we may assume m < k −1 and \|A\| = r −1. Observe that \|A\| = r −1 is only possible if for every i = 1,2,...,s, \|Ai+1\| = \|Ai\|+1. We will assume, without loss of generality, that among all possible semi-paths of maximal length, P is one for which \|e1\U\| is minimal. There are two cases: Case 1: There exists an index j ≥d, such that A j intersects (eij\U), let j′ be the first such index, then there is another index d −1 ≤q < j′ such that (ei′ j\U)∩(eiq\U) ̸= / 0, and let u be an element in this intersection. If iq < ij′ −1 then viq ̸∈A from the minimality of j′, and by Lemma 4.2.8, NH(viq) ⊆F so A∪{viq} is a set of vertices, of size r, incident with at most k−1 hyperedges, hence Lemma 4.2.2 holds. If d < iq = i j′ −1 then by applying Lemma 4.2.6 to the maximal semi-path, eiq−1,viq−2,eiq−2,viq−3,...,v2,e2,v1,e1,viq,eiq,u,eiq+1,viq+1,eiq+2,...,et,vt we get NH(viq−1) ⊆F, since viq−1 is a non-defining vertex in the first hyperedge. Also we have viq−1 ̸∈A from the minimality of j′, hence A∪{viq−1} is a set of r vertices, incident with at most k −1 hyperedges and therefore Lemma 4.2.2 holds. Suppose d = iq = i j′ −1. Since \|Ad+1\| = \|Ad\|+1, Ad+1 = Ad ∪(eid\U)∪{vid−1} and vid−1 / ∈Ad, we have (ed+1\U) ⊆(e1\U), otherwise Ad+1 would have at least two new elements, but by the minimality of \|e1\U\|, we have (ed+1\U) = (e1\U). Fix any vertex vx, vx ∈U ∩(ed+1\e1). We need a similar lemma as Lemma 4.2.8. Claim 4.2.10. Suppose vj ∈e1 is such that (ej\U) intersects (ex\U) then Nh(vmax{j,x}−1) ⊆F. We skip the proof of Claim 4.2.10, since it is similar to the proof of Lemma 4.2.8. CEU eTD Collection 4.2 Hypergraphs with a circumference at most uniformity 69 Let (e1 ∩U) ∪{vx} = {vj0,vj1,...,vjs+1}, where 1 = j0 < j1 < ··· < js+1, define recursively the following sets B1 = e1\U, and for c = 1,2,...,s+1 let Bc+1 = Bc ∪ (evjc\U), if Bc ∩(evjc\U) = / 0, otherwise take Bc+1 = Bc ∪{vjc−1}. Finally Bs+2 has size at least r and is incident with at most k −1 hyperedges, therefore Lemma 4.2.2 holds. Case 2: For every index j ≥d, A j and (ei j\U) are disjoint. In this case, by construction, we have that r −1 = \|A\| = \|e1\U\| + (d −1) + \|eid\U\| + ··· + \|eis\U\|, this implies that ei j\U = 1 for every j, hence \|U\| ≥r−1, but since k−1 ≥\|U\|, we have that k = r and \|U\| = r −1. So there exists distinct vertices ud,ud+1,...,ur−1 such that ei = U ∪{ui} for each i ∈{d,d +1,...,r −1} and A = {v1,v2,...,vd−1,ud,ud+1,...,ur−1}. If d > 1, take the maximal semi-path, obtained from P, by changing the vertex vr−2 with v1 and the vertex v1 with ud, that is e1,ud,e2,v2,...,vr−3,er−2,v1,er−1,vr−1,...,vt. By Lemma 4.2.6, we have that NH(vr−2) ⊆F. Therefore A∪{vr−2} is a set of vertices of size r incident with at most k −1 hyperedges, thus Lemma 4.2.2 holds. Thus, we can suppose, that d = 1, and then each ui is vertex of degree one. We may also assume that the length of P is at least r, otherwise NH(vr−1) ⊂F, hence A∪{vr−1} is a vertex set of size r incident with at most k −1 hyperedges, therefore Lemma 4.2.2 holds. Claim 4.2.11. If there exists a hyperedge e, such that e ̸= er and e∩(U{vr−1}) ̸= / 0 then the vertices in e\U are only incident with e. Proof. Suppose without loss of generality v1 ∈e, otherwise we can rearrange the path. If e is a hyperedge of semi-path P, then e = ej for some j < r, otherwise we have a Berge-cycle length at least k, a contradiction. If e = ej, j < r then we already deduced that Claim 4.2.11 holds. If e is a non-defining hyperedge of semi-path P, then consider P′ obtain by replacing e1 in P with e, from Lemma 4.2.6, a vertex in v ∈e\U can only be incident with e,e2,e3,...,er−1, but if v is incident with one of these hyperedges from e2,e3,...,er−1 then e,e1,e2,...,er−1 together with the vertices v,v1,v2,...,vr−1 in some order would be a Berge-cycle of length r, a contradiction. Finally we have NH(e\U) = {e}, therefore Claim 4.2.11 holds. Let h1,h2,...,hp be the hyperedges incident with U\vr−1. If \|hi\U\| ≥2, for some i, then (e1∪e2∪···∪er−2∪hi)\U is a set of size at least r incident with k−1 hyperedges, hence Lemma 4.2.2 holds. Otherwise we have \|hi\U\| = 1, so these hyperedges form an r-star S with p ≥r −1 hyperedges. Every hyperedge from E(H){er,h1,h2,...,hp} CEU eTD Collection 70 The Structure of Hypergraphs avoiding long Berge-cycles can only intersect V(S) in vr−1, by setting K to be the r-graph induce from H by the vertices {vr−1}∪(V(H)\V(S)), we get the desired partition of H after removing the hyperedge er, therefore Lemma 4.2.2 holds. 4.2.3 Corollaries We note that as a corollary of Theorem 1.8.16 we obtain a slightly stronger version of Theorem 1.8.1. Let us recall Corollary 1.8.17. Corollary. Let k,n and r be positive integer with 3 ≤k ≤r. Then exr(n,BPk) = n r +1 (k −1)+1(r+1)N∗(n+1) Proof of Corollary 1.8.17. Let H be an n-vertex r-graph containing no Berge-path of length k. Define an (r +1)-graph H′ by adding a new vertex v to the vertex set of H and extending every hyperedge of H with v. If H′ is BC≥k-free, then from Theorem 1.8.16, we have e(H) = e(H′) ≤ n+1−1 r +1 (k −1)+1(r+1)N∗(n+1) = jn r k (k −1)+1(r+1)N∗(n+1). Suppose H′ contains a copy of a Berge-cycle v1,h1,v2 ...,hℓ−1,vℓ,hℓ,v1, of length ℓ, for some ℓ≥k. If v is one of the defining vertices, suppose without loss of generality v = v1, and let h′ i = hi{v} for each i = 1,2...ℓthen (h′ 1 ∪h′ k){v2,...,vk} ≥r +1−(k −1) ≥2 and that set intersects both h′ 1 and h′ k hyperedges. Therefore we can find two distinct vertices u ∈h′ 1 and u′ ∈h′ k different from all vi, i ∈{1,2,...,k} then u,h′ 1,v2,h′ 2,v3,...,h′ k−1,vk,h′ k,u′ is a Berge-path of length k in H, a contradiction. If v is not one of the defining vertices, then a similar argument leads us to contradiction. As a corollary of Theorem 1.8.20 we obtain a version of Theorem 1.8.1 with multiple hyperedges. Let us recall Corollary 1.8.21. Corollary. Let k,n and r be positive integer with 2 ≤k ≤r then exmulti r (n,BPk) = jn r k (k −1). CEU eTD Collection 4.2 Hypergraphs with a circumference at most uniformity 71 Proof of Corollary 1.8.21. This follows in a similar way as the previous corollary, by con-structing a BC≥k-free r-multi-graph H′. Hence, by Theorem 1.8.20, e(H) = e(H′) ≤ n+1−1 r +1−1 (k −1) = jn r k (k −1). CEU eTD Collection CEU eTD Collection Chapter 5 Connected Hypergraphs without long Berge-paths If I feel unhappy, I do mathematics to become happy. If I am happy, I do math-ematics to keep happy. Alfréd Rényi In this Chapter we prove the analogues of Theorem 1.3.7 for higher uniformity. Let us recall Theorem 1.3.7. Theorem (Kopylov , Balister, Gy˝ ori, Lehel, Schelp ). Let n > ℓ≥3 exconn(n,P ℓ) = max n e(Gn,ℓ,1),e(Gn,ℓ,⌊ℓ−1 2 ⌋) o . Extremal graphs are Gn,ℓ,1 or Gn,ℓ,⌊ℓ−1 2 ⌋, see Definition 1.3.6. In the following section we prove Theorem 1.8.4. Theorem (Gy˝ ori, Methuku, Salia, Tompkins, Vizer ). Let H′ n,k be a largest r-uniform connected n-vertex hypergraph with no Berge-path of length k, then lim k→∞lim n→∞ E(H′ n,k) kr−1n = 1 2r−1(r −1)!. We omit the proof of Theorem 1.8.10. CEU eTD Collection 74 Connected Hypergraphs without long Berge-paths If k is odd. If k is even. Fig. 5.1 The hypergraphs Hn,⌊k−1 2 ⌋and Hn,⌊k−1 2 ⌋,2. Theorem (Gy˝ ori, Salia , Zamora ). For all integers k,r with k ≥2r + 13 ≥18 there exists nk,r such that if n > nk,r, then we have • exconn r (n,BPk) = \|Hn,⌊k−1 2 ⌋\|, if k is odd, and • exconn r (n,BPk) = \|Hn,⌊k−1 2 ⌋,2\|, if k is even. Depending on the parity of k, the unique extremal hypergraph is Hn,⌊k−1 2 ⌋or Hn,⌊k−1 2 ⌋,2, (see Figure 5.1). Instead we prove a stability version of Theorem 1.8.10, in Section 5.2. 5.1 Proof of asymptotic Here we start to prove Theorem 1.8.4. We will use the following simple corollary of Theorem 1.5.4. Corollary 5.1.1. Let G be a connected graph on n vertices with no P k, then G has at most kr−1n 2r−1(r −1)! r-cliques if n ≥ck,r for some constant ck,r depending only on k and r. Proof. From Theorem 1.5.4, it follows that for large enough n, the number of r-cliques is at most n− k −1 2 k−1 2 r −1 + k−1 2 r + k−1 2 r −2 < n k 2 r −1 . Given an r-uniform hypergraph H we define the shadow graph of H, denoted ∂H to be the graph on the same vertex set with edge set: E(∂H) := {{x,y} : {x,y} ⊂e ∈E(H)}. CEU eTD Collection 5.1 Proof of asymptotic 75 Definition 5.1.2. If r = 3, then we call an edge e ∈E(∂H) fat if there are at least 2 distinct hyperedges h1,h2 with e ⊂h1,h2. If r > 3, then we call an edge e ∈E(∂H) fat if there are at least k distinct hyperedges h1,h2,...,hk in H with e ⊂hi for 1 ≤i ≤k. We call an edge e ∈E(∂H) thin if it is not fat. Thus, the set E(∂H) decomposes into the set of fat edges and the set of thin edges. We will refer to the graph whose edges consist of all fat edges in ∂H as the fat graph and denote it by F. Lemma 5.1.3. There is no P k in the fat graph F of the hypergraph H. Proof. Suppose we have such a P k with edges e1,e2,...,ek. For r = 3, if a hyperedge contains two edges from the path, then it must contain consecutive edges ei,ei+1. Select hyperedges h1,h2,...,hk where ei ⊂hi in such a way that hi+1 is different from hi for all 1 ≤i ≤k −1, and these edges yield the required Berge-path. Suppose now that r > 3, we will find a Berge-path of length k in H, with a greedy argument. For e1, select an arbitrary hyperedge h1 containing it. Suppose we have found a distinct hyperedge hi containing the fat edge ei for all 1 ≤i < i∗. Since the edge ei∗is fat, there are at least k different hyperedges h1 i∗,h2 i∗,...,hk i∗containing it. Select one of them, say h j i∗, which is not equal to any of h1,h2,...,hi∗−1. Thus, we may find distinct hyperedges h1,h2,...,hk where ei ⊂hi for 1 ≤i ≤k, and thus, we have a Berge-path of length k. We call a hyperedge h ∈E(H) fat if h contains no thin edge. Let F denote the hypergraph on the same set of vertices as H consisting of the fat hyperedges, then Lemma 5.1.4. If r = 3, then \|E(H\F)\| ≤(k −1)n 2 . If r > 3, then \|E(H\F)\| ≤(k −1)2n 2 . Proof. Arbitrarily select a thin edge from each h ∈H \ F. Let G be the graph consisting of the selected thin edges. We know that each edge in G was selected at most once if r = 3 and at most k −1 times in the r > 3. Thus, we have that \|H\F\| ≤\|E(G)\| for r = 3 and \|H \F\| ≤(k −1)\|E(G)\| for r > 3. Moreover, G is P k-free since a P k in G would imply a Berge P k in H by considering any hyperedge from which each edge was selected. It follows by Theorem 1.3.1 that \|E(G)\| ≤(k−1)n 2 , so \|H\F\| ≤(k−1)n 2 if r = 3, and \|H\F\| ≤(k−1)2n 2 if r > 3. CEU eTD Collection 76 Connected Hypergraphs without long Berge-paths Any hyperedge of F contains only fat edges, so it corresponds to a unique r-clique in F. This implies the following. Observation 5.1.5. The number of hyperedges in E(F) is at most the number of r-cliques in the fat graph F. To this end we will upper bound the number of r-cliques in F, by making use of the following important lemma. Lemma 5.1.6. There are no two disjoint cycles of length at least k/2+1 in the fat graph F. Proof. Let C and D be two such cycles. By connectivity, there are vertices v ∈V(C) and w ∈V(D) and a Berge-path from v to w in H containing no additional vertices of C or D as defining vertices. This path can be extended using the hyperedges containing the edges of C and D to produce a Berge-path of length k in H (note that here we used that the edges of C and D are fat), a contradiction. Assume that F has connected components C1,C2,...,Ct. Trivially, Nr(F) = t ∑ i=1 Nr(Ci). (5.1) If \|V(Ci)\| ≤k/2, then trivially Nr(Ci) ≤ \|V(Ci)\| r ≤\|V(Ci)\|r r! ≤kr−1 \|V(Ci)\| 2r−1(r −1)!. So we can assume \|V(Ci)\| ≥k/2. By Lemma 5.1.6, we have that for all but at most one i, Ci does not contain a cycle of length at least k/2+1. So by Corollary 1.5.5, for all but at most one i, say i0, we have Nr(Ci) ≤\|V(Ci)\|−1 k/2−2 k/2−1 r ≤kr−1 \|V(Ci)\| 2r−1(r −1)! +O(kr−2). If \|V(Ci0)\| ≥ck,r, then by Lemma 5.1.3 and by Corollary 5.1.1 we have Nr(Ci0) ≤kr−1 \|V(Ci)\| 2r−1(r −1)!. Otherwise, Nr(Ci0) ≤ \|V(Ci0)\| r = o(n). Therefore, by (5.1), we have Nr(F) = t ∑ i=1 Nr(Ci) ≤ CEU eTD Collection 5.2 Proof of stability 77 ≤ t ∑ i=1 kr−1 \|V(Ci)\| 2r−1(r −1)! +O(kr−2) +o(n) ≤ kr−1n 2r−1(r −1)! +O(kr−2)n+o(n). Therefore, by Observation 5.1.5, \|E(F)\| ≤Nr(F) ≤ kr−1n 2r−1(r −1)! +O(kr−2)n+o(n). (5.2) Since \|E(H)\| = \|E(H\F)\|+\|E(F)\|, adding up the upper bounds in (5.2) and Lemma 5.1.4, we obtain the desired upper bound on \|E(H)\|. 5.2 Proof of stability Let us start with recalling Definition 1.8.9. Definition 5.2.1. For integers n,a ≥1 and b1,...,bt ≥2 with n ≥2a+∑t i=1 bi let us denote by Hn,a,b1,b2,...,bt the following r-uniform hypergraph. • Let the vertex set of Hn,a,b1,b2,...,bt be A∪L∪St i=1 Bi, where A,B1,B2,...,Bt and L are pairwise disjoint sets of sizes \|A\| = a, \|Bi\| = bi (i = 1,2,...,t) and \|L\| = n−a−∑t i=1 bi. • Let the hyperedges of Hn,a,b1,b2,...,bt be A r ∪ t [ i=1 A∪Bi r ∪ {c}∪A′ : c ∈L,A′ ∈ A r −1 . Fig. 5.2 The hypergraph Hn,a,b1,b2,...,bt. The result here provides a stability version of Theorem 1.8.10 and also an extension of Theorem 1.9.2 for uniformity at least 3. CEU eTD Collection 78 Connected Hypergraphs without long Berge-paths First we state a theorem for hypergraphs with minimum degree at least 2, and then in full generality. In the proof, the hypergraphs Hn, k−3 2 ,3 and Hn, k−3 2 ,2,2 will play a crucial role in case k is odd, while if k is even, then the hypergraphs Hn,⌊k−3 2 ⌋,4, Hn,⌊k−3 2 ⌋,3,2 and Hn,⌊k−3 2 ⌋,2,2,2 will be of importance (Definition 1.8.9), note that all of them are n-vertex, maximal, BPk-free hypergraphs. In both cases, the hypergraph listed first contains the largest number of hyperedges. This number gives the lower bound in the following theorem. We recall Theorem 1.9.3. Theorem (Gerbner, Nagy, Patkós, Salia, Vizer ,). For any ε > 0 there exist integers q = qε and nk,r such that if r ≥3, k ≥(2+ε)r +q, n ≥nk,r and H is a connected n-vertex, r-uniform hypergraph with minimum degree at least 2, without a Berge-path of length k, then we have the following. • If k is odd and \|H\| > \|Hn, k−3 2 ,3\| = (n−k+3 2 ) k−3 2 r−1 + k+3 2 r , then H is a sub-hypergraph of Hn, k−1 2 . • If k is even and \|H\| > \|Hn,⌊k−3 2 ⌋,4\| = (n −⌊k+5 2 ⌋) ⌊k−3 2 ⌋ r−1 + ⌊k+5 2 ⌋ r , then H is a sub-hypergraph of Hn,⌊k−1 2 ⌋,2 or H+ n,⌊k−1 2 ⌋. Let H′ n′,a,b1,b2,...,bt be the class of hypergraphs that can be obtained from Hn,a,b1,b2,...,bt for some n ≤n′ by adding hyperedges of the form A′ j ∪D j, where the D j’s partition [n′][n], all Dj’s are of size at least 2 and A′ j ⊆A for all j. Let us define H+ n′,⌊k−1 2 ⌋analogously. We recall Theorem 1.9.4. Theorem 5.2.2 (Gerbner, Nagy, Patkós, Salia, Vizer ,). For any ε > 0 there exist integers q = qε and nk,r such that if r ≥3, k ≥(2+ε)r +q, n ≥nk,r and H is a connected n-vertex, r-uniform hypergraph without a Berge-path of length k, then we have the following. • If k is odd and \|H\| > \|Hn, k−3 2 ,3\|, then H is a sub-hypergraph of some H′ ∈H′ n, k−1 2 . • If k is even and \|H\| > \|Hn,⌊k−3 2 ⌋,4\|, then H is a sub-hypergraph of some H′ ∈H′ n,⌊k−1 2 ⌋,2 or H+ n,⌊k−1 2 ⌋. We start the proof of Theorem 1.9.3 with a technical lemma that will be crucial later. Lemma 5.2.3. Let H be a connected r-uniform hypergraph with minimum degree at least 2 and with longest Berge-path and Berge-cycle of length ℓ−1. Let C be a Berge-cycle of length ℓ−1 in H, with defining vertices V = {v1,v2,...,vℓ−1} and defining edges E(C) = {e1,e2,...,eℓ−1} with vi,vi+1 ∈ei (modulo ℓ−1). Then, we have CEU eTD Collection 5.2 Proof of stability 79 (i) every hyperedge h ∈H\C contains at most one vertex from V(H)\V. (ii) If u,v are not necessarily distinct vertices from V(H) \V, then there cannot exist distinct hyperedges h1,h2 ∈H\C and an index i with v,vi ∈h1 and u,vi+1 ∈h2. (iii) If there exists a vertex v ∈V(H)\V and there exist different hyperedges h1,h2 ∈H\C with v,vi−1 ∈h1 and v,vi+1 ∈h2, then there exists a cycle of length ℓ−1 not containing vi as a defining vertex. Proof. We prove (i) by contradiction. Suppose h ∈H\C contains two vertices fromV(H)\V. We distinguish two cases. Case 1. Hyperedge h contains a vertex u ̸∈V and a different vertex v ∈ei \V for some i ≤ℓ. Then vi+1,ei+1,vi+2,...,vℓ,eℓ,v1,e1,...,vi,ei,v,h,u is a path of length ℓ, a contradiction. Case 2. Hyperedge h contains two vertices u and v from V(H)\V(C). We consider the hypergraph H′ obtained from H by removing a hyperedge h. Case 2.1. There is a Berge-path in H′ from {v,u} to the cycle C, in particular, to a defining vertex of C. Then let P be a shortest such path, let us assume P is from v to vi. Without loss of generality we may suppose that P does not contain ei as a defining hyperedge, (it is possible P contains ei−1 as a defining hyperedge). Then u,h,v,P,vi,ei,vi+1,...,ei−2,vi−1 is a Berge-path of length at least ℓ, contradicting the assumption that the longest path in H is of length ℓ−1. Case 2.2. Suppose there is no Berge-path from the vertex v to the cycle C in H′. However by connectivity of H, there is a shortest path P from v to a defining vertex of C, say vi and it does not use any defining hyperedge of C but possibly ei−1. Also, h is not a hyperedge of P. There exists a hyperedge h′ ̸= h containing v, as the minimum degree is at least 2 in H. Note that h′ is not a hyperedge of the path P, even more all vertices of h′ different from v are not defining vertices of P or C. Fix a vertex u′ ∈h′ \ {v}. Then u′,h′,P,ei,vi+1,...,ei−2,vi−1 is a Berge-path of length at least ℓ, a contradiction. To prove (ii), assume first that u = v. Then one could enlarge C by removing ei and adding h1,v,h2 to obtain a longer cycle, a contradiction. Assume now u ̸= v. Then removing ei and adding h1,v and h2,u, one would obtain a path of length ℓ, a contradiction. Finally to show (iii), we can replace ei−1,vi,ei in C by h1,v,h2 to obtain the desired cycle. We say that an r-uniform hypergraph H has the set degree condition, if for any set X of vertices with \|X\| ≤k/2, we have \|E(X)\| ≥\|X\| ⌊k−3 2 ⌋ r−1 , i.e., the number of those hyperedges CEU eTD Collection 80 Connected Hypergraphs without long Berge-paths that are incident to some vertex in X is at least \|X\| ⌊k−3 2 ⌋ r−1 . We first prove Theorem 1.9.3 for such hypergraphs. Proof of Theorem 1.9.3 for hypergraphs having the set degree condition. Let a hypergraph H be an n-vertex BPk-free with the set degree condition. Also, assume \|H\| is as claimed in the statement of the theorem. However, for the most part of the proof, we will only use the set degree condition. Claim 5.2.4. Let P be a longest Berge-path in H with defining vertices U = {u1,...,uℓ} and defining hyperedges F = {f1, f2,..., fℓ−1} in this given order. Suppose P minimizes x1 +xℓ among longest Berge-paths of H, where xi for i ∈[ℓ], denotes the number of hyperedges in F incident to ui. Then the sizes of NH\F(u1) and NH\F(uℓ) are at least k−3 2 . Proof of Claim 5.2.4. Observe that the statement is trivially true for r ≥4 and for arbitrary longest path, as by the set degree condition, there exist at least ⌊k−3 2 ⌋ r−1 −k +1 hyperedges in H \F incident to u1. This is strictly greater than ⌊k−5 2 ⌋ r−1 if r ≥4 and k ≥(2+ε)r +q, for large enough q, hence \|NH\F(u1)\| > k−5 2 , finishing the proof for r ≥4. Thus we can assume that r = 3. Let P be a longest Berge-path in H, minimizing x1 +xℓ. First we claim that if u1 ∈fi then xi ≥x1. Note that the Berge-path ui, fi−1,ui−1, fi−2,ui−2,...,u2, f1,u1, fi,ui+1, fi+1,...,uℓ, fℓ−1,uℓ is also a longest Berge-path, with the same set of defining vertices and defining hyperedges and endpoint xℓ, hence by the minimality of the sum x1 +xℓ, the number of hyperedges from F incident to ui is at least x1. This means that if we consider all possible Berge-paths obtained from P by the way described above (including itself), then the number of pairs (u, f), where u ∈U, f ∈F and u ∈f, is at least x2 1. On the other hand, this number is upper bounded by 3\|F\| = 3(ℓ−1), hence we have x2 1 ≤3(ℓ−1) ≤3(k −1), therefore x1 ≤ p 3(k −1). The same holds for the other end vertex uℓand so for xℓby symmetry. Since the degree of u1 is at least ⌊k−3 2 ⌋ 2 , out of which at most p 3(k −1) of the hyper-edges are defining hyperedges, the degree of u1 in H\F is at least k−3 2 2 − p 3(k −1) > k−3 2 −1 2 , if k ≥21. Thus \|NH\F(u1)\| ≥ k−3 2 and in the same way we have \|NH\F(ul)\| ≥ k−3 2 . CEU eTD Collection 5.2 Proof of stability 81 Claim 5.2.5. Let ℓ−1 be the length of the longest Berge-path in H. Then ℓ≥k −3 and H contains a Berge-cycle of length ℓ−1. Proof of Claim 5.2.5. Let u1, f1,u2, f2,...,uℓ−1, fℓ−1,uℓbe a longest Berge-path given by Claim 5.2.4 with defining hyperedges F = { f1, f2,..., fℓ−1} and defining vertices U = {u1,u2,...,uℓ}. For E ⊆E(H) and an integer j with 1 ≤j ≤ℓ, let S j,E denote the set of indices of vertices in U ∩NH\E(uj), and we simply denote Sj,F by S j. In particular Sj denotes the set of indices i such that there is a hyperedge of H that contains both ui and u j and is not a defining hyperedge of the path. For any set S of integers let S−:= {a : a > 0, a+1 ∈S}, S−−= (S−)−. The operations + and ++ are defined analogously. To start the proof, observe first that H cannot contain a Berge-cycle C of length ℓ. Indeed, the hyperedges of such a cycle contain at most ℓ(r −1) vertices. Therefore there is a vertex v ∈V(H)\ V(C), then as H is connected, there exists a path from v to C and we obtain a path of length at least ℓ, contradicting our assumption on the length of the longest path. If ℓ∈S1 or equivalently 1 ∈Sℓ, then a hyperedge showing this, together with F forms a Berge-cycle of length ℓin H. So we can assume S1,Sℓ⊆{2,...,ℓ−1} and so S− 1 ⊆{1,2,...,ℓ−1}. If S− 1 ∩Sℓ̸= / 0 (or symmetrically S1 ∩S+ ℓ̸= / 0), then H contains a Berge-cycle of length ℓ. Indeed, if i ∈S− 1 ∩Sℓ, then there are hyperedges e and e′ in H \ F with u1,ui ∈e and uℓ,ui−1 ∈e′. Then ui−1, fi−2,ui−2,..., f2,u2, f1,u1,e,ui, fi+1,ui+2,..., fℓ−1,uℓ,e′ is a Berge-cycle of length ℓ. (Note that e and e′ are distinct hyperedges as ℓ̸∈S1.) Note that by Claim 5.2.4, we have \|Sℓ\|,\|S− 1 \| ≥ k−3 2 . So to avoid S− 1 ∩Sℓ̸= / 0, we have ℓ≥k −3. The exact same argument shows that if S−− 1 ∩Sℓ̸= / 0 or symmetrically S1 ∩S++ ℓ ̸= / 0, then H contains a Berge-cycle of length ℓ−1 and we are done in this case. For two indices x < y ∈Sℓ, let us introduce the relation x ∼y if S1 ∩(x,y] = / 0. Clearly, ∼is an equivalence relation. Assume Sℓhas m1 equivalence classes. Also, we say that a maximal subset of consecutive integers in Sℓis an interval of Sℓ. As S+ ℓ∩S1 = / 0 by the above, elements of the same interval belong to the same equivalence class. Let m2 be the number of intervals in Sℓ. If H does not contain cycles of length ℓand ℓ−1, then for the maximal element z of each equivalence class, we have that z+1,z+2 / ∈S1 and so by the definition of equivalence classes z+1,z+2 / ∈Sℓ. Moreover, if an element z belongs both to S1 and Sℓ, then z is the smallest element of an equivalence class. Also if z is the largest element of an interval that is not the rightmost interval in an equivalence class, then z+1 / ∈S1 ∪Sℓ. These CEU eTD Collection 82 Connected Hypergraphs without long Berge-paths observations show that 2⌊k−3 2 ⌋+m1 −2+(m2 −m1) ≤ℓ−2 holds. As ℓ≤k, we must have m2 ≤4. Similarly as in the proof of Claim 5.2.4 we can see that for any j ∈S− 1 , the vertex u j is the endpoint of a longest path Fj with other end vertex uℓand with defining vertex set U. Observe that the neighborhood Sℓof uℓwith respect to the non-defining hyperedges of F and Fj is the same, as the single hyperedge h ∈Fj \F contains u1 and therefore cannot contain uℓwithout creating a cycle of length ℓ. Therefore Sj,F j ⊆U and similarly as above if [(S− ℓ∪S−− ℓ )∩(S+ ℓ∪S++ ℓ )]∩S j,F j ̸= / 0, then H contains a Berge-cycle of length ℓor ℓ−1. Let S∗:= (S− ℓ∪S−− ℓ )∩(S+ ℓ∪S++ ℓ ) then \|S∗\| ≥\|Sℓ\|−2m2 ≥ k−3 2 −8. Let US− 1 := {ui : i ∈S− 1 } and consider E(US− 1 ). Observe that all but one of the defining hyperedges of Fi are in F, thus there are at most \|F\|+\|US− 1 \| ≤k −1+\|S− 1 \| hyperedges altogether in E(US− 1 ) that are defining hyperedges of F or an Fi. From above, all other hyperedges in E(US− 1 ) are completely in U \S∗, thus we have E(US− 1 ) ⊆ U \S∗ r ∪F ∪ [ x∈S− 1 Fx. By the set degree condition and the above, we must have \|S− 1 \| k−3 2 r −1 ≤\|E(US− 1 )\| ≤ k − k−3 2 +8 r +k −1+\|S− 1 \|. (5.3) Using k−3 2 ≤\|S− 1 \|, a r = a r a−1 r−1 and (a+1 r−1) ( a r−1) = a+1 a−r+2 ≤ a a−r, and writing k = αr we have k − k−3 2 +8 r = k − k−3 2 +8 r k − k−3 2 +7 r −1 ≤ k/2+9 r k+17 2 r −1 = α 2 +9/r k+17 2 r −1 ≤ α 2 +9/r α α −2 10 k−3 2 r −1 . (5.4) Therefore (5.3), (5.4) and k −1+\|S− 1 \| ≤2k = 2αr implies αr/2−2 ≤(α 2 +9/r)( α α−2)10 + 2α. This shows that for any ε > 0, there is an r0 such that if r > r0, then α < 2 +ε must hold, a contradiction. For the finitely many smaller values of r, the above inequality gives an upper bound βr for α = k/r, which might be larger than 2+ε. In that case we can choose qε := maxr≤r0 βrr. Then we have k > qε ≥αr = k, a contradiction. CEU eTD Collection 5.2 Proof of stability 83 Note that the cycleC given by Claim 5.2.5 is a longest Berge-cycle in H and let its defining vertices and defining hyperedges be V := {u1,u2,...,uℓ−1} and E(C) := {e1,e2,...,eℓ−1}, respectively, with ui,ui+1 ∈ei. We have ℓis either k −3, k −2, k −1 or k by Claim 5.2.5. Let us call ui−1 and ui+1 the neighbors of ui on C. 5.2.1 Preliminary technical claims By Lemma 5.2.3 (i), for any vertex w ∈V(H) \V we have NH\C(w) ⊆V. For any vertex w ∈V(H)\V, we partition NH\C(w) into two parts the following way: let Mw denote the set of vertices v ∈V such that there exists exactly one hyperedge in H\C containing both w and v, and let Dw denote the set of those vertices v ∈V for which there exist at least 2 hyperedges in H \C containing both v and w. Claim 5.2.6. For any w and w′ with w,w′ ∈V(H)\V and not necessarily distinct, we have (i) If u j ∈NH\C(w), u j+1 ∈NH\C(w′), then w = w′, u j,u j+1 ∈Mw and there exists a non-defining hyperedge h with w,u j,uj+1 ∈h. (ii) If u j ∈NH\C(w), u j+2 ∈Dw, then there exists a cycle C′ of length ℓ−1 in H such that the defining vertices of C′ are those of C but uj+1 replaced by w. Fig. 5.3 Sketch of the proof of Claim 5.2.6. Proof. Let uj ∈NH\C(w), uj+1 ∈NH\C(w′). If w ̸= w′, then for the hyperedges h,h′ ∈H\C with u j,w ∈h and u j+1,w′ ∈h′, we have h ̸= h′, from Lemma 5.2.3 (i). But then w′,h′,u j+1,ej+1,uj+2,...,uℓ−1,eℓ−1,u1,e1,...,u j,h,w CEU eTD Collection 84 Connected Hypergraphs without long Berge-paths is a Berge-path of length ℓ, see Figure 5.3, a contradiction. So w = w′, and if there exist h ̸= h′ with uj,w ∈h and u j+1,w ∈h′, then the Berge-path presented above is in fact a Berge-cycle that is longer than C, a contradiction. This proves (i). For the second part of the claim, observe that if u j ∈NH\C(w) and uj+2 ∈Dw, then there exist two distinct hyperedges h,h′ ∈H \C such that uj,w ∈h and uj+2,w ∈h′, so in C we can replace ej,uj+1,ej+1 by h,w,h′ to obtain desired cycle C′, see Figure 5.3. Claim 5.2.7. Suppose ui−1,ui+1,u j ∈Dw are three distinct vertices for some w ∈V(H)\V and let w∗∈V(H)\V be a vertex distinct from w. Then we have the following. (i) There is no hyperedge h ∈H\C with ui,u j−1 ∈h nor with ui,u j+1 ∈h. (ii) If uj+2 ∈NH\C(w), then ei−1,ei do not contain uj+1. (iii) Hyperedges ei−1 and ei are not incident with the vertices w,w∗. (iv) Suppose ut+1 ∈Dw∗or ut−1 ∈Dw∗for some t ̸= i. Then there is no h ∈H\C incident to ui and ut. (v) The hyperedges ej−1,ej are not incident with ui. Fig. 5.4 Sketch of the proof of Claim 5.2.7. Proof. We start with the proof of (i), see Figure 5.4 (i). Suppose by contradiction that ui,uj−1 ∈h ∈H \C. Then by Claim 5.2.6 (i), we have w / ∈h (as otherwise ui,ui−1 ∈Mw, CEU eTD Collection 5.2 Proof of stability 85 contradicting ui−1 ∈Dw). Furthermore, as ui+1,u j ∈Dw, there exist two distinct hyperedges h′,h′′ ∈H\C with ui+1,w ∈h′ and u j,w ∈h′′. Using the fact that u j−1 and ui+1 are different vertices as there can not be neighboring vertices in Dw by Lemma 5.2.3 (ii), we have that ui−1,ei−1,ui,h,u j−1,ej−2,u j−2,...,ui+1,h′,w,h′′,uj,ej,u j+1,...,ei−2 is a Berge-cycle longer than C, a contradiction. Similarly we can extend the cycle C if ui,u j+1 ∈h ∈H \ C. This proves (i). To show (ii) see Figure 5.4 (ii), it is enough to get a contradiction if ei contains u j+1, since the other case ei−1 contains u j+1 is symmetric. We have two non-defining distinct hyperedges, a hyperedge h′′ incident to w and u j+2 and a hyperedge h′ incident to w and ui+1 as ui+1 ∈Dw. Then ui,ei,uj+1,ej,uj,ej−1,...,ei+1,ui+1,h′,w,h′′,u j+2,ej+2,...,ui−2,ei−2,ui−1,ei−1 is a Berge-cycle longer than C, a contradiction. To show statement (iii), suppose first w∗∈ei. Then for a non-defining hyperedge h incident to w and ui+1, we have that w∗,ei,ui,ei−1,ui−1,...,ui+1,h,w is a path of length ℓ- a contradiction. If w∗∈ei−1, then similarly, for a non-defining hyperedge h incident to w and ui−1, we have that w∗,ei−1,ui,ei,ui+1,...,ei−2,ui−1,h,w is a path of length ℓ- a contradiction. If w ∈ei−1, then we have a contradiction since there exists a cycle longer than C, which is obtained from C by exchanging the edge ei−1 with h,w,ei−1, where h is a non-defining hyperedge incident to w and ui−1. Similarly we get a contradiction if w ∈ei. To prove (iv) by a contradiction, suppose that we have a non-defining hyperedge h of C incident to ui and ut. Assume without loss of generality that ut−1 ∈Dw∗since the other case is symmetrical. Then there exists a non-defining hyperedge h′ different from h, incident to ut−1 and w∗. Also there are two distinct non-defining hyperedges h′′,h′′′ with w,ui−1 ∈h′′ and w,ui+1 ∈h′′′. At first note that hyperedge h is distinct from h′′ and h′′′ by Claim 5.2.6 (i). From Lemma 5.2.3 (i) we have that hyperedges h′′ and h′′′ distinct from h′. Finally we have a contradiction since the following is a Berge-path of length ℓ w∗,h′,ut−1,et−2,···ui+1,h′′′,w,h′′,ui−1,ei−2,··· ,ut+1,et,ut,h,ui. To prove (v) suppose by a contradiction that ej contains ui. There are distinct non-defining hyperedges h,h′ with w,u j ∈h and w,ui−1 ∈h′. Then uj+1,ej,ui,ei,ui+1,ei+1,...,ej−1,u j,h,w,h′,ui−1,ei−2,ui−2,...,ej+2,u j+2,ej+1 CEU eTD Collection 86 Connected Hypergraphs without long Berge-paths is a Berge-cycle of length longer than C. This contradiction proves (v). The proof for the case ui ∈ej−1 is analogous. By Claim 5.2.6 and the set degree condition ⌊k−3 2 ⌋ r −1 ≤\|Mw\|+ min nj ℓ−1−\|Mw\| 2 k ,\|Dw\| o r −1 must hold for all w ∈V(H)\V(C). At first we observe that \|Mw\| ≤3 as otherwise ℓ−1− \|Mw\| ≤k −5. Therefore, we have \|Dw\| ≥⌊k−3 2 ⌋, if k ≥11. We say that a vertex ui ∈V is replaceable by w, if ui−1,ui+1 ∈Dw, and we denote by Rw the set of vertices that are replaceable by w. A vertex is called replaceable, if it is replaceable by w for some w ∈V(H) \V. For a replaceable vertex w′, we define Dw′ and Mw′ as for vertices in V(H)\V. For a vertex w ∈V(H)\V let us call a maximal set I of consecutive defining vertices of C in V \Dw a missing interval for w (or just missing intervals, if w is clear from the context), if its size is at least two. Let I1,I2,...,Is be the missing intervals of C for w and let us denote by I1,I2,...,Is the same intervals without the terminal vertices (it is possible that Ij = / 0). We have ∑s i=1(\|Ii\| −1) = ℓ−1 −2\|Dw\|. In particular, as \|Dw\| ≥ k−3 2 by the set degree condition and Lemma 5.2.3 (i), we have s ≤3, if k is even and s ≤2, if k is odd. Let us consider a hyperedge ej ∈C such that u j or uj+1 is from a missing interval. The number of such hyperedges is ∑s i=1(\|Ii\|+1), which is at most 9, if k is even, and at most 6, if k is odd. Our next technical claim is about missing intervals. Claim 5.2.8. Suppose that ui,ui+1,...,ui+t form a missing interval for some w ∈V(H)\V. Then (i) ei−1 and ei+t do not contain vertices w∗∈V(H)\V; and (ii) if ui−1 ∈Dw′ (resp. ui+t+1 ∈Dw′) for some w′ ̸= w, then ei−1 (resp. ei+t) does not contain a vertex from Rw. Proof. To prove (i) observe that there exists a Berge-path starting with the vertex w, a non-defining hyperedge h, the vertex ui−1, going around C with defining vertices and hyperedges and finishing with a vertex ui. Such h exists since ui−1 does not belong to the missing interval, so ui−1 ∈Dw. Note that we did not use a hyperedge ei−1 which contains w∗. If w = w∗, then ei−1 closes a Berge-cycle longer than C, a contradiction, while if w ̸= w∗, then finishing with ei−1,w∗we obtain a Berge-path of length ℓ, a contradiction. This contradiction proves (i). Similar argument shows the statement for the hyperedge ei+t. We omit the proof of part (ii) since the same argument will provide the desired result after changing a replaceable vertex with w. CEU eTD Collection 5.2 Proof of stability 87 Here we will show that \|Dw∗\| ≥ k−3 2 holds even for vertices w∗∈V(C)\V, therefore we have \|Dw′\| ≥ k−3 2 for all w′ ∈V(H)\V. By Claim 5.2.7 (iii) and Claim 5.2.8 (i), if w∗∈V(C)\V and ui ∈Dw, then w∗/ ∈ei−1,ei. Therefore the number of defining hyperedges that may contain w∗is at most 3. So Claim 5.2.6 and the set degree condition implies ⌊k−3 2 ⌋ r −1 ≤3+\|Mw∗\|+ min nj ℓ−1−\|Mw∗\| 2 k ,\|Dw∗\| o r −1 . Just as for w ∈V(H)\V(C), in two steps we obtain \|Dw\| ≥⌊k−3 2 ⌋for k large enough. Before continuing with a give possible embeddings of H into some Hn,a,b1,...,bs let us state a last technical claim that will be used several times. Let us recall that a terminal vertex v is a vertex of a missing interval that is adjacent to a vertex from Dw. Claim 5.2.9. Suppose Dw = Dw′ for some w′ ∈V(H)\V with w′ ̸= w. (i) There does not exist h ∈H\C such that h contains terminal vertices of two distinct missing intervals of w. (ii) If {ui,ui+1,ui+2} and {uj,uj+1} form missing intervals of w and there exists h ∈H\C with ui+1,u j ∈h or ui+1,u j+1 ∈h, then there does not exist h′ ∈H\C, with ui,ui+2 ∈h′. Proof. We prove (i) by contradiction. Suppose {ui,ui+1,...,ui+t} and {uj,uj+1,...,u j+z} are two distinct missing intervals of w. • Suppose first ui,u j+z ∈h ∈H\C. We have ui+t+1,ui−1,u j+z+1 ∈Dw, therefore there are three different hyperedges hw, h′ w and hw′, such that hw is incident to w and ui+t+1, h′ w is incident to w and ui−1 and hw′ is incident to u j+z+1 and w′. Note that all those hyperedges are different from h by Claim 5.2.6 (i). Then we have a contradiction since the following Berge-path is of length ℓ, as it contains all the ℓ−1 defining vertices of C and w and w′: ui+t,...,ui,h,u j+z,ej+z−1,...,ui+t+1,hw,w,h′ w,ui−1,ei−2,...,uj+z+1,hw′,w′. • If ui+t,uj+z ∈h ∈H\C, then the Berge-path of length ℓ(using similar ideas as in the previous bullet) is ui,...,ui+t,h,u j+z,ej+z−1,...,ui+t+1,hw,w,h′ w,ui−1,ei−2,...,uj+z+1,hw′,w′, and we are done with the proof of (i). In (ii) we can assume that ui+1,uj+1 ∈h holds since the case ui+1,u j ∈h is identical. The proof of this part is similar, at first we observe from part (i) that we have h ̸= h′. Then CEU eTD Collection 88 Connected Hypergraphs without long Berge-paths the following Berge-path of length ℓgives us a contradiction: ui,h′,ui+2,ei+1,ui+1,h,uj+1,ej,uj,ej−1,...,ui+3,hw,w,h′ w,ui−1,ei−2,...,u j+2,hw′,w′. 5.2.2 Possible embeddings of Hypergraph Now we are in the situation to be able to give possible embeddings of H into some Hn,a,b1,...,bs. In this subsection, we gather all the information that we know about these embeddings so far and in the next subsection, we analyze further the different cases to finish the proof. Let us fix w ∈V(H)\V with Dw of maximum size and let H∗denote the subhypergraph of H that we obtain by removing those defining hyperedges ei of C for which at least one of ui or ui+1 is a vertex of a missing interval for w. By the above, \|H\| ≤\|H∗\|+9. If we are in a case when for all w′ ∈V(H)\V we have Dw′ ⊆Dw, then let A = Dw, Bi = Ii for i = 1,2,...,s and L = V(H)(Dw ∪s i=1 Ii). Let us summarize the findings of the technical claims and enumerate the types of different hyperedges in H\Hn,a,b1,b2,...,bs in this scenario. Summary 5.2.10. If h ∈H\Hn,a,b1,b2,...,bs is not a defining hyperedge of C (i.e., h ∈H\C), then 1. either there exists v ∈(V(H)\V)∪Rw such that h{v} ⊆Dw∪Ss i=1 Ii and h∩Ss i=1 Ii ̸= / 0; We refer to these hyperedges as type 1 hyperedges in what follows. 2. h ⊆V \ Rw and h contains vertices from at least two distinct missing intervals. We refer to these hyperedges as type 2 hyperedges in what follows. If ei ∈H\Hn,a,b1,b2,...,bs is a defining hyperedge of C, then 3. either ei ∈H\H∗; or 4. ui or ui+1 belongs to Rw, ei {ui,ui+1} ⊆Dw ∪Ss i=1 Ii and ei ∩Ss i=1 Ii ̸= / 0. Proof. Suppose first that h is not a defining hyperedge of C and h contains a vertex v ∈ (V(H)\V)∪Rw. We claim that h cannot contain any v′ ∈V(H)\V with v′ ̸= v. Indeed, if v / ∈V, then it follows from Lemma 5.2.3 (i). If v ∈Rw and v′ = w, then w can be inserted to obtain a longer cycle than C, while if w ̸= v′, then using h, the defining vertices and hyperedges of C one can create a Berge-path of length ℓfrom v′ to w. We also claim that h cannot contain a neighbor of a vertex in Dw on C. Indeed, if v / ∈V, then it follows from Lemma 5.2.3 (ii) and (iii). If v ∈Rw, then it follows from Claim 5.2.7 (i). CEU eTD Collection 5.2 Proof of stability 89 Therefore, h cannot contain other vertices of Rw, nor terminal vertices of missing intervals. This gives possibility 1. Otherwise if h ∈H\Hn,a,b1,b2,...,bs is not a defining hyperedge of C, then we must have h ⊆V \Rw. As all hyperedges in A∪Ij r belong to Hn,a,b1,b2,...,bs, there must exist two distinct missing intervals meeting h. This gives possibility 2. Let ei ∈H \ Hn,a,b1,b2,...,bs be a defining hyperedge of C. If at least one of ui or ui+1 belongs to a missing interval, then ei ∈H\H∗by definition of H∗. This gives possibility 3. Note that we have more information on some of these hyperedges by Claim 5.2.8. Otherwise ui or ui+1 belongs to Rw. By Claim 5.2.7 (ii), ei does not contain any other vertex from Rw, and by Claim 5.2.7 (iii) ei cannot contain any vertex from V(H)\V. This gives us possibility 4. Even more, if the unique element of ei ∩Rw is also replaceable by some w′ ̸= w, then ei cannot contain w either. If we are in a case when we have vertices w,w′ ∈V(H)\V with Dw ̸⊆Dw′ and Dw′ ̸⊆ Dw, then as k−3 2 ≤\|Dw\|,\|Dw′\|, we will have k−1 2 ≤\|Dw ∪Dw′\|. Since the elements of Dw ∪Dw′ cannot be neighbors on C by Claim 5.2.6 (i) and \|C\| ≤k −1, we must have \|Dw ∪Dw′\| = k−1 2 . If \|C\| = 2 k−3 2 +2, then we will embed H to Hn,⌊k−1 2 ⌋, with A = Dw ∪Dw′ and all the other vertices are going to L. If \|C\| = 2 k−3 2 +3, then we will embed H∗to Hn,⌊k−1 2 ⌋,2 with A = Dw ∪Dw′, the unique missing interval goes to B1 and all the remaining vertices are going to L. Summary 5.2.11. If for w,w′ ∈V(H)\V we have Dw ̸⊆Dw′ and Dw′ ̸⊆Dw, then 1. there is no hyperedge h ∈H\C with h ∈H\Hn,⌊k−1 2 ⌋or h ∈H\Hn,⌊k−1 2 ⌋,2 depending on whether \|V\| = 2 k−3 2 +2 or \|V\| = 2 k−3 2 +3; and 2. if ui−1,ui+1 ∈Dw ∪Dw′, then ei−1 {ui},ei {ui} ⊆Dw ∪Dw′ ∪I, where I is the unique possible interval u j,u j+1 of size two disjoint with Dw ∪Dw′. Furthermore, if ui is replaceable by either w or w′, then ei−1 {ui},ei {ui} ⊆Dw ∪Dw′. Proof. Note that every u ∈V \ (Dw ∪Dw′) has a neighbor on C in Dw ∪Dw′. Therefore, if v ∈h ∈H \C with v ∈V(H) \V, then Claim 5.2.6 (i) yields h \ {v} ⊆Dw ∪Dw′. So we only have to consider hyperedges h ⊂V. If ui is replaceable by either w or w′ and ui ∈h ∈H \C, then Claim 5.2.7 (i) and (iv) yield h{ui} ⊆Dw ∪Dw′. Finally, if u j,uj+1 form the unique interval of V \ (Dw ∪Dw′), and ui is neither replaceable by w nor by w′, then one of ui−1,ui+1 belong to Dw, the other to Dw′. Suppose that ui,uj ∈h ∈H \C, the CEU eTD Collection 90 Connected Hypergraphs without long Berge-paths other case ui,uj+1 ∈h ∈H \C is symmetric. Then uj−1 ∈Dw∗and ui−1 ∈Dw∗∗for some w∗,w∗∗∈{w,w′}. Therefore w∗,h′,u j−1,ej−2,...,ui+1,ei,ui,h,uj,ej,uj+1,...,ei−2,ui−1,h′′,w∗∗ is either a cycle (if w∗= w∗∗) or a path (if w∗̸= w∗∗) of length k. Such distinct hyperedges h′,h′′ exist from the definition of Dw∗,Dw∗∗as well as they are different from the hyperedge h since h ⊂V. This settles part 1. For part 2, let us consider defining hyperedges ei−1,ei of C with ui−1,ui+1 ∈Dw ∪Dw′. Observe first that all but at most one of the ui’s are replaceable either by w or by w′. If ui is indeed replaceable by w or by w′, then Claim 5.2.7 (iii) yields ei−1{ui},ei{ui} ⊆Dw∪Dw′. For the at most one exception ui, we have that one of ui−1,ui+1 is in Dw, the other one is in Dw′ and by Claim 5.2.7 (v) we are done. 5.2.3 Case-by-case analysis We finish the proof with a case-by-case analysis according to the length of the longest Berge-cycle C and subcases will be defined according to the size of Dw. Let us remind the reader that the length of the cycle C, ℓ−1, might take the values 2 k−3 2 , 2 k−3 2 +1, 2 k−3 2 +2 or 2 k−3 2 +3, and in the last case k is even. In each case we will use the summaries from the previous subsection. CASE I ℓ−1 = 2 k−3 2 . As \|Dw\| ≥ k−3 2 , then by Claim 5.2.6 (i), Dw must consist of every second vertex of V, so there are no missing intervals. Summary 5.2.10 implies H ⊆Hn,⌊k−3 2 ⌋thus \|H\| ≤\|Hn,⌊k−3 2 ⌋\| < \|Hn,⌊k−3 2 ⌋,3\|, which contradicts the assumption on \|H\|. CASE II ℓ−1 = 2 k−3 2 +1. \|Dw\| ≥ k−3 2 and Claim 5.2.6 (i) imply that, after a possible relabelling we have Dw = {u1,u4,...,u2⌊k−3 2 ⌋} and thus {u2,u3} is the only missing interval for w, and all other vertices in V \Dw are in Rw. As all vertices in V \Dw are neighbors to some vertex in Dw, by Summary 5.2.10, all hyperedges in H\C belong to Hn,⌊k−3 2 ⌋,2. To consider the defining hyperedges of C, let us analyze those that contain an ui ∈Rw. Observe that by Claim 5.2.6 (i) a vertex in Dw cannot be a neighbor on C of a vertex in Dw′ for some w′ ∈V(H) \V, so Dw = Dw′ for any two w,w′ ∈V(H) \V. Hence we have that CEU eTD Collection 5.2 Proof of stability 91 ei−1 and ei cannot contain any of u2 and u3 by Claim 5.2.6 (i) applied to the cycle C′ we obtain by Claim 5.2.6 (ii). Therefore by Summary 5.2.10 we have that H∗⊆Hn,⌊k−3 2 ⌋,2, thus \|H\| ≤\|Hn,⌊k−3 2 ⌋,2\|+3 < \|Hn,⌊k−3 2 ⌋,3\|, which contradicts the assumption on \|H\|. CASE III ℓ−1 = 2 k−3 2 +2. The three subcases below cover this case. CASE III/A There exists w ∈V(H)\V with \|Dw\| = k−3 2 +1. Then there is no missing interval for w, and so V \Dw ⊆Rw, so by Summary 5.2.10 we have H = H∗⊆Hn,⌊k−1 2 ⌋. CASE III/B There exists w ∈V(H) \V, for which there are two missing intervals, {ui,ui+1} and {uj,u j+1}. Note that there is no type 1 hyperedge of H\C, as each vertex of the missing intervals is terminal. Observe that all the vertices in V \Dw have neighbors in Dw, therefore the fact that \|Dw′\| ≥ k−3 2 , together with \|Dw\| = k−3 2 and Claim 5.2.6 (i) imply Dw = Dw′ for all w,w′ ∈V(H)\V. This enables us to conclude that - by Claim 5.2.9 (i), there is no hyperedge h ∈H\C of type 2; and - by Claim 5.2.7 (v), if ul ∈Rw, then el−1,el do not contain vertices of missing intervals. So by Summary 5.2.10 we have H∗⊆Hn,⌊k−3 2 ⌋,2,2 and thus \|H\| ≤\|Hn,⌊k−3 2 ⌋,2,2\|+6 < \|Hn,⌊k−3 2 ⌋,3\|, contradicting the assumption on \|H\|. CASE III/C For all w ∈V(H)\V, there is only one missing interval containing three vertices {ui(w),ui(w)+1,ui(w)+2}. If there exist two vertices w,w′ ∈V(H)\V with i(w) ̸= i(w′), then Dw ∪Dw′ must contain every second vertex of C. So by Summary 5.2.11, we have H ⊆Hn,⌊k−1 2 ⌋as claimed by the theorem. So we can assume that the Dws are the same and without loss of generality suppose that for every w ∈V(H)\V, the missing interval is {u1,u2,u3}. Moreover, as every replaceable vertex ui is replaceable by any w ∈V(H)\V, replaceable vertices and defining hyperedges ei−1,ei behave as vertices in V(H)\V and hyperedges in H\C. By Summary 5.2.10 and the above, we have to deal with type 1 hyperedges of H\C and H\H∗= {e2⌊k−3 2 ⌋+2,e1,e2,e3}. CEU eTD Collection 92 Connected Hypergraphs without long Berge-paths • At first suppose that there exists a type 1 hyperedge of H\C, i.e., h ∈H\C with v,u2 ∈h for some v ∈(V(H) \V) ∪Rw. Without loss of generality we may assume v ∈V(H) \V. Then we claim that there is no hyperedge h′ ∈H with u1,u3 ∈h′. Suppose by a contradiction that such h′ exists, then observe that h′ ̸= h, as otherwise we would have v,u1,u3 ∈h′ that is not possible by Summary 5.2.10. Also, either h′ / ∈{e1,e3} or h′ / ∈{e2⌊k−3 2 ⌋+2,e2}, so we may assume h′ / ∈{e1,e3} without loss of generality. Since u2⌊k−3 2 ⌋+2 ∈Dv, there is a hyperedge h′′ different from the hyperedges h and h′, incident to the vertices v and u2⌊k−3 2 ⌋+2. We have a contradiction since the following is a longer Berge-cycle than C, containing all defining vertices of C and v: v,h,u2,e1,u1,h′,u3,e3,u4,··· ,u2⌊k−3 2 ⌋+2,h′′. As no hyperedge contains both u1 and u3, we obtained H ⊆Hn,⌊k−1 2 ⌋in this case. • Suppose next that there is no type 1 hyperedge of H \C, i.e., by Summary 5.2.10, we have H∗⊆Hn,⌊k−3 2 ⌋,3. Observe that e2⌊k−3 2 ⌋+2 and e3 do not contain vertices from (V(H)\V)∪Rw by Claim 5.2.8 (i) and (ii). If the same holds for e1,e2, then H ⊆Hn,2⌊k−3 2 ⌋,3 contradicting the assumption \|H\| > \|Hn,2⌊k−3 2 ⌋,3\|. So we can assume that e2 contains a vertex v ∈(V(H)\V)∪Rw. Then we claim that there is no hyperedge h ∈H\C with u1,u3 ∈h. In here we get a contradiction as in the previous settings with a longer Berge-cycle, therefore we omit the proof. We obtained the following contradiction \|H\| ≤2+\|Hn,⌊k−3 2 ⌋,3\|− k−3 2 r −2 < \|Hn,⌊k−3 2 ⌋,3\|. CASE IV ℓ−1 = 2 k−3 2 +3. Note that in this case k is even and the length of C is k −1. We again distinguish several subcases. CASE IV/A \|Dw\| = k−1 2 . Then as Dw does not contain neighboring vertices on C, after relabelling, we can suppose that we have Dw = {u1,u4,u6,...,uk−2}. So there is one missing interval {u2,u3}, therefore there does not exist a type 1 or type 2 hyperedge h ∈H\C. If ui ∈Rw, then by Claim 5.2.7 (iii) ei−1 and ei do not contain vertices from V(H)\V. We claim that ei−1 and ei do not contain vertices from the missing interval {u2,u3}. Indeed, if there exists w∗̸= w with u1 ∈NH\C(w∗) and u2 ∈ei or ei−1, then the following is a Berge-path of length k: ui,ei (or ei−1), u2,e2,u3,...,ui−1,h,w,h′,ui+1,ei+1,...,uk−1,ek−1,u1,h′′,w∗. CEU eTD Collection 5.2 Proof of stability 93 Here h and h′ exist and are distinct as ui is in Rw and h′′ exists by the choice of w∗. Similarly, if there exists w∗∗̸= w with u4 ∈NH\C(w∗), then ei−1,ei cannot contain u3. As all Dw∗is of size at least k−3 2 , the only cases when we are not yet done is when u1 / ∈NH\C(w∗) and Dw∗= {u4,u6,...,u2⌊k−3 2 ⌋+2} or u4 / ∈NH\C(w∗) and Dw∗= {u6,u8,...,u2⌊k−3 2 ⌋+2,u1}. By symmetry, we can assume the first. But then any replaceable ui but u2⌊k−3 2 ⌋+3 can be replaced with some w∗̸= w, and the above arguments applied to the new cycle C′ show that any ui ∈h ∈H \C′ (in particular, it applies to ei and ei−1!) cannot contain u3, and by Summary 5.2.10, we already know that ei−1,ei cannot contain u2⌊k−3 2 ⌋+3. Therefore setting A = Dw {u1}, B1 = {u2⌊k−3 2 ⌋+3,u1,u2,u3} we have that H is a subfamily of Hn,⌊k−3 2 ⌋,4 apart from e2⌊k−3 2 ⌋+2,e2⌊k−3 2 ⌋+3,e1,e2,e3 and the hyperedges containing both w and u1. On the other hand, there cannot exist h ∈H \C with u2⌊k−3 2 ⌋+3,u2 ∈h nor with u2⌊k−3 2 ⌋+3,u3 ∈h as in the former case w,h′,u1,e2⌊k−3 2 ⌋+3,u2⌊k−3 2 ⌋+3,h,u2,e2,...,u2⌊k−3 2 ⌋+2,h′′,w∗, while in the latter case w,h′,u1,e1,u2,e2,u3,h,u2⌊k−3 2 ⌋+3,e2⌊k−3 2 ⌋+2,u2⌊k−3 2 ⌋+2,...,e4,u4,h′′,w∗ is a Berge-path of length k. So we have \|H\| ≤\|Hn,⌊k−3 2 ⌋,4\|+5+ k−3 2 r −2 −2 k−3 2 r −2 < \|Hn,⌊k−3 2 ⌋,4\|, contradicting the assumption on \|H\|. So we obtained that ei−1,ei cannot contain u2,u3 and thus so far by Summary 5.2.10 we have H∗⊆Hn,⌊k−1 2 ⌋,2. Now let us concentrate on the hyperedges in H\H∗. So {u2,u3} is the unique missing interval (all other vertices of V \Dw are in Rw), and thus H\H∗contains three hyperedges: e1,e2 and e3. Observe that by Claim 5.2.8 (i), e1 and e3 do not contain any w′ ∈V(H)\V. By Claim 5.2.7 (v), e1 and e3 do not contain any vertex in Rw. • If e2 does not contain any vertex in Rw ∪(V(H) \V), then we are done, since H ⊆ Hn,⌊k−1 2 ⌋,2. • If e2 does contain a vertex from Rw ∪(V(H)\V), then there does not exist any other hyperedge h that contains both u2 and u3. Indeed, if e2 contained w, then w could be inserted in between u2 and u3 in the Berge-cycle C to form a longer cycle than C, a contradiction. If e2 contains some w′ ̸= w from V(H)\V, then we can reach a contradiction as before: we CEU eTD Collection 94 Connected Hypergraphs without long Berge-paths would find a Berge-path of length k starting with w′,e2,u2,h,u3, then going through C and ending with u1,h′,w as u1 ∈Dw. Finally, if e2 contains a replaceable ui, then at least one of u1,u4 belongs to Dw′ for some w′ ∈V(H) \V with w′ ̸= w, since Dw′ ⊆Dw from Claim 5.2.6 (i) and \|Dw \ Dw′\| ≤1. By symmetry, we may assume that u1 ∈Dw′. Then we have a contradiction since the following Berge-path has length k. The Berge-path is ui,e2,u2,h,u3,u4,... that goes around the cycle C, replaces ui by w and finishes with u1,hw′,w′, such hw′ exists from the definition of Dw′. Therefore, if e2 does contain a vertex from Rw ∪(V(H) \V), then there does not exist any other hyperedge h that contains both u2 and u3. Hence, H ⊆H+ n,⌊k−1 2 ⌋with A = Dw, L = V(H)\Dw and e2 being the unique hyperedge of H+ n,⌊k−1 2 ⌋that contains less than r −1 vertices of A. CASE IV/B For all w′ ∈V(H)\V, we have \|D′ w\| = \|Dw\| = k−3 2 . As the length of C is k −1, k is even and vertices of Dw are not neighbors on C, we have at most three missing intervals. If there are three missing intervals, then each of them contains two vertices. If there are two missing intervals, then they contain two and three vertices and if there is only one missing interval, then it contains 4 vertices. According to this structure, we are going to consider the following three subcases. -CASE IV/B/1 There exists w ∈V(H)\V with V \Dw containing 3 intervals of length 2. Observe that as all the missing intervals are of size 2, we do not have type 1 hyperedges h ∈H \C. As all vertices in V \ Dw have neighbors in Dw, we obtain that for any w′ ∈ V(H)\V we have Dw = Dw′. So Claim 5.2.9 (i) implies that there does not exist any type 2 hyperedges h ∈H\C. Finally, Claim 5.2.7 (v) implies that defining hyperedges of C, apart from those in H\H∗, are in Hn,⌊k−3 2 ⌋,2,2,2. So we obtained a contradiction as \|H\| ≤9+\|Hn,⌊k−3 2 ⌋,2,2,2\| < \|Hn,⌊k−3 2 ⌋,4\|. CASE IV/B/2 For all w ∈V(H)\V, the number of missing intervals is at most 2 and there exist w,w′ ∈V(H)\V with Dw ̸= Dw′. By relabeling, we can assume that {u2,u3} forms the unique missing interval for both w and w′, i.e., the unique interval of length more than 1 in V \ (Dw ∪Dw′). According to Summary 5.2.11, if every ui / ∈Dw ∪Dw′ ∪{u2,u3} is replaceable, then we have H \ {e1,e2,e3} ⊆Hn,⌊k−1 2 ⌋,2, while if there is ui ∈V (Dw ∪Dw′) (i ̸= 2,3) that is not in Rw ∪Rw′, then we know ei−1 {ui},ei {ui} ⊆Dw ∪Dw′ ∪{u2,u3}. CEU eTD Collection 5.2 Proof of stability 95 • At first we suppose that there exists a u ∈Dw ∪Dw′ such that \|{w∗∈(V(H) \V) ∪ Rw ∪Rw′ : u ∈NH\C(w∗)}\| = 1. In that case the unique w∗must be either w or w′, say w. Consider the hypergraph H \Hn,⌊k−3 2 ⌋,3 with Hn,⌊k−3 2 ⌋,3 having A = Dw ∪Dw′ {u} = Dw′ and B1 = {u,u2,u3}. Then, by Summary 5.2.11, the hyperedges left are incident with the vertex u, thus the number of hyperedges is at most ⌊k−3 2 ⌋ r−2 +5. Here the first term is an upper bound for those hyperedges that are incident with both u and w, while the second term is 5 for {ei−1,ei,e1,e2,e3}. So we have a contradiction as \|H\| ≤ k−3 2 r −2 +5+\|Hn,⌊k−3 2 ⌋,3\| < \|Hn,⌊k−3 2 ⌋,4\|. • Suppose now that for all u ∈Dw ∪Dw′, \|{w∗∈(V(H)\V)∪Rw ∪Rw′ : u ∈NH\C(w∗)}\| ≥2. At first we show that u2,u3 / ∈ei−1,ei if ui ∈V (Dw ∪Dw′) (i ̸= 2,3). This holds by Summary 5.2.11, if ui is replaceable by either w or w′. Therefore without loss of generality we may assume ui+1 ∈Dw \ Dw′ and ui−1 ∈Dw′ \ Dw. Note that Dw = (Dw ∪Dw′) \ {ui−1} and Dw′ = (Dw ∪Dw′){ui+1}. Because of symmetry, it is enough to show a contradiction only if u2 ∈ei, the three remaining cases are similar to this one. The following is a Berge-path of length k ui,ei,u2,e2,u3,e3,...,ui−1,h,w′,h′,u1,ek−1,uk−1,ek−2 ...,ei+1,ui+1,h′′,w, a contradiction. The hyperedges h,h′,h′′ can be chosen distinct as u1,ui−1 ∈Dw′ and ui+1 ∈ Dw and by Lemma 5.2.3 (i), h∗∈H\C cannot contain distinct vertices from outside V. By Claim 5.2.8 (i) and (ii), e1 and e3 are not incident with vertices in V(H) \V or in Rw ∪Rw′. Even more, they are not incident with ui either, since otherwise if ui ∈e1, the following path is of length k, a contradiction: ui,e1,u2,e2,u3,e3,...,ui−1,h,w′,h′,u1,ek−1,uk−1,...,ei+1,ui+1,h′′,w. An analogous argument shows ui / ∈e3. Finally, if e2 contains any vertex from V(H) \ (Dw ∪Dw′), then similarly to previous cases a hyperedge e2 ̸= h ∈H containing both u2,u3 would lead to a Berge-path of length k. So if no such hyperedge h exists, then H ⊆Hn,⌊k−1 2 ⌋,2. Otherwise, we have H ⊆H+ n,⌊k−1 2 ⌋. Both possibilities are as claimed by the theorem. CEU eTD Collection 96 Connected Hypergraphs without long Berge-paths CASE IV/B/3 For all w′ ∈V(H)\V ∪Rw, the number of missing intervals is at most 2 and Dw = Dw′. As Dw = Dw′ for all w,w′ ∈V(H) \V, it follows that we do not have to distinguish between vertices in V(H)\V and vertices in Rw. Also, anything that we prove for hyperedges h ∈H\C is valid for all ei,ei−1 if ui ∈Rw, by Claim 5.2.6 (ii). CASE IV/B/3/1 Let us consider first the case when for every v ∈V(H) \V ∪Rw, the missing intervals for v are {u2,u3,u4} and {ui,ui+1} for some 6 ≤i ≤k −2, after possible relabeling. By Summary 5.2.10 and Claim 5.2.9 (i), we need to consider the 7 hyperedges in H \H∗, the hyperedges in H\C containing u3,ui or u3,ui+1 and the hyperedges in H\C containing u3 and some v ∈V(H)\V ∪Rw. • If there are no hyperedges in H \C containing u3,ui or u3,ui+1 or u3 and some v ∈ V(H) \V ∪Rw, then H∗⊆Hn,⌊k−3 2 ⌋,3,2, with embedding A = Dw, B1 = {u2,u3,u4}, B2 = {ui,ui+1} and \|H\| ≤\|H∗\|+7 ≤\|Hn,⌊k−3 2 ⌋,3,2\|+7 < \|Hn,⌊k−3 2 ⌋,4\|, contradicting the assumption on \|H\|. • If there are no hyperedges in H \C containing u3 and some v ∈V(H) \V ∪Rw, but there exist a hyperedge h ∈H \C containing u3,ui or u3,ui+1, then by Claim 5.2.9 (ii), there is no hyperedge containing both u2 and u4. In particular, with embedding A = Dw, B1 = {u2,u3,u4}, B2 = {ui,ui+1} we have \|Hn,⌊k−3 2 ⌋,3,2 \H\| ≥ ⌊k−3 2 ⌋ r−2 . Also, by Summary 5.2.10, the hypergraph H\Hn,⌊k−3 2 ⌋,3,2 may contain the 7 hyperedges of H\H∗and at most 2 ⌊k−3 2 ⌋ r−2 + ⌊k−3 2 ⌋ r−3 hyperedges containing ui or/and ui+1 and u3. So we have \|H\| ≤\|Hn,⌊k−3 2 ⌋,3,2\|+7+2 k−3 2 r −2 + k−3 2 r −3 − k−3 2 r −2 < \|Hn,⌊k−3 2 ⌋,4\|, which contradicts the assumption on \|H\|. • Suppose that there is a hyperedge h ∈H\C containing u3 and some v ∈V(H)\V ∪Rw. There is no h′ ∈H\C incident with u2 and u4. Indeed, otherwise v,u3,e2,u2,h′,u4,e4,...,u1,hw,w is a Berge-path of length k, a contradiction. By the above, Summary 5.2.10 and Claim 5.2.9 (i), we have that H∗⊆Hn,⌊k−1 2 ⌋,2 with embedding A = Dw ∪{u3}, B1 = {ui,ui+1}. Even more, since Dv = Dw ∋u1,u5, by Lemma 5.2.3 (iii) there exist cycles C2,C4 with v replacing u2 and u4, respectively. Observe that the set Dw∗does not change when we apply these changes from C to C2 and C to C4. In C2, CEU eTD Collection 5.2 Proof of stability 97 e1,e2 are not defining hyperedges, while in C4, e3,e4 are not defining hyperedges. Therefore, applying Lemma 5.2.3 (ii), we obtain that e1,e2 do not contain u4,ui,ui+1 and e3,e4 do not contain u2,ui,ui+1. Hence hyperedges e1,e2,e3,e4 are also from Hn,⌊k−1 2 ⌋,2 by Summary 5.2.10. By Claim 5.2.8 (i) and Claim 5.2.9 (i), we have that the hyperedges ei−1 and ei+1 are also from Hn,⌊k−1 2 ⌋,2. Finally, if ei does not contain any vertex from (V(H)\V)∪Rw ∪{u3}, then we have H ⊆Hn,⌊k−1 2 ⌋,2. Otherwise, as in Case IV/A, one can see that there does not exist h ̸= ei with ui,ui+1 ∈h and thus H = H+ n,⌊k−1 2 ⌋with A = Dw ∪{u3} and ei being the unique hyperedge with less than r −1 elements in A. CASE IV/B/3/2 For all v ∈V(H)\V ∪Rw, after possible relabelling the only missing interval consists of {u2,u3,u4,u5}. By Summary 5.2.10, we need to handle hyperedges e1,e2,e3,e4,e5 and those h ∈H\C that contain a v ∈V(H)\V ∪Rw and u3 and/or u4. • If there are no such hyperedges and e1,e2,e3,e4,e5 ⊆Dw ∪{u2,u3,u4,u5}, then H ⊆ Hn,⌊k−3 2 ⌋,4 contradicting the assumption on \|H\|. • Suppose next there is no h ∈H\C with a vertex from V(H)\V ∪Rw containing u3 or u4, but some ei (i = 1,2,3,4,5) does contain a vertex from outside V. By Claim 5.2.8 (i), it is neither e1 nor e5. If ei contains a vertex v from outside V, then there cannot exist h ∈H\C with u2,ui+1 ∈h, as then v,ei,ui,ei−1,...,u2,h,ui+1,ei+1,ui+2,ei+2,...,uk−1,ek−1,u1,h′ is a Berge-cycle of length k. For the existence of h′ we used Dv = Dw ∋u1. Therefore we have \|H\| ≤3+\|Hn,⌊k−3 2 ⌋,4\|− k−3 2 r −2 < \|Hn,⌊k−3 2 ⌋,4\|, contradicting the assumption on \|H\|. • If there exists a hyperedge h ∈H\C incident with some vertex v ∈V(H)\V ∪Rw and u3, then there is no h′ ̸= h, h′ ∈H \C incident with some vertex from V(H) \V ∪Rw and u4, by Claim 5.2.6 (i). Even more, there is no h′′ ∈H\C with u2,u4 ∈h′′. The argument is the same as if e3 contained v from the previous bullet. Similarly one can get that there is no hyperedge h′′ ∈H\C with u2,u5 ∈h′′. Observe that there should exist at least two distinct v1,v2 ∈V(H) \V ∪Rw for which hyperedges hv1,hv2 with v1,u3 ∈hv1 and v2,u3 ∈hv2 exist. Indeed, otherwise using that there is no non-defining edge incident to u2,u4, we have \|H\| ≤5+1+\|Hn,⌊k−3 2 ⌋,4\|− k−3 2 r −2 < \|Hn,⌊k−3 2 ⌋,4\|. CEU eTD Collection 98 Connected Hypergraphs without long Berge-paths We will show that either H ⊆Hn,⌊k−1 2 ⌋,2 or H ⊆H+ n,⌊k−1 2 ⌋with A = Dw ∪{u3} and B1 = {u4,u5}. Let w∗denote an arbitrary vertex w∗∈V(H)\V with w∗̸= v1,v2. We will use that u1,u6 ∈Dw = Dw∗= Dv1 = Dv2 thus there exists a hyperedge that is not a defining hyperedge of C and is different from hv1 and hv2, containing either u1 or u6 together with v1 or v2 or w∗. We need to prove that u4,u5 / ∈e1,e2 and u2 / ∈e3,e5. In each of the cases we present a Berge-path of length k below, which is a contradiction. If u4 ∈e1, then the path is v1,hv1,u3,e2,u2,e1,u4,e4,u5,...,uk−1,ek−1,u1,h,w∗. If u4 ∈e2, then the path is u2,e2,u4,e4,u5,e5,...,uk−1,ek−1,u1,h,v1,hv1,u3,hv2,v2. If u5 ∈e1 or e2 , then the path is u2,e1 or e2,u5,e4,u4,e3,u3,hv1,v1,h,u6,e6,...,uk−1,ek−1,u1,h′,w∗. If u2 ∈e3, then the path is v1,hv1,u3,e2,u2,e3,u4,e4,u5,...,uk−1,ek−1,u1,h,w∗. If u2 ∈e5, then the path is u2,e5,u5,e4,u4,e3,u3,hv1,v1,h,u6,e6,...,uk−1,ek−1,u1,h′,w∗. From here, one can conclude to H ⊆Hn,⌊k−1 2 ⌋,2 or H ⊆H+ n,⌊k−1 2 ⌋as in Case IV/A, depending on whether e4 ⊆Dw ∪{u3,u4,u5} or not. The above case-by-case analysis concludes the proof of Theorem 1.9.3 under the set degree condition, i.e., for any set X of vertices with \|X\| ≤k/2 the number of hyperedges incident with some vertex in X, \|E(X)\|, is at least \|X\| ⌊k−3 2 ⌋ r−1 . Let n′ k,r denote the threshold such that the statement of Theorem 6 holds for hypergraphs with the set degree condition if n ≥n′ k,r. We are now ready to prove the general statements. CEU eTD Collection 5.2 Proof of stability 99 Proof of Theorem 1.9.3 and Theorem 1.9.4. Let H be a connected n-vertex r-uniform hyper-graph without a Berge-path of length k, and suppose that if k is odd, then \|H\| > \|Hn,⌊k−3 2 ⌋,3\| = n−k +3 2 ⌊k−3 2 ⌋ r −1 + ⌊k+3 2 ⌋ r , while if k is even, then \|H\| > \|Hn,⌊k−3 2 ⌋,4\| = n− k +5 2 ⌊k−3 2 ⌋ r −1 + ⌊k+5 2 ⌋ r . We obtain a subhypergraph H′ of H using a standard greedy process: as long as there exists a set S of vertices with \|S\| ≤k/2 such that \|E(S)\| < \|S\| ⌊k−3 2 ⌋ r−1 , we remove S from H and all hyperedges in E(S). Let H′ denote the subhypergraph at the end of this process. Claim 5.2.12. There exists a threshold n′′ k,r, such that if \|V(H)\| ≥n′′ k,r, then H′ is connected and contains at least n′ k,r vertices. Proof. To see that H′ is connected, observe that every component of H′ possesses the set degree condition. Therefore Claim 5.2.5 yields that every component contains a cycle of length at least k −4. Therefore, as H is connected, H contains a Berge-path with at least 2k −8 vertices from two different components of H′, a contradiction as k ≥9. Suppose to the contrary that H′ has less than n′ k,r vertices. Observe that, by definition of the process, \|E(H′)\|−\|V(H′)\| ⌊k−3 2 ⌋ r−1 strictly increases at every removal of some set X of at most k vertices. Therefore if n > n′ k,r +k n′ k,r r = n′′ k,r and \|V(H′)\| < n′ k,r, then at the end we would have more hyperedges than those in the complete r-uniform hypergraph on \|v(H′)\| vertices, a contradiction. By Claim 5.2.12 and the statement for hypergraphs with the set degree property, we know that H′ has n1 ≥n′ k,r vertices, and H′ ⊆Hn1,⌊k−1 2 ⌋if k is odd, and H′ ⊆Hn1,⌊k−1 2 ⌋,2 or H+ n1,⌊k−1 2 ⌋if k is even. Then for any hyperedge h ∈E(H)\E(H′) that contain at least one vertex from V(H)\V(H′) with degree at least two, we can apply Lemma 5.2.3 (i) to obtain that all such h must meet the A of H′ in r−1 vertices. This shows that if the minimum degree of H is at least 2, then H ⊆Hn2,⌊k−1 2 ⌋if k is odd, and H ⊆Hn2,⌊k−1 2 ⌋,2 or H ⊆H+ n2,⌊k−1 2 ⌋if k is even, where n2 ≤n is the number of vertices that are contained in a hyperedge of H that is either in H′ or has a vertex in V(H)\V(H′) with degree at least 2. This finishes the proof of Theorem 1.9.3. 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On some problems in graph theory, combinatorial analysis and combinatorial number theory. Graph Theory and Combinatorics (Cambridge, 1983), Academic Press, London, pages 1–17. Erd˝ os, P., Rényi, A., and Sós, V. T. (1966). On a problem of graph theory. Studia Sci. Math. Hungar, 1:215–235. Erd˝ os, P. and Stone, A. (1946). On the structure of linear graphs. Bull. Amer. Math. Soc, 52(1087-1091):1. Erd˝ os, P. (1938). On sequences of integers no one of which divides the product of two others and on some related problems. Inst. Math. Mech. Univ. Tomsk, 2:74–82. Erd˝ os, P. (1964). Extremal problems in graph theory. In Theory of graphs and its applications. Smolenice Publishing House of the Czechoslovak Academy of Science, Prague. Erd˝ os, P. and Gallai, T. (1959). On maximal paths and circuits of graphs. Acta Mathematica Academiae Scientiarum Hungarica, 10(3-4):337–356. CEU eTD Collection References 103 Erd˝ os, P., Gy˝ ori, E., and Simonovits, M. (1992). How many edges should be deleted to make a triangle-free graph bipartite. In Sets, graphs and numbers, Colloq. Math. Soc. János Bolyai, volume 60, pages 239–263. Erd˝ os, P. and Simonovits, M. (1965). A limit theorem in graph theory. In Studia Sci. Math. Hung. Citeseer. Erd˝ os, P. and Simonovits, M. (1982). Compactness results in extremal graph theory. Combinatorica, 2(3):275–288. Ergemlidze, B., Gy˝ ori, E., and Methuku, A. (2019a). Asymptotics for Turán numbers of cycles in 3-uniform linear hypergraphs. Journal of Combinatorial Theory, Series A, 163:163–181. Ergemlidze, B., Gy˝ ori, E., Methuku, A., and Salia, N. (2019b). A note on the maximum number of triangles in a C5-free graph. Journal of Graph Theory, 90(3):227–230. Ergemlidze, B., Gy˝ ori, E., Methuku, A., Salia, N., and Tompkins, C. (2020a). On 3-uniform hypergraphs avoiding a cycle of length four. arXiv preprint arXiv:2008.11372. Ergemlidze, B., Gy˝ ori, E., Methuku, A., Salia, N., Tompkins, C., and Zamora, O. (2020b). Avoiding long Berge cycles: the missing cases k = r +1 and k = r +2. Combi-natorics, Probability and Computing, 29(3):423–435. Ergemlidze, B. and Methuku, A. (2018). Triangles in C5-free graphs and hypergraphs of girth six. arXiv preprint arXiv:1811.11873. Euler, L. (1758). Elementa doctrinae solidorum. Novi commentarii academiae scien-tiarum Petropolitanae, pages 109–140. Faudree, R. J. and Schelp, R. H. (1975). Path Ramsey numbers in multicolorings. Journal of Combinatorial Theory, Series B, 19(2):150–160. Frankl, P. and Füredi, Z. (1987). Exact solution of some turán-type problems. Journal of Combinatorial Theory, Series A, 45(2):226–262. Frankl, P. and Rödl, V. (1986). Large triangle-free subgraphs in graphs without K4. Graphs and Combinatorics, 2(1):135–144. Füredi, Z. (1996). On the number of edges of quadrilateral-free graphs. Journal of Combinatorial Theory, Series B, 68(1):1–6. Füredi, Z. (2014). Linear trees in uniform hypergraphs. European Journal of Combina-torics, 35:264–272. Füredi, Z. and Gunderson, D. (2015). Extremal numbers for odd cycles. Combinatorics, Probability and Computing, 24(4):641–645. Füredi, Z., Kostochka, A., and Luo, R. (2018). Avoiding long Berge cycles ii, exact bounds for all n. arXiv preprint arXiv:1807.06119. Füredi, Z., Kostochka, A., and Luo, R. (2019a). Avoiding long Berge cycles. Journal of Combinatorial Theory, Series B, 137:55–64. CEU eTD Collection 104 References Füredi, Z., Kostochka, A., and Luo, R. (2019b). On 2-connected hypergraphs with no long cycles. arXiv preprint arXiv:1901.11159. Füredi, Z., Kostochka, A., and Verstraëte, J. (2016). Stability in the Erd˝ os–Gallai theorems on cycles and paths. Journal of Combinatorial Theory, Series B, 121:197–228. Füredi, Z. and Özkahya, L. (2017). On 3-uniform hypergraphs without a cycle of a given length. Discrete Applied Mathematics, 216:582–588. Füredi, Z. and Simonovits, M. (2013). The history of degenerate (bipartite) extremal graph problems. In Erd˝ os Centennial, pages 169–264. Springer. Gerbner, D. (2019). On Berge-Ramsey problems. arXiv preprint arXiv:1906.02465. Gerbner, D., Gy˝ ori, E., Methuku, A., and Vizer, M. (2017). Generalized Turán problems for even cycles. arXiv preprint arXiv:1712.07079. Gerbner, D., Methuku, A., Omidi, G., and Vizer, M. (2020a). Ramsey problems for Berge hypergraphs. SIAM Journal on Discrete Mathematics, 34(1):351–369. Gerbner, D., Methuku, A., and Palmer, C. (2020b). General lemmas for Berge–turán hypergraph problems. European Journal of Combinatorics, 86:103082. Gerbner, D., Nagy, D., Patkós, B., Salia, N., and Vizer, M. (2020c). Stability of extremal connected hypergraphs avoiding Berge-paths. arXiv preprint arXiv:2008.02780. Gerbner, D. and Palmer, C. (2017). Extremal results for Berge hypergraphs. SIAM Journal on Discrete Mathematics, 31(4):2314–2327. Gerbner, D. and Palmer, C. (2019). Counting copies of a fixed subgraph in F-free graphs. European Journal of Combinatorics, 82:103001. Gerbner, D. and Palmer, C. (2020). Some exact results for generalized Turán problems. arXiv preprint arXiv:2006.03756. Grósz, D., Methuku, A., and Tompkins, C. (2020). Uniformity thresholds for the asymptotic size of extremal Berge-f-free hypergraphs. European Journal of Combinatorics, page 103109. Grzesik, A. (2012). On the maximum number of five-cycles in a triangle-free graph. Journal of Combinatorial Theory, Series B, 102(5):1061–1066. Gyárfás, A. (2019). The turán number of Berge K4 in triple systems. SIAM Journal on Discrete Mathematics, 33(1):383–392. Gyárfás, A., Lehel, J., Sárközy, G. N., and Schelp, R. H. (2008). Monochromatic Hamil-tonian Berge-cycles in colored complete uniform hypergraphs. Journal of Combinatorial Theory, Series B, 98(2):342–358. Gyárfás, A., Rousseau, C. C., and Schelp, R. H. (1984). An extremal problem for paths in bipartite graphs. Journal of graph theory, 8(1):83–95. CEU eTD Collection References 105 Gyárfás, A. and Sárközy, G. N. (2011). The 3-colour Ramsey number of a 3-uniform Berge cycle. Combinatorics, Probability & Computing, 20(1):53. Gy˝ ori, E. (2006). Triangle-free hypergraphs. Combinatorics, Probability and Comput-ing, 15(1-2):185–191. Gy˝ ori, E. (2021). Personal communication. Gy˝ ori, E., Katona, G. Y., and Lemons, N. (2016). Hypergraph extensions of the Erd˝ os-Gallai theorem. European Journal of Combinatorics, 58:238–246. Gy˝ ori, E. and Lemons, N. (2009). Hypergraphs with no odd cycle of given length. Electronic Notes in Discrete Mathematics, 34:359–362. Gy˝ ori, E. and Lemons, N. (2012a). 3-uniform hypergraphs avoiding a given odd cycle. Combinatorica, 32(2):187–203. Gy˝ ori, E. and Lemons, N. (2012b). Hypergraphs with no cycle of a given length. Combinatorics, Probability & Computing, 21(1-2):193. Gy˝ ori, E., Lemons, N., Salia, N., and Zamora, O. (2020). The structure of hypergraphs without long Berge cycles. Journal of Combinatorial Theory, Series B. Gy˝ ori, E. and Li, H. (2012). The maximum number of triangles in C2k+1-free graphs. Combinatorics, Probability & Computing, 21(1-2):187. Gy˝ ori, E., Methuku, A., Salia, N., Tompkins, C., and Vizer, M. (2018a). On the maxi-mum size of connected hypergraphs without a path of given length. Discrete Mathematics, 341(9):2602–2605. Gy˝ ori, E., Pach, J., and Simonovits, M. (1991). On the maximal number of certain subgraphs in Kr-free graphs. Graphs and Combinatorics, 7(1):31–37. Gy˝ ori, E., Salia, N., Tompkins, C., and Zamora, O. (2018b). The maximum number of P ℓcopies in P k-free graphs. Discrete Mathematics & Theoretical Computer Science, 21(1). Gy˝ ori, E., Salia, N., Tompkins, C., and Zamora, O. (2019). Turán numbers of Berge trees. arXiv preprint arXiv:1904.06728. Gy˝ ori, E., Paulos, A., Salia, N., Tompkins, C., and Zamora, O. (2020). Generalized planar Turán numbers. Gy˝ ori, E., Salia, N., and Zamora, O. (2021). Connected hypergraphs without long berge-paths. European Journal of Combinatorics, 96:103353. Hall, P. (2009). On representatives of subsets. In Classic Papers in Combinatorics, pages 58–62. Springer. Hanson, D. and Toft, B. (1991). k-saturated graphs of chromatic number at least k. Ars Combinatoria, 31:159–164. CEU eTD Collection 106 References Hatami, H., Hladk y, J., Král, D., Norine, S., and Razborov, A. (2013). On the number of pentagons in triangle-free graphs. Journal of Combinatorial Theory, Series A, 120(3):722– 732. Jackson, B. (1981). Cycles in bipartite graphs. Journal of Combinatorial Theory, Series B, 30(3):332–342. Jensen, J. L. W. V. et al. (1906). Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta mathematica, 30:175–193. Kopylov, G. (1977). On maximal paths and cycles in a graph. In Doklady Akademii Nauk, volume 234, pages 19–21. Russian Academy of Sciences. Korándi, D., Roberts, A., and Scott, A. (2020). Exact stability for Turán’s theorem. arXiv preprint arXiv:2004.10685. Kostochka, A., Lavrov, M., Luo, R., and Zirlin, D. (2020a). Conditions for a bigraph to be super-cyclic. arXiv preprint arXiv:2006.15730. Kostochka, A., Lavrov, M., Luo, R., and Zirlin, D. (2020b). Longest cycles in 3-connected hypergraphs and bipartite graphs. arXiv preprint arXiv:2004.08291. Kostochka, A. and Luo, R. (2020). On r-uniform hypergraphs with circumference less than r. Discrete Applied Mathematics, 276:69–91. Kostochka, A., Luo, R., and Zirlin, D. (2020c). Super-pancyclic hypergraphs and bipartite graphs. Journal of Combinatorial Theory, Series B, 145:450–465. K˝ ovári, P., Sós, V. T., and Turán, P. (1954). On a problem of Zarankiewicz. In Colloquium Mathematicum, volume 3, pages 50–57. Polska Akademia Nauk. Krivelevich, M., Kronenberg, G., and Mond, A. (2019). Turán-type problems for long cycles in random and pseudo-random graphs. arXiv preprint arXiv:1911.08539. Lazebnik, F. and Verstraëte, J. (2003). On hypergraphs of girth five. the electronic journal of combinatorics, 10(1):R25. Letzter, S. (2019). Many H-copies in graphs with a forbidden tree. SIAM Journal on Discrete Mathematics, 33(4):2360–2368. Lu, L. and Wang, Z. (2020). On the cover Ramsey number of Berge hypergraphs. Discrete Mathematics, 343(9):111972. Luo, R. (2018). The maximum number of cliques in graphs without long cycles. Journal of Combinatorial Theory, Series B, 128:219–226. Magnant, C. (2019). Colored complete hypergraphs containing no rainbow Berge triangles. Theory and Applications of Graphs, 6(2):1. Maherani, L. and Shahsiah, M. (2018). Turán numbers of complete 3-uniform Berge-hypergraphs. Graphs and Combinatorics, 34(4):619–632. Mantel, W. (1907). Problem 28. Wiskundige Opgaven, 10(60-61):320. CEU eTD Collection References 107 McLennan, A. (2005). The Erd˝ os-Sós conjecture for trees of diameter four. Journal of Graph Theory, 49(4):291–301. Mubayi, D. and Suk, A. (2020). A survey of hypergraph Ramsey problems. In Discrete Mathematics and Applications, pages 405–428. Springer. Palmer, C., Tait, M., Timmons, C., and Wagner, A. Z. (2019). Turán numbers for Berge-hypergraphs and related extremal problems. Discrete Mathematics, 342(6):1553– 1563. Ramsey, F. P. (1930). On a problem of formal logic. Proceedings of the London Mathematical Society, 2(1):264–286. Salia, N., Tompkins, C., Wang, Z., and Zamora, O. (2018). Ramsey numbers of Berge-hypergraphs and related structures. arXiv preprint arXiv:1808.09863. Salia, N., Tompkins, C., and Zamora, O. (2019). An Erd˝ os-Gallai type theorem for vertex colored graphs. Graphs and Combinatorics, 35(3):689–694. Samotij, W. (2014). Stability results for random discrete structures. Random Structures & Algorithms, 44(3):269–289. Simonovits, M. (1968). A method for solving extremal problems in graph theory, stability problems. In Theory of Graphs (Proc. Colloq., Tihany, 1966), pages 279–319. Simonovits, M. (1969). Extrém gráfok struktúrájáról (on the structure of extremal graphs, in hungarian). CSc Thesis, Eötvös Loránd University, Budapest. Solymosi, J. (2019). Personal communication. Sudakov, B. and Verstraëte, J. (2017). The extremal function for cycles of length ℓ mod k. The Electronic Journal of Combinatorics, pages P1–7. Swanepoel, K. J. (2009). Unit distances and diameters in Euclidean spaces. Discrete & Computational Geometry, 41(1):1–27. Tuite, J., Erskine, G., and Salia, N. (2021). Turán problems for k-geodetic digraphs. Turán, P. (1941). On an external problem in graph theory. Mat. Fiz. Lapok, 48:436– 452. Woodall, D. (1976). Maximal circuits of graphs. i. Acta Mathematica Academiae Scientiarum Hungarica, 28(1-2):77–80. Zarankiewicz, K. (1951). Problem p 101. In Colloq. Math, volume 2, page 5. Zykov, A. A. (1949). On some properties of linear complexes. Matematicheskii sbornik, 66(2):163–188. CEU eTD Collection CEU eTD Collection Index (s,k −1)-block tree, 15 Gn,k,s, 10 S(r) n , 15 ex(n,H,F), 12 exc(n,T), 12 exconn(n,H), 10 Hn,a,b1,b2,...,bt, 22 H′ n′,a,b1,b2,...,bt, 31 k-shadow, 14 f(n,k,r,a), 21 f ∗(n,k,r,a), 21 Mantel’s theorem, 3 Berge-hypergraphs, 14 Brown-Erd˝ os C4-free graph, 6 Erd˝ os-Simonovits Theorem, 7 Erd˝ os-Gallai Theorems, 7 Erd˝ os-Rényi-Sós Theorem, 6 Erd˝ os-Sós Conjecture, 9 Erd˝ os-Stone-Simonovits Theorem, 4 Extremal Graph theory, 3 Extremal graphs, 3 Finite geometry constructions, 6 Hypergraphs, 14 K˝ ovári-Sós-Turán Conjecture, 6 K˝ ovári-Sós-Turán Theorem, 5 The Exact C4 result of Füredi, 6 The generalized Turán number, 12 The Turán number of a family H, 3 The Turán number of a graph H, 3 The Turán number of Berge-hypergraphs, 17 Turán graph, 4 Turán number, 3 Turán’s theorem, 4 Woodall’s Theorem, 9 Zarankiewicz problem, 5 CEU eTD Collection
2182	https://flexbooks.ck12.org/cbook/ck-12-college-precalculus/section/9.2/primary/lesson/two-dimensional-vectors-c-precalc/	Two-Dimensional Vectors \| CK-12 Foundation AI Teacher Tools – Save Hours on Planning & Prep. Try it out! Skip to content What are you looking for? Search Math Grade 6 Grade 7 Grade 8 Algebra 1 Geometry Algebra 2 PreCalculus Science Earth Science Life Science Physical Science Biology Chemistry Physics Social Studies Economics Geography Government Philosophy Sociology Subject Math Elementary Math Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Interactive Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Conventional Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Probability & Statistics Trigonometry Math Analysis Precalculus Calculus What's the difference? 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Learn. Interact. eXplore. CCSS Math Concepts and FlexBooks aligned to Common Core NGSS Concepts aligned to Next Generation Science Standards Certified Educator Stand out as an educator. Become CK-12 Certified. Webinars Live and archived sessions to learn about CK-12 Other Resources CK-12 Resources Concept Map Testimonials CK-12 Mission Meet the Team CK-12 Helpdesk FlexLets Know the essentials. Pick a Subject Donate Sign InSign Up Start Practice 9.2 Two-Dimensional Vectors FlexBooks 2.0> CK-12 College Precalculus> Two-Dimensional Vectors Written by:CK-12 \|Terri Germain-Williams \| +1 more Fact-checked by:The CK-12 Editorial Team Last Modified: Jul 02, 2025 Introduction An airplane being pushed off course by wind and a swimmer's movement across a flowing river are both examples of vectors in action. [Figure 1] Points in the coordinate plane describe a location. Vectors, on the other hand, have no location and indicate only magnitude and direction. Vectors can describe the strength of forces like gravity or speed and the direction of a ship at sea. Vectors are extremely useful in modeling complex situations in the real world. What are other differences between points and vectors? Vectors One way to define avectoris as a line segment with a direction.Vectors can be represented graphically using an arrow that points from the vector's initial point, called thetail, to its terminal point, called thehead. [Figure 2] The two defining characteristics of a vector are its magnitude and its direction. The magnitude is shown graphically by the length of the arrow, and the direction is indicated by the angle in which the arrow is pointing. The direction of a vector on the Cartesian coordinate plane is determined by its counter-clockwise rotation from the positive side of the x-axis. This can be seen in the images below: [Figure 3] [Figure 4] Notice how the vector below is shown multiple times on the same coordinate plane. This image emphasizes that the location on the coordinate plane does not matter and is not unique. Each representation of the vector has identical direction and magnitude. [Figure 5] There are a few different ways to symbolize a vector,v: v,v→,or v.⇀ There are also a few ways to describe a specific vector. First, you can describe its angle and magnitude. Second, you can describe it as an ordered pair or in component form,⟨x,y⟩. Note that when discussing vectors,you should use brackets ⟨⟩instead of parentheses to help avoid confusion between a vector and a point. Vectors can be two-dimensional, three-dimensional, or n-dimensional, where n is an integer greater than 3. The following video discusses vectors and vector vocabulary: When a vector is angled in a certain direction, it is often useful to break the vector into individual components. Thecomponent of a vector depicts the influence of that vector in a given direction. A two-dimensional vector can be thought of as having an influence in two different directions: the horizontal direction (x-axis) and the vertical direction (y-axis). For example, the vector on the left in the image below has components in both the x-direction,ABx, and the y-direction,ABy. Here,ABx= 4 andABy= 3 because point B is 4 units to the right and 3 units up from point A. Similarly, vectorDhas componentsDx= 3 andDy= –1.25. The negative sign inDy indicates that the y component of vectorDis downward, in the negativ e y-direction. As with other numbers, we usually only include the negative sign explicitly. [Figure 6] The following video further explains how to find the component form of a vector given the vector's magnitude and direction: To determine the vector between two points, start with the terminal point and subtract the initial point. Suppose A(u,v)and B(x,y). Then: A B→=⟨x−u,y−v⟩B A→=⟨u−x,v−y⟩. Vectors are said to be equal if they have the same magnitude and the same direction. The absolute value of a vector\|v→\|is the same as the length or magnitude of the vector.Magnitude can be found by using the Pythagorean Theorem or the distance formula. Play, Learn, and Explore with two-dimensional vectors using the example of a river ferry and currents: www.ck12.org/a/2137380. Examples Example 1 Consider the points A(1,3),B(−4,−6),C(5,−13). Find the vectors in component form of A B→,B A→,A C→,C B→. Solution: A B→=⟨−5,−9⟩B A→=⟨5,9⟩A C→=⟨4,−16⟩C B→=⟨−9,7⟩ Example 2 What are the x and y components of the vectors shown in the diagram? Solution: \| \| AB \| CD \| EF \| GH \| IJ \| KL \| MN \| --- --- --- --- \| \| xcomponent \| 2.25 \| -1.5 \| 0 \| -1.5 \| 2.0 \| 1.5 \| -2.25 \| \| ycomponent \| 0.25 \| -2.0 \| -2.25 \| -2.0 \| -1.5 \| 2.0 \| 0 \| In the diagram, each division is 0.25 units. All vectors that point toward the left have negative x components, and those that point downward have negative y c omponents. Notice that for the horizontal vector, MN, the y component is equal to 0. Likewise, for the vertical vector, EF, the x component is equal to 0. Example 3 1) Which of the vectors in Example 2 is equal to vectorCD? Solution: VectorGH=CD.Both vectors have the same length and the same direction or orientation. 2) Which vector is equal to–CD? Solution: VectorKL=–CD. Both vectors have the same length, and the two vectors point in opposite directions. Example 4 Return to the problem from the Introduction: W hat are other differences between points and vectors? Solution: There are many differences between points and vectors. Points are specific locations, and vectors are made up of distance and angles. Parentheses are used for points, and ⟨⟩are used for vectors. Without the starting point, the vector could start from anywhere. Example 5 A marine traffic controller sees a ship traveling NNW at 17 knots (nautical mph) on radar. Describe the ship's movement in component form using a vector. Solution: The marine traffic controller is using cardinal direction which is based on the orientation of a compass. [Figure 7] NNW is halfway between NW and N. When you describe ships at sea, it is best to use a bearing that has 0∘as due North and 270∘as due West. This makes NW equal to 315∘ and NNW equal to 337.5∘. [Figure 8] When the problem is depicted above, it becomes a basic trig question to find the x and y components of the vector. Note that the reference angle the vector makes with the negative portion of the x axis is 67.5∘. sin⁡67.5∘=y 17,cos⁡67.5∘=x 17≈<−6.5,15.7> Note:270∘ clockwise is equivalent to 90∘counterclockwise. This problem can also be worked using this angle measurement. Example 6 A father and daughter are sledding. After going downhill, the father pulls his daughter up the hill using a rope attached to the sled. The sled sits on the ground.The hill has a 20∘incline. The rope makes a 39∘angle with the slope. Draw a force diagram showing how these forces act on the daughter's center of gravity: a. The force of gravity. b. The force holding the daughter in the sled to the ground. c. The force pulling the daughter backward down the slope. d. The force of the father pulling the daughter up the slope. Solution: The girl's center of gravity is represented by the black dot. The force of gravity is the black arrow straight down. The green arrow is the effect of gravity pulling the girl down the slope. The red arrow is the effect of gravity pulling the girl straight into the slope. The blue arrow represents the force the father is exerting as he pulls the girl up the hill. [Figure 9] Notice that the father's force vector (blue) is longer than the force pulling the girl down the hill. This means that, over time, father and daughter will make progress and ascend the hill. Also, note that the father is wasting some of his energy by lifting rather than just pulling. If he could pull at an angle directly opposing the force pulling the girl down the hill, he would be using all his energy efficiently. Example 7 Center the force diagram from the previous question onto the origin, and identify the angle between each consecutive force vector. Solution: T he x- and y-axes are included as a reference; note that the gravity vector overlaps with the negative y axis. To find each angle, you must use your knowledge of supplementary, complementary, and vertical angles and all the clues from the question. To check, see if all the angles sum to be 360∘. [Figure 10] Summary Avectoris a line segment defined by its direction and its magnitude. The tail of a vector is the initial point where the vector starts. The head of a vector is the terminal point where it ends. Magnitude refers to the length of the vector and is associated with measurements such as the strength of a force or the speed of an object. Thecomponentof a vector depicts the influence of that vector in a given direction. Review Describe what a vector is and give a real-life example of something that a vector could model. Consider the points A(3,5),B(−2,−4),C(1,−12),D(−5,7). Find the vectors in component form of: A B→ 3.B A→ 4.A C→ 5.C B→ 6.A D→ D A→ For each of the following vectors, draw the vector on a coordinate plane starting at the origin, and find its magnitude: 8.⟨3,7⟩ 9.⟨−3,4⟩ 10.⟨−5,10⟩ 11.⟨6,−8⟩ Can the x or y component of a vector ever have a greater magnitude than the vector itself? If two vectors have magnitudes that are not equal, can the sums of their magnitudes ever be zero? 14.A ship is traveling SSW at 13 knots. Describe this ship's movement in a vector. A vector that describes a ship's movement is ⟨5 2,5 2⟩. In what direction is the ship traveling, and what is its speed in knots? If a boat is being motored perpendicularly at 35 km/hr across a stream that is flowing at 25 km/hr, how can the direction and speed it travels be clearly shown using vectors? Review (Answers) Please see the Appendix. 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2183	https://testbook.com/ugc-net-commerce/slope-of-the-demand-curve	Get Started Exams SuperCoaching Live Classes FREE Test Series Previous Year Papers Skill Academy Pass Pass Pass Pro Pass Elite Pass Pass Pro Pass Elite Rank Predictor IAS Preparation More Free Live Classes Free Live Tests & Quizzes Free Quizzes Previous Year Papers Doubts Practice Refer and Earn All Exams Our Selections Careers IAS Preparation Current Affairs Practice GK & Current Affairs Blog Refer & Earn Our Selections The slope of the demand curve is a critical concept in microeconomics that reflects the responsiveness of quantity demanded to changes in price. It illustrates how the quantity demanded of a good or service changes in response to variations in its price, showcasing the degree of elasticity or inelasticity in consumer behaviour. The slope is a key factor in understanding market dynamics, pricing strategies, and the overall sensitivity of consumers to price fluctuations. Understanding the slope of demand curve is crucial for analyzing consumer reactions to market changes. Slope of demand curve is a very important topic to be studied for the commerce related exams such as the UGC-NET Commerce Examination. In this article, the readers will be able to know about the slope of demand curve along with some other related topics in detail. In this article, you will learn about the following: Meaning of Demand Aggregate Demand What is the Slope of Demand Curve? Slope of the Aggregate Demand Curve Factors Influencing the Slope of Demand Curve Significance of the Slope of Demand Curve Real-World Examples Explaining the Slope of Demand Curve The Reason Behind Downward Slope of Demand Curve Also, read about factors-affecting-law-of-demand. 📚 Exclusive Free UGC NET Created by Our Experts \| Subjects \| PDF Link \| --- \| \| Download Free UGC NET Paper 1Important Qs Important PDF Created by UGC NET Experts \| Download Link \| \| Grab the Free UGC NET Commerce Important Qs used by UGC NET Students \| Download Link \| \| Download Free UGC NET Political Science Important Qs Created by UGC NET Experts \| Download Link \| \| Exclusive Free History Important Questions crafted by top mentors \| Download Link \| \| Exclusive Free Geography Important Questions crafted by top mentors \| Download PDF \| \| Download Free UGC NET Education Important Qs Created by UGC NET Experts \| Download PDF \| \| Exclusive Free Sociology Important Questions crafted by top mentors \| Download PDF \| \| Download Free UGC NET English Important Qs Created by UGC NET Experts \| Download PDF \| \| Exclusive Free Economics Important Questions crafted by top mentors \| Download PDF \| \| Download Free UGC NET Home Science Important Qs Created by UGC NET Experts \| Download PDF \| \| Exclusive Free Psychology Important Questions crafted by top mentors \| Download PDF \| Click Here to Download UGC NET Paper 1 Important Questions PDF Meaning of Demand In economics, demand refers to the quantity of a good or service that consumers are willing and able to purchase at various prices during a specific period. It is a fundamental concept in microeconomics and is a key component of the supply and demand model. Also, read about Objective-of-firm-demand-analysis. UGC NET/SET Course Online by SuperTeachers: Complete Study Material, Live Classes & More Get UGC NET/SET SuperCoaching @ just ₹25999₹9915 Your Total Savings ₹16084 Explore SuperCoaching Want to know more about this Super Coaching ? People also like Prev Next UGC NET/SET & Assistant Professor/Lectureship (Combo) ₹47999 (57% OFF) ₹20999 (Valid for 6 Months) Explore this Supercoaching CUET PG ₹13999 (59% OFF) ₹5833 (Valid Till June 2025 Exam) Explore this Supercoaching Assistant Professor / Lectureship (UGC) ₹26999 (57% OFF) ₹11666 (Valid for 3 Months) Explore this Supercoaching Aggregate Demand Aggregate demand is a macroeconomic concept that represents the total quantity of goods and services demanded by all sectors of an economy at various price levels and in a given period. It encompasses the combined spending of households, businesses, government, and foreign buyers on a nation's output of goods and services. Also, read about Elasticity-of-demand-and-its-measurement. What is the Slope of Demand Curve? The slope of a demand curve measures the rate at which the quantity demanded of a good or service changes in response to a change in price. It is a critical concept in economics and is expressed as the ratio of the change in quantity demanded to the change in price. The slope of demand curve is a visual representation of the law of demand in action The general formula for the slope (M) of a demand curve is given by: M=Change in Quantity Demanded/Change in Price Mathematically, the slope is represented as the derivative of the demand function with respect to price. In the context of a linear demand equation (where the demand curve is a straight line), the slope is constant along the entire curve. Interpretation of Slope Whether steep or flat, the slope of demand curve explains how prices influence consumer purchases. The interpretation has been stated below. Negative Slope: In most cases, demand curves slope downward from left to right. This negative slope indicates that as the price of a good decreases, the quantity demanded increases, and vice versa. Magnitude of the Slope: The steeper the slope, the less responsive quantity demanded is to changes in price, and the more inelastic the demand. A flatter slope implies greater responsiveness and more elastic demand. Perfectly Elastic and Perfectly Inelastic: A perfectly elastic demand curve has an infinite slope, indicating that consumers are extremely responsive to any change in price. On the other hand, a perfectly inelastic demand curve is vertical, suggesting that quantity demanded remains constant regardless of price changes. Unitary Elasticity: When the percentage change in quantity demanded is exactly equal to the percentage change in price, the demand is said to be unitary elastic. In this case, the slope is equal to -1. Understanding the slope of the demand curve is crucial for businesses and policymakers. It provides insights into how consumers respond to changes in prices and helps in making informed decisions regarding pricing strategies, market dynamics, and economic policy. Read about Law-of-variable-proportion. Slope of the Aggregate Demand Curve The slope of the aggregate demand curve reflects the relationship between the price level and the overall quantity of goods and services demanded in an economy, just as in the case of the individual demand curve. The slope is normally presented in a macroeconomic framework, and different things can influence the value of the slope. The aggregate demand (AD) curve is downward-sloping, indicating an inverse relationship between the general price level and the quantity of real GDP demanded. As the price level decreases, the quantity of real GDP demanded tends to increase, and conversely. The slope of demand curve formula presents a quantitative way to evaluate price sensitivities. The formula for the slope (M) of the aggregate demand curve is conceptually similar to the slope of an individual demand curve: M=Change in Real GDP Demanded/Change in Price Level However, the aggregate demand curve considers changes in the overall price level and real GDP for an entire economy. Study about Price-Discrimination. Factors Influencing the Slope of Demand Curve Understanding the slope of demand curve is crucial in microeconomics and for students preparing for competitive exams like UGC NET Commerce. The slope essentially represents how sensitive the quantity demanded is to a change in price. The slope of demand curve is not constant—it varies based on several influencing factors. These factors help economists and business managers analyze consumer behavior, forecast revenue, and set prices effectively. Let’s break down what affects this slope and why it matters. Availability of Substitutes When close substitutes are available, demand becomes more elastic, and the slope of demand curve is steeper. A small change in price can cause a significant shift in demand. For example, in the smartphone market, if the price of Brand A rises, buyers may quickly switch to Brand B. Necessities vs. Luxuries Necessities (e.g., basic groceries): Show inelastic demand, leading to a flatter demand curve. Luxuries (e.g., designer clothes): Show elastic demand, leading to a steeper curve. So, the slope of demand curve is less steep for necessities and steeper for luxury items. Time Horizon Short-term: Consumers have fewer alternatives → less responsive to price changes → flatter demand curve. Long-term: Consumers adjust preferences → more responsive → steeper slope. Proportion of Income Spent Goods that take a larger share of income (e.g., electronics, cars) tend to have steeper demand curves, as price changes strongly influence purchasing behavior. Items that represent a small fraction of income (like salt or toothpaste) have flatter curves. Significance of the Slope of Demand Curve The slope of a demand curve is analyzed by economists to assess the responsiveness of buyers in various sectors. For this reason, understanding the slope of demand curve matters very much for economic and business decision-making: Market equilibrium- The slope determines the sharpness with which the demand curve intersects the supply curve. Demand curve with a steep slope means that small changes in supply will cause substantial changes in price. Price strategies- The firms use an estimate of the slope of the demand curve to measure how much any price increase or decrease will affect sales quantity and revenue. The slope of the demand curve can be of great help for policy analysts in making forecasts concerning market behavior. Example: Price increase for an inelastic product would not significantly affect demand, while the elastic ones have to be treaded very carefully. Consumer surplus- Consumer surplus also depends on the slope of the demand curve. A steep curve often means that the consumer surplus is low, indicating unwillingness of consumers to pay high prices. Real-World Examples Explaining the Slope of Demand Curve Example 1: Gasoline Nature: Necessity Demand: Inelastic Slope: Flatter demand curve Even if the price rises, consumers still buy due to the lack of alternatives. Example 2: Luxury Watches Nature: Luxury Demand: Elastic Slope: Steeper curve Buyers can delay or skip purchase if prices increase. Example 3: Generic Medications Nature: Essential but substitutable Demand: Relatively elastic Slope: Steep but not vertical Consumers are highly responsive to minor price changes due to budget constraints. The Reason Behind Downward Slope of Demand Curve The downward slope of the demand curve is a fundamental concept in economics and is explained by the law of demand. The law of demand states that, all else being equal, as the price of a good or service decreases, the quantity demanded for that good or service increases, and vice versa. There are several reasons behind this inverse relationship: Substitution Effect: When the price of a good decreases, it becomes more attractive relative to other goods that haven't experienced a price change. Consumers are more likely to switch from relatively more expensive goods to the now relatively cheaper goods, leading to an increase in quantity demanded. Income Effect: A decrease in the price of a good increases the real purchasing power of consumers' income. As the price drops, consumers effectively feel richer, allowing them to buy more of the goods with the same amount of money. Conversely, an increase in price has the opposite effect on real purchasing power. Diminishing Marginal Utility: The principle of diminishing marginal utility suggests that as individuals consume more units of a good, the additional satisfaction (utility) derived from each additional unit decreases. Therefore, consumers are generally willing to pay a higher price for the first unit of a good compared to subsequent units. As the price decreases, consumers are willing to buy more units, leading to an increase in quantity demanded. Law of Scarcity: Resources are limited, and individuals and businesses must make choices about how to allocate their resources. A lower price for a good makes it more affordable and encourages consumers to allocate more of their resources (money) to that good. Expectations of Future Price Changes: Consumers often consider future price changes when making purchasing decisions. If consumers expect prices to decrease in the future, they may delay purchases, reducing current demand. Conversely, if they anticipate future price increases, they may increase current demand. Read about Pricing-Strategies. \| \| \| Major Takeaways for UGC NET Aspirants: What is the Slope of Demand Curve? The slope of the demand curve shows how much the quantity demanded changes in response to changes in price. Definition of Slope of Demand Curve:It is the ratio of the change in quantity demanded to the change in price, representing consumer responsiveness. Importance of Slope of Demand Curve: Understanding the slope helps economists and businesses analyze market behavior and set better pricing strategies. Interpretation of the Slope: A downward slope means demand increases as price decreases, and the degree of steepness indicates elasticity. Perfectly Elastic vs. Perfectly Inelastic: Perfect elasticity means any price change alters demand drastically, while perfect inelasticity means demand doesn’t change at all. Slope of Aggregate Demand Curve: The aggregate demand curve also slopes downward, showing an inverse link between price level and total demand in the economy. Formula for Slope of Demand Curve: The slope is calculated as change in quantity demanded divided by change in price, which remains constant in linear curves. Factors Influencing the Slope: The slope varies based on availability of substitutes, nature of the good (necessity or luxury), time frame, and income proportion. Availability of Substitutes: More substitutes make demand more elastic and create a flatter slope due to easier switching by consumers. Necessities vs. Luxuries: Necessities have flatter slopes due to inelastic demand, while luxuries have steeper slopes indicating higher sensitivity. Time Horizon: Short-run demand is less elastic (flatter slope), while long-run demand is more elastic (steeper slope) as consumers adjust. Proportion of Income Spent: Items taking a larger share of income tend to show steeper slopes, indicating more elastic behavior. Significance for Pricing Strategy: Firms use the slope to predict how price changes affect sales and revenue, aiding in pricing decisions. Market Equilibrium Impact: The slope determines how sharply the demand and supply curves intersect, affecting price stability. Consumer Surplus Insight: A steep slope implies lower consumer surplus as buyers are less responsive to price drops. Real-World Example – Gasoline: Demand is inelastic with a flatter slope because consumers still need fuel even when prices rise. Real-World Example – Luxury Watches: Demand is elastic with a steeper slope as buyers may delay purchases due to price increases. Real-World Example – Generic Medicines: Demand is relatively elastic with a noticeable slope due to budget-conscious consumers. Reason for Downward Slope – Substitution Effect: Lower prices make a product more appealing than its substitutes, increasing its demand. Reason for Downward Slope – Income Effect:When prices fall, consumers feel wealthier and buy more, boosting demand. Reason for Downward Slope – Diminishing Marginal Utility: As consumers buy more of a product, the added satisfaction drops, so lower prices are needed to motivate further purchase. Reason for Downward Slope – Scarcity and Resource Allocation: People allocate more of their limited income to cheaper goods, increasing demand when prices fall. Reason for Downward Slope – Price Expectations: If people expect prices to rise soon, they buy now, increasing current demand and vice versa. \| Conclusion The slope of demand curve is a primary determinant of microeconomics. Indeed, consumer demand is indicative of price with respect to quantity demanded, revealing the preferred or desired market behaviour of consumers. Differentiating between elastic, inelastic, or unitary slopes helps to gauge the responsiveness of consumers to changes in price levels. It is one of the most salient covert factors for businesses and policymakers while making decisions in pricing strategies and market dynamics. A detailed study of the slope of demand curve equips commerce students with tools to crack UGC NET Commerce exams. Slope of the demand curve is a vital topic as per several competitive exams. It would help if you learned other similar topics with the Testbook App. More Articles for UGC NET Commerce Notes the law of diminishing marginal product and the law of variable proportions What is Owner’s Equity Elasticity Demand Formula Marginal Cost Formula Notes for UGC-NET Commerce Examination Understanding Abnormal Loss in Business - Causes & Accounting Understanding Marginal Revenue and Price Elasticity of Demand Marginal Propensity to Save Outstanding Expenses Marginal Propensity to Consume Slope of Demand Curve FAQs What does a steeper demand curve signify? A steeper demand curve indicates a more inelastic demand, meaning consumers are less responsive to price changes. Quantity demanded changes relatively less in response to changes in price. How is elasticity related to the slope of the demand curve? The elasticity of demand is inversely related to the slope of the demand curve. A flatter slope indicates more elastic demand, while a steeper slope suggests less elastic or more inelastic demand. What is the significance of a perfectly elastic demand curve? A perfectly elastic demand curve has an infinite slope, indicating that consumers are extremely responsive to any change in price. In this scenario, the quantity demanded drops to zero if there is any increase in price. How does the slope of the demand curve impact pricing strategies? Businesses use the information about the slope of the demand curve to determine optimal pricing strategies. Understanding whether demand is elastic or inelastic helps in setting prices that maximise revenue and profit. Can the slope of the demand curve change over time? Yes, the slope of the demand curve can change based on factors such as consumer preferences, income levels, availability of substitutes, and market conditions. Technological advancements and shifts in consumer behaviour may contribute to changes in slope. Report An Error Important Links OverviewInflation Accountingaudit of financial statements and audit reportCost and Management AccountingHolding Company AccountsScope and Importance of International BusinessEconomic Monetary Policiesactivity based costingEnergy Audit NotesIssue Forfeiture And Reissue Of SharesLiquidation of CompanyVerification & Valuation of Assets & LiabilitiesEconomic System in Business EnvironmentFEMA (Foreign Exchange Management Act)Costing for Decision MakingEconomic Fiscal PoliciesTarget CostingSafety Audit NotesFinancial Accounting, and Management AccountingMerger and AmalgamationMacro and Micro Economic Policy Download the Testbook APP & Get 5 days Pass Pro Max @₹5 10,000+ Study Notes Realtime Doubt Support 71000+ Mock Tests Rankers Test Series more benefits Download App Now Track your progress, boost your prep, and stay ahead Download the testbook app and unlock advanced analytics. Scan this QR code to Get the Testbook App ✕
2184	https://econeye.blog/2022/04/09/how-do-we-calculate-consumer-surplus-and-producer-surplus/	How Do We Calculate Consumer Surplus and Producer Surplus? – Econ Eye Skip to content Econ Eye Official blog of the Bade-Parkin economics textbook series Menu Ask a Question Eye On Our Texts Topic Map How Do We Calculate Consumer Surplus and Producer Surplus? April 9, 2022 April 13, 2022 ~ EconEye Admin A student asks … How do we calculate consumers surplus and producer surplus from a graph and from a data table? What is consumer surplus? When you buy something for less than it is worth to you, you receive consumer surplus. Consumer surplus is the excess of the benefit received from a good or service over the amount paid for it. What is producer surplus? When a firm sells a good or service at a price greater than the good’s marginal cost, the firm receives producer surplus. Producer surplusis the excess of the amount received from the sale of a good or service over the cost of producing it. How do we calculate consumer surplus and producer surplus? We calculate consumer surplus as the marginal benefit of a good minus its price, summed over the quantity bought. And we calculate producer surplus as the price received minus the marginal cost, summed over the quantity sold. How do we calculate consumer surplus and producer surplus from a graph? Two steps are required to calculate consumer surplus and producer surplus from a graph. First, identify the area on the graph that represents the surplus. Second, calculate the size of the area. Work through this example. How do we calculate consumer surplus and producer surplus from a data table? A demand schedule gives the marginal benefit of buying a good or service, and a supply schedule gives the marginal cost of selling it. In either case, we calculate the relevant surplus using the steps given above and summing across the quantity traded. Work through this example. If I’m given a table on a test, should I use a graph anyway? I am a TA for Principles of Economics and I have worked with hundreds of students. After years of watching the mistakes students commonly make, I recommend graphing the demand and supply curves and calculating the surpluses from your graph. Work these questions to solidify your understanding of the lesson and get instant feedback. Answer the following questions to check your understanding of the story. 1) Daniel is willing to pay $5 for a coffee and $8 for two coffees. The price of a coffee is $3. How much consumer surplus does Daniel receive from buying two coffees? a) $4 b) $0 c) $8 d) $2 Daniel receives $2 of consumer surplus from buying the first coffee ($5 – $3 = $2), but he does not receive as much consumer surplus from buying the second coffee. Daniel does not receive any consumer surplus from buying the second coffee. But how much consumer surplus does he receive from buying the first coffee? Daniel’s total willingness to pay for 2 coffees is $8, but he pays $6 for them. Daniel receives $2 of consumer surplus from buying the first coffee ($5 – $3 = $2) and $0 of consumer surplus from buying the second coffee ($3 – $3 = $0). 2) The table below gives the demand and supply schedules for ice cream cones. The consumer surplus is __ and the produce surplus is __. a) $25 an hour; $50 an hour b) $50 an hour; $50 hour c) $25 an hour; $100 an hour d) $50 an hour; $50 an hour Did you forget to divide consumer surplus by 2 in your calculation? In equilibrium, 50 cones an hour are traded at $2 a cone. But this revenue is not all surplus, and the surplus is not divided equally between consumers and producers. Did you forget to divide producer surplus by 2 in your calculation? In equilibrium, 50 cones an hour are traded at $2 a cone. Consumer surplus is ($3 – $2) × (50 – 0) ÷ 2 = $25 an hour and producer surplus is ($2 – $0) × (50 – 0) ÷ 2 = $50 an hour. 0% Close Quiz Share this: Click to share on Facebook (Opens in new window)Facebook Like this: Like Loading... Posted in Student Question Post navigation ‹ Previous Unemployment Hits Pandemic Low in March Next ›The Elasticity of Supply of Cocoa Leave a ReplyCancel reply This site uses Akismet to reduce spam. Learn how your comment data is processed. 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Does it Work for the Demand for all Goods? Calculating Chained-dollar Real GDP when the Base Year is Later When is the Tax on a Good Paid Entirely by Buyers or Sellers? How Do We Calculate Consumer Surplus and Producer Surplus? Why is a PPF bowed outward? More Questions … Econ Help Desk Ask a Question Econ Instructor Assigning Economics in the News for Homework, Test, or Quiz in MyLab Economics About Econ Eye About Us Our Team Archives Archives Search for: © Copyright, Bade Editorial Management Service Inc. (2022-2024) Discover more from Econ Eye Subscribe now to keep reading and get access to the full archive. Type your email… Subscribe Continue reading %d
2185	https://www.engineeringcivil.com/critical-depth-of-open-channel-flow.html	Awarded as the best online publication by CIDC Critical Depth of Open-Channel Flow For a given value of specific energy, the critical depth gives the greatest discharge, or conversely, for a given discharge, the specific energy is a minimum for the critical depth. For rectangular channels, the critical depth, dc ft (m), is given by dc =[Q2/b2g]1/3 where dc =critical depth, ft (m) Q= quantity of flow or discharge, ft3/s (m3/s) B= width of channel, ft (m) Kanwarjot Singh Related Posts Knowing Expansion and Contraction of Piping Systems Buttress Dams and its Classifications Artificial Recharge of Groundwater ROOF TOP RAIN WATER HARVESTING Riverfront Development Of Karha River Baramati City Dampness of Wall And The Necessity Of Kerb Concrete In Sunken Areas If you have a query, you can ask a question here. 10 comments on "Critical Depth of Open-Channel Flow" Hi, It is good……… this is what i’ve been looking for.. simple and straightforward! thanks that was short and sharp What does q equal if its a wide channel ? q = Q/b ; (m³s-1/m) b = the channel width ; (m) Q = flow rate ; (m3/s) The concept of critical depth for engineer is like you wrote here. That is great way. But the meaning critical depth is that water molucule upstream and downstream of the critical depth wont comunicate. It is a point of discontinity. if we introduce more formula for tripoziod channel it would be useful thaks this is what iam searching for.. The specific energy of triangle channel for critical depth If water is pouring out of an horizontal canal into the void, the depth at the last section (before pouring) is the critical depth Leave a Reply Cancel reply Your email address will not be published. Required fields are marked Comment Name Email Save my name, email, and website in this browser for the next time I comment. Δ More Articles What is Civil Engineering Top Contributors Knowledge Center Journals Books & Softwares Knowledge Center Civil Engineering Branches
2186	https://www.nature.com/articles/s41540-025-00524-y	An in silico approach deciphering the commensal dynamics in the cutaneous milieu \| npj Systems Biology and Applications Your privacy, your choice We use essential cookies to make sure the site can function. We also use optional cookies for advertising, personalisation of content, usage analysis, and social media. By accepting optional cookies, you consent to the processing of your personal data - including transfers to third parties. Some third parties are outside of the European Economic Area, with varying standards of data protection. See our privacy policy for more information on the use of your personal data. Manage preferences for further information and to change your choices. Accept all cookies Skip to main content Thank you for visiting nature.com. You are using a browser version with limited support for CSS. 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Advertisement View all journals Search Search Search articles by subject, keyword or author Show results from Search Advanced search Quick links Explore articles by subject Find a job Guide to authors Editorial policies Log in Explore content Explore content Research articles Reviews & Analysis News & Comment Collections Follow us on Twitter Sign up for alerts RSS feed About the journal About the journal Aims & Scope Content types Journal Information About the Editors Contact Open Access Calls for Papers Article Processing Charges Editorial policies Journal Metrics About the Partner Publish with us Publish with us For Authors and Referees Language editing services Open access funding Submit manuscript Sign up for alerts RSS feed nature npj systems biology and applications articles article An in silico approach deciphering the commensal dynamics in the cutaneous milieu Download PDF Download PDF Article Open access Published: 07 May 2025 An in silico approach deciphering the commensal dynamics in the cutaneous milieu Sumathi Kalankariyan1, Anjana Thottapillil1, Abha Saxena1, Manoj Srivatsn S1, Vinitha Kadamkode2, Renu Kapoor2, Rupak Mitra2, Janhavi Raut2& … KV Venkatesh1,3 Show authors npj Systems Biology and Applicationsvolume 11, Article number:42 (2025) Cite this article 1745 Accesses 2 Citations Metrics details Abstract The skin microbiota, particularly coagulase-negative staphylococci (CoNS) such as S. epidermidis, plays a crucial role in maintaining skin health and immunity. S. epidermidis, a predominant commensal species, interacts intimately with keratinocytes to regulate immune responses and antimicrobial defence mechanisms. Metabolic byproducts like short-chain fatty acids (SCFAs) influence keratinocyte activation, while cell wall components engage Toll-like receptors (TLRs) to modulate inflammation. These interactions are fundamental for preserving skin homeostasis and combating pathogenic invaders. Our comprehensive mathematical model, integrating commensal dynamics, immune responses, and skin microenvironment variables, provides insights into these intricate interactions. The model delves into the complexities of skin scenarios and perturbations, aiming to understand the colonization dynamics of S. epidermidis and its influence on skin barrier functions. It examines how disruptions in key factors such as AMP, growth factor-mediated repair pathways, and filaggrin mutations influence the behaviour of the system. The study depicts the skin microenvironment as a highly dynamic one, highlighting the critical role of S. epidermidis and capturing its role in barrier dysfunction caused by internal and external factors. By offering insights into skin barrier function and immune responses, the model illuminates key interactions of commensals within the skin microenvironment which can ultimately benefit skin health. Similar content being viewed by others Staphylococcus epidermidis and its dual lifestyle in skin health and infection Article 30 August 2022 Discovery and engineering of the antibody response to a prominent skin commensal Article 11 December 2024 Conversation between skin microbiota and the host: from early life to adulthood Article Open access 01 April 2025 Introduction Skin serves as the outermost barrier against pathogenic invasion and plays an important role in adapting the systemic responses to changing environments. Epidermal barrier consists of several specialized cellular constructs (desmosomes, tight junctions and structural proteins) which interconnect and maintain tightly regulated and balanced interface. Epidermal barrier integrity relies on the concerted actions of these components along with the resident immune cells and microbiota and contributes to its defense mechanisms1,2. The innate immune response recruit’s neutrophils, phagocytic cells, to combat skin pathogens through multiple mechanisms such as induction of proinflammatory cytokines3,4,5. Antimicrobial peptides (AMPs) actively secreted by the keratinocytes such as hBDs and S100A7, fortify the skin against pathogens, and their expression is modulated by cues from the skin microbiota6. A constant interaction between host cells and skin microbes plays a vital role in fostering immune tolerance and shaping the development of host immune cells from early life. Commensal skin microbiota establishes a mutualistic relationship with the skin’s immune system, enhancing the immune barrier through controlled inflammation7. Among the diverse population of the resident commensals, S. epidermidis is one of the abundant skin commensals extensively studied for its beneficial effects on skin barrier maintenance such as its role in inflammation and wound healing8. S. epidermidis enhances antimicrobial defense by the upregulation of specific AMP as well as contributes to skin hydration by enhancing lipid mediators and tight junctions in keratinocytes8,9,10,11. In contrast, studies also indicate S. epidermidis behaving as opportunistic pathogen thereby has been called as “Pathobiont”, often existing along a spectrum between mutualism and pathogenicity12. The cause and consequences on the factors driving its transition from commensal to pathogenic state postulates the question on the precise role of S. epidermidis on skin and the key facilitators of its commensal behaviour. S. epidermidis is identified as the second most common microbe isolated from lesions affected by Atopic dermatitis (AD)8, a prevalent skin condition characterized by a dysfunctional skin barrier, with filaggrin mutations serving as key contributing factors. Filaggrin contributes to the structural integrity of the skin by forming aggregates with the keratin13. In the upper-most layer of the skin, stratum corneum, Filaggrin is broken down into free amino acids, pyrrolidone carboxylic acid, and urocanic acid (UCA), which are key components of the skin’s natural moisturizing factors (NMF)14. These molecules help regulate skin hydration, pH and have important immunomodulatory effects. Maintaining skin pH is essential for metabolism and organization of the lipids of the extracellular matrix and anti-microbial activities13. In addition, component of NMF also play a role in photoprotection15. The deficit of Filaggrin in skin causes altered organization of the skin barrier resulting in an elevated pH and protease activity, making skin susceptible to infections. Loss of function variants in Filaggrin and other critical barrier proteins disrupts the normal epidermal functions and leads to increased water loss (TEWL)15,16,17 making host susceptible to many skin diseases, including Atopic dermatitis18. The mechanism underpinning the AD pathogenesis, particularly in relation to skin barrier dysfunction and filaggrin mutations, is well established18, but the interplay between S. epidermidis and these factors is poorly understood. Gaining a clearer understanding into the filaggrin mutation and its influence on commensal could be valuable for designing novel therapeutic strategies for AD. Mathematical models have emerged as a significant advancement in understanding the skin milieu in recent years, offering a valuable tool to overcome the limitations of traditional in vitro methods19,20. A dearth of computational models is reported in the study by Elea et al. who has developed an ODE based model addressing the interaction between two microbial populations and has well stated the limitation in the number of models even on addressing the bacterial population and skin interaction21. This sets our model unique and stand -alone on having a deeper look on the commensal S epidermidis dynamics aiming to understand the aspects that preserve healthy levels of commensals on the skin and their interactions with skin cells to sustain cutaneous immune homeostasis under both normal and compromised conditions. The model also attempts to provide a basis for “pathobiont” state of S. epidermidis, focusing on how the system calibrates a critical threshold of commensal growth beyond which pathogenicity ensues. This study investigates the commensal behaviour of S. epidermidis and its relationship with epidermal barrier responses in conditions such as compromised skin immunity or filaggrin mutations, both of which are associated with the development of skin atopy. More importantly, the role of S. epidermidis in dysfunctional skin barrier conditions, as addressed through our in silico approach, underscores its relevance in exacerbating inflammatory dermatoses and contributing to various pathological skin conditions. This highlights the potential applications and translational value of the model in advancing skin health research. By employing approaches such as phase plane plots, parameter perturbation, and sensitivity analysis, we have elucidated key aspects in the mechanisms driving host-microbe interactions. The in silico skin model on commensal-host interactions Our mathematical model represents the interactions and pathways in the skin consisting of microbial signaling, immune response, hydration, and barrier repair. Mechanistic interactions in the skin microenvironment captured in our meticulously developed network include 21 nodes and 44 edges (Fig. 1). Directed edges emphasize the functional importance of each interaction, with 12 edges dedicated to commensal bacterial product signaling, 11 to immune signaling, and 8 to hydration-related components, while 10 are linked to epidermal damage and repair signaling pathways. Our model has captured the essential features of signaling pathways such as non-linear interactions and feedback loops. Fig. 1: A holistic in silico model of skin: the model illustrates the skin microenvironment as a complex network of interactions. In the model, commensal bacteria trigger an immune response via T cells, which activate keratinocytes. These cells sustain both commensal and damaged keratinocytes by synthesizing growth factors, neutrophils, and antimicrobial peptides (AMPs). AMPs, while promoting skin health, also result in damage when the pathogenic community increases. Growth factors, along with filaggrin and desmoglein, an important desmosomal protein, help repair this damage. Filaggrin synthesis is influenced by the microbial population, which in turn regulates Natural Moisturizing Factor. The model visually represents these interactions and pathways like microbial signaling (Purple), immune response (Green), hydration (Blue), and barrier repair (Red). Short Chain Fatty Acids – SCFA, Anti-Microbial Peptides-AMP, Desmoglein- DES, Filaggrin – FLG, Natural Moisturizing Factor – NMF, Total Water Content – TW, Damaged Keratinocytes – Dke. Full size image Results Model Benchmarking and Model Validation The model development and validation leveraged all relevant published datasets, along with in-house growth curve experiments with commensal on keratinocytes (Supplementary Table 1). The model was benchmarked in stages, wherein the first stage drew upon in vitro experimental data capturing commensal growth in presence and absence of keratinocytes starting with an initial concentration of 1 × 10 5. Note that in such a scenario, only the variable pertaining to short chain fatty acids (SCFA) and antimicrobial peptides (AMP) were accounted for. The other variables relating to immune response, filaggrin, NMF, damaged keratinocytes and growth factors etc. in the system were not considered. The parameters pertaining to commensal’s growth curve, AMP levels, neutrophils and SCFA were benchmarked using dynamic data sets until the saturation of commensal growth. As shown in Fig. 2a, the blue curve represents the stabilized growth rate of S. epidermidis without keratinocyte interactions, indicating isolated growth. In contrast, the red curve demonstrates keratinocyte-mediated interactions, showing a lag due to the negative feedback from AMP. The model equations were then simulated to match the immune interactions in keratinocytes, from available literature data focusing on AMP22. In the second stage, parameters pertaining to skin microenvironment, such as immune and growth factors and filaggrin, (equations 6–21, Mathematical model; ODEs, Supplementary information) were fitted using literature data, while the parameters pertaining to equation 1–5 were fixed as estimated in stage one). Data from the literature showed that the presence of S. epidermidis elicits an immune response in the skin microenvironment. The model recapitulation of the increase in interleukin-1 (IL-1)11, interleukin-8 (IL-8)23 and filaggrin levels24 (Fig. 2b) was achieved by fine-tuning the model parameters. These variables had two time points while certain variables for e.g., interleukin-17 (IL-17), growth factors, desmoglein25 and neutrophils where only the fold changes were available under two physiological conditions, in such cases, parametric values were set to match the overall physiological trends. Fig. 2: Model benchmarking and validation of S. epidermidis growth dynamics and AMP, IL1, IL8 responses. aS. epidermidis (B1) growth rate in the presence (red) and absence of keratinocytes (blue) with respect to the experimental data (red and purple dot) given at 24 th hour and 48th hour b Comparison between the predicted (Red) fold change and the observed (Blue) fold change in AMP, IL1, and Filaggrin c Model Validation for the variables: Comparison between the predicted (Red) fold change and the observed (Blue) fold change in AMP, IL1 and IL8. Plot Representation: (a)Blue- Absence of S. epidermidis, Red -Presence of S. epidermidis. Blue-Observed, Red -Predicted. Full size image Following this iterative process of benchmarking to converge on the relevant parametric space accommodating all the above constraints, the model was refined further by quantitative validation against literature datasets of key epidermal markers, including IL-123 (3-fold), AMP26 (10-fold), and IL-823 as seen in Fig. 2c. The model could also capture the trends like elevated IL-1711,22, neutrophil levels11 and growth factors26 along with a slight decrease in desmoglein24. In addition to quantitative validations, the model successfully predicted qualitative phenotypic observations. Total water content (TW) and natural moisturizing factor (NMF) were maintained at optimal levels due to the predicted increase in filaggrin, while desmoglein remained stable. The initial condition and the parametric value in the model represent the normal (healthy) state of the commensal interaction with the keratinocyte. Supplementary Fig. 1 and 2 showcase the model’s ability to stabilize over time and reach a healthy state, demonstrating the model’s robustness. Parameter sensitivity analysis The certainty of model predictions regarding underlying mechanisms was quantified using parameter sensitivity analysis, which estimates the relative change in system outputs based on input parameters. The method section details the complete metric used for this estimation. Parameters with sensitivity coefficient (S) of 0 indicate that a 100% change in input resulted in no change in output, suggesting no effect on the system. Conversely, S = 1 signifies that the output changed proportionally to the input, indicating a substantial impact. Parameters with a S> 0.5, regardless of the sign, were considered sensitive, indicating that a 100% change in the parameter value resulted in a 50% or greater change in the output. It was observed that for S> 0.5 the physiologically relevant impact on the output variables were observed, such as in the AMP synthesis, therefore, a threshold value of S = 0.5 was chosen as the sensitivity cut off. The observed results shown in Supplementary Fig. 3 indicate that the growth of S. epidermidis was seen to be highly sensitive (S = −1.02) when parameters such as Rate of AMP synthesis by activated keratinocytes (k6 AMP), Rate of IL8 synthesis by activated keratinocytes (k14 IL8), and Rate of Neutrophil activation by B1 (k7 Neut) were increased which highlighted the antimicrobial activity of AMP and neutrophils against S. epidermidis. This has further led to a decline in Growth factors (S = −1.01, S = −1.01, S = −0.88), filaggrin (S = −0.85, S = −0.74,) and thereby an increase in damaged keratinocytes (S = 1.01, S = 1.01, S = 0.91) indicating that these skin immune components associated parameters are notably sensitive for various outputs through the anti-microbial and immune activity. However, certain parameters such as Rate of Desmoglein synthesis by keratinocytes (k20 Des,), Rate of IFNγ synthesis by activated keratinocytes (k13 IFN γ) and Rate of CD8 activation by B1and IFNγ (k9 CD8) did not change the growth of S. epidermidis when increased, but their decrease significantly reduced the commensal growth (S = 1.02; S = 0.94; S = 0.83) indicating the non-linearity of the system. The nonlinear relationship of the parameters k20 Des, k13 IFN γ , k9 CD8 was also observed in neutrophil (S = 0.93; S = 0.83; S = 0.69), AMP (S = 0.65; S = 0.53) levels which decreased, while damaged keratinocytes (S = −1.02; S = −1.01; S = −1.01) and filaggrin (S = −0.60; S = −1.02; S = −0.96) showed an increase. Conversely, the parameters demonstrated linearity with the growth factor levels (S = 0.98; S = 1.02). This revealed a crucial effect of k20 Des, k13 IFN γ, k9 CD8 which influences S. epidermidis growth, growth factor production via CD8-IFNγ cross-talk and filaggrin production through IL17 via CD8-IL17 cross talk. Other sensitive parameters include Rate of FLG synthesis by keratinocytes (k22 FLG). It significantly impacts NMF (S = 0.94) by affecting filaggrin synthesis and therefore hydration of skin. Thus, the sensitivity analysis revealed that parameters could elicit differential effects in outputs, demonstrating the robustness of our model in simulating a living system, wherein several biological functions are maintained in rigidness and some processes adapt as per the changes. This could be observed in our skin model where some parameters are very sensitive while others do not change significantly, which represents a homeostatic state in a healthy skin barrier. Simulation of Skin microenvironment in response to the commensal S. epidermidis The healthy state of the model was perturbed by removing the commensal interactions to examine the commensal role in healthy skin. The observed changes were given in terms of fold change in comparison to a healthy state (Fig. 3). In the absence of S. epidermidis, the decrease in the activation of keratinocytes to 0.46-fold results in a massive increase in damaged keratinocytes. Due to the lack of microbial interactions, there is a decline in CD8 + T cells and IL17 levels, nearly reaching zero, as depicted in Supplementary Fig. 4a. The synthesis of growth factors, mediated by IL17 and CD8 + T cells, was reduced by 99% relative to the presence of commensal (Supplementary Fig. 4b). The reduction significantly hampered the repair of damaged keratinocytes, resulting in a continuous accumulation of damaged keratinocytes by 1017-fold over time (Fig. 3 column A, Supplementary Fig. 4e). With the loss of microbial interaction, activated neutrophils decreased and consequently reduced AMP levels to 0.12-fold. In response to the loss of S protease interaction, desmoglein synthesis increased by 13% and a slight increase in filaggrin and subsequent water content, which can be an attempt to revive the repair process. This disruption underscored the impaired epidermal homeostasis in the absence of commensal microorganism, highlighting the crucial role of commensal-keratinocyte interactions, mediated through the immune response. This demonstrated that S. epidermidis exerts a beneficial effect on the skin through upregulation of pro inflammatory signals and antimicrobial peptides respectively. These physiological response in the skin niche has been captured both quantitatively and qualitatively in our model, showing the alignment with real world data22,27. Fig. 3: Heat map showing steady-state responses of key variables to variations in S. epidermidis growth rate. A Heat map representing the steady-state values of variables when only keratinocyte-induced conditions are present. Varying S. epidermidis growth rate parameter (k1 B1) to a fold change of (B) 0.19 (C) 0.56 (D) 1.48 (E) 1.67. The growth rate parameter (k1 B1) value for healthy condition was 0.27 h−1. Plot Representation: (Fold change relative to healthy state, Kera - Activated Keratinocytes, Neut - Activated Neutrophil, GFr – Growth Factor, DKe – Damaged Keratinocytes, B1 – S. epidermidis, B1p – Penetrated S. epidermidis, Spr – S. epidermidis Protease). Full size image The impact of S. epidermidis growth rate on skin microenvironment As observed in the previous results section, the presence of commensal microorganisms plays a vital role in maintaining the delicate homeostasis of the skin environment. This elicits the question on the bacterial load necessary to sustain this equilibrium and how deviations from these levels can disrupt homeostasis. To understand this impact, we perturbed the growth rate parameter (k1 B1) within the range of 0.05 – 0.45 h−1 (0.19–1.67-fold), as depicted in the respective columns of Fig. 3. Fig. 3 column B shows a lowered growth rate (k1 B1) of 0.19-fold, reduced the activated keratinocytes (0.73-fold) thereby decreasing the growth factors (0.14-fold) and increasing the damaged keratinocytes (4.82-fold). We observed in Fig. 3 column C that increasing growth rate parameter by 0.56-fold does not significantly differ from the healthy state, indicating commensal behaviour. But an increased S. epidermidis proliferation to around 3.85 and 7.43 folds (Fig. 3 column D & E) heightened the protease activity (5.23 and 13.26 folds) followed by an increase of 22 and 85 folds in the penetrated population (Fig. 3 column D & E). This surge in protease activity exacerbated the tissue damage (23-fold). This indicated that only at a certain range of growth rate, S. epidermidis exhibits commensal properties. To predict the critical bounds of the range, the growth rate values against the corresponding steady-state values of Damaged keratinocytes and S protease were checked as depicted in Fig. 4. The data shows that when the growth rate falls below 0.1 h−1 (0.4-fold), damage increases even with minimal AMP. This is due to the loss of activation of keratinocytes via microbial interaction, which is essential for synthesizing repair components like growth factors. A growth rate above 0.33 h−1 (1.2-fold) facilitates higher penetration through excessive protease, potentially transitioning the S. epidermidis to a pathogen. These overall observations highlight a key finding of our study: S. epidermidis maintains its role as a commensal when its growth is kept within a critical window. Fig. 4: The steady-state values of damaged keratinocytes and S protease in relation to the corresponding commensal growth rate (k 1B1) values. Plot Representation: (Fold change relative to healthy state, (a) Blue- Damaged keratinocytes, Red – S Protease). Full size image The model provides a basis for the dual role as explored in the reviews by Brown & Horswill, 2020 and Nguyen et al., 20178,12. The impact of AMP and immune killing of S. epidermidis on Skin Microenvironment Conditions such as immunocompromised states can significantly weaken the immune response to commensal microorganisms, altering the dynamics and mechanisms that sustain commensal populations. To investigate the effects of such conditions on commensal growth, an immunocompromised condition was simulated in the model by reducing antimicrobial peptide (AMP) production and immune-mediated killing by 10-fold. Due to the reduced immune activity, the healthy commensal population is now perturbed, surging up to 10 5-fold. (Supplementary Fig. 5a). The microbial protease from the commensal targets desmoglein, decreasing its levels and resulting in deeper skin infiltration. The increase in IL-8 and IL-1 levels (Supplementary Fig. 5f, Supplementary Fig. 5c) further intensifies the skin inflammation and the epidermal damage (Supplementary Fig. 5e). However, the system doesn’t settle at these states, indicating absence of monotonic progression from the initial to a steady state instead, exhibits oscillatory behaviour. Generally, the characteristics and amplitude of oscillations depend on negative feedback and time delays. The tuning mechanism for these sustained oscillations involved the neutrophil negative feedback, yet the key mechanism was unclear. To understand feedback mechanisms, the phase plane stimulation-response curve (Fig. 5a) was studied, which represents the steady-state response of neutrophils to the activated keratinocytes. As shown in Fig. 5a, the system starts at a value of 1 and eventually progresses towards cyclic behavior, indicating that an increase in activated keratinocytes leads to an increase in neutrophils. The delay in this feedback from the neutrophil loop prevents the system from settling into a steady state, causing a cyclic pattern hence maintaining a continuum of oscillations. As a physiological response this validates the point that neutrophils play a particularly important role in the immune response during S. epidermidis associated infection8. Notably, the activation of keratinocytes is initiated by the commensal, which triggers additional reinforcement mechanisms, such as growth factor production that gradually reduces damage. Consequently, the response of the commensal to damaged keratinocytes, as depicted in Fig. 5b, also exhibits limit cycle characteristics. This visualization captures the interplay between the skin’s immune system and the maintenance of commensal populations at optimal levels, mediated by neutrophils and the innate immune response, alongside the ongoing damage. Additionally, conditions like immunocompromise, which can lead to sepsis, further exacerbate this damage. Overall, the delayed negative feedback is the primary driver of the system’s spikes and sustained oscillations indicating the delay in immune response is critical to balancing the transition between sepsis and homeostasis. This approach allowed the investigation of how the skin’s immune response influences the maintenance of commensal populations under impaired immune function. Fig. 5: Phase plane analysis of the state variables and their amplitude responses under varying immune sensitivity. Phase plane plot of (a) neutrophil levels and activated keratinocytes (b) S. epidermidis and damaged keratinocytes. The maximum and minimum amplitude values of damaged keratinocytes and S Protease corresponding to commensal growth rate (k 1 B1) values (c) S protease response for reduced immune sensitivity against commensal, (d) damaged keratinocyte response for reduced immune sensitivity against commensal. Plot Representation: ((d)Blue- Damaged keratinocytes, (c) Red – S Protease; Solid line- Maximum amplitude, Dotted lines – Minimum amplitude). Full size image The oscillatory behaviour exhibited by the immunocompromised system when checked for the range of growth rate showed a maximum amplitude of 10 7-fold for protease (Fig. 5c) and the resultant damage to 10 2 -10 3-fold (Fig. 5d) respectively. The lower amplitude of the same is indicated by dotted lines. Comparing with the healthy condition (Fig. 4), it is evident that under an immunocompromised condition, even a minimal growth rate of commensal is sufficient to shift its behaviour to a pathogenic characteristic causing damage to the skin microenvironment. Conversely, increased immune sensitivity, such as in autoimmune conditions, results in the killing of commensal leading to a no- commensal state (Supplementary Fig. 6). Enhanced immune sensitivity does not bring any oscillations within the system, as the commensal population drops too low to restore balance, highlighting the need for homeostasis within the environment. The initial damage that occurs with the excessive immune condition persists at the 7-fold higher state (Supplementary Fig. 6) as with loss in repair mechanism. These predictions point out that conditions such as reduced immunity (higher growth rate) or excessive immune reaction (lesser growth rate) can transition S. epidermidis to either an infectious pathogen and cause loss of balance in epithelial flora. Thus, through the in silico skin model, we could get insights into the fine regulation of the commensal and the immune response to maintain the symbiotic relationship in the skin microenvironment. A comparative evaluation of epidermal barrier integrity with and without S. epidermidis A dysregulated skin barrier marks the onset for the sequelae of pathological skin conditions. Perturbations to the barrier can range from minor cues to extensive cutaneous damage. The complex dynamic system of the skin actively works to restore its integrity. To visualize the cellular interaction involving the commensal participation in the restoration process, a disrupted barrier scenario was set up in the model by increasing the damaged keratinocytes of the model (Dke) to a concentration of 10 4 (Supplementary Fig. 7a). The increased damage has been done to simulate a physiological scenario where barrier integrity is compromised as in a mechanical abrasion which disturbs the barrier integrity triggering a response process, as evidenced by multiple studies in vitro28. The damaged barrier disruption resulted in a temporary rise in the penetrated population (penetrated population B1) as depicted in Supplementary Fig. 7f, and a subsequent decline in commensal growth. In the presence of commensal, several changes occur such as an increase in activated keratinocytes (Kera) as shown in Fig. 6, along with filaggrin29 (10.47-folds), and AMP30 (2.29-folds) as a response to mitigate the damage levels and eventually bringing it down to 8.86-fold within 2 h (Fig. 6 column A) ensuring a healthy steady state (Fig. 6 column C, Supplementary Fig. 7a). Supplementary Figs. 7c and g, demonstrate the dynamics of filaggrin and AMP, both of which rise in response to the barrier damage and finally attain a steady state along with other variables. The commensal population eventually stabilizes within 40 to 60 h (Supplementary Fig. 7b). In contrast, the absence of commensal causes the AMP and growth factors to decrease by 0.11 and 0.01-folds (Fig. 6 column D), attributed to the loss of CD8-mediated interaction and neutrophil activation (Fig. 6 column D). Consequently, restoration of barrier integrity is significantly delayed (Fig. 6 column B) thereby leaving the damaged cells 1017-fold higher (Fig. 6 column D, Supplementary Fig. 7a) than a healthy state. The high epidermal damage is also accompanied by severe inflammation as seen by an increase in cytokines such as IL1, IL8 (4.76-fold), and the protease target, desmoglein (14.25-fold) in an overexpressed state (Fig. 6 column D), aggravating the barrier damage. Figs. S8b and d validate the observed delay in response to varying degrees of barrier disruption. They demonstrate a 1000-fold difference in the time required for damaged keratinocytes to reach a steady state, whereas, in the presence of commensal, the maximum time observed was 9 h (Supplementary Fig. 8a, c). The growth factor-mediated mechanism of tissue restoration was seen to be crucial and dominant in this process. Supplementary Fig. 9 provides data for this observation. Thus, the model has captured the beneficial influence of S. epidermidis in maintaining barrier integrity, when the barrier breach occurs further affirming its strong role in maintaining skin functionality as evidenced by multiple studies22,27 and indicated the alignment of the model to real world trends. Fig. 6: Heat map showing steady state values of variables for barrier disruption condition in presence and absence of S. epidermidis. A Value at 2nd hour in presence of S. epidermidis (B) Value at 2nd hour in absence of S. epidermidis (C) Steady State Value in presence of S. epidermidis (D) Steady State Value in absence of S. epidermidis. Plot Representation: (Fold change relative to healthy state, Kera - Activated Keratinocytes, Neut - Activated Neutrophil, GFr – Growth Factor, DKe – Damaged Keratinocytes, B1 – S. epidermidis, B1p – Penetrated S. epidermidis, Spr – S. epidermidis Protease). Full size image The effect of filaggrin mutation on epidermal barrier repair Filaggrin is a well-studied protein that contributes to epidermal barrier integrity and skin hydration, and its mutation is known to elevate the risk of epidermal damage16,17. To capture the mechanism by which the filaggrin mutation influences the commensal dynamics in the skin, a perturbation of the filaggrin knockout is replicated by making FLG zero in the model. Filaggrin-deficient skin becomes susceptible to commensal penetration, increasing it by 43.07- -fold (Fig. 7 column A) and the resultant protease (3.54-fold), resulting in a 128-fold increase in damaged keratinocytes (Fig. 7 column A, Supplementary Fig. 10e). Furthermore, the commensal population in the skin was eliminated by AMP action (Fig. 7 column A, Supplementary Fig. 10c) subsequently reducing the neutrophil levels to 0.28-fold. (Fig. 7 column A). Loss of microbial interaction with CD8 cells causes the growth factor decline (Supplementary Fig. 10f), leading to the continuous accumulation of damaged cells (Supplementary Fig. 10e, Fig. 7 column A). With filaggrin reduction, a significant decline in total water content to 0.47-fold (Fig. 7 column A, Supplementary Fig. 10a) consistent with loss of hydration due to the complete deterioration of Natural Moisturizing Factor (Supplementary Fig. 10b, Fig. 7 column A) within 3–4 h aligning with the findings from De & Handa, 2012; Jakasa et al., 2011; Kezic et al., 201116,18,29,31. Overall, it is evident that with filaggrin knockdown, barrier functionality and skin immune response are severely compromised. Through model perturbation, we were able to simulate the physiological effect of filaggrin loss of skin. Fig. 7: Heat map showing steady-state variable profiles at 72 hours under filaggrin mutation conditions. A Steady-state values of variables at 72 hours under normal skin conditions with filaggrin mutation (B) Steady-state values of biological at 72 hours under damaged skin conditions with filaggrin mutation. Plot Representation: (Fold change relative to healthy state, Kera - Activated Keratinocytes, Neut - Activated Neutrophil, GFr – Growth Factor, DKe – Damaged Keratinocytes, B1 – S. epidermidis, B1p – Penetrated S. epidermidis, Spr – S. epidermidis Protease). Full size image The effect of filaggrin mutation on damaged skin (Dke=10 4) was evaluated as well. The damaged keratinocytes remained elevated with filaggrin mutation (Supplementary Fig. 11a, Fig. 7 column B). This is because of the decline in commensal (Supplementary Fig. 11b, Fig. 7 column B), leading to a reduction in AMP (0.7-fold) and growth factor (0.09-fold), through the loss of CD8 (0.01-fold) and IL17 (0.03-fold) (Fig. 7 column B). This left the damage unresolved and in a heightened state. Additionally, Supplementary Figs. 11e, f, and Fig. 7 column B indicate reduced natural moisturizing factors and water content, signifying skin dryness. This underscores the crucial role of filaggrin in maintaining a healthy, moisturized state of the skin and its impact on commensal dynamics. Discussion Epidermal skin barrier is constantly exposed to environmental stress. The skin microbiome exerts a symbiotic relationship by relaying multiple signals to maintain barrier integrity. Among a myriad of microbial species inhabiting the skin, S. epidermidis is one of the extensively studied commensals for its beneficial effects on the skin microenvironment32,33,34. To understand the unique role of commensals in the skin, the present study investigated the effect of commensal growth dynamics on epidermal barrier integrity and response in immunocompromised and weakened skin barrier with the filaggrin mutation. Through the in silico model, S. epidermidis- keratinocytes interactions and the consequences of commensal growth were simulated. For this, the model was validated using S. epidermidis population data from growth curve experiments (Supplementary Table 1) followed by an accurate depiction of skin immune responses, highlighting S. epidermidis’s pivotal role in priming the system (Fig. 2a). From the literature data, an increased hBD (human beta-defensin) expression in keratinocytes in the presence of S. epidermidis was observed which matched with our in silico model prediction of increased antimicrobial peptides (AMPs) as seen in Fig. 2b in the Results section ‘Advancing Model Accuracy: Model Benchmarking and Model Validation’. The qualitative observations in the model such as enhanced inflammatory signals and growth factors, indicate immune activation, while also upregulating hydration markers like filaggrin, thereby supporting epidermal homeostasis in the presence of commensals. These align with observation in recent studies such as Ommori et al., 2024; Williams et al., 2024 highlighting role of commensal in the skin’s defense against external insults or injury. These findings further support the model’s ability to capture the dynamic interplay between the skin, immune responses, and growth factors in maintaining a healthy epidermal barrier, characterized by basal levels of growth factors, cytokines, and antimicrobial peptides (AMPs) in the presence of S. epidermidis. In a healthy condition, commensals co-exist with the resident immune cells of the maintaining overall skin health by maintaining the barrier integrity. These model observations are demonstrated in Results section ‘Simulation of Skin microenvironment in response to the commensal S. epidermidis’ Figs. 3 and 4. The tolerance towards the commensal colonization is complexly regulated by regulatory T cells (Treg) within a threshold, which allows the maintenance of the relative abundance of S. epidermidis in an optimal level32,33,34. But certain conditions such as barrier disruption or genetic mutations such as filaggrin can lower the threshold for commensal population to overgrow resulting in impaired barrier function16,35. A recent study by Khadka et al.36, confirms that the commensal exacerbates inflammation in a barrier disruption model thus emphasizing the host barrier context and the permissive window of commensal growth. Our model reflects the transitional state beyond which commensal becomes a pathogen as observed in Fig. 4. Such increased abundance of S. epidermidis correlates with diseases like Atopic dermatitis, where it can elicit inflammatory responses24,37. A greater understanding of this threshold can help in identifying skin types which are infection prone and can help in designing tailored skin regimen for sensitive skin conditions. Further, our skin model investigated the effect of commensal growth dynamics in an immunocompromised condition, the results of which can be seen in Results section ‘The impact of AMP and immune killing of S. epidermidis on Skin Microenvironment’, Fig. S5a–e. As per the findings of (Natsis & Cohen, 2018), an immunocompromised condition makes skin vulnerable leading to infections by the commensal S. epidermidis itself38. The model was able to capture the detrimental effects of S. epidermidis. Disruption to the intact skin is a conducive environment for the transition of commensal to pathogen through higher colonization causing damage to the keratinocytes. In a study by Oh et al., 2013, an increased permissiveness and microbial alterations is observed in patient immunocompromised skin leading to greater microbial colonization39. The system tries to counter this overgrowth by the feedback interactions from the immune cells in the skin which leads to oscillatory behaviour starting with a transient sudden increase of immune cells infiltration followed by dampening of the response (Fig. 5a–d). Immune response is characterized by oscillatory behaviour starting with a transient sudden increase of immune cells infiltration followed by dampening of the response when the pathogen load clears40. The autoimmune disorders model by Valeyev et al.41 demonstrates oscillatory immune cell behavior, supporting our findings that a compromised skin barrier triggers pathogenic disruptions and inflammatory responses, offering valuable insights into the mechanisms driving cutaneous flares. The increase in neutrophils and the delayed negative feedback towards the rising commensal level leads to cyclic trends. Studying the aberrant oscillatory dynamics in immune cells and other skin components can provide greater diagnostic significance for chronic skin diseases like Atopic dermatitis and psoriasis where skin immune is derailed. The model also demonstrated a scenario of heightened immune response by neutrophils and AMPs on commensal. This parallels with auto-immune skin diseases such as psoriasis42, characterized by a higher expression of AMPs which triggers multiple cutaneous inflammatory pathways resulting in skin barrier damage further exacerbated by the loss of S. epidermidis interactions that help maintain immune balance. Hence, the study effectively captures the key cross-talk among the skin components and the commensal highlighting the dual behavior of S. epidermidis depending on the skin microenvironmental changes. In a healthy condition, commensals co-exist with the resident cells of the immune system maintaining overall skin health by maintaining the barrier integrity. The mechanisms ensuring the immune tolerance are exceedingly complex, involving the interplay of diverse regulatory mechanisms. The threshold represents the level of immunogenic stimulation required to elicit an immune response43 which is observed in Fig. 4, where beyond a critical point, commensal transitions into pathogenic behavior. An immunocompromised condition makes the skin vulnerable, reducing the threshold of immunogenic response. For e.g., presence of a filaggrin mutation disrupts the stratum corneum integrity lowering skin inflammatory thresholds leading to higher antigen penetration24. A better understanding of immune regulatory thresholds may hold greater value in designing better therapies for skin barrier conditions. Filaggrin mutations have long been linked to compromised barrier function and increased susceptibility to various diseases such as Atopic dermatitis13,44,45. Our investigation into the impact of filaggrin mutations on both healthy and barrier-disrupted skin showed that the antimicrobial activity decreased with filaggrin mutation as reported by Nath et al.17. We simulated the effect of filaggrin mutation by a virtual knockdown of FLG. This led to a loss of moisturizing effect as observed in decreased NMF in the Results section ‘The effect of Filaggrin Mutation on epidermal barrier repair’, Fig. 7. This led to more penetration due to protease action present in the skin barrier leading to a barrier breach triggering AMPs for anti-microbial action against the commensal population. Our model predicts that a filaggrin deficit skin characterized by a dry skin and higher protease action becomes hostile for normal commensal colonization, triggering a higher epidermal damage as seen in Fig. 7 A. The greater penetration and abnormal commensal -immune response is also seen in an in vivo murine study where filaggrin null mice exhibit an abnormal early life skin response in barrier disruption scenario46,47. The present work demonstrated the unique features of commensal-host interrelations and how coordinated regulation of commensal S. epidermidis, and the skin cells such as neutrophils, cytokines and growth factors cross talks are required for a balanced and functional skin environment. The model not only effectively captured essential biological responses to commensal colonization, such as AMP production and inflammation, but also provided key predictive hypotheses which can be studied further experimentally. The impact of filaggrin mutation on commensal behaviour has not been previously studied, though several studies on the impact of filaggrin loss on S. aureus colonization exist. Our findings reveal a dysregulated immune response driven by increased barrier permeability due to filaggrin loss, underscoring the need for experimental validation. Similarly, though oscillations in inflammation have been modelled in other diseases like rheumatoid Arthritis48 and psoriasis49, our model predicts that aberrant oscillations are generated in an immunocompromised skin which can again be experimentally validated for better understanding of cutaneous inflammatory responses. Future experiments focusing on quantifying threshold dynamics and commensal behavior based on our model predictions can enhance our understanding the commensal behaviour and interplay along with other pathogens like Staphylococcus aureus and Cutibacterium acnes in inflammatory skin conditions like Atopic dermatitis and acne respectively. While many studies have examined the impact of microbes on skin health, a systems biology approach such as this may help in deciphering the complex dynamics within the skin environment. We believe that this model can be utilized to explore skin dysbiosis which is the basis of many cutaneous diseases and also in the rational designing of therapeutic strategies for skin. Methods Mathematical modelling and simulation The quantitative description of the model was set by ordinary differential equations (ODEs) which were formulated using mass balance and Hill’s equation representing the multiple interactions and the feedback within the system (detailed in the Supplementary information). It was subsequently visualized using Ode45 solver in MATLAB. Following parameter optimization, the model’s suitability was evaluated by achieving a controlled, non-zero steady state that reflects the healthy condition. Advancing model accuracy: parameter sensitivity analysis In our study, we worked with approximately 129 model parameters, some of which were fitted based on experimental data, while others were assumed based on known physiological outcomes. Each parameter within the system was analyzed to understand its distinct role in the overall operation. The comprehensive list of all parameters and their corresponding values can be found in the (Supplementary Tables 3 and 4). A defined quantitative metric to evaluate the change in the system variables, namely, S. epidermidis, Growth Factor, AMP, Neutrophil, Damaged Keratinocytes, Filaggrin (termed as outputs, O) with a change in the parameter values (termed as Input, I) ranging from 10% to 100%, considering both increase and decrease facilitated the understanding on the sensitive parameters along with the relationship between input and output. The sensitivity coefficient ‘S’ was quantified by comparing the relative change in the output (O) to its mean value with the relative change in the input (I) parameter to its mean value. This metric measures the system’s responsiveness to variations in the input parameter. Mathematically ‘S’ is expressed as: $$\frac{{\rm{\partial }}O}{O}\,=\,\frac{\varDelta O}{\frac{{O}{f}+{O}{i}}{2}}=\frac{{o}{f}-\,{o}{i}}{\frac{{O}{f}+{O}{i}}{2}},\,\frac{{\rm{\partial }}I}{I}=\,\frac{\varDelta I}{\frac{{I}{f}+{I}{i}}{2}}\,=\frac{{I}{f}-\,{I}{i}}{\frac{{I}{f}+{I}{i}}{2}}$$ $$S=\frac{\partial O}{\partial I\,}.\frac{\bar{I}}{\bar{O}}$$ therefore, $$S=\frac{{o}{f}-\,{o}{i}}{{o}{f}+\,{o}{i}}\,.\frac{{I}{f}+\,{I}{i}}{{I}{f}-{I}{i}}$$ The subscript i indicates the initial value, and f represents the final value obtained after the parameter change. Cell culture experiments Culturing of HaCaT HaCaT cells (Cat No. T0020001, Addexbio) were grown and maintained in high glucose DMEM (Cat. No. AL219A, Himedia) supplemented with 0.03 mM Calcium chloride (C5670, Sigma), 1 mM sodium pyruvate, and 10% FBS (Fetal Bovine Serum) at 37 °C with 5%CO2. 80% confluent cells were trypsinized and plated in 24 well plates at a density of 80,000 cells/well. Post confluency, the cells were differentiated for 72 h by adding a differentiation medium containing 1.4 mM Calcium chloride. Culturing of S. epidermidis S. epidermidis (ATCC 12228 ) was subcultured from glycerol stock on a TSA plate and incubated at room temperature at 37 °C for 20 h, the OD was adjusted accordingly and used for growth experiments. Growth of S. epidermidis on HaCaT After differentiation of HaCaT, OD of S. epidermidis was adjusted to 0.2 and the cells were further diluted and 10 4 CFU was added on to HaCat wells and in wells with no HaCaT cells, the plates were incubated at 37 °C with 5% CO2 for 24 and 48 h, the growth of S. epidermidis was enumerated by plating them on TSA (BD Cat log- 236950) plates. The colonies after 24 h were counted, and number of CFU/ml of S. epidermidis was plotted against the time of incubation. Data availability All data generated or analyzed during this study are included in this published article [and its supplementary information files]. References Abdallah, F., Mijouin, L. & Pichon, C. Skin immune landscape: inside and outside the organism. Mediators Inflamm.2017, 5095293 (2017). ArticlePubMedPubMed CentralGoogle Scholar Park, Y. J., Kim, C. W. & Lee, H. K. Interactions between host immunity and skin-colonizing staphylococci: no two siblings are alike. Int. J. Mol. Sci.20, 718 (2019). ArticleCASPubMedPubMed CentralGoogle Scholar Ann, Q. Neutrophil Extracellular Traps: As Antimicrobial Peptides. Oral. Rehabilitation Dent. (2019). Ohnemus, U. et al. Regulation of epidermal tight-junctions (TJ) during infection with exfoliative toxin-negative Staphylococcus strains. J. Investig. Dermatol.128, (2008). Piipponen, M., Li, D. & Landén, N. X. The immune functions of keratinocytes in skin wound healing. Int. J. Mol. Sci.21, 8790 (2020). ArticleCASPubMedPubMed CentralGoogle Scholar Harder, J., Gläser, R. & Schröder, J. M. Human antimicrobial proteins – effectors of innate immunity. J. Endotoxin Res.13, 317–338 (2007). 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Staphylococcus epidermidis isolates from atopic or healthy skin have opposite effect on skin cells: potential implication of the AHR pathway modulation. Front. Immunol.14, 1098160 (2023). ArticleCASPubMedPubMed CentralGoogle Scholar Ommori, R., Shinkuma, Satoru & Asada, Hideo Staphylococcus epidermidis augments human β-defensin-3 synthesis through the transforming growth factor alpha-epidermal growth factor receptor cascade. J. Dermatol. Sci.116, 34–40 (2024). ArticleCASPubMedGoogle Scholar Lai, Y. et al. Activation of TLR2 by a small molecule produced by staphylococcus epidermidis increases antimicrobial defense against bacterial skin infections. J. Investig. Dermatol.130, 2211–2221 (2010). ArticleCASPubMedGoogle Scholar Hughes, A. J., Tawfik, S. S., Baruah, K. P., O’Toole, E. A. & O’Shaughnessy, R. F. L. Tape strips in dermatology research. Br. J. Dermatol.185, 26–35 (2021). ArticleCASPubMedGoogle Scholar Archer, N. K. et al. Injury, dysbiosis, and filaggrin deficiency drive skin inflammation through keratinocyte IL-1α release. J. Allergy Clin. Immunol.143, 1426–1443 (2019). ArticleCASPubMedGoogle Scholar Pfalzgraff, A. et al. Antimicrobial endotoxin-neutralizing peptides promote keratinocyte migration via P2X7 receptor activation and accelerate wound healing in vivo. Br. J. Pharmacol.175, 3581–3593 (2018). ArticleCASPubMedPubMed CentralGoogle Scholar Kezic, S. et al. Levels of filaggrin degradation products are influenced by both filaggrin genotype and atopic dermatitis severity. Allergy.: Eur. J. Allergy. Clin. Immunol.66, 934–940 (2011). ArticleCASGoogle Scholar Scharschmidt, T. C. et al. A Wave of Regulatory T Cells into Neonatal Skin Mediates Tolerance to Commensal Microbes. Immunity43, 1011–1021 (2015). ArticleCASPubMedPubMed CentralGoogle Scholar Scharschmidt, T. C. Establishing tolerance to commensal skin bacteria: timing is everything. Dermatol. Clin.35, 1–9 (2017). ArticleCASPubMedPubMed CentralGoogle Scholar Abt, M. C. et al. Commensal Bacteria Calibrate the Activation Threshold of Innate Antiviral. Immun. Immun.37, 158–170 (2012). CASGoogle Scholar Parlet, C. P., Brown, M. M. & Horswill, A. R. Commensal staphylococci influence Staphylococcus aureus skin colonization and disease. Trends Microbiol.27, 497–507 (2019). ArticleCASPubMedPubMed CentralGoogle Scholar Khadka, V. D., Markey, L., Boucher, M. & Lieberman, T. D. Commensal Skin Bacteria Exacerbate Inflammation and Delay Skin Barrier Repair. J. Investig. Dermatol.144, 2541–2552 (2024). ArticleCASPubMedGoogle Scholar Byrd, A. L. et al. Staphylococcus aureus and Staphylococcus epidermidis strain diversity underlying pediatric atopic dermatitis. Sci. Transl. Med. 9, eaal4651 (2017). Natsis, N. E. & Cohen, P. R. Coagulase-negative Staphylococcus skin and soft tissue infections. Am. J. Clin. Dermatol.19, 671–677 (2018). ArticlePubMedGoogle Scholar Oh, J. et al. The altered landscape of the human skin microbiome in patients with primary immunodeficiencies. Genome Res23, 2103–2114 (2013). ArticleCASPubMedPubMed CentralGoogle Scholar Della Marca, R., Machado Ramos, M., da, P., Ribeiro, C. & Soares, A. J. Mathematical modelling of oscillating patterns for chronic autoimmune diseases. Math. Methods Appl. Sci.45, 7144–7161 (2022). ArticleGoogle Scholar Valeyev, N. V. et al. A systems model for immune cell interactions unravels the mechanism of inflammation in human skin. PLoS Comput. Biol.6, e1001024 (2010). ArticlePubMedPubMed CentralGoogle Scholar Chang, H. W. et al. Alteration of the cutaneous microbiome in psoriasis and potential role in Th17 polarization. Microbiome6, 154 (2018). ArticlePubMedPubMed CentralGoogle Scholar Guram, K. et al. A threshold model for T-cell activation in the era of checkpoint blockade immunotherapy. Front. Immunol.10, 491 (2019). ArticleCASPubMedPubMed CentralGoogle Scholar Scharschmidt, T. C. et al. Filaggrin deficiency confers a paracellular barrier abnormality that reduces inflammatory thresholds to irritants and haptens. J. Allergy Clin. Immunol.124, 496–506 (2009). ArticleCASPubMedPubMed CentralGoogle Scholar Clausen, M. L. et al. Association of disease severity with skin microbiome and filaggrin gene mutations in adult atopic dermatitis. JAMA Dermatol154, 293–300 (2018). ArticlePubMedPubMed CentralGoogle Scholar Gonzalez, J. R. & Scharschmidt, T. C. Elucidating the Role of the Skin Barrier Protein Filaggrin in the Adaptive Immune Response to Commensals. FASEB J.34, (2020). Gonzalez, J. R. et al. FLG Deficiency in Mice Alters the Early-Life CD4+ T-Cell Response to Skin Commensal Bacteria. J. Investig. Dermatol.143, 790–800 (2023). ArticleCASPubMedGoogle Scholar Moise, N. & Friedman, A. Rheumatoid arthritis — a mathematical model. J. Theor. Biol.461, 17–33 (2019). ArticlePubMedGoogle Scholar Özköse, F. Modeling of psoriasis by considering drug influence: a mathematical approach with memory trace. Comput. Biol. Med.168, 107791 (2024). ArticlePubMedGoogle Scholar Download references Acknowledgements The research findings presented in this manuscript were funded by Unilever Research Ltd., Bangalore to MetFlux Research Private Limited, India. We acknowledge the contributions of Vijayalakshmi AB and Umesh Vazirani in data mining. Author information Authors and Affiliations MetFlux Research Private Limited, Bengaluru, India Sumathi Kalankariyan,Anjana Thottapillil,Abha Saxena,Manoj Srivatsn S&KV Venkatesh Unilever R & D, Bangalore, India Vinitha Kadamkode,Renu Kapoor,Rupak Mitra&Janhavi Raut Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai, Maharashtra, India KV Venkatesh Authors 1. Sumathi KalankariyanView author publications Search author on:PubMedGoogle Scholar 2. Anjana ThottapillilView author publications Search author on:PubMedGoogle Scholar 3. Abha SaxenaView author publications Search author on:PubMedGoogle Scholar 4. Manoj Srivatsn SView author publications Search author on:PubMedGoogle Scholar 5. Vinitha KadamkodeView author publications Search author on:PubMedGoogle Scholar 6. Renu KapoorView author publications Search author on:PubMedGoogle Scholar 7. Rupak MitraView author publications Search author on:PubMedGoogle Scholar 8. Janhavi RautView author publications Search author on:PubMedGoogle Scholar 9. KV VenkateshView author publications Search author on:PubMedGoogle Scholar Contributions M.S.S. developed the model and codes for it. S.K. performed model validation, simulation and analysis, manuscript writing and editing. A.T. manuscript writing, review and editing. A.S. supervised and managed the project. V.K., R.K., R.M., J.R. conceptualization, experimental results and manuscript review. K.V.V. conceptualization, supervised the model validation and analysis, manuscript review. Corresponding author Correspondence to KV Venkatesh. Ethics declarations Competing interests The authors declare no competing interests. Additional information Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Supplementary information Supplementary_Information Rights and permissions Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. 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Copy shareable link to clipboard Provided by the Springer Nature SharedIt content-sharing initiative Subjects Dynamic networks Nonlinear dynamics Download PDF Sections Figures References Abstract Introduction The in silico skin model on commensal-host interactions Results Discussion Methods Data availability References Acknowledgements Author information Ethics declarations Additional information Supplementary information Rights and permissions About this article Advertisement Fig. 1: A holistic in silico model of skin: the model illustrates the skin microenvironment as a complex network of interactions. View in articleFull size image Fig. 2: Model benchmarking and validation of S. epidermidis growth dynamics and AMP, IL1, IL8 responses. View in articleFull size image Fig. 3: Heat map showing steady-state responses of key variables to variations in S. epidermidis growth rate. View in articleFull size image Fig. 4: The steady-state values of damaged keratinocytes and S protease in relation to the corresponding commensal growth rate (k 1B1) values. View in articleFull size image Fig. 5: Phase plane analysis of the state variables and their amplitude responses under varying immune sensitivity. View in articleFull size image Fig. 6: Heat map showing steady state values of variables for barrier disruption condition in presence and absence of S. epidermidis. View in articleFull size image Fig. 7: Heat map showing steady-state variable profiles at 72 hours under filaggrin mutation conditions. View in articleFull size image Abdallah, F., Mijouin, L. & Pichon, C. Skin immune landscape: inside and outside the organism. Mediators Inflamm.2017, 5095293 (2017). ArticlePubMedPubMed CentralGoogle Scholar Park, Y. J., Kim, C. W. & Lee, H. K. Interactions between host immunity and skin-colonizing staphylococci: no two siblings are alike. Int. J. Mol. Sci.20, 718 (2019). ArticleCASPubMedPubMed CentralGoogle Scholar Ann, Q. Neutrophil Extracellular Traps: As Antimicrobial Peptides. Oral. Rehabilitation Dent. (2019). Ohnemus, U. et al. Regulation of epidermal tight-junctions (TJ) during infection with exfoliative toxin-negative Staphylococcus strains. J. Investig. Dermatol.128, (2008). Piipponen, M., Li, D. & Landén, N. X. The immune functions of keratinocytes in skin wound healing. Int. J. Mol. Sci.21, 8790 (2020). ArticleCASPubMedPubMed CentralGoogle Scholar Harder, J., Gläser, R. & Schröder, J. M. Human antimicrobial proteins – effectors of innate immunity. J. Endotoxin Res.13, 317–338 (2007). ArticleCASPubMedGoogle Scholar Nakamizo, S. et al. Commensal bacteria and cutaneous immunity. Semin. Immunopathol.37, 73–80 (2015). ArticleCASPubMedGoogle Scholar Nguyen, T. H., Park, M. D. & Otto, M. Host response to Staphylococcus epidermidis colonization and infections. Front. Cell. Infect. Microbiol.7, 90 (2017). ArticlePubMedPubMed CentralGoogle Scholar Zheng, Y. et al. Commensal Staphylococcus epidermidis contributes to skin barrier homeostasis by generating protective ceramides. Cell Host Microbe30, 301–313 (2022). ArticleCASPubMedPubMed CentralGoogle Scholar Nakatsuji, T. et al. Staphylococcus aureus Exploits Epidermal Barrier Defects in Atopic Dermatitis to Trigger Cytokine Expression. J. Investig. Dermatol.136, 2192–2200 (2016). ArticleCASPubMedGoogle Scholar Williams, M. R. et al. Staphylococcus epidermidis activates keratinocyte cytokine expression and promotes skin inflammation through the production of phenol-soluble modulins. Cell Rep.42, 113024 (2023). ArticleCASPubMedPubMed CentralGoogle Scholar Brown, M. M. & Horswill, A. R. Staphylococcus epidermidis-Skin friend or foe? PLoS Pathog.16, e1009026 (2020). ArticleCASPubMedPubMed CentralGoogle Scholar Stefanovic, N. & Irvine, A. D. Filaggrin and beyond: new insights into the skin barrier in atopic dermatitis and allergic diseases, from genetics to therapeutic perspectives. Ann. Allergy Asthma Immunol.132, 187–195 (2024). ArticleCASPubMedGoogle Scholar Riethmuller, C. et al. Filaggrin breakdown products determine corneocyte conformation in patients with atopic dermatitis. J. Allergy Clin. Immunol.136, 1573–1580 (2015). ArticleCASPubMedPubMed CentralGoogle Scholar Armengot-Carbo, M., Hernández-Martín, A. & Torrelo, A. The role of filaggrin in the skin barrier and disease development. Actas Dermosifiliogr.106, 86–95 (2015). ArticleCASPubMedGoogle Scholar De, D. & Handa, S. Filaggrin mutations and the skin. Indian J. Dermatol. Venereol. Leprol.78, 545–551 (2012). ArticlePubMedGoogle Scholar Nath, S. et al. Dysbiotic Lesional Microbiome With Filaggrin Missense Variants Associate With Atopic Dermatitis in India. Front. Cell Infect. Microbiol.10, 570423 (2020). ArticleCASPubMedPubMed CentralGoogle Scholar Jakasa, I. et al. Skin barrier function in healthy subjects and patients with atopic dermatitis in relation to filaggrin loss-of-function mutations. J. Invest. Dermatol.131, 540–542 (2011). ArticleCASPubMedGoogle Scholar Tanaka, R. J. & Ono, M. Skin disease modeling from a mathematical perspective. J. Investig. Dermatol.133, 1472–1478 (2013). ArticleCASPubMedGoogle Scholar Yue, R. & Dutta, A. Computational systems biology in disease modeling and control: review and perspectives. npj Syst. Biol. Appl.8, 37 (2022). ArticleCASPubMedPubMed CentralGoogle Scholar Thibault Greugny, E., Fages, F., Radulescu, O., Szmolyan, P. & Stamatas, G. N. A skin microbiome model with AMP interactions and analysis of quasi-stability vs stability in population dynamics. Theor. Comput. Sci.983, 114294 (2024). ArticleGoogle Scholar Naik, S. et al. Commensal-dendritic-cell interaction specifies a unique protective skin immune signature. Nature520, 104–108 (2015). ArticleCASPubMedPubMed CentralGoogle Scholar Duckney, P. et al. The role of the skin barrier in modulating the effects of common skin microbial species on the inflammation, differentiation and proliferation status of epidermal keratinocytes. BMC Res Notes6, 474 (2013). ArticlePubMedPubMed CentralGoogle Scholar Cau, L. et al. Staphylococcus epidermidis protease EcpA can be a deleterious component of the skin microbiome in atopic dermatitis. J. Allergy Clin. Immunol.147, 955–966 (2021). ArticleCASPubMedGoogle Scholar Landemaine, L. et al. Staphylococcus epidermidis isolates from atopic or healthy skin have opposite effect on skin cells: potential implication of the AHR pathway modulation. Front. Immunol.14, 1098160 (2023). ArticleCASPubMedPubMed CentralGoogle Scholar Ommori, R., Shinkuma, Satoru & Asada, Hideo Staphylococcus epidermidis augments human β-defensin-3 synthesis through the transforming growth factor alpha-epidermal growth factor receptor cascade. J. Dermatol. Sci.116, 34–40 (2024). ArticleCASPubMedGoogle Scholar Lai, Y. et al. Activation of TLR2 by a small molecule produced by staphylococcus epidermidis increases antimicrobial defense against bacterial skin infections. J. Investig. Dermatol.130, 2211–2221 (2010). ArticleCASPubMedGoogle Scholar Hughes, A. J., Tawfik, S. S., Baruah, K. P., O’Toole, E. A. & O’Shaughnessy, R. F. L. Tape strips in dermatology research. Br. J. Dermatol.185, 26–35 (2021). ArticleCASPubMedGoogle Scholar Archer, N. K. et al. Injury, dysbiosis, and filaggrin deficiency drive skin inflammation through keratinocyte IL-1α release. J. Allergy Clin. Immunol.143, 1426–1443 (2019). ArticleCASPubMedGoogle Scholar Pfalzgraff, A. et al. Antimicrobial endotoxin-neutralizing peptides promote keratinocyte migration via P2X7 receptor activation and accelerate wound healing in vivo. Br. J. Pharmacol.175, 3581–3593 (2018). ArticleCASPubMedPubMed CentralGoogle Scholar Kezic, S. et al. Levels of filaggrin degradation products are influenced by both filaggrin genotype and atopic dermatitis severity. Allergy.: Eur. J. Allergy. Clin. Immunol.66, 934–940 (2011). ArticleCASGoogle Scholar Scharschmidt, T. C. et al. A Wave of Regulatory T Cells into Neonatal Skin Mediates Tolerance to Commensal Microbes. Immunity43, 1011–1021 (2015). ArticleCASPubMedPubMed CentralGoogle Scholar Scharschmidt, T. C. Establishing tolerance to commensal skin bacteria: timing is everything. Dermatol. Clin.35, 1–9 (2017). ArticleCASPubMedPubMed CentralGoogle Scholar Abt, M. C. et al. Commensal Bacteria Calibrate the Activation Threshold of Innate Antiviral. Immun. Immun.37, 158–170 (2012). CASGoogle Scholar Parlet, C. P., Brown, M. M. & Horswill, A. R. Commensal staphylococci influence Staphylococcus aureus skin colonization and disease. Trends Microbiol.27, 497–507 (2019). ArticleCASPubMedPubMed CentralGoogle Scholar Khadka, V. D., Markey, L., Boucher, M. & Lieberman, T. D. Commensal Skin Bacteria Exacerbate Inflammation and Delay Skin Barrier Repair. J. Investig. Dermatol.144, 2541–2552 (2024). ArticleCASPubMedGoogle Scholar Byrd, A. L. et al. Staphylococcus aureus and Staphylococcus epidermidis strain diversity underlying pediatric atopic dermatitis. Sci. Transl. Med. 9, eaal4651 (2017). Natsis, N. E. & Cohen, P. R. Coagulase-negative Staphylococcus skin and soft tissue infections. Am. J. Clin. Dermatol.19, 671–677 (2018). ArticlePubMedGoogle Scholar Oh, J. et al. The altered landscape of the human skin microbiome in patients with primary immunodeficiencies. Genome Res23, 2103–2114 (2013). ArticleCASPubMedPubMed CentralGoogle Scholar Della Marca, R., Machado Ramos, M., da, P., Ribeiro, C. & Soares, A. J. Mathematical modelling of oscillating patterns for chronic autoimmune diseases. Math. Methods Appl. Sci.45, 7144–7161 (2022). ArticleGoogle Scholar Valeyev, N. V. et al. A systems model for immune cell interactions unravels the mechanism of inflammation in human skin. PLoS Comput. Biol.6, e1001024 (2010). ArticlePubMedPubMed CentralGoogle Scholar Chang, H. W. et al. Alteration of the cutaneous microbiome in psoriasis and potential role in Th17 polarization. Microbiome6, 154 (2018). ArticlePubMedPubMed CentralGoogle Scholar Guram, K. et al. A threshold model for T-cell activation in the era of checkpoint blockade immunotherapy. Front. Immunol.10, 491 (2019). ArticleCASPubMedPubMed CentralGoogle Scholar Scharschmidt, T. C. et al. Filaggrin deficiency confers a paracellular barrier abnormality that reduces inflammatory thresholds to irritants and haptens. J. Allergy Clin. Immunol.124, 496–506 (2009). ArticleCASPubMedPubMed CentralGoogle Scholar Clausen, M. L. et al. Association of disease severity with skin microbiome and filaggrin gene mutations in adult atopic dermatitis. JAMA Dermatol154, 293–300 (2018). ArticlePubMedPubMed CentralGoogle Scholar Gonzalez, J. R. & Scharschmidt, T. C. Elucidating the Role of the Skin Barrier Protein Filaggrin in the Adaptive Immune Response to Commensals. FASEB J.34, (2020). Gonzalez, J. R. et al. FLG Deficiency in Mice Alters the Early-Life CD4+ T-Cell Response to Skin Commensal Bacteria. J. Investig. Dermatol.143, 790–800 (2023). ArticleCASPubMedGoogle Scholar Moise, N. & Friedman, A. Rheumatoid arthritis — a mathematical model. J. Theor. Biol.461, 17–33 (2019). ArticlePubMedGoogle Scholar Özköse, F. Modeling of psoriasis by considering drug influence: a mathematical approach with memory trace. Comput. Biol. Med.168, 107791 (2024). 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2187	https://aithor.com/essay-examples/idealism-and-realism	AI Detector AI Humanizer Essay Editor Pricing Support Home›Essay Examples›Idealism and Realism Idealism and Realism Topics:Education, Philosophy, Ethics This paper was proofread by: Mateusz Brodowicz 14 min readPublished: April 19, 2024 Aithor writes your essay in 2 min, not days AI undetectable Original References Topic finder Suggested topics 1. Introduction The term "international relations" refers to the exchange of thoughts, goods, and warfare between different nations. The most significant and complicated problem in international relations is to find more peaceful solutions in a world that is still in turmoil. To tackle these problems, many political scientists have attempted to combine the use of idealism and realism. Their usage is often used in attempts to solve an issue, more often though, idealism is most recommended. The concepts of idealism and realism are extensively referred to in the theories, opinions, and working history of each political scientist relevant today. Writers who use idealism point toward highly efficient solutions with a positive outcome. While realists may state that these idealistic views are not realistic and it would be far more effective to develop solutions that provide a satisfactory ending utilizing minimal resources. The two ideas of international relations are particularly based on theory. The theory of realism is based on the world as it is, using force and a great amount of power to perform a specific task and obtain an outcome. Such a theory can be related to many moments in history as well as today's methods of warfare and other forms of power use. Realists are greatly pessimistic about world affairs and cite this as a reason for the use of power. Idealism is based on the hope for a better world. Idealists are very optimistic about world affairs and usually cite a minimal use of power as a means to a fruitful end. Ideally, a perfect world results while using minimal power is the desired goal. Today, these two theories are still present in the efforts of international relations. Often, more pragmatic solutions usually turn toward realism, whereas deeply humanitarian efforts are idealistic in nature. 1.1 Definition of Idealism Idealism is a term with several related meanings. It comes via idea from the Greek idein which means to see. Idealism is the thesis that reality is known through ideas. It asserts that reality as it appears is a product of the understanding. Idealists tend to think that reality, as accessible to the human understanding, is in some way constructed by the mind. Furthermore, it holds that the mind is the most real thing that we can know; it is the fount of everything experiential. This thesis can be held explicitly or implicitly. Some beliefs are clearly idealist. Bishop Berkeley, for example, held that the esse of things is percipi, directly asserting that there is no reality to anything apart from our experience of it. Other thinkers may not make any explicit assertions about the nature of reality, but when they reflect upon the nature of what we can know it becomes clear that their epistemology and their internalized model of reality are bursting with idealist convictions. Idealism can also be distinguished according to metaphysical and epistemological strains. During the modern period it tended to be a metaphysical idealism, but this is not a necessary relation. The basic thrust of idealism is the same whether it asserts that reality is mind dependent because it is simply the content of the cognitive process, or because there is a divine creator whose thoughts articulate everything that is real. Finally, idealism has a normative aspect. It can be idealistic about what is real in contrast to what is merely actual, holding that reality is not to be found in the present state of things, but in some higher state towards which present existence should be measured. This is change idealism and is prevalent in political and ethical thought. 1.2 Definition of Realism Definition of realism. Realism, like idealism, has more than one meaning. However, the essential features of it may be identified. In opposition to the other-worldliness of idealism, realism holds that the world of matter, or the universe, is primary. The ultimate reality is the world of physical objects. Idealists are sometimes understood as denying the existence of the external world, and maintaining that the whole of reality is mental. This is an error. In Plato's theory of forms, sometimes called Platonic idealism, the objects of the physical world are often considered as "copies" of the perfect, real forms. The reality of idealism, therefore, does not necessarily exclude the reality of the physical world. Yet idealism does make the reality of the external world a dependent reality, a secondary reality. "Reality" in any case is a metaphysical question. What are we asking when we ask if something is real? What is reality? Although it is very difficult to define, a realist would say that the primary reality is what we can sense, feel, and touch. A realist equates reality with existence. Therefore, a realist will define that as being real with which we can interact, be it objectively or with tools. A second point concerning reality is that for an object to exist, it must have an effect. This is to say that there must be something that differentiates between a thing existing and it not existing. If there is no difference, then there is no difference in the two states and the thing that we are discussing can be considered to not exist. Finally, the reality of a thing lies in the space that it occupies. Idealists and even common sense folk attempt to perform an act of division between an object and the space that it occupies. Realists, however, say that the only separation between the two is a mental one. Therefore, an object's reality lies in the space that it occupies, in its effect, and in its characteristics whether or not we observe them. 2. Key Differences Between Idealism and Realism Idealists believe in a more abstract reality that is unseen and mainly in the form of ideas. In terms of Plato's divided line, the realists would be said to believe in the world of the senses rather than the world of the forms. Plato himself was an idealist, as well as many who came after him. The realist theory of reality, at its most basic level, holds that 'reality' is the state of things as they actually exist rather than in the mind. This means that if the external reality in question is a tree, the only place the tree exists is where the tree actually is, and the tree cannot exist in another place or time because there is only one actual place and time of the event or thing. This is known as the principle of 'uni-location'. The tree will continue to exist until it no longer exists, and therefore it does not cease to be real while nothing changes about it. A realist views the truth as something that is established as real or existent. The concept of the truth being 'established' implies that the truth is certain and sure. An idea can only be established as true when it is an idea that can be seen to exist, and there is no doubt as to its existence. This concerns the concept of 'correspondence', where a statement is said to be true if it corresponds to a real state of affairs. It is clear that the realists' view is in line with our possible natural understanding of what is real and what is true. The implications of this are that the realists' view is functional. By this, it is meant that there is no reason or use for anything that does not influence the actual state of affairs. This indicates that certain aspects of reality can be more or less 'high level', where higher level is defined by what has more influence on the state of affairs. The final point to make on the realists' view of reality is that reality itself is interconnected and provides a setting or arena where events occur. This is as far as we need to go with realist theory, and we can see that there are many aspects of reality that we cannot see, mainly due to the fact that interactions in the arena can be very complex and it would take too long to find out what happened! 2.1 Perspective on the Nature of Reality There are several key differences between idealism and realism. The first of these is in their perspective on the nature of reality. Idealism is an old philosophy, beginning in ancient India in the East, and in the ancient civilizations of Greece and China in the West. Early western forms were not metaphysical, but tended towards political theory. The philosophy in this form says that there is an ultimate reality in another world than the empirical one we live in. In the 18th century, idealism was developed by a man named Immanuel Kant. Kant's idealism is a form of metaphysical idealism in which the ultimate reality is in the empirical world that we know, but the nature of it is made what it is by the way we perceive it, with space, time, and causality being the structures in terms of which we understand and thus construct the world. In general, however, idealism refers to the belief that reality is based upon thought or ideas. Realism, on the other hand, is a view of the external world as it is, with emphasis on seeing things as they are, and the belief in things that neither matter to the senses nor can be imagined. This is a contrast from the idealist position that the reality of the world is dependent on the mind. Idealists would say that things such as justice or beauty are real in that they are ideas. For Plato, these ideas in the intellect are the only things that are ultimately real. The forms are unchanging, perfect, and can only be mimicked by anything in the empirical realm. For Plato, the only way to attain true knowledge was to study these forms. Aristotelian realism diverges from Plato at this point by suggesting that form and matter are the same for substances. This is to say that the idea of what the substance is does not exist independently from the thing itself. Still, Aristotle held the belief that there was a reality separate from the empirical world in which we lived, but that it was attained through a rational understanding of the empirical world. 2.2 Approach to Knowledge and Truth Idealists believe that reality is a creation of the mind. It is that which man gives meaning to. Idealism goes back to ancient Greece. Plato, a student of Socrates and teacher of Aristotle, has been the most well-known western idealist. Plato believed that the phenomenal world was an imperfect representation of the real world. It was these ideas that caused Aristotle to split from the teachings of his master and begin his own school in Athens. Aristotle turned his focus on the things of this world, studying the natural sciences, developing his own system of logic, and theory of knowledge. Realism is descended from his work and is a greater influence on modern-day philosophy. His most well-known successor in ancient times was Plotinus, but after that, his work was forgotten in the west until the Renaissance. 2.3 Implications for Ethics and Morality Both idealism and realism are based on the perception of the world around us. Idealism is based on the view that reality is a world within a person's mind. Realism is based on the view of a world separate from our minds. The main issue, or what can be seen as the only issue, with idealism is that it is highly implausible. Most people believe that the chair they are sitting on is a material object outside of their own mind. To the idealist, the chair only exists as a mental construct. To many this is a difficult claim to accept. So difficult in fact, that even some idealists attempt to interpret the nature of reality in ways similar to realists due to the overwhelming implausibility. "Berkeley is a case in point. In grappling with the problem of the material world, he ended up conceding at least half the battle to his realist opponents" (Curd & Cover 682). This is incredibly damaging to idealism. The entire point of philosophy is to provide insight beyond that of common belief, if some form of a lead has to be taken from realism then it severely discredits idealism as a philosophy. If the nature of reality is defined by material objects outside of a person's own mind, is it possible to argue that there are no material objects at all? This argument would be almost impossible, for if there really are material objects, the argument would have to say there is some non-perceivable way that we are being deceived into thinking we perceive an object in a certain way, and it is this way. This is not only complex and confusing but is far from the certainty and simplicity sought by most philosophers. However, if the argument is successful, it would be a major victory for idealism in a philosophical sense, yet it would yield no real world implications. 3. Applications and Criticisms of Idealism and Realism There is a long history of idealist thought regarding the relationship between the ideal and the material. Plato was an early idealist who believed that the material world is not a perfect sphere, and that the real world is comprised of intangible "forms," such as the form of the good or the form of beauty. Other famous idealists include St. Augustine, Thomas Aquinas, and Descartes. Modern idealism has come from the classical idealists, but more from its criticism. It seemed to empiricists that the knowledge of man can only be of the world, the whole is the same as the parts and bad is only privation of good. Idealists believed that reality is changeless and that bad is not a thing in itself. Locke's distinction between primary and secondary qualities has also been an issue of discussion for idealists as they thought a thing is not as it appears and therefore appearance is reality. Idealists believe in the power of mind, as opposed to the belief of empiricists in the impotence of man. Kant's Copernican revolution has also had an immense influence for idealists in the philosophy of the mind and epistemology. The quote "percepts without concepts are blind" shows Kant's idea of the mind's active role in knowledge and it has become an idealistic catch-cry. Modern idealism still issues from the criticism of empiricism and is seen more as a viable philosophy due to advances in modern science and logic which have seemed to support the idealists' critique of empiricism. For instance, modern physics has shown reality to be quite different from the world that is portrayed by the senses and logicians have well demonstrated the difficulties and contradictions of the dualism of a "private language" and a "public language" for the world. Idealism has been a prevalent philosophy in the history of Education, whether teachers know it or not. Rousseau was an educational idealist who believed in "natural education" and who thought that the goal of education is the realization of the ideal. Transpose by the Wordsworth and Coleridge and the romanticists affirmed idealist principles in education. Modern education theory is still largely based on idealistic views. 3.1 Idealism in Philosophy and Education After discussing the general concept of idealism, knowledge, and reality, we move to talk about idealism in philosophy and education. The reason for discussing them together is because Macmurray has mentioned that the scope of philosophy being too wide would take another paper, and the nature of idealism as a theory is best revealed in terms of educational practice. Now, what idealists envision is to seek the highest possible aims and to make the fullest possible realization of human potentialities. To them, education is a dynamic process which is primarily concerned with the betterment and self-realization of the pupil, and it is not merely instruction or the preparation for adult life. Hence, education can be seen as the means and the pupil the growing point towards the realization of cultural aims and personal ideal. Idealists see the aim of education to be the fostering of intelligence and wisdom in order to manifest the better aspects of human nature and to transcend through a higher level of consciousness. This would eventually lead to a transformation in the present culture and a qualitative change in the human condition. By this, idealists mean the release of the potential goodness and intelligence of man from its present state of subservience to bad impulses and the force of circumstance. This change is crucial for the survival of man's intellect as it is necessary for solving the problems that face humanity today. In doing so, education should also endeavor to provide a cultural heritage to future generations in the form of classics of literature and art. This is important because culture is the revelation of deeper secular meaning in man's common life and a thing of beauty which is transcendent to everyday experience. Finally, culture takes place in an organic society; hence, the ideal aim of education is to train and educate future builders of a better society. Education in an idealist society will reflect the philosophy in every way. First is the issue of the curriculum, which must be carefully planned with human values and the highest ideals in view. The subject matter of courses must be selected for its intrinsic value and such that it can stimulate and engage the active minds of students. Hence, there is likely to be a greater emphasis on humanities and the classics with students only specializing at a later age. The methods of a teacher would be to present a living example of the life of the mind to students and to seek to inspire in them by communicated enthusiasm and passionately held beliefs. The teacher is also a key figure in the guidance of students and transmission of culture; hence, there are implications that the teacher's role will be greatly enhanced. An idealist method of education is the Socratic Method, a form of cooperative argumentative dialogue between individuals, based on asking and answering questions to stimulate critical thinking and to draw out ideas and underlying presumptions. High-quality education institutions and well-paid teachers would also be ideal in a realist society, but obviously, this is extremely hard to achieve in any society. 3.2 Realism in International Relations At the international level of analysis, the position of realism and idealism has a long series of proponents and opponents. Notes Morgenthau, "international politics are struggle for power". He defines power in terms of national interest defined in terms of power. He says that the concept of interest is consolidated in categories or moral values. National interest provides an objective standard by which the foreign policies of states can be compared and a common measure for the solution of international problems. Realism is based on the thinking of Thucydides, the father of realism. According to him "the standard for discriminating between a good and a bad statesman is the ability to discern reality". This phrase emphasizes the superiority of one who uses his five senses and experience to understand and adjust to the world as it is, rather than the one who bases his judgments on what the world ought to be according to some set of abstract principles. This becomes a clear rejection of the idealist analysis and the Utopian approach to international relations. Utopians are those who believe in a world government which will provide security by laws and institutions. This is believed to be attainable by creating a moral and rational public opinion which will in turn shift the policies of the states from power to legalistic and moralistic ones. But the realists call this a very naïve effort. They argue that there are no effective agencies to protect the rules of law, the world is anarchical and states will only try to preserve, and improve their positions relative to each other. Ideally Utopians appear to have the stronger foundations for the argument but realists have time and time again gotten the better of the discussion. The failure of Utopian thinking to arrive with any practical solution to the problem of war and security has been a clear defeat. Utopians are unable to argue against the critics who point out that their plan requires harmonizing the divergent and often competing interests of states into a single set of laws and institutions which is just not possible. The bare existence of Utopian thought is known as the liberal program which will become the topic of the next essay. 3.3 Criticisms of Idealism and Realism Idealism and the theory that underpins it has been subjected to a battery of criticism. Most criticisms are of the very general form that idealism is merely a nice dream. The general criticisms and the discussion here have taken place with reference to idealism in foreign policy. However, it should be noted that the distinction between the various 'types' of idealism is significant when examining the criticisms. Simple idealism, or the idealism of the 'law of the heart' has often been said to be a noble and moral approach to the world, but one which is hopelessly impractical. This is an off the cuff dismissal, but it does identify a weakness. If the simple idealist were to claim that his approach was the only moral approach he may well be able to support his claim. The idealism with a more articulated theory comes under more fire. Interdependence/liberal idealism is often said to be a recent development in world politics and one which is fast becoming out of date. The claim is that the theory is essentially a product of western postwar exuberance. In the war and preceding years, western man had seen two things which had horrified him; the continuing development of means of destruction and the recession from humanity which was often present in the dictators' 'reasons of state'. This had been confirmation of older fears concerning the nature of the world. One need only mention the 'war to end all wars' and the League of Nations which was felt would prevent a similar occurrence. The recent horror had driven home the lesson that man could no longer afford to associate in power blocs to the end that the ends of state, given a Machiavellian interpretation, would always be pursued. This had led modern man to look back into his own just war tradition and the classical theories of how to make the world a better place. They were found to be strangely relevant. The Cold War and the development of hot wars in the third world witnessed attempts to apply liberal ideals to pacify situations. Regardless of the failure of these actions, they revealed a further thing to liberal idealists. Wars were not simply caused by the malevolence of men or groups; they were caused by the conditions under which people politics believed that altering. If the right conditions were set, it was usually possible to avoid conflict between states, notably by concerning differences through international law. This last part of the modern idealist's argument is the one which has been most harshly attacked. The belief that social and legal progress can resolve all international problems and eventually do away with war has been called a misconceived interpretation of both history and human nature. Related articles View all articles ## How to Cite Othello in MLA Format William Shakespeare’s Othello is one of the most frequently studied plays in literature courses. 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The goal is to clearly show the reader whic... 3 minutes ## كيف يمكن لمبادرات التدوير في الجامعات تقليل الهدر مقدمة تعد مبادرات التدوير في الجامعات من المبادرات الحيوية التي تتوجه نحو تحقيق استدامة بيئية واجتماعية، حيث تشكل جزءًا أساسيًا من الجهود الرامية إلى تقليل الهدر وتحقيق الاستفادة القصوى من الموارد المتاحة. تُعرف مبادرات التدوير بأنها عمليات جمع، وفرز، ومعالجة المواد القابلة لإعادة الاستخدام، بهدف تقليص حجم المخلفات التي تنتهي إلى المدافن أو المحارق، وبالتالي الحد من الآثار السلبية المترتبة على البيئة والصحة العامة. في سياق الجامعات، التي تعد مراكز للمعرفة والابتكار، تتبنى العديد من المؤسسات ... 2 minutes ## كيف يمكن للمكتبات الرقمية أن تسد فجوة الوصول إلى المعرفة مقدمة تُعدُّ المكتبات الرقمية من الاحتياجات الأساسية التي تمثل تطوراً نوعياً في مجال تيسير الوصول إلى المعرفة، خاصة في ظل التحول الرقمي الذي يشهده العالم في كافة القطاعات. فهي ليست فقط مصدراً لمجموعة ضخمة من المصادر المعلوماتية والموارد التعليمية، وإنما تمثل منصة متاحة على مدار الساعة، تُوفر فرصة للمستخدمين من مختلف الفئات والبيئات الجغرافية للاستفادة منها دون قيود المكان أو الزمان. في سياق سعي المجتمعات إلى تحقيق التنمية المستدامة، برزت المكتبات الرقمية كوسيلة فعالة لتجاوز الحواجز التقليدية... 2 minutes ## كيف يمكن للمدن العربية الحد من التلوث باستخدام النقل العام النظيف مقدمة تُعَّدُ التلوث الهوائي أحد التحديات الكبرى التي تواجهها المدن العربية، حيث يتسبب انبعاث الغازات الضارة والجسيمات الدقيقة من وسائل النقل التقليدية في انتشار الأمراض التنفسية والقلق البيئي، مما يهدد جودة الحياة وسلامة السكان. يُسهم النقل العمومي الملوث في تصاعد معدلات تلوث الهواء نتيجة الاعتماد المفرط على المركبات الخاصة التي تستهلك الوقود الأحفوري وتُطلق غازات الدفيئة، الأمر الذي يتفاقم بسبب ضعف البنية التحتية للنقل العام وتردي كفاءتها وغياب الاستراتيجيات المستدامة. لذلك، فإن تحويل نظم ... 2 minutes ## كيف يمكن للمدن الصحراوية أن تصبح أكثر استدامة مقدمة تعد المدن الصحراوية من الظواهر الحضرية التي ظهرت نتيجة لتداخل العديد من العوامل البيئية، والاجتماعية، والاقتصادية، مما أدى إلى تكوين فضاءات سكنية واقتصادية تتكيف مع الظروف القاسية التي تفرضها البيئة الصحراوية. فهي تجمع بين التحديات المرتبطة بندرة الموارد المائية، وارتفاع درجات الحرارة، وتأثيرات العوامل الطبيعية الأخرى، إلى جانب الضغوط البشرية التي تؤدي إلى تزايد معدلات النمو السكاني وتوسع العمران بسرعة غير مسبوقة. إن هذا الواقع يتطلب إعادة تقييم منهجيات التخطيط العمراني، مع تبني استرات... 2 minutes ## كيف يمكن استخدام الألعاب التعليمية لتحفيز التعلم مقدمة تُعد الألعاب التعليمية من الأدوات الفعالة التي تسهم في إحداث فارق نوعي في عملية التعلم، إذ تجمع بين عناصر الترفيه والتفاعل العلمي لتوفير بيئة تعليمية محفزة وفعالة. يُنظر إليها كوسيلة حديثة تتجاوز الطُرق التقليدية، إذ تتيح للمتعلمين فرصة التفاعل المباشر مع المحتوى المعرفي من خلال أنشطة تفاعلية تتسم بالمتعة، مما يعزز من مستوى الاهتمام ويُحفز الدافعية للتعلم. تُعتمد الألعاب التعليمية على مبدأ أن التعلم يصبح أكثر فعالية عندما يكون ممتعًا، حيث يتمكن المتعلم من استيعاب المفاهيم بشكل أعمق من ... 2 minutes ## كيف يغير التعليم التعاوني طريقة التفكير في الفصول الدراسية مقدمة تتجه الأبحاث التربوية الحديثة نحو تبني استراتيجيات تعليمية تفاعلية تركز على تعزيز مشاركة الطلاب وتنمية مهارات التفكير النقدي والإبداعي، ومن بين هذه الاستراتيجيات يبرز التعليم التعاوني كوسيلة فاعلة لتحقيق هذه الأهداف. يُعرف التعليم التعاوني بأنه نمط من أنماط التدريس ينطلق من التعاون بين الطلاب ضمن مجموعات صغيرة، حيث يتشاركون في استكمال المهام وتحليل المشكلات بشكل جماعي، مما يعزز من قدراتهم على التفكير بصورة أكثر عمقاً واستقلالية. تتجلى أهمية هذا النهج في قدرته على توفير بيئة تعليمية محف... 2 minutes Start your journey with Aithor Personalized papers are done in minutes. 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2188	https://www.maths.tcd.ie/~dwilkins/Courses/MAU23302/EuclidStudyNotes/StudyNote_EuclidElementsBk01Prop08.pdf	Study Note—Euclid’s Elements, Book I, Proposition 8 David R. Wilkins © Trinity College Dublin 2022 Proposition 8 of Book I of Euclid’s Elements of Geometry encapsulates the SSS Congruence Rule. The proposition ensures that if the sides BC, CA and AB of a triangle ABC are respectively equal to the sides EF, FD and DE of a triangle DEF, then the angles of the first triangle at vertices A, B and C are respectively equal to the angles of the second triangle at D, E and F. Euclid supposes that the first triangle could in principle be moved, with-out changing the lengths of the sides, or the angles at the vertices, so as move the vertices B and C of the first triangle to the locations of the vertices E and F of the second triangle, whilst ensuring that the vertex C of the first triangle is moved to a location on the same side of the side EF of the second triangle as the vertex D of that second triangle. A B C D E F G On applying the preceding Proposition 7, it follows that, when the first triangle ABC is moved in this fashion, the vertex A is moved to the posi-tion where the vertex D of the second triangle is located. Consequently the first triangle can be moved, without altering the lengths of its sides and the angles at its vertices, so as to coincide exactly with the second triangle. Con-sequently the angles of the first triangle must be equal to the corresponding angles of the second. 1 The Alternative Proof of Proposition 8 attributed to Philo of Byzantium. Proclus, in his commentary on Proposition 8 of the first book of Euclid’s Elements of Geometry supplies an alternative proof of Proposition 8, which he attributes to Philo of Byzantium (who lived in the third century before the common era). Philo’s proof runs as follows. Let ABC and DEF be triangles for which the sides BC, CA and AB are equal in length to the sides EF, FD and DE respectively of the second triangle, as indicated in the following figure: E F D B C A We must show that the angles of the first triangle at A, B and C are respectively equal to the angles of the second triangle at D, E and F. The proof strategy attributed to Philo of Byzantium requires a triangle to be placed on the side EF of the second triangle, and on the opposite side to the vertex D of the second triangle, so that the triangle GEF so placed is equal to in all respects to the triangle ABC. It should thus be possible to move the triangle ABC, without altering the lengths of its sides, or the angles of its vertices, so that the resulting triangle has vertices at G, E and F, where the sides EF, FG and GE of the new triangle are respectively equal in length to the respective sides BC, CA and AB of the first given triangle, and the angles of the new triangle at G, E and F are respectively equal to the angles of the first given triangle at A, B and C. The points D and G are then joined by a finite straight line. Five cases arise, depending on the location of the point where the finite straight line DG intersects the infinite straight line that passes through the points E and F: (i) the first case arises when the straight line DG passes though the point F; (ii) the second case arises when the straight line DG passes though a point on EF lying between E and G; 2 (iii) the third case arises when the straight line DG intersects the infinite straight line that passes through E and F at a point on the opposite side of the point F to the point E; (iv) the fourth case arises when the straight line DG passes through the point E, and the proof in this case is exactly analogous to that in the first case; (v) the fifth case arises when the straight line DG intersects the infinite straight line that passes through E and F at a point on the opposite side of the point E to the point F, and the proof in this case is exactly analogous to that in the third case. The first three cases are discussed by Proclus in his commentary on Propo-sition 8 of Book I of Euclid’s Elements of Geometry. The results in the last two cases may be deduced immediately from those in the first and third cases on interchanging the labels on the vertices so that vertices B, C, E and F are relabelled C, B, F and E respectively. 3 E F D G E F D G E F D G E F D G E F D G (i) (ii) (iii) (iv) (v) 4 We first prove the result in case (i). The configuration in this case is as follows. E F D G B C A Thus sides BC and EF are equal to one another, sides AC, DF and GF are equal to one another, sides AB, DE and GE are equal to one another, angles CAB and FGE are equal to one another, angles ABC and GEF are equal to one another, and angles BCA and EFG are equal to one another. Now ADG is an isosceles triangle with base DG and equal sides AD and AG. It follows from the Isosceles Triangle Theorem (Euclid, Elements, I.5, also known as the Pons Asinorum), that the angles EDG and EGD (coloured in yellow) at the base of this isosceles triangle are equal to one another, and consequently the angle EDF is equal to the angle EGF (given that, in this case, the point F lies on the finite straight line joining the points D and G. But the angle BAC is also equal to the angle EGF, and angles that are equal to the same angle are equal to one another (applying the first of Euclid’s common notions). Consequently the angles BAC and EDF are equal to one another. 5 E F D G B C A We now apply the SAS Congruence Rule (Euclid, Elements, I.4) to the triangles BAC and EDF. The sides BA and AC are equal to the sides ED and DF respectively. Moreover we have shown that the angle BAC enclosed by the first pair of sides is equal to the angle EDF enclosed by the second pair of sides. E F D G B C A Applying the SAS Congruence Rule (Euclid, Elements, I.4), we conclude that the angles of the triangle BAC are equal to the corresponding angles of the triangle EDF and thus the angles ABC and DEF are equal to one another, and also the angles BCA and EFD are equal to one another. This completes the proof of Proposition 8 in case (i) in which the straight line DG passes through the point F. 6 We next prove the result in case (ii). In this case the straight lines DG and EF intersect at a point that lies between E and F. Thus the configuration in this case is as follows. This configuration is constructed so that the sides and angles of the tri-angle ABC are equal to the corresponding sides and angles of the triangle GEF, so that the sides BC, CA and AB of the triangle ABC are respectively equal to the sides EF, FG and GA of the triangle GEF, and the angles of the first triangle at A, B and C are respectively equal to the angles of the second triangle at G, E and F. E F D G B C A The triangle EDG is then an isosceles triangle with base DG and equal sides ED and EG. The angles EDG and EGD of this triangle at vertices D and G are therefore equal to one another by the Isosceles Triangle Theorem (Euclid, Elements, Proposition 5). E F D G 7 Similarly the triangle FDG is an isosceles triangle with base DG. There-fore the angles FDG and FGD are equal to one another. E F D G Now the angle EDF is the sum of the angles EDG and FDG, and the angle EGF is the sum of the angles EGD and FGD. We have shown that the angles EDG and FDG are respectively equal to the angles EGD and FGD. Also, by the second of Euclid’s common notions, when equals are added to equals, the sums are themselves equal. It therefore follows that the angles EDF and EGF (coloured in yellow) are equal to one another. E F D G B C A We now apply the SAS Congruence Rule (Euclid, Elements, I.4) to the triangles BAC and EDF. The sides BA and AC are equal to the sides ED and DF respectively. Moreover we have shown that the angle BAC enclosed by the first pair of sides is equal to the angle EDF enclosed by the second pair of sides. Applying the SAS Congruence Rule (Euclid, Elements, I.4), we conclude that the angles of the triangle BAC are equal to the corresponding angles 8 E F D G B C A of the triangle EDF and thus the angles ABC and DEF are equal to one another, and also the angles BCA and EFD are equal to one another. This completes the proof of Proposition 8 in case (ii) in which the straight lines DG and EF intersect at a point lying between the points E and F. Finally we prove the result in case (iii). In this case the straight line DG intersects the infinite straight line that passes through the points E and F as some intersection point that lies on the opposite side of the point F to the point E. Thus the configuration in this case is as follows. This configuration is constructed so that the sides and angles of the tri-angle ABC are equal to the corresponding sides and angles of the triangle GEF, so that the sides BC, CA and AB of the triangle ABC are respectively equal to the sides EF, FG and GA of the triangle GEF, and the angles of the first triangle at A, B and C are respectively equal to the angles of the second triangle at G, E and F. E F D G B C A The triangle EDG is then an isosceles triangle with base DG and equal 9 sides ED and EG. The angles EDG and EGD of this triangle at vertices D and G are therefore equal to one another by the Isosceles Triangle Theorem (Euclid, Elements, Proposition 5). E F D G Similarly the triangle FDG is an isosceles triangle with base DG. There-fore the angles FDG and FGD are equal to one another. E F D G Now the angle EDF is the difference of the angles EDG and FDG, and the angle EGF is the difference of the angles EGD and FGD. We have shown that the angles EDG and FDG are respectively equal to the angles EGD and FGD. Also, by the third of Euclid’s common notions, when equals are subtracted from equals, the remainders are themselves equal. It therefore follows that the angles EDF and EGF (coloured in yellow) are equal to one another. As in the proof for case (ii), we now apply the SAS Congruence Rule (Euclid, Elements, I.4) to the triangles BAC and EDF. The sides BA and AC are equal to the sides ED and DF respectively. Moreover we have shown 10 E F D G B C A that the angle BAC enclosed by the first pair of sides is equal to the angle EDF enclosed by the second pair of sides. E F D G B C A Applying the SAS Congruence Rule (Euclid, Elements, I.4), we conclude that the angles of the triangle BAC are equal to the corresponding angles of the triangle EDF and thus the angles ABC and DEF are equal to one another, and also the angles BCA and EFD are equal to one another. This completes the proof of Proposition 8 in case (iii) in which the straight line DG intersects the infinite straight line that passes through the points E and F as some intersection point that lies on the opposite side of the point F to the point E. As noted before, the required result in cases (iv) and (v) follows on ap-plying the resultd in cases (i) and (iii) with vertices appropriately relabelled. We have therefore completed the discussion of the alternative proof of Propo-sition 8 of Book I of Euclid’s Elements of Geometry attributed to Philo of Byzantium. 11
2189	https://math.stackexchange.com/questions/1591126/number-of-ways-to-color-n-objects-with-3-colors-if-colors-must-be-used-once	combinatorics - Number of ways to color n objects with 3 colors if colors must be used once - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR == Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… 1. 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You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Number of ways to color n objects with 3 colors if colors must be used once Ask Question Asked 9 years, 8 months ago Modified9 years, 8 months ago Viewed 10k times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. \begingroup I am aware this combinatoric problem (which I got from Discrete Mathematics Elementary and Beyond) has been answered on here before, but from what I can tell the solution I have come up with is different than the answers on other posts but I am not sure so I would like to see where my thought process is wrong. My solution was originally: 3!\cdot3^{n-3} My rationale was: The first one has the option of 3 colors. Second, two, and the third, one. Each one after that can have any 3 colors since we've met our initial requirements. The below 'diagram' shows how many options each object has for colors. ``` \| 3 \| \| 2 \| \| 1 \| \| 3 \| \| 3 \| \| 3 \| ----- ----- ----- ----- ----- ----- .... ``` To me this makes sense for the answer, similarly to string combination problems. However it appears that my answer yields numbers different to the correct answer people arrive at using the Inclusion-Exclusion principal formula. Could someone explain how this is yields the wrong answer? I am assuming I am not accounting for all cases in some way. I am seeing the following as a correct answer in the case of distinctly labeled answers 3^n - (2^n)\cdot{{3}\choose{2}} + 1^n\cdot{{3}\choose{1}} Which for this specific problem reduces to: 3^n - 2^n\cdot3 + 3 combinatorics discrete-mathematics combinations coloring Share Cite Follow Follow this question to receive notifications edited Apr 13, 2017 at 12:21 CommunityBot 1 asked Dec 28, 2015 at 1:35 Dominic FarolinoDominic Farolino 531 1 1 gold badge 8 8 silver badges 16 16 bronze badges \endgroup Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 3 Save this answer. Show activity on this post. \begingroup The number of ways to color n objects with 3 colors is 3^n. Now we need to subtract those colorings which did not use one of the three colors. The number of ways to color with just 2 colors is 2^n and since there are three colors that can be omitted we have 3^n-3\cdot 2^n but now we have removed all colorings with an given color twice so we need to add those back again and we get a final answer: 3^n-3\cdot 2^n +3 This is an example of the inclusion-exclusion principle. Share Cite Follow Follow this answer to receive notifications answered Dec 28, 2015 at 1:48 Rene SchipperusRene Schipperus 40.5k 2 2 gold badges 38 38 silver badges 84 84 bronze badges \endgroup 1 \begingroup Ok this makes sense...How would this apply for the problem of coloring 5 objects with 4 different colors where all 4 need to be used? Would the number be 4^5-{{4}\choose{3}}\cdot3^n+{{4}\choose{2}}\cdot2^n\endgroup –Dominic Farolino Commented Dec 28, 2015 at 4:10 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. \begingroup Your solution relies on the assumption that the first three objects must be different colors, which is not the case. It becomes combinatorially cumbersome to account for all the different possibilities for which objects are differently colored, which is why the principle of inclusion/exclusion is so useful here. Share Cite Follow Follow this answer to receive notifications answered Dec 28, 2015 at 1:47 kccukccu 21.2k 1 1 gold badge 25 25 silver badges 43 43 bronze badges \endgroup 1 \begingroup I believe you, I just need to read about that principal then haha\endgroup –Dominic Farolino Commented Dec 28, 2015 at 1:57 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions combinatorics discrete-mathematics combinations coloring See similar questions with these tags. 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2190	https://www.healthline.com/health/how-many-teeth-do-adults-have	Health Conditions All + Breast Cancer + Cancer Care + Caregiving for Alzheimer's Disease + Chronic Kidney Disease + Chronic Obstructive Pulmonary Disease (COPD) + Digestive Health + Eye Health + Heart Health + Menopause + Mental Health + Migraine + Multiple Sclerosis (MS) + Parkinson’s Disease + Psoriasis + Rheumatoid Arthritis (RA) + Sleep Health + Type 2 Diabetes + Weight Management ### Condition Spotlight All + Controlling Ulcerative Colitis + Navigating Life with Bipolar Disorder + Mastering Geographic Atrophy + Managing Type 2 Diabetes ### Wellness Topics All + CBD + Fitness + Healthy Aging + Hearing + Mental Well-Being + Nutrition + Parenthood + Recipes + Sexual Health + Skin Care + Sleep Health + Vitamins and Supplements + Women's Wellness ### Product Reviews All + At-Home Testing + Men's Health + Mental Health + Nutrition + Sleep + Vitamins and Supplements + Women's Health ### Featured Programs All + Your Guide to Glucose Health + Inflammation and Aging + Cold & Flu Season Survival Guide + She’s Good for Real ### Featured Video Series Pill Identifier FindCare Drugs A-Z ### Lessons All + Crohn’s and Ulcerative Colitis Essentials + Diabetes Nutrition + High Cholesterol + Taming Inflammation in Psoriasis + Taming Inflammation in Psoriatic Arthritis ### Newsletters All + Anxiety and Depression + Digestive Health + Heart Health + Migraine + Nutrition Edition + Type 2 Diabetes + Wellness Wire ### Lifestyle Quizzes Find a Diet Find Healthy Snacks Weight Management How Well Do You Sleep? 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Join Bezzy on the web or mobile app. All + Breast Cancer + Multiple Sclerosis + Depression + Migraine + Type 2 Diabetes + Psoriasis ### Follow us on social media Can't get enough? Connect with us for all things health. How Many Teeth Should I Have? Medically reviewed by Christine Frank, DDS — Written by Taylor Norris — Updated on March 30, 2019 Depending on if all of your adult teeth came in, or if you’ve ever had teeth removed or damaged, adults typically have 32 teeth. Babies and children have fewer teeth. Do you know how many teeth you have? Depending on if all of your adult teeth came in, or if you’ve ever had teeth removed or damaged, all adults have roughly the same number of teeth. Teeth are an important part of both your bone structure and your digestion. Every tooth has three layers: the enamel, dentin, and pulp. Enamel. Enamel is the visible, white, outer layer. This hard surface protects the inner layers of each tooth from the damage of decay or injury. Enamel is the hardest tissue in the whole body. Dentin. This is the middle layer of the tooth, which is the most similar to bone tissue. Dentin makes up the majority of the tooth structure. It has millions of tiny tubes connecting it to the life-source of the tooth: pulp. Pulp. The pulp is the living core of each tooth, and the innermost layer. The pulp is made up of blood and nerves. The part of the tooth above the gumline is called the crown. And the part of the tooth below the gumline is called the root, which attaches the tooth to your jawbone. How many teeth do babies have? On average, babies first start getting new teeth around 6 months. But it’s not unheard of to see a 3 month old with a tooth, or a 1 year old with just one tooth still. All a child’s “baby teeth” should be in between 2-3 years old. Baby teeth are also called primary, or deciduous teeth, because they’re temporary and they fall out. A full set of baby teeth is 20 teeth: 10 on top and 10 on bottom. We get baby teeth because as a child, our mouths aren’t big enough for a full set of adult teeth, but kids still need teeth to chew. So all people are born with both full sets of teeth in their jaw. First come the baby teeth and later, as kids grow older, they lose them and gain their larger, adult teeth one by one. Even though baby teeth are “temporary,” it’s important that they’re kept clean so that they’re healthy, to maintain lifelong oral health. Tooth decay in childhood can adversely affect adult teeth. Brush your child’s baby teeth for 2 whole minutes, just as you do your own. How to care for baby teeth Start a brushing routine as soon as your baby’s first tooth pops up. Use a clean baby towel with warm water to rub each tooth. You can rub to clean gums, too. Let your child chew on a cold, wet towel. This can soothe teething soreness. When your toddler has most of their teeth in, you can switch to brushing with a kid’s toothbrush (usually one with soft bristles). Be sure to use one with a small head so that you can comfortably and effectively brush all their teeth. How many teeth do adults have? People start losing their baby teeth and getting their adult set as early as 5 years old. Adults have 32 teeth. You should have this full set of adult teeth by your late teens. Adult teeth include incisors, canines, premolars, and molars: 8 incisors. Your four front teeth on the top and bottom are sharp for holding and cutting food. Incisors also help you sense the texture and kind of food you eat. 4 canines or cuspids. The pointed teeth on the top and bottom are called canine teeth, or cuspids. They have cusps for grabbing and tearing food. 8 premolars. These teeth are between the cuspids and molars both physically and in form. Premolars look like molars but they have two cusps and are sometimes called bicuspids. Premolars cut and tear food. 12 molars. You have eight molars on top and bottom. They have broad chewing surfaces to grind down food before it’s finally swallowed. This includes wisdom teeth, your third set of molars, which can show up as late as your early 20s and are often removed. Not everyone can comfortably fit all 32 adult teeth in their mouth. Science showsTrusted Source that human jaws began shrinking around the time humans transitioned from hunter-gather societies to sedentary farmers. This could be because the new foods that humans could eat were cooked softer and easier to chew, and thus eating to survive didn’t require a big strong jaw. Having too many teeth, or overcrowding, can cause: misaligned teeth increased decay impacted wisdom teeth risk for periodontal disease This is why many people have their wisdom teeth removed. Take care of your teeth You get two full sets of teeth over your lifetime. As a baby, you have 20 teeth, and as an adult you should have 32 teeth. Among the 32 teeth, each has its own function in the chewing and eating process. Take good care of your teeth and keep your gums healthy in order to avoid cavities and other overall health issues. ADVERTISEMENT Erectile dysfunction treatment at a more affordable cost Hims offers doctor-trusted generic medications around 95% cheaper than brand name alternatives. Visit online to find the right treatment for you with ease. Hard Mints - starting at less than $2/dose GET STARTED No insurance needed 200K+ people treated Discreet delivery How we reviewed this article: Healthline has strict sourcing guidelines and relies on peer-reviewed studies, academic research institutions, and medical journals and associations. We only use quality, credible sources to ensure content accuracy and integrity. You can learn more about how we ensure our content is accurate and current by reading our editorial policy. Anatomy of teeth. (n.d.). Eruption charts. (n.d.). Pinhasi R, et al. (2015). Incongruitybetween affinity patterns based on mandibular and lower dental dimensionsfollowing the transition to agriculture in the near east, anatolia and europe. Our experts continually monitor the health and wellness space, and we update our articles when new information becomes available. Current Version Mar 30, 2019 Written By Taylor Norris Edited By Judy Lee May 18, 2018 Medically Reviewed By Christine Frank, DDS Share this article Medically reviewed by Christine Frank, DDS — Written by Taylor Norris — Updated on March 30, 2019 Was this article helpful? Yes No Read this next What Are the Different Types of Teeth Called? Medically reviewed by Christine Frank, DDS Do you know the names of all your teeth? We’ll go over all the different types of teeth in both children and adults, including canines, incisors… READ MORE Does Everyone Have Wisdom Teeth? Medically reviewed by Jennifer Archibald, DDS Wisdom teeth are four molars that emerge later in life, typically young adulthood. But does everyone have these extra molars? READ MORE When Do Baby Teeth Fall Out and Adult Teeth Come In? Medically reviewed by Karen Gill, M.D. Baby teeth start falling out around age 6. Here's a general timeline regarding what you can expect. READ MORE Wisdom Teeth Medically reviewed by Christine Frank, DDS Why do we have wisdom teeth, if they’re always getting removed? Once essential for an early human diet of roots, leaves, meat, and nuts, wisdom teeth… READ MORE What You Should Know About an Impacted Wisdom Tooth Medically reviewed by Christine Frank, DDS What does it mean to have an impacted wisdom tooth? We’ll explain what an impacted wisdom tooth is and how it’s treated. READ MORE Treatment Options for Tooth Decay in Children Medically reviewed by Jennifer Archibald, DDS Tooth decay is a common concern in children. The use of fluoride, dental sealants, and regular dental cleanings can help treat and prevent tooth decay. READ MORE How Long Does It Take to Recover from Dry Socket, and How Long Are You at Risk? Medically reviewed by Christine Frank, DDS With treatment, dry socket symptoms usually resolve within 7 days. You’re at the greatest risk for dry socket the first week following tooth… READ MORE One-Sided Jaw Pain: Likely Causes and Treatment Medically reviewed by the Healthline Medical Network Likely causes of jaw pain include dental problems, TMJ disorders, and sinusitis. Rarely, pain on one side of the jaw can be an early sign of heart… READ MORE Byte Aligners Review Medically reviewed by Christine Frank, DDS Everything you need to know about Byte aligners and how to get your own. READ MORE Home Remedies for Dry Socket Medically reviewed by Christine Frank, DDS Dry socket after a tooth extraction is preventable but can lead to severe pain and discomfort. Home remedies such as honey, black tea, and warm salt… READ MORE
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2192	https://pubmed.ncbi.nlm.nih.gov/36318248/	REBASE: a database for DNA restriction and modification: enzymes, genes and genomes - 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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REBASE: a database for DNA restriction and modification: enzymes, genes and genomes Richard J Roberts1,Tamas Vincze1,Janos Posfai1,Dana Macelis1 Affiliations Expand Affiliation 1 New England Biolabs, 240 County Road, Ipswich, MA 01938 USA. PMID: 36318248 PMCID: PMC9825431 DOI: 10.1093/nar/gkac975 Item in Clipboard REBASE: a database for DNA restriction and modification: enzymes, genes and genomes Richard J Roberts et al. Nucleic Acids Res.2023. Show details Display options Display options Format Nucleic Acids Res Actions Search in PubMed Search in NLM Catalog Add to Search . 2023 Jan 6;51(D1):D629-D630. doi: 10.1093/nar/gkac975. Authors Richard J Roberts1,Tamas Vincze1,Janos Posfai1,Dana Macelis1 Affiliation 1 New England Biolabs, 240 County Road, Ipswich, MA 01938 USA. PMID: 36318248 PMCID: PMC9825431 DOI: 10.1093/nar/gkac975 Item in Clipboard Full text links Cite Display options Display options Format Abstract REBASE is a comprehensive and extensively curated database of information about the components of restriction-modification (RM) systems. It is fully referenced and provides information about the recognition and cleavage sites for both restriction enzymes and DNA methyltransferases together with their commercial availability, methylation sensitivity, crystal and sequence data. All completely sequenced genomes and select shotgun sequences are analyzed for RM system components. When PacBio sequence data is available, the recognition sequences of many DNA methyltransferases (MTases) can be determined. This has led to an explosive growth in the number of well-characterized MTases in REBASE. The contents of REBASE may be browsed from the web rebase.neb.com and selected compilations can be downloaded by FTP (ftp.neb.com). Monthly updates are also available via email. © The Author(s) 2022. Published by Oxford University Press on behalf of Nucleic Acids Research. PubMed Disclaimer Similar articles REBASE--a database for DNA restriction and modification: enzymes, genes and genomes.Roberts RJ, Vincze T, Posfai J, Macelis D.Roberts RJ, et al.Nucleic Acids Res. 2015 Jan;43(Database issue):D298-9. doi: 10.1093/nar/gku1046. Epub 2014 Nov 5.Nucleic Acids Res. 2015.PMID: 25378308 Free PMC article. REBASE--enzymes and genes for DNA restriction and modification.Roberts RJ, Vincze T, Posfai J, Macelis D.Roberts RJ, et al.Nucleic Acids Res. 2007 Jan;35(Database issue):D269-70. doi: 10.1093/nar/gkl891.Nucleic Acids Res. 2007.PMID: 17202163 Free PMC article. REBASE--restriction enzymes and DNA methyltransferases.Roberts RJ, Vincze T, Posfai J, Macelis D.Roberts RJ, et al.Nucleic Acids Res. 2005 Jan 1;33(Database issue):D230-2. doi: 10.1093/nar/gki029.Nucleic Acids Res. 2005.PMID: 15608184 Free PMC article. Specificity of restriction endonucleases and DNA modification methyltransferases a review (Edition 3).Kessler C, Manta V.Kessler C, et al.Gene. 1990 Aug 16;92(1-2):1-248. doi: 10.1016/0378-1119(90)90486-b.Gene. 1990.PMID: 2172084 Review. [DNA restriction-modification systems].Glatman LI, Kravets AN.Glatman LI, et al.Antibiot Khimioter. 1989 Dec;34(12):932-8.Antibiot Khimioter. 1989.PMID: 2560908 Review.Russian.No abstract available. See all similar articles Cited by Emerging methylation-based approaches in microbiome engineering.Won C, Yim SS.Won C, et al.Biotechnol Biofuels Bioprod. 2024 Jul 10;17(1):96. doi: 10.1186/s13068-024-02529-x.Biotechnol Biofuels Bioprod. 2024.PMID: 38987811 Free PMC article.Review. Enzyme Databases in the Era of Omics and Artificial Intelligence.Prešern U, Goličnik M.Prešern U, et al.Int J Mol Sci. 2023 Nov 29;24(23):16918. doi: 10.3390/ijms242316918.Int J Mol Sci. 2023.PMID: 38069254 Free PMC article.Review. Phage defence loci of Streptococcus thermophilus-tip of the anti-phage iceberg?Kelleher P, Ortiz Charneco G, Kampff Z, Diaz-Garrido N, Bottacini F, McDonnell B, Lugli GA, Ventura M, Fomenkov A, Quénée P, Kulakauskas S, de Waal P, van Peij NNME, Cambillau C, Roberts RJ, van Sinderen D, Mahony J.Kelleher P, et al.Nucleic Acids Res. 2024 Oct 28;52(19):11853-11869. doi: 10.1093/nar/gkae814.Nucleic Acids Res. 2024.PMID: 39315705 Free PMC article. Revised whole genome and DNA methylome of Mycobacterium marinum type strain ATCC 927 T.Savijoki K, Deptula P, Roberts RJ, Hammarén M, Yli-Kauhaluoma J, Varmanen P, Parikka M.Savijoki K, et al.Microbiol Resour Announc. 2024 Apr 11;13(4):e0101623. doi: 10.1128/mra.01016-23. Epub 2024 Feb 28.Microbiol Resour Announc. 2024.PMID: 38415640 Free PMC article. Methylation of foreign DNA overcomes the restriction barrier of Flavobacterium psychrophilum and allows efficient genetic manipulation.Sloboda S, Ge X, Jiang D, Su L, Wiens GD, Beveridge CA, Duchaud E, McBride MJ, Rochat T, Zhu Y.Sloboda S, et al.Appl Environ Microbiol. 2025 Feb 19;91(2):e0144824. doi: 10.1128/aem.01448-24. Epub 2025 Jan 10.Appl Environ Microbiol. 2025.PMID: 39791877 Free PMC article. See all "Cited by" articles References Roberts R.J., Vincze T., Posfai J., Macelis D.. REBASE–a database for DNA restriction and modification: enzymes, genes and genomes. Nucleic Acids Res. 2015; 43:D298–D299. - PMC - PubMed Benson D.A., Clark K., Karsch-Mizrachi I., Lipman D.J., Ostell J., Sayers E.W.. GenBank. Nucleic Acids Res. 2014; 42:D32–D37. - PMC - PubMed Kim D. Pruitt K.D., Tatusova T., Brown G.R., Maglott D.R.. NCBI reference sequences (RefSeq): current status, new features and genome annotation policy. Nucleic Acids Res. 2012; 40:D130–D135. - PMC - PubMed Roberts R.J., Belfort M., Bestor T., Bhagwat A.S., Bickle T.A., Bitinaite J., Blumenthal R.M., Degtyarev S.K., Dryden D.T.F., Dybvig K.et al. .. A nomenclature for restriction enzymes, DNA methyltransferases, homing endonucleases and their genes. Nucleic Acids Res. 2003; 31:1805–1812. - PMC - PubMed Anton B.P., Roberts R.J.. Beyond restriction modification: epigenomic roles of DNA methylation in prokaryotes. Annu. Rev. Microbiol. 2021; 75:129–149. - PubMed Show all 8 references Publication types Research Support, Non-U.S. Gov't Actions Search in PubMed Search in MeSH Add to Search MeSH terms DNA / genetics Actions Search in PubMed Search in MeSH Add to Search DNA Methylation Actions Search in PubMed Search in MeSH Add to Search DNA Modification Methylases / metabolism Actions Search in PubMed Search in MeSH Add to Search DNA Restriction Enzymes / metabolism Actions Search in PubMed Search in MeSH Add to Search DNA Restriction-Modification Enzymes / genetics Actions Search in PubMed Search in MeSH Add to Search Databases, Factual Actions Search in PubMed Search in MeSH Add to Search Substances DNA Restriction Enzymes Actions Search in PubMed Search in MeSH Add to Search DNA Modification Methylases Actions Search in PubMed Search in MeSH Add to Search DNA Actions Search in PubMed Search in MeSH Add to Search DNA Restriction-Modification Enzymes Actions Search in PubMed Search in MeSH Add to Search Related information MedGen PubChem Compound (MeSH Keyword) LinkOut - more resources Full Text Sources Europe PubMed Central PubMed Central Other Literature Sources The Lens - Patent Citations Database Molecular Biology Databases BioCyc REBASE - The Restriction Enzyme Database Full text links[x] Free PMC article [x] Cite Copy Download .nbib.nbib Format: Send To Clipboard Email Save My Bibliography Collections Citation Manager [x] NCBI Literature Resources MeSHPMCBookshelfDisclaimer The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). 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2193	http://stat.wharton.upenn.edu/~waterman/Teaching/IntroMath99/Class04/Notes/node12.htm	5.3 Examples Next:5.4Up:5.Previous:5.2 5.3 Examples How long does it take a principle amount to double if the interest rate is ? Principle is P 0. Principle has doubled when amount at time t equals twice the principle. That is . So the question is, what value of t (how long) makes P t double P 0? The formula for continuous compounding states that , so the question is now ``what value of t makes ''? Some simplifications (trying to get at t): So the doubling time is ln(2)/r and this does not depend on the principle P 0. In particular for r equal to 4% or 0.04 we get ln(2)/0.04 = 17.329. Always check the answers: Say P 0 is 1000. Then So, it has indeed doubled (up to rounding error). Next example: A invests $1000 at 5% per annum continuously compounded. B invests $200 at 20% per annum continuously compounded. Questions? Does B ever catch up? How long does it take? B catches up if the difference between them becomes zero. Alternatively if the ratio of their amounts becomes 1. Equality can be expressed as a difference of zero or as a ratio of one. Draw a picture using a graphing package . When does it happen: either approximate from the graph or work out explicitly once and for all. A has at time t, . B has at time t, . The equality condition can be stated as: Solve this for t again. So it takes about 11 years for B to catch up. Next:5.4Up:5.Previous:5.2Richard Waterman 1999-05-14
2194	https://emedicine.medscape.com/article/1923010-overview	Eye Globe Anatomy Updated: Nov 27, 2024 Author: Hon-Vu Quang Duong, MD; Chief Editor: Vinay K Kapoor, MBBS, MS, FRCSEd, FICS, FAMS more...;) Share Print Feedback Overview Overview For the purpose of this article on eye globe anatomy, the eye will be divided into two sections: the extraocular (i.e., structures outside of the globe) and the ocular (i.e., the globe and the intraocular structures). Throughout the years, in the absence of ancillary or diagnostic tools, many descriptive phrases, clichés, or analogies have been used to describe the eye, such as "the eye is the window to the soul," the "eye works like a camera," and "the eye is the only structure that allows us to physically see blood vessels and nerve fibers." We use our eyes daily and in almost every activity we perform (e.g., reading, driving, watching). At the foundation, the eye is a sensory organ that detects specifically photons. Photons within the visible spectrum (i.e., light) enter the eye first by passing through the cornea, a clear, dome-shaped structure at the anterior post portion of the globe. Light passing through the cornea is converged (bent) where it passes through the anterior chamber and the pupil, a circular opening regulating the amount of light entering the eye. Light is further converged by the crystalline lens located posterior to the iris. Light continues through the vitreous humor and the light converges on the retina, specifically the fovea centralis of the macula. Within the layers of the retina, photons trigger a series of electrical and chemical reactions, ultimately sending electrical signals by way of the optic nerve, along with visual pathway to the occipital cortex. Within the occipital cortices, these electrical signals are processed and interpreted (i.e., "seen") by the brain as a visual image. The image below depicts a sagittal section of the eye. Eye, sagittal section. View Media Gallery) Next: Extraocular Structures Several structures exist within the category of extraocular structures, including the orbit, extraocular muscles, conjunctiva, lacrimal system, and eyelids. The functions associated with these structures include protection and lubrication. Orbit The orbit is a cone-shaped (four-sided pyramidal cavity) structure consisting of a base (the orbital margin) that opens into the midline of the face, the apex, the narrow end pointed posteriorly into the head, and the four walls. In adults, the orbit is formed by seven bones: (1) frontal, (2) zygoma, (3) maxilla, (4) ethmoid, (5) sphenoid, (6) lacrimal, and (7) palatine. The frontal, ethmoid, and sphenoid are the only three unpaired bones of the orbit. The orbital margin (base) is formed by the maxillary, zygomatic, frontal, and lacrimal bones. The lesser wing of the sphenoid and frontal bones form the roof of the orbit, whereas the maxillary, zygomatic, and palatine bones form the floor of the orbit. The medial wall is formed by the sphenoid, ethmoid, lacrimal, and maxillary bones. The lateral wall is formed by the greater wing of the sphenoid and zygomatic bones. The orbit has a volume of 30 mL, measures 4 cm horizontally and 3.5 cm vertically, and has a depth (anteroposteriorly) of 4.5 cm. Associated with the orbit are the foramina and fissures (see Table 1, below), which are important in transmitting nerves, arteries, and veins. The primary function of the orbit is to protect the eye from physical injuries. Other functions include the housing and support of the globe and associated structures, the facilitation of eye movement through the attachment of extraocular muscles, and the provision of pathways for neurovascular structures. Table 1. Contents of Orbital Foramens and Fissures (Open Table in a new window)) \| \| \| \| \| \| --- --- \| \| Nerve(s) \| Artery(ies) \| Vein(s) \| Other Structure(s) \| \| Supraorbital foramen \| Supraorbital nerve (from frontal branch of cranial nerve [CN] V1) \| Supraorbital \| \| \| \| Superior orbital fissure \| 1. Lacrimal branch of CN V1 2. Frontal branch of CN V1 3. Superior and inferior branches of oculomotor nerve (CN III) 4. Trochlear (CN IV) 5. Nasociliary branch of CN V1 6. Abducens (CN VI) 7. Sympathetic root ganglion \| \| Superior ophthalmic \| \| \| Inferior orbital fissure \| 1. Infraorbital branch of CN V2 2. Zygomatic branch of CN V2 \| Infraorbital \| Pterygoid plexus communication with inferior ophthalmic \| \| \| Optic canal \| 1. Optic (CN II) 2. Sympathetic fibers from internal carotid plexus \| Ophthalmic \| \| \| \| Anterior ethmoidal foramen \| Anterior ethmoidal \| Anterior ethmoidal \| Anterior ethmoidal \| \| \| Posterior ethmoidal foramen \| Posterior ethmoidal \| Posterior ethmoidal \| Posterior ethmoidal \| \| \| Zygomatic foramen \| Zygomatic branch of CN V2 (zygomaticofacial and zygomaticotemporal branches) \| Zygomatic \| \| \| \| Lacrimal fossa \| \| \| \| Lacrimal gland \| \| Fossa for lacrimal sac \| \| \| \| Lacrimal sac and nasolacrimal duct \| Extraocular Muscles The extraocular muscles control the movements of the eye (see Table 2, below). Six extraocular muscles exist: four rectus muscles (superior, inferior, medial, and lateral) and two oblique muscles (superior and inferior). The extraocular muscles are innervated by three CNs: oculomotor (CN III), trochlear (CN IV), and abducens nerves (CN VI). Oculomotor nerve (CN III) - Superior, medial, and inferior rectus; inferior oblique; levator palpebrae superioris Trochlear nerve (CN IV) - Superior oblique Abducens nerve (CN VI) - Lateral rectus The levator palpebrae superioris is the other extraocular muscle that, although not directly involved in eye movement, is primarily responsible for elevating the upper eyelid. The origin of the four rectus muscles is the common ring tendon (annulus of Zinn) at the orbital apex. The rectus muscles insert on the sclera anterior to the equator. The insertions of the rectus muscles are not equidistant from the limbus. The imaginary line connecting the insertions of the four rectus muscles is known as the spiral of Tillaux. The origin of the superior oblique is the sphenoid bone and has a long (10 + 10 mm) tendinous insertion. It inserts superior and temporal to the posterior pole and is covered by the superior rectus. The origin of the inferior oblique is the maxilla and has a muscular insertion temporal and inferior to the geometric posterior pole and covers the inferior rectus. Table 2. Extraocular Muscles (Open Table in a new window)) \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| \| Muscle \| Origin \| Insertion \| Distance From the Limbus \| Innervation \| Primary Function \| Length of Active Muscle (mm) \| Length of Tendon (mm) \| Direction of Pull (degree) \| Arc of Contact (mm) \| \| Superior rectus \| Common ring tendon \| Anterior and superior surface \| 7.7 mm \| Superior branch of cranial nerve [CN] III \| Elevation, intorsion, adduction \| ±40 \| 6 \| 23 \| 6.5 \| \| Lateral rectus \| Common ring tendon \| Anterior and lateral surface \| 7.0 mm \| Abducens nerve (CN VI) \| Abduction \| ±40 \| 7 \| 90 \| 12 \| \| Inferior rectus \| Common ring tendon \| Anterior and inferior surface \| 6.5 mm \| Inferior branch of CN III \| Depression, extorsion, adduction \| ±40 \| 7 \| 23 \| 6.5 \| \| Medial rectus \| Common ring tendon \| Anterior and medial surface \| 5.5 mm \| Inferior branch of CN III \| Adduction \| ±39 \| 4.5 \| 90 \| 7 \| \| Superior oblique \| Sphenoid, orbit apex above common ring tendon \| Posterior, temporosuperior quadrant \| \| Trochlear nerve (CN IV) \| Intorsion, depression, abduction \| ±32 \| 26 \| 51 \| 7-8 \| \| Inferior oblique \| Maxilla, behind the lacrimal fossa \| Posterior, temporoinferior quadrant \| \| Inferior branch of CN III \| Extorsion, elevation abduction \| ±37 \| 1 \| 51 \| 15 \| \| Levator palpebrae superioris \| Orbit apex above common ring tendon \| \| \| Superior branch of CN III \| Lid elevation \| \| \| \| \| \| Relative to the visual axis in the primary position. \| \| \| \| \| \| \| \| \| \| The extraocular muscles have a large ratio of nerve fibers to muscle fibers (1:3 to 1:5) compared to other skeletal muscles (1:50 to 1:125). This allows for precise control of eye movements. Conjunctiva The conjunctiva is a thin, transparent mucous membrane overlying the anterior-most portion of the sclera and lining the inner surfaces of the eyelids. The conjunctiva is divided into the limbal, bulbar, forniceal, and palpebral regions. [2, 5, 6] Limbal conjunctiva - The transition zone between the cornea and the bulbar conjunctiva Bulbar conjunctiva - Covers the anterior portion of the eyeball, extending from the corneoscleral limbus onto the surface of the globe Forniceal conjunctiva - Forms the superior and inferior fornices, which are deep recesses where the bulbar and palpebral conjunctiva meet Palpebral (tarsal) conjunctiva - Lines the posterior surface of the eyelids The conjunctival epithelium varies in structure depending on its location: Near the limbus - Columnar epithelium Closer to the fornix - Squamous epithelium The epithelium is non-keratinized and typically consists of 5-7 cell layers. Beneath the epithelium lies the substantia propria, a fibrovascular connective tissue that is thicker in the fornix and thinner at the limbus. Associated with the conjunctiva are specialized cellular components: [2, 6, 7] Goblet cells - Secretory cells that produce mucin, a crucial component of the tear film. Mucus secretion from these cells is vital for maintaining epithelial integrity and supporting the tear film. They are more numerous in the inferior and medial portions of the bulbar conjunctiva and in the fornix. Melanocytes - Pigment-producing cells found in the basal layer of the epithelium Langerhans cells - Antigen-presenting cells, important for immune responses Lymphocytes - Part of the conjunctiva-associated lymphoid tissue (CALT), contributing to ocular surface defense The conjunctiva also bears eccrine glands: the conjunctival glands (of Krause) and the accessory lacrimal glands (of Wolfring). The conjunctival glands (of Krause) are concentrated in the upper fornix, whereas the accessory lacrimal glands (of Wolfring) are associated with the tarsus. The conjunctiva serves several important functions: [2, 7] Allows smooth movement of the eyelids over the eyeball Keeps the ocular surface moist Protects the eye from external injuries Contains blood vessels that supply oxygen and nutrients to the ocular surface Plays a role in the immune defense of the eye Studies highlight the conjunctiva's lymphoid layer, also known as CALT, which includes T and B lymphocytes, resembling mucosal-associated lymphoid tissue found in other areas, such as the gut and bronchi. In this layer, Langerhans cells, specialized antigen-presenting immune cells, are distributed, especially in the tarsal conjunctiva, decreasing in number with aging. [7, 8, 9] The fibrous layer, located beneath the lymphoid layer, houses the blood vessels, nerves, and accessory lacrimal glands of the conjunctiva, including the glands of Krause and Wolfring. The glands of Krause, concentrated in the upper fornix, and Wolfring, associated with the tarsus, contribute to tear secretion, complementing the main lacrimal gland's function. These glands are essential in sustaining ocular moisture, with their structures resembling primary lacrimal glands. Additionally, melanocytes are frequently seen in the conjunctival limbus and plica semilunaris, and they can give the conjunctiva a brownish tint in some individuals. Lacrimal Gland and the Nasolacrimal System The lacrimal gland is an exocrine gland nestled within the fossa of the frontal bone located in the anterior superotemporal quadrant of the orbit. The lacrimal gland is composed of acini, myoepithelial cells, and intralobular and interlobular ducts. It produces the aqueous component of tears, which contains water, electrolytes, and proteins. The gland is divided into the orbital lobe and the palpebral lobe by the tendon of the levator palpebrae superioris. Ducts from both lobes traverse through the palpebral lobe and empty into the conjunctival fornix temporally. The nasolacrimal system consists of the following structures: Lacrimal puncta - Two small openings located on the medial aspect of each eyelid margin Lacrimal canaliculi - Two narrow canals that extend from the puncta; they have a vertical segment and a horizontal segment Common canaliculus - Formed by the union of the upper and lower canaliculi Lacrimal sac - A dilated structure located in the lacrimal fossa; it's divided into a fundus superiorly and a body inferiorly Nasolacrimal duct - Extends from the lacrimal sac to the inferior nasal meatus See the following image. The drainage of tears involves several structures starting with the lacrimal puncta, tiny openings near the inner edges of the eyelids, through which tears enter the canaliculi. From the canaliculi, tears pass into the lacrimal sac at the medial canthus before moving through the nasolacrimal duct to the inferior meatus of the nasal cavity. A key structure in tear drainage is the valve of Hasner, located at the end of the nasolacrimal duct, which prevents the backflow of air and fluid into the tear drainage system from the nasal cavity. [2, 13] Tears from the lacrimal gland provide essential moisture, nutrients, and immune protection to the eye surface. They mix with secretions from the meibomian glands and goblet cells to form the tear film, a three-layer structure essential for smooth vision and ocular comfort. The tear film, consisting of the lipid, aqueous, and mucin layers, spreads across the ocular surface during blinking, protecting the cornea from environmental irritants and infections. Studies highlight that the lacrimal gland contains stem/progenitor cells that can potentially regenerate damaged tissue, offering possibilities for treating dry eye disease. The nasolacrimal system is now recognized to have an important immunological function, containing lacrimal drainage-associated lymphoid tissue (LDALT) that plays a role in ocular surface defense. [15, 16] Blood supply to the lacrimal gland and the nasolacrimal system primarily comes from the lacrimal artery, a branch of the ophthalmic artery, while additional supply to the nasolacrimal duct comes from the facial artery. The lacrimal gland receives both parasympathetic innervation, which drives tear secretion, and sympathetic innervation in response to emotional or physical stimuli. The sensory innervation of the lacrimal gland is through the lacrimal nerve (a branch of the ophthalmic nerve). The lacrimal apparatus also has lymphatic connections to the preauricular and submandibular lymph nodes, crucial for immune defense and fluid balance within the ocular structures. [2, 17] Eye and lacrimal duct, anterior view. View Media Gallery) Eyelids The eyelids are designed to protect, nourish, and sustain the cornea and the anterior sclera. Anatomically, the eyelid is divided into two lamellae, anterior and posterior, demarcated by the gray line, which represents the muscle of Riolan, an extension of the orbicularis oculi muscle. The anterior lamella is composed of the epithelium and orbicularis oculi. This muscle has a palpebral segment, further subdivided into pretarsal and preseptal regions, both contributing to eyelid movement and tear distribution. [2, 18] The orbicularis oculi muscle has a dual role, as it includes both voluntary fibers for forced eyelid closure and involuntary fibers responsible for the blink reflex. The lacrimal part (pars lacrimalis) of this muscle, known as Horner’s muscle, attaches near the lacrimal crest, aiding in tear drainage through a pumping action. [2, 18] The orbital septum, an extension of the periosteum, acts as a barrier that prevents the spread of infection between the anterior lamella (superficial structures) and the orbit. The septum helps differentiate between preseptal and orbital infections, which have different clinical implications and treatments. [2, 18] The posterior lamella is primarily composed of the tarsus and palpebral conjunctiva. The tarsus is a dense connective tissue structure supporting the eyelid's shape and housing meibomian glands. These glands secrete oils critical for tear film stability and preventing tear evaporation. Meibomian gland orifices open along the eyelid margin just posterior to the gray line, ensuring an oily tear layer across the eye. [2, 18] Muscles of eyelid: [2, 18] Orbicularis oculi - Main protractor (closure) muscle Levator palpebrae superioris - Main retractor of upper eyelid Muller's muscle - Accessory retractor Innervation: [19, 20] Sensory - Ophthalmic (V1) and maxillary (V2) branches of trigeminal nerve Motor - Facial nerve (CN VII) for orbicularis, oculomotor nerve (CN III) for levator Previous Next: Intraocular Structures Tunica Fibrosa Sclera The sclera is a dense, fibrous tissue that forms the outermost layer of the eye (see the image below). It protects the eyes and provides for extraocular muscle insertion. Posteriorly, the sclera is perforated by the optic nerve at the lamina cribrosa. The peripapillary sclera (PPS) surrounding the optic nerve head has been identified as a critical biomechanical structure, with its properties potentially influencing glaucoma susceptibility. Scleral thickness is not uniform. Anteriorly, the sclera is 0.6 mm thick, 0.3 mm thick at the insertion of the rectus muscles, 0.5 mm thick at the equator, and 1.0 mm thick at the posterior pole. Externally, the sclera is covered by the episclera, containing episcleral vessels, and the anterior and posterior plexus. The sclera is understood to have a complex, hierarchical structure composed of collagen fibrils organized into lamellae. These lamellae demonstrate more branching and interweaving compared to the cornea, varying with tissue depth and anatomical location. Scleral composition includes approximately 68% water, 28% collagens, and smaller amounts of proteoglycans, elastin, and other proteins. Imaging techniques have revealed that scleral collagen fiber organization varies regionally, likely adapting to local mechanical stresses. Studies have highlighted the importance of scleral biomechanics in myopia development and progression. Eye, sagittal section. View Media Gallery) Cornea The cornea is a clear and transparent layer anterior to the eye. It is the eye's main refracting surface. This layer is avascular and exhibits the following five layers: The epithelium is a stratified squamous nonkeratinized epithelium (5-6 layers of cells); it is highly sensitive due to numerous nerve endings and has excellent regenerative power. The corneal epithelium plays a crucial role in maintaining corneal homeostasis through the production of various growth factors and cytokines. Research has highlighted the importance of limbal stem cells, located at the corneal-scleral junction, in maintaining the corneal epithelium. [22, 23] The Bowman membrane is structureless and acellular. It separates the epithelium from the stroma. The substantia propria (stroma) forms 90% of the cornea's thickness; fibrils of the stroma criss-cross at 90° angles, and these fibrils are of types I, III, V, and VII collagen. The Descemet membrane is structureless, homogeneous, and measures 3-12 microns; it is composed of the anterior banded zone and the posterior nonbanded zone; the Descemet membrane is rich in type IV collagen fibers. The endothelium is a single layer of simple cuboidal and hexagonal cells that line the inner surface of the cornea. The endothelium is derived from the neural crest and functions to transport fluid from the anterior chamber to the stroma. Because the cornea is avascular, its nutrients are derived mainly from diffusion from the endothelium layer. Imaging techniques, such as in vivo confocal microscopy, have allowed for better visualization and quantification of corneal nerves, revealing their importance in corneal sensation and wound healing. Studies have focused on the corneal nerves' role in maintaining corneal health, with new surgical techniques such as corneal neurotization being developed to address neurotrophic keratopathy. [24, 25] Tunica vasculosa (uveal tract) Choroid The choroid is a spongy, brown membrane with extensive venous plexuses, which has the following four layers: The epichoroid layer bridges the space between the sclera and choroid. It contains collagen fibers, melanocytes, and nerve fibers. The vessels layer forms the bulk of the choroid layer and contains melanocytes, blood vessels, and connective tissue. It provides structural integrity and pigmentation. The choriocapillaris is a layer of capillaries lined by fenestrated type II endothelium that supplies nutrition to the outer portion of the retina. The Bruch membrane is a shiny, homogeneous membrane that lies between the choriocapillaris and retina. It ensures selective permeability for waste removal and nutrient exchange. The choroid plays a crucial role in supplying oxygen and nutrients to the outer retinal layers and regulating ocular temperature. Ciliary body The ciliary body is a ring-shaped structure located between the choroid and iris. The ciliary body is the thickest part of the tunica vasculosa. It encircles the eye, anterior to the ora serrata. The ciliary processes are radiating folds of smooth muscles. The bulk of the ciliary body is made up of smooth muscle arranged in meridional, radial, and equatorial bands. They function to contract and relax the zonule fibers, which results in altered tension on the capsule of the lens. The ciliary body is divided into two regions: [27, 28] Pars plicata - The anterior, folded portion containing the ciliary processes Pars plana - The posterior, flatter region transitioning to the retina Functions of the ciliary body include: [27, 28] Aqueous humor production Accommodation Supporting the lens via zonular fibers Contributing to uveoscleral outflow Iris The iris is the most anterior part of the uvea. It has a central aperture, the pupil. Peripherally, the iris is attached to the ciliary body, and anteriorly, it rests against the anterior surface of the lens, thus separating the anterior chamber from the posterior chamber. The anterior surface is irregular with crypts and furrows; posteriorly, the surface shows shallow furrows and is uniformly black due to the two layers of pigmented epithelium. The iris has both a sphincter and a dilator pupillary muscle. The sphincter pupillae muscle lies as a ring of smooth muscle at the pupillary margin and is supplied by the parasympathetic fibers of CN III. The dilator pupillae muscle is thin and radially oriented; it is supplied by the sympathetic fibers. Lens The lens is a crystalline structure, biconvex, and covered by a lens capsule. Attached to it are the zonular fibers that pass to the ciliary body as the suspensory ligament. The lens is avascular and derives its nutrients from the aqueous humor. It is elastic and transparent and held in place by suspensory ligament or zonule. The lens continues to grow throughout life, with new fiber cells being added to its outer layers. This growth process is tightly regulated to maintain optical clarity. The lens is composed of three main parts: the capsule, the epithelium, and the fiber cells. The fiber cells make up the bulk of the lens and are arranged in concentric layers. Imaging techniques have shown that the human lens has a gradient refractive index, with the highest index at the center. This gradient contributes significantly to the lens's optical power. Research has revealed that the lens contains its own circadian clock machinery, which may play a role in regulating lens homeostasis and growth. Chambers of the Eye The anterior chamber is a space bounded anteriorly by the posterior surface (endothelium) of the cornea and posteriorly by the lens, iris, and anterior surface of the ciliary body. Circumferentially, the lateral border of the anterior chamber is occupied by the trabecular meshwork, through which aqueous humor is drained into the scleral venous sinus (canal of Schlemm). The posterior chamber is bounded anteriorly by the iris, posteriorly by the lens and zonule fibers, and peripherally by the ciliary processes. Aqueous Humor The aqueous humor is a watery fluid that fills both the anterior and posterior chambers of the eye. It is secreted partially by the ciliary epithelium and partially by diffusion from the capillaries in the ciliary processes. The aqueous humor contains diffusible materials from the blood plasma but has a low protein content. The aqueous humor contains various growth factors and cytokines that play crucial roles in maintaining ocular health. Outflow pathways for aqueous humor include the trabecular outflow, uveoscleral, and lymphatic routes. [31, 32] Scleral Venous Sinus (Canal of Schlemm) The scleral venous sinus, or canal of Schlemm, is an annular vessel encircling the eye. The canal is lined by endothelium and its function is to drain the aqueous humor. Trabecular Meshwork The trabecular meshwork is a spongelike tissue that is interposed between the anterior chamber and the scleral venous sinus. The trabeculae are made up of a core of collagenous fibers that are covered by endothelium. Vitreous Body The vitreous body is a clear, transparent gel that fills the space between the retina and the lens that adheres to the retina. Its function is to maintain the shape and turgidity of the eye and to permit the passage of light rays to the retina. Refractive Media Refractive media include all transparent structures through which light rays must pass to reach the retina, such as the cornea, anterior chamber, lens, and vitreous body. Retina The retina is the innermost layer of the eyeball, which is composed of photoreceptor cells. In the posterior pole, a shallow depression is termed the fovea centralis (see the following image). This area is the point of greatest visual acuity. This area is composed of only cones. Around the fovea is an area containing yellow pigment termed the macula lutea. Eye, fundoscopic view. View Media Gallery) The layers of the retina are as follows: Pigment epithelium (the layer closest to the choroid layer) Layer of rods and cones External limiting membrane Outer nuclear layer Outer plexiform layer Inner nuclear layer Inner plexiform layer Ganglion cell layer Optic nerve fiber layer Internal limiting membrane (the layer closest to the vitreous body) Pigment epithelium The pigment epithelium is a single layer of polygonal cells regular in shape. The nuclei are spherical and lie toward the cell base. Numerous melanin granules are seen. Functionally, the pigment epithelium absorbs light and prevents reflection, and it is also involved in the nutrition of photoreceptors. The pigment epithelium is essential for the formation of rhodopsin and its movement by storing and releasing vitamin A, a rhodopsin precursor. Layer of rods and cones Rods are slender, cylindrical cells and number about 130 x 106. Cones have a flask shape and number about 67 x 106. Nuclei of the cones are larger than those of the rods, and they are also less dense. They do not contain rhodopsin but have a pigment that is sensitive to blue, green, and red light. The fovea, located in the central retina, is known to contain about 200,000 cones per sq mm, providing high visual acuity. External and internal limiting membrane The external limiting membrane supports the photoreceptor cells. The internal limiting membrane is the basal lamina of the Muller cells, separating the retina from the vitreous body. Muller glia cells have been found to act as optical fibers, guiding light through the retina to photoreceptors. Outer and inner nuclear layers The outer nuclear layer is composed of the nucleated portion of the rod and cone cells. The inner nuclear layer contains the cell bodies and nuclei of the bipolar neurons as well as supporting cells named Muller cells. Outer and inner plexiform layers The outer plexiform layer contains synapses made between the rod and cone cells and the bipolar cells. The inner plexiform layer contains the synapses between the bipolar neurons and the ganglionic cells. Ganglion cell layer The ganglion cell layer contains cell bodies and nuclei of the ganglion cells. Neuroglia are also present. Optic nerve fibers The optic nerve fibers contain axons of the ganglion cells that pass radially to form the optic nerve. Previous References Clemente CD, Gray H. Gray's Anatomy of the Human Body. 30th ed. Philadelphia, Pa: Lea & Febiger; 1985. Standring S. Gray's Anatomy: The Anatomical Basis of Clinical Practice. 42nd ed. 2021. Chapter 14 and 15:Pages:233-268, 17:Pages:273-291, Section 3 and 4:Pages:386-635, Chapters 44 and 45:Pages:769-790. The Actions and Innervation of Extraocular Muscles. Purves D, Augustine GJ, Fitzpatrick D, et al. Neuroscience. 2nd ed. Sunderland (MA): Sinauer Associates; 2001. [Full Text]. Titova A, Nikolaev S, Bilyalov A, et al. Extreme Tolerance of Extraocular Muscles to Diseases and Aging: Why and How?. Int J Mol Sci. 2024 May 3. 25 (9):4985. [QxMD MEDLINE Link]. [Full Text]. Ang LP, Tan DT. Ocular surface stem cells and disease: current concepts and clinical applications. Ann Acad Med Singap. 2004 Sep. 33 (5):576-80. [QxMD MEDLINE Link]. [Full Text]. Conjunctival Epithelium Neoplasms. EyeWiki. Available at 2023; Accessed: 12 November 2024. Mudhar HS. Update on conjunctival pathology. Indian J Ophthalmol. 2017 Sep. 65 (9):797-807. [QxMD MEDLINE Link]. [Full Text]. Knop N, Knop E. Conjunctiva-Associated Lymphoid Tissue in the Human Eye. Invest Ophthalmol Vis Sci. 2000. 41(6):1270-1279. [Full Text]. Siebelmann S, Gehlsen U, Hüttmann G, et al. Development, alteration and real time dynamics of conjunctiva-associated lymphoid tissue. PLoS One. 2013. 8 (12):e82355. [QxMD MEDLINE Link]. [Full Text]. Kondapalli SS, Venkateswaran N, Noh A. Unraveling the Role of Accessory Lacrimal Glands. 2024. Available at Mohd Zaki A. The role of mast cells, T cells and their cytokines in Allergic Conjunctivitis. 2019. Available at Dartt DA. Neural regulation of lacrimal gland secretory processes: relevance in dry eye diseases. Prog Retin Eye Res. 2009 May. 28 (3):155-77. [QxMD MEDLINE Link]. [Full Text]. Orbit, Eyelids, and Lacrimal System. [Full Text]. Kopacz D, Niezgoda L, Sikorska E, Nowak-Szwed A, Maciejewicz P. Tear Film – Physiology and Disturbances in Various Diseases and Disorders. 2020. [Full Text]. Garg A, Zhang X. Lacrimal gland development: From signaling interactions to regenerative medicine. Dev Dyn. 2017 Dec. 246 (12):970-980. [QxMD MEDLINE Link]. [Full Text]. Knop E, Knop N. Lacrimal drainage-associated lymphoid tissue (LDALT): a part of the human mucosal immune system. Invest Ophthalmol Vis Sci. 2001 Mar. 42 (3):566-74. [QxMD MEDLINE Link]. [Full Text]. O’Sullivan NL, Montgomery PC. Ocular Mucosal Immunity. Mucosal Immunology. 2015. 1873–97. [Full Text]. Gooris PJJ, Cornelius CP. Anatomy of the Orbit: Overall Aspects of the Peri- and Intra Orbital Soft Tissues. Gooris PJ, Mourits MP, Bergsma J. Surgery in and around the Orbit. Cham; 2023. [Full Text]. Nerve Supply of the Eyelids. EyeWiki. Available at Accessed 12 November 2024. 2024; Palermo EC. Anatomy of the periorbital region. Surg Cosmet Dermatol. 2013. 5(3):24556. [Full Text]. Boote C, Sigal IA, Grytz R, Hua Y, Nguyen TD, Girard MJA. Scleral structure and biomechanics. Prog Retin Eye Res. 2020 Jan. 74:100773. [QxMD MEDLINE Link]. [Full Text]. Yu FS, Yin J, Xu K, Huang J. Growth factors and corneal epithelial wound healing. Brain Res Bull. 2010 Feb 15. 81 (2-3):229-35. [QxMD MEDLINE Link]. [Full Text]. Secker GA, Daniels JT. Limbal epithelial stem cells of the cornea. StemBook. Cambridge (MA): Harvard Stem Cell Institute; 2008. [Full Text]. Cruzat A, Pavan-Langston D, Hamrah P. In vivo confocal microscopy of corneal nerves: analysis and clinical correlation. Semin Ophthalmol. 2010 Sep-Nov. 25 (5-6):171-7. [QxMD MEDLINE Link]. [Full Text]. Dragnea DC, Krolo I, Koppen C, Faris C, Van den Bogerd B, Ní Dhubhghaill S. Corneal Neurotization-Indications, Surgical Techniques and Outcomes. J Clin Med. 2023 Mar 13. 12 (6):2214. [QxMD MEDLINE Link]. [Full Text]. Nickla DL, Wallman J. The multifunctional choroid. Prog Retin Eye Res. 2010 Mar. 29 (2):144-68. [QxMD MEDLINE Link]. [Full Text]. Bassnett S. Zinn's zonule. Prog Retin Eye Res. 2021 May. 82:100902. [QxMD MEDLINE Link]. [Full Text]. Delamere NA. Ciliary Body and Ciliary Epithelium. Adv Organ Biol. 2005 Jan 1. 10:127-148. [QxMD MEDLINE Link]. [Full Text]. Liu Z, Wang R, Lin H, Liu Y. Lens regeneration in humans: using regenerative potential for tissue repairing. Ann Transl Med. 2020 Nov. 8 (22):1544. [QxMD MEDLINE Link]. [Full Text]. Lim JC, Li I, Donaldson PJ, Poulsen RC. Shedding new light on the role of the lens in regulating circadian rhythms. Invest Ophthalmol Vis Sci. 2020. 61(7):2851. [Full Text]. Cheng Y, Zheng S, Pan CT, et al. Analysis of aqueous humor concentrations of cytokines in retinoblastoma. PLoS One. 2017. 12 (5):e0177337. [QxMD MEDLINE Link]. [Full Text]. Lee JY, Akiyama G, Saraswathy S, et al. Aqueous humour outflow imaging: seeing is believing. Eye (Lond). 2021 Jan. 35 (1):202-215. [QxMD MEDLINE Link]. Kolb H, Nelson RF, Ahnelt PK, Ortuño-Lizarán I, Cuenca N. The Architecture of the Human Fovea. Kolb H, Fernandez E, Jones B, et al. Webvision: The Organization of the Retina and Visual System [Internet]. Salt Lake City (UT): University of Utah Health Sciences Center; 2020. [Full Text]. Franze K, Grosche J, Skatchkov SN, et al. Muller cells are living optical fibers in the vertebrate retina. Proc Natl Acad Sci U S A. 2007 May 15. 104 (20):8287-92. [QxMD MEDLINE Link]. [Full Text]. Eye, sagittal section. Eye, fundoscopic view. Eye and lacrimal duct, anterior view. of 3 encoded search term (Eye Globe Anatomy) and Eye Globe Anatomy What to Read Next on Medscape
2195	https://artofproblemsolving.com/wiki/index.php/Rearrangement_Inequality?srsltid=AfmBOorPz69YoUSEZFPu5uU32lEIfEjk87Q8k5kxVfENIE8g6OCoQetQ	Art of Problem Solving Rearrangement Inequality - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Rearrangement Inequality Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Rearrangement Inequality The Rearrangement Inequality states that, if is a permutation of a finiteset (in fact, multiset) of real numbers and is a permutation of another finite set of real numbers, the quantity is maximized when and are similarly sorted (that is, if is greater than or equal to exactly of the other members of , then is also greater than or equal to exactly of the other members of ). Conversely, is minimized when and are oppositely sorted (that is, if is less than or equal to exactly of the other members of , then is also greater than or equal to exactly of the other members ). Contents [hide] 1 Introductory 2 Intermediate 2.1 Proof of the Rearrangement Inequality 3 Uses 4 See Also 5 External Links Introductory Consider the following simple application: suppose you are involved in the hold-up of a convenience store. You note, as you are emptying the register, that there are different numbers of each denomination (penny, nickel, dime, quarter, dollar bill, five dollar bill, ten dollar bill and twenty dollar bill) in the register. When would your take be maximized? Certainly, you would hope that there would be the largest number of twenty dollar bills, then the next largest number of tens, etc. Meanwhile, you would find yourself very disappointed if there were more pennies than nickels, more nickels than dimes, and so on. This is a simple application of the rearrangement inequality. It is also an application of the greedy algorithm, so one possible interpretation of the rearrangement inequality is that sometimes, the greedy algorithm works. Intermediate Proof of the Rearrangement Inequality The proof of the Rearrangment Inequality can be handled with proof by contradiction. Only the maximization form is proved here; the minimization proof is virtually identical. Before we begin the proof properly, it is useful to consider the case where . Without loss of generality, sort and so that and . By hypothesis, . Expanding and taking some terms to the other side of the inequality, we get , as desired. Now for the general case. Again, without loss of generality, set and ; and suppose the grouping that maximizes the desired sum of products is not the one that pairs with , with , and so on. This means that there is at least one instance where is paired with while is paired with , where and . However, using the technique seen above to prove the inequality for , we can see that the sum of products can only increase if we instead pair with and with (unless both a's or both b's are equal, in which case either we can choose another pair of products or note that the current arrangement is actually identical to the desired one), which contradicts our assumption that the arrangement we had was already the largest one. Note: The minimization equality can be very easily proved by noting that if we have the set , ordered in decreasing order and the set , ordered in increasing order, then the maximum sum is just . Thus, by negating all values the inequality follows. Uses The Rearrangement Inequality has a wide range of uses, from MathCounts level optimization problems to Olympiad level inequality problems. A relatively simple example of its use in solving higher-level problems is found in the proof of Chebyshev's Inequality. It is particularly useful in that it does not require any terms of either sequence to be positive or negative, unlike the power-mean family of inequalities. See Also Chebyshev's Inequality Power Mean Inequality External Links The Rearrangement Inequality by Dragos Hrimiuc Retrieved from " Categories: Algebra Inequalities Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
2196	https://brainly.com/question/37003248	[FREE] The pKa of arginine's α-Carboxyl group, α-Amino group, and side chain are 1.8, 9.0, and 12.5, respectively. - 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 +75,4k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +43k Ace exams faster, with practice that adapts to you Practice Worksheets +8,2k Guided help for every grade, topic or textbook Complete See more / Chemistry Textbook & Expert-Verified Textbook & Expert-Verified The pKa of arginine's α-Carboxyl group, α-Amino group, and side chain are 1.8, 9.0, and 12.5, respectively. Calculate the isoelectric point. 1 See answer Explain with Learning Companion NEW Asked by Jaelynnn53281 • 09/06/2023 0:00 / -- Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 313870 people 313K 0.0 0 Upload your school material for a more relevant answer The pI of arginine is calculated as the average of the pKa values of the amino and side chain groups, which is (9.0 + 12.5)/2 = 10.75. Explanation The isoelectric point (pI) is the pH at which a molecule carries no net charge. To calculate the pI of an amino acid like arginine, which has a basic side chain, we need to take into account the pKa values of the carboxyl, amino, and side chain groups. In the case of arginine, the pKa values are 1.8 (α-Carboxyl group), 9.0 (α-Amino group), and 12.5 (side chain). For amino acids with basic side chains, the pI is calculated as the average of the pKa values of the amino and side chain groups. Thus, the pI of arginine is (9.0 + 12.5)/2 = 10.75. Learn more about Isoelectric Point here: brainly.com/question/34714032 SPJ11 Answered by CallumArcher •2.8K answers•313.9K people helped Thanks 0 0.0 (0 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 313870 people 313K 0.0 0 Biochemistry Free For All - Kevin Ahern & Indira Rajagopal Fundamentals of Biochemistry - Henry Jakubowski and Patricia Flatt Fundamentals of General Organic and Biological Chemistry - Mcmurry Et Al. Upload your school material for a more relevant answer The isoelectric point of arginine is calculated using the pKa values of its α-Amino group and side chain. The pI is found to be 10.75, indicating the pH at which arginine has no net charge. This calculation helps understand how arginine behaves in different pH environments. Explanation To calculate the isoelectric point (pI) of arginine, we need to consider the pKa values of its ionizable groups. Arginine has three relevant pKa values: α-Carboxyl group: pKa = 1.8 α-Amino group: pKa = 9.0 Side chain (guanidinium group): pKa = 12.5 The isoelectric point (pI) is defined as the pH at which the molecule has no net charge. For amino acids, this is calculated by averaging the pKa values of the two groups that are closest to each other and straddle the pI. In the case of arginine, since it has a basic side chain, we take the pKa of the α-Amino group (9.0) and the pKa of the side chain (12.5). We can calculate the pI using the formula: p I=2 p K a1+p K a2 Substituting the values: p I=2 9.0+12.5 Calculating this results in: p I=2 21.5=10.75 Therefore, the isoelectric point of arginine is 10.75. This means at pH 10.75, arginine exists predominantly in a zwitterionic form, having equal positive and negative charges. Examples & Evidence For instance, in a biochemical experiment where arginine is used, maintaining the environment at pH 10.75 ensures that arginine remains neutral, which is important for its function in proteins and enzymes. The calculation of the isoelectric point is based on well-established principles in biochemistry, particularly concerning amino acid behavior in solution as described in textbooks and peer-reviewed articles. Thanks 0 0.0 (0 votes) Advertisement Jaelynnn53281 has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Chemistry solutions and answers Community Answer The pka's of arginine's a-carboxyl group, a-amino group, and side chain are 1.8, 9.0, and 12.5, respectively. what is arginine's pi? Community Answer 4.2 19 A drink that contains 4 1/2 ounces of a proof liquor… approximately how many drinks does this beverage contain? Community Answer 5.0 7 Chemical contamination is more likely to occur under which of the following situations? When cleaning products are not stored properly When dishes are sanitized with a chlorine solution When raw poultry is stored above a ready-to-eat food When vegetables are prepared on a cutting board that has not been sanitized Community Answer 4.3 189 1. Holding 100mL of water (ebkare)__2. Measuring 27 mL of liquid(daudgtear ldnreiyc)____3. Measuring exactly 43mL of an acid (rtube)____4. Massing out120 g of sodium chloride (acbnela)____5. Suspending glassware over the Bunsen burner (rwei zeagu)____6. Used to pour liquids into containers with small openings or to hold filter paper (unfenl)____7. Mixing a small amount of chemicals together (lewl letpa)____8. Heating contents in a test tube (estt ubet smalcp)____9. Holding many test tubes filled with chemicals (estt ubet karc) ____10. Used to clean the inside of test tubes or graduated cylinders (iwer srbuh)____11. Keeping liquid contents in a beaker from splattering (tahcw sgasl)____12. A narrow-mouthed container used to transport, heat or store substances, often used when a stopper is required (ymerereel kslaf)____13. Heating contents in the lab (nuesnb bneurr)____14. Transport a hot beaker (gntos)____15. Protects the eyes from flying objects or chemical splashes(ggloges)____16. Used to grind chemicals to powder (tmraor nda stlepe) __ Community Answer Food waste, like a feather or a bone, fall into food, causing contamination. Physical Chemical Pest Cross-conta Community Answer 8 If the temperature of a reversible reaction in dynamic equilibrium increases, how will the equilibrium change? A. It will shift towards the products. B. It will shift towards the endothermic reaction. C. It will not change. D. It will shift towards the exothermic reaction. Community Answer 4.8 52 Which statements are TRUE about energy and matter in stars? Select the three correct answers. Al energy is converted into matter in stars Only matter is conserved within stars. Only energy is conserved within stars. Some matter is converted into energy within stars. Energy and matter are both conserved in stars Energy in stars causes the fusion of light elements Community Answer 4.5 153 The pH of a solution is 2.0. Which statement is correct? Useful formulas include StartBracket upper H subscript 3 upper O superscript plus EndBracket equals 10 superscript negative p H., StartBracket upper O upper H superscript minus EndBracket equals 10 superscript negative p O H., p H plus P O H equals 14., and StartBracket upper H subscript 3 upper O superscript plus EndBracket StartBracket upper O upper H superscript minus EndBracket equals 10 to the negative 14 power. Community Answer 5 Dimensional Analysis 1. I have 470 milligrams of table salt, which is the chemical compound NaCl. How many liters of NaCl solution can I make if I want the solution to be 0.90% NaCl? (9 grams of salt per 1000 grams of solution). The density of the NaCl solution is 1.0 g solution/mL solution. Community Answer 8 For which pair of functions is the exponential consistently growing at a faster rate than the quadratic over the interval mc015-1. Jpg? mc015-2. Jpg mc015-3. Jpg mc015-4. Jpg mc015-5. Jpg. New questions in Chemistry Consider the chemical equations shown here. C H 4(g)+2 O 2(g)→C O 2(g)+2 H 2O(g)Δ H 1=−802 k J 2 H 2O(g)→2 H 2O(l)Δ H 2=−88 k J Which equation shows how to calculate Δ H r x n for the equation below? C H 4(g)+2 O 2(g)→C O 2(g)+2 H 2O(l) What is Δ H r x n for the overall reaction? Consider the chemical equations shown here. 2 H 2(g)+O 2(g)→2 H 2O(g)Δ H 1=−483.6 k J+2=−241.8 k J/m o l 3 O 2(g)→2 O 3(g)Δ H 2=284.6 k J+2=142.3 k J/m o l What is the overall enthalpy of reaction for the equation shown below? 3 H 2(g)+O 3(g)→3 H 2O(g)□ kJ Consider the chemical equations shown here. P 4(s)+3 O 2(g)→P 4O 6(s)Δ H 1=−1,640.1 k J P 4O 10(s)→P 4(s)+5 O 2(g)Δ H 2=2,940.1 k J What is the overall enthalpy of reaction for the equation shown below? Round the answer to the nearest whole number. P 4O 6(s)+2 O 2(g)→P 4O 10(s) Prepare a "Coal and Petroleum" fact file. What is a double bond? A. A covalent bond between 2 atoms where each atom contributes 2 electrons. B. A covalent bond between 2 atoms where each atom contributes 4 electrons. C. An ionic bond between 2 atoms where 2 electrons have been transferred. D. A covalent bond between 2 atoms where each atom contributes 1 electron. 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2197	https://pressbooks.bccampus.ca/douglasphys1108/chapter/5-2-centripetal-force/	Skip to content Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices. 27 5.2 Centripetal Force Summary Calculate coefficient of friction on a car tire. Calculate ideal speed and angle of a car on a turn. Any force or combination of forces can cause a centripetal or radial acceleration. Just a few examples are the tension in the rope on a tether ball, the force of Earth’s gravity on the Moon, friction between roller skates and a rink floor, a banked roadway’s force on a car, and forces on the tube of a spinning centrifuge. Any net force causing uniform circular motion is called a centripetal force. The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration. According to Newton’s second law of motion, net force is mass times acceleration: net F = ma. For uniform circular motion, the acceleration is the centripetal acceleration— a = ac. Thus, the magnitude of centripetal force Fc is [latex]\boldsymbol{F_{\textbf{c}}=ma_{\textbf{c}}}.[/latex] By using the expressions for centripetal acceleration ac from [latex]\boldsymbol{a_{\textbf{c}}=\frac{v^2}{r}}[/latex] we get an expression for the centripetal force Fc in terms of mass, velocity, and radius of curvature: [latex]\boldsymbol{F_{\textbf{c}} = \dfrac{mv^2}{r}}[/latex] The centripetal force Fc is always perpendicular to the path and pointing to the center of curvature, because ac is perpendicular to the velocity and pointing to the center of curvature. Note that if you solve the first expression for r, you get [latex]\boldsymbol { r\: = \frac{mv^2} {F_{\textbf{c}} } }.[/latex] This implies that for a given mass and velocity, a large centripetal force causes a small radius of curvature—that is, a tight curve. Example 1: What Coefficient of Friction Do Care Tires Need on a Flat Curve? (a) Calculate the centripetal force exerted on a 900 kg car that negotiates a 500 m radius curve at 25.0 m/s. (b) Assuming an unbanked curve, find the minimum static coefficient of friction, between the tires and the road, static friction being the reason that keeps the car from slipping (see Figure 2). Strategy and Solution for (a) We know that [latex]\boldsymbol{{F}_{\textbf{c}}=\frac{mv^2}{r}}.[/latex] Thus, [latex]\boldsymbol{ {F}_{\textbf{c}}\:= \frac{mv^2}{r}}= \frac{(900\textbf{ kg})(25.0\textbf{ m/s})^2}{(500\textbf{ m})} =\:1125\textbf{ N} }.[/latex] Strategy for (b) Figure 2 shows the forces acting on the car on an unbanked (level ground) curve. Friction is to the left, keeping the car from slipping, and because it is the only horizontal force acting on the car, the friction is the centripetal force in this case. We know that the maximum static friction (at which the tires roll but do not slip) is μsN, where μs is the static coefficient of friction and N is the normal force. The normal force equals the car’s weight on level ground, so that N = mg. Thus the centripetal force in this situation is [latex]\boldsymbol{F_{\textbf{c}}=f=\mu_{\textbf{s}}\textbf{N}=\mu_{\textbf{s}}mg}.[/latex] Now we have a relationship between centripetal force and the coefficient of friction. Using the first expression for Fc from the equation [latex]\begin{array}{c} \boldsymbol{F_{\textbf{c}}=m\frac{v^2}{r}} \ \boldsymbol{F_{\textbf{c}}=mr\omega^2} \end{array}[/latex] [latex]\boldsymbol{m \frac{v^2}{r} = \mu_{\textbf{s}}mg. }[/latex] We solve this for μs, noting that mass cancels, and obtain [latex]\boldsymbol{ \mu_{\textbf{s}}\:= \frac{v^2}{rg}} .[/latex] Solution for (b) Substituting the knowns, [latex]\boldsymbol{ \mu_{\textbf{s}} \: = \frac{(25.0\textbf{ m/s})^2}{(500\textbf{m})(9.80\textbf{ m/s}^2) } =\:0.13} .[/latex] (Because coefficients of friction are approximate, the answer is given to only two digits.) Discussion The coefficient of friction found in part (b) is much smaller than is typically found between tires and roads. The car will still negotiate the curve if the coefficient is greater than 0.13, because static friction is a responsive force, being able to assume a value less than but no more than μsN. A higher coefficient would also allow the car to negotiate the curve at a higher speed, but if the coefficient of friction is less, the safe speed would be less than 25 m/s. Note that mass cancels, implying that in this example, it does not matter how heavily loaded the car is to negotiate the turn. Mass cancels because friction is assumed proportional to the normal force, which in turn is proportional to mass. If the surface of the road were banked, the normal force would be less as will be discussed below. Figure 2. This car on level ground is moving away and turning to the left. The centripetal force causing the car to turn in a circular path is due to friction between the tires and the road. A minimum coefficient of friction is needed, or the car will move in a larger-radius curve and leave the roadway. Optional: Banked Curves Let us now consider banked curves, where the slope of the road helps you negotiate the curve. See Figure 3. The greater the angle θ, the faster you can take the curve. Race tracks for bikes as well as cars, for example, often have steeply banked curves. In an “ideally banked curve,” the angle θ is such that you can negotiate the curve at a certain speed without the aid of friction between the tires and the road. We will derive an expression for θ for an ideally banked curve and consider an example related to it. For ideal banking, the net external force equals the horizontal centripetal force in the absence of friction. The components of the normal force N in the horizontal and vertical directions must equal the centripetal force and the weight of the car, respectively. In cases in which forces are not parallel, it is most convenient to consider components along perpendicular axes—in this case, the vertical and horizontal directions. Figure 3 shows a free body diagram for a car on a frictionless banked curve. If the angle θ is ideal for the speed and radius, then the net external force will equal the necessary centripetal force. The only two external forces acting on the car are its weight w and the normal force of the road N. (A frictionless surface can only exert a force perpendicular to the surface—that is, a normal force.) These two forces must add to give a net external force that is horizontal toward the center of curvature and has magnitude mv2/r. Because this is the crucial force and it is horizontal, we use a coordinate system with vertical and horizontal axes. Only the normal force has a horizontal component, and so this must equal the centripetal force—that is, [latex]\boldsymbol{N\textbf{sin} \;\theta\: = \frac{mv^2}{r} }.[/latex] Because the car does not leave the surface of the road, the net vertical force must be zero, meaning that the vertical components of the two external forces must be equal in magnitude and opposite in direction. From the figure, we see that the vertical component of the normal force is N cos θ, and the only other vertical force is the car’s weight. These must be equal in magnitude; thus, N cos θ = m g Now we can combine the last two equations to eliminate N and get an expression for θ, as desired. Solving the second equation for N = mg/(cos θ), and substituting this into the first yields [latex]\boldsymbol{mg \frac{\textbf{sin} \;\theta}{\textbf{cos} \;\theta}} = \frac{mv^2}{r}}[/latex] [latex]\boldsymbol{ mg \;\textbf{tan}(\theta)\:= \frac{mv^2}{r}}[/latex] [latex]\boldsymbol{\textbf{tan}\theta\:= \frac{v^2}{rg}.}[/latex] Taking the inverse tangent gives [latex]\boldsymbol{\theta=\textbf{tan}^{-1} (\frac{v^2}{rg})}[/latex] (ideally banked curve, no friction). This expression can be understood by considering how θ depends on v and r. A large θ will be obtained for a large v and a small r. That is, roads must be steeply banked for high speeds and sharp curves. Friction helps, because it allows you to take the curve at greater or lower speed than if the curve is frictionless. Note that θ does not depend on the mass of the vehicle. Figure 3. The car on this banked curve is moving away and turning to the left. Example 2: What Is the Ideal Speed to Take a Steeply Banked Tight Curve? Curves on some test tracks and race courses, such as the Daytona International Speedway in Florida, are very steeply banked. This banking, with the aid of tire friction and very stable car configurations, allows the curves to be taken at very high speed. To illustrate, calculate the speed at which a 100 m radius curve banked at 65.0° should be driven if the road is frictionless. Strategy We first note that all terms in the expression for the ideal angle of a banked curve except for speed are known; thus, we need only rearrange it so that speed appears on the left-hand side and then substitute known quantities. Solution Starting with [latex]\boldsymbol{\textbf{tan}\theta\:= \frac{v^2}{rg}}[/latex] we get [latex]\boldsymbol{v=(rg\:\textbf{tan}\:\theta)^{1/2}}.[/latex] Noting that tan 65.0º = 2.14, we obtain [latex]\begin{array}{lcl} \boldsymbol{v} & = & \boldsymbol{[(100\textbf{ m})(9.80\textbf{ m/s}^2)(2.14)]^{1/2}} \ \boldsymbol{} & = & \boldsymbol{45.8}\textbf{ m/s.} \end{array}[/latex] Discussion This is just about 165 km/h, consistent with a very steeply banked and rather sharp curve. Tire friction enables a vehicle to take the curve at significantly higher speeds. Calculations similar to those in the preceding examples can be performed for a host of interesting situations in which centripetal force is involved—a number of these are presented in this chapter’s Problems and Exercises. TAKE-HOME EXPERIMENT Ask a friend or relative to swing a golf club or a tennis racquet. Take appropriate measurements to estimate the centripetal acceleration of the end of the club or racquet. You may choose to do this in slow motion. PHET EXPLORATIONS: GRAVITY AND ORBITS Move the sun, earth, moon and space station to see how it affects their gravitational forces and orbital paths. Visualize the sizes and distances between different heavenly bodies, and turn off gravity to see what would happen without it. Figure 4. Gravity and Orbits Section Summary Centripetal force Fc is any force causing uniform circular motion. It is a “center-seeking” force that always points toward the center of rotation. It is perpendicular to linear velocity v and has magnitude [latex]\boldsymbol{F_{\textbf{c}}=ma_{\textbf{c}}},[/latex] which can also be expressed as [latex]\boldsymbol{F_{\textbf{c}}=m\frac{v^2}{r}}[/latex] Conceptual Questions Conceptual Questions 1: If you wish to reduce the stress (which is related to centripetal force) on high-speed tires, would you use large- or small-diameter tires? Explain. 2: Define centripetal force. Can any type of force (for example, tension, gravitational force, friction, and so on) be a centripetal force? Can any combination of forces be a centripetal force? 3: If centripetal force is directed toward the center, why do you feel that you are ‘thrown’ away from the center as a car goes around a curve? Explain. 4: Race car drivers routinely cut corners as shown in Figure 5. Explain how this allows the curve to be taken at the greatest speed. Figure 5. Two paths around a race track curve are shown. Race car drivers will take the inside path (called cutting the corner) whenever possible because it allows them to take the curve at the highest speed. 5: A number of amusement parks have rides that make vertical loops like the one shown in Figure 6. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if: (a) The car goes over the top at faster than this speed? (b)The car goes over the top at slower than this speed? Figure 6. Amusement rides with a vertical loop are an example of a form of curved motion. 7: What is the direction of the force exerted by the car on the passenger as the car goes over the top of the amusement ride pictured in Figure 6 under the following circumstances: (a) The car goes over the top at such a speed that the gravitational force is the only force acting? (b) The car goes over the top faster than this speed? (c) The car goes over the top slower than this speed? 8: As a skater forms a circle, what force is responsible for making her turn? Use a free body diagram in your answer. 9: Suppose a child is riding on a merry-go-round at a distance about halfway between its center and edge. She has a lunch box resting on wax paper, so that there is very little friction between it and the merry-go-round. Which path shown in Figure 7 will the lunch box take when she lets go? The lunch box leaves a trail in the dust on the merry-go-round. Is that trail straight, curved to the left, or curved to the right? Explain your answer. Figure 7. A child riding on a merry-go-round releases her lunch box at point P. This is a view from above the clockwise rotation. Assuming it slides with negligible friction, will it follow path A, B, or C, as viewed from Earth’s frame of reference? What will be the shape of the path it leaves in the dust on the merry-go-round? 10: Do you feel yourself thrown to either side when you negotiate a curve that is ideally banked for your car’s speed? What is the direction of the force exerted on you by the car seat? 11: Suppose a mass is moving in a circular path on a frictionless table as shown in figure. In the Earth’s frame of reference, there is no centrifugal force pulling the mass away from the centre of rotation, yet there is a very real force stretching the string attaching the mass to the nail. Using concepts related to centripetal force and Newton’s third law, explain what force stretches the string, identifying its physical origin. Figure 8. A mass attached to a nail on a frictionless table moves in a circular path. The force stretching the string is real and not fictional. What is the physical origin of the force on the string? Problems & Exercises 1: A 22.0 kg child is riding a playground merry-go-round is moving at a speed of 0.0698 m/s when she is 1.25 m from the center. What is the centripetal force exerted on her? Compare this force to her weight. 2: What is the ideal banking angle for a gentle turn of 1.20 km radius on a highway with a 105 km/h speed limit (about 65 mi/h), assuming everyone travels at the limit? 3: What is the ideal speed to take a 100 m radius curve banked at a 20.0° angle? 4: (a) What is the radius of a bobsled turn banked at 75.0° and taken at 30.0 m/s, assuming it is ideally banked? (b) Calculate the centripetal acceleration. (c) Does this acceleration seem large to you? 5: Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen in Figure 9. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the centripetal force), and the vertical normal force (which must equal the system’s weight). (a) Show that θ (as defined in the figure) is related to the speed v and radius of curvature r of the turn in the same way as for an ideally banked roadway—that is, θ = tan-1 v2/rg. (b) Calculate θ for a 12.0 m/s turn of radius 30.0 m (as in a race). Figure 9. A bicyclist negotiating a turn on level ground must lean at the correct angle—the ability to do this becomes instinctive. The force of the ground on the wheel needs to be on a line through the center of gravity. The net external force on the system is the centripetal force. The vertical component of the force on the wheel cancels the weight of the system while its horizontal component must supply the centripetal force. This process produces a relationship among the angle θ, the speed v, and the radius of curvature r of the turn similar to that for the ideal banking of roadways. 6: Integrated Concepts If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads). (a) Calculate the ideal speed to take a 100 m radius curve banked at 15.0º. (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km/h? 7: Modern roller coasters have vertical loops like the one shown in Figure 11. The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, keeping the passengers pressed firmly into their seats. What is the speed of the roller coaster at the top of the loop if the radius of curvature there is 15.0 m and the downward acceleration of the car is 1.50 g? Figure 11. Teardrop-shaped loops are used in the latest roller coasters so that the radius of curvature gradually decreases to a minimum at the top. This means that the centripetal acceleration builds from zero to a maximum at the top and gradually decreases again. A circular loop would cause a jolting change in acceleration at entry, a disadvantage discovered long ago in railroad curve design. With a small radius of curvature at the top, the centripetal acceleration can more easily be kept greater than g so that the passengers do not lose contact with their seats nor do they need seat belts to keep them in place. 8: Unreasonable Results (a) Calculate the minimum coefficient of friction needed for a car to negotiate an unbanked 50.0 m radius curve at 30.0 m/s. (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent? Glossary centripetal force : any net force causing uniform circular motion ideal banking : the sloping of a curve in a road, where the angle of the slope allows the vehicle to negotiate the curve at a certain speed without the aid of friction between the tires and the road; the net external force on the vehicle equals the horizontal centripetal force in the absence of friction ideal speed : the maximum safe speed at which a vehicle can turn on a curve without the aid of friction between the tire and the road ideal angle : the angle at which a car can turn safely on a steep curve, which is in proportion to the ideal speed banked curve : the curve in a road that is sloping in a manner that helps a vehicle negotiate the curve Solutions Problems & Exercises 1: (a) 483 N (b) 2.24 times her weight 2: 4.14° 4: 24.6 m (b) 36.6 m/s2 (c) ac =3.73 g. This does not seem too large, but it is clear that bobsledders feel a lot of force on them going through sharply banked turns. 6: (a) 16.2 m/s (b) 0.234 8: (a)1.84 (b) A coefficient of friction this much greater than 1 is unreasonable . (c) The assumed speed is too great for the tight curve. License Douglas College Physics 1108 Custom Textbook Winter 2021 current Copyright © by OpenStax is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. Share This Book
2198	https://stats.libretexts.org/Bookshelves/Applied_Statistics/Natural_Resources_Biometrics_(Kiernan)/03%3A_Hypothesis_Testing/3.05%3A_Hypothesis_Test_about_a_Variance	Skip to main content 3.5: Hypothesis Test about a Variance Last updated : Jan 8, 2024 Save as PDF 3.4: Hypothesis Test for a Population Proportion 3.6: Putting it all Together Using the Classical Method Page ID : 2887 Diane Kiernan SUNY College of Environmental Science and Forestry via OpenSUNY ( \newcommand{\kernel}{\mathrm{null}\,}) Hypothesis Test about a Variance When people think of statistical inference, they usually think of inferences involving population means or proportions. However, the particular population parameter needed to answer an experimenter’s practical questions varies from one situation to another, and sometimes a population’s variability is more important than its mean. Thus, product quality is often defined in terms of low variability. Sample variance s2 can be used for inferences concerning a population variance σ2. For a random sample of n measurements drawn from a normal population with mean μ and variance σ2, the value s2 provides a point estimate for σ2. In addition, the quantity (n−1)s2σ2 follows a Chi-square(χ2) distribution, with df=n–1. The properties of Chi-square (χ2) distribution are: Unlike Z and t distributions, the values in a chi-square distribution are all positive. The chi-square distribution is asymmetric, unlike the Z and t distributions. There are many chi-square distributions. We obtain a particular one by specifying the degrees of freedom (df=n–1) associated with the sample variances s2. One-sample (χ2) test for testing the hypotheses: Null hypothesis: H0:σ2=σ20(constant) Alternative hypothesis: Ha:σ2>σ20(one-tailed), reject H0 if the observed χ2>χ2U(upper-tail value at α). Ha:σ2<σ20 (one-tailed), reject H0 if the observed χ2<χ2L(lower-tail value at α). Ha:σ2≠σ20 (two-tailed), reject H0 if the observed χ2>χ2Uor χ2<χ2Lat α/2. where the χ2 critical value in the rejection region is based on degrees of freedom df=n–1 and a specified significance level of α. Test statistic: χ2=(n−1)S2σ20(3.5.1) As with previous sections, if the test statistic falls in the rejection zone set by the critical value, you will reject the null hypothesis. Example 3.5.1: A forester wants to control a dense understory of striped maple that is interfering with desirable hardwood regeneration using a mist blower to apply an herbicide treatment. She wants to make sure that treatment has a consistent application rate, in other words, low variability not exceeding 0.25 gal./acre (0.06 gal.2). She collects sample data (n = 11) on this type of mist blower and gets a sample variance of 0.064 gal.2 Using a 5% level of significance, test the claim that the variance is significantly greater than 0.06 gal.2 H0:σ2=0.06 H1:σ2>0.06 The critical value is 18.307. Any test statistic greater than this value will cause you to reject the null hypothesis. The test statistic is χ2=(n−1)S2σ20=(11−1)0.0640.06=10.667 We fail to reject the null hypothesis. The forester does NOT have enough evidence to support the claim that the variance is greater than 0.06 gal.2 You can also estimate the p-value using the same method as for the student t-table. Go across the row for degrees of freedom until you find the two values that your test statistic falls between. In this case going across the row 10, the two table values are 4.865 and 15.987. Now go up those two columns to the top row to estimate the p-value (0.1-0.9). The p-value is greater than 0.1 and less than 0.9. Both are greater than the level of significance (0.05) causing us to fail to reject the null hypothesis. Software Solutions Minitab (referring to Ex. 3.5.1) Test and CI for One Variance \| Method \| \| \| \| Null hypothesis \| Sigma-squared \| = 0.06 \| \| Alternative hypothesis \| Sigma-squared \| > 0.06 \| The chi-square method is only for the normal distribution. Tests \| \| \| \| \| \| Method \| Statistic \| DF \| P-Value \| \| Chi-Square \| 10.67 \| 10 \| 0.384 \| Excel Excel does not offer 1-sample χ2 testing. 3.4: Hypothesis Test for a Population Proportion 3.6: Putting it all Together Using the Classical Method
2199	https://www.cuemath.com/worksheets/lcm-and-gcf-worksheets/	LCM and GCF Worksheets \| Printable Online LCM and GCF Worksheet PDFs We use cookies to improve your experience. Learn more OK Loading... Sign up LCM and GCF Worksheets LCM And GCF Worksheets is great for practicing finding the Least Common Multiple and Greatest Common Factor of number sets. Understanding LCM's and GCFs are very important for working with fraction problems. Benefits of LCM and GCF Worksheets Lcm And Gcf Worksheets help kids to understand the least common multiple and find the least common multiple between two or three numbers. It helps to develop the math skills students at the elementary level might also learn a variety of strategies to find the factors. Printable PDFs for LCM and GCF Worksheets Lcm And Gcf Worksheets help kids increase their confidence and their problem-solving skills will be improved. These math worksheets are a helpful guide for kids as well as their parents to see and review their answer sheets. Children can download the pdf format of these easily accessible lcm and gcf worksheets to practice and solve questions for free. LCM and GCF Worksheet - 1Download PDF LCM and GCF Worksheet - 2Download PDF LCM and GCF Worksheet - 3Download PDF LCM and GCF Worksheet - 4Download PDF ☛ Check Grade wise LCM and GCF Worksheets 6th Grade LCM Worksheets Grade 6 GCF Worksheets Explore math program Math worksheets and visual curriculum Sign up FOLLOW CUEMATH Facebook Youtube Instagram Twitter LinkedIn Tiktok MATH PROGRAM Online math classes Online Math Courses online math tutoring Online Math Program After School Tutoring Private math tutor Summer Math Programs Math Tutors Near Me Math Tuition Homeschool Math Online Solve Math Online Curriculum NEW OFFERINGS Coding SAT Science English MATH ONLINE CLASSES 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math ABOUT US Our Mission Our Journey Our Team QUICK LINKS Maths Games Maths Puzzles Our Pricing Math Questions Blogs Events FAQs MATH TOPICS Algebra 1 Algebra 2 Geometry Calculus math Pre-calculus math Math olympiad Numbers Measurement MATH TEST CAASPP CogAT STAAR NJSLA SBAC Math Kangaroo AMC 8 MATH CURRICULUM 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math FOLLOW CUEMATH Facebook Youtube Instagram Twitter LinkedIn Tiktok MATH PROGRAM Online math classes Online Math Courses online math tutoring Online Math Program After School Tutoring Private math tutor Summer Math Programs Math Tutors Near Me Math Tuition Homeschool Math Online Solve Math Online Curriculum NEW OFFERINGS Coding SAT Science English MATH CURRICULUM 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math MATH TEST CAASPP CogAT STAAR NJSLA SBAC Math Kangaroo AMC 8 ABOUT US Our Mission Our Journey Our Team MATH TOPICS Algebra 1 Algebra 2 Geometry Calculus math Pre-calculus math Math olympiad Numbers Measurement QUICK LINKS Maths Games Maths Puzzles Our Pricing Math Questions Blogs Events FAQs MATH ONLINE CLASSES 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math Terms and ConditionsPrivacy Policy

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