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12300	https://www.chemistrylearner.com/lucas-reagent.html	# Chemistry Learner It's all about Chemistry Inorganic Chemistry Chemical Bond Chemical Reactions Materials Chemistry Organic Chemistry Periodic Table Periodic Trends Periodic Table Groups How to Read Periodic Table Worksheets Naming Covalent Compounds Worksheets Net Ionic Equation Worksheets Types of Chemical Reactions Worksheets Word Equations Worksheets Valence Electrons Worksheets Periodic Trends Worksheets Graphing Periodic Trends Worksheets Periodic Trends Ionization Energy Worksheets Atomic Structure Worksheets Atomic Structure And Isotopes Worksheets Home / Organic Chemistry / Lucas Reagent Lucas Reagent Table of Contents Lucas reagent is a solution of anhydrous zinc chloride (Lewis acid) in concentrated hydrochloric acid. It is used as a reagent to test alcohols and classify them in accordance to their reactivity. The reaction is a substitution reaction where the chloride of the zinc chloride gets replaced by the hydroxyl group of the alcohol. The reactivity of the alcohol with Lucas Reagent is measured by the degree of turbidity which may vary from colorless to turbid. The formation of turbid solution happens due to the formation of chloroalkane. This experiment was done in 1930. Since then, it is utilized as a standard technique in organic chemistry experiments. But now this method is not widely used as large number of spectroscopic and chromatographic analytical methods have replaced it. Lucas Reagent Formula Lucas reagent is basically a solution which is formed by the combination of HCl and ZnCl2 Lucas Test This test is more often used to categorize the different types of alcohols based on the time taken to form a turbid solution or precipitation using the Lucas Reagent namely: Primary alcohol: Here no visible reaction is observed and the solution remains colorless e.g. 1-Pentanol Secondary alcohol: Here the solution turns turbid or cloudy in 5-20 minutes with slight heating e.g. 2-Pentanol Tertiary alcohol: Here the solution turns turbid or cloudy rapidly with the formation of two separate layers at room temperature e.g. 2-Methyl-2-butanol In Lucas test, zinc chloride acts as a catalyst. The classification of the alcohols is usually done based on the difference in reaction with concentrated hydrochloric acid. A simple reaction is given below: ZnCl2 ROH + HCl—>RCl + H2O The tertiary alcohol undergoes the most stable reaction and the primary alcohol undergoes the least stable reaction. This test can be conducted only with those alcohols which are soluble in Lucas reagent and with lower molecular weight. Alcohols generally with more than six carbon atoms cannot be tested. Lucas Reagent Preparation The Lucas reagent can be prepared by the following steps: Pour the concentrated HCl into a 50 ml graduated cylinder. Measure out 47 ml of concentrated HCl and pour it into the 100 ml beaker Place the 100 ml beaker in the ice bath to absorb the heat generated during the dissolution of the ZnCl2 Weigh out 62.5 g of anhydrous ZnCl2 and allow it to dry in an oven for at least two hours. Cool the anhydrous ZnCl2 in a dessicator to prevent air contact. Add the ZnCl2 to the hydrochloric acid in the beaker slowly to avoid the mixture overflowing the sides of the small beaker. Stir the mixture until the ZnCl2 dissolves completely to form the Lucas Reagent. Store the reagent in a cool, dry place for later use. Lucas Reagent Mechanism The reaction which normally occurs is a SN1 nucleophilic substitution which is a two steps reaction. Alcohols which have a capability to form carbocation intermediates exhibit this reaction. Only secondary and tertiary alcohols exhibit the SN1 nucleophilic mechanism. The two steps which are generally followed in this reaction: In the first step the proton (H+) from Hydrochloric acid (HCl) will protonate the OH– group of the alcohol. Water (H2O) attached to the carbon is a weaker nucleophile than Cl (Chloride). Thus nucleophile Cl– replaces the H2O group forming a carbocation as its present in excess. In the second step the Cl– attacks the carbocation and thus forms alkyl chloride. Here the first step is generally the slowest step and is the rate-determining step. As the tertiary carbocation is much stabilized, they are the ones that undergo reaction and form a turbid solution. The opposite of it happens in the case of primary alcohols. Thus on the basis of the reactions and the rate of all the three reactions, we can not only measure the reactivity but also characterize the three groups of alcohols. Lucas Reagent MSDS Lucas reagent is highly toxic and corrosive and should be handled carefully while conducting the experiment. The toxicity and corrosiveness arise as a result of the constituents. Hydrochloric acid can irritate the skin. The vapors should not be inhaled as they might affect the respiratory system. Zinc Chloride is highly corrosive and may cause damage to the skin and the respiratory system. Apart from the constituents, the vapors of the alcohol are slightly irritating to the eyes and nose. It should not be inhaled as it may turn fatal. References1. 2. 3. 4. 5. 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12301	https://www.trinadeboreeteachingandlearning.com/math-blog/place-value-games	Place Value: Math is Fun: 3 Games to Teach Place Value in the Classroom — Trina Deboree Teaching and Learning 0 Readers in the Making is under construction! We are working on making our site a better experience for you! Skip to Content Listen One Tired Teacher Read Blog Learn Mastering makerspace-sales page JOIN THE WORKSHOP! Open Menu Close Menu Listen One Tired Teacher Read Blog Learn Mastering makerspace-sales page JOIN THE WORKSHOP! Open Menu Close Menu Folder:Listen Folder:Read Folder:Learn JOIN THE WORKSHOP! Back One Tired Teacher Back Blog Back Mastering makerspace-sales page Place Value: Math is Fun: 3 Games to Teach Place Value in the Classroom Math May 28 Written By Trina Deboree Are you ready to make teaching place value fun in 2nd grade? I know what you are thinking. How can teaching place value to 2nd graders be fun?! For some kids, math is anything but enjoyable. Making math and place value fun and engaging can help your most reluctant math students jump for joy. With the right tools of the trade, it is possible to get everyone excited to better understand two-digit and three-digit numbers. Whether 1st graders or 2nd graders are learning about the ones place, the tens place, the hundreds place, or the thousands place, there are plenty of engaging ways for kids to practice their skills in the place value of the digits. Check out these top 3 FREE place value games designed to help students understand the place value system in a classroom setting! Top 3 Games to Teach Place Value in the Classroom: 1. Base Ten Bingo Base Ten Bingo is a fun online game that uses good ole fashion bingo to help students master place value. You can set your kids up on ABCya.com. Students choose between 2-digit numbers, 3-digit numbers, or more. Then they play BINGO! You collect Bingo bugs as you play. Kids think this game is so fun! The best part of this game is how you can differentiate. Even 1st graders can play with two-digit numbers and will learn the correct places for numbers. Students use a traditional bingo card. The base ten blocks are shown, and students have to use these blocks to count out the number. If they have the total on their bingo sheet, they click the number that represents the model. Students quickly learn that numbers have different values based on the position of a digit. Get ready to hear lots of shouts or whispers of -BINGO - in your classroom! 2. Base Ten Bingo Base Ten Bingo is a fun online game that uses good ole fashion bingo to help students master place value. You can set your kids up on ABCya.com. Students choose between 2-digit numbers, 3-digit numbers, or more. Then they play BINGO! You collect Bingo bugs as you play. Kids think this game is so fun! The best part of this game is how you can differentiate. Even 1st graders can play with two-digit numbers and will learn the correct places for numbers. Students use a traditional bingo card. The base ten blocks are shown, and students have to use these blocks to count out the number. If they have the total on their bingo sheet, they click the number that represents the model. Students quickly learn that numbers have different values based on the position of a digit. Get ready to hear lots of shouts or whispers of -BINGO - in your classroom! 3. Number Beat-It Number Beat-It is a fun math game for kids to play while listening to the song Beat It by Michael Jackson! (If you are an 80s kid, you know what I'm talking about!) Kids take turns trying tobeatthe larger numbers. Strategy is at play when kids can move the cards around and try to make the bigger number. Students can compare the place value of each digit with their partner's given number increasing place value understanding. For example, if students are playing for the hundredths place, they will compare the face value of a digit to see who has the large number. Children in kindergarten and 1st grade have a great deal of experience with the digit value. Most children will be familiar with recording whole numbers on paper and comprehending the positional values while also placing numbers in the correct position to the tens place. As 2nd grade teachers, we can take it from there! Yet, sometimes place value feels tricky for kids. This game of Number Beat-It will allow your students to practice place value in a fun and engaging manner. Use this game as a warm-up in your math block, or place it inside a math center to use again and again. Laminate the cards and board to last longer. (Add white cubes or an orange rod or orange squares to help students understand the one-to-one relationships to numbers if needed. Math manipulatives can often help solidify a concept.) Directions: Shuffle the cards and place the deck number-side down on the table. Each player uses one row of boxes on the Beat-It Mat. (Choose hundreds mat or thousands mat.) In each round, players take turns turning over the top card from the deck and placing it number-side up on any empty boxes. Each player turns 3 or 4 (depending on hundreds or thousands). Each player sets their 3-4 cards on their row of the Beat-It mat. At the end of the round, players read their numbers aloud and compare them. Each player records the comparison on their Beat-It record sheet. The player with the more significant number for the round scores 1 point. Play five rounds for the game. Shuffle the deck between matches. The player with the most points wins the game. Strategy comes in when students move the cards around to make the largest target number and BEAT the other number. Grab Number Beat-It For Free! If you're looking for a way to make place value more fun for your students, look no further! These three games are sure to get them engaged and excited about learning. The best part is they can be easily implemented in any classroom. So what are you waiting for? Give your students the gift of learning place value with these top 3 games. Plus,Number Beat-Itis available as a free download - so you have nothing to lose! And you get to rock out to some 80s music while the kids play! Enjoy! mathplace valueplace value games2nd gradefirst grade1st gradesecond gradeteacher Trina Deboree Previous Previous Place Value: Math is Super Fun With Tic-Tac-Toe for 2nd Graders --------------------------------------------------------------- JOIN ME IN SPREADING A LOVE OF READING! Let's do it! We won't send you spam. Unsubscribe at any time. 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12302	https://journals.lww.com/10.4103/aja202527	Asian Journal of Andrology Log in or Register Submit a Manuscript ### Secondary Logo Enter your Email address: Privacy Policy ### Journal Logo Articles Advanced Search Toggle navigation RegisterLogin Browsing History Home Current Issue Previous Issues Archive 2014 Onwards Archive 1999-2013 For Authors Submit a Manuscript Information for Authors Published Ahead-of-Print Journal Info About the Journal Editorial Board Advertising Subscriptions AJA Club Reprints Rights and Permissions News Articles Advanced Search - Volume - Issue Previous Article Next Article Outline INTRODUCTION PATIENTS AND METHODS Patients Study design Statistical analyses Ethical approval RESULTS DISCUSSION CONCLUSIONS AUTHOR CONTRIBUTIONS COMPETING INTERESTS ACKNOWLEDGMENTS REFERENCES Images Slideshow Gallery Export PowerPoint file Download PDF EPUB Cite Copy Export to RIS Export to EndNote Share Email Facebook X LinkedIn Favorites Permissions More Cite Permissions Image Gallery Article as EPUB Export All Images to PowerPoint FileAdd to My Favorites Email to Colleague Colleague's E-mail is Invalid Your Name: Colleague's Email: Separate multiple e-mails with a (;). Message: Your message has been successfully sent to your colleague. Some error has occurred while processing your request. Please try after some time. Export to End Note Procite Reference Manager [x] Save my selection ORIGINAL ARTICLE Trend in testicular volume change after orchiopexy in 854 children with cryptorchidism He, Ying-Ying 1,2,; Ke, Zhi-Cong 2,; Li, Shou-Lin 2; Guo, Hui-Jie 2; Zhang, Pei-Liang 2; Chen, Peng-Yu 2; Xu, Wan-Hua 2; Sun, Feng-Hao 2; Yang, Zhi-Lin 2 Author Information 1 Department of Urology, Shenzhen Children’s Hospital, China Medical University, Shenzhen 518000, China 2 Department of Urology and Laboratory of Pelvic Floor Muscle Function, Shenzhen Children’s Hospital, Shenzhen 518000, China. Correspondence: Dr. ZL Yang (yangzhilin207@163.com) These authors contributed equally to this work. Asian Journal of Andrology ():10.4103/aja202527, July 08, 2025. \| DOI: 10.4103/aja202527 Open PAP Abstract The aim of this study was to investigate the trend in testicular volume changes after orchiopexy in children with cryptorchidism. The clinical data of 854 children with cryptorchidism who underwent orchiopexy between January 2013 and December 2016 in Shenzhen Children’s Hospital (Shenzhen, China) were retrospectively analyzed. The mean (standard deviation) age of the patients was 2.8 (2.5) years, and the duration of follow-up ranged from 1 year to 5 years. Ultrasonography was conducted preoperatively and postoperatively. The variables analyzed included age at the time of surgery, type of surgical procedure, laterality, preoperative testicular position, preoperative and postoperative testicular volumes, and the testicular volume ratio of them. The average testicular volumes preoperatively and at 1 year, 2 years, 3 years, and 5 years postoperatively were 0.27 ml, 0.38 ml, 0.53 ml, 0.87 ml, and 1.00 ml, respectively (P< 0.001). The corresponding testicular volume ratios were 0.67, 0.76, 0.80, 0.83, and 0.84 (P< 0.001). The mean volume of the undescended testes was significantly smaller than the mean normative value (P< 0.001, lower than the 10 th percentile). The postoperative testicular volumes in children with cryptorchidism were generally lower than those in healthy boys but were still greater than the 10 th percentile and exhibited an increasing trend. The older the child is at the time of surgery, the larger the gap in volume between the affected and normal testes. Although testicular volume tends to gradually increase after orchiopexy for cryptorchidism, it could not normalizes. Earlier surgery results in affected testicular volumes closer to those of healthy boys. INTRODUCTION Cryptorchidism, known as undescended testis (UDT), is one of the most common congenital abnormalities in boys and is among the few well-known risk factors for testicular cancer.1 In addition to male infertility, which is the most important consequence of cryptorchidism, boys with UDT have a significantly greater risk of developing testicular tumors or testicular torsion. In addition, these boys are also more prone to testicular trauma than boys with normal testicles.2 Cryptorchidism affects 1.0% to 4.6% of full-term neonates and 1.1% to 4.5% of preterm neonates.3 For normal spermatogenesis, the testes should be properly positioned within the scrotum at birth to maintain the necessary temperature. Effective screening and early surgical intervention are crucial for optimizing fertility in adulthood and significantly reducing the risk of testicular cancer. The standard surgical treatment for cryptorchidism is orchiopexy, which improves future fertility prospects and facilitates cancer surveillance.4 The choice of surgical technique, whether open or laparoscopic, primarily depends on the location and accessibility of the undescended testis.5 Given that 80% to 90% of the testis comprises seminiferous tubules, testicular volume is closely linked to the semen profile and testicular function.6 Thus, it is essential to assess testicular volume. However, research on changes in testicular volume after orchiopexy is limited, and the trend of testicular development following orchiopexy remains unclear. To address this gap, we analyzed preoperative and long-term postoperative testicular volumes in a large cohort and examined trends in testicular volume changes. PATIENTS AND METHODS Patients In this study, we used the clinical data of children who were diagnosed with cryptorchidism and underwent orchiopexy at Shenzhen Children’s Hospital (Shenzhen, China), between January 2013 and December 2016. The inclusion criteria were as follows: (1) a preoperative diagnosis of cryptorchidism confirmed through neurological examination and ultrasonography; (2) age between 0 and 14 years; (3) orchiopexy performed at our hospital; and (4) complete clinical, medical, and surgical records. The exclusion criteria included: (1) a follow-up period less than 1 year; (2) incomplete medical records or follow-up data; or (3) the presence of disorders of sexual development or associated chromosomal or gonadal abnormalities. Additionally, 1607 healthy boys aged 0 to 14 years were studied as a control group.7 Study design The study indices included age at the time of surgery, surgical approach, laterality of cryptorchidism, preoperative testicular position, preoperative and postoperative testicular volumes, and the testicular volume ratio. We also collected ultrasound data preoperatively and at 1 year, 2 years, 3 years, and 5 years postoperatively and measured the length, width, and height of the testes to calculate testicular volume using the following formula: testicular volume = 0.52 × length × width × height.8 To establish a baseline for comparison, we calculated the median and the 10 th and 90 th percentiles of testicular volumes for age-matched control groups of healthy boys, allowing us to chart the natural progression of testicular growth relative to age. The primary outcome was testicular volume, and the secondary outcome was the testicular volume ratio. The children with cryptorchidism were categorized into seven age groups according to their age at the time of surgery: 0 to 1 year (0 < age < 1 year), 1 year to 2 years (1 year ≤ age < 2 years), 2 years to 3 years (2 years ≤ age < 3 years), 3 years to 4 years (3 years ≤ age < 4 years), 4 years to 6 years (4 years ≤ age < 6 years), 6 years to 8 years (6 years ≤ age < 8 years), and 8 years to 14 years (8 years ≤ age ≤ 14 years). The testicular volume ratio was defined as the ratio of the volume of the affected testis to that of the contralateral testis. Statistical analyses All the statistical analyses were performed using the Statistical Package for the Social Sciences (SPSS, version 29.0.1.0; IBM Corp., Armonk, NY, USA). Statistical data were analyzed using the t-test, Mann‒Whitney U test, or one-way analysis of variance (ANOVA), whereas categorical data were analyzed using the Chi-squared test. One-way ANOVA was conducted to compare the preoperative testicular volumes of children with cryptorchidism to those at 1 year, 2 years, 3 years, and 5 years postoperatively, as well as across different age groups and surgical procedures. We conducted supplementary normality tests for testicular volume and volume ratios via the Kolmogorov‒Smirnov (K‒S) and Shapiro‒Wilk (S‒W) tests. Normally distributed data are presented as the mean (standard deviation [s.d.]), whereas nonnormally distributed data are reported as median (interquartile range [IQR]). P< 0.05 indicated statistical significance. Ethical approval This study was approved by the Ethics Committee of Shenzhen Children’s Hospital (Approval No. 202209402). Informed consent for surgery was obtained from all parents or guardians of the participating children. Permission to use publicly available data has been obtained. RESULTS Eight hundred and fifty-four children diagnosed with cryptorchidism between January 2013 and December 2016 were included in this study. The observation period was from January 2014 to December 2022. The median follow-up time was 41.0 (range: 12.0–76.0) months. Among them, 311 had left-sided cryptorchidism, 407 had right-sided cryptorchidism, and 136 had bilateral cryptorchidism. Additionally, 118 had low cryptorchidism, 650 had intermediate cryptorchidism, and 86 had high cryptorchidism. Table 1 presents the basic characteristics of all the children with cryptorchidism. A normality distribution test revealed that the testicular volume ratios from the 2 nd postoperative year were nonnormally distributed. Table 1: Basic characteristics of 854 children with cryptorchidism The average age of the patients was 2.8 years. The mean (s.d.) testicular volume on the affected side was 0.27 (0.17) ml preoperatively, 0.38 (0.39) ml at 1 year postoperatively, 0.53 (0.75) ml at 2 years postoperatively, 0.87 (1.50) ml at 3 years postoperatively, and 1.00 (1.62) ml at 5 years postoperatively. The median testicular volume ratios were 0.6 preoperatively, 0.7 at 1 year postoperatively, 0.8 at 2 years postoperatively, 0.8 at 3 years postoperatively, and 0.9 at 5 years postoperatively (Table 2 and Figure 1). The testicular volume was significantly higher after orchiopexy than that before orchiopexy (P = 0.001). These findings indicate that surgical treatment has a definite effect on the development of cryptorchid testes. As the patient grows, the volume of the affected testis becomes closer to the contralateral testis, and the testicular ratios increase (P = 0.001), indicating that with increasing age, the affected testis tends to “catch up” with the contralateral testis. Table 2: Preoperative and postoperative testicular volume and testicular volume ratio in children with cryptorchidism Figure 1: Trends in testicular volume and the testicular volume ratio preoperatively and at 1 year, 2 years, 3 years, and 5 years postoperatively in children with cryptorchidism. (a) Trend in testicular volume at different stages. (b) Trend in the testicular volume ratio. Testicular volume ratio: the ratio of the affected testicular volume to the contralateral testicular volume. In this study, we compared the preoperative and 1-year postoperative testicular volumes of cryptorchid children with those of healthy boys across all age groups. The 1-year postoperative testicular volume was significantly greater than the preoperative testicular volume in all age groups (all P< 0.05; Table 3 and Figure 2). The preoperative testicular volume of patients with cryptorchidism was markedly lower than that of healthy boys and lower than the 10 th percentile. One year after orchiopexy, the testicular volume of patients with cryptorchidism was lower than the 50 th percentile but was higher than the 10 th percentile, close to the average volume of healthy boys. This finding indicates that although the testicular volume increases after surgery, it still does not fully “catch up” to that of healthy boys. However, as surgery was delayed, the volume of the affected testis lagged behind that of healthy boys (Figure 3). The older the child is at the time of surgery, the greater the gap in testicular development between boys with cryptorchidism and healthy boys. Table 3: Comparison of preoperative and 1-year postoperative testicular volume across different age groups Figure 2: Preoperative and 1-year postoperative testicular volume in children with cryptorchidism, categorized by age group. Age group 1: age <1 year; 2: 1 year ≤ age <2 years; 3: 2 years ≤ age <3 years; 4: 3 years ≤ age <4 years; 5: 4 years ≤ age <6 years; 6: 6 years ≤ age <8 years; 7: 8 years ≤ age ≤14 years. Figure 3: Comparison of testicular volume between cryptorchid patients and normal boys. Red curve: preoperative testicular volume of cryptorchid patients. Pink curve: postoperative testicular volume (1 year postoperatively) in the same cohort. P10: the 10 th percentile of testicular volume in normal boys; P50: the median testicular volume in normal boys; P90: the 90 th percentile of testicular volume in normal boys. DISCUSSION This is a large-scale clinical study of postoperative testicular volume changes in children with cryptorchidism and a long follow-up period. Our study revealed that the preoperative testicular volume in patients with cryptorchidism was significantly lower than that in healthy boys. Postoperatively, the testicular volume of the affected side tended to “catch up” but did not reach the normal volume in healthy boys. Age at the time of surgery is a critical factor influencing postoperative testicular development. Earlier surgery results in testicular volumes closer to those of healthy boys, whereas delayed surgery results in a larger gap between the affected and normal testicular volumes. Decreased testicular volume due to atrophy impacts future fertility and hormonal function, especially when orchiopexy is delayed. Given this, incorporating preoperative volume measurements as a baseline could strengthen the model’s predictive accuracy. A recently article on laparoscopic orchiopexy indicates that testicular volume recovery can signify successful intervention and improved vascularization postoperatively.9 Therefore, using testicular volume as both a prognostic marker and an indicator of model effectiveness could enhance patient monitoring protocols. In this study, we observed that the testicular volume and testicular volume ratio increased following orchiopexy, indicating that the volume of the affected testis tended to “catch up” with the volume of contralateral healthy testis after the operation. Ajiki et al.10 studied 93 children with cryptorchidism and reported that the post-orchiopexy testicular volume on the cryptorchid side was significantly greater than the preoperative volume. Tseng et al.11 included 182 children with cryptorchidism, with a median follow-up of 34 months, and most children had good postoperative testicular growth. Similarly, Marret et al.12 investigated 55 children and reported that surgical treatment led to good testicular development. Another study also reported significantly increased testicular volumes after orchiopexies. In that study, the testicular growth rate was higher than normal.13 Histological analysis revealed that the increase in testicular development after orchiopexy was due to an increase in the number of germ cells per tubule and interstitial cells.14,15 Laparoscopic orchidopexies were associated with a lower risk of complications such as testicular atrophy and wound infection and greater success in maintaining the position of the testicle in the middle or lower part of the scrotum without subsequent retraction.2 Therefore, orchiopexy positively affects testicular development in children with cryptorchidism, and the change in testicular volume during the follow-up period is one of the most important criteria for surgical success.16–19 We observed that the preoperative testicular volume in children with cryptorchidism was lower than that in healthy boys. Although the testicular volume tended to “catch up” after orchiopexy, it remained lower than that in healthy boys, and this disparity did not normalize by the end of the study’s observation period. This phenomenon may be related to cryptorchidism. Testicular hypoplasia, characterized by reduced size due to poor testicular development and a decrease in or absence of germ cells, is a known consequence of this condition.20 Tasian et al.15 studied the relationship between histopathological changes in the testes and reduced fertility potential in children with cryptorchidism. They reported that each additional month of undescended testes was associated with a significant reduction in germ cells and mesenchymal stromal cells, which are often moderate to severe. Additionally, undescended testes are prone to severe germ cell depletion, which may be exacerbated by elevated temperatures in the inguinal canal that increase the levels of reactive oxygen species, promote germ cell apoptosis, and reduce testicular weight.21,22 Early orchiopexy in children with cryptorchidism may help reduce the risk of testicular cancer, preserve fertility, and improve testicular development.23 The optimal age for surgical intervention in children with cryptorchidism remains debated, with current recommendations suggesting surgery between 6 months and 12 months or up to 18 months after birth.14,23–25 The evidence increasingly suggests the benefits of early surgery for promoting testicular health and fertility.26 Despite early diagnosis in many cases, most children are referred for surgery after their 1 st year.27 In our study, most children underwent surgeries between 1 year and 2 years of age, and many patients were still older than 2 years, possibly because their parents lacked medical knowledge or because they did not pay attention to the disease. We categorized children into age groups on the basis of their age at surgery and revealed that earlier surgery was associated with a smaller difference in testicular volume between boys with cryptorchidism and healthy boys in the same age group. Conversely, later surgery resulted in a more pronounced difference. These findings suggest that early surgery is crucial for promoting optimal testicular development in children with cryptorchidism. Kollin et al.28 compared the growth of testes in children with spontaneously descending cryptorchidism to that of normal and surgically treated testes. They reported that, compared with their scrotal counterparts, spontaneously descending testes were growth impaired from birth. The longer the testes remained untreated, the more significant the growth impairment was. Tseng et al.25 studied 134 children with unilateral cryptorchidism, with a median follow-up of 3.9 years. They reported that testicular growth was more rapid in children who underwent surgery before 1 year of age, suggesting that orchiopexy be performed within the 1 st year after birth. Conversely, another study by Tseng et al.11 analyzed 183 children with cryptorchidism, with a median follow-up of 34 months, suggesting that performing surgery before 2 years of age is associated with an increased risk of postoperative testicular atrophy. These conflicting conclusions from the same center may be due to variations in the criteria for postoperative testicular comparisons and potential biases in case data. Similarly, Marret et al.12 examined 55 children with cryptorchidism to assess the incidence of testicular atrophy following surgery before 1 year of age and concluded that orchiopexy before 1 year did not increase the risk of testicular atrophy and was associated with improved testicular development. However, Niedzielski et al.29 concluded that there were significant increases in testicular volume and reductions in testicular atrophy indices across all treatment modalities, with no significant differences between age groups or treatment types. This study has several advantages. First, the patients’ testicular volumes were accurately measured using ultrasound, which is more objective and precise than assessing testicular volume through physical examination. Second, this study included a long and regular postoperative follow-up period. The follow-up time in most studies is approximately 1 year after orchiopexy. Our extended follow-up provides more comprehensive data for assessing the growth trends of testicular volume. There are some limitations in this study. First, despite our efforts to collect comprehensive data, some information was missing, leading to the exclusion of patients with incomplete data. Additionally, variations in the experience levels of ultrasonographers and pediatric urologists and differences in technique standardization may have introduced variability. Furthermore, we recognize that because the study was conducted in a single center, the generalizability of the findings may be limited. A multicenter study would provide a more comprehensive view and findings with broader applicability. We will consider this for future research to strengthen the external validity and influence of our study. CONCLUSIONS The testicular volume in children with cryptorchidism tends to increase after orchiopexy but could not actually normalizes. The age at the time of surgery is a critical factor influencing postoperative testicular development. Earlier surgery results in affected testicular volumes closer to those of healthy boys. AUTHOR CONTRIBUTIONS YYH and ZLY designed this study. YYH and ZCK acquired, analyzed, and interpreted the data and drafted the manuscript. YYH, HJG, PLZ, SLL, and PYC recorded and checked the data. ZCK, WHX, and FHS performed the follow-up procedures. ZLY revised the manuscript. All authors read and approved the final manuscript. COMPETING INTERESTS All authors declare no competing interests. ACKNOWLEDGMENTS This study was funded by the Guangdong High-level Hospital Construction Fund and Shenzhen Fund for Guangdong Provincial High-level Clinical Key Specialties (SZXK035). We appreciate all the doctors and nurses in the Urology Department of Shenzhen Children’s Hospital (Shenzhen, China) for their contributions and thank all the children who were enrolled in this study. REFERENCES 1.Gurney JK, McGlynn KA, Stanley J, Merriman T, Signal V, et al. Risk factors for cryptorchidism. Nat Rev Urol 2017; 14: 534–48. Cited Here \| Google Scholar 2.Anand S, Krishnan N, Pogorelic Z. Utility of laparoscopic approach of orchiopexy for palpable cryptorchidism: a systematic review and meta-analysis. Children (Basel) 2021; 8: 677. 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Cited Here \| Google Scholar View full references list Keywords: children; cryptorchidism; orchiopexy; testicular volume; trends ©The Author(s)(2025) View full article text Source Trend in testicular volume change after orchiopexy in 854 children with cryptorchidism Asian Journal of Andrology : July 08, 2025 Full-Size Email Favorites Export View in Gallery Email to Colleague Colleague's E-mail is Invalid Your Name: Colleague's Email: Separate multiple e-mails with a (;). Message: Your message has been successfully sent to your colleague. Some error has occurred while processing your request. Please try after some time. Readers Of this Article Also Read In vitro effects of antidepressants on human sperm function Pharmacological actions of the bioactive compounds of Epimedium on the male... Role of pericytes in regulating penile angiogenesis and nerve regeneration Gene regulation and signaling transduction in mediating the self-renewal,... 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12303	https://openstax.org/books/university-physics-volume-3/pages/6-5-de-broglies-matter-waves	Skip to Content Go to accessibility page Keyboard shortcuts menu Log in University Physics Volume 3 6.5 De Broglie’s Matter Waves University Physics Volume 3 6.5 De Broglie’s Matter Waves Search for key terms or text. Learning Objectives By the end of this section, you will be able to: Describe de Broglie’s hypothesis of matter waves Explain how the de Broglie’s hypothesis gives the rationale for the quantization of angular momentum in Bohr’s quantum theory of the hydrogen atom Describe the Davisson–Germer experiment Interpret de Broglie’s idea of matter waves and how they account for electron diffraction phenomena Compton’s formula established that an electromagnetic wave can behave like a particle of light when interacting with matter. In 1924, Louis de Broglie proposed a new speculative hypothesis that electrons and other particles of matter can behave like waves. Today, this idea is known as de Broglie’s hypothesis of matter waves. In 1926, De Broglie’s hypothesis, together with Bohr’s early quantum theory, led to the development of a new theory of wave quantum mechanics to describe the physics of atoms and subatomic particles. Quantum mechanics has paved the way for new engineering inventions and technologies, such as the laser and magnetic resonance imaging (MRI). These new technologies drive discoveries in other sciences such as biology and chemistry. According to de Broglie’s hypothesis, massless photons as well as massive particles must satisfy one common set of relations that connect the energy E with the frequency f, and the linear momentum p with the wavelength λ.λ. We have discussed these relations for photons in the context of Compton’s effect. We are recalling them now in a more general context. Any particle that has energy and momentum is a de Broglie wave of frequency f and wavelength λ:λ: E=hf E=hf 6.53 λ=hp. λ=hp. 6.54 Here, E and p are, respectively, the relativistic energy and the momentum of a particle. De Broglie’s relations are usually expressed in terms of the wave vector →k,k⃗ , k=2π/λ,k=2π/λ, and the wave frequency ω=2πf,ω=2πf, as we usually do for waves: E=ℏω E=ℏω 6.55 →p=ℏ→k. p⃗ =ℏk⃗ . 6.56 Wave theory tells us that a wave carries its energy with the group velocity. For matter waves, this group velocity is the velocity u of the particle. Identifying the energy E and momentum p of a particle with its relativistic energy mc2mc2 and its relativistic momentum mu, respectively, it follows from de Broglie relations that matter waves satisfy the following relation: λf=ωk=E/ℏp/ℏ=Ep=mc2mu=c2u=cβ λf=ωk=E/ℏp/ℏ=Ep=mc2mu=c2u=cβ 6.57 where β=u/c.β=u/c. When a particle is massless we have u=cu=c and Equation 6.57 becomes λf=c.λf=c. Example 6.11 How Long Are de Broglie Matter Waves? Calculate the de Broglie wavelength of: (a) a 0.65-kg basketball thrown at a speed of 10 m/s, (b) a nonrelativistic electron with a kinetic energy of 1.0 eV, and (c) a relativistic electron with a kinetic energy of 108keV.108keV. Strategy We use Equation 6.57 to find the de Broglie wavelength. When the problem involves a nonrelativistic object moving with a nonrelativistic speed u, such as in (a) when β=u/c≪1,β=u/c≪1, we use nonrelativistic momentum p. When the nonrelativistic approximation cannot be used, such as in (c), we must use the relativistic momentum p=mu=m0γu=E0γβ/c,p=mu=m0γu=E0γβ/c, where the rest mass energy of a particle is E0=mc2E0=mc2 and γγ is the Lorentz factor γ=1/√1−β2.γ=1/1−β2−−−−−√. The total energy E of a particle is given by Equation 6.53 and the kinetic energy is K=E−E0=(γ−1)E0.K=E−E0=(γ−1)E0. When the kinetic energy is known, we can invert Equation 6.18 to find the momentum p=√(E2−E20)/c2=√K(K+2E0)/c and substitute in Equation 6.57 to obtain λ=hp=hc√K(K+2E0). 6.58 Depending on the problem at hand, in this equation we can use the following values for hc: hc=(6.626×10−34J·s)(2.998×108m/s)=1.986×10−25J·m=1.241eV·μm Solution For the basketball, the kinetic energy is K=mu2/2=(0.65kg)(10m/s)2/2=32.5J and the rest mass energy is E0=mc2=(0.65kg)(2.998×108m/s)2=5.84×1016J. We see that K/(K+E0)≪1 and use p=mu=(0.65kg)(10m/s)=6.5J·s/m: λ=hp=6.626×10−34J·s6.5J·s/m=1.02×10−34m. 2. For the nonrelativistic electron, E0=mc2=(9.109×10−31kg)(2.998×108m/s)2=511keV and when K=1.0eV, we have K/(K+E0)=(1/512)×10−3≪1, so we can use the nonrelativistic formula. However, it is simpler here to use Equation 6.58: λ=hp=hc√K(K+2E0)=1.241eV·μm√(1.0eV)[1.0eV+2(511keV)]=1.23nm. If we use nonrelativistic momentum, we obtain the same result because 1 eV is much smaller than the rest mass of the electron. 3. For a fast electron with K=108keV, relativistic effects cannot be neglected because its total energy is E=K+E0=108keV+511keV=619keV and K/E=108/619 is not negligible: λ=hp=hc√K(K+2E0)=1.241eV·μm√108keV[108keV+2(511keV)]=3.55pm. Significance We see from these estimates that De Broglie’s wavelengths of macroscopic objects such as a ball are immeasurably small. Therefore, even if they exist, they are not detectable and do not affect the motion of macroscopic objects. Check Your Understanding 6.11 What is de Broglie’s wavelength of a nonrelativistic proton with a kinetic energy of 1.0 eV? Using the concept of the electron matter wave, de Broglie provided a rationale for the quantization of the electron’s angular momentum in the hydrogen atom, which was postulated in Bohr’s quantum theory. The physical explanation for the first Bohr quantization condition comes naturally when we assume that an electron in a hydrogen atom behaves not like a particle but like a wave. To see it clearly, imagine a stretched guitar string that is clamped at both ends and vibrates in one of its normal modes. If the length of the string is l (Figure 6.18), the wavelengths of these vibrations cannot be arbitrary but must be such that an integer k number of half-wavelengths λ/2 fit exactly on the distance l between the ends. This is the condition l=kλ/2 for a standing wave on a string. Now suppose that instead of having the string clamped at the walls, we bend its length into a circle and fasten its ends to each other. This produces a circular string that vibrates in normal modes, satisfying the same standing-wave condition, but the number of half-wavelengths must now be an even number k,k=2n, and the length l is now connected to the radius rn of the circle. This means that the radii are not arbitrary but must satisfy the following standing-wave condition: 2πrn=2nλ2. 6.59 If an electron in the nth Bohr orbit moves as a wave, by Equation 6.59 its wavelength must be equal to λ=2πrn/n. Assuming that Equation 6.58 is valid, the electron wave of this wavelength corresponds to the electron’s linear momentum, p=h/λ=nh/(2πrn)=nℏ/rn. In a circular orbit, therefore, the electron’s angular momentum must be Ln=rnp=rnnℏrn=nℏ. 6.60 This equation is the first of Bohr’s quantization conditions, given by Equation 6.36. Providing a physical explanation for Bohr’s quantization condition is a convincing theoretical argument for the existence of matter waves. Figure 6.18 Standing-wave pattern: (a) a stretched string clamped at the walls; (b) an electron wave trapped in the third Bohr orbit in the hydrogen atom. Example 6.12 The Electron Wave in the Ground State of Hydrogen Find the de Broglie wavelength of an electron in the ground state of hydrogen. Strategy We combine the first quantization condition in Equation 6.60 with Equation 6.36 and use Equation 6.38 for the first Bohr radius with n=1. Solution When n=1 and rn=a0=0.529Å, the Bohr quantization condition gives a0p=1·ℏ⇒p=ℏ/a0. The electron wavelength is: λ=h/p=h/ℏ/a0=2πa0=2π(0.529Å)=3.324Å. Significance We obtain the same result when we use Equation 6.58 directly. Check Your Understanding 6.12 Find the de Broglie wavelength of an electron in the third excited state of hydrogen. Experimental confirmation of matter waves came in 1927 when C. Davisson and L. Germer performed a series of electron-scattering experiments that clearly showed that electrons do behave like waves. Davisson and Germer did not set up their experiment to confirm de Broglie’s hypothesis: The confirmation came as a byproduct of their routine experimental studies of metal surfaces under electron bombardment. In the particular experiment that provided the very first evidence of electron waves (known today as the Davisson–Germer experiment), they studied a surface of nickel. Their nickel sample was specially prepared in a high-temperature oven to change its usual polycrystalline structure to a form in which large single-crystal domains occupy the volume. Figure 6.19 shows the experimental setup. Thermal electrons are released from a heated element (usually made of tungsten) in the electron gun and accelerated through a potential difference ΔV, becoming a well-collimated beam of electrons produced by an electron gun. The kinetic energy K of the electrons is adjusted by selecting a value of the potential difference in the electron gun. This produces a beam of electrons with a set value of linear momentum, in accordance with the conservation of energy: eΔV=K=p22m⇒p=√2meΔV. The electron beam is incident on the nickel sample in the direction normal to its surface. At the surface, it scatters in various directions. The intensity of the beam scattered in a selected direction φ is measured by a highly sensitive detector. The detector’s angular position with respect to the direction of the incident beam can be varied from φ=0° to φ=90°. The entire setup is enclosed in a vacuum chamber to prevent electron collisions with air molecules, as such thermal collisions would change the electrons’ kinetic energy and are not desirable. Figure 6.19 Schematics of the experimental setup of the Davisson–Germer diffraction experiment. A well-collimated beam of electrons is scattered off the nickel target. The kinetic energy of electrons in the incident beam is selected by adjusting a variable potential, ΔV, in the electron gun. Intensity of the scattered electron beam is measured for a range of scattering angles φ, whereas the distance between the detector and the target does not change. When the nickel target has a polycrystalline form with many randomly oriented microscopic crystals, the incident electrons scatter off its surface in various random directions. As a result, the intensity of the scattered electron beam is much the same in any direction, resembling a diffuse reflection of light from a porous surface. However, when the nickel target has a regular crystalline structure, the intensity of the scattered electron beam shows a clear maximum at a specific angle and the results show a clear diffraction pattern (see Figure 6.20). Similar diffraction patterns formed by X-rays scattered by various crystalline solids were studied in 1912 by father-and-son physicists William H. Bragg and William L. Bragg. The Bragg law in X-ray crystallography provides a connection between the wavelength λ of the radiation incident on a crystalline lattice, the lattice spacing, and the position of the interference maximum in the diffracted radiation (see Diffraction). The lattice spacing of the Davisson–Germer target, determined with X-ray crystallography, was measured to be a=2.15Å. Unlike X-ray crystallography in which X-rays penetrate the sample, in the original Davisson–Germer experiment, only the surface atoms interact with the incident electron beam. For the surface diffraction, the maximum intensity of the reflected electron beam is observed for scattering angles that satisfy the condition nλ=asinφ (see Figure 6.21). The first-order maximum (for n=1) is measured at a scattering angle of φ≈50° at ΔV≈54V, which gives the wavelength of the incident radiation as λ=(2.15Å)sin50°=1.64Å. On the other hand, a 54-V potential accelerates the incident electrons to kinetic energies of K=54eV. Their momentum, calculated from Equation 6.61, is p=2.478×10−5eV·s/m. When we substitute this result in Equation 6.58, the de Broglie wavelength is obtained as λ=hp=4.136×10−15eV·s2.478×10−5eV·s/m=1.67Å. The same result is obtained when we use K=54eV in Equation 6.61. The proximity of this theoretical result to the Davisson–Germer experimental value of λ=1.64Å is a convincing argument for the existence of de Broglie matter waves. Figure 6.20 The experimental results of electron diffraction on a nickel target for the accelerating potential in the electron gun of about ΔV=54V: The intensity maximum is registered at the scattering angle of about φ=50°. Figure 6.21 In the surface diffraction of a monochromatic electromagnetic wave on a crystalline lattice structure, the in-phase incident beams are reflected from atoms on the surface. A ray reflected from the left atom travels an additional distance D=asinφ to the detector, where a is the lattice spacing. The reflected beams remain in-phase when D is an integer multiple of their wavelength λ. The intensity of the reflected waves has pronounced maxima for angles φ satisfying nλ=asinφ. Diffraction lines measured with low-energy electrons, such as those used in the Davisson–Germer experiment, are quite broad (see Figure 6.20) because the incident electrons are scattered only from the surface. The resolution of diffraction images greatly improves when a higher-energy electron beam passes through a thin metal foil. This occurs because the diffraction image is created by scattering off many crystalline planes inside the volume, and the maxima produced in scattering at Bragg angles are sharp (see Figure 6.22). Figure 6.22 Diffraction patterns obtained in scattering on a crystalline solid: (a) with X-rays, and (b) with electrons. The observed pattern reflects the symmetry of the crystalline structure of the sample. Since the work of Davisson and Germer, de Broglie’s hypothesis has been extensively tested with various experimental techniques, and the existence of de Broglie waves has been confirmed for numerous elementary particles. Neutrons have been used in scattering experiments to determine crystalline structures of solids from interference patterns formed by neutron matter waves. The neutron has zero charge and its mass is comparable with the mass of a positively charged proton. Both neutrons and protons can be seen as matter waves. Therefore, the property of being a matter wave is not specific to electrically charged particles but is true of all particles in motion. Matter waves of molecules as large as carbon C60 have been measured. All physical objects, small or large, have an associated matter wave as long as they remain in motion. The universal character of de Broglie matter waves is firmly established. Example 6.13 Neutron Scattering Suppose that a neutron beam is used in a diffraction experiment on a typical crystalline solid. Estimate the kinetic energy of a neutron (in eV) in the neutron beam and compare it with kinetic energy of an ideal gas in equilibrium at room temperature. Strategy We assume that a typical crystal spacing a is of the order of 1.0 Å. To observe a diffraction pattern on such a lattice, the neutron wavelength λ must be on the same order of magnitude as the lattice spacing. We use Equation 6.61 to find the momentum p and kinetic energy K. To compare this energy with the energy ET of ideal gas in equilibrium at room temperature T=300K, we use the relation K=32kBT, where kB=8.62×10−5eV/K is the Boltzmann constant. Solution We evaluate pc to compare it with the neutron’s rest mass energy E0=940MeV: p=hλ⇒pc=hcλ=1.241×10−6eV·m10−10m=12.41keV. We see that p2c2≪E20 so K≪E0 and we can use the nonrelativistic kinetic energy: K=p22mn=h22λ2mn=(6.63×10−34J·s)2(2×10−20m2)(1.66×10−27kg)=1.32×10−20J=82.7meV. Kinetic energy of ideal gas in equilibrium at 300 K is: KT=32kBT=32(8.62×10−5eV/K)(300K)=38.8meV. We see that these energies are of the same order of magnitude. Significance Neutrons with energies in this range, which is typical for an ideal gas at room temperature, are called “thermal neutrons.” Example 6.14 Wavelength of a Relativistic Proton In a supercollider at CERN, protons can be accelerated to velocities of 0.75c. What are their de Broglie wavelengths at this speed? What are their kinetic energies? Strategy The rest mass energy of a proton is E0=m0c2=(1.672×10−27kg)(2.998×108m/s)2=938MeV. When the proton’s velocity is known, we have β=0.75 and βγ=0.75/√1−0.752=1.134. We obtain the wavelength λ and kinetic energy K from relativistic relations. Solution λ=hp=hcpc=hcβγE0=1.241eV·μm1.134(938MeV)=1.16fm K=E0(γ−1)=938MeV(1/√1−0.752−1)=480.1MeV Significance Notice that because a proton is 1835 times more massive than an electron, if this experiment were performed with electrons, a simple rescaling of these results would give us the electron’s wavelength of (1835)0.77fm=1.4pm and its kinetic energy of 480.1MeV/1835=261.6keV. Check Your Understanding 6.13 Find the de Broglie wavelength and kinetic energy of a free electron that travels at a speed of 0.75c. Previous Next 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. Authors: Samuel J. Ling, Jeff Sanny, William Moebs Publisher/website: OpenStax Book title: University Physics Volume 3 Publication date: Sep 29, 2016 Location: Houston, Texas Book URL: Section URL: © Jul 8, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.
12304	https://www.sciencedirect.com/science/article/pii/S1879522614000633	Maintenance bacillus Calmette–Guérin therapy prolongs recurrence-free survival in non-muscle-invasive bladder cancer: A real-world experience - ScienceDirect Skip to main contentSkip to article Journals & Books ViewPDF Download full issue Search ScienceDirect Outline Abstract Keywords 1. Introduction 2. Patients and methods 3. Results 4. Discussion Conflicts of interest Sources of Funding Acknowledgments References Show full outline Cited by (2) Figures (2) Tables (5) Table 1 Table 2 Table 3 Table 4 Table 5 Urological Science Volume 26, Issue 2, June 2015, Pages 96-100 Original article Maintenance bacillus Calmette–Guérin therapy prolongs recurrence-free survival in non-muscle-invasive bladder cancer: A real-world experience Author links open overlay panel Chung-Yi Liu, Cheng-Keng Chuang, Ying-Hsu Chang, Heng-Chang Chuang, Kai-Jie Yu, Po-Hung Lin, See-Tong Pang Show more Outline Add to Mendeley Share Cite rights and content Under a Creative Commons license Open access Abstract Objective We studied the benefit of bacillus Calmette–Guérin (BCG) maintenance therapy to determine the ideal maintenance therapy schedule. Methods We retrospectively reviewed non-muscle-invasive bladder cancer patients who underwent transurethral resection of bladder tumors and BCG instillation treatment at Chang-Gung Memorial Hospital, Linkou, Taiwan, from January 1997 to December 2009. All patients in the study had non-muscle-invasive urothelial carcinoma of the bladder or carcinoma in situ. We compared the recurrence-free rate of patients who received induction alone and with maintenance BCG therapy sessions. In addition, we analyzed the best number of maintenance therapy sessions that gave the lowest cancer recurrence. Results This study included 427 patients with a mean age of 64 years. The median number of BCG treatments was 11, and the ratio of male to female was 3:1. Receiving an induction dose alone was a significant factor for tumor recurrence with a hazard ratio of 3.77. The recurrent risk rate of patient who received BCG therapy 13–15 times had lower recurrence rate than other groups. Conclusion A maintenance dose gave patients a significant benefit over those who just received induction therapy. BCG maintenance therapy worked best if given 13–15 times in our study. Previous article in issue Next article in issue Keywords bacillus Calmette–Guérin maintenance non-muscle-invasive bladder cancer 1. Introduction Approximately 75–85% of patients with bladder cancer present with the disease confined to the mucosa1 or submucosa (stage T1).2Transurethral resection (TUR) of bladder tumors, with or without intravesical adjuvant therapy, is still the primary treatment for non-muscle-invasive bladder tumor. According to many published results of randomized trials, it has become clear that intravesical bacillus Calmette–Guérin (BCG) is a better therapeutic choice for high risk Ta, T1 papillary carcinomas as well as for carcinoma in situ (CIS)3 compared with TUR alone4, 5 or chemotherapy.6, 7 For optimal efficacy, an induction course followed by maintenance BCG is recommended. Lamm et al8 reported in the Southwest Oncology Group study that 3-year BCG maintenance therapy markedly prolonged the recurrence-free survival and time to disease progression in comparison with conventional induction therapy. The maintenance therapy significantly prolonged the post-TUR recurrence-free survival compared with BCG induction therapy alone.9 However, patients may not complete the entire BCG maintenance regimen due to several reasons. Some adverse events may occur after treatment and present as the most frequent conditions for therapy withdrawal. The longer the instillations are given, the more likely a severe toxicity will develop in the patients.8 Some studies reported decreased recurrence and progression with maintenance therapy. Herr et al10 reported that BCG treatment without maintenance in patients with high-risk non–muscle-invasive bladder cancer compared favorably with trials in which comparable patients received maintenance BCG. Most practice guidelines recommend maintenance BCG for 1–3 years.2, 8, 11 However, only few studies commented on the duration of dose of maintenance BCG used. Among them, the largest study was reported by Lamm et al that 3-year BCG maintenance therapy may prolong the recurrent free survival.8 The aim of our study was to prove that BCG maintenance therapy is more beneficial than induction only and to investigate the optimal duration for maintenance therapy. 2. Patients and methods A retrospective cohort study was performed, with 427 consecutive patients with bladder cancer were evaluated from 1997 to 2009. They underwent TUR and were found to have non-muscle-invasive bladder cancer (Ta, T1, and/or Tis). Subsequently, they received six weekly instillations of Connaught strain (81 mg) BCG therapy as induction therapy and were evaluated for response after 3 months by cystoscopy, urine cytology, and TUR biopsy. Patients then received three weekly maintenance therapies every 3–6 months and up to 21 times if possible. The total therapy course was nearly 2 years. Patients were also followed every 3–6 months with cystoscopy, repeated TUR as needed, and urine cytology if possible. After finishing the BCGintravesical instillation therapy, there was no other intravesical chemotherapy during follow-up duration. Patients with previous bladder cancer histories will receive the intravesical chemotherapy, such as mitomycin C or epirubincin. We excluded patients who had previously received BCG intravesical instillation therapy. Each patient's data were entered into a database, and clinical information was recorded from charts. Patients upstaged to muscle invasion (T2) at the beginning of TUR, undergoing immediate cystectomy without receiving BCG, receiving incomplete induction BCG therapy, or missing follow-up within 1 month were all excluded from this analysis. The end point was defined as local recurrence or progression. Tumor recurrence was defined as any tumor on biopsy or positive urine cytology during follow-up examinations. Progression was defined as a muscle-invasive tumor or metastasis. Pathological staging was based on the TNM classification and tumor grade was determined in accordance with the WHO classification. Patients who died from other causes were defined as censored data. Follow-up duration was calculated by the subtraction of local recurrence or progression date (expired date, the last clinics follow-up date) and the last date of BCG therapy. Statistical analysis was performed using SPSS 17.0 (SPSS Inc., Chicago, IL, USA). Baseline characteristics were represented as mean±standard deviation for continuous data and n (%) for categorical data (Table 1). Originally, we divided patients into two groups, the induction group (with 6 BCG treatments) and the maintained group (>6 BCG treatment). Chi-square test was used to compare categorical variables, and two-sample independent t test for continuous variable. Recurrence-free plot was based on the Kaplan–Meier method with the log rank test. Furthermore, we divided patients into several groups according to the numbers of BCG treatment. The first group completed six BCG treatments, followed by maintenance groups from one to three treatments, from four to six treatments, from seven to nine treatments, and ≥10 treatments. Cox proportional hazard model was used to estimate the hazard ratios of recurrence. A p value <0.05 was considered significant. Table 1. Clinicodemographic characteristics. \| Baseline characteristics \| Statistics \| --- \| \| Age at BCG treatment (y) \| 64.4±12.0 \| \| Sex \| \| Male \| 312 (73.1) \| \| Female \| 115 (26.9) \| \| BCG treatments \| \| 6 (induction) \| 137 (32.1) \| \| >6 (maintenance) \| 290 (67.9) \| \| Multiplicity \| \| 1 \| 230 (55.0) \| \| ≥2 \| 188 (45.0) \| \| Category \| \| Urothelial carcinoma \| 391 (92.7) \| \| Concomitant carcinoma in situ \| 31 (7.3) \| \| Grade \| \| Low \| 179 (43.9) \| \| High \| 229 (56.1) \| \| Stage \| \| Ta \| 136 (46.7) \| \| T1 \| 155 (53.3) \| \| Bladder cancer history \| 280 (65.6) \| \| Had received intravesical therapy \| 62 (14.5) \| \| Mean follow-up (mo) \| 43.10 \| \| Median follow-up (mo) \| 31.7 (0.23–160.46) \| \| Treatment times \| 10.8±4.5 \| Data are presented as n (%) or mean±SD. BCG=bacillus Calmette–Guérin. 3. Results From January 1997 to December 2009, 427 patients were evaluated. Among them, 312 were males and 115 females. There were 106 patients with recurrent and 321 patients without recurrent tumor. Mean age was 64.4 years and mean follow-up time was 31.7 months. Median number of treatments was 11. There were 280 (65.6%) patients with history of bladder tumor. Most patients had urothelial carcinoma and only 31 patients (7.3%) were concomitant with CIS. Of the 427 patients, 290 entered maintenance therapy and 137 patients received induction therapy only. Main characteristics of the patients are given in Table 1. Table 2 shows the characteristics of the patients with induction therapy only and those who entered maintenance therapy. More patients in the maintenance group had high grade, advanced stage, and multiplicity than the induction group. Also, the maintenance group had better recurrence-free rates than the induction group. The recurrent free rates were 64.6% versus 89.6%, 52.1% versus 82.6%, 48.9 versus 80.0% in 1 year, 3 years, and 10 years respectively (Table 3). Table 2. Comparison of bacillus Calmette-Guérin (BCG) induction therapy and maintenance therapy. \| Empty Cell \| BCG induction (n=137) \| BCG maintenance (n=290) \| p \| --- \| Age at BCG instillation (y)a, \| 67.15±11.47 \| 63.17±12.03 \| 0.001 \| \| Sexb \| \| \| 0.981 \| \| Male \| 100 (73.0) \| 212 (73.1) \| \| \| Female \| 37 (27.0) \| 78 (26.9) \| \| \| History of bladder tumorb \| 88 (64.2) \| 192 (66.2) \| 0.689 \| \| Smokingb \| 49 (36.6) \| 100 (35.3) \| 0.806 \| \| Exposure to dying agentsb \| 8 (7.1) \| 17 (7.4) \| 0.917 \| \| Multiplicityb \| \| \| 0.421 \| \| 1 piece \| 71 (52.2) \| 159 (56.4) \| \| \| ≥2 pieces \| 65 (47.8) \| 123 (43.6) \| \| \| Unknown cases \| 1 \| 8 \| \| \| Size (the biggest diameter, cm)a \| 2.17±1.58 \| 1.93±1.16 \| 0.139 \| \| Categoryb \| \| \| 0.232 \| \| UC \| 129 (94.9) \| 262 (91.6) \| \| \| UC concomitant CIS \| 7 (5.1) \| 24 (8.4) \| \| \| Unknown cases \| 1 \| 4 \| \| \| Gradeb \| \| \| 0.288 \| \| Low \| 62 (47.7) \| 117 (42.1) \| \| \| High \| 68 (52.3) \| 161 (57.9) \| \| \| Unknown cases \| 10 \| 12 \| \| \| Stageb \| \| \| 0.823 \| \| Ta \| 42 (47.7) \| 94 (46.3) \| \| \| T1 \| 46 (52.3) \| 109 (53.7) \| \| \| Unknown cases \| 49 \| 87 \| \| \| Previous intravesical chemotherapyb \| 27 (19.7) \| 35 (12.1) \| 0.036 \| \| Median follow-up (mo) \| 20.240 \| 37.850 \| <0.001 \| Data are presented as n (%) or mean±SD. p<0.05, significant. CIS=carcinoma in situ; UC=urothelial carcinoma. a Two-sample independent t test, mean±standard deviation. b Chi-square test, n (%). Table 3. Cumulative recurrence-free rate of bacillus Calmette–Guérin (BCG) treatment times.a \| Cumulative recurrence-free rate \| n \| Follow-up after BCG treatment (%) \| --- \| 1 y \| 5 y \| 10 y \| \| Overall \| 427 \| 81.7 \| 72.9 \| 70.0 \| \| 6 BCG treatment times (induction) \| 137 \| 64.6 \| 52.1 \| 48.9 \| \| >6 BCG treatment times (maintenance) \| 290 \| 89.6 \| 82.6 \| 80.0 \| a Log-rank test for recurrence-free rate between the induction and maintenance therapies is p<0.001. Fig.1 shows the significantly lower recurrent free rate in patients who received induction therapy only by Kaplan–Meier test. 1. Download: Download high-res image (158KB) 2. Download: Download full-size image Fig.1. Recurrence-free rate between bacillus Calmette–Guérin (BCG) induction and maintenance therapy. Patients on maintenance BCG therapy had higher recurrence-free rate than those who just received induction therapy. The characteristic of each group is shown in Table 4. Table 5 reveals the significantly lower recurrence rates for each of the further classified subgroups of maintenance therapy compared with the patients of the induction group, whereas the group with BCG maintenance therapy from seven to nine times had the lowest hazard ratio (Fig.2). The recurrence-risk for patients who received BCG maintenance therapy from seven to nine times was 84% lower compared with the patients who only received induction BCG therapy. The other groups were lower by 67%, 71%, and 83% after BCG maintenance therapy from one to three times, from four to six times, and ≥10 times respectively. The group with BCG maintenance therapy from seven to nine times had the lowest recurrence-risk rate. Table 4. Clinicodemographic characteristics of each group. \| Empty Cell \| BCG induction \| BCG maintenance:1–3 (n=76) \| BCG maintenance: 4–6 (n=79) \| BCG maintenance: 7–9 (n=84) \| BCG maintenance: ≥ 10 (n=51) \| --- --- \| Age at BCG instillation (y) \| 67±11 \| 65±13 \| 64±12 \| 62±12 \| 61±12 \| \| Sex \| \| Male \| 100 (73.0) \| 56 (73.7) \| 58 (73.4) \| 63 (75.0) \| 35 (68.6) \| \| Female \| 37 (27.0) \| 20 (26.3) \| 21 (26.6) \| 21 (25.0) \| 16 (31.4) \| \| History of bladder tumor \| 88 (64.2) \| 52 (68.4) \| 45 (57.0) \| 63 (75.0) \| 32 (62.7) \| \| Smoking \| 49 (36.6) \| 22 (29.7) \| 27 (35.1) \| 29 (34.9) \| 22 (44.9) \| \| Multiplicity \| \| 1 piece \| 71 (52.2) \| 44 (58.7) \| 45 (59.2) \| 42 (50.6) \| 28 (58.3) \| \| ≥2 pieces \| 65 (47.8) \| 31 (41.3) \| 31 (40.8) \| 41 (49.4) \| 20 (41.7) \| \| Size (cm) \| 2±2 \| 2±1 \| 2±1 \| 2±1 \| 2±1 \| \| Category \| \| UC concomitant CIS \| 3 (02.2) \| 3 (03.9) \| 2 (02.5) \| 3 (03.7) \| 2 (04.0) \| \| UC \| 129 (94.9) \| 69 (90.8) \| 71 (89.9) \| 76 (93.8) \| 46 (92.0) \| \| Grade \| \| Low \| 62 (47.7) \| 29 (40.3) \| 30 (39.0) \| 33 (41.3) \| 25 (51.0) \| \| High \| 68 (52.3) \| 43 (59.7) \| 47 (61.0) \| 47 (58.8) \| 24 (49.0) \| \| Stage \| \| Ta \| 42 (47.7) \| 30 (54.5) \| 22 (38.6) \| 30 (50.8) \| 12 (37.5) \| \| T1 \| 46 (52.3) \| 25 (45.5) \| 35 (61.4) \| 29 (49.2) \| 20 (62.5) \| \| Previous intravesical chemotherapy \| 27 (19.7) \| 15 (19.7) \| 14 (17.7) \| 5 (06.0) \| 1 (02.0) \| Data are presented as n (%) or mean±SD. BCG=bacillus Calmette–Guérin; CIS=carcinoma in situ; UC=urothelial carcinoma. Table 5. Estimated hazard ratio (HR) for numbers of maintenance bacillus Calmette–Guérin therapy sessions. \| BCG treatment group \| Model I \| Model IIa \| --- \| Estimated HR (95% CI) \| p \| Estimated HR (95% CI) \| p \| \| Group 1 \| 1 \| \| 1 \| \| \| Group 2 vs. Group 1 \| 0.33 (0.19–0.58) \| >0.001 \| 0.26 (0.13–0.54) \| >0.001 \| \| Group 3 vs. Group 1 \| 0.29 (0.16–0.52) \| >0.001 \| 0.28 (0.14–0.56) \| >0.001 \| \| Group 4 vs. Group 1 \| 0.16 (0.08–0.33) \| >0.001 \| 0.12 (0.05–0.32) \| >0.001 \| \| Group 5 vs. Group 1 \| 0.17 (0.07–0.39) \| >0.001 \| 0.10 (0.02–0.41) \| 0.001 \| \| Sex \| \| \| 1.47 (0.86–2.52) \| 0.156 \| \| Age \| \| \| 1.00 (0.98–1.02) \| 0.932 \| \| Grade \| \| \| 1.45 (0.83–2.52) \| 0.193 \| \| Stage \| \| \| 1.14 (0.68–1.93) \| 0.616 \| \| Mutiplicity (≥2 vs. 1) \| — \| \| 2.27 (1.36–3.79) \| 0.002 \| \| Chemotherapy \| — \| \| 1.94 (1.09–3.45) \| 0.025 \| CI=confidence interval; Group 1=BCG induction; Group 2=BCG maintenance from one to three times; Group 3=BCG maintenance from four to six times; Group 4=BCG maintenance from seven to nine times; Group 5=BCG maintenance≥10 times. a Model II adjusted by sex, age, grade, stage, multiplicity, and chemotherapy. 1. Download: Download high-res image (151KB) 2. Download: Download full-size image Fig.2. Recurrence-free rate between different maintenance therapy courses. The group given bacillus Calmette–Guérin (BCG) maintenance from seven to nine times had a higher recurrence-free rate than the other groups. 4. Discussion Herr et al12 reported that maintenance BCG does not appear superior to initial BCG treatments in preventing or delaying tumor progression. Herr et al10 also revealed that BCG treatment without maintenance for patients with high-risk non–muscle-invasive bladder cancer compared favorably with trials in which comparable patients received maintenance BCG.10 Nevertheless, many studies have found BCG therapy to be effective in reducing the risk of disease progression only when maintenance schedules were applied.3 Hinotsu et al9 demonstrated that BCG intravesical instillation maintenance therapy was able to prolong post-TUR recurrence-free survival significantly in patients with recurrent or multiple, stage Ta or T1, bladder cancer. Also, a number of reports have shown that BCG must be given in a maintenance schedule for optimal efficacy.2 Our study revealed that patients who received maintenance BCG therapy had significantly higher recurrence-free rate. Two-year recurrence-free rates of approximately 80% and 60% were estimated for the maintenance group and the nonmaintenance group, respectively in the study by Lamm et al.8 In addition, Saint et al13 carried out a clinical study that administered the BCG Connaught strain according to a 3-year maintenance schedule, and they reported a 2-year recurrence-free rate of 84.9%. Furthermore, van der Meijden et al14 reported an estimated 2-year recurrence-free rate of 70% with maintenance therapy that employed the BCG Tice strain. Hinotsu et al9 also reported a 2-year recurrent free rate of 92.7%. In our study, the 2-year recurrent free rates were 88.3% and 63.5% for the maintenance group and the nonmaintenance group, respectively, demonstrating findings similar to the results of the previous studies. Although maintenance dose can reduce the recurrence rate, no large-scaled studies have shown the optimal times of standard maintenance dose. European Association of Urology guidelines still suggest that BCG should be given on a maintenance schedule in recent reports.15, 16, 17, 18 In addition, Bohle and Bock18 reported that administration of BCG maintenance therapy for at least 1 year resulted in significantly superior suppression on the risk of disease progression. Moreover, Sylvester et al17 reported that the BCG group showed statistically significant suppression of disease progression and, for patients who received some form of BCG maintenance therapy for at least 1 year, striking efficacy was demonstrated. In our study, patients who received BCG maintenance therapy from seven to nine times had the lowest recurrence rate in patients with non-muscle-invasive bladder tumor. However, this trend was not seen after adjusting by sex, age, tumor grade, tumor stage, and tumor number. The major limitation of our study was the small amount of patients included, although nearly 500, as compared with other studies. Patient classification bias was also another limitation in our study due to retrospective study. It needed to have large number patients and a prospective study to confirm our result. In conclusion, this study demonstrated that BCG intravesical instillation maintenance therapy may significantly increase post-TUR recurrence free rate in patients with bladder cancer. The duration of maintenance BCG therapy is important in preventing the recurrent of non–muscle-invasive bladder cancer. However, the optimal duration of maintenance therapy needs further refinement and validation. Conflicts of interest All contributing authors declare no conflicts of interest. Sources of Funding No funding was received for the work described in the article. Acknowledgments The authors are grateful to Drs Chee-Jen Chang and Hsiao-Jung Tseng of the Biostatistical Center for Clinical Research for providing assistance with the statistical analyses. Recommended articles References 1W.T. Abraham Cardiac resynchronization therapy for the management of chronic heart failure Am Heart Hosp J, 1 (2003), pp. 55-61 CrossrefView in ScopusGoogle Scholar 2M. Babjuk, W. Oosterlinck, R. Sylvester, E. Kaasinen, A. Böhle, J. Palou-Redorta, et al. EAU guidelines on non–muscle-invasive urothelial carcinoma of the bladder Eur Urol, 54 (2008), pp. 303-314 View PDFView articleView in ScopusGoogle Scholar 3A.P. van der Meijden, R.J. Sylvester, W. Oosterlinck, W. Hoeltl, A.V. Bono, EORTC Genito-Urinary Tract Cancer Group Maintenance Bacillus Calmette–Guérin for Ta T1 bladder tumors is not associated with increased toxicity: results from a European Organisation for Research and Treatment of Cancer Genito-Urinary Group Phase III Trial Eur Urol, 44 (2003), pp. 429-434 View PDFView articleView in ScopusGoogle Scholar 4D.L. Lamm, D.E. Thor, S.C. Harris, J.A. Reyna, V.D. Stogdill, H.M. Radwin Bacillus Calmette–Guérin immunotherapy of superficial bladder cancer J Urol, 124 (1980), pp. 38-43 CrossrefView in ScopusGoogle Scholar 5H.W. Herr, C.M. Pinsky, W.F. Whitmore Jr., H.F. Oettgen, M.R. Melamed Effect of intravesical bacillus Calmette-Guérin (BCG) on carcinoma in situ of the bladder Cancer, 51 (1983), pp. 1323-1326 View in ScopusGoogle Scholar 6P.U. Malmström, H. Wijkström, C. Lundholm, K. Wester, C. Busch, B.J. Norlén 5-year follow-up of randomized prospective study comparing mitomycin C and BCG in patients with superficial bladder carcinoma J Urol, 161 (1999), pp. 1124-1127 View PDFView articleView in ScopusGoogle Scholar 7D.L. Lamm, B.A. Blumenstein, E.D. Crawford, J.E. Montie, P. Scardino, H.B. Grossman, et al. A randomized trial of intravesical doxorubicin and immunotherapy with bacille Calmette–Guérin for transitional cell carcinoma of the bladder N Engl J Med, 325 (1991), pp. 1205-1209 View in ScopusGoogle Scholar 8D.L. Lamm, B.A. Blumenstein, J.D. Crissman, J.E. Montie, J.E. Gottesman, B.A. Lowe, et al. Maintenance bacillus Calmette–Guérin immunotherapy for recurrent TA, T1 and carcinoma in situ transitional cell carcinoma of the bladder: a randomized Southwest Oncology Group Study J Urol, 163 (2000), pp. 1124-1129 View PDFView articleCrossrefView in ScopusGoogle Scholar 9S. Hinotsu, H. Akaza, S. Naito, S. Ozono, Y. Sumiyoshi, S. Noguchi, et al. Maintenance therapy with bacillus Calmette–Guérin Connaught strain clearly prolongs recurrence-free survival following transurethral resection of bladder tumour for non-muscle-invasive bladder cancer Br J Urol International, 108 (2010), pp. 187-195 Google Scholar 10H.W. Herr, G. Dalbagni, S.M. Donat Bacillus Calmette–Guérin without maintenance therapy for high-risk non–muscle-invasive bladder cancer Eur Urol, 60 (2011), pp. 32-36 View PDFView articleView in ScopusGoogle Scholar 11M.C. Hall, S.S. Chang, G. Dalbagni, R.S. Pruthi, J.D. Seigne, E.C. Skinner, et al. Guideline for the management of nonmuscle invasive bladder cancer (stages Ta,T1 and Tis): 2007 update J Urol, 178 (2007), pp. 2314-2330 View PDFView articleCrossrefView in ScopusGoogle Scholar 12H.W. Herr Is maintenance bacillus Calmette–Guérin really necessary? Eur Urol, 54 (2008), pp. 971-973 View PDFView articleView in ScopusGoogle Scholar 13F. Saint, J.J. Patard, J. Irani, L. Salomon, A. Hoznek, P. Legrand, et al. Leukocyturia as a predictor of tolerance and efficacy of intravesical BCG maintenance therapy for superficial bladder cancer Urology, 57 (2001), pp. 617-621 discussion 621–2 View PDFView articleView in ScopusGoogle Scholar 14A.P. van der Meijden, M. Brausi, V. Zambon, W. Kirkels, C. de Balincourt, R. Sylvester, et al. Intravesical instillation of epirubicin, bacillus Calmette–Guérin and bacillus Calmette–Guérin plus isoniazid for intermediate and high risk Ta, T1 papillary carcinoma of the bladder: a European Organization for Research and Treatment of Cancer genito-urinary group randomized phase III trial J Urol, 166 (2001), pp. 476-481 View PDFView articleView in ScopusGoogle Scholar 15A. Bohle, D. Jocham, P.R. Bock Intravesical bacillus Calmette–Guérin versus mitomycin C for superficial bladder cancer: a formal meta-analysis of comparative studies on recurrence and toxicity J Urol, 169 (2003), pp. 90-95 View PDFView articleCrossrefView in ScopusGoogle Scholar 16P.U. Malmström, R.J. Sylvester, D.E. Crawford, M. Friedrich, S. Krege, E. Rintala, et al. An individual patient data meta-analysis of the long-term outcome of randomised studies comparing intravesical mitomycin C versus bacillus Calmette–Guérin for non–muscle-invasive bladder cancer Eur Urol, 56 (2009), pp. 247-256 View PDFView articleView in ScopusGoogle Scholar 17R.J. Sylvester, A.P. van der Meijden, D.L. Lamm Intravesical bacillus Calmette–Guérin reduces the risk of progression in patients with superficial bladder cancer: a meta-analysis of the published results of randomized clinical trials J Urol, 168 (2002), pp. 1964-1970 View PDFView articleView in ScopusGoogle Scholar 18A. Bohle, P. Bock Intravesical bacillus Calmette-Guerin versus mitomycin C in superficial bladder cancer: formal meta-analysis of comparative studies on tumor progression Urology, 63 (2004), pp. 682-687 View in ScopusGoogle Scholar Cited by (2) A prospective comparative study to assess the efficacy and tolerability of 2 different doses of intravesical bacillus Calmette-Guerin (BCG) in patients with non–muscle-invasive bladder cancer 2020, Urologic Oncology Seminars and Original Investigations Citation Excerpt : The AEs observed in both the groups were mild in frequency and subsided within 2 to 3 days with analgesics, antibiotics, and fluid therapy. Although the results of this study are similar to earlier published data, these findings should further encourage urologists in considering and formulating the treatment protocol of patients with NMIBC [20-22]. As per the comprehensive meta-analysis, further studies need to look into the efficacy of low-dose BCG based on their risk factors . Show abstract Bacillus Calmette-Guerin (BCG) is widely used as an immunotherapeutic agent and recommended in management of non–muscle-invasive bladder cancer (NMIBC). There is no consensus on the optimal dose of the BCG. However, dose reduction has been assessed to decrease the side effects following instillation of BCG. This study compared the efficacy and safety of 80 and 120 mg doses of Sii Onco BCG (Moscow I, Russian strain) in patients with NMIBC. Patients with histologically confirmed, completely resected solitary or multiple Ta or T1 (with or without carcinoma in situ), grade 1 to 3 urothelial carcinoma of the bladder were included. After transurethral resection of the tumor, repeated intravesical instillations with Sii Onco BCG (80 or 120 mg) were administered, following the induction and 3 weekly maintenance schedule (at 3, 6, 9, 15, 21, 27, and 33 months). Recurrence and progression of the tumor were monitored at scheduled time intervals using cystoscopy. A total of 104 eligible patients were enrolled to receive 80 mg (n = 51) dose or 120 mg dose (n = 53) of Sii Onco BCG. On completion of 3 years follow-up, recurrence-free survival rate of 84.31% and 86.79% and progression-free survival rate of 84.31% and 94.34% were observed for 80 and 120 mg groups, respectively; difference being statistically nonsignificant. Both, 80 and 120 mg doses of Sii Onco BCG are effective and safe for prophylaxis and management of NMIBC. ### Editorial comment on "Maintenance bacillus Calmette-Guérin therapy for nonmuscle-invasive bladder cancer: A need for more randomized trials" 2015, Urological Science Copyright © 2014 Published by Elsevier Taiwan LLC. 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12305	https://www.smartick.com/blog/mathematics/measurements-and-data/measurement-british-us/	Loading [MathJax]/extensions/tex2jax.js Try it for free! Try it for free! Accelerate your child’s learning Smartick is a fun way to learn math! Start your 7-day free trial! May21 Systems of Measurement: British Imperial vs U.S. Customary Units In today’s post we’re going to take another look at the differences we find in mathematics in different parts of the world. This time, we’re going to compare the British Imperial and U.S. Customary systems of measurement and find out why their equivalences are different. Two systems of measurement have traditionally been used in Anglophone countries: The U.S. Customary System of Units, a system of measurement used in the United States of America. The British Imperial System, a system implemented the United Kingdom, its territories and former colonies. The names of the units and relations between them are generally the same in both systems, but the sizes of the units vary, sometimes considerably. This can generate plenty of confusion, so we’ll explain the differences in detail. Origins of the Imperial and U.S. Systems Both systems of measurements are derived from English systems previously used in the Middle Ages, which in turn were the result of combining local Anglo-Saxon units inherited from German tribes and Roman units. Due to this common heritage, the two systems are fairly similar, but there are certain differences, above all found in the measurements of units of volume. Equivalences of the units of capacity and volume Fluid Ounce The British Imperial fluid ounce is equal to 28.413 milliliters, while in the U.S. Customary System it is equivalent to 29.573 ml. Pint A pint in the British Imperial System is 568.261 milliliters(or 20 fluid ounces), while a U.S. pint is just 473.176 ml (or 16 fluid ounces). Quart A British Imperial quart is equal to 1.13 liters (or 40 fluid ounces), whereas a quart in the U.S. Customary System is 0.94 l (or 32 fluid ounces). Gallon A gallon in the British Imperial System is equal is 4.54 liters (or 160 fluid ounces) while a U.S. gallon is equal to 3.78 liters (or 128 fluid ounces). Remember that these are only a few examples of the most common measurements of volume in Anglophone countries. If you want to learn more about interesting mathematical differences like these, and practice many others, log in to Smartick and try our online math learning method for free. Fun is our brain’s favorite way of learning Diane Ackerman Smartick is a fun way to learn math 15 fun minutes a day Adapts to your child’s level Millions of students since 2009 Start your 7-day free trial Author Recent Posts Smartick Content Creation Team. A multidisciplinary and multicultural team made up of mathematicians, teachers, professors and other education professionals! They strive to create the best math content possible. Latest posts by Smartick (see all) Attention Games: The Key to Fostering Sustained Concentration in Children - 01/31/2025 Mathematical Formulas: What Are They, How Are They Made and Types of Formulas? - 11/29/2024 The Language of Functions and Graphs - 07/01/2024 Learn More: Learn about the Metric System and Measurements of Volume and Mass Conversion Capacity Problems in the Metric System Practicing with Units of Measurement in the Metric System Review of All Units of Measurements Dimensions: Length, Width, and Height of an Object Add a new public comment to the blog: Cancel reply The comments that you write here are moderated and can be seen by other users. For private inquiries please write to hello@smartick.com
12306	https://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/lindep/lindep.html	Testing for Linear Dependence of Vectors There are many situations when we might wish to know whether a set of vectors is linearly dependent, that is if one of the vectors is some combination of the others. Two vectors u and v are linearly independent if the only numbers x and y satisfying xu+yv=0 are x=y=0. If we let then xu+yv=0 is equivalent to If u and v are linearly independent, then the only solution to this system of equations is the trivial solution, x=y=0. For homogeneous systems this happens precisely when the determinant is non-zero. We have now found a test for determining whether a given set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero determinant. The set is of course dependent if the determinant is zero. Example The vectors <1,2> and <-5,3> are linearly independent since the matrix has a non-zero determinant. Example The vectors u=<2,-1,1>, v=<3,-4,-2>, and w=<5,-10,-8> are dependent since the determinant is zero. To find the relation between u, v, and w we look for constants x, y, and z such that This is a homogeneous system of equations. Using Gaussian Elimination, we see that the matrix in row-reduced form is Thus, y=-3z and 2x=-3y-5z=-3(-3z)-5z=4z which implies 0=xu+yv+zw=2zu-3zv+zw or equivalently w=-2u+3v. A quick arithmetic check verifies that the vector w is indeed equal to -2u+3v. [Vector Calculus Home] [Math 254 Home] [Math 255 Home] [Notation] [References] Copyright © 1996 Department of Mathematics, Oregon State University If you have questions or comments, don't hestitate to contact us.
12307	https://www.mathway.com/popular-problems/Algebra/679366	Find the Domain f(x) = log base b of x \| Mathway Enter a problem... [x] Algebra Examples Popular Problems Algebra Find the Domain f(x) = log base b of x f(x)=log b(x)f(x)=log b(x) Set the base in log b(x)log b(x) greater than 0 0 to find where the expression is defined. b>0 b>0 Set the argument in log b(x)log b(x) greater than 0 0 to find where the expression is defined. x>0 x>0 Set the base in log b(x)log b(x) equal to 1 1 to find where the expression is undefined. b=1 b=1 The domain is all values of b b that make the expression defined. Interval Notation: (0,1)∪(1,∞)(0,1)∪(1,∞) Set-Builder Notation: {b\|b>0,b≠1}{b\|b>0,b≠1} f(x)=l o g b x f(x)=l o g b⁡x log b x 3 log b⁡x 3 log b x 2 log b⁡x 2 log b x 4 log b⁡x 4 ( ( ) ) \| \| [ [ ] ] √ √   ≥ ≥           7 7 8 8 9 9       ≤ ≤           4 4 5 5 6 6 / / ^ ^ × ×     ∩ ∩ ∪ ∪   1 1 2 2 3 3 - - + + ÷ ÷ < <     π π ∞ ∞  , , 0 0 . . % %  = =     Cookies & Privacy This website uses cookies to ensure you get the best experience on our website. More Information ⎡⎢⎣x 2 1 2√π∫x d x⎤⎥⎦[x 2 1 2 π∫⁡x d x] Please ensure that your password is at least 8 characters and contains each of the following: a number a letter a special character: @$#!%?& Do Not Sell My Personal Information When you visit our website, we store cookies on your browser to collect information. 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12308	https://anzoteh96.wordpress.com/2014/01/21/tournaments-of-towns-fall-2013-senior-o-level/	Tournaments of Towns, Fall 2013, Senior O-Level. \| anzoteh96 anzoteh96 ========= Anything. Everything. --------------------- MenuSkip to content About Maths Project Insight Search for: Tournaments of Towns, Fall 2013, Senior O-Level. January 21, 2014Mathematicsanzo960504 The second post stated that I took part in International Mathematical Tournaments of Towns (IMTOT), Fall 2013. The official solution is now available on the official website: Have a look at the problems: TT2013F_SOsolutions And I was surprised by my ability of doing maths after 4-month-break: Solving all itself was not any big deal, but you would feel great if you can finish within half of the time given. The problems below are sorted by the order of attempt. 17/12/2013: Senior O-Level. Time given: 4 Hours Problem 1 (3 marks).As usual, it’s easier to prove the existence of things compared to its opposite. Moreover, when the last digit is 2, 4, 6, 8, 0, 5 the number is composite (multiple of 2 or 5). Isn’t it a good strategy to move all to the right? Yes, it is! The ~~hard ~~ (nontrivial: this is not hard!) part is to deal with 1, 3, 7, 9.Using alternating digit argument we have 1+9=3+7, 11 divides 9317 and we can have 9317246850. However, this was not my first idea, bu I checked for divisibility of 7 using 9, 3, 1, 0, and it thankfully succeeded at 9317 😀 Problem 3 (4 marks).OK let’s start from the beginning, lcm (n, n+1)=n(n+1) is for sure, and by problem definition lcm (n, n+2)=n(n+2) is impossible, so it must be n(n+2)/2,… Ah huh! The idea is lcm (n, n+i)=n(n+i)/i (obvious, isn’t it?) This means n is a multiple of 1, 2, ,…, 35 and same goes for 36=49. This settles everything. Problem 4 (5 marks).A classical trick used before: bijection. Move one column, and two rows for each rook. Oh yeah, biject the coordinates and they will be neither on same row nor same column. Problem 2 (4 marks).Despite being the second problem, it was not as easy as problems mentioned above. How to prove something parallel? I drew few diagrams, oh wait! I saw something: YBX is similar to ABC by ratio and angles defined in the problem! What’s next? I wanted to avoid cases, and used directed angle to prove 2 lines parallel. That was a lot of work! Great! I solved the first 4 problems in an hour. Now turn to the last challenge of the day: Nasty problem 5. Problem 5 (6 marks).The answer turned out to be a paradox. At the first glance, we should always be able to determine the answer, right? I tried to prove that, but in vain attempt. After a thorough thought, and knowing that 3D diagrams are harder to draw and imagine, I tried to overlap a sphere and a cube with same centre, and here comes my solution: draw six circles on each face of cube which are tangent to sides of square (for each square face) and we found the desired path. Written solution, however, was in algebra (as expected). — After finish writing, I had a shot on the clock: elapsed time was 1 hour and 45 minutes. Perfect! I could have a good time checking my solutions and have a rest. After all papers have been collected, I had a survey with the other seniors: 4 of us had it all, with some others solved 4 out of 5. An easy paper, indeed! (lol I realised that I could have avoided nasty directed angle for P2 by proving AY=XZ and AZ=XY, which was what Zi Song did). The A-Level paper was harder as usual, but overall it was a decent effort for me. I will upload it on my next post. Share this: Click to share on X (Opens in new window)X Click to share on Facebook (Opens in new window)Facebook Like Loading... Post navigation ← Joke to conclude the year 2013.Tournament of Towns Fall 2013, Senior A-Level → Leave a comment Cancel reply Δ Archives Archives Categories Categories Blog at WordPress.com. Comment Reblog SubscribeSubscribed anzoteh96 Sign me up Already have a WordPress.com account? Log in now. anzoteh96 SubscribeSubscribed Sign up Log in Copy shortlink Report this content View post in Reader Manage subscriptions Collapse this bar %d Design a site like this with WordPress.com Get started
12309	https://open.oregonstate.education/cellbiology/chapter/cell-cycle-and-mitosis/	Skip to content 8 The Cell Cycle and Mitosis chapter 8 chapter 8 The Cell Cycle and Mitosis Introduction For most of this textbook, we have looked at each structure and process separately so that we can better understand them. We have focused on how the underlying chemistry of structures is essential to understanding the function of each organelle and each individual process. Here we are finally able to integrate all of the different bits of the cell and see how it all comes together as a whole and drives complex cellular processes. More specifically, we will be exploring the process of cell growth and division. We assume that this is not your first exposure to the concepts of cell growth and division (also known as mitosis and cytokinesis), and we approach the topic from that lens. If you need a refresher on the basics, we encourage you to return to the introduction and explore the review material. Cell growth and division are moments within a larger process known as the cell cycle. The cell cycle impacts every aspect of cellular function. Every organelle gets involved in some way or another at every step of the way. Growth requires protein and membrane synthesis, energy production, transport of materials, and more. Selecting the correct moment to undergo cell division requires careful coordination of both the internal and external environment of the cell. Cells that divide at the wrong moment are unlikely to survive the process, so it must be carefully timed and controlled. Mitosis itself requires a complete and total disruption of cellular function as well as a complete but temporary rearrangement of the cell’s contents. This, too, requires precise coordination of the organelles. In this, the final chapter of this textbook, we will explore how the cell coordinates all of the various processes we’ve learned about so far in order to grow and then, at just the right moment, divide into two daughter cells. Topic 8.1: Regulating the Cell Cycle: Checkpoint Control Learning Goals Review the stages of the cell cycle, including the checkpoints, and identify the key features of each stage. Describe how specific protein modifications (e.g., phosphorylation and ubiquitination) result in activation/deactivation of cyclin-CDK complexes to regulate cell cycle checkpoints. Explain how the activation of the cyclin-CDK complexes results in the start of the next phase of the cell cycle. Use examples from both M- and S-cyclin-CDK complexes to explain this. Detail how fluorescence-activated cell sorting (FACS) can be used to identify the stage of the cell cycle for a population of cells. Introduction to the Cell Cycle and Checkpoints A discussion of the cell cycle and mitosis is a very good way to end this book, as it is a wonderful example of how the concepts we’ve covered in this book are interconnected. The progression of the cell from interphase to cell division is precisely regulated, and it involves every other cellular component in some way. The cell cycle is defined as the events that enable cells to proceed from one cell division event to the next. Cell division itself consists of the overlapping processes of mitosis (nuclear division) and cytokinesis (division of the cytoplasm). The cell cycle is divided up into four separate phases based on the primary event that is taking place in that stage: G1 (gap or growth 1) phase: This is the “gap” between the end of cytokinesis and the start of DNA synthesis. A lot of the work of this phase involves cell growth so that it can support itself and also have the resources it needs for the next phase. S (synthesis) phase: This phase is defined by the initiation and termination of DNA synthesis. G2 (gap or growth 2) phase: This second “gap” phase lasts from the end of DNA synthesis to the onset of mitosis. The cell continues to grow but also prepares for what’s to come in the next phase. M (mitosis) phase: This is the phase in which cell division occurs. Figure 08-01 shows an overview of the stages of the cell cycle. Collectively, we consider G1, S, and G2 to be interphase (i.e., the phases “in between” M phase). It is important to keep in mind that while cells do need a mechanism to control growth, that is not always their primary concern. There must also be provision for cells to “step away” from the cell cycle for a time and into other developmental pathways (mating, meiosis, differentiation). Some cell types will hit a point in their development where division isn’t a good option anymore for one reason or another, so they withdraw from the continuous cycle of growth and replication. Whether the cell leaves the cell cycle on a temporary or permanent basis is also highly controlled, and we will discuss how the cell makes these kinds of decisions later in this topic. (Hint: Signaling proteins are involved.) Signaling and regulation are at the heart of the proper progression of the cell cycle. Using these tools, the cell ensures that discrete events, such as DNA synthesis and mitosis, do not occur before the cell is ready, and they occur in the right order the cell pauses DNA replication or mitosis when errors are identified and repairs are attempted only cells that should divide are allowed to do so Some cells must undergo terminal differentiation, stall temporarily or permanently, or even go through programmed cell death (i.e., apoptosis). These are all cell cycle decisions. growth is coordinated with cell division so that the size of cells is maintained over the many cellular generations Some cells are designed to get bigger or smaller over several cell cycles. Some even divide asymmetrically so that one large and one small daughter cell is produced. All of these will require precise cell cycle control. One of the most important controls placed on the progression of the cell cycle is a series of checkpoints in which the cell is required to meet certain criteria before it is allowed to proceed. In this way, these checkpoints act as a form of quality control. The control of the checkpoints and the mechanism of when/how cell division takes place is a perfect example of how the cell uses signaling to understand its environment and to effect internal change. Cells do not pass through these checkpoints randomly. They are constantly receiving cues from the exterior (such as growth factors, for example) and from the interior that help them decide exactly when and how to divide. Each of these checkpoints is controlled by one or more “gatekeeper proteins” that respond to cellular conditions and will only allow the cell to move forward into the next phase of the cell cycle if conditions are “right.” As a result, the checkpoints help ensure that specific criteria are met before the cell cycle is allowed to continue. There are a few different checkpoints, but the ones we are going to focus on are the following: the G1/S checkpoint, which allows the cell to pass into S phase, and the G2/M checkpoint, which controls when the cell enters mitosis. We will also look at the checkpoint that is in the middle of mitosis, which ensures that the mitotic spindle is set up correctly prior to chromosome separation. We call this the metaphase checkpoint, but we have seen many other names used as well (spindle assembly checkpoint, M phase checkpoint, M/G1 checkpoint, etc.). Cells can only move through the checkpoint and into the next stage of the cell cycle when they have met the required conditions. For example, if biotin, a vitamin, is missing from the growth medium, yeast cells will not pass the G1/S checkpoint even if all other conditions are perfect. Low nutrient levels reduce the growth rate, which, if severe enough, can make it so a newly divided cell will not survive. Thus, having the ability to confirm that everything is in place before dividing is key to survival for the cell. G1/S Checkpoint The first checkpoint a new cell will encounter is the G1/S checkpoint (also sometimes known as the “restriction” point or “Start”). As its name implies, this checkpoint marks the transition from G1 to S phase. Since S phase involves the replication of DNA, it is important that the cell and the environment are both ready before replication starts. Not only is this an energy- and nutrient-intensive process, but replication is when any preexisting errors in DNA become permanent mutations, so the DNA must be in good shape before the cell starts this process. Once again, making an error with the timing of replication could result in the death of the cell, so the checkpoint plays a key role here. Cell signaling is important for ensuring that conditions are ideal for cell division. The cell responds to internal and external cues in order to “decide” when to divide. Some of the conditions that must be met include the following: Proper nutrients (carbon source, energy source, inorganic phosphate, nitrogen, vitamins, etc.) must be present at specific concentrations. Sister chromatid separation (from the previous mitosis) must be complete. There must be no detectable DNA damage. The cell must have reached a critical threshold size. Additionally, external factors must also be appropriate. For example, in yeast, if the appropriate mating factor is present in the environment, the cells cannot proceed to S phase and are switched instead into an alternative pathway (called the sexual pathway). Similarly, in mammalian cells, appropriate growth factors must be present to allow cells to pass this checkpoint. If not, the cell remains in G1. In some cases, cells are stalled for extended periods of time…maybe indefinitely. We say that these cells have removed themselves from the cell cycle and that they are in G0 phase. This could be due to a long-term deprivation of nutrients or other resources required for cell division. More commonly, however, this is a normal part of the development of certain cell types. For example, Stem cells for specific tissues will enter G0 for short periods of time until replacement cells are required. This allows tissues to grow to a certain size and then stop growing and maintain a relatively stable size and distribution. Your blood cell system (called the hematopoietic system) does this. New blood cells are grown only when specific cell types are needed or cells are lost through injury. Some cell types undergo terminal differentiation, which means that when these cells reach maturity, they do not need to go through mitosis anymore. Good examples of this are muscle cells (multiple muscle cells fuse together at maturity to produce multinucleate muscle fibers), neurons (with their extremely long axons), and osteocytes (bone cells, which are intricately embedded in the calcified matrix of the bone). G2/M Checkpoint The second is the G2/M checkpoint, which stops the cell from entering mitosis before its ready. It is also sometimes called “CD” or “Commitment to Division,” as the cell cannot stop the process of mitosis once it has passed this point. As its name suggests, this checkpoint controls the transition from G2 to M phase. Some conditions that must be met here are the following: DNA replication must be complete and accurate. No DNA damage can be detected (through a robust biochemical surveillance system) or this checkpoint cannot be passed. This is the most important factor for passing the G2/M checkpoint. The cell must also have reached a certain minimum size so that it is big enough that, when split in two, the two daughter cells will also be large enough to survive. It is considered exceedingly rare that cells would stall and enter G0 from G2. One would think that there is not really much point in doing all of the work of replication unless there is an intention for the cell to complete mitosis. However, since this is biology, there are examples of chromosomes called polytene chromosomes that can have thousands of sister chromatids, instead of simply two. These cells undergo repeated rounds of replication without moving forward to mitosis. It is thought to help increase the number of copies of genes in the cell, which can significantly impact gene expression. While this is thought to be relatively common in some cell types, such as the salivary glands of certain insects (like flies), and examples can be found throughout eukaryotes, it is still considered to be a relatively rare occurrence overall. Metaphase Checkpoint This checkpoint marks the halfway point of mitosis, but it’s also the point right before the actual division of the genetic material, so it makes sense that this would be a point that the cell would verify before proceeding. To separate the sister chromatids of the chromosome, the mitotic spindle, composed of microtubules and associated motor proteins, must be assembled, and the chromosomes must be properly attached. If the spindle is not assembled correctly, an entire chromosome could get destroyed or mislocalized. Considering how important the genetic material is to the cell, you can imagine how bad this kind of “mitotic misfire” would be; both daughter cells would likely die, if they could even complete mitosis. Interestingly, you can see this checkpoint if you watch closely as cells divide under the microscope (Video 08-01). The chromosomes line up at the metaphase plate and then wait at the checkpoint for a little while until suddenly the sister chromatids split and move to opposite ends of the spindle. Cell Cycle Checkpoint Control: Cyclins and CDKs Like all things in cell biology, there was a time when the details of the cell cycle were not known to scientists. Mitosis was observed extremely early (as early as the 1600s), as even the most rudimentary light microscopes could be used to observe it. The fact that cells arose from other cells was identified in the 1800s, but it wasn’t until the 1950s that the rest of the cell cycle was suggested. By the late 1960s, scientists believed that “something” in the cytoplasm was controlling the cell cycle, but they had no proof. Dr. Yoshio Masui, a Japanese Canadian researcher at the University of Toronto, ran some experiments while working as a postdoctoral fellow at Yale University that provided the evidence needed. In these experiments, Dr. Masui extracted cytosol from frog’s eggs that were in mitosis and then injected it into a cell that was stalled at the end of interphase. (Aside: This is a normal part of frog oocyte maturation.) As a control, they compared this with oocytes that were injected with cytosol from another oocyte also in interphase. They found that when the cytosol from a mitotic oocyte was injected into the interphase oocyte, a mitotic spindle would begin to form. This did not happen in the control, in which interphase cytosol was injected into an interphase oocyte. The results of these experiments provided evidence of the following: The active agent that promotes the next stage of the cell cycle is in the cytosol. Control of DNA synthesis and mitosis is positive—that is, the active agent promotes mitosis in the recipient cell. Cells can be advanced into the next stage before they planned it by adding the appropriate factors to their cytosol. After this initial discovery, it was replicated using cytosol from mitotic cells from many different species and by taking cytosol from one species and using it to induce mitosis in another species. This meant that the “factor” in the cytosol was universal and helped promote the “maturation” of the cell. As a result, they called this molecule the maturation promoting factor (MPF). It was called a factor, and not a protein, because at this point, no one knew what it was. About 20 more years of research were required to figure out that the “factor” in question was actually a protein. MPF = Activated Cyclin-CDK Complex We now know that MPF is actually a set of proteins that work together to control the checkpoints and thus the entire cell cycle. The active agent is a protein called cyclin-dependent kinase (CDK). As you remember from Chapter 7, a kinase is an enzyme that specifically adds phosphate groups to other proteins. This particular kinase is only active in the presence of a second protein known as cyclin. Cyclin binds to CDK (Figure 08-02), and acts as a regulatory unit; CDK can only perform its function as a kinase when cyclin is bound. If cyclin is removed from CDK, then CDK is inactivated. This is also the source of its name…cyclin-dependent kinase. These two proteins combine to produce a single enzyme called the cyclin-CDK complex. The cyclin-CDK system is somewhat unique due to the use of cyclins to regulate CDK activity. The cell controls the activity of CDKs by controlling the synthesis and destruction of cyclins. CDK concentrations in the cell remain constant throughout the cell cycle, but they are not always active. On the other hand, cyclin concentrations show a cyclical pattern. They increase as the cell moves through the cell cycle, which coincides with increasing enzymatic activity of CDK, peaking at the appropriate point in the cell cycle (usually a checkpoint). After the relevant checkpoint is passed, cyclin concentrations crash down to almost nothing. Once again, this change in cyclin concentration coincides with the end of the CDK enzymatic activity. It is this “cycling” of the cyclin concentrations that regulates CDK activity, allowing the cell to progress through checkpoints of the cell cycle. Since the initial studies last century, we have learned that there are actually several classes of cyclins and CDKs, and each of these classes is responsible for controlling a specific part of the cell cycle. The four major classes of cyclins are listed in Table 08-01. Table 08-01: The different classes of mitotic cyclins \| Mitotic Cyclin Classes \| \| \| Cyclin class \| Function \| \| G1 cyclins \| Unique cyclins that are thought to help the cell respond to external signals to leave G0 and initiate cell division. We won’t discuss these any further in this book. \| \| G1/S cyclins \| Cyclins that control the G1/S checkpoint and control the transition from G1 to S phase. \| \| S cyclins \| Cyclins that activate at the start of S phase (by the G1/S cyclins) and directly induce replication of DNA. The concentration of these cyclins remains high right through to M phase. \| \| M cyclins \| Cyclins that control the G2/M checkpoint. They remain active in the first half of mitosis until their destruction is signaled by the anaphase-promoting complex (APC). \| Increasing cyclin concentrations occur directly prior to a cell passing a cell cycle checkpoint. This is because the cyclin-CDK activity must hit a certain level before the checkpoint can be passed, and cyclins are required components of CDK activity. In some cases, such as the M cyclin shown in Figure 08-03, the activity of the CDK decreases rapidly not long after the checkpoint has been passed, but in others, like the G1-cyclin shown here, the activity of the CDK is activated at one checkpoint and then stays high throughout the rest of the cycle. Phosphorylation Controls Cyclin-CDK Enzymatic Activity The activity of the CDKs must be very tightly controlled. The consequences of a cell moving to the next stage of the cell cycle before its ready could be disastrous. As such, cyclin-CDKs are at the heart of a complex signaling pathway involving potentially hundreds of enzymes that are fighting with each other to either activate or deactivate the cyclin-CDK complex. Figure 08-04 shows an extremely simplified version of a regulatory pathway that includes two different cyclin-CDKs (highlighted by yellow arrows). In particular, take note of how the pathways labeled cell growth (labeled as cell proliferation in the figure) and programmed cell death (also known as apoptosis) are connected to each other. This gives a strong hint about how important it is to get this right. If it goes wrong, the cell has the option to initiate apoptosis and die if necessary. Because CDK activation must be tightly controlled, there are multiple layers of regulation. Note that while the binding of cyclin to CDK is necessary for CDK activity, it is not sufficient for activation of the CDK on its own. (Remember our discussion of necessary and sufficient from Chapter 3.) This means that other factors are also needed to activate the CDK complex. In this case, the CDK-cyclin complex itself must also be phosphorylated (by other kinases). Interestingly, phosphorylation of CDK can also be used to keep the CDK inactive as well, depending on the location of the phosphate addition on the protein. The activation of CDKs follows a set pattern, which requires both the presence of cyclin and proper phosphorylation of the CDK. We’ll use the activation of the M-CDK/cyclin complex as an example, which is illustrated by Figure 08-05 and Video 08-02. In a nutshell, the binding of cyclin initiates the activation process. The signaling proteins upstream activate two kinases in particular: Wee1 and CAK. Both phosphorylate the CDK, but for different purposes. The phosphate group that Wee1 adds is considered to be an inhibitory phosphate, which limits the activity of the CDK. On the other hand, CAK (which stands for cyclin-activating kinase) adds a phosphate that is necessary for the CDK to become active. Once everything else is ready and it’s time for the checkpoint to be passed, a third protein called cdc25 comes in and removes the inhibitory phosphate (thus cdc25 is a phosphatase), and the CDK becomes fully active. You can think about this a little bit like a race car at the starting line of a race. By the time you get to the starting line, the car is all gassed up (cyclin) and running (activating phosphate). The drivers might even be revving the engines a bit, but with the break on (inhibitory phosphate) so that the car can’t go anywhere until the signal is received that the race has started. By being ready to go before the race starts, they can leave the starting line as quickly as possible once the signal to start is received. In yeast, Wee1 and Cdc25 are key regulators of M-CDK. The antagonistic (i.e., opposing) relationship of Wee1 and Cdc25 was discovered using genetic experiments in which the “dosage” of the genes was experimentally manipulated. This simply means that the amount of protein present in the cell was increased or decreased depending on the mutation. Increasing the gene dosage (i.e., a gain-of-function mutation) increases the concentration of enzyme in the cell, whereas decreasing gene dosage (i.e., a loss-of-function mutation) leads to a decrease in enzyme concentration. The effects of these mutations on cell division are summarized in Table 08-02. Table 08-02: The effects of mutations in WEE1 and CDC25 on yeast cell growth and division \| The Effects of Mutations in WEE1 and CDC25 on Yeast Cell Growth and Division \| \| \| \| Gene \| Gain-of-Function Mutation \| Loss-of-Function Mutation \| \| WEE1—inhibitory kinase \| Cells divide later at a larger-than-normal size \| Cells divide early at a smaller-than-normal size (i.e., they’re “wee”!) \| \| CDC25—activating phosphatase \| Cells divide early at a smaller-than-normal size \| Cells divide later at a larger-than-normal size \| \| i.e., too much protein produced or produced all of the time without regulation i.e., no protein produced \| \| \| Specific Cyclins and Their Role in the Cell Cycle As mentioned earlier in this topic, we are focusing on three specific checkpoints in the cell cycle. Of those, two are controlled directly by cyclin-CDKs: the G1/S checkpoint and the G2/M checkpoint. Each of these checkpoints is controlled by its own cyclin-CDK complexes. Cyclins and CDKs involved in the start of mitosis are called M cyclins. The progression into S phase is more complicated, and multiple cyclin/CDK complexes are needed (see Figure 08-03). Interestingly, the metaphase checkpoint is indirectly controlled by cyclin/CDK complexes. The M cyclin-CDK complex also activates the process by which the metaphase checkpoint is set up and then passed. We’ll look at how the different checkpoints work here. Control of the G1/S Checkpoint and S Phase Progression The transition from G1 to S phase is controlled by multiple cyclin-CDK complexes, including a G1 cyclin-, an S-cyclin-, and a G1/S cyclin-CDK complex (Figure 08-03). The G1/S cyclin-CDK complex will be de-activated once the checkpoint is passed, but the others will continue to function. Once the G1/S checkpoint is passed, DNA replication will begin. One of the most important criteria for the passing of the G1/S checkpoint is that there can be no detectable DNA damage. The protein that manages checking for DNA damage is a transcription regulator called p53. p53 is sometimes called “the guardian of the genome” because of the vital role it plays in ensuring that DNA remains intact and undamaged. It also functions in a way that is somewhat counterintuitive—in its inactive state, p53 is constantly translated and then immediately degraded (Figure 08-06). When DNA damage is detected, p53 gets phosphorylated, which stops it from being degraded. It then binds to the promoter sequence for a CDK inhibitor called p21, thereby activating its transcription and subsequent translation. p21 blocks the activity of the G1/S cyclin-CDK complex, which will stop the cell from passing the G1/S checkpoint. It is also interesting to note that p53 is mutated in as many as 50% of all cancers, which is a clear indication of just how important this protein is in the proper function of the cell cycle. If you lose p53 function in a cell, you lose your ability to delay the cell cycle so that there’s time to repair DNA damage. At that point, damage is less likely to get fixed before replication, and mutations accumulate at a much faster rate, with each new round of replication. Once the G1/S checkpoint has been passed successfully, the active cyclin-CDK complexes help initiate S phase via phosphorylation of the replication machinery, which, in turn, helps it assemble on the DNA. DNA helicase—responsible for DNA unwinding—is also activated via phosphorylation, allowing replication to begin. While there is much focus on how S phase is initiated, it is equally important to consider how it is terminated. DNA replication machinery must be deactivated at the end of replication as well. This ensures that the entire genome is replicated once, and only once, during S phase. Again, the cyclin-CDKs control this by phosphorylating key enzymes, which will lead to the eventual shutdown of replication. Prior to the start of S phase, each chromosome in the genome was made of a single helix of DNA that is in a complex with histones to form a chromatin fiber. (Review Chapter 3 if needed.) It has its own telomeres at each end of the chromosome and a centromere near the middle of the DNA. During DNA replication, the DNA double helix is duplicated, and the chromosome now includes two sister chromatids. This requires that a whole new set of histones is synthesized and imported into the nucleus to form the new chromatin fiber. Also, the two sister chromatids must be held together until the point at which they separate, during mitosis. This is done using a protein we’ve seen before (in Chapter 3) known as cohesin (Figure 08-07). Cohesin attaches the two chromatids together along their entire length at specific sites on the DNA called cohesin attachment regions (or CARs). One interesting final point: S-CDK has been shown to remain active right until the start of mitosis (Figure 08-03) despite the fact that S phase is over long before that. The reasons for this are not entirely clear, but there is some evidence that S-cyclin-CDK helps with the activation of M-CDKs. This provides further continuity in the cell cycle and shows that the different cyclins are influenced by each other. M-CDK Controls the G2/M Checkpoint and Reentry into G1 The transition from G2 phase to M phase is complex. It requires an almost complete rearrangement of the cytoplasm, including shutting down all transcription and translation, preparing all of the organelles for separation (which often means deconstructing them), building a second microtubule organizing center (MTOC) for the mitotic spindle, and completely rearranging the cytoskeleton to allow for division. M-CDK-cyclin becomes enzymatically active at the end of the G2 phase and is the primary control for this transition. M cyclin concentrations peak in metaphase, before crashing, which deactivates the M-CDK and marks the beginning of the transition back to G1. As a kinase, the role of M-CDK is to phosphorylate other proteins, which, in turn, will activate or deactivate them depending on the protein. Some of the targets of the activated M cyclin-CDK complex include the following: Histone H1—Phosphorylating H1 leads to changes in chromatin configuration and, in conjunction with other proteins, leads to the tighter packing of chromatin required for mitosis. Condensins—These are a class of DNA-binding proteins that bind to chromatin to help with higher-order chromosome condensation. They work to loop up the chromatin fiber into the tightly coiled mitotic chromosome. Nuclear lamins—Phosphorylated lamins have a lower affinity for each other, and as such, the nuclear lamina falls apart. Disassembly of nuclear lamina results in the breakup of the nuclear envelope. (We explored this in detail in Chapter 3.) Structural proteins of the nucleolus—Since DNA from multiple chromosomes is used to form the nucleolus, it must be taken apart prior to mitosis. Phosphorylation of the structural proteins results in dispersion of the nucleolar proteins and disintegration of the nucleolus. A variety of protein kinases that regulate the cytoskeleton—In order for mitosis to happen, the cytoskeleton needs to be completely taken apart and rebuilt. A number of proteins are involved in this, including microtubule-associated proteins (MAPs) and even some actin-binding proteins (ABPs), and must be activated directly or indirectly by the M-CDK. cdc25, the M-CDK-activating phosphatase—This creates a positive feedback loop, resulting in further activation of M-CDK-cyclin. As a result, M-CDK activity rises increasingly rapidly as more M-CDK becomes active (Figure 08-08). Anaphase-promoting complex (APC)—This protein is key to passing the metaphase checkpoint. At the beginning of anaphase, APC degrades the cohesin proteins that bind sister chromatids together, releasing the daughter chromosomes. Interestingly, APC also activates enzymes that tag M cyclin for degradation. This ensures that M-CDK will be properly deactivated when its task is done. The cell cannot complete mitosis and return the cytoplasm to its interphase state unless M-CDK is inactive. We will be breaking down the events of mitosis and how CDKs and other proteins drive that process in the next topic of this chapter. The Metaphase Checkpoint Ensures Proper Mitotic Spindle Formation Unlike the other two checkpoints, this checkpoint is not directly controlled by a cyclin-CDK complex. However, that doesn’t mean that the CDKs and cyclins have no role to play. This checkpoint is in the middle of M phase and ensures that the mitotic spindle is formed properly before mitosis is allowed to proceed. The M cyclins and their associated CDKs are involved in this checkpoint, as they activate the proteins required to create the mitotic spindle. Passing the metaphase checkpoint not only allows mitosis to proceed but also sets in motion a series of events that will eventually shut down M phase, which we will talk about in the next section. Proper spindle assembly is essential to the metaphase checkpoint, so it stands to reason that the proteins involved in regulating this checkpoint would be interacting directly with the spindle itself. At some point during G2 phase, a large protein complex known as a kinetochore assembles at the centromere of each sister chromatid in the chromosome. This complex will help capture microtubules during the assembly of the mitotic spindle so that the sister chromatids can be separated. Several proteins are also assembled at the kinetochore, the most important of which is one called anaphase-promoting complex, or APC, and they will remain there until microtubules are properly attached to the kinetochores of both sister chromatids. Once that happens, the checkpoint proteins assembled at the kinetochore, including APC, are released and activated. Once that happens, the cohesins holding the sister chromatids together are degraded and anaphase begins. This is not the end of APC’s role in mitosis, however. Deactivation of the Cyclin-CDK Complex Just like in S phase, the cell must eventually end M phase, put everything back where it was, and allow the new daughter cells to reenter G1. To do this, the M cyclin-CDK must be deactivated so that it stops phosphorylating its target proteins and the cell can exit M phase. APC is directly involved in the degradation of M cyclins, which deactivates the M-CDK. Incidentally, that means that M cyclin-CDK not only creates a positive feedback loop to activate itself, through cdc25 (Figure 08-08), but it also creates a negative feedback loop, using APC, which will result in its eventual deactivation. This is a fascinating case study on the complexity of cell signaling and how it regulates cellular function. Like most signaling events, the cyclin-CDK complex must be deactivated once its job is complete. This is done by a combination of shutting down the transcription and translation of new cyclin and degrading the cyclin proteins that already exist. Negative feedback loops are often built into signaling pathways, and CDKs are no different. The mechanism is illustrated in Figure 08-09. APC is phosphorylated by M-CDK so that it is active and controls the metaphase checkpoint. Unlike many of the regulatory proteins we’ve seen, APC is not a kinase or a phosphatase. APC is a protein that transfers a small protein tag called ubiquitin to other proteins. (Review Chapter 4 if needed.) APC tags the cyclin with ubiquitin, which results in it being sent to the proteasome for degradation. Once the cyclin is degraded, the CDK is no longer active, and all of the phosphorylation targets of the CDK can then be dephosphorylated and returned to their original state. The result is that the concentration of cyclin in the cell drops rapidly, and the CDK is also deactivated, and the cell can start the process of putting the cytosolic contents back together again in the new daughter cells, complete cytokinesis, and return to G1. For its part, APC is deactivated in G1. It is one of the phosphorylation targets of the activated G1/S-cyclin-CDK complex. Studying Cells: Experimental Techniques to Identify the Cell Cycle Initially, we knew two things about the cell cycle: there was mitosis, and there was the time in between mitosis (i.e., interphase). We could observe this in the most rudimentary light microscopes roughly 200 years before we knew that DNA was the molecule that stored genetic information. As we now know, there are four major stages to the life cycle of a cell, and only one of them is mitosis. When researchers study the cell cycle, it is important that they can differentiate among the four stages. DNA content is a useful cue, since the amount of DNA changes in predictable ways throughout the cycle. In addition, sometimes researchers will need to work with a population of cells that are all in the same stage of development. In this section, we look at two ways to identify, and possibly synchronize, a population of cells based on cell cycle. We will explore two different techniques and discuss each in turn: Chemically synchronizing cells in a population: This technique is the starting point for many experiments to ensure all cells are starting at the same stage in the cell cycle. Fluorescence-activated cell sorting (FACS): This is a subtype of a more commonly known procedure called flow cytometry, where cells are monitored for certain properties and grouped based on these properties. Chemically Synchronizing Cells in a Population When studying cells as they progress through the cell cycle, it’s common to want to take measurements based on a particular parameter. We can watch cells go through mitosis using a simple light microscope, but some of the other phases are more difficult to detect. As such, we may want to work with a population of cells that we know are all at the same phase. However, that generally does not happen naturally in a population of cells. Just like your classmates in elementary and high school hit their growth spurts at slightly different times even though you were all roughly the same age, cells will not naturally align their cell cycles. In order to synchronize a population of cells, the scientist must apply some kind of block that stalls the cell cycle at a known step in the cell cycle. The two most common blocks are based on biological concepts we’ve seen before in this book. In Chapter 4, we discovered the existence of temperature-sensitive mutations, and how they could be used to study the essential proteins of the secretory pathway. Most cell cycle proteins are equally essential, as a cell that cannot progress through its cell cycle cannot divide. So temperature-sensitive mutants are useful in this context as well. In Chapter 6, we also saw that chemical inhibitors could be used to disrupt cytoskeletal function temporarily. This, too, is a strategy that can be applied to the cell cycle. There are a few known chemical inhibitors that block further progression in the cell cycle. DNA replication is often targeted by these compounds, and the cell stalls at the start of S phase as a result of the chemical inhibition. When these blocks are applied, the cells will continue to progress through the cell cycle until they hit the blockage, and then they can go no farther. You can think of this like construction on a major bridge or roadway. The cars are able to move freely elsewhere as they work toward their destination, but once they arrive at the construction, they stop and stay there until the construction block is lifted so that the stopped traffic can again begin to move forward. Fluorescence-Activated Cell Sorting (FACS) This final technique is, in some ways, explained by its name. When we do FACS, we fluorescently label cells and then use that fluorescence to differentiate between cells that are in different states (Figure 08-10). The machine (called a flow cytometer) that measures the fluorescence is also able to separate a mixed population of cells based on this fluorescence (i.e., the cells are “sorted” into different groups based on the measured fluorescence). When using FACS to learn about the stage of the cell cycle that each cell in a population is in, a live fluorescent DNA stain like DAPI is commonly used. Since the amount of DNA changes as the cell cycle progresses, this is a very simple way to separate the cells in a sample. The result of this technique is twofold: You now have synchronized populations of cells without using chemical inhibitors or temperature-sensitive mutations. These can be used for further experiments. Cells in G1 will have one “full set” of their DNA, while cells in G2 and M phase will have duplicated their DNA (i.e., two “full sets”). Cells in S phase will have more than one set but less than two sets, as replication has started in S phase but is not yet complete. Additional visual separation may be required to differentiate between G2 and M phase, but this can be done relatively easily, as a cell undergoing mitosis can be identified using a standard light microscope. The flow cytometer also produces a graphical readout (Figure 08-10) that summarizes how many cells were found with each “amount” of fluorescent material (DNA in this case). These readouts can be used to learn important information about your population of cells. Interpreting FACS Readouts A simple FACS readout is shown in Figure 08-10. It is very much like a histogram, in which the y-axis represents the number of cells counted at each point on the x-axis (often listed as a percentage of the total population), and the x-axis measures the amount of DNA. (Note: This is usually done indirectly, by measuring the fluorescence intensity.) The amount of fluorescence intensity correlates to the amount of DNA in a given cell. Since the amount of DNA in a cell correlates with the phase of the cell cycle, this readout can help us determine how many of the cells in our sample are in each stage of the cell cycle at the moment at which they were measured. We can compare cells at different time points, which might tell us whether our cells are able to progress through the cell cycle or whether they have stalled. The following are some things to remember about FACS and the readout in Figure 08-10: The readout indicates “replicated” and “unreplicated” DNA, not whether it’s in G1, S, G2, or M phase. This is an important limitation of this technique. There are two phases in which the DNA has been replicated and as such are lumped together in the second peak (G2 and M phase). FACS cannot differentiate between these, since the DNA content is the same for both. Depending on the question you’re trying to answer, this might be key to correctly interpreting your results. To separate G2 from M phase in your sample, you would need to do something additional, like look at the cells in a microscope, for example. The part in between the peaks is more important than it might seem. In this area, we have more than one “full set” of DNA but less than two. There’s only one stage of the cell cycle where the amount of DNA goes from one set to two: S phase. Cells that are in the process of undergoing S phase will be found between the peaks that represent G1 cells and G2/M cells. Like many of the techniques we’ve seen so far (FRAP, SDS-PAGE, etc.), FACS is a way to quantify what’s going on in your samples. It’s not actually the experiment itself. The experiment would be done before you put your samples through the FACS machine. If, for example, you wanted to explore the impact of a newly discovered mutation, you could compare the FACS readout from a wild-type (i.e., unmutated) and a mutated sample of cells and see how similar or different their FACS readouts are. Changes in the FACS readout, even really strange and unexpected changes that don’t make a lot of sense at first glance, can tell us very important things about the samples and treatments we choose to study. Like all of the experiments we’ve covered in this book, understanding what the technique can, and cannot, reveal is essential to being able to accurately interpret the data you receive from your experiment. Topic 8.2: Mitosis and Cell Division Learning Goals Relate the progression of mitosis to the activation and deactivation of proteins by the mitotic CDK-cyclin complex. Explain the role of the cytoskeleton (and associated motor proteins) during mitosis and cytokinesis, including how dynamic instability contributes to the formation and function of the mitotic spindle. Describe how signaling events are used to end mitosis, starting at anaphase and followed by the transition of the cell back into G1. The stages of mitosis are likely something you have been learning about since grade school, so it might seem odd at first that we are going to revisit them here. However, the reality of what actually has to happen, at the cellular level, in order for mitosis to occur is incredibly complex. As you may have guessed by now, nothing in the cell just happens by chance. It’s all driven by proteins acting on other molecules in the cell. For a cell to go through mitosis, here’s just a short list of cellular events that occur: The entire transcription and translation machinery needs to be shut down so that the DNA can be replicated, condensed, accurately divided, and then set up again in the new cell. The microtubule network in the cell needs to be dismantled and completely rearranged so that it can help with DNA separation. The actin network also needs to be rearranged to help with cytokinesis (division of the cytosol). This means that all of the regular work of the cytoskeleton during interphase (transport, organelle positioning, etc.) is also disrupted. Every organelle and structure in the cell must be duplicated and/or distributed into the two daughter cells so that both of the new cells have everything they need to survive. This often requires a complete dismantling of larger organelles (like the endomembrane system or the nucleus). A whole series of signaling events needs to be started up to initiate all of this change, and then, at the end, it all needs to be shut down again so that everything can transition back to the interphase and regular cellular function can be rebooted. This incomplete list of events highlights the complexity of the process and also how vulnerable the cell is while this takes place. As long as the cell is focused on division, it cannot respond to changes in the environment or defend itself, metabolize new foods, or deal with damage. The cell needs to be efficient and get this work done quickly so that it reduces the time that it is vulnerable. The work also needs to be done very accurately, as any mistake in the process of mitosis is very likely going to kill one or both daughter cells. Mitosis is a highly regulated dance that includes every part of the cell. This final topic of the book works to highlight some of the key elements of that dance and expand your understanding of the beautiful complexity involved in M phase. The Hypercondensation of Chromosomes for Mitosis Waaaay back in Chapter 3, we discussed the structure of the genome in detail, including how the DNA that forms each of the chromosomes is organized and compacted with the help of histone proteins. In this chapter, we will look at how the cell transforms the organization of the DNA in the interphase nucleus into the hypercompacted mitotic chromosomes that are required for cell division. Just like we did in Chapter 3, we want to remind you that this is an area of active research that is extremely difficult to study, so we don’t have all of the answers for how this process works. We will do our best to tell you what science currently believes to be true and also what we still have yet to discover. There are two parts to chromosome management during mitosis: The first is the actual condensation of the chromosomes at the start of mitosis. The second is the maintenance of the structure of the mitotic chromosome throughout the stages of mitosis, including the separation of the sister chromatids at anaphase. Remember that chromosomes are, in essence, extremely long strands of DNA that could easily become a tangled mess in the dynamic environment of the cell during mitosis. Based on what we currently know, it appears to use at least some (but not all) of the same proteins that are used to maintain interphase packing of DNA. We’ll focus primarily on the formation of the mitotic chromosomes and highlight what we know about the maintenance of these structures as we discuss the proteins involved. Condensation of the chromosome for mitosis is thought to require only five additional proteins beyond the histones used to form the chromatin fiber. They are condensin I and II, cohesins, a kinesin known as Kif4A, and an enzyme called DNA topoisomerase II alpha. The roles of Kif4A and DNA topoisomerase II alpha are not entirely clear, even though it is clear that without them the mitotic chromosome is unable to form. From the histones, a key player in chromosome condensation for mitosis appears to be histone H1, also known as the linker histone. We’ll start our discussion with the H1 histone, as it connects the most directly with what we learned about interphase chromatin in Chapter 3. Phosphorylation of Histone H1 Promotes Higher-Order Chromatin Packing While the structure of histone H1 is slightly different from the core histones, it still has some key features that are common to all histones. For example, it has two tail regions, which are sites where posttranslational chemical modifications (phosphorylation, acetylation, methylation) commonly occur. Histone H1 is a key phosphorylation target for M-CDKs. Phosphorylation is thought to have a few different effects on the structure of the chromatin, which helps it begin the process of disassembling the interphase organization so that the mitotic chromosome structure can take shape. Our discussion of H1 in Chapter 3 showed us that H1 is often used to help pack up the chromatin more tightly, which reduces access for gene expression, so the involvement of histone H1 in preparing the DNA for mitosis makes quite a bit of sense. Condensin I and Condensin II Work Together to Pack the DNA into a Tight Column The major work of packing the DNA up into the tight column that forms the mitotic chromosomes falls to two proteins that are part of the condensin family. Condensin I and II each have their own role to play in the process (Figure 08-11 and Video 08-03). Cohesins Hold the Sister Chromatids Together Interestingly, while cohesins have an important role in organizing interphase chromatin (they help form topologically associated domains, or TADs), they are removed from much of the DNA as the genome prepares for mitosis. Condensins pick up the slack and help loop up the DNA during the extreme condensation required for mitosis. However, cohesins still have an important function at this stage. As we saw earlier, cohesins are used to hold the sister chromatids together after replication, right until they are degraded during anaphase so that chromatid separation can take place (Figure 08-11). The end result of all of this packing is an extremely condensed chromosome that is as much as 20,000 times more compacted than the original DNA strand. When observed in electron microscopy (Figure 08-12), the looped fibers of DNA can be observed as well as the sister chromatids and the centromere. This level of packing is so extreme that no transcription is possible. M Phase: Mitosis and Cytokinesis M phase consists of two separate events: Mitosis: This is the process of division of the nucleus and its contents (i.e., the DNA!). Cytokinesis: This is where the rest of the cytoplasm gets divided into two new cells. Both events are managed and controlled by a variety of proteins, which are activated and deactivated by the CDKs. It’s also important to note that the cytoskeleton plays a key role in both events. To start, we’re going to take a step back and look at the process of mitosis through a microscope (Video 08-04). It is likely that you have already learned the names of the stages of mitosis, as they are taught in elementary and/or high school. However, they are summarized in Figure 08-13. We are going to look at each of these stages in turn. Our primary focus will be the rearrangements of the cytoskeleton that control the cellular events and the key signaling events that are responsible for triggering different parts of the process. Mitosis Stage 1: Prophase (and Prometaphase) Prophase is the first stage of mitosis. At this point, the cell has passed the G2/M checkpoint, and its M-CDK is fully activated, so it begins to prepare itself in earnest for mitosis. Each of these events is the result of M-CDK phosphorylating one or many of the targets mentioned earlier in this chapter. Chromosomes condense as H1 and nucleolin are phosphorylated. The nuclear lamins depolymerize, resulting in the nuclear envelope breaking apart. The mitotic spindle will begin to form. The first step in mitotic spindle formation is the production of two centralized microtubule organizing centers (MTOCs) in the cell, which will become the opposite poles of the spindle. The process of spindle formation actually begins well before the start of M phase. As early as the transition to G2 phase, some of the components of the mammalian MTOC (also called the centrosome) will duplicate. At the onset of prophase, the centrosome splits in two, and the two MTOC will begin to migrate to either side of the nucleus. Interestingly, not very much is currently known about how plant cells transition from a decentralized gamma-tubulin distribution to the highly organized mitotic spindle. There is more than one model for how the mitotic spindle is formed. Here we are going to focus on one called the “search and capture” model (Figure 08-14). In this model, the formation of the mitotic spindle relies very heavily on the dynamic instability of the microtubules. As microtubules grow outward from the spindle pole, they “search” the cytosol for things to grab onto (chromosomes, the plasma membrane, or microtubules from the other pole of the spindle). As you will remember from Chapter 6, microtubules are highly unstable unless capped or stabilized in some way. The formation of the mitotic spindle uses this characteristic to its advantage—a microtubule that doesn’t “capture” anything in the cytosol will depolymerize. As such, microtubules will grow and shrink at random until they encounter the proper components to form the mitotic spindle. As the components of the spindle are stabilized and come together, three types of microtubule structures begin to form (Figure 08-15): If microtubules from opposite poles happen to interact with each other, they will stabilize each other (with the help of motor proteins), thus forming a bridge between the two spindle poles. These are aptly named interpolar microtubules, as they span the two poles. Conversely, if they happen to interact with a chromosome’s kinetochore during their random growth and shrinkage, they will attach and become a kinetochore microtubule. Astral microtubules, the third type of microtubule, are anchored at the plasma membrane (also with the help of motor proteins). Though there is much that is still unknown about the formation of the mitotic spindle, scientists do know that motor proteins are absolutely crucial to this process. Several motor proteins have been identified as having a role in the formation of the mitotic spindle, including the following: Kinesins are found at the kinetochore and help the chromosomes stay attached to the kinetochore microtubules as they disassemble in anaphase. Dyneins hold the astral microtubules and also help pull the spindle poles apart in anaphase Interpolar microtubules have a special double-headed kinesin, which walks along each of the interpolar microtubules in a set. They also help to push the spindle poles apart in anaphase B. There is a distinction sometimes made between the beginning and end of prophase. We often say that once the nuclear envelope breaks down, we have entered prometaphase. The breakdown of the nuclear envelope is an important event, as the mitotic spindle cannot be completed until that happens and the condensed chromosomes are free in the cytosol. It is only then that the “search and capture” part of spindle formation can begin in earnest. By the end of prometaphase, each of the chromosomes is held at the center of the mitotic spindle by microtubules attached to the kinetochore. The kinetochore is a specialized region of the chromosome associated with the centromere. As you may recall, the centromere is a DNA sequence that is the site of assembly of the kinetochore structure. The kinetochore is a plaque or button-shaped structure composed of proteins. There is one kinetochore associated with each of the chromatids, and the microtubules get inserted into them. They face in opposite directions on the two sides of the chromosome. Mitosis Stage 2: Metaphase After the nuclear envelope breaks down and the chromosomes are attached to the mitotic spindle by the microtubules, we transition to metaphase. Notably in this phase, the chromosomes align on the metaphase plate—an imaginary plane equidistant from the two spindle poles. To accomplish the chromosomal alignment, chromosomes are pulled simultaneously toward both spindle poles by the kinetochore microtubules. Even tension on the chromosome from both sides of the spindle is required for everything to stabilize and stay in place. The proper tension is only achieved when the chromosome is properly aligned at the metaphase plate. It is known that even though both ends of the microtubules are being held in the mitotic spindle, the state of kinetochore microtubules is still very dynamic—tubulin units are continually being added and removed at both ends. This is intriguing and unusual, considering that each microtubule is held at the minus end by the MTOC (presumably with the help of gamma-tubulin), and the plus end is held by the kinetochore. While it is not entirely clear how the minus end can both be held by the MTOC and allow for dynamic instability, the story of how this works at the kinetochore is taking shape. Video 08-05 summarizes how we think it works. Something worth noting is the release of the metaphase checkpoint signal, which is the signal to activate the anaphase-promoting complex (APC) and initiate anaphase. This event leads us to the next stage of mitosis. Mitosis Stage 3: Anaphase Sister chromatids do not remain associated with each other by accident—they are held together by cohesins. At anaphase, the cohesins are dissolved, which allows the sister chromatids to separate. The abrupt destruction of the cohesin linkage between sister chromatids is brought about by activation of APC by M-CDK. Remember that APC’s function is to tag proteins with ubiquitin, marking them for degradation via the proteasome (Figure 08-16). APC first acts on a regulatory protein called securin, which then releases an enzyme called separase. Separase cleaves the cohesins, allowing the sister chromatids to separate. The movement of kinetochores toward the spindle poles in anaphase takes place through the action of motor proteins. The motor proteins that helped with spindle assembly now work to separate the sister chromatids from each other. In addition, the kinetochore microtubules are depolymerizing rapidly from their plus end. These two forces work together to separate the sister chromatids and move them to the opposite poles. Anaphase movement of chromosomes is separable into two phases with different mechanisms: anaphase A and anaphase B (Figure 08-17): Anaphase A. During the initial part of anaphase, kinetochore microtubules shorten and move the chromosomes toward the opposite poles. This requires a combination of depolymerization of kinetochore microtubules and the action of motor proteins at the kinetochore that help keep the chromosome connected to the shrinking microtubule. Anaphase B. During anaphase B, the length of the kinetochore microtubules (MTs) remains more or less constant. Now it is the other MTs of the mitotic spindle that begin to do the work. Again, there is much that is still unknown about this process, but three separate forces have been shown to play a role in anaphase B: Tubulin subunits are added to the plus ends of the interpolar microtubules, making them longer. Motor proteins move the overlapping interphase microtubules from the two poles apart to make the spindle longer. This requires the help of the kinesins that are holding the paired interpolar microtubules together. Dyneins, attached to the astral microtubules and the cell cortex, walk toward the spindle poles on either side. This results in a shortening of the distance between the poles of the spindle and the plasma membrane. It also helps pull the spindle poles farther apart from each other. Mitosis Stage 4: Telophase By telophase, the division of the nuclear contents is now complete; thus the cell needs to shut down mitosis and prepare for entry into G1. Several processes occur: The nuclear lamina must reassemble so that the nuclear envelope can be rebuilt around the chromosome. The mitotic spindle disintegrates, and the cytoskeleton reassembles into its interphase conformation. The chromosomes must decondense and reform their interphase arrangement, which includes the reassembly of the nucleolus. If you examine the above list, you should notice that this is the reverse of what happened in preparation for mitosis and prophase. Since preparation for mitosis requires the activation of M-CDK, it makes sense that all of these changes to prepare for G1 are a result of the deactivation of the same M-CDK. APC shuts down M-CDK by tagging M cyclin with ubiquitin. This results in the destruction of M cyclin by the proteasome (Figure 08-09). Once the cyclin is gone, the CDK can no longer function and is turned off. Because of this, the balance between active kinases and phosphatases is drastically shifted in favor of the phosphatases, which remove the phosphates that were previously added to M-CDK and its target molecules. This allows the proteins to return to their pre–M phase state, which allows the cell cycle to reenter G1. As a direct result of the M-CDK shutdown, the nuclear lamina gets dephosphorylated and begins to reform around the chromosomes (Figure 08-18). The chromosomes retain their connection to the nuclear lamina throughout the cell cycle. As such, the chromosomes themselves provide a template for nuclear envelope reformation, ensuring that the nucleus reforms without excluding any of the chromosomes. APC continues to be active throughout the end of M phase and into G1, when it gets deactivated by the G1/S-cyclin-CDK complex. M Phase Final Stage: Cytokinesis Once the division of the nuclear contents is complete, the rest of the cytoplasm also needs to be divided between daughter cells. This occurs during cytokinesis, which is also the final stage of M phase (Figure 08-19). This process happens quite differently in plants, animals, and yeast and other fungi. However, in all cases, actin is the cytoskeletal fiber that plays the most important role. Cytokinesis in Mammalian Cells In mammalian cells, actin arranges itself in an antiparallel array right underneath the plasma membrane. Together with myosin motors, they help form a contractile ring that pulls the plasma membrane inward and eventually splits the cell in two. Like many other parts of M phase, there is still much that is uncertain about the process of cytokinesis. However, the contractile ring has been found to always form around the midzone of the mitotic spindle, where the original metaphase plate was found. Thus, it is believed that the mitotic spindle itself plays a role in determining where the contractile ring will form. As the myosin walks along the actin filaments of the ring, the ring contracts around what’s left of the mitotic spindle (namely, the interpolar MTs). The organelles have been either broken down into vesicles (ER and Golgi) or duplicated (mitochondria) to ensure that there will be more or less equal quantities of them in each of the daughter cells. Thus, dividing the cytoplasm also divides the organelles. At the same time, vesicles fuse rapidly with the plasma membrane, which helps ensure that there is enough plasma membrane to fit around each daughter cell. Cytokinesis in Plant Cells In plants, the cell not only must divide its cytoplasm, but a new cell wall must be produced between the two daughter cells. As a result, the Golgi apparatus, a key synthesizer of cell wall components, remains functional throughout. The many small mobile plant Golgi do not vesiculate (i.e., break into small pieces) like they do in animal cells but instead divide by fission prior to mitosis. Electron microscopy shows that they congregate on either side of the site of the new cell wall. The interpolar microtubules also help guide this process through the formation of a structure known as the phragmoplast in the center of the cell. The Golgi secrete cell wall compounds very rapidly to the center of the phragmoplast to build a cell plate. Actin has an important role to play in forming the phragmoplast and directing the vesicles to the cell plate, though the exact mechanism is still unclear. The cell plate grows from the center of the cell toward the outer walls and then fuses with the side walls of the dividing cell. At that point, the new daughter cells have been fully formed with their full complement of organelles and DNA. Mitosis and Cytokinesis in Other Eukaryotes The information in this chapter covers the most commonly discussed forms of cytokinesis—namely, plants and animals. It’s important to remember that much like every other process in the cell, Eukaryotes are a hugely diverse set of organisms, which means that mitosis and cytokinesis can vary quite a bit from what is described in this chapter. In addition, specific tissues within organisms will vary. While we don’t have time to go into all of the different ways that mitosis happens, we did want to end this chapter with a few interesting examples of how different it can be: Plant and animal cells undergo what’s called open mitosis, which means that the nuclear envelope breaks down completely in order to form the mitotic spindle. There are also two other forms: Partially closed mitosis occurs when the nuclear envelope remains intact and the spindle is formed both inside and outside of the nuclear envelope. Budding yeast (Saccharomyces cerevisiae) uses a partially closed format, which is especially interesting considering how often they are used as a model organism for all Eukaryotes. A third form is closed mitosis, which happens when the nuclear envelope not only remains intact, but the nuclear pores are plugged to close the nucleus off completely from the rest of the cell. This is more common in protists, fungi, and other single-celled organisms. Many cell types don’t divide evenly at mitosis. Lots of plant and animal cells undergo asymmetric division, which results in one daughter cell that is larger than the other. Additionally, lots of cells “grow” their daughter cell off of one side and move the new daughter nuclei into the bud once the bud is large enough. Saccharomyces cerevisiae is called “budding yeast” for this very reason. There are many additional cells that do this, especially single-celled algae, fungi, and other protists. Chapter Summary As you can tell now that you are at the end of this chapter and the end of the book, the cell cycle is truly a great process to integrate all the knowledge that you have gained thus far. In addition, we hope that you appreciate the additional layers of complexity required to regulate one of the most important cellular processes. The cell cycle is an intricate dance that requires the cooperation of most of the cellular systems. We’ve emphasized a few in this chapter. But we challenge you to think of even more connections to the other units in the book. Signaling: The cell cycle requires integrated information from the extracellular and intracellular environments through signaling. The balance of activating forces versus inactivating forces helps ensure that cells divide only when the time is right. Cytoskeleton: The cytoskeleton is highly involved, especially in M phase. It’s clear that dynamic instability, a hallmark of the cytoskeletal system, is vital for the formation of the mitotic spindle but also the correct alignment of the chromosomes at the metaphase plate. DNA packing: The molecular packers and organization help unpack all the DNA during synthesis and do the opposite in M phase, completely packaging the DNA. Other organelles: Of course, these organelles need to be properly divided between daughter cells, but that does not end their contribution to this process. Prior to shutdown for mitosis, they need to make sure that everything the cell will need is available. This will include a great deal of energy, in the form of ATP and GTP, membrane, vesicles, and even polysaccharides if a wall needs to be made between the two daughter cells. With that, we’ve come to the end of the book. We congratulate you for getting to the end! No matter your starting place, we hope that you have at least a little more appreciation for the complex beauty of the cell. Review Questions Note on usage of these questions: Some of these questions are designed to help you tease out important information within the text. Others are there to help you go beyond the text and begin to practice important skills that are required to be a successful cell biologist. We recommend using them as part of your study routine. We have found them to be especially useful as talking points to work through in group study sessions. Topic 8.1: Regulating the Cell Cycle: Checkpoint Control Make a list of the four phases of the cell cycle, and describe in detail what has to happen in each phase before it is complete. Be as specific as you can. Describe how to use FACS is used in experiments to understand the regulation of the cycle. Explain the concept of checkpoint control of the cell cycle and give examples of what types of conditions should be met before S and M phases proceed. Explain why those conditions need to be met before the cell moves to the next phase. Describe the sequence of events leading to CDK activation and deactivation. Define negative and positive feedback. How are they thought to play a role in cell cycle control? Discuss the role of CDK-cyclin complexes in regulation of the cell cycle. How do they contribute to the cell “knowing” the conditions are right for the next stage of the cell cycle? Compare and contrast the roles of CDKs that control entry into S phase versus entry into M phase. Make a list of the different examples of proteins that they are thought to interact with and their effects. Explain the role of phosphorylation in regulation of the cell cycle. Explain how phosphorylation can affect protein folding and how that might result in changes in protein activity. Draw a FACS graph for the following groups of cells: Cells arrested in G1 Cells arrested in G2 Cells arrested in S phase Cells arrested in M phase Topic 8.2: Mitosis and Cell Division Describe the role of dynamic instability in the formation and maintenance of the mitotic spindle. Illustrate with labeled diagrams the relationship between chromatids and chromosomes at each of the stages of the cell cycle. Explain how each stage of mitosis is ultimately controlled by the activation and deactivation of CDKs. Compare and contrast the following stages: prophase and prometaphase prophase and telophase anaphase A and B What is cytokinesis, and how does it differ in plants and animals?
12310	https://www.easa.europa.eu/en/document-library/easy-access-rules/online-publications/easy-access-rules-large-aeroplanes-cs-25?page=6	Easy Access Rules for Large Aeroplanes (CS-25) Revision from January 2023 Last updated date 30 January 2023 More info and download PDF Easy Access Rules for Large Aeroplanes (CS-25) Revision from January 2023 SUBPART B – FLIGHT PERFORMANCE CS 25.101 General ED Decision 2016/010/R (See AMC 25.101) (a) Unless otherwise prescribed, aeroplanes must meet the applicable performance requirements of this Subpart for ambient atmospheric conditions and still air. (b) The performance, as affected by engine power or thrust, must be based on the following relative humidities: (1) 80%, at and below standard temperatures; and (2) 34%, at and above standard temperatures plus 28°C (50°F). Between these two temperatures, the relative humidity must vary linearly. (c) The performance must correspond to the propulsive thrust available under the particular ambient atmospheric conditions, the particular flight condition, and the relative humidity specified in sub-paragraph (b) of this paragraph. The available propulsive thrust must correspond to engine power or thrust, not exceeding the approved power or thrust, less – (1) Installation losses; and (2) The power or equivalent thrust absorbed by the accessories and services appropriate to the particular ambient atmospheric conditions and the particular flight condition. (See AMCs No 1 and No 2 to CS 25.101(c).) (d) Unless otherwise prescribed, the applicant must select the take-off, en-route, approach, and landing configuration for the aeroplane. (e) The aeroplane configurations may vary with weight, altitude, and temperature, to the extent they are compatible with the operating procedures required by sub-paragraph (f) of this paragraph. (f) Unless otherwise prescribed, in determining the accelerate-stop distances, take-off flight paths, take-off distances, and landing distances, changes in the aeroplane’s configuration, speed, power, and thrust, must be made in accordance with procedures established by the applicant for operation in service. (g) Procedures for the execution of balked landings and missed approaches associated with the conditions prescribed in CS 25.119 and 25.121(d) must be established. (See AMC 25.101(g)) (h) The procedures established under sub-paragraphs (f) and (g) of this paragraph must – (1) Be able to be consistently executed in service by crews of average skill, (2) Use methods or devices that are safe and reliable, and (3) Include allowance for any time delays in the execution of the procedures, that may reasonably be expected in service. (See AMC 25.101(h)(3).) (i) The accelerate-stop and landing distances prescribed in CS 25.109 and 25.125, respectively, must be determined with all the aeroplane wheel brake assemblies at the fully worn limit of their allowable wear range. (See AMC 25.101(i).) [Amdt 25/2] [Amdt 25/18] AMC 25.101 General ED Decision 2003/2/RM The test aeroplane used in the determination of the scheduled performance should be in a condition which, as far as is reasonably possible, is representative of the average new production aeroplane. Where the test aeroplane differs from this standard (e.g. with regard to engine idle thrust settings, flap rigging, etc.) it will be necessary to correct the measured performance for any significant performance effects of such differences. AMC No. 1 to CS 25.101(c) Extrapolation of Performance with Weight ED Decision 2003/2/RM The variation of take-off, climb and landing performance with weight may be extrapolated without conservatism to a weight greater, by up to 10%, than the maximum weight tested and to a weight lower, by up to 10%, than the lowest weight tested. These ranges may not be applicable if there are significant discontinuities, or unusual variations, in the scheduling of the relevant speeds with weight, in the weight ranges covered by extrapolation. AMC No. 2 to CS 25.101(c) General ED Decision 2003/2/RM 1 GENERAL - CS 25.101 1.1 Explanation - Propulsion System Behaviour. CS 25.101(c) requires that aeroplane “performance must correspond to the propulsive thrust available under the particular ambient atmospheric conditions, the particular flight condition, . . .” The propulsion system’s (i.e., turbine engines and propellers, where appropriate) installed performance characteristics are primarily a function of engine power setting, airspeed, propeller efficiency (where applicable), altitude, and ambient temperature. The effects of each of these variables must be determined in order to establish the thrust available for aeroplane performance calculations. 1.2 Procedures. 1.2.1 The intent is to develop a model of propulsion system performance that covers the approved flight envelope. Furthermore, it should be shown that the combination of the propulsion system performance model and the aeroplane performance model are validated by the takeoff performance test data, climb performance tests, and tests used to determine aeroplane drag. Installed propulsion system performance characteristics may be established via the following tests and analyses: a. Steady-state engine power setting vs. thrust (or power) testing. Engines should be equipped with adequate instrumentation to allow the determination of thrust (or power). Data should be acquired in order to validate the model, including propeller installed thrust, if applicable, over the range of power settings, altitudes, temperatures, and airspeeds for which approval is sought. Although it is not possible to definitively list or foresee all of the types of instrumentation that might be considered adequate for determining thrust (or power) output, two examples used in past certification programmes are: (1) engine pressure rakes, with engines calibrated in a ground test cell, and (2) fan speed, with engines calibrated in a ground test cell and the calibration data validated by the use of a flying test bed. In any case, the applicant should substantiate the adequacy of the instrumentation to be used for determining the thrust (or power) output. b. Lapse rate takeoff testing to characterise the behaviour of power setting, rotor speeds, propeller effects (i.e., torque, RPM, and blade angle), or gas temperature as a function of time, thermal state, or airspeed, as appropriate. These tests should include the operation of an Automatic Takeoff Thrust Control System (ATTCS), if applicable, and should cover the range of power settings for which approval is sought. i. Data for higher altitude power settings may be acquired via overboost (i.e., operating at a higher than normal power setting for the conditions) with the consent of the engine and propeller (when applicable) manufacturer(s). When considering the use of overboost on turbopropeller propulsion system installations to simulate higher altitude and ambient temperature range conditions, the capability to achieve an appropriate simulation should be evaluated based on the engine and propeller control system(s) and aircraft performance and structural considerations. Engine (gearbox) torque, rotor speed, or gas temperature limits, including protection devices to prohibit or limit exceedences, may prevent the required amount of overboost needed for performance at the maximum airport altitude sought for approval. Overboost may be considered as increased torque, reduced propeller speed, or a combination of both in order to achieve the appropriate blade angle for the higher altitude and ambient temperature range simulation. Consideration for extrapolations will depend on the applicant’s substantiation of the proper turbopropeller propulsion system simulated test conditions. ii. Lapse rate characteristics should be validated by takeoff demonstrations at the maximum airport altitude for which takeoff approval is being sought. Alternatively, if overboost (see paragraph (i) above) is used to simulate the thrust setting parameters of the maximum airport altitude for which takeoff approval is sought, the takeoff demonstrations of lapse rate characteristics can be performed at an airport altitude up to 915 m (3,000 feet) lower than the maximum airport altitude. c. Thrust calculation substantiation. Installed thrust should be calculated via a mathematical model of the propulsion system, or other appropriate means, adjusted as necessary to match the measured inflight performance characteristics of the installed propulsion system. The propulsion system mathematical model should define the relationship of thrust to the power setting parameter over the range of power setting, airspeed, altitude, and temperature for which approval is sought. For turbojet aeroplanes, the propulsion system mathematical model should be substantiated by ground tests in which thrust is directly measured via a calibrated load cell or equivalent means. For turbopropeller aeroplanes, the engine power measurements should be substantiated by a calibrated dynamometer or equivalent means, the engine jet thrust should be established by an acceptable engine model, and the propeller thrust and power characteristics should be substantiated by wind tunnel testing or equivalent means. d. Effects of ambient temperature. The flight tests of paragraph 1.2.1.a. above will typically provide data over a broad range of ambient temperatures. Additional data may also be obtained from other flight or ground tests of the same type or series of engine. The objective is to confirm that the propulsion system model accurately reflects the effects of temperature over the range of ambient temperatures for which approval is being sought (operating envelope). Because thrust (or power) data can usually be normalised versus temperature using either dimensionless variables (e.g., theta exponents) or a thermodynamic cycle model, it is usually unnecessary to obtain data over the entire ambient temperature range. There is no need to conduct additional testing if: i. The data show that the behaviour of thrust and limiting parameters versus ambient temperature can be predicted accurately; and ii. Analysis based upon the test data shows that the propulsion system will operate at rated thrust without exceeding propulsion system limits. 1.2.2 Extrapolation of propulsion system performance data to 915 m (3,000 feet) above the highest airport altitude tested (up to the maximum takeoff airport altitude to be approved) is acceptable, provided the supporting data, including flight test and propulsion system operations data (e.g., engine and propeller control, limits exceedence, and surge protection devices scheduling), substantiates the proposed extrapolation procedures. Considerations for extrapolation depend upon an applicant's determination, understanding, and substantiation of the critical operating modes of the propulsion system. This understanding includes a determination and quantification of the effects that propulsion system installation and variations in ambient conditions have on these modes. 2 Expansion of Takeoff and Landing Data for a Range of Airport Elevations. 2.1 These guidelines are applicable to expanding aeroplane Flight Manual takeoff and landing data above and below the altitude at which the aeroplane takeoff and landing performance tests are conducted. 2.2 With installed propulsion system performance characteristics that have been adequately defined and verified, aeroplane takeoff and landing performance data obtained at one field elevation may be extrapolated to higher and lower altitudes within the limits of the operating envelope without applying additional performance conservatisms. It should be noted, however, that extrapolation of the propulsion system data used in the determination and validation of propulsion system performance characteristics is typically limited to 915 m (3,000 feet) above the highest altitude at which propulsion system parameters were evaluated for the pertinent power/thrust setting. (See paragraph 1 of this AMC for more information on an acceptable means of establishing and verifying installed propulsion system performance characteristics.) 2.3 Note that certification testing for operation at airports that are above 2438 m (8,000 feet) should also include functional tests of the cabin pressurisation system. Consideration should be given to any other systems whose operation may be sensitive to, or dependent upon airport altitude, such as: engine and APU starting, passenger oxygen, autopilot, autoland, autothrottle system thrust set/operation." AMC 25.101(g) Go-around ED Decision 2020/024/R 1. General CS 25.101(g) requires that procedures must be established for the execution of go-arounds from landing configurations (identified as ‘balked landings’ in this AMC) and from approach configurations (identified as ‘missed approaches’ in this AMC) associated with the conditions prescribed in CS 25.119 and CS 25.121(d). Also, as required by CS 25.1587(b)(4), each AFM must contain the procedures established under CS 25.101(g), including any relevant limitations or information. The landing climb gradient determined under the CS 25.119 conditions, the approach climb gradient determined under the CS 25.121(d) conditions, and the additional operating limitations regarding the maximum landing weight established in accordance with CS 25.1533(a)(2) must be consistent with the established balked landing and missed approach procedures (CS 25.101(g)) provided in the aeroplane flight manual (AFM). In order to demonstrate the acceptability of the recommended missed approach and balked landing procedures, the applicant should conduct demonstrations (by flight test or pilot‑in‑the‑loop simulator tests) to include a one engine inoperative go-around at a weight, altitude, temperature (WAT)-limited or simulated WAT‑limited thrust or power condition. The applicant should conduct the demonstrations at WAT-limited conditions that result in the greatest height loss and/or longest horizontal distance to accelerate to the scheduled approach climb speed. Alternatively, the applicant may conduct testing at simulated WAT-limited conditions (with reduced thrust or power on the operating engine) and use the resulting time delays for each crew action in a subsequent off-line simulation/analysis in accordance with the procedures below. Although compliance with CS 25.101(g) and (h) and CS 25.121(d) is not directly linked with the criteria for the approval of weather minima for approach, the minimum decision height for initiating a go-around is dependent upon the weather minima to be approved. In addition, a steeper climb gradient and the associated lower WAT-limited landing weight may be associated with CAT II operations. As such, if CAT II weather minima approval is expected, the applicant should conduct the go-around demonstration and/or analysis consistent with both CAT I and II operations for the associated decision height and WAT-limited thrust or power condition (or a critical combination thereof). 2. Procedures The go-around demonstration specified in Chapter 1 of this AMC can be conducted at an altitude above the normal decision height/altitude (for test safety), with the height loss in the manoeuvre used to show that ground contact prior to the runway threshold would not occur if the manoeuvre was initiated at the decision height/altitude. Flight testing, simulation and/or analysis at a range of (WAT limit or simulated WAT limit) conditions throughout the approved envelope should be conducted to assess the height loss relative to the decision height/altitude consistent with the criteria for the weather minima to be approved (or higher as constrained by AFM limitations). At least one flight test or pilot-in-the-loop simulator test should be conducted at a WAT-limited condition to assess the OEI go-around procedure and establish the time delays used for any subsequent analysis/simulation. In addition, the assessment of the go-around procedure should include consideration of the horizontal distance (based upon the minimum go-around trajectory) needed to establish the minimum engine‑out climb gradient required by CS 25.121(d) or a steeper gradient as required by specific weather minima operational criteria. It should be shown by flight test, simulation and/or analysis that the aeroplane would remain above the profile illustrated in Figure 1 below when the go-around is evaluated at the critical WAT limit condition (up to the structural maximal landing weight) and flown in accordance with the one-engine-inoperative (OEI) go‑around procedure. This provides a minimum design standard trajectory for a missed approach with one engine inoperative and does not constitute a means to ensure obstacle clearance. It does not preclude additional missed approach procedures that may be developed to satisfy operational requirements, including special or complex missed approach path requirements. The operator should seek approval from their national aviation authority to use the additional procedures and data. (a) In accordance with CS 25.101(h), the established procedures for executing balked landings and missed approaches must: (i) be able to be consistently executed in service by crews of average skill, (ii) use methods or devices that are safe and reliable, and (iii) include allowance for any time delays in the execution of the procedures that may reasonably be expected in service (including the recovery of full go-around thrust or power if equipped with a reduced go-around (RGA) thrust or power function that requires a manual override), but should not be less than one second between successive flight crew actions, except for movements of the primary flying controls. (b) The flight test demonstration(s), simulation and/or analysis should be made with: (i) all engines operating (AEO) and the thrust or power initially set for a 3-degree approach, and the configuration and final approach airspeed consistent with the AEO landing procedure (not more than VREF + 5 kt) in zero wind conditions, (ii) application of the available go-around thrust or power at the selected go-around height (initially the RGA thrust or power level, if so equipped, followed by either automatic or manual selection of full go-around thrust or power in accordance with the established missed approach and engine failure AFM procedures) with simultaneous failure of the critical engine (or with a simulated engine failure, including the effects on dependent systems), and (iii) the high-lift system, pitch attitude, engine/propeller controls and airspeed adjusted to achieve the conditions consistent with CS 25.121(d), in accordance with the established missed approach and engine failure AFM procedures. The landing gear should be selected to the ‘up’ position only after a positive rate of climb is achieved. If the use of automatic features (autopilot, auto-throttle, flight director, etc.) is included in the procedure, these features should be considered during the demonstration. Figure 1. Trajectory Assessment for OEI Go-around Segment A: From the initiation of go-around at the decision height/altitude to the runway threshold – remain above a 1:50 (2.0 %) plane extended to the runway threshold for clearance of airport obstacles. Segment B: From the runway threshold plus a distance defined by 40 seconds VT_appr, not more than the distance indicated in the table below – remain above ground height. \| \| \| --- \| \| Field Elevation (ft) \| Distance (ft) \| \| 0-3 048 m (0-10 000 ft) \| 3 048 m (10 000 ft) \| \| >3 048 m (> 10 000 ft) \| = Field Elevation \| Segment C: A straight line from the end of Segment B at ground height with a gradient defined by CS 25.121(d)(1) or a steeper gradient as required by specific weather minima operational criteria, up to a height, H1 – remain above the line. Where: VT_appr is the true airspeed for the normal recommended AEO approach speed in zero wind at the flight condition being assessed (not more than VREF + 9.3 km/h (5 kt) CAS). H1 is the height above the runway elevation where the aeroplane has achieved the approach climb configuration and stabilised on the approach climb speed out of ground effect (1x the wingspan), not less than the height at which the go-around was initiated. [Amdt 25/13] [Amdt 25/26] AMC 25.101(h)(3) General ED Decision 2003/2/RM CS 25.109(a) and (b) require the accelerate-stop distance to include a distance equivalent to 2 seconds at V1 in addition to the demonstrated distance to accelerate to V1 and then bring the aeroplane to a full stop. This additional distance is not intended to allow extra time for making a decision to stop as the aeroplane passes through V1, but is to account for operational variability in the time it takes pilots to accomplish the actions necessary to bring the aeroplane to a stop. It allows for the typical requirement for up to three pilot actions (i.e. brakes – throttles – spoilers) without introducing additional time delays to those demonstrated. If the procedures require more than three pilot actions, an allowance for time delays must be made in the scheduled accelerate-stop distance. These delays, which are applied in addition to the demonstrated delays, are to be 1 second (or 2 seconds if a command to another crew member to take the action is required) for each action beyond the third action. This is illustrated in Figure 1. FIGURE 1. ACCELERATE-STOP TIME DELAYS where:– VEF is the calibrated airspeed selected by the applicant at which the critical engine is assumed to fail. The relationship between VEF and V1 is defined in CS 25.107. ∆tact 1 = the demonstrated time interval between engine failure and activation of the first deceleration device. This time interval is defined as beginning at the instant the critical engine is failed and ending when the pilot recognises and reacts to the engine failure, as indicated by the pilot’s application of the first retarding means during accelerate-stop tests. A sufficient number of demonstrations should be conducted using both applicant and Agency test pilots to assure that the time increment is representative and repeatable. The pilot’s feet should be on the rudder pedals, not the brakes, during the tests. For AFM data expansion purposes, in order to provide a recognition time increment that can be executed consistently in service, this time increment should be equal to the demonstrated time or 1 second, whichever is greater. If the aeroplane incorporates an engine failure warning light, the recognition time includes the time increment necessary for the engine to spool down to the point of warning light activation, plus the time increment from light ‘on’ to pilot action indicating recognition of the engine failure. ∆tact 2 = the demonstrated time interval between activation of the first and second deceleration devices. ∆tact 3 = the demonstrated time interval between activation of the second and third deceleration devices. ∆tact 4→n = the demonstrated time interval between activation of the third and fourth (and any subsequent) deceleration devices. For AFM expansion, a 1-second reaction time delay to account for in-service variations should be added to the demonstrated activation time interval between the third and fourth (and any subsequent) deceleration devices. If a command is required for another crew member to actuate a deceleration device, a 2-second delay, in lieu of the 1-second delay, should be applied for each action. For automatic deceleration devices that are approved for performance credit for AFM data expansion, established systems actuation times determined during certification testing may be used without the application of the additional time delays required by this paragraph. AMC 25.101(i) Performance determination with worn brakes ED Decision 2003/2/RM It is not necessary for all the performance testing on the aircraft to be conducted with fully worn brakes. Sufficient data should be available from aircraft or dynamometer rig tests covering the range of wear and energy levels to enable correction of the flight test results to the 100% worn level. The only aircraft test that should be carried out at a specific brake wear state is the maximum kinetic energy rejected take-off test of CS 25.109(i), for which all brakes should have not more than 10% of the allowable brake wear remaining. CS 25.103 Stall speed ED Decision 2016/010/R (See AMC 25.103) (a) The reference stall speed VSR is a calibrated airspeed defined by the applicant. VSR may not be less than a 1-g stall speed. VSR is expressed as: where – VCLMAX = Calibrated airspeed obtained when the loadfactor-corrected lift coefficient is first a maximum during the manoeuvre prescribed in sub-paragraph (c) of this paragraph. In addition, when the manoeuvre is limited by a device that abruptly pushes the nose down at a selected angle of attack (e.g. a stick pusher), VCLMAX may not be less than the speed existing at the instant the device operates; nzw = Load factor normal to the flight path at VCLMAX; W = Aeroplane gross weight; S= Aerodynamic reference wing area; and q= Dynamic pressure. (b) VCLMAX is determined with: (1) Engines idling, or, if that resultant thrust causes an appreciable decrease in stall speed, not more than zero thrust at the stall speed; (2) Propeller pitch controls (if applicable) in the take-off position; (3) The aeroplane in other respects (such as flaps, landing gear, and ice accretions) in the condition existing in the test or performance standard in which VSR is being used; (4) The weight used when VSR is being used as a factor to determine compliance with a required performance standard; (5) The centre of gravity position that results in the highest value of reference stall speed; and (6) The aeroplane trimmed for straight flight at a speed selected by the applicant, but not less than 1.13 VSR and not greater than 1.3 VSR. (See AMC 25.103(b)) (c) Starting from the stabilised trim condition, apply the longitudinal control to decelerate the aeroplane so that the speed reduction does not exceed 0.5 m/s2 (one knot per second). (See AMC 25.103(b) and (c)). (d) In addition to the requirements of sub-paragraph (a) of this paragraph, when a device that abruptly pushes the nose down at a selected angle of attack (e.g. a stick pusher) is installed, the reference stall speed, VSR, may not be less than 3,7 km/h (2 kt) or 2%, whichever is greater, above the speed at which the device operates. (See AMC 25.103(d)) [Amdt 25/3] [Amdt 25/18] AMC 25.103(b) Stalling speed ED Decision 2003/2/RM The airplane should be trimmed for hands-off flight at a speed 13 percent to 30 percent above the anticipated VSR with the engines at idle and the airplane in the configuration for which the stall speed is being determined. Then, using only the primary longitudinal control for speed reduction, a constant deceleration (entry rate) is maintained until the airplane is stalled, as defined in CS 25.201(d). Following the stall, engine thrust may be used as desired to expedite the recovery. The analysis to determine VCLMAX should disregard any transient or dynamic increases in recorded load factor, such as might be generated by abrupt control inputs, which do not reflect the lift capability of the aeroplane. The load factor normal to the flight path should be nominally 1.0 until VCLMAX is reached. AMC 25.103(c) Stall speed ED Decision 2003/2/RM The stall entry rate is defined as the mean rate of speed reduction (in m/s2 (knots CAS/second)) in the deceleration to the stall in the particular stall demonstration, from a speed 10% above that stall speed, i.e. AMC 25.103(d) Stall speed ED Decision 2003/2/RM In the case where a device that abruptly pushes the nose down at a selected angle of attack (e.g. a stick pusher) operates after CLMAX, the speed at which the device operates, stated in CS 25.103(d), need not be corrected to 1g. Test procedures should be in accordance with AMC 25.103(b) to ensure that no abnormal or unusual pilot control input is used to obtain an artificially low device activation speed. CS 25.105 Take-off ED Decision 2015/008/R (a) The take-off speeds prescribed by CS 25.107, the accelerate-stop distance prescribed by CS 25.109, the take-off path prescribed by CS 25.111, the take-off distance and take-off run prescribed by CS 25.113, and the net take-off flight path prescribed by CS 25.115, must be determined in the selected configuration for take-off at each weight, altitude, and ambient temperature within the operational limits selected by the applicant - (1) In non-icing conditions; and (2) In icing conditions, if in the configuration used to show compliance with CS 25.121(b), and with the most critical of the “Take-off Ice” accretion(s) defined in Appendices C and O, as applicable, in accordance with CS 25.21(g): (i) The stall speed at maximum take-off weight exceeds that in non-icing conditions by more than the greater of 5.6 km/h (3 knots) CAS or 3% of VSR; or (ii) The degradation of the gradient of climb determined in accordance with CS 25.121(b) is greater than one-half of the applicable actual-to-net take-off flight path gradient reduction defined in CS 25.115(b). (b) No take-off made to determine the data required by this paragraph may require exceptional piloting skill or alertness. (c) The take-off data must be based on: (1) Smooth, dry and wet, hard-surfaced runways; and (2) At the option of the applicant, grooved or porous friction course wet, hardsurfaced runways. (d) The take-off data must include, within the established operational limits of the aeroplane, the following operational correction factors: (1) Not more than 50% of nominal wind components along the take-off path opposite to the direction of take-off, and not less than 150% of nominal wind components along the take-off path in the direction of take-off. (2) Effective runway gradients. [Amdt 25/3] [Amdt 25/16] CS 25.107 Take-off speeds ED Decision 2016/010/R (a) V1 must be established in relation to VEF as follows: (1) VEF is the calibrated airspeed at which the critical engine is assumed to fail. VEF must be selected by the applicant, but may not be less than VMCG determined under CS 25.149(e). (2) V1, in terms of calibrated airspeed, is selected by the applicant; however, V1 may not be less than VEF plus the speed gained with the critical engine inoperative during the time interval between the instant at which the critical engine is failed, and the instant at which the pilot recognises and reacts to the engine failure, as indicated by the pilot’s initiation of the first action (e.g. applying brakes, reducing thrust, deploying speed brakes) to stop the aeroplane during accelerate-stop tests. (b) V2MIN, in terms of calibrated airspeed, may not be less than – (1) 1·13 VSR for – (i) Two-engined and three-engined turbo-propeller powered aeroplanes; and (ii) Turbojet powered aeroplanes without provisions for obtaining a significant reduction in the one-engineinoperative power-on stall speed; (2) 1·08 VSR for – (i) Turbo-propeller powered aeroplanes with more than three engines; and (ii) Turbojet powered aeroplanes with provisions for obtaining a significant reduction in the one-engine-inoperative power-on stall speed: and (3) 1·10 times VMC established under CS 25.149. (c) V2, in terms of calibrated airspeed, must be selected by the applicant to provide at least the gradient of climb required by CS 25.121(b) but may not be less than – (1) V2MIN; (2) VR plus the speed increment attained (in accordance with CS 25.111(c)(2)) before reaching a height of 11 m (35 ft) above the takeoff surface; and (3) A speed that provides the manoeuvring capability specified in CS 25.143(h). (d) VMU is the calibrated airspeed at and above which the aeroplane can safely lift off the ground, and continue the take-off. VMU speeds must be selected by the applicant throughout the range of thrust-to-weight ratios to be certificated. These speeds may be established from free air data if these data are verified by ground take-off tests. (See AMC 25.107(d).) (e) VR, in terms of calibrated air speed, must be selected in accordance with the conditions of sub-paragraphs (1) to (4) of this paragraph: (1) VR may not be less than – (i) V1; (ii) 105% of VMC; (iii) The speed (determined in accordance with CS 25.111(c)(2)) that allows reaching V2 before reaching a height of 11 m (35 ft) above the take-off surface; or (iv) A speed that, if the aeroplane is rotated at its maximum practicable rate, will result in a VLOF of not less than - (A) 110% of VMU in the allengines-operating condition, and 105% of VMU determined at the thrust-to-weight ratio corresponding to the one-engine-inoperative condition; or (B) If the VMU attitude is limited by the geometry of the aeroplane (i.e., tail contact with the runway), 108% of VMU in the all-engines-operating condition and 104% of VMU determined at the thrust-to-weight ratio corresponding to the one-engine-inoperative condition. (See AMC 25.107(e)(1)(iv).) (2) For any given set of conditions (such as weight, configuration, and temperature), a single value of VR, obtained in accordance with this paragraph, must be used to show compliance with both the one-engine-inoperative and the all-engines-operating take-off provisions. (3) It must be shown that the one-engine-inoperative take-off distance, using a rotation speed of 9.3 km/h (5 knots) less than VR established in accordance with sub-paragraphs (e)(1) and (2) of this paragraph, does not exceed the corresponding one-engine-inoperative take-off distance using the established VR. The take-off distances must be determined in accordance with CS 25.113(a)(1). (See AMC 25.107(e)(3).) (4) Reasonably expected variations in service from the established take-off procedures for the operation of the aeroplane (such as over-rotation of the aeroplane and out-of-trim conditions) may not result in unsafe flight characteristics or in marked increases in the scheduled take-off distances established in accordance with CS 25.113(a). (See AMC No. 1 to CS 25.107(e)(4) and AMC No. 2 to CS 25.107(e)(4).) (f) VLOF is the calibrated airspeed at which the aeroplane first becomes airborne. (g) VFTO, in terms of calibrated airspeed, must be selected by the applicant to provide at least the gradient of climb required by CS 25.121(c), but may not be less than – (1) 1.18 VSR; and (2) A speed that provides the manoeuvring capability specified in CS 25.143(h). (h) In determining the take-off speeds V1, VR, and V2 for flight in icing conditions, the values of VMCG, VMC, and VMU determined for non-icing conditions may be used. [Amdt 25/3] [Amdt 25/18] AMC 25.107(d) Take-off speeds ED Decision 2003/2/RM 1 If cases are encountered where it is not possible to obtain the actual VMU at forward centre of gravity with aeroplanes having limited elevator power (including those aeroplanes which have limited elevator power only over a portion of the take-off weight range), it will be permissible to test with a more aft centre of gravity and/or more than normal nose-up trim to obtain VMU. 1.1 When VMU is obtained in this manner, the values should be corrected to those which would have been attained at forward centre of gravity if sufficient elevator power had been available. The variation of VMU with centre of gravity may be assumed to be the same as the variation of stalling speed in free air with centre of gravity for this correction. 1.2 In such cases where VMU has been measured with a more aft centre of gravity and/or with more than normal nose-up trim, the VR selected should (in addition to complying with the requirements of CS 25.107(e)) be greater by an adequate margin than the lowest speed at which the nose wheel can be raised from the runway with centre of gravity at its most critical position and with the trim set to the normal take-off setting for the weight and centre of gravity. NOTE: A margin of 9,3 km/h (5 kt) between the lowest nose-wheel raising speed and VR would normally be considered to be adequate. 2 Take-offs made to demonstrate VMU should be continued until the aeroplane is out of ground effect. The aeroplane pitch attitude should not be decreased after lift-off. AMC 25.107(e)(1)(iv) Take-off speeds ED Decision 2003/2/RM VMU Testing for Geometry Limited Aeroplanes. 1 For aeroplanes that are geometry limited (i.e., the minimum possible VMU speeds are limited by tail contact with the runway), CS 25.107(e)(1)(iv)(B) allows the VMU to VLOF speed margins to be reduced to 108% and 104% for the all-engines-operating and one-engineinoperative conditions, respectively. The VMU demonstrated must be sound and repeatable. 2 One acceptable means for demonstrating compliance with CS 25.107(d) and 25.107(e)(1)(iv) with respect to the capability for a safe lift-off and fly-away from the geometry limited condition is to show that at the lowest thrust-to-weight ratio for the all-engines-operating condition: 2.1 During the speed range from 96 to 100% of the actual lift-off speed, the aft under-surface of the aeroplane should be in contact with the runway. Because of the dynamic nature of the test, it is recognised that contact will probably not be maintained during this entire speed range, and some judgement is necessary. It has been found acceptable for contact to exist approximately 50% of the time that the aeroplane is in this speed range. 2.2 Beyond the point of lift-off to a height of 11m (35 ft), the aeroplane’s pitch attitude should not decrease below that at the point of lift-off, nor should the speed increase more than 10%. 2.3 The horizontal distance from the start of the take-off to a height of 11 m (35 ft) should not be greater than 105% of the distance determined in accordance with CS 25.113(a)(2) without the 115% factor. AMC 25.107(e)(3) Take-off speeds ED Decision 2003/2/RM In showing compliance with CS 25.107(e)(3) – a. Rotation at a speed of VR-9,3 km/h (5 kt) should be carried out using, up to the point of lift-off, the same rotation technique, in terms of control input, as that used in establishing the one-engine-inoperative distance of CS 25.113(a)(1); b. The engine failure speed used in the VR-9,3 km/h (5 kt) demonstration should be the same as that used in the comparative take-off rotating at VR; c. The tests should be carried out both at the lowest practical weight (such that VR-9,3 km/h (5 kt) is not less than VMCG) and at a weight approaching take-off climb limiting conditions; d. The tail or tail skid should not contact the runway. AMC No. 1 to CS 25.107(e)(4) Take-off speeds ED Decision 2003/2/RM Reasonably expected variations in service from established take-off procedures should be evaluated in respect of out-of-trim conditions during certification flight test programmes. For example, normal take-off should be made with the longitudinal control trimmed to its most adverse position within the allowable take-off trim band. AMC No. 2 to CS 25.107(e)(4) Take-off speeds ED Decision 2003/2/RM 1 CS 25.107(e)(4) states that there must be no marked increase in the scheduled take-off distance when reasonably expected service variations, such as over-rotation, are encountered. This can be interpreted as requiring take-off tests with all engines operating with an abuse on rotation speed. 2 The expression ‘marked increase’ in the take-off distance is defined as any amount in excess of 1% of the scheduled take-off distance. Thus the abuse test should not result in a field length more than 101% of the scheduled field length. 3 For the early rotation abuse condition with all engines operating and at a weight as near as practicable to the maximum sea-level take-off weight, it should be shown by test that when the aeroplane is rotated rapidly at a speed which is 7% or 19 km/h (10 kt), whichever is lesser, below the scheduled VR speed, no ‘marked increase’ in the scheduled field length would result. CS 25.109 Accelerate-stop distance ED Decision 2016/010/R (See AMC 25.109) (a) (See AMC 25.109(a) and (b).) The accelerate-stop distance on a dry runway is the greater of the following distances: (1) The sum of the distances necessary to – (i) Accelerate the aeroplane from a standing start with all engines operating to VEF for take-off from a dry runway; (ii) Allow the aeroplane to accelerate from VEF to the highest speed reached during the rejected take-off, assuming the critical engine fails at VEF and the pilot takes the first action to reject the take-off at the V1 for take-off from a dry runway; and (iii) Come to a full stop on a dry runway from the speed reached as prescribed in sub-paragraph (a)(1)(ii) of this paragraph; plus (iv) A distance equivalent to 2 seconds at the V1 for take-off from a dry runway. (2) The sum of the distances necessary to – (i) Accelerate the aeroplane from a standing start with all engines operating to the highest speed reached during the rejected take-off, assuming the pilot takes the first action to reject the take-off at the V1 for take-off from a dry runway; and (ii) With all engines still operating, come to a full stop on a dry runway from the speed reached as prescribed in sub-paragraph (a)(2)(i) of this paragraph; plus (iii) A distance equivalent to 2 seconds at the V1 for take-off from a dry runway. (b) (See AMC 25.109(a) and (b).) The accelerate-stop distance on a wet runway is the greater of the following distances: (1) The accelerate-stop distance on a dry runway determined in accordance with sub-paragraph (a) of this paragraph; or (2) The accelerate-stop distance determined in accordance with sub-paragraph (a) of this paragraph, except that the runway is wet and the corresponding wet runway values of VEF and V1 are used. In determining the wet runway accelerate-stop distance, the stopping force from the wheel brakes may never exceed: (i) The wheel brakes stopping force determined in meeting the requirements of CS 25.101(i) and sub-paragraph (a) of this paragraph; and (ii) The force resulting from the wet runway braking coefficient of friction determined in accordance with subparagraphs (c) or (d) of this paragraph, as applicable, taking into account the distribution of the normal load between braked and unbraked wheels at the most adverse centre of gravity position approved for take-off. (c) The wet runway braking coefficient of friction for a smooth wet runway is defined as a curve of friction coefficient versus ground speed and must be computed as follows: (1) The maximum tyre-to-ground wet runway braking coefficient of friction is defined as (see Figure 1): \| \| \| --- \| \| Tyre Pressure (psi) \| Maximum Braking Coefficient (tyre-to-ground) \| \| 50 \| µt/gMAX = −0⋅0350 + 0⋅306 − 0⋅851 + 0⋅883 \| \| 100 \| µt/gMAX = −0⋅0437 + 0⋅320 − 0⋅805 + 0⋅804 \| \| 200 \| µt/gMAX = −0⋅0331 + 0⋅252 − 0⋅658 + 0⋅692 \| \| 300 \| µt/gMAX = −0⋅0401 + 0⋅263 − 0⋅611 + 0⋅614 \| Figure 1 where: Tyre Pressure = maximum aeroplane operating tyre pressure (psi) µt/gMAX = maximum tyre-to-ground braking coefficient V = aeroplane true ground speed (knots); and Linear interpolation may be used for tyre pressures other than those listed. (2) (See AMC 25.109(c)(2) The maximum tyre-to-ground wet runway braking coefficient of friction must be adjusted to take into account the efficiency of the anti-skid system on a wet runway. Anti-skid system operation must be demonstrated by flight testing on a smooth wet runway and its efficiency must be determined. Unless a specific anti-skid system efficiency is determined from a quantitative analysis of the flight testing on a smooth wet runway, the maximum tyre-to-ground wet runway braking coefficient of friction determined in sub-paragraph (c)(1) of this paragraph must be multiplied by the efficiency value associated with the type of anti-skid system installed on the aeroplane: \| \| \| --- \| \| Type of anti-skid system \| Efficiency value \| \| On-off \| 0⋅30 \| \| Quasi-modulating \| 0⋅50 \| \| Fully modulating \| 0⋅80 \| (d) At the option of the applicant, a higher wet runway braking coefficient of friction may be used for runway surfaces that have been grooved or treated with a porous friction course material. For grooved and porous friction course runways, (1) 70% of the dry runway braking coefficient of friction used to determine the dry runway accelerate-stop distance; or (2) (See AMC 25.109(d)(2).) The wet runway braking coefficient of friction defined in sub-paragraph (c) of this paragraph, except that a specific anti-skid efficiency, if determined, is appropriate for a grooved or porous friction course wet runway and the maximum tyre-to-ground wet runway braking coefficient of friction is defined as (see Figure 2): \| \| \| --- \| \| Tyre Pressure (psi) \| Maximum Braking Coefficient (tyre-to-ground) \| \| 50 \| µt/gMAX = 0⋅147 −1⋅05 + 2⋅673 − 2⋅683 + 0⋅403 + 0⋅859 \| \| 100 \| µt/gMAX = 0⋅1106 − 0⋅813 + 2⋅13 − 2⋅20 + 0⋅317 + 0⋅807 \| \| 200 \| µt/gMAX = 0⋅0498 − 0⋅398 +1⋅14 −1⋅285 + 0⋅140 + 0.701 \| \| 300 \| µt/gMAX = 0⋅0314 − 0⋅247 + 0⋅703 − 0⋅779 − 0⋅00954 + 0⋅614 \| Figure 2 where: Tyre Pressure = maximum aeroplane operating tyre pressure (psi) µt/gMAX = maximum tyre-to-ground braking coefficient V = aeroplane true ground speed (knots); and Linear interpolation may be used for tyre pressures other than those listed. (e) Except as provided in sub-paragraph (f)(1) of this paragraph, means other than wheel brakes may be used to determine the accelerate-stop distance if that means – (1) Is safe and reliable; (2) Is used so that consistent results can be expected under normal operating conditions; and (3) Is such that exceptional skill is not required to control the aeroplane. (f) The effects of available reverse thrust – (1) Must not be included as an additional means of deceleration when determining the accelerate-stop distance on a dry runway; and (2) May be included as an additional means of deceleration using recommended reverse thrust procedures when determining the accelerate-stop distance on a wet runway, provided the requirements of sub-paragraph (e) of this paragraph are met. (See AMC 25.109(f).) (g) The landing gear must remain extended throughout the accelerate-stop distance. (h) If the accelerate-stop distance includes a stopway with surface characteristics substantially different from those of the runway, the take-off data must include operational correction factors for the accelerate-stop distance. The correction factors must account for the particular surface characteristics of the stopway and the variations in these characteristics with seasonal weather conditions (such as temperature, rain, snow and ice) within the established operational limits. (i) A flight test demonstration of the maximum brake kinetic energy accelerate-stop distance must be conducted with not more than 10% of the allowable brake wear range remaining on each of the aeroplane wheel brakes. [Amdt 25/18] AMC 25.109(a) and (b) Accelerate-stop distance ED Decision 2003/2/RM Propeller pitch position. For the one-engine-inoperative accelerate-stop distance, the critical engine’s propeller should be in the position it would normally assume when an engine fails and the power levers are closed. For dry runway one-engine-inoperative accelerate-stop distances, the high drag ground idle position of the operating engines’ propellers (defined by a pitch setting that results in not less than zero total thrust, i.e. propeller plus jet thrust, at zero airspeed) may be used provided adequate directional control is available on a wet runway and the related operational procedures comply with CS 25.109(f) and (h). Wet runway controllability may either be demonstrated by using the guidance available in AMC 25.109(f) at the appropriate power level, or adequate control can be assumed to be available at ground idle power if reverse thrust credit is approved for determining the wet runway accelerate-stop distances. For the all-engines-operating accelerate-stop distances on a dry runway, the high drag ground idle propeller position may be used for all engines (subject to CS 25.109(f) and (h)). For criteria relating to reverse thrust credit for wet runway accelerate-stop distances, see AMC 25.109(f). AMC 25.109(c)(2) Accelerate-stop distance: anti-skid system efficiency ED Decision 2003/2/RM CS 25.109(c)(2) identifies 3 categories of anti-skid system and provides for either the use of a default efficiency value appropriate to the type of system or the determination of a specific efficiency value. Paragraph 1 of this AMC gives a description of the operating characteristics of each category to enable the classification of a particular system to be determined. Paragraph 2 gives an acceptable means of compliance with the requirement for flight testing and use of default efficiency values in accordance with CS 25.109(c)(2). These values are appropriate where the tuning of the anti-skid system is largely qualitative and without detailed quantitative analysis of system performance. Where detailed data recording and analysis is used to optimise system tuning, an efficiency value somewhat higher than the default value might be obtained and determined. Typically, a value of 40% might be achieved with an On/Off system. The quasi-modulating category covers a broad range of systems with varying performance levels. The best quasi-modulating systems might achieve an efficiency up to approximately 80%. Fully modulating systems have been tuned to efficiencies greater than 80% and up to a maximum of approximately 92%, which is considered to be the maximum efficiency on a wet runway normally achievable with fully modulating digital anti-skid systems. Paragraph 3 gives an acceptable means of compliance with CS 25.109(c)(2) where the applicant elects to determine a specific efficiency value. In Paragraph 4 of this AMC, guidance is given on the use of 2 alternative methods for calculating antiskid system efficiency from the recorded data. One method is based on the variation of brake torque throughout the stop, while the other is based on wheel speed slip ratio. Finally, Paragraph 5 gives guidance on accounting for the distribution of the normal load between braked and unbraked wheels. 1 Classification of anti-skid system types 1.1 For the purposes of determining the default anti-skid efficiency value under CS 25.109(c)(2), anti-skid systems have been grouped into three broad classifications; on/off, quasi-modulating and fully modulating. These classifications represent evolving levels of technology and performance capabilities on both dry and wet runways. 1.2 On/off systems are the simplest of the three types of anti-skid systems. For these systems, fully metered brake pressure (as commanded by the pilot) is applied until wheel locking is sensed. Brake pressure is then released to allow the wheel to spin back up. When the system senses that the wheel is accelerating back to synchronous speed (i.e. ground speed), full metered pressure is again applied. The cycle of full pressure application/complete pressure release is repeated throughout the stop (or until the wheel ceases to skid with brake pressure applied). 1.3 Quasi-modulating systems attempt to continuously regulate brake pressure as a function of wheel speed. Typically, brake pressure is released when the wheel deceleration rate exceeds a preselected value. Brake pressure is re-applied at a lower level after a length of time appropriate to the depth of skid. Brake pressure is then gradually increased until another incipient skid condition is sensed. In general, the corrective actions taken by these systems to exit the skid condition are based on a pre-programmed sequence rather than the wheel speed time history. 1.4 Fully modulating systems are a further refinement of the quasi-modulating systems. The major difference between these two types of anti-skid systems is in the implementation of the skid control logic. During a skid, corrective action is based on the sensed wheel speed signal, rather than a preprogrammed response. Specifically, the amount of pressure reduction or reapplication is based on the rate at which the wheel is going into or recovering from a skid. Also, higher fidelity transducers and upgraded control systems are used, which respond more quickly. 1.5 In addition to examining the control system differences noted above, a time history of the response characteristics of the anti-skid system during a wet runway stop should be used to help identify the type of anti-skid system. Comparing the response characteristics from wet and dry runway stops can also be helpful. Figure 1 shows an example of the response characteristics of a typical on-off system on both wet and dry runways. In general, the on-off system exhibits a cyclic behaviour of brake pressure application until a skid is sensed, followed by the complete release of brake pressure to allow the wheel to spin back up. Full metered pressure (as commanded by the pilot) is then re-applied, starting the cycle over again. The wheel speed trace exhibits deep and frequent skids (the troughs in the wheel speed trace), and the average wheel speed is significantly less than the synchronous speed (which is represented by the flat topped portions of the wheel speed trace). Note that the skids are deeper and more frequent on a wet runway than on a dry runway. For the particular example shown in Figure 1, the brake becomes torque-limited toward the end of the dry runway stop and is unable to generate enough torque to cause further skidding. FIGURE 1. ANTI-SKID SYSTEM RESPONSE CHARACTERISTICS On-Off System The effectiveness of quasi-modulating systems can vary significantly depending on the slipperiness of the runway and the design of the particular control system. On dry runways, these systems typically perform very well; however, on wet runways their performance is highly dependent on the design and tuning of the particular system. An example of the response characteristics of one such system is shown in Figure 2. On both dry and wet runways, brake pressure is released to the extent necessary to control skidding. As the wheel returns to the synchronous speed, brake pressure is quickly increased to a pre-determined level and then gradually ramped up to the full metered brake pressure. On a dry runway, this type of response reduces the depth and frequency of skidding compared to an on-off system. However, on a wet runway, skidding occurs at a pressure below that at which the gradual ramping of brake pressure occurs. As a result, on wet runways the particular system shown in Figure 2 operates very similarly to an on-off system. FIGURE 2. ANTI-SKID SYSTEM RESPONSE CHARACTERISTICS Quasi-Modulating System FIGURE 3. ANTI-SKID SYSTEM RESPONSE CHARACTERISTICS Fully Modulating System When properly tuned, fully modulating systems are characterised by much smaller variations in brake pressure around a fairly high average value. These systems can respond quickly to developing skids and are capable of modulating brake pressure to reduce the frequency and depth of skidding. As a result, the average wheel speed remains much closer to the synchronous wheel speed. Figure 3 illustrates an example of the response characteristics of a fully modulating system on dry and wet runways. 2 Demonstration of anti-skid system operation when using the anti-skid efficiency values specified in CS 25.109(c)(2) 2.1 If the applicant elects to use one of the anti-skid efficiency values specified in CS 25.109(c)(2), a limited amount of flight testing must still be conducted to verify that the anti-skid system operates in a manner consistent with the type of anti-skid system declared by the applicant. This testing should also demonstrate that the anti-skid system has been properly tuned for operation on wet runways. 2.2 A minimum of one complete stop, or equivalent segmented stops, should be conducted on a smooth (i.e. not grooved or porous friction course) wet runway at an appropriate speed and energy to cover the critical operating mode of the anti-skid system. Since the objective of the test is to observe the operation (i.e. cycling) of the anti-skid system, this test will normally be conducted at an energy well below the maximum brake energy condition. 2.3 The section of the runway used for braking should be well soaked (i.e. not just damp), but not flooded. The runway test section should be wet enough to result in a number of cycles of anti-skid activity, but should not cause hydroplaning. 2.4 Before taxy and with cold tyres, the tyre pressure should be set to the highest value appropriate to the take-off weight for which approval is being sought. 2.5 The tyres and brakes should not be new, but need not be in the fully worn condition. They should be in a condition considered representative of typical in-service operations. 2.6 Sufficient data should be obtained to determine whether the system operates in a manner consistent with the type of anti-skid system declared by the applicant, provide evidence that full brake pressure is being applied upstream of the anti-skid valve during the flight test demonstration, determine whether the anti-skid valve is performing as intended and show that the anti-skid system has been properly tuned for a wet runway. Typically, the following parameters should be plotted versus time: (i) The speed of a representative number of wheels. (ii) The hydraulic pressure at each brake (i.e. the hydraulic pressure downstream of the anti-skid valve, or the electrical input to each anti-skid valve). (iii) The hydraulic pressure at each brake metering valve (i.e. upstream of the anti-skid valve). 2.7 A qualitative assessment of the anti-skid system response and aeroplane controllability should be made by the test pilot(s). In particular, pilot observations should confirm that: (i) Anti-skid releases are neither excessively deep nor prolonged; (ii) The gear is free of unusual dynamics; and (iii) The aeroplane tracks essentially straight, even though runway seams, water puddles and wetter patches may not be uniformly distributed in location or extent. 3 Determination of a specific wet runway anti-skid system efficiency 3.1 If the applicant elects to derive the anti-skid system efficiency from flight test demonstrations, sufficient flight testing, with adequate instrumentation, must be conducted to ensure confidence in the value obtained. An anti-skid efficiency of 92% (i.e. a factor of 0·92) is considered to be the maximum efficiency on a wet runway normally achievable with fully modulating digital anti-skid systems. 3.2 A minimum of three complete stops, or equivalent segmented stops, should be conducted on a wet runway at appropriate speeds and energies to cover the critical operating modes of the anti-skid system. Since the objective of the test is to determine the efficiency of the anti-skid system, these tests will normally be conducted at energies well below the maximum brake energy condition. A sufficient range of speeds should be covered to investigate any variation of the anti-skid efficiency with speed. 3.3 The testing should be conducted on a smooth (i.e. not grooved or porous friction course) runway. 3.4 The section of the runway used for braking should be well soaked (i.e. not just damp), but not flooded. The runway test section should be wet enough to result in a number of cycles of anti-skid activity, but should not cause hydroplaning. 3.5 Before taxy and with cold tyres, the tyre pressure should be set to the highest value appropriate to the take-off weight for which approval is being sought. 3.6 The tyres and brake should not be new, but need not be in the fully worn condition. They should be in a condition considered representative of typical in-service operations. 3.7 A qualitative assessment of anti-skid system response and aeroplane controllability should be made by the test pilot(s). In particular, pilot observations should confirm that: (i) The landing gear is free of unusual dynamics; and (ii) The aeroplane tracks essentially straight, even though runway seams, water puddles and wetter patches may not be uniformly distributed in location or extent. 3.8 The wet runway anti-skid efficiency value should be determined as described in Paragraph 4 of this AMC. The test instrumentation and data collection should be consistent with the method used. 4 Calculation of anti-skid system efficiency 4.1 Paragraph 3 above provides guidance on the flight testing required to support the determination of a specific anti-skid system efficiency value. The following paragraphs describe 2 methods of calculating an efficiency value from the data recorded. These two methods, which yield equivalent results, are referred to as the torque method and the wheel slip method. Other methods may also be acceptable if they can be shown to give equivalent results. 4.2 Torque Method Under the torque method, the anti-skid system efficiency is determined by comparing the energy absorbed by the brake during an actual wet runway stop to the energy that is determined by integrating, over the stopping distance, a curve defined by connecting the peaks of the instantaneous brake force curve (see figure 4). The energy absorbed by the brake during the actual wet runway stop is determined by integrating the curve of instantaneous brake force over the stopping distance. FIGURE 4. INSTANTANEOUS BRAKE FORCE AND PEAK BRAKE FORCE Using data obtained from the wet runway stopping tests of paragraph 3, instantaneous brake force can be calculated from the following relationship: where: Fb = brake force Tb = brake torque α = wheel acceleration I = wheel moment of inertia; and Rtyre = tyre radius For brake installations where measuring brake torque directly is impractical, torque may be determined from other parameters (e.g. brake pressure) if a suitable correlation is available. Wheel acceleration is obtained from the first derivative of wheel speed. Instrumentation recording rates and data analysis techniques for wheel speed and torque data should be well matched to the anti-skid response characteristics to avoid introducing noise and other artifacts of the instrumentation system into the data. Since the derivative of wheel speed is used in calculating brake force, smoothing of the wheel speed data is usually necessary to give good results. The smoothing algorithm should be carefully designed as it can affect the resulting efficiency calculation. Filtering or smoothing of the brake torque or brake force data should not normally be done. If conditioning is applied, it should be done in a conservative manner (i.e. result in a lower efficiency value) and should not misrepresent actual aeroplane/system dynamics. Both the instantaneous brake force and the peak brake force should be integrated over the stopping distance. The anti-skid efficiency value for determining the wet runway accelerate-stop distance is the ratio of the instantaneous brake force integral to the peak brake force integral: where: η = anti-skid efficiency; and s = stopping distance The stopping distance is defined as the distance travelled during the specific wet runway stopping demonstration, beginning when the full braking configuration is obtained and ending at the lowest speed at which anti-skid cycling occurs (i.e. the brakes are not torque limited), except that this speed need not be less than 19 km/h (10 kt). Any variation in the anti-skid efficiency with speed should also be investigated, which can be accomplished by determining the efficiency over segments of the total stopping distance. If significant variations are noted, this variation should be reflected in the braking force used to determine the accelerate-stop distances (either by using a variable efficiency or by using a conservative single value). 4.3 Wheel Slip Method At brake application, the tyre begins to slip with respect to the runway surface, i.e. the wheel speed slows down with respect to the aeroplane’s ground speed. As the amount of tyre slip increases, the brake force also increases until an optimal slip is reached. If the amount of slip continues to increase past the optimal slip, the braking force will decrease. Using the wheel slip method, the anti-skid efficiency is determined by comparing the actual wheel slip measured during a wet runway stop to the optimal slip. Since the wheel slip varies significantly during the stop, sufficient wheel and ground speed data must be obtained to determine the variation of both the actual wheel slip and the optimal wheel slip over the length of the stop. A sampling rate of at least 16 samples per second for both wheel speed and ground speed has been found to yield acceptable fidelity. For each wheel and ground speed data point, the instantaneous anti-skid efficiency value should be determined from the relationship shown in Figure 5: FIGURE 5. ANTI-SKID EFFICIENCY – WHEEL SLIP RELATIONSHIP WSR = wheel slip ratio = OPS = optimal slip ratio; and ηi = instantaneous anti-skid efficiency To determine the overall anti-skid efficiency value for use in calculating the wet runway accelerate-stop distance, the instantaneous anti-skid efficiencies should be integrated with respect to distance and divided by the total stopping distance: where: η = anti-skid efficiency; and s = stopping distance The stopping distance is defined as the distance travelled during the specific wet runway stopping demonstration, beginning when the full braking configuration is obtained and ending at the lowest speed at which anti-skid cycling occurs (i.e. the brakes are not torque limited), except that this speed need not be less than 19 km/h (10 kt). Any variation in the anti-skid efficiency with speed should also be investigated, which can be accomplished by determining the efficiency over segments of the total stopping distance. If significant variations are noted, this variation should be reflected in the braking force used to determine the accelerate-stop distances (either by using a variable efficiency or by using a conservative single value). The applicant should provide substantiation of the optimal wheel slip value(s) used to determine the anti-skid efficiency value. An acceptable method for determining the optimal slip value(s) is to compare time history plots of the brake force and wheel slip data obtained during the wet runway stopping tests. For brake installations where measuring brake force directly is impractical, brake force may be determined from other parameters (e.g. brake pressure) if a suitable correlation is available. For those skids where wheel slip continues to increase after a reduction in the brake force, the optimal slip is the value corresponding to the brake force peak. See Figure 6 for an example and note how both the actual wheel slip and the optimal wheel slip can vary during the stop. FIGURE 6. SUBSTANTIATION OF THE OPTIMAL SLIP VALUE 4.4 For dispatch with an inoperative anti-skid system (if approved), the wet runway acceleratestop distances should be based on an efficiency no higher than that allowed by CS 25.109(c)(2) for an on-off type of anti-skid system. The safety of this type of operation should be demonstrated by flight tests conducted in accordance with Paragraph 2 of this AMC. 5 Distribution of normal load between braked and unbraked wheels In addition to taking into account the efficiency of the anti-skid system, CS 25.109(b)(2)(ii) also requires adjusting the braking force for the effect of the distribution of the normal load between braked and unbraked wheels at the most adverse centre of gravity position approved for take-off. The stopping force due to braking is equal to the braking coefficient multiplied by the normal load (i.e. weight) on each braked wheel. The portion of the aeroplane’s weight being supported by the unbraked wheels (e.g. unbraked nose wheels) does not contribute to the stopping force generated by the brakes. This effect must be taken into account for the most adverse centre of gravity position approved for take-off, considering any centre of gravity shifts that occur due to the dynamics of the stop. The most adverse centre of gravity position is the position that results in the least load on the braked wheels. AMC 25.109(d)(2) Accelerate-stop distance: anti-skid efficiency on grooved and porous friction course (PFC) runways. ED Decision 2003/2/RM Properly designed, constructed and maintained grooved and PFC runways can offer significant improvements in wet runway braking capability. A conservative level of performance credit is provided by 25.109(d) to reflect this performance improvement and to provide an incentive for installing and maintaining such surfaces. In accordance with CS 25.105(c) and 25.109(d), applicants may optionally determine the acceleratestop distance applicable to wet grooved and PFC runways. These data would be included in the AFM in addition to the smooth runway accelerate-stop distance data. The braking coefficient for determining the accelerate-stop distance on grooved and PFC runways is defined in CS 25.109(d) as either 70% of the braking coefficient used to determine the dry runway accelerate-stop distances, or a curve based on ESDU 71026 data and derived in a manner consistent with that used for smooth runways. In either case, the brake torque limitations determined on a dry runway may not be exceeded. Using a simple factor applied to the dry runway braking coefficient is acceptable for grooved and PFC runways because the braking coefficient’s variation with speed is much lower on these types of runways. On smooth wet runways, the braking coefficient varies significantly with speed, which makes it inappropriate to apply a simple factor to the dry runway braking coefficient. For applicants who choose to determine the grooved/PFC wet runway accelerate-stop distances in a manner consistent with that used for smooth runways, CS 25.109(d)(2) provides the maximum tyre-to-ground braking coefficient applicable to grooved and PFC runways. This maximum tyre-to-ground braking coefficient must be adjusted for the anti-skid system efficiency, either by using the value specified in CS 25.109(c)(2) appropriate to the type of anti-skid system installed, or by using a specific efficiency established by the applicant. As anti-skid system performance depends on the characteristics of the runway surface, a system that has been tuned for optimum performance on a smooth surface may not achieve the same level of efficiency on a grooved or porous friction course runway, and vice versa. Consequently, if the applicant elects to establish a specific efficiency for use with grooved or PFC surfaces, anti-skid efficiency testing should be conducted on a wet runway with such a surface, in addition to testing on a smooth runway. Means other than flight testing may be acceptable, such as using the efficiency previously determined for smooth wet runways, if that efficiency is shown to be representative of, or conservative for, grooved and PFC runways. The resulting braking force for grooved/PFC wet runways must be adjusted for the effect of the distribution of the normal load between braked and unbraked wheels. This adjustment will be similar to that used for determining the braking force for smooth runways, except that the braking dynamics should be appropriate to the braking force achieved on grooved and PFC wet runways. Due to the increased braking force on grooved and PFC wet runways, an increased download on the nose wheel and corresponding reduction in the download on the main gear is expected. AMC 25.109(f) Accelerate-stop distance: credit for reverse thrust. ED Decision 2003/2/RM In accordance with CS 25.109(f), reverse thrust may not be used to determine the accelerate-stop distances for a dry runway. For wet runway accelerate-stop distances, however, CS 25.109(f) allows credit for the stopping force provided by reverse thrust, if the requirements of CS 25.109(e) are met. In addition, the procedures associated with the use of reverse thrust, which CS 25.101(f) requires the applicant to provide, must meet the requirements of CS 25.101(h). The following criteria provide acceptable means of demonstrating compliance with these requirements: 1 Procedures for using reverse thrust during a rejected take-off must be developed and demonstrated. These procedures should include all of the pilot actions necessary to obtain the recommended level of reverse thrust, maintain directional control and safe engine operating characteristics, and return the reverser(s), as applicable, to either the idle or the stowed position. These procedures need not be the same as those recommended for use during a landing stop, but must not result in additional hazards, (e.g., cause a flame out or any adverse engine operating characteristics), nor may they significantly increase flightcrew workload or training needs. 2 It should be demonstrated that using reverse thrust during a rejected take-off complies with the engine operating characteristics requirements of CS 25.939(a). No adverse engine operating characteristics should be exhibited. The reverse thrust procedures may specify a speed at which the reverse thrust is to be reduced to idle in order to maintain safe engine operating characteristics. 3 The time sequence for the actions necessary to obtain the recommended level of reverse thrust should be demonstrated by flight test. The time sequence used to determine the accelerate-stop distances should reflect the most critical case relative to the time needed to deploy the thrust reversers. For example, on some aeroplanes the outboard thrust reversers are locked out if an outboard engine fails. This safety feature prevents the pilot from applying asymmetric reverse thrust on the outboard engines, but it may also delay the pilot’s selection of reverse thrust on the operable reversers. In addition, if the selection of reverse thrust is the fourth or subsequent pilot action to stop the aeroplane (e.g., after manual brake application, thrust/power reduction, and spoiler deployment), a one second delay should be added to the demonstrated time to select reverse thrust. (See figure 1 of AMC 25.101(h)(3).) 4 The response times of the affected aeroplane systems to pilot inputs should be taken into account. For example, delays in system operation, such as thrust reverser interlocks that prevent the pilot from applying reverse thrust until the reverser is deployed, should be taken into account. The effects of transient response characteristics, such as reverse thrust engine spin-up, should also be included. 5 To enable a pilot of average skill to consistently obtain the recommended level of reverse thrust under typical in-service conditions, a lever position that incorporates tactile feedback (e.g., a detent or stop) should be provided. If tactile feedback is not provided, a conservative level of reverse thrust should be assumed. 6 The applicant should demonstrate that exceptional skill is not required to maintain directional control on a wet runway with a 19 km/h (ten knot) crosswind from the most adverse direction. For demonstration purposes, a wet runway may be simulated by using a castering nosewheel on a dry runway. Symmetric braking should be used during the demonstration, and both all-engines-operating and critical-engine-inoperative reverse thrust should be considered. The brakes and thrust reversers may not be modulated to maintain directional control. The reverse thrust procedures may specify a speed at which the reverse thrust is reduced to idle in order to maintain directional controllability. 7 To meet the requirements of CS 25.101(h)(2) and 25.109(e)(1) the probability of failure to provide the recommended level of reverse thrust should be no greater than 1 per 1000 selections. The effects of any system or component malfunction or failure should not create an additional hazard. 8 The number of thrust reversers used to determine the wet runway accelerate-stop distance data provided in the AFM should reflect the number of engines assumed to be operating during the rejected take-off along with any applicable system design features. The all-engines-operating accelerate-stop distances should be based on all thrust reversers operating. The one-engine-inoperative accelerate-stop distances should be based on failure of the critical engine. For example, if the outboard thrust reversers are locked out when an outboard engine fails, the one-engine-inoperative accelerate stop distances can only include reverse thrust from the inboard engine thrust reversers. 9 For the engine failure case, it should be assumed that the thrust reverser does not deploy (i.e., no reverse thrust or drag credit for deployed thrust reverser buckets on the failed engine). 10 For approval of dispatch with one or more inoperative thrust reverser(s), the associated performance information should be provided either in the Aeroplane Flight Manual or the Master Minimum Equipment List. 11 The effective stopping force provided by reverse thrust in each, or at the option of the applicant, the most critical take-off configuration, should be demonstrated by flight test. Flight test demonstrations should be conducted to substantiate the accelerate-stop distances, and should include the combined use of all the approved means for stopping the aeroplane. These demonstrations may be conducted on a dry runway. 12 For turbo-propeller powered aeroplanes, the criteria of paragraphs 1 to 11 above remain generally applicable. Additionally, the propeller of the inoperative engine should be in the position it would normally assume when an engine fails and the power lever is closed. Reverse thrust may be selected on the remaining engine(s). Unless this is achieved by a single action to retard the power lever(s) from the take-off setting without encountering a stop or lockout, it must be regarded as an additional pilot action for the purposes of assessing delay times. If this is the fourth or subsequent pilot action to stop the aeroplane, a one second delay should be added to the demonstrated time to select reverse thrust. CS 25.111 Take-off path ED Decision 2015/008/R (See AMC 25.111) (a) The take-off path extends from a standing start to a point in the take-off at which the aeroplane is 457 m (1500 ft) above the take-off surface, or at which the transition from the take-off to the en-route configuration is completed and VFTO is reached, whichever point is higher. In addition – (1) The take-off path must be based on the procedures prescribed in CS 25.101(f); (2) The aeroplane must be accelerated on the ground to VEF, at which point the critical engine must be made inoperative and remain inoperative for the rest of the take-off; and (3) After reaching VEF, the aeroplane must be accelerated to V2. (b) During the acceleration to speed V2, the nose gear may be raised off the ground at a speed not less than VR. However, landing gear retraction may not be begun until the aeroplane is airborne. (See AMC 25.111(b).) (c) During the take-off path determination in accordance with sub-paragraphs (a) and (b) of this paragraph – (1) The slope of the airborne part of the take-off path must be positive at each point; (2) The aeroplane must reach V2 before it is 11 m (35 ft) above the take-off surface and must continue at a speed as close as practical to, but not less than V2 until it is 122 m (400 ft) above the take-off surface; (3) At each point along the take-off path, starting at the point at which the aeroplane reaches 122 m (400 ft) above the take-off surface, the available gradient of climb may not be less than – (i) 1·2% for two-engined aeroplanes; (ii) 1·5% for three-engined aeroplanes; and (iii) 1·7% for four-engined aeroplanes, (4) The aeroplane configuration may not be changed, except for gear retraction and automatic propeller feathering, and no change in power or thrust that requires action by the pilot may be made, until the aeroplane is 122 m (400 ft) above the take-off surface, and (5) If CS 25.105(a)(2) requires the take-off path to be determined for flight in icing conditions, the airborne part of the take-off must be based on the aeroplane drag: (i) With the most critical of the “Take-off Ice” accretion(s) defined in Appendices C and O, as applicable, in accordance with CS 25.21(g), from a height of 11 m (35 ft) above the take-off surface up to the point where the aeroplane is 122 m (400 ft) above the take-off surface; and (ii) With the most critical of the “Final Take-off Ice” accretion(s) defined in Appendices C and O, as applicable, in accordance with CS 25.21(g), from the point where the aeroplane is 122 m (400 ft) above the take-off surface to the end of the take-off path. (d) The take-off path must be determined by a continuous demonstrated take-off or by synthesis from segments. If the take-off path is determined by the segmental method – (1) The segments must be clearly defined and must relate to the distinct changes in the configuration, power or thrust, and speed; (2) The weight of the aeroplane, the configuration, and the power or thrust must be constant throughout each segment and must correspond to the most critical condition prevailing in the segment; (3) The flight path must be based on the aeroplane’s performance without ground effect; and (4) The take-off path data must be checked by continuous demonstrated take-offs up to the point at which the aeroplane is out of ground effect and its speed is stabilised, to ensure that the path is conservative to the continuous path. The aeroplane is considered to be out of the ground effect when it reaches a height equal to its wing span. (e) Not required for CS-25. [Amdt 25/3] [Amdt 25/16] AMC 25.111 Take-off path ED Decision 2003/2/RM The height references in CS 25.111 should be interpreted as geometrical heights. AMC 25.111(b) Take-off path ED Decision 2003/2/RM 1 Rotation speed, VR, is intended to be the speed at which the pilot initiates action to raise the nose gear off the ground, during the acceleration to V2; consequently, the take-off path determination, in accordance with CS 25.111(a) and (b), should assume that pilot action to raise the nose gear off the ground will not be initiated until the speed VR has been reached. 2 The time between lift-off and the initiation of gear retraction during take-off distance demonstrations should not be less than that necessary to establish an indicated positive rate of climb plus one second. For the purposes of flight manual expansion, the average demonstrated time delay between lift-off and initiation of gear retraction may be assumed; however, this value should not be less than 3 seconds. CS 25.113 Take-off distance and take-off run ED Decision 2016/010/R (See AMC 25.113) (a) Take-off distance on a dry runway is the greater of – (1) The horizontal distance along the take-off path from the start of the take-off to the point at which the aeroplane is 11 m (35 ft) above the take-off surface, determined under CS 25.111 for a dry runway; or (2) 115% of the horizontal distance along the take-off path, with all engines operating, from the start of the take-off to the point at which the aeroplane is 11 m (35 ft) above the take-off surface, as determined by a procedure consistent with CS 25.111. (See AMC 25.113(a)(2), (b)(2) and (c)(2).) (b) Take-off distance on a wet runway is the greater of – (1) The take-off distance on a dry runway determined in accordance with sub-paragraph (a) of this paragraph; or (2) The horizontal distance along the take-off path from the start of the take-off to the point at which the aeroplane is 4,6 m (15 ft) above the take-off surface, achieved in a manner consistent with the achievement of V2 before reaching 11 m (35 ft) above the take-off surface, determined under CS 25.111 for a wet runway. (See AMC 25.113(a)(2), (b)(2) and (c)(2).) (c) If the take-off distance does not include a clearway, the take-off run is equal to the take-off distance. If the take-off distance includes a clearway – (1) The take-off run on a dry runway is the greater of – (i) The horizontal distance along the take-off path from the start of the takeoff to a point equidistant between the point at which VLOF is reached and the point at which the aeroplane is 11 m (35 ft) above the take-off surface, as determined under CS 25.111 for a dry runway; or (ii) 115% of the horizontal distance along the take-off path, with all engines operating, from the start of the take-off to a point equidistant between the point at which VLOF is reached and the point at which the aeroplane is 11 m (35 ft) above the take-off surface, determined by a procedure consistent with CS 25.111. (See AMC 25.113(a)(2), (b)(2) and (c)(2).) (2) The take-off run on a wet runway is the greater of – (i) The horizontal distance along the take-off path from the start of the takeoff to the point at which the aeroplane is 4,6 m (15 ft) above the take-off surface, achieved in a manner consistent with the achievement of V2 before reaching 11 m (35 ft) above the take-off surface, determined under CS 25.111 for a wet runway; or (ii) 115% of the horizontal distance along the take-off path, with all engines operating, from the start of the take-off to a point equidistant between the point at which VLOF is reached and the point at which the aeroplane is 11 m (35 ft) above the take-off surface, determined by a procedure consistent with CS 25.111. (See AMC 25.113(a)(2),(b)(2) and (c)(2)) [Amdt 25/9] [Amdt 25/18] AMC 25.113(a)(2), (b)(2) and (c)(2) Take-off distance and take-off run ED Decision 2003/2/RM In establishment of the take-off distance and take-off run, with all engines operating, in accordance with CS 25.113(a), (b) and (c), the flight technique should be such that – a. A speed of not less than V2 is achieved before reaching a height of 11 m (35 ft) above the take-off surface, b. It is consistent with the achievement of a smooth transition to a steady initial climb speed of not less than V2 + 19 km/h (10 kt) at a height of 122 m (400 ft) above the take-off surface. CS 25.115 Take-off flight path ED Decision 2003/2/RM (a) The take-off flight path must be considered to begin 11 m (35 ft) above the take-off surface at the end of the take-off distance determined in accordance with CS 25.113(a) or (b) as appropriate for the runway surface condition. (b) The net take-off flight path data must be determined so that they represent the actual take-off flight paths (determined in accordance with CS 25.111 and with sub-paragraph (a) of this paragraph) reduced at each point by a gradient of climb equal to – (1) 0·8% for two-engined aeroplanes; (2) 0·9% for three-engined aeroplanes; and (3) 1·0% for four-engined aeroplanes. (c) The prescribed reduction in climb gradient may be applied as an equivalent reduction in acceleration along that part of the take-off flight path at which the aeroplane is accelerated in level flight. CS 25.117 Climb: general ED Decision 2003/2/RM Compliance with the requirements of CS 25.119 and 25.121 must be shown at each weight, altitude, and ambient temperature within the operational limits established for the aeroplane and with the most unfavourable centre of gravity for each configuration. CS 25.119 Landing climb: all engines operating ED Decision 2016/010/R (See AMC 25.119) In the landing configuration, the steady gradient of climb may not be less than 3·2%, with the engines at the power or thrust that is available 8 seconds after initiation of movement of the power or thrust controls from the minimum flight idle to the go-around power or thrust setting; and (a) In non-icing conditions, with a climb speed of VREF determined in accordance with CS 25.125(b)(2)(i); and (b) In icing conditions with the most critical of the “Landing Ice” accretion(s) defined in Appendices C and O, as applicable, in accordance with CS 25.21(g), and with a climb speed of VREF determined in accordance with CS 25.125(b)(2)(i). [Amdt 25/3] [Amdt 25/16] [Amdt 25/18] AMC 25.119 Landing climb: all-engines-operating ED Decision 2020/024/R In establishing the thrust specified in CS 25.119, either – a. Engine acceleration tests should be conducted using the most critical combination of the following parameters: i. Altitude; ii. Airspeed; iii. Engine bleed; iv. Engine power off-take; likely to be encountered during an approach to a landing airfield within the altitude range for which landing certification is sought; or b. The thrust specified in CS 25.119 should be established as a function of these parameters. For aeroplanes equipped with a reduced go-around (RGA) thrust or power function, the climb requirements specified in CS 25.119 are applicable with the RGA function active. During the determination of the maximum thrust or power specified in AMC 25.119 a. and b. the thrust or power controls should be moved to the RGA thrust or power setting. This is consistent with an AFM all‑engines-operating go-around procedure which recommends the use of an RGA function (see AMC 25.143(b)(4)). In exceptional circumstances such as in the presence of wind shear or of unplanned obstacles, the flight crew may elect to use go-around thrust or power that exceeds the RGA setting. However, the applicant is not required to provide AFM climb gradient performance for this situation. If an AFM go-around procedure is approved to use thrust or power above the RGA setting, then the climb requirements of CS 25.119 will apply at the higher thrust or power setting. [Amdt 25/3] [Amdt 25/26] CS 25.121 Climb: one-engine-inoperative ED Decision 2015/008/R (See AMC 25.121) (a) Take-off; landing gear extended. (See AMC 25.121(a).) In the critical take-off configuration existing along the flight path (between the points at which the aeroplane reaches VLOF and at which the landing gear is fully retracted) and in the configuration used in CS 25.111 but without ground effect, the steady gradient of climb must be positive for two-engined aeroplanes, and not less than 0·3% for three-engined aeroplanes or 0·5% for fourengined aeroplanes, at VLOF and with – (1) The critical engine inoperative and the remaining engines at the power or thrust available when retraction of the landing gear is begun in accordance with CS 25.111 unless there is a more critical power operating condition existing later along the flight path but before the point at which the landing gear is fully retracted (see AMC 25.121(a)(1)); and (2) The weight equal to the weight existing when retraction of the landing gear is begun determined under CS 25.111. (b) Take-off; landing gear retracted. In the take-off configuration existing at the point of the flight path at which the landing gear is fully retracted, and in the configuration used in CS 25.111 but without ground effect, (1) the steady gradient of climb may not be less than 2·4% for two-engined aeroplanes, 2·7% for three-engined aeroplanes and 3·0% for four-engined aeroplanes, at V2 with – (i) The critical engine inoperative, the remaining engines at the take-off power or thrust available at the time the landing gear is fully retracted, determined under CS 25.111, unless there is a more critical power operating condition existing later along the flight path but before the point where the aeroplane reaches a height of 122 m (400 ft) above the take-off surface (see AMC 25.121(b)(1)(i)) ; and (ii) The weight equal to the weight existing when the aeroplane’s landing gear is fully retracted, determined under CS 25.111. (2) The requirements of sub-paragraph (b)(1) of this paragraph must be met: (i) In non-icing conditions; and (ii) In icing conditions with the most critical of the “Take-off Ice” accretion(s) defined in Appendices C and O, as applicable, in accordance with CS 25.21(g), if in the configuration used to show compliance with CS 25.121(b) with this “Take-off Ice” accretion: (A) The stall speed at maximum take-off weight exceeds that in non-icing conditions by more than the greater of 5.6 km/h (3 knots) CAS or 3% of VSR; or (B) The degradation of the gradient of climb determined in accordance with CS 25.121(b) is greater than one-half of the applicable actual-to-net take-off flight path gradient reduction defined in CS 25.115(b). (c) Final take-off. In the en-route configuration at the end of the take-off path determined in accordance with CS 25.111: (1) the steady gradient of climb may not be less than 1·2% for two-engined aeroplanes, 1·5% for three-engined aeroplanes, and 1·7% for four-engined aeroplanes, at VFTO and with – (i) The critical engine inoperative and the remaining engines at the available maximum continuous power or thrust; and (ii) The weight equal to the weight existing at the end of the take-off path, determined under CS 25.111. (2) The requirements of sub-paragraph (c)(1) of this paragraph must be met: (i) In non-icing conditions; and (ii) In icing conditions with themost critical of the “Final Take-off Ice” accretion(s) defined in Appendices C and O, as applicable, in accordance with CS 25.21(g), if in the configuration used to show compliance with CS 25.121(b) with this “Take-off Ice” accretion: (A) The stall speed at maximum take-off weight exceeds that in non-icing conditions by more than the greater of 5.6 km/h (3 knots) CAS or 3% of VSR; or (B) The degradation of the gradient of climb determined in accordance with CS 25.121(b) is greater than one-half of the applicable actual-to-net take-off flight path gradient reduction defined in CS 25.115(b). (d) Approach. In a configuration corresponding to the normal all-engines-operating procedure in which VSR for this configuration does not exceed 110% of the VSR for the related all-engines-operating landing configuration: (1) steady gradient of climb may not be less than 2·1% for two-engined aeroplanes, 2·4% for three-engined aeroplanes and 2·7% for four-engined aeroplanes, with – (i) The critical engine inoperative, the remaining engines at the go-around power or thrust setting; (ii) The maximum landing weight; (iii) A climb speed established in connection with normal landing procedures, but not more than 1·4 VSR; and (iv) Landing gear retracted. (2) The requirements of sub-paragraph (d)(1) of this paragraph must be met: (i) In non-icing conditions; and (ii) In icing conditions with the most critical of the Approach Ice accretion(s) defined in Appendices C and O, as applicable, in accordance with CS 25.21(g). The climb speed selected for non-icing conditions may be used if the climb speed for icing conditions, computed in accordance with sub-paragraph (d)(1)(iii) of this paragraph, does not exceed that for non-icing conditions by more than the greater of 5.6 km/h (3 knots) CAS or 3%. [Amdt 25/3] [Amdt 25/16] AMC 25.121 Climb: One-engine-inoperative ED Decision 2003/2/RM 1 In showing compliance with CS 25.121 it is accepted that bank angles of up to 2° to 3° toward the operating engine(s) may be used. 2 The height references in CS 25.121 should be interpreted as geometrical heights. AMC 25.121(a) Climb: One-engine-inoperative ED Decision 2003/2/RM The configuration of the landing gear used in showing compliance with the climb requirements of CS 25.121(a) may be that finally achieved following ‘gear down’ selection. AMC 25.121(a)(1) Climb: One-engine-inoperative ED Decision 2003/2/RM A ‘power operating condition’ more critical than that existing at the time when retraction of the landing gear is begun would occur, for example, if water injection were discontinued prior to reaching the point at which the landing gear is fully retracted. AMC 25.121(b)(1)(i) Climb: One-engine-inoperative ED Decision 2007/010/R A ‘power operating condition’ more critical than that existing at the time the landing gear is fully retracted would occur, for example, if water injection were discontinued prior to reaching a gross height of 122 m (400 ft). [Amdt 25/3] CS 25.123 En-route flight paths ED Decision 2015/008/R (See AMC 25.123) (a) For the en-route configuration, the flight paths prescribed in sub-paragraphs (b) and (c) of this paragraph must be determined at each weight, altitude, and ambient temperature, within the operating limits established for the aeroplane. The variation of weight along the flight path, accounting for the progressive consumption of fuel and oil by the operating engines, may be included in the computation. The flight paths must be determined at a selected speed not less than VFTO, with – (1) The most unfavourable centre of gravity; (2) The critical engines inoperative; (3) The remaining engines at the available maximum continuous power or thrust; and (4) The means for controlling the engine-cooling air supply in the position that provides adequate cooling in the hot-day condition. (b) The one-engine-inoperative net flight path data must represent the actual climb performance diminished by a gradient of climb of 1·1% for two-engined aeroplanes, 1·4% for three-engined aeroplanes, and 1·6% for four-engined aeroplanes. (1) In non-icing conditions; and (2) In icing conditions with the “En-route Ice” accretion defined in Appendices C and O, as applicable, in accordance with CS 25.21(g), if: (i) A speed of 1.18VSR with the most critical of the “En-route Ice ” accretion(s) exceeds the en-route speed selected in non-icing conditions by more than the greater of 5.6 km/h (3 knots) CAS or 3% of VSR, or (ii) The degradation of the gradient of climb is greater than one-half of the applicable actual-to-net flight path reduction defined in sub-paragraph (b) of this paragraph. (c) For three- or four-engined aeroplanes, the two-engine-inoperative net flight path data must represent the actual climb performance diminished by a gradient climb of 0·3% for three-engined aeroplanes and 0·5% for four-engined aeroplanes. [Amdt 25/3] [Amdt 25/16] AMC 25.123 En-route flight paths ED Decision 2003/2/RM If, in showing compliance with CS 25.123, any credit is to be taken for the progressive use of fuel by the operating engines, the fuel flow rate should be assumed to be 80% of the engine specification flow rate at maximum continuous power, unless a more appropriate figure has been substantiated by flight tests. CS 25.125 Landing ED Decision 2016/010/R (a) The horizontal distance necessary to land and to come to a complete stop from a point 15 m (50 ft) above the landing surface must be determined (for standard temperatures, at each weight, altitude and wind within the operational limits established by the applicant for the aeroplane): (1) In non-icing conditions; and (2) In icing conditions with the most critical of the “Landing Ice” accretion(s) defined in Appendices C and O, as applicable, in accordance with CS 25.21(g), if VREF for icing conditions exceeds VREF for non-icing conditions by more than 9.3 km/h (5 knots) CAS at the maximum landing weight (b) In determining the distance in (a): (1) The aeroplane must be in the landing configuration. (2) A stabilised approach, with a calibrated airspeed of not less than VREF, must be maintained down to the 15 m (50 ft) height. (i) In non-icing conditions, VREF may not be less than: (A) 1.23 VSR0; (B) VMCL established under CS25.149(f); and (C) A speed that provides the manoeuvring capability specified in CS 25.143(h). (ii) In icing conditions, VREF may not be less than: (A) The speed determined in sub-paragraph (b)(2)(i) of this paragraph; (B) 1.23 VSR0 with the most critical of the "Landing Ice" accretion(s) defined in Appendices C and O, as applicable, in accordance with CS 25.21(g), if that speed exceeds VREF for non-icing conditions by more than 9.3 km/h (5 knots) CAS; and (C) A speed that provides the manoeuvring capability specified in CS 25.143(h) with the most critical of the “ Landing ice accretion(s) defined in appendices C and O, as applicable, in accordance with CS 25.21(g). (3) Changes in configuration, power or thrust, and speed, must be made in accordance with the established procedures for service operation. (See AMC 25.125(b)(3).) (4) The landing must be made without excessive vertical acceleration, tendency to bounce, nose over or ground loop. (5) The landings may not require exceptional piloting skill or alertness. (c) The landing distance must be determined on a level, smooth, dry, hard-surfaced runway. (See AMC 25.125(c).) In addition – (1) The pressures on the wheel braking systems may not exceed those specified by the brake manufacturer; (2) The brakes may not be used so as to cause excessive wear of brakes or tyres (see AMC 25.125(c)(2)); and (3) Means other than wheel brakes may be used if that means – (i) Is safe and reliable; (ii) Is used so that consistent results can be expected in service; and (iii) Is such that exceptional skill is not required to control the aeroplane. (d) Reserved. (e) Reserved. (f) The landing distance data must include correction factors for not more than 50% of the nominal wind components along the landing path opposite to the direction of landing, and not less than 150% of the nominal wind components along the landing path in the direction of landing. (g) If any device is used that depends on the operation of any engine, and if the landing distance would be noticeably increased when a landing is made with that engine inoperative, the landing distance must be determined with that engine inoperative unless the use of compensating means will result in a landing distance not more than that with each engine operating. [Amdt 25/3] [Amdt 25/16] [Amdt 25/18] AMC 25.125(b)(3) Change of Configuration ED Decision 2007/010/R No changes in configuration, addition of thrust, or nose depression should be made after reaching 15 m (50 ft) height. [Amdt 25/3] AMC 25.125(c) Landing ED Decision 2007/010/R 1 During measured landings, if the brakes can be consistently applied in a manner permitting the nose gear to touch down safely, the brakes may be applied with only the main wheels firmly on the ground. Otherwise, the brakes should not be applied until all wheels are firmly on the ground. 2 This is not intended to prevent operation in the normal way of automatic braking systems which, for instance, permit brakes to be selected on before touchdown. [Amdt 25/3] AMC 25.125(c)(2) Landing ED Decision 2007/010/R To ensure compliance with CS 25.125(c)(2), a series of six measured landings should be conducted on the same set of wheel brakes and tyres. [Amdt 25/3] PreviousNext
12311	https://www.unitsconverters.com/en/Kv/M-To-V/M/Utu-4235-4232	kV/m to V/m \| Kilovolt per Meter to V/m Units Converters AccelerationAngleAreaEnergyForceLengthPowerPressureSpeedTemperatureTimeVolumeWeight Percentage error Subtract fraction LCM of three numbers Discover more Unit Converter kV/m to V/m (Kilovolt per Meter to Volt per Meter) -10%Copy+10%-10%Copy+10% = ⇄ kV/m to μV/m \| kV/m to mV/m \| kV/m to N/C \| kV/m to V/m \| kV/m to V/mm \| kV/m to V/μm More 👎) 👍 Converting 2X of 1 ▶1/2X of 1 ▶ 5X of 1 ▶1/5X of 1 ▶ 8X of 1 ▶1/8X of 1 ▶ Abvolt per Centimeter Smallest Volt per Meter Base Volt Per Micrometer Biggest Result 1 kV/m is equivalent to 1000 V/m Discover more Unit Converter Home » Electric Field Strength » kV/m to V/m Formula Used 1 Volt per Meter = 0.001 Kilovolt per Meter ∴ 1 Kilovolt per Meter = 1000 Volt per Meter Other kV/m Conversions kV/m to V/μm⇄ [Kilovolt per Meter to Volt Per Micrometer⇄] (Biggest) kV/m to kV/cm⇄ [Kilovolt per Meter to Kilovolt per Centimeter⇄] kV/m to V/mil⇄ [Kilovolt per Meter to Volt per Mil⇄] kV/m to kV/in⇄ [Kilovolt per Meter to Kilovolt per Inch⇄] kV/m to statV/cm⇄ [Kilovolt per Meter to Statvolt per Centimeter⇄] kV/m to statV/in⇄ [Kilovolt per Meter to Statvolt per Inch⇄] kV/m to V/cm⇄ [Kilovolt per Meter to Volt per Centimeter⇄] kV/m to V/in⇄ [Kilovolt per Meter to Volt per Inch⇄] kV/m to kV/μm⇄ [Kilovolt per Meter to Kilovolt per Micrometer⇄] kV/m to kV/mm⇄ [Kilovolt per Meter to Kilovolt per Millimeter⇄] kV/m to kV/nm⇄ [Kilovolt per Meter to Kilovolt per Nanometer⇄] kV/m to MV/cm⇄ [Kilovolt per Meter to Megavolt per Centimeter⇄] kV/m to MV/in⇄ [Kilovolt per Meter to Megavolt per Inch⇄] kV/m to MV/m⇄ [Kilovolt per Meter to Megavolt per Meter⇄] kV/m to MV/μm⇄ [Kilovolt per Meter to Megavolt per Micrometer⇄] kV/m to MV/mm⇄ [Kilovolt per Meter to Megavolt per Millimeter⇄] kV/m to MV/nm⇄ [Kilovolt per Meter to Megavolt per Nanometer⇄] kV/m to µV/cm⇄ [Kilovolt per Meter to Microvolt per Centimeter⇄] kV/m to µV/in⇄ [Kilovolt per Meter to Microvolt per Inch⇄] kV/m to µV/μm⇄ [Kilovolt per Meter to Microvolt per Micrometer⇄] kV/m to µV/mm⇄ [Kilovolt per Meter to Microvolt per Millimeter⇄] kV/m to µV/nm⇄ [Kilovolt per Meter to Microvolt per Nanometer⇄] kV/m to mV/cm⇄ [Kilovolt per Meter to Millivolt per Centimeter⇄] kV/m to mV/in⇄ [Kilovolt per Meter to Millivolt per Inch⇄] kV/m to mV/μm⇄ [Kilovolt per Meter to Millivolt per Micrometer⇄] kV/m to mV/mm⇄ [Kilovolt per Meter to Millivolt per Millimeter⇄] kV/m to mV/nm⇄ [Kilovolt per Meter to Millivolt per Nanometer⇄] kV/m to N/C⇄ [Kilovolt per Meter to Newton per Coulomb⇄] kV/m to V/m⇄ [Kilovolt per Meter to Volt per Meter⇄] (You are Here) (Base Unit) kV/m to V/mm⇄ [Kilovolt per Meter to Volt per Millimeter⇄] kV/m to V/nm⇄ [Kilovolt per Meter to Volt per Nanometer⇄] kV/m to mV/m⇄ [Kilovolt per Meter to Millivolt per Meter⇄] kV/m to μV/m⇄ [Kilovolt per Meter to Microvolt per Meter⇄] kV/m to abV/cm⇄ [Kilovolt per Meter to Abvolt per Centimeter⇄] (Smallest) Discover more Unit Converter kV/m to V/m Conversion The abbreviation for kV/m and V/m is kilovolt per meter and volt per meter respectively. 1 kV/m is 1000 times bigger than a V/m. To measure, units of measurement are needed and converting such units is an important task as well. unitsconverters.com is an online conversion tool to convert all types of measurement units including kV/m to V/m conversion. Kilovolt per Meter to V/m Check our Kilovolt per Meter to V/m converter and click on formula to get the conversion factor. When you are converting electric field strength from Kilovolt per Meter to V/m, you need a converter that is elaborate and still easy to use. All you have to do is select the unit for which you want the conversion and enter the value and finally hit Convert. kV/m to Volt per Meter The formula used to convert kV/m to Volt per Meter is 1 Kilovolt per Meter = 1000 Volt per Meter. Measurement is one of the most fundamental concepts. Note that we have Volt Per Micrometer as the biggest unit for length while Abvolt per Centimeter is the smallest one. Convert kV/m to V/m How to convert kV/m to V/m? Now you can do kV/m to V/m conversion with the help of this tool. In the length measurement, first choose kV/m from the left dropdown and V/m from the right dropdown, enter the value you want to convert and click on 'convert'. Want a reverse calculation from V/m to kV/m? You can check our V/m to kV/m converter. kV/m to V/m Converter Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like electric field strength finds its use in a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions. unitsconverters.com helps in the conversion of different units of measurement like kV/m to V/m through multiplicative conversion factors. When you are converting electric field strength, you need a Kilovolt per Meter to Volt per Meter converter that is elaborate and still easy to use. Converting kV/m to Volt per Meter is easy, for you only have to select the units first and the value you want to convert. If you encounter any issues to convert Kilovolt per Meter to V/m, this tool is the answer that gives you the exact conversion of units. You can also get the formula used in kV/m to V/m conversion along with a table representing the entire conversion. Discover more Unit Converter HindiFrenchSpanishMarathiPortugueseGermanPolishDutchItalianRussianGujaratiPunjabiTurkishKorean Percentage calculatorFraction calculatorLCM HCF Calculator About Contact Disclaimer Terms of Use Privacy Policy © 2016-2025. A softUsvista venture! Let Others Know ✖ Facebook Twitter Reddit LinkedIn Email WhatsApp Copied!
12312	https://www.ck12.org/flexi/cbse-math/area-of-regular-and-irregular-polygons/how-can-the-area-of-a-polygon-be-calculated/	Flexi answers - How can the area of a polygon be calculated? \| CK-12 Foundation Subjects Explore Donate Sign InSign Up All Subjects CBSE Math Area of Regular and Irregular Polygons Question How can the area of a polygon be calculated? Flexi Says: The formula for finding the area of a regular polygon is as follows: Area=1 2⋅Perimeter⋅apothem The distance from the center of a regular polygon to the side at a right angle represents the height of the triangles. This distance is called the apothem. To learn more about the area of polygons, click here! Analogy / Example Try Asking: How do you solve the area of regular polygons?A square piece of construction paper has sides that are 6 inches long. What is the piece of paper's area? (Answer in square inches.)Find the area of a regular nonagon, with a side length of 6 ft. and the apothem equal to 4 ft. How can Flexi help? By messaging Flexi, you agree to our Terms and Privacy Policy
12313	https://www.quora.com/What-is-the-definition-of-the-locus-of-a-point-that-moves-with-a-specific-distance-ratio-between-two-fixed-points	Something went wrong. Wait a moment and try again. Ratio and Proportion Mathematical Terms and De... Analytic Geometry Math Definitions 5 What is the definition of the locus of a point that moves with a specific distance ratio between two fixed points? Allen Ries Math Major University of Alberta · Author has 25.1K answers and 9.7M answer views · May 2 What is the definition of the locus of a point that moves with a specific distance ratio between two fixed points? It is the locus of an infinite number of points. It is not a ratio of distances . It is the total (sum) of the distances from the fixed points to the ‘locus of points’. Refer to diagram . P; P’ and P’ are 3 examples from the locus of points that meet the criteria of an ellipse. What is the definition of the locus of a point that moves with a specific distance ratio between two fixed points? It is the locus of an infinite number of points. It is not a ratio of distances . It is the total (sum) of the distances from the fixed points to the ‘locus of points’. Refer to diagram . P; P’ and P’ are 3 examples from the locus of points that meet the criteria of an ellipse. Related questions How do I find the locus of a point which moves so that its distance from the point (2,1) is double its distance (1,2)? What is the locus of a point whose distance from two fixed points is equal to their sum or difference? What is the locus of points (M), the sum of whose distance from two given points is a constant? A point 𝑃 moves such that its distance from point 𝐴 (-3,4) is always twice its distance from point 𝐵 (6,-2). How to find the equation of locus and how to determine whether the locus intersect the y-axis? A point moves so that its distance from the point (-2,3) is always three times its distance from the point (0,3). What is the equation of its locus? Pramodkumar Tandon Retired as Prof. & Head at Institute of Engineering and Rural Technology (1965–present) · Author has 2.1K answers and 1.2M answer views · Sep 7 Related What is the equation of the locus of the point that moves such that its distance from point A(3,1) is twice the distance from the point B(5,1)? Sayan Banerjee SME (2012–present) · Author has 1.3K answers and 1.3M answer views · May 2 Definition of locus: The path traced out by a point that moves under a condition(s). In the case that you have asked, it is a conic section. The conic section could be parabola, ellipse, hyperbola or circle , depending on the value of the ratio. Ration will be SP:SM, (distance from point: distance from fixed line) If the ratio is 1, it is a parabola. If the ratio is less than 1, it is an ellipse. If the ration is more than 1, it is a hyperbola. Pradeep Hebbar Many years of Structural Engineering & Math enthusiasm · Author has 9.2K answers and 6.2M answer views · Sep 8 Related What is the equation of the locus of the point that moves such that its distance from point A(3,1) is twice the distance from the point B(5,1)? Given two points A(3,1) and B(5,1) Point C(x,y) moves such that its distance from A is twice the distance from B Let us define a circle centered at A with radius 2k (x−3)2+(y−1)2=4k2…(1) Further, we define a circle centered at B with radius k (x−5)2+(y−1)2=k2…(2) From eqn. (1), (x−3)2+(y−1)2=4[(x−5)2+(y−1)2] 3x2−34x+3y2−6y+94=0 Locus of point C is a circle with this equation. Given two points A(3,1) and B(5,1) Point C(x,y) moves such that its distance from A is twice the distance from B Let us define a circle centered at A with radius 2k (x−3)2+(y−1)2=4k2…(1) Further, we define a circle centered at B with radius k (x−5)2+(y−1)2=k2…(2) From eqn. 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If the distance of a point from the fixed line x=-1 is twice its distance from the point (1,0), what is the locus of the point? What is the process for finding the locus of a point based on its distance from a line and other given points? Dave Benson trying to make maths easy. · Author has 6.1K answers and 2.1M answer views · Jul 31 Related How do I find the locus of a point which moves so that its distance from the point (2,1) is double its distance (1,2)? How do I find the locus of a point which moves so that its distance from the point (2,1) is double its distance (1,2)? Equate distances: √{(x-2)²+(y-1)²} = 2√{(x-1)²+(y-2)²} Answer simplify: x²-4x+4+y²-2y+1 = 4(x²-2x+1+y²-4y+4) ⟹ 3x²+3y²-4x-14y+15 = 0 ⟹ x²+y²-4x/3–14y/3+5 = 0 Answer is Circle general form compare x²+y²+2gx+2fy+c = 0 Centre(-g,-f) = (2/3,7/3) & (radius)² = g²+f²-c = 16/9+49/9 -5 = 8/9 So standard form: (x-2/3)²+(y-7/3)² = 8/9 Answer A D Telang Former Faculty at Goa Engineering College (1983–2010) · Author has 1.1K answers and 1M answer views · 4y Related What is the equation of the locus of a moving point so that its distance from the point (2,3) is half of the distance from x-axis resultant? Say the coordinates of moving point are (x,y), then it's distance from the point (2,3) is d=√(x−2)2+(y−3)2 Therefore, the equation of the locus is y2=d i.e. y2=√(x−2)2+(y−3)2 This gives the equation of the locus as, x2+3 y24− 4 x− 6 y+ 13=0 Sponsored by Grammarly Is your writing working as hard as your ideas? Grammarly’s AI brings research, clarity, and structure—so your writing gets sharper with every step. George Ivey Former Math Professor at Gallaudet University · Author has 23.7K answers and 2.6M answer views · May 2 Such a locus is the perpendicular bisector of the line segment having the two fixed points as endpoints. Gary Ward MaEd in Education & Mathematics, Austin Peay State University (Graduated 1997) · Author has 4.9K answers and 7.6M answer views · Jan 27 Related What is the equation of the locus of a point which moves so that its ordinate is thrice its abscissa? What is the equation of the locus of a point which moves so that its ordinate is thrice its abscissa? Thrice is an interesting word that I don’t see very often. y = 3x What is the equation of the locus of a point which moves so that its ordinate is thrice its abscissa? Thrice is an interesting word that I don’t see very often. y = 3x Sponsored by CDW Corporation How can AI help your teams make faster decisions? CDW’s AI solutions offer retrieval-augmented generation (RAG) to expedite info with stronger insights. Scott Rodham Past Head Teacher Mathematics (1995–present) · 4y Related What is the locus of the points whose distances to two orthogonal lines have a constant ratio? Let the original line have the original equation y=mx+c And the orthogonal line be Then their intersection is the point A Now let the ratio be a:b where ‘a’ is parallel to the original line and ‘b’ parallel to the orthogonal line. Now there are 4 positions of the “locus” point according to the absolute value equations above. Using the formula for the distance of a point from a line Let the original line have the original equation y=mx+c And the orthogonal line be Then their intersection is the point A Now let the ratio be a:b where ‘a’ is parallel to the original line and ‘b’ parallel to the orthogonal line. Now there are 4 positions of the “locus” point according to the absolute value equations above. Using the formula for the distance of a point from a line Kaushal Kishore Works at L&T MHI Boilers (Indian company) · Author has 86 answers and 298K answer views · 6y Related What is the locus of a point whose distance from the origin is three times its distance from the plane 2x-y+2z=3? Gordon M. Brown Math Tutor at San Diego City College (2018-Present) · Author has 6.2K answers and 4.3M answer views · 3y Related What is the equation of the locus of a point which moves so that it's distance from the point (4,5) is always three times its distance from a point (0,3)? The equation is 2x^2 + 2y^2 + 2x - 11y + 10 = 0. The locus is a circle defined by this equation. The exercise is straightforward—so much high-school algebra, in fact: Let (x, y) be any point on the locus; set up two occurrences of the distance formula, and multiply the second distance by 3; equate these; square both sides; and simplify. That’s all there is to it. (Click on either image below to expand it as necessary.) The equation is 2x^2 + 2y^2 + 2x - 11y + 10 = 0. The locus is a circle defined by this equation. The exercise is straightforward—so much high-school algebra, in fact: Let (x, y) be any point on the locus; set up two occurrences of the distance formula, and multiply the second distance by 3; equate these; square both sides; and simplify. That’s all there is to it. (Click on either image below to expand it as necessary.) Gopal Menon B Sc (Hons) in Mathematics, Indira Gandhi National Open University (IGNOU) (Graduated 2010) · Author has 10.2K answers and 15.2M answer views · 7y Related A point moves so that its distance from the point (-2,3) is always three times its distance from the point (0,3). What is the equation of its locus? Let point (x,y) be on the locus. ⇒ The distance of this point from (−2,3) is three times the distance of this point from (0,3). ⇒(x+2)2+(y−3)2=9[(x−0)2+(y−3)2]. ⇒x2+4x+4+y2−6y+9=9x2+9y2−54y+81 ⇒8x2−4x+8y2−48y+68=0. ⇒8x2−4x+12+8y2−48y+72−92=0. ⇒8(x2−12x+116)+8(y2−6y+9)=92. ⇒(x2−12x+116)+(y2−6y+9)=916. [math]\Rightarrow \qquad \left(x-\frac{1}{4}\right)^2+[/math] Let point (x,y) be on the locus. ⇒ The distance of this point from (−2,3) is three times the distance of this point from (0,3). ⇒(x+2)2+(y−3)2=9[(x−0)2+(y−3)2]. ⇒x2+4x+4+y2−6y+9=9x2+9y2−54y+81 ⇒8x2−4x+8y2−48y+68=0. ⇒8x2−4x+12+8y2−48y+72−92=0. ⇒8(x2−12x+116)+8(y2−6y+9)=92. ⇒(x2−12x+116)+(y2−6y+9)=916. ⇒(x−14)2+(y−3)2=(34)2. ⇒ The equation of the locus is (x−14)2+(y−3)2=(34)2. This is a circle with centre (14,3) and radius 34. Pradeep Hebbar Many years of Structural Engineering & Math enthusiasm · Author has 9.2K answers and 6.2M answer views · Mar 27 Related What is the equation to the locus of a moving point which is equidistant from the point (2a,2b) and (2c,2D)? Let A and B respectively be the points having given coordinates (2a,2b) and (2c,2d) For a point P to be equidistant from these points, it must lie on the perpendicular bisector of AB . Let m be its slope. It s equation has the form, y=mx+k…(1) Let M(p,q) be the midpoint of AB p=2a+2c2=a+c q=2b+2d2=b+d Slope of perpendicular bisector is the negative reciprocal of slope of AB m=−1(2d−2b2c−2a)=2a−2c2d−2b For passage of perpendicular bisector through M, q=mp+k b+d=m(a+c)+k k=b+d−m(a+c) From eqn. (1), y=mx+b+d−m(a+c) y=mx+b+d−am−ac Let A and B respectively be the points having given coordinates (2a,2b) and (2c,2d) For a point P to be equidistant from these points, it must lie on the perpendicular bisector of AB . Let m be its slope. It s equation has the form, y=mx+k…(1) Let M(p,q) be the midpoint of AB p=2a+2c2=a+c q=2b+2d2=b+d Slope of perpendicular bisector is the negative reciprocal of slope of AB m=−1(2d−2b2c−2a)=2a−2c2d−2b For passage of perpendicular bisector through M, q=mp+k b+d=m(a+c)+k k=b+d−m(a+c) From eqn. (1), y=mx+b+d−m(a+c) y=mx+b+d−am−ac Related questions How do I find the locus of a point which moves so that its distance from the point (2,1) is double its distance (1,2)? What is the locus of a point whose distance from two fixed points is equal to their sum or difference? What is the locus of points (M), the sum of whose distance from two given points is a constant? A point 𝑃 moves such that its distance from point 𝐴 (-3,4) is always twice its distance from point 𝐵 (6,-2). How to find the equation of locus and how to determine whether the locus intersect the y-axis? A point moves so that its distance from the point (-2,3) is always three times its distance from the point (0,3). What is the equation of its locus? What is the locus of the point of whose distances from the point (4,0) the point (0,3) equal? What is the locus of a point such that the sum of its distance from the point (0,2) and (o-2) is 6? Does a general formula exist for the locus of a point whose distance from 3 other points is fixed? If the distance of a point from the fixed line x=-1 is twice its distance from the point (1,0), what is the locus of the point? What is the process for finding the locus of a point based on its distance from a line and other given points? What is the equation of a locus of a moving point P such that its distance from point (2,3) is always double its distanceFrom (5, 0)? What does the aerial distance of two points indicate? What is the equation of the locus of the point that moves such that its distance from point A(3,1) is twice the distance from the point B(5,1)? What is the locus of point which moves such that it's distance from the y-axis is equal to its distance from the point (2,0)? What is the equation of the locus of a point which moves so that it's distance from the point (4,5) is always three times its distance from a point (0,3)? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12314	https://pmc.ncbi.nlm.nih.gov/articles/PMC9154640/	Accelerating the Uptake of WHO Recommendations for Mass Drug Administration Using Ivermectin, Diethylcarbamazine, and Albendazole - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Advanced Search Journal List User Guide New Try this search in PMC Beta Search View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer \| PMC Copyright Notice Am J Trop Med Hyg . 2022 Mar 15;106(5 Suppl):26–28. doi: 10.4269/ajtmh.21-0972 Search in PMC Search in PubMed View in NLM Catalog Add to search Accelerating the Uptake of WHO Recommendations for Mass Drug Administration Using Ivermectin, Diethylcarbamazine, and Albendazole Jonathan D King Jonathan D King 1 World Health Organization, Geneva, Switzerland; Find articles by Jonathan D King 1,, Julie Jacobson Julie Jacobson 2 Bridges to Development, Seattle, Washington; Find articles by Julie Jacobson 2, Alison Krentel Alison Krentel 3 School of Epidemiology and Public Health, University of Ottawa, Ottawa, Canada Find articles by Alison Krentel 3 Author information Article notes Copyright and License information 1 World Health Organization, Geneva, Switzerland; 2 Bridges to Development, Seattle, Washington; 3 School of Epidemiology and Public Health, University of Ottawa, Ottawa, Canada Address correspondence to Jonathan D. King, World Health Organization, 20 Avenue Appia, 1211 Geneva, Switzerland. E-mail: kingj@who.int Disclaimer: The authors alone are responsible for the views expressed in this article and they do not necessarily represent the views, decisions, or policies of the funders or the institutions with which they are affiliated. Financial support: Publication costs for this supplement were funded by the Bill & Melinda Gates Foundation. Authors’ addresses: Jonathan D. King, World Health Organization, Geneva, Switzerland, E-mail: kingj@who.int. Julie Jacobson, Bridges to Development, Seattle, WA, E-mail: jjacobson@bridgestodevelopment.org. Alison Krentel, School of Epidemiology and Public Health, University of Ottawa, Ottawa, Canada, E-mail: akrentel@bruyere.org. The World Health Organization holds the copyright and has granted the Publisher permission for the reproduction of this article. Received 2021 Sep 9; Accepted 2022 Jan 10; Issue date 2022 May. © The American Society of Tropical Medicine and Hygiene This is an open-access article distributed under the terms of the Creative Commons Attribution (CC-BY) License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. PMC Copyright notice PMCID: PMC9154640 PMID: 35292578 ABSTRACT. Triple therapy with ivermectin, diethylcarbamazine, and albendazole (IDA) for the elimination of lymphatic filariasis (LF) represents a compelling example of accelerating the timeline from development to introduction and impact. Previous articles outlined how the clinical development process was able to compress timelines and provide the evidence needed for the WHO to issue guidelines on the use of IDA for mass drug administration for LF. We explored the drivers for the rapid and successful introduction of IDA in the early-adopter countries. Lessons from this experience highlight five key elements for moving from WHO recommendations to program uptake after the publication of the guideline: 1) early engagement with stakeholders to create partnerships to coordinate and plan for implementation; 2) recognition by countries and partners of the potential of IDA to improve efforts to eliminate LF; 3) high-level commitment and coordination at regional levels and, most importantly, at the country level; 4) understanding of the perspectives among people living in LF-endemic communities where mass drug administration is warranted; and 5) affirmation of the feasibility of IDA through sharing lessons learned. INTRODUCTION In October 2017, the WHO produced a guideline on alternative mass drug administration (MDA) regimens targeting the neglected tropical disease lymphatic filariasis (LF) that included a recommendation for the use of triple-therapy MDA with ivermectin, diethylcarbamazine, and albendazole (IDA) 1—a newly discovered combination of existing medicines. Between publishing the guideline in October 2017 and a global meeting in July 2019 (Global Review of Initial Use of Triple-Therapy Mass Drug Administration and Planning for Accelerated Elimination of Lymphatic Filariasis, Bangkok, Thailand), eight countries (Samoa, American Samoa, Kenya, India, Papua New Guinea, Timor Leste, Malaysia, and Sao Tome and Principe) adopted IDA as an MDA regimen in national neglected tropical disease (NTD) programs (Table 1). Table 1. First countries implementing IDA \| 2018 \| 2019 \| :---: \| \| American Samoa \| Egypt \| \| Fiji \| Guyana \| \| Kenya \| Malaysia \| \| India \| Sao Tome and Principe \| \| Papua New Guinea \| Timor Leste \| \| Samoa \| \| Open in a new tab During this period, the WHO supported multinational and national consultations to guide countries on the implementation of the 2017 guidelines that helped move IDA, within 2 years, from WHO policy to national policy to program implementation. Reflecting on the early experiences of IDA implementation, there were several important factors that contributed to the quick and successful transition of global policy to country action, but five key elements were required: 1) early engagement with stakeholders to create partnerships to coordinate and plan for implementation; 2) recognition by countries and partners of the potential of IDA to improve efforts to eliminate LF; 3) high-level commitment and coordination at regional levels and, most importantly, at the country level; 4) understanding of the drivers of acceptance of IDA among people living in communities where the regimen is warranted according to WHO recommendations; and 5) affirmation of the feasibility of IDA by sharing lessons learned from the first countries to use IDA. STAKEHOLDER ENGAGEMENT AND COORDINATION Early planning and partner coordination were key to the successful introduction of IDA. The WHO planned a series of workshops inviting eligible countries, their partners, and global experts to familiarize countries with IDA and to begin planning for how IDA could be used in their local context to accelerate LF elimination. Preparatory work for these meetings began early in anticipation of the release of the new guideline to ensure uptake could start quickly. The initial planning and peer-to-peer sharing through these meetings between countries and partners was an essential element to getting well-thought-out strategic plans for IDA use. Country programs established national task forces comprised of representatives of national experts in NTDs, pharmacovigilance, primary health care, community groups, and government officials. These task forces met frequently and provided a mechanism to move forward the planning and coordination for IDA introduction in each country. The meetings allowed discussion of the guideline and an open forum to address questions and concerns from all stakeholders. The task forces reviewed the evidence upon which the recommendations were based, including the data on the efficacy, safety, and acceptability of IDA from the field trials. 2–5 Importantly, the meetings provided an opportunity for the program to be open to new information and emerging research that could help guide the use of IDA at the national level, and to work collaboratively with partners and other departments to support rollout. COUNTRY AND PARTNER WILLINGNESS Countries first adopting IDA viewed the novelty of the regimen as a potential to bring vigor into established programs and, in some settings, to revitalize aging routine programs. The greater efficacy of IDA meant a potential for stopping transmission with fewer MDA rounds. The extended benefits of ivermectin against other helminths and ectoparasites was also motivating. We learned through the experiences of the early-adopter countries and early qualitative studies that simply introducing the height-based dosing poles in some locations represented something new and interesting in the community, particularly in those areas where MDA for LF had previously used age-based dosing tables. Programs were hopeful that these new aspects and increased interest would result in greater coverage. Through the meetings and consultations, program staff and health officials discussed the willingness to adopt IDA, but also concerns, and planned for how to address issues as they arose. For example, in American Samoa, as part of the United States, a concern about regulatory approval of IDA was raised that was addressed by applying for and receiving Investigational New Drug approval from the U.S. Food and Drug Administration. Each country raised questions about the potential impact of fear of the number of tablets or of adverse events. This presented an opportunity to establish or improve collaboration between NTD programs and national pharmacovigilance agencies to ensure communities of the availability of care for and close monitoring of adverse events. Presenting data from the safety trial and the acceptability study reassured stakeholders and programs about community response to IDA and levels of adverse events. 1,2 Early-adopter countries used novel techniques such as telephone hot lines and Facebook pages to provide community members with access to health personnel during the MDA. HIGH-LEVEL COMMITMENT AND COORDINATION Obtaining support from high-level leadership proved crucial to the successful introduction of IDA in early-adopter countries. Coordination on the introduction and implementation of IDA occurred at the national and regional levels, involving multiple sectors of government, including high-level officials and ministers. The South East Asia Region of the WHO sponsored a regional ministerial meeting (Keeping the Promise: Ending NTDs on Time in the SEA Region, April 2017) to sensitize ministers of new strategies to accelerate elimination of NTDs including IDA. 6 Recommendations from the national task forces were submitted to the ministers of health facilitating the official adoption of IDA into national LF elimination programs. Involving ministers and public health officials in operational decision-making and requesting their participation in public launching events raised the visibility of the LF elimination program. 7 The first countries using IDA were successful in garnering support of local politicians in IDA MDA launching events. This was important to demonstrate to the community that celebrities and leaders can also have their height measured and take all the tablets safely. For those countries with prior rounds of MDA, this demonstration of renewed support for LF elimination across stakeholders and partners helped signal to community members the importance of MDA participation. UNDERSTANDING PERSPECTIVES OF THE COMMUNITY BY INCORPORATING SOCIAL RESEARCH INTO PLANNING During the community safety studies and the first implementation units where IDA was used in national LF elimination programs, social research was conducted concurrently, providing opportunities to expand understanding on its use. Community acceptability studies yielded important insights and helped inform how IDA would be perceived by communities. 3 These studies helped to identify where there had been challenges with MDA reach in the past, and how new messaging and tailored approaches to MDA may yield better coverage with IDA. Specifically, this included community feedback on the number of pills involved in IDA, the perception of adverse events, and any additional benefits of ivermectin. Efforts to coordinate operational research, specifically around acceptance of IDA, allowed results from initial studies to be used to help subsequent programs plan their social mobilization and messaging. SHARING EARLY LESSONS LEARNED Multinational meetings in 2019 provided a space to share documented experiences among early-adopter countries, build partnerships, and identify best practices across countries and regions. Case studies from the first program implementation experiences presented at these meetings helped expand understanding of IDA use outside the research trial setting. This process generated confidence in the feasibility of implementation, affirmed the safety and acceptability of the new MDA regimen, and informed planning for potential next-wave adopters. During these meetings, countries presented their new or modified approaches to community engagement and social mobilization. Programs reported on the methods of distribution used, dosing strategy (height, weight, or age), and monitoring of directly observed therapy such as finger-marking with indelible ink. Countries that had not yet started IDA were interested in the additional, improved efforts invested in monitoring for adverse events. Introducing microplanning as a tool to improve MDA was discussed as well as new aspects of drug distributor training and innovative interpersonal communication. For programs considering the introduction of IDA, hearing from the experiences of their peers fostered reassurance as they witnessed practical and creative ways of incorporating IDA successfully. CONCLUSION The transition from WHO policy to integration of IDA into country NTD programs has been successful in a relatively short timeline. By the end of 2019, 11 countries spanning all five LF-endemic regions had implemented IDA, treating more than 45 million people. 8 This was possible as a result of five elements: 1) early engagement with stakeholders, 2) the potential of IDA being recognized by countries and partners , 3) high-level commitment and coordination at regional and country levels, 4) inclusion of the end user through the incorporation of social research, and 5) sharing experiences between countries. The onset of the coronavirus pandemic in 2020 has slowed progress in the implementation of IDA. Since the end of 2019, only three additional countries have adopted IDA and only six countries have been able to implement second rounds of IDA MDA in the initial areas covered. 9 This article highlights briefly some important shared factors in the successful transition of policy to action at the national level. More in-depth firsthand experience from IDA implementation leaders in a set of early-adopter countries has been documented. 10,11 ACKNOWLEDGMENTS The uptake of IDA would not be possible without the leadership from National Governments, efforts from WHO Country and Regional Offices and partners of the Global Programme to Eliminate Lymphatic Filariasis. REFERENCES World Health Organization , 2017. Guideline: Alternative Mass Drug Administration Regimens to Eliminate Lymphatic Filariasis. Available at: Accessed May 25, 2021. [PubMed] Weil GJ et al. 2019. The safety of double- and triple-drug community mass drug administration for lymphatic filariasis: a multicenter, open-label, cluster-randomized study. PLoS Med 16: e1002839. [DOI] [PMC free article] [PubMed] [Google Scholar] Krentel A. et al. , 2021. A multicenter, community-based, mixed methods assessment of the acceptability of a triple drug regimen for elimination of lymphatic filariasis. PLoS Negl Trop Dis 15: e0009002. [DOI] [PMC free article] [PubMed] [Google Scholar] King CL et al. 2018. A trial of a triple-drug treatment for lymphatic filariasis. N Engl J Med 379: 1801–1810. [DOI] [PMC free article] [PubMed] [Google Scholar] Bjerum CM Ouattara AF Aboulaye M Kouadio O Marius VK Andersen BJ Weil GJ Koudou BG King CL , 2020. Efficacy and safety of a single dose of ivermectin, diethylcarbamazine, and albendazole for treatment of lymphatic filariasis in Côte d’Ivoire: an open-label randomized controlled trial. Clin Infect Dis 71: e68–e75. [DOI] [PMC free article] [PubMed] [Google Scholar] World Health Organization Regional Office for South-East Asia , 2019. Finish the Task of Eliminating Neglected Tropical Diseases (NTDs) and Other Diseases on the Verge of Elimination: A More Responsive WHO in the South-East Asia Region: Our Journey Together: Our Journey Ahead. Available at: Accessed May 28, 2021. The END Fund , n.d. In Pictures: Launch of IDA Drug Therapy Pilot Program in Kenya. Available at: Accessed January 16, 2022. World Health Organization , 2020. Global programme to eliminate lymphatic filariasis: progress report 2019. Wkly Epidemiol Rec 95: 509–524. [Google Scholar] World Health Organization , 2021. Global programme to eliminate lymphatic filariasis: progress report 2020. Wkly Epidemiol Rec 96: 497–508. Available at: Accessed October 15, 2021. [Google Scholar] Rainima-Qaniuci M et al. 2022. The importance of partnership in the rollout of triple-drug therapy to eliminate lymphatic filariasis in the Pacific. Am J Trop Med Hyg 106 (Suppl 5): 39–47. [DOI] [PMC free article] [PubMed] [Google Scholar] Tripathi B Roy N , Dhingra N, 2022. Introduction of triple-drug therapy for accelerating lymphatic filariasis elimination in India: lessons learned. Am J Trop Med Hyg 106 (Suppl 5): 29–38. [DOI] [PMC free article] [PubMed] [Google Scholar] Articles from The American Journal of Tropical Medicine and Hygiene are provided here courtesy of The American Society of Tropical Medicine and Hygiene ACTIONS View on publisher site PDF (439.5 KB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page ABSTRACT. INTRODUCTION STAKEHOLDER ENGAGEMENT AND COORDINATION COUNTRY AND PARTNER WILLINGNESS HIGH-LEVEL COMMITMENT AND COORDINATION UNDERSTANDING PERSPECTIVES OF THE COMMUNITY BY INCORPORATING SOCIAL RESEARCH INTO PLANNING SHARING EARLY LESSONS LEARNED CONCLUSION ACKNOWLEDGMENTS REFERENCES Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
12315	https://www.yourdictionary.com/implicant	Implicant Definition & Meaning \| YourDictionary Dictionary Thesaurus Sentences Grammar Vocabulary Usage Reading & Writing Articles Vocabulary Usage Reading & Writing Sign in Menu Word Finder Words with Friends Cheat Wordle Solver Word Unscrambler Scrabble Dictionary Anagram Solver Wordscapes Answers Sign in with Google Dictionary Thesaurus Sentences Grammar Vocabulary Usage Reading & Writing Word Finder Word Finder Words with Friends Cheat Wordle Solver Word Unscrambler Scrabble Dictionary Anagram Solver Wordscapes Answers Dictionary DictionaryThesaurusSentencesArticlesWord Finder Make Our Dictionary Yours Sign up for our weekly newsletters and get: Grammar and writing tips Fun language articles WordOfTheDay and quizzes Sign in with Google By signing in, you agree to our Terms and Conditions and Privacy Policy. Success! We'll see you in your inbox soon. Thank you! Undo Home Dictionary Meanings Implicant Definition Implicant Definition Meanings Definition Source [x] All sources [x] Wiktionary Word Forms Noun Filter(0) noun (propositional calculus) The hypothesis of an implication. Wiktionary (electrical engineering) On a Karnaugh map: a set of 1's (whose quantity is a power of two) which are related by adjacency (i.e., the set is connected, if the Karnaugh map is considered to be a graph which "wraps around" its edges, like a torus; and, all elements of the subgraph induced by the set have the same degree). Equivalently (in terms of Boolean algebra), a product term which, when true, always implies that the given Boolean function is true. Wiktionary 1/1 Skip Ad Continue watching after the adVisit Advertiser websiteGO TO PAGE Other Word Forms of Implicant Noun Singular: implicant Plural: implicants Related Articles Difference Between Connotation vs. Denotation What Is a Research Abstract? 3 Effective Examples Definition of Academic Writing With Examples 15 Types of Essays (and What You Need To Know About Them) What's a Zoomer? A Look at the Informal Term for Gen Z What Is a Banana Republic? Explanation and Examples Implicant Is Also Mentioned In prime-implicant non-essential-prime-implicant essential-prime-implicant Find Similar Words Find similar words to implicant using the buttons below. Words Starting With IIMIMP Words Ending With TNTANT Unscrambles implicant Words Starting With I and Ending With T Starts With I& Ends With TStarts With IM& Ends With TStarts With I& Ends With NT Word Length 9 Letter Words9 Letter Words Starting With I9 Letter Words Ending With T Advertisement Words Near Implicant in the Dictionary implementor implements impletion implex implexion impliable implicant implicate implicated implicates implicating implication Filter Random Word Learn a new word now! Get a Random Word Copyright © 2025 LoveToKnow Media. All Rights Reserved Features Dictionary Thesaurus Sentences Grammar Vocabulary Usage Reading & Writing Company About Us Contact Us Privacy Policy Editorial Policy Cookie Settings Terms of Use Suggestion Box Do Not Sell My Personal Information Random Word Learn a new word now! Get a Random Word Follow Us Connect Contact Us Suggestion Box Follow Us LinkedIn Facebook Instagram TikTok Do Not Sell My Personal Information Copyright © 2025 LoveToKnow Media. All Rights Reserved
12316	https://math.libretexts.org/Bookshelves/Arithmetic_and_Basic_Math/Basic_Math_(Grade_6)/04%3A_Dividing_Fractions/24%3A_Fractions_in_Lengths_Areas_and_Volumes/24.02%3A_Rectangles_with_Fractional_Side_Lengths	24.2: Rectangles with Fractional Side Lengths - 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 Section 24: Fractions in Lengths, Areas, and Volumes 4: Dividing Fractions { } { "24.01:_Fractional_Lengths" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "24.02:_Rectangles_with_Fractional_Side_Lengths" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "24.03:_Fractional_Lengths_in_Triangles_and_Prisms" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "24.04:_Volume_of_Prisms" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } { "21:_Making_Sense_of_Division" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "22:_Meanings_of_Fraction_Division" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "23:_Algorithm_for_Fraction_Division" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "24:_Fractions_in_Lengths_Areas_and_Volumes" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "25:_Let\'s_Put_It_to_Work" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } Sun, 27 Mar 2022 16:11:59 GMT 24.2: Rectangles with Fractional Side Lengths 40250 40250 admin { } Anonymous Anonymous 2 false false [ "article:topic", "license:ccby", "licenseversion:40" ] [ "article:topic", "license:ccby", "licenseversion:40" ] 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. Arithmetic and Basic Math 4. Basic Math (Grade 6) 5. 4: Dividing Fractions 6. Section 24: Fractions in Lengths, Areas, and Volumes 7. 24.2: Rectangles with Fractional Side Lengths Expand/collapse global location Basic Math (Grade 6) Front Matter 1: Area and Surface Area 2: Introducing Ratios 3: Unit Rates and Percentages 4: Dividing Fractions 5: Arithmetic in Base Ten 6: Expressions and Equations 7: Rational Numbers 8: Data Sets and Distributions 9: Putting it All Together Back Matter 24.2: Rectangles with Fractional Side Lengths Last updated Mar 27, 2022 Save as PDF 24.1: Fractional Lengths 24.3: Fractional Lengths in Triangles and Prisms picture_as_pdf Full Book Page Downloads Full PDF Import into LMS Individual ZIP Buy Print Copy Print Book Files Buy Print CopyReview / Adopt Submit Adoption Report Submit a Peer Review View on CommonsDonate Page ID 40250 ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Lesson 1. Summary Practice Lesson Let's explore rectangles that have fractional measurements. Exercise 24.2.1: Areas of Squares Figure 24.2.1 What do you notice about the areas of the squares? Kiran says “A square with side lengths of 1 3 inch has an area of 1 3 square inches.” Do you agree? Explain or show your reasoning. Exercise 24.2.2: Areas of Squares and Rectangles Your teacher will give you graph paper and a ruler. On the graph paper, draw a square with side lengths of 1 inch. Inside this square, draw another square with side lengths of 1 4 inch. Use your drawing to answer the questions. 1. How many squares with side lengths of 1 4 inch can fit in a square with side lengths of 1 inch? 2. What is the area of a square with side lengths of 1 4 inch? Explain or show your reasoning. On the graph paper, draw a rectangle that is 3 1 2 inches by 2 1 4 inches. For each question, write a division expression and then find the answer. 1. How many 1 4-inch segments are in a length of 3 1 2 inches? 2. How many 1 4-inch segments are in a length of 2 1 4 inches? Use your drawing to show that a rectangle that is 3 1 2 inches by 2 1 4 inches has an area of 7 7 8 square inches. Exercise 24.2.3: Areas of Rectangles Each of these multiplication expressions represents the area of a rectangle. 2⋅4 2 1 2⋅4 2⋅4 3 4 2 1 2⋅4 3 4 All regions shaded in light blue have the same area. Match each diagram to the expression that you think represents its area. Be prepared to explain your reasoning. Figure 24.2.2: Four vertically oriented rectangles labeled A, B, C, D. Rectangle A has a horizontal dotted line about three quarters of the way down its width. The top portion is shaded blue. Rectangle B has a horizontal dotted line about three quarters of the way down its width and a vertical dotted line about three quarters of the way to the right of its length. The top left portion of the rectangle is shaded blue. Rectangle C is shaded blue. Rectangle D has a vertical dotted line about three quarters of the way to the right of its length. The left portion of the rectangle is shaded blue. Use the diagram that matches 2 1 2⋅4 3 4 to show that the value of 2 1 2⋅4 3 4 is 11 7 8. Are you ready for more? The following rectangles are composed of squares, and each rectangle is constructed using the previous rectangle. The side length of the first square is 1 unit. Figure 24.2.3: A sequence of rectangles composed of squares. The first is a unit square. The second is composed of two unit squares, stacked vertically. The third is composed of two unit squares, stacked vertically, and a 2 by 2 unit square on the right. The fourth is composed of two unit squares, stacked vertically, a 2 by 2 unit square on the right, and a 3 by 3 unit square at the bottom. The fifth is composed of two unit squares, stacked vertically, a 2 by 2 unit square on the right, a 3 by 3 unit square at the bottom, and a 4 by 4 unit square on the left. Draw the next four rectangles that are constructed in the same way. Then complete the table with the side lengths of the rectangle and the fraction of the longer side over the shorter side. \| short side \| long side \| long side short side \| --- \| 1 \| \| \| \| 1 \| \| \| \| 2 \| \| \| \| 3 \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| Table 24.2.1 Describe the values of the fraction of the longer side over the shorter side. What happens to the fraction as the pattern continues? Exercise 24.2.4: How Many Would it Take? (Part 2) Noah would like to cover a rectangular tray with rectangular tiles. The tray has a width of 11 1 4 inches and an area of 50 5 8 square inches. Find the length of the tray in inches. If the tiles are 3 4 inch by 9 16 inch, how many would Noah need to cover the tray completely, without gaps or overlaps? Explain or show your reasoning. Draw a diagram to show how Noah could lay the tiles. Your diagram should show how many tiles would be needed to cover the length and width of the tray, but does not need to show every tile. Summary If a rectangle has side lengths a units and b units, the area is a⋅b square units. For example, if we have a rectangle with 1 2-inch side lengths, its area is 1 2⋅1 2 or 1 4 square inches. Figure 24.2.4: A large square evenly divided into 4 smaller squares. The large square has bottom horizontal side length of 1 inch. Of the four smaller squares, the top left square is shaded blue. It has side lengths labeled one half inch. This means that if we know the area and one side length of a rectangle, we can divide to find the other side length. Figure 24.2.5 If one side length of a rectangle is 10 1 2 in and its area is 89 1 4 in 2, we can write this equation to show their relationship: 10 1 2⋅?=89 1 4 Then, we can find the other side length, in inches, using division: 89 1 4÷10 1 2=? Practice Exercise 24.2.5 Find the unknown side length of the rectangle if its area is 11 m 2. Show your reasoning. Figure 24.2.6: A rectangle that has area labeled 11 meters squared. The side length of one side of the rectangle is labeled three and two thirds meters and the side length of the other side is labeled with a question mark. Each angle has a right angle symbol indicated. Check your answer by multiplying it by the given side length (3 2 3). Is the resulting product 11? If not, revise your previous work. Exercise 24.2.6 A worker is tiling the floor of a rectangular room that is 12 feet by 15 feet. The tiles are square with side lengths 1 1 3 feet. How many tiles are needed to cover the entire floor? Show your reasoning. Exercise 24.2.7 A television screen has length 16 1 2 inches, width w inches, and area 462 square inches. Select all the equations that represent the relationship of the side lengths and area of the television. w⋅462=16 1 2 16 1 2⋅w=462 462÷16 1 2=w 462÷w=16 1 2 16 1 2⋅462=w Exercise 24.2.8 The area of a rectangle is 17 1 2 in 2 and its shorter side is 3 1 2 in. Draw a diagram that shows this information. What is the length of the longer side? Exercise 24.2.9 A bookshelf is 42 inches long. How many books of length 1 1 2 inches will fit on the bookshelf? Explain your reasoning. A bookcase has 5 of these bookshelves. How many feet of shelf space is there? Explain your reasoning. (From Unit 4.4.1) Exercise 24.2.10 Find the value of 5 32÷25 4. Show your reasoning. (From Unit 4.3.2) Exercise 24.2.11 How many groups of 1 2 3 are in each of these quantities? 1 5 6 4 1 3 5 6 (From Unit 4.2.3) Exercise 24.2.12 It takes 1 1 4 minutes to fill a 3-gallon bucket of water with a hose. At this rate, how long does it take to fill a 50-gallon tub? If you get stuck, consider using a table. (From Unit 2.4.4) 24.2: Rectangles with Fractional Side Lengths is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by LibreTexts. Back to top 24.1: Fractional Lengths 24.3: Fractional Lengths in Triangles and Prisms Was this article helpful? Yes No Recommended articles 24.4: Volume of Prisms 24.3: Fractional Lengths in Triangles and Prisms 24.1: Fractional Lengths Section 21: Making Sense of Division Section 22: Meanings of Fraction Division Article typeSection or PageLicenseCC BYLicense Version4.0 Tags This page has no tags. © 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. Accessibility Statement.For more information contact us atinfo@libretexts.org. Support Center How can we help? Contact Support Search the Insight Knowledge Base Check System Status× contents readability resources tools ☰ 24.1: Fractional Lengths 24.3: Fractional Lengths in Triangles and Prisms Complete your gift to make an impact
12317	https://www.wolframalpha.com/widgets/view.jsp?id=ba4c9d41e1d28a6d43143de56c2074f7	Wolfram\|Alpha Widgets: "Extrema Calculator w/ Domain" - Free Mathematics Widget HOMEABOUTPRODUCTSBUSINESSRESOURCES Wolfram\|Alpha WidgetsOverviewTourGallerySign In Extrema Calculator w/ Domain Extrema Calculator w/ Domain Function Lower Bound Upper Bound Submit Computing... Get this widget Build your own widget»Browse widget gallery»Learn more»Report a problem»Powered by Wolfram\|Alpha Terms of use Share a link to this widget: More Embed this widget» Added May 20, 2014 by mtyacorn in Mathematics Outputs extrema of a function within given domain. Domain is initially {x\|-∞≤x≤∞}. Send feedback\|Visit Wolfram\|Alpha SHARE Email Twitter FacebookShare via Facebook » More... Share This Page Digg StumbleUpon Delicious Reddit Blogger Google Buzz Wordpress Live TypePad Tumblr MySpace LinkedIn URL EMBED Make your selections below, then copy and paste the code below into your HTML source. For personal use only. Theme Output Type Lightbox Popup Inline [x] Widget controls displayed [x] Widget results displayed Output Width px Output Height px To add the widget to Blogger, click here and follow the easy directions provided by Blogger. To add the widget to iGoogle, click here. On the next page click the "Add" button. You will then see the widget on your iGoogle account. To embed this widget in a post on your WordPress blog, copy and paste the shortcode below into the HTML source: For self-hosted WordPress blogs To embed this widget in a post, install the Wolfram\|Alpha Widget Shortcode Plugin and copy and paste the shortcode above into the HTML source. To embed a widget in your blog's sidebar, install the Wolfram\|Alpha Widget Sidebar Plugin, and copy and paste the Widget ID below into the "id" field: To add a widget to a MediaWiki site, the wiki must have the Widgets Extension installed, as well as the code for the Wolfram\|Alpha widget. To include the widget in a wiki page, paste the code below into the page source. Save to My Widgets Build a new widget About Pro Products Mobile Apps Business Solutions For Developers Resources & Tools Blog Community Participate Contact Connect © 2025 Wolfram Alpha LLC—A Wolfram Research Company Terms Privacy Sign Up for Wolfram\|Alpha News Name (optional) Organization (optional) Email Message (optional) Thank You We appreciate your interest in Wolfram\|Alpha and will be in touch soon. —The Wolfram\|Alpha Team
12318	https://artofproblemsolving.com/wiki/index.php/Set?srsltid=AfmBOop1NTrl3IR2-XT1Gh-2TGjG3xNnIyo1duZTZ6dt2izqTK2xD2jq	Art of Problem Solving Set - 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 Set Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Set The notion of a set is one of the fundamental notions in mathematics that is difficult to precisely define. Of course, we have plenty of synonyms for the word "set," like collection, ensemble, group, etc., but those names really do not define the meaning of the word set; all they can do is replace it in various sentences. So, instead of defining what sets are, one has to define what can be done with them or, in other words, what axioms the sets satisfy. These axioms are chosen to agree with our intuitive concept of a set, on one hand, and to allow various, sometimes quite sophisticated, mathematical constructions on the other hand. For the full collection of these axioms, see Zermelo-Fraenkel Axioms. In this article we shall present just a brief discussion of the most common properties of sets and operations related to them. Contents 1 Rough Definition 2 Relation of Belonging 3 Specifying a Set 3.1 Listing its Elements 3.2 Stating the Common Property of its Elements 3.3 Set Notation 4 Subsets 5 Power Sets 6 Operations 6.1 Union and Intersection 6.2 Cartesian Product 7 Empty Set 8 Infinite Sets 9 Cardinality 10 Problems 10.1 Introductory 10.2 Intermediate 10.3 Olympiad 11 See Also 12 External Links Rough Definition A set is a collection of objects. The objects can be anything: numbers, letters, libraries that have at least 20 male staff, or absolutely nothing. Order does not matter. What does matter is what is in the set. There might be a finite number of objects in the set, in which case it is called a finite set. Otherwise we call it an infinite set. The objects in a set are called the elements of the set. A common misconception is that a set can have multiple indistinct elements, such as the following: Such an entity is actually called a multiset. Relation of Belonging The most important property of sets is that, for every object and a set , we can say whether belongs to (written as ), or not (written as ). Two sets and are equal if they include the same objects, i.e., if for every object , we have if and only if . Specifying a Set There are many ways to specify a set, using different notation. Listing its Elements This means that in order to identify a particular set, it suffices to tell which objects belong to this set. If the set contains just several such objects, all you need to do is list them. So, you can specify the set consisting of the numbers , and , for example. (The standard notation for this set is . Note that the order in which the terms are listed is completely unimportant: we have to follow some order when writing things in one line, but you should actually imagine those numbers flowing freely inside those curly braces with no preference given to any of them. What matters is that these four numbers are in the set and everything else is out). But how do you specify sets that have very many (maybe infinitely many) elements? You cannot list them all even if you spend your entire life writing! Stating the Common Property of its Elements Another way to specify a set is to use some property to tell when an object belongs to this set. For instance, we may try to think (alas, only try!) of the set of all objects with green hair. In this case, we do not even try to list all such objects. We just decide that something belongs to this set if it has green hair and doesn't belong to it otherwise. This is a wonderful way to describe a set. Unfortunately, this method has several potential pitfalls. It turns out, counter-intuitively, that not every collection of objects with a certain property is a set. The most famous example of this problem is Russell's Paradox: consider the property, "is a set and does not contain itself." (Remember that, given a set, we should be able to tell about every object whether it belongs to this set or not; in particular, we can ask this question about the set itself). The set specified by this property can neither belong, nor not belong to itself. There are a variety of ways to resolve this paradox, but the problem is clear: this way to describe sets should be used with extreme caution. One way to avoid this problem is as follows: given a property , choose a known set . Then the collection of elements of which have property will always be a set. (In particular, for our previous example to lead to a paradox, we would need to choose . However, it turns out that it can be proven that the set of all sets does not exist -- the collection of all sets is too big to be a set.) Set Notation There is a notation used just for sets: That symbolizes the set of all reals not equal to 0. This is probably the fastest way of describing a large set. Also, the empty set can be specified using set notation: Since there are no reals such that the square of it is less than 0, that set is the empty set. Subsets We say that a set is a subset of a set if every object that belongs to also belongs to . This is denoted or . For example, the sets and are subsets of the set , but the set is not. Thus we can say that two sets are equal if and only if each is a subset of the other. A special kind of subset is the empty set. Power Sets Main article: power set The power set of a set , denoted is defined as the set of all its subsets. For example, the power set of is . If a is a finite set of size then has size . Operations Union and Intersection The union of two or more sets is the set of all objects that belong to one or more of the sets. The union of A and B is denoted . For example, the union of and is . Unions can also be represented just as sums and products can be. would be the union of all sets that satisfy the statement. So, for example, would be the set of all natural numbers . The intersection of two or more sets is the set of all objects that belong to all of the sets. The intersection of A and B is denoted . For example, the intersection of and is . Just like unions, intersections can be represented as such: . For example, , or the empty set defined next. Cartesian Product The Cartesian Product of two sets and is defined as the set of Ordered Pairs such that and Empty Set Main article: empty set An empty set, denoted is a set with no elements. An empty set has some special properties: It is a subset of every other set. The union of any other set and an empty set is the original set. The intersection of any other set and an empty set is an empty set. Infinite Sets An infinite set can be defined as a set that has the same cardinality as one of its proper subsets. Alternatively, infinite sets are those which cannot be put into correspondence with any set of the form . Cardinality The cardinality of a set , denoted , is (informally) the size of the set. For a finite set, the cardinality is simply the number of elements. The empty set has cardinality 0. iff there is a bijective function meaning that there is a function that maps all elements of to all the elements of with one-to-one correspondence. iff there exists an injective function meaning there is a function that maps all elements of to (not necessarily all) elements of . can be defined similarily by expressing it as . iff there exists an injective function and there is no bijective function meaning but . is defined similarly. Although showing that and implies that is easy to prove when using finite sets, it is more complicated when using infinite sets. This theorem is called the Cantor-Bernstein-Schröder theorem and was proven by Georg Cantor, Felix Bernstein, and Ernst Schröder. Problems Introductory The regular 5-point star is drawn and in each vertex, there is a number. Each and are chosen such that all 5 of them came from set . Each letter is a different number (so one possible way is ). Let be the sum of the numbers on and , and so forth. If and form an arithmetic sequence (not necessarily in increasing order), find the value of . (Source) Intermediate Let set be a 90-elementsubset of and let be the sum of the elements of Find the number of possible values of (Source) Olympiad Let be a fixed positive integer, and let be an infinite family of sets, each of size , no two of which are disjoint. Prove that there exists a set of size that meets each set in . (Source) See Also Set theory Function External Links Naive Set Theory by Paul R. Halmos. Set Theory and Logic by Robert Roth Stoll. 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12319	https://www.cdc.gov/dpdx/diagnosticprocedures/other/aspirates.html	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. Other Specimens – Aspirates Duodenal aspirates may be useful in demonstrating Giardia duodenalis or Strongyloides stercoralis larvae. Material collected following intubation through the nose and stomach into the upper small intestine may be submitted to the laboratory. Centrifuge the specimen at 500 × g for 2 to 3 minutes and examine the wet mount. An unfixed specimen can be examined immediately or if the specimen cannot be examined within 1 to 2 hours after collection, it should be preserved in 10% formalin. Sigmoidoscopy material and abscesses of the liver and lung may demonstrate amebic trophozoites. Material from the mucosal surface or from visible lesions should be aspirated. This material can be examined immediately in a 0.85% saline wet mount preparation (or part of this material could be placed in formalin) or can be fixed in PVA. Once fixed in PVA, the material can be stained using trichrome stain and examined for trophozoites of Entamoeba histolytica. A real-time PCR test is available for confirmation of amebiasis that detects E. histolytica at the species level. If molecular diagnosis is necessary, specimens should be unpreserved and kept refrigerated or frozen. Lymph node material, bone marrow, and spleen may be examined for the presence of motile trophozoites of Trypanosoma brucei gambiense or Trypanosoma brucei rhodesiense. For Leishmania donovani infections, material obtained by needle aspiration from bone marrow or spleen can be used to demonstrate amastigote stages. Smears can then be prepared by fixing in methanol and staining with Giemsa stain. Skin ulcers may demonstrate the amastigote stages in cutaneous and mucocutaneous leishmaniasis. Permanent stained smears made with these specimens by fixing in methanol and staining with Giemsa stain. For additional information about aspirates, call the Division of Parasitic Diseases, at (404) 718-4110. DPDx is an educational resource designed for health professionals and laboratory scientists. For an overview including prevention, control, and treatment visit www.cdc.gov/parasites/. To receive email updates about this page, enter your email address:
12320	https://www.khanacademy.org/math/cc-third-grade-math/intro-to-multiplication/imp-multiplication-intro/v/multiplication-as-repeated-addition	Multiplication as repeated addition (video) \| Khan Academy Skip to main content If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. Explore Browse By Standards Explore Khanmigo Math: Pre-K - 8th grade Math: High school & college Math: Multiple grades Math: Illustrative Math-aligned Math: Eureka Math-aligned Math: Get ready courses Test prep Science Economics Reading & language arts Computing Life skills Social studies Partner courses Khan for educators Select a category to view its courses Search AI for Teachers FreeDonateLog inSign up Search for courses, skills, and videos Help us do more We'll get right to the point: we're asking you to help support Khan Academy. 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Skip to lesson content 3rd grade math Course: 3rd grade math>Unit 1 Lesson 1: Multiplication as equal groups Equal groups Equal groups Introduction to multiplication Understand equal groups as multiplication Multiplication as repeated addition Relate repeated addition to multiplication Math> 3rd grade math> Intro to multiplication> Multiplication as equal groups © 2025 Khan Academy Terms of usePrivacy PolicyCookie NoticeAccessibility Statement Multiplication as repeated addition CCSS.Math: 3.OA.C.7 Google Classroom Microsoft Teams About About this video Transcript Let's explore the idea of multiplication being the same as repeated addition. It shows how to express multiplication problems using repeated addition and highlights that multiplication is commutative, meaning the order of the numbers doesn't change the outcome. Skip to end of discussions Questions Tips & Thanks Want to join the conversation? Log in Sort by: Top Voted 𝓜𝓪𝓱𝓮𝓼𝓱 𝓜𝓪𝓱𝓮𝓷𝓭𝓻𝓪𝓴𝓪𝓻 5 years ago Posted 5 years ago. Direct link to 𝓜𝓪𝓱𝓮𝓼𝓱 𝓜𝓪𝓱𝓮𝓷𝓭𝓻𝓪𝓴𝓪𝓻's post “At 2:54, Sal was saying t...” more At 2:54 , Sal was saying that "you can view multiplication as repeated addition." so is multiplication just reapeated addition? (48 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Chase WP 5 years ago Posted 5 years ago. Direct link to Chase WP's post “Multiplication, in a way,...” more Multiplication, in a way, can be viewed as repeated addition. It's basically the exact same thing, but since repeated addition would take a lot longer, multiplication is much easier to do and remember. In short, yes, it is basically repeated addition. (52 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... Griffin 2 years ago Posted 2 years ago. Direct link to Griffin's post “why did you use avocodo? ...” more why did you use avocodo? its kinda funny Answer Button navigates to signup page •16 comments Comment on Griffin's post “why did you use avocodo? ...” (32 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Griffin 2 years ago Posted 2 years ago. Direct link to Griffin's post “i mean at 0:03” more i mean at 0:03 8 comments Comment on Griffin's post “i mean at 0:03” (12 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... annayay 2 years ago Posted 2 years ago. Direct link to annayay's post “POV: you love avacados.. ...” more POV: you love avacados.. sal be like. next thing you know hes uses that for every multiplacation video lol Answer Button navigates to signup page •3 comments Comment on annayay's post “POV: you love avacados.. ...” (12 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer tomiwajakinpelu 2 years ago Posted 2 years ago. Direct link to tomiwajakinpelu's post “so two plus two plus blah...” more so two plus two plus blah blah blah blah is repeated addition but if you don't want to do repeated addition you can just do 6x2 or 2x6 its the same thing Answer Button navigates to signup page •1 comment Comment on tomiwajakinpelu's post “so two plus two plus blah...” (8 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer 🏝️🌴Sufi🌴🏝️ 3 months ago Posted 3 months ago. Direct link to 🏝️🌴Sufi🌴🏝️'s post “Yes it is that’s correct ...” more Yes it is that’s correct but if you do want to do repeated addition what would you do well you can do 6+6 or 2+2+2+2+2+2 right? Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... just chilling a year ago Posted a year ago. Direct link to just chilling's post “how do you get where you ...” more how do you get where you get the number of points Answer Button navigates to signup page •1 comment Comment on just chilling's post “how do you get where you ...” (5 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer C/-/®️!$👑💸💍 a year ago Posted a year ago. Direct link to C/-/®️!$👑💸💍's post “you have to go on math b....” more you have to go on math b.e.s.t florida and answer questions or watch videos 3 comments Comment on C/-/®️!$👑💸💍's post “you have to go on math b....” (2 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... Drako2131 18 days ago Posted 18 days ago. Direct link to Drako2131's post “3x4 is 4x3 so 3x4 is 12” more 3x4 is 4x3 so 3x4 is 12 Answer Button navigates to signup page •Comment Button navigates to signup page (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer TheReal3A 17 days ago Posted 17 days ago. Direct link to TheReal3A's post “Ah, that's right! It does...” more Ah, that's right! It doesn't matter which way the numbers are in a multiplication - they lead to the same result. 3x4 = 12 AND 4x3 = 12. Comment Button navigates to signup page (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more helclaroscampos 5 months ago Posted 5 months ago. Direct link to helclaroscampos's post “how do you add” more how do you add Answer Button navigates to signup page •Comment Button navigates to signup page (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer ❤Neva🍄Samara🪻Nemeroff🌺🩷 4 months ago Posted 4 months ago. Direct link to ❤Neva🍄Samara🪻Nemeroff🌺🩷's post “you add by putting the n...” more you add by putting the numbers together, for example 😀+😀=😀😀 because if you have one smiley face in one hand and one on the other you have two smiley faces in all. Comment Button navigates to signup page (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... mykuromi450 a year ago Posted a year ago. Direct link to mykuromi450's post “what if 3x4=12 4x3=12 the...” more what if 3x4=12 4x3=12 then 6x2=12 2x6=12 my goal here, what if there were four the same answer Answer Button navigates to signup page •1 comment Comment on mykuromi450's post “what if 3x4=12 4x3=12 the...” (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer TheReal3A a year ago Posted a year ago. Direct link to TheReal3A's post “Woah, you're discovering ...” more Woah, you're discovering factors! For a number, a factor is a smaller number that can be multiplied to get the original number. A factor is always paired up with another. For example, 6: since 23 = 6, a factor pair of 6 is 2,3 (2 and 3 are both factors of 6). You can find all the factors of a number by making all the possible multiplications to make that number. You just found that 3,4 and 2,6 are factor pairs for 12. By the way, there's another - 1,12, because 112 = 12. Bonus: 223 = 12, you see this when you break up the 4 (in 43 = 12) into 22. Happy learning :D 2 comments Comment on TheReal3A's post “Woah, you're discovering ...” (2 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Leaf 2 months ago Posted 2 months ago. Direct link to Leaf's post “Can we add mentally for m...” more Can we add mentally for multiplication? If we can, can you write some examples? Answer Button navigates to signup page •Comment Button navigates to signup page (2 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer erin.buliga 2 months ago Posted 2 months ago. Direct link to erin.buliga's post “The only time you can rea...” more The only time you can really add mentally is using time tables. Basically, as you get older, you will have to memorize all multiplication up to 12x12. It might seem tedious at your age, but trust me, it will help you so much in the future. For example, you will be able to mentally know 2x4, 9x12, 11x6, etc. Above 12x12, you have to either do long multiplication or use a calculator. Comment Button navigates to signup page (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Hunter's Herporium_YT a year ago Posted a year ago. Direct link to Hunter's Herporium_YT's post “he eats a lot of avocado...” more he eats a lot of avocados I think I am allergic :) Answer Button navigates to signup page •Comment Button navigates to signup page (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer keelsonkidz a month ago Posted a month ago. Direct link to keelsonkidz's post “maybe,me too” more maybe,me too Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... Video transcript [Instructor] So as some of you already know, I really enjoy eating a good avocado, which, despite its appearance that it looks like a vegetable, but it's actually a fruit. And let's say that I eat two avocados per day, and I eat two avocados per day for six days. Now, there's a couple of ways that I could think about how many avocados did I eat? I could say, hey, I eat two a day, and I'm going to do that for six days, so I'm gonna add six twos together. So it'll be two, plus two, plus two, plus two, plus two, plus two, I have six twos right over there. And then I can add them together. And we could say two plus two is four. You add another two, you get to six. You add another two, you get to eight. Yet another two to get to 10. Yet another two, you get to 12. And that all is fine, but there's an easier way to express this repeated addition. One way is to view it as multiplication. Instead of just writing out six twos and adding them together, mathematicians have come up with a neater way of writing that. Let's say, okay, we're going to add up a bunch of twos. How many twos are we going to add up? We're going to have six of those twos, and we need to come up with some type of a symbol for it. So we will use this X-looking thing. And so, six times two can be viewed as repeated addition in exactly this same way. So six times two would be equal to 12. And we could go the other way around. If someone were to ask you, what is four times three? Pause this video and see if you can write it out as repeated addition, like we saw up here. Well, one way to interpret this is to say, this is four threes, so we could say this is equal to three, plus three, plus three, plus three. And three plus three is six, six plus three is nine, nine plus three is equal to 12. You might be familiar with skip counting, and you would say three, six, nine, 12. Just out of curiosity, what do you think three times four is going to be? Pause this video and try to represent it as repeated addition, and then see what you come up with. Well, we can interpret this as three fours. And so, we could say this is going to be four, plus four, plus four. And if we skip count fours, we'd have four, eight, 12, (chuckles) I was about to go to 16, four, eight, 12. So this is going to be 12. So this is interesting, at least for this example, for these two examples, I got to the same thing. Four times three got me the same result as three times four, interesting. I wonder if that's always true. But anyway, big picture from this video is that you can view multiplication as repeated addition. Creative Commons Attribution/Non-Commercial/Share-AlikeVideo on YouTube Up next: exercise Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. 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12321	https://www.drivecms.com/uploads/mcelroytutoring.com/GRE%20Vocab%20Capacity%202017%20Edition_%20Over%201,30%20-%20Brian%20McElroy.pdf	GRE Vocab Capacity 2017 Edition Powerful Memory Tricks and Mnemonics to Learn GRE Vocabulary Words Now with more than 1,300 mnemonics! By Brian McElroy and Vince Kotchian Contents Why This Book Is Different Mnemonic Examples Word Root Examples How To Use This Book Other Tips The Mnemonics Appendix: Word Roots Index Acknowledgments Why This Book Is Different There are tons of books, apps, and websites designed to help you learn GRE words. However, if you’ve tried typical vocabulary study methods, they might not have worked very well for you. The problem with most vocabulary products is that the sentences in the books are boring! Your brain might not naturally form connections to the meanings of words if they’re not presented to you in a memorable, creative way. GRE Vocab Capacity is different. We’ve not only clearly deﬁned the words but we’ve also created sentences designed to help you remember the words through a variety of unusual associations - using mnemonics. Mnemonic Examples A mnemonic is just a memory device. It works by creating a link in your brain to something else, so that recall of one thing helps recall of the other. This can be done in many ways – but the strongest links are through senses, emotions, rhymes, and patterns. Consider this example: quash (verb): to completely stop from happening. Think: squash. The best way to quash an invasion of ants in your kitchen is simple: squash them. Now your brain has a link from the word quash (which it may not have known) to the word squash (which it probably knows). Both words sound and look the same, so it’s easy to create a visual and aural link. If you picture someone squashing ants (and maybe get grossed out), you have another visual link as well as an emotional link. Here’s another example: eschew (verb): to avoid. Think: ah-choo! Eschew people who say "ah-choo!" unless you want to catch their colds. The word eschew sounds similar to a sneeze (ah-choo!), so your brain will now link the two sounds. If you picture yourself avoiding someone who is about to sneeze in your face, even better! Again, the more connections you make in your brain to the new word, the easier it will be for you to recall it. Word Root Examples Word roots are parts of words that often mean the same thing. For example, the root chron pretty much always has something to do with time: synchronize, chronological, etc. So knowing what word roots mean can be useful in helping you learn words. They can also provide a hint for the meaning of words that you don’t know. However, English is a funny language, and roots don’t always have the same meaning - it can depend on what word they’re in. So what you shouldn’t expect from roots is that they’ll be reliable to help you determine the deﬁnition of a word you don’t know. Keep in mind that knowing your word roots is not a substitute for knowing the actual deﬁnitions of words. We’ve included an appendix with this book that lists many common word roots along with their usual deﬁnitions, and an easy example word that uses the root. For example: chron: time. Think: chronological: arranged in order of time. We recommend learning all the word roots if you have time. It may be helpful to learn all the word roots ﬁrst and then begin learning words you don’t know, looking for instances of the roots to help you learn the new words. Word Root Tip: the more letters you can match from the beginning of a word you do know to a word you don’t know, the more likely it is they have similar meanings. Matching ﬁve or more letters is a good benchmark. For example, if you know that paciﬁst has to do with being peaceful, you’d be right in guessing that paciﬁc has a similar meaning since they both start with paciﬁ. How To Use This Book One tool we recommend is periodic review of ﬂashcards. The "tactile learning" (in other words, learning by doing) aspect of making your own cards is very effective. Buying pre-made ﬂashcards skips this step, and you miss out on an opportunity to "write it down in your head" by physically writing the words and deﬁnitions down yourself. Here’s an example of what a ﬂashcard might look like: Notice that the front of the card just has the word with its pronunciation. Using the word when talking to someone that day will help you learn it. The back of the card contains a brief deﬁnition, a mnemonic (which you can invent or ﬁnd in this book), a sentence using the word in a way that calls to mind its meaning, and any synonyms of the word you’d also like to learn (try an online dictionary like m-w.com to look up synonyms of a word). Here’s a method and chart that describe the order with which to study your cards: 1) Create 50 custom vocabulary ﬂashcards: a) Side A shows the word and its pronunciation. b) Side B shows the deﬁnition, part-of-speech, mnemonic, sample sentence, and synonym(s). 2) Label the cards with numbers 1-50 and stack them in order. 3) Use the table below to decide which cards to study and review each day. Other Learning Tips For troublesome words - or for any word you want to be sure of - we recommend writing down the word’s deﬁnition in your own words, then making up your own sentence using the word. Until you can explain something in your own words, you probably don’t fully understand it, and your own deﬁnitions and sentences will often be more memorable than ours. Some people go straight to the most bizarre looking words in the book, but don’t overlook the words that you “kind of” know but can’t easily deﬁne. That goes for words outside this book, too. If you can’t easily deﬁne a word you see in the newspaper, for instance, look it up! The context you read it in will help you learn it. For words that just won’t stick in your brain, try associating a movement with the word. Making a speciﬁc gesture with your body every time you study the word will provide an additional connection in your memory. A few more suggestions: Modify it. If you don't like our mnemonic or think that it can be improved, then use your own. They are your own customized cards to do with what you like. Even draw a picture or make a collage if it helps. Shufﬂe. When studying a group of cards, don’t always study them in the same order, so your brain won’t be able to associate one card with another. Study before bed. Studies have shown that memorization-type tasks are best studied during the hours preceding a full night's sleep. One popular theory to explain this phenomenon is that when we sleep, our mind organizes the day's events, starting with the most recent ones. One last tip: use it or lose it! The more you can work your new vocabulary into your daily speech and writing, the more you’ll remember it. You might sound nerdy, but it’s worth it. Our hope is that this book not only helps you improve your vocabulary, but also inspires you to start creating your own mnemonics! - Brian and Vince P.S. - We encourage you to write a review of GRE Vocab Capacity on Amazon.com to tell others about your experience with the book. Please contact us directly for suggestions that you might have so we can improve the book for future readers. We’re also available for private tutoring of the GRE, as well as the GMAT, SAT, ACT, and ISEE - either in person (in San Diego) or online (via Skype). mcelroy@post.harvard.edu (Brian McElroy) vince@vincekotchian.com (Vince Kotchian) www.McElroyTutoring.com or vincekotchian.com The Mnemonics abase (verb): to humiliate or degrade. “uh BASE” Think: give up a base. When you’re making out with someone, if you give up a base too quickly, then you just abase yourself. abashed (adjective): embarrassed. “uh BASHED” Think: Bashful the dwarf. When Snow White kisses him, Bashful gets so abashed that he blushes. abate (verb): to reduce. “uh BATE” Think: rebate. It may be annoying to have to mail it in, but the rebate on the new cell phone will abate its cost. aberration (noun): an exception or departure from the norm. “ah (rhymes with “nah”) burr A shun” Think: a bare Asian. Seeing a bare Asian would be an aberration – most people in Asia wear clothes. abeyance (noun): temporary inactivity; suspension. “uh BAY ants” Think: “obey” ends. When our lieutenant’s command to obey ends, our work plans are held in abeyance because we’re lazy. abhor (verb): to hate. “ab WHORE” Think: ab-whore. Daria abhors the tube-top-wearing blonde who stole her boyfriend and refers to her as an "ab-whore". abject (adjective): miserable; wretched. “ab-JEKT” Think: rejects. If she rejects my marriage proposal, I’ll be abject, with nothing to live for. abnegate (verb): to give up something; to deny oneself something. “ab nuh GATE” Think: abs negated. If you abnegate food, the fat covering your abs will get negated. abomination (noun): something awful. “uh BOM in A shun” Think: bomb a nation. It is an abomination to bomb a nation: civilians get killed. aboriginal (adjective): existing since the beginning. “AB or IDGE in ul” Think: original. In Australia, the original natives are the Aborigines - they are aboriginal since they were its ﬁrst inhabitants. abort (verb): to end prematurely. “uh BORT” Think: abortion. An abortion can abort a pregnancy. abound (verb): to be numerous. “uh BOUND” Think: abundant. Kangaroos abound in Australia; they’re abundant, bouncing around wherever you look. abrasive (adjective): causing irritation. “Uh BRAY sive” Think: braying donkey. Adopting a homeless donkey seemed great until I realized it would wake me up every morning with its abrasive braying. abridge (verb): to shorten. “uh BRIJ” Think: a bridge. A bridge would abridge my commute, which involves driving around the canyon. abrogate (verb): to get rid of; to abolish. “AB roh gate” Think: a broken gate. After my 120 lb. Mastiff decided to abrogate the barrier to the kitchen and eat from the garbage, we were left with a broken gate. abscission (noun): the shedding of leaves, ﬂowers, or fruits. “ab SIZH un” Think: scissors. Instead of waiting for the grapes to drop off of the vines, speed up the abscission by getting out there with a pair of scissors. absolute (adjective) complete and total. “ab so LUTE” Think: Absolut Vodka. The reason Absolut Vodka is more expensive than most brands is its superior purity; it is literally absolute vodka. absolve (verb): to free from guilt; to forgive. “ub SOLVE” Think: dissolve. Catholics believe that confessing to a priest will dissolve their guilt and absolve them from sin. abstemious (adjective): sparing; moderate. “ab STEM ee us” Think: abstinence. The health teacher knew that if he told students to be abstemious, some of them would still get pregnant, so he urged them to practice abstinence. abstruse (adjective): hard to comprehend. “ab STROOS” Think: abstract and confusing. The abstract strudel directions will confuse the new cook because they are abstruse. abysmal (adjective): awful. “uh BIZ mull” Think: Pepto-Bismol. When I had food poisoning, my stomach felt so abysmal that I had to drink a bottle of Pepto-Bismol. accede (verb): to express approval for; to give into. “uh SEED” Think: agreed. Since we all accede to the plan to seed the garden, it looks like we're agreed. accolade (noun): an expression of praise. “AK oh lade” Think: Escalade. I received many accolades for my service, but my favorite was the gift of a brand-new Cadillac Escalade. accretion (noun): growth via a gradual buildup. “uh CREE shun” Think: creeps up on. Gaining weight creeps up on a lot of people since they don’t notice the slow accretion of fat. accumulate (verb): to gradually increase. “Ah KYOOM you late” Think: cumulus clouds. We better pack up this picnic and leave - those cumulus clouds are accumulating and I think there’s gonna be a thunderstorm soon. acerbic (adjective): harsh; biting. “uh SIR bick” Think: acidic. On American Idol, Simon Cowell’s criticism was acerbic to the point of being acidic. acidulous (adjective): somewhat harsh. “uh SID you luss” Think: acid-ish. I like Sour Patch Kids because their acidulous taste is acid-ish without being too painful. acme (noun): the highest point of something “ACK me” Think: acne. In high school, I was plagued by acne: the acme of my nose was often a giant zit. acquisitive (adjective): eager to acquire and possess; greedy. “uh QUIZ zit tive” Think: a squid visited. An acquisitive squid visited my house and wrapped his arms around all of my valuable Chinese porcelain. acrimonious (adjective): bitter. “ak rih MOAN ee us” Think: a crime on us. Committing a crime on us makes us acrimonious. acumen (noun): insightfulness. “AK you min” Think: accurate men. In business, accurate men usually have acumen. adamant (adjective): stubborn; unyielding. “AD uh mint” Think: Adam…damn it! God was adamant that Adam not return to the Garden of Eden: “I said no, damn it!" adept (adjective): very skilled. “uh DEPPED” Think: adapt. Mountain lions can adapt to almost any climate and environment; they’re adept at survival. adequate (adj): appropriate; good enough. “AD uh quit” Think: had to quit. I had to quit at mile 15 of my marathon, but now that I think about it, 15 miles is plenty adequate for a solid workout. adhere (verb): to stick to something. “ad HERE” Think: adhesive. The adhesive on the Band-Aid made it adhere to my ﬁnger. admonished (verb): warned to do what's best. “ad MON isht” Think: add Monistat. "Add Monistat to your body if you're suffering from a vaginal yeast infection," the ad admonished. adorned (adjective): decorated. “uh DORNED” Think: add ornaments. If you adore Christmas, then you probably enjoy adorning your home by adding ornaments to your tree. adroit (adjective): skillful. “uh DROIT” (rhymes with “Detroit”) Think: a Droid. A Droid is an adroit cell phone since it can do so much. adulation (noun): excessive admiration. “ad joo LAY shun” Think: adult adoration. The grown adult’s adoration of role-playing video games could only be called adulation. adulterate (verb): to corrupt; to make impure. “uh DULT er ate” Think: adultery. In The Bible, God said, "Thou shalt not commit adultery." because an affair will adulterate a marriage. adversary (noun): an enemy or rival. “AD ver sare ee” Think: adversity Israel faces more adversity than most countries; it is nearly completely bordered with adversaries. advocate (noun:) supporter, ally. “AD voh kit” Think: I voted. I voted in the election, proving I’m an advocate of our democracy. aegis (noun): protection. “EE gis” Think: egg us Go ahead, try to egg us - our house has the aegis of the police, since my dad’s a cop. aesthetic (adjective): relating to beauty. “es THE tick” (rhymes with “pathetic”) Think: athletic body. If you're athletic, then you're likely to have a body that is aesthetically pleasing. affable (adjective): friendly. “AFF uh bull” Think: laughable. Since they want tourists to feed them, zoo giraffes are so affable that it's laughable. affectation (noun): an artiﬁcial way of behaving. “aff eck TAY shun” Think: a fake ﬁction. Madonna's phony English accent is an affectation; it is a fake ﬁction. afﬁliated (adjective): related to, intertwined. “uh FILL ee ate ed” Think: Philly I ate. When I went to Philly I ate a cheesesteak, because cheesesteaks are afﬁliated with Philadelphia. affront (noun): an insult or offense. “uh FRONT” Think: afro in front. I have a huge afro, so if I sit in front of people at a movie, they often take it as an affront. aggrandized (verb): made greater; enhanced. “uh GRAND ized” Think: a grand-sized. I aggrandized my social status by throwing a lavish party - it gave me a grand-sized reputation. aghast (adjective): struck by fear or amazement. “uh GASSED” Think: a ghost. I was aghast when I looked in the mirror and saw a ghost standing next to me. alacrity (noun): cheerful promptness. “uh LACK crih tee” Think: electricity. Electricity has alacrity, since it only takes a millisecond for the light to come on after I ﬂip the switch. algorithm (noun): a mathematical formula or procedure. “AL guh rhythm” Think: Al Gore’s rhythm. It's an inconvenient truth that, on the dance ﬂoor, Al Gore's rhythm is as dull and predictable as a computer algorithm. alleviate (verb): to soothe or lessen the severity of. “uh LEAVE ee ate” Think: Aleve. Allison’s headache was so bad that she took four Aleve pills to alleviate the pain. allusion (noun): an indirect reference. “al LEW shun” Think: A lewd sin. “Adjusting the antenna” is one of the funnier allusions to what some might consider a lewd sin. altruistic (adjective): unselﬁsh concern for others. “al true IST ick” Think: always true stick. My wingman is altruistic: he’s always true to me and will stick by my side when I hit on chicks - even if he’s not interested in any of them. amalgamate (verb): unify; join parts into a whole. “uh MAL gum ate” Think: gum. After breaking the vase, Malcolm used gum to amalgamate the pieces back together. ambiguity (noun): The state of being unclear or ambiguous. “am big YOU it ee” Think: a big “U” for undecided. When it came time to indicate her political party on the ballot, Virginia checked neither a big “D” for Democrat, nor a big “R” for Republican, but instead, a big “U” for undecided due to her ambiguity. ambivalence (noun): contradictory feelings toward something. “am BIV ull ents” Think: valence electron. The valence electron was ambivalent about which electrons he wanted to pair off with. According to his mother, he was unsure and afraid of commitment. ameliorated (verb): made better. “uh MEAL ee or ate id” Think: Emilio rated. Emilio rated my pasta as a 10 out of 10, which ameliorated my fear that I had ruined it. amenable (adjective): willing; cooperative. “uh MEN uh BULL” Think: amen-able. After she shouted "amen!", I was able to tell that she was amenable to my plan. amicable (adjective): friendly. “AM ick uh bull” Think: hammock-able. When Amy reminded me that her hammock was able to hold two people, I knew that she was amicable. amortize (verb): to gradually pay off or reduce. “am MORT eyes” Think: a mortgage. Unless you have a ton of money, when you buy a house, you probably amortize the loan with a mortgage. ample (adjective): enough or more than enough. “AM pull” Think: Apple. With about $200 billion in cash reserves (in 2015), Apple has ample resources. anachronism (noun): something belonging to a different time period. “an NACK ron is im” Think: inaccurate chronology. The movie has something inaccurate about its chronology: a caveman wearing a watch - a huge anachronism. analogue (noun): something similar. “an al LOG” Think: Analog vs. Digital. My audiophile cousin swears that analog is way better than digital, but to me they sound pretty similar. anathema (noun): something hated; a curse. “uh NATH em uh” Think: a nasty enema. If a patient is constipated, then a nasty enema may follow, which can be anathema for the nurse. anile (adjective): senile. “ANN ile” (rhymes with “dial”) Think: senile. I knew my Aunt Ann was anile to the point of being senile when she asked to go swimming in the Nile. animosity (noun): hatred; hostility. “an ih MOSS ih tee” Think: enemy city. During the war, I accidentally parachuted into the enemy city and was met with animosity. annotation (noun): a comment or note on a literary work. “ann oh TAY shun” Think: a notation. There are lots of annotations in my copy of Hamlet; I made a notation every time I needed to deﬁne an unfamiliar term. annul (verb): to cancel. “ann ULL” Think: null set. In mathematics, the null set means “a set that contains nothing.” If you annul (cancel) your marriage, you end it. anodyne (noun): a pain-reliever. “ann oh DINE” Think: am not dying. I have the ﬂu, but my doctor-prescribed anodyne ﬁnally has made me feel like I am not dying. anomaly (noun): something unusual. “uh NOM uh lee” Think: abnormally knobby knee. I have an abnormally knobby knee; my doctor tells me it’s an anomaly. antedate (verb): to come before. “AN tuh date” Think: auntie ante- (before) date. Chances are that your auntie has a birth date that antedates yours. antediluvian (adjective): ancient; primitive. “ann tee die LOUVE ee in” Think: anti-dildo-lovin’. Only someone with antediluvian views on sex would be anti-dildo-lovin'. antipode (noun): the exact opposite. “ANN tih pode” Think: anti-pole. The North Pole is the antipode to the South Pole - you might say they're "anti-poles." antithesis (noun): opposite. “ann TITH uh sis” Think: anti-thesis. You got a "D" on your essay because your examples argued for the antithesis of your introduction's thesis. apace (adverb): quickly. “uh PACE” Think: keep pace. The Indy 500 racer's pit crew changed his tires apace so he could keep pace with the leaders. apartheid (noun): the policy of separating groups based on race. “uh PAR thighed” Think: apart to hide. In South Africa, apartheid kept blacks apart to hide them from racist whites. aplomb (noun): conﬁdence. “uh PLOM” Think: the bomb. If you have aplomb, you think you're the bomb. apocryphal (adjective): of doubtful truthfulness. “uh POCK rih full” Think: apocalypse predictions. The prediction that the apocalypse would happen in 2012 turned out to be apocryphal. apoplectic (adjective): enraged. “ah puh PLEK tick” Think: Apu epileptic. In the Simpsons, when Nelson robbed his Kwik-E-Mart, Apu shook with apoplectic rage as if he was having an epileptic seizure. apostle (noun): a supporter. “uh PAH sill” Think: A posse. The famous rapper was known to roll deep with his many apostles – his posse, that is. apothegm (noun): a short, wise remark. “APP uh THEM” Think: pocket the gem. "Pocket the gem!" is a good apothegm to remember if you’re training to be a jewelry store robber. apotheosis (noun): a perfect example. “uh POTH ee oh sis” Think: a potent thesis. My professor said he gave me only A in the class because my paper was the apotheosis of a persuasive essay: it had a potent thesis. appease (verb): to soothe, satisfy or pacify. “uh PEAS” Think: please with peas. I appease and please my baby daughter by hiding her peas inside of her mashed potatoes. apportion (verb): to divide and distribute. “uh POOR shun” Think: a portion. If you want a portion of lunch, go ask the lunch lady - she apportions it to everyone. apposite (adjective): appropriate. “APP uh sit” Think: a positive site. Wikipedia is a positive site because it’s apposite for all kinds of research. approbation (noun): approval; praise. “app pro BAY shun” Think: approve probation. Maybe the best approbation I ever received was when the judge ﬁnally approved me for probation. apropos (adjective): relevant. “app pro POH” Think: appropriate. It’s apropos and appropriate that we’re talking about posing because I was just discovered and contracted to be a model! arbitrary (adjective): done without reason; random. “ARE bit TRARE ee” Think: varies a bit. When I order a pizza, the amount of toppings I get often seems pretty arbitrary – it always varies a bit. arcane (adjective): mysterious; known only to a few. “are CANE” Think: Ark of the Covenant. Indiana Jones understood the arcane Ark of the Covenant; the Nazis did not, which is why they perished. arch (adjective): sassy. “arch” Think: arched eyebrow. Her playful, arch comment made me arch my eyebrow. archaic (adjective): no longer current; outdated. “are KAY ick” Think: arch age. I knew you time-traveled here from the Roman Empire because your archaic expressions sound like you’re from the arch age. arduous (adjective): strenuous; difﬁcult. “ARE joo us” Think: hard for us. Clearing out that hoarder’s house is arduous; it’s hard for us because he kept every piece of junk mail he ever received. arid (adjective): very dry. “AIR rid” Think: Arrid Extra Dry. “Get a little closer; don't be shy! Get a little closer, with Arrid Extra Dry deodorant (which keeps your armpits arid)!” arrogate (verb): to unrightfully take or claim. “ARE uh gate” Think: a rogue ate. My liege – a rogue ate my rations – may I have more since he arrogated what was rightfully mine? articulate (adjective): using clear, expressive language. “are TICK you lit” Think: article. Oscar Wilde was so articulate that his conversational speech could be used as a newspaper article without any editing. artiﬁce (noun): deception; trickery. “ART ih ﬁss” Think: artiﬁcial. In The Hunger Games, Efﬁe Trinket tries to win people over with artiﬁce, but it doesn’t work because her sweetness is so artiﬁcial. artless (adjective): simple; without cunning. “ART less” Think: art-less ﬂirting. Flirting is an art I use less than most people; it’s deﬁnitely pretty artless when I just go up to a girl and tell her I like her. ascendancy (noun): governing or controlling inﬂuence. “Uh SEND and see”. Think: ascend and see. In battles, armies strive to take hills: ascending to higher ground makes it easier to see one’s enemy and leads to tactical ascendancy. ascetic (adjective): practicing self-denial. “uh SET ick” Think: asset? ick! The ascetic Buddhist monk, when offered the chance to take money or another asset, said “ick!” ashen (adjective): very pale. “ASH in” Think: ash. When he saw the ghost, his complexion became so ashen that his face was the color of ash. askew (adjective): slanted to one side. “uh SKEW” Think: skew. Your picture is askew because the earthquake skewed it from hanging evenly. asperity (noun): bad temper. “ass PEAR it ee” Think: a spear in me. I have asperity because I have a spear in me – can you blame me? aspersion (noun): a false claim intended to harm. “ass SPUR shun” Think: asp poison. Her aspersions about what I did last night felt like asp poison. aspiration (noun): a hope or ambition. “asp ear A shun” Think: as a pirate. As a pirate, I’ll be able to fulﬁll my aspiration of sailing the high seas and robbing the rich. assail (verb): to attack violently. “ass SALE” Think: ass sail. Come at me, bro: if I assail you, I'll make your ass sail out the window. assiduous (adjective): hardworking; dedicated. “ass SID you us” Think: assist us. Assiduous Sid worked his ass off to assist us. assuage: (verb): to make less severe. “uh SWAJ” Think: massage. astray: (adj): away from the correct path; into error. “ass TRAY” Think: a stray dog. A stray dog has gone astray from its family. astute (adjective): clever. “ass TOOT” Think: SAT student. I had an SAT student named Stu who was so astute that he got a 1600 on the SAT. attenuate (verb): to reduce. “at TEN you ate” Think: ten to eight. If you go from ten drinks a week to eight drinks a week, then you’ve attenuated your number of beverages. audacious (adjective): fearlessly bold; arrogantly bold. “awe DAY shus” Think: awed us. Walking up to Obama and swiping his pen as he was about to sign the bill was so audacious that it awed us. augment (verb): to increase the size of or to improve. “awwg MEANT” Think: Aug. meant. The arrival of Aug. meant that the colonists could augment their food storage by harvesting maize. august (adjective): majestic. “awe GUST” Think: Augustus Caesar. Augustus Caesar, the ﬁrst Roman emperor, was so august that they named a month after him. auspicious (adjective): favorable. “awe SPISH us” Think: suspicious I’m awesome. Dude, the chances she'll go out with me are auspicious or "awe-spicious" because I'm suspicious that I'm awesome. austere (adjective): plain; strict; serious; cold. “awe STEER” Think: Austria stern. Life among the Alps in Austria is stern and austere - it's hard to party when there's a wind chill of -20. authoritative (adjective): having impressive knowledge about a subject; conﬁdent. “Auth OR it tay tive”. Think: author. The reason we can speak authoritatively about GRE Vocab Capacity is that we wrote it: we’re the authors. automaton (noun): one who acts in a robotic way. “awe tah mah tahn” (rhymes with “on”) Think: automation. Working on assembly line where automation has replaced creativity can make you feel like an automaton. autonomous (adjective): operating independently. “awe tahn nom us” Think: Auto no mo’ us. With the advent of self-driving autos like the Google car, the cars won’t be needing us no mo’. avaricious (adjective): greedy. “ave uh RISH us” Think: have our riches. My boy band and I don't trust you as an agent - you're avaricious and just want to have our riches. aver (verb): to state conﬁdently; to declare. “uh VAIR” Think: verify. After I verify that the blood sample from the crime scene matches your DNA, I’ll aver that you are the killer. aversion (noun): A tendency to avoid or dislike. “uh VER shun” Think: cover versions. I have an aversion to cover versions of songs – I almost always prefer the original tune. avuncular (adjective): like an uncle. “uh VUNK you lure” Think: uncle. The avuncular professor was like an uncle to him, dispensing well-intentioned advice. badger (verb): to annoy or pester. “BAD jur” Think: bad jerk. Good jerks can get laughs, but a bad jerk will just badger you with his attempts at humor. baleful (adjective): threatening harm. “BALE full” Think: Christian Bale. I’m not a big Christian Bale fan – he always has that baleful look on his face, like he wants to start a ﬁght with you. banal (adjective): unoriginal. “buh NALL” Think: ban all. The banal librarian thought there were enough books already and wanted to ban all the new ones. base (adjective): not honest or good; having low quality or value. (rhymes with “face”) Think: basement. I live in my mom’s basement, but I don’t list that on my OkCupid proﬁle: there’s an unfortunate stereotype that people who live in basements tend to be base. battery (noun): a large group of similar things. “BAT er ee” Think: batter - y. I thought that my ﬁrst mix of cupcake batter tasted a little too batter-y, so I put it through a battery of taste tests before baking my ﬁnal batch. bauble (noun): a small, inexpensive piece of jewelry or toy. “BAW bull” Think: bobblehead. The bauble that my favorite baseball player gave me was a bobblehead of himself. baying (verb): shouting. “BAY ing” Think: Michael Bay movie. Even though I was in the other room, I could tell my roommates were watching a Michael Bay movie, like Transformers, because of all the baying from the T.V. beatiﬁc (adjective): extremely happy. “bee TIH ﬁck” Think: beautiful! terriﬁc! If you feel beatiﬁc, you probably walk around exclaiming, “beautiful! terriﬁc!” all day. beatify (verb): to bless; to make happy. “BEE tih ﬁe” Think: beautiful home = happiness. The makeover will beautify your home and beatify your family. becalm (verb): to make motionless; to soothe. “buh KAHM” Think: be calm! When my 3-year-old is running around causing havoc, I usually whisper “be calm!” to becalm him. bedlam (noun): a state of uproar and confusion. “BED lum” Think: bed lamb. It was complete bedlam when I entered my hotel room and saw that the bed had a lamb sleeping in it. beguile (verb): to trick. “buh GILE” (rhymes with “dial”) Think: be gullible. Be gullible, and you'll be easy to beguile. behemoth (noun): something huge. “buh HE mith” Think: beast mammoth. One really large beast was the woolly mammoth, a behemoth that lived during the Ice Age. beleaguered (adjective): weary, tired, bothered. “bee LEE gerd” Think: B - Leaguer. “I’m sick and tired of being a B-leaguer instead of an A-Leaguer,” said the B-movie actor. belied (verb): contradicted. “buh LIED” Think: lied. The used car salesman's smooth manner was belied by his sweaty handshake and made me think, "He lied!” belittle (verb): to put down; to disparage. “bee LITTLE” Think: be little. When you say “Good boy!” and pat me on head, you belittle me and make me feel as if I be little. bellicose (adjective): warlike; inclined to ﬁght. “BELL ih kose” Think: belly bellow. “ARRGHH!” When I heard the beast’s belly bellow, I knew it was bellicose. bemoan (verb): to mourn over; to express grief for. “bee MOAN” Think: moan. I be moanin’ about the new laws restricting what we can smoke - my friends bemoan the legislation, too. beneﬁcence (noun): the quality of being kind or charitable. “buh NIF uh sense” Think: beneﬁt sent. Through the beneﬁcence of musicians like Paul McCartney and Sting, the beneﬁt concert sent millions of dollars to starving children. benign (adjective): harmless. “Bee NINE” Think: be nice. It would be nice if the lump on my arm is benign instead of cancerous. bereft (adjective): deprived or robbed of something. “bee REFT” Think: he left. After he left her at the altar and crushed her dreams, she felt completely bereft. beseech (verb): to beg or ask. “bee SEACH” Think: screech. Forget fancy language – the best way to beseech someone is to screech at him. bifurcated (verb): split in two. “BY fur kated” Think: by forking. By forking, the road bifurcated into the popular road and the road less traveled by. bilious (adjective): bad-tempered. “BILL ee iss” Think: bully us. We goth kids are only bilious because the jocks like to bully us. blase (adjective): apathetic; unconcerned. “blah SAY” Think: blah say. I'm a rock star, so I'm blase and "blah blah blah" is all I say even when blazingly hot girls try to talk to me. blithe (adjective): happy, casual, unconcerned. “blythe” Think: glide. He just glides through life – he’s so blithe. bloviated (verb): was wordy/windy when speaking. “BLOW vee ate ed” Think: blow hot air. In Harry Potter, Gilderoy Lockhart bloviated; he would blow a lot of hot air without much meaning. bludgeon (verb): to hit forcefully. “BLUJ in” Think: Bludgers. In Quidditch, the Bludgers are 10-inch, black, iron balls that ﬂy around and sometimes bludgeon players. bonhomie (noun): a pleasant and friendly mood. “bahn nom EE” Think: abundance of homies. When I have an abundance of homies, I have bonhomie. boon (noun): a beneﬁt. “boon” Think: booing. One boon of booing is that it unites an audience in mutual unappreciation. boor (noun): a crude person with rude, clumsy manners. “boar” Think: boar manners. The boor had table manners like a wild boar and ate directly off the plate with his mouth. bootless (adjective): useless. “BOOT less” Think: booty-less. A booty-less pirate is probably a bootless pirate. bowdlerize (verb): to cut out all the offensive parts of a book. “BOWED lure eyes” Think: boulder-ize. Originally, they would bowdlerize Huckleberry Finn so much that they might as well have let boulders roll over the book and tear out half the pages. bravado (noun): a false show of bravery; swagger. “bruh VAH doe” Think: brave avocado. Though its trash-talking seemed brave, the avocado and its bravado didn’t scare me, since I knew it was just a piece of fruit. brazen (adjective): shamelessly bold. “BRAY zen” Think: blazin’. Blazin’ up a joint during class is certainly brazen, but it’ll get you expelled 100 out of 100 times. brevity (noun): shortness of duration. “BREV it ee” Think: abbreviate. I know your speech is brief but abbreviate it even more - this professor actually awards points for brevity. bromide (noun): a cliché or tired saying. “BRO myed” Think: bro lied. My bro on the lacrosse team told me to "give 110 percent," but the next day my math teacher told me that was impossible. Bro lied in his bromide. brusque (adjective): abrupt; curt; harsh. “brusk” Think: brushed off. I tried to make friends with the club's bouncer, but he was brusque and brushed me off. bucolic (adjective): rustic; rural. “byoo CAA lick” Think: blue collar. I’m just a bucolic broccoli farmer - a blue collar worker - I don’t understand what those suits are talking about! bugbear (noun): something to fear. “BUG bear” Think: bug a bear. If you bug a bear, you'll soon have a very serious bugbear. bulwark (adjective): a strong support or protection. “BOOL work” Think: bull work. In a bullﬁghting arena, the barrier to protect the spectators from the bull better work; it has to be a bulwark. bumptious (adjective): assertive in a loud, arrogant way. “BUMP shus” Think: bump us. You’re the type of guy who would push past us in a crowd and bump us and not say you’re sorry – you’re bumptious. bungle (verb): to screw up. “BUNG gull” Think: bunghole. I bungled the job so many times that they started calling me a “bunghole”. buoyant (adjective): happy; conﬁdent. “BOY ent” Think: boo-yah! If you hear someone yell “boo-yah!” then you can bet she’s feeling buoyant. burdensome (adjective): oppressive; causing difﬁculty or worry. “BIRD den sum” Think: bird dim sum. The decor of this Chinese restaurant is nice, except for the giant vulture circling our table. The burdensome feeling that bird gives me makes it hard for me to enjoy my dim sum. burgeoning (adjective): growing. “BURJ un ing” Think: burgers. If you eat too many burgers, your waistline will be burgeoning. buttress (noun): a support. “BUT ress” Think: butt rest. The stone column is both a buttress and a butt rest for tired people to lean against. bygone (adjective): past. “BY gone” Think: bye gone. The bygone days of my childhood are days I’ve said bye to cause they’re gone. byzantine (adjective): devious; complicated. “BIZ in teen” Think: busy ant. Only the busy ant will be able to make its way through the byzantine maze you've created. cache (noun): a secure storage place or something in that place. “kah SHAY” Think: cash hiding place. The drug dealer kept his cash in a cache under the bed - he didn't trust banks. cacophony (noun): a harsh, unharmonious sound. “kuh KAW fun ee” Think: cough symphony. The sounds from the tuberculosis ward were a cacophony - an unpleasant cough symphony. cadge (verb): to beg or get via begging. “cadj” Think: locked in a cage. If you’re locked in a cage, you’ll cadge for food and water. cajole (verb): to coax. “ka JOLE” Think: cage hole. At the vet, I have to cajole my cat out of the cage hole so he can get examined. calamitous (adjective): related to a terrible event. “ka LAM it us” Think: calamari vomit. It's calamitous when you eat undercooked calamari, become vomitous, and puke on your date. callous (adjective): unsympathetic; hard-hearted. “KAL us” Think: callus. The callous dictator thought nothing of executing his rivals; he must have had a callus on his soul. callow (adjective): inexperienced; immature. “KAL owe” Think: shallow. Popping her gum while reading Cosmo, the callow teenager was shallow only because she hadn’t seen much of the world yet. calumnious (adjective): slanderous, defamatory, an untrue statement intended to injure one's reputation. "kuh-LUM-nee-us" Think: gossip column. The author of the famous gossip column was less concerned with provoking lawsuits through his calumnious statements than he was with attracting hordes of readers through salacious headlines. camaraderie (noun): togetherness. “com uh ROD er ee” Think: comrades. The Russian camera factory workers shared a sense of camaraderie, calling each other comrades. canard (noun): a false report or rumor. “cuh NARD” Think: Qatar. There’s a widespread canard that Qatar bribed FIFA to host the World Cup. canny (adjective): clever. “CAN knee” Think: can knee. I’m canny because I can use my knee to drive my car when I need both hands for something else. canonize (verb): to make a saint (literal) or to put someone beyond reproach (ﬁgurative). “CAN nun eyes” Think: can on ice. In order to properly canonize St. Patrick, one must keep a beer can on ice at all times. capacious (adjective): spacious. “ka PAY shus” Think: Batman’s cape is spacious. Batman is a big guy, so his cape is spacious and capacious. capitulate (verb): to surrender. “ka PIT chew late” Think: capsized? it’s too late. Once your boat has capsized, it's too late to think about winning the race: capitulate and just try not to drown. capricious (adjective): impulsive; done without forethought. “ka PRISH us” Think: capri pants. Jenny made the capricious decision to buy ﬁve pairs of capri pants, which she later regretted when they went out of style. captious (adjective): overly critical. “CAP shus” Think: red CAPS. Our English teacher is captious: our papers come back with lots of red writing that’s in all CAPS. cardinal (adjective): of main importance. “KAR din ull” Think: cardinal bird. You'd think the bright red male cardinal (noun) was the most cardinal (adjective) bird because of its vivid color. caricatured (verb): distorted, often comically. “Care ick cah chured” Think: cartoon character. Check out any political cartoon character and you’ll see someone caricatured: any cartoon of Trump will have enormous hair. castigate (verb): to criticize severely. “KAS tig ate” Think: castrate. The worst way for a Maﬁa boss to castigate someone is to castrate him. caterwaul (verb): to cry or to complain loudly. “CAT tur wall” Think: cat in wall. If there’s a cat in your wall it will probably caterwaul since it wants to get out. causal (adjective): relating to the cause of something or causing something. “KAW zul” Think: cause. There is a causal link between laziness and poor grades; being lazy causes less studying and therefore lower marks. (Don’t confuse causal with casual). celerity (noun): quickness. “suh lear it ee” Think: accelerate. After Cee Lo switched to an all-celery diet, he lost 30 pounds and his ability to accelerate increased, as did his celerity. censure (verb): to criticize harshly. “SEN sure” Think: censor. If you really wanted to censure your rival’s editorial you could just censor it completely. cerebral (adjective): intellectual. “suh REE brul” Think: cerebrum. Einstein was so cerebral that they studied the cerebrum of his brain after he died. chagrin (noun): distress caused by disappointment. “shuh GRIN” Think: chuggin’ tragic. You won't believe this, but to my chagrin, Chad is chuggin' a bottle of mouthwash right now – this is a tragic date. champion (verb): to ﬁght for. “CHAM pee in” Think: champion (noun). If you champion (verb) that turtle in the turtle race and cheer for her really loudly, it’s more likely she’ll become the champion (noun). chary (adjective): very cautious. “cherry” Think: chair-wary. My brothers were always pulling my chair away as I was about to sit down, so now I’m chary, or chair-wary. chicanery (noun): trickery. “shi CAN er ee" Think: chick-gain-ery Your frat brother’s feigned interest in that cute girl's paintings was clearly chicanery; his motive was "chick-gain-ery". choleric (adjective): irritable. “CALL er ick” Think: cholera. I'd be choleric too if someone’s fecal matter made me get cholera. chronological (adjective): ordered by time. “kron oh LAH ji kull” Think: chron = time. The Houston Chronicle is a newspaper that, like any good journal, reports events in chronological order. All its reporters wear chronometers (watches) to keep track of their deadlines. churlish (adjective): rude; difﬁcult. “CHURL ish” Think: church lush. The church lush usually showed up to mass stumbling drunk, inviting us to call him churlish. cinematic (adjective): suggestive of a movie. “Sin im AT tick” Think: cinema. The video you shot on your iPhone is cinematic enough to be shown in a cinema. circuitous (adjective): roundabout; not direct. “sir KYOO it us” Think: circuit-ish. The crooked cabdriver took a circuitous route; his path was circuit-ish to increase the fare. circumscribed (adjective): restricted. “SIR kum skribed” Think: circumference scribe. The evil scribe drew a magical circumference around our campsite, which circumscribed our movement to that circle. circumspect (adjective): cautious. “sir kum SPECT” Think: circle inspect. When you rent a car, walk in a circle to inspect it for dents; if you’re not circumspect now, they may charge you later. circumvents (verb): avoids; gets around something. “sir kum VENTS” Think: circumference vents. She circumvents the guards by crawling around the enemy base's circumference through the vents. clairvoyant (adjective): able to see the future. “clare VOY int” Think: clear voyage. My trusted psychic, Miss Cleo, assured me that rowing a boat from California to Hawaii would work out just ﬁne. "I see a clear voyage in your future," said the clairvoyant woman. clandestine (adjective): secret. “klan DEST in” Think: clan of destiny. Because we’re the clan of destiny, we have to keep our meetings clandestine – otherwise, the empire will kill us all. clangorous (adjective): loud; noisy. “KLAYNG er us” Think: clang. The disinterested 3rd graders in the school band clanged on their instruments as hard as they could, producing a clangorous racket. clemency (noun): mercy. “KLEM in see” Think: Clemens mercy. The pitcher Roger Clemens was shown mercy by the jury and found not guilty - an act of clemency, since he was accused of taking steroids. climatic (adjective): pertaining to climate and weather. Think: dramatic. Due to global warming, climatic events such as hurricanes and ﬂoods have been much more dramatic in recent years. (Don’t confuse with “climactic”, which refers to the climax of a work of art.) climax (noun): the most intense, exciting, or important part of something. “CLY max” Think: climb ax. The climax of our ascent of Mt. Everest was deﬁnitely reaching the summit; our climb ax enabled us to scramble up the ﬁnal few feet. cloying (adjective): gross because it's too much. “KLOY ing” Think: annoying. Always talking baby-talk to each other, the couple was so annoying that they were cloying. coalesce (verb): to unite into a whole. “koh uh LESS” Think: coal essence. Coal, in essence, is just carbon – if you squeeze it hard enough it will coalesce into a diamond. cocksure (adjective): overconﬁdent. “KOCK (rhymes with “dock”) sure” Think: cocky and sure. The baseball rookie was so cocky and sure that he'd hit a home run during his ﬁrst at-bat that he was cocksure. coddle (adjective): to treat with excessive care. “KOD ul” Think: cuddle. The mother dog would coddle and cuddle her puppy so much that I thought it would never learn to fend for itself. coerced (verb): forced. “co URSED” Think: cooperate by force. I didn't want to leave the bar, but the bouncer coerced me to cooperate by using force. coeval (adjective): existing at the same time, contemporary. “Coh EE vill” Think: co-evil. Hitler, Mussolini, and Franco, three of history’s most evil rulers, were coeval because they all lived during the same era. cognizant (adjective): aware; informed. “KOG nih zent” Think: recognize. If you're cognizant of our theory, you must recognize where our solution came from. coherence (noun): the quality of being understandable. “Co HERE ents” Think: can hear it. Public Speaking 101 taught me that the ﬁrst rule of coherence when giving a speech is speaking loudly - make sure your audience can hear it. cohesive (adjective): holding together well. “co HEESE ive” Think: adhesive. A cohesive argument holds together even when attacked – as if it’s strengthened by an adhesive. cohort (noun): a friend or companion. “KOH hort (rhymes with “short”)” Think: co-heart. My cohort and I are so close that it feels more like we’re co-hearts. coin (verb): to invent a new word or phrase. “KOIN” Think: coin (noun). Just as the U.S. mint molds metal into a new coin (noun), so we can coin (verb) new expressions. collusion (noun): the process of working together to deceive, often illegally by businesses. “cuh LUGE un” Think: co-illusion. When the world’s two main exporters of oil decided to create the co-illusion of scarcity when there was none, the media accused them of price gouging and collusion. commensurate (adjective): equal or proportionate. “kuh MEN sur it” Think: co-measure it. Our estimates of the carbon content of this dinosaur bone will be commensurate if we co-measure it. commiserate (verb): to sympathize with. “co MISS ur ate” Think: misery loves company. Come commiserate with us - misery loves company. companionable (adjective): sociable; friendly. “kum PAN yin uh bull” Think: companion-able. Most dogs are companionable and love people; that’s why they’re so able to be companions. complicit (adjective): involved in a crime. “kum PLISS it” Think: accomplice. Though I robbed the bank and my accomplice just drove me there, he was considered complicit by the law. composure (noun): calmness. “kum POSE ure” Think: composer’s calm pose. Even though he was performing his music for kings and queens, the composer’s calm pose showed his composure. compunction (noun): regret, remorse. “com PUNK shun” Think: punctured balloon. I felt compunction after accidentally puncturing the child’s birthday balloon and making him cry. concession (noun): admitting partial or total defeat. “kuhn SESH in” Think: confession. After it became apparent that my opponent would win the election, my concession speech was basically just a confession that I lost. concoct (verb): to make; to invent to deceive. “kuhn COCKED” Think: con cocked I’ve concocted a safer plan – we’ll con the bank teller by showing her a cocked (but unloaded) pistol. concomitant (adjective): accompanying, especially in a less important way. “con COM it ent” Think: can come with it. Drinking too much carries the concomitant risk of depression that can come with it. concord (noun): harmony. “CON kord” Think: concurred. We all concurred that we should go into the grape jelly business, so it’s no surprise that our company is enjoying a feeling of concord. concupiscence (noun): strong desire, esp. sexual desire. “con KYOOP uh sense” Think: Cupid’s essence. If you have concupiscence, you have Cupid's essence running through your veins. condign (adjective): deserved; appropriate. “con DINE” Think: can dig. I can dig the murderer's conviction because it was condign. condones (verb): allows something that is bad. “con DONES” Think: con done. The con done it because the lazy warden condones misbehavin’. conferred (verb): given to. “kuhn FURD” Think: fur coat. My great-grandmother’s fur coat was conferred to me in her will. conﬁscate (verb): to take something away. “KAHN ﬁss kate” Think: can frisk. If I can frisk you and feel that you’re carrying a weapon, then I’ll conﬁscate it. conﬂagration (noun): a ﬁre. “con FLUH gray shun” Think: burning ﬂag nation. There is a debate about the ﬂag in our nation - is it legal to use the Stars and Stripes for a conﬂagration? conﬂate (verb): to confuse. “con FLATE” Think: con inﬂated. The con artist inﬂated the value of the racehorse by grooming it, making me conﬂate sleek appearance with speed. confound (verb): to confuse (a person) or mix up (a thing), or as an exclamation (“confounded” only). “kun FOUND” Think: can’t ﬁnd. I can’t ﬁnd my keys anywhere and I’m confounded as to where they may be. Where the heck are my confounded keys? conglomerate (verb): to gather into a whole. “kuhn GLAH merr it” Think: can gobble it. My cookie got smashed into a thousand pieces – I’ll have to conglomerate them so I can gobble it. conniving (verb): secretly plotting to do bad things. “cun NIVE ing” Think: mean girls’ knives. Mean girls act nice, but don't be conned: they're conniving to stick knives in your back. connoisseur (noun): an expert; one who knows the subtleties of a subject. “con iss URE” Think: can know sir. When it comes to breakdancing, call me “can know sir” because I’m a connoisseur of the art. conscientious (adjective): driven by the urge to do what's right; careful. “con she ENT shus” Think: consciences. Conscientious people usually are driven to do good deeds by their consciences. conspicuous (adjective): noticeable. “kun SPICK you us” Think: can pick on us. Since you’re new at this school, here’s a tip: don’t wear anything conspicuous. If the jocks see us wearing anything that stands out, they can pick on us more easily. consternation (noun): confusion; agitation; dismay. “con ster NAY shun” Think: concerned nation. On 9/11/01, a concerned nation stood in consternation watching the aftermath of terrorism. contumacious (adjective): stubbornly disobedient; rebellious. “con too MAY shus” Think: contrary tummy. I have a contrary tummy – it’s contumacious and gives me indigestion if I try to eat spicy food. conundrum (noun): a difﬁcult problem. “con NUN drum” Think: nun drum. Building a nun drum is a conundrum because nuns don't like loud noises. conversant (adjective): familiar with. “con VERSE int” Think: converse it. If you’re conversant with something, you can converse about it intelligently. copious (adjective): plentiful. “KOPE ee us” Think: copy us. If the zombie apocalypse happens and we survive, let’s hope cloning can copy us and make humans more copious. cordial (adjective): affectionate. “KORD jull” Think: cordial (noun). The alcohol in the cordial (noun) made me act more cordial (adjective). cordon (verb): to enclose, either to restrict or to protect. “KORD un” Think: cord on. Cordon off that area by tying a cord on and around all the trees so people know it’s off limits. corroborate (verb): to support with evidence. “kuh ROB er ate” Think: co-robber. When robbing a bank, use a co-robber who will corroborate your story. cosmopolitan (adjective): sophisticated. “kos muh POL it in” Think: Cosmo. After reading the dating advice in Cosmo, the 14-year-old thought she was quite cosmopolitan. covert (adjective): not openly shown. “co VIRT” Think: covered. The CIA agent was on a covert mission, so he covered his true identity. cowed (adjective): intimidated. “COWD” Think: coward. The bully was at heart a coward: as soon as I stood up to him he was cowed into silence. craven (adjective): cowardly. “KRAY vin” Think: Wes Craven. My craven roommate refused to go to the Wes Craven movie – it was way too scary for her. credence (adjective): belief. “KREED ints” Think: Creed is. If you tell me that Creed is your favorite band, then I won’t give any credence to your musical judgements. credulous (adjective): too ready to believe things. “KREDGE uh liss” Think: cradle us. When we are young children and our parents cradle us, we tend to be quite credulous – we believe anything they tell us (just ask Santa). crepuscular (adjective): related to twilight. “cru PUS q lur” Think: creepy muscular. In the movie Twilight, creepy, muscular vampires prowl during the crepuscular hours of the evening. crestfallen (adjective): dejected; depressed. “CREST fall en” Think: Crest fallen. When I saw that my Crest toothpaste had fallen off my brush into the sink, I was crestfallen since that was a waste of perfectly good toothpaste. cryptic (adjective): having an unclear or hidden meaning. “KRYP tick” Think: crypt. Scrawled in blood on the wall of the mummy’s crypt, the cryptic hieroglyphics both confused and frightened us. cull (verb): to select; to get rid of what’s unneeded. “KUHL” Think: kill. We’re in a famine, so we need to cull the herd and kill the sick cattle. culminate (verb): to reach a point of highest development. “CULL min ate” Think: coal mine diamond. I remember the old days, when you were just a lump of coal in the mine with everybody else. Now, thousands of years of pressure from the Earth’s crust have culminated in you becoming a diamond! culpable (adjective): deserving blame. “KULP uh bull” Think: culprit. Unsurprisingly, the cop thought the culprit he had arrested was culpable. cumbersome (adjective): awkward due to large size. “KUM bir some” Think: cucumber. It felt cumbersome to walk down the beach with a gigantic cucumber down the front of my Speedo. cunning (adjective): cleverly forethought, often in a tricky or deceptive way. “KUN ing” Think: cunning kung-fu. In my opinion, the best kind of kung-fu is cunning kung-fu, where you seek to defeat your opponent through deception instead of just physical skill. cupidity (noun): greedy desire for. “cue PID it ee” Think: Cupid. After being shot by Cupid's arrow, Sarah developed such cupidity for her valentine that she called him daily. curmudgeon (noun): a grumpy old man. “kur MUDGE in” Think: curse mud. Only a curmudgeon would curse the mud in the garden on this sunny spring day. cursory (adjective): hasty; superﬁcial. “CURSE uh ree” Think: curse sorry. I only gave my rental car a cursory inspection, which led me to curse and be sorry later when I noticed a huge dent. curtail (verb): to lessen. “kur TAIL” Think: cut off your tail. If you really want to win this lizard beauty pageant, you've got to be shorter. curtail your length - cut off your tail. cynical (adjective): believing that people are generally selﬁsh and dishonest. “SIN ick ull” Think: sin ick. I used to be optimistic, but now that I’m older and more cynical, I expect that, given the chance, most people will take whatever they want even if they have to sin. Ick! cynosure (noun): something that guides or stands out. “SINE uh sure” Think: sign to be sure. Polaris (the North Star) was a cynosure for ancient sailors, a sign they could be sure of. daunt (verb): to intimidate or discourage. “DAUNT” Think: don’t! My mean old aunt Mildred would often daunt me when I was younger by screaming, “don’t!” whenever I got too loud. dearth (noun): lack. “DEARTH” Think: dead earth. Due to the dead earth of our farmland, there will be a dearth of food this winter. debacle (noun): a complete disaster. “duh BAHK ul” Think: da bottle. I’m an alcoholic, so when I hit da bottle, the night usually becomes a debacle. debased (adjective): lowered in value or reputation. “dee BASED” Think: de-base. Milk chocolate is a crime against the cacao bean. Confectioners start off with a base of pure dark chocolate, but then they debase it by adding milk powder and tons of sugar. debauchery (noun): extreme indulgence in pleasure. “duh BOTCH er ee” Think: the bachelor party. During the bachelor party, the wolf pack in The Hangover participated in some serious debauchery. debilitate (verb): to weaken. “duh BILL it ate” Think: decrease ability. Cancer will often debilitate its victims and can decrease their ability to be active. decadent (adjective): decaying; self-indulgent. “DEK uh dent” Think: decayed. In WALL-E, the decadent passengers of the spaceship have decayed into overweight, lazy, passive lumps. decimate (verb): to destroy a large part of. “DESS uh mate” Think: decimal remains. At the start of our campaign, all of our soldiers were healthy, but attacks and disease have decimated the ranks so that only a decimal remains alive. declaimed (verb): spoke loudly and self-importantly. “dee CLAIMED” Think: “I declare!” "Well, I declare!" the Southern belle declaimed. decorous (adjective): well-behaved. “DECK or us” Think: the chorus. Kids in the chorus are usually not rebels - they’re often decorous. decrepit (adjective): worn-out; run-down. “duh CREP it” Think: scrap it. Your decrepit old car looks like crap; you should scrap it. decried (verb): expressed strong disapproval about. “duh CRIED” Think: cried. After my boss decried my work in front of everyone, I went home and cried. defamatory (adjective): something that hurts someone's reputation. “duh FAM ih tory” Think: de-fame. The defamatory Enquirer story will "de-fame" that actor; he'll lose his fame. defenestrate (verb): to quickly throw out. “duh FEN eh strate” Think: defense demonstrate. If you defenestrate a burglar through a plate-glass window, your home defense is demonstrated. defunct (adjective): no longer existing. “duh FUNKED” Think: de-function. When I can ﬂy in my dreams, the law of gravity seems to be defunct, like it has been "de-functioned". degenerate (verb, adjective) verb: to move backward or decay, adjective: decayed. “duh JENN er it” Think: Jenner ate my dust. 1976 Olympic decathlon champion Caitlyn Jenner ate my dust when I challenged her to a footrace; I guess that her speed has degenerated with age. delectable (adjective): delightful; delicious. “duh LECT uh bull” Think: delicious electable. Ryan Gosling should run for president since most women think he's delectable and delicious enough to be electable. deleterious (adjective): harmful. “duh luh TEER ee us” Think: deletes. Using that old computer could be deleterious to your grade since it randomly deletes ﬁles. delimit (verb): to determine the limit, boundary or extent of something. “dee LIM it” Think: determine limit. Partying until 3 am with your friends is a great way to determine your limits, but the next-morning hangover might lead you delimit your alcohol intake the next time you go out. delineate (verb): to outline; to describe in detail. “Dee LIN ee ate” Think: the line. The strip of masking tape I put down is the line that clearly delineates the two halves of this dorm room - keep your stuff on your side! demagogue (noun): a leader who gains power by trickery. “dem a GOG” Think: demigod. The cult was led by a demagogue; he manipulated followers into thinking he was a demigod. demarcate (verb): to deﬁne; to set apart. “de MARK ate” Think: mark it. If you want to demarcate your side of the dorm room, mark it with a long piece of masking tape. demean (verb): to lower in character, status, or reputation.“dee MEAN” Think: mean. Thanks to the jocks’ demeaning comments to him, the new kid went home after his ﬁrst day at our school and told his mom that we’re all mean. demeanor (noun): one’s appearance and behavior. “duh ME ner” Think: meaner personality. Not only has my ambition diminished with age, but so has my demeanor: I used to have a meaner personality. demotic (adjective): popular; common. “duh MOTT ick” Think: democratic. Obama uses demotic language in his speeches to seem more democratic. demur (verb): to object. “duh MURE” Think: murmur. Though no one has spoken up yet, the murmur from the class suggests they demur to my idea that they do more homework. denigrate (verb): to attack the reputation of or to put down. “DEN ih grate” Think: deny I’m great. If you deny I'm great, you denigrate me. denizen (noun): inhabitant; one who is often at a place. “DEN uh zen” Think: den citizen. One of the denizens of the caves in my woods is a black bear - he's a bear den citizen. denuded (verb): stripped bare. “duh NOOD id” Think: nude. Loggers denuded the forested rise, felling trees and trampling undergrowth until it was just an nude hill of earth. depiction (noun): a representation. “Dee PICK shun” Think: the picture. The picture I drew of myself in kindergarten was a crude depiction of a human being: my self-portrait had no torso. deplete (verb): to use up. “duh PLEET” Think: delete. Depleted uranium has had some of its radioactivity deleted. deplore (verb): to hate. “duh PLORE (rhymes with “ﬂoor”)” Think: deep lore. I deplore (hate) when my uncle likes to tell that campﬁre ghost story like it’s part of some deep lore that goes back generations: the truth is that he’s just repeating the plot of his favorite horror movie. depredate (verb): to take by force; to ravage; to ruin. “DEH pruh date” Think: predator. The predators in the forest will depredate your village’s livestock if you don’t build a really good fence and get guard dogs. deride (verb): to make fun of. “duh RIDE” Think: Dee’s ride. We all deride Dee's ride - it's a brown 1987 Buick with ghetto rims. derivative (adjective): lacking originality. “duh RIH vuh tiv” Think: derivative relatives. My father likes to claim that his recipes are unique, but the truth is that he learned everything he knows about cooking from Aunt Jean. In other words, his recipes are derivative of his relative. descry (verb): to catch sight of; to discover. “dih SCRY” Think: describe. Ok, now that I descry the iceberg that we’re sailing towards, I can describe it to you. desecrate (verb): to violate something sacred. “DEH suh krate” Think: de-sacred. If you peed on an altar, you would desecrate it, or “de-sacred” it - it would no longer be sacred. desiccated (adjective): dried out. “DEH si kate id” Think: desert sick. The desert made me sick because the dry heat desiccated my body. despoiled (verb): stripped of value. “duh SPOILED” Think: spoiled. Desperate for oil, the U.S. drilled in Alaska and despoiled the land, and act which spoiled it for future generations. despot (noun): an all-powerful ruler. “DES put” Think: despicable. History has shown us that despots - like Kim Jong Il -are often despicable human beings. desuetude (noun): disuse. “DES wuh tude” Think: disuse attitude. The unnecessary security guard at the knitting store had an air of lazy desuetude about him - kind of a disuse attitude. deteriorate (verb): to worsen over time. “duh TIER ee or eight” Think: the terrier ate. He’s really cute, but the terrier ate all the sofa cushions while we were gone – he’s really making our decor deteriorate! devoid (adjective): completely lacking. “duh VOID” Think: the void. The void of deep space is devoid of air, warmth, or life. devolve (verb): to become less advanced over time. “duh VAHLV” Think: the Volvo. When I bought the Volvo in 1988, it was state-of-the-art, but since then it has slowly devolved into a hunk of junk. devout (adjective): deeply religious or loyal. “duh VOUT (rhymes with “out”)” Think: devote. I’d say I’m devout – I have devoted my entire life to studying the bible. dexterity (adjective): skill; good coordination. “dex TERR it ee” Think: Dexter. The ﬁctional serial killer Dexter has a grisly dexterity about the way he kills people. diabolical (adjective): devilish. “dia BALL ih cull” Think: die abolish. Your law that makes cigarettes part of school lunches is diabolical and will cause children to die... abolish it! diaphanous (adjective): so ﬂimsy as to be see-through. “die APH in us” Think: Diana’s fan. Princess Diana's delicate rice-paper fan was diaphanous. diatribe (noun): an angry speech. “DIE a tribe” Think: die tribe. I didn't understand the words of his diatribe, but I guessed the native said I'd die from his tribe killing me. dichotomy (noun): two-part, polarity, contrast. “die KOTT uh me” Think: Thy cot, oh my. Thy cot, oh my – it’s so comfortable when I’m sleeping in it, but my back hurts so much when I get up. didactic (adjective): designed to teach. “die DAKT ick” Think: dictionary tactic. The deﬁnitions in a dictionary use the tactic of explaining words clearly in order to be didactic. difﬁdent (adjective): timid. “DIFF uh dent” Think: difﬁcult dentures. I’m difﬁdent when in public because I’m self-conscious about how weird my difﬁcult dentures look. digression (noun): a departure from the main topic. “duh GRESH in” Think: Dig Russians. “Have I ever mentioned to you that I dig White Russians?” said Lebowski, trying to change the subject when confronted about Bunny’s failed rescue. dilatory (adjective): tending to procrastinate. “DILL a tor ee” Think: delay later. The dilatory gator liked to delay things until later. dilettante (noun): a dabbler; one with superﬁcial knowledge of an area. “DILL uh taunt” Think: diluted. The dilettante’s knowledge of the subject was, understandably, diluted. dilute (verb): to lessen the concentration of. “duh LOOT” Think: dilution is the solution. If your drink is too strong, then dilution is the solution: just add ice! dint (noun): force; power. “DINT” Think: Hulk’s dent. The Incredible Hulk made a dent in the car by dint of his enormous strength. dire (adjective): desparate. “DAHYER (rhymes with “ﬁre”)” Think: die. If you are afraid that you might die, then the situation is dire. discomﬁt (verb): to embarrass or confuse. “dis KUM ﬁt” Think: discomfort. Realizing one's suit had been replaced with a too-tight Speedo would discomfort and discomﬁt anyone. disconcert (verb): to confuse or frustrate. “dis KUN sirt” Think: diss the concert. To liven up recitals, I disconcert the musicians by dissing the concert. discreet (adjective): having or showing self-restraint and good judgment. “dis KREET” Think: this secrET. I’m pregnant - but please, be discreET and keep this secrET - if my parents ﬁnd out, they’ll kill me. discrepancy (noun): a difference, divergence, or disagreement. “dis KREP in see” Think: this crepe vs. Nancy’s. There seems to be a large discrepancy between the size this crepe of mine and that of Nancy’s... I wonder whether she took a bite of mine while I wasn’t looking. discrete (adjective): individually distinct; separate. “dis KREET” Think: Crete. The Greek island of Crete is discrete because it doesn’t touch any other land. discriminate (verb): to notice subtle variations. “dis KRIM in ate” Think: one meaning is criminal; one is neutral. Discriminate (verb), so you'll know when "discriminate" is about prejudice and when it's about noticing. disgruntled (adjective): displeased. “dis GRUNT ulled” Think: grunted. The fat warthog grunted to show he was disgruntled with his small dinner. dismantle (verb): to take apart or destroy. “Dis MAN till” Think: Mickey Mantle. A 16-time baseball All-Star, Mickey Mantle often dismantled opposing teams with his brilliant hitting. dismissive (adjective): showing rejection and contempt for. “dis MISS ive” Think: dismiss. When she sings “Call Me Maybe”, Carly Rae Jepsen is dismissive because she dismissed all the other boys who tried to chase her. disparage (verb): to insult or put down. “dis PARRIAGE (rhymes with “marriage”) Think: despair and rage. He felt despair and rage because the rapper liked to diss and disparage him. disparate (adjective): distinct; different. “DISS per it” Think: This parrot vs. that pirate. This parrot is disparate (different) from that pirate on whose shoulder it is sitting. They are disparate species, after all...even if they do look a bit alike. dispassionate (adjective): not passionate / interested. “diss PASH in it” Think: not passionate. Dis = not, so dispassionate = not passionate (not interested). dispatch (noun): speed; efﬁciency. “DIS patch” Think: dispatcher. If you want a job as a dispatcher - using the radio to direct police - you'd better have dispatch. displacing (verb): removing from the usual place. “Dis place” Think: dis place to dat place. Don’t think of your demotion as me displacing you - I’m just moving your desk from dis place to dat place. disputatious (adjective): inclined to argue. “dis pyoo TAY shus” Think: dispute. After being pulled over, the disputatious lawyer unwisely disputed the accuracy of the cop's radar gun. dissemble (verb): to mislead, hide or conceal. “dis EM bull” Think: disassemble gun. The terrorist tried to dissemble his plan by disassembling his gun before trying to smuggle it through airport security. disseminated (verb): spread out. “dis EM in nate id” Think: diss 'em, Nate. His dad advised to “diss ‘em, Nate”, so Nate disseminated ﬂyers all over the school that criticized his opponents in the election. distension (noun): swelling. “dis TEN shun” Think: dis-tension. A belly showing distension after a huge meal might be because the person has weak abs with no muscle tension. dither (verb): to stress out from indecision. “DITH ur” Think: ditz do either. You’re such a ditz – you’d do either and it’s making you dither. diurnal (adjective): daily; of the daytime. “die URN ul” Think: The urinal. My use of the urinal is diurnal – I pee every day. divergent (adjective): moving in different directions. “duh VERJE int” Think: Two roads diverged. “Two roads diverged in a yellow wood,” begins the famous Robert Frost Poem, “The Road Not Taken.” divisive (adjective): creating disunity. “di VIE sive” Think: divide. Yoko Ono had a divisive effect on The Beatles, dividing the group into two parts. docile (adjective): Calm, even-tempered. “DOSS ill” Think: doctor. The docile doctor remained calm even though his patient was clinging to life by a thread. doctrinaire (noun): rigid and dogmatic. “DOCK trih NAIR (rhymes with “hair”)” Think: Doctorate in Air. I once met an academic with a Doctorate (Ph.D) in Air, and I asked him whether oxygen molecules always travel in pairs. “Yes, they do,” he said, “...with zero exceptions.” doggedness (noun): stubborn determination. “DOG ed ness” Think: dog-ness. The ﬁghter's doggedness, even after he was knocked down, was like that of a fearless Bulldog. doggerel (noun): poorly written verse. “DOG ur ul” Think: dog verse. Most Valentine's Day card poems are such doggerel that it seems as though dogs wrote the verse. dogmatic (adjective): stubborn; inﬂexible. “dog MATICK” Think: dog bath. My dog automatically becomes dogmatic if you try to give him a bath, since he hates water. dolorous (adjective): sad; mournful. “DOLL ur us” Think: Dolores’s doldrums. I’d be dolorous and in the doldrums too if my name were Dolores. dormant (adjective): temporarily inactive. “DOOR munt” Think: doorman. If you work as a doorman, you know that most of the time you’re just standing there, dormant. dour (adjective): gloomy; stern. “DOUR” Think: sour. The teacher's dour expression made her pupils feel sour. draconian (adjective): cruelly strict. “druh KONY en” Think: Draco Malfoy. If Draco Malfoy had taught the Gryfﬁndor students, I'm sure he would have been a draconian instructor. droll (adjective): funny. “DROLL” Think: roll (with laughter). Droll humor makes me roll with laughter. dubious (adjective): doubtful. “DOO bee us” Think: dubious doob. That is a dubious doob, my friend – it looks like oregano if you ask me. dudgeon (noun): a tantrum caused by being offended. “DUDGE in” Think: dungeon grudge. I was in high dudgeon after they threw me in the dungeon for jaywalking, and I held a grudge. dupe (verb): to trick. “DOOP” Think: dope. A dope is easy to dupe. duplicitous (adjective): deceptive. “due PLISS it us” Think: duplicate-ness. Politicians try to make everyone like them, but their two-faced duplicate-ness is duplicitous. dwindle (verb): to gradually become smaller. “DWIN dull” Think: candle. Our excitement at exploring the cave quickly turned to fear when we saw that our candle had burned low, dwindling into a stub that wouldn’t last for long. dyspeptic (adjective): grumpy. “diss PEP tick” Think: Pepto-Bismol. This Pepto-Bismol will prevent indigestion and the resulting dyspeptic mood. ebullient (adjective): excitedly enthusiastic. “uh BOOL ee int” Think: Red Bull. After I chugged a giant Red Bull, I felt extremely ebullient. eclectic (adjective): varied. “ek LEK tick” Think: selection collection. If your musical tastes are eclectic, I can probably name any style selection and you'll say it’s in your collection. effaced (verb): made less visible. “uh FACED” Think: erased. On old nickels, Thomas Jefferson's face is often effaced to the point of almost being erased. effete (adjective): without strength or vitality; weak; soft. “uh FET” Think: feeble. The former athlete became effete and feeble from years of just sitting on the couch. efﬁcacious (adjective): effective. “eff ick A (sounds like the letter) shus” Think: effectiveness. If you have senioritis, a brief vacation is an efﬁcacious way to increase your effectiveness. efﬂorescence (noun): blossoming. “eff lerr REH since” Think: ﬂorist. Efﬂorescence is all around me – I’m a ﬂorist. efﬂuvium (noun): an invisible, often harmful, vapor. “uh FLOOVE lee um” Think: ﬂu. The efﬂuvium coming from the ﬂu patient’s mouth infected the nurse. effrontery (noun): shameless boldness. “uh FRONT ur ree” Think: fronting homies. What’s with these homies dissin’ my girl? Why do they gotta front? (Because they have effrontery, Weezer.) effusive (adjective): extremely expressive. “uh FYUSE ive” Think: fussy. Imagine if Nicki Minaj was your grandma? She's so effusive she'd make a fuss over your every accomplishment. egalitarian (adjective): based on the belief in human equality. “ee gal ih TAIR ee in” Think: equal eagle. In the U.S., our egalitarian belief that all men are created equal is symbolized by the bald eagle. egregious (adjective): bad in an obvious way. “uh GREE jis” Think: outrageous. Her saying that she had to wash her hamster was such an egregious and outrageous excuse that it made me say "Jesus!" eldritch (adjective): weird; eerie. “el DRISH” Think: elf-witch. The elf-witch Galadriel in The Lord Of The Rings was eldritch because of her ability to speak inside our heads. emancipate (verb): to free. “eh MAN sih payt” Think: Emancipation Proclamation. President Abraham Lincoln issued the Emancipation Proclamation on January 1, 1863. It declared “that all persons held as slaves” within the rebellious states “are, and henceforward shall be free.” embellish (verb): to decorate. “em BELL ish” Think: bells. Hanging little bells all over your home is one (weird) way to embellish it. embroiled (verb): in a difﬁcult situation. “em BROY uled” Think: on broil. I was embroiled in a dangerous situation when I got locked in an oven set on "broil". embryonic (adjective): in an early stage. “em bree ON ick” Think: embryo. It's pretty obvious that a human embryo is embryonic when compared to an adult human. eminent (adjective): respected, famous, well-known. “EM ih nint” Think: Eminem. If asked to name the most eminent white rapper, I wouldn’t think twice: Eminem is an easy choice. (Sorry Macklemore.) emollient (adjective): soothing. “uh MOLE ee int” Think: emo. Listening to emo music has an emollient effect on my emotions because it’s so sensitive. emphatic (adjective): forceful. “em FAT ick” Think: emphasize. When I yell at people, I emphasize every word to be more emphatic about my demands. empirical (adjective): observed, ﬁrsthand. “em PEER ih cuhl” Think: miracle. If you’re claiming that there was a miracle here, then I would like to see some empirical evidence that it happened. encomium (noun): praise. “en COH mi um” Think: in Comic-Con. In Comic-Con, the entertainment convention, nerds give encomium to the latest comic-book movies. encompass (verb): to include. “en KUM pess” Think: compass. Use this compass to draw a circle around the things you want to encompass. encroaching (verb): gradually invading one's rights or property. “en KROACH ing” Think: roaches. My apartment's roaches are encroaching upon my space: they now occupy the kitchen. enervating (adjective): tiring. “EN ur vating” Think: renovating. Renovating their kitchen by themselves not only got on the couple's nerves, but also was extremely enervating. enigmatic (adjective): mysterious, unpredictable. “en igg MAT ick” Think: dark matter. Dark matter comprises most of the universe but remains enigmatic to scientists. enmity (noun): hatred. “EN mit ee” Think: enemy. I have enmity for my enemy - what else would you expect? ennui (noun): dissatisfaction resulting from boredom. “ON we” Think: ennui there yet? Take a seven-year-old on a long car ride, and you’ll hear the ennui in his voice when he repeatedly asks, “ennui there yet?” ensorcelled (adjective): bewitched; enchanted. “EN sir celled” Think: sorcerer. The sorcerer ensorcelled the adventurers with a powerful spell that made them forget who they were. entitled (adjective): pompous, conceited. “en TIGHT ild” Think: N possible titles. If your name begins with “Count,” “Duke,” “Prince” or any other of N possible titles, then you’re probably entitled. entreat (verb): to plead. “en TREAT” Think: in retreat. In retreat, the ﬂeeing general entreated us to spare his soldiers’ lives. ephemeral (adjective): ﬂeeting; short-lived. “Eh FEM er ul” Think: FM for all. Since satellite radio is ten times better than normal radio, the days of FM for all are ephemeral. epitome: adjective, the purest or best example of something. “uh PIT uh me” Think: epic tome When it comes to classic literature, "Moby Dick" is the epitome of an epic tome. equivocal (adjective): intentionally unclear. “uh QUIV oh cull” Think: equally vocal. The equivocal politician was equally vocal about both sides of the issue. eradicate (verb): to wipe out. “ee RAD ih kate” Think: radiate. You can radiate food to eradicate the bacteria in it. ersatz (adjective): being an artiﬁcial and inferior substitute. “AIR sotts” Think: Er, Saltz? When I tried using Saltz by Pﬁtzer (TM) instead of actual table salt, I found it to be an inferior substitute. erstwhile (adjective): former. “ERST while” Think: Hearse after a while. It’s important to prepare for the future, but also to live for the moment. One must remember that our lives will eventually become our erstwhile lives, because we’re all destined for the hearse after a while. erudite (adjective): knowledgeable (from studying). “ERR ooh dite” Think: he read it. My English professor is so erudite: every time I bring up a great book I don’t think he knows about, it turns out that he read it. eschew (verb): to avoid. “us CHOO” Think: ah-choo! Eschew people who say "ah-choo!" unless you want to catch their colds. esoteric (adjective): known by only a few people. “ess oh TERR (rhymes with “err”) ick” Think: isolated terriﬁc. Einstein’s esoteric knowledge isolated him from most of his peers since his acumen was so terriﬁc. espouse (verb): to support or to give loyalty to. “ess POUZE” Think: spouse. Chances are that you will espouse your spouse - you married her, so you probably have her back. espy (verb): to glimpse; to catch sight of. “uh SPY” Think: I spy. I spy something blue - do you espy it, too? estimable (adjective): worthy. “ESS tim uh ble” Think: esteem-able. If someone is estimable they are "esteem-able", i.e., they're deserving of your positive regard. estranged (verb): separated in a negative way. “uh STRANGED” Think: stranger. Gotye's estranged girlfriend cut him out and treated him like a stranger and it felt so rough. ethereal (adjective): delicate; heavenly; insubstantial. “uh THEER ee ul” Think: other than real. My God, Joyce - your meringue cookies are ethereal - so light, so delicious - they must be something other than real. etiolated (verb): made pale; weakened. “EE tea uh late id” Think: toilet-ed. Keeping my goldﬁsh in the bleach-containing toilet tank violated his trust and etiolated him so much that he turned white. euphemism (noun): an inoffensive term used in place of an offensive one. “YOOF ih mism” Think: use feminism. Use feminism if you’re a guy and want to create a euphemism for PMS – otherwise you might get yourself killed. eurytopic (adjective): tolerant of many different environments. “you ree TOP ick” Think: Europe tropics. That plant is eurytopic because it grows both in cold, rainy Europe and in the hot, humid tropics. evanescent (adjective): ﬂeeting; lasting only brieﬂy. “eh van ESS sint” Think: vanish scent. The cologne’s fragrance will vanish soon; its scent is evanescent. evinced (verb): revealed. “uh VINSED” Think: evidence. Vince evinced the villain by providing evidence. exacerbated (verb): made more severe; aggravated. “eggs ZASS er bait id” Think: exasperated. I’m exasperated - not only did you get us lost in the woods, but you also exacerbated the situation by dropping our phone in that swamp. exact (verb): to take. “egg ZACKT” Think: Exacto knife. I’m going to exact my share of the cookie with an Exacto knife. exacting (adjective): requiring strict attention to detail. “egg ZAKT ing” Think: exact. Our exacting architecture professor demanded that our model be drawn exactly to scale. exaggerate (verb): to overestimate or underestimate, stretch the truth or hyperbolize. “eks ADGE urr ate” Think: ex’s age rate. My ex likes to exaggerate his age rate in the reverse direction: last year he told people he was 39, but now he’s claiming that he’s 38. excise (verb): to take or cut out. “eck SIZE” Think: ex-size. During liposuction, doctors excise fat – your current size will be an ex-size and you’ll be skinny. excoriated (verb): strongly condemned. “ex KOR ee ate id” Think: scoured. Simon Cowell’s criticism on American Idol excoriated the contestant – she felt as if she’d been scoured by a rough dish pad. exculpated (verb): freed from blame. “ex CULL pate id” Think: ex-culprit. If you commit a crime but have a clever lawyer, you'll be exculpated and be an "ex-culprit". execrable (adjective): detestable; awful. “ex eh CRUH bull” Think: excrement. Your unfunny jokes about excrement are execrable – they’re shit. exigent (adjective): requiring immediate and/or signiﬁcant action. “EX ih gent” Think: exit gent. When there’s an exigent problem, Clark Kent becomes an exit gent and returns as Superman. exodus (noun): the departure of many people. “EX uh duss” Think: exit us! During the Syrian civil war, there was a mass exodus of refugees who must have been thinking, “exit us!” exorbitant (adjective): excessive. “ex ORB it int” Think: extra for orbit. The fancy space hotel charged exorbitant fees due to the extra costs needed to orbit the earth. expatiate (verb): to speak or write about in detail. “ex PAY she ate” Think: paid she ate = food critic. Sarah loved to eat and to write, so she decided that she watned to expatiate as a food critic and get paid when she ate. expatriate (noun): one who has moved to a foreign country. “ex PAY tree ate” Think: ex-patriot. We said our expatriate friend was an anti-American ex-patriot since he moved to France. expedient (adjective): helpful in a practical way. “ex PEED ee int” Think: speedy. To be speedy, I booked my ﬂight on Expedia.com; it was more expedient than calling the airline. explicate (verb): to explain, analyze, or develop an idea. “EX plih kate” Think: looks like “explain”. To explain is to make explicit, or explicate. exponent (noun): a supporter of something. “EX poe nint” Think: ex-opponent. Upon Romney's nomination, McCain became his exponent for the greater good of the GOP and therefore was his ex-opponent. expunge (verb): to get rid of. “ex PUNJ” Think: ex with sponge. The best way to make a spill an “ex-spill” is to use a sponge to expunge the mess. expurgate (verb): to cleanse. “EX per gate” Think: purge. By the time they were done expurgating the “offensive” parts from Huckleberry Finn, there was almost nothing left; they purged almost everything. extant (adjective): present or existing (opposite of extinct). “EX tint” Think: existing ant. Since he was about to get stepped on, "I exist!" exclaimed the ant to the elephant. extemporaneous (adjective): done without preparation. “ex tem per RAIN ee us” Think: ex= without, tempo= time. If you don’t have time to prepare, then your speech will have to be extemporaneous. extenuating (adjective): less serious due to a partial excuse. “ex TEN you ate ing” Think: extension. I got an extension on my paper because there were extenuating circumstances – I got trampled by an elephant. extirpate (verb): to get rid of completely. “EK stir pate” Think: exterminate. If you have pests in your house that you want to extirpate, call someone who will exterminate them. extol (verb): to praise highly. “ek STOLL” Think: ex-toll. I extol this highway because it used to charge a toll, but now it’s an ex-toll road. extraneous (adjective): not important. “eks TRAIN ee us” Think: extra strain. Just give me the facts, ma’am. All these extraneous details are putting an extra strain on my memory. extrapolate (verb) to infer, conclude or draw a conclusion based on another observation or fact. “eks TRAP oh late” Think: Extra police = we’ll be late Due to the fact that there are extra police on the highway today, and trafﬁc is at a standstill, I’m guessing that there was a big accident. Hence, I can extrapolate that we’ll be late to work today. exult: verb, to show great happiness. “eggs ult” Think: ultimate ex I exulted in the fact that my ex still plays ultimate Frisbee with me, since ultimate has always been my one true love. fabricate (verb): to make up in order to deceive. “FAB rick ate” Think: fabric background. The movie set background was fabricated, woven from fabric to resemble a mountain range. facetious (adjective): playfully funny. “fa SEE shus” Think: face “E”. The facetious comedian made us smile so much that our faces looked like we were constantly saying “E”. (try it!) fallible (adjective): capable of making an error. “FAL (rhymes with “pal”) ih bull” Think: fail-able. Jenkins! The rookie agent you picked is fallible – for him, the mission is extremely fail-able. fanatic (adjective): full of extreme enthusiasm. “fuh NAT ick” Think: fan lunatic. The fanatic Green Bay Packers fan - a lunatic - painted his face green and wore a cheesehead hat every day of the year. farce (noun): a comical, unrealistic, mocking display or show. “FARSE” Think: farts. I know your play is a farce because of how many times the characters fart. fastidious (adjective): having very picky standards. “fuh STID ee us” Think: fast to tidy up. My roommate is fastidious about cleaning; she gets mad if I am not fast to tidy up the apartment. fatuous (adjective): lazily foolish. “fat SHOE us” Think: fat ass. If you’re fatuous about nutrition, you might end up with a fat ass. fawning (verb) kissing up to. “FON ing” Think: fawn (baby deer). The little fawn's only hope to get the bear to spare its life was by using fawning behavior. feckless (adjective): weak; worthless; irresponsible. “FEK liss” Think: F in class. If you get an F in class, your study habits were probably feckless. fecund (adjective): fruitful; inventive. “FEE kund” Think: feces under. Spreading manure, i.e., feces, under your crops as fertilizer will make your harvest fecund. feign (verb): to fake. “FAYN (rhymes with “rain”)” Think: faint. Here’s the plan: we feign illness by pretending to faint as soon as the ﬁnal exam begins so we won’t have to take it. felicity (noun): happiness. “fu LISS it ee” Think: feline city. If you're a cat person, then a feline city would bring you great felicity. Meow. ferret (verb): to bring to light; to uncover. “FERR it” Think: ferret (the animal). If you keep a ferret as a pet, it will ferret out all your lost earrings since they can crawl under anything. fervor (noun): passion. “FURR ver” Think: fever. The lovers’ fervor for each other was so great that their skin felt fever-hot. festoon (verb): to decorate. “feh STOON” Think: festival. The harvest moon festival is coming! Time to festoon the barn for the big dance! fetid (adjective): bad-smelling. “FED id” Think: feet. Feet are often fetid. ﬁasco (noun): a disaster. “fee ASS koh” Think: ﬂask. Smuggling that ﬂask into the football game was a ﬁasco; we got kicked out in the ﬁrst quarter. ﬁlial (adjective): like a son or daughter. “FIL ee ul” Think: afﬁliated. It was the son's ﬁlial duty to care for his dying mother since he was afﬁliated with her by blood. ﬁllip (noun): stimulus, impetus, ﬂick. “PHIL ipp” Think: ﬁll up my tank. If you could please ﬁll up my tank of gas, then that would be a ﬁllip to my transportation abilities. ﬁnagled (verb): obtained, often through trickery or indirect methods. “ﬁh NAY gulled” Think: ﬁnagle a bagel. Even though I had lost my wallet, I ﬁnagled a bagel from the bagel lady by claiming I had invented cream cheese. ﬁnicky (adjective): difﬁcult to please. “FIN ick ee” Think: nit-picky. The princess was nit-picky – she was so ﬁnicky that she refused to sleep on the mattress with a pea under it. ﬁtful (adjective): irregular; intermittent. “FIT full” Think: ﬁt-full. Our new baby only sleeps ﬁtfully – the night seems full of his crying ﬁts. ﬂagrant (adjective): shockingly bad. “FLAY grint” Think: fragrant vagrant. No matter what your feelings towards homeless people are, you can’t deny that man is a fragrant vagrant - his B.O. is so bad it’s ﬂagrant. ﬂeeting (adjective): short-lived. “FLEET ing” Think: ﬂee. My stay in the village was by necessity ﬂeeting: a dragon attacked, and I had to ﬂee. ﬂippant (adjective): lacking respect or seriousness. “FLIP int” Think: ﬂip off. If you're ﬂippant, you probably ﬂip people off on a regular basis. ﬂorid (adjective): overly decorated; reddish. “FLOR id” Think: ﬂowered. The 12-year-old girl’s room was ﬂowered with hundreds of red-hued decorations - her style was ﬂorid. ﬂotilla (noun): a ﬂeet of ships. “ﬂo TILL uh” Think: ﬂoating around Godzilla. Floating around Godzilla was a ﬂotilla from the Japanese navy. ﬂotsam (noun): ﬂoating debris. “FLOT sim” Think: ﬂoat. After the Titanic sank, Rose was able to survive by climbing onto a piece of ﬂoating ﬂotsam from the wreckage. ﬂounder (verb): to act clumsily or ineffectively. “FLOUN dir” Think: ﬂop under. Bad dancers ﬂounder (verb) through clubs like ﬂounders (noun) that ﬂop under the seat of the boat once they’re caught. ﬂourish (verb): to thrive. “FLERR ish” Think: ﬂorist. My ﬂowers always ﬂourish, because I have 20 years experience as a ﬂorist. ﬂouted (verb): treated without respect. “FLOUT id” Think: ﬂung out. The rebel ﬂouted the rules so badly that he ﬂung them out the window. ﬂuctuate (verb): to change or go back and forth. “FLUCK chew ate” Think: ﬂocked. Sometimes customers have ﬂocked to our restaurant, and other times we are half empty: demand seems to ﬂuctuate. ﬂummoxed (adjective): confused. “FLUM ixed” Think: ﬂume ox. At the water park, I was completely ﬂummoxed when, in the ﬂume ride, I saw an ox swimming along. foible (noun): a minor weakness of character. “FOY bull” Think: foil-able. Your plan to take over the world is foil-able because you have many foibles. foment (verb): to encourage the growth of. “FOH ment” Think: form it! In Star Wars, Princess Leia fomented the Rebel group by telling the Rebels to “form it!” forage (verb, noun) v: to search for food, n: food, a search. “FORE ige” Think: for aged cheese. Being French, I have been foraging our supermarkets and farmer’s markets far and wide for aged cheese that reminds me of home. forbearance (noun): patience; tolerance. “four BEAR ints” Think: bear tolerance. I'm usually harsh on people who borrow my money, but for bears I have more tolerance and practice forbearance since they scare me. foreground (verb): to highlight. “FOUR ground” Think: foreground (noun). That boy is magic! Foreground (verb) his talent by making sure he's in the foreground (noun) of the stage! forestall (verb): to delay, hinder, or prevent. “four STALL” Think: for stall. The booby traps I surrounded my fort with will forestall invaders – they’re for stalling. formidable (adjective): serious, respectable, worthy. “FOR mid ih bull” Think: forbid. The cliff face was formidable; it seemed to forbid us from even attempting to climb. fortitude (noun): strength. “FORT ih tude” Think: fortress. The Bulgarian weightlifter's mental fortitude during training gave him a body that looked like a fortress. fortuitous (adjective): lucky. “for TWO it iss” Think: fortunate for us. It was fortuitous and fortunate for us that the polar bear we encountered had just eaten a seal and was too full to eat us. fracas (noun): a noisy brawl. “FRAH kiss” Think: frat ruckus. The frat brothers often caused a ruckus by getting into a drunken fracas. fractious (adjective): cranky. “FRAK shis” Think: fracture us. The fractious football player is best avoided: if his team loses, he gets mad enough to fracture us. fraternize (verb): to be friendly with. “FRAH turn eyes” Think: frat. We frat brothers fraternize with all the freshman chicks so they'll come to our parties. frenetic (adjective): wildly excited or active. “fruh NET ick” Think: frenzy of energy. Have you ever watched a Pug play? It’s frenetic - like a chubby little frenzy of energy. froward (adjective): stubbornly disobedient. “FRO werd” Think: afro. I try to straighten my hair but it’s froward - after a few hours, it’s a fro again. frugal (adjective): thrifty; inclined to save money. “FROO gull” Think: fructose corn syrup gal. I’m too frugal to use healthy sweeteners - I’m a high-fructose corn syrup gal. fruition (noun): a productive result. “froo ISH un” Think: grow fruit. My plans to grow my own oranges came to fruition when my orange tree produced fruit. fudge (verb): to fake or falsify. “FUDJ” Think: “oh, fudge!” If the salesperson fudges the facts about the used car you buy, you’ll be saying “oh, fudge!” later when it breaks down. fuliginous (adjective): obscure; murky; dark. “few LIDGE in is” Think: full of gin. After he was full of gin, James Joyce composed poetry so moody and fuliginous that few could appreciate it. fulsome (adjective): abundant, sometimes disgustingly so. “FULL some” Think: full of some. In the U.S., we are full of some crops - for instance, corn here is so fulsome that we put it in nearly every food product. funereal (adjective): like a funeral. “fyoo NERR ee ul” Think: funeral. Your gothic style is so funereal it looks as though you're headed to a funeral instead of the mall. furor (noun): an outburst of rage or excitement. “FURE rir” Think: furious. The governor's use of the Fuhrer’s (Hitler's) image in an ad made many furious and created a political furor. furtive (adjective): done by stealth. “FIR tive” Think: furtive fart. Watch out for that kid - he will fart in class but it's so furtive that he never gets blamed. gadﬂy (noun): someone who annoys by being very critical. “GAD ﬂy” Think: egad, ﬂy! Brad is such a gadﬂy about my outﬁts that I want to say “Egad, ﬂy!” and hit him with a ﬂyswatter. gaffe (noun): a social mistake. “GAFF” Think: laugh. It’s deﬁnitely a gaffe to bring your pet giraffe to the party - everyone will laugh. gainsay (verb): to deny. “GAIN say” Think: against say. Those who gainsay us are against what we say. gallant (adjective): courageous; noble. “GAL int” Think: galloping knight. She’s still single because she’s waiting for a gallant knight to come galloping in on his horse and sweep her away. gambit (noun): a move made to try to gain an advantage. “GAM bit” Think: gamble it. When you play chess, sometimes you have to gamble it and use a gambit by sacriﬁcing a piece for a better position. gamboled (verb): danced around happily; frolicked. “GAM bulled” Think: game ball. After she scored three goals and led the team to victory, the coach awarded her the game ball and she gamboled all over the place. garble (verb): to make hard to understand. “GAR bull” Think: gargle. A bad connection can garble a voicemail to the point that the message just sounds like someone mid-gargle. gargantuan (adjective): enormous. “gar (rhymes with “far”) GAN shoo-in” Think: gigantic. The gargantuan orangutan was so gigantic that it needed a special enclosure at the zoo. garrulous (adjective): talkative, chatty, prone to discussing trivial things. “GARE rule luss” Think: girls rule us. The reason those girls rule us is that they have a talent for being garrulous – we can barely get a word in during conversation. gauche (adjective): awkward. “GOSH” Think: go douche. The gauche thing about Summer's Eve commercials is that they're basically telling you to go douche. gaudy (adjective): ﬂashy in a tasteless way. “GODDY” Think: gawd ugly. The rapper's inch-thick gold chain was so gaudy that even his fans said, "gawd that's ugly!" genial (adjective): good-natured. “JEAN ee ul” Think: genie. If you sign up to be a genie and to grant people wishes, you’re probably by nature genial. germane (adjective): relevant; appropriate; ﬁtting. “jer MAIN” Think: yer main. Enough digressions! Stick to yer main point; unless your remarks are germane, I get distracted. germinate (verb): to grow or to cause to grow. “JERM in ate” Think: germ in Nate. After entering his nose, the germ in Nate was able to germinate into a cold because he was so run down. ghastly (adjective): horrid. “GASSED lee” Think: ghostly. Looking in the mirror and seeing a ghostly ﬁgure behind me was ghastly. gild (verb): to make attractive, often deceptively. “GILLED” Think: gold. I’m a terrible painter, so instead I usually gild my vases with gold so they look okay. glacial (adjective): slow and/or cold. “GLAY shull” Think: glacier. My answer had a glacial (slow) pace, and the interviewer gave me a glacial (cold) look that made me feel like I was on a glacier. glancing (adjective): indirect. “GLANSE ing” Think: glance (verb). The knight only glanced sideways at his opponent; as a result, his lance’s blow was glancing and didn’t inﬂict any damage. glaring (adjective): obvious; harshly bright. “GLARE ing” Think: glare. My resume had such a glaring typo that my interviewer just sat there and glared at me until I left. glowered (verb): looked at with anger. “GLAH werd” Think: glow RRR. The scary, frowning jack-o'-lantern glowered at us - its glow seemed to say "RRRRRRRR!" glut (noun): too much of something. “GLUT” Think: glutton. Since my dog is a glutton for dog treats, I have a glut of Snausages in my house. goosebumps (noun): small bumps on the skin caused by fear or excitement. “Goose bumps” Think: goose bumps. Pluck the feathers off the skin of a goose, and you’ll see the same little bumps that appear on your forearms when you get scared. gossamer (adjective): delicate; ﬂimsy. “GAWS a murr” Think: goose feather. Wafting through the air, the goose feather was gossamer and felt soft to the touch when it landed on my palm. grandiloquent (adjective): loud; colorful; egotistical. “gran DILL oh quent” Think: grand eloquent. If you're grandiloquent, you’re grand and eloquent with your speech so everyone notices you. grandiose (adjective): affecting grandness by showing off or exaggerating. “GRAN dee ose” Think: grand ideas. I have a lot of grand ideas: for example, my grandiose plan to jump the Grand Canyon with my rocket car. grandstand (verb): to show off. “GRAND stand” Think: handstand. If you’re doing a handstand, it’s probably to grandstand for an audience. grasping (adjective): excessively greedy. “GRASP ing” Think: Mr. Burns’ grasping. The Simpsons' Mr. Burns is a grasping (adjective) tycoon who is always grasping (verb) at any new source of proﬁt. grating (adjective): irritating. “GREAT ing” Think: grater. Reading Facebook election posts is grating; I'd almost rather rub a cheese grater on myself. gravitas (noun): powerful seriousness. “GROV it oss” Think: gravity. As the judge entered, his gravitas was like gravity, drawing everyone's eyes to him and silencing the room. gregarious (adjective): social. “gruh GAIR ee us” Think: congregate. If you're gregarious, you like to congregate with others whenever possible. grisly (adjective): horriﬁc; disgusting. “GRIS lee” Think: grizzly death. If you piss off a grizzly bear, it may give you a grisly death. grouse (verb): to complain. “GRAHWSE (rhymes with “house”)” Think: Grouch. Oscar the Grouch likes to grouse about everyone else on Sesame Street. grovel (verb): to act like an unworthy servant by crawling or lowering oneself. “GRAH vuhl” Think: gravel. Grovel to Her Majesty by putting your face in the gravel, slave! gumption (noun): drive; initiative. “GUMP shun” Think: Forrest Gump. Forrest Gump showed gumption by playing football, co-founding a shrimp business, and running across the country. guttural (adjective): strange and unpleasant sounding. “GUT ur ul” Think: gutter roar. You'd have the guttural gutter roar of a homeless man if you spent the night sleeping in the gutter. hackneyed (adjective): trite or overused. “HACK need” Think: hacked knees. The veteran soccer player had hacked knees; his knees were hackneyed from overuse. haggard (adjective): worn-out looking. “HAH gerd” Think: hag. After months of partying with little sleep, Lindsay Lohan began to look haggard and worried people would think she was an old hag. halcyon (adjective): happy; peaceful; prosperous. “HAL see yon” Think: hell’s she on? In her halcyon years, people would ask "What the hell's she on?" because she was constantly happy. hallowed (adjective): sacred. “HAL owed” Think: halo-ed. The cemetery where saints are buried is so hallowed it's practically "halo-ed". hapless (adjective): unlucky. “HAP less” Think: happy less. The hapless are often happy less because of their rotten luck. haptic (adjective): related to the sense of touch. “HAP tick” Think: half ticked. I am only half ticked off that my haptic senses are fading with age, since my resistance to pain has also increased. harangue (noun): a ranting lecture. “huh RANG” Think: her ears rang. Her ears rang so much after the loud harangue that she joked she'd rather hang than listen to it again. harbinger (noun): something that shows what will happen in the future. “har BINJ er” Think: bringer. The superstitious woman thought the black cat crossing her path was a harbinger of bad luck and a bringer of misfortune. hardscrabble (adjective): involving struggle and hard work. “HARD skrab bull” Think: hard and scrappy scrabble champ. Despite her hardscrabble upbringing, Lucinda was a hard and scrappy scrabble champ; she excelled not because of her genius, but through hard work and a superior will to win. harmonious (adjective): free from disagreement; forming a pleasing whole. “Har MOAN ee us” Think: harp money. Playing the harp brings me as much money as it does because the way I play is so harmonious. harried (adjective): harassed. “HAH (rhymes with “NAH”) reed” Think: hurried. Being harried by your teacher and hurried to ﬁnish your test - just because you’re the last one in the room - is terrible. harrow (verb): to torment or greatly distress. “HAH (rhymes with “NAH”) rowed” Think: hair arrow. Not two days after I’d grown the perfect afro, my friend decided to harrow me by shooting me in the hair with an arrow. haughty (adjective): proud in a way that looks down on others. “HOT ee” Think: stuck-up hottie. Unfortunately, that senior class hottie is usually haughty when you talk to her. headlong (adjective): done without adequate thinking; rash. “HEAD long” Think: headﬁrst. If you dived headﬁrst into a shallow pool, it would be a headlong decision. hector (verb): to bully or harass. “HEK tir” Think: heckle. I tried to hector the comedian by heckling him, but he made fun of me, so I stopped. hegemony (noun): dominance. “HEJ uh moany” Think: huge money. The country with huge amounts of money enjoyed hegemony over its neighbors because it could afford an immense army. heinous (adjective): wicked; hateable. “HAIN us” Think: anus. I called you an anus because of your heinous deeds – you cheated on me! herald (noun, verb) n: messenger, indicator, omen v: to signal the arrival of. “HAIR uhld” Think: Harold the Herald. “Harold the Herald” would be a great name for a psychic who heralds the future... if people had more capacious vocabularies. hermetic (adjective): protected from outside inﬂuence. “her MET ick” Think: hermit. The hermit lived in a hermetic cave that was only reachable via a treacherous mountain path. heterodox (adjective): unorthodox; unconventional. “HETT er oh docks” Think: hetero = different. If you are heterosexual, then you prefer the opposite (different) sex. If you are heterodox, then you prefer an unconventional (different) lifestyle and/or philosophy. heterogeneous (adjective): made of dissimilar parts. “heh te-ro JEAN ee-us” Think: heterosexual. Heterosexual sex is more heterogeneous than homosexual sex since it involves a wider variety of body parts. heyday (noun): one's best time period. “HAY day” Think: hey day. During my heyday, when I was the starting quarterback and had a 4.0, all the girls said hey to me every day. hiatus (noun): an interruption or break. “hi A (sounds like the letter A) tus” Think: Hyatt. The Hawaiian Hyatt ad urged us to take a hiatus from work to stay at its luxurious hotel for a few days. hidebound (adjective): inﬂexible; ultra-conservative. “HIDE bound” Think: hide-bound. The hidebound extremists were bound in animal hides and unsurprisingly were against gay marriage. hirsute (adjective): hairy. “her STOOT” Think: hair suit. I saw an old guy in the locker room who was so hirsute that he looked like he was wearing a hair suit. histrionic (adjective): overly emotional for effect. “hiss tree ON ick” Think: hysterical. Her hysterical laughter was designed to get attention and was therefore histrionic. hodgepodge (noun): a jumble of different things. “HAHJ pahj” Think: garage. If your garage is anything like mine, it’s a hodgepodge of tools, old papers, junk, and who knows what else. holistic (adjective): dealing with something as a whole. “ho LIST ick” Think: whole list. Holistic medicine treats the whole list of body issues instead of just addressing one symptom. homespun (adjective): simple; unpretentious. “home SPUN” Think: home spun. Her clothes are pretty homespun, but then again, they actually are home spun - her mom weaves them at home on a loom. homogeneous (adjective): having the same composition throughout. “huh MOJ in us” Think: homogenized milk. Milk is homogenized to mix in the cream and make a homogeneous liquid. hortatory (adjective): intended to urge action. “HORT uh tore ee” Think: horror story. The horror story about the spread of Ebola had a hortatory effect on us; we began washing our hands after touching anything. hubris (noun): excessive pride or self-conﬁdence. “HYU bris” Think: huge breasts. If a girl gets implants and suddenly has huge breasts, she may develop hubris from all the male attention. humbuggery (noun): nonsense; rubbish. “hum BUG ur ee” Think: bah, humbug! Ebenezer Scrooge said, "bah, humbug!" so much because he thought Christmas was humbuggery. humdrum (adjective): boring. “HUM drum” Think: hums and drums. That was a humdrum band – it was just one guy who would hum and another guy beating a drum. husbandry (noun): careful management. “HUZ bun dree” Think: husband. In the 1950s, a woman's husband usually practiced husbandry of their ﬁnances. iconoclast (noun): someone who goes against society. “eye KON oh klast” Think: clashed. The iconoclast had beliefs that clashed with most people's views. ideological (adjective): related to belief, sometimes at the expense of the practical. “eye dee uh LODGE ih cuhl” Think: idea. The ideological candidate refused to compromise on his ideas about what was right, winning some support but ultimately losing to his more practical opponent. idyllic (adjective): pleasingly, naturally simple. “eye DILL ick” Think: ideal. The idyllic forest grove, with its sunbeams, babbling brook, and butterﬂies, seemed an ideal campsite. idiosyncrasy (noun): a weird trait. “id ee oh SIN kra see” Think: ‘N SYNC-cracy. I might seem idiotic to suggest an 'N SYNC-cracy where 'N SYNC rules our nation, but it's just my idiosyncrasy. ignominy (noun): deep disgrace. “IG no min-ee” Think: ignored many. Joe Paterno ignored many of the crimes that were being committed at Penn State; his legacy is now one of ignominy. illiberal (adjective): narrow-minded. “ill LIB ur ul” Think: ill liberal. Unlike his fellow open-minded Democrats, Jack was so illiberal that people thought he must be a mentally ill liberal. illusory (adjective): not real. “ill LOO sir ree” Think: illusion. The mirage of an oasis in the desert was an illusion; it was therefore illusory. imbroglio (noun): complicated situation. “im BRO glee oh” Think: igloo bro! I knew my friend was in an imbroglio after getting the text, "I just woke up and I'm in an igloo, bro!" imminent (adjective): about to happen. “IMM un nent” Think: in a moment. The evil-looking storm clouds told us a downpour was imminent – it would happen in a moment. immure (verb): to enclose or imprison. “imm YOUR” Think: in manure. We never should have tried to drive through this cow pasture: our car is immured in manure. immutable (noun): unchangeable. “im MUTE uh bull” Think: im-mutate-able. They poured radioactive chemicals on me to try to make me into a mutant, but it was impossible: I’m immutable, so I’m im-mutate-able. impassive (adjective): unemotional. “im PASS ive” Think: I’m passive. I’m passive, and I remained impassive so the bully who stole my Dippin’ Dots wouldn’t hit me. impeccable (adjective): ﬂawless. “im PECK uh bull” Think: im-peckable. Due to the impeccable net you covered my apple tree with, the crows can’t get at the fruit – it’s im-peckable. impecunious (adjective): poor. “imm peh Q nee us” Think: I’m pecking (pecuniary = related to money). I’m so impecunious that, at dinnertime, I’m pecking at the ground like a chicken to look for bugs to eat. impeded (verb): blocked. “im PEED id” Think: stampede. As I walked across the fruited plain, a buffalo stampede impeded my progress. imperative (adjective): very important. “imm PEAR uh tiv” Think: I’m parenting! I’m parenting here, and I expect respect! Of course it’s imperative that you clean your room! imperious (adjective): dominant in a kingly way. “im PEER ee us” Think: emperor. When we went out to dinner with the emperor, he was so imperious that he ordered all of our meals. imperturbable (adjective): unable to be upset or excited; calm. “im purr TUR buh bull” Think: im (not) disturbable. A phone rang as I swung at the golf ball, but my imperturbable nature kept me from being disturbed and slicing the shot. impetuous (adjective): impulsive; spontaneous. “im PET chew iss” Think: impatient us. Impatient people like us make impetuous decisions like betting on horses with cool names without researching them ﬁrst. impetus (noun): driving force, incentive, stimulus. “IMM pet us” Think: pet us. As dogs, our impetus to obey commands is that people will pet us as a reward. impenetrable (adjective): unable to be penetrated (literal), unable to be understood or overcome (metaphorical). “im PEN it truh bull” Think: pennant. In order to win the pennant for his team, the manager tried to make sure that his gameplan was impenetrable to the opposing team. impinge (verb): to trespass on one’s freedoms. “im PINJ” Think: I’m pinched. I’m pinched on the butt every time I go to that biker bar – it impinges on my dignity. implacable (adjective): unable to be satisﬁed or pleased. “im PLAK uh bull” Think: im-plaque-able My dental hygenist was implacable; no matter how much I brushed and ﬂossed, she kept telling me that my mouth was full of plaque. implication (noun): a conclusion, hint, suggestion, connection or insinuation (not directly stated). “ihm (rhymes with “him”) plih KAY shun” Think: implying = suggesting. When Mike’s date told him that she was tired and it was late, implying that was that it was time to go home, the implication was obvious to everyone but him. “OK, want to get some coffee then?” he asked cluelessly. implicit (adjective): suggested but not directly expressed. “im PLISS it” Think: implied. It became implicit that the evening was over when my date implied that if she didn’t leave now she would be too tired to work the next day. imploring (verb): begging. “Im PLOR ing” Think: I’m poor. I implore you to lend me a few bucks since I’m poor. importune (verb): to nag; to persistently insist. “im por TOON” Think: “I’m poor” tune. The homeless man at the end of my block always importunes us for money with his little “I’m poor” tune. impregnable (adjective): unconquerable; impenetrable. “im PREG nuh bull” Think: impossible to get pregnant. Impregnable metal chastity belts in the Middle Ages made it impossible for women who wore them to get pregnant. imprimatur (noun): ofﬁcial approval. “im prim a TURE” Think: imprint. In Game of Thrones, a king conveys his imprimatur with an imprint of his crest on a scroll's wax seal. impromptu (adjective): without preparation. “im PROMPT ooh” Think: improvise. If you forget your lines, I'm not going to prompt you, so just improvise and make some impromptu remarks. impudence (noun): rudeness. “IM pew dense” Think: in puberty. Give those 12-year-olds a break. They’re still in puberty - that’s why they’re so impudent to the substitute teacher. impugn (verb): to attack verbally. “im PYOON” Think: imply ugly. Your insults impugn me; they imply ugly things. inalienable (adjective): impossible to take away or give up. “In ALIEN uh bull” Think: in (not) alien. You say I’m an illegal alien, but I have a green card that gives me the inalienable right to live in the U.S. inane (adjective): lacking meaning; silly. “in AIN (rhymes with “gain”)” Think: insane. Saying you "like stuff" to describe your interests is inane, and it might make people think you're insane. incandescent (adjective): bright; brilliant. “in can DESS sent” Think: candle sent. The candle sent incandescent light throughout the tomb, revealing a sleeping vampire. incensed (adjective): extremely angry. “in SENSED” Think: incense. If you're incensed, smoke may be wafting off of your head as if you were a giant stick of burning incense (noun). incessant (adjective): never-ending, constant. “in SESS int” Think: cease = to stop To cease is to stop (i.e, cease and desist), so if something is incessant, then it is never-ending. For example, when I was in college, I listened to Bob Dylan’s “Basement Tapes” incessantly. inchoate (adjective): incomplete; formless. “in COH it” Think: inches of chow. The pile of chow on the hungry man’s Thanksgiving dinner place was eight inches high -- and created an inchoate blob of food. incipient (adjective): beginning. “in SIP ee ent” Think: sippy cup. When your child’s toddlerhood is incipient (beginning), it helps to have a good sippy cup to minimize the number of spills. incisive (adjective): sharp; direct. “in SICE ive” Think: incision. Luckily, the surgeon was incisive - she only had seconds to make an incision before the patient's appendix burst. incoherent (adjective): unclear. “in co HERE ent” Think: I couldn’t hear it. Your slurred voicemail to me at 2:30 A.M. was incoherent - I couldn’t hear it. incorporate (verb): to include or take in. “in KOR poor ate” Think: carp I ate. I don’t eat much seafood, but after all that delicious carp I ate at the cookout, I think I should start incorporating more ﬁsh into my diet. incorrigible (adjective): unable to be reformed. “in CORE ij uh bull” Think: in-correctable. Despite his teachers’ best efforts to make him sit still, the hyperactive little boy seemed incorrigible and in-correctable. inculcate (verb): to teach by constant repetition and warning. “IN cull kate” Think: in cult. In the cult of Scientology, they inculcated Tom Cruise until he was brainwashed. incumbent (adjective): not optional; obligatory. “in COME bent” Think: income bent. If your income is bent in the direction of a half-million dollars a year or more, then it is incumbent upon you to make sure to donate some of your money to worthy causes. indefatigable (adjective): tireless, persistent. “in duh FAT ig uh bull” Think: in-defeatable. Because of his indefatigable work ethic, Michael Phelps is nearly in-defeatable in the pool. indictment (noun): a criticism or accusation. “in DAHYT (rhymes with “night”) ment Think: dictaphone meant. The fact that the defendant had illegally used a dictaphone to record private conversations with her employer, which she later shared online, meant that there were grounds for an indictment by the court. indigenous (adjective): native to an area. “in DIJIN us” Think: Indian dig in U.S. The archaeologist found arrowheads during her Indian dig in the U.S. and concluded that Native Americans were indigenous to the area. indignant (adjective): offended or angered by perceived unfair treatment. “in DIG nint” Think: ain't diggin' it! When I was passed over for the promotion at work, I was indignant. I told my boss, "I ain't diggin' it!" indomitable (adjective): unconquerable. “in DOM it a bull” Think: in-dominate-able. Spain's national soccer team is so good that they're indomitable or "in-dominate-able" - they're unable to be dominated. industrious (adjective): hard-working. “in DUSS tree us” Think: maids dusting. Succeeding in the cleaning industry means only hiring industrious maids who are really good at dusting. ineffable (adjective): that which cannot be described in words. “in EFF uh bull” Think: in-F-able. There’s a word beginning with “F” that you’re not supposed to say, so if you can’t describe something, it’s ineffable - like that word is “in-F-able”. ineluctable (adjective): inevitable, bound to happen, certain. “in ee LUCKED a bull” Think: unelectable. It is ineluctable that a sex scandal on the eve of the election would render the candidate unelectable. inestimable (adjective): too great to calculate. “in EST imm uh bull” Think: in-estimate-able. If something is inestimable, it’s in-estimate-able – you can’t estimate it. inexorable (adjective): unstoppable. “in EX ur a bull” Think: in-x-out-able. The ﬁghter's inexorable rise made it impossible to cross his name off the contender list; he was "in-x-out-able". inﬁnitesimal (adjective): incredibly tiny. “in ﬁn ih TESS ih mull” Think: inﬁnitely small. Electrons are pretty much inﬁnitely small - they’re so inﬁnitesimal that observing them changes them. inﬂux (noun): the arrival of many things or people. “IN ﬂucks” Think: in ﬂood. The inﬂux of college students to Boston every September is like a ﬂood. ingenious (adjective): extremely clever. “in JEAN yiss” Think: genie genius. The genie granted me one wish, which I used to wish for unlimited wishes. “You’re a genius!” he said. I know, I know. ingenuous (adjective): completely sincere; naive. “in JEN you us” Think: genius without the “I” is no genius at all. The young actress, being an innocent ingenue, was too ingenuous to realize the director was trying to seduce her. ingrained (adjective): deeply worked into something. “in GRAYND” Think: in grain. Pesticides sprayed on wheat will become ingrained into the grain. ingratiate (verb): to make someone like you. “in GRAY she ate” Think: gratitude grated. The new guy’s excessive gratitude grated and seemed like an attempt to ingratiate himself to us. inimical (adjective): unfriendly; hostile. “in IM ih kull” Think: enemy. Of course the other beauty contestant hid your lipstick! She’s your enemy; it’s no surprise she’ll be inimical. inimitable (adjective): not capable of being imitated. “in IM it a bull” Think: in-imitate-able. Michelangelo's art is inimitable and in-imitate-able; it has a magic that cannot be reproduced. innate (adjective): existing since birth; inherent. “in NATE” Think: in natal. The ability of a spider to spin a web is not learned but innate; it’s in it even in the natal stage before being born. innocuous (adjective): harmless. “in NOCK you us” Think: innocent. My dog will bark at you once you come in but it’s innocent - he’s innocuous. inordinate (adjective): exceeding reasonable limits. “in ORD in it” Think: not ordinary. Joey Chestnut consumed an inordinate number of hot dogs; it's not ordinary that he ate 62 of them. Inscrutable (adjective): impossible to understand or interpret. “In SCREW tah bull” Think: in screw table. If I told you “in a table, there’s a screw”, you’d understand me, but if I said “in a screw, there’s a table”, I’d no doubt be inscrutable to you. insinuate (verb): to hint or imply; to subtly introduce. “in SIN you ate” Think: in sin you ate. Pop culture insinuates that all women should be skinny, as if to say “in sin you ate that piece of cake”. insipid (adjective): bland; dull. “in SIP id” Think: in sippy. FYI: if your drinks are served in sippy cups, you're probably a baby - that's why they feed you insipid, mushy foods. insolence: noun, rudeness, insensitivity. “IN suh lince” Think: in silence To punish me for my insolence, my kindergarten teacher forced me to sit in the corner in silence. insular (adjective): narrow-minded. “IN suh lur” Think: insulated. The hermit’s outlook was so insular because his cave insulated him from the rest of the world. integrate (verb): to unite into a whole. “IN teh great” Think: interstate. The new interstate highway will integrate our town with the one in the next state since travel between the two will be easier. interloper (noun): one who intrudes. “in tir LOPE ur” Think: interrupt elope. The interloper interrupted them from eloping when the priest said, "Speak now or forever hold your peace." intimate (verb): to hint at. “IN tim it” Think: intimate apparel. I like my girlfriend’s intimate (adjective) apparel because it intimates (verb) at the shape of her body without looking slutty. intrepid (adjective): extremely brave. “in TREP id” Think: entrap it! Instead of running from the attacking polar bear, our intrepid guide handed us nets, shouting, “entrap it!” intrinsic (adjective): inherent. “in TRIN zik” Think: twins. Having similar personalities is something that is comes naturally for most identical twins; it’s intrinsic. intrusive (adjective): causing disruption or annoyance by being unwelcome. “in TRUE sive” Think: intruder. We all found it to be intrusive when an intruder interrupted our Christmas morning by breaking into our house and stealing presents from under the tree. inundated (adjective): ﬂooded. “IN un date id” Think: nuns date. After the church allowed nuns to date, they inundated Match.com. inveigh (verb): to protest or complain bitterly. “in VAY” Think: weigh in. When a person with a German accent weighs in on a topic which upsets him, he is inveighing. inveigle (verb): to entice, lure (a person), acquire or win (a thing) through deception or ﬂattery. "in-VAY-gull" Think: inveigle a bagel. I was able to inveigle a bagel by impressing the bagel store owner with my ﬂuent Polish. invidious (adjective): causing envy. “in VID ee us” Think: envious. I knew marrying a supermodel would make my friends envious – it’s unfortunately an invidious thing to do. inviolate (adjective): pure; intact. “in VIE oh let” Think: unviolated. The virgin tract of rainforest was inviolate; it had not yet been violated by greedy loggers. irascible (adjective): easily angered. “ir RASS ih bull” Think: irritable rascal. My grandfather is an irritable old rascal; he’s so irascible that he yells at every waiter we ever get. irk (verb): to annoy. “ERRK” Think: jerk. Of course he irks you – he’s a jerk! ironic (adjective): the opposite of what one would expect. “Eye RON ick” Think: “Hi, Ron” = ick. Ron’s internal dialogue: “It’s ironic that when that girl I’ve been crushing on ﬁnally said “Hi, Ron,” I just then started to lose interest in her. Sometimes I disgust myself with my self-sabotaging ways. Ick.” irresolute (adjective): not ﬁrm or determined. ”ear REZ oh loot” Think: error in resolutions. Looking back to January, I made an error in making New Year’s resolutions; I’m too irresolute to accomplish anything besides playing video games. jargon (noun): specialized language used by a particular group of people. “JAR gun” Think: Jar Jar Binks. Part of the problem with Star Wars’ Jar Jar Binks was the confusing jargon he used when talking to Anakin Skywalker. jejune (adjective): dull; juvenile. “jih JOON” Think: juvenile. My frat brothers' fart jokes are so jejune that you could almost call them juvenile or “jejune-venile”. jettison (verb): to get rid of or to reject something. “Jet ih son” Think: jet engine. After the jet engine failed, the pilot jettisoned fuel so the plane would be light enough to make it to the airport. jingoism (noun): extreme nationalism, belligerent foreign policy. “JIN go ism” Think: Ringo-ism. The British man’s jingoism went so far as to make him campaign for the Beatles’ Ringo Starr to rule the free world. jocose (adjective): given to joking. “juh KOSE” Think: joke coach. It was no surprise that the jocose high school student grew up to be a joke coach. judicious (adjective): having good judgment. “joo DISH us” Think: judgment. The Beatles' song "Hey Jude" says to be judicious, to use good judgment, and to "let her into your heart". juggernaut (noun): something very powerful. “JUG ur not” Think: juggler-knot. That juggler tied that huge knot by juggling six balls of yarn - it’ll be a juggernaut to untie. juvenescence (noun): the state of being youthful or growing young. “JOOV in ess ense” Think: juvenile adolescent. Creating juvenescence by partying in Vegas for his 40th birthday made the man feel like a juvenile adolescent again. juxtapose (verb): to contrast. “JUCKS tuh pohz” Think: Huxtable. Cliff Huxtable – Bill Cosby’s warm, good-natured character on The Cosby Show – was juxtaposed with his real-life persona after he was accused of rape on multiple occasions. kindle (verb): to start; to stir up. “KIN dull” Think: kindling. You can’t just light a log on ﬁre - to kindle the campﬁre, you need some kindling: twigs, paper, dried grass, etc. kindred (adjective): closely related. “KIN drid” Think: kind. I’m so kind to you because we’re kindred spirits. kismet (noun): fate. “KISS met” Think: kiss met. I knew it was kismet that I'd marry her because we kissed as soon as we met each other. kowtow (verb): to kiss up to. “COW TOWE (rhymes with “ow”) Think: cow toes. If you want to kowtow to a farmer, bow and offer to give a pedicure to his cow's toes. lachrymose (adjective): tearful; mournful. “lack ri MOSE” Think: lack Christmas. If you lack Christmas presents, I don't blame you for being lachrymose. lackadaisical (adjective): without energy or spirit. “lack a DAYS ih cull” Think: like a daze. Lackadaisical people are lazy, like a daze has come over them. laconic (adjective): using few words. “luh CON ick” Think: lacking kick. His personality was lacking kick; he was so laconic that he barely even said hello to us. lampoon (verb): to mock or satirize. “lam POON” Think: laugh harpoon. The Onion lampooned Kanye so skillfully that its article was like a laugh harpoon. languid (adjective): lazy; lacking energy. “lan GWID” Think: laying squid. The laying squid was languid because it just lay on the bottom of the ocean all day. largess (noun): generosity. “large ESS” Think: large-ness. Due to his wealthy parents' largess and the large-ness of their generosity, the college student lived pretty large and drove a Ferrari. lassitude (noun): tiredness; laziness. “LASS ih tude” Think: lazy attitude. Your lassitude is caused by your lazy attitude and your belief that Lassie will come save you if you need help. latent (adjective): existing but unseen or inactive. “LAY tent” Think: lay tent. You claim to like hiking, but your desire must be latent since you just lay in the tent when we camp. laudable (adjective): worthy of praise. “LOD ih bull” Think: applaudable. Something that's laudable is applaudable. lax (adjective): loose; not strict. “LACKS” Think: lacks. His diet plan was lax because he lacks the discipline to avoid junk food. legerdemain (noun): sleight of hand; a display of skill. “leh jer da-MANE” Think: Ledger’s domain. Heath Ledger's domain was the silver screen; his acting legerdemain captivated audiences. lenient (adjective): forgiving, not strict. “LEEN ee ent” Think: loan lent. My bank is really lenient – I have terrible credit, but I asked for a loan and they lent one to me. levity (noun): lightheartedness. “LEV ih tee” Think: levitate. The comedian's levity put us in such a good mood that our spirits felt as if they were levitating. licentious (adjective): lacking restraint. “lie SEN shus” Think: license-ish. Flappers were thought to be licentious, since they acted as if they had a license to do whatever they wanted. lionized (verb): treated with great interest. “LIE un ized” Think: lion-ized. The cute little meerkat was so lionized by the zoo's visitors that he felt like a lion. listless (adjective): having little interest or energy. “LIST liss” Think: list-less. If you've never made a to-do list, you're list-less and probably listless. logorrhea (noun): excessive wordiness. “log uh REE uh” Think: diarrhea. I thought a long speech would help my grade, but my teacher said my logorrhea was like verbal diarrhea. loquacious (adjective): very talkative. “luh QUAY shus” Think: quack quack. The loquacious duck just wouldn’t shut up: “quack quack, I’m a duck, quack quack, blah blah blah.” lovelorn (adjective): without love. “LOVE lorn” Think: love torn. After his wife died in an accident, the man felt lovelorn, as though he'd had his love torn from him. lucid (adjective): clear; intelligible. “LOO sid” Think: Luz = light in Spanish. I’m not fully lucid in the morning until the sun rises and the lug (light) comes through the window – I can’t think clearly while it’s still dark outside because I’m still in dreamland. lucre (noun): money; proﬁt. “LOO kurr” Think: lucrative. When you put in the years of training necessary to secure a lucrative career, lucre is your reward. ludicrous (adjective): ridiculous. “LOOD ih kris” Think: Ludacris ridiculous. The rapper Ludacris is known for his ridiculous, ludicrous lines like “I got hoes in different area codes.” lugubrious (adjective): mournful or gloomy. “luh GOO bree us” Think: lug Brian. I became lugubrious when I realized I would have to lug the unconscious Brian up the stairs. lumber (verb): to move with clumsiness. “LUM bir” Think: lumber (noun). Frankenstein would lumber (verb) around as if his limbs were made of lumber (noun). luminary (noun): one regarded for his brilliant achievements. “LOOM in airy” Think: illuminate. It would take a luminary like Stephen Hawking to illuminate quantum physics for me. lurid (adjective): sensational; shocking. “LURR id” Think: lure in. The strip club’s lurid neon silhouette of a naked woman was designed to lure in lonely gentlemen. macabre (adjective): gruesome; horrible. “muh COBB” Think: massacre. The massacre of the tourists by jungle cannibals was truly macabre. macerate (verb): to weaken, break down, or make soft. “MAH sir ate” Think: mace. Spraying a mugger in the face with mace (tear gas) will hopefully macerate him. machination (noun): a crafty scheme. “MOCK in ae shun” Think: machine nation. I don't trust C-3PO and R2D2; I bet they have machinations designed to create a machine nation in which we are slaves. maelstrom (noun): something violently powerful; a whirlpool. “MALE strum” Think: mail storm. Spam emails ﬂock to my inbox like a maelstrom; reading the mail storm would suck up all my time. magisterial (adjective): having strong authority; kingly. “mah gist STEER ee ul” Think: Majesty. If people are greeting you by saying "Your Majesty", you're probably looking magisterial – wear that outﬁt again! magnanimous (adjective): generous. “mag NAN ih muss” Think: magnet for animals. The “Feed the Birds” lady in Mary Poppins was a magnet for animals because she was so magnanimous to them. magnate (noun): a powerful or inﬂuential person. “MAG nate” Think: chick magnet. You’d be a chick magnet, too, if you were an oil magnate like me. makeshift (adjective): serving as a temporary substitute. “MAKE shift” Think: break shift, make shift. If you break the shift gears on your bike, then you might have to improvise something makeshift until you can get to a repair shop. malevolent (adjective): evil. “muh LEV uh lent” Think: violent male. Malevolent criminals are usually violent males; most serial killers are men. malfeasance (noun): misdeed, violation. “mal FEE zance” Think: mal=bad, ﬂeas. The ﬂeas and ants in my house commit malfeasances daily; I think of them as mal (bad) ﬂeas / ants. malign (verb): to speak evil of. “muh LINE” Think: malignant. The evil witch not only maligned her enemies but also cast spells designed to give them malignant tumors. malinger (verb): to fake sickness to avoid working. “muh LIN gur” Think: linger. Those who malinger often linger in bed, pretending to have the ﬂu. malleable (adjective): able to be shaped. “MAL ee uh bull” Think: mallet-able. 24-karat gold is so malleable that you can dent it with a wooden hammer -it’s “mallet-able.” manacle (verb): to restrain. “MAN uh cuhl” Think: man shackle. These military rules manacle us just as surely as if they’d put man shackles on our wrists. mandate (noun): an order or command. “MAN date” Think: mandatory. The captain's mandate was obviously mandatory – so swab the deck! manifold (adjective): diverse; varied. “MAN ih fold” Think: many folds. The surface of the brain is manifold because it has many folds. marginal (adjective): very limited. “MARGE in ul” Think: margins. I only had marginal success in deciphering the ancient manuscript because the only legible parts were the margins. marshal (verb): to gather and organize. “MAR shul” Think: ﬁre marshal. The ﬁre marshal’s job is to marshal the volunteer ﬁremen if there’s a ﬁre alarm. maudlin (adjective): overly sentimental. “MAWD lin” Think: Maude’s violin. Maude played emotional violin music every time she made an entrance, so we called her maudlin. mawkish (adjective): overly sentimental. “MOCK ish” Think: Ma’s awkward kiss. Ma is awkward because she has to kiss us every time we leave the house -she’s mawkish. meager (adjective): very small; inadequate. “MEE grr” Think: me grr. These meager portions on this pirate ship make me grr. meddle (verb): to become involved with another’s affairs. “MED uhl” Think: middle. It’s annoying when you meddle with us – you always jump into the middle of any quarrel we have. meld (verb): to merge; to blend. “MELD” Think: melt. If your ice cream cup is half vanilla, half chocolate and it melts, the ﬂavors will meld. melliﬂuous (adjective): having a sweet, smooth, rich ﬂow. “muh LIFF ﬂu iss” Think: melody ﬂow. Adele’s melliﬂuous voice lets a melody ﬂow from her lips like honey. melodramatic (adjective): overly dramatic. “MEH low druh MATT ick” Think: dramatic melody. It’s melodramatic to hire a violinist to follow you around and play a dramatic melody when you enter a room. mendacity (noun): dishonesty. “men DAH sit ee” Think: mend the city The former mayor of Providence, Buddy Cianci, promised that he would mend the city and its underhanded ways, but his mendacity became apparent when he himself was arrested for corruption. mendicant (noun): a beggar. “MEND ih kint” Think: mend? I can’t! If you tell a mendicant to sew up the holes in his clothes, he’d probably say, “mend? I can’t! They’re about to fall apart.” menial (adjective): a task suitable to a servant. “MEAN ee ul” Think: me kneel. Tasks that make me kneel, like scrubbing the ﬂoor, are aptly called menial. mephitic (adjective): foul-smelling. “meh FIT ick” Think: meth breath. I bet the devil Mephistopheles has mephitic breath, like that of a meth user. mercenary (adjective): motivated by money. “MURR sin erry” Think: merchant. I knew the merchant’s compliments were insincere since he was clearly mercenary. mercurial (adjective): having rapidly changing moods. “murr CURE ee ul” Think: Mercury. Marie Curie was notorious for her mercurial moods, which revolved as fast as the planet Mercury. meretricious (adjective): falsely attractive. “merr (rhymes with “err”) uh TRISH us” Think: merit tricks us. The sparkle of pyrite, or fool’s gold, is meretricious because its merit tricks us into thinking it’s a precious stone. meterological (adj): pertaining to weather patters. “Me tee or oh loj ih cull” Think: meteor logical. Because it’s unlikely that meteorlogical events could have killed every living dinosaur, a meteor striking Earth’s surface is a more logical explanation for their extinction. meticulous (adjective): extremely detail-conscious. “Meh TICK u luss” Think: ticks on us. Hiking in the woods is fun, but we need to be meticulous when checking our skin to make sure that there aren’t any Lyme-disease-carrying ticks on us. mettle (noun): strength; stamina. “MET ul” Think: made of metal. In Gladiator, Russell Crowe was so full of mettle he might have been made of metal. miasma (noun): an unhealthy atmosphere. “mi AS muh” Think: my asthma. The smog in Los Angeles is a miasma that worsens my asthma. microcosm (noun): a small thing representing a larger thing. “MY crow kos um” Think: micro-cosmos. The glow-in-the-dark stars on my ceiling are a microcosm of the universe -they're a "micro cosmos.” milieu (noun): setting or environment. “mill YOU” Think: my loo. I prefer to use my loo in my own milieu - other people's bathrooms are gross! milquetoast (noun): a timid person. “MILK toast” Think: Milhouse / milky toast. Milhouse is a milquetoast - he's about as tough as a piece of soggy, milky toast, since Bart Simpson bosses him around. mimetic (adjective): something that imitates or mimics. “mih MET ick” Think: mimes mimic. You may think mimes are annoying, but their mimetic abilities are pretty cool when they mimic what it would look like to be trapped in a glass box. minatory (adjective): threatening. “MIN a tor ee” Think: minotaur. The minotaur, a creature that is half-man and half-bull, is minatory by nature. minion (noun): a servant; a follower. “MIN yun” Think: mini-one. In Austin Powers, Mini-Me is Dr. Evil's minion; he is a mini-one of Dr. Evil. misanthrope (noun): one who hates people. “MISS en throwpe” Think: mistake to be an anthropologist. It's a huge mistake to be an anthropologist and study people all day long if you're a misanthrope. miscreant (noun): a person who behaves badly or in a way that breaks the law. “MISS kree int” Think: mistake of creation. The harsh judge believed the miscreant was a mistake of creation. miserly (adjective): stingy or cheap with money. “MY zir lee” Think: miserable Scrooge. Scrooge was miserable at making friends because he was too miserly to ever chip in for the dinner tab. misnomer (noun): a wrong or inappropriate name. “miss NO murr” Think: mis-name. “The Battle of Bunker Hill” is a misnomer: it mis-names the battle, which was actually fought on the nearby Breed’s Hill. mitigate (verb): to lessen or make less severe. “MITT ih gate” Think: mitt gate. The thief wore oven mitts to climb the spiked gate of the mansion to mitigate the pain in his hands. modicum (noun): a small amount. “MODD ih kum” Think: modest amount. My pet mouse is cheap to feed because a modicum, or modest amount, of food will ﬁll up his little belly. modish (adjective): fashionable. “MODE ish” Think: model-ish. Modish brands like Burberry and Prada are model-ish because only models seem to actually wear them. monastic (adjective): strict; secluded; austere. “muh NAST ick” Think: monastery. If you’re a monk and you live in a monastery, your life is probably monastic – no partying for you. morass (noun): a situation that makes you stuck. “muh RASS” Think: molasses. Don’t tailgate a molasses truck - if you run into it and it spills on you, you’ll be in a morass. morbid (adjective): obsessed with or overly focused on death. “MORE bid” Think: more bit (bite the bullet). If you’re more obsessed with biting the bullet than most, then you are probably a morbid person. mordant (adjective): bitingly harsh, often in a funny way. “MORE dint” Think: Mordor. Those evil creatures who live in Mordor are especially fond of mordant jokes. mores (noun): customary rules and standards. “MORES (rhymes with “snores”)” Think: morals. Our society’s mores include morals like helping others who are less fortunate. moribund (adjective): dying. “MORE ih bunned” Think: morbid end. If someone is moribund, they’re probably headed toward a morbid end, i.e., death. morose: (adjective): gloomy. “muh ROWSE (rhymes with “ghost”)” Think: no rose. “The Bachelor” contestant was morose because after the ceremony was over she still had no rose. motile (adjective): having the ability to move. “MOH till” Think: mobile. Now that my one-year-old can walk, he’s so motile that I have to be really mobile just to catch up to him. motley (adjective): made up of several different parts. “MOT lee” Think: Motley Crue. The band Motley Crue has a motley history of parties, drugs, and other types of craziness. multifaceted (adjective): having many aspects or parts. “Mull tee FASS ih ted” Think: multi (many) facets. Multifaceted diamonds sparkle because light reﬂects off of their many facets. mundane (adjective): commonplace. “MUN dane” Think: Mondays. Asking someone if they have a “case of the Mondays” is such a mundane saying that it’s not funny anymore. muniﬁcent (adjective): generous or giving. “moon IF uh sint” Think: money sent. The money sent to us by our grandparents every year makes us consider them to be muniﬁcent. myopic (adjective): shortsighted. “my OP ick” Think: my old pic. Putting my old pic on Match.com was myopic; in person, people said I was older than they'd thought I'd be. myriad (noun): a large number. “MERE (rhymes with “here”) ee id” Think: merry ads. The myriad of merry ads during the holidays tries to persuade people to spend money on presents. nadir (noun): the lowest point. “NAY dur” Think: nads. A dude’s nads are literally the nadir of his reproductive system. naïveté (noun): lack of experience, wisdom or judgement. “nye eve uh TAY” Think: Adam and Eve. Some people take the creation parable of Adam and Eve literally, but as a scientist, I attribute that to naïveté. nascent (adj): coming into existence; new. “NAH sint” Think: new car scent. I jumped into the nascent BMW while it was still on the assembly line and breathed in the best new car scent I’ve ever smelled. nebulous (adjective): vague. “NEB yule us” Think: Nebula. The Horsehead Nebula is so many light-years away that we only have a nebulous idea of what it's like. neophyte (noun): a beginner. “ne YO ﬁght” Think: Neo ﬁght. When Neo has his ﬁrst ﬁght with an agent in The Matrix, he is a neophyte and gets his ass kicked. nepotism (noun): unfairly hiring family. “NEP oh TIS um” Think: nephew favoritism. They said I practiced nephew favoritism and accused me of nepotism when I promoted my 22-year-old nephew to vice president of the company. nettle (verb): to irritate. “NET ul” Think: needle. Poking someone with a needle is a quick way to nettle him. newfangled (adj): new, often needlessly so. “nu FANG ulled” Think: new tangled. This newfangled yo-yo is so new to me that it’s tangled around my entire body. noisome (adjective): stinky. “NOICE um” Think: nose poison. The boys’ locker room is noisome; going in there is like taking nose poison. non sequitur (n): something unrelated. “non SECK quit ur” Think: not sequence. Bringing up koala bears after my girlfriend asked me about our relationship was a non sequitur; it was not in the right sequence. nonchalant (adjective): acting casual or disinterested. “non shuh LAUNT (rhymes with “ﬂaunt”)” Think: non-challenge I’m nonchalant when I ask out a girl; it’s really a non-challenge for me since I’ve got so much game. nondescript (adjective): plain. “non duh SCRIPPED” Think: no description. The little desert island was so nondescript that it had no description in our guidebook. nonpareil (adjective): having no equal. “non pear ELL” Think: no parallel. The master parachutist had nonpareil skill; he truly had no parallel in the parachuting ﬁeld. nonplussed (adj): bewildered or confused. “non PLUSSED” Think: no plus. The calculator you loaned me made me nonplussed during the test because it had no plus button. nontrivial (adjective): not unimportant. “non TRIV ee ul” Think: trivia. Most of the questions they ask during trivia night at the bar are rather trivial if you ask me...but my pop-culture-loving roommate ﬁnds them nontrivial. normative (adjective): prescribing a standard or model. “NORM a tive” Think: normal. Are you normal? If you said, “yes”, then are you buying into someone else’s normative rules of behavior? nostalgia (noun): a bittersweet longing for the past. “nuh STAL guh” Think: nose tampon. My nostalgia for my glory days got so bad that I had to use a nose tampon for my constant snifﬂes. nostrum (noun): a questionable medicine or remedy. “NAH strum” Think: nostril rum. The Simpsons' Dr. Nick's nostrum was nostril rum - rum meant to be snorted to clear the sinuses. notorious (adjective): famous for being bad. “no TORY us” Think: Notorious B.I.G. The Notorious B.I.G. got away with calling himself notorious since he sold crack as a youth. novel (adjective): strikingly new. “NAW vul” Think: novice. When I was an internet novice, the idea of email was novel to me. novitiate (noun): a beginner. “no VISHY it” Think: novice initiate. The novitiate is a novice frat brother, so they’ll initiate him by making him run around campus wearing a thong. noxious (adjective): harmful. “NOCK shus” Think: toxic. Burning plastic releases noxious fumes that are toxic to living things. nuance (noun): a subtle or slight distinction. “NEW aunts (rhymes with “ﬂaunts”)” Think: new ants. Forgive me for not picking up on the nuance of today’s experiment - I couldn’t tell it had new ants compared to yesterday’s - ants all look the same to me. nugatory (adjective): unimportant. “NOOG a-tory” Think: negative nuggets. Eating chicken nuggets is nugatory for good health; their health beneﬁts could be said to be negative. obdurate (adj): hardened; stubbornly persistent. “ob DUR it “ Think: obstacle durable. That 10-ton boulder you blocked my front door with is an obstacle that’s durable - I tried to push it, but it’s obdurate. obeisance (n): a gesture to show respect. “oh BEY since” Think: obey stance. When the natives bowed to the conqueror in obeisance, it was like an "obey stance". obfuscated (verb): made unclear. “awb fusk KATED” Think: obstacles confused. The professor barely spoke English: the obstacles his speech created confused us and obfuscated his message. objectionable (adjective): undesirable; offensive. “Obb JECT shin a bull” Think: Objection! Objection, your honor! The plantiff’s attorney just mooned me - behavior which is clearly objectionable. objective (adjective): not inﬂuenced by personal perspective. “ob JECT ive” Think: object. It’s easy to have an objective opinion about an object like a rock – there’s not much debate about what a rock is. objurgation (noun): a harsh reprimand or criticism. “ob jur GAY shun” Think: failed obligation. When I got married I took on the obligation to be faithful, so I wasn’t surprised when my wife gave me a severe objurgation after I cheated on her. obloquy (noun): harshly critical statements; being discredited. “OB la quee” Think: lob queries. The prosecutor kept lobbing queries at the witness in an attempt to discredit him through obloquy. obsequious (adj): too eager to help or obey. “ob SEE quee us” Think: obscene kiss ass. I can’t believe that obsequious guy got the promotion – he’s just an obscene kiss ass. obstinate (adjective): stubborn. “OB stin-nit” Think: obstacle in it. The obstinate horse behaved as though there was an obstacle to movement in it. obstreperous (adjective): stubbornly resistant to control; noisy. “ob STREP er-us” Think: strep. The bacteria that cause strep throat are so obstreperous that many people take antibiotics for the condition. obtrusive (adjective): noticeable in an unwelcome way. “obb TRUE sive” Think: O.B.T. You’re right – I’m not really listening to you. But it’s because there’s an O.B.T. (Orange Baboon Tarantula) under the table that keeps trying to crawl up my leg in an obtrusive way. occluded (adjective): closed up; blocked. “awk LEWDED” Think: Octomom cluttered. Octomom cluttered the hospital's nursery with her eight babies and occluded it so no other infants were admitted. odious (adj): arousing or deserving hatred. “OH dee-us” Think: odorous onions. Only eating onions is odious because they make one's breath so odorous. ofﬁcious (adjective): bossy. “oh FISH us” Think: vicious ofﬁcial. The ofﬁcial was vicious enough to measure every ﬁsh we caught with a ruler to make sure it was legal. offset (verb): to have the opposite effect; to balance. “off SET” Think: offs it. Seeing a beautiful woman light up a cigarette offsets my interest in her; it effectively offs it since smoking is gross. ogle (verb): to stare at in a way that shows excessive desire. “OH gull” Think: Google. Half of my Google searches are for swimsuit models since I like to ogle them. omission (noun): something not done or missed. “oh MISSION” Think: miss My Oscar speech was marred by my omission of my agent; I can’t believe I missed thanking him. omniscient (adjective): all-knowing. “om NISH ent” Think: owns his science. I'm not surprised to hear that Jesus got an A in AP Chem; he owns his science classes because he's omniscient. onus (noun): burden; obligation. “OWN us” Think: on us. Since we broke the vase, the onus is on us to pay for it. opaque (adjective): something that is cloudy, blurry, or difﬁcult to understand. “oh PAKE” Think: an opaque lake. If you don’t want to get sick, then I don’t recommend swimming in an opaque lake. openhanded (adjective): generous. “OH pen HAN ded” Think: open your hand. To be openhanded, open your hand and share what you have with others instead of keeping it clenched in your ﬁst. opine (verb): to hold or state as an opinion. “OH pine” Think: opinion. If you opine about the election on Facebook, everyone gets to hear your opinion whether they like it or not. opportune (adjective): well-timed. “ah por TUNE” Think: opportunity. It's opportune that I got picked for this singing opportunity because a genie just granted my wish to sing perfectly. opprobrium (noun): public disgrace. “oh PRO bree-um” Think: Oprah opposes. Sometimes, Oprah brings people she opposes on her show to cause them opprobrium - it's like "Oprah-brium". opulent (adj): very expensive and comfortable. “OPP you lent” Think: opal I just ordered one thousand opals from the jeweler – I want my new ofﬁce to be opulent. ornate (adjective): elaborately or excessively decorated. “or NATE” Think: ornaments. Ornaments covered every inch of the Christmas tree due to the decorator’s ornate style. orthodox (adj): conventional; traditional. “orth oh DOCKS” Think: orthodontist. Look for an orthodox orthodontist; you don't want someone getting creative with your teeth. oscillate (verb): to vary; to go back and forth. “AW sill ate” Think: Ozzy. On MTV’s The Osbornes, Ozzy oscillates between sanity and insanity. ossiﬁed (verb): became hard or inﬂexible. “AW sih ﬁed” Think: fossil-ﬁed. The Velociraptor's bones could bend slightly, but after death, they ossiﬁed and turned into fossils. ostentatious (adjective): showy; pretentious; boastful. “aw sten TAY shus” Think: ostrich stunt. That ostrich stunt - when you showed up at the prom riding one like a horse - was ostentatious. ostracized (verb): excluded. “AW struh-sized” Think: ostrich-sized. That poor kid is ostrich-sized – he’ll be ostracized as soon as he starts high school. otiose (adjective): useless; lazy. “OH she ose” Think: tortoise fetch. Playing fetch with this tortoise is otiose. outmoded (adjective): out of date. “out MO ded” Think: Apple Pie a la mode. Apple pie a la mode (with vanilla ice cream on top) is a tasty but out-of-date dessert: I can see someone ordering it at a 1950s diner. outstrip (verb): to outrun or to exceed. “out STRIP” Think: out-strip. To be fair, that stripper gets more tips than you because her pole-dancing skills outstrip yours. overshadow (verb): to be more important than. “oh ver SHAH dow” Think: shadow over. Hanging out with LeBron is rough; not only do his skills overshadow mine on the court, but he’s so tall that even his shadow towers over me. overweening (adjective): arrogant. “oh ver WEEN ing” Think: weenie. The young actor's demands were so overweening that the movie crew started calling him a weenie behind his back. paciﬁc (adjective): soothing. “pa SIF ick” Think: paciﬁer. The infant’s paciﬁer had a paciﬁc effect, and she was soon asleep. painstaking (adjective): very careful. “PAINS taking” Think: taking pains. Taking pains to not infect the patient means being painstaking when you wash your hands before surgery. palatable (adjective): tasty. “PAL it ah-bull” Think: pal at table. Having my pal at the table for dinner seems to make the food more palatable since I’m in a good mood. palatial (adjective): magniﬁcent. “puh LAY shull” Think: palace. The palatial resort, with luxurious amenities and gourmet food, was like a palace. palimpsest (noun): something with multiple layers, changes, or meanings. “PA lim sest” Think: pa’s lamp set. Pa’s lamp set was more than just a bunch of lamps: it was also a piece of modern art with multiple shades of meaning. pall (noun): something that covers; a feeling of gloom. “Paul” Think: pallbearers. Looking back, it was a mistake to invite the pallbearers to our party. As soon as they entered, carrying that cofﬁn, a pall settled over the room. palliate (verb): to reduce the severity of. “PAL ee ate” Think: pill I ate. The prescription pill I ate should palliate my depression. pallid (adjective): lacking color. “PAL id” Think: pale. After playing video games in his mother's basement all winter, Al was so pale his friends described him as pallid. panacea (noun): a cure-all. “pan uh SEE uh” Think: pan of Vitamin C. The hippie advised me that eating a pan of vitamin C is a panacea for illness. pander: (verb): to appeal to someone's desire for selﬁsh reasons. “PAN dur” Think: panda logo. The American company that used a panda as its logo was accused of pandering to the Chinese market. pangs (noun): sharp feelings of pain. “Pangs” Think: pains. I read The Hunger Games for six hours, but then had to stop because of the pains from my own hunger pangs. panned (verb): sharply criticized. “Panned” Think: frying pan. My ﬁrst movie got panned so badly by critics that they might as well have hit me with a frying pan. paradigm (noun): an example used as a pattern or model. “par uh DIME” Think: pair of dimes. Those two girls are a pair of dimes since they’re both 10s – they’re paradigms for how to look hot. paragon (noun): a model of excellence. “PAR uh GONE” Think: Aragorn. If you're looking for a paragon in The Lord of the Rings, choose Aragorn: he became the king. pariah (noun): an outcast. “purr I uh” Think: Mariah’s anthem. Mariah Carey became a pariah after butchering the national anthem in front of 80,000 fans. parley (verb): to talk. “PAR lee” Think: parsley. If you're going to parley with someone you like, eat some parsley - it's good for your breath. parochial (adjective): narrow-minded. “purr RO key-ul” Think: parish yokel. Our church is led by a parish yokel who is so parochial that he believes women should be barefoot and pregnant. parody (noun): a mocking or satirical imitation. “par uh DEE” Think: parrot-y. Pro tip: a good way to make fun of someone is to repeat what he just said in a squawky parrot-y voice, to parody him. paroxysm: (noun): an attack, spasm, or outburst. “par ROX cism” Think: paradox spasm. If you drink ocean water, you’ll have a paroxysm – it’s a paradox that it’s water but it causes spasms if you swallow it. parsimonious (adjective): stingy. “par si MOAN nee us” Think: parsley money. My dad was so parsimonious he’d give us parsley money instead of lunch money. partiality (noun): an (often unfair) liking or bias toward something. “Par she AL it ee” Think: Porsche. If anyone is having trouble deciding what to get me for my birthday this year, I have a partiality toward Porsches. partisan (adjective): biased toward one side. “PART ih-sun” Think: Party’s son. The chairman of the Democratic Party's son was understandably partisan about politics. pastiche (noun): an imitation; something made of many things. “pass TICHE” Think: paste each. If you copy Wikipedia and paste each entry into your paper, it will be a pastiche. pathos (noun): a quality that evokes feelings of sadness or pity. “PATH owes Think: sympathy ohhhs. Your painting of a starving puppy has so much pathos that it always gets sympathy “ohhhs” from viewers. patois (noun): the speech/slang used by a certain group. “pa TWAH” Think: patio speech. Patio speech during barbecues is more likely to contain patois than speech in the ofﬁce. paucity (noun): lack of. “PAW city” Think: poor city. In the poor city, there was a paucity of resources. pedantic (adjective): teacherly; overly fussy. “peh DAN tick” Think: Dan ticked me off. Dan ticked me off when he was being overly pedantic, explaining every facet of his computer to me as if I were a child. pedestrian (adjective): dull; ordinary. “peh DEST ree en” Think: pedestrian (noun). You're pedestrian (adjective) because you're a pedestrian (noun) – cool kids drive to school. peevish (adjective): irritable; whiny; bratty. “PEE vish” Think: pet peeve. In the Harry Potter books, Peeves is a peevish ghost in Hogwarts whose pet peeve is happy students - so he tattletales on them. penchant (noun): a liking for something. “PEN shent” Think: pendant enchant. If a girl wears a low hanging pendant to enchant the boys, they’ll soon have a penchant for her. pendulous (adjective): hanging loosely; sagging. “PEN dew luss” Think: pendulum. As the naked old lady danced, her pendulous breasts swung back and forth like the pendulums of grandfather clocks. penitent (adjective): being sorry for one's actions. “PEN ih tent” Think: repent! The beggar's sign read, "repent! Do penance for your sins! Only the penitent will see God!" penurious (adjective): stingy. “peh NUR y us” Think: penny furious. The penny tip made the waiter furious; the customer must have been penurious. penury (noun): severe poverty. “PEN ur y” Think: penalty. The court's penalty was so large that the defendant suffered from penury to the point of only owning one penny. peons (noun): lower-class workers used by someone of a superior class. “PEE ins” Think: pee-ons. The nobleman so little respected his peons that he would pee on them. peregrinate (verb): to journey or travel from place to place. “PERR (rhymes with “err”) ih grin ate” Think: peregrin falcon. The fastest member of the animal kingdom is the peregrin falcon; it exceeds 200 m.p.h. while diving and can peregrinate speedily. peremptory (adjective): bossy, prone to cutting others off. “puh REMP tuh ree” Think: pre-empted. The Emperor pre-empted Luke's replies so much that even Darth Vader called him peremptory. perennial (adjective): constant; persistent; recurring. “purr ANY ul” Think: per annual. Per the annual tradition, it’s time to start the perennial search for the next American Idol. perﬁdy (noun): treachery; treason. “PURR-FIH-dee” Think: perforated ﬁdelity. When I realized my friend spread rumors about me, I felt like he had perforated ﬁdelity because of his perﬁdy. perfunctory (adj): showing little interest. “purr FUNKED ur y” Think: per function problem. I know I won’t get them right, so I only spend a perfunctory amount of time per function problem. peripatetic (adjective): wandering; traveling; constantly moving from place to place. “Peh ruh puh TET ick” Think: pitter-patter. The mouse in my house is peripatetic since I’m constantly hearing the pitter-patter of his little feet in the walls. peripheral (adjective): off to the side, external, related, tangential. “purr RIFF ur-ull” Think: Perry Farrell. Perry Farrell, the eccentric founder of Lollapalooza, has never made into the musical mainstream; he seems to prefer the peripheral genres. permeated (verb): spread through; penetrated. “PERM y ated” Think: perm he ate. The perm he ate will permeate his stomach lining and prove it’s stupid to eat human hair. permutation (noun): a transformation or rearrangement. “PERM you TAY shun” Think: mutation. A radioactive spider bite caused a permutation of Peter Parker’s genes and caused his mutation into Spiderman. pernicious (adjective): destructive; deadly. “per NISH iss (rhymes with “kiss”)” Think: piranhas vicious. Piranhas are vicious; lingering in waters they inhabit can be pernicious. perquisites (noun): privileges or bonuses. “PER quiz its” Think: perks. Hey suckahs - now that I’m CEO, I enjoy perquisites like a company helicopter and a gold wastebasket - perks you’ll never have. personiﬁed (verb): to be the perfect example of something; to represent a thing as a person. “Purr SAHN if eyed” Think: person. Look up “greatness” in the dictionary and the person in the picture is me; I’m greatness personiﬁed. perspicacious (adjective): sharp; clever. “purse puh-KAY shiss (rhymes with “kiss”)” Think: perspective for the aces. A perspicacious poker player uses her clear perspective to know who has the aces. pertinacious (adjective): stubbornly persistent. “purr tin aey shuss” Think: persistent and tenacious. My pertinacious defender was both persistent and tenacious; I had no open shots. perturb (verb): to disturb greatly. “purr TURB” Think: disturbed by turds. The turds your dog leaves on my beautiful lawn not only disturb me - they perturb me. pervasive (adjective): widespread, prevalent, omnipresent. “purr VAY sive (rhymes with “give”)” Think: perv invasive. That perv is so invasive – his inappropriate touching is pervasive and goes way over the line. perverse (adjective): bad; wrong; corrupt. “purr VERSE” Think: perverts. Perverts like Peeping Toms are perverse; they should be locked up. petulant (adjective): rude; irritable. “PETCH you lent” Think: petty aunt. My petty aunt is always whining about something or holding a grudge; she’s petulant. philander (verb): to have casual or unfaithful sex. “ﬁ LAND urr” Think: Zoolander. When you're ridiculously good-looking like Zoolander, it's easy to philander. phlegmatic (adjective): sluggish; unresponsive. “Fleg MATIC” Think: phlegm. When I had the ﬂu, I had so much phlegm clogging my respiratory system that I was completely phlegmatic. physiological (adjective): related to the body. “ﬁzz y yo LOG ih cull” Think: physical. I knew I wasn't just imagining I was ill because my physiological symptom, a fever of 103, was physical. picaresque (adjective): about someone's adventures. “pick a RESK” Think: Pixar-esque. Your novel is like Wall-E because of your hero's journey; it's both picaresque and Pixar-esque. picayune (adjective): unimportant; small-minded. “pick a YUNE” Think: picky one. The picayune bridezilla was quite the picky one, worrying about every single detail of her wedding. picturesque (adjective): lovely. “pick sure ESK” Think: pictures. The Grand Canyon at sunrise is so picturesque that you can't help but take pictures. piebald (adjective): many-colored; varied. “PIE bulled” Think: pies. The horse's coat was piebald, pebbled with blotches of so many colors that it looked like pies were thrown at it. pilfer (verb): to steal. “PILL fur” Think: pill for. Here’s an idea: we break into the pharmacy and pilfer pills for resale to drug addicts. pillory (verb): to publicly and harshly criticize. “PILL ur y” Think: Hillary. Republicans love to pillory Hillary Clinton. pinnacle (noun): the highest point of something. “PIN a cull” Think: Pinocchio. Pinocchio is the story of Geppetto, a woodworker, whose pinnacle of achievement is carving a wooden boy (Pinocchio) who can move and talk. pioneering (adj): earliest; original. “PIE-oh NEAR-ing” Think: pioneer. Lewis and Clark, the pioneers that led the ﬁrst American expedition to the Paciﬁc coast, were pioneering explorers. piquant (adjective): pleasantly spicy or tangy. “PEEK ent” Think: pickled ant. I never thought I’d like eating pickled ant, but it’s surprisingly piquant. pitfall (noun): drawback. “PIT fall” Think: pit fall. We prepared for lots of dangers before our jungle trek, but funnily enough, our biggest pitfall was an actual pit fall. pith (noun): the essential or central part. “PITH (rhymes with “with”)” Think: pit. If you’re a peach tree, the pith of your fruit is the pit, since that’s how you’ll reproduce. pittance (noun): a very small amount. “PIT ints” Think: pit ants. Pit ants are known for eating even the pittance of fruit that clings to discarded peach pits. pivotal (adjective): important, key. “PIV it ul” Think: pivot foot In basketball, knowing how to effectively use your pivot foot is pivotal to your success in the low post. Just ask Hakeem “The Dream” Olajuwon or Kevin McHale. placate (verb): to pacify or satisfy an angry or difﬁcult person. “PLAY kate” Think: play Kate. In order to placate my attention-loving sister, I told her that she could play Kate Winslet’s lead characer in our theater’s adaptation of Titanic. plaintive (adjective): melancholy. “PLANE tive (rhymes with “give”)” Think: complaint. When I hear the plaintive cry of a seagull, it always sounds like a complaint about the bird's woes or travails. platitude (noun): an overused expression. “PLAT (rhymes with “bat”) ih tude” Think: blah attitude. Dude, she’s giving you that blah attitude cause your pickup line was a platitude. platonic (adjective): not related to romance or sex. “Pluh TAWN ick” Think: Plato date. If you only talk to your date about the philosopher Plato, you'll end up as her platonic friend. plaudits (noun): approval. “PLAUD its” Think: applaud it. If you want to give plaudits to his work, applaud it. plausible (adjective): apparently true. “PLAWS ih bull” Think: applause-able. When the magician sawed the lady in half, it looked so plausible that it was applause-able. plebeian (adjective): common; low-class. “PLEE be in” Think: ﬂeas be in. I don't stay in plebeian motels 'cause ﬂeas be in 'em. plenipotentiary (adjective): fully empowered. “Plen a po TENT cha ry” Think: plenty of potency. If Romney wins the election, he will be plenipotentiary; being president will give him plenty of potency. pluck (noun): courage; spirit. “PLUK” Think: pluck a feather. You showed pluck by attempting to pluck a feather from a live ostrich; too bad it decided to peck you in the eye. plutocracy (noun): government controlled by the wealthy. “plue TOCK ra see” Think: Pluto vacation home. Hearing about the senator's vacation home on Pluto made me realize we're in a plutocracy. polarize (verb): to separate into two conﬂicting or opposite positions. “PO la rise” Think: Earth’s poles. Democrats and Republicans are so polarized that I’m surprised they don’t stay at the North and South Poles to keep as far apart as they can. polemic (noun): a harsh attack against a principle. “po LEM ick” Think: politician at a mic. Put a politician at a mic, and you’ll soon hear polemic as he attacks his opponent’s policies. politesse (noun): politeness. “paw lit ESS” Think: politeness. French maids are trained to show politeness at all times; their politesse is without equal. politic (adjective): shrewd; wise. “PAW lit ick” Think: politician. After reading The Prince by Machiavelli, the politician became much more politic and cleverly defeated his opponent. pomp (noun): ceremony and showy display. “PAWMP” Think: pump. The bodybuilder was all about pomp; before he went to the beach, he worked out to get a pump to show off his muscles. ponderous (adjective): heavy; dull. “PON dur us” Think: ponder. If a subject in school makes you ponder it for long periods of time, it could just be that it’s ponderous and is either heavy, or dull, or both. portentous (adjective): foreshadowing something bad; trying to seem important. “Poor TENT shis” Think: important tent. It’s important that we set up our tent; those thunderclouds look portentous. poseur (noun): one who pretends to be something he is not. “po ZERR (rhymes with “err”)” Think: poser. The poseur pretended to be interested in literature to impress girls, but he was exposed as a poser who didn’t even know who Shakespeare was. posit (verb): to assume to be true; to suggest. “PAUSE it” Think: positive. Positive about her ﬁndings, the scientist ﬁnally agreed to posit the existence of extraterrestrial life in a journal article. posthumous (adjective): after death. “PAWST (rhymes with “lost”) hyu miss” Think: post-human. It’s a small comfort to be posthumously awarded a medal – you’re post-human at that point, i.e., dead. pragmatic (adjective): practical. “prag MATICK” Think: practical automatic. To be practical, buy an automatic car instead of a stick shift - it’s more pragmatic for city driving. prattle (noun): meaningless talk. “PRAT ul” Think: baby rattle. Her prattle about reality TV was as exciting as listening to a baby shake its rattle. precarious (adjective): dangerously unstable. “pruh CARE y us” Think: preach carefulness. Preach carefulness to people who are standing on precarious rock ledges. precocious (adjective): very talented at a young age. “pruh CO shus” Think: pre-coaching. Sadie was precocious at piano pre-coaching; she taught herself to play Mozart at the age of two. precursor (noun): something that came before another thing. “PRE kur sir” Think: pre-cursor. When you’re typing, the word you just typed is literally pre-cursor; it’s before the cursor and is thus a precursor to it. predilection (noun): a preference. “pre dih LECT shun” Think: predicted direction. Google predicted the direction of my search when I typed “how to ﬁnd” by showing “how to ﬁnd love”, because it knows people have a predilection to seek romance. premonition (noun): hunch, intuition. “preh mo NISH UN” Think: pre-mention. I knew you were going to say that! I had a premonition pre (before) you mentioned it. prescient (adjective): seeing the future; well-planned and thought out. “PRESS y ent” Think: knowing about present you sent. Since I’m prescient, I already know what’s in the present you sent me. pretext (noun): a fake excuse. “PRE text” Think: pee text. She pretends to have to pee and leaves on the pretext of using the restroom so she can text without getting caught. prevarication (noun): a lie. “pruh VARE uh kay shun” Think: pre-veriﬁcation. Pre-veriﬁcation, your story about getting chased by a bear was believable, but your friend just conﬁrmed the animal was a squirrel, exposing your prevarication. primed (adjective): ready. "PRYMED" Think: primed for prime time. When a television news anchor has paid her dues, you might say that she’s primed for prime time. primordial (adjective): original; existing since the beginning. “Pry MORDY ul” Think: primary order. The Big Bang is primordial because it has the primary position in the order of events. pristine (adjective): pure. “prih STEEN” Think: Listerine. Listerine mouthwash tastes bad but it kills bacteria and makes your mouth pristine. proclivity (noun): a tendency, inclination, or predisposition toward a particular activity. “pro CLIVE ih tee” Think: pro-cliff tee I’m guessing that the guy with the pro-mountain climbing t-shirt has a proclivity for extreme sports, since he climbs cliffs. prodigal (adj): wasteful. “PROD ih gull” Think: Prada gal. The Prada gal was prodigal: she spent all her money on designer clothes. prodigious (adjective): impressively large; extraordinary. “prah DIJ us” Think: prodigy. The child prodigy could multiply prodigious numbers in his head. profane (adjective): sacrilegious; vulgar; improper. “proh FAIN” Think: profanity. Using profanity in church is obviously profane. proﬂigacy (noun): reckless wastefulness. “prah FLIG a-see” Think: proﬁts ﬂing. If your proﬁts ﬂing out the window, you're probably following a course of proﬂigacy. profound (adjective): deep. “pro FOUND” Think: professor found. My professor found the cure for cancer because his decades of study gave him a profound understanding of the disease. profuse (adjective): plentiful; abundant. “pro FYOOS” Think: professors use. Professors use books with a profuse amount of information to make reading assignments take forever. progenitors (noun): direct ancestors. “pro JEN ih ters (rhymes with “hers”)” Think: produced from genitals. You were produced from the genitals of your progenitors. prognosticate (verb): to predict. “Prog NAWS tih-kate” Think: professional knows. Punxsutawney Phil, the groundhog who prognosticates whether winter will last for six more weeks, is obviously a professional that knows the future. proliferate (verb): to grow or multiply quickly. “pro LIFF ur ate” Think: pro-life-rate. The pro-life-rate of births is higher than the pro-choice rate; pro-life people proliferate because they don’t get abortions. proliﬁc (adjective): abundantly productive. “pro LIFF ick” Think: pro-life-ic. That state is anti-abortion, and they’re proliﬁc baby-makers because of their pro-life-ic stance on the subject. prolix (adjective): too long / wordy. “PRO licks” Think: proliﬁc. After writing dozens of 1000+ page books, the proliﬁc author was often criticized for his prolix writing style. prominent (adjective): well-known; standing out. “PRAH min int” Think: Prom King. The Prom King is usually not the shy boy that no one knows; he’s often a prominent, popular kid. promulgate (verb): to make known. “PROM ul gate” Think: promote mullet. The hillbilly hairstylist would often promote mullets by promulgating about them to new clients. pronounced (adjective): noticeable. “pruh NOUNCED” Think: pronounced (verb). My British friend enunciates very loudly and distinctly; his pronunciation is quite pronounced. propagate (verb): breed, grow, promote, publicize. “PROP a gate” Think: proper gait. The ﬁrst step to propagating your DNA is to cultivate the proper gait: in other words, you have to learn how to look good while walking. propensity (noun): a tendency, inclination, or predisposition toward a particular activity. “pro PEN si-tee” Think: vaporizer pen. The newest generation of smokers has a propensity to use vaporizer pens since they’re purported to be safer than traditional cigarettes, but experts aren’t sure whether they actually are less harmful. prophetic (adjective): that which foretells the future. “prah FET ick” Think: prophet. In the bible, Jesus is considered to be a prophet because many of his prophetic claims actually happened - like predicting that Peter would deny him three times. propitious (adjective): favorable; promising. “pro PISH us” Think: prop it up. In "A Charlie Brown Christmas", Linus thought the little tree was propitious, so he decided to prop it up. propriety (noun): the quality of being proper or appropriate. “pro PRY it y” Think: proper. For the sake of propriety, use proper manners and eat your salad with the salad fork, not the dinner fork. prosaic (adjective): dull or boring. “pro SAY ick” Think: pros say ick. I was going to watch the new Adam Sandler movie, but the movie critic pros say, “ick - the ﬁlm is prosaic”. protean (adjective): varied; versatile. “PRO tee-in” Think: proteins. Since they can be formed from a vast number of combinations of 500 different amino acids, proteins are protean. providential (adjective): favorable, fortunate, timely. “prah vih DEN shull” Think: Pro-V Dental. The friendly representative from Pro-V Dental insurance company helped me create a policy just one week before I accidentally chipped my tooth. It was quite providential. (providence = fate) provincial (adjective): narrow-minded. “prah VIN shull” Think: province. If you never leave your Canadian province, your worldview will probably be somewhat provincial. prowess (noun): exceptional bravery and/or skill. “PROW (rhymes with “how”) ess” Think: prowl lioness. While on the prowl, the lioness displayed her prowess by bringing down a woolly mammoth. proximity (noun): closeness. “prock SIM ih tee” Think: approximately. Approximately means "close to"; proximity means closeness. prudent (adjective): wise. “PROO dent” Think: prude student. In high school, prude students are prudent, since it's not a great idea to be 16 and pregnant. puerile (adjective): childish. “PURE ul” Think: puberty. The high school freshman’s puerile sense of humor was typical of a boy who was going through puberty. pugnacious (adjective): wanting to ﬁght. “pug NAY shus” Think: Pug nation. Imagine how pugnacious a Pug nation would be - those little dogs deﬁnitely would be ﬁghting all the time. pulchritude (noun): beauty. “PULK rih tude” Think: poll: Christ, dude. She had so much pulchritude that the most common response about her looks in the bros' poll was just, "Christ, dude!" punctilious (adjective): marked by following the rules strictly. “punk TILL y-us” Think: punctual. The teacher's pet was both punctilious and punctual, but most wanted to punch him. pungent (adjective): strongly scented. “PUN jent” Think: punch scent. The boxer's body odor was so pungent it was like getting hit by a punch of scent. punitive (adjective): involving punishment. “PYOON ih tive (rhymes with “give”)” Think: punish. The punitive damages in the O.J. Simpson murder case were clearly designed to punish the defendant. purist (noun): traditionalist, literalist. “PURE rissed” Think: pure wrist. I’m a purist, so I don’t think you need a fancy set of titanium golf clubs to be a good golfer: a good golf swing is pure wrist and hips. pusillanimous (adjective): cowardly. “pyoo suh LAN ih-muss” Think: pussycat. The pussycat is an animal that is pusillanimous when scared - hence the expression “scaredy-cat”. putrid (adjective): foul or rotten. “PYOO trid” Think: puked. The dead mouse smelled so putrid that I almost puked while getting rid of it. quagmire (noun): a difﬁcult situation. “KWAG mire (rhymes with “ﬁre”)” Think: quicksand mire. Quicksand can mire you if you step in it, and the more you struggle, the worse the quagmire becomes. quail (verb): to pull back in fear. “kwale” Think: quail (the bird). I feel bad for quail (noun) - those poor birds quail (verb) as soon as they see people because they’re often hunted for sport. quandary (noun): a situation that makes you confused about what to do. “kwan duh rye” Think: wandering. I’m wandering around aimlessly because I’m in a quandary about where to go next. quash (verb): to completely stop from happening. “kwash” Think: squash. The best way to quash an invasion of ants in your kitchen is simple: squash them. querulous (adjective): whiny; complaining. “KWER uh luss” Think: quarrel us. We'd invite you over more, but you're so querulous that you always end up in a quarrel with us! quiescent (adjective): at rest. “KWEE ess ent” Think: quiet. The hibernating bear was both quiet and quiescent. quintessential (adjective): the most typical; the purest. “kwin teh SEN shull” Think: essential. Watching a Red Sox game at Fenway Park is essential to get the quintessential Boston experience. quixotic (adjective): idealistic; impractical. “kwicks OT ick” Think: quick exotic. It's quixotic to think that you should earn some quick cash by becoming an exotic dancer. quizzical (adjective): questioning; teasing. “KWIZ ih cull” Think: quizzing. When I started asking my date about the periodic table, her quizzical expression seemed to be quizzing me about why I’d brought up such an awkward topic. quotidian (adjective): daily. “kwoh TID (rhymes with “did”) y-en” Think: quota. The meter maid met her daily quota of parking tickets by her quotidian patrolling of the streets. raconteur (noun): a good storyteller. “rack on TURR” Think: recount. Jack White named one of his bands The raconteurs because they were so good at recounting stories via song. ragamufﬁn (noun): a dirty, poor person or child. “RAG a mufﬁn” Think: rags on mufﬁn. The rags on your little mufﬁn make him look like a ragamufﬁn - shop at Baby Gap next time. raiment (noun): clothing. “RAY ment” Think: rain meant. In the nudist colony, a forecast of rain meant they'd actually have to don some raiment. ramiﬁcation (noun): the result of an action or decision. “RAM if a CAY shun” Think: Dodge Ram. One ramiﬁcation of trading in my Prius for a Dodge Ram is that I am spending a lot more money on gas. rampant (adjective): widespread; uncontrolled. “RAM pent” Think: rampage. If you're dumb enough to take bath salts, the destruction after your rampage will be rampant. rancorous (adjective): hateful. “RAN kur us” Think: Star Wars rancor. In Return Of The Jedi, the rancor under Jabba The Hutt's palace is undoubtedly rancorous for having been imprisoned. rankled (verb): irritated. “RAN kulled” Think: wrinkled. Tim Gunn told the “Project Runway” contestant to "make it work", so the wrinkled dress she made rankled him. rapacious (adjective): greedy; predatory; ravenous. “ruh PAY shus” Think: rapes us. The rapacious new tax law takes so much of our earnings that it effectively rapes us. rapt (adjective): completely interested. “RAPPED” Think: wrapped. The audience was held rapt by the master violinist’s performance; they were completely wrapped up in it. rapturous (adjective): full of wonderful feelings; ecstatic. “RAP shur us” Think: raptor saw us. The raptor saw us being lowered into his cage and felt rapturous since he was hungry. rareﬁed (adjective): lofty; reserved only for a select few. “RARE uh ﬁed” Think: rare ﬁnd. The trufﬂe your pig dug up is a rare ﬁnd, peasant - you dare not eat such a rareﬁed delicacy - save it for his Majesty. rash (adjective): hasty; incautious. “RASH” Think: rash (noun). If you make the rash (adjective) decision to have unprotected sex with that NBA player, you might get a rash (noun). raucous (adjective): noisy; disorderly. “RAW kus” Think: rocks us. The Beastie Boys' raucous track, "Fight For Your Right to Party", rocks us. raze (verb): to completely destroy. “RAZE” Think: rays blaze. The powerful laser's rays are making a blaze that will raze the old building to make room for the new one. readily (adjective): with preparation / enthusiasm. “RED il-ly” Think: ready. If you are ready to answer a question, then you can answer it readily. realization (noun): the making of something into reality. “Real liz a shun” Think: real Z nation. After a grueling, 15-year guerrilla war against the ruling forces of Rhodesia and its conservative white minority, the nation of Zimbabwe ofﬁcially became independent in 1980, the realization of a longtime dream for independence. reap (verb): to gather or obtain. “REEP” Think: Grim Reaper. If there’s a knock on the door and you see the Grim Reaper through the peephole, don’t answer: he has come to reap your life. rebuttal (noun): a response. “re BUTT ul” Think: butt. After I lectured the college sophomore about the dangers of binge drinking, his rebuttal was to moon me – he showed me his butt. recalcitrant (adjective): difﬁcult to manage or change. “ruh CAL sih trant” Think: calc rant. The calc worksheet made Alex rant because it was so recalcitrant. recant (verb): to formally deny a former position. “RE cant” Think: really I can’t. I know I said I would move to Canada if we elected Obama, but really I can’t, so I recant that statement. recapitulated (verb): summarized. “RE kuh PIT u lated” Think: recap. His recap of the news nicely recapitulated the day's events. recidivist (noun): someone who relapsed into crime. “ruh SID uh vist” Think: Sid's division. Sid had two sides to his personality: the law-abiding side and the recidivist. reclusive (adjective): characterized by hiding and avoiding society. “ruh CLUE sive” Think: brown recluse. Luckily for us, the deadly poisonous brown recluse spider is reclusive. recondite (adjective): not easily understood. “REH cun dite” Think: reckoned it. I couldn’t understand my professor’s recondite lecture, but I reckoned it had something to do with the fourth dimension. recrudescent (adjective): reactivating. “reh kru DES sent” Think: recruits sent. The conﬂict in Afghanistan must be recrudescent since more recruits are sent there daily. rectitude (noun): extreme integrity. “RECK tih tude” Think: correct attitude. Since he was a church rector, Paul considered the correct attitude to be rectitude. redouble (verb): to greatly increase the size or amount of something. “re DUB ull” Think: double. Football practice was brutal today! Coach made us double our efforts, but then that wasn’t enough for him, so we had to redouble them. redress: to set right. “ruh DRESS” Think: re-dress Lady Gaga. Lady Gaga's fashion choices are so wrong that the only way to redress her style is to literally re-dress her. reductive (adjective): related to making something smaller or simpler. “ruh DUCT ihve” Think: reduce. Reductive Spark Notes reduce brilliant works of literature into basic summaries. redundant (adjective): needlessly repetitive. “ruh DUN dent” Think: re-done. Duh... that has already been done well - it will be redundant if you decide it needs to be redone. refracted (verb): distorted or changed from an initial direction. “re FRAK tid” Think: reﬂected fractured. The prism refracted the white light and reﬂected it, fractured, into a rainbow of colors. refractory (adjective): stubborn; unmanageable. “ruh FRAK turry” Think: re-fracture. The refractory athlete insisted on playing despite his broken toe; unsurprisingly, he re-fractured it. refulgent (adjective): brightly shining. “ruh FOOL jent” Think: refuels it. The campﬁre gets refulgent after he refuels it. refute (verb): to speak against or disprove. “ruh FYOOT” Think: refuse. My dog refutes my argument that he needs a bath by adamantly refusing to get in the tub. rejuvenated (verb): gave new life to. “ruh JUVE in ated” Think: re-juvenile. His plans for the new year rejuvenated the middle-aged man so much that he felt like a juvenile again. relinquish (verb): to give up to or return to. “ruh LEN quish” Think: release anguish. When I went to prison, I had to relinquish my baby boy to social services which gave me release anguish. relish (verb): to enjoy; to savor. “REH lish” Think: delish. I relish (verb) eating hot dogs with relish (noun) because they taste delish. remedial (adjective): intended to correct at a basic level. “ruh MEED y ul” Think: remedy. If you are terrible at math, the only remedy might be to take a remedial arithmetic class. reminiscent (adjective): similar to, evoking. “rem (rhymes with “gem”) in IS int” Think: remind scent. The smell of baking bread is reminiscent of my youth; the scent reminds me of my grandmother’s kitchen. remiss (adjective): careless. “ruh MISS” Think: re-miss. If you are remiss in your study technique, you'll miss the point the ﬁrst time you read then re-miss it the 2nd time. remunerated (verb): compensated or paid for. “ruh MOON ur ated” Think: re-moneyed. It cost me $300 to remove the rat from my apartment, but my landlord remunerated / "re-moneyed" me. renowned (adjective): famous in a good way. “ruh NOUNED” Think: re-known. Renowned celebrities are often known in their era then re-known on reality T.V. shows several years later. repertoire (noun): “bag of tricks,” canon. “ryep uh TWAR” Think: repeat. When I followed the Grateful Dead around for a summer, I realized their repertoire was ﬁnite; their set list repeated most of the time. replete (adjective): full. “ruh PLEET” Think: replace completely. Replace your energy completely after your workout so your body stays replete with energy. reprehensible (adjective): deserving blame. “rep ree HEN sih bull” Think: pretend hens. Heyyy... you sold me pretend hens instead of real ones - that’s reprehensible. reprobate (adjective): evil. “REP roh bait” Think: re-probed it. The aliens who gave Cartman an anal probe on “South Park” would be even more reprobate if they re-probed it. reprove (verb): to gently criticize or correct. “reh PROVE” Think: reps at the gym. Some bodybuilder types can’t help but reprove everyone else’s technique while they doing reps at the gym. repudiate (verb): to refuse to accept; to reject. “ruh PYOO dee ate” Think: refuse poo I ate. If I were to eat poo, my stomach would refuse the poo I ate and repudiate it by vomiting uncontrollably. repugnant (adjective): gross. “ruh PUG nent” Think: ugly Pug. Although some people think Pugs' upturned faces and wheezing are cute, many ﬁnd the breed to be repugnant. requisite (adjective): necessary. “RECK wiz it” Think: requires it. If you fail English, your school requires it to be re-taken; it’s requisite that you have four years of English. resigned (adjective): reluctantly accepting of a bad situation. “ruh ZINED” Think: resignation. After being implicated in Watergate, Nixon was resigned and offered his resignation from ofﬁce. resilient (adjective): sturdy; ﬂexible. “ruh ZILL y ent” Think: Brazil nut. Have you ever tried to crack open a Brazil nut? Their shells are resilient! resolute (adjective): ﬁrmly determined. “rez oh LOOT” Think: resolution. It's no use to make a New Year's resolution if you're not resolute enough to follow through with it. resonant (adjective): creating sonic vibrations (literal), connecting on a deep level (metaphorical). “REZ uh nent” Think: resin on it. Not only is Nina Simone’s voice literally resonant, with a booming force that can stun a live crowd, but it is also ﬁguratively resonant in that she is able to form a deep connection with her audience. Simone can sing or with as much grace or grit as the occasion requires, with the ability to take her usual tone and sprinkle some resin on it. respite (adjective): a short rest. “RESS pit” Think: rest it. Don't overwork your respiratory system; if you take a respite and rest it, your lungs will thank you. resplendent (adjective): shining brilliantly. “re SPLEN dint” Think: splendid. Cinderella was resplendent in a sequined, white ball gown; she looked absolutely splendid. restitution (noun): the act of making up for something bad. “rest ih TOO shun” Think: rest of tuition. My college's restitution for allowing prostitution was paying the rest of our tuition. restive (adjective): restless; ﬁdgety. “rest ive (rhymes with “give”)” Think: rest on stove. Good luck taking a rest on a stove - you’ll feel too restive to sleep because you’ll worry it will turn on. resurgence (noun): a comeback. “ruh SURGE ince” Think: re-surge. When LL Cool J said, "Don't call it a comeback", he meant that his re-surging to the top wasn't a resurgence. reticent (adjective): reserved; quiet. “RET uh sent” Think: ready but hesitant. If you have to recite a speech and you're technically ready but hesitant, you might be reticent. retiring (adjective): shy. “re TIRE ing” Think: retire from parties. The shy girl was so retiring that she decided that she would retire from going to parties. retrenchment (noun): a reduction. “ruh TRENCH ment” Think: return to trench. For a WWI soldier, a retrenchment of the attack plans meant he could return to his trench and lay low for a while. retrospection (noun): the act of thinking about the past. “reh tro SPEC shun” Think: retro-inspection. Retrospection about the 1980s is a retro-inspection that can lead to wearing neon clothes and leg warmers. revamp (verb): to revise, improve, or make over. “re VAMP” Think: return as a vampire. If you’re sucked dry by a vampire, don’t worry - you’ll die, but then be revamped as a strong, new member of the undead. revanche (noun): revenge. “RE van shay” Think: revenge. Motivated by revenge, the French monarch ordered her general to take revanche on those who had captured the island. reverberate (verb): to echo. “ruh VERB er ate” Think: re-vibrate. I yodeled in the empty concert hall, and the echoes reverberated and re-vibrated as they bounced off the walls. reverent (adjective): having deep respect for. “REV ur ent” Think: reverend. During church, the reverend reminded them to be reverent to Jesus. revile (verb): to abuse verbally. “ruh VILE” Think: evil, vile. My disillusionment with the army began when I tripped, causing the drill sergeant to revile me with the most evil, vile insults I’ve ever heard. revulsion (noun): disgust. “ruh VUL shun” Think: revolt and shun. When the king barfed then ate the barf, I felt such revulsion that I wanted to revolt and shun him. rhapsodize (verb): to enthusiastically praise. “RAP sa dyes” Think: rapture. The rapper Sisqo felt so much rapture when looking at women wearing thongs that he rhapsodized about them in “The Thong Song”. rhetorical (adjective): hypothetical, related to rhetoric (communication style). “rhuh TORE ih cull” Think: Slick Rick’s rhetoric. Rapper Slick Rick is a smooth talker; in other words, he has slick rhetoric. rickety (adjective): weak. “RIK it y” Think: rickets. Rickets, a disease that weakens the bones, makes its sufferers rickety. rift (noun): a break or split. “RIFT” Think: ripped. The rift in our friendship was so deep that it felt as though our bond had been ripped. riposte (noun): a comeback. “rih PAWST” Think: rip post. After being mocked, the blogger would rip into his critic’s post with a brutal riposte. risible (adjective): funny; inclined to laugh. “RIZ uh bull” Think: get a rise. If you like to get a rise out of people by being a class clown, you're probably risible. risque (adjective): almost improper or indecent. “riss K” Think: risky. Making a risque joke the ﬁrst time you meet your girlfriend's parents is risky. roborant (noun): an invigorating drug. “ROB uh rent” Think: robo-ant. After I gave him a roborant, my ant felt as strong as a robo-ant. robust (adjective): healthy; strong; rich; full. “roh BUST” Think: robots. Humans wouldn't last long on Mars due to the extreme cold - we sent robots since they're more robust. rotund (adjective): round; full; plump. “roh TUND” Think: round tummy. Your pet hippo’s tummy has grown so rotund that it's literally round at this point. row (noun): a disagreement. “ROW (rhymes with “plow”)” Think: rrrr...ow! The row between the two boys started with growling: “rrrr!” and was quickly followed by an “ow!” as one punched the other. rudimentary (adjective): basic; primitive. “rude ih MENT uh ry” Think: rude elementary. Rude elementary school kids are impolite only because their knowledge of social graces is rudimentary. rufﬁan (noun): a brutal person. “RUFFY en” Think: rough. The club hired a rufﬁan as a bouncer because he was strong enough to be rough with misbehaving drunks. ruminate (verb): to carefully reﬂect on. “ROOM in ate” Think: Ramen marinate. To ruminate means to think about something for at least as long as it takes your Top Ramen to marinate. saccharine (adjective): sweet in a fake way. “SACK a rin” Think: saccharin. The beauty contestant's personality was so saccharine that there must have been Sweet and Low (saccharin) in her veins. sacrosanct (adjective): holy. “SACK ro SANKED” Think: sacred sanctuary. The temple was a sacred sanctuary and was declared sacrosanct to protect it from real estate developers. salacious (adjective): appealing to sexual desire. “suh LA shus” Think: salivate. All the girls read Fifty Shades of Gray because the salacious details make them salivate. salient (adjective): very important or noticeable. “SAY lee ent” Think: saline. If you’re dehydrated, getting saline into your bloodstream is your most salient concern. salubrious (adjective): good for your health. “Sal OOH bree us” Think: salute. The kale smoothie I just drank was so salubrious that my stomach would salute me if it could. salutary (adjective): beneﬁcial. “SAL (rhymes with “pal”) u tary” Think: salute. Sal’s cooking has such a salutary effect on me that I salute him. sangfroid (noun): coolness and composure. “sang FRWA (it’s a French word)” Think: sang frog. “You don’t scare me!” sang the frog when he saw the fox - he had sangfroid in spades. sanguine (adjective): optimistic. “SAN gwin” Think: Penguin sang win. The penguin sang that he would win; he was sanguine. sap (verb): to weaken. “SAP” Think: tree sap. Cutting your initials into a tree can sap (verb) its vitality because it will make the sap (noun) leak out. sapid (adjective): ﬂavorful. “SAP id” Think: maple sap. We make maple syrup from the sap of maple trees because their sap is naturally sapid. sapient (adjective): wise. “SAP y ent” Think: Homo sapiens. Be proud that you're a member of Homo sapiens; you're more sapient than any other animal on the planet. sardonic (adjective): mocking (in a mean way). “sar DON ick” Think: sarcastic sardines. When the seniors saw I ate sardines for lunch every day, they made sardonic, sarcastic comments. sashayed (verb): strutted or walked in a showy or ﬂashy way. “sah SHAYED” Think: Miss America sash. Miss America sashayed across the stage, showing off her ﬁrst-place sash. satiated (adjective): satisﬁed. “SAY she ated” Think: say she ate. If you say she ate, she must be satiated. scanty (adjective): barely sufﬁcient; minimal. “SKAN tee” Think: scanty panty. Thong underwear is basically just a really scanty panty. scapegoat (noun): one that takes the blame. “SKAPE goat” Think: escaped goat. Even though the dog ate some of the vegetables in the garden, the escaped goat became the scapegoat. scathing (adjective): sharply critical. “SKAY thing” Think: scythe. Getting killed by the Grim Reaper’s sharp, hooked scythe is as about as scathing a criticism as one can get. schadenfreude (noun): enjoyment from others' troubles. “SHA den froy dah” Think: shady Freud. If your psychologist giggles about your divorce he has schadenfreude and is a shady Freud. schism (noun): a separation into opposing groups; a divide. “skism” Think: schizophrenic. The schizophrenic patient underwent a schism that gave him multiple personalities. scintillating (adjective): sparkling; brilliant. “SIN tull ating” Think: squint. Her sequined shirt was so scintillating that I had to squint to see it. sclerotic (adjective): rigid; reluctant to adapt or compromise. “sclear OTT ick” Think: arthosclerosis. When plaque builds up inside someone’s arteries, he can develop arthosclerosis – a dangerous condition in which those blood vessels become sclerotic. scofﬂaw (noun): a contemptuous law-breaker. “SKOF law” Think: scoff at the law. A scofﬂaw will scoff at the law he just broke since he has no respect for it. scotch (verb): to put a sudden end to; to injure. “Skotch” Think: scratch. Well, the boss just scotched our plan to bring our cats to work, so scratch that idea. scrupulous (adjective): having integrity, or being exact. “SKRUP u luss” Think: scrape the poop. If you are scrupulous, you will scrape your dog's poop off my lawn. scrutinize (verb): to examine carefully. “SKROO tin eyes” Think: desire to screw in eyes. I’m an 18-year-old cheerleader - when a dirty old man scrutinizes me, I see the desire to screw in his eyes. scufﬂe (noun): a brief ﬁght. “SKUFF ul” Think: scuff. The scufﬂe was no big deal, but I did scuff my new pair of shoes. scurrilous (adjective): obscenely abusive. “SKURR a lus” Think: scurvy curses. After the pirate developed scurvy, his curses became even more scurrilous. scuttle (verb): to destroy; to scrap. “SKUT ul” Think: it’s cut. Scuttle the launch of that Space Shuttle! It's cut from the space program as of 2011. secretes (verb): forms and gives off. “suh CREETS” Think: secret sea creature. The octopus is a sea creature that stays secret when it secretes an inky cloud. sectarian (adjective): narrow-minded. “sek TEAR y-en” Think: sector. Sectarian views are shallow because they only consider one sector of the whole issue. secular (adjective): not related to the spiritual or religious. “SEK u lur” Think: sex u later. “If ur religious, I am not interested, but if ur secular I might want to sex you later,” said the poorly written Tinder proﬁle. sedentary (adjective): inactive; lazy. “suh DENT a ry” Think: sofa dent. Sedentary people make sofa dents because they sit on the cushions for hours at a time. sedulous (adjective): careful; hardworking; diligent. “SED u lus” Think: schedule us. Our sedulous hairstylist is always able to schedule us since she's so efﬁcient. segue (noun): a transition. “seg WAY” Think: Segway. A good way to make sure your friends go along with your conversational segue is to barge in riding a Segway. self-styled (adjective): self-proclaimed. “Self-sty-ulled” Think: selﬁe style. The Kardashians are self-styled experts on fashion as evidenced by how many selﬁes they take of their style. semblance (noun): an outward appearance; an image. “sem BLENSE” Think: resemblance. The lie fooled me because it had the semblance of honesty, a slight resemblance to the truth. seminal (adjective): important; original. “SEM in ul” Think: seminar. If a book is seminal, you're probably gonna have to read it in your freshman year literature seminar. sententious (adjective): using quotable or preachy sayings. “sen TEN shus” Think: sentences. The Reverend Jesse Jackson is sententious because many people quote his sentences. sentient (adjective): having sense perception; conscious. “SENTY ent” Think: sensed it. I knew the alien life form was sentient after I pricked it with a pin and it moved: it sensed it. sequacious (adjective): something that imitates another's idea. “suh QUAY shus” Think: sequel. Your movie is so sequacious of mine that it feels like a sequel. sere (adjective): dried; withered. “SEER” Think: sear. If you sear those vegetables on the grill too long they'll become sere. serendipity (noun): luck. “ser en DIP ih tee” Think: Sara ended pity. After winning the lottery, Sara ended her pity toward herself because of her amazing serendipity. servile (adjective): submissive. “sir VILE” Think: servant. The servant was so servile that he wouldn't make eye contact. sham (noun): a trick that deceives. “SHAM” Think: shame. Your story about being a doctor is a sham...shame on you! shard (noun): a broken piece of something fragile. “SHARD” Think: sharp. Be careful of the shard of glass on the ﬂoor; it's really sharp. shelve (verb): to put aside or postpone. “SHELVE” Think: shelf. I shleved my plan to sabotage my rival and put my notes for it back on the shelf once I learned I got the promotion. shirk (verb): to avoid a duty. “SHIRK” Think: shark. If the beach lifeguard shirks his duties, then you might want to keep a look out for sharks. showy (adjective): designed to attract attention. “SHOWY” Think: show. Donald Trump’s showy, gold-plated toilet was clearly designed to show off his wealth. shrewd (adjective): clever. “SHROOD” Think: sued. The shrewd attorney sued as many people as she could; she knew her superior knowledge of the law would make her win. simper (verb): to smile in a silly way. “SIMPER” Think: smile chimp. Have you ever seen a smile on a chimp? They simper in a way that cracks me up. simulacrum (noun): an image or representation of something. “sim u LAY crum” Think: simulation. Coachella audiences saw a simulacrum of Tupac: a hologram that was an incredible simulation of him. sinuous (adjective): having many curves. “SIN u is” Think: sine wave. Unsurprisingly, if you graph a sine wave on your calculator it's going to look sinuous. skittish (adjective): restless; easily frightened. “SKIT ish” Think: Skittles. After I ate a 54 oz. bag of Skittles by myself, the sugar high made me skittish. skulduggery (noun): tricky or sneaky behavior. “skul DUG er y” Think: skull he dug. The skull he dug up from the local cemetery proved he was a witch doctor who practiced skulduggery. skulk (verb): to hide or be stealthy. “SKULK” Think: skunks lurk. Skunks lurk and skulk until it’s dark enough for them to eat from your garbage cans. slake (verb): to quench or satisfy. “SLAKE” Think: lake. If you're a zebra, you probably can't operate a water fountain: slake your thirst at the lake. slander (noun): a false statement intended to hurt someone’s reputation. “SLAN der” Think: slammed her. To try to steal voters from Hillary Clinton, Donald Trump repeatedly slammed her in interviews, ﬁguring that enough slander against her might make voters forget how deplorable he was. slatternly (adjective): untidy or promiscuous. “SLA tern ly” Think: slutty. If you want to say she's slutty but use a bit more ﬂattery, call her slatternly. slipshod (adjective): careless; sloppy. “SLIP shod” Think: slip shoddy. I slip when I walk on your shoddy living room ﬂoor because its construction is really slipshod. slothful (adjective): lazy. “SLOTH ful” Think: sloth. My pet sloth is too slothful to move even when he’s really hungry. slovenly (adjective): untidy; sloppy. “SLOV en lee” Think: sloppy. Charlie Brown’s friends make fun of Pig-Pen because of his sloppy, slovenly appearance. sojourn (noun): a temporary stay. “SO jern” Think: journey. If you journey somewhere, it’s probably for a sojourn unless you bought a one-way ticket. solecism (noun): a blunder. “SO luh sism” Think: sole is in. If you put your foot in your mouth - like if you ask a woman her age - it's a solecism - your sole is in your mouth. solicitous (adjective): concerned for. “so LISS it us” Think: solely listened to us. I knew the man was solicitous because he solely listened to us. solidarity (adjective): unity, agreement, mutual support. “Solid AIR ity” Think: solid dare. Normally, I’m not one to accept a dare, but due to our solid friendship and solidarity, I will accept your challenge to run a marathon for charity. solipsistic (adjective): being extremely self-centered. “sah lip SIS tick” Think: sold lipstick. The model whose image sold lipstick became solipsistic due to all the compliments she received. somnolent (adjective): sleepy. “SAWM nuh lint” Think: insomnia. If you have insomnia you're probably somnolent from lack of sleep. sonorous (adjective): having a deep, rich sound. “SAWN er us” Think: Tyranno-sonorous rex. Tyrannosaurus rex had a sonorous roar that could be heard for miles. sophistry (noun): deceptive reasoning. “SO ﬁs tree” Think: sophisticated trickery. Sophocles’ sophistry was so sophisticated that his trickery made his character Oedipus kill his dad and marry his mom. sophomoric (adjective): immature. “sof uh MOR ick” Think: sophomore-onic. Sophomores act moronic since they’re immature and are more sophomoric than seniors. soporiﬁc (adjective): causing sleep. “sop or IF ick” Think: sleepover-iﬁc. That boring movie is perfect for our slumber party - it's sleepover-iﬁc because it's soporiﬁc. sordid (adjective): ﬁlthy; foul; morally degraded. “SORE did” Think: sorry I did. If you are a normal person with a conscience and you do something sordid, you’ll be thinking, “sorry I did that” before long. soupcon (noun): a little bit. “SOUP sawn” Think: soup can. After surviving the apocalypse, we only had a soupcon of food left: in fact, we only had one Campbell's soup can. sovereign (adjective): independent. “SAW vern” Think: reign. If you’re sovereign, you reign over your world and no one else does. sparing (adjective): not using or giving a lot of something. “Spare ing” Think: spare ring. I’m a sparing guy, so when I proposed to my girlfriend, I asked her if she had a spare ring I could use as the engagement ring. sparse (adjective): simple, unadorned, austere. “SPARSE” Think: sparks. When I saw how simple her apartment was, sparks ﬂew: I have always been attracted to minimalism. specious (adjective): seeming true but actually false. “SPEE shus” Think: suspicious McLovin. It’s understandable the cashier in Superbad is suspicious when she sees Fogell’s specious license that identiﬁes him as “McLovin”. spendthrift (noun): someone who spends wastefully. “SPEND thrift” Think: spend before thrift. Spendthrift means someone for whom spending comes before being thrifty. splenetic (adjective): bad-tempered. “spluh NET ick” Think: spleen anger. In medieval times, people thought anger came from one's spleen; "splenetic" was coined to describe an angry person. spurious (adjective): false. “SPUR y us” Think: spur curious. His spur-of-the-moment explanation made me curious whether his story was spurious. squalid (adjective): ﬁlthy. “SQUA lid” Think: squat lid. If the bathroom stall is squalid, squat over the lid when you pee. squelch (verb): to crush or silence. “SKWELSH” Think: squash and squish. Squelch the ant uprising! Squash them! Squish them! stalwart (adjective): loyal; strong supporter. “STAL wert” Think: tall war hero. George Washington was a tall war hero and all-around stalwart: he was a strong, loyal supporter of the American Revolution. stanch (verb): to stop the ﬂowing of. “STANCH” Think: ‘stache. I thought my ‘stache was sexy, but in fact it stanched the ﬂow of all females to my bedroom. staple (noun): something commonly used; an essential. “STA pull” Think: maple. At IHOP, maple syrup is a staple since they serve about a billion pancakes a year. statuesque (adjective): attractively tall. “stah choo ESK” Think: statue Esquire. I wanted to make a statue of the Esquire model because she was so statuesque. staunch (adjective): ﬁrm; true; strong. “STONCH” Think: stay unchanged. I’m a staunch supporter of Justin Bieber, so my support for him will stay unchanged even if he does something really stupid. steadfast (adjective): loyal; immovable. “STEAD fast” Think: stayed fastened. The fallen soldier's dog was so steadfast that it stayed fastened to the ground near his grave. stigmas (noun): marks of shame. “STIG mas” Think: Stick-mas instead of Christmas. One of our stigmas growing up was that we celebrated “Stick-mas” instead of Christmas – we were too poor for any presents but sticks. stilted (adjective): overly formal; stiff. “STILL tid” Think: stilts. The soldier's manner of walking was so stilted that it looked like his legs were actually wooden stilts. stolid (adjective): unemotional. “STOW lid” Think: solid. The stolid butler was solid and expressionless; he never broke down and cried. storied (adjective): having an interesting/celebrated history. “STOH reed” Think: stories. The most interesting man in the world's storied history makes people tell his stories. stratagem (noun): a clever scheme. “STRAT a gem” Think: strategy gem. The general's battleﬁeld strategy was such a gem that most historians call it a stratagem. streamlined (adjective): simpliﬁed; modernized. “STREAM lined” Think: stream line. The streamlined shape of a trout lets it swim through even a rushing stream in a straight line. strenuous: (adjective): requiring lots of energy. “STREN u us” Think: strain on us. The strenuous hike up Mt. Whitney was a strain on us. stricture (noun): a restraint; a criticism. “STRICT sure” Think: restrict. The tourniquet around my arm stopped me from bleeding to death, but the stricture restricted any circulation and they almost had to amputate my limb. strident (adjective): harsh; loud. “STRI dent” Think: Stridex. Stridex commercials are as strident as the salicylic acid in the pads, in an effort to hold teens’ interest. stringent (adjective): strict. “STRIN jent” Think: strict gent. Our architecture professor is a strict gent: he’s so stringent that if your drawing has any eraser marks, he’ll dock you a full letter grade. stultify (verb): to make ineffective. “STULT ih fy” Think: stupid dolt. If you stultify yourself by punching yourself in the skull, you'll become a stupid dolt. stupeﬁed (adjective): stunned. “STOOP uh fyed” Think: stupid. Hermione cast the stupefy spell on Crabbe, who became so stupeﬁed that he looked stupid. subjective (adjective): personal; unaffected by the outside world. “sub JEKT ive” Think: king’s subject. His Majesty considers me to be his subject and his subjective opinion is that I’m a peasant even though I’m of noble birth. sublime (adjective): awesome. “sub LIME” Think: the band “Sublime”. The band Sublime has spawned several cover bands, a good sign that it made sublime music. subsequent (adjective): following, next. “SUB suh kwent” Think: sub-sequence = next in the sequence. A subsequence is the next occurrence in a sequence: that which follows. subsidy (noun): government aid to keep a price low. “SUB sih dee” Think: sub city. In order to prevent itself from becoming a sub-city in the wake of its bankruptcy, the city of Detroit had to rely on subsidies from the federal government. substantiate (verb): to support with proof or evidence. “sub STANCH ee ate” Think: substance. You won’t be able to substantiate your claim that I ate your lunch without evidence that has more substance. subversive (adjective): seeking to undermine or disturb. “sub VERSE ive” Think: subversive verses. The political poet was detained by government ofﬁcials for her "subversive verses." subvert (verb): to weaken or ruin. “sub VERT” Think: sub hurt. Captain: the torpedo from that sub hurt our ship and subverted our morale. succor (noun): aid. “SUCK er” Think: supper. If you’re starving and stranded in a snowstorm, hopefully your succor will include some sort of supper. succumb (verb): to give in to a superior force. “suh KUM” Think: suck under. Do not succumb to the deadly pull of the quicksand or it will suck you under. sullen (adjective): sad, gloomy, resentful. “SULL en” Think: mullet. The barber gave me a mullet – that’s why I’m so sullen. sumptuary (adjective): made to prevent overindulgence. “SUMP tyoo air y” Think: consumption-ary. Consumption of harmful things, like cigarettes or alcohol, can be limited with a sumptuary tax. superﬁcial (adjective): on the surface. “soup er FISH ul” Think: super ofﬁcial website. The shiny new website for the family restaurant looked super ofﬁcial, but the truth was that it was starving for customers: so far, the success of the business was only superﬁcial. supplant (verb): replace. “suh PLANT” Think: up plant. After you pull up a plant out of the soil, you should supplant it with another one to help preserve the environment. surly (adjective): in a foul mood, ill-tempered. “SUR ly” Think: swirly. The school bully gave me a swirly (he stuck my head in a toilet and ﬂushed it) – that’s why I’m surly. surmise (verb): to guess. “sur MISE” Think: summarize. Since a police report will only summarize what happened, one usually has to surmise the actual events of a crime. surpassing (adjective): really, really great. “sur PASS ing” Think: super pass. If you’re super at running, you’ll pass everyone due to your surpassing speed. surreptitious (adjective): sneaky or stealthy. “sur ep TISH us” Think: reptiles. Reptiles like snakes are good at camouﬂage because they’re surreptitious. susceptible (adjective): easily affected or inﬂuenced by something. “suh SEPT ih bull” Think: suggest-able. I’m susceptible to pranks because I’m so suggest-able – I’ll follow any suggestion. swathe (noun, verb) a cover or wrap / to cover or wrap. “SWAYTHE” Think: swat the (mosquitos). When camping near standing water, I would rather swathe myself in mosquito repellent than swat the pesky pests away all day. sybarite (noun): one devoted to pleasure. “SIB a right” Think: sit at a bar. If you go sit at a bar every night to watch sports and drink beer, you might be a sybarite. sycophant (noun): one who ﬂatters for self gain. “SICK a fent” Think: sick of elephant. The animals were sick of the elephant because he was a sycophant who kissed up to the zookeeper. synergy (noun): combined action that produces mutually helpful results. “SIN ur gee” Think: ‘N SYNC energy. By forming a boy band and using synergy, ‘N SYNC created more energy than they would had they all gone solo. synoptic (adjective): giving a summary. “sin OP tick” Think: synopsis. The synoptic nature of SparkNotes provides a synopsis of a novel's plot at the expense of the novel's beauty. syntax (noun): linguistics, use of language. “SIN tax” Think: sin tax. The conservative legislature once tried to impose a “sin tax” on all gay marriages, but was forced to change its syntax due to a lawsuit from the ACLU. taciturn (adjective): not talkative. “TASS it turn” Think: takes his turn. If she's passive and taciturn at the debate and just politely takes her turn when speaking, she'll never win. tangible (adjective): able to be touched. “TAN jib ul” Think: tango-ble. If you can dance the tango with someone – if she’s tango-ble – then she’s perforce tangible. taxonomic (adjective): related to the process of categorization. “Taks oh NOM ik” Think: taxes are not my thing. Because I am a creative type, doing my taxes each year, along with all the requisite classiﬁcations and categorizations of various personal and business expenses, is a taxonomic activity that is clearly not my thing. Time to hire an accountant. temerity (noun): recklessness. “tuh MERR (rhymes with “err”) uh tee” Think: team error. If you have temerity, maybe you should join team error because I bet you make a lot of mistakes. temperance (noun): moderation. “TEM per ence” Think: temper ants. At the picnic, I didn't lose my temper over the ants, because I possess the quality of temperance. tempestuous (adjective): stormy. “tem PEST you us” Think: tempers. Our hot tempers make us have a tempestuous relationship. temporal (adjective): relating to time. “TEM puh rull” Think: temporary. Technically, diamonds aren't forever; in a temporal sense, they're only temporary and will turn to dust one day. tenable (adjective): able to be defended; workable. “TEN uh bull” Think: ten able. The scientist's theory was tenable because it was “ten-able”, worthy of being rated a 10 out of 10. tendentious (adjective): biased. “ten DEN shus” Think: tendency. Don’t let him judge the beauty contest: he’s tendentious and has a tendency to vote for the contestants that ﬂirt with him the most. tensile (adjective): related to tension. “TENSE I’ll” Think: dense tile. The tensile strength of that dense old tile on the kitchen counter is quite impressive, which is why I’ve been having such a hard time removing it during renovations. tenuous (adjective): lacking substance or strength. “TEN you us” Think: tentative. At the debate, the tentative speaker's argument was unsurprisingly judged to be tenuous. tepid (adjective): lukewarm, apathetic. “TEH pid” Think: “tap it” = tap the tepid keg. “Should I tap this keg now?” asked the overzealous fraternity brother. “Um, sure, I guess you could tap it” was my tepid response: it was full of tepid beer. terse (adjective): brief and abrupt. “TURSE” Think: terse verse. Haikus are verses / That are as terse as the lives / Of gentle fruit ﬂies. timorous (adjective): fearful. “TIM uh riss” Think: timid of us. Tim felt timid around us since he was timorous. tirade (noun): a long angry speech. “TIE raid” Think: tired of rage. If someone gives you a tirade, you’ll probably be tired of the rage after a few minutes. titular (adjective): relating to a title. “TITCH u lur” Think: title. The titular character in Harry Potter is Harry Potter because his name is also the title of the book. tonic (noun): something helpful. “TAWN ick” Think: gin and tonic. Drinking a gin and tonic before my speech was a tonic for my anxiety. toothsome (adjective): tasty; appealing. “TOOTH some” Think: tooth some. The food looked so toothsome that I wanted to give my tooth some. torpid (adjective): sluggish. “TOR pid” Think: tar pit. Once I walked into the sticky tar pit, my pace became torpid. tortuous (adjective): winding. “TOR tyoo us” Think: tortoise. The streets of Boston are so tortuous that you have to drive at tortoise’s speed. totalitarian (adjective): relating to a government with total power. “TOE tal (rhymes with “gal”) ih TEAR y en” Think: total power. Our totalitarian dictator uses his total power to make us eat Total cereal daily - he's a control freak. touted (verb): praised publicly. “TOUT ed” Think: shouted. Guinness Stout is highly touted; I know this because the guy drinking it next to me shouted its praises in my ear. tranquil (adjective): calm. “TRAN quil” Think: NyQuil. Taking NyQuil before bed made me so tranquil that I slept for 12 hours. transgression (noun): a violation of a rule. “TRANS gression” Think: trans aggression. Some states have passed laws that make using a bathroom different than your biological gender a transgression due to fear of trans aggression. transitory (adjective): existing only brieﬂy. “TRANS ih tory” Think: transit story. I found romance on the subway, but alas, our love was transitory: it was a public transit story that only lasted until her stop. treacly (adjective): overly sweet or sentimental. “TREAK (rhymes with leak) lee” Think: trickle-y tears. The scene with a homeless puppy is so treacly it seems designed to make tears trickle down one's face. tremulous (adjective): fearful. “TREM you luss” Think: tremble. I felt so tremulous when I saw a shark swim underneath me that I began to tremble. trepidation (noun): fear. “treh pid AY shun” Think: trap. The haunted house ﬁlled me with trepidation; I feared a trap would be sprung on me at any moment. triﬂing (adjective): unimportant, inconsequential. “TRY ﬂing” Think: riﬂing through drawers. If shadowy henchmen are riﬂing through your drawers for some reason, then it’s probably more than a triﬂing matter. truculent (adjective): ready to ﬁght. “TRUCK you lent” Think: truce you lent. The armies should write their own peace treaty, because they're still truculent after that truce you lent them. truncated (adjective): shortened. “TRUN kated” Think: trunk ate. The elephant's trunk ate so many branches that the tree was truncated. tumid (adjective): swollen. “TYOO mid” Think: tumor-ed. The cancer patient's large tumor caused his abdomen to be tumid. tumultuous (adjective): disorderly; like a riot. “tuh MULT you us” Think: tumbled us. The mosh pit was so tumultuous that it tumbled us around like a dryer. turbid (adjective): stirred up and made unclear or muddy. “TURR (rhymes with “purr”) bid” Think: tar bed. The lake became turbid when storms disturbed particles from the tar bed underneath its waters. turgid (adjective): swollen. “TURR (rhymes with “purr”) jid” Think: turkey in. After Thanksgiving dinner, my belly was so turgid that it looked like I had eaten the whole turkey. turpitude (noun): vile or immoral behavior. “TURP ih tude” Think: turd attitude. His turd attitude made him engage in turpitude. ubiquitous (adjective): existing everywhere. “ooh BICK quit us” Think: you big Quidditch. You big Quidditch fans have made the Harry Potter sport ubiquitous on college campuses. umbrage (noun): offense; annoyance. “UM bridge” Think: umbrella rage. Someone who takes umbrage at his umbrella probably felt rage when it broke during a storm. unadorned (adjective): plain. “un uh DORNED” Think: unadored. Since your girlfriend’s hand has no ring and is unadorned, I assume she’s unadored: if you like it then you should put a ring on it. unassuming (adjective): modest. “un uh SOOM ing” Think: un-assume. The millionaire's unassuming car deﬁnitely didn't make us assume he was wealthy. unbridled (adjective): not restrained. “un BRIDE ulled (rhymes with “culled”)” Think: un-bridle. After I took off my horse's bridle, he became so unbridled that I had no control over him. unconscionable (adjective): unreasonable; not guided by conscience. “un CON shin uh bull” Think: un-conscience. It would be unconscionable to leave your two-year-old alone at home -you’d have to have no conscience - an “un-conscience”. unconventional (adjective): not typical. “Un con VENT shin ull” Think: Republican Convention. The most unconventional thing about the Republican Convention was its candidate, Donald Trump. unctuous (adjective): smooth in a fake way. “UNK shis” Think: skunk-tous. Pepe Le Pew, the smooth-talking, playboy skunk, acts unctuous to charm the ladies. uncultivated (adjective): not grown or used, (of a person) not reﬁned. “Un KULT iv ate id” Think: cult avoided. Because of his simple, uncultivated nature, he was, somewhat ironically, able to avoid the siren call of the radical cult that was full of “intelligent” people from his town. undermine (verb): to weaken in a sneaky way. “UN der mine” Think: under mine. Under the ground lay a land mine designed to undermine the army's advance. underscore (verb): to highlight. “UN der score” Think: to score = to write. To score a composition is to write a composition; to underscore something on a piece of paper, you write under it (underline). understated (adjective): downplayed; made to seem less than it actually is. “UN der stated” Think: understatement. “I have no complaints” is an understated way to respond if you’re wealthy and asked how much money you make; it’s an understatement. undulate (verb): to move in a smooth, wavelike way. “UN dyoo late” Think: undo lace. You’ll deﬁnitely turn your lover on if you undo your lace lingerie while slowly undulating your body. uniform (adjective): always the same. “YOON if orm” Think: Army uniform. Throughout the U.S., the uniform (noun) that Army soldiers wear is uniform (adjective). unkempt (adjective): untidy. “un KEMT” Think: un-kept hair. If you had just kept up with your personal hygiene, your hair wouldn’t be so unkempt and birds wouldn’t have nested in it. unpretentiousness (noun): the state of being unassuming, modest, or natural. “Un pree TEN shish niss” Think: un-pretend us Ness. Ness did not try to pretend to be more glamorous or important than she truly was around us; hence, we found her personality refreshing and unpretentious. unpropitious (adjective): unfavorable, not a good sign or omen. “Un pruh PISH us”. Think: un propped. Grandpa’s crutches prop him up; it’s unpropitious that they’re lost, since he’s now un-propped and might fall. unruly (adjective): difﬁcult to discipline or manage. “un RULE y” Think: un-rule-able. My two-year-old is unruly; he is un-rule-able and says “No!” to me every time I tell him to do something. unsavory (adjective): unpleasant, esp. morally unpleasant. “un SAVE or y” Think: un-savor. The icky memory of the unsavory used car salesman was not one I wanted to save or savor. untenable (adjective): not workable, indefensible, weak, shaky. “un TEN uh bull” Think: un tent-able. If you are stuck in the woods during a rainstorm, and your tent is un tent-able, then you’ve got an untenable situation on your hands. untoward (adjective): improper; troublesome. “un TOE ward” Think: undertow. The beach's dangerous undertow was untoward, dragging the girl underwater and loosening her bikini. unwieldy (adjective): awkward; cumbersome. “un WEILD (rhymes with “ﬁeld”) y” Think: unable to wield. The ogre dropped his giant club and I picked it up, but it was too unwieldy to wield against him in battle. upbraided (verb): criticized severely. “up BRAID ed” Think: upside braid. The hippie upbraided me so much that I was afraid she was going to slap me upside the head with her giant braid. urbane (adjective): sophisticated; polite and polished. “ur BAIN” Think: urban. The farmboy moved to a hip urban city and became so urbane that he threw away his straw hat. usurious (adjective): a rip-off. “you SIR y us” Think: u serious? I know that I have bad credit, but the usurious rate on my credit card made me say “U serious?” usurp (verb): to illegally take by force. “ooh SURP” Think: u slurp. I know you're an anteater, but if you usurp my ant farm and u slurp up my ants, I'll be really angry. utilitarian (adjective): useful. “you till-it TEAR y-en” Think: utilize. The military likes to buy utilitarian tools that can be utilized for many different tasks. utopian (adjective): of a perfect society, ideal. “you TOPE y-en” Think: You Tokin’. In an utopian world, you could be tokin’ all the time, but that doesn’t really work out in real life unless your name is Snoop Dog. vacuous (adjective): stupid. “VACK you us” Think: vacuum. The beauty pageant contestant's answer was so vacuous that the judges thought her brain had been vacuumed out of her head. vainglorious (adjective): boastful. “vane GLOR y us” Think: vain. The evil queen in Snow White is vainglorious - because she's vain and thinks she's glorious. vanquished (adjective): defeated. “Van KWISHT” Think: van squished. If a van squished the ant crossing the road, then you could say that the ant has been vanquished. vapid (adjective): dull; air-headed. “VAH pid” Think: vapor. All vapor and no substance, MTV is so vapid that it makes me want to take a nap. variegated (adjective): varied. “VAH ree GAIT (rhymes with “wait”) ed” Think: varied. The autumn leaves in Vermont are known for their variegated colors; last year, they varied from red to yellow to orange. vaunted (adjective): widely praised. “VON ted” Think: vaulted well. The gymnast vaulted so well that she was vaunted by the judges. vehement (adjective): strongly emotional. “VE huh ment” Think: he meant it. His warning was so vehement that we knew he meant it. venal (adjective): corrupt or corruptible. “VEE nil” Think: venereal disease. Nuns with venereal disease are, most likely, venal: they broke their oaths of chastity. veracious (adjective): full of truth (veracity). “vur A shus” Think: verify If you can verify something, then it is veracious (truthful). (Not to be confused with “voracious”) verbose (adjective): wordy. “vur BOES (rhymes with “toes”)” Think: verb boss. They call me a verb boss since I am verbose and know a zillion different words. verboten (adjective): forbidden. “fur BO tin” Think: verb eatin’. In North Korea, verb eatin' - instead of speaking one's mind - is common since many topics are verboten. verisimilar (adjective): seeming to be true. “VEH ree sim il er” Think: very similar. The conman's verisimilar story almost tricked me since it was very similar to the truth. vernacular (noun): the way a certain group uses language. “ver NACK u ler” Think: verb knack. Once you develop a knack for the way we use verbs, you’ll have become familiar with our vernacular. vertiginous (adjective): dizzy or producing dizziness. “ver TIJ en is” Think: vertigo. Standing on the edge of the skyscraper made me feel really vertiginous because I have vertigo. vestige (noun): last remains. “VEST idge (rhymes with “fridge”)” Think: vest. The explosion blew off most of my three-piece suit – the only vestige left was the vest. vex (verb): to annoy. “VEKS” Think: hex. In Harry Potter, casting a hex, or spell designed to cause pain, on someone will deﬁnitely vex him. vicarious (adjective): felt by imagining the experience of another. “vie CARE y us” Think: bi-curious. The bi-curious woman preferred to keep her fantasy vicarious, so she just watched. vigilant (adjective): watchful; alert. “VIJ i lent” Think: vigilante. If you want some street justice, hire a vigilante - they are vigilant by nature. vilify (verb): to speak ill of. “VILL if-i” Think: villain-fy. The dumpee decided to vilify her ex-boyfriend so the other girls would think he was a villain. vindicate (verb): to prove correct; to free from blame. “VIN di kate” Think: Vin indicated. Judge Vincent indicated that the DNA evidence had fully vindicated the falsely accused defendant. vindictive (adjective): wanting revenge. “ vin DICT ive” Think: Vin Diesel. Vin Diesel often plays vindictive characters since he has been typecast as a tough guy. virtuoso (noun): someone highly skilled at something. “vurr tyoo OH so” Think: virtues (oh so many). Virtues? I have oh so many, because I’m a gosh-darned virtuoso. virulent (adjective): infectious; harmful; hostile. “VIE roo lent” Think: virus. The swine ﬂu virus is so virulent that it can kill a previously healthy person. viscous (adjective): syrupy. “VISS kiss” Think: sticks to us. The viscous Bisquick pancake batter sticks to us. vitiate (verb): to impair or degrade. “VIH she ate” Think: wish you ate. If you eat Taco Bell, it will vitiate your stomach and make you wish you ate something else. vitriolic (adjective): full of hatred. “vit ri OL ic” Think: vitriolic alcoholic. Some people are just plain mean when they drink; there is nothing worse than a vitriolic alcoholic. vituperated (verb): criticized harshly. “vie TOOP ur ated” Think: viper. He was vituperated so badly that he felt like he had been bitten by a viper. vivacious (adjective): lively. “viv A shus” Think: Viva la Vida. The Coldplay song “Viva la Vida” means “long live life” and makes me want to be vivacious. vocation (noun): job. “vo KAY shun” Think: afford a vacation. If you want to afford a vacation get a vocation. vociferous (adjective): loud. “vo SIF er us” Think: voice for us. The announcer's loud voice, for us, was too vociferous. volatile (adjective): unstable, dangerous. “VOL uh tull” Think: volcano isle. This may look like a peaceful island, but it’s volatile – it’s a volcano isle that could still erupt. volition (noun): a conscious choice. “vo LISH un” Think: volunteer. No one forced him to volunteer for the mission; he did it of his own volition. voluminous (adjective): large or numerous. “vo LUM in us” Think: 18 volume diary. I gave up on reading her diary after realizing how voluminous it was - it had 18 volumes! voracious (adjective): having a huge appetite. “vo RAY shus” Think: carnivore ate us. The carnivore ate us because of its voracious appetite. voyeur (noun): pervert, watcher, “Peeping Tom”. “vo YER” Think: Foyer. Last night I caught a voyeur hanging out in the foyer (entrance) of my apartment building who was trying to spy into people’s windows. Joke’s on him: I was eating ice cream and wearing sweatpants. wafﬂe (verb): to go back and forth. “WAF ul” Think: should I get wafﬂes? When I go out to brunch, I wafﬂe (verb) between getting wafﬂes (noun) and getting eggs. wan (adjective): sick-looking. “WON” Think: old Obi-Wan. In Star Wars, Obi-Wan Kenobi looked wan even though he was a Jedi master because he was old. wane (verb): to decrease in size, amount, length, or quantity. (rhymes with “pain”) Think: Lil Wayne. The rapper Lil Wayne is only 5’5”; some might think his height has waned, but he has been that little since high school. wanting (adjective): lacking or absent. “ WONTING” Think: wanting a boyfriend. Wanting (verb) a boyfriend is normal if the romance in your life is wanting (adjective). waspish (adjective): irritable. “WOSP ish” Think: wasp-ish. The trouble with keeping them as pets is that wasps are almost always waspish – they’ll sting you if you look at them the wrong way. watershed (noun): a turning point. “WAH ter shed” Think: Watergate. Nixon's involvement in the Watergate scandal was a watershed for his public opinion and led to his resignation. wax (verb): to increase; to grow. “WAKS” Think: ear wax. Thanks to your body’s glands, your sticky, orange-brown ear wax will wax daily even if you use Q-tips. welter (verb): to be in turmoil; to get tossed around. “WEL turr” Think: welts. When I surf, I welter in the waves and my board hits me; I come out covered in welts. whet (verb): to sharpen; to make more intense. “WET” Think: wet mouth. If you’re starving and I show you a picture of a cheeseburger, it will whet your appetite and your mouth will water and get wet. whimsical (adjective): playful; random; fanciful. “WIM sih kull” Think: whim. The princess’s whimsical ideas included her sudden whim to travel to Antarctica. willful (adjective): stubborn, insistent. “WILL full” Think: will-full. The willful horse was so will-full that he refused to be trained or ridden. wily (adjective): clever; sly. “WHILE (rhymes with “trial”) y” Think: Wile E. Coyote. Wile E. Coyote was not quite wily enough to catch the Roadrunner despite his clever traps. winnow (verb): to separate the useful from the not-useful. “WIN no” Think: minnows. Winnow the minnows from your catch of ﬁsh; they're too small to eat. winsome (adjective): charming and pleasing. “WIN sum” Think: win some hearts. She'll probably win some hearts at the dance due to her winsome manner. wistful (adjective): sadly wishing for. “WIST ful” Think: wishful. The “Forever Alone” meme guy feels wistful because he is still alone after weeks of being wishful for a girlfriend. wizened (adjective): shrunken and wrinkled, usually due to age. “WI zend” Think: wizard. Wizards like Gandalf and Dumbledore are usually wizened since they’re really old. wont (adjective): accustomed. “WONT” Think: want. It makes sense that you want to do things you are wont to doing, as opposed to trying risky new activities. woo (verb): to seek or pursue romantically. “WOO” Think: woo-hoo! If you spend all night watching sports and exclaiming “woo-hoo!” then your chances of wooing your date drop dramatically. workmanlike (adjective): good but not great. “WORK mun like” Think: workman’s design. A workman will produce a workmanlike house design, but hire an architect if you want originality. worldly (adjective): not spiritual; sophisticated; experienced. “WORLD lee” Think: world traveler. I’ve been all around the world and I, I, I, I can’t ﬁnd my baby (but I’m worldly now). wrongheaded (adjective): having ideas that are wrong. “RONG head ed” Think: wrong foot. I think that we may have gotten off on the wrong foot when I made that wrongheaded remark about your fashion choices. wry (adjective): cleverly and/or ironically humorous. “RYE” Think: PB&J on rye. Surprising me by serving a PB&J sandwich on rye bread is a good example of my mother's wry humor. zealous (adjective): passionate. “ZELL us” Think: jealous. Zoe was so zealous about her ﬁrst boyfriend that she became jealous of every other girl he knew. zenith (noun): the highest point. “ZEE nith” Think: beneath. Once you reach the zenith, everything else is beneath it. zephyr (noun): a gentle breeze. “ZEF ur” Think: zebra fur. The summer evening zephyr was as soft as zebra fur. Appendix: Word Roots Authors’ Note: Word Roots are a helpful way to deduce the meaning of words that you don’t know. They are not, however, foolproof: the English language is unpredictable, and full of words of which the standard meaning of the root is either ﬂipped on its head (for example, the word “invaluable” means “valuable”) or simply inaccurate. In other cases, the word root could have multiple languages of origin and thus the meaning of the word is unclear. This is why mnemonic devices are generally superior to word roots with regard to recalling the precise deﬁnition of a word. That being said, having incomplete information on a word is better than having none at all! We suggest that when creating your own mnemonics, you make sure to take special note whenever the word root goes against its most common interpretation. a: without. Think: amoral - without morality. ab: away or apart from. Think: abnormal - away from being normal. ac: sharp; biting. Think: acid - something that can chemically burn. ad: toward or near. Think: adjacent - next to. ag: to do. Think: agent - something that acts. al: other. Think: alien - something foreign. am: love. Think: amour - a love affair. amb: to walk. Think: amble - to walk slowly. ambi: both. Think: ambidextrous - able to use both hands. anim: life. Think: animate - to give life to. ante: before. Think: antechamber - the entryway before the main room. anthro: human. Think: anthropology - the study of man. anti: against. Think: antifreeze - chemical used against freezing. apt: skill. Think: aptitude - ability. arch: the biggest. Think: archenemy - the biggest enemy. auto: self. Think: autonomy - independence of the self. be: to have. Think: befriend - to become friends with. bell: war. Think: belligerent - warlike. ben: good. Think: beneﬁt - an advantage. bi: two. Think: bisexual - having both male and female sexuality. bon: good. Think: bonus. brev: short. Think: abbreviate - to shorten. cand: to burn. Think: candle. cap: head. Think: captain - a leader. card: heart. Think: cardiac - of the heart. carn: ﬂesh. Think: carnivore - a meat-eating animal. chron: time. Think: chronology - sequence of events. circu: around. Think: circumference - the distance around a circle. cis: to cut. Think: scissors. clu: close. Think: conclusion. claim: to declare or shout. Think: exclaim - to shout out. cli: to lean. Think: recline - to lean back. col: together. Think: collaborate - to work together. con: together. Think: connect. cre: to grow. Think: increase. cred: to believe. Think: credibility - believability. cryp: hide. Think: cryptic - having an unclear or hidden meaning. culp: blame. Think: culprit - one who is guilty. de: reversal. Think: defame - to take away the fame of (through slander). dem: people. Think: democracy - rule by the people. dict: to say. Think: diction - choice of words. dign: worth. Think: dignity - worthiness. dis: reversal. Think: disarm - to take away an armament (like a gun). dac: to teach. Think: didactic - intended to teach. dog: belief. Think: dogma - established beliefs. dox: opinion. Think: orthodox - adhering to established opinion. dol: suffer. Think: condolences - sympathy for another’s suffering. don: to give. Think: donate. dub: doubt. Think: dubious - doubtful. duct: to lead. Think: orchestra conductor. dur: hard. Think: durable. dys: faulty or broken. Think: dysfunctional. enni: year. Think: centennial - a 100th anniversary. epi: on. Think: epidermis - the layer of the skin on the dermis. equ: equal. Think: equal. err: to wander. Think: error. eu: good. Think: eulogy - a praising speech. ex: out Think: exclude. extra: outside of. Think: extraterrestrial - outside of Earth. fac: to make. Think: factory. fer: to bring. Think: transfer. ferv: to burn. Think: fervor - passion. ﬁd: faith. Think: ﬁdelity - faithfulness. ﬁn: end. Think: ﬁnal. ﬂam: to burn. Think: ﬂame. ﬂex: to bend. Think: ﬂexible. ﬂict: to hit. Think: conﬂict - ﬁghting. ﬂu: to ﬂow. Think: ﬂuid. fore: before. Think: foreshadow - to hint at the future. fort: chance. Think: fortune-teller. frac: to break. Think: fracture. found: bottom. Think: foundation. fus: to pour. Think: blood transfusion - transferring blood into someone. gen: type. Think: gender - sex. gn: know. Think: recognize. grand: large. Think: Grand Canyon. grat: pleasing. Think: grateful. grav: heavy. Think: gravity. greg: group. Think: congregrate - to group together. hes: to stick. Think: adhesive. hetero: different. Think: heterosexual - sexual with a sex different than one’s own. hom: same. Think: homosexual - sexual with one’s own sex. hyper: over. Think: A hyperactive little kid. hypo: under. Think: hypothermia – body temperature below normal. im: not. Think: impossible. in: not. Think: insane. inter: between. Think: interstate highway. intra: within. Think: intravenous - within a vein. ject: to throw. Think: eject. junct: to join. Think: junction. lect: to choose. Think: elect. lev: lift. Think: elevator. log: speech. Think: dialogue. lum: light. Think: illuminate. mag: big. Think: magnify. mal: bad. Think: malicious. man: hand. Think: manual labor. min: small. Think: minimum. mit: to send. Think: transmit. misc: mixed. Think: miscellaneous. morph: shape. Think: amorphous - without shape. mort: death. Think: immortal - without death. mut: change. Think: mutate. nox: harm. Think: obnoxious. nym: name. Think: synonym. nov: new. Think: novice - a beginner. omni: all. Think: omnipotent - all powerful. pac: peace. Think: paciﬁer. pan: all. Think: panoramic - taking in all the scenery. par: equal. Think: disparity - difference. para: next to. Think: parallel. path: feeling. Think: empathy. ped: child. Think: pediatrician - child doctor. ped: foot. Think: pedal. pen: to pay. Think: compensation - payment. pend: to hang. Think: pendulum. peri: around. Think: perimeter. pet: to strive. Think: compete. phil: love. Think: bibliophile - one who loves books. phone: sound. Think: megaphone. plac: to please. Think: placate - to calm down or appease. ple: to ﬁll. Think: complete. pos: to place. Think: deposit. port: to carry. Think: import. post: after. Think: posthumous - after death. pov: poor. Think: poverty. pre: before. Think: preview. prehend: to get. Think: comprehend. pro: a lot. Think: profuse - large in quantity. prob: to test. Think: probe. pug: to ﬁght. Think: pugilist - a boxer. punct: to prick. Think: puncture. quis: to search for. Think: inquisitive - seeking knowledge. qui: quiet. Think: quiet. rid: to laugh. Think: ridicule. rog: to ask. Think: interrogate - to question intensely. sacr: holy. Think: sacred. sci: to know. Think: conscious. scribe: to write. Think: scribble. se: apart. Think: separate. seq: to follow. Think: sequence. sens: to be aware. Think: sense. sol: to loosen. Think: dissolve. spec: to look. Think: spectator. sta: to be still. Think: static - still. sua: smooth. Think: suave. sub: below. Think: submarine. super: above. Think: supersonic - faster than sound. tac: silent. Think: tacit - understood without words. tain: to hold. Think: contain. tens: to stretch. Think: tension. theo: God. Think: atheist - without belief in God. tort: to twist. Think: contort - to bend severely. tract: to pull. Think: attract. trans: across. Think: transport. ut: to use. Think: utensil. ver: truth. Think: verify. vi: life. Think: viable - able to survive. vid: to see. Think: video. vok: to call. Think: invoke - to summon. vol: to wish. Think: voluntary - of one’s own wish. Index of Words abase abashed abate aberration abeyance abhors abject abnegate abomination aboriginal abort abound abrasive abridge abrogate abscission absolute absolve abstemious abstruse abysmal accede accolade accretion accumulate acerbic acidulous acme acquisitive acrimonious acumen adamant adept adequate adhere admonished adorned adroit adulation adulterate adversary advocate aegis aesthetic affable affectation afﬁliated affront aggrandized aghast alacrity algorithm alleviate allusion altruistic amalgamate ambiguity ambivalence ameliorated amenable amicable amortize ample anachronism analogue anathema anile animosity annotation annul anodyne anomaly antedate antediluvian antipode antithesis apace apartheid aplomb apocryphal apoplectic apostle apothegm apotheosis appease apportion apposite approbation apropos arbitrary arcane arch archaic arduous arid arrogate articulate artiﬁce artless ascendancy ascetic ashen askew asperity aspersion aspiration assail assiduous assuage astray astute attenuate audacious augment august auspicious austere authoritative automaton autonomous avaricious aver aversion avuncular badger baleful banal base battery bauble baying beatiﬁc beatify becalm bedlam beguile behemoth beleaguered belied belittle bellicose bemoan beneﬁcence benign bereft beseech bifurcated bilious blasé blithe bloviated bludgeon bonhomie boon boor bootless bowdlerize bravado brazen brevity bromide brusque bucolic bugbear bulwark bumptious bungle buoyant burdensome burgeoning buttress bygone byzantine cache cacophony cadge cajole calamitous callous callow calumnious camaraderie canard canny canonize capacious capitulate capricious captious cardinal caricatured castigate caterwaul causal celerity censure cerebral chagrin champion chary chicanery choleric chronological churlish cinematic circuitous circumscribed circumspect circumvents clairvoyant clandestine clangorous clemency climatic climax cloying coalesce cocksure coddle coerced coeval cognizant coherence cohesive cohort coin collusion commensurate commiserate companionable complicit composure compunction concession concoct concomitant concord concupiscence condign condones conferred conﬁscate conﬂagration conﬂate confound conglomerate conniving connoisseur conscientious conspicuous consternation contumacious conundrum conversant copious cordial cordon corroborate cosmopolitan covert cowed craven credence credulous crepuscular crestfallen cryptic cull culminate culpable cumbersome cunning cupidity curmudgeon cursory curtail cynical cynosure daunt dearth debacle debased debauchery debilitate decadent decimate declaimed decorous decrepit decried defamatory defenestrate defunct degenerate delectable deleterious delimit delineate demagogue demarcate demean demeanor demotic demur denigrate denizen denuded depiction deplete deplore depredate deride derivative descry desecrate desiccated despoiled despot desuetude deteriorate devoid devolve devout dexterity diabolical diaphanous diatribe dichotomy didactic difﬁdent digression dilatory dilettante dilute dint dire discomﬁt disconcert discreet discrepancy discrete discriminate disgruntled dismantle dismissive disparage disparate dispassionate dispatch displacing disputatious dissemble disseminated distension dither diurnal divergent divisive docile doctrinaire doggedness doggerel dogmatic dolorous dormant dour draconian droll dubious dudgeon dupe duplicitous dwindle dyspeptic ebullient eclectic effaced effete efﬁcacious efﬂorescence efﬂuvium effrontery effusive egalitarian egregious eldritch emancipate embellish embroiled embryonic eminent emollient emphatic empirical encomium encompass encroaching enervating enigmatic enmity ennui ensorcelled entitled entreat ephemeral epitome equivocal eradicate ersatz erstwhile erudite eschew esoteric espouse espy estimable estranged ethereal etiolated euphemism eurytopic evanescent evinced exacerbated exact exacting exaggerate excise excoriated exculpated execrable exigent exodus exorbitant expatiate expatriate expedient explicate exponent expunge expurgate extant extemporaneous extenuating extirpate extol extraneous extrapolate exult fabricate facetious fallible fanatic farce fastidious fatuous fawning feckless fecund feign felicity ferret fervor festoon fetid ﬁasco ﬁlial ﬁllip ﬁnagled ﬁnicky ﬁtful ﬂagrant ﬂeeting ﬂippant ﬂorid ﬂotilla ﬂotsam ﬂounder ﬂourish ﬂouted ﬂuctuate ﬂummoxed foible foment forage forbearance foreground forestall formidable fortitude fortuitous fracas fractious fraternize frenetic froward frugal fruition fudge fuliginous fulsome funereal furor furtive gadﬂy gaffe gainsay gallant gambit gamboled garble gargantuan garrulous gauche gaudy genial germane germinate ghastly gild glacial glancing glaring glowered glut goosebumps gossamer grandiloquent grandiose grandstand grasping grating gravitas gregarious grisly grouse grovel gumption guttural hackneyed haggard halcyon hallowed hapless haptic harangue harbinger hardscrabble harmonious harried harrow haughty headlong hector hegemony heinous herald hermetic heterodox heterogeneous heyday hiatus hidebound hirsute histrionic hodgepodge holistic homespun homogeneous hortatory hubris humbuggery humdrum husbandry iconoclast ideological idyllic idiosyncrasy ignominy illiberal illusory imbroglio imminent immure immutable impassive impeccable impecunious impeded impenetrable imperative imperious imperturbable impetuous impetus impinge implacable implication implicit imploring importune impregnable imprimatur impromptu impudence impugn inalienable inane incandescent incensed inchoate incipient incisive incoherent incorporate incorrigible inculcate incumbent indefatigable indictment indigenous indignant indomitable industrious ineffable ineluctable inestimable inexorable inﬁnitesimal inﬂux ingenious ingenuous ingrained ingratiate inimical inimitable innate innocuous inordinate inscrutable insinuate insipid insolence insular integrate interloper intimate intrepid intrinsic intrusive inundated inveigh inveigle invidious inviolate irascible irk ironic irresolute jargon jejune jettison jingoism jocose judicious juggernaut juvenescence juxtapose kindle kindred kismet kowtow lachrymose lackadaisical laconic lampoon languid largess lassitude latent laudable lax legerdemain lenient levity licentious lionized listless logorrhea loquacious lovelorn lucid lucre ludicrous lugubrious lumber luminary lurid macabre macerate machination maelstrom magisterial magnanimous magnate makeshift malevolent malfeasance malign malinger malleable manacle mandate manifold marginal marshal maudlin mawkish meager meddle meld melliﬂuous melodramatic mendacity mendicant menial mephitic mercenary mercurial meretricious meterological meticulous mettle miasma microcosm milieu milquetoast mimetic minatory minion misanthrope miscreant miserly misnomer mitigate modicum modish monastic morass morbid mordant mores moribund morose motile motley multifaceted mundane muniﬁcent myopic myriad nadir naivete nascent nebulous neophyte nepotism nettle newfangled noisome non sequitur nonchalant nondescript nonpareil nonplussed nontrivial normative nostalgia nostrum notorious novel novitiate noxious nuance nugatory obdurate obeisance obfuscated objectionable objective objurgation obloquy obsequious obstinate obstreperous obtrusive occluded odious ofﬁcious offset ogle omission omniscient onus opaque openhanded opine opportune opprobrium opulent ornate orthodox oscillate ossiﬁed ostentatious ostracized otiose outmoded outstrip overshadow overweening paciﬁc painstaking palatable palatial palimpsest pall palliate pallid panacea pander pangs panned paradigm paragon pariah parley parochial parody paroxysm parsimonious partiality partisan pastiche pathos patois paucity pedantic pedestrian peevish penchant pendulous penitent penurious penury peons peregrinate peremptory perennial perﬁdy perfunctory peripatetic peripheral permeated permutation pernicious perquisites personiﬁed perspicacious pertinacious perturb pervasive perverse petulant philander phlegmatic physiological picaresque picayune picturesque piebald pilfer pillory pinnacle pioneering piquant pitfall pith pittance pivotal placate plaintive platitude platonic plaudits plausible plebeian plenipotentiary pluck plutocracy polarize polemic politesse politic pomp ponderous portentous poseur posit posthumous pragmatic prattle precarious precocious precursor predilection premonition prescient pretext prevarication primed primordial pristine proclivity prodigal prodigious profane proﬂigacy profound profuse progenitors prognosticate proliferate proliﬁc prolix prominent promulgate pronounced propagate propensity prophetic propitious propriety prosaic protean providential provincial prowess proximity prudent puerile pugnacious pulchritude punctilious pungent punitive purist pusillanimous putrid quagmire quail quandary quash querulous quiescent quintessential quixotic quizzical quotidian raconteur ragamufﬁn raiment ramiﬁcation rampant rancorous rankled rapacious rapt rapturous rareﬁed rash raucous raze readily realization reap rebuttal recalcitrant recant recapitulated recidivist reclusive recondite recrudescent rectitude redouble redress reductive redundant refracted refractory refulgent refute rejuvenated relinquish relish remedial reminiscent remiss remunerated renowned repertoire replete reprehensible reprobate reprove repudiate repugnant requisite resigned resilient resolute resonant respite resplendent restitution restive resurgence reticent retiring retrenchment retrospection revamp revanche reverberate reverent revile revulsion rhapsodize rhetorical rickety rift riposte risible risque roborant robust rotund row rudimentary rufﬁan ruminate saccharine sacrosanct salacious salient salubrious salutary sangfroid sanguine sap sapid sapient sardonic sashayed satiated scanty scapegoat scathing schadenfreude schism scintillating sclerotic scofﬂaw scotch scrupulous scrutinize scufﬂe scurrilous scuttle secretes sectarian secular sedentary sedulous segue self-styled semblance seminal sententious sentient sequacious sere serendipity servile sham shard shelve shirk showy shrewd simper simulacrum sinuous skittish skulduggery skulk slake slander slatternly slipshod slothful slovenly sojourn solecism solicitous solidarity solipsistic somnolent sonorous sophistry sophomoric soporiﬁc sordid soupcon sovereign sparing sparse specious spendthrift splenetic spurious squalid squelch stalwart stanch staple statuesque staunch steadfast stigmas stilted stolid storied stratagem streamlined strenuous stricture strident stringent stultify stupeﬁed subjective sublime subsequent subsidy substantiate subversive subvert succor succumb sullen sumptuary superﬁcial supplant surly surmise surpassing surreptitious susceptible swathe sybarite sycophant synergy synoptic syntax taciturn tangible taxonomic temerity temperance tempestuous temporal tenable tendentious tensile tenuous tepid terse timorous tirade titular tonic toothsome torpid tortuous totalitarian touted tranquil transgression transitory treacly tremulous trepidation triﬂing truculent truncated tumid tumultuous turbid turgid turpitude ubiquitous umbrage unadorned unassuming unbridled unconscionable unconventional unctuous uncultivated undermine underscore understated undulate uniform unkempt unpretentiousness unpropitious unruly unsavory untenable untoward unwieldy upbraided urbane usurious usurp utilitarian utopian vacuous vainglorious vanquished vapid variegated vaunted vehement venal veracious verbose verboten verisimilar vernacular vertiginous vestige vex vicarious vigilant vilify vindicate vindictive virtuoso virulent viscous vitiate vitriolic vituperated vivacious vocation vociferous volatile volition voluminous voracious voyeur wafﬂe wan wane wanting waspish watershed wax welter whet whimsical willful wily winnow winsome wistful wizened wont woo workmanlike worldly wrongheaded wry zealous zenith zephyr Acknowledgments We mainly used the online version of the Merriam-Webster dictionary (www.m-w.com) for deﬁnitions. We’d like to thank Edwin Kotchian for his editing, feedback, and creative suggestions.
12322	https://www.youtube.com/watch?v=CeAFS8nLK-g	Evaluating Expressions & Using Nested Absolute Value Bars & Nested Parenthesis Cole's World of Mathematics 45800 subscribers 9 likes Description 623 views Posted: 15 Oct 2021 This video goes through several examples of Evaluating Algebraic Expressions, which also includes examples with Nested Absolute Value Bars and Nested Parenthesis. evaluateexpressions #nestedabsolutevaluebars #nestedparenthesis Math Tutorials on this channel are targeted at college-level mathematics courses including calculus, pre-calculus, college algebra, trigonometry, probability theory, TI-84 tutorials, introductory college algebra topics, and remedial math topics from algebra 1 and 2 for the struggling college adult (age 18 and older). Don’t forget guys, if you like this video please “Like” and “Share” it with your friends to show your support - it really helps me out! 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My Enterprise Membership provides me with Commercial Rights to the videos I create using Doodly, including Royalty Free Music Tracks provided by Doodly, in order to promote myself or my business on any platform, including uploading my created videos to Facebook, my website, YouTube etc. 2 comments Transcript: if it's an algebraic expression that means it's gonna have some variables in there it'll have some numbers in there add subtract multiply and divide positive negative numbers and they always have to tell you what the variables are so maybe say negative absolute value of x plus parentheses negative y plus a 3x all right and if they tell you that they want you to evaluate this algebraic expression they have to tell you what the variables are so when x equals negative 4 and y equals negative 6. all right so this is your beginning algebra early early algebra one things okay you are evaluating an algebraic expression so every time you see an x you're going to plug in a negative four every time you see a y you're going to plug in a negative 6. and then we just definitely want to make sure that the negatives that are floating around there in front that we don't drop them or anything like that if you need to so to help you see it there is a negative 1 being multiplied by the absolute value some people like to put them in there then they know they're not going to drop it same thing here this y is a variable negative 1 is the coefficient negative 1 times y that helps some people remember okay so let's do a negative one i'm gonna actually put the ones in so you can see them absolute value i'm gonna replace the x with a negative four i'm gonna go ahead and put a plus sign there this inside this brackets right here is going to be a negative 1 times a y so negative 1 i'm going to change it to a dot because that means multiplication as well and my y is a negative six all right and if it would make you happy you can't put a set of parentheses around all of it because i'm doing that's the math that's going inside you don't have to have it plus three times whatever x is so i'm gonna do my 3 i'm going to choose a dot i'm going to replace it with a negative 4. all right whether or not you use dots here i definitely wouldn't use parentheses here if you're going to put parentheses around that whole thing you could put parentheses there it's a matter of choice all right again on the absolute value absolute value of the negative four is a positive four it's being multiplied by the negative one that's sitting in front i'm just going to go one thing at a time all the way across here if i'm multiplying like signs answer is positive that's going to give me a 6. i'm multiplying unlike signs answer's going to give me a negative go ahead and multiply this that means i really have a negative 4 sitting there then i have a 6 then i have a negative 12. i'm going to do these from left to right across the page or i choose to because addition just left to right so that's going to be a 2 plus a negative 12. it's going to give me a negative 10. when you do these types of problems in mymathlab you really should like be showing your work that sort of thing obviously you're not turning any of that in but if you get it wrong you can try them as you have seen multiple times until you get it right if you do the question help you might be able to find in your work where you made your mistake okay on our written exams and on the final exams your work must look like this okay so you do have to show this on the written exam so if you practice it on your homework then it'll be standard there when you do it on the written exams and final exams all right let's do another example in this little corner right here all right any of these algebraic expressions can also have lots of absolute value bars in it maybe not just one so i could do absolute value a minus parentheses and negative b when i hand write them sometimes i don't tell i don't make one higher than the other that's clearly a minus sign this is clearly a negative sign usually when it's typed this one will be higher all right this one should will be lower in the problem okay again absolute value bars around the whole entire thing when a equals negative 6 equals 10. okay so again if we're showing all of our work you're just going to do on that first step you're going to replace a and b with what they are so we're keeping the absolute value bars because we're not taking any absolute value yet i'm going to replace a with a negative 6. that minus sign right there is going to come down if you don't want to drop the negative there you're going to remember that there's really a negative 1 being multiplied by whatever b is okay so negative 1 is being multiplied by my b which is 10. you can go ahead and keep those parentheses around it if you want definitely multiplying in here first keep those absolute value bars negative 6 minus a negative 10 so i am subtracting a negative right here okay this is where i do the plus plus thing it's minus a negative so i do plus plus i make two little quick marks on there plus plus again you may do it any way you want absolute value of four final answer four okay so hopefully this is this is an okay thing we've done this a lot even if it's been a while hopefully it's not too bad you plug this stuff back in it kind of kind of comes now can you have nested parentheses and nested absolute value bars sure all right and again these are algebraic expressions just in kind of like in the form of just basic polynomial there no quotients in these okay let's look at some nested absolute value bars all right and or nested parentheses nested absolute value bars or nested parentheses nested just means one you know a set and then another set on the inside no said on the inside another set on the inside all right they can get kind of silly with them let's say we've got some absolute value bars i'm going to try to make mine a little bit different size here and then let's say inside here i've maybe got absolute value of a negative four i'm going to go back to just all numbers in here because i just want the concept of the nesting going on in here plus maybe say 2 absolute value of a 3. let's do that so i tried to make this these two a little bit higher and larger and then clearly something in here so the idea of nested i've got absolute value bars inside a set of absolute value bars this one is by itself single number i can take the absolute value there the outside bars are going to stay this becomes always positive so four when you take the absolute value bars the dark bars when you take the absolute value the bars drop so i took it made it positive here this one you can probably do in your head absolute value of three is three three times two is six add on the inside absolute value of ten ten okay so that nested idea of nested absolute value bars we could also do nested parentheses and come up with some of those maybe say a 4 minus a 6. quantity squared minus three parentheses five minus eight and then square the whole quantity i could have used like i did i chose to use curvy brackets right there you might even see a set of square brackets means the same thing all right so let's kind of do that in yellow highlighter just to show all right if sometimes textbooks will intentionally make this set square and then this set curvy so that you can clearly tell the difference these two would go together these two would go together obviously the absolute value bars goes together all right so it doesn't matter whether they are curvy or square either one they're all the same all right i'm going to work inside here i think i'll switch and do the square brackets all the way down right here is going to be a negative 2. the only thing i'm doing is inside a set of brackets and squaring i'm going to keep going here this is absolute value let's show some more work some of you hopefully everybody in the room can just look at that and tell me it's negative three okay but if you needed to see steps that's the same thing as a five plus a negative eight if you needed to see that step all right so there's the negative three throwing in that extra step just for anybody that might need it negative two squared negative 2 times negative 2 is going to give me a positive 4. keep the 3. negative 3 inside here i don't want to do too much let's keep the bars around it i got a little sloppy the way i wrote that okay now at this point some people will take a look because that minus sign just kind of glares at you and then they try to subtract and they put a 1 in here all right however this right here that's really three times the absolute value of the negative three multiplication has to be done before add and subtract okay so the correct way to do it would be to keep the four and the minus sign there absolute value of a negative three is positive three three times three square brackets and squared stays there the whole time again showing every line of work which is what i'm doing right now 3 times 3 is 9. again subtraction 4 minus 9 is going to give me a negative 5 inside there and it's okay that i have the square brackets and then with square brackets all the way down that's fine when i square negative 5. i get a wipe out okay do i necessarily expect you to show that many lines no all right but i'm doing it for anybody that needs it okay in case people need it you
12323	https://www.youtube.com/watch?v=PKXjV5mtxy8	What is the Averaging Principle? [FULL PROOF & MORE] (Maths for Computer Science) PageWizard Games, Learning & Entertainment 833 subscribers 5 likes Description 130 views Posted: 21 Dec 2022 In today's lesson, we learn about the averaging principle, which is related to the pigeonhole principle. We derive a special case of it, then prove it more generally! This theorem is especially handy for deriving lower bounds and finding out if some element is at least some number (less than the average), if the average is small! Time Stamps: 0:00 Opening, a re-formulation of the generalized pigeonhole principle 13:40 Averaging principle (with proof) 24:40 Closing Have a beautiful day! Supporters (to date of publication, by tier (top to bottom)): Patreon Supporters (General Support): Draikou Patreon Supporters (Basic Support): Patreon Supporters (Supporter Access!): Eric R Become a supporter today! 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Please donate and contribute to support my work for more content: PATREON: SUBSCRIBESTAR: PAYPAL: Follow also at: FACEBOOK: TWITTER: QUORA: TWITCH: ComputerScience mathematics combinatorics Transcript: Opening, a re-formulation of the generalized pigeonhole principle so up to this point we've talked about the pigeonhole principle we talked about the generalized pigeonhole principle now I want to show you a reformulation that generalizes over the generalized pigeonhole principle it's going to be called It's called The averaging principle some people like to think of this as a precursor or prototype for the probabilistic method so if you've ever heard of that before this is going to feel like kind of like a baby step towards that thank you so we're going to consider a reformulation so I want to show you just just to convince you of the averaging principle for integers so if I have an integer number of objects an integer number of bins I would like to show you where and how we can get something that's going to approximate the averaging principle more broadly so I want to just show you how we could take the generalized prison whole principle turn it into what we're going to see is called the averaging principle so consider a reef a reformulation of our previous Counting Concepts from the pigeonhole now suppose that we're given suppose that we are given a non-empty so that means that we have at least one element in it it's non-empty and finite finite collection so it means that you're allowed to have duplicates in here that's just why I'm using the word collection as opposed to a set but it's still true for set too a collection a of n numbers so I'm using the word numbers here more explicitly so here's sort of a broader conceptual question is there an element in a that is at least some number we're going to call MU now it's a little bit of a hint where we're going with this if you know if you know you know if you see the word average and you hear a word see a symbol like Mew you might know where we're going with this but uh and I'm gonna stress that this may which may depend on a so it's not just a constant or something it's going to be something you could calculate based on a so I want to try to see if we could figure out what mu might be now so notice that this is slightly different in the way I formulated it now it's a collection of numbers and I would like to know if there's some number in that collection that is at least some something called mu so let's try let's try to determine mu in the special case when a has positive integers so let's just let's just work with positive inters we assume that those numbers are positive integers for now so we're going to do something kind of crafty here so pay attention very closely so so so for example here suppose that I give you a collection like this three two and five some people call that a tuple so it's a three Tuple it has three numbers in it these are all positive integers and I'm going to try to come up with a way to figure out that there's some number at least at least uh that depends on a so there's gonna be something in here that I could guarantee is at least mu so watch this carefully is I'm going to visualize this a little bit differently than I would otherwise I'm going to view each position in a as a bin so for example there I have three three objects sorry not three object three bends one for each position I have three positions three bins so I want to view each one of them as a bin now now each number a in a itself in the collection as a objects so example um three is three objects so when all of a is considered foreign all of a is considered it totals to the sum over all of the elements in a of their values many objects so there's that many objects and the reason why I'm allowed to do this is because all of them are positive integers so please keep that in mind so the this whole idea the way it works is as long as I could take the the numbers and I can break them into discrete sizes uh this trick works so integers it's rather trivial because I could just say oh the number itself is that so what I'm trying to say and let me try to visualize this um is three is like having three objects two is like having two objects five is like having five objects and now these might occupy these bins as a potential placement of where these values could be if I were like a skeksis or something like from The Dark Crystal I would say this is the essence the essence of the of the of the numbers so what we're going to do is we're going to Trend the essence from from the numbers by taking all of these and bucking them all up into one one big super bucket now it's not a bucket like these these are bins I'm just gonna collect all all of the objects and since this actually adds up to 10 we should have one two three four five six seven eight nine ten so there's ten objects in total and I have three bins foreign so now how many so so if now if I look at this in terms of this now I have 10 objects three bins if I were to imagine like a house of skexis I could like take the life force from one and just give it to another and say one of these bins like I drained the essence from one of these and I put it in one of these bins what is what can I say about there being like for one of these bins for some bin if I have three bins and ten objects how many must be at least in some bin well the generalized pigeonhole principle tells us this it tells us the answer to this question no get back here get back here I'm not going to drain your essence so let's let's go back over here so the generalized foreign principle baits States some bin States some bin has at least how many well remember how many objects did we have we have the sum over all of the numbers in the collection has at least this many right so that's how many objects I have this is how many bins I have it has that many objects in it that's what the generalized pigeonhole pencil would tell you so now now this is where it gets a little bit weird because now we got to translate this back to our original question so remember the bins represent the the positions the numbers can occupy so now when we think about a bin we're actually thinking about the number that might be sitting there and the number of objects is its value so translating translating this back there is a number a number in a remember that's among all of the bins because there are three bins for one for each position those are the positions in a that is at least at least what we have above but this is the ceiling of this right you would agree with me I could just drop the ceiling and the value either stays the same or gets smaller right so I'm just going to write it like this to make it a little easier for us to parse what's going on and I'm saying that this value here this is Mu but what's that that's literally the average that is literally the average so this brings us to a this is the average right this is the average of a so remember the argument that we've employed to derive this argument here this relies on the fact that they're positive integers you can actually prove a statement that's far more General using the same kind of techniques we've actually used for the pigeonhole principle and this will work even for real numbers so if I give you n positions in my collection and the numbers are real numbers then I could still prove that actually there's going to be some element or some number in the collection that's going to be at least the average and it must be this guy likewise you can actually prove also another fact and this is what makes it sort of a neat concept other than just the just the pigeonhole principle is that there's also an element that is at most the average so there's two parts to it the proof is actually surprisingly simple once you've seen how you can prove like the generalized pigeonhole principle because they're based on very similar ideas just we need to make sure when we prove the claim we don't make any assumptions that they're integers Averaging principle (with proof) so theorem so this is what we're going to refer to as the averaging principle so given a non-empty finite and finite collection a of n numbers foreign my purple presume that they're real numbers but you'll find that the arguments as long as you're it it it should be fine enough for us of n real numbers the average of a is Mu which is going to be equal to 1 over n times the sum over all of the elements in a of a so here's what the real part heart of the averaging principle is about there exists this number X in a that is at least mu and there is there is a number Y in a that is at most you most most meal I must stress as an important note that X could be equal to y so X and Y foreign so X and Y can be the same for example if your collection has all the same elements in every single one of the spots like all the elements are all ones then what the averaging principle would say is that oh yeah all the average is one that means that X there is an element that is at least at least this and it most that it's namely one for example so let's prove this so let's do a proof together so we're going to prove it in two parts because there are two State there are two parts to this this theorem the first one is that this part about the saying that there's an element that is at least mu and then there's an element that is at most to me those are two parts so we're going to prove them separately you're going to find that the arguments used are very similar to each other so much so that I actually changed it up because you could use potentially the same idea in both directions well not directions but most both parts be a bit more precise so my apologies to prove the we prove the two parts of the theorem separately so the question is which one should we go for first I think we're going to go with the at least mu part I don't know what you think I think that's a pretty natural one to start with because that's the first thing I stated here so first part foreign first we prove there is a number that is at least mu now we're going to use contradiction because it makes things a little bit fun with us so I'm going to assume otherwise then that means something I want you to think about what that means so we're going to assume otherwise for the sake of a contradiction so assume otherwise so assume or otherwise so it says that there exists a number that is at least mu that means that every number has to be what at most or at most or is it what it's less than mu right if if it was equal to Mu then it's at least mu that means that every number has to be less than mu all right it must be on every number in a is less than mu now watch this watch this just watch watch how this works then then let's write out the sum of all the numbers and the sum from over all of the elements in a all those numbers you'd agree with me that this has to be strictly less than if I took and replaced each a with mu right because every element is less than mu but what is this there are n elements or n numbers in my collection so I'm adding new n times that is the same as me writing n times mu but what is that this is n times the average of a times 1 over n times the sum of a right that's what mu is but do you see anything interesting going on here look we got n times 1 over n right let's see we're going to annihilate each other and we end up with just the sum this sum right do you see a problem with this where this seems a little nonsensical can you see why what's going on here guys this looks a little funky what did I say over there that the sum over all the elements in the collection is less than this right but we show that that is in fact equal to this so the sum over all the elements in the collection is less than the sum of the all the elements in the collection that's nonsense right because they have to be equal to each other literally the same thing which is a contradiction that's all possible now that brings us to the second part we're going to use the same same strategy now I'm not stress you don't have to go about this way but you'll find that this is just as fun I'm just gonna write it like this just to change it up next we prove that there is an element there's an element or number let's write number just be a bit more precise about our language there's a number that is at most mu now we're going to assume again for the sake of a contradiction we're going to assume otherwise so so again what does that mean if it's at there's going to be some element that's at most mu that means that if I make every if then that means that if that's not true that means that every element has to be greater than mu assume otherwise foreign number in a is greater than mu and we could do something quite similar to what we did before I'm just going to shake it up a little bit I'm just going to do a little dance to Shake It Up so mu the average of a is just this right right that's the definition of mu this has to be strictly bigger than 1 over n times the sum over all the elements in the collection where I replace a with mu right that has to be the case because every element in a is strictly bigger than mu but what's this well that's one over n we add mu n times this is the same as me writing this and times mu because that's just we're adding mu n times but what is this n times 1 over n that's supposed to cancel so I just end up with mu again we end up with a nonsensical situation because mu is now greater than mu that time okay he says because it's saying that mu is equal to Mu but mu is not equal to Mu it doesn't make a whole lot of sense like this part is saying mu is not equal to Mu which is a contradiction and that completes the proof so that's your proof for the averaging principle so the averaging principle just to summarize it tells you that if you give me a collection of numbers and so this is a statement about a collection of numbers so you're allowed to have duplicates in there or everything like that um what it says is that there always will exist an element in the collection that is at least the average of the collection and there exists an element Closing that is going to be at most the average of the collection
12324	https://howellkb.uah.edu/public_html/DEtext/Part4/PW_Def_Fcts.pdf	28 Piecewise-Deﬁned Functions and Periodic Functions At the start of our study of the Laplace transform, it was claimed that the Laplace transform is “particularly useful when dealing with nonhomogeneous equations in which the forcing func-tions are not continuous” . Thus far, however, we’ve done precious little with any discontinuous functions other than step functions. Let us now rectify the situation by looking at the sort of discontinuous functions (and, more generally, “piecewise-deﬁned” functions) that often arise in applications, and develop tools and skills for dealing with these functions. We will also take a brief look at transforms of periodic functions other than sines and cosines. As you will see, many of these functions are, themselves, piecewise deﬁned. And ﬁnally, we will use some of the material we’ve recently developed to re-examine the issue of resonance in mass/spring systems. 28.1 Piecewise-Deﬁned Functions Piecewise-Deﬁned Functions, Deﬁned When we talk about a “discontinuous function f ” in the context of Laplace transforms, we usually mean f is a piecewise continuous function that is not continuous on the interval (0, ∞) . Such a function will have jump discontinuities at isolated points in this interval. Computationally, however, the real issue is often not so much whether there is a nonzero jump in the graph of f at a point t0 , but whether the formula for computing f (t) is the same on either side of t0 . So we really should be looking at the more general class of “piecewise-deﬁned” functions that, at worst, have jump discontinuities. Just what is a piecewise-deﬁned function? It is any function given by different formulas on different intervals. For example, f (t) =        0 if t < 1 1 if 1 < t < 2 0 if 2 < t and g(t) =        0 if t ≤1 t −1 if 1 < t < 2 1 if 2 ≤t are two relatively simple piecewise-deﬁned functions. The ﬁrst (sketched in ﬁgure 28.1a) is discontinuous because it has nontrivial jumps at t = 1 and t = 2 . However, the second 555 556 Piecewise-Deﬁned and Periodic Functions (a) (b) T T 1 1 1 1 2 2 0 0 Figure 28.1: The graphs of two piecewise-deﬁned functions. function (sketched in ﬁgure 28.1b) is continuous because t −1 goes from 0 to 1 as t goes from 1 to 2 . There are no jumps in the graph of g . By the way, we may occasionally refer to the sort of lists used above to deﬁne f (t) and g(t) as conditional sets of formulas or sets of conditional formulas for f and g , simply because these are sets of formulas with conditions stating when each formula is to be used. Do note that, in the above formula set for f , we did not specify the values of f (t) when t = 1 or t = 2 . This was because f has jump discontinuities at these points and, as we agreed in chapter 24 (see page 496), we are not concerned with the precise value of a function at its discontinuities. On the other hand, using the formula set given above for g , you can easily verify that lim t→1−g(t) = 0 = lim t→2+ g(t) and lim t→1−g(t) = 1 = lim t→2+ g(t) ; so there is not a true jump in g at these points. That is why we went ahead and speciﬁed that g(1) = 0 and g(2) = 1 . In the future, let us agree that, even if the value of a particular function f or g is not explicitly speciﬁed at a particular point t0 , as long as the left- and right-hand limits of the function at t0 are deﬁned and equal, then the function is deﬁned at t0 and is equal to those limits. That is, we’ll assume f (t0) = lim t→t0−f (t) = lim t→t0+ f (t) whenever lim t→t0−f (t) = lim t→t0+ f (t) . This will simplify notation a little, and may keep us from worrying about issues of continuity when those issues are not important. Step Functions, Again Most people would probably consider the step functions to be the simplest piecewise-deﬁned functions. These include the basic step function, step(t) = ( 0 if t < 0 1 if 0 ≤t (sketched in ﬁgure 28.2a), as well as the step function at a point α , stepα(t) = step(t −α) = ( 0 if t < α 1 if α < t Piecewise-Deﬁned Functions 557 (a) (b) α T T 0 0 1 1 Figure 28.2: The graphs of (a) the basic step function step(t) and (b) a shifted step function stepα(t) with α > 0 . (sketched in ﬁgure 28.2b). We will be dealing with other piecewise-deﬁned functions, but, even with these other func-tions, we will ﬁnd step functions useful. Step functions can be used as ‘switches’ — turning on and off the different formulas in our piecewise-deﬁned functions. In this regard, let us quickly observe what we get when we multiply the step function by any function/formula g(t) : g(t) stepα(t) = ( g(t) · 0 if t < α g(t) · 1 if α < t ) = ( 0 if t < α g(t) if α < t . Here, the step function at α ‘switches on’ g(t) at t = α . For example, t2 step3(t) = ( 0 if t < 3 t2 if 3 < t and sin(t −4) step4(t) = ( 0 if t < 4 sin(t −4) if 4 < t . This fact will be especially useful when applying Laplace transforms in problems involving piecewise-deﬁned functions, and we will ﬁnd ourselves especially interested in cases where the formula being multiplied by stepα(t) describes a function that is also translated by α (as in sin(t −4) step4(t) ). The Laplace transform of stepα(t) was computed in chapter 24. If you don’t recall how to compute this transform, it would be worth your while to go back to review that discussion. It is also worthwhile for us to look at a differential equation involving a step function. !◮Example 28.1: Consider ﬁnding the solution to y′′ + y = step3 with y(0) = 0 and y′(0) = 0 . Taking the Laplace transform of both sides: L y′′ + y s = L step3 s ֒ → L y′′ s + L[y]\|s = 1 s e−3s ֒ → s2Y(s) −sy(0) −y′(0) + Y(s) = 1 s e−3s 558 Piecewise-Deﬁned and Periodic Functions ֒ → s2 + 1 Y(s) = 1 s e−3s ֒ → Y(s) = 1 s(s2 + 1)e−3s . Thus, y(t) = L−1 1 s(s2 + 1)e−3s t . Here, we have the inverse transform of an exponential multiplied by a function whose inverse transform can easily be computed using, say, partial fraction. This would be a good point to pause and discuss, in general, what can be done in such situations.1 28.2 The “Translation Along the T-Axis” Identity The Identity As illustrated in the above example, we may often ﬁnd ourselves with L−1 e−αs F(s) t where α is some positive number and F(s) is some function whose inverse Laplace transform, f = L−1[F] , is either known or can be found with relative ease. Remember, this means F(s) = L[ f (t)]\|s = Z ∞ 0 f (t)e−st dt . Consequently, e−αs F(s) = e−αs Z ∞ 0 f (t)e−st dt = Z ∞ 0 f (t)e−αse−st dt = Z ∞ 0 f (t)e−s(t+α) dt . Using the change of variables τ = t + α (thus, t = τ −α ), and being careful with the limits of integration, we see that e−αs F(s) = · · · = Z ∞ t=0 f (t)e−s(t+α) dt = Z ∞ τ=α f (τ −α)e−sτ dτ . (28.1) This last integral is almost, but not quite, the integral for the Laplace transform of f (τ −α) (using τ instead of t as the symbol for the variable of integration). And the reason it is not is that this integral’s limits start at α instead of 0 . But that is where the limits would start if the function being transformed were 0 for τ < α . This, along with observations made a page or so ago, suggests viewing this integral as the transform of f (t −α) stepα(t) = ( 0 if t < α f (t −α) if α ≤t . 1 The observant reader will note that y can be found directly using convolution. However, beginners may ﬁnd the computation of the needed convolution, sin(t) ∗step3(t) , a little tricky. The approach being developed here reduces the need for such convolutions, and can be applied when convolution cannot be used. Still, convolutions with piecewise-deﬁned functions can be useful, and will be discussed in section 28.4. The “Translation Along the T-Axis” Identity 559 After all, Z ∞ τ=α f (τ −α)e−sτ dτ = Z ∞ t=α f (t −α)e−st dt = Z α t=0 f (t −α) · 0 · e−st dt + Z ∞ t=α f (t −α) · 1 · e−st dt = Z α t=0 f (t −α) stepα(t)e−st dt + Z ∞ t=α f (t −α) stepα(t)e−st dt = Z ∞ t=0 f (t −α) stepα(t)e−st dt = L f (t −α) stepα s . Combining the above computations with equation set (28.1) then gives us e−αs F(s) = · · · = Z ∞ τ=α f (τ −α)e−sτ dτ = · · · = L f (t −α) stepα(t) s . Cutting out the middle, we get our second translation identity: Theorem 28.1 (translation along the T–axis) Let F(s) = L[ f (t)]\|s where f is any Laplace transformable function. Then, for any positive constant α , L f (t −α) stepα(t) s = e−αs F(s) . (28.2a) Equivalently, L−1 e−αs F(s) t = f (t −α) stepα(t) . (28.2b) Computing Inverse Transforms The Basic Computations Computing inverse transforms using the translation along the T –axis identity is usually straight-forward. !◮Example 28.2: Consider ﬁnding the inverse Laplace transform of e−2s s2 + 1 . Applying the identity, we have L−1 e−2s s2 + 1 t = L−1h e−2s 1 s2 + 1 \| {z } F(s) i t = L−1 e−2s F(s) t = f (t −2) step2(t) . Here the inverse transform of F is easily read off the tables: f (t) = L−1[F(s)]\|t = L−1 1 s2 + 1 t = sin(t) . 560 Piecewise-Deﬁned and Periodic Functions 1 PSfrag 2 T 2 + π 2 + 2π Figure 28.3: The graph of sin(t −2) step(t −2) . So, for any X , f (X) = sin(X) . Using this with X = t −2 in the above inverse transform computation then yields L−1 e−2s s2 + 1 t = f (t −2) step2(t) = sin(t −2) step2(t) . Keep in mind that sin(t −2) step2(t) = ( 0 if t < 2 sin(t −2) if 2 < t . The graph of this function is sketched in ﬁgure 28.3. Observe that, as illustrated in ﬁgure 28.3, the graph of L−1 e−αs F(s) t = f (t −α) stepα(t) is always zero for t < α , and is the graph shifted by α of f (t) on [0, ∞) on α ≤t . Remembering this can simplify graphing these types of functions. Describing Piecewise-Deﬁned Functions Arising From Inverse Transforms Let us start with a simple, but illustrative, example. !◮Example 28.3: Consider computing the inverse Laplace transform of F(s) = 1 s2 e−s − 1 s2 e−2s . Going to the tables, we see that G(s) = 1 s2 H ⇒ g(t) = t . Using this, along with linearity and the second translation identity, we get f (t) = L−1[F(s)]\|t = L−1 1 s2 e−s −1 s2 e−2s t = L−1 1 s2 e−1s t −L−1 1 s2 e−2s t = (t −1) step1(t) −(t −2) step2(t) . The “Translation Along the T-Axis” Identity 561 Note that the step functions tell us that ‘signiﬁcant changes’ occur in f (t) at the points t = 1 and t = 2 . While the above is a valid answer, it is not a particularly convenient answer. It would be much easier to graph and see what f really is if we go further and completely compute f (t) on the intervals having t = 1 and t = 2 as endpoints: For t < 1 , then f (t) = (t −1) step1(t) \| {z } 0 −(t −2) step2(t) \| {z } 0 = 0 −0 = 0 . For 1 < t < 2 , then f (t) = (t −1) step1(t) \| {z } 1 −(t −2) step2(t) \| {z } 0 = (t −1) −0 = t −1 . For 2 < t , then f (t) = (t −1) step1(t) \| {z } 1 −(t −2) step2(t) \| {z } 1 = (t −1) −(t −2) = 1 . Thus, f (t) =        0 if t < 1 t −1 if 1 < t < 2 1 if 2 < t . (This is the function sketched in ﬁgure 28.1b on page 555.) As just illustrated, piecewise-deﬁned functions naturally arise when computing inverse Laplace transforms using the second translation identity. Typically, use of this identity leads to an expression of the form f (t) = g0(t) + g1(t) stepα1(t) + g2(t) stepα2(t) + g3(t) stepα3(t) + · · · (28.3) where f is the function of interest, the gk(t)’s are various formulas, and the αk’s are positive constants. This expression is a valid formula for f , and the step functions tell us that ‘signiﬁcant changes’ occur in f (t) at the points t = α1 , t = α2 , t = α3 , . . . . Still, to get a better picture of the function f (t) , we will want to obtain the formulas for f (t) over each of the intervals bounded by the αk’s . Assuming we were reasonably intelligent and indexed the αk’s so that 0 < α1 < α2 < α3 < · · · , we would have For t < α1 , f (t) = g0(t) + g1(t) stepα1(t) \| {z } 0 + g2(t) stepα2(t) \| {z } 0 + g3(t) stepα3(t) \| {z } 0 + · · · = g0(t) + 0 + 0 + 0 + · · · = g0(t) . 562 Piecewise-Deﬁned and Periodic Functions For α1 < t < α2 , f (t) = g0(t) + g1(t) stepα1(t) \| {z } 1 + g2(t) stepα2(t) \| {z } 0 + g3(t) stepα3(t) \| {z } 0 + · · · = g0(t) + g1(t) + 0 + 0 + · · · = g0(t) + g1(t) . For α2 < t < α3 , f (t) = g0(t) + g1(t) stepα1(t) \| {z } 1 + g2(t) stepα2(t) \| {z } 1 + g3(t) stepα3(t) \| {z } 0 + · · · = g0(t) + g1(t) + g2(t) + 0 + · · · = g0(t) + g1(t) + g2(t) . And so on. Thus, the function f described by formula (28.3), above, is also given by the conditional set of formulas f (t) =              f0(t) if t < α1 f1(t) if α1 < t < α2 f2(t) if α2 < t < α3 . . . where f0(t) = g0(t) , f1(t) = g0(t) + g1(t) , f2(t) = g0(t) + g1(t) + g2(t) , . . . . Computing Transforms with the Identity The translationalongthe T –axisidentityis alsohelpful incomputingthe transforms of piecewise-deﬁned functions. Here, though, the computations typically require little more care. We’ll deal with fairly simple cases here, and develop this topic further in the next section. !◮Example 28.4: Consider ﬁnding L[g(t)]\|s where g(t) = ( 0 if t < 3 t2 if 3 < t . Remember, this function can also be written as g(t) = t2 step3(t) . Plugging this into the transform and applying our new translation identity gives L[g(t)]\|s = L t2 step3(t) s = L f (t −3) step3(t) s = e−3s F(s) Rectangle Functions 563 where f (t −3) = t2 . But we need the formula for f (t) , not f (t −3) , to compute F(s) . To ﬁnd that that formula, let X = t −3 (hence, t = X + 3 ) in the formula for f (t −3) . This gives f (X) = (X + 3)2 . Thus, f (t) = (t + 3)2 = t2 + 6t + 9 , and F(s) = L[ f (t)]\|s = L t2 + 6t + 9 s = L t2 s + 6L[t]\|s + 9L\|s = 2 s3 + 6 s2 + 9 s . Plugging this back into the above formula for L[g(t)]\|s gives us L[g(t)]\|s = e−3s F(s) = e−3s 2 s3 + 6 s2 + 9 s . 28.3 Rectangle Functions and Transforms of More Complicated Piecewise-Deﬁned Functions Rectangle Functions “Rectangle functions” are slight generalizations of step functions. Given any interval (α, β) , the rectangle function on (α, β) , denoted rect(α,β) , is the function given by rect(α,β)(t) =        0 if t < α 1 if α < t < β 0 if β < t . The graph of rect(α,β) with −∞< α < β < ∞has been sketched in ﬁgure 28.4. You can see why it is called a rectangle function — it’s graph looks rather “rectangular” , at least when α and β are ﬁnite. If α = −∞or β = ∞, the corresponding rectangle functions simplify to rect(−∞,β)(t) = ( 1 if t < β 0 if β < t and rect(α,∞)(t) = ( 0 if t < α 1 if α < t . And if both a = −∞and b = ∞, then we have rect(−∞,∞)(t) = 1 for all t . 564 Piecewise-Deﬁned and Periodic Functions α β T 1 Figure 28.4: Graph of the rectangle function rect(α,β)(t) with −∞< α < β < ∞. All of these rectangle functions can be written as simple linear combinations of 1 and step functions at α and/or β , with, again, the step functions acting as ‘switches’ — switching the rectangle function ‘on’ (from 0 to 1 at α ), and switching it ‘off’ (from 1 back to 0 at β ). In particular, we clearly have rect(−∞,∞)(t) = 1 and rect(α,∞)(t) = stepα(t) . Somewhat more importantly (for us), we should observe that, for −∞< α < β < ∞, 1 −stepβ(t) = ( 1 −0 if t < β 1 −1 if β < t ) = ( 1 if t < β 0 if β < t ) = rect(−∞,β)(t) , and stepα(t) −stepβ(t) =        0 −0 if t < α 1 −0 if α < t < β 1 −1 if β < t        =        0 if t < a 1 if α < t < β 0 if β < t        = rect(α,β)(t) . In summary, for −∞< α < β < ∞, rect(α,β)(t) = stepα(t) −stepβ(t) , (28.4a) rect(−∞,β)(t) = 1 −stepβ(t) (28.4b) and rect(α,∞)(t) = stepα(t) . (28.4c) These formulas allow us to quickly compute the Laplace transforms of rectangle functions using the known transforms of 1 and the step functions. !◮Example 28.5: L rect(3,4)(t) s = L rect(3,4)(t) s = L step3(t) −step4(t) s = L step3(t) s −L step4(t) s = 1 s e−3s −1 s e−4s . Rectangle Functions 565 Transforming More General Piecewise-Deﬁned Functions To help us deal with more general piecewise-deﬁned functions, let us make the simple observa-tions that g(t) rect(a,b)(t) =        g(t) · 0 if t < a g(t) · 1 if a < t < b g(t) · 0 if b < t        =        0 if t < a g(t) if a < t < b 0 if b < t , and g(t) rect(−∞,b)(t) = ( g(t) · 1 if t < b g(t) · 0 if b < t ) = ( g(t) if t < b 0 if b < t . So functions of the form f (t) =        0 if t < a g(t) if a < t < b 0 if b < t and h(t) = ( g(t) if t < b 0 if b < t can be rewritten, respectively, as f (t) = g(t) rect(a,b)(t) and h(t) = g(t) rect(−∞,b)(t) . More generally, it should now be clear that anything of the form f (t) =              g0(t) if t < α1 g1(t) if α1 < t < α2 g2(t) if α2 < t < α3 . . . (28.5a) can be rewritten as f (t) = g0(t) rect(−∞,α1)(t) + g1(t) rect(α1,α2)(t) + g2(t) rect(α2,α3)(t) + · · · . (28.5b) The second form (with the rectangle functions) is a bit more concise than the “conditional set of formulas” used in form (28.5a), and is generally preferred by typesetters. Of course, there is a more important advantage of form (28.5b): Assuming f is piecewise continuous and of exponential order, its Laplace transform can now be taken by expressing the rectangle functions in formula (28.5b) as the linear combinations of 1 and step functions given in equation set (28.4), and then using linearity and what we learned in the previous section about taking transforms of functions multiplied by step functions. !◮Example 28.6: Consider ﬁnding F(s) = L[ f (t)]\|s when f (t) = ( t2 if t < 3 0 if 3 < t . From the above, we see that f (t) = t2 rect(−∞,3)(t) = t2 1 −step3(t) = t2 −t2 step3(t) . 566 Piecewise-Deﬁned and Periodic Functions So F(s) = L[ f (t)]\|s = L t2 −t2 step3(t) s = L t2 s −L t2 step3(t) s . The Laplace transform of t2 is in the tables, while the transform of t2 step3(t) just happened to have been computed in example 28.4 a few pages ago. Using these transforms, the above formula for F becomes F(s) = 2 s3 −e−3s 2 s3 + 6 s2 + 9 s . !◮Example 28.7: Consider ﬁnding F(s) = L[ f (t)]\|s when f (t) =        0 if t < 2 e3t if 2 < t < 4 0 if 4 < t . From the above, we see that f (t) = e3t rect(2,4)(t) = e3t step2(t) −step4(t) = e3t step2(t) −e3t step4(t) . Thus, L[ f (t)]\|s = L e3t step2(t) s −L e3t step4(t) s . (28.6) Both of the transforms on the right side of our last equation are easily computed via the translation identity developed in this chapter. For the ﬁrst, we have L e3t step2(t) s = L g(t −2) step2(t) s = e−2sG(s) where g(t −2) = e3t . Letting X = t −2 (so t = X + 2 ), the last expression becomes g(X) = e3(X+2) = e3X+6 = e6e3X . So g(t) = e6e3t and G(s) = L[g(t)]\|s = L e6e3t s = e6L e3t s = e6 1 s −3 . This, along with the ﬁrst equation in this paragraph, gives us L e3t step2(t) s = e−2sG(s) = e−2se6 1 s −3 = e−2(s−3) s −3 . The transform of e3t step4(t) can be computed in the same manner, yielding L e3t step4(t) s = e−4(s−3) s −3 . Rectangle Functions 567 (The details of this computation are left to you.) Finally, combining the formulas just obtained for the transforms of e3t step2(t) and e3t step4(t) with equation (28.6), we have L[ f (t)]\|s = L e3t step2(t) s −L e3t step4(t) s = e−2(s−3) s −3 −e−4(s−3) s −3 . !◮Example 28.8: Let’s ﬁnd the Laplace transform F(s) of f (t) =        2 if t < 1 e3t if 1 < t < 3 t2 if 3 < t . Toapplythe Laplace transform, we ﬁrst convert the above toanequivalent expressioninvolving step functions: f (t) = 2 rect(−∞,1)(t) + e3t rect(1,3)(t) + t2 rect(3,∞)(t) = 2 1 −step1(t) + e3t step1(t) −step3(t) + t2 step3(t) = 2 −2 step1(t) + e3t step1(t) −e3t step3(t) + t2 step3(t) . Using the tables and methods already discussed earlier in this chapter (as in examples 28.7 and 28.4), we discover that L\|s = 2 s , L 2 step1(t) s = 2e−s s , L e3t step1(t) s = e−(s−3) s −3 , L e3t step3(t) s = e−3(s−3) s −3 and L t2 step3(t) s = e−3s 2 s3 + 6 s2 + 9 s . Combining the above and using the linearity of the Laplace transform, we obtain F(s) = L[ f (t)]\|s = L 2 −2 step1(t) + e3t step1(t) −e3t step3(t) + t2 step3(t) s = 2 s −2e−s s + e−(s−3) s −3 −e−3(s−3) s −3 + e−3s 2 s3 + 6 s2 + 9 s . 568 Piecewise-Deﬁned and Periodic Functions 28.4 Convolution with Piecewise-Deﬁned Functions Take another look at example 28.1 on page 557. As noted in the footnote,we could have by-passed much of the Laplace transform computation by simply observing that y(t) = sin(t) ∗step3(t) and computing that convolution. But in the footnote, it was claimed that computing such con-volutions can be “a little tricky” . Well, to be honest, it’s not all that tricky. It’s more an issue of careful bookkeeping. When computing a convolution h ∗f in which f is piecewise deﬁned, you need to realize that the resulting convolution will also be piecewise deﬁned,with (as you will see in the examples) the formula for h ∗f changing at the same points where the formula for f changes. Hence, you should compute h ∗f separately over the different intervals bounded by these points. Moreover, in computing the corresponding integrals, you will also need to account for the piecewise-deﬁned nature of f , and break up the integral appropriately. To simplify all this, it is strongly recommended that you compute the convolution h ∗f using the integral formula h ∗f (t) = Z t 0 h(t −x) f (x) dx (and not with the integrand h(x) f (t −x) ). One or two examples should clarify matters. !◮Example 28.9: Let’s compute sin(t) ∗step3(t) . Since step3 is piecewise deﬁned, we will, as suggested, use the integral formula sin(t) ∗step3(t) = Z t 0 sin(t −x) step3(x) dx First, we compute the integral assuming t < 3 . This one is easy: sin(t) ∗step3(t) = Z t 0 sin(t −x) step3(x) \| {z } = 0 since x<t<3 dx = Z t 0 sin(t −x) · 0 dx = 0 . So, sin(t) ∗step3(t) = 0 if t < 3 . (28.7) On the other hand, if 3 < t , then the interval of integration includes x = 3 , the point at which the value of step3(x) radically changes from 0 to 1. Thus, we must break up our integral at the point x = 3 in computing h ∗f : sin(t) ∗step3(t) = Z t 0 sin(t −x) step3(x) dx = Z 3 0 sin(t −x) step3(x) \| {z } = 0 since x<3 dx + Z t 3 sin(t −x) step3(x) \| {z } = 1 since 3<x dx = Z 3 0 sin(t −x) · 0 dx + Z t 3 sin(t −x) · 1 dx Convolution with Piecewise-Deﬁned Functions 569 = 0 + cos(t −t) −cos(t −3) = 1 −cos(t −3) . Thus, sin(t) ∗step3(t) = 1 −cos(t −3) if 3 < t . (28.8) Combining our two results (formulas (28.7) and (28.7)), we have the complete set of conditional formulas for our convolution, sin(t) ∗step3(t) = ( 0 if t < 3 1 −cos(t −3) if 3 < t . Glance back at the above example and observe that, immediately after the computation of sin(t) ∗step3(t) for each different case ( t < 3 and 3 < t ), the resulting formula for the convolution was rewritten along with the values assumed for t (formulas (28.7) and (28.7), respectively). Do the same in your own computations! Always rewrite any derived formula for your convolution along with the values assumed for t . And write this someplace safe where you can easily ﬁnd it. This is part of the bookkeeping, and helps ensure that you do not lose parts of your work when you compose the full set of conditional formulas for the convolution. One more example should be quite enough. !◮Example 28.10: Let’s compute e−3t ∗f (t) where f (t) =        t if t < 2 2 if 2 < t < 4 0 if 4 < t . For this convolution, we can do a little “pre-computing” to simplify later steps: e−3t ∗f (t) = Z t 0 e−3(t−x) f (x) dx = Z t 0 e−3t+3x f (x) dx = e−3t Z t 0 e3x f (x) dx . Now, if t < 2 , e−3t ∗f (t) = e−3t Z t 0 e3x f (x) \|{z} = x since x < t < 2 dx = e−3t Z t 0 e3xx dx . This integral is easily computed using integration by parts, yielding e−3t ∗f (t) = e−3t t 3e3t −1 9e3t + 1 9 = 1 9 3t −1 + e−3t . Thus, e−3t ∗f (t) = 1 9 3t −1 + e−3t . (28.9) 570 Piecewise-Deﬁned and Periodic Functions On the other hand, when 2 < t < 4 , e−3t ∗f (t) = e−3t Z t 0 e3x f (x) dx = e−3t Z 2 0 e3x f (x) \|{z} = x since x < 2 dx + Z t 2 e3x f (x) \|{z} = 2 since 2 < x < t < 4 dx = e−3t Z 2 0 e3xx dx + Z t 2 e3x · 2 dx = · · · = e−3t 1 9 5e6 + 1 + 2 3 e3t −e6 = · · · = 2 3 + 1 9 1 −e6 e−3t . Thus, e−3t ∗f (t) = 2 3 + 1 9 1 −e6 e−3t for 2 < t < 4 . (28.10) Finally, when 6 < t , e−3t ∗f (t) = e−3t Z t 0 e3x f (x) dx = e−3t Z 2 0 e3x f (x) \|{z} = x since x < 2 dx + Z 4 2 e3x f (x) \|{z} = 2 since 2 < x < 4 dx + Z t 4 e3x f (x) \|{z} = 0 since 4 < x dx = e−3t Z 2 0 e3xx dx + Z 4 2 e3x · 2 dx + Z t 4 e3x · 0 dx = · · · = e−3t 1 9 5e6 + 1 + 2 3 e12 −e6 + 0 = · · · = 1 9 6e12 + 1 −e6 e−3t . Thus, e−3t ∗f (t) = 1 9 6e12 + 1 −e6 e−3t for 4 < t . (28.11) Putting it all together, equations (28.9), (28.10) and (28.11) give us e−3t ∗f (t) =            1 9 3t −1 + e−3t if t < 2 2 3 + 1 9 1 −e6 e−3t if 2 < t < 4 1 9 6e12 + 1 −e6 e−3t if 4 < t . Periodic Functions 571 (a) (b) T T 1 1 1 1 2 2 3 3 4 4 Figure 28.5: Two periodic functions: (a) a basic saw function, and (b) a basic square wave function. 28.5 Periodic Functions Basics Often, a function of interest f is periodic with period p for some positive value p . This means that the graph of the function remains unchanged when shifted to the left or right by p . This is equivalent to saying f (t + p) = f (t) for all t . (28.12) You are well-acquainted with several periodic functions — the trigonometric functions, for example. In particular, the basic sine and cosine functions sin(t) and cos(t) are periodic with period p = 2π . But other periodic functions, such as the “saw” function sketched in ﬁgure 28.5a and the “square-wave” function sketched in ﬁgure 28.5b can arise in applications. Strictly speaking, a truly periodic function is deﬁned on the entire real line, (−∞, ∞) . For our purposes, though, it will sufﬁce to have f “periodic on (0, ∞) ” with period p . This simply means that f is that part of a periodic function along the positive T–axis. What f (t) is for t < 0 is irrelevant. Accordingly, for functions periodic on (0, ∞) , we modify requirement (28.12) to f (t + p) = f (t) for all t > 0 . (28.13) In what follows, however, it will usually be irrelevant as to whether a given function is truly periodic or merely periodic on (0, ∞) , In either case, we will refer to the function as “periodic” , and specify whether it is deﬁned on all of (−∞, ∞) or just (0, −∞) only if necessary. A convenient way to describe a periodic function f with period p is by f (t) = ( f0(t) if 0 < t < p f (t + p) in general . The f0(t) is the formula for f over the base period interval (0, p) . The second line is simply telling us that the function is periodic and that equation (28.12) or (28.13) holds and can be used to compute the function at points outside of the base period interval. (The value of f (t) at t = 0 and integral multiples of p are determined — or ignored — following the conventions for piecewise-deﬁned functions discussed in section 28.1.) 572 Piecewise-Deﬁned and Periodic Functions !◮Example 28.11: Let saw(t) denote the basic saw function sketched in ﬁgure 28.5a. It clearly has period p = 1 , has jump discontinuities at integer values of t , and is given on (0, ∞) by saw(t) = ( t if 0 < t < 1 saw(t + 1) in general . In this case, the formula for computing saw(τ) when 0 < τ < 1 is saw0(τ) = τ . So, for example, saw 3/4 = 3/4 . On the other hand, to compute saw(τ) when τ > 1 (and not an integer), we must use saw(t + 1) = saw(t) repeatedly until we ﬁnally reach in a value t in the base period interval (0, 1) . For example, saw 8 3 = saw 5 3 + 1 = saw 5 3 = saw 2 3 + 1 = saw 2 3 = 2 3 . Often, the formula for the function over the base period interval is, itself, piecewise deﬁned. !◮Example 28.12: Let sqwave(t) denote the square-wave function in ﬁgure 28.5b. This function has period p = 2 , and, over its base period interval (0, 2) , is given by sqwave(t) = ( 1 if 0 < t < 1 0 if 1 < t < 2 . So, sqwave(t) =        1 if 0 < t < 1 0 if 1 < t < 2 sqwave(t −2) in general . Before discussing Laplace transforms of periodic functions, let’s make a couple of observa-tions concerning a function f which is periodic with period p over (0, ∞) . We won’t prove them. Instead, you should think about why these statements are “obviously true” . 1. If f is piecewise continuous over (0, p) , then f is piecewise continuous over (0, ∞) . 2. If f is piecewise continuous over (0, p) , then f is of exponential order s0 = 0 . Transforms of Periodic Functions Suppose we want to ﬁnd the Laplace transform F(s) = L[ f (t)]\|s = Z ∞ 0 f (t)e−st dt Periodic Functions 573 when f is piecewise continuous and periodic with period p . Because f (t) satisﬁes f (t) = f (t + p) for t > 0 , we should expect to (possibly) simplify our computations by partitioning the integral of the transform into integrals over subintervals of length p , F(s) = Z ∞ 0 f (t)e−st dt = Z p 0 f (t)e−st dt + Z 2p p f (t)e−st dt + Z 3p 2p f (t)e−st dt + Z 4p 3p f (t)e−st dt + Z 5p 4p f (t)e−st dt + · · · . For brevity, let’s rewrite this as F(s) = ∞ X k=0 Z (k+1)p kp f (t)e−st dt . (28.14) Now consider using the substitution τ = t −kp in the kth term of this summation. Then t = τ + kp , e−st = e−s(τ+kp) = e−kpse−sτ , and, by the periodicity of f , f (τ + p) = f (τ) f (τ + 2p) = f ([τ + p] + p) = f (τ + p) = f (τ) f (τ + 3p) = f ([τ + 2p] + p) = f (τ + 2p) = f (τ) . . . f (τ + kp) = · · · = f (τ) . So, Z (k+1)p t=kp f (t)e−st dt = Z (k+1)p−kp τ=kp−kp f (τ + kp)e−s(τ+kp) dt = Z p 0 f (τ)e−kpse−sτ dt = e−kps Z p 0 f (τ)e−sτ dt . Note that the last integral does not depend on k . Consequently, combining the last result with equation (28.14), we have F(s) = ∞ X k=0 e−kps Z p 0 f (τ)e−sτ dt = " ∞ X k=0 e−kps # Z p 0 f (τ)e−sτ dτ . Here we have an incredible stroke of luck, at least if you recall what a geometric series is and how to compute its sum. Assuming you do recall this, we have ∞ X k=0 e−kps = ∞ X k=0 e−psk = 1 1 −e−ps . (28.15) 574 Piecewise-Deﬁned and Periodic Functions We also have this if you do not recall about geometric series, but will you certainly want to go to the addendum on page 576 to see how we get this equation. Whether or not you recall about geometric series, equation (28.15) combined with the last formula for F (along with the observations made earlier regarding piecewise continuity and periodic functions) gives us the following theorem. Theorem 28.2 Let f be apiecewise continuousandperiodic functionwithperiod p . Thenits Laplace transform F is given by F(s) = F0(s) 1 −e−ps for s > 0 where F0(s) = Z p 0 f (t)e−st dt . There are at least two alternative ways of describing F0 in the above theorem. First of all, if f is given by f (t) = ( f0(t) if 0 < t < p f (t + p) in general , then, of course, F0(s) = Z p 0 f0(t)e−st dt . Also, using the fact that Z p 0 f0(t)e−st dt = Z ∞ 0 f0(t) rect(0,p)(t) e−st dt , we see that F0(s) = L f0(t) rect(0,p)(t) s or, equivalently, that F0(s) = L f (t) rect(0,p)(t) s . Whether any of alternative descriptions of F0(s) is useful may depend on what transforms you have already computed. !◮Example 28.13: Let’s ﬁnd the Laplace transform of the saw function from example 28.11 and sketched in ﬁgure 28.5a, saw(t) = ( t if 0 < t < 1 saw(t + 1) in general . Here, p = 1 , and the last theorem tells us that L[saw(t)]\|s = F0(s) 1 −e−1·s = F0(s) 1 −e−s for s > 0 Periodic Functions 575 where (using each of the formulas discussed for F0 ) F0(s) = Z 1 0 saw(t)e−st dt (28.16a) = Z 1 0 te−st dt (28.16b) = L t rect(0,1)(t) s . (28.16c) Had the author been sufﬁciently clever, L t rect(0,1)(t) would have already been computed in a previous example, and we could write out the ﬁnal result using formula (28.16c). But he wasn’t, so let’s just compute F0(s) using formula (28.16b) and integration by parts: F0(s) = Z 1 0 te−st dt = −t s e−st 1 t=0 − Z 1 0 −1 s e−st dt = −1 s e−s·1 + 0 − 1 s2 e−s·1 −e−s·0 = 1 s2 1 −e−s −se−s . Hence, L[saw(t)]\|s = F0(s) 1 −e−s = 1 s2 · 1 −e−s −se−s 1 −e−s = 1 s2 1 − se−s 1 −e−s = 1 s2 −1 s · e−s 1 −e−s . This is our transform. If you wish, you can apply a little algebra and ‘simplify’ it to L[saw(t)]\|s = 1 s2 −1 s · 1 es −1 , though you may prefer to keep the formula with 1 −e−ps in the denominator to remind you that this transform came from a periodic function with period p . Just for fun, let’s go even further using the fact that e−s 1 −e−s = e−s 1 −e−s · 2es/2 2es/2 = 1 2 · 2e−s/2 es/2 −e−s/2 = 1 2 · e−s/2 sinh(s/2) . Thus, the above formula for the Laplace transform of the saw function can also be written as L[saw(t)]\|s = 1 s2 − 1 2s · e−s/2 sinh(s/2) . This is signiﬁcant only in that it demonstrates why hyperbolic trigonometric functions are sometimes found in tables of transforms. 576 Piecewise-Deﬁned and Periodic Functions Addendum: Verifying Equation (28.15) Equation (28.15) gives a formula for adding up ∞ X k=0 e−kps assuming p and s are positive values. To derive that formula, we start with the N th partial sum of the series, SN = N X k=0 e−kps = e−0ps \| {z } =1 + e−1ps + e−2ps + e−3ps + · · · + e−Nps . Multiplying this by e−ps , we get e−psSN = e−ps 1 + e−1ps + e−2ps + e−3ps + · · · + e−Nps = e−1ps + e−2ps + e−3ps + · · · + e−Nps + e−(N+1)ps . The similarity between SN and e−psSN naturally leads us to compute their difference, 1 −e−ps SN = SN −e−psSN = 1 + e−1ps + e−2ps + e−3ps + · · · + e−Nps − e−1ps + e−2ps + e−3ps + · · · + e−Nps + e−(N+1)ps = 1 −e−(N+1)ps . Dividing through by 1 −e−ps then yields SN = 1 −e−(N+1)ps 1 −e−ps . The above formula for SN holds for any choice of p and s , but if p > 0 and s > 0 , as assumed here, then lim N→∞e−(N+1)ps = 0 , and thus, ∞ X k=0 e−kps = lim N→∞ N X k=0 e−kps = lim N→∞SN = lim N→∞ 1 −e−(N+1)ps 1 −e−ps = 1 −0 1 −e−ps = 1 1 −e−ps , conﬁrming equation (28.15). An Expanded Table of Identities 577 Table 28.1: Commonly Used Identities (Version 2) In the following, F(s) = L[ f (t)]\|s . h(t) H(s) = L[h(t)]\|s Restrictions f (t) Z ∞ 0 f (t)e−st dt eαt f (t) F(s −α) α is real f (t −α) stepα(t) F(s) e−αs α > 0 d f dt sF(s) −f (0) d2 f dt2 s2F(s) −s f (0) −f ′(0) dn f dtn snF(s) −sn−1 f (0) −sn−2 f ′(0) −sn−3 f ′′(0) −· · · −f (n−1)(0) n = 1, 2, 3, . . . t f (t) −dF ds tn f (t) (−1)n dn F dsn n = 1, 2, 3, . . . Z t 0 f (τ) dτ F(s) s f (t) t Z ∞ s F(σ) dσ f ∗g(t) F(s)G(s) f is periodic with period p R p 0 f (t)e−st dt 1 −e−ps 28.6 An Expanded Table of Identities For reference, let us write out a new table of Laplace transform identities containing the identities listed in our ﬁrst table of Laplace transform identities, table 25.1 on page 514, along with some of the more important identities derived after making that table. Our new table is table 28.1. 578 Piecewise-Deﬁned and Periodic Functions y(t) Y m 0 f Figure 28.6: A mass/spring system with mass m and an outside force f acting on the mass. 28.7 Duhamel’s Principle and Resonance The Problem Now is a good time to re-examine some of those “forced” mass/spring systems originally dis-cussed in chapters 17 and 22, and diagramed in ﬁgure 28.6. Recall that this system is modeled by m d2y dt2 + γ dy dt + κy = f where y = y(t) is the position of the mass at time t (with y = 0 being the “equilibrium” position of the mass when f = 0 ), m is the mass of the object attached to the spring, κ is the spring constant, γ is the damping constant, and f = f (t) is the sum of all forces acting on the spring other than the damping friction and the spring’s reaction to being stretched and compressed ( f was called Fother in chapter 17 and F in chapter 22). Remember, also, that m and κ are positive constants. Our main interest will be in the phenomenon of resonance in an undamped system. Accord-ingly, we will assume γ = 0 , and restrict our attention to solving m d2y dt2 + κy = f . (28.17) Ultimately, we will further restrict our attention to cases in which f is periodic. But let’s wait on that, and derive some basic formulas without assuming this periodicity. Solutions Using Arbitrary f The General Solution As you know quite well by now, the general solution to our differential equation,equation (28.17), is y(t) = yp(t) + yh(t) where yh is the general solution to the corresponding homogeneous differential equation, and yp is any particular solution to the given nonhomogeneous differential equation. The formula for yh is already known. In chapter 17, we found that yh(t) = c1 cos(ω0t) + c2 sin(ω0t) where ω0 = r κ m . Duhamel’s Principle and Resonance 579 Recall that ω0 is the natural angular frequency of the mass/spring system, and is related to the system’s natural frequency ν0 and natural period p0 via ν0 = ω0 2π and p0 = 1 ν0 = 2π ω0 . For future use, note that yh is a periodic function with period p0 ; hence yh(t + p0) −yh(t) = 0 for all t . That leaves ﬁnding a particular solution yp . Let’s take this to be the solution to the initial-value problem m d2y dt2 + κy = f with y(0) = 0 and y′(0) = 0 . This is easily found by either applying the Laplace transform and using the convolution identity in taking the inverse transform, or by appealing directly to Duhamel’s principle. Either way, we get yp(t) = h ∗f (t) = Z t 0 h(t −x) f (x) dx where h(τ) = L−1 1 ms2 + κ τ = 1 m L−1 1 s2 + κ/m τ . Since ω0 = √κ/m , h(τ) = 1 m L−1 1 s2 + (ω0)2 τ = 1 ω0m sin(ω0τ) . Thus, the above integral formula for yp can be written as yp(t) = 1 ω0m Z t 0 sin(ω0[t −x]) f (x) dx . (28.18) The Difference Formula and First Theorem For our studies, we will want to see how any solution y varies “over a cycle” (i.e., as t increases by p0 ). This variance in y over a cycle is given by the difference y(t + p0) −y(t) , and will be especially meaningful when the forcing function is periodic with period p0 . For now, let’s consider the difference y(t + p0)−y(t) assuming y = yp+yh is any solution to our differential equation. Of course, the yh term is irrelevant because of its periodicity, y(t + p0) −y(t) = yp(t + p0) + yh(t + p0) − yp(t) + yh(t) = yp(t + p0) −yp(t) + yh(t + p0) −yh(t) \| {z } 0 . Now, using formula (28.18) for yp , we see that yp(t + p0) = 1 ω0m Z t+p0 0 sin(ω0[(t + p0) −x]) f (x) dx = 1 ω0m Z t+p0 0 sin(ω0[t −x] + ω0 p0 \|{z} 2π ) f (x) dx 580 Piecewise-Deﬁned and Periodic Functions = 1 ω0m Z t+p0 0 sin(ω0[t −x]) f (x) dx = 1 ω0m Z t 0 sin(ω0[t −x]) f (x) dx + 1 ω0m Z t+p0 t sin(ω0[t −x]) f (x) dx . But the ﬁrst integral in the last line is simply the integral formula for yp(t) given in equation (28.18). So the above reduces to y(t + p0) = y(t) + 1 ω0m Z t+p0 t sin(ω0[t −x]) f (x) dx . (28.19) To further “reduce” our difference formula, let us use a well-known trigonometric identity: Z t+p0 t sin(ω0[t −x]) f (x) dx = Z t+p0 t sin(ω0t −ω0x) f (x) dx = Z t+p0 t [sin(ω0t) cos(ω0x) −cos(ω0t) sin(ω0x)] f (x) dx = sin(ω0t) Z t+p0 t cos(ω0x) f (x) dx −cos(ω0t) Z t+p0 t sin(ω0x) f (x) dx . Combining this result with the last equation for y(t + p0) and recalling the previous results derived in this section then yields: Theorem 28.3 Let m and κ bepositiveconstants, andlet f beanypiecewisecontinuousfunctionofexponential order. Then, the general solution to m d2y dt2 + κy = f is y(t) = yp(t) + c1 cos(ω0t) + c2 sin(ω0t) where ω0 = r κ m and yp(t) = 1 ω0m Z t 0 sin(ω0[t −x]) f (x) dx . Moreover, y(t + p0) −y(t) = 1 ω0m [IS(t) sin(ω0t) + IC(t) cos(ωt)] for t ≥0 where IS(t) = Z t+p0 t cos(ω0x) f (x) dx and IC(t) = − Z t+p0 t sin(ω0x) f (x) dx . Duhamel’s Principle and Resonance 581 T 0 p0 p0 p0 a a + p0 Figure 28.7: Illustration for lemma 28.4. Resonance from Periodic Forcing Functions A Useful Fact Take a look at ﬁgure 28.7. It shows the graph of some periodic function g with period p0 , and with two regions of width p0 “greyed in” in two shades of grey. The darker grey region is between the graph and the T –axis with 0 < t < p0 . The lighter grey region is between the graph and the T –axis with a < t < a + p0 for some real number a . Note the similarity in the shapes of the regions. In particular, note how the pieces of the lighter grey region can be rearranged to perfectly match the darker grey region. Consequently, the areas in each of these two regions, both above and below the T –axis, are the same. Add to this the relationship between “integrals” and “area” , and you get the useful fact stated in the next lemma. Lemma 28.4 Let g be a periodic, piecewise continuous function with period p0 . Then, for any t , Z t+p0 t g(x) dx = Z p0 0 g(x) dx . If you wish, you can rigorously prove this lemma using some basic theory from elementary calculus. ?◮Exercise 28.1: Prove lemma 28.4. A good start would be to show that d dt Z t+p0 t g(x) dx = 0 . Resonance Now consider the formulas for IS(t) and IC(t) from theorem 28.3, IS(t) = Z t+p0 t cos(ω0x) f (x) dx and IC(t) = − Z t+p0 t sin(ω0x) f (x) dx . If f is also periodic with period p0 , then the products in these integrals are also periodic, each with period p0 . Lemma 28.4 then tells us that IS(t) = Z t+p0 t cos(ω0x) f (x) dx = Z p0 0 cos(ω0x) f (x) dx 582 Piecewise-Deﬁned and Periodic Functions and IC(t) = − Z t+p0 t sin(ω0x) f (x) dx = − Z p0 0 sin(ω0x) f (x) dx . Thus, if f is periodic with period p0 , the difference formula in theorem 28.3 reduces to y(t + p0) −y(t) = 1 ω0m [IS sin(ω0t) + IC cos(ωt)] where IS and IC are the constants IS = Z p0 0 cos(ω0x) f (x) dx and IC = − Z p0 0 sin(ω0x) f (x) dx . Using a little more trigonometry (see the derivation of formula (17.8b) on page 363), we can reduce this to the even more convenient form given in the the next theorem. Theorem 28.5 (resonance in undamped systems) Let m and p be positive constants, and f a periodic piecewise continuous function. Assume further that f has period p0 , the natural period of the mass/spring system modeled by m d2y dt2 + κy = f . That is, period of f = p0 = 2π ω0 with ω0 = r κ m . Also let IS = Z p0 0 cos(ω0x) f (x) dx and IC = − Z p0 0 sin(ω0x) f (x) dx . Then, for any solution y to the above differential equation, and any t > 0 , y(t + p0) −y(t) = A cos(ω0t −φ) (28.20) where A = 1 ω0m q (IS)2 + (IC)2 and with φ being the constant satisfying 0 ≤φ < 2π , cos(φ) = IC p (IS)2 + (IC)2 and sin(φ) = IS p (IS)2 + (IC)2 . To see what all this implies, assume f , y , etc. are as in the theorem, and look at what the difference formula tells us about y(tn) when τ is any ﬁxed value in [0, p0) , and tn = τ + np0 for n = 1, 2, 3, . . . . The value of y(τ) can be computed using the integral formula for yp in theorem 28.3. To compute each y(tn) , however, it is easier to use this computed value for y(τ) along with difference formula (28.20) and the fact that, for any integer k , cos(ω0tk −φ) = cos(ω0[τ + kp0] −φ) = cos(ω0τ −φ + k ω0 p0 \|{z} 2π ) = cos(ω0τ −φ) . Duhamel’s Principle and Resonance 583 Doing so, we get y(t1) = y(τ + p0) = y(τ) + A cos(ω0τ −φ) , y(t2) = y(τ + 2p0) = y(τ + p0) + A cos(ω0[τ + p0] −φ) = [y(τ) + A cos(ω0t −φ)] + A cos(ω0τ −φ) = y(τ) + 2A cos(ω0τ −φ) , y(t3) = y(τ + 3p0) = y(τ + 2p0) + A cos(ω0[τ + 2p0] −φ) = [y(τ) + 2A cos(ω0t −φ)] + A cos(ω0τ −φ) = y(τ) + 3A cos(ω0τ −φ) , and so on. In general, y(tn) = y(τ) + nA cos(ω0τ −φ) . (28.21) Clearly, if A ̸= 0 and ω0τ −φ is neither π/2 or 3π/2 , then y(tn) →±∞ as n →∞ . This is clearly “runaway resonance” . Thus, it is the A in difference formula (28.20) that determines if we have “runaway reso-nance” . If A ̸= 0 , the solution contains an oscillating term with a steadily increasing amplitude. On the other hand, if A = 0 , then the solution y is periodic, and does not “blow up” . By the way, for graphing purposes it may be convenient to use the periodicity of the cosine term and rewrite equation (28.21) as y(tn) = y(τ) + nA cos(ω0tn −φ) . Replacing tn with t , and recalling what n and τ represent, we see that this is the same as saying y(t) = y(τ) + nA cos(ω0t −φ) (28.22) where n is the largest integer such that np0 ≤t and τ = t −np0 . !◮Example 28.14: Let us use the theorems in this section to analyze the response of an undamped mass/spring system with natural period p0 = 1 to a force f given by the basic saw function sketched in ﬁgure 28.8a, f (t) = saw(t) = ( t if 0 < t < 1 saw(t −1) if 1 < t . The corresponding natural angular frequency is ω0 = 2π p0 = 2π . The actual values of the mass m and spring constant κ are irrelevant provided they satisfy 2π = ω0 = r κ m . 584 Piecewise-Deﬁned and Periodic Functions (a) (b) T T 1 1 1 2 2 3 3 4 4 5 Y Figure 28.8: (a) A basic saw function, and (b) the corresponding response of an undamped mass/spring system with natural period 1 over 6 cycles. Also, since the solution to the corresponding homogeneous differential equation was pretty much irrelevant in the discussion leading to our last theorem, let’s assume our solution satisﬁes y(0) = 0 and y′(0) = 0 , so that the solution formula described in theorem 28.3 becomes y(t) = yp(t) = 1 2πm Z t 0 sin(ω0[t −x]) saw(x) dx . In particular, if 0 ≤x ≤t < 1 , then saw(x) = x and we can complete our computations of y(t) using integration by parts: y(t) = 1 2πm Z t 0 sin(2π[t −x]) x dx = 1 2πm x 2π cos(2π[t −x]) t x=0 − Z t 0 1 2π cos(2π[t −x]) dx = 1 2πm t 2π cos(2π[t −t]) − 0 2π cos(2π[t −0]) + 1 (2π)2 sin(2π[t −t]) − 1 (2π)2 sin(2π[t −0]) . This simpliﬁes to y(t) = 1 8π3m [2πt −sin(2πt)] when 0 ≤t < 1 . (28.23) In a similar manner, we ﬁnd that IS = Z p0 0 cos(ω0x) f (x) dx = Z 1 0 cos(2πx) x dx = · · · = 0 and IC = − Z p0 0 sin(ω0x) f (x) dx = − Z 1 0 sin(2πx) x dx = · · · = 1 2π . Thus, A = 1 2πm q (IS)2 + (IC)2 = 1 4π2m . Since A ̸= 0 , we have resonance. There is an oscillatory term whose amplitude steadily increases as t increases. Additional Exercises 585 To actually graph our solution, we still need to ﬁnd the phase, φ , which (according to our last theorem) is the value in [0, 2π) such that cos(φ) = IC p (IS)2 + (IC)2 = 1 and sin(φ) = IS p (IS)2 + (IC)2 = 0 . Clearly φ = 0 . So let t ≥0 . Then, employing formula (28.22) (derived just before this example), y(t) = y(τ) + nA cos(ω0t −φ) = 1 8π3m [2πτ −sin(2πτ)] + n 4π2m cos(2πt) where (since p0 = 1 ) n is the largest integer with n ≤t and τ = t −n . This is the function graphed in ﬁgure 28.8b. Additional Exercises 28.2. Using the ﬁrst translation identity or one of the differentiation identities, compute each of the following: a. L e4t step6(t) s b. L t step6(t) s 28.3. Compute (using the translation along the T –axis identity) and then graph the inverse transforms of the following functions: a. e−4s s3 b. e−3s s + 2 c. √πs−3/2e−s d. π s2 + π2 e−2s e. e−4s (s −5)3 f. (s + 2)e−5s (s + 2)2 + 16 28.4. Finish solving the differential equation in example 28.1. 28.5. Compute and then graph the inverse transforms of the following functions (express your answers as conditional sets of formulas): a. 1 −e−s s2 b. e−s + e−3s s c. 2 s3 −2 + 4s s3 e−2s d. π 1 + e−s s2 + π2 e. (s + 4)e−12 −8e−3s s2 −16 f. e−2s −2e−4s + e−6s s2 28.6. Find and graph the solution to each of the following initial-value problems: a. y′ = step3(t) with y(0) = 0 b. y′ = step3(t) with y(0) = 4 c. y′′ = step2(t) with y(0) = 0 and y′(0) = 0 d. y′′ = step2(t) with y(0) = 4 and y′(0) = 6 e. y′′ + 9y = step10(t) with y(0) = 0 and y′(0) = 0 586 Piecewise-Deﬁned and Periodic Functions 28.7. Compute the Laplace transforms of the following functions using the translation along the T –axis identity. (Trigonometric identities may also be useful for some of these.) a. f (t) = ( 0 if t < 6 e4t if 6 < t b. g(t) =    0 if t < 4 1 √ t −4 if 4 < t c. t step6(t) d. te3t step2(t) e. t2 step6(t) f. sin(2(t −1)) step1(t) g. sin(2t) stepπ/2(t) h. sin(2t) stepπ/4(t) i. sin(2t) stepπ/6(t) 28.8. For each of the following choices of f : i. Graph the given function over the positive T –axis. ii. Rewrite the function in terms of appropriate rectangle functions,and then rewrite that in terms of appropriate step functions. iii. Then ﬁnd the Laplace transform F(s) = L[ f (t)]\|s . a. f (t) = ( e−4t if t < 6 0 if 6 < t b. f (t) = ( 2t −t2 if t < 2 0 if 2 < t c. f (t) = ( 2 if t < 3 2e−4(t−3) if 3 < t d. f (t) = ( sin(πt) if t < 1 0 if 1 < t e. f (t) = ( t2 if t < 3 9 if 3 < t f. f (t) =        0 if t < 2 3 if 2 < t < 4 0 if 4 < t g. f (t) =        1 if t < 1 2 if 2 < t < 3 4 if 3 < t h. f (t) =        0 if t < 1 sin(πt) if 1 < t < 2 0 if 2 < t i. f (t) =        0 if t ≤1 (t −1)2 if 1 < t < 3 4 if 3 ≤t j. f (t) =        t if t ≤2 4 −t if 2 < t < 4 0 if 3 ≤t 28.9. The inﬁnite stair function, stair(t) , can be described in terms of rectangle functions by stair(t) = ∞ X n=0 (n + 1) rect(n,n+1)(t) = 1 rect(0,1)(t) + 2 rect(1,2)(t) + 3 rect(2,3)(t) + 4 rect(3,4)(t) + · · · . Additional Exercises 587 Using this: a. Sketch the graph of stair(t) over the positive T –axis, and rewrite the formula for stair(t) in terms of step functions. b. Assuming the linearity of the Laplace transform holds for inﬁnite sums as well as ﬁnite sums, ﬁnd an inﬁnite sum formula for L[stair(t)]\|s . c. Recall the formula for the sum of a geometric series, ∞ X n=0 Xn = 1 1 −X when \|X\| < 1 . Using this, simplify the inﬁnite sum formula for L[stair(t)]\|s which you (we hope) obtained in the previous part of this exercise. 28.10. Find and graph the solution to each of the following initial-value problems: a. y′ = rect(1,3)(t) with y(0) = 0 b. y′′ = rect(1,3)(t) with y(0) = 0 and y′(0) = 0 c. y′′ + 9y = rect(1,3)(t) with y(0) = 0 and y′(0) = 0 28.11. Compute each of the following convolutions: a. t2 ∗step3(t) b. 1 ∗step4(t) c. cos(t) ∗rect(0,π)(t) d. 2e−2t ∗rect(1,3)(t) e. e−2t ∗ e5t rect(1,3)(t) f. sin(t) ∗ sin(t) rect(2π,3π)(t) g. t ∗f (t) where f (t) = ( √ t if 0 ≤t < 4 2 if 4 < t h. sin(t) ∗f (t) where f (t) =        1 if t < 2π cos(t) if 2π < t < 3π −1 if 3π < t 28.12. Each function listed below is at least periodic on (0, ∞) . Sketch graph of each, and then ﬁnd its Laplace transform using the methods developed in section 28.5. a. f (t) = ( e−2t if 0 < t < 3 f (t −3) if t > 3 b. f (t) = sqwave(t) (from example 28.12) c. f (t) =        1 if 0 < t < 1 −1 if 1 < t < 2 f (t −2) if t > 2 d. f (t) = ( 2t −t2 if t < 2 f (t −2) if 2 < t (see exercise 28.8 a) 588 Piecewise-Deﬁned and Periodic Functions (a) (b) T T 1 1 1 1 2 2 0 3 4 Figure 28.9: (a) The square wave function for exercise 28.13 a, and (b) the rectiﬁed sine function for exercise 28.13 b. e. f (t) =        t if 0 < t < 2 4 −t if 2 < t < 4 f (t −4) if t > 4 (see exercise 28.8 i) f. f (t) = \|sin(t)\| 28.13. In each of the following exercises, you are given the natural period p0 and a forcing function f for an undamped mass/spring system modeled by m d2y dt2 + κy = f . Analyze the corresponding resonance occurring in each system. In particular, let y be any solution to the modeling differential equation and: i. Compute the difference y(t + p0) −y(t) . ii. Compute the formula for y(t) assuming y(0) = 0 and y′(0) = 0 . (Express your answer using τ and n where n is the largest integer such that np0 ≤t and τ = t −np0 .) iii. Using the formula just computed for part ii along with your favorite computer math package, sketch the graph of y over several cycles. (For convenience, assume m is a unit mass.) a. p0 = 2 and f is basic squarewave sketched in ﬁgure 28.9a. That is, f (t) =        1 if 0 < t < 1 0 if 1 < t < 2 f (t + 2) in general . b. p0 = 1 and f (t) = \|sin(πt)\| , the “rectiﬁed sine function” sketched in ﬁgure 28.9b. c. p0 = 1 and f (t) = sin(4πt) .
12325	https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0289870	Outcomes of minimal change disease without nephrotic range proteinuria \| PLOS One Skip to main content Advertisement plos.org Create account Sign in Publish About Browse Searchadvanced search Browse Topics Browse Subject Areas ? Click through the PLOS taxonomy to find articles in your field. For more information about PLOS Subject Areas, click here. 0 Save Total Mendeley and Citeulike bookmarks. 1 Citation Paper's citation count computed by Dimensions. 1,425 View PLOS views and downloads. 0 Share Sum of Facebook, Twitter, Reddit and Wikipedia activity. Open Access Peer-reviewed Research Article Outcomes of minimal change disease without nephrotic range proteinuria Hyung Eun Son,Roles Visualization, Writing – original draft Affiliations Department of Internal Medicine, Chung-Ang University Gwangmyeong Hospital, Gwangmyeong, Korea, Department of Internal Medicine, Chung-Ang University College of Medicine, Seoul, Korea ⨯ Giae Yun,Roles Data curation Affiliation Department of Internal Medicine, Seoul National University Bundang Hospital, Seong-nam, Korea ⨯ Eun-Jeong Kwon,Roles Data curation Affiliation Department of Internal Medicine, Seoul National University Bundang Hospital, Seong-nam, Korea ⨯ Seokwoo Park,Roles Investigation Affiliations Department of Internal Medicine, Seoul National University Bundang Hospital, Seong-nam, Korea, Department of Biomedical Sciences, Seoul National University College of Medicine, Seoul, Korea ⨯ Jong Cheol Jeong,Roles Methodology, Software Affiliation Department of Internal Medicine, Seoul National University Bundang Hospital, Seong-nam, Korea ⨯ Sejoong Kim,Roles Supervision Affiliations Department of Internal Medicine, Seoul National University Bundang Hospital, Seong-nam, Korea, Department of Internal Medicine, Seoul National University College of Medicine, Seoul, Korea ⨯ Ki Young Na,Roles Supervision Affiliations Department of Internal Medicine, Seoul National University Bundang Hospital, Seong-nam, Korea, Department of Internal Medicine, Seoul National University College of Medicine, Seoul, Korea ⨯ Jin Ho Paik,Roles Data curation, Investigation Affiliations Department of Pathology, Seoul National University Bundang Hospital, Seong-nam, Korea, Department of Pathology, Seoul National University College of Medicine, Seoul, Korea ⨯ Ho Jun ChinRoles Conceptualization, Formal analysis, Writing – review & editing E-mail:mednep@hanmail.net Affiliations Department of Internal Medicine, Seoul National University Bundang Hospital, Seong-nam, Korea, Department of Internal Medicine, Seoul National University College of Medicine, Seoul, Korea ⨯ Outcomes of minimal change disease without nephrotic range proteinuria Hyung Eun Son, Giae Yun, Eun-Jeong Kwon, Seokwoo Park, Jong Cheol Jeong, Sejoong Kim, Ki Young Na, Jin Ho Paik, Ho Jun Chin x Published: August 17, 2023 Article Authors Metrics Comments Media Coverage Peer Review Abstract Introduction Materials and methods Results Discussion Supporting information References Reader Comments Figures Abstract Minimal change disease (MCD) is characterized by edema and nephrotic range proteinuria (NS). However, the fate of MCD without nephrotic proteinuria requires elucidation. We retrospectively reviewed 79 adults diagnosed with primary MCD at their initial renal biopsy at a tertiary hospital between May 2003 and June 2017. Clinicopathologic features were compared between patients with and without NS. The frequency of flaring to nephrotic proteinuria and renal outcomes were assessed during follow-up. There were 20 and 59 patients in the Non-NS and NS groups, respectively. The Non-NS group had a lower frequency of acute kidney injury (AKI) during the follow-up period [5.0% vs. 59.3%, p <0.001]. The response rate to steroid treatment was 100% in the Non-NS group and 92.3% in the NS group (p = 1.000). Except for one patient, the Non-NS group was treated with steroids when their proteinuria increased to a nephrotic level. There were no differences in the frequency of the first relapse or the number of relapses among patients with initial remission from nephrotic range proteinuria. At the final visit, the complete remission rate was 73.4%. The estimated glomerular filtration rate during follow-up was significantly better in the NS group than the Non-NS group, given the higher rates of AKI at renal biopsy. The rates of renal events, end-stage renal disease, and mortality did not differ between the groups. Adult MCD patients with nephrotic and non-nephrotic range proteinuria showed similar outcomes. Accordingly, this population must be carefully managed, regardless of the amount of proteinuria at renal biopsy. Figures Citation:Son HE, Yun G, Kwon E-J, Park S, Jeong JC, Kim S, et al. (2023) Outcomes of minimal change disease without nephrotic range proteinuria. PLoS ONE 18(8): e0289870. Editor:Maria Lourdes Gonzalez Suarez, Mayo Clinic Rochester, UNITED STATES Received:September 4, 2022; Accepted:July 27, 2023; Published: August 17, 2023 Copyright: © 2023 Son et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability:All relevant data are within the paper and its Supporting Information files. Funding:The authors received no specific funding for this work. Competing interests: The authors have declared that no competing interests exist. Introduction The primary nephrotic syndrome presents with heavy proteinuria, hypoalbuminemia, hyperlipidemia, and clinical symptoms such as edema or pleural effusion. Chronic glomerulonephritis is reportedly the third highest cause of progression to end-stage renal disease (ESRD) in South Korea . Minimal change disease (MCD) comprises approximately 11–20% of adult primary nephrotic syndrome cases . Among 21,426 adults that underwent a kidney biopsy in South Korea, 9.17% were diagnosed with MCD , making it the third most prevalent primary glomerulonephritis following IgA nephropathy and membranous nephropathy . MCD often presents with abrupt onset of edema and full nephrotic syndrome. Hypertension, microhematuria, and acute renal failure may also develop. Compared with children, a smaller percentage of adults with MCD show features of nephrotic syndrome. Diagnosis of MCD in adults is based on histopathological characteristics, and adequate samples that include more than 10 glomeruli are needed for a 95% chance of detection . In adults, MCD exhibits a relatively large level of proteinuria compared with other primary glomerulonephritis. Until this study, most studies have focused on heavy proteinuria diagnosed with MCD. Therefore, the long-term prognosis of adult MCD without nephrotic syndrome as an initial manifestation is unclear. To fully understand MCD, the prognosis of MCD with typical pathologic findings needs to be defined in terms of clinical characteristics, response rates, and relapse rates in patients with and without nephrotic syndrome. In this study, we compared the clinicopathologic features and outcomes between MCD patients with and without heavy proteinuria for approximately 5 years. Materials and methods Patients We retrospectively reviewed 1,516 adult candidates aged ≥ 18 years who underwent renal biopsy between May 2003 and June 2017 at a single center in South Korea: Seoul National University Bundang Hospital. Renal biopsy was performed using the ultrasonography-guided percutaneous gun biopsy technique. We collected the pathological findings on light, immunofluorescence, and electron microscopy (EM), and all biopsy specimens were initially evaluated by an independent renal pathologist blinded to patients’ outcomes. The detailed histology reports and biopsy slides of patients with MCD were re-reviewed. We defined MCD pathologically as near-normal findings on light microscopy, except for the mild expansion of the mesangium or global glomerulosclerosis, which could also be seen as nonspecific findings. Segmental sclerosis was not seen. There were no complement or immunoglobulin deposits on immunofluorescence microscopy. Diffuse effacement of podocyte foot processes was seen on EM. We excluded patients with biopsy samples that included fewer than 10 glomeruli; patients with possible secondary causes, such as malignancy or lupus nephritis; patients who had immunosuppressive treatment before renal biopsy; patients with fewer than three proteinuria tests after the renal biopsy; and patients followed for < 3 months after the renal biopsy (Fig 1). In total, 79 patients were included in this study. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Selection of patients. MCD: Minimal change disease, SLE: Systemic lupus erythematosus. Data collection The patient’s clinical characteristics, laboratory results, renal pathology results, and medication prescriptions during each follow-up period were collected from their electronic health records (EHRs) from the period of renal biopsy to the final follow-up visit, with the primary query being the patients’ identification number. The baseline data collected during the biopsy included age, sex, systolic and diastolic blood pressures, comorbidities, and medications. Comorbid conditions included a history of hypertension, diabetes, coronary heart disease, and cerebrovascular disease. Hypertension was defined as systolic blood pressure of ≥140 mmHg, diastolic blood pressure of ≥90 mmHg, or taking anti-hypertensive medications. We collected laboratory data, including serum cholesterol levels, glucose, total protein, albumin, hemoglobin, isotope dilution mass spectrometry (IDMS)-traceable creatinine, and spot urine protein to creatinine ratio (UPCR). The estimated glomerular filtration rate (eGFR) was calculated using the Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation . Acute kidney injury (AKI) was defined according to the Kidney Disease Improving Global Outcomes (KDIGO) criteria . Specifically, AKI was diagnosed based on the change in serum creatinine concentration from the lowest value during the follow-up period to the value measured on the biopsy day. The stages of AKI were as follows stage 1, ≥0.3 mg/dL absolute or 1.5- to 2.0-fold relative increase in serum creatinine; stage 2, > 2.0- to 3.0-fold increase in serum creatinine; and stage 3, >3.0-fold increase in serum creatinine or serum creatinine ≥4.0 mg /dL with an acute rise of >0.5 mg/dL. Nephrotic range proteinuria was defined as over 3.0 g/g in a spot urine sample. Complete remission (CR) of proteinuria, relapse of proteinuria, and steroid dependency was defined according to the revised KDIGO guideline for glomerulonephritis (Table 1). Microscopic hematuria was defined as ≥5 erythrocytes/high power field on urine sediment microscopy. ESRD events were collected from the ESRD registry of the Korean Society of Nephrology, and mortality events from our hospital’s database or the Ministry of the Interior and Safety Korea database after merging with our data using patients’ unique identification numbers. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Definitions of MCD courses. Kidney pathology As previously described , pathologic diagnosis contains the following contents. Glomerular lesions such as global sclerosis, segmental sclerosis, glomerular ischemic change, and crescentic changes were reported as a proportion of the total glomeruli in an evaluated specimen. A semi-quantitative assessment of changes in mesangial cellularity, mesangial matrix, tubular atrophy, and interstitial inflammation and fibrosis was performed, with the results classified as normal, mild, moderate, moderate to severe, and severe. Vascular abnormality was defined as the presence of arteriolar hyalinosis and arteriosclerosis. The immunofluorescence study used the classic direct technique with antibodies against eight proteins (IgG, IgM, IgA, C3, C1q, fibrinogen, kappa, and lambda chains). The results were reported semi-quantitatively as negative (0), trace (0.5), and 1–3 positive (1–3). After EM analysis was done in all biopsy samples, findings on EM were described as follows; the presence of electron-dense deposits in the area of the mesangium, subendothelium, and subepithelium; and severity of foot process effacement of the podocytes, which were reported as none, focal (mild, moderate, moderate to severe, severe), or diffuse. Outcomes Renal events were defined as any decrease in eGFR by more than 50% during a follow-up visit compared with that at renal biopsy, eGFR <15 ml/min/1.73 m 2, or development of ESRD during the follow-up period. The slope of GFR change per year (ml/min/1.73 m 2/year) was defined as the difference between eGFR measured on the day of the biopsy and the last measurement, divided by the number of years of follow-up. We compared the achievement of CR, the number of relapses of proteinuria after remission, the slope of eGFR change during the follow-up period, steroid dependency rates, and the development of renal events such as AKI, ESRD, and mortality between the groups. Exposure Patients were divided into two groups according to the UPCR measured on the day of the renal biopsy. Patients with UPCR ≥3.0 g/g were classified as the nephrotic group (NS group), and those with UPCR <3.0 g/g were classified as the non-nephrotic group (non-NS group). Statistical analysis Data are presented as mean ± standard deviation (SD) for continuous variables and number (percentages) for categorical variables. Continuous variables were compared using the Mann-Whitney U test, and categorical variables using Pearson’s Chi-square test or Fisher’s Exact test according to the number of cells. The relationship between variables was assessed by linear and logistic regression tests for continuous and dichotomized variables. Kaplan-Meier analysis was used to evaluate survival curves for CR, relapse of proteinuria, renal events, ESRD, and mortality. Risk factors for CR, relapse of proteinuria, renal events, ESRD, or mortality were identified using adjusted Cox proportional models. Statistical significance was set at a p<0.05. All the analyses were performed using SPSS Statistics for Macintosh, Version 22 (IBM, Armonk, NY) and Stata software, Version 17 (Stata Corp LLC., College Station, TX). Ethical approval This study was approved by the institutional review board (IRB) of Seoul National University Bundang Hospital (IRB approval No. B-1910-572-304). The IRB waived written consent because of the study’s retrospective nature. All data were anonymized entirely before access to the database. Results Baseline characteristics Among the 79 patients included, the mean age at renal biopsy was 53.7 ± 19.2 (range: 18.5–99.0) years, and there were 38 men (48.1%). The highest level of UPCR within the 6 months before the renal biopsy was 9.04 ± 6.34 (range: 0.38–36.66) g/g creatinine. There were 16 patients (20.3%) with a pre-biopsy UPCR <3.00 g/g creatinine, among whom two exhibited UPCRs <0.30 g/g creatinine at admission for renal biopsy. Another patient with nephrotic range UPCR before renal biopsy (1/63) also had a UPCR <0.30 g/g creatinine at admission. This suggests that spontaneous remission of proteinuria without immunosuppressive therapy occurred in three patients during the 6 months wait for renal biopsy. The patients were divided into Non-NS (n = 20, 25.3%) and NS-groups (n = 59, 74.7%) according to the presence of proteinuria ≥ 3g/g on the day of the biopsy. The groups’ demographics and clinic-pathologic findings are summarized in Table 2. The groups had no differences in demographic features, underlying medical illnesses, and blood pressures at renal biopsy. However, there was a difference in the degree of proteinuria. UPCR within 6 months before renal biopsy was 11.22 ± 6.25 g/g creatinine in the NS group and 2.63 ± 1.92 g/g creatinine in the Non-NS group (p <0.001). The UPCR at renal biopsy was also higher in the NS group (10.16 ± 6.21 vs. 1.36 ± 1.00 g/g creatinine, p <0.001). The eGFR was lower (75 ± 36 vs. 98 ± 27 ml/min/1.73 m 2, p = 0.012), and the prevalence of AKI at admission for renal biopsy was higher in the NS group (59.3% vs. 5.0%, p <0.001). Across both groups, there were 14 patients with stage 1 AKI, nine with stage 2, and 13 with stage 3 at admission for renal biopsy. Dialysis was only needed in five patients (8.3%) in the NS group. Serum protein and albumin levels were lower in the NS group when the total cholesterol level was higher (p <0.001, Table 2). Four out of 20 patients in the Non-NS group had a max UPCR ≥ 3g/g creatinine before the biopsy. When dividing patients according to the maximal amount of proteinuria 6 months before biopsy by 3g/g creatinine, 16 patients had < 3 g/g creatinine of proteinuria, and the other 63 patients had > 3g/g creatinine of proteinuria. When comparing groups according to a max UPCR before biopsy, the two groups showed a high UPCR, lower GFR, and a higher proportion of AKI (S1 Table). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Characteristics of patients with minimal change disease according to the amount of proteinuria. The renal specimens obtained for pathological examination included enough glomeruli for MCD diagnosis (10–86 glomeruli). Notably, most pathologic findings were similar between the groups, except for interstitial changes and podocyte effacement. The presence of interstitial fibrosis and inflammation was more prevalent in the NS group (p <0.05, S2 Table), as was diffuse effacement of the podocytes under EM examination (93.2% vs. 40.0%, p <0.001). The severity of podocyte effacement was positively correlated to the amount of proteinuria and the presence of nephrotic range proteinuria according to the adjusted linear and logistic regression models, respectively (S3 Table). Clinical course During follow-up after renal biopsy (mean: 62.9 ± 50.8 months, range: 3.0–201.7 months), a UPCR test was conducted 28.5 ± 19.0 (range: 3–142) times; at a mean of 0.64 ± 0.49 times/month. Among the 20 patients in the Non-NS group, seven (35.0%) experienced a UPCR <0.30 g/g creatinine at least once during follow-up. One of seven had recorded UPCRs <0.30, and the other six had recorded 0.3–2.99 g/g creatinine at admission for renal biopsy, respectively. In the Non-NS group, eight out of 20 patients (40.0%) had experienced a relapse of proteinuria, UPCR ≥3.0 g/g creatinine at any point during follow-up, after renal biopsy. These eight patients received steroid treatment without other immunosuppression therapy. One patient was treated with steroids only to address their UPCR of 0.30–2.99 g/g creatinine. The maximum dose of prednisolone administered was 0.92 ± 0.16 (range: 0.5–1.0) mg/kg/day. The remission of proteinuria was achieved in all nine patients. The number of proteinuria relapses was 0.23 ± 0.33 times/year during the follow-up period in the Non-NS group. Steroids were administered to 93.2% of patients (55/59) in the NS group. The maximum dose of prednisolone administered to the 55 patients in the NS group was 0.88 ± 0.17 (range: 0.4–1.2) mg/kg/day. The proteinuria spontaneously improved to a UPCR <0.3 g/g creatine in four patients. Among these, two patients remained in remission; one increased in UPCR to 0.30–0.29 g/g creatinine, and one experienced a relapse of proteinuria. Thirty-seven out of 55 patients received steroids only, while 14 patients received steroids and calcineurin inhibitors, and four received steroids and cyclophosphamide. Among them, remission of proteinuria was achieved in 51 patients. Eventually, 55 out of 59 patients in the NS group achieved the first remission of proteinuria. Four (6.8%) achieved the first remission without any immunosuppressive treatment. Among 51 patients who used steroids, 34 patients (61.8%) used steroids only, while 13 patients (23.6%) with calcineurin inhibitors, and four patients (7.3%) with cyclophosphamide. Of these, 25 patients ultimately experienced a relapse. The number of proteinuria relapses was 0.25 ± 0.41 times/year during the follow-up period in the NS group. In 10 patients, the proteinuria remained at UPCR <0.3 g/g creatinine after the first remission without relapse throughout the follow-up period. Four patients were treated with steroids and cyclophosphamide. They all head heavy proteinuria over at least 6 g/g at initial manifestation. Podocyte effacement on kidney pathology was diffuse and wide in all four patients. Because they all showed severe nephrotic syndrome, they needed intense immunosuppressive therapy. They achieved the first CR without steroid dependency during withdrawal, and AKI in two developed and improved. The response rate to steroid treatment in patients with UPCR ≥3.00 g/g creatinine at any point during follow-up was 100% (8/8 patients) in the Non-NS group and 92.3% (51/55 patients) in the NS group (p = 1.000). The frequency of first relapses in patients with UPCR <0.30 g/g creatinine (p = 0.782) and the number of relapses (p = 0.830) did not differ between the groups. At the final visit, the CR rate was 73.4% (58/79), showing no significant difference between the groups (Fig 2 and Table 3). The days to first remission were 10.7 ± 16.5 days in the non-NS group and 8.7 ± 20.2 days in the NS group (p = 0.701) (Fig 3A). Except for nine patients who did not achieve CR, the days to the first relapse were 35.5 ± 49.0 days in the non-NS group and 29.6 ± 43.6 days in the NS group (p = 0.637) (Fig 3B). The eGFR during the observation period was significantly better in the NS group than the Non-NS group because of the higher incidence of AKI at renal biopsy. The ratio of last serum creatinine by the lowest serum creatinine 6 months before the biopsy was 1.47 (± 0.50) in the non-NS group and 1.59 (± 0.82) in the NS group (p = 0.213). The delta of the last serum creatinine to the lowest serum creatinine 6 months before biopsy was 0.32 (± 0.50) in the non-NS group and 0.36 (± 0.60) in the NS group, respectively (p = 0.679). Overall, the incidence rate of renal events, ESRD events, or mortality was not different between the groups (Table 3). Clinical outcomes were not different when analyzing patients divided by the maximal amount of proteinuria 6 months before biopsy (S4 Table). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Progression of proteinuria according to initial amount and treatment. IS: Immunosuppressive treatment, pt: Patient. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Survival time from the kidney biopsy to the first remission (A) and time from the first remission to the first relapse (B). The linear line indicates the Kaplan-Meier graph line in the Non-NS group. The dashed line indicates that in the NS group. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Outcomes of minimal change disease. Factors associated with the outcomes of MCD We analyzed the relationship between nephrotic range proteinuria and each event, including AKI, the first CR in patients with UPCR ≥3.00 g/g creatinine at any time, relapse after the first CR, CR at the last visit, and renal events. In the multiple logistic regression analysis, younger age, nephrotic range proteinuria, and lower eGFR at admission were related to AKI’s presence at MCD diagnosis. The Cox proportional hazards analysis found that the presence of nephrotic range proteinuria at diagnosis of MCD was not associated with the first remission of proteinuria in patients with UPCR ≥3.00 g/g creatinine at any time, relapse after the first remission of proteinuria, the number of relapses, CR at the last visit, or development of renal events during the observation period (S5 Table). Discussion We analyzed the differences in clinical course between nephrotic and non-nephrotic MCD. The long-term outcome of MCD without nephrotic syndrome was rarely reported before. This report focused on the clinical courses during about 5 years of MCD without heavy proteinuria at renal biopsy. The results showed similar remission and relapse rates and response to steroids rates in both groups. On the other hand, AKI was more common in patients with nephrotic syndrome. Interestingly, podocyte effacement on renal pathological examination did not correspond to the amount of proteinuria. The initial AKI rate differed between the two groups: 59.3% in the NS group compared with just 5.0% in the Non-NS group. Notably, previous reports had found that the incidence of AKI in MCD was 17–38% [10,11], which is lower than that observed in our study. Compared with other studies, we reported AKI in older adults and followed them for longer periods. Furthermore, unlike previous reports, our study’s criteria for diagnosing AKI followed the recent guideline. Our study showed similar incidences of relapse and changes in serum creatinine, irrespective of AKI. A previous study in children reported that AKI was unrelated to long-term renal function outcomes , despite the children exhibiting severe proteinuria. However, a previous study reported higher relapse rates in patients with acute renal failure than those without. They had defined acute renal failure as a 50% decrease in serum creatinine, which is a stricter criterion than that in our study. Compared with this previous study, we observed more prominent tubular atrophy or interstitial inflammation in patients with MCD with AKI. However, the final serum creatinine assessment during follow-up was not different between the NS and Non-NS groups. This reversible AKI in our study could be explained by glomerular hypoperfusion related to hypoalbuminemia with severe podocyte effacement and decreased slit membrane function . Increased tubular endothelin-1 expression in kidney biopsy samples might be involved in the mechanism of reversible tubular damage in MCD complicated with AKI . The clinical course, including remission and relapse, did not differ between patients in both groups. The final CR rate was also equivalent in the NS and Non-NS groups. In a previous follow-up period similar to our study , the CR rate at the final follow-up visit was 87% in the NS group, compared to the 73.4% observed in our study. During the follow-up period, there was no difference in the rates of first CR and development of first relapse, regardless of the presence of nephrotic syndrome at the time of kidney biopsy. The eGFR decreased to a greater extent in the Non-NS group, although the mean levels of eGFR and serum creatinine at the last visit were similar in the two groups. Patients in the Non-NS group had better renal function and higher serum albumin levels on the biopsy day than those in the NS group. Half of the patients that did not have nephrotic syndrome received steroids after developing proteinuria during follow-up. All patients in the Non-NS group who received steroids achieved CR. Spontaneous remission from nephrotic proteinuria before admission was observed in one patient. Four of the 59 patients with nephrotic syndrome achieved CR spontaneously without immunosuppressive agents. Notably, previous studies reported spontaneous proteinuria remission in 8–33% of untreated patients with MCD [16,17]. In one study, half of the patients who received no immediate immunosuppressive treatment achieved spontaneous remission, showing delayed amelioration of nephrotic syndrome. Two patients in the NS group and one in the Non-NS group showed no podocyte effacement. Similar to our study, the severity of podocyte effacement was not correlated with the prognosis of MCD in previous reports . However, others have found a positive correlation between the amount of proteinuria and the severity of podocyte effacement [19,20]. These studies included several types of glomerulonephritis besides MCD, while ours focused on a relatively large number of MCD patients. Sampling error could be another cause for the discrepancy in findings, where samples may have been obtained from regions distant from the location of podocyte effacement. On the other hand, this unclear association between podocyte effacement and the severity of proteinuria might suggest the presence of unknown mechanisms in MCD. Proteinuria in MCD is usually induced by T-cell dysfunction and, eventually, a B-cell pathway . In addition, spontaneous relief of heavy proteinuria in MCD cannot be explained by conventional mechanisms, such as enlarged slit pores or a negative charge in the glomerular basement membrane. Furthermore, several recent studies have presented putative molecular mechanisms involving angiopoietin like-4, interleukin-8, and hemopexin . There are some limitations to this study. This was a retrospective study, and the confounders were not strictly controlled. Therefore, we only included patients without previous immunosuppressive treatment. In addition, using strict inclusion criteria meant that many patients were excluded from the study. As a result, the number of patients included was small. Second, only adults were included in this study, despite MCD being prevalent in children. However, the clinical course of MCD differs in various aspects between adults and children. Therefore, it was meant to describe the prognosis of MCD in adults specifically. In the same sense, further studies considering the initial amount of proteinuria are required for patients of all ages. Third, the pathologic findings in some patients were away from classic MCD in our study. Findings were EM findings, such as < 50% of podocyte effacement, and IM findings, such as mild deposits. Tubulointerstitial damage to differentiate between MCD and unsampled FSGS was not thoroughly investigated. This might be because MCD is sometimes manifested atypically. However, we tried sampling adequate glomeruli to maintain sampling adequacy. Our study suggested no definite correlation between pathologic severity and clinical features such as the amount of proteinuria; however, we still need further findings to prevail this. In conclusion, there was no overall clinicopathological difference in the prognosis of MCD between patients with and without heavy proteinuria. Therefore, patients who show non-nephrotic proteinuria as an initial manifestation should be monitored as closely as those with nephrotic MCD without spontaneous remission. Future studies involving recently developed immunosuppressive treatments might help elucidate the mechanisms underlying MCD relapse. Supporting information Characteristics of patients with minimal change disease according to the highest amount of proteinuria during 6 months before renal biopsy. Showing 1/6: pone.0289870.s001.docx Skip to figshare navigation sorry, we can't preview this file 1 / 6 Share Download figshare S1 Table.Characteristics of patients with minimal change disease according to the highest amount of proteinuria during 6 months before renal biopsy. Creatinine before biopsy: The lowest value of serum creatinine 6months before renal biopsy, DM: Diabetes mellitus, CHD: Coronary heart disease, SBP: Systolic blood pressure, DBP: Diastolic blood pressure, Cr: Creatinine, GFR: Estimated glomerular filtration rate by CKD-EPI equation, AKI: Acute kidney injury based on the lowest creatinine value during the follow-up period, Max UPCR before biopsy: The highest value of UPCR during the 6 months before biopsy, UPCR: Spot urine protein to creatinine ratio (g/g cr). (DOCX) S2 Table.Renal pathologic findings of patients according to the amount of proteinuria. LM findings: Light microscopy findings, Mes: Mesangial, IF findings: Immunofluorescent microscopy findings, EM findings: Electron microscopy findings P-values determined using the Mann-Whitney test, uc: Uncountable. (DOCX) S3 Table.Effect of podocyte effacement severity on proteinuria. For the amount of proteinuria, a multiple linear regression model that adjusted for age, sex, blood pressures, and pathologic findings of podocyte effacement (none, focal, and diffuse) was used. For the presence of NS, a multiple logistic regression model that adjusted for age, sex, serum glucose, pathologic findings of interstitial fibrosis, interstitial inflammation, tubular atrophy, and podocyte effacement (none, focal, and diffuse) was used. NS: Nephrotic-range proteinuria, CI: Confidence interval, RR, relative risk. (DOCX) S4 Table.Outcomes of minimal change disease according to the highest amount of proteinuria during 6 months before renal biopsy. Min UPCR after biopsy: The lowest value of UPCR during the follow-up period starting 1 month after biopsy, Max UPCR after biopsy: The highest value of UPCR during the follow-up period starting 1 month after biopsy, FU duration after biopsy: Follow-up duration between renal biopsy and the last test of UPCR, RAS: Renin-angiotensin-system, CR: Complete remission of proteinuria <0.3 g/g creatinine, relapse: UPCR >3.0 g/g creatinine after achieving CR of UPCR, SD: Presence of steroid dependency in a patient where relapse of UPCR occurred during steroid tapering or within 2 weeks after cessation of steroids, renal event: Any decrease of GFR of more than 50% during a follow-up visit compared to that at renal biopsy, GFR <15 ml/min/1.73 m 2, or development of ESRD, The first CR (n, %) among patients with UPCR >3.00 g/g cr: In the Non-NS group, patients who had increased proteinuria of >3.00 g/g cr during observation period. The first relapse among patients with UPCR <0.30 g/g cr: In the Non-NS group, patients who had UPCR of <0.30 g/g cr at any time of period. (DOCX) S5 Table.Effect of presence of nephrotic range proteinuria at diagnosis of MCD on each event. For presence of AKI at diagnosis, a multiple logistic regression model adjusted for age, sex, serum albumin, eGFR, blood pressures, and severity of podocyte effacement was used. For the first remission in patients with UPCR >3.0 g/g cr: A Cox proportional hazards model adjusted for age; sex; and pathologic findings of deposition of IgA, IgG, lambda chains, or interstitial inflammation, was used. For the first relapse in patients with UPCR <0.3 g/g cr: A Cox proportional hazards model adjusted for age, sex, and steroid treatment was used. For the number of relapses in patients with UPCR <0.3 g/g cr: A multiple linear regression model adjusted for age, sex, serum cholesterol, and steroid treatment was used. For renal events, a Cox proportional hazards model adjusted for age, sex, presence of AKI at renal biopsy, presence of hypertension, eGFR, pathologic deposition of IgA, RAS blockade medication, anti-hypertensive drugs, anti-diabetic drugs, immunosuppressive drugs, and number of relapses of nephrotic range proteinuria was used. For the remission of proteinuria at the last visit, a Cox proportional hazards model adjusted for age; sex; serum albumin; presence of diabetes mellitus; first remission of proteinuria; relapse of proteinuria; and pathologic findings of changes in the mesangial matrix, interstitial inflammation, and presence of atherosclerosis, was used. AKI: Acute kidney injury, eGFR: Estimated glomerular filtration rate, cr: Creatinine, RAS: Renin-angiotensin system, UPCR: Urine protein/creatinine ratio, MCD: Minimal change disease. (DOCX) S1 File.A minimal anonymized data set. (ZIP) References Hong YA, Ban TH, Kang CY, Hwang SD, Choi SR, Lee H, et al. 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Steroid responsiveness and frequency of relapse in adult-onset minimal change nephrotic syndrome. Am J Kidney Dis. 2002;39(3):503–12. Available from: pmid:11877569 View Article PubMed/NCBI Google Scholar van den Berg JG, van den Bergh Weerman MA, Assmann KJ, Weening JJ, Florquin S. Podocyte foot process effacement is not correlated with the level of proteinuria in human glomerulopathies. Kidney Int. 2004;66(5):1901–6. Available from: pmid:15496161 View Article PubMed/NCBI Google Scholar Huh W, Kim DJ, Kim MK, Kim YG, Oh HY, Ruotsalainen V, et al. Expression of nephrin in acquired human glomerular disease. Nephrol Dial Transplant. 2002;17(3):478–84. Available from: pmid:11865096 View Article PubMed/NCBI Google Scholar Koop K, Eikmans M, Baelde HJ, Kawachi H, De Heer E, Paul LC, et al. Expression of podocyte-associated molecules in acquired human kidney diseases. J Am Soc Nephrol. 2003;14(8):2063–71. Available from: pmid:12874460 View Article PubMed/NCBI Google Scholar Kim SH, Park SJ, Han KH, Kronbichler A, Saleem MA, Oh J, et al. Pathogenesis of minimal change nephrotic syndrome: an immunological concept. Korean J Pediatr. 2016;59(5):205–11. Available from: pmid:27279884 View Article PubMed/NCBI Google Scholar Cara-Fuentes G, Clapp WL, Johnson RJ, Garin EH. Pathogenesis of proteinuria in idiopathic minimal change disease: molecular mechanisms. Pediatr Nephrol. 2016;31(12):2179–89. Available from: pmid:27384691 View Article PubMed/NCBI Google Scholar Download PDF Citation XML Print Share Reddit Facebook LinkedIn Mendeley Bluesky Email Advertisement Subject Areas ? For more information about PLOS Subject Areas, click here. We want your feedback. Do these Subject Areas make sense for this article? Click the target next to the incorrect Subject Area and let us know. Thanks for your help! BiopsyIs the Subject Area "Biopsy" applicable to this article? Yes No Thanks for your feedback. 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12326	https://brilliant.org/wiki/integration-of-logarithmic-functions/	Integration of Logarithmic Functions \| Brilliant Math & Science Wiki HomeCourses Sign upLog in The best way to learn math and computer science. Log in with GoogleLog in with FacebookLog in with email Join using GoogleJoin using email Reset password New user? Sign up Existing user? Log in Integration of Logarithmic Functions Sign up with FacebookorSign up manually Already have an account? Log in here. Aditya Virani, A Former Brilliant Member, Satyajit Mohanty, and 5 others Samir Khan Mahindra Jain User 123 Calvin Lin Jimin Khim contributed The derivative of the logarithm ln⁡x \ln x ln x is 1 x \frac{1}{x} x 1, but what is the antiderivative? This turns out to be a little trickier, and has to be done using a clever integration by parts. The logarithm is a basic function from which many other functions are built, so learning to integrate it substantially broadens the kinds of integrals we can tackle. Contents Integrating ln⁡x \ln x ln x Integrating Functions of ln⁡x \ln x ln x Proof of ∫ln⁡x d x=x ln⁡x−x+C\int\ln x\, dx=x\ln x-x+C ∫ln x d x=x ln x−x+C using Taylor Series Integrating ln⁡x \ln x ln x ∫ln⁡(x)d x=x ln⁡(x)−x+C\int \ln(x)\ dx = x\ln (x) - x + C∫ln(x)d x=x ln(x)−x+C In the equation above, C C C is the constant of integration, and this notation C C C will be used throughout the wiki. For this solution, we will use integration by parts: ∫f(x)g′(x)d x=f(x)g(x)−∫f′(x)g(x)d x.\int f(x) g'(x)\ dx = f(x) g(x) - \int f'(x) g(x)\ dx.∫f(x)g′(x)d x=f(x)g(x)−∫f′(x)g(x)d x. We use f(x)=ln⁡(x)f(x)=\ln(x)f(x)=ln(x) and g′(x)=1 g'(x)=1 g′(x)=1, which means that g(x)=x g(x)=x g(x)=x. Plugging these into our integration by parts formula, we get ∫1⋅ln⁡(x)d x=x ln⁡(x)−∫(ln⁡(x))′x d x=x ln⁡(x)−∫x x d x=x ln⁡(x)−x+C. \begin{aligned} \int 1 \cdot \ln(x)\ dx&=x \ln(x) - \int \big(\ln(x)\big)' x\ dx\&=x \ln(x) - \int \frac{x}{x}\ dx\&=x \ln(x) - x + C. \end{aligned} ∫1⋅ln(x)d x=x ln(x)−∫(ln(x))′x d x=x ln(x)−∫x xd x=x ln(x)−x+C. We can factorize a bit and get the desired formula ∫ln⁡(x)d x=x(ln⁡(x)−1)+C.□\int \ln(x)\ dx = x\big(\ln(x) - 1\big) + C.\ _\square∫ln(x)d x=x(ln(x)−1)+C.□ This shows that an unlikely application of an integration technique can actually be the right way forward! Now that we know how to integrate this, let's apply the properties of logarithms to see how to work with similar problems. Evaluate ∫ln⁡2 x d x \displaystyle{\int \ln 2x \, dx} ∫ln 2 x d x. According to the properties of logarithms, we know that ln⁡2 x=ln⁡x+ln⁡2,\ln 2x=\ln x+\ln2,ln 2 x=ln x+ln 2, and thus ∫ln⁡2 x d x=∫(ln⁡x+ln⁡2)d x=∫ln⁡x d x+∫ln⁡2 d x=x ln⁡x−x+x ln⁡2+C.□\begin{aligned} \int\ln2x~dx&=\int\left(\ln x+\ln2\right)~dx\ &=\int\ln x~dx+\int\ln2~dx\ &=x\ln x-x+x\ln2+C.\ _\square \end{aligned}∫ln 2 x d x=∫(ln x+ln 2)d x=∫ln x d x+∫ln 2 d x=x ln x−x+x ln 2+C.□ Evaluate ∫log⁡x d x.\displaystyle{\int\log x~dx}.∫lo g x d x. According to the properties of logarithms, we have log⁡x=ln⁡x ln⁡10.\log x=\frac{\ln x}{\ln10}.lo g x=ln 10 ln x. Hence the given integral can be rewritten as ∫log⁡x d x=∫ln⁡x ln⁡10 d x=1 ln⁡10 x(ln⁡x−1)+C.□\int\log x~dx=\int\frac{\ln x}{\ln10}~dx=\frac{1}{\ln10}x(\ln x-1)+C.\ _\square∫lo g x d x=∫ln 10 ln xd x=ln 10 1x(ln x−1)+C.□ When integrating the logarithm of a polynomial with at least two terms, the technique of u u u-substitution is needed. The following are some examples of integrating logarithms via U-substitution: Evaluate ∫ln⁡(2 x+3)d x\displaystyle{ \int \ln (2x+3) \, dx} ∫ln(2 x+3)d x. For this problem, we use u u u-substitution. Let u=2 x+3.u=2x+3.u=2 x+3. Then we have d u=2 d x,du=2dx,d u=2 d x, or d x=1 2 d u,dx=\frac{1}{2}du,d x=2 1d u, and the given integral can be rewritten as follows: ∫ln⁡(2 x+3)d x=1 2∫ln⁡u d u=1 2 u(ln⁡u−1)+C=2 x+3 2(ln⁡(2 x+3)−1)+C.□\begin{aligned} \int\ln(2x+3)~dx&=\frac{1}{2}\int\ln u~du\ &=\frac{1}{2}u(\ln u-1)+C\ &=\frac{2x+3}{2}\big(\ln(2x+3)-1\big)+C.\ _\square \end{aligned}∫ln(2 x+3)d x=2 1∫ln u d u=2 1u(ln u−1)+C=2 2 x+3(ln(2 x+3)−1)+C.□ Evaluate ∫ln⁡(x−2)3 d x \displaystyle{\int \ln (x-2)^3 \, dx} ∫ln(x−2)3 d x. According to the properties of logarithms, we have ∫ln⁡(x−2)3 d x=3∫ln⁡(x−2)d x.\int \ln (x-2)^3 \, dx=3\int\ln(x-2)~dx.∫ln(x−2)3 d x=3∫ln(x−2)d x. Now let u=x−2.u=x-2.u=x−2. Then we have d u=d x,du=dx,d u=d x, and the integral can be rewritten as follows: ∫ln⁡(x−2)3 d x=3∫ln⁡(x−2)d x=3∫ln⁡u d u=3 u(ln⁡u−1)+C=3(x−2)(ln⁡(x−2)−1)+C.□\begin{aligned} \int \ln (x-2)^3 \, dx&=3\int\ln(x-2)~dx\ &=3\int\ln u~du\ &=3u(\ln u-1)+C\ &=3(x-2)\big(\ln(x-2)-1\big)+C.\ _\square \end{aligned}∫ln(x−2)3 d x=3∫ln(x−2)d x=3∫ln u d u=3 u(ln u−1)+C=3(x−2)(ln(x−2)−1)+C.□ Integrating Functions of ln⁡x \ln x ln x We now look at examples where we're integrating functions of ln⁡x \ln x ln x. These problems often require familiarity with integration by parts, u u u-substitution and ln⁡∣f∣\ln \|f\|ln∣f∣ form. Evaluate ∫x ln⁡x d x \displaystyle{\int x \ln x \, dx} ∫x ln x d x. To solve this, we use the principle of integration by parts. Let u=ln⁡x u=\ln x u=ln x and v′=x.v'=x.v′=x. Then we have ∫x ln⁡x d x=∫v′u d x=u v−∫v u′d x=1 2 x 2 ln⁡x−∫1 2 x 2⋅(ln⁡x)′d x=1 2 x 2 ln⁡x−∫1 2 x d x=1 2 x 2 ln⁡x−1 4 x 2+C.□\begin{aligned} \int x\ln x~dx&=\int v'u~dx\ &=uv-\int vu'~dx\ &=\frac{1}{2}x^2\ln x-\int\frac{1}{2}x^2\cdot\left(\ln x\right)'~dx\ &=\frac{1}{2}x^2\ln x-\int\frac{1}{2}x~dx\ &=\frac{1}{2}x^2\ln x-\frac{1}{4}x^2+C.\ _\square \end{aligned}∫x ln x d x=∫v′u d x=uv−∫v u′d x=2 1x 2 ln x−∫2 1x 2⋅(ln x)′d x=2 1x 2 ln x−∫2 1x d x=2 1x 2 ln x−4 1x 2+C.□ More generally, ∫x m ln⁡x d x=x m+1(ln⁡x m+1−1(m+1)2)+C. \int x^m \ln x \, dx = x^{m+1} \left( \frac{ \ln x } { m+1 } - \frac{1}{ (m+1)^2 } \right) + C. ∫x m ln x d x=x m+1(m+1 ln x−(m+1)2 1)+C. The proof of this is similar to the above. Evaluate ∫ln⁡x x d x \displaystyle{\int \frac{ \ln x } { x} \, dx} ∫x ln xd x. To solve this, we use u u u-substitution. Let u=ln⁡x.u=\ln x.u=ln x. Then since d u=1 x d x,du=\frac{1}{x}dx,d u=x 1d x, we know that ∫ln⁡x x d x=∫u x d x=∫u d u=1 2 u 2+C=1 2(ln⁡x)2+C.□\int \frac{ \ln x } { x} \, dx=\int \frac{u}{x}~dx=\int u~du=\frac{1}{2}u^2+C=\frac{1}{2}(\ln x)^2+C.\ _\square∫x ln xd x=∫x ud x=∫u d u=2 1u 2+C=2 1(ln x)2+C.□ More generally, ∫(ln⁡x)n x d x=(ln⁡x)n+1 n+1,n≠−1. \int \frac{ (\ln x)^n }{x} \, dx = \frac{ (\ln x ) ^ { n + 1 } } { n+1} , \quad n \neq -1. ∫x(ln x)nd x=n+1(ln x)n+1,n=−1. The proof of this is similar to the above. Show that ∫1 x ln⁡x d x=ln⁡∣ln⁡x∣+C \displaystyle{\int \frac{ 1} { x \ln x } \, dx = \ln \lvert \ln x \rvert+C} ∫x ln x 1d x=ln∣ln x∣+C. Perceive 1 x ln⁡x\frac{1}{x\ln x}x l n x 1 as 1 x ln⁡x.\frac{\frac{1}{x}}{\ln x}.l n x x 1. Then since (ln⁡x)′=1 x,(\ln x)'=\frac{1}{x},(ln x)′=x 1, the given integral will give the ln⁡∣f∣\ln\lvert f\rvert ln∣f∣ form as follows: ∫1 x ln⁡x d x=ln⁡∣ln⁡x∣+C.□\int \frac{ 1} { x \ln x } \, dx=\ln\lvert\ln x\rvert+C.\ _\square∫x ln x 1d x=ln∣ln x∣+C.□ Alternative Solution: We can also use the substitution u=ln⁡x.u=\ln x.u=ln x. As d u=1 x d x,du=\frac{1}{x}dx,d u=x 1d x, we know that ∫1 x ln⁡x d x=∫1 ln⁡x⋅1 x d x=∫1 u d u=ln⁡∣u∣+C=ln⁡∣ln⁡x∣+C.□\int \frac{ 1} { x \ln x } \, dx=\int\frac{1}{\ln x}\cdot\frac{1}{x}~dx=\int\frac{1}{u}~du=\ln\lvert u\rvert+C=\ln\lvert\ln x\rvert+C.\ _\square∫x ln x 1d x=∫ln x 1⋅x 1d x=∫u 1d u=ln∣u∣+C=ln∣ln x∣+C.□ Evaluate ∫(ln⁡x)2 d x \displaystyle{\int ( \ln x ) ^2 \, dx} ∫(ln x)2 d x. Let u=(ln⁡x)2 u=(\ln x)^2 u=(ln x)2 and v′=1.v'=1.v′=1. Then since v=x v=x v=x and u′=2 ln⁡x x u'=\frac{2\ln x}{x}u′=x 2 l n x we have ∫(ln⁡x)2 d x=∫u v′d x=u v−∫u′v d x=x(ln⁡x)2−∫2 ln⁡x x⋅x d x=x(ln⁡x)2−∫2 ln⁡x d x.\begin{aligned} \int ( \ln x ) ^2 \, dx&=\int uv'~dx\ &=uv-\int u'v~dx\ &=x(\ln x)^2-\int\frac{2\ln x}{x}\cdot x~dx\ &=x(\ln x)^2-\int2\ln x~dx. \end{aligned}∫(ln x)2 d x=∫u v′d x=uv−∫u′v d x=x(ln x)2−∫x 2 ln x⋅x d x=x(ln x)2−∫2 ln x d x. Using the formula ∫ln⁡x d x=x ln⁡x−x+C,\int\ln x~dx=x\ln x-x+C,∫ln x d x=x ln x−x+C, which we have learned above, we have x(ln⁡x)2−∫2 ln⁡x d x=x(ln⁡x)2−2 x ln⁡x+2 x+C.□x(\ln x)^2-\int2\ln x~dx=x(\ln x)^2-2x\ln x+2x+C.\ _\square x(ln x)2−∫2 ln x d x=x(ln x)2−2 x ln x+2 x+C.□ Some problems can be very complex so that they require using both integration by parts and u u u-substitution. Evaluate ∫sin⁡(ln⁡x)d x \displaystyle{\int \sin ( \ln x ) \, dx} ∫sin(ln x)d x. We use the substitution t=ln⁡x.t=\ln x.t=ln x. Then since d t=1 x d x,dt=\frac{1}{x}dx,d t=x 1d x, or d x=x d t,dx=xdt,d x=x d t, we have ∫sin⁡(ln⁡x)d x=∫x sin⁡t d t.\int \sin ( \ln x ) \, dx=\int x\sin t~dt.∫sin(ln x)d x=∫x sin t d t. From the relationship t=ln⁡x t=\ln x t=ln x we know that e t=x.e^t=x.e t=x. Hence ∫x sin⁡t d t=∫e t sin⁡t d t.\int x\sin t~dt=\int e^t\sin t~dt.∫x sin t d t=∫e t sin t d t. Now let u=sin⁡t u=\sin t u=sin t and v′=e t.v'=e^t.v′=e t. Then since u′=cos⁡t u'=\cos t u′=cos t and v=e t,v=e^t,v=e t, we have ∫e t sin⁡t d t=∫u v′d t=u v−∫u′v d t=e t sin⁡t−∫e t cos⁡t d t.(1)\begin{aligned} \int e^t\sin t~dt&=\int uv'~dt\ &=uv-\int u'v~dt\ &=e^t\sin t-\int e^t\cos t~dt. \qquad(1) \end{aligned}∫e t sin t d t=∫u v′d t=uv−∫u′v d t=e t sin t−∫e t cos t d t.(1) Using integration by parts once again, we have ∫e t cos⁡t d t=e t cos⁡t+∫e t sin⁡t d t,\int e^t\cos t~dt=e^t\cos t+\int e^t\sin t~dt,∫e t cos t d t=e t cos t+∫e t sin t d t, substituting which into (1)(1)(1) gives ∫e t sin⁡t d t=e t sin⁡t−∫e t cos⁡t d t=e t sin⁡t−e t cos⁡t−∫e t sin⁡t d t⇒2∫e t sin⁡t d t=e t sin⁡t−e t cos⁡t∫e t sin⁡t d t=e t(sin⁡t−cos⁡t)2+C.\begin{aligned} \int e^t\sin t~dt&=e^t\sin t-\int e^t\cos t~dt\&=e^t\sin t-e^t\cos t-\int e^t\sin t~dt\ \Rightarrow 2\int e^t\sin t~dt&=e^t\sin t-e^t\cos t\ \int e^t\sin t~dt&=\frac{e^t(\sin t-\cos t)}{2}+C. \end{aligned}∫e t sin t d t⇒2∫e t sin t d t∫e t sin t d t=e t sin t−∫e t cos t d t=e t sin t−e t cos t−∫e t sin t d t=e t sin t−e t cos t=2 e t(sin t−cos t)+C. Now we finally have ∫sin⁡(ln⁡x)d x=e t(sin⁡t−cos⁡t)2+C=e ln⁡x(sin⁡(ln⁡x)−cos⁡(ln⁡x))2+C.□\begin{aligned} \int \sin ( \ln x ) \, dx &=\frac{e^t(\sin t-\cos t)}{2}+C\ &=\frac{e^{\ln x}\big(\sin(\ln x)-\cos(\ln x)\big)}{2}+C.\ _\square \end{aligned}∫sin(ln x)d x=2 e t(sin t−cos t)+C=2 e l n x(sin(ln x)−cos(ln x))+C.□ Proof of ∫ln⁡x d x=x ln⁡x−x+C\int\ln x\, dx=x\ln x-x+C ∫ln x d x=x ln x−x+C using Taylor Series If we don't want to use integration by parts, we can also solve our original integral using Taylor expansion. We know that the Taylor series expansion of ln⁡x\ln x ln x is ln⁡x=(x−1)−(x−1)2 2+(x−1)3 3−(x−1)4 4+⋯.(1)\ln x=(x-1)-\frac{(x-1)^2}{2}+\frac{(x-1)^3}{3}-\frac{(x-1)^4}{4}+\cdots. \qquad(1)ln x=(x−1)−2(x−1)2+3(x−1)3−4(x−1)4+⋯.(1) By integrating both sides, we get ∫ln⁡x d x=(x−1)2 2−(x−1)3 6+(x−1)4 12−(x−1)5 20+⋯.(2)\int\ln x~dx=\frac{(x-1)^2}{2}-\frac{(x-1)^3}{6}+\frac{(x-1)^4}{12}-\frac{(x-1)^5}{20}+\cdots.\qquad(2)∫ln x d x=2(x−1)2−6(x−1)3+12(x−1)4−20(x−1)5+⋯.(2) We want to compare this with the Taylor series expansion of x ln⁡x−x.x\ln x-x.x ln x−x. Multiplying both sides of (1)(1)(1) by x−1 x-1 x−1 gives (x−1)ln⁡x=(x−1)2−(x−1)3 2+(x−1)4 3−(x−1)5 4+⋯.(x-1)\ln x=(x-1)^2-\frac{(x-1)^3}{2}+\frac{(x-1)^4}{3}-\frac{(x-1)^5}{4}+\cdots.(x−1)ln x=(x−1)2−2(x−1)3+3(x−1)4−4(x−1)5+⋯. Then we add (1)(1)(1) to both sides to obtain x ln⁡x=(x−1)+(x−1)2 2−(x−1)3 6+(x−1)4 12−(x−1)5 20+⋯.x\ln x=(x-1)+\frac{(x-1)^2}{2}-\frac{(x-1)^3}{6}+\frac{(x-1)^4}{12}-\frac{(x-1)^5}{20}+\cdots.x ln x=(x−1)+2(x−1)2−6(x−1)3+12(x−1)4−20(x−1)5+⋯. Finally, subtracting x x x from both sides gives x ln⁡x−x=−1+(x−1)2 2−(x−1)3 6+(x−1)4 12−(x−1)5 20+⋯,x\ln x-x=-1+\frac{(x-1)^2}{2}-\frac{(x-1)^3}{6}+\frac{(x-1)^4}{12}-\frac{(x-1)^5}{20}+\cdots,x ln x−x=−1+2(x−1)2−6(x−1)3+12(x−1)4−20(x−1)5+⋯, which is identical with (2)(2)(2) except for the constant term (which is irrelevant, since there is always a constant of integration). Therefore we can conclude that ∫ln⁡x d x=x ln⁡x−x+C.□\int\ln x~dx=x\ln x-x+C.\ _\square∫ln x d x=x ln x−x+C.□ Cite as: Integration of Logarithmic Functions. Brilliant.org. Retrieved 00:29, September 29, 2025, from Join Brilliant The best way to learn math and computer science.Sign up Sign up to read all wikis and quizzes in math, science, and engineering topics. 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12327	https://www.youtube.com/watch?v=dGdcdJTWFxo	The locus of the midpoints of the focal chords of the parabola y^(2)=4ax is \| 12 \| PARABOLA \| M... Doubtnut 3940000 subscribers 4 likes Description 149 views Posted: 5 Feb 2022 The locus of the midpoints of the focal chords of the parabola y^(2)=4ax is Class: 12 Subject: MATHS Chapter: PARABOLA Board:IIT JEE You can ask any doubt from class 6-12, JEE, NEET, Teaching, SSC, Defense and Banking exam on Doubtnut App or You can Whatsapp us at - 8400400400 Link - Join our courses to improve your performance and Clear your concepts from basic for Class 6-12 School and Competitive exams (JEE/NEET) - Contact Us: 👉 Have Any Query? Ask Us. 🤙 Call: 01247158250 💬 WhatsApp: 8400400400 📧 Email: info@doubtnut.com 🌐 Website: Welcome to Doubtnut. Doubtnut is World’s Biggest Platform for Video Solutions of Physics, Chemistry, Maths and Biology Doubts with over 5 million+ Video Solutions. 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12328	http://www.gutenberg.org/ebooks/author/797	Donate About Project Gutenberg Contact Us History & Philosophy Kindle & eReaders Help Pages Offline Catalogs Donate Frequently Downloaded Main Categories Reading Lists Search Options Frequently Downloaded Main Categories Books by De Quincey, Thomas (sorted by popularity) Sort Alphabetically by Title Sort by Release Date Alias Quincey, Thomas de See also: en.wikipedia Displaying results 1–25 \| Next Confessions of an English Opium-Eater Thomas De Quincey 2515 downloads "De Bello Gallico" and Other Commentaries Julius Caesar 2141 downloads The Posthumous Works of Thomas De Quincey, Vol. 1 Thomas De Quincey 1655 downloads The Collected Writing of Thomas De Quincey, Vol. II Thomas De Quincey 982 downloads Narrative and Miscellaneous Papers Thomas De Quincey 602 downloads The Uncollected Writings of Thomas de Quincey, Vol. 2 Thomas De Quincey 508 downloads The Uncollected Writings of Thomas de Quincey—Vol. 1 Thomas De Quincey 493 downloads The English Mail-Coach and Joan of Arc Thomas De Quincey 458 downloads The Campaner Thal, and Other Writings Jean Paul 456 downloads The Posthumous Works of Thomas De Quincey, Vol. 2 Thomas De Quincey 440 downloads Miscellaneous Essays Thomas De Quincey 414 downloads The Footpath Way: An Anthology for Walkers 371 downloads The Lock and Key Library: Classic Mystery and Detective Stories: Old Time English 296 downloads De Quincey's Revolt of the Tartars Thomas De Quincey 294 downloads Stories of Intellect Nathaniel Hawthorne, Charles Dickens, Rebecca Harding Davis, Edgar Allan Poe, Baron Edward Bulwer Lytton Lytton, Thomas De Quincey, and Harriet Elizabeth Prescott Spofford 292 downloads Autobiographic Sketches Thomas De Quincey 263 downloads The Caesars Thomas De Quincey 260 downloads Walladmor, Vol. 2 (of 2) Willibald Alexis 199 downloads De l'assassinat considéré comme un des Beaux-Arts (French) Thomas De Quincey 197 downloads Theological Essays and Other Papers — Volume 1 Thomas De Quincey 197 downloads Narrative and Miscellaneous Papers — Volume 2 Thomas De Quincey 186 downloads Memorials and Other Papers — Complete Thomas De Quincey 181 downloads Biographical Essays Thomas De Quincey 178 downloads Narrative and Miscellaneous Papers — Volume 1 Thomas De Quincey 176 downloads Theological Essays and Other Papers — Volume 2 Thomas De Quincey 157 downloads Displaying results 1–25 \| Next
12329	https://www.youtube.com/watch?v=RxcEvolRdvI	Laplace Transform of tf(t) Dr Chris Tisdell 93600 subscribers 414 likes Description 62875 views Posted: 5 Jun 2009 Free ebook The video presents a simple proof of an result involving the Laplace transform of tf(t). In particular it is shown that the Laplace transform of tf(t) is -F'(s), where F(s) is the Laplace transform of f(t). The proof involves an application of Leibniz rule for differentiating integrals. I also give an example at the end illustrating how to apply the proven result. Laplace transforms find important applications in solving ordinary differential equations with discontinuities. Such ideas are seen in 2nd-year university mathematics courses. 50 comments Transcript: hi again everyone today we're going to look at a problem from llast transforms okay now llas transforms are used in Applied Mathematics and Engineering to solve ordinary differential equations with discontinuities okay now up here this is the notation we use for the lass transform of a function of T little F of T right and it's an integral transform you can see that we take little F of T multiply by this exponential function and then integrate over this half line now notice here the t is the dummy variable okay so so um it it disappears after we do any Integrations now the S here is the real variable though it's the independent variable right so you can think of this as being a function of s big F of s so the notation we generally use is little F of T is some function of T and the capital big F of s is the transform of little F okay so we're asked to show that the transform of T a function of T is negative big f- of s so how do we do that well the basic idea is to take this differentiate both sides with respect to S and then rearrange now at the heart of the method is an idea called liet's Rule okay and lits rule allows us to differentiate under the integral sign so let's start with this and differentiate both sides okay this and [Music] this right so we differentiate here so um we're going to get DF DS which I'm going to denote by f- of s equals the derivative of this integral okay so we want to work on this and liet Rule rule allows us to move this inside the integral sign and you change the straight D's to curly D's so so here I've I've pushed this inside change the straight D's to curly D's why because this is a function of two variables s and t all right so here I've used the socalled oh liet rule okay so all that really remains now is to take this derivative okay so because we're differentiating with respect to X s partially a minus t will come to the front and the rest will be unchanged okay so now if we rearrange this okay I can pull that minor sign out the front and put the T over here here minus all right what do we have well there any difference between this and this is that we have t f of T So this must be the lass transform of t f of T don't forget the minus sign out here okay all right so what have we proved there we've proved that F big f- of s equals the minus of the transform of t t little F of T okay so we just pull the mon sign put it up here and we get exactly what we're looking for okay so this example it really highlights where liet's rule comes in handy why would you want to differentiate an integral sign well here's a good example okay now where can this be actually used when you're calculating transforms well you might want to take the transform the plus transform of T s of T so here your little F of T would be S of T what you would do is work out the transform of sin T take the derivative and then put a minus sign in there okay so um theast transform of sin t is 1 on S2 + 1 2 okay so this is a function of s so F Das of s is going to be - 2 s over S 2 + 1 or 2 so to apply this the laast transform of this will be well just add a minus sign in here okay so that's a good example of where this identity can be used in calculating theast transforms
12330	https://www.njlm.net/articles/PDF/2785/66477_CE[Ra1]_F(SHU)_QC(KK_RDW_SHU)_PF1(AP_SS)_PFA(SHU)_PB(AP_OM)_PN(SHU).pdf	1 Pathology Section External Auditory Canal Granulations of Varying Aetiology Other than Cholesteatoma: A Series of Seven Cases National Journal of Laboratory Medicine. 2024 Jan, Vol-13(1): PS01-PS04 DOI: 10.7860/NJLM/2024/66477.2785 Case Series INTRODUCTION Granulation tissue is a new connective tissue with angiogenesis that develops over the surface of a healing wound. It has a red, beefy appearance due to abundant newly formed capillaries. Granulation tissue in the EAC is a diagnostic finding in certain pathologies. For example, granulation tissue in the junction of the bony and cartilaginous portions of the EAC is almost always seen in malignant otitis externa . Aural granulation masses can occur due to various causes such as inflammation, neoplastic growths (both benign and malignant), infection (e.g., furuncle with granulation tissue), trauma, Langerhans cell histiocytosis, malignant otitis externa, skin adnexal tumours of the EAC, rhabdomyosarcoma, osteoma, complications of CSOM, and relapsing polychondritis . These aural masses can present as granulation tissue, aural polyps, or growths with persistent ear discharge, ear fullness, ear pain, and hearing impairment . EAC granulations are frequently observed in Ear, Nose, Throat (ENT) Outpatient Departments (OPDs). However, many of them are due to the unsafe variety of CSOM, which can lead to cholesteatoma formation and subsequent bone erosion, with granulation tissue present at the site of bone healing. In the present case, the author described a series of cases where the aetiology of EAC granulations was caused by factors other than CSOM. These other causes include tubercular otitis media, malignant otitis externa, and benign tumours such as osteomas, exostosis, pleomorphic adenoma, tumours from the ceruminous and sebaceous glands, and malignant tumours like squamous cell carcinoma and rhabdomyosarcoma. The treatment modalities for these cases differ, as does their prognosis. A thoughtful insight into the differential diagnoses of these cases will lead to early diagnosis and prompt treatment. CASE SERIES This hospital-based case series included all the patients who came to the GMC hospital for the treatment of EAC granulations over a six-month period. Patients with CSOM as the cause of EAC granulation were excluded from the study. Patients of all age groups were included, but only those with a definite diagnosis as the cause of EAC granulations were included in the study. The author had seven cases in this six-month period where the cause of EAC granulation differed from CSOM. The age of patients ranged from 12 years to 71 years, and the male-to-female ratio was 5:2. Granulation was found in the left ear in four patients, while in three patients, it was seen in the right ear [Table/ Fig-1]. All patients had a history of ear discharge, out of which three had blood-tinged discharge. Facial palsy was observed in one patient with malignant otitis externa. Hearing loss was reported by four patients. Otalgia, associated with infective aetiology, was observed in only two patients. During clinical examination, the tympanic membrane was not visible in four patients. Manas Ranjan Satpathy1, Rashmi Rekha Mahapatra2, KCKDN Hembram3 Keywords: Diagnoses, Granulation, Malignant otitis externa ABSTRACT External Auditory Canal (EAC) granulations can occur due to various causes, each with different presenting features, treatment options, and prognoses. They can arise from trivial situations, such as a neglected Foreign Body (FB), or indicate serious pathology like squamous cell carcinoma. The most common causes of EAC granulations are otitis media, including squamous and mucosal types. However, clinical practice encounters numerous other rare and common causes. The present case series describes seven patients with EAC granulations resulting from different aetiologies apart from Chronic Suppurative Otitis Media (CSOM). The study was conducted over a six-month period at a Government Medical College (GMC) in North Odisha. It aims to assist clinicians in considering these situations when managing patients with EAC granulation. Proper diagnosis and treatment of these patients resulted in successful outcomes. Therefore, the author recommends considering these underlying causes whenever EAC granulations do not respond to appropriate medical treatment with antibiotics. Radiological imaging, fine needle aspiration cytology, and histopathological studies can help exclude many of these causes from the list of differential diagnoses. S. No. Age/Sex Side Symptoms Clinical findings Radiological findings Final diagnosis 1 71/M Left Bleeding, tinnitus Granulation tissue filling the EAC. HRCT showed bone erosion Squamous cell carcinoma 2 62/M Right Discharge, hearing loss, tinnitus Granulation in posterior part of EAC, FB (rice grain) seen during probing of the granulation. HRCT showed normal mastoid FB with granulation 3 58/M Left Aural fullness, tinnitus, hearing loss, discharge, painless Pale granulation in floor of EAC, Attic normal. HRCT showed granulation in EAC and middle ear Tuberculosis of EAC 4 67/M Right Discharge, facial palsy, tinnitus Uncontrolled diabetic, CKD, Lt. LMN facial palsy. HRCT showed bone erosion. Malignant otitis externa 5 13/F Left Bleeding, mass, hearing loss Granulation tissue in EAC, CHL. HRCT normal Post-traumatic granulation tissue Manas Ranjan Satpathy et al., External Auditory Canal Granulations Other than Cholesteatoma www.njlm.net National Journal of Laboratory Medicine. 2024 Jan, Vol-13(1): PS01-PS04 2 conservative treatment. On otoscopic examination, pale-looking granulation tissue was observed on the floor of the EAC. A biopsy of the granulation was performed under local anaesthesia. The tissue was sent for histopathological study, which revealed the presence of caseating granulomas along with Langhans giant cells. Based on the patient’s contact history, clinical and histopathological findings, the diagnosis of tuberculosis involving the EAC was made [Table/Fig-5]. The patient was treated with antitubercular therapy. [Table/Fig-2]: Squamous cell carcinoma of left ear: a) CT scan image showing bone erosion of the left EAC; b) Photomicrograph showing islands of malignant squamous cells with keratin pearl formation (H&E stain, 100X). [Table/Fig-3]: CT scan of temporal bone showing granulations present in external and middle ear secondary to retained foreign body in axial and sagittal section. [Table/Fig-4]: Otoscopic appearance of resolution of granulation after removal of the foreign body. [Table/Fig-5]: Tubercular lesion: a) Otoscopic picture of granulation in the EAC; b) Photomicrograph showing granulomas consisting of clusters of epithelioid cells (H&E stain, 100X). Case 4 Malignant otitis externa: A 67-year-old male with diabetes and Chronic Kidney Disease (CKD) presented with a discharge from his right ear accompanied by tinnitus for a duration of two months. Recently, the patient developed facial palsy on the right side of his face. Upon examination, granulation tissue was found in the right EAC, along with purulent discharge. A pus culture was conducted, revealing the growth of Pseudomonas. HRCT of the temporal bone showed bone erosion in the EAC and facial canal. A biopsy of the granulation tissue was performed to rule out malignancy. The patient was treated with intravenous piperacillin with tazobactam for a period of 10 days, followed by oral levofloxacin once daily for one month. 6 17/F Right Pain, blood-tinged discharge, hearing loss Granulation tissue with purulent discharge from posterior wall of EAC. HRCT normal Organised furuncle with granulation tissue 7 12/M Left Discharge, intermittent pain, swelling Granulation arising from posterior wall. HRCT showed normal middle ear and mastoid Infected EAC epidermoid cyst [Table/Fig-1]: Summary of all clinical findings of the cases with final diagnosis. EAC: External auditory canal; HRCT: High resolution computed tomography; FB: Foreign body; CKD: Chronic kidney disease; LMN: Lower motor neuron; CHL: Conductive hearing loss Case 1 Squamous cell carcinoma: A 71-year-old male came to the ENT-OPD with a history of bleeding from the left ear for two months, accompanied by tinnitus. On examination, granulation tissue was found filling the left EAC. High-Resolution Computed Tomography (HRCT) of the temporal bone revealed bone erosion in the left EAC. A biopsy was performed, and the sample was sent for histopathological study, which revealed a well-differentiated squamous cell carcinoma [Table/Fig-2]. The epithelium exhibited marked keratinisation and the formation of keratin pearls. The patient was treated with chemoradiation as he was not fit and not willing to undergo subtotal temporal bone resection. Case 2 Foreign body with granulation: A 62-year-old male presented with complaints of discharge, hearing loss, and tinnitus in the right ear. On examination, granulation tissue was observed in both the EAC and middle ear. High-Resolution Computed Tomography (HRCT) of the temporal bone showed granulation in the external and middle ear [Table/Fig-3]. During an examination of the ear under a microscope, a FB was seen impacted in the granulation, which was subsequently removed. The patient was managed with conservative treatment, and the granulations completely disappeared within one month following the removal of the foreign body [Table/Fig-4]. Case 3 Tuberculosis of EAC: A 58-year-old male presented with symptoms of aural fullness, tinnitus, hearing loss, and painless discharge from the left ear for five months. The symptoms did not subside with www.njlm.net Manas Ranjan Satpathy et al., External Auditory Canal Granulations Other than Cholesteatoma National Journal of Laboratory Medicine. 2024 Jan, Vol-13(1): PS01-PS04 3 Case 5 Post-traumatic EAC granulation: A 13-year-old female child presented to the OPD with complaints of bleeding from the left ear and hearing loss. Upon examination, granulation tissue was found filling the left EAC. During a detailed history-taking, the mother revealed that the child had experienced trauma to the left ear in a road traffic accident one month prior, which was treated at a nearby local hospital. A CT scan of the temporal bone revealed granulation tissue in the cartilaginous EAC, with a normal bony canal, middle ear, and mastoid [Table/Fig-6]. The histopathological examination confirmed a diagnosis of post-traumatic granulation, and the patient underwent surgical treatment [Table/Fig-7]. The granulation tissue was removed, and meatoplasty was performed to prevent future canal stenosis. was completely excised and sent for histopathological examination, which revealed a cyst lining composed of stratified squamous epithelium containing keratin flakes within the cyst [Table/Fig-8]. This finding led to the diagnosis of an epidermoid cyst of the EAC. [Table/Fig-6]: CT scan of the patient with post-traumatic granulation showing granulation tissue filling the cartilaginous EAC with normal middle ear and mastoid. [Table/Fig-7]: Post-traumatic granulation tissue: a) Granulation tissue in the EAC with scar mark of the injury around the left ear; b) Photomicrograph showing capillaries lined by reactive endothelial cells, plump fibroblasts, and a mixed inflammatory infiltrate (H&E stain, 100X). [Table/Fig-8]: Photomicrograph of epidermoid cyst lined by stratified squamous epithelium with cyst contents composed of keratin flakes (H&E stain, 100X). Case 6 Organised furuncle with granulation tissue: A 17-year-old female complained of pain, blood-tinged discharge, and hearing loss in her right ear. Upon examination, granulation tissue with purulent discharge was observed on the posterior wall of the right ear canal. The discharge was cleaned from the ear canal, and upon probing, a furuncle was discovered in the deep posterior wall. Following a course of oral antibiotics, the furuncle was scooped out under local anaesthesia. The ear canal was regularly dressed for two weeks, and the ear was inspected periodically to check for the recurrence of granulation tissue. There was no reappearance, and the patient was completely cured within one month. Case 7 Infected epidermoid cyst with granulation: A 12-year-old male child presented with complaints of discharge, intermittent pain, and swelling in the left EAC. On palpation with a probe, granulation tissue was observed to originate from the posterior wall of the EAC. Initially, cholesteatoma was suspected, but a HRCT of the temporal bone revealed a normal middle ear and mastoid. During exploration in the operation theatre, a cystic swelling was observed. The cystic mass DISCUSSION Aural granulation tissues are often thought to be the result of CSOM only. However, the present case series illustrates that other causes should be considered, and it is prudent to actively rule out other causes by taking a detailed history. When the CT scan shows a normal middle ear and mastoid without any involvement by the disease process, other pathologies of the EAC need to be considered . Aural granulation tissues are typically of inflammatory origin and can provide clues to the underlying pathology when examined under a microscope, indicating whether they are a result of CSOM or an indicator of some rare pathological process . The proper diagnosis of EAC granulations relies on a detailed history, meticulous clinical evaluation, and adequate investigations. HRCT of the temporal bone is a useful addition to the battery of investigations, serving several purposes. Firstly, it provides a three-dimensional view of the anatomy of the middle ear, mastoid, and EAC. Secondly, it helps in determining the extent and spread of the disease process into the surrounding areas. Thirdly, it aids in predicting impending complications. When the otoscopy is unable to visualise the tympanic membrane due to overlying granulation tissue, HRCT is the best method to assess the condition of the middle ear . In the present study, HRCT was performed in all cases, assisting the authors in establishing a final diagnosis. Similarly, cytology and histopathology also aided in establishing a pathological diagnosis [5,6]. The EAC is an uncommon site for malignant neoplasms. When it does occur, squamous cell carcinoma is the most frequent, followed by basal cell carcinoma in terms of occurrence rate . Squamous cell carcinomas of the temporal bone most commonly affect individuals in their fifth or sixth decade of life . A study conducted by Kishore P et al., revealed that a neglected retained foreign body in the ear can lead to common ear symptoms such as hearing loss, pain, ear discharge, and a sensation of fullness in the ear. They can also cause EAC granulations and aural polyps. These conditions do not respond to conservative management and require early diagnosis and surgical intervention for foreign body removal . Tuberculosis of the EAC is a rare condition but can be observed in developing countries like India where tuberculosis still poses a significant burden. Therefore, when a patient with long-standing ear discharge is resistant to routine antibiotics and is found to have granulations in the EAC, suspicion of tuberculosis should be considered . Manas Ranjan Satpathy et al., External Auditory Canal Granulations Other than Cholesteatoma www.njlm.net National Journal of Laboratory Medicine. 2024 Jan, Vol-13(1): PS01-PS04 4 PARTICULARS OF CONTRIBUTORS: 1. Assistant Professor, Department of ENT, FM Medical College, Balasore, Odisha, India. 2. Assistant Professor, Department of Pathology, PRM Medical College, Baripada, Odisha, India. 3. Assistant Professor, Department of ENT, PRM Medical College, Baripada, Odisha, India. PLAGIARISM CHECKING METHODS: [Jain H et al.] • Plagiarism X-checker: Jul 12, 2023 • Manual Googling: Aug 25, 2023 • iThenticate Software: Aug 31, 2023 (11%) Etymology: Author Origin NAME, ADDRESS, E-MAIL ID OF THE CORRESPONDING AUTHOR: Dr. Rashmi Rekha Mahapatra, Assistant Professor, Department of Pathology, PRM Medical College, Baripada, District-Mayurbhanj-757107, Odisha, India. E-mail: rashmimahapatra123@gmail.com Date of Submission: Jul 11, 2023 Date of Peer Review: Aug 04, 2023 Date of Acceptance: Sep 01, 2023 Date of Publishing: Jan 01, 2024 Author declaration: • Financial or Other Competing Interests: None • Was informed consent obtained from the subjects involved in the study? Yes • For any images presented appropriate consent has been obtained from the subjects. Yes Emendations: 6 Malignant otitis externa occurs in immunocompromised or diabetic and renal failure patients. It carries a bad prognosis due to the compromised health status of these patients. Cranial nerve palsies are frequently associated with this condition, which in turn leads to high morbidity and mortality. Therefore, early diagnosis and management of malignant otitis externa are crucial to reduce complications and mortality rates . According to Kumar A et al., trauma to the EAC can injure the skin, expose the cartilage, and result in the formation of granulation tissue. Infection and retained foreign bodies in such cases can further complicate the disease process in this blind-ended pouch. Infections and granulation can be managed with appropriate medical therapy, but surgical intervention is necessary to prevent and treat post-traumatic canal stenosis, which is common in these cases . Furunculosis is the localised form of external otitis characterised by infection in a single hair follicle of the cartilaginous EAC. If left untreated or undertreated, it can progress and become an organised furuncle of the EAC. Chronic infection leads to the formation of granulation tissue at the site where the perichondrium of the cartilaginous EAC is exposed. Epidermoid cysts of the EAC primarily occur in its cartilaginous portion. This is because the bony EAC lacks skin appendages such as hair follicles and sebaceous glands. Epidermoid cysts are not very common in the EAC and are mainly observed in paediatric patients, being rare in adults . They can cause obstruction, resulting in decreased hearing, pain, and infection, leading to discharge and granulations . In the study of EAC lesions conducted by Chatra PS, it was observed that various types of lesions, including inflammatory, neoplastic, congenital, and traumatic lesions, can affect the EAC . Similarly, in the present study, we also found different causes of EAC granulations, including inflammatory, infective, neoplastic, and traumatic causes. The treatment of these patients was guided by the final diagnosis. Squamous cell carcinoma was treated with chemoradiation. Patients with foreign bodies and granulation underwent microscopic examination and removal of the foreign body. They were followed up until the granulation tissue disappeared through conservative treatment. Tubercular otitis externa was treated with antitubercular therapy. Malignant otitis externa was initially treated with intravenous antibiotics, followed by oral Levofloxacin. Post-traumatic granulation was managed with debridement and regular dressing. The CT scan of the patient with granulations in the cartilaginous portion of the EAC but normal bony canal and middle ear showed no abnormalities. Granulation tissue associated with furuncle was excised under local anaesthesia, followed by antibiotic dressing. The patient with granulation associated with an infected epidermoid cyst initially underwent conservative treatment with debridement and antibiotics, followed by complete excision of the cyst after one month. CONCLUSION(S) The authors suggested considering all possible differential diagnoses whenever a case of EAC granulation does not respond to medical management. In such situations, patients should undergo a comprehensive evaluation that includes a detailed medical history, a thorough clinical examination, and assessments through radiological, microbiological, and pathological investigations. In these cases, conducting a histopathological study is imperative to reach a definitive diagnosis. REFERENCES Handzel O, Halperin D. Necrotizing (malignant) external otitis. Am Fam Physician. 2003;68(2):309-12. PMID: 12892351. Abdel Tawab HM, Kumar VR, Tabook SMS. A surprising finding after external ear polypectomy in a deaf mute patient. Case Rep Otolaryngol. 2015;2015:401708. Kim JR, Im H, Chae SW, Song JJ. Clinical features of benign tumors of the external auditory canal according to pathology. Ann Otolaryngol Rhinol. 2017;4(3):1169. Friedmann I. Pathological lesions of the external auditory meatus: A review. Journal of the Royal Society of Medicine. 1990;83(1):34-37. Vanneste F, Casselman J, Lemahieu SF, Wilms G. High resolution CT findings in diseases of the external auditory canal. A review of 31 cases. J Belg Radiol. 1989;72(3):199-205. Gerber C, Zimmer G, Linder T, Schuknecht B, Betts DR, Walter R. Primary pleomorphic adenoma of the external auditory canal diagnosed by fine needle aspiration cytology. Acta Cytol. 1999;43(3):489-91. org/10.1159/ 000331106. Allanson BM, Low TH, Clark JR, Gupta R. Squamous cell carcinoma of the external auditory canal and temporal bone: An update. Head Neck Pathol. 2018;12(3):407-18. Doi: 10.1007/s12105-018-0908-4. Epub 2018 Aug 1. PMID: 30069837; PMCID: PMC6081282. Lobo D, Llorente JL, Suárez C. Squamous cell carcinoma of the external auditory canal. Skull Base. 2008;18(3):167-72. Doi: 10.1055/s-2007-994290. PMID: 18978963; PMCID: PMC2459330. Kishore P , Amalanathan S, Rajendiran A, Colbert KAR. An unusual presentation of the external auditory canal mass. Egypt J Otolaryngol. 2023;39(1):03. https:// doi.org/10.1186/s43163-022-00366-z. Swain SK, Sahu MC. Tubercular otitis externa in an elderly male- A case report. Iran J Otorhinolaryngol. 2019;31(103):127-30. PMID: 30989081; PMCID: PMC6449525. Karaman E, Yilmaz M, Ibrahimov M, Haciyev Y, Enver O. Malignant otitis externa. J Craniofac Surg. 2012;23(6):1748-51. Doi: 10.1097/SCS.0b013e31825e4d9a. PMID: 23147298. Kumar A, Aniruddha S, Shailendra K. Post-traumatic external auditory canal stenosis causing conductive hearing loss, case in point. Clinics in Otology. 2016;3(3):174-76. Suzuki T, Taki M, Shibata T, Matsunami T, Sakaguchi H, Yamamoto S, et al. Epidermal cyst of the bony external auditory canal. Otolaryngol Head Neck Surg. 2007;136(1):155-56. Doi: 10.1016/j.otohns.2006.02.028. PMID: 17210360. Kim GW, Park JH, Kwon OJ, Kim DH, Kim CW. Clinical characteristics of epidermoid cysts of the external auditory canal. J Audiol Otol. 2016;20(1):36-40. Doi: 10.7874/jao.2016.20.1.36. Epub 2016 Apr 21. PMID: 27144232; PMCID: PMC4853893. Chatra PS. Lesions in the external auditory canal. Indian J Radiol Imaging. 2011;21(4):274-78. Doi: 10.4103/0971-3026.90687. PMID: 22223939; PMCID: PMC3249942.
12331	https://sites.millersville.edu/bikenaga/math-proof/divisibility/divisibility.pdf	3-27-2020 Divisibility Deﬁnition. If a and b are integers, then a divides b if an = b for some integer n. In this case, a is a factor or a divisor of b. The notation a \| b means “a divides b”. The notation a ̸ \| b means a does not divide b. Notice that divisibility is deﬁned in terms of multiplication — there is no mention of a “division” operation. The deﬁnition agrees with ordinary usage: For example, 12 divides 48, because 12 · 4 = 48. It does have the following peculiar consequence: 0 \| 0 because (for instance) 0 · 417 = 0. You may object that “you can’t divide by 0”, but that is a diﬀerent use of the word “divide” — it refers to multiplying by the multiplicative inverse, and it’s true that 0 doesn’t have a multiplicative inverse in any “reasonable” number system. If this still troubles you, there’s no great harm in adding the condition “a ̸= 0” to the deﬁnition above. Here are some things to keep in mind when writing proofs involving divisibility: (a) It’s often useful to translate divisibility statements (like a \| b) into equations using the deﬁnition. (b) Do not use fractions or the division operation (“/” or “÷”) in your proofs! Proposition. Let a, b, and c be integers. (a) If a \| b and b \| c, then a \| c. (b) If a \| x and a \| y, then a \| (bx + cy) for all b, c ∈Z. Proof. (a) Suppose a \| b and b \| c. Now a \| b means that am = b for some m and b \| c means that bn = c for some n. Hence, amn = c, so a \| c. (b) Suppose a \| x and a \| y. Now a \| x means that am = x for some m and a \| y means that an = y for some n. Then bx + cy = bam + can = a(bm + cn), so a \| bx + cy. An expression of the form bx + cy is called a linear combination of x and y. Here are some special cases of part (b): 1. If a divides x and y, then a divides x + y and x −y. 2. If a divides x, then a divides bx for all b. In the ﬁrst case, apply (b) with b = 1 and c = 1 (and b = 1 and c = −1). In the second case, apply (b) c = 0. Example. Suppose n is an integer. Prove that the only positive integer that divides both 2n + 3 and 3n + 4 is 1. Suppose k is a positive integer and k \| 2n + 3 and k \| 3n + 4. Then by part (b) of the preceding proposition, k divides any linear combination of 2n + 3 and 3n + 4. I’ll choose a combination so that the n-terms cancel: k \| 3 · (2n + 3) −2 · (3n + 4) = 1. So k is a positive integer which divides 1 — but the only positive integer which divides 1 is 1. Hence, k = 1. 1 Deﬁnition. An integer n > 1 is prime if the only positive divisors of n are 1 and n. An integer n > 1 which is not prime is composite. For example, the ﬁrst few primes are 2, 3, 5, 7, 11, 13, . . .. On the other hand, the ﬁrst few composite numbers are 4, 6, 8, 9, 10, . . .. Proposition. If n is composite, then there are integers a and b such that 1 < a, b < n and n = ab. Proof. Since n is composite, it is not prime. Therefore, n has a positive divisor a other than 1 and n. Suppose n = ab. I still have to show that 1 < a, b < n. Note that if b = 1, then a = n (contradiction), and if b = n, then a = 1 (contradiction). So a and b are both diﬀerent from 1 and n. Suppose on the contrary that a > n. Since b > 1, it follows that n = ab > n · 1 = n. This is a contradiction. Likewise, if b > n, then since a > 1, I have n = ab > 1 · n = n. This is a contradiction. Now I know that a and b are positive integers which are not greater than n, and neither is 1 or n. This implies that 1 < a, b < n. Proposition. Every integer n > 1 has a prime factor. Proof. I’ll use induction, starting with n = 2. In fact, 2 has a prime factor, namely 2. Suppose that n > 2, and that every integer k less than n has a prime factor. I must show that n has a prime factor. If n is prime, then n has a prime factor, namely itself. So assume n is composite. By the last lemma, there are integers a and b such that 1 < a, b < n and n = ab. If either a or b is prime, then I have a prime factor of n. Suppose then that a and b are both composite. In this case, since a < n, I know that a must have a prime factor, by induction. But a prime factor of a is a prime factor of n, by transitivity of divisibility. This completes the induction step, and the proof. I sketched the proof of the following result when I discuss proof by contradiction. Having proved the last two results, the proof is now complete — but I’ll repeat it here. It is essentially the proof in Book IX of Euclid’s Elements. Theorem. There are inﬁnitely many primes. Proof. Suppose on the contrary that there are only ﬁnitely many primes p1, p2, . . . , pn. Consider the number p1p2 · · · pn + 1. When this number is divided by p1, p2, . . . , pn, it leaves a remainder of 1. Therefore, it has no prime factors. This contradicts the preceding lemma. Hence, there must be inﬁnitely many primes. 2 The situation changes greatly if you consider primes of a restricted form. For example, it’s not known whether there are inﬁnitely many Mersenne primes — primes of the form 2n −1, where n > 1. Proposition. If n is composite, then it has a prime factor p such that p ≤√n. Proof. First, an earlier result shows that there are integers a and b such that 1 < a, b < n and n = ab. If a > √n and b > √n, then n = ab > (√n)(√n) = n, which is a contradiction. Therefore, a and b can’t both be greater than √n. Suppose without loss of generality that a ≤√n. Then either a is prime or a has a prime factor, by the preceding lemma. In either case, I have a prime less than or equal to √n which divides a, and hence divides n. Example. Use trial division to determine whether 163 is prime. By the last lemma, to test whether n is prime, divide n in succession by the primes less than √n. If no such prime divides n, then n is prime. I have √ 163 = 12.76715 . . .. By trial, I ﬁnd that 2 ̸ \| 163, 3 ̸ \| 163, 5 ̸ \| 163, 7 ̸ \| 163, 11 ̸ \| 163. Since these are all the primes less than √ 163, it follows that 163 is prime. There are simple tests for divisibility by small numbers based on the decimal representation of a number. If an an−1 . . . a1 a0 is the decimal representation of a number, its digital sum is D(an an−1 . . . a1 a0) = an + an−1 + · · · + a1 + a0. That is, D(x) is the sum of the digits of x. For example, D(119) = 1 + 1 + 9 = 11, D(247) = 2 + 4 + 7 = 13. Proposition. (a) A number is even (divisible by 2) if and only if its units digit is 0, 2, 4, 6, or 8. (b) A number is divisible by 5 if and only if its unit digit is 0 or 5. (c) A number is divisible by 3 if and only if its digital sum is divisible by 3. (d) A number is divisible by 9 if and only if its digital sum is divisible by 9. Proof. Suppose x = an an−1 . . . a1 a0 is the decimal representation of a positive integer x. Then x = an · 10n + an−1 · 10n−1 + · · · + a1 · 10 + a0. All of the results can be proved by using this representation (and where appropriate, the digital sum). For example, here’s a sketch of the proof of (a). Note that since 2 \| 10, 2 \| an · 10n, 2 \| an−1 · 10n−1, . . . , 2 \| a1 · 10. Thus, for some integer m, x = 2m + a0. From this, it follows that 2 \| x if and only if a0 is 0, 2, 4, 6, or 8. I’ll let you write out the details. 3 For (c) and (d), note that x −D(x) = (an · 10n + an−1 · 10n−1 + · · · + a1 · 10 + +a0) −(an + an−1 + · · · + a1 + a0) = an(10n −1) + an−1(10n−1 −1) + · · · + a1(10 −1). Each term of the form 10k −1 is 999 . . .9 (k −1 nines), so the right side is divisible by 3 and by 9. Thus, x −D(x) is divisible by 3 and 9, so x is divisible by 3 or 9 if and only if D(x) is. If you’re uncomfortable about using the decimal representation of 10k −1 as 999 . . .9, you can note that for k ≥1, 10k −1 = (10 −1)(10k−1 + 10k−2 + · · · + 10 + 1) = 9(10k−1 + 10k−2 + · · · + 10 + 1). Hence, 9 \| 10k −1. For example, 9183 is divisible by 3, since 9 + 1 + 8 + 3 = 21 is divisible by 3. And 725 is not divisible by 9, because 7 + 2 + 5 = 14 is not divisible by 9. Remark. The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed as a product of powers of primes, and this expression is unique up to the order of the factors. For example, 720 = 24 · 32 · 5. Here is a sketch of the proof that every positive integer greater than 1 can be expressed as a product of powers of primes. Do a generalized induction: n = 2 is a product of a single prime (namely 2), and that is the basis step. Take an integer n > 2, and suppose every integer greater than 1 and less than n can be written as a product of powers of primes. If n is prime, we’re done (since a prime is product of a single prime, namely itself). If n is not prime, an earlier result show it can be factored as n = ab, where 1 < a, b < n. By the induction hypothesis, factor a and b into products of powers of primes. Then putting their factorizations together shows n factors into a product of powers of primes. The proof that a factorization into a product of powers of primes is unique up to the order of factors uses additional results on divisibility (e.g. Euclid’s lemma), so I will omit it. While this result is very important, overuse of the Fundamental Theorem in divisibility proofs often results in sloppy proofs which obscure important ideas. Try to write your proofs in other ways. Deﬁnition. Let m and n be integers, not both 0. The greatest common divisor (m, n) of m and n is the largest integer which divides both m and n. The reason for not deﬁning “(0, 0)” is that any integer divides both 0 and 0 (e.g. 4571 \| 0 because 4571 · 0 = 0), so there is no largest integer which divides both 0 and 0. Here are some numerical examples: (6, 8) = 2, (−15, 10) = 5, (77, 0) = 77. Proposition. Let m and n be integers, not both 0. (a) (m, n) ≥1. (b) (m, n) = (\|m\|, \|n\|). (c) (m, n) \| m and (m, n) \| n. (d) If a and b are integers, then (m, n) \| am + bn. Proof. (a) 1 is a common divisor of any two integers m and n. Since (m, n) is the greatest common divisor, (m, n) ≥1. 4 (In particular, (m, n) must be positive.) (b) First, I’ll show m and \|m\| have the same divisors. If k \| m, then m = ak for some integer a. So −m = (−a)k, and hence k \| −m. But \|m\| is either m or −m, and hence k \| \|m\|. Conversely, suppose k \| \|m\| — say \|m\| = ak for some integer a. If \|m\| = m, then k \| m. And if \|m\| = −m, then ak = −m, so (−a)k = m, and k \| m. Thus, m and \|m\| have the same divisors, and likewise n and \|n\| have the same divisors. It follows that the common divisors of m and n are the same as the common divisors of \|m\| and \|n\|. Since the sets of common divisors are the same, their largest elements must be the same — that is, (m, n) = (\|m\|, \|n\|). (This means that when you compute the greatest common divisor of two numbers, you can take absolute values to get two positive numbers.) (c) This follows from the deﬁnition: (m, n) (is the largest integer which) divides both m and n. (d) Since (m, n) \| m and (m, n) \| n, we have (m, n) \| am + bn by an earlier divisibility result. You might be able ﬁnd the greatest common divisor of two relatively small numbers by factoring. But what if the numbers are too big to be factored? The Euclidean algorithm gives a method for computing the greatest common divisor of two positive integers using only integer division. Example. Compute (271, 113). I’ll arrange the computations in a table with two columns. Begin the ﬁrst column with the larger number ﬁrst. Divide 271 by 113: 271 = 2 · 113 + 45. Put the quotient 2 next to the 113 and the reaminder 45 below the 113. (Note that there is a blank space (marked with a “-”) next to 271. This isn’t important here, but later you may see a third column added to this table for the Extended Euclidean Algorithm, in which case the blank space is important.) Continue in the same fashion: Divide 113 by 45: 113 = 2 · 45 + 23. Put the quotient 2 next to 45 and the remainder 23 below 45. 271 − 113 2 45 2 23 1 22 1 1 22 The table stops when you get an a ﬁrst column number “divides evenly into” the one above it. The the remainder is 0, and since you can’t divide by 0, the process must stop. The last nonzero remainder is the greatest common divisor. So (271, 113) = 1. You can at least see from this example why the process has to stop. When you divide, the remainder is always less that the thing you divided by. So the remainders in the ﬁrst column are positive numbers that keep getting smaller — and since the process can’t go on forever (reason: the Well-Ordering Axiom for the positive integers), it must end in the only way possible with a remainder of 0. I won’t prove that this algorithm gives the greatest common divisor, but here’s the idea: The greatest common divisor of any two consecutive numbers in the ﬁrst column remains the same. Check it yourself in the table above. 5 Example. Show that if n is an integer, then (n, n + 2) is either 1 or 2. (n, n + 2) divides both n and n + 2, so it divides any linear combination of n and n + 2. In particular, (n, n + 2) \| (n + 2) −n = 2. Now (n, n + 2) ≥1; the only positive integers which divide 2 are 1 and 2. Therefore, (n, n + 2) is either 1 or 2. Notice that both of these cases can occur: If n = 1, then (n, n + 2) = (1, 3) = 1, and if n = 2, (n, n + 2) = (2, 4) = 2. Proposition. If am + bn = 1 for some a, b ∈Z, then (m, n) = 1. Proof. (m, n) \| m and (m, n) \| n, so (m, n) \| am + bn = 1. But (m, n) ≥1, and the only positive integer which divides 1 is 1. Therefore, (m, n) = 1. Deﬁnition. Let m, n ∈Z. m and n are relatively prime if (m, n) = 1. Thus, the last lemma says that if some linear combination of m and n equals 1, then m and n are relatively prime. Example. Prove that for all n ∈Z, 4n + 3 and 6n + 4 are relatively prime. Two integers are relatively prime if their only (positive) common factor is 1. Thus, this problem says that 1 is the only common factor of 4n + 3 and 6n + 4. The table below shows the values of 4n+3 and 6n+4 for −5 ≤n ≤5. The result seems plausible based on the evidence. n -5 -4 -3 -2 -1 0 1 2 3 4 5 4n + 3 -17 -13 -9 -5 -1 3 7 11 15 19 23 6n + 4 -26 -20 -14 -8 -2 4 10 16 22 28 34 To prove it, I’ll use part (a). I want numbers a and b such that a(4n + 3) + b(6n + 4) = 1. Since there are no n’s on the right side, I want to choose a and b to make the n’s on the left cancel out. One way to do this is 3 · (4n + 3) + (−2)(6n + 4) = 1. This linear combination is equal to 1, so by (a), (4n + 3, 6n + 4) = 1. As the example shows, one way of showing that two integers are relatively prime is to ﬁnd a linear combination of them that equals 1. The converse is true: If two integers are relatively prime, then some linear combination of the integers equals 1. In fact, more is true: The greatest common divisor (m, n) of m and n can always be written as a linear combination am+bn of m and n. An extended version of the Euclidean algorithm ﬁnds a linear combination am+bn such that (m, n) = am+bn. You’ll probably see this result in a course in abstract algebra or number theory; I won’t prove it here. c ⃝2020 by Bruce Ikenaga 6
12332	https://themindsjournal.com/find-the-missing-number-in-this-math-puzzle/	Find The Missing Number In 60 Seconds! Can You? 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Time Limit – 60 Seconds Only! Author : Kerin Stark Are you ready to jump into a playful math riddle? These puzzles are tests for your intelligence quotient and logical thinking as much as they are about numbers. Can you find the missing number in just 60 seconds? Give it a try and let us see how bright your numerical talent shines! Math problems with missing numbers are excellent for improving IQ and logical reasoning skills. They make you think deeper by analyzing information, recognizing patterns, and using deductive thinking to find solutions. So go ahead, exercise your brain, and have fun while at it! Find The Missing Number: Can You Replace The Question Mark? Calling all puzzlers! Prepare yourself for an electrifying challenge that will set your gray matter on fire. Presenting the ultimate test of wit and wisdom: “Can You Find The Correct Answer in 60 Seconds?” Do you think you can handle it? If you can understand number patterns, break down relationships between them, and solve everything within the time ticking away, congratulations! You belong among the world’s most brilliant mathematicians who ever lived. Math puzzles are not only about math but also require attention to detailcoupled with strong reasoning abilities, which would allow one to arrive at correct answers easily. Does your brain love untangling numerical mysteries with lightning-speed accuracy too? Well, then let’s see if you can beat our test by measuring quick thinking combined with mathematical skill—tick-tock goes the clock! Are you ready to show off just how smart you really are? Read more here: Emoji Quiz: Determine the Value of Smiling Face Emojis in 15 Seconds! Can You? Have you ever considered doing those “find the missing number” puzzles regularly? They’re not only fun games but also great exercises for your mind. Seriously speaking, these things work memory like nothing else does; they teach people how to pay close attention to even the smallest things around them—a critical skill useful in every area of life, basically. Apart from that, math becomes much easier once its concepts become clear through such activities. Scroll Below To Check Math Tricky Riddles With Answers To start with, we go 4 times 2 minus 1, and that equals seven. Next, take the result, which is seven (7), and multiply it by two (2), then subtract one (1). This gives us thirteen (13). Now do it again: take thirteen (13), multiply by two (2), and subtract one (1), giving twenty-five (25). We repeat this process of multiplying by two and subtracting one from twenty-five. This will give forty-nine. We take forty-nine now, multiply by two, and then subtract one. This will give you ninety-seven, so the missing number is indeed ninety-seven. What It Means Definitely! If you can find the correct answer, it suggests that there’s a challenge or question being asked where someone is expected to figure it out or give the right solution. It implies that there is a solution or missing information waiting to be found through reasoning, deduction, or problem-solving skills. On the other hand, if you can’t find the missing number, it means that the person may not be able to solve this problem because either they don’t know how to do math or it’s too hard for them. Read more here: Pattern Optical Illusion: Can You Spot the Odd Pattern in the Image? You Only Have 5 Seconds! Hey everyone, we’ve got a math puzzle for you to share with your friends and family! Can they solve it in just 60 seconds? Let’s kick off some friendly competition! Take a look at this tricky puzzle and see if you can crack it in under a minute. Share it with your loved ones, and challenge them to test their math skills too! Remember, the clock is ticking, so put your thinking caps on and let the fun begin! May the best puzzler win! ?⏳ Published On: April 22, 2024 Last updated on: February 20, 2025 Quiz brain test, math puzzle, minds journal, Quiz --- Share --- Share on Pinterest Share on Facebook Email this Page Share on LinkedIn Share on Telegram Share on Tumblr Share on Reddit Print this Page Email this Page Share on Pocket Share on WhatsApp Kerin Stark I am a person who loves to craft engaging narratives. As an extrovert, I thrive on connecting with audiences through my words. When I’m not immersed in writing, I find joy in dance, where movement becomes my language. Poetry and storytelling hold a special place in my heart. Let’s embark on a journey of words together. Disclaimer: The informational content on The Minds Journal have been created and reviewed by qualified mental health professionals. 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We’ll pick 15+ winners to be featured on our… Latest Articles The Season of the Dead: Eclipse Season Horoscope According to Vedic Astrology Self-Soothing: The Kind We Bring From Childhood 9 Creepy Portals To The Spirit World Lurking Inside Your Home Your Spiritual Guidance For October 2025: Channeled Messages For Your Soul It’s Time to Harvest: Full Moon Horoscope for 12 Zodiac Signs 12 Zodiac Girls and Their Most Attractive Qualities 11 Magic Phrases That Instantly Build Confidence in Your Child Bed Rotting: The Self-Care Myth You Need to Know About What Your Smile Says About You: The Personality Test Everyone’s Talking About October 2025 Monthly Horoscope: Predictions For Each Zodiac Sign Are you ready to jump into a playful math riddle? These puzzles are tests for your intelligence quotient and logical thinking as much as they are about numbers. Can you find the missing number in just 60 seconds? Give it a try and let us see how bright your numerical talent shines! Math problems with missing numbers are excellent for improving IQ and logical reasoning skills. They make you think deeper by analyzing information, recognizing patterns, and using deductive thinking to find solutions. So go ahead, exercise your brain, and have fun while at it! Find The Missing Number: Can You Replace The Question Mark? Calling all puzzlers! Prepare yourself for an electrifying challenge that will set your gray matter on fire. Presenting the ultimate test of wit and wisdom: “Can You Find The Correct Answer in 60 Seconds?” Do you think you can handle it? If you can understand number patterns, break down relationships between them, and solve everything within the time ticking away, congratulations! You belong among the world’s most brilliant mathematicians who ever lived. Math puzzles are not only about math but also require attention to detailcoupled with strong reasoning abilities, which would allow one to arrive at correct answers easily. Does your brain love untangling numerical mysteries with lightning-speed accuracy too? Well, then let’s see if you can beat our test by measuring quick thinking combined with mathematical skill—tick-tock goes the clock! Are you ready to show off just how smart you really are? Read more here: Emoji Quiz: Determine the Value of Smiling Face Emojis in 15 Seconds! Can You? Have you ever considered doing those “find the missing number” puzzles regularly? They’re not only fun games but also great exercises for your mind. Seriously speaking, these things work memory like nothing else does; they teach people how to pay close attention to even the smallest things around them—a critical skill useful in every area of life, basically. Apart from that, math becomes much easier once its concepts become clear through such activities. Scroll Below To Check Math Tricky Riddles With Answers To start with, we go 4 times 2 minus 1, and that equals seven. Next, take the result, which is seven (7), and multiply it by two (2), then subtract one (1). This gives us thirteen (13). Now do it again: take thirteen (13), multiply by two (2), and subtract one (1), giving twenty-five (25). We repeat this process of multiplying by two and subtracting one from twenty-five. This will give forty-nine. We take forty-nine now, multiply by two, and then subtract one. This will give you ninety-seven, so the missing number is indeed ninety-seven. What It Means Definitely! If you can find the correct answer, it suggests that there’s a challenge or question being asked where someone is expected to figure it out or give the right solution. It implies that there is a solution or missing information waiting to be found through reasoning, deduction, or problem-solving skills. On the other hand, if you can’t find the missing number, it means that the person may not be able to solve this problem because either they don’t know how to do math or it’s too hard for them. Read more here: Pattern Optical Illusion: Can You Spot the Odd Pattern in the Image? You Only Have 5 Seconds! Hey everyone, we’ve got a math puzzle for you to share with your friends and family! Can they solve it in just 60 seconds? Let’s kick off some friendly competition! Take a look at this tricky puzzle and see if you can crack it in under a minute. Share it with your loved ones, and challenge them to test their math skills too! Remember, the clock is ticking, so put your thinking caps on and let the fun begin! May the best puzzler win! ?⏳ Published On: April 22, 2024 Last updated on: February 20, 2025 Quiz brain test,math puzzle,minds journal,Quiz --- Share --- Share on Pinterest Share on Facebook Email this Page Share on LinkedIn Share on Telegram Share on Tumblr Share on Reddit Print this Page Email this Page Share on Pocket Share on WhatsApp Kerin Stark I am a person who loves to craft engaging narratives. As an extrovert, I thrive on connecting with audiences through my words. When I’m not immersed in writing, I find joy in dance, where movement becomes my language. Poetry and storytelling hold a special place in my heart. Let’s embark on a journey of words together. Leave A Comment Leave a Comment Cancel reply Comment Name Email Website - [x] Save my name, email, and website in this browser for the next time I comment. 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12333	https://www.kidney-international.org/article/S0085-2538(15)48976-8/pdf	Kidney International, Vol. 63 (2003), pp. 1066–1071 Effect of vitamin C supplements on urinary oxalate and pH in calcium stone-forming patients ALESSANDRA CALA ´ BRIA BAXMANN , C LAUDIA DE O.G. M ENDONC ¸ A , and I TA PFEFERMAN HEILBERG Nephrology Division, Universidade Federal de Sa ˜o Paulo, UNIFESP, Brazil Effect of vitamin C supplements on urinary oxalate and pH in Humans cannot synthesize vitamin C because they do calcium stone-forming patients. not have the last enzyme in the biosynthetic pathway . Background. The contribution of ascorbate to urinary oxa- The current recommended dietary allowance (RDA) for late is controversial. The present study aimed to determine vitamin C is 60 mg/day, but it is fairly common to find whether urinary oxalate and pH may be affected by vitamin subjects who because of health fads or the advice of C supplementation in calcium stone-forming patients. Methods. Forty-seven adult calcium stone-forming patients others, take megadoses of vitamin C on a daily basis in received either 1 g ( N 23) or 2 g ( N 24) of vitamin C addition to the amount provided by their usual diet . supplement for 3 days and 20 healthy subjects received 1 g. A Vitamin C can be metabolized to oxalate, which could 24-hour urine sample was obtained both before and after vita- increase oxalate excretion and hence the risk of calcium min C for calcium, oxalate, magnesium, citrate, sodium, potas-oxalate stone formation [3, 4]. Therefore, stone formers sium, and creatinine determination. The Tiselius index was used as a calcium oxalate crystallization index. A spot fasting are frequently advised to avoid vitamin C supplements. morning urine sample was also obtained to determine the uri- Most of the oxalate found in the urine is usually formed nary pH before and after vitamin C. endogenously as a metabolic end product of glycoxylic Results. Fasting urinary pH did not change after 1 g (5.8 acid (50% to 70%) and ascorbic acid (30% to 50%), 0.6 vs. 5.8 0.7) or 2 g vitamin C (5.8 0.8 vs. 5.8 0.7). A significant increase in mean urinary oxalate was observed in with minor contributions by gelatin, tryptophan, phenyl-calcium stone-forming patients receiving either 1 g (50 16 vs. alanine, tyrosine, aspartic acid, creatinine, and purines . 31 12 mg/24 hours) or 2 g (48 21 vs. 34 12 mg/24 hours) Several studies have been performed in the last 40 of vitamin C and in healthy subjects (25 12 vs. 39 13 mg/24 years in an attempt to determine the contribution of hours). A significant increase in mean Tiselius index was ob-high-dose vitamin C intakes to urinary oxalate excretion served in calcium stone-forming patients after 1 g (1.43 0.70 vs. 0.92 0.65) or 2 g vitamin C (1.61 1.05 vs. 0.99 0.55) [6–17]. However, the data from various investigations are and in healthy subjects (1.50 0.69 vs. 0.91 0.46). Ancillary contradictory, in part because of the difficulties regarding analyses of spot urine obtained after vitamin C were performed oxalate assay techniques [13–17]. In assays used before in 15 control subjects in vessels with or without ethylenedi- 1987, artificial oxalate elevation occurred due to inadver-aminetetraacetic acid (EDTA) with no difference in urinary oxalate between them (28 23 vs. 26 21 mg/L), suggesting tent ascorbate conversion to oxalate in stored samples, that the in vitro conversion of ascorbate to oxalate did not occur. either because of the requirement of heating the urine Conclusion. These data suggest that vitamin C supplementa- or because of the use of alkaline eluents [18, 19]. tion may increase urinary oxalate excretion and the risk of In healthy subjects, three short-term prospective stud-calcium oxalate crystallization in calcium stone-forming pa-ies conducted in the last 6 years, with appropriate precau-tients. tions to prevent nonenzymatic conversion of ascorbate to oxalate, have determined the effects of ascorbate in-take on oxaluria using doses ranging from 400 mg/day Vitamin C (ascorbic acid, ascorbate) is an essential micronutrient involved in many biological and biochemi- to 4 g/day [14–16]. Levine et al reported a significant cal functions, acting as an electron donor or reducing increase of 33% in urinary oxalate after 1 g vitamin C agent in chemical reactions . supplement, whereas Liebman et al and Auer, Auer, and Rodgers found that ascorbate doses from 2 to 4 g per day did not increase urinary oxalate. In a large Key words: vitamin C, ascorbate, oxalate, urinary pH, kidney stone, urinary calculi. epidemiologic study based on completion of dietary ques-tionnaires, Curhan et al did not observe a positive Received for publication June 24, 2002 association between consumption of vitamin C supple-and in revised form August 29, 2002 Accepted for publication October 16, 2002 ments and the risk of kidney stones in women. There are few studies focusing on the effect of vitamin  2003 by the International Society of Nephrology 1066 Baxmann et al: Vitamin C and oxalate excretion 1067 C supplementation on oxalate excretion in calcium-stone or 2 g (1000 mg twice a day) of vitamin C supplement for 3 days. The 24-hour urine sample (post) was collected forming patients [2, 3, 20, 21]. Three studies reported a significant increase from 31% to 100% in urinary oxalate during day 3 of vitamin C supplementation. A morning spot urine sample was also obtained after a 12-hour fast after vitamin C supplement at doses from 0.5 to 2 g per day [2, 3, 20]. On the other hand, Heckers et al to determine the urinary pH both at baseline and in the morning following day 3 of vitamin C supplementation found no increase in urinary oxalate in seven calcium stone-forming patients taking 1 g/day vitamin C. in the calcium stone-forming group. Twenty healthy sub-jects (8 men and 12 women) received 1 g (500 mg twice Although it has been suggested that vitamin C has some effect on urinary acidification, reducing the urinary a day) vitamin C supplement for 3 days and a 24-hour urine sample was obtained from them before and after pH, the role of vitamin C as a urinary acidifier is still controversial [22–30]. the supplementation. Since few studies focusing on the effect of vitamin C Effect of acid preservation on urinary on urinary oxalate excretion in a population of calcium oxalate determination stone formers have been performed, the objective of the present study was to determine whether urinary oxalate It has been suggested that, for a more reliable urinary oxalate measurement, urine must be acidified in order excretion and pH could be affected by vitamin C supple-mentation in calcium stone-forming patients. to ensure the complete dissolution of calcium oxalate crystals, with hydrochloric acid (HCl) being used as a preservative for urine collection. However, when acid is METHODS added to the plastic container before urine collection, Protocol other parameters such as uric acid, sodium, and potas-sium cannot be determined in the same urine sample Forty-seven adult calcium stone-forming patients (24 men and 23 women) participated in the study. Patients . To test whether the addition of acid after the urine specimen was delivered to the laboratory would interfere with diabetes, hyperparathyroidism, abnormal renal function, or those taking drugs that could affect calcium with the urinary oxalate results, an additional group con-sisting of 40 healthy subjects was submitted to two collec-metabolism were excluded. All patients were referred to the Renal Lithiasis Unit of the Nephrology Division, tions of 24-hour urine on different occasions. The first sample was obtained with acid preservation (HCl 6N, Universidade Federal de Sa ˜ o Paulo, Brazil, and were sequentially enrolled in the study after a diagnosis of 20 mL/L) and the second in a dry plastic container, with HCl added as soon as the urine sample was received at renal stone has been established. The diagnosis of stone disease was based on at least one of the following criteria: the laboratory. The mean oxalate excretion in these 40 control samples was similar for specimens previously (1) renal colic with confirmed hematuria, ( 2) voiding of a calculus, ( 3) previous surgical or endoscopic removal acidified or not (27 14 vs. 29 12 mg/24 hours, P 0.05, Wilcoxon test). Based on these results for the con-of stone(s), and/ ( 4) or radiographic (intravenous urogra-phy or ultrasonography) evidence of stone(s). A written trol samples, 24-hour urine samples obtained from the 47 calcium stone-forming patients pre- and post-vitamin consent was obtained from all patients and the local Ethics Committee approved the study. A 24-hour urine C supplement intake were then collected into a dry plas-tic container, with HCl added as soon as the urine sample sample was obtained from the 47 calcium stone-forming patients both before (pre) and after (post) vitamin C was delivered to the laboratory. However, since in vitro conversion of ascorbate to oxalate could be further in-supplementation for determination of calcium, oxalate, sodium, potassium, urea, creatinine, magnesium, uric duced by a nonacid environment, the 47 calcium stone-forming patients and the 20 controls were also asked to acid, and citrate. Calcium, oxalate, magnesium, and ci-trate data were used to calculate the risk of calcium repeat the entire protocol, taking vitamin C supplements for 3 additional days, and to collect another 24-hour oxylate crystallization by the Tiselius index, calculated according to the formula that follows: 1.9 calcium 0.84 urine sample on day 3 in a vessel containing HCl. oxalate magnesium 0.12 citrate 0.22 volume 1.03 Effect of EDTA preservation on urinary . Patients were instructed to abstain from consuming oxalate determination oxalate-rich and vitamin C–rich foods (a listing of these foods was provided), as well as dairy products for the It has been suggested that urine should be collected with EDTA for ascorbate stabilization and to inhibit its two 24-hour periods of urine collection (baseline and after vitamin C). The purpose of these measures was conversion to oxalate [16, 17, 33, 34]. To assess whether our results could have been biased by an in vitro conver-to avoid any influence of diet (with respect to oxalate, ascorbate, or calcium intake) on oxalate excretion other sion of ascorbate to oxalate, we performed an additional experiment by collecting a spot urine from 15 control than vitamin C supplementation. Patients were then ran-domly selected to receive either 1 g (500 mg twice a day) subjects 6 hours after intake of the vitamin C supplement Baxmann et al: Vitamin C and oxalate excretion 1068 Table 1. Mean urinary parameters before and after vitamin C supplements in calcium stone-forming patients and healthy subjects Calcium stone forming Healthy subjects Calcium stone forming Vitamin C, 1 g ( N 23) Vitamin C, 1 g ( N 20) Vitamin C, 2 g ( N 24) Pre Post Post a Pre Post a Pre Post Post aCalcium mg /24 hours 153 58 169 80 188 102 155 99 159 69 129 44 155 98 157 81 Oxalate mg /24 hours 31 12 50 16 b 49 19 b 25 13 c 39 13 b,c 34 12 48 21 b 59 19 bSodium mEq /24 hours 209 95 203 83 ND 172 57 ND 194 83 197 98 ND Potassium mEq /24 hours 59 25 61 25 ND 56 18 ND 58 23 54 27 ND Urea g/24 hours 20 6 19 9 20 8 20 8 21 9 16 8 22 4 18 9Creatinine mg /24 hours 1413 346 1335 423 1342 481 1262 536 1257 441 1404 404 1354 424 1403 512 Uric acid mg /24 hours 608 219 415 168 b ND 465 139 c ND 484 187 305 109 ND Magnesium mg /24 hours 81 34 81 32 87 37 70 31 73 21 69 28 70 23 91 42 bCitrate mg /24 hours 347 208 380 246 502 287 432 227 423 225 376 249 331 197 392 219 Tiselius index 0.91 0.65 1.43 0.7 b 1.41 0.8 b 0.91 0.46 1.5 0.69 b 0.98 0.55 1.61 1.05 b 1.62 1.0 bVolume 1997 817 1924 618 1899 578 1250 669 1308 528 1634 565 1649 674 2104 772 bX SD, a Collected under acid preservation. b vs. pre; c vs. calcium stone-forming P 0.05 (1 g) and dividing the samples into two aliquots, with or magnesium, and citrate and the calcium oxalate crystalli-zation Tiselius index before and after the vitamin C sup-without EDTA (final concentration, 0.01 mol/L). The results showed no difference in oxalate concentration plements in calcium stone-forming patients and healthy subjects. Calcium stone-forming patients presented sig-between the samples with and without EDTA (28 23 vs. 26 21 mg/L, respectively, P 0.05, Wilcoxon test) nificantly higher mean values of urinary oxalate than healthy subjects, both before (31 12 vs. 25 13 mg/24 hours, P 0.05) and after the vitamin C supplements METHODS (50 16 vs. 39 13 mg/24 hours, P 0.05). Calcium Urinary parameters stone-forming patients also presented higher urinary uric acid than healthy subjects (608 219 vs. 465 139 mg/24 Urinary calcium was determined by atomic absorption hours, P 0.05). Among healthy subjects, a significant spectrophotometry (Perkin-Elmer Atomic Spectropho-increase in mean urinary oxalate level (39 13 vs. 25 tometer 290-B, Norwalk, CT, USA), oxalate by an enzy-13 mg/24 hours, P 0.05) and in mean Tiselius index matic reaction using the Sigma Oxalate Diagnostic Kit (1.5 0.69 vs. 0.91 0.46, P 0.05) was observed after (Sigma Chemical Co., St. Louis, MO, USA), sodium, the vitamin C supplement in comparison to baseline, and potassium by flame photometry (Celm Fc-130), urea with no changes in the remaining urinary parameters. In by an enzymatic ultraviolet test, creatinine by Jaffe’s calcium stone-forming patients, a significant increase in method , uric acid by an automated colorimetric en-mean urinary oxalate was observed after supplementa-zymatic method (ABA VP), and citrate by an enzymatic tion with either 1 g (50 16 vs. 31 12 mg/24 hours) assay using citrate lyase . The urinary pH was ob-or 2 g (48 21 vs. 34 12 mg/24 hours) vitamin C tained in the second micturition after a 12-hour fast, and compared to baseline, representing a 61% and 41% in-determined with a pH meter. crement in mean oxaluria, respectively. This increase Statistical analysis was also observed in urine samples further collected un-der acid preservation for calcium stone-forming patients Results are reported as mean standard deviation taking 1 g (49 19 vs. 31 12 mg/24 hours) or 2 g (SD). The Wilcoxon test was used to compare the results vitamin C (59 19 vs. 34 12 mg/24 hours). There was obtained after the vitamin C supplement to those ob-a significant reduction of uric acid (415 168 vs. 608 tained before the supplement in the same group. The 219 mg/24 hours) in calcium stone-forming patients tak-Mann-Whitney test was used to compare the differences ing 1 g, but this reduction has occurred within normal between the calcium stone-forming and healthy subject limits, and was not observed among the patients taking groups. The level of significance was defined as P 0.05. 2 g. Similarly, a significant increase in urinary magnesium (91 42 vs. 69 28 mg/24 hours) in calcium stone- RESULTS forming patients taking 2 g, but not in these taking 1 g Mean age (37 14 vs. 37 16 years) and body mass was observed but again, such increase occurred within index (27 5 vs. 24 4 kg/m 2) did not differ between the range of normal limits. A significant increase in mean calcium stone-forming patients and healthy subjects Tiselius index was observed after supplementation with (data not shown in tables). either 1 g (0.92 0.65 vs. 1.43 0.70) or 2 g vitamin C Table 1 shows the mean values of urinary calcium, (0.99 0.55 vs. 1.61 1.05). The remaining parameters remained unchanged after vitamin C. oxalate, sodium, potassium, urea, creatinine, uric acid, Baxmann et al: Vitamin C and oxalate excretion 1069 Fig. 1. Urinary pH pre- and post-vitamin C supplements. Mean values are indicated by horizontal bars. Figure 1 shows that the mean urinary pH values before vs. after1 or 2 g vitamin C supplementation in calcium stone-forming patients were not significantly different. The distribution of the urinary oxalate values in per- Fig. 2. Box plots of urinary oxalate pre- and post-vitamin C with or centiles is shown in Figure 2A and B. without acid preservation in calcium stone-forming patients taking sup-plements of either 1 gram ( A) or 2 grams ( B). The horizontal lines in the box denote the 25th, 50th, and 75th percentile values. The error DISCUSSION bars denote the 5th and 95th percentile values. The symbols below the 5th percentile error bar denote the zero and 1st percentile values and The assumption that the intake of high doses of vita- the symbols above the 95th percentile error bar denote the 99th and min C may be a major causative factor in the formation 100th percentile. of renal calcium oxalate stones is old, dating back to the finding that oxalate is one of the metabolic end products of ascorbic acid excreted in the urine . Several studies have been conducted on healthy subjects to examine the ascorbic acid in four calcium stone-forming patients and with Chalmers, Cowley, and Brown who observed a effect of vitamin C on urinary oxalate excretion using different doses and periods of supplementation [6–17]. 79% increase in mean urinary oxalate after 2 g vitamin C in 17 calcium stone-forming patients. Differences re-On the other hand, a reduced number of studies has been performed to examine the effect of vitamin C on urinary garding the magnitude of urinary oxalate increase be-tween the present data and these other studies may be oxalate in calcium stone-forming patients [2, 3, 20, 21]. However, the data from various investigations are con- ascribed to different oxalate assays and duration of sup-plementation. Urivetzky, Kessaris, and Smith , using tradictory, in part because of difficulties regarding oxa-late assay techniques. Despite newer assays that mitigate the same assay as ours, also observed increases of 38% and 107% in oxaluria following doses of 1 g and 2 g per the in vitro conversion of ascorbic acid to oxalate, contro-versy still remains, with some studies suggesting that day in 15 calcium stone-forming patients, respectively. Conversely, no increase in urinary oxalate after a single vitamin C leads to an increase in oxaluria , whereas others do not support this observation [13, 15, 16]. dose of 1 g vitamin C was detected by Heckers et al , who regrettably did not indicate the assay used for In the present study, a significant increase of 61% and 41% was observed in mean urinary oxalate after urinary oxalate determination. In the present series, the significant increase in urinary supplementation with 1 or 2 g vitamin C, respectively, in calcium stone-forming patients. The lack of a further oxalate observed even in the urine specimens collected in a gallon containing acid preservative minimizes the increase on urinary oxalate after 2 g vitamin C may be ascribed to a saturable transport mechanism leading to possibility that the increase in oxaluria after vitamin C was due to the in vitro nonenzymatic conversion of ascor-a reduced relative absorption capacity with increasing intakes of the compound [5, 38]. Our findings are in bate to oxalate since HCl reduces the urinary pH to values around 1. According to Auer, Auer, and Rodgers , accordance with Tiselius et al who observed a 48% increase in oxaluria following a dose of 1 g per day of the possibility that urinary oxalate measurements may be Baxmann et al: Vitamin C and oxalate excretion 1070 falsely elevated by the presence of high urinary ascorbate allowance. Vitamin C does not seem to be an efficient urinary acidifier in calcium stone-forming patients. must be considered. However, Liebman et al re-ported that whereas 2 g vitamin C supplement produced increments in mean urinary ascorbate concentration in ACKNOWLEDGMENTS six healthy subjects ranging from 100 to 540 mg/L, the Research supported by grants from Coordenac ¸a ˜ o de Aperfeic ¸oame-nto Pessoal de Nı ´vel Superior (CAPES), Conselho Nacional de Desen-increases in urinary oxalate were less than 1.0 mg/L, volvimento Cientı ´fico e Tecnolo ´ gico (CNPq), and Fundac ¸a ˜ o Oswaldo suggesting that urinary oxalate data did not appear to be Ramos. The authors wish to express their thanks to Silvia Regina confounded by the potential interference of ascorbate. Moreira for technical assistance. It has been suggested that the addition of disodium Reprint requests to Ita Pfeferman Heilberg, M.D., Universidade, Fed-EDTA stabilizes ascorbate in urine and inhibits its con- eral de Sa ˜o Paulo, Nephrology Division, Rua: Botucatu, 740 Vila Clem- version to oxalate [16, 17, 33, 34]. However, when EDTA entino, Sa ˜o Paulo, SP, Brazil 04023-900. E-mail: ipheilberg@nefro.epm.br is added to the container before urine collection, other parameters such as calcium and sodium cannot be deter-mined in the same urine sample. Therefore, to assess REFERENCES whether the addition of EDTA could prevent the in 1. Levine M, Rumsey SC, Daruwala R , et al : Criteria and recommen-dations for vitamin C intake. JAMA 281:1415–1423, 1999 vitro conversion of ascorbate to oxalate, we performed 2. Urivetzky M, Kessaris D, Smith AD : Ascorbic acid overdosing: ancillary analyses in spot urine obtained from 15 healthy A risk factor for calcium oxalate nephrolithiasis. J Urol 149:1215– volunteers with or without EDTA after the intake of the 1218, 1992 3. Chalmers A, Cowley D, Brown J : A possible etiological role for vitamin C supplement and found no difference between ascorbate in calculi formation. Clin Chem 32:333–336, 1986 them. These findings minimize the possibility that our 4. Curhan GC, Williet WC, Speizer FE, Stampfer MJ : Intake of oxaluria results were due to an in vitro conversion of vitamins B 6 and C and the risk of kidney stones in women. J Am Soc Nephrol 10:840–845, 1999 ascorbate to oxalate. 5. Gerster H : No contribution of ascorbic acid to renal calcium In the present study, calcium stone-forming patients oxalate stones. Ann Nutr Metab 41:269–282, 1997 presented significantly higher mean values of urinary 6. Takiguchi H, Furuyama S, Shimazono N : Urinary oxalic acid excretion by man following ingestion of large amounts of ascorbic oxalate than the healthy subjects both at baseline and acid. J Vitaminol 12:307–312, 1966 after the vitamin C supplements. Higher baseline values 7. Takenouchi K, Aso K, Kawase K , et al : On the metabolites of of urinary oxalate among stone formers when compared ascorbic acid, especially oxalic acid, eliminated in urine, following the administration of large amounts of ascorbic acid. J Vitaminol to normal volunteers have also been observed by other 12:49–58, 1966 investigators [39–41], including previous studies by our 8. Hatch M, Mulgrew S, Bourke E , et al : Effect of megadoses of group . It is possible that this increase in oxaluria ascorbic acid on serum and urinary oxalate. Eur Urol 6:166–169, 1980 results from lower levels of intestinal colonization by 9. Hughes C, Dutton S, Truswell AS : High intakes of ascorbic Oxalobacter formigenes , an oxalate-degrading bacterium acid and urinary oxalate. J Hum Nutr 35:274–280, 1981 leading to less oxalate degradation in the intestinal lu- 10. Schmidt KH, Hagmaier V, Hornig DH , et al : Urinary oxalate excretion after large intakes of ascorbic acid in man. Am J Clin men, as suggested by Sihdu et al . Nutr 34:305–311, 1981 Some studies have reported that ascorbic acid reduces 11. Fituri N, Allawi N, Bentley M, Costello J : Urinary and plasma urinary pH [23–25, 29, 30], whereas others have found oxalate during ingestion of pure ascorbic acid: a re-evaluation. Eur Urol 9:312–315, 1983 the agent to be ineffective as a urinary acidifier [22, 12. Tsao CS, Salimi SL : Effect of large intake of ascorbic acid on 26–28]. Some of these conflicting results may be ascribed urinary and plasma oxalic acid levels. Internat J Vit Nutr Res 54:245– to the type of urine collection, usually consisting of 24-hour 249, 1984 13. Wandzilak TR, D’Andre SD, Davis PA, Williams HE: Effect urine samples, which are subjected to diet interference of high dose vitamin C on urinary oxalate levels. J Urol 151:835– and eventual bacterial contamination. In the present 837, 1994 study, urinary pH measured after a 12-hour fast (except 14. Levine M, Conry-Cantilena C, Wang Y , et al : Vitamin C pharma-cokinetics in healthy volunteers: Evidence for a recommended for vitamin C supplement intake) in the second micturi- dietary allowance. Proc Natl Acad Sci USA 93:3704–3709, 1996 tion showed no significant change from baseline values. 15. Liebman M, Chai W, Harvey E, Boenisch L : Effect of supplemen-These findings are in agreement with other investigators tal ascorbate and orange juice on urinary oxalate. Nutr Res 17:415– 425, 1997 [22, 28]. 16. Auer BL, Auer D, Rodgers AL : The effect of ascorbic acid ingestion on the biochemical and physicochemical risk factors asso-ciated with calcium oxalate kidney stone formation. Clin Chem CONCLUSION Lab Med 36:143–148, 1998 17. Auer BL, Auer D, Rodgers AL : Relative hyperoxaluria, crystallu-In conclusion, the intake of vitamin C supplements of ria and haematuria after megadose ingestion of vitamin C. Eur J 1 or 2 g per day may produce a significant increase in Clin Invest 28:695–700, 1998 urinary oxalate, elevating the risk of calcium oxalate 18. Rose GA : Assay of oxalate and glycolate in urine, in Oxalate Metabolism in Relation to Urinary Stone , London, Springer-Verlag, crystallization. As a consequence, patients with a history 1988, pp 1–26 of stone disease should be discouraged from taking a 19. Mazzachi BC, Teubner JK, Ryall RL : Factors affecting measure-ment of urinary oxalate. Clin Chem 30:1339–1343, 1984 vitamin C amount exceeding the recommended daily Baxmann et al: Vitamin C and oxalate excretion 1071 Tiselius HG, Almgard LE : The diurnal urinary excretion of oxa- and non-acid urine collection on urinary oxalate determination. Arch Ital Urol Androl 73:78, 2001 late and the effect of pyridoxine and ascorbate on oxalate excretion. 33. Mingen GL, Madappally MM : Rapid enzymatic determination Eur Urol 3:41–46, 1977 of urinary oxalate. Clin Chem 35:2330–2333, 1989 21. Heckers H, Wagner I, Schmelz E, Trenkel A : Zur diatetischen 34. Chalmers AH, Cowley DM, McWhinney BC : Stability of ascor-Therapie und Pravention von Calcium-Oxalat-Nierensteinen. Er- bate in urine: Relevance to analyses for ascorbate and oxalate. nahrungs-umschau 40:416–420, 1993 Clin Chem 31:1703–1705, 1985 22. Hetey SK, Kleinberg ML, Parker WD, Johnson EW : Effect of 35. McFate RP, Cohn C, Eichelberger L , et al : Symposium on azote-ascorbic acid on urine pH in patients with injured spinal cords. mia. Am J Clin Pathol 24:511–71, 1954 Am J Hosp Pharm 37:235–237, 1980 36. Rebelo MA, Schor N : Dosagem enzima ´ tica do citrato urina ´ rio. 23. McDonald DF, Murphy GP : Bacteriostatic and acidifying effects J Bras Nefrol 12:71–76, 1990 of methionine, hydrolyzed casein and ascorbic acid on the urine. 37. Hellman L, Burns JJ: Metabolism of l-ascorbic acid 1-14C in man . J Biol Chem 230:923–930, 1958 N Engl J Med 261:803–805, 1959 38. Toggenburger G, Hausermann , M utsch B, et al : Na dependent, 24. Travis LB, Dodge WF, Mintz AA : Urinary acidification with potential-sensitive l-ascorbate transport across brush border mem-ascorbic acid. J Pediatr 67:1176–1178, 1965 brane vesicles from kidney cortex. Biochim Biophys Acta 646:433– 25. Murphy FJ, Zelman S : Ascorbic acid as a urinary acidifying agent: 443, 1981 1. Comparison with the ketogenic effect of fasting. J Urol 94:297– 39. Tiselius HG, Almgard LE : The diurnalurinary excretion of oxa-299, 1965 late and the effect of pyridoxine and ascorbate on oxalate excreton. 26. Murphy FJ, Zelman S, Mau W : Ascorbic acid as a urinary acidi- Eur Urol 3:41–46, 1977 fying agent: Its adjunctive role in chronic urinary infections. J Urol 40. Gambaro G, Petrarulo M, N ardelotto A, et al : Erythrocyte 94:300–305, 1965 transmembrane flux and renal clearance of oxalate in idiopathic 27. Houston JB, Levy G : Modification of drug biotransformation by calcium nephrolithiasis. Kidney Int 48:1549–1552, 1995 41. Baggio B, Gambaro G, Favaro S , et al : Prevalence of hyperoxal-vitamin C in man. Nature 255:78–79, 1965 uria in idiopatic calcium oxalate kidney stone disease. Nephron 28. Wall I, Tiselius HG : Long-term acidification of urine in patients 35:11–14, 1983 treated for infected renal stones. Urol Int 45:336–341, 1990 42. Mendonca COG, Martini LA, B axmann AC, et al : Effects of 29. Coe F : Treatment and prevention of renal stones. Consultant 18:47– an oxalate load upon urinary oxalate excretion in calcium stone 50, 1978 formers. J Renal Nutr , 2002 30. Tagasaki E : Observations on composition and recurrence of uri- 43. Sidhu H, S chmidt ME, C ornelius JG, et al : Direct correlation nary calculi. Urol Int 30:228–236, 1975 between hyperoxaluria/oxalate stone disease and the absence of 31. Tiselius HG : Aspects on estimation of the risk of calcium oxalate the gastrointestinal tract-dwelling bacterium Oxalobacter formi- crystallization in urine. Urol Int 47:255–259, 1991 genes : Possible prevention by gut recolonization or enzyme replace-ment therapy. J Am Soc Nephron 10(Suppl 1):334–340, 1999 32. Gandolpho L, Nishiura JL, Gomes SA , et al : The effect of acid
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Register Oct 05 Ace Arithmetic on the GMAT Focus Edition 11:00 AM IST 01:00 PM IST Attend this session to evaluate your current skill level, learn process skills, and solve through tough Arithmetic questions. Save Now! Sep 22 Special Offer: Get 25% Off Target Test Prep GMAT Plans 12:00 PM EDT 11:59 PM EDT The Target Test Prep GMAT Flash Sale is LIVE! Get 25% off our game-changing course and save up to $450 today! Use code FLASH25 at checkout. This special offer expires on September 30, so grab your discount now! Back to Forum Create Topic Reply Dr. A: The new influenza vaccine is useless at best and possibly dange sjgmat sjgmat Joined: 22 Apr 2008 Last visit: 07 Feb 2010 Posts: 20 Posts: 20 Post URL25 Jul 2008, 04:12 Show timer 00:00 Start Timer Pause Timer Resume Timer Show Answer a 11% b 45% c 24% d 8% e 12% A B C D E Hide Show History My Mistake Official Answer and Stats are available only to registered users.Register/Login. Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)! Difficulty: 95% (hard) Question Stats: 24% (01:50) correct 76%(02:01) wrong based on 3231 sessions History Date Time Result Not Attempted Yet Dr. A: The new influenza vaccine is useless at best and possibly dangerous. I would never use it on a patient. Dr. B: But three studies published in the Journal of Medical Associates have rated that vaccine as unusually effective. Dr. A: The studies must have been faulty because the vaccine is worthless. In which of the following is the reasoning most similar to that of Dr. A? (A) Three of my patients have been harmed by that vaccine during the past three weeks, so the vaccine is unsafe. (B) Jerrold Jersey recommends this milk, and I don’t trust Jerrold Jersey, so I won’t buy this milk. (C) Wingzz tennis balls perform best because they are far more effective than any other tennis balls. (D) I’m buying Vim Vitamins. Doctors recommend them more often than they recommend any other vitamins, so Vim Vitamins must be good. (E) Since University of Muldoon graduates score about 20 percent higher than average on the GMAT, Sheila Lee, a University of Muldoon graduate, will score about 20 percent higher than average when she takes the GMAT. Show Hide Answer Official Answer Official Answer and Stats are available only to registered users.Register/Login. 16 Kudos Add Kudos 159 Bookmarks Bookmark this Post Most Helpful Reply zonk zonk Current Student Joined: 12 Jul 2008 Last visit: 10 Nov 2013 Posts: 366 Schools:Wharton Posts: 366 Post URL25 Jul 2008, 10:24 sjgmat Dr. A: The new influenza vaccine is useless at best and possibly dangerous. I would never use it on a patient. Dr. B: But three studies published in the Journal of Medical Associates have rated that vaccine as unusually effective. Dr. A: The studies must have been faulty because the vaccine is worthless. In which of the following is the reasoning most similar to that of Dr. A? (A) Three of my patients have been harmed by that vaccine during the past three weeks, so the vaccine is unsafe. (B) Jerrold Jersey recommends this milk, and I don’t trust Jerrold Jersey, so I won’t buy this milk. (C) Wingzz tennis balls perform best because they are far more effective than any other tennis balls. (D) I’m buying Vim Vitamins. Doctors recommend them more often than they recommend any other vitamins, so Vim Vitamins must be good. (E) Since University of Muldoon graduates score about 20 percent higher than average on the GMAT, Sheila Lee, a University of Muldoon graduate, will score about 20 percent higher than average when she takes the GMAT. Any opinoins? Can someone guide me with general logic/checkpoints to tackle such questions? 90% of time I select wrong solution Thanks Show more Answer is C. The stimulus shows circular reasoning. A: I think the vaccine is bad. B: The vaccine is not bad because of these data. A: The data must be wrong, because the vaccine is bad. C also shows circular reasoning: Wingzz tennis balls are the best because they are the best. 36 Kudos Add Kudos 11 Bookmarks Bookmark this Post goalsnr goalsnr Joined: 03 Apr 2007 Last visit: 17 Oct 2012 Posts: 630 Products: Posts: 630 Post URL25 Jul 2008, 09:51 sjgmat Dr. A: The new influenza vaccine is useless at best and possibly dangerous. I would never use it on a patient. Dr. B: But three studies published in the Journal of Medical Associates have rated that vaccine as unusually effective. Dr. A: The studies must have been faulty because the vaccine is worthless. In which of the following is the reasoning most similar to that of Dr. A? (A) Three of my patients have been harmed by that vaccine during the past three weeks, so the vaccine is unsafe. (B) Jerrold Jersey recommends this milk, and I don’t trust Jerrold Jersey, so I won’t buy this milk. (C) Wingzz tennis balls perform best because they are far more effective than any other tennis balls. (D) I’m buying Vim Vitamins. Doctors recommend them more often than they recommend any other vitamins, so Vim Vitamins must be good. (E) Since University of Muldoon graduates score about 20 percent higher than average on the GMAT, Sheila Lee, a University of Muldoon graduate, will score about 20 percent higher than average when she takes the GMAT. Any opinoins? Can someone guide me with general logic/checkpoints to tackle such questions? 90% of time I select wrong solution Thanks Show more C Dr A says the vaccine is useless without giving reasons why it is useless. In C - its stated tennis balls are best without giving reasons 12 Kudos Add Kudos Bookmarks Bookmark this Post General Discussion snaps snaps Joined: 11 Mar 2008 Last visit: 27 Jan 2012 Posts: 54 Location: Canada Posts: 54 Post URL02 Sep 2008, 12:25 sjgmat Dr. A: The new influenza vaccine is useless at best and possibly dangerous. I would never use it on a patient. Dr. B: But three studies published in the Journal of Medical Associates have rated that vaccine as unusually effective. Dr. A: The studies must have been faulty because the vaccine is worthless. In which of the following is the reasoning most similar to that of Dr. A? (A) Three of my patients have been harmed by that vaccine during the past three weeks, so the vaccine is unsafe. (B) Jerrold Jersey recommends this milk, and I don’t trust Jerrold Jersey, so I won’t buy this milk. (C) Wingzz tennis balls perform best because they are far more effective than any other tennis balls. (D) I’m buying Vim Vitamins. Doctors recommend them more often than they recommend any other vitamins, so Vim Vitamins must be good. (E) Since University of Muldoon graduates score about 20 percent higher than average on the GMAT, Sheila Lee, a University of Muldoon graduate, will score about 20 percent higher than average when she takes the GMAT. Show more Info from the text: “The vaccine is useless” because “the vaccine is worthless” i.e. no reason is given A) “The vaccine is unsafe” because “three patients have been harmed” some sort of reasoning is present B) JJ recommends this milk BUT SINCE i don’t trust this ninja i won’t buy this milk – a reason is there: this ninja is not trustworthy source for info C) WingzZ balls perform better because they are “far more effective than others” i.e. better. So WingzZ get to perform Better because they are Better. D) I buy VVs because doctor recommends them more often because blah blah -- I bought a Yugo car because my best friend recommended it. E) Reason is given Thus C is the answer. 11 Kudos Add Kudos 1 Bookmarks Bookmark this Post KnewtonAlex KnewtonAlex Knewton GMAT Representative Joined: 27 Oct 2009 Last visit: 18 Feb 2010 Posts: 11 Affiliations: Knewton, Inc. Posts: 11 Post URL29 Oct 2009, 06:27 I agree with Hades that this type of question is much more likely to appear on the LSAT than on the GMAT. In fact, on the LSAT, the question stem would ask "which of the following displays FLAWED reasoning most similar to that of Dr. A?" That said, C is the best answer. Dr. A is taking as a given (evidence) that the vaccine is useless, and then basing his or her conclusion about the studies on that evidence. This is flawed because the studies should be judged to be true or false regardless of any preconceived notion about the vaccine. Likewise, answer choice C is claiming as evidence that Wingzz tennis balls are the most effective, and then basing its conclusion, "they perform best," on that evidence; the same backwards reasoning shown in the prompt. The performance of tennis balls is not BASED on their effectiveness, it is the thing that tells us how effective they are, just like the studies should tell us whether the vaccine works or not. The only other viable choice, B, is incorrect because it displays a different distinct flaw, common on the LSAT: attacking a speaker rather than a speaker's ideas. There is no analogue to "Jerrold Jersey" (a person whose opinion should not be trusted) in the prompt. In the prompt, the attack on the studies is impersonal, based on a generalization. In B, the attack on Jerrold Jersey IS the generalization. Alex Knewton Verbal Developer Kudos Add Kudos Bookmarks Bookmark this Post elevinty elevinty Joined: 24 Jan 2010 Last visit: 23 Oct 2013 Posts: 102 Status:Mesmerized Location: UAE, Dubai Posts: 102 Post URL14 Oct 2010, 00:50 this is circular reasoning, i.e., someone is stating something is effective because he thinks it's effective. That someone didn't mention why it's good or effective, and his only reasoning for it is because he thinks it's good. In this questions we need to look for reasoning similar to that. In C there is no mentioning why he thinks the balls are good; he just thinks they are effective. so the answer is C 2 Kudos Add Kudos Bookmarks Bookmark this Post nandeta nandeta Joined: 16 Jan 2020 Last visit: 02 Apr 2020 Posts: 52 Posts: 52 Post URL29 Jan 2020, 22:36 The crux of this argument is that the logical reasoning of this argument is at fault. So we need to find an answer option who logical reasoning is at fault. Dr. A: The studies must have been faulty because the vaccine is worthless. The vaccines can be faulty and that is why the studies is worthless.. nor the other way around. (A) Three of my patients have been harmed by that vaccine during the past three weeks, so the vaccine is unsafe. Logical (B) Jerrold Jersey recommends this milk, and I don’t trust Jerrold Jersey, so I won’t buy this milk. Logical (C) Wingzz tennis balls perform best because they are far more effective than any other tennis balls. Logical (D) I’m buying Vim Vitamins. Doctors recommend them more often than they recommend any other vitamins, so Vim Vitamins must be good. Logical (E) Since University of Muldoon graduates score about 20 percent higher than average on the GMAT, Sheila Lee, a University of Muldoon graduate, will score about 20 percent higher than average when she takes the GMAT. Illogical. Gmat scores depends on your personal capability and not in your college name. Kudos Add Kudos Bookmarks Bookmark this Post aarushi4101 aarushi4101 Joined: 14 Jul 2020 Last visit: 21 Aug 2020 Posts: 4 Posts: 4 Post URL12 Aug 2020, 09:06 ERROR.. I couldn't understand these kinda questions ...totally confused in all the given options. Even after reading this question thrice ,my mind goes with option B and not C...IS IT WISE TO LEAVE SUCH QUESTIONS IN EXAM??? Kudos Add Kudos Bookmarks Bookmark this Post richirish richirish Joined: 25 May 2020 Last visit: 09 Feb 2021 Posts: 131 Schools:Kenan-Flagler'23(D) Schools:Kenan-Flagler'23(D) Posts: 131 Post URL13 Aug 2020, 05:26 aarushi4101 ERROR.. I couldn't understand these kinda questions ...totally confused in all the given options. Even after reading this question thrice ,my mind goes with option B and not C...IS IT WISE TO LEAVE SUCH QUESTIONS IN EXAM??? Show more If you don't want to score above V40 then yes, otherwise a big NO. 1 Kudos Add Kudos Bookmarks Bookmark this Post CRACKGMATNUT CRACKGMATNUT Joined: 23 Jul 2020 Last visit: 26 May 2024 Posts: 150 Location: India Concentration: Entrepreneurship, Marketing Schools:Ivey'24(A) GMAT 1:700 Q49 V35 Schools:Ivey'24(A) GMAT 1:700 Q49 V35 Posts: 150 Post URL25 Aug 2020, 12:45 Hi. Can anyone help explaining why ain't the reasoning Strawman instead ? Dr. A try to falsify the argument with no clear understanding, which can be easily broken by DR. B Can anyone explain how is Stimulus circular reasoning? Posted from my mobile device Kudos Add Kudos Bookmarks Bookmark this Post Fdambro294 Fdambro294 Joined: 10 Jul 2019 Last visit: 20 Aug 2025 Posts: 1,351 Posts: 1,351 Post URL28 Aug 2021, 20:03 Dr A is performing what is known as an Informal Logical Fallacy nicknamed “Begging the Question.” Basically this occurs when the author’s Premises aren’t used as support for the Conclusion: the argument’s premises already assume that the conclusion is true. The author has already assumed the truth of his conclusion, rather than providing any support for the conclusion. “Running helps you lose weight because running leads to weight loss.” In the above example, there is no support for the claim that running helps you lose weight. The conclusion is already assumed in the premises. The above is a more extreme example but it basically occurs when the facts that are used as support already assume that the conclusion is true. Dr A says: “the studies must have been faulty because the vaccine is worthless. Dr A is out to prove that the studies proving that the vaccine might have some value must be wrong because the vaccines does not have any value. That is essentially the pattern we are looking for. (B) There is support offered in the argument pattern in B. Because the author doesn’t trust Jerrold Jersey’s milk recommendations, the author won’t buy the milk. This is a separate logical fallacy in which you dismiss someone’s argument because of something personal about the author: not on the basis of the facts presented. It does not fallow the same pattern. (C) “Wingzz tennis balls perform the best because they are far more effective than any other.” Just as Dr. A did in his argument, this author is already assuming the Conclusion within the premises. Effectively, because the balls are the most effective ———-> they are better than any other ball. There is a slim opening to say “what does effective really mean.” Maybe the ball’s effectiveness is some kind of industry jargon that somehow indicates the balls are better and it can be relied on as actual evidence. However, common sense would tell us that the author is basically repeating the conclusion in his premises, just as Dr A did in the passage. I believe this answer best illustrates the “Begging the Question” fallacy. Posted from my mobile device Kudos Add Kudos Bookmarks Bookmark this Post drabhinabasaha drabhinabasaha Joined: 30 Oct 2024 Last visit: 24 Jun 2025 Posts: 1 Location: India Posts: 1 Post URL13 Nov 2024, 10:39 It's like saying JOCKEY or nothing! sjgmat Dr. A: The new influenza vaccine is useless at best and possibly dangerous. I would never use it on a patient. Dr. B: But three studies published in the Journal of Medical Associates have rated that vaccine as unusually effective. Dr. A: The studies must have been faulty because the vaccine is worthless. In which of the following is the reasoning most similar to that of Dr. A? (A) Three of my patients have been harmed by that vaccine during the past three weeks, so the vaccine is unsafe. (B) Jerrold Jersey recommends this milk, and I don’t trust Jerrold Jersey, so I won’t buy this milk. (C) Wingzz tennis balls perform best because they are far more effective than any other tennis balls. (D) I’m buying Vim Vitamins. Doctors recommend them more often than they recommend any other vitamins, so Vim Vitamins must be good. (E) Since University of Muldoon graduates score about 20 percent higher than average on the GMAT, Sheila Lee, a University of Muldoon graduate, will score about 20 percent higher than average when she takes the GMAT. Show more 1 Kudos Add Kudos Bookmarks Bookmark this Post raviteja00seven raviteja00seven Joined: 14 Nov 2024 Last visit: 24 May 2025 Posts: 5 Posts: 5 Post URL21 Dec 2024, 09:51 ninjas !!??:dazed you sure make learning fun snaps sjgmat Dr. A: The new influenza vaccine is useless at best and possibly dangerous. I would never use it on a patient. Dr. B: But three studies published in the Journal of Medical Associates have rated that vaccine as unusually effective. Dr. A: The studies must have been faulty because the vaccine is worthless. In which of the following is the reasoning most similar to that of Dr. A? (A) Three of my patients have been harmed by that vaccine during the past three weeks, so the vaccine is unsafe. (B) Jerrold Jersey recommends this milk, and I don’t trust Jerrold Jersey, so I won’t buy this milk. (C) Wingzz tennis balls perform best because they are far more effective than any other tennis balls. (D) I’m buying Vim Vitamins. Doctors recommend them more often than they recommend any other vitamins, so Vim Vitamins must be good. (E) Since University of Muldoon graduates score about 20 percent higher than average on the GMAT, Sheila Lee, a University of Muldoon graduate, will score about 20 percent higher than average when she takes the GMAT. Info from the text: “The vaccine is useless” because “the vaccine is worthless” i.e. no reason is given A) “The vaccine is unsafe” because “three patients have been harmed” some sort of reasoning is present B) JJ recommends this milk BUT SINCE i don’t trust this ninja i won’t buy this milk – a reason is there: this ninja is not trustworthy source for info C) WingzZ balls perform better because they are “far more effective than others” i.e. better. So WingzZ get to perform Better because they are Better. D) I buy VVs because doctor recommends them more often because blah blah -- I bought a Yugo car because my best friend recommended it. E) Reason is given Thus C is the answer. Show more Kudos Add Kudos Bookmarks Bookmark this Post shankk shankk Joined: 18 Mar 2024 Last visit: 28 September 2025 Posts: 4 Location: India Posts: 4 Post URL25 Sep 2025, 21:52 Does anyone have links to similar questions? Really need to practice this type. Kudos Add Kudos Bookmarks Bookmark this Post Bunuel Bunuel Math Expert Joined: 02 Sep 2009 Last visit: 28 Sep 2025 Posts: 104,347 Products: Expert Expert reply Active GMAT Club Expert! Tag them with @ followed by their username for a faster response. Posts: 104,347 Post URL25 Sep 2025, 23:45 shankk Does anyone have links to similar questions? Really need to practice this type. Show more Check Similar Reasoning questions. New to the GMAT Club? • Questions: Question Bank \| Bunuel's Signature Collection • Tests: GMAT Club Tests \| Forum Quiz • Guides: Quantitative \| Verbal \| Ultimate Quantitative Collection \| All You Need for Quant • Rules: Quantitative \| Verbal Signature Read More Kudos Add Kudos Bookmarks Bookmark this Post NEW TOPIC POST REPLY Question banks Downloads My Bookmarks Important topics Reviews Similar topics Similar Topic Author Kudos Replies Last Post Until now, only injectable vaccines against influenza have been availa solidcolor by: solidcolor 29 Aug 2025, 23:20 17 278 Until now only injectable vaccines against influenza have been availab anilnandyala by: anilnandyala 30 Jun 2025, 12:27 9 75 For the first time since influenza vaccines became popular, the demand souvik101990 by: souvik101990 07 Feb 2025, 23:24 15 65 P: Complying with the new safety regulations is useless. Even if the Bunuel by: Bunuel 18 Oct 2022, 09:31 1 P: Complying with the new safety regulations is useless. 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12336	https://en.wikipedia.org/wiki/Fixed-target_experiment	Jump to content Fixed-target experiment Add links From Wikipedia, the free encyclopedia A fixed-target experiment in particle physics is an experiment in which a beam of accelerated particles is collided with a stationary target. The moving beam (also known as a projectile) consists of charged particles such as electrons or protons and is accelerated to relativistic speed. The fixed target can be a solid block or a liquid or a gaseous medium. These experiments are distinct from the collider-type experiments in which two moving particle beams are accelerated and collided. The famous Rutherford gold foil experiment, performed between 1908 and 1913, was one of the first fixed-target experiments, in which the alpha particles were targeted at a thin gold foil. Explanation [edit] The energy involved in a fixed target experiment is 4 times smaller compared to that in collider with the dual beams of same energy. More over in collider experiments energy of two beams is available to produce new particles, while in fixed target case a lot of energy is just expended in giving velocities to the newly created particles. This clearly implies that fixed target experiments are not helpful when it comes to increasing the energy scales of experiments. The targeted source also wears down with number of strikes and usually require a regular replacement. Current day fixed-target experiments try to use highly resistant materials but the damage cannot be avoided entirely. The fixed target experiments have a significant advantage for experiments that require higher luminosity (rate of interaction). The High Luminosity Large Hadron Collider, which is an upcoming upgraded version of the Large Hadron Collider (LHC) at CERN, will attain total integrated luminosity of around in its run. While luminosity scale of about have already been approached by older fixed target experiments such at the E288 led by Leon Lederman at Fermilab. Another advantage for fixed-target experiments is that they are easier and cheaper to build compared to the collider accelerators. Experimental facilities [edit] Rutherford's gold foil experiment that led to the discovery that mass and positive charge of an atom was concentrated in a small nucleus was probably the first fixed-target experiment. Later half of the 20th century saw the rise of particle and nuclear physics facilities such as CERN's Super Proton Synchrotron (SPS) and Fermilab's Tevatron where number of fixed-target experiments led to new discoveries. 43 fixed-target experiments were conducted at the Tevatron during its run period from 1983 to 2000. While proton and other beams from SPS are still used by fixed target experiments such as NA61/SHINE and COMPASS collaboration. A fixed-target facility at the LHC, called AFTER@LHC, is also being planned. Physics at fixed-target experiments [edit] The fixed-target experiments are mainly implemented for the intensive studies of the rare processes, dynamics at high Bjorken x, diffractive physics, spin-correlations, and numerous nuclear phenomena. The experiments at Fermilab's Tevatron facility covered wide range of physics domains such as testing the theoretical predictions of quantum chromodynamics theory, studies of structure of proton, neutron and mesons, and studies of heavy quarks such as charm and bottom. Several experiments looked into CP symmetry tests. Few collaborations also studied the hyperons and the neutrinos created at fixed-target setups. NA61/SHINE at the SPS is studying the phase transitions in strongly interacting matter and physics related to onset of confinement. While the COMPASS experiment investigates the structure of the hadrons. AFTER@LHC aims at the studies of gluon and quark distribution inside protons and neutrons using fixed-target facilities. There are possibilities to observe the W and Z bosons as well. Observation and studies of the Drell-Yan pair production and quarkonium are also being looked into. Thus the number of options available to explore extreme and rare physics at the fixed-target experiments are numerous. See also [edit] Collider List of fixed-target experiments External links [edit] Fixed-target experiments at CERN References [edit] ^ Jump up to: a b "The Particle Adventure \| How do we experiment with tiny particles? \| Fixed-target experiments". particleadventure.org. Retrieved 2021-07-16. ^ "Detectors, Fixed-Target \| Encyclopedia.com". encyclopedia.com. Retrieved 2021-07-16. ^ Jump up to: a b c "Fixed-target physics". ed.fnal.gov. Retrieved 2021-07-16. ^ "Fixed target, striking physics". CERN Courier. 2019-03-11. Retrieved 2021-07-21. ^ Jump up to: a b c "Fixed Target vs Collider Experiments (with discussion) \| Matt Evans". mtdevans.com. Retrieved 2021-07-22. ^ Lincoln, Don (2013-08-02). "Fixed-target vs. collider". News. Archived from the original on 2022-01-21. Retrieved 2021-07-20. ^ "Fixed Target and Colliding Beam Accelerators". www.hep.ucl.ac.uk. Retrieved 2021-07-22. ^ Lawhun, Sarah (11 April 2018). "Right on target". symmetry magazine. Retrieved 2021-07-22. ^ "Chapter 4 Accelerators and collider experiments" (PDF). ^ Brodsky, S.J.; Fleuret, F.; Hadjidakis, C.; Lansberg, J.P. (2013-01-01). "Physics opportunities of a fixed-target experiment using LHC beams". Physics Reports. 522 (4): 239–255. arXiv:1202.6585. Bibcode:2013PhR...522..239B. doi:10.1016/j.physrep.2012.10.001. ISSN 0370-1573. S2CID 53312294. ^ Topilskaya, Nataliya; Kurepin, Alexey (2019). Bondarenko, S.; Burov, V.; Malakhov, A. (eds.). "Some proposed fixed target experiments with the LHC beams". EPJ Web of Conferences. 204: 03002. Bibcode:2019EPJWC.20403002T. doi:10.1051/epjconf/201920403002. ISSN 2100-014X. ^ Jump up to: a b Loginov, Andrey Borisovich (2006). Search for anomalous production of events with a high energy lepton and photon at the Tevatron (Thesis). arXiv:hep-ex/0703011. doi:10.2172/900361. OSTI 900361. ^ Jump up to: a b c "Physics at a Fixed-Target Experiment Using the LHC Beams". Hindawi. Retrieved 2021-07-24. ^ Jump up to: a b c Trzeciak, B.; Da Silva, C.; Ferreiro, E. G.; Hadjidakis, C.; Kikola, D.; Lansberg, J. P.; Massacrier, L.; Seixas, J.; Uras, A.; Yang, Z. (September 2017). "Heavy-Ion Physics at a Fixed-Target Experiment Using the LHC Proton and Lead Beams (AFTER@LHC): Feasibility Studies for Quarkonium and Drell–Yan Production". Few-Body Systems. 58 (5): 148. arXiv:1703.03726. Bibcode:2017FBS....58..148T. doi:10.1007/s00601-017-1308-0. ISSN 0177-7963. S2CID 119054649. ^ Gutierrez, Gaston; Reyes, Marco A. (2014-11-10). "Fixed target experiments at the Fermilab Tevatron". International Journal of Modern Physics A. 29 (28): 1446008. arXiv:1409.8243. Bibcode:2014IJMPA..2946008G. doi:10.1142/S0217751X14460087. ISSN 0217-751X. S2CID 118569968. ^ Küchler, D.; O’Neil, M.; Scrivens, R.; Thomae, R. (February 2014). "Preparation of a primary argon beam for the CERN fixed target physics". Review of Scientific Instruments. 85 (2): 02A954. Bibcode:2014RScI...85bA954K. doi:10.1063/1.4854275. ISSN 0034-6748. PMID 24593533. ^ "Experiments \| CERN". home.cern. Retrieved 2021-07-24. ^ Brodsky, S.J.; Fleuret, F.; Hadjidakis, C.; Lansberg, J.P. (January 2013). "Physics opportunities of a fixed-target experiment using LHC beams". Physics Reports. 522 (4): 239–255. arXiv:1202.6585. Bibcode:2013PhR...522..239B. doi:10.1016/j.physrep.2012.10.001. S2CID 53312294. Retrieved from " Categories: Experimental particle physics Particle experiments Fixed-target experiments
12337	https://www.schoolmykids.com/learn/periodic-table/si-silicon	Silicon (Si) - Atomic, Physical & Chemical Properties, Uses, and Periodic Table Trends Silicon Element Information, Facts, Properties, Trends, Uses, Comparison with other elements Element 14 of Periodic table is Silicon with atomic number 14, atomic weight 28.0855. Silicon, symbol Si, has a Tetrahedral Packing structure and Gray color. Silicon is a Metalloid element. It is part of group 14 (carbon family). Discover everything about Silicon Facts, Physical Properties, Chemical Properties, Electronic configuration, Atomic and Crystal Structure. In this comprehensive guide, you'll learn about Silicon's unique chemical and physical properties, trends in the periodic table, isotopes, and its historical significance. We'll also cover its abundance, crystal structure, electron configuration, and health & safety guidelines. Explore how Silicon compares with other elements and discover its many uses. Silicon is a chemical element with symbol Si and atomic number 14. It is a tetravalent Metalloid, more reactive than germanium, the Metalloid directly below it in the table. Controversy about silicon's character dates to its discovery. It belongs to group 14 of the periodic table having trivial name tetrels, crystallogens. You can also download Printable Periodic Table of Elements Flashcards for Silicon in a PDF format. Silicon Facts Read key information and facts about element Silicon \| \| \| --- \| \| Name \| Silicon \| \| Atomic Number \| 14 \| \| Atomic Symbol \| Si \| \| Atomic Weight \| 28.0855 \| \| Phase \| Solid \| \| Color \| Gray \| \| Appearance \| crystalline, reflective with bluish-tinged faces \| \| Classification \| Metalloid \| \| Natural Occurance \| Primordial \| \| Group in Periodic Table \| 14 \| \| Group Name \| carbon family \| \| Period in Periodic Table \| period 3 \| \| Block in Periodic Table \| p-block \| \| Electronic Configuration \| [Ne] 3s2 3p2 \| \| Electronic Shell Structure (Electrons per shell) \| 2, 8, 4 \| \| Melting Point \| 1687 K \| \| Boiling Point \| 3173 K \| \| CAS Number \| CAS7440-21-3 \| Neighborhood Elements 14 Si \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- --- --- \| \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 \| 11 \| 12 \| 13 \| 14 \| 15 \| 16 \| 17 \| 18 \| \| 1 \| \| \| 2 8 4 14 Si Silicon Element Details Element Properties Classification Metalloid State at STP Solid Atomic Mass 28.0855 Electronegativity 1.9 Pauling Scale Melting Point 1687 K Boiling Point 3173 K First Ionisation Energy 786.5 kJ/mol Atomic Radius 111 pm Covalent Radius 111 pm Thermal Conductivity 150 W/(m K) Specific Heat 710 J/(kg K) Heat Fusion 50.2 kJ/mol Heat Evaporation 359 kJ/mol Molar Volume 12.054 cm3/mol Lattice Constant 543.09 pm Abundance Discovered 1823 Electrons per Shell 2,8,4 \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| 2 He Helium 4.003 \| \| 2 \| 3 Li Lithium 6.941 \| 4 Be Beryllium 9.012 \| 5 B Boron 10.811 \| 6 C Carbon 12.011 \| 7 N Nitrogen 14.007 \| 8 O Oxygen 15.999 \| 9 F Fluorine 18.998 \| 10 Ne Neon 20.180 \| \| 3 \| 11 Na Sodium 22.990 \| 12 Mg Magnesium 24.305 \| 13 Al Aluminium 26.982 \| 14 Si Silicon 28.085 \| 15 P Phosphorus 30.974 \| 16 S Sulfur 32.065 \| 17 Cl Chlorine 35.453 \| 18 Ar Argon 39.948 \| \| 4 \| 19 K Potassium 39.098 \| 20 Ca Calcium 40.078 \| 21 Sc Scandium 44.956 \| 22 Ti Titanium 47.867 \| 23 V Vanadium 50.941 \| 24 Cr Chromium 51.996 \| 25 Mn Manganese 54.938 \| 26 Fe Iron 55.845 \| 27 Co Cobalt 58.933 \| 28 Ni Nickel 58.693 \| 29 Cu Copper 63.546 \| 30 Zn Zinc 65.409 \| 31 Ga Gallium 69.723 \| 32 Ge Germanium 72.640 \| 33 As Arsenic 74.922 \| 34 Se Selenium 78.960 \| 35 Br Bromine 79.904 \| 36 Kr Krypton 83.798 \| \| 5 \| 37 Rb Rubidium 85.468 \| 38 Sr Strontium 87.620 \| 39 Y Yttrium 88.906 \| 40 Zr Zirconium 91.224 \| 41 Nb Niobium 92.906 \| 42 Mo Molybdenum 95.940 \| 43 Tc Technetium 98 \| 44 Ru Ruthenium 101.070 \| 45 Rh Rhodium 102.906 \| 46 Pd Palladium 106.420 \| 47 Ag Silver 107.868 \| 48 Cd Cadmium 112.411 \| 49 In Indium 114.818 \| 50 Sn Tin 118.710 \| 51 Sb Antimony 121.760 \| 52 Te Tellurium 127.600 \| 53 I Iodine 126.904 \| 54 Xe Xenon 131.293 \| \| 6 \| 55 Cs Cesium 132.905 \| 56 Ba Barium 137.327 \| 57 - 71 La - Lu Lanthanides \| 72 Hf Hafnium 178.490 \| 73 Ta Tantalum 180.948 \| 74 W Tungsten 183.840 \| 75 Re Rhenium 186.207 \| 76 Os Osmium 190.230 \| 77 Ir Iridium 192.217 \| 78 Pt Platinum 195.078 \| 79 Au Gold 196.967 \| 80 Hg Mercury 200.590 \| 81 Tl Thallium 204.383 \| 82 Pb Lead 207.200 \| 83 Bi Bismuth 208.980 \| 84 Po Polonium 209 \| 85 At Astatine 210 \| 86 Rn Radon 222 \| \| 7 \| 87 Fr Francium 223 \| 88 Ra Radium 226 \| 89 - 103 Ac - Lr Actinides \| 104 Rf Rutherfordium 261 \| 105 Db Dubnium 262 \| 106 Sg Seaborgium 266 \| 107 Bh Bohrium 264 \| 108 Hs Hassium 269 \| 109 Mt Meitnerium 268 \| 110 Ds Darmstadtium 281 \| 111 Rg Roentgenium 272 \| 112 Cn Copernicium 285 \| 113 Nh Nihonium 284 \| 114 Fl Flerovium 289 \| 115 Mc Moscovium 288 \| 116 Lv Livermorium 292 \| 117 Ts Tennessine 294 \| 118 Og Oganesson 294 \| \| \| Lanthanides \| \| \| 57 La Lanthanum 138.905 \| 58 Ce Cerium 140.116 \| 59 Pr Praseodymium 140.908 \| 60 Nd Neodymium 144.240 \| 61 Pm Promethium 145 \| 62 Sm Samarium 150.360 \| 63 Eu Europium 151.964 \| 64 Gd Gadolinium 157.250 \| 65 Tb Terbium 158.925 \| 66 Dy Dysprosium 162.500 \| 67 Ho Holmium 164.930 \| 68 Er Erbium 167.259 \| 69 Tm Thulium 168.934 \| 70 Yb Ytterbium 173.040 \| 71 Lu Lutetium 174.967 \| \| \| Actinides \| \| \| 89 Ac Actinium 227 \| 90 Th Thorium 232.038 \| 91 Pa Protactinium 231.036 \| 92 U Uranium 238.029 \| 93 Np Neptunium 237 \| 94 Pu Plutonium 244 \| 95 Am Americium 243 \| 96 Cm Curium 247 \| 97 Bk Berkelium 247 \| 98 Cf Californium 251 \| 99 Es Einsteinium 252 \| 100 Fm Fermium 257 \| 101 Md Mendelevium 258 \| 102 No Nobelium 259 \| 103 Lr Lawrencium 262 \| Explore our interactive periodic table Download Periodic Table Flash Cards - Silicon Element Table of Content History Abundance Physical Properties Thermal Properties Crystal Structure Atomic & Orbital Properties Chemical Properties Isotopes Databases Health and Safety Parameters and Guidelines Compare Silicon with other elements How to Locate Silicon on Periodic Table Periodic table is arranged by atomic number, number of protons in the nucleus which is same as number of electrons. The atomic number increases from left to right. Periodic table starts at top left ( Atomic number 1) and ends at bottom right (atomic number 118). Therefore you can directly look for atomic number 14 to find Silicon on periodic table. Another way to read periodic table and locate an element is by using group number (column) and period number (row). To locate Silicon on periodic table look for cross section of group 14 and period 3 in the modern periodic table. Explore Interactive Periodic Table to Understand and Learn Cool Trends Silicon History The element Silicon was discovered by J. Berzelius in year 1823 in Sweden. Silicon was first isolated by J. Berzelius in 1823. Silicon derived its name from the Latin silex, 'flint' (originally silicium). \| \| \| --- \| \| Discovered By \| J. Berzelius \| \| Discovery Date \| 1823 in Sweden \| \| First Isolation \| 1823 \| \| Isolated by \| J. Berzelius \| Humphry Davy thought in 1800 that silica was a compound, not an element, and in 1808 suggested the present name. In 1811 Louis-Joseph Gay-Lussac and Louis-Jacques Thénard probably prepared impure silicon, but Berzelius is credited with the discovery for obtaining the pure element in 1823. Download printable flash card for Silicon periodic table PDF Silicon Uses Silicon is used majorly in the semiconductor industry in solid-state electronics. To use it there, the silicon has to be doped with boron, gallium, phosphorus, or arsenic. Semiconductors: Silicon is used in the production of semiconductors, making it crucial for the electronics industry, including computers and smartphones. Solar Panels: Silicon is used in the manufacturing of solar cells for converting sunlight into electricity. Construction Materials: Silicon is used in the production of cement and concrete as it is a key component in the formation of sand and other building materials. Silicon Presence: Abundance in Nature and Around Us The table below shows the abundance of Silicon in Universe, Sun, Meteorites, Earth's Crust, Oceans and Human Body. \| \| ppb by weight (1ppb =10^-7 %) \| ppb by atoms (1ppb =10^-7 %) \| --- \| Abundance in Universe \| 700000 \| 30000 \| \| Abundance in Sun \| 900000 \| 40000 \| \| Abundance in Meteorites \| 140000000 \| 100000000 \| \| Abundance in Earth's Crust \| 270000000 \| 200000000 \| \| Abundance in Oceans \| 1000 \| 220 \| \| Abundance in Humans \| 260000 \| 58000 \| Crystal Structure of Silicon The solid state structure of Silicon is Tetrahedral Packing. The Crystal structure can be described in terms of its unit Cell. The unit Cells repeats itself in three dimensional space to form the structure. Unit Cell Parameters The unit cell is represented in terms of its lattice parameters, which are the lengths of the cell edges Lattice Constants (a, b and c) \| a \| b \| c \| --- \| 543.09 pm \| 543.09 pm \| 543.09 pm \| and the angles between them Lattice Angles (alpha, beta and gamma). \| alpha \| beta \| gamma \| --- \| π/2 \| π/2 \| π/2 \| The positions of the atoms inside the unit cell are described by the set of atomic positions ( xi, yi, zi) measured from a reference lattice point. The symmetry properties of the crystal are described by the concept of space groups. All possible symmetric arrangements of particles in three-dimensional space are described by the 230 space groups (219 distinct types, or 230 if chiral copies are considered distinct. \| \| \| --- \| \| Space Group Name \| Fd_ 3m \| \| Space Group Number \| 227 \| \| Crystal Structure \| Tetrahedral Packing \| \| Number of atoms per unit cell \| \| The number of atoms per unit cell in a simple cubic, face-centered cubic and body-centred cubic are 1,4,2 respectively. Silicon Atomic and Orbital Properties Silicon atoms have 14 electrons and the electronic shell structure is [2, 8, 4] with Atomic Term Symbol (Quantum Numbers) 3P0. \| \| \| --- \| \| Atomic Number \| 14 \| \| Number of Electrons (with no charge) \| 14 \| \| Number of Protons \| 14 \| \| Mass Number \| 28 \| \| Number of Neutrons \| 14 \| \| Shell structure (Electrons per energy level) \| 2, 8, 4 \| \| Electron Configuration \| [Ne] 3s2 3p2 \| \| Valence Electrons \| 3s2 3p2 \| \| Valence (Valency) \| 4 \| \| Main Oxidation States \| -4, 4 \| \| Oxidation States \| -4, -3, -2, -1, 0, 1, 2, 3, 4 \| \| Atomic Term Symbol (Quantum Numbers) \| 3P0 \| Bohr Atomic Model of Silicon - Electrons per energy level \| n \| \| s \| p \| d \| f \| --- --- --- \| \| 1 \| K \| 2 \| \| \| \| \| 2 \| L \| 2 \| 6 \| \| \| \| 3 \| M \| 2 \| 2 \| \| \| Ground State Electronic Configuration of Silicon - neutral Silicon atom Abbreviated electronic configuration of Silicon The ground state abbreviated electronic configuration of Neutral Silicon atom is [Ne] 3s2 3p2. The portion of Silicon configuration that is equivalent to the noble gas of the preceding period, is abbreviated as [Ne]. For atoms with many electrons, this notation can become lengthy and so an abbreviated notation is used. This is important as it is the Valence electrons 3s2 3p2, electrons in the outermost shell that determine the chemical properties of the element. Unabbreviated electronic configuration of neutral Silicon Complete ground state electronic configuration for the Silicon atom, Unabbreviated electronic configuration 1s2 2s2 2p6 3s2 3p2 Electrons are filled in atomic orbitals as per the order determined by the Aufbau principle, Pauli Exclusion Principle and Hund’s Rule. As per the Aufbau principle the electrons will occupy the orbitals having lower energies before occupying higher energy orbitals. According to this principle, electrons are filled in the following order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p… The Pauli exclusion principle states that a maximum of two electrons, each having opposite spins, can fit in an orbital. Hund's rule states that every orbital in a given subshell is singly occupied by electrons before a second electron is filled in an orbital. Atomic Structure of Silicon Silicon atomic radius is 111 pm, while it's covalent radius is 111 pm. \| \| \| --- \| \| Atomic Radius Calculated \| 111 pm(1.11 Å) \| \| Atomic Radius Empirical \| 110 pm (1.1 Å) \| \| Atomic Volume \| 12.054 cm3/mol \| \| Covalent Radius \| 111 pm (1.11 Å) \| \| Van der Waals Radius \| 210 pm \| \| Neutron Cross Section \| 171 \| \| Neutron Mass Absorption \| 0.0002 \| Spectral Lines of Silicon - Atomic Spectrum of Silicon A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to identify atoms and molecules. Spectral lines are the result of interaction between a quantum system and a single photon. A spectral line may be observed either as an emission line or an absorption line. Spectral lines are highly atom-specific, and can be used to identify the chemical composition of any medium. Several elements, including helium, thallium, and caesium, were discovered by spectroscopic means. They are widely used to determine the physical conditions of stars and other celestial bodies that cannot be analyzed by other means. Emission spectrum of Silicon Absorption spectrum of Silicon Silicon Chemical Properties: Silicon Ionization Energies and electron affinity The electron affinity of Silicon is 133.6 kJ/mol. \| \| \| --- \| \| Valence \| 4 \| \| Electronegativity \| 1.9 \| \| ElectronAffinity \| 133.6 kJ/mol \| Ionization Energy of Silicon Ionization energy is the amount of energy required to remove an electron from an atom or molecule.in chemistry, this energy is expresed in kilocalories per mole (kcal/mol) or kilojoules per mole (kJ/mol). Refer to table below for Ionization energies of Silicon Here are the ionization energies of Silicon (Si) both in electron volts (eV) and in kilojoules per mole (kJ/mol). \| Ionization energy number \| Enthalpy in kJ/mol \| Energy (eV) \| --- \| 1st \| 786.5 \| 8.152 \| \| 2nd \| 1577.1 \| 16.346 \| \| 3rd \| 3231.6 \| 33.493 \| \| 4th \| 4355.5 \| 45.142 \| \| 5th \| 16091 \| 166.772 \| \| 6th \| 19805 \| 205.265 \| \| 7th \| 23780 \| 246.463 \| \| 8th \| 29287 \| 303.539 \| \| 9th \| 33878 \| 351.122 \| \| 10th \| 38726 \| 401.368 \| \| 11th \| 45962 \| 476.364 \| \| 12th \| 50502 \| 523.418 \| \| 13th \| 235196 \| 2437.643 \| \| 14th \| 257923 \| 2673.193 \| The conversion from kJ/mol to eV is done using the formula: Energy (kJ/mol) = Energy (eV) x 96.485 Energy (kJ/mol)=Energy (eV)x96.485 where 1 eV = 96.485 kJ/mol. 1 electronvolt (eV) is equal to 96.485 kilojoules per mole (kJ/mol) Silicon Physical Properties Refer to below table for Silicon Physical Properties \| \| \| --- \| \| Density \| 2.33 g/cm3(when liquid at m.p density is $2.57 g/cm3) \| \| Molar Volume \| 12.054 cm3/mol \| Elastic Properties \| \| \| --- \| \| Young Modulus \| 47 \| \| Shear Modulus \| Bulk Modulus \| 100 GPa \| \| Poisson Ratio Hardness of Silicon - Tests to Measure of Hardness of Element \| \| \| --- \| \| Mohs Hardness \| 6.5 MPa \| \| Vickers Hardness \| Brinell Hardness Silicon Electrical Properties Electrical resistivity measures element's electrical resistance or how strongly it resists electric current.The SI unit of electrical resistivity is the ohm-metre (Ω⋅m). While Electrical conductivity is the reciprocal of electrical resistivity. It represents a element's ability to conduct electric current. The SI unit of electrical conductivity is siemens per metre (S/m). Silicon is a Semiconductor. Refer to table below for the Electrical properties of Silicon \| \| \| --- \| \| Electrical conductors \| Semiconductor \| \| Electrical Conductivity \| 1000 S/m \| \| Resistivity \| 0.001 m Ω \| \| Superconducting Point Silicon Heat and Conduction Properties \| \| \| --- \| \| Thermal Conductivity \| 150 W/(m K) \| \| Thermal Expansion \| 0.0000026 /K \| Silicon Magnetic Properties \| \| \| --- \| \| Magnetic Type \| Diamagnetic \| \| Curie Point \| Mass Magnetic Susceptibility \| -1.6e-9 m3/kg \| \| Molar Magnetic Susceptibility \| -4.49e-11 m3/mol \| \| Volume Magnetic Susceptibility \| -0.00000373 \| Optical Properties of Silicon \| \| \| --- \| \| Refractive Index Acoustic Properties of Silicon \| \| \| --- \| \| Speed of Sound \| 2200 m/s \| Silicon Thermal Properties - Enthalpies and thermodynamics Refer to table below for Thermal properties of Silicon \| \| \| --- \| \| Melting Point \| 1687 K(1413.85 °C, 2576.930 °F) \| \| Boiling Point \| 3173 K(2899.85 °C, 5251.730 °F) \| \| Critical Temperature \| Superconducting Point Enthalpies of Silicon \| \| \| --- \| \| Heat of Fusion \| 50.2 kJ/mol \| \| Heat of Vaporization \| 359 kJ/mol \| \| Heat of Combustion \| -9055 J/(kg K) \| Silicon Isotopes - Nuclear Properties of Silicon Silicon has 23 isotopes, with between 22 and 44 nucleons. Silicon has 3 stable naturally occuring isotopes. Isotopes of Silicon - Naturally occurring stable Isotopes: 28Si, 29Si, 30Si. \| Isotope \| Z \| N \| Isotope Mass \| % Abundance \| T half \| Decay Mode \| --- --- --- \| 22Si \| 14 \| 8 \| 22 \| Synthetic \| \| \| \| 23Si \| 14 \| 9 \| 23 \| Synthetic \| \| \| \| 24Si \| 14 \| 10 \| 24 \| Synthetic \| \| \| \| 25Si \| 14 \| 11 \| 25 \| Synthetic \| \| \| \| 26Si \| 14 \| 12 \| 26 \| Synthetic \| \| \| \| 27Si \| 14 \| 13 \| 27 \| Synthetic \| \| \| \| 28Si \| 14 \| 14 \| 28 \| 92.2297% \| Stable \| \| \| 29Si \| 14 \| 15 \| 29 \| 4.6832% \| Stable \| N/A \| \| 30Si \| 14 \| 16 \| 30 \| 3.0872% \| Stable \| N/A \| \| 31Si \| 14 \| 17 \| 31 \| Synthetic \| \| \| \| 32Si \| 14 \| 18 \| 32 \| Synthetic \| \| \| \| 33Si \| 14 \| 19 \| 33 \| Synthetic \| \| \| \| 34Si \| 14 \| 20 \| 34 \| Synthetic \| \| \| \| 35Si \| 14 \| 21 \| 35 \| Synthetic \| \| \| \| 36Si \| 14 \| 22 \| 36 \| Synthetic \| \| \| \| 37Si \| 14 \| 23 \| 37 \| Synthetic \| \| \| \| 38Si \| 14 \| 24 \| 38 \| Synthetic \| \| \| \| 39Si \| 14 \| 25 \| 39 \| Synthetic \| \| \| \| 40Si \| 14 \| 26 \| 40 \| Synthetic \| \| \| \| 41Si \| 14 \| 27 \| 41 \| Synthetic \| \| \| \| 42Si \| 14 \| 28 \| 42 \| Synthetic \| \| \| \| 43Si \| 14 \| 29 \| 43 \| Synthetic \| \| \| \| 44Si \| 14 \| 30 \| 44 \| Synthetic \| \| \| Regulatory and Health - Health and Safety Parameters and Guidelines The United States Department of Transportation (DOT) identifies hazard class of all dangerous elements/goods/commodities either by its class (or division) number or name. The DOT has divided these materials into nine different categories, known as Hazard Classes. \| \| \| --- \| \| DOT Numbers \| 1346 \| \| DOT Hazard Class \| 4.1 \| Flammable solids, self-reactive substances and solid desensitized explosives NFPA 704 is a Standard System for the Identification of the Hazards of Materials for Emergency Response. NFPA is a standard maintained by the US based National Fire Protection Association. The health (blue), flammability (red), and reactivity (yellow) rating all use a numbering scale ranging from 0 to 4. A value of zero means that the element poses no hazard; a rating of four indicates extreme danger. \| \| \| \| --- \| NFPA Fire Rating \| 0 \| Will not burn \| \| NFPA Health Rating \| 1 \| Flash Points Above 93.3°C (200°F) \| \| NFPA Reactivity Rating \| 0 \| Will not burn \| \| NFPA Hazards \| \| \| 0 1 NFPA Rating \| \| \| --- \| \| Autoignition Point \| 150 °C \| \| Flashpoint Database Search List of unique identifiers to search the element in various chemical registry databases \| Database \| Identifier number \| --- \| \| CAS Number - Chemical Abstracts Service (CAS) \| CAS7440-21-3 \| \| RTECS Number \| RTECSVW0400000 \| \| CID Number \| CID5461123 \| \| Gmelin Number \| NSC Number Compare Silicon with other elements Compare Silicon with Group 14, Period 3 and Metalloid elements of the periodic table. Compare Silicon with all Group 14 elements Compare Silicon with Carbon Compare Silicon with Germanium Compare Silicon with Tin Compare Silicon with Lead Compare Silicon with Flerovium Compare Silicon with all Period 3 elements Compare Silicon with Sodium Compare Silicon with Magnesium Compare Silicon with Aluminium Compare Silicon with Phosphorus Compare Silicon with Sulfur Compare Silicon with Chlorine Compare Silicon with Argon Compare Silicon with all Metalloid elements Silicon vs Boron Comparison Silicon vs Germanium Comparison Silicon vs Arsenic Comparison Silicon vs Antimony Comparison Silicon vs Tellurium Comparison Silicon vs Polonium Comparison Explore our interactive periodic table Periodic Table Element Comparison Frequently Asked Questions (FAQ) Find the answers to the most frequently asked questions about Silicon The electronic configuration of Silicon is 1s2 2s2 2p6 3s2 3p2. The abbreviated electronic configuration of Silicon is [Ne] 3s2 3p2. To form abbreviated notation of electronic configuration, the completely filled subshells are replaced by the noble gas of the preceding period in square brackets. Symbol of Silicon is Si. Silicon is a chemical element with symbol Si and atomic number 14. Silicon is a chemical element with the symbol Si and atomic number 14. Silicon is the 14 element on the periodic table. It is located in group 14 and period 3 in the modern periodic table. Silicon is the 14 element on the periodic table. Silicon is located in group 14 and period 3 in the modern periodic table. The atomic number of Silicon is 14. Silicon is of Gray color. The element Silicon was discovered by J. Berzelius in year 1823 in Sweden. Silicon was first isolated by J. Berzelius in 1823. Silicon has 4 valence electrons. Silicon has 14 electrons out of which 4 valence electrons are present in the 3s2 3p2 outer orbitals of atom. Melting Point of Silicon is 1687 K. Boiling Point of Silicon is 3173 K. Melting Point of Silicon in Kelvin is 1687 K. Boiling Point of Silicon in Kelvin is 3173 K. Melting Point of Silicon in Celsius is 1413.85 °C. Boiling Point of Silicon in Celsius is 2899.85 °C. Melting Point of Silicon in Fahrenheit is 2576.93 °F. Boiling Point of Silicon in Fahrenheit is 5251.73 °F. The electronic configuration of Silicon will be 1s2 2s2 2p6 3s2 3p2. The electronic configuration of Silicon will be 1s2 2s2 2p6 3s2 3p2. On the periodic table, elements are listed in order of increasing atomic number. Silicon is the 14 element on the periodic table. Silicon is located in group 14 and period 3 in the modern periodic table. We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our advertising and analytics tools. Consent Setting
12338	https://www.maths.nottingham.ac.uk/plp/pmzcw/download/fnt_chap5.pdf	Further Number Theory G13FNT cw '11 5 Gaussian Integers and sums of squares Aims of this chapter: to discover things about the arithmetic of Z by passing to larger number rings. The Gaussian integers Denition. The set of Gaussian integers is Z[i] = {a + b i \| a, b ∈Z}. Remark. Z[i] is closed under addition and multiplication, and contains Z: it is a subring of C. It is not a eld: dividing one Gaussian integer by another results in an element of Q(i) with rational real and imaginary parts. Questions: What does Z[i] look like? Does it have an \arithmetic" like that of Z? What are \Gaussian primes"? Remark. Note that (1 + i)(1 −i) = 2, so the number 2 is \not prime" in Z[i]. Neither is 5, since 5 = (1 + 2i)(1 −2i). What about 3 = (−1)(−3) = i(−3i)? Denition. An element α ∈Z[i] is a unit, or invertible element, if there exists a β ∈Z[i] such that α · β = 1. Two elements α and β in Z[i] are called associate to each other if α = γβ for some unit γ. To answer the above questions properly we rst need to decide what the units of Z[i] are. As well as ±1 there are also ±i since i(−i) = 1. Are there any more? To decide this we'll introduce a function on Z[i] called the norm. Denition. The function N: Z[i] →Z, called the norm, is dened by N(a + bi) = (a + bi)(a −bi) = a2 + b2, so N(α) = α · α. Lemma 5.1 (Properties of the norm). a). N(α) = 0 if and only if α = 0; b). N(α · β) = N(α) · N(β); c). N(α) = 1 if and only if α is a unit in Z[i]; d). {1, i, −1, −i} is the complete set of units of Z[i]. Proof. a). is obvious. b). We have N(α · β) = (α · β) · α · β = α · α · β · β = N(α) · N(β) . c). If N(α) = 1 then α · α = 1 and since α is also in Z[i], we must have that α is a unit. Conversely, if α · β = 1 for some β ∈Z[i], then N(α) · N(β) = 1 and since both N(α) and N(β) are positive integers, we have N(α) = N(β) = 1. d). We nd all the units by solving N(α) = 1. If α = a+b i, then a2 +b2 = 1 gives that either a or b must be 0 and the other ±1. Further Number Theory G13FNT cw '11 Theorem 5.2 (Euclidian division in Z[i]). Given α and β ̸= 0 in Z[i], there exists κ and ρ in Z[i] such that α = κ · β + ρ and N(ρ) < N(β) . We call κ the κuotient and ρ the ρemainder. Proof. The vector from 0 to iβ is perpendicular to the vector from 0 to β in the complex plane C = R2. So the set β · Z[i] = {κ · β \| κ ∈Z[i]} forms a lattice of squares with side length \|β\| = p N(β). Our given α belongs to at least one of these squares. Let κ · β be a closest corner of this square, i.e. an element in β · Z[i] of smal-lest distance to α. Put ρ = α − κβ ∈Z[i]. So \|ρ\| is smaller or equal than half the diagonal of the square. So p N(ρ) = \|ρ\| ⩽ √ 2 2 · \|β\| < p N(β) . Denition. We say that α in Z[i] divides β in Z[i], denoted by α \| β if there is a γ ∈Z[i] such that β = γ · α. Denition. An element δ ∈Z[i] is called a a greatest common divisor of α and β, if δ is an element in Z[i] of maximal norm such that δ \| α and δ \| β. Note that if ε is a unit in Z[i] and δ a greatest common divisor of α and β then ε · δ is also a greatest common divisor. A greatest common divisor can be computed with the Euclidian algorithm using the previous theorem. See the example below. The algorithm also yields two Gaussian integers ξ and η such that a chosen greatest common divisor δ can be written as δ = ξα + ηβ. Conversely to the above, any two gcd(α, β) are obtained by multiplying with a unit. See problem sheet. Let α = 1 −8 i and β = 5 + 5 i. So N(α) = 65 and N(β) = 50. If κ = −1 −i, then ρ = 1 + 2 i with N(ρ) = 5 < N(β). α = 1 −8 i = (−1 −i) · β + (1 + 2 i) . In the next step we try to divide β by ρ = 1 + 2 i. But actually β lies on the lattice ρZ[i]. We nd β = (3 −i) · ρ + 0 . Hence 1 + 2i is a greatest common divisor of α and β. Denition. An element π ∈Z[i] is called a Gaussian prime if N(π) > 1 and the following holds: if, for any α and β ∈Z[i] such that π divides α · β, then π divides α or β. Lemma 5.3. Let 0 ̸= π ∈Z[i]. The following are equivalent • π is a Gaussian prime Further Number Theory G13FNT cw '11 • If, for some α and β ∈Z[i] we have π = α · β, then α or β is a unit. Proof. ⇓: If π = α · β, then π \| α · β. Without loss of generality, we may assume that there is γ ∈Z[i] such that α = πγ. Then π = πγβ, so γβ = 1 shows that β is a unit and α is not a unit because π is not. ⇑: Suppose π divides α · β. Let δ be a gcd(α, π). So there is a γ such that π = γδ. By assumption, either δ or γ is a unit. If γ is a unit then πγ−1 = δ divides α. So π divides α. Otherwise δ is a unit. As δ = ξα + ηβ for some ξ, η ∈Z[i], we get that π divides δβ and hence β. Lemma 5.4. If π ∈Z[i] is such that N(π) is a prime number then π is a Gaussian prime Proof. If π = α · β then N(α) · N(β) = N(π). So either N(α) = 1 or N(β) = 1. Example. 1 + i is a Gaussian prime of norm 2. Also 1 + 2 i of norm 5 is a Gaussian prime. So 5 = (1 + 2 i) · (1 −2 i) is not a Gaussian prime. But q = 3 or q = 7 are Gaussian primes: Lemma 5.5. Let q be a prime number with q ≡3 (mod 4). Then q ∈Z[i] is a Gaussian prime. Proof. If q = α · β for α = a + b i and β ∈Z[i], then q2 = N(q) = N(α) · N(β). But N(α) = a2 + b2 = q ≡3 (mod 4) is not possible for a, b ∈Z. So either N(α) = 1 or N(β) = 1. Lemma 5.6. Let p be a prime number with p ≡1 (mod 4). Then there exists a Gaussian prime π such that p = π · π. Proof. By quadratic reciprocity, p ≡1 (mod 4) implies ( −1 p ) = +1. So there is a c ∈Z such that c2 ≡−1 (mod p). Hence p divides (c−i)(c+i) in Z[i]. But p does not divide c+i or c−i. Therefore p is not a Gaussian prime. Hence there is α · β, both non-units, with p = α · β. By p2 = N(p) = N(α) · N(β), we must have N(α) = p and hence π = α is a Gaussian prime. And p = N(π) = ππ. Proposition 5.7. Up to associates, the Gaussian primes are the following : • 1 + i is a Gaussian prime of norm 2. • For each prime number p ≡1 (mod 4) there are exactly two Gaussian primes π and π of norm p. • Each prime number q ≡3 (mod 4) is a Gaussian prime of norm q2. Proof. All in the list are Gaussian primes. Let α be a Gaussian prime. Then there is a prime p dividing N(α). In the above list we nd a Gaussian prime π dividing p, so π \| p \| αα. So either π or the Gaussian prime π divides α, and hence is associate to it. So α is in the above list. A complete set of non-associate Gaussian primes Pi is a set of Gaussian primes such that for each Gaussian prime π there is exactly one of the four associates in Pi. Further Number Theory G13FNT cw '11 Theorem 5.8. Let Pi be a complete set of non-associate Gaussian primes. Every 0 ̸= α ∈Z[i] can be written as α = in · Y π∈Pi πaπ for some 0 ⩽n < 4 and aπ ⩾0. All but a nite number of aπ are zero and aπ = ordπ(α) is the highest power of π dividing α. Proof. Existence is proved by induction on N(α). If N(α) = 1 then α = in. If N(α) > 1, then there is a Gaussian prime π dividing α, so α = πβ for some β ∈Z[i]. By induction hypothesis β has a factorisation, then so does α. Suppose now α had two distinct factorisation of the above form α = in · Y k πak k = in′ · Y j πbj j for some 0 ⩽n, n′ < 4, ak ⩾0 and bj ⩾0. If a Gaussian prime from our set of non-associate Gaussian primes appears on both sides of this equation, we may divide by it. Therefore, we may assume that each Gaussian prime only appears on one side, i.e. ak · bk = 0. Suppose there is still a Gaussian prime πk dividing the left-hand side, i.e. ak > 0 and bk = 0. So πk divides the right-hand side and hence divides one of its factors. It cannot divide in′ as πk is not a unit. So it divides a πbj j with bj > 0, so k ̸= j. So it divides πj, so there is a γ such that γ · πk = πj. Since πj is a Gaussian prime, either γ or πk is a unit. Hence πk and πj are associate. Contradiction. Hence, after this simplication each side is a power of i. So the factorisation of α is unique. Pythagorean triples In this section we'll apply the arithmetic of Z[i] to solve a classical problem: nding all Pythagorean Triples. Denition. A Pythagorean triple is a triple (x, y, z) where x, y, z ∈N and x2 + y2 = z2. Examples: 32 + 42 = 52 and 52 + 122 = 132, so (3, 4, 5) and (5, 12, 13) are Pythagorean triples. How do we nd all Pythagorean triples? Note that x2+y2 = N(x+i y), so Pythagorean triples come from Gaussian integers of square norm. The easiest way to get a Gaussian integer of square norm is to take the square of a Gaussian integer: α = (2 + i)2 = 3 + 4 i has norm 32 + 42 = (22 + 12)2 = 52, and β = (3 + 2 i)2 = 5 + 12 i has norm 52 + 122 = (32 + 22)2 = 132. More generally, taking the norm of α = (a + b i)2 = (a2 −b2) + 2ab i gives (a2 −b2)2 + (2ab)2 = (a2 + b2)2, so (a2 −b2, 2ab, a2 + b2) is a Pythagorean triple whenever a > b > 0. Our aim is to show that these are essentially all Pythagorean triples. Note that if (x, y, z) is a Pythagorean triple then so is (kx, ky, kz) for all k ⩾1, so we may as well only look for primitive Pythagorean triples with gcd(x, y, z) = 1, or equivalently gcd(x, y) = gcd(x, z) = gcd(y, z) = 1. Secondly, in a primitive Pythagorean triple (x, y, z) we cannot have both x and y odd, since that would imply z2 = x2 + y2 ≡1 + 1 = 2 (mod 4) which is impossible. So we might as well assume that x is odd and y is even (interchanging x and y if necessary). Further Number Theory G13FNT cw '11 Theorem 5.9. Let (x, y, z) be a primitive Pythagorean triple with y even. Then there exist coprime integers a, b with a > b > 0 and a ̸≡b (mod 2) such that x = a2 −b2, y = 2ab, z = a2 + b2. Proof. Let α = x + y i ∈Z[i], so N(α) = x2 + y2 = z2. The idea is to show that α is a square in Z[i]; writing α = (a + b i)2 then gives the result. We have z2 = N(α) = α · α = (x + y i)(x −y i). We next show that the factors x + y i and x −y i are coprime in Z[i]. If a Gaussian prime π divides both x + y i and x −y i then it divides 2 x = (x + y i) + (x −i y) and also divides 2y i = (x + y i) −(x −i y); since x and y are coprime it then divides 2, so π = 1 + i (times a unit). But 1 + i does not divide x + y i, since x ̸≡y (mod 2). Hence x+y i and x−y i are coprime in Z[i]. As their product is a square, unique factorisation in Z[i] implies that each is a square times a unit; using −1 = i2, each must be either a square or i times a square. Finally, x + y i = (a + b i)2 leads to x = a2 −b2 and y = 2ab, while x + y i = i(a + b i)2 leads to x = −2ab and y = a2 −b2. Since x, y > 0 and x is odd we must be in the rst case with a > b > 0, giving the result as stated. The conditions that gcd(a, b) = 1 and a ̸≡b (mod 2) both follow from gcd(x, y) = 1. Sums of two squares In this lecture we will investigate the following related questions: (i). For which n ∈N can we solve n = x2 + y2 with x, y ∈Z? (ii). Given n ∈N, how many solutions does the equation n = x2 + y2 have? These questions can be rephrased in terms of Gaussian integers α = x + y i, since N(α) = N(x + y i) = x2 + y2: (i). Which n ∈N are norms of Gaussian integers α ∈Z[i]? (ii). Given n ∈N, how many Gaussian integers α ∈Z[i] have norm n? Lemma 5.10. Any prime p ≡1 (mod 4) can be written as a sum of two squares. Proof. By lemma 5.6 we have p = ππ for some Gaussian prime π = x + y i. Then p = x2 + y2. Theorem 5.11. Let n = a · b2 be an integer with a square-free. Then n can be written as a sum of two squares if and only if no prime q ≡3 (mod 4) divides a. Proof. ⇐: For every prime p dividing a, there is a Gaussian prime πp of norm p by Proposi-tion 5.7. Put x + iy = b · Q p\|a πp. Then x2 + y2 = n. ⇒: Let n = x2 + y2 = (x + i y)(x −i y). If a prime q ≡3 (mod 4) divides n then, as it is a Gaussian prime by Lemma 5.5, it divides x + i y or x −i y. So q divides x and y, hence q2 divides n. The statement can now be proved by induction on b. Further Number Theory G13FNT cw '11 We will use L-functions to solve the second question, making further use of the Dirichlet character χ1 introduced in Chapter 4: χ1(n) =      +1 if n ≡1 (mod 4); −1 if n ≡3 (mod 4); 0 if n is even. The connection with Z[i] can be seen if we recall how ordinary prime numbers p factorise in Z[i], which depends on p modulo 4. Theorem 5.12. For each n ∈N, the number of integral solutions x, y to the equation n = x2 + y2 is given by 4 P d\|n χ1(d). The number of solutions with x > 0 and y ⩾0 is P d\|n χ1(d). Remark. To each solution (x, y) corresponding to α = x + y i ∈Z[i], we nd three more corresponding to the associates iα, −α and −iα, namely (−y, x), (−x, −y), (y, −x). Also, if x ̸= y then there are four more solutions coming from α and its associates, obtained by interchanging x and y. Examples: If n is 9, then P d\|9 χ1(d) = χ1(1) + χ1(3) + χ1(9) = 1 −1 + 1 = 1, and the single solution with x > 0 and y ⩾0 is (x, y) = (3, 0). If n = 25, then P d\|25 χ(d) = χ1(1) + χ1(5) + χ1(25) = 1 + 1 + 1 = 3, and solutions are (x, y) = (3, 4), (4, 3), (5, 0). Suppose now that n = p is a prime number. Then P d\|p χ1(d) = χ1(1) + χ1(p) = 1 + χ1(p), which equals 1 when p = 2 (solution: (1, 1)); equals 0 when p ≡3 (mod 4) (no solutions); and equals 2 when p ≡1 (mod 4) (two solutions, diering only in the order of x and y, for example 61 = 52 + 62 = 62 + 52 only). Example. Find all ways of writing n = 130 as a sum of two squares. Proof of Theorem 5.12. Let an be the number of solutions to n = x2 +y2 with x, y ∈Z, which is the number of elements α = x + y i ∈Z[i] with norm n. Then X n⩾1 an ns = X 0̸=α∈Z[i] 1 N(α)s . By unique factorisation in Z[i], the latter sum has an Euler product expansion: X 0̸=α∈Z[i] 1 N(α)s = 4 Y π∈Pi 1 1 −N(π)−s , where the product is over all prime elements π of Z[i], choosing one from each set of four associate primes, and the factor of 4 allows for the unit factor in the factorisation of each α. Now we look at the factors for each type of Gaussian prime in turn: (i). π = 1 + i with N(π) = 2 contributes a factor 1/(1 −2−s). (ii). each π with N(π) = p ≡1 (mod 4) contributes a factor 1/(1 −p−s), and there are two such Gaussian primes for each prime p ≡1 (mod 4). (iii). each π = q ≡3 (mod 4) with N(π) = q2 contributes a factor of 1/(1 −q−2s). Hence our product is Y π∈Pi 1 1 −N(π)−s = 1 1 −2−s   Y p≡1 mod 4 1 (1 −p−s)2     Y q≡3 mod 4 1 1 −q−2s  = ζ(s) · L(s, χ1) Further Number Theory G13FNT cw '11 using an exercise from problem sheet 4 at the end. So we have 1 4 X n⩾1 an ns = ζ(s) · L(s, χ1) =  X m⩾1 1 ms    X d⩾1 χ(d) ds  . Comparing coecients: 1 4an = X md=n χ1(d) = X d\|n χ1(d). Remark. The function ζ(s, Z[i]) = 1 4 X 0̸=α∈Z[i] 1 N(α)s is the zeta function of Z[i]; it is analogous to the Riemann zeta function ζ(s) = 1 2 X 0̸=n∈Z 1 \|n\|s . Remark. Since an ⩾0, the formula we proved has the consequence that for each n ∈N, the number of divisors of n which are congruent to 1 modulo 4 is greater or equal the number of divisors which are congruent to 3 modulo 4. Sums of more squares In the previous section, we found a formula for the number of integral solutions to n = x2+y2 for any given n ∈N. In particular this equation has a solution whenever n = p ≡1 (mod 4). Now we know that not every integer is a sum of two squares, can we do any better by taking sums of three squares? Now 3 = 12 + 12 + 12 and 6 = 12 + 12 + 22, but 7 is not a sum of three squares. In fact, no integer n ≡7 (mod 8) is a sum of three squares, since all squares are congruent to 0, 1 or 4 modulo 8. Instead of answering the question \exactly which positive integers are sums of three squares" (which turns out to be quite dicult) we'll move on to four squares, where there is a classical result. Theorem 5.13 (Lagrange, 1770). Every positive integer is a sum of four squares. In other words, for every n ∈N there exist x, y, z, w ∈Z (including zero) such that n = x2 + y2 + z2 + w2. Theorem 5.14 (Jacobi). Let n ⩾1 be an integer. Let An be the number of solutions x, y, z, w ∈ Z to the equation x2 + y2 + z2 + w2 = n. Then An = ( 8 P d\|n d if n is odd and 24 P 2∤d\|n d if n is even.
12339	https://dynamiclearningmaps.org/sites/default/files/documents/Math_EEs/M.EE.6.NS.5-8_Instructions.pdf	Mini-Map for M.EE.6.NS.5-8 Subject: Mathematics The Number System (NS) Grade: 6 DLM Essential Element: M.EE.6.NS.5-8 Page 1 of 4 © 2021 Accessible Teaching, Learning, and Assessment Systems (ATLAS) Learning Outcome DLM Essential Element Grade-Level Standard M.EE.6.NS.5-8 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero). M.6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. M.6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. M.6.NS.7 Understand ordering and absolute value of rational numbers. M.6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. DLM Essential Element: M.EE.6.NS.5-8 Page 2 of 4 Linkage Level Descriptions Initial Precursor Distal Precursor Proximal Precursor Target Successor Communicate understanding of "separateness" by recognizing objects that are not joined together. Communicate understanding of set by recognizing a group of objects sharing an attribute. Count all objects in a set to communicate the total number of objects in that set. Identify sets having the same number of objects. Identify a set containing a different number of objects than the other two sets. Recognize a set containing more or fewer objects than the other set. Communicate understanding that opposite numbers are equidistant from zero but in opposite directions, or that when two opposite numbers are added together they yield a sum of zero (e.g., 3 + (- 3) = 0, thus 3 and -3 are opposite numbers). Demonstrate use of positive and negative numbers in real-world contexts such as temperature, elevation, credits, and debits (e.g., representing a debit of 500 dollars as -500 dollars). Communicate understanding of inequalities in real-world contexts (e.g., -3 degrees > -7 degrees means that -3 degrees is warmer than -7 degrees). Communicate the meaning of zero in relation to positive and negative numbers in real-world contexts (e.g., recognize that no elevation, or 0 feet, means "at sea level"; positive elevation, for example, 200 feet, means "above sea level"; and negative elevation, for example, -200 feet, means "below sea level"). DLM Essential Element: M.EE.6.NS.5-8 Page 3 of 4 Initial Precursor and Distal Precursor Linkage Level Relationships to the Target How is the Initial Precursor related to the Target? How is the Distal Precursor related to the Target? In order to use positive and negative numbers, students need to gain experience with creating sets. Educators can help students learn this by providing students with opportunities to take a set of objects (e.g., tiles, linking cubes, buttons) and separate them based on a given characteristic (e.g., shape, color, size) into two distinct sets. Then encourage them to separate them again based on another characteristic. As students begin to develop the understanding of sets and numbers, educators will highlight the differences between sets on the basis of overall area or discrete number using the words same, different, fewer and more. Provide students with multiple opportunities to count and compare a wide variety of sets with an increasing number of items, label the set (e.g., eight ball, 12 bears, 15 blocks), and move items in and out of the sets, labeling and counting them again (e.g., "You just said this set has 11 cubes; if I take two cubes, how many will you have?"). Instructional Resources Released Testlets See the Guide to Practice Activities and Released Testlets. Using Untested (UN) Nodes See the document Using Mini-Maps to Plan Instruction. DLM Essential Element: M.EE.6.NS.5-8 Page 4 of 4 Link to Text-Only Map M.EE.6.NS.5-8 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero).
12340	https://www.pas.rochester.edu/~arijit/c02.pdf	Diﬀerentiation Formulas d dx k = 0 (1) d dx [f(x) ± g(x)] = f ′(x) ± g′(x) (2) d dx [k · f(x)] = k · f ′(x) (3) d dx [f(x)g(x)] = f(x)g′(x) + g(x)f ′(x) (4) d dx f(x) g(x) = g(x)f ′(x) −f(x)g′(x) [g(x)]2 (5) d dx f(g(x)) = f ′(g(x)) · g′(x) (6) d dx xn = nxn−1 (7) d dx sin x = cos x (8) d dx cos x = −sin x (9) d dx tan x = sec2 x (10) d dx cot x = −csc2 x (11) d dx sec x = sec x tan x (12) d dx csc x = −csc x cot x (13) d dx ex = ex (14) d dx ax = ax ln a (15) d dx ln \|x\| = 1 x (16) d dx sin−1 x = 1 √ 1 −x2 (17) d dx cos−1 x = −1 √ 1 −x2 (18) d dx tan−1 x = 1 x2 + 1 (19) d dx cot−1 x = −1 x2 + 1 (20) d dx sec−1 x = 1 \|x\| √ x2 −1 (21) d dx csc−1 x = −1 \|x\| √ x2 −1 (22) Integration Formulas Z dx = x + C (1) Z xn dx = xn+1 n + 1 + C (2) Z dx x = ln \|x\| + C (3) Z ex dx = ex + C (4) Z ax dx = 1 ln aax + C (5) Z ln x dx = x ln x −x + C (6) Z sin x dx = −cos x + C (7) Z cos x dx = sin x + C (8) Z tan x dx = −ln \| cos x\| + C (9) Z cot x dx = ln \| sin x\| + C (10) Z sec x dx = ln \| sec x + tan x\| + C (11) Z csc x dx = −ln \| csc x + cot x\| + C (12) Z sec2 x dx = tan x + C (13) Z csc2 x dx = −cot x + C (14) Z sec x tan x dx = sec x + C (15) Z csc x cot x dx = −csc x + C (16) Z dx √ a2 −x2 = sin−1 x a + C (17) Z dx a2 + x2 = 1 a tan−1 x a + C (18) Z dx x √ x2 −a2 = 1 a sec−1 \|x\| a + C (19)
12341	https://pmc.ncbi.nlm.nih.gov/articles/PMC7924338/	Imaging Techniques in Alzheimer’s Disease: A Review of Applications in Early Diagnosis and Longitudinal Monitoring - 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 Int J Mol Sci . 2021 Feb 20;22(4):2110. doi: 10.3390/ijms22042110 Search in PMC Search in PubMed View in NLM Catalog Add to search Imaging Techniques in Alzheimer’s Disease: A Review of Applications in Early Diagnosis and Longitudinal Monitoring Wieke M van Oostveen Wieke M van Oostveen 1 Faculty of Science, Leiden University, Einsteinweg 55, 2333 CC Leiden, The Netherlands; w.m.van.oostveen@umail.leidenuniv.nl Find articles by Wieke M van Oostveen 1, Elizabeth C M de Lange Elizabeth C M de Lange 2 Division of Systems Biomedicine and Pharmacology, Leiden Academic Centre of Drug Research, Leiden University, Einsteinweg 55, 2333 CC Leiden, The Netherlands Find articles by Elizabeth C M de Lange 2, Editor: Arkadiusz Orzechowski Author information Article notes Copyright and License information 1 Faculty of Science, Leiden University, Einsteinweg 55, 2333 CC Leiden, The Netherlands; w.m.van.oostveen@umail.leidenuniv.nl 2 Division of Systems Biomedicine and Pharmacology, Leiden Academic Centre of Drug Research, Leiden University, Einsteinweg 55, 2333 CC Leiden, The Netherlands Correspondence: ecmdelange@lacdr.leidenuniv.nl; Tel.: +31-71-527-6330 Roles Arkadiusz Orzechowski: Academic Editor Received 2020 Dec 15; Accepted 2021 Feb 16; Collection date 2021 Feb. © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( PMC Copyright notice PMCID: PMC7924338 PMID: 33672696 Abstract Background. Alzheimer’s disease (AD) is a progressive neurodegenerative disorder affecting many individuals worldwide with no effective treatment to date. AD is characterized by the formation of senile plaques and neurofibrillary tangles, followed by neurodegeneration, which leads to cognitive decline and eventually death. Introduction. In AD, pathological changes occur many years before disease onset. Since disease-modifying therapies may be the most beneficial in the early stages of AD, biomarkers for the early diagnosis and longitudinal monitoring of disease progression are essential. Multiple imaging techniques with associated biomarkers are used to identify and monitor AD. Aim. In this review, we discuss the contemporary early diagnosis and longitudinal monitoring of AD with imaging techniques regarding their diagnostic utility, benefits and limitations. Additionally, novel techniques, applications and biomarkers for AD research are assessed. Findings. Reduced hippocampal volume is a biomarker for neurodegeneration, but atrophy is not an AD-specific measure. Hypometabolism in temporoparietal regions is seen as a biomarker for AD. However, glucose uptake reflects astrocyte function rather than neuronal function. Amyloid-β (Aβ) is the earliest hallmark of AD and can be measured with positron emission tomography (PET), but Aβ accumulation stagnates as disease progresses. Therefore, Aβ may not be a suitable biomarker for monitoring disease progression. The measurement of tau accumulation with PET radiotracers exhibited promising results in both early diagnosis and longitudinal monitoring, but large-scale validation of these radiotracers is required. The implementation of new processing techniques, applications of other imaging techniques and novel biomarkers can contribute to understanding AD and finding a cure. Conclusions. Several biomarkers are proposed for the early diagnosis and longitudinal monitoring of AD with imaging techniques, but all these biomarkers have their limitations regarding specificity, reliability and sensitivity. Future perspectives. Future research should focus on expanding the employment of imaging techniques and identifying novel biomarkers that reflect AD pathology in the earliest stages. Keywords: Alzheimer’s disease, imaging techniques, early diagnosis, longitudinal monitoring, amyloid-β, tau, MRI, PET 1. Introduction Alzheimer’s disease (AD) is a progressive neurodegenerative disorder resulting in memory loss, cognitive impairment, behavioural changes and eventually death . AD is the most common cause of dementia and is predicted to affect more than 152 million people in 2050 . The disease is neuropathologically characterized by the deposition of abnormal protein resulting in the formation of extracellular senile plaques and intracellular neurofibrillary tangles (NFTs) [3,4]. The senile plaques contain primarily neurotoxic amyloid-β (Aβ) , whereas NFTs consist of abnormal hyperphosphorylated tau aggregates [6,7]. Although the contribution of abnormal protein deposition to AD is recognized, the exact pathogenesis of AD is complex , and definitive diagnosis can only be assured post-mortem by histology staining of the brain . Currently, AD is the only cause of death in the top ten deaths globally for which no effective therapeutic treatment is available, and there are no registered drugs to slow down disease progression . Therefore, much effort is put into understanding the pathogenesis of AD for the development of therapeutic agents . In AD, neuropathological changes occur up to thirty years before clinical manifestation of the disease . The initial pathological event in AD is Aβ deposition, which contributes to the formation of senile plaques. Likewise, hyperphosphorylation results in NFTs, leading to neuronal loss, brain atrophy, neurotoxicity, and ultimately cognitive decline . In 1991, Braak and Braak characterized the spread of NFTs across the brain and defined six different stages . These Braak stages correspond with the expansion of NFTs from transentorhinal regions (stage I/II) to limbic areas (stage III/IV) and neocortical regions (stage V/VI) as AD progresses. The above listed events succeed and overlap each other and, therefore, AD is seen as a continuum with pathological changes and clinical symptoms corresponding to the disease stage (Figure 1). Since damage inflicted by these events can surpass a certain neuropathological threshold beyond which any treatment will be unsuccessful, it has been suggested that therapeutic agents should focus on halting neurodegeneration in the silent phase of AD before it becomes too severe [14,15,16]. Therefore, sensitive and specific methods are needed to diagnose AD in the early or preclinical stage . Nowadays, the field of research focuses on identifying so-called biomarkers, which are physiological, chemical or anatomical parameters called biomarkers that effectively reflect certain pathopsychological processes in AD . These biomarkers can be categorized into three different classes based on the type of pathophysiology the biomarker tracks. In this so-called “A/T/N” system, “A” refers to biomarkers measuring Aβ deposition, “T” indicates biomarkers sensitive for tau and “N” the value of biomarkers perceptive for neurodegeneration . This framework is adaptable and can continuously be expanded if new biomarkers become available . Figure 1. Open in a new tab The Alzheimer’s disease continuum with corresponding pathological changes, biomarkers and clinical diagnosis. Figure adapted from Yoshiyama et al. . An ideal biomarker is inexpensive, easy to monitor and non-invasive and, therefore, will barely harm a patient. Moreover, a good biomarker has high sensitivity and predictive qualities for the specific pathological event . Eventually, biomarkers could offer a diagnostic tool to detect the disease in early stages, thereby providing the opportunity to delay disease progression or even impede the clinical manifestation of the disease . Additionally, monitoring these biomarkers over time could give insight into disease progression and be utilized to track the effectiveness of disease-modifying therapeutics. In this review, we focus on biomarkers that can be tracked with structural or functional neuroimaging techniques, such as magnetic resonance imaging (MRI) and positron emission tomography (PET), respectively. The aim of this review is to give an overview of established biomarkers for the early diagnosis and longitudinal monitoring of AD and discuss their feasibility and potential drawbacks. First, we shed light on current biomarkers for the early diagnosis of AD and longitudinal monitoring of disease progression. These biomarkers are reviewed based on their diagnostic utility, benefits and limitations. Subsequently, we introduce new biomarkers and applications of imaging techniques that show promising results for the early diagnosis or longitudinal monitoring of Alzheimer’s disease. Finally, we summarize our findings and provide future perspectives. 2. Contemporary Early Diagnosis of AD with Imaging Techniques Since the number of patients with AD is increasing due to an aging population, much effort has been put into the detection of the disease as early as possible. Many methods have been tested, ranging from cognitive tests, MRI scans and sampling of cerebral fluid . In this section, we focus on what imaging techniques and associated biomarkers are applied in the early diagnosis of AD. The first applications of imaging techniques in AD were computed topography (CT) and MRI, but these techniques were used to exclude other causes of dementia rather than to diagnose AD in an early stage . Later, imaging techniques were utilized as positive support to confirm the clinical diagnosis of AD. These techniques focused on the neuronal injury and degeneration aspects of AD . Nowadays, imaging modalities focus on either identifying amyloid deposition or identifying neurodegeneration . 2.1. Structural MRI 2.1.1. Background The pathology of AD follows a typical spreading pattern through the brain in which certain areas are among the first affected, while other regions will only be impaired in severe stages of AD [3,25]. In this so-called topographic pattern that characterizes AD, the earliest changes are found in the medial temporal lobe structures, the entorhinal and perirhinal cortex and the hippocampus (Figure 2). This typical pattern of disease progression opened possibilities for the early diagnosis of AD by investigating these brain parts with imaging techniques. Figure 2. Open in a new tab Regions affected by Alzheimer’s disease. Figure created with www.BioRender.com (accessed on 12 February 2021). 2.1.2. Findings Since neuronal damage in the hippocampus is manifested as decreased hippocampal volume , a widely accepted method for assessing AD pathology is volumetric MRI scans of the hippocampus (Figure 3) . These scans are T1-weighted images from which hippocampal atrophy can be measured with either manual or automated segmentation . According to a study of Bobinski et al., MRI provided a powerful tool in assessing the hippocampal volume and predicted volumes that correlated strongly with neuronal numbers, suggesting the anatomic validity of volumetric MRI measurements . Moreover, another study found that volume reductions in the hippocampus are early indications for AD pathology, measurable with MRI . Figure 3. Open in a new tab Neuroimages of the healthy versus the Alzheimer’s disease (AD) brain. Neuroimaging with (a) structural MRI, (b) FDG-PET, (c) amyloid-PET with PiB and (d) tau PET with 18 F-AV1451 in both healthy and AD brains. Figure created with www.BioRender.com (accessed on 14 February 2021). In addition to the hippocampus, other limbic brains regions that can be studied with MRI are the entorhinal cortex and amygdala (Figure 2). Although it is believed that the entorhinal cortex is among the regions affected first in AD [30,31,32] and the accuracy of entorhinal cortex volumetry being slightly higher , several cross-sectional studies suggested that entorhinal cortex measurements are unlikely to offer additional benefits over hippocampal volumetry in AD patients when compared to healthy controls [33,34,35]. Moreover, high variability in methods to assess the entorhinal cortex volume due to anatomic ambiguity in the cortex’s boundaries eliminates the slight superiority of the entorhinal cortex over the hippocampus [29,34,36]. In addition to the use of structural MRI in assessing volume reductions, another application of this imaging technique is to detect cortical thickness reduction in certain brain areas, such as the temporal, orbitofrontal and parietal regions . Detailed study has demonstrated the effect of AD on cortical thickness and led to the suggestion of a so-called AD “disease signature” in which certain brain regions known to be affected by AD show cortical thinning . Assessment of the cortical thickness is believed to be a useful biomarker in the early diagnosis of AD, since subtle changes in areas known to be affected by AD can be detected . Furthermore, a study into region- and phase-specific changes has linked disease severity to cortical thickness , thereby coupling the clinical dementia ranking stages to a level of cortical thinning. Additionally, cortical thickness correlates strongly with cognitive impairment in the clinical stages of AD [41,42]. Over time, volumetric MRI of the hippocampus has been seen to be the best-established biomarker for AD [1,33,36,43], especially as a diagnostic marker in the mild cognitive impairment stage (MCI) . Additionally, one major benefit of MRI is the availability of appliances in hospitals and research centres . Moreover, MRI is safe and is seen as non-invasive, since it involves no ionizing radiation. 2.1.3. Limitations However, structural MRI as an imaging technique for AD has its limitations. First, decreased hippocampal volume is not an AD-specific measure . An extensive study by Geuze et al. reviewed more than 420 records reporting the assessment of hippocampal volume with MRI . In addition to AD, other neurodegenerative diseases are characterized with diminished hippocampal volume as well such as Parkinson’s disease , epilepsy and Huntington’s disease . Additionally, volume reduction has also been observed after cardiac arrest , chronic alcohol abuse and survivors of low birth weight . Moreover, recent study has demonstrated that hippocampal texture predicts conversion from MCI to AD with higher accuracy than the hippocampal volume, although these results have to be validated with histological data . Lastly, manual segmentation of T1-weighted images is time-consuming , requires specialistic training and can result in high levels of variability in the measurements , due to different protocols for assessing the measurements . Fortunately, in the last decade, much effort has been put into establishing methods for automated segmentation, resulting in more accurate data from MRI images in less time [53,55,56,57]. One major drawback of structural MRI in general is the impossibility to directly observe the effect of amyloid plaques or NFTs in the brain. Atrophy is downstream of the pathological event and not disease specific . Moreover, several studies demonstrated that in atypical forms of AD, the hippocampus is spared [58,59]. Therefore, structural MRI in atypical manifestations of AD might be not able to identify the disease in an early stage. 2.2. FDG-PET 2.2.1. Background Multiple diseases affecting the central nervous system (CNS) are associated with impaired glucose uptake by neurons . With fluorodeoxyglucose positron emission tomography (FDG-PET), it is possible to measure the resting state cerebral metabolic rates of glucose as a proxy of neuronal activity, without the requirement of cognitive activity [61,62]. FDG-PET measures the uptake of a radiolabeled glucose analogue which correlates with cerebral metabolism and synaptic activity (Figure 3) [23,43]. Since reduced cerebral metabolism is associated with age, healthy age-matched individuals show corresponding cerebral metabolism patterns . The comparison of FDG-PET scans of AD patients with healthy individuals of the same age revealed patterns of metabolic abnormalities in AD, leading to a so-called FDG-PET endophenotype . This endophenotype is seen as a characteristic of AD in which certain brain regions or areas are affected in a spatial pattern . In AD, hypometabolism occurs first in the temporoparietal areas of the brain, including the precuneus and posterior cingulate cortex [1,61] (Figure 2). Moreover, as the disease progresses, the metabolic deficits are gradually aggravated . 2.2.2. Findings Among the first studies that successfully applied FDG-PET in studying Alzheimer’s disease was a research project by Benson et al. in 1983 in which both AD patients and patients with multi-infarct dementia were studied . The results from this study revealed that in AD patients, almost all brain areas demonstrate reduced glucose metabolism, but the primary motor and sensory cortex are spared. This work inspired other researchers and led to an increase in studies investigating the effect of AD on glucose metabolism in the brain [64,65,66]. However, all these studies used patients with diagnosed AD in mild to severe stages of the disease and did not use FDG-PET to diagnose the patients. In the 1990s, automated methods to standardize the evaluation of PET scans increased, leading to more consistency in the evaluation of FDG-PET images obtained in different research centers or with different equipment [67,68]. A large study by Silverman et al. used FDG-PET as a diagnostic tool for differentiating healthy individuals from patients with AD symptoms. In the study, the sensitivity and specificity of FDG-PET were addressed, in which sensitivity reflects the ability to identify AD subjects among all individuals, whereas specificity addresses the ability to correctly identify subjects as non-AD. FDG-PET was able to detect AD subjects with a sensitivity of 94% and a 73% specificity. Additionally, in patients diagnosed with questionable or mild dementia, the sensitivity was 95% with a specificity of 71% . These results indicated that FDG-PET is a sensitive indicator of AD and can also be used to assess early-stage dementia. The findings were underlined with other research studies with sensitivity ranging from 84% to 93% and specificity between 63% and 74% [70,71]. Furthermore, reviews based on meta-analyses of articles regarding the identification of AD patients among healthy individuals resulted in pooled sensitivities up to 96% with specificities up to 90% [72,73,74]. Finally, Panegyres et al. demonstrated that FDG-PET is able to differentiate between different types of dementia up to 95% . Over the years, FDG-PET emerged to be a relevant and highly specific biomarker for the early diagnosis of AD and other types of major neurodegenerative diseases [43,75]. It is seen as a robust and reliable biomarker in the in vivo diagnosis of early stages of AD [23,36,43]. Moreover, compared to structural MRI of the hippocampus and entorhinal cortex, FDG-PET is diagnostically superior to volumetry measures . Additionally, according to the hypothetical model of dynamic biomarkers proposed by Jack and colleagues, abnormal FDG-PET precedes changes detectable with MRI [77,78] (Figure 1), suggesting an FDG-PET of higher value than structural MRI in the early diagnosis of AD. 2.2.3. Limitations However, FDG-PET has its limitations. PET scanners are not widely available and considered as relatively expensive [23,36]. FDG-PET requires the intravenous injection of a radiolabeled agent and is, therefore, more invasive than MRI. Moreover, hypometabolism is a result of neurodegeneration and, therefore, it might not be suitable to detect signs of AD in the earliest stages before neuronal loss occurs . By the time hypometabolism is measurable with FDG-PET, damage inflicted to neurons might be too severe to benefit from therapies. Additionally, increasing evidence suggests that FDG-PET shows the consumption of glucose by astrocytes, rather than by neurons and, therefore, hypometabolism can be ascribed to decreased astrocyte function . Lastly, it is important to keep in mind that atypical clinical manifestations of AD may have the same pathophysiology as typical AD, but can show distinct metabolic patterns . This heterogeneity in changes in metabolic patterns among the distinct AD subtypes can reduce the diagnostic accuracy of FDG-PET. Since neurodegeneration is a pathological event preceded by amyloid plaques and NFTs, biomarkers sensitive for these two events might be more suitable for the diagnosis of AD in the early stages than FDG-PET. 2.3. Amyloid-PET 2.3.1. Background Many researchers believe that the first pathological event in AD is Aβ accumulation, leading to the formation of senile plaques [7,20,82,83]. However, this belief is still subject to debate [84,85]. To detect and design therapies for plaques, it is important to find out what causes plaque formation. Therefore, senile plaques were intensively studied, but due to their insolubility, the attempts to identify their composition in many studies failed . Finally, in the mid-1980s, researchers were able to identify the Aβ protein as a primary component of the plaques. This protein, with an average chain length of forty-two amino residues, results from the cleavage of the larger amyloid precursor protein (APP) by β- and γ-secretase (Figure 4) [36,86]. At first, it was believed that Aβ was an abnormal protein, but the presence of Aβ in culture medium, cerebrospinal fluid (CSF) and plasma revealed that Aβ is a normal product of APP metabolism [12,83]. This understanding led to a new hypothesis: the amyloid cascade hypothesis. This hypothesis states that a dysregulation in the production and clearance of Aβ in the brain leads to the accumulation of Aβ in oligomers, protofibrils and eventually mature fibrils , ultimately leading to neurodegeneration and dementia. All other disease characteristics, such as the formation of NFTs out of hyperphosphorylated tau, and neurodegeneration, are seen a result of this accumulation [12,83,88]. Figure 4. Open in a new tab Accumulation of amyloid-β in AD. Figure adapted from Patterson et al. and created with www.BioRender.com (accessed on 16 November 2020). 2.3.2. Findings Because Aβ deposition in the brain is commonly seen as the earliest hallmark of AD, many studies have focused on identifying biomarkers that differentiate healthy controls from individuals with the first pathophysiological signals of AD. Currently, two distinct biomarkers are used to assess Aβ pathology: the concentration of Aβ42 in the CSF and amyloid-PET [1,77,90]. In this section, we discuss how PET imaging of the Aβ accumulation can contribute to the early diagnosis of AD. 11 C-PiB In 2004, a novel 11 C radiotracer named Pittsburgh compound-B (PiB) was applied in a study containing mild AD patients and a control group . The study showed that PiB retention time was equivalent in both groups in brain regions known to be relatively unaffected by Aβ deposits. However, compared to controls, the individuals with mild AD showed considerable retention of PiB in areas known to contain substantial Aβ accumulation in AD (Figure 3). These areas included cortical areas, such as the frontal cortex and neocortex (Figure 2). The findings suggested that PET neuroimaging with PiB could provide quantitative information about Aβ deposition in living patients in early (mild AD) stages of the disease. Rabinovici et al. demonstrated that amyloid-PET imaging with PiB was able to distinguish AD subjects from patients with other forms of dementia, such as frontotemporal dementia (FTD) . All AD subjects (7/7) had positive PiB-PET scans, while in FTB patients and healthy controls, respectively, 8/12 and 7/8 scans were negative. These findings were confirmed in a study with AD subjects and patients suffering from FTD in which the retention time of PiB in FTD patients was measured. In total, 8/10 FTD patients showed a significantly lower retention time compared to AD subjects, indicating that PiB might be a tool in differentiating FTD from AD . In addition to differentiating between different types of dementia, amyloid-PET with PiB was able to identify the different stages of the AD continuum. In a study by Lowe et al., PiB-PET was able to significantly differentiated healthy controls from non-amnestic MCI and amnestic MIC, and AD . In another study, PiB-PET clearly differentiated AD patients from MCI and healthy subjects . Moreover, both studies suggested that the diagnostic value of PiB-PET increases when combined with FDG-PET, since information obtained from both techniques might be complementary. In 2014, Leuzy et al. published a paper concerning the increased PiB retention restricted to specific brain regions associated with higher levels of Aβ deposition . This pattern was histological confirmed by the comparison of imaging data with immunohistochemical exams post-mortem [97,98]. On a molecular level, PiB is believed to bind insoluble Aβ fibrils . Another study reports the strongest PiB binding to Aβ42 fibrils, followed by significant binding to Aβ42 oligomers and protofibrils , but compared to the fibril binding, this binding to protofibrils and oligomers is increasingly lower. Additionally, increasing evidence suggests insoluble Aβ being only a fraction of total Aβ in the brain and a more prominent role of soluble protofibrils in the pathogenesis of Aβ . Aβ-Pet with PiB may, therefore, be more a reflection of a fraction of insoluble Aβ than an image of total Aβ pathology in the brain. Over the years, amyloid-PET with PiB has emerged as the gold standard in Aβ imaging . Nevertheless, PiB has its limitations and drawbacks as a radiotracer in Aβ imaging. First, the short half-life (twenty minutes) of the 11 C isotope requires a nearby cyclotron for clinical usage [86,103]. Second, as previously mentioned, PiB has the tendency to only bind to the fibrillar form of Aβ and has low affinity for soluble oligomeric Aβ , while it is believed that in some genetic forms of AD, oligomeric Aβ plays a significant role in the disease manifestations [105,106]. PiB-PET might fail as a diagnostic tool in identifying these types of AD. Lastly, the selectivity of a positive Aβ scan obtained with PiB as a biomarker for AD is relatively low, because elevated PiB uptake has also been found in 30% of healthy controls without cognitive disorders . 18 F-Labelled Radiotracers The short half-life of 11 C in PiB led to the development of 18 F-labelled radiotracers, and in 2008, the first study reported successful imaging with a fluorinated radiotracer in humans . Currently, three 18 F-labelled radiotracers for assessing Aβ deposition are approved: florbetapir, flutemetamol and florbetapen . Florbetapir was the first fluorinated radiotracer and had retention ratios strongly associated with PiB [103,108]. A detailed meta-analysis into the three 18 F-labelled radiotracers revealed no apparent differences between the diagnostic accuracy of the radiotracers . However, compared to PiB, the fluorinated tracers showed higher levels of non-specific uptake in the white matter due to the more lipophilic nature of both radiotracer and white matter [110,111], resulting in more background noise . Due to this extra noise, the typical white matter pattern caused by cortical amyloid plaque is lost . 2.3.3. Limitations Altogether, Aβ-PET collected the first in vivo evidence of earliest protein deposition . PET has become a powerful tool in the detection of Aβ deposition and can contribute to the early diagnosis of AD. However, to fully employ its opportunities, some obstacles must be resolved. One difficulty to overcome is finding a consensus on methods to quantify amyloid-PET scans . There is an urgent need for a tool to discover even the smallest Aβ deposits, and cut-off levels need to be defined in order to make studies comparable . Furthermore, most studies have used radiolabeled tracers in the typical form of the AD spectrum, resulting in much knowledge about Aβ accumulation in typical AD, while the atypical, non-amnestic type has remained understudied . An increasing body of evidence suggests the utility of Aβ-PET to diagnose patients with atypical manifestations of AD, such as posterior cortical atrophy (PCA) and logopenic-variant primary progressive aphasia (LvPPA) [104,116]. To maximally benefit from the diagnostic accuracy of Aβ-PET, more study into Aβ accumulation in atypical subtypes of AD is required. Lastly, the amyloid cascade hypothesis is still a hypothesis, and although Aβ deposition is an early event in the pathogenesis of AD, it may not be the direct cause of neurodegeneration and cognitive decline . 2.4. Tau-PET 2.4.1. Background Since multiple attempts for developing anti-amyloid drugs have failed in clinical trials, interest has shifted from treating Aβ accumulation towards development of PET radiotracers for identifying tau aggregates. This shift of interest is accompanied by the thought that tau protein aggregates are more closely related with cognitive impairment [111,117]. Furthermore, increasing evidence suggests a role for oligomeric Aβ and tau species in the early stages of AD rather than Aβ plaques and NFTs [118,119]. Tau is a microtubule-associated protein with six isoforms and is abundantly expressed in the CNS where it stabilizes the microtubules of axons (Figure 5) . Several posttranslational processes can modify tau, such as acetylation, glycosylation, methylation and phosphorylation, which affect the ultrastructural conformation of tau . Although normal phosphorylation of tau is required for its role in cytoskeletal plasticity during early development , hyperphosphorylation combined with decreased dephosphorylation leads to soluble hyperphosphorylated tau that rapidly aggregates into so-called tauopathies . In AD, aggregation of tau results in paired helical fragments (PHFs), and these PHFs can further accumulate into intracellular NTFs [117,123]. Even though the exact mechanism of tau aggregation is still unclear, the accumulation of tau in considered to play a major role in the neurodegenerative aspect of AD . Figure 5. Open in a new tab Tau protein aggregation leads to formation of neurofibrillary tangles in AD. (a) Role of tau protein in healthy brain; (b) role of tau protein in Alzheimer’s disease brain. NFT: neurofibrillary tangles; PHF: paired helical filaments. Figure adapted from “Pathology of Alzheimer’s Disease”, by BioRender.com (2020). Retrieved from (accessed on 16 November 2020). 2.4.2. Findings Just like Aβ accumulation, NFTs spread through the brain as AD progresses. This spreading pattern initiates in the entorhinal cortex (Figure 2), and as the disease progresses, NFTs spread to the limbic (stage III-IV) and isocortical (V-VI) association areas (Figure 6) . However, in atypical variants of AD, the exact spreading pattern may be distinct from typical AD, and these differences in spreading patterns characterize atypical variants in early stages . Although post-mortem quantification of tauopathies in the brain remains the gold standard, growing evidence suggests a role for tau-PET imaging with radiotracers in vivo for the clinical evaluation of the disease [124,126]. There are several challenges in the development of radiotracers for tau-PET. First of all, PET tracers must be able to pass the blood–brain barrier (BBB) . Second, the intracellular location of NTFs poses a second barrier for the radiotracer to overcome [4,117,127]. Moreover, rapid clearance from the blood and high sensitivity are desired . Since Aβ deposits and NFTs both compromise beta sheets and Aβ concentrations are remarkably higher, high affinity for tau over Aβ is required . Figure 6. Open in a new tab Tau spreading pattern in each Braak stage. Spreading pattern of tau throughout the brain from Braak stage I-II to stage III-IV (limbic regions) and stage V-VI (isocortical areas). Figure adapted from Goedert and created with www.BioRender.com (accessed on 11 February 2021). In AD, tau aggregates are most prominently present in the ultrastructural PHF form and therefore most attempts in developing tau-PET tracers have focused on imaging these PHFs . Based on their structures, the currently available tau tracers can be divided into four groups: the nonselective tracer 18 F-FDDNP, quinoline derivatives, pyrido−indole derivatives and PBB3 [123,124]. Computational modeling of tau fibril using cryo-EM structures of PHFs and straight filaments has identified four high-affinity binding sites for tau tracers . Three binding sites are buried within the core of the fibril, whereas one site is located on the surface. The next section discusses several tau PET tracers and their binding to tau at the molecular level based on this computational model. 18 F-FDDNP 18 F-FDDNP is a fluorinated naphthyl-ethylidene derivative and was the first tracer applied in PET imaging of tauopathy in the brain . This tracer is able to bind both extracellular amyloid-β plaques and intracellular NFTs due to the presence of β sheets in these proteins [131,132]. 18 F-FDDNP seems to favor the core sites of the tau fibril for binding due to hydrophobic interactions. In addition to Aβ plaques and NFTs, 18 F-FDDNP also binds prion plaques and is used to assess chronic traumatic encephalopathy suspicion [127,130]. Since 18 F-FDDNP favors binding to amyloid-β over tau [127,129,132], screening of β sheet-binding small molecules was performed on a large scale to identify more suitable and specific tau tracers. Quinoline Derivatives The first selective tau PET tracers were based on quinoline and benzimidazole derivatives [111,133] and focused on the imaging of PHF tau . A study by Okamura et al. synthesized three new compounds, BF-126, BF-158 and BF-170, as possible probes for in vivo tau-PET imaging in the brain . The compounds showed good brain uptake combined with rapid clearance from brain tissue. Additionally, in the neuropathological exam, the three compounds were able to visualize NFTs and PHF-type neuritis, suggesting that quinoline and benzimidazole derivatives might be potential tracers for tau-PET. With these findings in mind, the search for selective tau tracers continued with the development of 18 F-THK5105 and 18 F-THK5117. These two compounds were developed to enhance the binding affinity to PHF-tau and demonstrated binding affinity and selectivity to PHF-tau over amyloid-β in AD . Similar to 18 F-FDDNP, these two radiotracers favor the hydrophobic core sites of tau over the surface site according to the computational model . Moreover, clinical PET studies revealed that these radiotracers were able to differentiate brains of AD subjects from brains of healthy controls [135,136]. A drawback of these radiotracers was the high non-negligible binding to white matter caused by the β sheet conformation of myeline. To solve this problem, 18 F-THK5351 was developed. This new compound exhibited rapid clearance from the white matter . Additionally, 18 F-THK5351 showed higher specific binding to tau-associated regions than 18 F-THK5117 . As a result, 18 F-THK5351 occurs to be the most promising arylquinoline radiotracer for the early detection of tau-associated pathology in AD subjects . Pyrido−Indole Derivatives 18 F−T808 and flortaucipir, also known as 18 F-AV-1451 or 18 F-T807, are both fluorinated pyrido-indole derivatives with high selectivity for tau over Aβ deposits [4,111]. Although 18 F−T808 exhibited high tau affinity, rapid uptake and clearance, a disadvantage of this compound was de defluorination followed by bone uptake of 18 F . On the other hand, flortaucipir showed over 25-fold selectivity for tau against Aβ plaques combined with low levels of white matter uptake . Moreover, the uptake of flortaucipir corresponds well with the expected spatial pattern of tau pathology in the brain of AD subjects (Figure 3) [140,141]. Furthermore, it is believed that flortaucipir binds with high affinity to all three isoforms of tau when in the classical PHF form , which is likely due to its high affinity for more than one binding site of the tau fibril . However, flortaucipir exhibited low affinity for tau aggregates consisting of primarily straight tau filaments (Figure 5), indicating that flortaucipir might not be a suitable radiotracer in diseases other than AD. PBB3 The last group of tau-PET tracers consists of PBB3, a 11 C-labelled radiotracer that is able to detect both AD and non-AD tauopathies . The compound exhibited up to 50-fold higher binding affinity for tau than for Aβ, binds to a wide range of tau isoforms and has affinity for tau at a binding site differently than other radiotracers, which might explain its wide binding range . The uptake of the compounds is elevated in the hippocampus and spreads to the association cortex as disease progresses. The drawbacks of PBB3 are the usage of short half-life 11 C in the radiotracer and the ability of its major metabolite to cross the BBB [143,144]. 2.4.3. Limitations The development of novel tau tracers is an ongoing process in which several pharmaceutical companies are trying to improve the pharmacokinetics and pharmacodynamics of the tracers . Compared to Aβ, the development of tau-PET tracers is still behind, and clinical validation of the tracers is required . Nevertheless, tau-PET poses another neuroimaging tool for the early diagnosis of AD. 2.5. Summary Although the applications of imaging techniques in the early diagnosis of AD are on the rise, an impeccable biomarker that can diagnose AD in the earliest stage is still not available. All current techniques have their limitations (Table 1), and most importantly, there is a significant amount of protein deposition or atrophy needed for detection. Since AD is known to have decades of pathological changes before the clinical onset of disease and disease-modifying treatments may be the most beneficial before certain thresholds of protein levels or atrophy are passed, current imaging biomarkers may diagnose AD in an overly progressed stage in which therapies will inevitably fail. Table 1. Advantages and limitations of imaging techniques currently used in early diagnosis and longitudinal monitoring of AD. \| Technique \| \| Early Diagnosis \| Longitudinal Monitoring \| :---: :---: \| \| Structural MRI \| Advantages \| Powerful in predicting volumes \| Changes in atrophy closely related to changes in cognitive abilities High atrophy rates predict cognitive decline \| \| MRI scanners widely available Safe \| \| Limitations \| Direct observation of Aβ plaques or NFTs not possible \| Findings based on small population sizes and limited number of scans \| \| Decreased hippocampal volume not AD-specific measure Atrophy patterns differ among AD subtypes \| \| FDG-PET \| Advantages \| Extensive research led to an FDG-PET endophenotype usable for comparison Highly sensitive and specific \| Differences in metabolism patterns able to predict risk to convert to AD Diminished FDG uptake precedes clinical manifestation Heterogeneity in topographic progression of reduced metabolism may predict AD variant \| \| Limitations \| \| Reduced glucose metabolism caused by other diseases or injuries \| \| Rather reflection of glucose consumption by astrocytes Invasive due to injection and radiolabelled tracer Expensive and not widely available \| \| Amyloid-PET \| Advantages \| Aβ plaques seen as earliest hallmark of AD Retention time of radiotracers matches spreading pattern of Aβ plaques \| PiB retention time able to predict conversion from MCI to AD \| \| Limitations \| Exact role of Aβ accumulation in AD still unknow Elevated PiB uptake also found in healthy controls No standard method for quantifying Aβ plaques Not much known about Aβ accumulation in atypical forms \| Weak correlation between Aβ deposition and disease severity Aβ accumulation stabilizes in later stages of AD Choice of reference region subject to debate \| \| Invasive due to injection and radiolabelled tracer \| \| Tau-PET \| Advantages \| Tau accumulation believed to be closely related to cognitive impairment Radiotracer uptake matches spreading pattern of tau Radiotracers have high affinity for PHF tau \| Strong relationship between neurofibrillary pathology and neurodegeneration Accumulation rates consistently increase throughout the brain \| \| Limitations \| Most tracers low affinity for straight filaments \| High level of heterogeneity in tau topography between AD subtypes \| \| Still new field of research Invasive due to injection and radiolabelled tracer \| Open in a new tab 3. Contemporary Longitudinal Monitoring of AD with Imaging Techniques Since AD is a progressive chronic disease with no clinical endpoint, the longitudinal monitoring of disease progression and variations in biomarker levels can give insight into the pathogenesis and prognosis of the disease . In correspondence with the early diagnosis of the disease, to date, there is no consensus on the biomarkers, techniques or tests that are the most clinically relevant in monitoring the disease in the long-term. In this section, we discuss the imaging techniques and associated biomarkers that are applied in the longitudinal study of AD. 3.1. Structural MRI 3.1.1. Background Whilst structural MRI might not be the most suitable technique for the early diagnosis of AD, many studies have used structural MRI to monitor disease progression, because rates of change in multiple structural measures are closely associated with changes in cognitive abilities . 3.1.2. Findings In an early study by Fox et al. in 1999, whole-brain atrophy was linked to increased disease severity . Patients with untreated, probable AD were age-matched with a group of healthy controls to assess the relationship between disease severity and atrophy progression within the subject. Each individual underwent at least two MRI scans and both groups also participated in mini-mental state examinations (MMSE) on the MRI scans’ dates. The scans revealed that AD subjects had a mean rate of whole-brain atrophy of 2.4 ± 1.4% per year, while the control group had a mean loss of 0.4 ± 0.7%. Additionally, the MMSE scores demonstrated a significant difference in the mean rate of decline between AD patients and healthy individuals, indicating that rate of cerebral atrophy is strongly correlated with decline in cognitive ability. Two other studies also investigated the correlation between whole brain atrophy rates and cognitive performance and underlined the finding that whole-brain atrophy is strongly associated with cognitive decline, making cerebral atrophy an interestingly and clinically relevant biomarker for tracking AD progression [148,149]. Moreover, one of the studies implicated patients with MCI and found that a higher rate of brain atrophy per year was associated with an elevated risk of developing dementia . In addition to whole-brain atrophy, rates of atrophy have also been evaluated in other structural regions of the brain. Cardenas et al. focused on identifying spatial patterns of brain atrophy associated with cognitive performance and possible future cognitive decline . The study used deformation-based morphometry (DBM) in which every voxel is spatially normalized to a template brain. This enables the comparison between subjects with different rates of disease progression . Atrophy rates in the hippocampus and entorhinal cortex (ERC) of non-demented elderly with different levels of cognitive performance were combined with several neuropsychological tests. Smaller volumes of hippocampus and ERC were strongly correlated with memory function at baseline and also predicted memory decline . These results suggest that baseline volumes of these regions may predict cognitive decline due to aging, pathology or both. Thompson et al. created maps of hippocampal and ventricular change over a longer period with a goal to visualize the spatial progression of AD and the rate of change . Over time, the hippocampal volume decreased, while the ventricular volumes expanded (Figure 7). Interestingly, the spreading patterns were different between aging and dementia. Temporal horn expansion in the ventricles turned out to be a promising marker for disease progression and corresponded well with rates of cognitive decline. These results suggest that visualizations of hippocampal atrophy and ventricular expansion rates may provide a promising marker to monitor AD progression. In a study by Jack et al., the above structural MRI measures were combined and evaluated on their ability to predict disease progression . A group of 160 individuals was recruited based on their profile to meet either the criteria for cognitively unimpaired, MCI or AD. All subjects had a series of MRI scans of the whole brain, hippocampus, entorhinal cortex and ventricles. Similar to the previously described studies, the change in cognitive performance was assessed with multiple tests. Over time, subjects could remain stable or shift to a more cognitive impaired group. In all brain regions, the atrophy rates were higher among subjects that converted to a more severe disease profile and supported the applicability of rates of change from longitudinal MRI measures as markers for AD progression. Figure 7. Open in a new tab AD leads to hippocampal atrophy and ventricle enlargement. Healthy brain (left) versus AD brain (right). AD leads to decreased hippocampal volume, shrinkage of cerebral cortex and ventricle enlargement. MTA: medial temporal lobe atrophy; MTA = 0: no atrophy in medial temporal lobe; MTA = 4: severe volume loss of hippocampus. Figure created with www.BioRender.com (accessed on 14 February 2021). 3.1.3. Limitations Although from these results, changes in atrophy rates of several brain regions may seem promising for the longitudinal monitoring of AD, these methods also have their disadvantages. Lawrence et al. found that most studies that monitor disease progression have small sample sizes with regularly below a hundred participants, probably due to high costs associated with repeated MRI measures . Additionally, there is a lack of studies that implement more than one follow-up scan, and most studies have limited time between the two scans, while AD progression is protracted. Furthermore, whole-brain atrophy rates and hippocampal volume reduction are not AD-specific measures, and since MMSE is not sensitive enough to diagnose AD , atrophy rates and declined cognitive performance might be wrongly attributed to AD. In similarity with sMRI in the early diagnosis of AD, it is important to note that spatial patterns of atrophy differ per AD subtype . In typical AD, key regions of atrophy are the hippocampus and ERC, whilst in atypical AD, such as the previously mentioned LvPPA and PCA, but also the dysexecutive/behavioral variant , these regions undergo slower rates of change [58,59]. It is, therefore, important to discriminate between the different types of AD before longitudinal assessments of atrophy rates are made. 3.2. FDG-PET 3.2.1. Background In the previously reported study of Panegyres et al., FDG-PET was listed as a promising technique in the early diagnosis of AD and other types of early-onset dementia . Although in this study, longitudinal clinical follow-up was included, this was primarily conducted as a diagnostic reference standard . There are, however, other studies that have assessed the clinical relevance of FDG-PET as a tool for disease progression in AD. 3.2.2. Findings The first study to longitudinally monitor changes in metabolism patterns with FDG-PET was a follow-up study by Drzezga et al. in 2003. MCI subjects underwent two FDG-PET scans with an interval of one year to identify typical patterns of cerebral metabolism . Since patients suffering from MCI have a high risk to convert to AD within one year, FDG-PET scans of these MCI patients may give insight into the pathophysiology of AD. Converter MCI patients showed decreased glucose metabolism in the temporoparietal and posterior cingulate cortex at baseline (Figure 2). After one year, the glucose metabolism also decreased in prefrontal areas, along with a further diminished metabolism in the posterior cingulate cortex, while these regions were spared in stable MCI patients. The results indicated that metabolic change rates within one patient group can differ over time as disease progression differs. Differences in cognitive decline are correlated to different spatial patterns of decreased glucose metabolism and can be used to predict one’s risk to convert to AD. Fouquet et al. expanded this study by also taking into account the metabolic characteristics that distinguish converters to AD from stable MCI patients . Amnestic MCI (aMCI) patients were recruited and had two FDG-PET scans with an eighteen month interval. All aMCI subjects had progressive metabolic decline over the follow-up period in the temporoparietal cortex and posterior medial parietal areas. Moreover, two medial prefrontal areas, i.e., the anterior cingulate cortex and subgenual area (Figure 2), had significantly greater decline in converters than stable aMCI subjects. This contrasts with the findings of Drzezga et al., in which lateral prefrontal regions were areas of hypometabolism. However, multiple studies support the assumption of Fouquet et al., reporting decreased metabolism in the same two medial prefrontal regions [3,157] Altogether, these findings highlight the potential of FDG-PET for the longitudinal monitoring of AD progression. The previously described studies did not discriminate between patients with early and late MCI. This distinction in the MCI stage has been proposed by Alzheimer’s Disease Neuroimaging Initiative (ADNI), a consortium focusing on the development of standardized biomarker procedures and use of imaging techniques in healthy, MCI and mild AD subjects . The ADNI criteria classify subjects into MCI based on the scores from different tests, such as MMSE, WMS-R Logical Memory II and Clinical Dementia Rating (CDR). Classification into early MCI or late MCI is solely based on the outcome of the WMS-R Logical Memory II test, and the ADNI refers to early MCI subjects as patients that meet all the criteria for aMCI, but are in an earlier, and, therefore, less severe, point on the clinical spectrum [158,159]. With this discrepancy in mind, another research project focused on investigating differences in hypometabolism patterns and neuropsychological characteristics between early and late MCI . Evaluation of the baseline scans and tests with the follow-up tests suggested that early MCI patients differ in patterns of hypometabolism and associated cognitive deficits compared to late MCI subjects. A major limitation of this study is the inclusion of only one FDG-PET scan at baseline instead of scans at every follow-up to track hypometabolism progression. There are several more studies that have identified FDG-PET as a promising tool to predict one’s risk to convert from cognitively unimpaired to MCI and from MCI to AD [161,162,163,164]. In a more recent study, patients already converted to AD were followed for three years to observe longitudinal changes in cortical glucose metabolism in amnestic and non-amnestic subjects with sporadic AD . FDG-PET images at baseline demonstrated different regions with diminished glucose metabolism in amnestic and non-amnestic subjects. Similar progression patterns of metabolic reduction were observed in most regions, except for a higher rate of decline in anterior cortices in non-amnestic forms. Glucose decline progressed from anterior to posterior in amnestic patients, while in non-amnestic subjects, decline progressed along a posterior-to-anterior axis. Additionally, the non-amnestic early-onset AD patients presented more rapid and severe decline in glucose metabolism than amnestic subjects. The differences found in the spatial distribution and temporal trajectory of hypometabolism between amnestic and non-amnestic early-onset AD suggested the treatment of these two forms of sporadic AD as two separate entities. A limitation of this study was the high amount of attrition due to disease severity at baseline, which is a characteristic of early-onset AD. Ishibashi et al. recruited healthy individuals from an ongoing longitudinal study of cognition and aging. These controls were compared with two female subjects that were diagnosed with AD during the study and with a group of fifteen patients in the early stage of AD . Female subject A had a glucose reduction rate of 9.41% over nine years, whereas female subject B’s glucose metabolism decreased with 9.07% over twelve years. In contrast, the rate of FDG reduction in the control group was 2.2% over ten years. Based on these data, the researchers estimated that diminished FDG uptake started four and two years, respectively, before clinical indications of cognitive decline in subject A and B. These differences in time between glucose hypometabolism and onset of memory loss between subject A and B are probably due to heterogeneity in the characteristics of sporadic AD and inherited AD. Lastly, several studies have reported the usefulness of heterogeneity in hypometabolism patterns in distinguishing different variants of AD [167,168,169]. Additionally, this heterogeneity turned out to be of value in predicting progression to different forms of dementia in the prodromal MCI phase . 3.2.3. Limitations Although FDG-PET is thoroughly studied and seen as a robust biomarker of neurodegeneration , similar to other imaging techniques, FDG-PET has its limitations for the longitudinal monitoring of AD. Reduced glucose metabolism is not an AD-specific characteristic, but does also occur in a broad range of other diseases. For instance, FDG uptake is also diminished in certain brain regions after a stroke or other brain injuries . Since approximately 30% of elderly people suffer from a silent infarct, lacking any clinical manifestations , alterations in cerebral glucose metabolism are not surprising in this aged population . It is, therefore, suggested to consider FDG-PET as an independent biomarker rather than a biomarker of neurodegeneration in the “A/T/N” framework, because, as mentioned before, FDG uptake is likely to reflect the glucose consumption by astrocytes instead of neurons [80,174]. In conclusion, FDG-PET is an effective technique in monitoring glucose metabolism in the brain, but it is a tool to measure glucose uptake rather than a biomarker for longitudinal monitoring of the neurodegenerative progression in AD. 3.3. Amyloid-PET 3.3.1. Background Although a fundamental role of Aβ deposition in the pathogenesis of AD is widely accepted, the relationship between plaque density and disease severity is weak [175,176]. Multiple longitudinal studies have investigated the correlation in plaque density and cognitive decline. 3.3.2. Findings The goal of the study of Villemagne et al. was to visualize the longitudinal deposition of Aβ and to investigate the relationship between Aβ deposits and cognitive decline. Therefore, AD patients, MCI patients and age-matched healthy controls were recruited, and all subjects underwent PET imaging with PiB at baseline and follow-up. Low increased PiB retention at follow-up was found in AD and MCI patients, and in healthy controls with high retention at baseline. MCI subjects with high PiB retention had a higher chance to convert to AD than MCI patients with low PiB, and healthy controls with more PiB retention were at a higher risk to become MCI subjects than controls with low PiB. Although high levels of Aβ accumulation predicted one’s risk to convert to MCI or AD, the small increases in PiB retention only partly explained the cognitive decline, suggesting a more prominent role for other downstream factors. In a similar study by Koivonen et al., changes in Aβ burden were evaluated over a period of two years . In line with the previous described study, at baseline, MCI patients had higher PiB retention compared to controls. Additionally, uptake was elevated in MCI subjects that later converted to AD than non-converters, indicating that PiB retention time can predict conversion to AD. Another study consolidated the earlier findings by stating that a positive PiB scan is a strong indication for progression of MCI into AD . Interestingly, during follow-up, the PiB uptake ratio increased in non-converters, while the retention time did not increase in converters, suggesting that PiB uptake only modesty changes once converted to AD. To further assess this assumption, longer follow-up time is needed. Intensive longitudinal research in a group of two hundred participants revealed that Aβ deposition progresses slowly, likely to be prolonged for more than twenty years . Additionally, Aβ seemed to slow down as the disease proceeded. Therefore, it is believed that as AD progresses, the Aβ accumulation will reach a plateau (Figure 1), while the cognitive decline will intensify . This finding was first reported by Jack et al. in a study that modelled the temporal trajectory of Aβ deposition with PET imaging . Over time, Aβ deposition followed a sigmoidal-shaped trajectory, indicating that at high Aβ load, an equilibrium is reached. In other words, Aβ accumulation precedes cognitive decline, and once a quantitative plateau is reached, the disease will become more severe. This statement is further underlined by the hypothetical model of dynamic biomarkers [77,78]. Therefore, Aβ accumulation may be a promising biomarker for predicting one’s risk to convert from cognitively unimpaired to MCI or from MCI to AD (phase 1) but may be less useful as a marker to track disease progression once a patient has established AD (phase 2-5) (Figure 8). If Aβ accumulation is no longer a dynamic marker of disease progression in the late stages of AD, it is assumed that other downstream factors are responsible for the observed associations between Aβ deposition and altered brain structures . Figure 8. Open in a new tab Spreading pattern of amyloid-β accumulation throughout the brain. Amyloid-β accumulation starts in frontal areas and spreads to other regions as disease progresses, leading to a plateau in amyloid-β as disease progresses. Figure adapted from Goedert and created with www.BioRender.com (accessed on 11 February 2021). 3.3.3. Limitations In addition to the limited value of Aβ deposition in the longitudinal monitoring of already established AD, there are other limitations associated with Aβ imaging for longitudinal monitoring of AD. For patients with already severe AD, it might be too difficult to lie still during the time needed to obtain the scans. Furthermore, PET imaging uses radiotracers, such as PiB and florbetapir. These radiotracers are known to target predominately neuritic plaques . Since these types of plaques are only scarcely presented in the cerebellum, this region is often used as a reference region for many Aβ-PET studies . However, multiple studies question the reliability of the cerebellum as a reference region for normalization in longitudinal Aβ deposition, since it can affect the quantitative outcome in these longitudinal studies and lead to additional variability . It has, therefore, been suggested to combine multiple reference regions into one more alike to the longitudinal changes to minimize potential variations . 3.4. Tau-PET 3.4.1. Background As mentioned above, it is hypothesized that Aβ burden reaches a plateau as AD progresses [180,184], making longitudinal tracking of Aβ accumulation not the most promising biomarker for disease progression. On the other hand, tau levels slowly increase during the AD continuum with a steep increment approximately eight to nine years before disease onset (Figure 1) . Therefore, longitudinal monitoring of tau accumulation may be a more powerful tool in monitoring disease progression. Moreover, increasing evidence suggests a close relationship between cognitive decline and tauopathy, making tau an interesting target for longitudinal monitoring of AD . Since only recently the focus of research has shifted from Aβ deposition modifying therapies to tau associated treatment, longitudinal research with tau-PET is still in the early stages compared to decades-long studies into Aβ burden. 3.4.2. Findings In 2015, Ishiki et al. performed a longitudinal study into tauopathy with the then novel radiotracer 18 F-THK-5117 . 18 F-THK5117 was significantly increased in middle and temporal gyri as well in the fusiform gyrus of AD patients. Higher levels of tau load were found in patients with more severe AD compared patients with mild AD. Additionally, these tauopathies were more widely spread across the cortical regions. Furthermore, uptake of 18 F-THK5117 was strongly associated with the rate of cognitive decline, indicating a strong relationship between neurofibrillary pathology and neurodegenerative decline. This relationship may be useful in the longitudinal assessment of disease progression and the efficacy of therapies. The assumption is underlined by a large study investigating the relationship between tau accumulation, Aβ deposition and cognitive impairment. The study included cognitively unimpaired subjects with normal Aβ levels, subjects with no cognitive impairment, but abnormal Aβ, and cognitively impaired subjects with abnormal Aβ and an amnestic phenotype . The cognitively unimpaired group with normal Aβ had no detectable tau accumulation throughout the brain, whereas the unimpaired abnormal Aβ subjects had low, but significant rates (0.5% per year) of accumulation in multiple regions of the brain. This is in contrast to a study by Harrison et al. in which healthy adults with normal Aβ showed observable tau accumulation associated with brain atrophy . This might be due to the different radiotracers used in the studies. The cognitively impaired abnormal Aβ subjects exhibited an increment in tau of 3% per year . The accumulation rates differed only slightly from each other and during disease progression, accumulation rates increased consistently throughout the different brain areas. This indicated that tau accumulation is not restricted to one region at a time. Furthermore, the early increment in tau was not limited to the ERC, but rather widespread. Altogether, the study found that disease progression can be measured by increasing tau burden, and, therefore, tau accumulation rates may be useful as a clinical outcome for disease-modifying therapies. The spreading of tau throughout the brain was further studied by Cho et al. in a research study into longitudinal changes in tau accumulation in cortical regions. In contrast to Jack et al., this study reported hierarchical spreading of tau from the entorhinal cortex to other brain regions (Figure 2) . This typical topography of tau accumulation is believed to initiate in the ERC and with further neuronal degeneration and more cognitive decline, it spreads to other brain areas, such as the limbic regions and association cortices . Moreover, tau accumulation rates in the ERC decrease as AD progresses to higher Braak stages. Sintini et al. addressed the relationship between tau-PET uptake and brain atrophy in atypical AD. Interestingly, the regional patterns of tau accumulation and atrophy differed from one another in atypical AD . High levels of tau accumulation were found in the frontal lobe, whereas atrophy rates were the greatest in temporoparietal areas. This difference suggested a temporal lag between tau deposition and the progression of neurodegeneration. This assumption has been previously proposed by other studies as well [114,115]. Furthermore, the research found that age has a negative effect on disease progression, since younger patients had higher rates of tau accumulation and atrophy. Lastly, there was a close relationship found between tau-PET uptake and gray matter volume. 3.4.3. Limitations While tau accumulation is probably the most promising biomarker for disease progression, the heterogeneity of tau topography between the different AD subtypes is currently its major disadvantage [169,190]. Whereas amyloid distribution is believed to be similar, tau distribution varies between AD subtypes [59,125,191]. Additionally, each AD phenotype expresses a unique longitudinal regional pattern, and this pattern differs across the AD phenotypes . It is, therefore, necessary to intensively study the dynamic patterns of AD biomarkers in atypical forms of AD in order to understand disease progression in these forms . With extensive research into the different spreading patterns across the dementia spectrum, this disadvantage can become an advantage, as heterogeneity contributes to accurate longitudinal monitoring of the different AD types [190,192]. 3.5. Summary Over the years, great progress has been made in identifying biomarkers that reflect the disease progression of AD (Table 1). However, the distinct phenotypes of AD with corresponding heterogeneity in topographic patterns pose a challenge in the longitudinal monitoring of AD. Comprehensive research into the different subtypes of AD is needed to recognize these different patterns, and after correct identification, these pattern differences can be of additional value, such as for heterogeneity in hypometabolism patterns. Subsequently, clinical endpoints for disease-modifying therapies can be identified. 4. Novel Methods, Applications of Imaging Techniques and Biomarkers in AD Research Since there is still no effective treatment for AD and no perfect biomarker to detect AD from the earliest stages up to severe disease manifestation, much effort is put into finding novel techniques, new biomarkers and the development of new methods to apply existing techniques in the early diagnosis and longitudinal monitoring of AD. In this section, we briefly discuss new processing methods for existing techniques, possible new biomarkers that can contribute to monitoring AD and novel applications of imaging techniques in the search to successfully diagnose and monitor AD (Table 2). Table 2. Summary of novel strategies in AD research. MethodologyVoxel-based morphometry Automated segmentation of brain tissues Comparison of voxels to measure concentration differences Deformation-based morphometry Transformation of all brain volumes to standard template brain Statistical analysis of deformation fields Tensor-based morphometry Uses regional differences in gradients of deformation Favored in large-scale MRI studies Pattern-based morphometry Able to extract multidimensional characteristics More research necessary for broad application Data-driven methods Large amounts of data can improve image quality More comparison between conventional and data-driven methods necessary Imaging techniqueDiffusion tensor imaging Measures displacement of water in three dimensions Needs more research to exploit full potential Functional MRI Uses BOLD signal for synaptic activity of neurons Not widely supported due to several limitations Optical coherence tomography Non-invasive and cheap technique to assess effect of AD in the eye Reliability still question of debate BiomarkerSV2A Reflects synaptic density in brain Large scale validation necessary for broad application RAGE Believed to regulate toxicity of Aβ Potentially powerful biomarker in early diagnosis Iron Relationship between Aβ and iron accumulation Detection of iron with QSM promising tool Open in a new tab SV2A: synaptic glycoprotein 2A; RAGE: receptor for advanced glycation end products; QSM: quantitative susceptibility mapping. 4.1. New Processing Methodologies for Existing Techniques One way to increase the clinical value of already existing techniques is to improve the methods that process the obtained data. In the field of neuroimaging with MRI, much more clinical value has been obtained by using different morphometry methods for data processing. Since neuroimaging data of MRI scans are generally stored as matrices of voxels, there are several methodologies to process this type of data. Additionally, a high focus has been put on establishing new data-driven methods for the processing of PET images. 4.1.1. Voxel-Based Morphometry The most commonly used data-driven method for T1-weighted MRI images is voxel-based morphometry (VBM), which automatically segmentizes brain tissue in white matter, gray matter and CSF . It transforms T1-weighted individual brain scans into a standard reference template. Subsequently, VBM measures differences in concentrations of the different brain tissues by comparing voxels of multiple brain regions . In AD, VBM is used to quantify atrophy and to automatically distinguish AD patients from MCI subjects and healthy controls . 4.1.2. Deformation-Based Morphometry Another, more biologically related method, which was previously mentioned, is DBM, in which all brain volumes are transformed into a standard template brain [36,196]. In contrast to VBM, with DBM, the high resolution of the MRI images is maintained . Instead of the voxels, the deformation fields that contain information about the spatial differences of the voxels between the imaged brain and the template brain are used for statistical analysis. Therefore, DBM is more sensitive for subtle changes in brain tissue composition than VBM. Additionally, data from multiple studies, imaging equipment and research centers can be processed without bias . 4.1.3. Tensor-Based Morphometry With tensor-based morphometry (TBM), regional differences in gradients of the deformation fields that line up the images into the template brain are measured . TBM can be used in a wide range of assessments, varying from the voxel level to analysis of the whole brain. Moreover, since TBM is an almost fully automated process, it is favored in large-scale MRI studies, such as clinical trials. 4.1.4. Pattern-Based Morphometry Another type of morphometry is pattern-based morphometry (PBM), a method with its origin in VBM and DBM . This data-driven method uses an algorithm based on sparse dictionary learning and is, in contrast to VBM, able to extract multidimensional patterns that characterize differences between groups, making PBM an interesting tool to compare different brain regions. Although PBM seems promising as a processing method for heterogenous disease, more research into robustness and extension to other types of neuroimaging is necessary for broad application. 4.1.5. Data-Driven Methods As technology improves, lately, much attention has been given to using large amounts of data to build data-driven models to improve the analysis of PET images. The effect of age on certain brain regions assessed with FDG-PET, for example, has been corrected using a data-driven approach . Moreover, data-driven analysis of tau PET images identified spatial patterns of radiotracer 18 F-AV1451 signal clusters compared to pathology-based methods, suggesting an advantage for data-driven methods in evaluating radiotracer data . To fully benefit from the advantages of data-driven methods in neuroimaging, extra studies comparing conventional methods with data-driven methods are necessary. 4.2. Novel Implications of Imaging Techniques In addition to new methodologies to increase the diagnostic value of already utilized imaging techniques, novel applications of other imaging techniques are increasing. Although structural MRI with T1-weighted images is still the gold standard in AD research with MRI, other MRI sequences seem to be promising. 4.2.1. Diffusion Tensor Imaging Diffusion tensor imaging (DTI) is an advanced type of diffusion MRI. This technique measures the displacement of water molecules in three dimensions to determine the integrity of the biological tissue [202,203]. In AD, DTI has been used to measure the integrity of brain regions by calculating the mean diffusivity . Additionally, DTI demonstrated to be of value in determining the architecture of white matter , and multiple studies have reported the relationship between white matter integrity and disease severity, suggesting the inclusion of white matter degeneration as a pathological biomarker of AD for early diagnosis [205,206,207]. In order to exploit the full potential of DTI as a diagnostic tool in AD, the exact relationship between disease severity and white fiber tracts has to be explored . 4.2.2. Functional MRI Functional MRI (fMRI) is an imaging technique that gives insight into the functional integrity of brain networks that support several cognitive domains in a non-invasive manner [23,208]. fMRI uses the blood-oxygen-level-dependent (BOLD) signal to measure the synaptic activity of neurons. fMRI can be used in two manners: resting state (rs) fMRI, which measures changes in BOLD signals during inactivity, or task-related fMRI in which patients perform several cognitive tasks . Since severely impaired patients may be too limited to perform these tasks, rsfMRI may be more feasible to monitor disease progression in later stages . Although fMRI has been demonstrated to be of clinical value in studying the default mode network [209,210,211,212], clinical use of fMRI is not widely supported due to limitations, such as low signal and noise . 4.2.3. Optical Coherence Tomography Another interesting imaging technique to be applied in AD research is optical coherence tomography (OCT), but to date, there is no consensus on the employment of this technique. In recent years, pathological changes in the retina have been linked to AD . These changes include Aβ plaques, thinning of the retinal nerve fiber layer (RNFL), ganglion cell loss and decreased vessel density. Since OCT is a non-invasive, fast and inexpensive technique , multiple studies have investigated the beneficial value of OCT in AD research. Although accumulation of Aβ in the lens, analysis of RNFL thickness and ganglion cell loss are proposed as diagnostic tools for AD [215,216,217,218], the reliability of these markers is still questioned due to possible other underlying diseases that cause these pathological changes, such as glaucoma. Nevertheless, the feasibility and cost effectiveness of OCT make it an interesting imaging technique to further investigate for applications in AD. 4.3. New Biomarkers For many years, the focus of AD research has been on atrophy, glucose metabolism and imaging of Aβ deposition and tau burden. However, since none of these biomarkers stands out as a faultless biomarker for the diagnosis of AD and disease progression, research focuses on identifying novel biomarkers that reflect the progression of AD. Over the years, several new biomarkers have been introduced. 4.3.1. Synaptic Vesicle Glycoprotein 2A One marker suggested to be of clinical value in AD is synaptic vesicle glycoprotein 2A (SV2A), which reflects the synaptic density . This protein is located in the cell membrane of secretory vesicles, and since SV2A is ubiquitously expressed throughout the brain, lower levels of SV2A may be a promising biomarker of synaptic loss in AD. 18 F-UCB-J is a PET radiotracer considered to be sensitive for synaptic loss, because altered uptake of 18 F-UCB-J in the gray matter was correlated to altered expression of SV2A and lower synaptic density [219,220]. Although these results seem promising, large scale validation of this and other radiotracers is necessary to further exploit the clinical possibilities of SV2A in AD . 4.3.2. Receptor for Advanced Glycation End Products There is increasing evidence that the receptor for advanced glycation end products (RAGE) regulates the neurotoxicity of Aβ in AD . The binding of RAGE to Aβ results in the release of reactive oxygen species that contribute to the formation of senile plaques and NFTs. Moreover, RAGE levels are significantly higher in AD subjects than in cognitively healthy controls . Therefore, it has been suggested that in the early stages of AD, RAGE is a potent biomarker. 11 C-FPS-ZM1 is a radiotracer for PET imaging of RAGE in the brain . Since RAGE overexpression is believed to precede the formation of Aβ plaques, PET imaging of RAGE with 11 C-FPS-ZM1 may be a powerful tool in the early diagnosis of AD . 4.3.3. Iron Excessive accumulation of iron in specific brain parts is increasingly related to AD . Although iron is required for maintaining homeostasis and plays a key role in many biological processes, abnormal accumulation of iron in subcortical and deep gray matter nuclei has been associated with AD . Multiple studies have used quantitative susceptibility mapping (QSM) to quantify the local magnetic susceptibility derived from MRI images caused by deposits containing both Aβ and iron [224,226,227]. QSM application in the detection of iron has been demonstrated to be of clinical value in assessing the relationship between Aβ accumulation and iron burden . Therefore, the detection of iron with QSM may be of potential aid in imaging Aβ in the early diagnosis of AD . 5. Conclusions and Future Perspectives In this review, we focused on the applications of imaging techniques in the early diagnosis and longitudinal monitoring of AD. AD is a neurodegenerative disease in which pathological changes occur decades before disease manifestation. The disease is characterized by the formation of senile plaques, NFTs and subsequent synaptic loss and neurodegeneration. Although AD affects a major part of the population worldwide, to date, there is no therapy to cure AD. Since disease-modifying therapies may be the most beneficial in early stages of the disease, it is important to diagnose AD as early as possible. Additionally, longitudinal monitoring of disease progression is crucial to gain a better understanding of the pathogenesis and to set clinical endpoints for potential treatment. To date, several biomarkers have been proposed for the early diagnosis and longitudinal monitoring of AD, but all these biomarkers have their limitations regarding specificity, reliability and sensitivity. Aβ deposition is among the earliest hallmarks of AD, but to date, there is no consensus on exactly how Aβ accumulation contributes to the pathogenesis of AD. Moreover, detailed study into Aβ over time has revealed that Aβ levels reach an equilibrium, making Aβ a questionable biomarker for monitoring disease progression. Tau accumulation, on the other hand, is believed to be more biologically related to the symptoms associated with neurodegeneration in AD. Imaging studies with tau PET-tracers have demonstrated promising results, but compared to Aβ-PET imaging, large-scale validation of these tracers must be performed to make tau-PET imaging a reliable tool in AD. Moreover, longitudinal studies into different phenotypes of AD revealed heterogeneity in the topographic patterns of tau accumulation throughout the brain. This heterogeneity can be of additional value, but first, more detailed study in these different tau spreading patterns is required. More general imaging techniques, such as FDG-PET and structural MRI, have been applied in AD research, but these techniques measure rather more common pathological changes than AD-specific characteristics. Brain atrophy, measured by structural MRI, is not restricted to AD pathology and is only detectable after a substantial amount of neurodegeneration. FDG-PET is used to measure the glucose uptake in the brain. Since synaptic loss in AD leads to hypometabolism, decreased glucose uptake is associated with AD. However, decreased glucose metabolism is not restricted to AD but can also occur after strokes and brain injury. Furthermore, an increasing body of evidence suggests that glucose uptake reflects astrocyte function rather than neuronal function. Altogether, to date, there is no perfect biomarker to detect AD in the early stages and to monitor disease progression over time. Furthermore, these biomarkers rely on neuroimaging techniques that require high-quality and expensive machinery, making them infeasible for large-scale examinations of greater populations. Hence, comprehensive and in-depth research into AD is crucial in the early diagnosis and longitudinal monitoring of AD. Since tau-PET appears to be the most promising tool for the diagnosis and tracking of disease progression, the field of research should focus on the validation and development of existing and new tau radiotracers. Furthermore, detailed research into new applications of other imaging techniques is necessary to overcome the limitations in the extensive scanning of large populations. Lastly, current tools require relatively high levels of protein accumulation or neurodegeneration to be detectable. Since higher levels are associated with higher disease severity and lower beneficial potential of therapies, identifying novel biomarkers that reflect the pathogenesis of AD in the earliest stages is essential for the development of disease-modifying therapies. Abbreviations AD Alzheimer’s disease NFT Neurofibrillary tangle Aβ Amyloid-β MRI Magnetic resonance imaging PET Positron emission tomography MCI Mild cognitive impairment CNS Central nervous system FDG-PET Fluorodeoxyglucose positron emission tomography APP Amyloid precursor protein CSF Cerebrospinal fluid PiB Pittsburgh compound-B FTD Frontotemporal dementia PHF Paired helical fragment BBB Blood-brain barrier MMSE Mini-mental state examination DBM Deformation-based morphometry ERC Entorhinal cortex MTA Medial temporal lobe atrophy aMCI Amnestic mild cognitive impairment ADNI Alzheimer’s Disease Neuroimaging Initiative VBM Voxel-based morphometry TBM Tensor-based morphometry fMRI Functional magnetic resonance imaging BOLD Blood-oxygen-level-dependent rsfMRI Resting state functional magnetic resonance imaging DTI Diffusion tensor imaging OCT Optical coherence tomography RNFL Retinal nerve fiber layer SV2A Synaptic vesicle glycoprotein 2A RAGE Receptor for advanced glycation end products QSM Quantitative susceptibility mapping Open in a new tab Author Contributions W.M.v.O., writing—original draft preparation; E.C.M.d.L., writing—review and editing; E.C.M.d.L., supervision. 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12342	https://www.youtube.com/watch?v=Bc0M5g0nOX8	Piecewise Functions - Linear, Absolute Value, Quadratic Mandy's Math World 2770 subscribers Description 1290 views Posted: 5 Nov 2021 Graph piecewise functions comprised of linear, absolute value and quadratic functions. Write formulas for piecewise functions and identify restricted domains. Access the notes and assignment to accompany the video from my TpT store at: Transcript: in today's lesson over piecewise functions we're going to deal with functions that are comprised of linear absolute value and quadratic functions so in yesterday's lesson we dealt with piecewise functions that were comprised of linear and constant functions so we're going to be adding these two functions to our list in this first example the directions say graph the piecewise function on the coordinate plane provided then evaluate the function for the given values so let's work through each of these you may recall that a piecewise function is made up of different pieces of functions that have restricted domains so let's start with this first one and i i shouldn't have put dots over each one of those because if you watched yesterday's lesson you know that i'm going to color coordinate my functions so let's start with this first one the absolute value of x minus 3 and it's restricted to the domain x is less than or equal to negative 3 which means i'm going to graph this function the absolute value of x minus 3 but i'm only going to graph the portion of it that's less than or equal to negative 3. so if you recall in your transformations of absolute value functions this right here is going to move my parent function down three units so i'm going to take that what would be at the origin and move it down three units and then you know an absolute value function it would go up positive slope that way negative slope this way right but because we're restricted to the domain x is less than or equal to negative three we only need this portion of our function right here we only need the part that is less than or equal to negative three and so i'm going to graph that part right here it's going to go up and out forever and ever and ever and ever and ever and i put a closed dot right here because we are including the point at negative 3 which is negative 3 0 that is included in this part of our function or in this piece of our function let's move on to the next piece of this function x squared minus 4. this is obviously a quadratic function that has been transformed or translated to be more specific down four units so if i take my quadratic function which if i were to graph it you know it takes a specific pattern i'm going to take it it would pass through the origin and i'm going to move it down one two three four units and then it's going to take that same pattern where from that origin which is now at 0 negative 4 it's going to go out 1 up 1 out two up four and out three up nine so from that um vertex right there i would go out one two three up there would be four five six seven eight nine i'm gonna put an open dot right there and then over here i'm gonna put a closed dot now why would i do that something in this domain lets me know that i have an open dot right here and it's this right here x is greater than negative three i am not including this point right here which is negative three five i am not including that point in this function okay i'm not including it however over here 3 5 is included in this piece of the function so now if i graph it it looks just like this there we go okay so it's not super pretty but that's how it's graphed and let's move on to the next piece of this function the next piece of the function is a linear function 2x minus 6 it would have a y-intercept of negative 6 and a slope of positive 2. but we're restricted to the domain x is greater than 3. so what you can do is you can start with your y-intercept and you can go up 2 over 1 up 2 over 1 or you could start with what i like to do is plug in 3 for x so instead of 2x minus six i'm going to plug in three for x two times three is six six minus six is zero which means the point three zero is where i can start from 1 2 right here but i'm going to put an open dot right there what would let you know that i'm going to have an open dot right there instead of a closed dot this right here x is greater than 3 i am not including this point in this piece of the function and that's good to know because 3 is actually included in the domain in that part of the function that we graphed in orange and now from that point i know that x is greater than three so i'm going to from there graph it with a slope of two which is up two over one up two over one and then i can connect it and graph that so there's what my function looks like when it's graphed on the coordinate plane and now we're going to evaluate the function for these values down here f of negative 3 f of 1 f of 5 and f of 3. and you may recall from yesterday's lesson that we need to make sure that whenever we're evaluating a piecewise function we plug in that value into the correct piece okay so f of negative three which piece of this function is it included in the one that's graphed in blue orange or pink the one that's graphed in blue negative three is included in this piece of the function let me see if i can highlight that right here x is less than or equal to negative three and in fact we already know what f of negative three is it's zero let's move on to f of one which piece of the function is f of one included in where is which domain is one included in is one less than or equal to negative three no let's look at the one in orange is it between negative three and three it sure is which means we're going to plug it into the piece that's graphed in orange that x squared minus four so f of one i can look at my graph and i can see that f of one is actually negative three i can also plug it into x squared minus four when i plug in one for x i get one minus four which is negative three let's move on to number three which is f of five which piece of the function is five included in the part that's graphed in pink because five is greater than three which means i'm going to plug in five into this function right here two x minus six two x minus six and i'm going to plug in 5 for x so 2 times 5 minus 6 2 times 5 is 10 and 10 minus 6 is 4. so f of 5 is 4 which means 5 4 is a point on this graph 1 2 3 4 5 4. and as you can see that point is located right here so we can see it that it's graphed now f of three which piece of the function is three included in the domain notice right here a lot of students kind of get tripped up on this it's included in the part that's graphed in orange i have an a closed dot right here on my graph where x is three the one that's graphed in pink has an open dot it is not included in that piece of the function which means i'm going to look at the middle function in this piecewise function which is x squared minus 4. and as you can see 3 5 we've already got right here we've shown that it's included in the piece that's graphed in orange which means f of 3 is five right what is what is y when x is three and it's five let's move on to our next example here we're given a piecewise function and we're asked to write the formula for the piecewise function and then define or identify the domain or define the domain for each interval then we're going to evaluate the function for the given values this says to include one linear one absolute value one constant and one quadratic function so let's do that let's start from left to right and we'll start with this constant function right here constant function is just going to be y equals what is y equal here y equals negative one two three four five negative six so negative six is my function right y equals f of x equals negative six what is the domain restriction here where we're going one two three four five six units to the left which means x is less than negative six now why would i put less than negative six and not less than or equal to because i have an open dot there which means i'm going to have i'm not including negative 6 in this particular piece of the function so like negative 6 negative 6 that point right there is not included in that piece of the function which means negative 6 will be included in the next piece of the function if this is a con or not a continuous function but um let's move on to the next piece of the function so the next piece of the function is our linear piece and let's see how we're going to do this when we're given this linear function right here and it's in this restricted domain let's first identify the slope which i'm just going to put some points on the graph i've got a point right there and a point right there and then a point right there i can clearly see which means my slope is up one two three over one so i have a slope of three and now let's identify a point on this function a point on this function would be negative one two three four five negative six negative four and you could include you could use any point on this particular graph i'm just using the point that i see that's clearly graphed and now we're going to write the equation for this linear function okay or for this line we're going to write the equation for the line using point-slope form of a linear equation which is y minus the y value y minus negative four is y plus four right i can combine those that minus a negative equals my slope which is three open parenthesis x minus the x value x minus negative 6 is x plus 6 and now let's simplify this y plus 4 equals let's distribute that 3 and i'm going to get 3x plus 18 when i subtract 4 from both sides i get 3x plus 14 and that's the equation for the linear piece of this piecewise function but now let's talk about what the what the domain of this particular um linear piece is we know that the furthest left it goes is negative 6. i'm going to write that just like that the furthest right it goes is 1 2 3 right there negative 3 x is everything in between that's how i do that so i'm going to write it down here now negative 6 to negative 3 x is everything in between now we've got our um our inequality symbols i know negative 6 is included in this piece negative 3 is not included so i've got a line right here underneath this inequality right x is greater than or equal to negative 6 x is less than negative 3. okay and i'm going to erase some of this off of here so there's not so much going on but i know this color designates that portion okay let's move on to the next piece of this piecewise function i clearly have a an absolute value function here i could have two linear pieces but i've got an absolute value function here so my absolute value function obviously has been flipped upside down and it's been translated up one two three four units which would look like this i've got a reflection across the x-axis so the negative absolute value of x plus four and my slope on both sides is just negative 1 and positive 1 right so i'm not i don't have any kind of vertical stretch or compression here now let's talk about the restricted domain it goes from that's the furthest left and that's the furthest right which is negative 3 and this is positive 3 x is everything in between i'm going to write it down here now negative 3 to positive 3 x is everything in between when x is everything in between my inequality symbols always look like that okay that kind of helps students they always look like that right because it's less than this number on the right and it's greater than this number on the left we're including that point on the left which means i'm going to have a line underneath it we're not including the number on the right so i'm not going to have a line underneath that so there's that portion of the graph let's move on um let's use green this right here it's curved so that's gonna be our quadratic piece and as i look at this i know from this vertex right here i'm gonna go out one up one out two up four so i know there's no stretch vertical stretcher compression here we're just translating this quadratic function down and to the right so what is that going to look like that vertex has moved one two three units to the right and four units down so 3 negative 4 is the point for that vertex how do we show this in vertex form of our quadratic equation well if we're moving right 3 units it's going to look like this x minus 3 squared right it's going to look opposite right x minus 3 squared and then we're moving down 4 which is minus 4. and now what's our restricted domain we're including that point right there at negative or at positive 3 when x is positive 3 which means x is greater than or equal to positive 3. 3 negative 4 is included in this point okay or in this piece of the function if i go up to this one up here we can see that 3 is not included in this piece of the function so let's move on and now we're going to evaluate the function for these given values all right so we've got f of 5 and we need to see which portion of the graph 5 is included in the domain in okay 5 is a number that is greater than or equal to 3 which means it's going to be included in this quadratic piece right here right so f of 5 i mean because technically i could plug in 5 into any piece right i could plug it into this piece i could plug into that piece but because we have these restricted domains we have to pay attention to the fact that 5 is only going to be included in this piecewise function in this function right here and this piece of it so if i plug in 5 into this function i could i'm just going to plug it in i'm going to show algebraically that it's 5 minus 3 squared minus four five minus three is two two squared minus four two squared is four and four minus four is zero so f of five equals zero five zero is a point in this piece of the function let's move on to f of 3. where is 3 included in my function 3 is included in the green piece of the function right right here it's greater than or equal to 3 it's not included in the purple piece and we already know that f of 3 is negative 4 right what is y when x is three it's negative four so it's really important that you make sure you plug into the correct piece of the function and that concludes your day two notes over piecewise functions i hope it was helpful
12343	https://math.stackexchange.com/questions/76457/check-if-a-point-is-within-an-ellipse	geometry - Check if a point is within an ellipse - 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 Check if a point is within an ellipse Ask Question Asked 13 years, 11 months ago Modified3 years, 11 months ago Viewed 102k times This question shows research effort; it is useful and clear 75 Save this question. Show activity on this post. I have an ellipse centered at (h,k)(h,k), with semi-major axis r x r x, semi-minor axis r y r y, both aligned with the Cartesian plane. How do I determine if a point (x,y)(x,y) is within the area bounded by the ellipse? geometry euclidean-geometry conic-sections Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Oct 27, 2011 at 23:40 J. M. ain't a mathematician 76.7k 8 8 gold badges 222 222 silver badges 347 347 bronze badges asked Oct 27, 2011 at 20:12 Dan Is Fiddling By FirelightDan Is Fiddling By Firelight 983 1 1 gold badge 9 9 silver badges 15 15 bronze badges 6 Which of the two solutions is more efficient (computationally-wise) assuming that in both cases the "\|x−h\|>rx" and "\|y−k\|>ry" rejections are implemented?Ricardo Sanchez-Saez –Ricardo Sanchez-Saez 2012-10-15 13:19:26 +00:00 Commented Oct 15, 2012 at 13:19 2 @rsanchezsaez Probably Srivatsan's because square roots are slow; if you're concerned about performance writing both and benching them is probably the best route. Alternately, you could try posing the question on Stack Overflow.Dan Is Fiddling By Firelight –Dan Is Fiddling By Firelight 2012-10-15 14:27:51 +00:00 Commented Oct 15, 2012 at 14:27 Thank for the tips Dan. I implemented Srivatsan's and seems to work fine. I don't want to spend much time in premature optimization. If in the future we run into performance issues we will profile to see if this is the bottleneck.Ricardo Sanchez-Saez –Ricardo Sanchez-Saez 2012-10-15 15:00:26 +00:00 Commented Oct 15, 2012 at 15:00 "Aligned with the Cartesian plane" means that the major and minor axes lie on the coordinate axes, right?rschwieb –rschwieb 2013-02-05 17:41:36 +00:00 Commented Feb 5, 2013 at 17:41 1 Check whether a point lies inside a rotated ellipseSen Jacob –Sen Jacob 2016-12-20 14:50:17 +00:00 Commented Dec 20, 2016 at 14:50 \|Show 1 more comment 4 Answers 4 Sorted by: Reset to default This answer is useful 133 Save this answer. Show activity on this post. The region (disk) bounded by the ellipse is given by the equation: (x−h)2 r 2 x+(y−k)2 r 2 y≤1. (x−h)2 r 2 x+(y−k)2 r 2 y≤1.(1) So given a test point (x,y)(x,y), plug it in (1)(1). If the inequality is satisfied, then it is inside the ellipse; otherwise it is outside the ellipse. Moreover, the point is on the boundary of the region (i.e., on the ellipse) if and only if the inequality is satisfied tightly (i.e., the left hand side evaluates to 1 1). Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Oct 27, 2011 at 20:20 SrivatsanSrivatsan 26.8k 7 7 gold badges 95 95 silver badges 148 148 bronze badges 7 12 I have just registered to upvote this. I am developing app for android and this has been EXACTLY what I was looking for. Thank u Eugene –Eugene 2013-04-04 07:56:35 +00:00 Commented Apr 4, 2013 at 7:56 1 What does h and k represent, is it the origin of the ellipse?Dave –Dave 2014-11-01 23:55:17 +00:00 Commented Nov 1, 2014 at 23:55 2 @Dave, yes h and k represent the origin of the ellipse.rein –rein 2015-01-22 20:22:20 +00:00 Commented Jan 22, 2015 at 20:22 1 I'm with @Eugene in that I'm trying to develop an app for iOS and Android. Neither seem to be able to work with ellipse bounds detection.Jason Foglia –Jason Foglia 2016-10-29 15:58:33 +00:00 Commented Oct 29, 2016 at 15:58 9 Note that if you're coding this, you should multiply both sides by r 2 x∗r 2 y r 2 x∗r 2 y (which can't be negative so it can't flip the inequality). Computers can multiply much more efficiently than they can divide.Peter Cordes –Peter Cordes 2017-03-15 10:11:42 +00:00 Commented Mar 15, 2017 at 10:11 \|Show 2 more comments This answer is useful 10 Save this answer. Show activity on this post. Another way uses the definition of the ellipse as the points whose sum of distances to the foci is constant. Get the foci at (h+f,k)(h+f,k) and (h−f,k)(h−f,k), where f=√r 2 x−r 2 y. The sum of the distances (by looking at the lines from (h,k+r y) to the foci) is 2√f 2+r 2 y=2 r x. So, for any point (x,y), compute √(x−(h+f))2+(y−k)2+√(x−(h−f))2+(y−k)2 and compare this with 2 r x. This takes more work, but I like using the geometric definition. Also, for both methods, if speed is important (i.e., you are doing this for many points), you can immediately reject any point (x,y) for which \|x−h\|>r x or \|y−k\|>r y. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Oct 28, 2011 at 5:37 marty cohenmarty cohen 111k 10 10 gold badges 88 88 silver badges 186 186 bronze badges 1 2 Right, it's easier to check if a point is within a rectangle. If it isn't in the rectangle, it certainly can't be in the ellipse.J. M. ain't a mathematician –J. M. ain't a mathematician 2011-10-28 07:16:22 +00:00 Commented Oct 28, 2011 at 7:16 Add a comment\| This answer is useful 9 Save this answer. Show activity on this post. 1) Consider the point as a vector p=[x y] 2) Consider the center of the ellipse as c=[h k] 3) Subtract the center of the ellipse p c e n t e r e d=p−c 4) Create a whitening matrix W=Λ−1/2 E T where Λ=[r x 0 0 r y] and E=[e m a j o r e m i n o r] where e m a j o r and e m a j o r are the unit vectors in the direction of the ellipse's major and minor axes. Since you're example is for a non-rotated matrix with major axis along x-axis and minor axis along y-axis e m a j o r=[1 0] and e m i n o r=[0 1] 5) Whiten the point p w h i t e=W p c e n t e r e d 6) Check if the length of the vector is less than 1. If it is, then the point is within the ellipse. Note: this is inspired by my experience with Covariance matrices. I'll try to update this answer with an intuitive relation b/w ellipses and covariance matrices. For now you can take a peak at Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered May 21, 2016 at 0:13 user3731622user3731622 239 2 2 silver badges 4 4 bronze badges Add a comment\| This answer is useful 4 Save this answer. Show activity on this post. I have another solution. In summary, transform everything so you can test whether a point is within a circle centered at (0,0). Since the ellipse is oriented orthogonally to the Cartesian plane, we can simply scale one of the dimensions by the quotient of the two axes. First subtract (h,k) from both points. (h,k) becomes (0,0). (x,y) becomes (x-h,y-k). Now we scale the second coordinate, normalizing it to the first. Scaling (x-h,y-k) we get (x−h,(y−k)∗r x r y) Now we simply need to test if \|(x−h,(y−k)∗r x r y)\|<=r x Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Nov 24, 2012 at 3:42 Victor EngelVictor Engel 247 1 1 silver badge 3 3 bronze badges 2 I apologize. The coding of the fractions didn't take. Perhaps someone can help me out with that and point me to a reference? I was using meta.math.stackexchange.com/questions/5020/… as a guide. Edit: I figured it out. I was missing the $$ tag.Victor Engel –Victor Engel 2012-11-24 03:45:11 +00:00 Commented Nov 24, 2012 at 3:45 This "move everything to 0,0" is a trick also heavily favored by my former math tutor, who was a mathematician in the age just prior to the explosion of graphing calculators. She was used to having to solve everything by hand on paper, so collected "quick and dirty" tricks for simplifying problems down to practical size.Tom –Tom 2023-09-28 22:28:59 +00:00 Commented Sep 28, 2023 at 22:28 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 geometry euclidean-geometry conic-sections See similar questions with these tags. 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12344	https://openstax.org/books/biology-2e/pages/46-2-energy-flow-through-ecosystems	Loading [MathJax]/jax/output/HTML-CSS/fonts/TeX/fontdata.js Skip to Content Go to accessibility page Keyboard shortcuts menu Log in Biology 2e 46.2 Energy Flow through Ecosystems Biology 2e 46.2 Energy Flow through Ecosystems Search for key terms or text. Learning Objectives By the end of this section, you will be able to do the following: Describe how organisms acquire energy in a food web and in associated food chains Explain how the efficiency of energy transfers between trophic levels affects ecosystem structure and dynamics Discuss trophic levels and how ecological pyramids are used to model them All living things require energy in one form or another. Energy is required by most complex metabolic pathways (often in the form of adenosine triphosphate, ATP), especially those responsible for building large molecules from smaller compounds, and life itself is an energy-driven process. Living organisms would not be able to assemble macromolecules (proteins, lipids, nucleic acids, and complex carbohydrates) from their monomeric subunits without a constant energy input. It is important to understand how organisms acquire energy and how that energy is passed from one organism to another through food webs and their constituent food chains. Food webs illustrate how energy flows directionally through ecosystems, including how efficiently organisms acquire it, use it, and how much remains for use by other organisms of the food web. How Organisms Acquire Energy in a Food Web Energy is acquired by living things in three ways: photosynthesis, chemosynthesis, and the consumption and digestion of other living or previously living organisms by heterotrophs. Photosynthetic and chemosynthetic organisms are both grouped into a category known as autotrophs: organisms capable of synthesizing their own food (more specifically, capable of using inorganic carbon as a carbon source). Photosynthetic autotrophs (photoautotrophs) use sunlight as an energy source, whereas chemosynthetic autotrophs (chemoautotrophs) use inorganic molecules as an energy source. Autotrophs are critical for all ecosystems. Without these organisms, energy would not be available to other living organisms and life itself would not be possible. Photoautotrophs, such as plants, algae, and photosynthetic bacteria, serve as the energy source for a majority of the world’s ecosystems. These ecosystems are often described by grazing food webs. Photoautotrophs harness the solar energy of the sun by converting it to chemical energy in the form of ATP (and NADP). The energy stored in ATP is used to synthesize complex organic molecules, such as glucose. Chemoautotrophs are primarily bacteria that are found in rare ecosystems where sunlight is not available, such as in those associated with dark caves or hydrothermal vents at the bottom of the ocean (Figure 46.9). Many chemoautotrophs in hydrothermal vents use hydrogen sulfide (H2S), which is released from the vents as a source of chemical energy. This allows chemoautotrophs to synthesize complex organic molecules, such as glucose, for their own energy and in turn supplies energy to the rest of the ecosystem. Figure 46.9 Swimming shrimp, a few squat lobsters, and hundreds of vent mussels are seen at a hydrothermal vent at the bottom of the ocean. As no sunlight penetrates to this depth, the ecosystem is supported by chemoautotrophic bacteria and organic material that sinks from the ocean’s surface. This picture was taken in 2006 at the submerged NW Eifuku volcano off the coast of Japan by the National Oceanic and Atmospheric Administration (NOAA). The summit of this highly active volcano lies 1535 m below the surface. Productivity within Trophic Levels Productivity within an ecosystem can be defined as the percentage of energy entering the ecosystem incorporated into biomass in a particular trophic level. Biomass is the total mass, in a unit area at the time of measurement, of living or previously living organisms within a trophic level. Ecosystems have characteristic amounts of biomass at each trophic level. For example, in the English Channel ecosystem the primary producers account for a biomass of 4 g/m2 (grams per square meter), while the primary consumers exhibit a biomass of 21 g/m2. The productivity of the primary producers is especially important in any ecosystem because these organisms bring energy to other living organisms by photoautotrophy or chemoautotrophy. The rate at which photosynthetic primary producers incorporate energy from the sun is called gross primary productivity. An example of gross primary productivity is shown in the compartment diagram of energy flow within the Silver Springs aquatic ecosystem as shown (Figure 46.8). In this ecosystem, the total energy accumulated by the primary producers (gross primary productivity) was shown to be 20,810 kcal/m2/yr. Because all organisms need to use some of this energy for their own functions (like respiration and resulting metabolic heat loss) scientists often refer to the net primary productivity of an ecosystem. Net primary productivity is the energy that remains in the primary producers after accounting for the organisms’ respiration and heat loss. The net productivity is then available to the primary consumers at the next trophic level. In our Silver Springs example, 13,187 of the 20,810 kcal/m2/yr were used for respiration or were lost as heat, leaving 7,633 kcal/m2/yr of energy for use by the primary consumers. Ecological Efficiency: The Transfer of Energy between Trophic Levels As illustrated in (Figure 46.8), as energy flows from primary producers through the various trophic levels, the ecosystem loses large amounts of energy. The main reason for this loss is the second law of thermodynamics, which states that whenever energy is converted from one form to another, there is a tendency toward disorder (entropy) in the system. In biologic systems, this energy takes the form of metabolic heat, which is lost when the organisms consume other organisms. In the Silver Springs ecosystem example (Figure 46.8), we see that the primary consumers produced 1103 kcal/m2/yr from the 3373 kcal/m2/yr of energy available to them from the primary producers. (The primary consumers used 3373 kcal/m2/yr from the 7618 kcal/m2/yr produced by the primary producers, as 4250 kcal/m2/yr goes to the decomposers.) The measurement of energy transfer efficiency between two successive trophic levels is termed the trophic level transfer efficiency (TLTE) and is defined by the formula: TLTE = production at present trophic levelproduction at previous trophic level × 100 In Silver Springs, the TLTE between the first two trophic levels was approximately 14.48 percent. The low efficiency of energy transfer between trophic levels is usually the major factor that limits the length of food chains observed in a food web. The fact is, after four to six energy transfers, there is not enough energy left to support another trophic level. In the Lake Ontario example shown in (Figure 46.6), only three energy transfers occurred between the primary producer, (green algae), and the apex consumer (Chinook salmon). Ecologists have many different methods of measuring energy transfers within ecosystems. Measurement difficulty depends on the complexity of the ecosystem and how much access scientists have to observe the ecosystem. In other words, some ecosystems are more difficult to study than others, and sometimes the quantification of energy transfers has to be estimated. Other parameters are important in characterizing energy flow within an ecosystem. Net production efficiency (NPE) allows ecologists to quantify how efficiently organisms of a particular trophic level incorporate the energy they receive into biomass; it is calculated using the following formula: NPE = net consumer productivityassimilation × 100 Net consumer productivity is the energy content available to the organisms of the next trophic level. Assimilation is the biomass (energy content generated per unit area) of the present trophic level after accounting for the energy lost due to incomplete ingestion of food, energy used for respiration, and energy lost as waste. Incomplete ingestion refers to the fact that some consumers eat only a part of their food. For example, when a lion kills an antelope, it will eat everything except the hide and bones. The lion is missing the energy-rich bone marrow inside the bone, so the lion does not make use of all the calories its prey could provide. Thus, NPE measures how efficiently each trophic level uses and incorporates the energy from its food into biomass to fuel the next trophic level. In general, cold-blooded animals (ectotherms), such as invertebrates, fish, amphibians, and reptiles, use less of the energy they obtain for respiration and heat than warm-blooded animals (endotherms), such as birds and mammals. The extra heat generated in endotherms, although an advantage in terms of the activity of these organisms in colder environments, is a major disadvantage in terms of NPE. Therefore, many endotherms have to eat more often than ectotherms to get the energy they need for survival. In general, NPE for ectotherms is an order of magnitude (10x) higher than for endotherms. For example, the NPE for a caterpillar eating leaves has been measured at 18 percent, whereas the NPE for a squirrel eating acorns may be as low as 1.6 percent. The inefficiency of energy use by warm-blooded animals has broad implications for the world's food supply. It is widely accepted that the meat industry uses large amounts of crops to feed livestock, and because the NPE is low, much of the energy from animal feed is lost. For example, it costs about $0.01 to produce 1000 dietary calories (kcal) of corn or soybeans, but approximately $0.19 to produce a similar number of calories growing cattle for beef consumption. The same energy content of milk from cattle is also costly, at approximately $0.16 per 1000 kcal. Much of this difference is due to the low NPE of cattle. Thus, there has been a growing movement worldwide to promote the consumption of nonmeat and nondairy foods so that less energy is wasted feeding animals for the meat industry. Modeling Ecosystems Energy Flow: Ecological Pyramids The structure of ecosystems can be visualized with ecological pyramids, which were first described by the pioneering studies of Charles Elton in the 1920s. Ecological pyramids show the relative amounts of various parameters (such as number of organisms, energy, and biomass) across trophic levels. Pyramids of numbers can be either upright or inverted, depending on the ecosystem. As shown in Figure 46.10, typical grassland during the summer has a base of many plants, and the numbers of organisms decrease at each trophic level. However, during the summer in a temperate forest, the base of the pyramid consists of few trees compared with the number of primary consumers, mostly insects. Because trees are large, they have great photosynthetic capability, and dominate other plants in this ecosystem to obtain sunlight. Even in smaller numbers, primary producers in forests are still capable of supporting other trophic levels. Another way to visualize ecosystem structure is with pyramids of biomass. This pyramid measures the amount of energy converted into living tissue at the different trophic levels. Using the Silver Springs ecosystem example, this data exhibits an upright biomass pyramid (Figure 46.10), whereas the pyramid from the English Channel example is inverted. The plants (primary producers) of the Silver Springs ecosystem make up a large percentage of the biomass found there. However, the phytoplankton in the English Channel example make up less biomass than the primary consumers, the zooplankton. As with inverted pyramids of numbers, this inverted pyramid is not due to a lack of productivity from the primary producers, but results from the high turnover rate of the phytoplankton. The phytoplankton are consumed rapidly by the primary consumers, thus, minimizing their biomass at any particular point in time. However, phytoplankton reproduce quickly, thus they are able to support the rest of the ecosystem. Pyramid ecosystem modeling can also be used to show energy flow through the trophic levels. Notice that these numbers are the same as those used in the energy flow compartment diagram in (Figure 46.8). Pyramids of energy are always upright, and an ecosystem without sufficient primary productivity cannot be supported. All types of ecological pyramids are useful for characterizing ecosystem structure. However, in the study of energy flow through the ecosystem, pyramids of energy are the most consistent and representative models of ecosystem structure (Figure 46.10). Visual Connection Figure 46.10 Ecological pyramids depict the (a) biomass, (b) number of organisms, and (c) energy in each trophic level. Pyramids depicting the number of organisms or biomass may be inverted, upright, or even diamond-shaped. Energy pyramids, however, are always upright. Why? Consequences of Food Webs: Biological Magnification One of the most important environmental consequences of ecosystem dynamics is biomagnification. Biomagnification is the increasing concentration of persistent, toxic substances in organisms at each trophic level, from the primary producers to the apex consumers. Many substances have been shown to bioaccumulate, including the pesticide dichlorodiphenyltrichloroethane (DDT), which was described in the 1960s bestseller, Silent Spring, by marine biologist Rachel Carson. DDT was a commonly used pesticide before its dangers became known. In some aquatic ecosystems, organisms from each trophic level consumed many organisms of the lower level, which caused DDT to increase in birds (apex consumers) that ate fish. Thus, the birds accumulated sufficient amounts of DDT to cause fragility in their eggshells. This effect increased egg breakage during nesting and was shown to have adverse effects on these bird populations. Carson's combination of scientific knowledge and illuminating writing helped raise awareness about overall environmental issues as well as the specifics of the pesticide. The use of DDT was banned in the United States in the 1970s. Other substances that biomagnify are polychlorinated biphenyls (PCBs), which were used in coolant liquids in the United States until their use was banned in 1979, and heavy metals, such as mercury, lead, and cadmium. These substances were best studied in aquatic ecosystems, where fish species at different trophic levels accumulate toxic substances brought through the ecosystem by the primary producers. As illustrated in a study performed by the National Oceanic and Atmospheric Administration (NOAA) in the Saginaw Bay of Lake Huron (Figure 46.11), PCB concentrations increased from the ecosystem’s primary producers (phytoplankton) through the different trophic levels of fish species. The apex consumer (walleye) has more than four times the amount of PCBs compared to phytoplankton. Also, based on results from other studies, birds that eat these fish may have PCB levels at least one order of magnitude higher than those found in the lake fish. Figure 46.11 This chart shows the PCB concentrations found at the various trophic levels in the Saginaw Bay ecosystem of Lake Huron. Numbers on the x-axis reflect enrichment with heavy isotopes of nitrogen (15N), which is a marker for increasing trophic level. Notice that the fish in the higher trophic levels accumulate more PCBs than those in lower trophic levels. (credit: Patricia Van Hoof, NOAA, GLERL) Other concerns have been raised by the accumulation of heavy metals, such as mercury and cadmium, in certain types of seafood. The United States Environmental Protection Agency (EPA) recommends that pregnant people and young children should not consume any swordfish, shark, king mackerel, or tilefish because of their high mercury content. These individuals are advised to eat fish low in mercury: salmon, tilapia, shrimp, pollock, and catfish. Biomagnification is a good example of how ecosystem dynamics can affect our everyday lives, even influencing the food we eat. Previous Next 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. Authors: Mary Ann Clark, Matthew Douglas, Jung Choi Publisher/website: OpenStax Book title: Biology 2e Publication date: Mar 28, 2018 Location: Houston, Texas Book URL: Section URL: © Jul 7, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.
12345	https://www.youtube.com/watch?v=-ZoutkxxqSM	Convert Ax+By = C (Standard form) to y= mx+b (Slope Intercept form) \| Tagalog Prof Math Wizard 18000 subscribers 247 likes Description 14329 views Posted: 8 May 2020 Convert Ax+By = C (Standard form) to y= mx+b (Slope Intercept form) \| Tagalog MathEasy #Grade8 46 comments Transcript: hello everyone this is teacher Mel and let us explore the world from math so fine ah ba ba convert from ax + B Y is equal to C form into y is equal to MX plus B form we have our first example for this one rewrite the linear equation 2x plus y is equal to 9 into the form Y is equal to MX plus B champion una nothing a good way is to write the equation what happens in whoa saying so for y is equal to MX plus b CX accompany coefficient AI NASA right side 9 question I'm God will be nothing but a into the form or into slope-intercept form sha you know x even moving Latins the right side of the equation or it a transpose snapping hyah it becomes y is equal to negative 2x plus 9 since positive is just the left side now it transposes the right side an equation again negative 2x i you see like not in a polka castle at y is equal to negative 2x plus 9 therefore y is equal to negative 2x plus 9 we write the linear equation 4x plus 2y is equal to 7 into the form Y is equal to MX plus sham process will let muna not been next I know your next step nothing since u MX I announce the right side an equation a lilypad nap and C for X to the right side of the equation so it becomes 2 y is equal to negative 4x last 7 rewrite long nothing to Y is equal to negative 4x plus 7 I never your final answer nothing in diba bucketing the bar sub is a form it should be y is equal to MX plus B but now well on coefficient C wife well about coefficient Jessie why mehran coefficient and why do I to angle go in added piracy wine Alana matira is we need to divide the whole equation by 2 edie divided not a new equation and 2 para cygwin along in madeira select side none in future whatever the coefficient of Y ending ID divided not in a number so whole equation okay so divide all of them by 2 this becomes Y as if 2y divided by 2 is just Y is equal to negative 2x plus 7 over 2 therefore your final answer nothing I Y is equal to negative 2x that's a negative 4 divided by 2 is negative 2x plus 7 over 2 thank you everyone if you have any question comment below I'm happy to answer any question please subscribe for more math videos once again this is teacher Mel [Music]
12346	https://artofproblemsolving.com/wiki/index.php/Symmetric_sum?srsltid=AfmBOopTaNYZCrGZhnz0cGYTbZBgSJtII3enn8coce568pH6xbwPzoWe	Art of Problem Solving Symmetric sum - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Symmetric sum Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Symmetric sum The symmetric sum of a function of variables is defined to be , where ranges over all permutations of . More generally, a symmetric sum of variables is a sum that is unchanged by any permutation of its variables. Any symmetric sum can be written as a polynomial of elementary symmetric sums. A symmetric function of variables is a function that is unchanged by any permutation of its variables. The symmetric sum of a symmetric function therefore satisfies Given variables and a symmetric function with , the notation is sometimes used to denote the sum of over all subsets of size in . See also Cyclic sum Muirhead's Inequality PaperMath’s sum This article is a stub. Help us out by expanding it. Retrieved from " Categories: Stubs Algebra Definition Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
12347	https://strimmerlab.github.io/publications/lecture-notes/MATH20802/maximum-likelihood-estimation.html	Skip to main content Statistical Methods: Likelihood, Bayes and Regression 3 Maximum likelihood estimation 3.1 Principle of maximum likelihood estimation 3.1.1 Outline The starting points in an ML analysis are the observed data D={x1,…,xn}D={x1,…,xn} with nn independent and identically distributed (iid) samples, with the ordering irrelevant, and a model FθFθ with corresponding probability density or probability mass function f(x\|θ)f(x\|θ) with parameters θθ From this we construct the likelihood function: Ln(θ\|D)=∏ni=1f(xi\|θ)Ln(θ\|D)=∏ni=1f(xi\|θ) Historically, the likelihood is also often interpreted as the probability of the data given the model. However, this is not strictly correct. First, this interpretation only applies to discrete random variables. Second, since the samples are iid even in this case one would still need to add a factor accounting for the multiplicity of possible orderings of the samples to obtain the correct probability of the data. Third, the interpretation of likelihood as probability of the data completely breaks down for continuous random variables because then f(x\|θ)f(x\|θ) is a density, not a probability. As we have seen in the previous chapter the origin of the likelihood function lies in its connection to relative entropy. Specifically, the log-likelihood function ln(θ\|D)=∑ni=1logf(xi\|θ)ln(θ\|D)=∑ni=1logf(xi\|θ) divided by sample size nn is a large sample approximation of the cross-entropy between the unknown true data generating model and the approximating model FθFθ. Note that the log-likelihood is additive over the samples xixi. The maximum likelihood point estimate ˆθML^θML is then given by maximising the (log)-likelihood ˆθML=arg maxln(θ\|D)^θML=arg maxln(θ\|D) Thus, finding the MLE is an optimisation problem that in practise is most often solved numerically on the computer, using approaches such as gradient ascent (or for negative log-likelihood gradient descent) and related algorithms. Depending on the complexity of the likelihood function finding the maximum can be very difficult. 3.1.2 Obtaining MLEs for a regular model In regular situations, i.e. when the log-likelihood function is twice differentiable with regard to the parameters, the maximum (peak) of the likelihood function lies inside the parameter space and not at a boundary, the parameters of the model are all identifiable (in particular the model is not overparameterised), and the second derivative of the log-likelihood at the maximum is negative and not zero (for more than one parameter: the Hessian matrix at the maximum is negative definite and not singular) then in order to maximise ln(θ\|D)ln(θ\|D) one may use the score function S(θ)S(θ) which is the first derivative of the log-likelihood function: Sn(θ)=dln(θ\|D)dθSn(θ)=∇ln(θ\|D)scalar parameter θ: first derivativeof log-likelihood functiongradient if θ is a vector(i.e. if there's more than one parameter)Sn(θ)=dln(θ\|D)dθSn(θ)=∇ln(θ\|D)scalar parameter θ: first derivativeof log-likelihood functiongradient if θ is a vector(i.e. if there's more than one parameter) A necessary (but not sufficient) condition for the MLE is that Sn(ˆθML)=0Sn(^θML)=0 To demonstrate that the log-likelihood function actually achieves a maximum at ˆθML^θML the curvature at the MLE must negative, i.e. that the log-likelihood must be locally concave at the MLE. In the case of a single parameter (scalar θθ) this requires to check that the second derivative of the log-likelihood function is negative: d2ln(ˆθML\|D)dθ2<0d2ln(^θML\|D)dθ2<0 In the case of a parameter vector (multivariate θθ) you need to compute the Hessian matrix (matrix of second order derivatives) at the MLE: ∇∇Tln(ˆθML\|D)∇∇Tln(^θML\|D) and then verify that this matrix is negative definite (i.e. all its eigenvalues must be negative). As we will see later the second order derivatives of the log-likelihood function also play an important role for assessing the uncertainty of the MLE. 3.1.3 Invariance property of the maximum likelihood The invariance principle states that the maximum likelihood is invariant against reparameterisation. Assume we transform a parameter θθ into another parameter ωω using some invertible function g()g() so that ω=g(θ)ω=g(θ). Then the maximum likelihood estimate ˆωML^ωML of the new parameter ωω is simply the transformation of the maximum likelihood estimate ˆθML^θML of the original parameter θθ with ˆωML=g(ˆθML)^ωML=g(^θML). The achieved maximum likelihood is the same in both cases. The reason why this works is that maximisation is a procedure that is invariant against transformations of the argument of the function that is maximised. Consider a function h(x)h(x) with a maximum at xmax=arg max h(x)xmax=arg max h(x). Now we relabel the argument using y=g(x)y=g(x) where gg is an invertible function. Then the function in terms of yy is h(g−1(y))h(g−1(y)). and clearly this function has a maximum at ymax=g(xmax)ymax=g(xmax) since h(g−1(ymax))=h(xmax)h(g−1(ymax))=h(xmax). The invariance property can be very useful in practise because it is often easier (and sometimes numerically more stable) to maximise the likelihood for a different set of parameters. See Worksheet L1 for an example application of the invariance principle. 3.1.4 Consistency of maximum likelihood estimates One important property of maximum likelihood is that it produces consistent estimates. Specifically, if the true underlying model FtrueFtrue with parameter θtrueθtrue is contained in the set of specified candidates models FθFθ Ftrue⏟true model⊂Fθ⏟specified modelsFtruetrue model⊂Fθspecified models then ˆθMLlarge n⟶θtrue^θMLlarge n⟶θtrue This is a consequence of DKL(Ftrue,Fθ)→0DKL(Ftrue,Fθ)→0 for Fθ→FtrueFθ→Ftrue, and that maximisation of the likelihood function is for large nn equivalent to minimising the relative entropy. Thus given sufficient data the MLE will converge to the true value. As a consequence, MLEs are asympotically unbiased. As we will see in the examples they can still be biased in finite samples. Note that even if the candidate model FθFθ is misspecified (i.e. it does not contain the actual true model) the MLE is still optimal in the sense in that it will find the closest possible model. It is possible to find inconsistent MLEs, but this occurs only in situations where the dimension of the model / number of parameters increases with sample size, or when the MLE is at a boundary or when there are singularities in the likelihood function. 3.2 Maximum likelihood estimation in practise 3.2.1 Likelihood estimation for a single parameter In the following we illustrate likelihood estimation for models with a single parameter. In this case the score function and the second derivative of the log-likelihood are all scalar-valued like the log-likelihood function itself. Example 3.1 Estimation of a proportion – maximum likelihood for the Bernoulli model: We aim to estimate the true proportion θθ in a Bernoulli experiment with binary outcomes, say the proportion of “successes” vs. “failures” or of “heads” vs. “tails” in a coin tossing experiment. Bernoulli model Ber(θ)Ber(θ): Pr("success")=θPr("success")=θ and Pr("failure")=1−θPr("failure")=1−θ. The “success” is indicated by outcome x=1x=1 and the “failure” by x=0x=0. We conduct nn trials and record n1n1 successes and n−n1n−n1 failures. Parameter: θθ probability of “success”. What is the MLE of θ? the observations D={x1,…,xn} take on values 0 or 1. the average of the data points is ˉx=1n∑ni=1xi=n1n. the probability mass function (PMF) of the Bernoulli distribution Ber(θ) is: p(x\|θ)=θx(1−θ)1−x={θif x=1 1−θif x=0 log-PMF: logp(x\|θ)=xlog(θ)+(1−x)log(1−θ) log-likelihood function: ln(θ\|D)=n∑i=1logf(xi\|θ)=n1logθ+(n−n1)log(1−θ)=n(ˉxlogθ+(1−ˉx)log(1−θ)) Note how the log-likelihood depends on the data only through ˉx! This is an example of a sufficient statistic for the parameter θ (in fact it is also a minimally sufficient statistic). This will be discussed in more detail later. Score function: Sn(θ)=dln(θ\|D)dθ=n(ˉxθ−1−ˉx1−θ) Maximum likelihood estimate: Setting Sn(ˆθML)=0 yields as solution ˆθML=ˉx=n1n With dSn(θ)dθ=−n(ˉxθ2+1−ˉx(1−θ)2)<0 the optimum corresponds indeed to the maximum of the (log-)likelihood function as this is negative for ˆθML (and indeed for any θ). The maximum likelihood estimator of θ is therefore identical to the frequency of the successes among all observations. Note that to analyse the coin tossing experiment and to estimate θ we may equally well use the binomial distribution Bin(n,θ) as model for the number of successes. This results in the same MLE for θ but the likelihood function based on the binomial PMF includes the binomial coefficient. However, as it does not depend on θ it disappears in the score function and has no influence in the derivation of the MLE. Example 3.2 Normal distribution with unknown mean and known variance: x∼N(μ,σ2) with E(x)=μ and Var(x)=σ2 the parameter to be estimated is μ whereas σ2 is known. What’s the MLE of the parameter μ? the data D={x1,…,xn} are all real in the range xi∈[−∞,∞]. the average ˉx=1n∑ni=1xi is real as well. Density: f(x\|μ)=1√2πσ2exp(−(x−μ)22σ2) Log-Density: logf(x\|μ)=−12log(2πσ2)−(x−μ)22σ2 Log-likelihood function: ln(μ\|D)=n∑i=1logf(xi\|μ)=−12σ2n∑i=1(xi−μ)2−n2log(2πσ2)⏟constant term, does not depend on μ, can be removed=−12σ2n∑i=1(x2i−2xiμ+μ2)+C=nσ2(ˉxμ−12μ2)−12σ2n∑i=1x2i⏟another constant term+C Note how the non-constant terms of the log-likelihood depend on the data only through ˉx! Score function: Sn(μ)=nσ2(ˉx−μ) Maximum likelihood estimate: Sn(ˆμML)=0⇒ˆμML=ˉx With dSn(μ)dμ=−nσ2<0 the optimum is indeed the maximum The constant term C in the log-likelihood function collects all terms that do not depend on the parameter. After taking the first derivative with regard to the parameter this term disappears thus C is not relevant for finding the MLE of the parameter. In the future we will often omit such constant terms from the log-likelihood function without further mention. Example 3.3 Normal distribution with known mean and unknown variance: x∼N(μ,σ2) with E(x)=μ and Var(x)=σ2 σ2 needs to be estimated whereas the mean μ is known What’s the MLE of σ2? the data D={x1,…,xn} are all real in the range xi∈[−∞,∞]. the average of the squared centred data ¯(x−μ)2=1n∑ni=1(xi−μ)2≥0 is non-negative. Density: f(x\|σ2)=(2πσ2)−12exp(−(x−μ)22σ2) Log-Density: logf(x\|σ2)=−12log(2πσ2)−(x−μ)22σ2 Log-likelihood function: ln(σ\|D)=ln(μ,σ2\|D)=n∑i=1logf(xi\|σ2)=−n2log(σ2)−12σ2n∑i=1(xi−μ)2−n2log(2π)⏟constant not depending on σ2=−n2log(σ2)−n2σ2¯(x−μ)2+C Note how the log-likelihood function depends on the data only through ¯(x−μ)2! Score function: Sn(σ2)=−n2σ2+n2σ4¯(x−μ)2 Note that to obtain the score function the derivative needs to be taken with regard to the variance parameter σ2 — not with regard to σ! As a trick, relabel σ2=v in the log-likelihood function, then take the derivative with regard to v, then backsubstitute v=σ2 in the final result. Maximum likelihood estimate: Sn(^σ2ML)=0⇒ ^σ2ML=¯(x−μ)2=1nn∑i=1(xi−μ)2 To confirm that we actually have maximum we need to verify that the second derivative of log-likelihood at the optimum is negative. With dSn(σ2)dσ2=−n2σ4(2σ2¯(x−μ)2−1) and hence dSn(^σ2ML)dσ2=−n2(^σ2ML)−2<0 the optimum is indeed the maximum. 3.2.2 Likelihood estimation for multiple parameters If there are several parameters likelihood estimation is conceptually no different from the case of a single parameter. However, the score function is now vector-valued and the second derivative of the log-likelihood is a matrix-valued function. Example 3.4 Normal distribution with mean and variance both unknown: x∼N(μ,σ2) with E(x)=μ and Var(x)=σ2 both μ and σ2 need to be estimated. What’s the MLE of the parameter vector θ=(μ,σ2)T? the data D={x1,…,xn} are all real in the range xi∈[−∞,∞]. the average ˉx=1n∑ni=1xi is real as well. the average of the squared data ¯x2=1n∑ni=1x2i≥0 is non-negative. Density: f(x\|μ,σ2)=(2πσ2)−12exp(−(x−μ)22σ2) Log-Density: logf(x\|μ,σ2)=−12log(2πσ2)−(x−μ)22σ2 Log-likelihood function: ln(θ\|D)=ln(μ,σ2\|D)=n∑i=1logf(xi\|μ,σ2)=−n2log(σ2)−12σ2n∑i=1(xi−μ)2−n2log(2π)⏟constant not depending on μ or σ2=−n2log(σ2)−n2σ2(¯x2−2ˉxμ+μ2)+C Note how the log-likelihood function depends on the data only through ˉx and ¯x2! Score function Sn, gradient of ln(θ\|D): Sn(θ)=∇ln(θ\|D)=(nσ2(ˉx−μ)−n2σ2+n2σ4(¯x2−2ˉxμ+μ2)) Maximum likelihood estimate: Sn(ˆθML)=0⇒ ˆθML=(ˆμML^σ2ML)=(ˉx¯x2−ˉx2) The ML estimate of the variance can also be written ^σ2ML=¯x2−ˉx2=¯(x−ˉx)2=1n∑ni=1(xi−ˉx)2. To confirm that we actually have maximum we need to verify that the eigenvalues of the Hessian matrix at the optimum are all negative. This is indeed the case, for details see Example 3.7. 3.2.3 Relationship of maximum likelihood with least squares estimation In Example 3.2 the form of the log-likelihood function is a function of the sum of squared differences. Maximising ln(μ\|D)=−12σ2∑ni=1(xi−μ)2 is equivalent to minimising ∑ni=1(xi−μ)2. Hence, finding the mean by maximum likelihood assuming a normal model is equivalent to least-squares estimation! Note that least-squares estimation has been in use at least since the early 1800s 4 and thus predates maximum likelihood (1922). Due to its simplicity it is still very popular in particular in regression and the link with maximum likelihood and normality allows to understand why it usually works well! 3.2.4 Bias and maximum likelihood estimates Example 3.4 is interesting because it shows that maximum likelihood can result in both biased and as well as unbiased estimators. Recall that x∼N(μ,σ2). As a result ˆμML=ˉx∼N(μ,σ2n) with E(ˆμML)=μ and ^σ2ML∼W1(s2=σ2n,k=n−1) (see Appendix) with mean E(^σ2ML)=n−1nσ2. Therefore, the MLE of μ is unbiased as Bias(ˆμML)=E(ˆμML)−μ=0 In contrast, however, the MLE of σ2 is negatively biased because Bias(^σ2ML)=E(^σ2ML)−σ2=−1nσ2 Thus, in the case of the variance parameter of the normal distribution the MLE is not recovering the well-known unbiased estimator of the variance ^σ2UB=1n−1n∑i=1(xi−ˉx)2=nn−1^σ2ML In other words, the unbiased variance estimate is not a maximum likelihood estimate! Therefore it is worth keeping in mind that maximum likelihood can result in biased estimates for finite n. For large n, however, the bias disappears as MLEs are consistent. 3.3 Observed Fisher information 3.3.1 Motivation and definition By inspection of some log-likelihood curves it is apparent that the log-likelihood function contains more information about the parameter θ than just the maximum point ˆθML. In particular the curvature of the log-likelihood function at the MLE must be somehow related the accuracy of ˆθML: if the likelihood surface is flat near the maximum (low curvature) then if is more difficult to find the optimal parameter (also numerically!). Conversely, if the likelihood surface is peaked (strong curvature) then the maximum point is clearly defined. The curvature is described by the second-order derivatives (Hessian matrix) of the log-likelihood function. For univariate θ the Hessian is a scalar: d2ln(θ\|D)dθ2 For multivariate parameter vector θ of dimension d the Hessian is a matrix of size d×d: ∇∇Tln(θ\|D) By construction the Hessian is negative definite at the MLE (i.e. its eigenvalues are all negative) to ensure the the function is concave at the MLE (i.e. peak shaped). The observed Fisher information (matrix) is defined as the negative curvature at the MLE ˆθML: Jn(ˆθML)=−∇∇Tln(ˆθML\|D) Sometimes this is simply called the “observed information”. To avoid confusion with the expected Fisher information introduced earlier IFisher(θ)=−EFθ(∇∇Tlogf(x\|θ)) it is necessary to always use the qualifier “observed” when referring to Jn(ˆθML). 3.3.2 Examples of observed Fisher information Example 3.5 Bernoulli model Ber(θ): We continue Example 3.1. Recall that ˆθML=ˉx=n1n and the score function Sn(θ)=n(ˉxθ−1−ˉx1−θ). The negative second derivative of the log-likelihood function is −dSn(θ)dθ=n(ˉxθ2+1−ˉx(1−θ)2) The observed Fisher information is therefore Jn(ˆθML)=n(ˉxˆθ2ML+1−ˉx(1−ˆθML)2)=n(1ˆθML+11−ˆθML)=nˆθML(1−ˆθML) The inverse of the observed Fisher information is: Jn(ˆθML)−1=ˆθML(1−ˆθML)n Compare this with Var(xn)=θ(1−θ)n for x∼Bin(n,θ). Example 3.6 Normal distribution with unknown mean and known variance: This is the continuation of Example 3.2. Recall the MLE for the mean ˆμML=1n∑ni=1xi=ˉx and the score function Sn(μ)=nσ2(ˉx−μ). The negative second derivative of the score function is −dSn(μ)dμ=nσ2 The observed Fisher information at the MLE is therefore Jn(ˆμML)=nσ2 and the inverse of the observed Fisher information is Jn(ˆμML)−1=σ2n For xi∼N(μ,σ2) we have Var(xi)=σ2 and hence Var(ˉx)=σ2n, which is equal to the inverse observed Fisher information. Example 3.7 Normal distribution with mean and variance parameter: This is the continuation of Example 3.4. Recall the MLE for the mean and variance: ˆμML=1nn∑i=1xi=ˉx ^σ2ML=1nn∑i=1(xi−ˉx)2=¯x2−ˉx2 with score function Sn(μ,σ2)=∇ln(μ,σ2\|D)=(nσ2(ˉx−μ)−n2σ2+n2σ4(¯x2−2μˉx+μ2)) The Hessian matrix of the log-likelihood function is ∇∇Tln(μ,σ2\|D)=(−nσ2−nσ4(ˉx−μ)−nσ4(ˉx−μ)n2σ4−nσ6(¯x2−2μˉx+μ2)) The negative Hessian at the MLE, i.e. at ˆμML=ˉx and ^σ2ML=¯x2−ˉx2 yields the observed Fisher information matrix: Jn(ˆμML,^σ2ML)=(n^σ2ML00n2(^σ2ML)2) Note that the observed Fisher information matrix is diagonal with positive entries. Therefore its eigenvalues are all positive as required for a maximum, because for a diagonal matrix the eigenvalues are simply the the entries on the diagonal. The inverse of the observed Fisher information matrix is Jn(ˆμML,^σ2ML)−1=(^σ2MLn002(^σ2ML)2n) Recall that x∼N(μ,σ2) and therefore ˆμML=ˉx∼N(μ,σ2n) Hence Var(ˆμML)=σ2n. If you compare this with the first diagonal entry of the inverse observed Fisher information matrix you see that this is essentially the same expression (apart from the “hat”). The empirical variance ^σ2ML follows a one-dimensional Wishart distribution ^σ2ML∼W1(s2=σ2n,k=n−1) (see Appendix) with variance Var(^σ2ML)=n−1n2σ4n. For large n this becomes Var(^σ2ML)a=2σ4n which is essentially (apart from the “hat”) the second diagonal entry of the inverse observed Fisher information matrix. 3.3.3 Relationship between observed and expected Fisher information The observed Fisher information Jn(ˆθML) and the expected Fisher information IFisher(θ) are related but also two clearly different entities: Both types of Fisher information are based on computing second order derivatives (Hessian matrix), thus both are based on the curvature of a function. The observed Fisher information is computed from the log-likelihood function. Therefore it takes the observed data D into account and explicitly depends on the sample size n. It contains estimates of the parameters but not the parameters themselves. While the curvature of the log-likelihood function may be computed for any point of the log-likelihood function the observed Fisher information specifically refers to curvature at the MLE ˆθML. It is linked to the (asymptotic) variance of the MLE as we have seen in the examples and will discuss in more detail later. In contrast, the expected Fisher information is derived directly from the log-density. It does not depend on the observed data, and thus does not depend on sample size. It can be computed for any value of the parameters. It describes the geometry of the space of the models, and is the local approximation of relative entropy. Assume that for large sample size n the MLE converges to ˆθML→θ0. It follows from the construction of the observed Fisher information and the law of large numbers that asymptotically for large sample size Jn(ˆθML)→nIFisher(θ0) (i.e. the expected Fisher information for a set of iid random variables, see Example 2.14). In a very important class of models, namely in an exponential family model, we find that Jn(ˆθML)=nIFisher(ˆθML) is valid also for finite sample size n. This is in fact the case for all the examples discussed above (e.g. see Examples 3.5 and 2.11 for the Bernoulli distribution and Examples 3.7 and 2.13 for the normal distribution). However, this is an exception. In a general model Jn(ˆθML)≠nIFisher(ˆθML) for finite sample size n. An example is provided by the Cauchy distribution with median parameter θ. It is not an exponential family model and has expected Fisher information IFisher(θ)=12 regardless of the choice the median parameter whereas the observed Fisher information Jn(ˆθML) depends on the MLE ˆθML of the median parameter and is not simply n2.
12348	https://pmc.ncbi.nlm.nih.gov/articles/PMC4625773/	Clinical manifestations and management of Gaucher disease - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. 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Learn more: PMC Disclaimer \| PMC Copyright Notice Clin Cases Miner Bone Metab . 2015 Oct 26;12(2):157–164. doi: 10.11138/ccmbm/2015.12.2.157 Search in PMC Search in PubMed View in NLM Catalog Add to search Clinical manifestations and management of Gaucher disease Silvia Linari Silvia Linari 1 Center for Bleeding Disorders, Department of Heart and Vessels, Careggi University Hospital, Florence, Italy Find articles by Silvia Linari 1,✉, Giancarlo Castaman Giancarlo Castaman 1 Center for Bleeding Disorders, Department of Heart and Vessels, Careggi University Hospital, Florence, Italy Find articles by Giancarlo Castaman 1 Author information Article notes Copyright and License information 1 Center for Bleeding Disorders, Department of Heart and Vessels, Careggi University Hospital, Florence, Italy ✉ Address for correspondence: Silvia Linari, MD, Center for Bleeding Disorders, Department of Heart and Vessels, Careggi University Hospital, Largo Brambilla 3, 50134 Florence, Italy, Phone: +39 055 7947587, Fax: +39 055 7947794, E-mail: linaris@aou-careggi.toscana.it Issue date 2015 May-Aug. Copyright © 2015, CIC Edizioni Internazionali PMC Copyright notice PMCID: PMC4625773 PMID: 26604942 Summary Gaucher disease is a rare multi-systemic metabolic disorder caused by the inherited deficiency of the lysosomal enzyme β-glucocerebrosidase, which leads to the accumulation of its normal substrate, glucocerebroside, in tissue macrophages with damage to haematological, visceral and bone systems. Anaemia, thrombocytopenia, enlargement of liver and/or spleen, skeletal abnormalities (osteopenia, lytic lesions, pathological fractures, chronic bone pain, bone crisis, bone infarcts, osteonecrosis and skeletal deformities) are typical manifestations of the most prevalent form of the disease, the so-called non-neuronopathic type 1. However, severity and coexistence of different symptoms are highly variable. The determination of deficient β-glucocerebrosidase activity in leukocytes or fibroblasts by enzymatic assay is the gold standard for the diagnosis of Gaucher disease. Comprehensive and reproducible evaluation and monitoring of all clinically relevant aspects are fundamental for the effective management of Gaucher disease patients. Enzyme replacement therapy has been shown to be effective in reducing glucocerebroside storage burden and diminishing the deleterious effects caused by its accumulation. Tailored treatment plan for each patient should be directed to symptom relief, general improvement of quality of life, and prevention of irreversible damage. Keywords: Gaucher disease, glucocerebroside, storage burden, activated macrophage, enzyme replacement therapy Etiology and pathogenesis Gaucher disease (GD) is an inherited lysosomal storage disease caused by an autosomal recessive defect of the gene encoding β-glucocerebrosidase, enzyme responsible for the accumulation of glucosylceramide into reticuloendothelial cells, rendering GD a multi-organ chronic disorder (1, 2). These lipid-laden cells are called Gaucher cells and are predominantly found in the spleen, liver, bone marrow and rarely lung. As a consequence, hepatosplenomegaly, pancytopenia, bone complications and, in a small number of patients, lung involvement with interstitial lung disease and pulmonary hypertension can occur (2, 3). Massive infiltration by Gaucher cells alone cannot explain the multifaceted characteristics of the disease. The accumulation leads to a secondary activation of macrophages, inducing the release of various cytokines and lysosomal proteins (4). The most striking seems to be the increased expression of chitotriosidase, which can be raised 1000-fold in Gaucher patients and is produced in Gaucher cells. The plasma concentration strongly correlates with the accumulation of Gaucher cells in the body (5). In an animal model, an inflammatory infiltration of several organ systems, B-cell stimulation and expression of TNF-α and IL-1β were observed (6). These observations may explain the increased occurrence of auto-antibodies, B-cell lymphomas, gammopathies and multiple myeloma in Gaucher patients (7, 8). Similarly, most of the Gaucher changes in the long bones can be explained by the release of cytokines by storage cells in the bone marrow. To date, the pathogenic etiology of neurological involvement in GD is still incomplete (9). Diagnosis of GD Residual levels of glucocerebrosidase in patients with GD have been variously estimated at 5–25% of normal activity. The measurement of β-glucocerebrosidase activity in leukocytes or in cultured fibroblasts obtained by skin biopsy is the gold standard for the diagnosis of GD (10). As for many other lysosomal enzymes, recently screening methods using dry blood spots have been developed also for β-glucocerebrosidase (11). The finding of a reduced β-glucocerebrosidase activity can be supplemented by detection of the genetic defect. The glucocerebrosidase gene (GBA) is located on chromosome 1q21; it contains 11 exons and 10 introns, covering 7.6 kilobases (kb) of sequence. There is a highly homologous pseudogene (psGBA), spanning 5.7 kb with the same exon and intron number as the GBA, located 16 kb downstream, sharing approximately 96% of the sequence in coding regions. Almost 300 mutations and polymorphisms in GBA have been identified. Most are point mutations, but insertions or deletions, splice site alterations, and recombinating alleles, also with the nearby pseudogene, have also been described (12). Recently mutations in gene of saposin C, that is the β-glucocerebrosidase activator, have been also reported in association with GD (13). Four mutations account for over 90% of disease alleles in Ashkenazi Jewish patients: N370S, 84GG, L444P and IVS2+1G (14). Non-Jewish patients exhibit a much wider range of genotypes, although two mutations (N370S and L444P) are common in both populations. Of note, homozygosis for L444P normally results in neuronopathic disease whereas the presence of a single mutant N370S allele usually prevents neurological involvement. However, genotype-phenotype correlations are of limited clinical value and the clinical diversity of this single gene disorder suggests that modifier genes and environmental factors must play an important role (15). Genetic diagnosis can be performed during pregnancy using amniocentesis or chorionic villi sampling, but this test is only useful in populations with a high gene frequency or in families already known to be affected (16). The detection of Gaucher cells in bioptic samples is not required for diagnostic purpose, even if it often represents the step triggering the diagnostic suspicion in adult patients. In fact, approximately 60% of all new GD diagnoses are still made by a bone marrow biopsy or histology of a surgically removed spleen. Epidemiology GD is pan-ethnic, but it has a particular high prevalence among Ashkenazi Jews (17). Within this population, non-neuronopathic forms of GD occur with a frequency of approximately 1:850 and carrier frequency is 1:17. Based on epidemiological investigations in Italy a prevalence of 1:40,000–1:86,000 is assumed in the general population. Acute neurological forms occur even less frequently. A prevalence estimate of 1:500000 may however be too low, as cases with fetal or neonatal manifestations (fetal hydrops, congenital ichthyosis) are not always properly diagnosed. Clinical classification Three major forms of GD have been clinically described. The most prevalent is the so-called non-neuronopathic form (type 1) characterized by anaemia, thrombocytopenia, enlargement of the spleen, skeletal abnormalities (18), and in a small number of patients, by lung involvement with interstitial lung disease (18) and pulmonary hypertension (19, 20). Type 1 is essentially a macrophage disorder, lacking primary central nervous system involvement. Patients with type 1 GD display a large variety of symptoms, ranging from asymptomatic subjects to those who display child-onset disease. Type 2 GD is an acute neuronopathic form with severe prognosis and survival limited to the first two or three years of life; it is characterized by neurological impairment in addition to visceral symptoms. The neurological symptoms start with oculomotor abnormalities followed by brainstem involvement. Type 3 GD is also characterized by neurological involvement but neurological symptoms generally appear later in life than in type 2 disease, and include abnormal eye movements, ataxia, seizures, and dementia, with patients surviving until their third or fourth decade (21). Recently, a clinical association has been reported between the presence of mutations in the β-glucocerebrosidase gene and Parkinsonism (22, 23). The identification of new phenotypes and appreciation that even patients with type 1 may develop some late-onset neurologic manifestations, may suggest is to consider GD as a continuum of disease states (Table 1) (24). Table 1. Clinical classification of the forms of GD. Recently the classic categories of types 1, 2 and 3 have blurry edges along a phenotypic continuum. Patients with GD can have a spectrum of symptoms, ranging from mild to severe neurological effects (24). \| \| Non-neuronopathic GD \| Acute neuronopathic GD \| Chronic neuronopathic GD \| :---: :---: \| \| Incidence \| 1:40000–1:60000 1:850 in Ashkenazi Jews \| <1:100000 \| <1:50000 to <1:100000 \| \| Ethnic group \| Panethnic, more common in Ashkenazi Jews \| Panethnic \| Panethnic \| \| Age at onset of disease \| Any age \| Infancy \| Childhood \| \| CNS symptoms \| − \| +++ \| + → +++ (progressive) \| \| Hepatosplenomegaly \| + → +++ \| ++ \| + → +++ \| \| Hematological symptoms \| + → +++ \| +++ \| + → +++ \| \| Bone symptoms \| − → +++ \| − \| ++ → +++ \| \| \| Open in a new tab Finally, a significant increased risk of haematological malignancies has been reported in GD. GD is frequently associated with immunologic abnormalities: polyclonal hypergammaglobulinemia may occur at diagnosis in 14–41% of adults (25–27); an increased prevalence of the pre-malignant condition monoclonal gammopathy of undetermined significance (MGUS), up to 25%, has been described. In GD the increased risk of MM is reported as ranging from 5.9 (28) to 51.1 times that of normal population. An increased risk of other hematological malignancies, such as amyloidosis or B-cell non-Hodgkin Lymphoma (NHL), has also been reported (29–31). Association between GD and solid organ malignancies is less clear but an increased incidence of hepatocellular carcinoma (HCC) has been described (29). Bone manifestations in GD Skeletal involvement has a high prevalence in adult type 1 GD patients and represents actually the major morbidity for its frequent association with considerable pain, limitations in mobility and an extremely negative impact on the quality of life. The bone manifestations of GD are multifaceted and can include bone marrow infiltration, severe acute “bone crises”, chronic intermittent bone pain, bone infarction, lytic lesions, Erlenmeyer flask deformity of the distal femur, osteopenia, osteoporosis, osteonecrosis, subchondral joint collapse, pathologic fractures of long bones and vertebrae, and growth retardation in children (2, 9, 32, 33). Bone disease has a very high prevalence in type 1 GD, with a radiological evidence described in 93% of patients. Data from the International Collaborative Gaucher Group (ICGG) Registry show that at diagnosis bone pain is present in 50% of type 1 GD patients, bone marrow infiltration in 82%, Erlenmeyer flask deformity in 60%, osteonecrosis in 30%, localized or generalized decrease of bone mineral density (BMD) in 49% and in 36%, respectively (34). The underlying pathology of bone disease is related to the accumulation of Gaucher cells that infiltrate the bone marrow compartment and lead directly or indirectly to localized bone defects, including cortical thinning, osteonecrosis and lytic lesions. The pattern and the progression of the bone marrow infiltration vary from patient to patient, but in general the infiltration seems to begin in the lumbar spine, before entering the metaphysis and diaphysis of the femur, and then being seen in the epiphysis in the later stages. Magnetic resonance imaging (MRI) is the most sensitive diagnostic procedure for the detection of bone marrow changes. Affected bone areas have pathologically reduced signal using T1-weighted spin echo sequences. The reduced signal is caused by displacement of the signal-rich yellow bone marrow (Figure 1) (35). Bone marrow infiltration has also been connected with abnormal bone remodeling. The Erlenmeyer flask deformity is a common remodeling disorder, which occurs in over 60% of the patients. This manifests as a deformity of the distal long bones and generally affects the distal femur (Figure 2) or proximal tibia (36). The metaphyseal regions are expanded and show incorrect bone remodeling, which seems to be connected with a relative failure of osteoclast activity. Another pathological skeletal change seen in GD patients with progressive bone disease is osteonecrosis, also known as avascular necrosis. Osteonecrosis is ischaemic death of bone tissue due to chronic bone infarction. The patient has often suffered many bone crises prior to the onset of the necrotic process. However, vascular obstruction is not necessarily the primary pathological mechanism. Gaucher cells can also damage the vessel wall through the release of lysosomal contents, with localized osteonecrosis that may be extensive. Osteonecrosis usually occurs bilaterally in the head of the femur, but some patients have repeated episodes in the same hip. Involvement of the femoral neck can precede the pathological changes in the head of the femur (37). Osteonecrosis can affect medullary as well as cortical bone. Medullary osteonecrosis may also be entirely asymptomatic. These lesions may cause major disability from pathological fractures and collapse of osseous endplates with joint disintegration and its replacement is often necessary to relieve pain and restore mobility. In children, the disorder is sometimes misdiagnosed as Legg-Calve-Perthes disease. One third of adult GD patients have a history of osteonecrosis (18); while specific genotypes, prior splenectomy, trauma, strenuous physical exertion, and pregnancies appear to be predisposing factors for the development of osteonecrosis, it is not possible to accurately predict in individual patients the future risk for its development. Bone crises can be a manifestation of the osteonecrotic process, when a sufficient proportion of the bone is affected. Extensive osteonecrosis can cause generalized systemic disease characterized by severe pain (due to edema in the bone cavity), high fever, shivering, complete disability, a high leukocyte count and an increased erythrocyte sedimentation rate. When a bone crisis occurs, it is the most incapacitating manifestation of GD, frequently requiring opiate analgesic. Differential diagnosis from septic osteomyelitis may be difficult. Patients with Gaucher crises have a negative blood culture. The best diagnostic option is a technetium-99m bone scan within 2–3 days of onset of bone crises. It is “cold” in Gaucher bone crises and “hot” in septic osteomyelitis (38). Although bone pain usually occurs in GD, there are not always specific radiological findings. The nature of the bone pain differs, in fact this symptom may be dull, sharp, non-specific or severe and localized, like joint pain. Patients with GD are highly likely to suffer fractures, which relates to the increased prevalence of osteopenia, osteonecrosis, infarction and bone crises. Fractures can occur anywhere in the skeleton. Delayed skeletal growth is very common in children with GD, particularly in those with severe type 1. Figure 1. Open in a new tab Coronal T1-weighted sequence of the femurs showing severe bone marrow infiltration with some focal areas. Figure 2. Open in a new tab A-P X-ray image of the distal femurs showing the Erlenmeyer flask deformity in the distal diametaphyseal transition zone. Gaucher cells can alternatively damage bone tissue since they are also activated macrophages which secrete several cytokines (TNFα, IL6, IL10, IL4 and monocyte chemo-attractant proteins) and/or interact with cells in their environment. The locally increased cytokines in bone stimulate production of osteoclast precursors, resulting in imbalances in bone remodeling to favour resorption over formation, leading to osteopenia or osteoporosis. A decreased bone mineralization can occur locally or diffusely and widespread, and can affect trabecular bone, as well as cortical bone (39). The lumbar spine represents an ideal location for the assessment of localized bone disease, as well as generalized osteoporosis. In fact, marrow infiltration is thought to begin in the lumbar vertebrae and then progress to the pelvis and the appendicular skeleton; moreover, the lumbar spine has a higher percentage of trabecular bone and may be more sensitive to changes in systemic bone mass than other sites where cortical bone predominates. Evaluations and monitoring for type 1 GD Challenges to patient care posed by clinical heterogeneity, variables progression rates, and potential permanent disability that can result from untreated or sub-optimally treated haematological, skeletal and visceral involvement dictate a need for comprehensive, serial monitoring. A consensus on minimum recommendations for effective monitoring of adult patients with type 1 GD had been developed by the ICGG Registry coordinators, with reproducible initial baseline and annual follow-up evaluations of all clinically relevant aspects of the disease. Assessment of disease severity includes blood tests, biochemical biomarkers, MRI or ultrasound for measurement of spleen and liver volume, a combination of modalities to evaluate trabecular and compact bone disease, including radiographs, bone densitometry by dual energy X-ray absorptiometry and MRI to assess the severity of bone marrow infiltration. Doppler echocardiogram, chest X-ray and electrocardiogram are useful to exclude pulmonary hypertension (Table 2) (40). Table 2. Minimum recommendations for monitoring patients with non-neuronopathic GD (40). \| \| Patients not on ERT \| Patients on ERT Not achieved therapeutic goals \| Patients on ERT achieved therapeutic goals \| Patients on ERT At time of dose change or significant clinical complications \| :---: :---: \| Frequency (every X months) \| 12 \| 12–24 \| 3 \| 12 \| 12–24 \| \| \| Comprehensive physical examination \| X \| \| \| X \| X (annual) \| \| \| SF-36 (QoL) survey \| X \| \| \| X \| X (annual) \| X \| \| Blood tests \| \| \| \| \| \| \| \| Hemoglobin \| X \| \| X \| \| X \| X \| \| Platelet count \| X \| \| X \| \| X \| X \| \| Biochemical markers \| X \| \| X \| \| X \| X \| \| Visceral \| \| \| \| \| \| \| \| Spleen volume (volumetric MRI or CT) \| \| X \| \| X \| X \| X \| \| Liver volume (volumetric MRI or CT) \| \| X \| \| X \| X \| X \| \| Skeletal \| \| \| \| \| \| \| \| MRI of entire femora (coronal; T1 and T2 weighted) \| \| X \| \| X \| X \| X \| \| DEXA (lumbar spine and femoral neck) \| \| X \| \| X \| X \| X \| Open in a new tab Different biochemical biomarkers are evaluated to quantify dynamic changes in the clinical course over the years: ferritin, immunoglobulin, angiotensin-converting enzyme (ACE), tartrate resistant acid phosphatase (TRAP) (41). At the present time plasma levels of the hydrolase chitotriosidase and the CC-chemokine ligand 18/pulmonary activated-related chemokine (CCL18/PARC) are considered useful tools. Both parameters correlate well with the body burden of Gaucher cells. Chitotriosidase activity in serum can increase up to 100–4000-fold over the normal values in GD, and is reduced by treatment. However, 5–6% of the general population is homozygous for the chitotriosidase gene mutation causing complete deficiency of the enzyme activity (42). Chitotriosidase measurement is unreliable in these situations. Serum CCL18/PARC, which is not affected by any known genetic abnormality, is increased 10–40 fold over the normal levels in GD and decreases during therapy with a pattern similar to chitotriosidase (43). Disease management GD is a multi-organ, chronic, heterogeneous disorder, requiring an individualized approach towards treatment. Many variables such as severity and rate of disease progression, concomitant pathological conditions, the impact of disease manifestations on quality of life and the phenotype/genotype relationship should be considered prior to the initiation of treatment in a patient with GD (10, 40, 44). At the present time, the therapeutic options for adult GD patients are enzyme replacement therapy (ERT) and substrate reduction therapy (SRT), although bone marrow transplantation and gene therapy have been applied in rare cases. It is generally accepted that a GD patient must be treated in the presence of complications such as anaemia, thrombocytopenia, bleeding tendency, skeletal disease, liver or lung involvement or organomegaly. Type 1 GD was the first lysosomal storage disorders for which an effective ERT was developed and it has become a prototype for treatments for related orphan diseases. After its introduction, in 1991, ERT has emerged as the standard of care for type 1 GD (45, 46). In order to establish the severity of disease and to tailor the initial and maintenance ERT dose, a classification in high- and low-risk type 1 GD patients has been suggested by a panel of experts (Tables 3, 4) (47). Table 3. Type 1 GD Highest Risk Patients- One or More of the following symptoms (47). Symptomatic skeletal disease Moderate to severe osteopenia Avascular necrosis Chronic bone pain Pathological fractures Bone crises Joint replacement(s) Impaired quality of life due to GD Cardiopulmonary disease, including pulmonary hypertension Platelet count <60,000mm3or abnormal bleedings Symptomatic anaemia or haemoglobin < 8g/dl Transfusion dependency Significant liver disease Severe hepatomegaly (>2.5xnormal) Infarcts Varices Portal hypertension Hepatitis Significant spleen disease Severe splenomegaly (>15xnormal) Infarcts Significant renal disease Open in a new tab Table 4. Type 1 GD Lower Risk Patients (47). Normal liver, cardiac, lung and renal function Minimal impairment of quality of life due to GD No obvious and recently rapid progression of disease manifestations Skeletal disease limited to mild osteopenia and Erlenmeyer flask deformity Haemoglobin>10.5 g/dl for females and >11.5 g/dl for males (or not more than 2 g/dl below lower limit of normal for age and sex) Platelet count >60,000mm3on three determinations Liver volume <2.5 × normal Spleen volume < 15 × normal Open in a new tab Over two decades since the introduction of therapy, it has become definitely clear that many of the symptoms and signs of visceral GD such as hepatosplenomegaly, as well as anaemia and thrombocytopenia, and often, skeletal or lung involvement, will respond adequately to ERT (48). Three different human recombinant enzymes have been approved; two of them are available in EU and USA: Imiglucerase (Cerezyme, Genzyme Corporation, Cambridge MA, USA) since 1994, and Velaglucerase alfa (VPRIV, Shire HGT, Cambridge MA, USA) since 2010. An additional preparation, Taliglucerase alfa (Elelyso, ProtalixBiotherapeutics, Carmiel, Israel) is available only in USA. The ICGG Registry has monitored the responses of a thousand patients world-wide to Imiglucerase (45). Decreased spleen and liver volumes and increased haemoglobin levels and platelet counts usually occur within 6 months of advent of therapy with every-other-week doses of 15–60 units/kg body weight. Platelet count in patients with massively enlarged spleens may require longer periods to respond, but dramatic improvements usually continue within the first 2–4 years of therapy (46, 49). Normalization or near-normalization of haemoglobin and platelet count as well as liver and spleen volume, with reduction in bone pains and prevention of irreversible skeletal complications are the ultimate end-points. Improvement in bone marrow and in the osseous skeleton in response to ERT has been observed to occur more slowly than the visceral and haematological responses. Increase of bone mineral density in response to ERT could take up to 8 years (50), and ERT is effective in ameliorating bone marrow infiltration only after years (Table 5) (51). Besides, pathological damage such as osteonecrosis, bone infarcts and fracture, once it has occurred, is irreversible, so the maximal benefit derived from ERT in terms of eliciting a response in the skeletal system may occur with early enzyme administration and preventive approaches. The use of biphosphonates can be an effective and safe mean to increase bone density and prevent complications (52). Supportive management for bone pains or bone crises is frequently required, and orthopedic surgery may be necessary in cases of pathologic fractures or osteonecrosis (53). Table 5. Therapeutic goals for ERT in GD patients (49). Therapeutic goals for anemia Increase hemoglobin levels within 12 to 24 months to ≥11g/dl for women and children 12g/dl for men Eliminate blood transfusion dependency and reduce fatigue, dyspnoea and angina Maintain improved hemoglobin values achieved after 12 to 24 months of therapy Therapeutic goals for thrombocytopenia Increase platelet count during the first year of therapy sufficiently, to prevent surgical, obstetrical and spontaneous bleeding Patients with splenectomy - normalization of platelet count by one year of treatment Moderate baseline thrombocytopenia - the platelet count should increase by 1.5- to 2-fold by year one and approach normal levels by year two Severe thrombocytopenia - the platelet count should increase by 1.5-fold by year one and continue to increase slightly during years two to five, but normalization is not expected Avoid splenectomy Maintain stable platelet counts to eliminate risk of bleeding Therapeutic goals for hepatomegaly and splenomegaly Reduce and maintain the liver volume to 1.0 to 1.5 times normal Reduce the liver volume by 20 to 30% within years one to two by 30 to 40% by years three to five Reduce and maintain spleen volume to less than two to eight times normal Reduce the spleen volume by reduce and maintain the 30 to 50% by year one and by 50 to 60% by years two to five Alleviate symptoms due to splenomegaly Eliminate hypersplenism Therapeutic goals for skeletal pathology Lessen or eliminate pain within one to two years Prevent bone crisis Prevent osteonecrosis and subchondral joint collapse Improve bone mineral density Increase trabecular bone mineral density by three to five years Pediatric patients: Attain normal or ideal peak skeletal mass Increase cortical and bone mineral density (BMD) by two years Therapeutic goals for growth in pediatric patients Normalise growth and achieve normal onset of puberty Therapeutic goals for pulmonary involvement Reverse hepatopulmonary syndrome and dependency on oxygen Ameliorate pulmonary hypertension (ERT+ adjuvant therapies) Improve functional status and quality of life Prevent sudden death Prevent pulmonary disease by timely initiation of ERT and avoidance of splenectomy Open in a new tab Substrate synthesis inhibition therapy is an alternative oral approach, based on reduced synthesis of glucosylceramide by inhibiting the appropriate synthetic enzyme (i.e. glucosylceramide synthase), resultanting in decreased production of this dangerous lipid and the ability of the residual enzyme activity to restablish a new steady state. This approach was first tried with the iminosugar N-butyldeoxynojirimycin, Miglu-stat (Zavesca, Actelion Corp) (54), approved by EMA in 2002 for patients with mild-to-moderate GD who are unsuitable for ERT and by FDA in 2003 for patients in whom ERT is not a therapeutic option. Clinical trials demonstrated effects on the visceral organs in type 1 GD with some shrinking of hepatosplenomegaly and improvement of haematologic findings. However, the substantial adverse events related to the use of miglustat, in particular significant diarrhea, and controversial tremor and paresthesias, limited drug’s acceptance. Ceramide analog of the substrate, Eliglustat (Genz-112638; Genzyme Corp) (55) is a novel agent with a better safety profile and higher potency than miglustat. Eliglustat was approved by FDA in August 2014 and approval procedures by EMA in January 2015. Conclusion Gaucher is a rare disease with heterogeneous multisystem involvement. The non-neuronopathic GD may show different symptoms at any age. Related to the predominant manifestation, GD patient can refer to different specialists (pediatrician, internist, hematologist for hematological changes, gastroenterologist for hepatosplenomegaly, rheumatologist or orthopedic for bone disease), but the rarity of the disease and nonspecific and heterogeneous nature of GD symptoms may impede its consideration in the differential diagnosis. Since diagnosis of GD by enzyme testing is unequivocal, performing the test may be convenient in patients with dubious pathological signs. A delay in diagnosis can cause the occurrence of irreversible complications, such as avascular necrosis or MM, in a rare disease for which an effective treatment is available. Different aspects of pathophysiology and in particular determination of disease severity remains incompletely understood, and the precise relationship between GD and certain co-morbidities, especially cancer, remains unclear. 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[DOI] [PubMed] [Google Scholar] Articles from Clinical Cases in Mineral and Bone Metabolism are provided here courtesy of CIC Edizioni Internazionali ACTIONS View on publisher site PDF (519.8 KB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Summary Etiology and pathogenesis Diagnosis of GD Epidemiology Clinical classification Bone manifestations in GD Evaluations and monitoring for type 1 GD Disease management Conclusion References Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
12349	https://physicsaholics.com/assets/pdf/uploads1636703622Solution%20DPP-9-Kinematics-%20Relative%20motion%20(Rain-Man%20problems).pdf	PHYSICSLIVE Use code PHYSICSLIVE to get 10% OFF on Unacademy PLUS. For Video Solution of this DPP, Click on below link Solution on Website:-Solution on YouTube:- Physics DPP DPP-9 Relative motion (Rain-Man problems) By Physicsaholics Team Join Unacademy PLUS Referral Code : Physicslive Q) A man standing on a road hold his umbrella at 30° with the vertical to keep the rain away. He throws the umbrella and starts running at 10 km/hr. He finds that raindrops are hitting his head vertically, the speed of raindrops with respect to the road will be: (a) 10 𝑘𝑚/ℎ𝑟 (b) 20 𝑘𝑚/ℎ𝑟 (c) 30 𝑘𝑚/ℎ𝑟 (d) 40 𝑘𝑚/ℎ𝑟 Ans. b Join Unacademy PLUS Referral Code : Physicslive Q) Rain is falling with a speed of 12 2m/s at angle of 450 with the vertical line. A man in glider going at a speed of u at an angle of 370 with respect to the ground. Find the speed of the glider so that rain appears to him falling vertically. Consider the motion of the glider and rain drops in the same vertical plane: (a) 15 𝑚/𝑠 (b) 30 𝑚/𝑠 (c) 10 𝑚/𝑠 (d) 20 𝑚/𝑠 Ans. a Join Unacademy PLUS Referral Code : Physicslive Q) A man is walking due east at the rate of 2 km/h. The rain appears to him to come down vertically at the rate of 2 km/h. The actual velocity and angle through which rain is falling with the vertical respectively are (a) 2 2 𝑘𝑚/ℎ, 450 (b) 1 2 𝑘𝑚/ℎ, 300 (c) 2 𝑘𝑚/ℎ, 00 (d) 2 𝑘𝑚/ℎ, 900 Ans. a Join Unacademy PLUS Referral Code : Physicslive Q) When a man moves down the inclined plane with a constant speed 5m/s which makes an angle of 370 with the horizontal, he finds that the rain is falling vertically downward. When he moves up the same inclined plane with the same speed, he finds that the rain makes an angle 𝜃= tan−1 7 8 with the horizontal. The speed of the rain is: (a) 116 𝑚/𝑠 (b) 32 𝑚/𝑠 (c) 5 𝑚/𝑠 (d) 73 𝑚/𝑠 Ans. b Join Unacademy PLUS Referral Code : Physicslive Q) A stationary person observes that rain is falling vertically down at 30km/hr. A cyclist is moving up on an inclined plane making an angle 300 with horizontal at 10km/hr. In what direction should the cyclist hold his umbrella to prevent himself from rain? (a) At an angle tan−1 2 7 with the vertical. (b) At an angle tan−1 3 7 with the horizontal (c) At an angle tan−1 3 7 with the vertical (d) At an angle tan−1 2 7 with the horizontal Ans. c Join Unacademy PLUS Referral Code : Physicslive Q) Rain is falling vertically downwards with a speed of 4 km/h. A girl moves on a straight road with a velocity of 3 km/h. The apparent velocity of rain with respect to the girl is: (a) 3 km/h (b) 4 km/h (c) 5 km/h (d) 7 km/h Ans. c Join Unacademy PLUS Referral Code : Physicslive Q) A man is cycling at 4 m/s On a horizontal rod. To him, rain appears to fall at 300 from vertical. If he doubles his velocity, rain appears to fall at 600 to vertical. Find the velocity of the rain: (a) 4 m/s (b) 5 m/s (c) 6 m/s (d) 4 3 m/s Ans. a 𝑉 𝑅= 4 𝑚/𝑠 Join Unacademy PLUS Referral Code : Physicslive Q) A man running on a horizontal road at 8 km/h finds the rain falling vertically. He increases his speed to 12 km/h and finds that the drops make angle 300 with the vertical. Angle of velocity of rain with vertical is: (a) tan−1 2 3 (b) tan−1 3 2 (c) tan−1 5 3 (d) cos−1 3 2 Ans. a Join Unacademy PLUS Referral Code : Physicslive Q) A man holds an umbrella at 300 with the vertical to keep himself dry. Then he runs at a speed of 10 m/s, and find the raindrops to be hitting vertically. Study the following statements and find the correct options: (1) Velocity of rain w.r.t. Earth is 20 m/s (2) Velocity of rain w.r.t. man is 10 3 m/s (3) Velocity of rain w.r.t. Earth is 30 m/s (4) Velocity of rain w.r.t. man is 10 2 m/s (a) Statement (2) and (3) are correct. (b) Statement (1) and (2) are correct. (c) Statement (3) and (4) are correct. (d) Statement (2) and (4) are correct. Ans. b Join Unacademy PLUS Referral Code : Physicslive Q) The path of one projectile as seen from another projectile is a: (a) Straight line (b) Parabola (c) Hyperbola (d) Circle Ans. a For Video Solution of this DPP, Click on below link Solution on Website:-Solution on YouTube:-
12350	https://www.quora.com/What-is-the-molar-mass-of-NaOH	Something went wrong. Wait a moment and try again. Molar Weight Sodium Hydroxide Inorganic Compounds Chemical Calculations Chemical Substances Elements and Compounds Chemistry Formulae 5 What is the molar mass of NaOH? Assistant Bot · 1y To calculate the molar mass of sodium hydroxide (NaOH), you need to sum the molar masses of its constituent elements: Sodium (Na): approximately 22.99 g/mol Oxygen (O): approximately 16.00 g/mol Hydrogen (H): approximately 1.01 g/mol Now, add these together: Molar mass of NaOH=Na+O+H=22.99g/mol+16.00g/mol+1.01g/mol=40.00g/mol Thus, the molar mass of NaOH is 40.00 g/mol. Related questions What is the molecular mass of NaOH? What is the molar mass of sodium hydroxide? What is the equivalent mass of NaOH? How would you prepare 500ml of approximately 0.20M NaOH using solid NaOH (molar mass is 40g/mol)? What is the mass of NaOH required to prepare 500ml of a .5 mole molar solution? Aryan Sarkar Studied Physics, Chemistry, and Mathematics (science grouping) & Computer Science (+2 Level) at Gyan Ganga Education Academy, Raipur (Graduated 2018) · 7y Originally Answered: What is the molecular mass of NaOH? · Let’s simplify the question first. NaOH : The formula clearly indicates that it constitutes of one atom each of Sodium(Na), Oxygen (O) and Hydrogen (H). Now you need to know the atomic weights of the above mentioned elements. Here they are: Sodium - 23 Oxygen - 16 Hydrogen - 1 So molecular weight of NaOH = 23 + 16+ 1 =40 Swomya Behera B.Tech from College of Engineering and Technology, Bhubaneswar (Graduated 2022) · 7y Common dude! Quora is a platform to exchange answers to those questions which have an impact on us for idk like days maybe but, asking homework questions wouldn't help. Just sum up the individual molar masses of all the elements (i.e Na= 23gms/mol , O=16gms/mol, H=1 gm/mol). The final answer becomes 40 gms/mol. Neerav Mullur Lived in Ontario, Canada · 8y Originally Answered: What is the molar mass of sodium hydroxide? · Molar mass is the mass of 1 mole of a substance, so we can add the masses of 1 mole of sodium, one mole of hydrogen and one mole of oxygen. 22.99+16.00+1.01=40 So, the molar mass of NaOH is 40 g/mol Related questions How is the 0.1 molar solution of NaOH prepared? How do I prepare a 1 molar solution of NaOH in 300ml water? How do you calculate the molarity of NaOH? What is 1 molar NaOH? How do I prepare 1 molar of NaOH? Bibhash Mallik BBA from Purbanchal University (Graduated 2021) · Author has 344 answers and 2.9M answer views · 3y We know that the formula of Sodium Hydroxide is NaOH. The compound contains one sodium atom, one oxygen atom and one hydrogen atom. Solution We know the atomic masses of all the atoms Na = 22.989 g/mol O = 15.999 g/mol H = 1.008 g/mol On Adding all the molar masses of what composes sodium hydroxide (NaOH) 22.989 g/mol + 15.999 g/mol + 1.008 g/mol= 39.996 g/ mol Salomon Abouganem Turned left in 2014, still going. · 10y Originally Answered: What is the molar mass of sodium hydroxide? · Very simple. Add all the molar masses of what composes sodium hydroxide (NaOH) Na x 1 = 22.989 g/mol O x 1 = 15.999 g/mol H x 1 = 1.008 g/mol Total molar mass of sodium hydroxide is 39.996 g/mol. Stuti Mehta 6y Originally Answered: What is the molecular mass of NaOH? · NaOH is constituted of Sodium ( Na ) , Oxygen (O), and Hydrogen (H). To find the molecular mass we need to add the atomic masses of all three elements So, Na = 23 O = 16 H= 1 Therefore the molecular mass will be 23 + 16 +1 = 40u Adithya Narayanan Rajesh 6y Mass of Na=23u Mass of O=16u Mass of H=1u Molar mass of NaOH=23u+16u+1u=40u Sally Urquhart Studied Chemistry & Mathematics (Graduated 1968) · 7y To find the mass of any compound, add up the masses of each element’s contribution to the overall mass. Rounding to the nearest whole number in this case 1 Na is 23g/mole, 1 O is 16 g/mole, and 1 H is 1 g/mol for a total of 40 g/mole for NaOH. चौधरी Nishant बालियान Software Engineer at Java (programming language) (2019–present) · 5y The atomic mass of Na is 23. the atomic mass of O is 16. The atomic mass of H is 1. NaOH= 123+116+11 =40 Gram Rohit Kodag Lives in B.K . Kangrali , Belagavi , Karnataka - 590010 · Author has 87 answers and 522.6K answer views · 7y Originally Answered: What is the molecular mass of NaOH? · Molecular mass : The sum of atoms of all the elements present in a compound. Mass of , Na - 23u O - 16u H - 1u NaOH = 23u + 16u + 1u = 40u Ravi Shankar Meghwar 5 years chemistry learning not teaching... · 10mo 40 g/mol Molar mass of sodium…23 g/mol Molar mass of Oxygen…16 g/mol Molar mass of hydrogen.1g/mol Molar mass of NaOH….+________ Naidu Mittireddy 6y The molecular molar mass consists of atomic masses of each atom . Sodium hydroxide is NaOH.its molar mass involves:mass of Na is-22.9899g/mole,mass of oxygen is 15.999g/mole and mass of hydrogen is 1.0079g/mole you add all these you get 39.996g/. Related questions What is the molecular mass of NaOH? What is the molar mass of sodium hydroxide? What is the equivalent mass of NaOH? How would you prepare 500ml of approximately 0.20M NaOH using solid NaOH (molar mass is 40g/mol)? What is the mass of NaOH required to prepare 500ml of a .5 mole molar solution? How is the 0.1 molar solution of NaOH prepared? How do I prepare a 1 molar solution of NaOH in 300ml water? How do you calculate the molarity of NaOH? What is 1 molar NaOH? How do I prepare 1 molar of NaOH? How will you prepare 2dm3 of 0.25 molar mass of NaOH solution (Na=23, O=16 and H=1)? How do I prepare a 0.1ml solution from 1 molar M stock cylinder NAOH? What is the process for preparing a higher molarity NaOH solution from concentrated NaOH? What is the mass of each of the following substances 2.4 mol of NaOH? What is the molarity of a 15% (m/v) NaOH solution? Related questions What is the molecular mass of NaOH? What is the molar mass of sodium hydroxide? What is the equivalent mass of NaOH? How would you prepare 500ml of approximately 0.20M NaOH using solid NaOH (molar mass is 40g/mol)? What is the mass of NaOH required to prepare 500ml of a .5 mole molar solution? How is the 0.1 molar solution of NaOH prepared? How do I prepare a 1 molar solution of NaOH in 300ml water? How do you calculate the molarity of NaOH? What is 1 molar NaOH? How do I prepare 1 molar of NaOH? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12351	https://math.stackexchange.com/questions/619805/construct-triangle-given-inradius-and-circumradius	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Construct triangle given inradius and circumradius Ask Question Asked Modified 11 years, 9 months ago Viewed 887 times 4 $\begingroup$ If we know the inradius $r$ of a triangle and the circumradius $R$ we can find out the distance between the incircle $I$ and the circumcircle $O$: $OI^2 = R^2-2Rr$. Therefore we can draw the incircle and the circumcircle, and their relative position is fixed. Making some drawings suggests that we can construct $ABC$ in a unique way starting from $R,r$. Is there only one triangle $ABC$ (up to isometry) which has inradius $r$ and circumradius $R$? If yes, how do we draw it? geometry triangles geometric-construction Share asked Dec 27, 2013 at 18:30 Beni BogoselBeni Bogosel 24k77 gold badges7272 silver badges135135 bronze badges $\endgroup$ 1 $\begingroup$ $$\frac{a+b+c}{abc}=\frac{x+y+z}{xyz}=\frac1{2rR}\qquad,\qquad{a,b,c}\neq{x,y,z}$$ $\endgroup$ Lucian – Lucian 2013-12-27 19:06:40 +00:00 Commented Dec 27, 2013 at 19:06 Add a comment \| 3 Answers 3 Reset to default 7 $\begingroup$ Here is a visual answer to my question (constructed painfully with Geogebra :) ): Share answered Dec 27, 2013 at 19:31 Beni BogoselBeni Bogosel 24k77 gold badges7272 silver badges135135 bronze badges $\endgroup$ 2 $\begingroup$ @Bogosei, could you please tell me how to construct it? $\endgroup$ user655800 – user655800 2019-10-08 15:11:21 +00:00 Commented Oct 8, 2019 at 15:11 $\begingroup$ How to construct the figure? Just construct two circles with $2r $\endgroup$ Beni Bogosel – Beni Bogosel 2019-10-09 21:28:34 +00:00 Commented Oct 9, 2019 at 21:28 Add a comment \| 4 $\begingroup$ The question you asked is the special case of so- called Poncelet's porism. It says, that if you have two conics on the plane and one can find a $n-$gon for which one conic is inscribed and the other one is circumscribed in it, then one can find infinitely many such $n-$ gons. See this wikipedia article for the details Poncelet's porism Share answered Dec 27, 2013 at 20:23 leshikleshik 4,99011 gold badge1717 silver badges2121 bronze badges $\endgroup$ Add a comment \| 3 $\begingroup$ This can't be true, since there is a three-dimensional space of triangles up to isometry (parametrized by their side-lengths, which only need to satisfy the triangle inequality), and you only give me two parameters. A better question is then: what is the family of triangles with the same in- and out- radii. Share answered Dec 27, 2013 at 18:33 Igor RivinIgor Rivin 26.4k11 gold badge2020 silver badges4040 bronze badges $\endgroup$ 3 $\begingroup$ I think I've also found a visual counterexample. $\endgroup$ Beni Bogosel – Beni Bogosel 2013-12-27 18:43:10 +00:00 Commented Dec 27, 2013 at 18:43 $\begingroup$ @BeniBogosel On the other hand, if the two centers coincide (so $R=2r$) do you get anything but the equilateral triangle? $\endgroup$ Igor Rivin – Igor Rivin 2013-12-27 18:46:52 +00:00 Commented Dec 27, 2013 at 18:46 $\begingroup$ A late answer: if $R=2r$ you can prove that the triangle is equilateral, so it is unique. $\endgroup$ Beni Bogosel – Beni Bogosel 2019-10-09 21:26:35 +00:00 Commented Oct 9, 2019 at 21:26 Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions geometry triangles geometric-construction See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Linked 1 maximum and minimum perimeter of triangle for fixed circumradius and inradius Related Why is the inradius of any triangle at most half its circumradius? 1 A problem on a triangle's inradius and circumradius . 4 Area of a triangle given its circumradius and inradius. 1 The circumradius of an isosceles triangle ABC is four times as that of inradius and A=B condition 1 Let $G, S, I$ be respectively centroid, circumcentre, incentre of triangle $ABC$. If $R, r$ are circumradius and inradius respectively then... 4 Prove that the incircle is the smallest circle which passes through the three sides of a triangle Construct a triangle given base, inradius and exradius Problem about ratio between circumradius and inradius 1 Relationship between inradius, circumradius, and semi-perimeter of a triangle. Hot Network Questions Why do universities push for high impact journal publications? Change default Firefox open file directory Can a state ever, under any circumstance, execute an ICC arrest warrant in international waters? Can a cleric gain the intended benefit from the Extra Spell feat? How can the problem of a warlock with two spell slots be solved? Clinical-tone story about Earth making people violent Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth? Are there any alternatives to electricity that work/behave in a similar way? 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12352	https://ocw.mit.edu/courses/2-29-numerical-marine-hydrodynamics-13-024-spring-2003/pages/lecture-notes/	Lecture Notes \| Numerical Marine Hydrodynamics (13.024) \| Mechanical Engineering \| MIT OpenCourseWare Browse Course Material Syllabus Calendar Lecture Notes Assignments Course Info Instructor Prof. Jerome Milgram Departments Mechanical Engineering As Taught In Spring 2003 Level Graduate Topics Engineering Mechanical Engineering Fluid Mechanics Ocean Engineering Hydrodynamics Systems Engineering Numerical Simulation Mathematics Differential Equations Mathematical Analysis Learning Resource Types notes Lecture Notes assignment Problem Sets assignment Programming Assignments Download Course menu search Give Now About OCW Help & Faqs Contact Us searchGIVE NOWabout ocwhelp & faqscontact us 2.29 \| Spring 2003 \| Graduate Numerical Marine Hydrodynamics (13.024) Menu More Info Syllabus Calendar Lecture Notes Assignments Lecture Notes All of the lecture notes may be downloaded as a single file (PDF - 5.6 MB). Week 1: Incompressible Fluid Mechanics Background (PDF) Particle Image Velocimetry Averaged Navier-Stokes Equations The Pressure Equation for an Incompressible Fluid The Vorticity Equation Inviscid Fluid Mechanics, Euler’s Equation Bernoulli Theorems for Inviscid Flow Vorticity Dynamics and Kelvin’s Circulation Theorem Potential Flows and Mostly Potential Flows Green Functions, Green’s Theorem and Boundary Integral Equations Example of Method Solution Interpretation of Boundary Integral Equation in Terms of Source and Dipole Layers The Kelvin-Neumann Problem The Kelvin-Neumann Green Function Source Only and Dipole Only Distributions Green’s Theorem in Two Dimensions Force on a Vortex Lift on a Vortex in a Cylinder Example: Design of 2D Airfoil Mean Line Using Dipoles and Vortices Week 2: Some Useful Results from Calculus (PDF) Derivation of Gauss’ Theorem Example of Use of Gauss Theorem: Froude Krylov Surge Force on a Ship The Transport Theorem Pressure Forces and Moments on an Object Week 3: An Application Using Complex Numbers (PDF) Example of Programming with Complex Numbers: Conformal Mapping of a Circle into an Airfoil Procedure to Compute Pressure Coefficient Week 4: Root Finding (PDF) Bisection Method Newton’s Method for Finding Roots of y(x) Review of Matrix Algebra Determinant of a Matrix Transpose of a Matrix, Calculating the Inverse of a Matrix Matrix Norms The Condition Number of a Matrix Gaussian Elimination Gaussian Elimination Operation Count for n Equations Errors in Numerical Solutions of Sets of Linear Equations, Scaled Partial Pivoting Rule Solution of Linear Equations by LU Decomposition Procedure for Factorization of A Week 5:Curve Fitting and Interpolation (PDF) Polynomial Approximation to a Function Lagrange Polynomials Example Week 6: Numerical Differentiation (PDF) Finite Difference Differentiation Week 7: Numerical Integration (PDF) Trapezoidal Rule Trapezoidal Rule Error Usual Trapezoidal Rule Numerical Integration Simpson’s Rule Week 8: Numerical Integration of Differential Equations (PDF) Euler’s Method, Modified Euler’s Method Fourth Order Runge Kutta Method Predictor-Corrector Methods Higher Order Differential Equations Review and Extension Week 9: Some Examples and Numerical Errors (PDF) Types of Numerical Hydrodynamics Problems, Example of Function Evaluation Example of Solution of Ordinary Differential Equation Example of Solution of Partial Differential Equation Cylindrical Coordinates Example of Discretized Integral Equation Stability Week 10: Panel Methods (PDF) Boundary Condition of Perturbation Potential, Three Dimensional Flows Interpretation of Green’s Theorem Arrangement of the Integral Equation Numerical Form of the Integral Equation Making the Numerical Equations Solution Steps Two Dimensional Panel Methods Numerical Form of the Two Dimensional Integral Equation Situations with the Generation of Lift Computation of Pressures and Forces Week 11: Boundary Layers (PDF - 1.3 MB) Two-Dimensional Steady Boundary Layer Equations Boundary Layer Parameters Mass Fluxes Example of Solution of Momentum Integral BL Equation Calculation of Turbulent Boundary Layer When Pressure Distribution is Known Laminar Closure Relations, Turbulent Closure Relations Sea Waves Example of Simulation Sea Spectra Fourier Transforms Computational FFT and IFFT of Real Numbers Simulation of Random Waves Review of Fourier Transforms, Inverse Fourier Transforms, FFT’s IFFT’s and Wave Simulation Generating Gaussian Random Numbers (Courtesy of Everett F. Carter Jr.) Wave Statistics Results from Theory Definition of a Gaussian Random Process Average Amplitude of the 1/n’th Highest Waves Extreme Waves Stiff Equations Dynamics of Horizontal Shallow Sag Cables in Water Week 12: Oscillating Rigid Objects (PDF) Potentials and Boundary Conditions Strip Theory Boundary Conditions on Hull Sway, Roll and Yaw Equations Simulations of Ship Motions in Random Seas Added Resistance and Drift Forces Gerritsma and Beukelman Theory for Added Resistance Nonlinear Wave Force Calculations Vertical Sea Loads Appendix: Further Material on Panel Methods and Strip Theory (Courtesy of Alexis Mantzaris) (PDF - 1.0 MB) Course Info Instructor Prof. Jerome Milgram Departments Mechanical Engineering As Taught In Spring 2003 Level Graduate Topics Engineering Mechanical Engineering Fluid Mechanics Ocean Engineering Hydrodynamics Systems Engineering Numerical Simulation Mathematics Differential Equations Mathematical Analysis Learning Resource Types notes Lecture Notes assignment Problem Sets assignment Programming Assignments Download Course Over 2,500 courses & materials Freely sharing knowledge with learners and educators around the world. 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12353	https://math.stackexchange.com/questions/3736580/show-that-for-n3-there-is-always-a-2-regular-graph-on-n-vertices-for-wh	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Show that for $n>3$, there is always a $2$-regular graph on $n$ vertices. For what values of $n>4$ will there be a 3-regular graph on n vertices? Ask Question Asked Modified 5 years, 3 months ago Viewed 312 times 0 $\begingroup$ Show that for $n>3$, there is always a $2$-regular graph on $n$ vertices. For what values of $n>4$ will there be a 3-regular graph on n vertices? I think this question is slightly out of my control. Can you please help me out with this question... For part two what I think is yes by handshaking I will exclude all the odd vertices as $3(2n+1)$ is not even number. So what should be the answer? All even number of vertices? Does that make sense? And for part 1 it is obviously true but how can I proceed to the answer? Thanks. graph-theory Share edited Jun 27, 2020 at 17:42 amWhy 211k198198 gold badges283283 silver badges505505 bronze badges asked Jun 27, 2020 at 14:13 Karan SinghKaran Singh 11999 bronze badges $\endgroup$ 4 1 $\begingroup$ For part 1, any cycle of length $n$ does the work $\endgroup$ DodoDuQuercy – DodoDuQuercy 2020-06-27 14:21:45 +00:00 Commented Jun 27, 2020 at 14:21 $\begingroup$ I know, like it is pretty obvious but how can I prove this! $\endgroup$ Karan Singh – Karan Singh 2020-06-27 14:22:36 +00:00 Commented Jun 27, 2020 at 14:22 1 $\begingroup$ I don't think there is much more to prove for part 1. If your $n$ vertices are $v_1, \dots, v_n$ you connect $v_i$ and $v_{i+1}$ for all $1\leq i \leq n-1$, as well as $v_1$ and $v_n$, and you just see that every vertex has degree $2$ $\endgroup$ DodoDuQuercy – DodoDuQuercy 2020-06-27 14:25:04 +00:00 Commented Jun 27, 2020 at 14:25 1 $\begingroup$ Okay makes sense $\endgroup$ Karan Singh – Karan Singh 2020-06-27 14:26:28 +00:00 Commented Jun 27, 2020 at 14:26 Add a comment \| 2 Answers 2 Reset to default 2 $\begingroup$ For part 1, consider the cycle of length $n$, so the statement is true. For part 2, the number of edges is $3n/2$ so the number of vertices must be even. It must be also at least 6 by assumption. For $n=4$ the square with two diagonals is the answer. For every even $n=2m\ge 6$ consider the dihedral group $D_m$. It is generated by 3 involutions. The Cayley graph corresponding to this generating set is 3-regular with $n$ verticed. Share edited Jun 27, 2020 at 16:51 answered Jun 27, 2020 at 14:24 markvsmarkvs 20k22 gold badges2121 silver badges3434 bronze badges $\endgroup$ 4 $\begingroup$ So basically the solution of part b will be same as I indicated in question? $\endgroup$ Karan Singh – Karan Singh 2020-06-27 14:28:51 +00:00 Commented Jun 27, 2020 at 14:28 $\begingroup$ Like I haven't studied that dihedral group or probably in my country it is named something else. Summing up, is whatever I said about part two is correct or not? $\endgroup$ Karan Singh – Karan Singh 2020-06-27 14:30:24 +00:00 Commented Jun 27, 2020 at 14:30 $\begingroup$ Thanks! For your time and help $\endgroup$ Karan Singh – Karan Singh 2020-06-27 14:31:11 +00:00 Commented Jun 27, 2020 at 14:31 $\begingroup$ Got it, thanks again $\endgroup$ Karan Singh – Karan Singh 2020-06-27 14:38:38 +00:00 Commented Jun 27, 2020 at 14:38 Add a comment \| 2 $\begingroup$ I don't know if this is what JCAA is referring to but for part 2, if you have an even number $2n$ ($n \geq 3$) of vertices $v_1, \dots, v_n$ and $u_1, \dots, u_n$, you can just consider the two cycles $v_1, \dots, v_n$ and $u_1, \dots, u_n$ and add the edges ${v_i,u_i}$ for all $1 \leq i \leq n$. The resulting graph will be 3-regular. As you mentioned, it obviously fails for odd number of vertices because the sum of degrees in a graph is always even since it is twice the number of edges. Share answered Jun 27, 2020 at 14:31 DodoDuQuercyDodoDuQuercy 1,79466 silver badges99 bronze badges $\endgroup$ 5 $\begingroup$ Sorry, but I didn't get your point. Like how can we consider two cycles in here? Isn't it mean that some of the vertices will have degree 2 while others will have degree 3? $\endgroup$ Karan Singh – Karan Singh 2020-06-27 14:35:39 +00:00 Commented Jun 27, 2020 at 14:35 1 $\begingroup$ You just split your $2n$ vertices into two sets of $n$ vertices $V = {v_1, \dots , v_n}$ and $U = {u_1, \dots, u_n}$, which you both turn separately into cycles of length $n$. Then you connect $v_i$ and $u_i$ for all $1 \leq i \leq n$. A vertex of the set $V$ will have two neighbors in $V$, and one neighbor in $U$. Similarly, a vertex of the set $U$ will have two neighbors in $U$, and one neighbor in $V$. The graph will thus be 3-regular. It really helps drawing the graph if you cannot clearly see it ;) $\endgroup$ DodoDuQuercy – DodoDuQuercy 2020-06-27 14:41:10 +00:00 Commented Jun 27, 2020 at 14:41 $\begingroup$ Never mind, I got your point. It makes sense too. $\endgroup$ Karan Singh – Karan Singh 2020-06-27 14:41:21 +00:00 Commented Jun 27, 2020 at 14:41 1 $\begingroup$ I really appreciate your help and work, thanks $\endgroup$ Karan Singh – Karan Singh 2020-06-27 14:41:43 +00:00 Commented Jun 27, 2020 at 14:41 $\begingroup$ It is also the Cayley graph of dihedral group but with respect to the two generators $a, b$: $a^m=b^2=1, bab= a^{-1} $. $\endgroup$ markvs – markvs 2020-06-27 14:43:29 +00:00 Commented Jun 27, 2020 at 14:43 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 graph-theory 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 Related 2 Wikipedia article about T-joins 2 a regular graph has 15 edges, how many vertices does it have? 2 Show that there are always two teams who played exactly the same number of games. 0 Prove $k$-regular graph with odd number of vertices has $\chi'(G) \geq k+1$ $k$-regular $(k-2)$-edge connected graph with even number of vertices and no perfect matching for any even $k \gt 3$ 2 Prove that, when k is odd, a k-regular graph must have an even number of vertices. 0 Determine all pairs (k, n) such that there exists a k-regular graph on n vertices 0 Formalizing graph theory proof that for every even number $n \geq 4$ there exists a graph with $n$ vertices, all of which have degree 3 How can I prove that there is not a 5-regular planar graph with 14 vertices. 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12354	https://www.mdpi.com/2077-0383/14/17/6160	Previous Article in Journal Clinical Predictors of Underlying Histologic Activity in Patients with Lupus Nephritis: A Focus on Urinary Soluble CD163 All articles published by MDPI are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited. For more information, please refer to Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications. 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Georgiana Barry, P. Padilla, V. on PubMed Dutta, S. Buciuc, A. Georgiana Barry, P. Padilla, V. /ajax/scifeed/subscribe Article Views 650 Table of Contents Abstract Introduction Materials and Methods Results Limitations Conclusions Author Contributions Funding Acknowledgments Conflicts of Interest References Altmetric share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles Need Help? Support Find support for a specific problem in the support section of our website. Get Support Feedback Please let us know what you think of our products and services. Give Feedback Information Visit our dedicated information section to learn more about MDPI. Get Information clear JSmol Viewer clear first_page Download PDF settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open AccessReview A Narrative Review on Toxidromes in the Psychiatric Population: Implications for Overdose Prevention by Sanjukta Dutta Sanjukta Dutta SciProfilesScilitPreprints.orgGoogle Scholar 1, Adela Georgiana Buciuc Adela Georgiana Buciuc SciProfilesScilitPreprints.orgGoogle Scholar 2,, Patrick Barry Patrick Barry SciProfilesScilitPreprints.orgGoogle Scholar 2 and Vanessa Padilla Vanessa Padilla SciProfilesScilitPreprints.orgGoogle Scholar 1 1 Miller School of Medicine, University of Miami, Miami, FL 33136, USA 2 Jackson Health System, University of Miami, Miami, FL 33136, USA Author to whom correspondence should be addressed. J. Clin. Med. 2025, 14(17), 6160; Submission received: 16 July 2025 / Revised: 18 August 2025 / Accepted: 29 August 2025 / Published: 31 August 2025 (This article belongs to the Section Mental Health) Download keyboard_arrow_down Download PDF Download PDF with Cover Download XML Download Epub Browse Figure Versions Notes Abstract Individuals with severe mental illness face a substantially higher risk of suicide compared with the general population, with drug overdose representing one of the most common and potentially lethal methods. This narrative review explores toxidromes frequently encountered in psychiatric populations, such as opioid, anticholinergic, and serotonergic toxicity, highlighting the clinical presentation in intentional overdose. Emphasis is placed on clinical recognition, antidote-based treatment, and systems-level strategies for the prevention of lethal overdose. We conducted a comprehensive literature search of PubMed, Google Scholar, and Web of Science for English-language articles using combinations of the following keywords: mental disorders; persons with psychiatric disorders; drug overdose; poisoning; serotonin syndrome; neuroleptic malignant syndrome; anticholinergic agents/poisoning; cholinergic antagonists/poisoning; psychotropic drugs/adverse effects; substance-related disorders; drug-related side effects and adverse reactions; polypharmacy; suicide, attempted; emergency service, hospital. By embedding toxidrome awareness into routine emergency and psychiatric practice, we aim to expedite treatment and improve patient outcomes. Keywords: suicide; mental illness; psychiatry; public health; poisoning; drug overdose; anticholinergic poisoning; serotonin syndrome; neuroleptic malignant syndrome; psychotropic drugs 1. Introduction Toxidromes are defined as “constellations of toxic effects comprising a set of clinical fingerprints for a group of toxic chemicals,” according to the U.S. Department of Health and Human Services . They serve as clinical tools to rapidly identify toxic exposures based on characteristic symptom patterns. In both emergency medicine and psychiatric settings, early recognition of these syndromes can enable timely, potentially life-saving interventions, including the administration of antidotes and the initiation of supportive care [1,2,3,4,5]. Suicide is a major contributor to premature mortality worldwide and a leading cause of death among adolescents and young adults [6,7,8]. Methods vary by availability, with self-poisoning among the most common methods utilized globally, alongside hanging and firearms [6,7,9,10]. In high-income countries, poisoning deaths often involve prescribed and over-the-counter medicines, frequently in combination, such as antidepressants, benzodiazepines, opioids, and non-opioid analgesics; in many low- and middle-income countries, pesticide ingestion predominates [6,9,11,12]. In a national, coroner-based study, approximately one-quarter of suicides involved poisoning, and psychotropic and sedative polypharmacy was common in poisoning deaths . In a review of more than 400,000 suicide attempts by drug overdose, opioids were implicated in 47.8% of fatal cases, followed by benzodiazepines and antidepressants . Severe mental illness (SMI) confers a markedly elevated suicide risk and distinct method profiles. A 2024 systematic review and meta-analysis found higher odds of drug overdose among those with major depressive disorder and substantial method differences by diagnosis . For example, higher rates of jumping as a method of suicide were observed in people with schizophrenia or bipolar disorder compared with people without SMI . Among individuals recently discharged from psychiatric inpatient care, fatal poisonings, particularly involving psychotropic medications, accounted for more than half of unnatural deaths, and intentional self-poisoning carried a higher relative risk than violent methods . The National Violent Death Reporting System data also links mental disorders and substance use disorders with an increased likelihood of poisoning as a suicide method . Disproportionate toxidrome risk in psychiatric populations reflects interacting clinical and systemic drivers. These include a higher likelihood of intentional overdose during acute crises; ready access to high-risk medications and complex regimens that raise iatrogenic toxicity; impaired insight, impulsivity, and cognitive deficits that compromise safe use; and discontinuous care—especially at transitions—undermining monitoring, adherence, and rapid intervention [2,3,4,9,10,15,16]. These patterns strain emergency and inpatient services and reveal gaps in mental health care, medication management, and suicide-prevention strategies. Addressing these challenges requires targeted interventions aimed at improving access, ensuring continuity of care, and enhancing pharmacological oversight for this high-risk population. This narrative review examines the intersection of mental illness, overdose risk, and toxidromic presentations. It summarizes the clinical features of common toxidromes encountered in medical settings, outlines diagnostic strategies and antidote-directed treatments, and highlights systems-level approaches for prevention and care coordination. By advancing point-of-care recognition and standardized response pathways in emergency and psychiatric practice, this review aims to enable timelier interventions and reduce preventable morbidity and mortality among people living with mental illness. 2. Materials and Methods A literature search was conducted using PubMed, Google Scholar, and Web of Science databases. The search included English-language articles published between 2005 and May 2025 and used combinations of the following MeSH keywords: mental disorders; persons with psychiatric disorders; drug overdose; poisoning; serotonin syndrome; neuroleptic malignant syndrome; anticholinergic agents/poisoning; cholinergic antagonists/poisoning; psychotropic drugs/adverse effects; substance-related disorders; drug-related side effects and adverse reactions; polypharmacy; suicide, attempted; emergency service, hospital. A total of 259,102 records were found using this search strategy, of which approximately 36,800 records were screened, and 300 full texts were reviewed. Inclusion criteria were studies that addressed adult patients in psychiatric or emergency settings with intentional overdose or clinically defined toxidromes. Eligible evidence included peer-reviewed original studies describing clinical features, diagnostics, management, or outcomes; systematic reviews; and authoritative clinical guidelines or toxicology references from recognized bodies, including the World Health Organization (WHO), American Heart Association (AHA), American Psychiatric Association (APA), and the U.S. Department of Health and Human Services. Exclusion criteria were studies with pediatric-only populations; non-human or in vitro studies; non-English publications; articles lacking clinically relevant toxidrome content; and gray literature (e.g., conference abstracts, preprints, dissertations). Titles and abstracts were screened independently by two reviewers; full texts were assessed and final inclusion decisions reached by consensus, with disagreements adjudicated by a third reviewer. As the included evidence ranged from case reports to clinical guidelines, formal risk-of-bias tools were not applicable; therefore, we opted for a narrative synthesis. No preregistered protocol, quantitative synthesis, or formal risk-of-bias assessment was undertaken. Literature was selected for inclusion based on its relevance to the following domains: the epidemiology of overdose in psychiatric populations, clinical presentations of toxidromes associated with psychotropic and illicit substances, diagnostic frameworks, therapeutic interventions, and systems-level strategies for risk mitigation. Given its non-systematic scope, we do not report PRISMA-style screening counts but instead specify search windows, databases/terms (including MeSH), inclusion/exclusion criteria, and reviewer procedures to ensure transparency. This review aims to provide a high-level synthesis to inform clinical practice and interdisciplinary collaboration. 3. Results 3.1. Epidemiology Individuals with SMI have markedly elevated suicide risk compared with the general population [4,7,9,14,17,18,19,20]. Meta-analyses and large registries estimate pooled suicide rates around 313 per 100,000 person-years in SMI, with particularly high rates of attempted suicide in individuals with major depressive disorder and bipolar disorder . Mood disorders overall confer approximately an 8- to 9-fold higher likelihood of death by suicide compared with individuals without SMI . While females with SMI attempt suicide at higher rates compared with males, males experience a higher rate of death by suicide . Among individuals with schizophrenia, the most severe form of psychiatric illness, suicide is the leading cause of early mortality . U.S. health-system data show adjusted odds of suicide mortality of 15.0 for individuals with schizophrenia spectrum disorders and 13.2 for individuals with bipolar disorder relative to controls, with elevated risk seen across anxiety and depressive disorders as well . Population studies similarly demonstrate 5- to 13-fold higher standardized mortality ratios for suicide across major psychiatric diagnoses, highest for psychotic and bipolar disorders . Although firearms account for the majority of suicide deaths in the general U.S. population, poisoning is disproportionately utilized by individuals with SMI, reflecting vulnerabilities in both access and clinical context [22,23]. Moreover, the proportion of fatal outcomes from suicide attempts in this population is substantially higher than in the general population [17,24,25]. The most salient epidemiologic facts include that suicide accounts for roughly 700,000–800,000 deaths globally each year, with an estimated worldwide mortality rate of approximately 10 per 100,000 [8,9,26,27]. There exists substantial regional variation in these figures, with approximately 80% of deaths occurring in low- and middle-income countries . In the United States, suicide mortality rose approximately 30–35% from 2000 to 2018 and contributed to stagnation in life expectancy; modest declines in 2019–2020 did not occur uniformly across demographic groups . Suicidality extends far beyond deaths: for each suicide completed, there are approximately 20 attempts, translating to ~16 million attempts annually worldwide, and an estimated 160 million people with suicidal thoughts [8,27]. In the United States, about 10.6 million adults (4.3%) report suicidal ideation within the past year, and 1.4 million report suicide attempts in a typical year [8,27]. These data collectively underscore that suicidality is a severe, multifaceted public health crisis requiring population-level and clinical interventions. 3.2. Risk Factors and Clinical Relevance in Psychiatric Populations The American Psychiatric Association (APA) emphasizes stratifying static risk factors (e.g., prior attempts, chronic severe mental illness, and medical comorbidity) and dynamic risk factors (e.g., suicidal intent/plan with accessible lethal means, intoxication, agitation, insomnia, worsening depression, treatment disengagement) when evaluating suicide risk . Overdose and poisoning risk in psychiatric populations is best understood as the product of converging pathways rather than any single precipitant. Large cohort analyses demonstrate that diverse psychiatric diagnoses independently elevate poisoning risk, which further escalates with psychiatric comorbidity, indicating additive pathways of vulnerability [29,30]. Four recurrent mechanisms of vulnerability emerge: state-dependent disinhibition and impaired judgment, pharmacologic complexity, ready access to psychoactive agents, and discontinuities in care. These mechanisms interact with social and structural disadvantages to create recurrent windows of vulnerability during which high-lethality ingestions are more likely to occur and more difficult to detect early. During acute psychiatric crises, heightened impulsivity and compromised judgment are common and strongly associated with self-injury, including overdose [21,31,32,33]. Trait impulsivity facilitates progression from suicidal ideation to attempt, particularly in adolescents and young adults [31,32,33]. Conditions such as major depressive disorder, bipolar disorder, and borderline personality disorder often involve affective lability, hopelessness, and emotional dysregulation, all of which are core risk factors for suicidality [33,34,35,36,37]. Across depressive disorders, cognitive rigidity and dichotomous thinking are consistently linked to suicidal ideation and attempts, in part due to impaired executive functions (set-shifting, cognitive flexibility) and ruminative brooding that narrow perceived options to a single “escape” and amplify hopelessness [36,37]. These thought patterns can foster the belief that suicide is the only escape from suffering. Furthermore, the presence of anhedonia and cognitive distortions can interfere with a patient’s ability to engage with protective factors like social support systems or treatment adherence. Meta-analytic and cohort findings show anhedonia predicts worse psychosocial and functional outcomes, consistent with reduced engagement with protective factors [38,39]. In borderline personality disorder, suicidal gestures may emerge in response to interpersonal conflict or perceived abandonment, often impulsive in nature, with little premeditation . In psychotic disorders, persecutory delusions, auditory hallucinations, or command hallucinations may directly incite suicidal behavior, particularly when the individual lacks insight into their illness or is not receiving adequate treatment [21,41]. Psychotropic polypharmacy is frequent when treating individuals with SMI, both inpatient and outpatient, with many patients receiving multiple concurrent psychotropics alongside other medications. Large observational datasets and cohort studies show high rates of concurrent use (often ≥3 psychotropics), with antipsychotic and antidepressant combinations particularly common [42,43]. In acute psychiatric settings, a majority of patients receive multiple drugs daily, and over half receive more than one psychotropic simultaneously . Community and national data similarly document substantial class and cross-class polypharmacy in severe and persistent mental illness [44,45]. Overlapping toxicodynamics are a central safety concern. This creates predictable syndromic pathways, as follows: combined opioids, benzodiazepines, gabapentinoids, and alcohol predispose to the opioid toxidrome with respiratory depression; cumulative anticholinergic load from tricyclic antidepressants, paroxetine, low-potency antipsychotics, diphenhydramine, and related agents increases the likelihood of hyperthermic anticholinergic delirium; and combining antidepressants with linezolid, tramadol, dextromethorphan, stimulants, or triptans raises the risk of serotonin syndrome. Narrow therapeutic index agents like lithium and QT-prolonging combinations add additional hazard, particularly around recent dose escalations, medication switches, or when multiple prescribers are involved without thorough reconciliation . Access to means further lowers the threshold for impulsive ingestion . Ready availability of prescribed and over-the-counter medications enables rapid, high-dose exposures before clinical intervention is possible [16,47]. Social and structural determinants such as loneliness, perceived burdensomeness, stigma, unemployment, housing instability, and limited access to care erode engagement and adherence, attenuating the buffering effect of outpatient care [48,49,50,51,52]. These pressures are magnified by systems-level discontinuities, such as limited access to crisis services, inconsistent medication management across settings, and especially during transition periods following ED or psychiatric discharge, when delayed follow-up and unclear clinical ownership are associated with repeated mixed ingestions and late recognition of evolving toxidromes [53,54]. Co-occurring substance use disorders are also prevalent among individuals with SMI and may contribute to both the method and lethality of suicide attempts . Substances such as alcohol, opioids, synthetic cannabinoids, and stimulants not only have the ability to induce psychotic episodes but also exacerbate impulsivity [56,57,58]. These substances may be used to potentiate the effects of prescribed psychotropics, sometimes intentionally as a means of self-harm [59,60,61]. The pharmacologic complexity of these cases increases the risk of toxidrome development and complicates clinical management. Recent data from the National Center for Drug Abuse Statistics (NCDAS) highlight the severity of this intersection: 96,700 drug overdose deaths occurred in one year in the U.S. (from March 2020 to March 2021), with opioids implicated in more than 72% of these fatalities . Among the most lethal substances, synthetic opioids continue to drive a sharp rise in overdose mortality, with the overall national overdose death rate at 216 per 100,000 residents . These patterns highlight the critical need for integrated mental health and substance use interventions to mitigate suicide risk. Beyond clinical and pharmacologic factors, fragmented care systems also contribute to suicide vulnerability. Individuals may face systemic barriers that increase suicide risk, such as limited access to crisis services, poor coordination between providers, medication mismanagement, and delayed interventions during critical times of vulnerability. 3.3. Comparison of Major Toxidromes in Psychiatry Table 1 provides a comparative overview of common toxidromes relevant to psychiatric practice, with emphasis on pathogenesis, clinical features, and evidence-based management strategies. It is intended as a practical reference for clinicians. Table 1. Overview of common toxidromes in psychiatry. Anticholinergic Toxidrome The anticholinergic toxidrome is a well-characterized clinical syndrome resulting from inhibition of muscarinic acetylcholine receptors. It commonly arises after an overdose or from drug interactions involving medications with antimuscarinic properties, which block the ability of acetylcholine to bind to muscarinic receptors. In psychiatric practice, this toxidrome is particularly relevant due to the frequent use of tricyclic antidepressants (TCAs), antipsychotics, antihistamines, and antiparkinsonian agents, all of which can cause central and peripheral anticholinergic effects [2,3,16,62,65]. Common agents associated with this syndrome include atropine, diphenhydramine, scopolamine, hyoscyamine, amitriptyline, phenothiazines, and antiparkinsonian medications, including benztropine and trihexyphenidyl [16,62,65]. Clinical features of this syndrome occur due to widespread inhibition of parasympathetic nervous system activity. Patients may present with agitation, visual and auditory hallucinations, stereotyped picking or grabbing movements, mydriasis, tachycardia, dry mucous membranes, warm and flushed skin, urinary retention, decreased bowel sounds, and hyperthermia [3,4,62,65]. Severe toxicity may result in delirium, seizures, or coma [3,4,62,65]. The widely cited mnemonic “mad as a hatter, blind as a bat, hot as a hare, dry as a bone, red as a beet” is clinically useful in recognizing the anticholinergic toxidrome . Diagnosis of anticholinergic toxicity relies on clinical recognition of its constellation of findings, supplemented by a focused medication history and exclusion of other toxidromes with overlapping features. Management of anticholinergic toxicity is largely supportive, with agitation and seizures typically treated using benzodiazepines [2,3,4,65]. In cases of moderate to severe central toxicity manifesting as delirium, physostigmine may be considered, as it is a reversible cholinesterase inhibitor that crosses the blood–brain barrier and can overcome the effects of anticholinergics [64,65]. The use of physostigmine is controversial in cases where TCA overdose is the etiology of anticholinergic toxicity, due to the risk of precipitating asystole or seizures in patients with cardiac conduction abnormalities; therefore, a toxicologist should be consulted prior to its initiation [62,63,64]. Early identification and treatment of anticholinergic toxicity is critical. Individuals living with SMI who present with acute agitation, delirium, or autonomic instability should be evaluated for this toxidrome, particularly in the context of polypharmacy or recent medication changes. 2. : Cholinergic Toxidrome The cholinergic toxidrome results from overstimulation of muscarinic and nicotinic acetylcholine receptors due to inhibition of acetylcholinesterase, the enzyme responsible for breaking down acetylcholine [2,3,4,62,68,69]. This syndrome most commonly occurs following exposure to organophosphates or carbamate insecticides, but can also result from overdose of therapeutic cholinesterase inhibitors used to treat conditions such as Alzheimer’s disease or myasthenia gravis [68,84]. Although less frequently encountered in psychiatric populations, cholinergic toxicity remains a relevant consideration in cases of undifferentiated poisoning or polypharmacy. Common offending agents include organophosphates (e.g., parathion, malathion), carbamates (e.g., carbaryl, physostigmine), and medications such as donepezil, rivastigmine, and galantamine [62,68,69]. Organophosphates phosphorylate acetylcholinesterase through an irreversible bond, thus permanently inactivating the enzyme; in contrast, carbamates form a reversible bond, which inhibits the enzyme temporarily, leading to short-lived toxicity lasting approximately 24 to 48 h [68,69]. Clinical features reflect both muscarinic and nicotinic receptor overstimulation. Muscarinic effects typically manifest as increased secretions and smooth muscle activity, such as excessive salivation, increased tear production, frequent urination, abdominal cramping, diarrhea, vomiting, pupil constriction, narrowing of the airways, and increased bronchial secretions; additional signs include bradycardia, diaphoresis, and hypotension [67,68,69]. Nicotinic effects may include fasciculations of striated muscle, cramping, weakness, and flaccid paralysis, including of the diaphragm in severe cases [67,68,69]. Both muscarinic and nicotinic receptors exist in the brain, thus contributing to combined central nervous system effects such as confusion, seizures, or coma in severe cases [62,69,85]. Most cases of death due to cholinergic toxicity are attributed to respiratory collapse or seizures . The diagnosis of cholinergic toxicity is clinical and should be suspected in any patient presenting with a combination of cholinergic signs, especially when there is a history of exposure to insecticides or cholinesterase inhibitors [67,69]. Miosis, bradycardia, and bronchorrhea are particularly helpful diagnostic clues. In cases with clinical suspicion and no known exposure, a urine test can be used to detect organophosphate metabolites, and an acetylcholinesterase assay may be used for confirmatory testing . In psychiatric patients, the cholinergic toxidrome may mimic other syndromes such as catatonia or serotonin toxicity, making a thorough exposure and medication history essential. Management of cholinergic toxicity involves immediate decontamination of clothing, initiation of supportive care, and administration of antidotes . Atropine, a competitive muscarinic antagonist, is the first-line treatment for cholinergic crisis, as it acts rapidly to prevent acetylcholine from binding at muscarinic receptors [67,68,69]. Although atropine has no effect on nicotinic receptors, it has shown efficacy in preventing lethal respiratory collapse . Glycopyrrolate, another antimuscarinic agent, has limited central nervous system penetration and is not typically used as monotherapy but may serve as an adjunct to atropine in select cases [67,68,69,70]. Oxime therapies such as pralidoxime (2-PAM) may be used to reactivate acetylcholinesterase, even after exposure to organophosphates; these are most effective when administered early, ideally within the first few hours of exposure [62,67,69,70]. Benzodiazepines should be used for seizures or agitation, and airway support is often necessary in cases with bronchorrhea or respiratory muscle paralysis [67,69,70]. Early recognition and aggressive management are critical to improving outcomes in cholinergic toxicity [67,69]. Although rare in psychiatry, clinicians should maintain awareness of this toxidrome when evaluating patients with pinpoint pupils, respiratory distress, or excessive secretions, particularly in the context of polypharmacy or overdose involving cognitive-enhancing agents. 3. : Opioid Toxidrome The opioid toxidrome is a commonly encountered clinical syndrome characterized by central nervous system (CNS) depression, respiratory depression, and miosis [2,62,66,86]. It results from activation of mu-opioid receptors, which leads to suppression of brainstem respiratory centers and decreased sympathetic tone [62,66,87,88]. This toxidrome is especially relevant in psychiatric populations due to the high prevalence of co-occurring opioid use disorder in patients with SMI, as well as the increasing incidence of stimulant substances being adulterated with opioids such as fentanyl [55,89]. Agents involved with this syndrome include prescription opioids such as oxycodone, hydrocodone, hydromorphone, morphine, methadone, and buprenorphine, as well as illicit opioids including heroin and synthetic analogs such as fentanyl and carfentanil [2,62,66,89]. The latter group has contributed substantially to the rising number of fatal overdoses because of their high potency and rapid onset of respiratory suppression [88,89]. Clinical features of opioid toxicity typically include decreased level of consciousness, bradypnea or apnea, miosis (classically described as pinpoint pupils), bradycardia, hypotension, and hypothermia [2,3,62,66,86]. In severe cases, non-cardiogenic pulmonary edema and respiratory failure may occur [88,90]. Although miosis is a characteristic finding, it may be absent in polysubstance overdoses or in patients with hypoxia due to advanced toxicity. Hyporeflexia with flaccid muscle tone may also be observed . Diagnosis of opioid toxicity is clinical and should be considered in any patient presenting with altered mental status and hypoventilation, particularly when there is a known or suspected history of opioid use [62,66]. Although not universally present, the triad of CNS depression, respiratory depression, and miosis is suggestive [2,62,66]. Confirmatory testing, such as a urine drug screen, may support the diagnosis but should not delay treatment. Management begins with airway stabilization and ventilatory support . The opioid antagonist naloxone is the treatment of choice and can be administered intranasally, intramuscularly, or intravenously depending on the clinical situation . Naloxone reverses opioid-induced respiratory and CNS depression by competitively binding to mu receptors [66,88,91]. Naloxone should be titrated to restore adequate respiratory effort rather than full arousal, as rapid reversal may precipitate acute withdrawal, agitation, or cardiovascular instability . In cases involving long-acting opioids or potent synthetic analogs, repeated administrations or continuous infusions of naloxone may be required due to its short half-life . In psychiatric settings, opioid toxicity may present either as an isolated overdose or as part of a polysubstance ingestion. Prompt recognition of this toxidrome is critical, especially in patients with a history of substance use, mood disorders, or prior suicide attempts. 4. : Sedative-Hypnotic Toxidrome The sedative-hypnotic toxidrome is characterized by generalized central nervous system (CNS) depression resulting from the enhancement of gamma-aminobutyric acid (GABA) activity at GABA-A receptors [71,72]. This toxidrome is most commonly associated with benzodiazepines, barbiturates, ethanol, and non-benzodiazepine hypnotics like Z-drugs [2,62,71,72]. In psychiatric populations, sedative-hypnotic agents are frequently prescribed for anxiety, insomnia, or agitation, and are also commonly co-ingested in intentional overdoses [62,71,72]. Common agents include benzodiazepines (e.g., diazepam, lorazepam), barbiturates (e.g., phenobarbital), ethanol, meprobamate, and the so-called “Z-drugs” such as zolpidem and zaleplon. These substances act via similar mechanisms, facilitating chloride influx through GABA-A receptors and leading to neuronal hyperpolarization and CNS depression . The sedative-hypnotic toxidrome is characterized by CNS depression with normal or decreased vital signs. Core findings include somnolence to coma, slurred speech, ataxia/incoordination, unsteady gait, nystagmus, impaired attention and memory (often anterograde amnesia), and respiratory depression with loss of airway reflexes in severe cases [71,72,93,94]. Pupils are typically normal or slightly constricted; skin is usually normal; bowel sounds are normal or decreased [72,93]. Respiratory depression is less common with isolated benzodiazepine overdose but may be pronounced in co-ingestion with other CNS depressants such as opioids or alcohol [71,72]. Complications include trauma from falls and motor vehicle crashes, hypoventilation, hypoxemia/hypercarbia, and coma; lethality rises with co-ingestants (opioids, ethanol) [71,93]. Diagnosis is clinical and should be considered in any patient with unexplained CNS depression, especially in the setting of a known psychiatric history or access to sedative medications. While serum drug levels can aid confirmation and diagnosis, treatment should not be delayed for laboratory results. Management of sedative-hypnotic toxidrome is primarily supportive. The American Heart Association (AHA) emphasizes airway compromise and hypoxemia as mechanisms of death in benzodiazepine overdose and recommends standard airway and ventilatory support as first-line management . Flumazenil may reverse sedation, but carries seizure and dysrhythmia risk in tolerant patients or mixed overdoses [5,73,95]. It should be avoided in patients with suspected co-ingestion of tricyclic antidepressants or in chronic benzodiazepine users due to the risk of precipitating seizures or acute withdrawal [73,95]. Dosing should be cautiously titrated with readiness to treat seizures. The AHA advises that in undifferentiated coma or mixed overdoses, the potential for flumazenil to precipitate refractory withdrawal, seizures (including in patients with epilepsy), and dysrhythmias outweighs its benefit; harms are amplified with co-ingested pro-convulsants (notably cyclic antidepressants) and hypoxia, and reversal may be incomplete in mixed sedative exposures . The AHA emphasizes use only in low-risk scenarios (iatrogenic oversedation, pediatric exploratory ingestion) after excluding benzodiazepine tolerance and dangerous co-ingestants . Patients presenting with sedative-hypnotic toxicity often have overlapping psychiatric or substance use disorders. A high index of suspicion and careful attention to medication history are essential for early identification and appropriate management. 5. : Sympathomimetic Toxidrome The sympathomimetic toxidrome results from excessive stimulation of adrenergic receptors by endogenous or exogenous catecholamines [2,3,4,62,74]. This syndrome is commonly associated with the use or overdose of central nervous system stimulants, such as amphetamines, cocaine, and synthetic cathinones . In psychiatric populations, this toxidrome is particularly relevant due to the prevalence of stimulant misuse, comorbid substance use disorders, and access to prescribed psychostimulants for conditions such as attention-deficit/hyperactivity disorder (ADHD) [74,96]. The most common agents include amphetamines, methamphetamine, cocaine, methylphenidate, synthetic cathinones (“bath salts”), and sympathomimetic decongestants such as pseudoephedrine, phenylephrine, and ephedrine [74,75]. These agents exert their effects by increasing synaptic concentrations of norepinephrine, dopamine, and serotonin through enhanced release, reuptake inhibition, or both. Clinical features are marked by autonomic hyperactivity and CNS excitation. Typical signs include hypertension, tachycardia, hyperthermia, agitation, paranoia, hallucinations, mydriasis, tremor, diaphoresis, and gastrointestinal distress such as nausea, vomiting, or abdominal pain [2,74]. The cardiovascular complications associated with sympathomimetic drugs are diverse and range from hemodynamic perturbations to life-threatening events . The American Heart Association (AHA) emphasizes that acute sympathomimetic toxicity can cause tachycardia, hypertension, hyperthermia-related metabolic derangements, and can precipitate malignant arrhythmias and cardiac arrest . Vasospasm-mediated ischemia can lead to myocardial infarction even with angiographically normal coronaries, and stress (takotsubo) cardiomyopathy has been reported . The AHA also notes dose-related blood pressure and heart rate elevations with therapeutic and over-the-counter sympathomimetics (e.g., pseudoephedrine, ephedrine), which may be clinically consequential in comorbid patients . Chronic or high-dose exposure to amphetamines/methamphetamines and related stimulants has been associated with dilated cardiomyopathy, heart failure, myocardial infarction, aortic dissection, pulmonary edema, arrhythmias, and sudden death [5,74]. Importantly, methamphetamine-associated cardiomyopathy may be at least partially reversible with cessation and guideline-directed therapy, per the AHA Scientific Statement on specific dilated cardiomyopathies . Catecholamine excess induces coronary vasospasm, supply–demand mismatch, mitochondrial dysfunction, oxidative stress, calcium overload, and myocyte death, underpinning ischemia, cardiomyopathy, and arrhythmogenesis . AHA statements on drug-induced arrhythmias also implicate stimulants (amphetamine/methamphetamine/MDMA) in atrial and ventricular arrhythmias via catecholamine excess . ADHD stimulants produce small mean increases in heart rate and blood pressure; rare reports include arrhythmia, nonischemic and takotsubo cardiomyopathy, and sudden death, though large-scale causal risk remains uncertain [97,101]. Diagnosis is clinical and based on the characteristic constellation of signs and symptoms. The presence of severe agitation, dilated pupils, tachycardia, and hyperthermia should prompt consideration of this toxidrome. Unlike anticholinergic toxicity, patients with sympathomimetic toxicity typically present with diaphoresis rather than dry skin. Management of sympathomimetic toxicity is centered on supportive care and sedation. Benzodiazepines are the first-line treatment to reduce agitation, lower sympathetic outflow, and prevent seizures . For patients with severe hypertension or tachyarrhythmias, alpha2-adrenergic agonists (such as dexmedetomidine or clonidine) or direct vasodilators (such as nitroprusside) may be required [5,74]. Beta-blockers should be avoided as monotherapy, particularly nonselective agents, due to the risk of unopposed alpha-adrenergic stimulation and worsening vasoconstriction . In cases of hyperthermia, active cooling measures should be initiated promptly. Sympathomimetic toxicity frequently occurs in the context of polysubstance use, often complicating the clinical picture . In psychiatric patients, prompt recognition and management are essential to prevent cardiovascular, neurologic, and metabolic complications. 6. : Neuroleptic Malignant Syndrome Neuroleptic malignant syndrome (NMS) is a severe, life-threatening toxicity resulting from dopamine receptor antagonism in the basal ganglia, hypothalamus, and brainstem, most commonly associated with overdose or adverse effects from typical and atypical antipsychotics [2,3,4,62,76,77]. Psychiatric patients are particularly vulnerable to NMS due to the widespread use of antipsychotic medications for conditions such as schizophrenia, bipolar disorder, and acute agitation, as well as for off-label indications including mood disorders and behavioral disturbances [76,77]. Common causative agents include first-generation (typical) antipsychotics such as haloperidol, fluphenazine, and chlorpromazine; second-generation (atypical) agents such as risperidone and olanzapine; and phenothiazine derivatives used as antiemetics (e.g., prochlorperazine, promethazine) [76,77]. These agents exert antagonistic effects at dopamine D2 receptors in the central nervous system, leading to characteristic movement-related and autonomic side effects, commonly referred to as extrapyramidal symptoms (EPS), or in severe cases, life-threatening NMS. It has been proposed that preexisting central dopaminergic hypoactivity, coupled with changes to the dopamine system from repeated stress exposure, is the primary mechanism of NMS . The most common clinical features of EPS include acute dystonia, trismus (jaw clenching), oculogyric crisis (involuntary eye movements), akathisia (inner restlessness), parkinsonism, bradykinesia, ataxia, and tremor. In some cases, hypotension may occur due to alpha-adrenergic blockade [76,77]. While EPS may be uncomfortable for patients, they are not typically life-threatening [76,77]. In contrast, NMS is a life-threatening response to dopamine antagonism and is distinguished by lead-pipe rigidity, altered mental status, autonomic dysregulation (including labile blood pressure and hyperthermia), and elevated creatine kinase [76,77]. These effects may be misinterpreted as worsening psychiatric symptoms, which can delay appropriate intervention [16,76,77]. NMS is typically associated with high-potency dopamine antagonists, psychomotor agitation, rapid dose titration, and can also occur after abrupt withdrawal of dopaminergic medications . Diagnosis of NMS is clinical and based on the presence of extrapyramidal signs and autonomic instability in the context of recent or ongoing antipsychotic exposure. Differentiation from other toxidromes, such as serotonin syndrome, is essential and can be guided by neuromuscular findings . Clonus and hyperreflexia, for example, are more characteristic of serotonergic toxicity, whereas rigidity and bradyreflexia suggest dopaminergic blockade . Management of mild to moderate EPS includes discontinuation of the offending agent and administration of anticholinergic medications such as benztropine or diphenhydramine [76,77,78]. In more severe cases, particularly those resembling anticholinergic toxicity, physostigmine may be used cautiously under toxicology consultation [76,77,78]. In the setting of NMS, dopamine agonists such as bromocriptine or amantadine are first-line treatments, along with dantrolene for muscle rigidity and hyperthermia [76,77,78]. Supportive care, including hydration, electrolyte correction, and intensive monitoring, is critical for recovery [16,62,76,77,78]. Recognizing NMS is essential for the timely reversal of symptoms and prevention of life-threatening complications. Psychiatric populations are especially at risk of this toxidrome due to chronic antipsychotic use, polypharmacy, and delays in diagnosis when symptoms mimic psychiatric deterioration [76,77]. 7. : Serotonergic Toxidrome (Serotonin Toxicity) Serotonin syndrome is a potentially life-threatening toxidrome caused by excessive serotonergic activity in the central and peripheral nervous systems. It is most often associated with the use or combination of two or more serotonergic agents, including antidepressants, opioids, and other medications with serotonergic properties [2,3,4,16,62,79,81,82]. Psychiatric patients are particularly at risk due to polypharmacy, treatment-resistant mood disorders, and the use of multiple serotonergic agents across therapeutic classes [81,82]. Common agents implicated in serotonin syndrome include selective serotonin reuptake inhibitors (SSRIs), serotonin–norepinephrine reuptake inhibitors (SNRIs), monoamine oxidase inhibitors (MAOIs), tricyclic antidepressants (TCAs), and atypical antidepressants such as trazodone and mirtazapine [79,80,81,82]. Additional contributors include opioid analgesics like meperidine, tramadol, and fentanyl; serotonergic anxiolytics such as buspirone; over-the-counter agents like dextromethorphan; and some synthetic cannabinoids and herbal products [79,80,81,82]. Clinical features of serotonin syndrome typically develop rapidly after a change in dose or the addition of a new serotonergic agent . Symptoms span three domains: mental status changes (agitation, confusion), autonomic instability (hyperthermia, hypertension, tachycardia, diaphoresis), and neuromuscular abnormalities (tremor, hyperreflexia, inducible or spontaneous clonus, ocular clonus, and myoclonus) [81,82]. Hyperreflexia and clonus are key distinguishing features from NMS and anticholinergic toxicity . Gastrointestinal symptoms such as diarrhea may also be present [80,81,82]. Diagnosis of serotonin syndrome is clinical and often guided by the Hunter Serotonin Toxicity Criteria, which require the presence of a serotonergic agent and specific neuromuscular findings such as inducible or spontaneous clonus in combination with other core features of serotonin toxicity [81,103]. As there are no biomarkers for serotonin syndrome, laboratory testing is nonspecific for diagnosis and is primarily used to assess complications such as rhabdomyolysis, acidosis, or renal injury . Management of mild to moderate serotonin syndrome involves immediate discontinuation of all serotonergic medications and supportive care [79,80,81,82]. Benzodiazepines are first-line agents for sedation, agitation, and seizure prevention [79,80,81,82]. For moderate to severe cases, cyproheptadine, a serotonin 2A antagonist, may be administered orally or via a nasogastric tube [81,82,104]. Alpha2-adrenergic agonists such as dexmedetomidine or clonidine may be used as adjunctive therapies in intensive care settings to manage autonomic symptoms when benzodiazepines alone are insufficient [80,83]. Hyperthermia in serotonin syndrome is believed to occur due to overstimulation of postsynaptic serotonin receptors, resulting in increased central catecholamine release and sympathetic nervous system activation [105,106,107]. As this is a centrally driven thermogenesis, antipyretics such as acetaminophen and ibuprofen are largely ineffective for treating hyperthermia caused by serotonin syndrome [79,80,81,82]. In cases of significant hyperthermia, aggressive external cooling and intensive care monitoring are essential [81,82]. Due to overlapping features with other syndromes, such as NMS, and pre-existing psychiatric symptoms, serotonin syndrome may be underrecognized. Given the widespread use of serotonergic medications, clinicians should maintain a high index of suspicion when encountering patients with agitation, hyperreflexia, or autonomic instability. 3.4. Clinical Approach to Overdoses A systematic and efficient approach is essential when evaluating a patient with suspected overdose . Psychiatric populations often present with diagnostic uncertainty, mixed ingestions, or nonspecific symptoms, thus underscoring the need for rapid identification of toxidromes and prompt initiation of supportive care . A comprehensive clinical assessment typically spans four domains: focused history, physical examination, diagnostic workup, and treatment [2,3,62,108,109]. Please see Figure 1 below for a framework outlining the clinical assessment of overdose. Figure 1. Clinical approach to suspected overdose. Focused History A thorough history is crucial when evaluating potential ingestion and should focus on the “what,” “when,” and “how much” of any substances involved. Key elements include identifying the specific agents and formulations—whether prescription, over-the-counter, or illicit—as well as the time and route of ingestion. It is important to determine the quantity of substance ingested, along with the individual’s intent, such as whether the event was a suicide attempt, recreational use, or therapeutic error. A comprehensive assessment should also consider any pre-existing medical conditions, particularly hepatic, renal, or cardiac disease, which may influence metabolism and toxicity. Additionally, evaluating access to medications within the home or care facility can provide critical context for risk assessment. Collateral information from family, emergency medical services, medication lists, or pill bottles can be invaluable in clarifying the clinical context of an overdose, particularly when patients have impaired insight, delirium, psychosis, or other conditions that limit the reliability of their history . Accordingly, history-gathering should be triangulated from collateral sources available at the point of care. High-yield elements include accounts from emergency personnel or bystanders to establish the time of last known well; pill counts and medication bottles retrieved from the scene; blister packs and pharmacy labels indicating dosing instructions and fill dates; and pharmacy or prescription drug monitoring program data to confirm recent dispenses and identify multiple prescribers. Family members, roommates, or caregivers can provide details about the patient’s baseline mental status, recent stressors, and access to over-the-counter products. Documenting the reliability of each source and explicitly linking collateral data to the exposure timeline both improve diagnostic accuracy and reduce anchoring on a primary psychiatric explanation. Clinicians should also obtain and verify documentation of prior suicide attempts and/or overdose history, as recurrent suicide attempts are associated with a substantially higher risk of severe or fatal overdose . 2. : Physical Examination The physical examination is a critical bedside tool that helps guide the differential diagnosis and often points toward a specific toxidrome. Particular attention should be given to vital signs, as abnormalities such as hyperthermia, bradycardia, or hypotension can offer important diagnostic clues . Neurologic findings, including level of consciousness, presence or absence of seizures, agitation, rigidity, clonus, and hyperreflexia, can help distinguish between various toxic syndromes; the Glasgow Coma Scale (GCS) can be used to identify patients in need of immediate intubation or resuscitation . Pupillary size and reactivity, as well as the moisture of mucous membranes, provide further insight into autonomic involvement, which may be helpful in differentiating opioid toxicity from the sympathomimetic toxidrome . Nystagmus may be present in patients with a variety of toxic ingestions, including phenytoin, phencyclidine, carbamazepine, lithium, ethanol, barbiturates, and sedative-hypnotics . Skin findings such as diaphoresis, dryness, flushing, or cyanosis can be telling features of specific poisonings. Additionally, gastrointestinal signs like altered bowel sounds or urinary retention may be present, and the cardiopulmonary exam may reveal tachyarrhythmias, wheezing, or rales suggestive of pulmonary edema, all of which contribute valuable information toward narrowing the diagnosis. The ability to recognize consistent patterns, such as the combination of miosis, bradypnea, and CNS depression in opioid toxicity or hyperthermia with clonus and agitation in serotonin syndrome, facilitates early presumptive diagnosis and intervention . 3. : Diagnostic Workup Laboratory and imaging studies play a key role in confirming suspected toxic overdoses, identifying complications, and monitoring the patient’s response to treatment. A standard initial workup should include measurement of electrolytes, renal function, and liver enzymes, along with serum glucose to detect hypoglycemia or hyperglycemia . An arterial or venous blood gas is essential to assess acid-base status, while creatine kinase should be checked to screen for rhabdomyolysis. Additional important labs include a coagulation panel and lactate, which can indicate systemic toxicity or poor perfusion. Acetaminophen and salicylate levels should be obtained in all cases of suspected overdose—even if not initially suspected—due to their potential for severe morbidity. In women of childbearing age, a urine pregnancy test is necessary; overdoses in pregnant patients often occur for suicidal or abortifacient reasons, especially in psychiatric patients or in settings where intimate partner violence is suspected [108,110]. Urine drug screens and urinalysis, though limited in scope, may aid in the identification of commonly abused substances and provide additional diagnostic clues . An electrocardiogram (ECG) should also be performed to assess cardiac effects such as QT prolongation, QRS widening, or bradyarrhythmias, which can guide both diagnosis and management. In cases involving altered mental status, seizures, or head trauma, neuroimaging such as a noncontrast CT scan of the head may be warranted . A chest radiograph can be obtained to evaluate for aspiration pneumonitis or pulmonary edema. 4. : Treatment and Antidote Administration The foundation of overdose management is supportive care, including airway stabilization, oxygenation, intravenous fluids, and correction of electrolyte or acid-base disturbances [2,108]. Close cardiac and respiratory monitoring is essential in all moderate to severe intoxications. In patients presenting with undifferentiated altered mental status, the empiric administration of low-risk, high-benefit interventions is recommended while the underlying cause is being determined. These interventions include dextrose to address potential hypoglycemia, oxygen to correct hypoxia or respiratory depression, naloxone if opioid overdose is suspected, and thiamine, particularly in individuals who are malnourished or have a history of alcohol use [2,3,62,108]. Once a specific toxidrome is identified, targeted antidotes should be promptly administered. Naloxone remains the antidote of choice for opioid toxicity, while flumazenil may be considered for benzodiazepine overdose only in carefully selected patients due to the risk of seizures. Physostigmine can be used in cases of severe anticholinergic toxicity, and cyproheptadine is indicated for serotonin syndrome. In cholinergic poisoning, such as from organophosphates, atropine and pralidoxime are the mainstays of treatment. For neuroleptic malignant syndrome, dantrolene and bromocriptine may be used to counteract the hyperthermia and dopamine blockade. Activated charcoal may be considered within one hour of ingestion in alert patients with a protected airway to reduce drug absorption. However, it has not been shown to significantly improve patient outcomes [108,111]. Whole bowel irrigation with polyethylene glycol is indicated in patients who have ingested particular substances, such as lithium, drug-filled packets, heavy metals, or sustained-release products . Lipid emulsion therapy and hemodialysis are reserved for selected poisonings with lipid-soluble or dialyzable agents [3,62,108]. Timely recognition of toxidromes, coupled with supportive and antidotal therapy, significantly improves outcomes in overdose management . Multidisciplinary collaboration with toxicology, psychiatry, and pharmacy is strongly recommended in complex or unclear cases. 3.5. Distinguishing Toxidromes from Psychiatric Syndromes Differentiating substance-induced alterations in mental status from primary psychiatric conditions depends on acuity, autonomic physiology, and neuromuscular findings, which primary psychiatric disorders rarely reproduce in a consistent cluster. Toxidromes typically emerge within minutes to hours after a plausible exposure, whereas exacerbations of mood or psychotic illness progress over days to weeks and are not tightly linked to a single dose. Autonomic features often provide the strongest basis for distinction. Patterns such as hyperthermia; marked tachycardia or hypertension; diaphoresis versus pronounced mucosal dryness; altered bowel motility; urinary retention; and characteristic pupillary changes (miosis in opioid toxicity, mydriasis in anticholinergic or sympathomimetic states) each increase the likelihood of a toxicologic process. Neuromuscular findings further refine the differential. Generalized or ocular clonus with diffuse hyperreflexia supports serotonin toxicity. “Lead-pipe” rigidity with bradykinesia and elevated creatine kinase suggests neuroleptic malignant syndrome. Agitated delirium with hot, dry skin, mydriasis, urinary retention, and diminished bowel sounds is characteristic of anticholinergic poisoning. Hypoventilation with pinpoint pupils strongly indicates opioid toxicity. Since phenomenology can overlap, such as when anticholinergic delirium mimics acute psychosis or manic agitation resembles sympathomimetic toxicity, clinicians should maintain a low threshold for bedside testing. High-yield evaluations include point-of-care glucose, core temperature, venous blood gas, and capnography when hypoventilation is suspected, as well as early electrocardiography (QRS widening in tricyclic antidepressant toxicity and marked QTc prolongation with certain antipsychotics or methadone). Therapeutic probes can both clarify diagnosis and accelerate treatment. Examples include naloxone for suspected opioid toxicity, carefully monitored physostigmine for anticholinergic delirium in appropriate settings, and a lorazepam challenge when malignant catatonia is considered. Routine broad urine toxicology panels are at best supportive and should not delay empiric, syndrome-directed care when a toxidrome is suspected. 3.6. Overdose Prevention and Longitudinal Systems-Level Interventions The long-term management of toxidromes in psychiatric populations must extend beyond acute stabilization to encompass longitudinal strategies that reduce the likelihood of recurrent overdose, minimize medication-related harm, and promote sustained recovery. While acute care focuses on toxidrome recognition and reversal, lasting risk reduction requires addressing the underlying behavioral, pharmacologic, and systemic contributors to toxic exposures. Psychiatric patients often face complex treatment regimens, fragmented access to care, and limited support during transitions between inpatient and outpatient settings, all of which heighten their vulnerability to toxic overdose [15,19]. A systems-based approach is essential to mitigate these risks, and we propose a four-pronged approach for providers that integrates the following: safer prescribing practices, robust follow-up protocols, interdisciplinary collaboration, and targeted public health interventions. When implemented collectively, these strategies can reduce the incidence of medication toxicity and overdose while improving long-term outcomes for individuals living with SMI. Restricting Access to High-Risk Medications Means restriction has emerged as a cornerstone of suicide prevention, aiming to reduce the availability of substances and tools commonly used in suicide attempts. A growing body of evidence supports the effectiveness of this approach in reducing suicide risk across diverse settings and populations [112,113,114,115,116,117,118]. A recent systematic review of suicide prevention strategies found that when patients are faced with a lack of access to their preferred method of suicide (e.g., firearms, chemicals), most do not turn to alternative methods, making restriction a valuable strategy capable of motivating behavioral change . Risk mitigation strategies can be implemented by reconciling prescriptions appropriately, limiting dispensed quantities of high-risk agents, using blister or unit-dose packaging, employing secure storage, and arranging disposal of unused medications [66,112,113,114,115,116,117,118]. Medication reconciliations should extend beyond simple list consolidation to include interaction checks, assessment of cumulative anticholinergic and serotonergic burden, and consolidation of prescribing and pharmacy oversight where feasible [112,113]. Psychotropic agents such as tricyclic antidepressants, monoamine oxidase inhibitors, and long-acting opioids carry significant risks when misused. Therefore, clinicians should consider safer alternatives when possible, especially for patients with a history of suicide attempts or substance use disorders . In high-risk cases, supervised administration or long-acting injectable formulations may further reduce the likelihood of overdose . Structured suicide safety assessments should always be paired with medication-specific means of safety. 2. : Enhanced Patient Education and Discharge Planning Patients treated for overdose and patients with SMI are at increased risk of recurrent admissions and carry a significantly increased rate of mortality, particularly within the first days and weeks after discharge . Therefore, timely outpatient follow-up, education, and communication between inpatient and community providers are essential to ensure safe care transitions . Scheduling psychiatric or primary care appointments within 72 h (goal ≤ 7 days) is shown to reduce recurrent presentations . Structured safety planning through the use of digital tools such as telepsychiatry visits, automated reminders, and virtual check-ins may improve post-discharge adherence [117,121]. These practices are consistent with the World Health Organization’s LIVE LIFE strategy, which emphasizes early identification, continuous monitoring, and follow-up of individuals affected by suicidal behavior . When opioids are present in the regimen or involved in the event, naloxone should be co-prescribed, and patients, along with their household members, should be educated on overdose recognition, prevention, and naloxone use [113,114]. Alcohol and stimulant use disorders should be addressed with counseling and facilitated referrals, with pharmacotherapy arranged when appropriate . Standardizing these measures, alongside compact toxidrome-guided recognition and antidote pathways, connects point-of-care diagnosis to prevention and can accelerate treatment. 3. : Pharmacist Involvement and Collaborative Care Models Pharmacists are critical to the safe prescribing and monitoring of psychotropic medications, particularly in patients with complex regimens. Integrating pharmacists into multidisciplinary psychiatric care teams has been shown to improve medication adherence, enhance patient education, and support safe prescribing practices [122,123,124]. Collaborative care models that include psychiatry, primary care, behavioral health, and pharmacy are especially effective for patients with co-occurring mental and physical illnesses, improving both safety and clinical outcomes [118,125,126]. Despite the increased upfront costs associated with establishing collaborative care models, there is growing evidence that they lead to long-term health care savings and higher patient satisfaction, with the added benefit of adapting to the unique mental health concerns of specific populations, such as students, geriatric patients, women, and individuals with substance use disorders [125,126]. 4. : Public Health and Policy Interventions Public health interventions can play a central role in reducing the incidence of overdose and suicide by improving awareness, reducing stigma, and encouraging recognition of risk within communities. The World Health Organization identifies effective measures such as community-based naloxone distribution, medication take-back programs, and safe storage campaigns to limit access to lethal means . These large-scale measures can inform state- and community-level policies that directly influence patient outcomes. Public education campaigns, training for healthcare providers, and responsible media reporting on suicide can reduce stigma and enhance early intervention [127,128]. Online suicide education programs can further enhance suicide literacy among participants without ready access to care, reduce self-stigma, and promote self-efficacy to seek support in psychologically difficult situations . Increasing the availability of accessible online public health resources may broaden reach and improve engagement in at-risk populations. In parallel, real-time public surveillance systems that track trends in poisoning and self-harm are critical for guiding resource allocation and informing the development of responsive, data-driven interventions [129,130]. 4. Limitations This selective narrative review has several limitations. We did not preregister a protocol, conduct a formal risk-of-bias assessment, or perform a meta-analysis, as substantial heterogeneity in study designs, exposure definitions, and outcomes precluded meaningful quantitative synthesis. Our search was limited to English-language, peer-reviewed sources and excluded gray literature, introducing potential publication and language bias. Because part of the search relied on Google Scholar, which yields non-replicable result sets, we report approximate screening counts rather than definitive study totals. 5. Conclusions Toxidromes represent a clinically essential yet often underutilized framework for identifying and managing toxicologic emergencies in psychiatric populations. Individuals with SMI are at elevated risk for overdose due to a convergence of factors, such as widespread access to high-risk medications, co-occurring substance use disorders, dynamic factors, and fragmented systems of care. As this narrative review has demonstrated, these vulnerabilities are compounded by the diagnostic complexity of mixed ingestions and the overlapping presentations of psychiatric syndromes and toxidromes. Accurate and timely recognition of toxidromes requires both structured clinical assessment and familiarity with characteristic symptom patterns. This narrative review provides a practical, point-of-care framework for overdose management in psychiatric settings. We connect psychiatric risk pathways to the toxicologic syndromes clinicians most often face and translate them into actions that fit emergency department and consultation-liaison psychiatry workflows. Diagnostic red flags include abrupt onset after a plausible exposure and autonomic instability out of proportion to primary psychiatric illness, with signature findings that accelerate decision-making. Opioid toxicity presents with miosis and hypoventilation; anticholinergic toxicity with hot, dry, agitated delirium, mydriasis, urinary retention, and decreased bowel sounds; and serotonin toxicity with spontaneous or inducible clonus, hyperreflexia, and hyperthermia. An early electrocardiogram helps detect tricyclic-related QRS widening. Prevention and systems strategies should be equally actionable. These include limiting the quantity of medications dispensed, ensuring secure storage, and facilitating proper disposal of unused medications. At every transition of care, rigorous medication reconciliation should be performed, including checks with the Prescription Drug Monitoring Program (PDMP) and screening for potential drug interactions. When opioids are prescribed, naloxone should be co-prescribed and accompanied by patient education. Antidotes for specific toxidromes (e.g., naloxone, carefully selected physostigmine, or cyproheptadine) should be rapidly administered when indicated, alongside appropriate supportive care. Treatment for substance use disorders should be initiated or bridged from the emergency department when appropriate. Rapid follow-up after discharge should be arranged, ideally within 72 h, with a goal of no more than 7 days post-discharge, and warm handoffs should be used to ensure continuity of care. Prescribing and pharmacy oversight should be consolidated, with pharmacists integrated into psychiatric care teams whenever possible. Embedding these steps can shorten the time to antidote-directed treatment and reduce preventable morbidity and mortality among people living with SMI. On a broader scale, public health interventions and policy reforms informed by surveillance data and grounded in harm reduction principles are essential. Bridging the gap between psychiatry and toxicology offers the opportunity to reduce preventable morbidity and mortality, improve medication safety, and foster a model of care that is more resilient, compassionate, recovery-oriented, evidence-based, person-centered, and responsive to the needs of individuals with mental illness. Author Contributions Conceptualization, V.P. and A.G.B.; methodology, V.P., S.D., P.B., and A.G.B.; validation, A.G.B. and V.P.; investigation, S.D., P.B., A.G.B., and V.P.; writing—original draft preparation, S.D.; writing—review and editing, S.D., A.G.B., P.B., and V.P.; supervision, V.P.; All authors have read and agreed to the published version of the manuscript. Funding This research received no external funding. Acknowledgments The authors have reviewed and edited the output and take full responsibility for the content of this publication. Conflicts of Interest The authors declare no conflicts of interest. References U.S. Department of Homeland Security. 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Clinical approach to suspected overdose. Table 1. Overview of common toxidromes in psychiatry. \| Toxidrome \| Pathogenesis \| Common Agents \| Clinical Features \| Diagnosis \| Management \| --- --- --- \| \| Anticholinergic \| Muscarinic acetylcholine receptor blockade causes central and peripheral Ach inhibition \| Atropine, diphenhydramine, hyoscyamine, TCAs, phenothiazines, benztropine, trihexyphenidyl, scopolamine \| Dry skin, mydriasis, urinary retention, ileus, delirium, hyperthermia, tachycardia, flushed skin, and hallucinations. Severe: delirium, seizures, coma \| Clinical: dry skin, mydriasis, altered mental status; absent diaphoresis (differentiates from sympathomimetic) \| Supportive care, benzodiazepines; physostigmine in severe cases [2,3,62,63,64,65] \| \| Opioid \| Mu-opioid receptor agonism \| Heroin, morphine, oxycodone, fentanyl, carfentanil, methadone \| Miosis, respiratory depression, bradycardia, hypotension, coma \| Clinical: classic triad—miosis, respiratory depression, loss of consciousness \| Naloxone, airway support [2,3,62,66] \| \| Cholinergic \| Excess acetylcholine due to acetylcholinesterase inhibition \| Organophosphates, carbamates, nerve agents, physostigmine \| Diarrhea, Urination, Miosis, Bradycardia, Bronchorrhea, Emesis, Lacrimation, Salivation, Sweating \| Clinical: muscarinic + nicotinic symptoms, bradycardia, wheezing, secretions \| Atropine, pralidoxime, benzodiazepines [2,3,62,67,68,69,70] \| \| Sedative-Hypnotic \| Potentiation of GABA-A receptor activity \| Benzodiazepines, barbiturates, zolpidem, ethanol \| CNS depression, slurred speech, ataxia, hypotension, respiratory depression \| Clinical: history + CNS depression without other findings (e.g., no miosis or clonus) \| Supportive care; flumazenil is rarely used due to risk of seizures or arrhythmia [2,3,62,71,72,73] \| \| Sympathomimetic \| Excessive stimulation of adrenergic receptors via increased catecholamine release or reuptake inhibition \| Amphetamines, methamphetamine, cocaine, methylphenidate, synthetic cathinones, pseudoephedrine, phenylephrine, ephedrine \| Hypertension, tachycardia, hyperthermia, agitation, paranoia, hallucinations, mydriasis, tremor, diaphoresis, seizures, rhabdomyolysis \| Clinical: agitation, mydriasis, hyperthermia, +diaphoresis (distinguishes from anticholinergic) \| Supportive care, benzodiazepines, cooling; avoid beta-blockers alone! [2,3,5,62,74,75] \| \| Neuroleptic \| Dopamine D2 receptor blockade, primarily in the basal ganglia and hypothalamus \| Haloperidol, fluphenazine, risperidone, olanzapine, prochlorperazine, promethazine \| Mild (EPS): dystonia, tremor, bradykinesia, akathisia; Severe (NMS): hyperthermia, lead-pipe rigidity, altered mental status, autonomic instability \| Clinical: rigidity, altered mental status, fever, increased creatine kinase \| Stop the offending agent! EPS: benztropine or diphenhydramine. NMS: bromocriptine, dantrolene, ICU support [2,3,62,76,77,78] \| \| Serotonergic \| Excess serotonergic activity, particularly at 5-HT2A receptors \| SSRIs, SNRIs, MAOIs, TCAs, trazodone, mirtazapine, tramadol, fentanyl, dextromethorphan, buspirone \| Agitation, clonus, hyperreflexia, mydriasis, hyperthermia, diarrhea, tremor, altered mental status \| Hunter Criteria: clonus + serotonergic agent; hyperreflexia and clonus are key signs \| Stop all serotonergic drugs; benzodiazepines, cyproheptadine, cooling; ICU monitoring if severe [2,3,62,79,80,81,82,83] \| \| \| \| --- \| \| \| Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. \| © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( Share and Cite MDPI and ACS Style Dutta, S.; Buciuc, A.G.; Barry, P.; Padilla, V. A Narrative Review on Toxidromes in the Psychiatric Population: Implications for Overdose Prevention. J. Clin. Med. 2025, 14, 6160. AMA Style Dutta S, Buciuc AG, Barry P, Padilla V. A Narrative Review on Toxidromes in the Psychiatric Population: Implications for Overdose Prevention. Journal of Clinical Medicine. 2025; 14(17):6160. Chicago/Turabian Style Dutta, Sanjukta, Adela Georgiana Buciuc, Patrick Barry, and Vanessa Padilla. 2025. "A Narrative Review on Toxidromes in the Psychiatric Population: Implications for Overdose Prevention" Journal of Clinical Medicine 14, no. 17: 6160. APA Style Dutta, S., Buciuc, A. G., Barry, P., & Padilla, V. (2025). A Narrative Review on Toxidromes in the Psychiatric Population: Implications for Overdose Prevention. Journal of Clinical Medicine, 14(17), 6160. Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here. Article Metrics Citations No citations were found for this article, but you may check on Google Scholar Article Access Statistics For more information on the journal statistics, click here. Multiple requests from the same IP address are counted as one view. Zoom \| Orient \| As Lines \| As Sticks \| As Cartoon \| As Surface \| Previous Scene \| Next Scene Back to TopTop
12355	https://forum.wordreference.com/threads/imperfect-or-preterite.39169/	Imperfect or Preterite \| WordReference Forums WordReference.comLanguage Forums ForumsRules/Help/FAQHelp/FAQ MembersCurrent visitors Interface Language Dictionary search: Log inRegister What's newSearch Search [x] Search titles and first posts only [x] Search titles only By: Search Advanced search… Rules/Help/FAQHelp/FAQ MembersCurrent visitors Interface Language Menu Log in Register Install the app Install How to install the app on iOS Follow along with the video below to see how to install our site as a web app on your home screen. Note: This feature may not be available in some browsers. Spanish-English / Español-Inglés Spanish-English Grammar / Gramática Español-Inglés Imperfect or Preterite Thread starterTate_Harmann Start dateJul 19, 2005 Tate_Harmann Senior Member St Paul, MN English / United States Jul 19, 2005 #1 Hello everybody, I have had it explained many times in many different ways, but I still am a little confused about when to use the imperfect over the preterite tense in Spanish. As a basic rule I understand that a person uses preterite for an action that only happened once, and imperfect for a continous past action. But in reading spanish literature I often find exceptions to the rule. For example: Roberto estaba mirando. Robert was looking. Isn't that something that only happened once, and therefore would be preterite? It's not a big deal when reading because I can still understand it, but it would be good to know for speaking purposes. For example: what would I use to ask somebody (that I know well) what they did today? Imperfect or Preterite? ¿Qu é hiciste hoy? o ¿Qu é hac ías hoy? I would think that it would be the first one, but I am not sure. Thank you for the help. jess oh seven Senior Member Scotland UK/US English Jul 19, 2005 #2 i think you could use both depending on what you want to say ¿Qué hiciste hoy? i think that would mean "what did you do today?" whereas ¿Qué hacías hoy? would be more like "what have you been doing today?"... i think.... maybe!! hehe. i suppose you could also use the perfect tense ie. "what have you done today?" W Whisky con ron Senior Member Scotland Venezuela / Español Jul 19, 2005 #3 Que hiciste hoy es lo mas común (what did you do today). Qué hacías hoy is at a determined point in time "qué hacías hoy cuando te llamé?". "Roberto estaba mirando" is a gerundio, therefore needs the verb "estar". Have to confess, I don't know which one is the imprefect or the preterite. Y just know which one is used in each phrase. Sorry can't give more proper grammar explanaition. Cheers A Artrella Banned BA Spanish-Argentina Jul 19, 2005 #4 Whisky con ron said: Que hiciste hoy es lo mas común (what did you do today). Qué hacías hoy is at a determined point in time "qué hacías hoy cuando te llamé?". "Roberto estaba mirando" is a gerundio, therefore needs the verb "estar". Have to confess, I don't know which one is the imprefect or the preterite. Y just know which one is used in each phrase. Sorry can't give more proper grammar explanaition. Cheers Click to expand... Whisky, the preterite is "hiciste" and the imperfect is "hacías" (imperfect because the action was still occurring when some other took place . When you say that an action is a "perfect" one, it means that it has begun and has finished, i.e. a complete action) HTH Tate_Harmann Senior Member St Paul, MN English / United States Jul 19, 2005 #5 Muchas gracias, I will use whatever is most common. But I suppose the perfect tense would be a proper way to say it also as jess oh seven said. Would it be: ¿Qué has hecho hoy? A Artrella Banned BA Spanish-Argentina Jul 19, 2005 #6 Tate_Harmann said: Muchas gracias, I will use whatever is most common. But I suppose the perfect tense would be a proper way to say it also as jess oh seven said. Would it be: ¿Qué has hecho hoy? Click to expand... This is correct Tate. In some countries like mine (Argentina) we tend to use the "imperfect" (mostly). But using the perfect tense is ok! C Christian Senior Member Los Angeles USA English Jul 19, 2005 #7 Robert was looking. Isn't that something that only happened once,[?]<<< I'd say that if Robert looked, it happened once. Robert was looking sounds imperfect to me--an action not necessarily completed. Tate_Harmann Senior Member St Paul, MN English / United States Jul 19, 2005 #8 Gracias, So, Roberto estaba mirando is correct because you don't know when the action stopped? I have many questions other about spanish because I live in the north midwest of the United States and don't often get opportunities to speak with spanish-speaking people. I don't want to throw them all out right away and upset anybody. But I have one more, it involves the difference between por and para. I was told that por is used with numbers and time, and para for all other things. But in reading I often find exceptions to this rule. If I wanted to say, "thank you for the reply" would it be: Gracias por su repuesta o Gracias para su repuesta Thank you again. W Whisky con ron Senior Member Scotland Venezuela / Español Jul 19, 2005 #9 Hey tate. The issue of "por" and "para" has been discussed over and over. I suggest you learn to use the "search" function at the top of the page. See, for instance, these posts: If it's of any consolation, por and para are quite difficult to understand. (and it is "gracias por tu respuesta") Saludos Tate_Harmann Senior Member St Paul, MN English / United States Jul 19, 2005 #10 Gracias por tu repuesta!, I will use the search first next time. Nos hablaremos tarde W Whisky con ron Senior Member Scotland Venezuela / Español Jul 19, 2005 #11 OK, buena suerte! Saludos O Outsider Senior Member Portuguese (Portugal) Jul 19, 2005 #12 Tate_Harmann said: Hello everybody, I have had it explained many times in many different ways, but I still am a little confused about when to use the imperfect over the preterite tense in Spanish. As a basic rule I understand that a person uses preterite for an action that only happened once, and imperfect for a continous past action. But in reading spanish literature I often find exceptions to the rule. For example: Roberto estaba mirando. Robert was looking. Isn't that something that only happened once, and therefore would be preterite? It's not a big deal when reading because I can still understand it, but it would be good to know for speaking purposes. For example: what would I use to ask somebody (that I know well) what they did today? Imperfect or Preterite? Click to expand... There are two main dimensions in which the preterite and the imperfect differ. One is completion: the preterite gives you the information that the event was completed at or before the time of which we speak; the imperfect does not imply completion (the event may have continued afterwards; in this case, Robert very likely continued 'looking' after that moment). The other dimension is duration: the preterite is used to speak of events that occur at a well defined, pointlike moment in time; the imperfect for events that spread out over a longer interval of time (or are repeated regularly over an interval). In my opinion, it's not accurate to say that the preterite is used for an action that only happened once. It may be used for actions that were repeated, but were definitely finished at or by the time of which we speak, and it may be used for actions that happened a few times in an irregular fashion. C Christian Senior Member Los Angeles USA English Jul 20, 2005 #13 Tate, Do yourself a favor and obtain "Spanish Verb Tenses" by Dorothy Richmond (Passport Books) from Amazon or someplace. It's a very clearly written guide to all tenses and moods, right through the dreaded subjunctive. Çan;t recommend it highly enough. You must log in or register to reply here. Share: BlueskyLinkedInWhatsAppEmailShareLink Spanish-English / Español-Inglés Spanish-English Grammar / Gramática Español-Inglés WR styleSystemLightDark English (EN-us) Contact us Terms and rules Privacy policy Help RSS Community platform by XenForo®© 2010-2025 XenForo Ltd. Back TopBottom
12356	https://www.johndcook.com/blog/2010/11/08/odd-perfect-numbers/	Odd perfect numbers Skip to content MATH PROBABILITY SIGNAL PROCESSING NUMERICAL COMPUTING SEE ALL … STATS EXPERT TESTIMONY WEB ANALYTICS FORECASTING RNG TESTING SEE ALL … PRIVACY HIPAA SAFE HARBOR CRYPTOGRAPHY DIFFERENTIAL PRIVACY PRIVACY FAQ WRITING BLOG RSS FEED TWITTER SUBSTACK ARTICLES TECH NOTES ABOUT CLIENTS ENDORSEMENTS TEAM SERVICES (832) 422-8646 Contact Odd perfect numbers Posted on 8 November 2010 by John Yesterday I wrote about even perfect numbers. What about odd perfect numbers? Well, there may not be any. I couldn’t care less about perfect numbers, even or odd. But I find the history and the mathematics surrounding the study of perfect numbers interesting. As soon as you define perfect numbers and start looking for examples, you soon realize that all your examples are even. So people have wondered about the existence of odd perfect numbers for at least 2300 years. No one has proved that odd perfect numbers do or do not exist. But people have proved properties that odd perfect number must have, if there are any. So far, although the requirements for odd perfect numbers have become more demanding, they are not contradictory and it remains logically possible that such numbers exist. However, most experts believe odd perfect numbers probably don’t exist. (Either odd perfect numbers exist or they don’t. How can one say they “probably” don’t exist? See an explanation here.) Wikipedia lists properties that odd perfect numbers must have. For example, an odd perfect number must have at least 300 digits. It’s interesting to think how someone determined that. In principle, you could just start at 1 and test odd numbers to see whether they’re perfect. But in practice, you just won’t get very far. A year is about 10^7.5 seconds (see here). If you had started testing a billion (10^9) numbers a second since the time of Euclid (roughly 10^3.5 years ago) you could have tested about 10^20 numbers by now. Clearly whoever came up with the requirement N > 10^300 didn’t simply use brute force. There may have been some computer calculation involved, but if so it had a sophisticated starting point. Related: Applied number theory Categories : Math Tags : MathNumber theory Bookmark the permalink Post navigation Previous PostEven perfect numbers Next PostDeleting the Windows recycle bin desktop icon 5 thoughts on “Odd perfect numbers” Edvin 9 November 2010 at 02:23 A short abstracts outline the strategy. I didn’t read the details. 2. Arnie Dris 11 December 2010 at 02:41 Hi John, I (humbly) invite you to take a look at It contains (among other things) an attempt to prove the OPN Conjecture via a “Tour de Force”. Do let me know if you have any questions! Arnie Dris 3. Steve Campbell 26 January 2012 at 21:15 The problem is, we only know that what our current boundry of numbers encompasses. 10To the 300th has been the extent of the search for odd perfect numbers. If you expand your search beyond the scope of current computer and mathematical capabilities, then maybe an odd perfect number does exist. Our current knowledge is so limited to what a computers can generate solutions, that maybe we should realize we are at the very smallest part of infinty. Mathematical tactics and theory can only cover a small part of the entire number set. This, of course, is a logical arguement, not a mathematical arguement. 4. Dan Eastwood 7 January 2019 at 17:21 I just rediscovered this post, and a little bit of searching turn up a newer, greater, bound. ODD PERFECT NUMBERS ARE GREATER THAN 101500 PASCAL OCHEM AND MICHAEL RAO ¨ Abstract. Brent, Cohen, and te Riele proved in 1991 that an odd perfect number N is greater than 10^300. We modify their method to obtain N > 10^1500. We also obtain that N has at least 101 not necessarily distinct prime factors and that its largest component (i.e. divisor p a with p prime) is greater than 10^62. Ochem, P., & Rao, M. (2012). Odd perfect numbers are greater than 10¹⁵⁰⁰. Mathematics of Computation, 81(279), 1869-1877. Still more searching give this, which I can’t access: 5. Dan Eastwood 7 January 2019 at 17:25 ^^^ found it, but it’s a deep read. Comments are closed. Search for: John D. Cook, PhD My colleagues and I have decades of consulting experience helping companies solve complex problems involving data privacy, applied math, and statistics. Let’s talk. We look forward to exploring the opportunity to help your company too. John D. Cook © All rights reserved. Search for: (832) 422-8646 EMAIL
12357	https://www.cuemath.com/questions/how-to-find-base-and-height-of-a-triangle-with-only-area/	How to find base and height of a triangle with only area? The area of a triangle is the amount of space enclosed between the sides of the triangle. Answer: If area and base are given: height = 2A/ b = (2 × Area) / base. If area and height are given: base = 2A/ h = (2 × Area) / height Let's find the base and height of a triangle when only the area is known. Explanation: Area of a triangle = 1/2 × Base × Height If area and base of the triangle are given: height = 2A/ b = (2 × Area) / base If area and height of the triangle are given: base = 2A/ h = (2 × Area) / height But for the equilateral triangle, height and base are interconnected: Area = √3/4 a2 height = √3/2 a So, from the area of the equilateral triangle calculate the side, then find the height of the triangle using the formula. Thus, we have seen how to calculate the base and height of a triangle with only the area given. Math worksheets and visual curriculum 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 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 MATH WORKSHEETS Kindergarten Worksheets 1st Grade Worksheets 2nd Grade Worksheets 3rd Grade Worksheets 4th Grade Worksheets 5th Grade Worksheets 6th Grade Worksheets 7th Grade Worksheets 8th Grade Worksheets 9th Grade Worksheets 10th Grade Worksheets Terms and ConditionsPrivacy Policy
12358	https://www.mathsisfun.com/geometry/parabola.html	Parabola When we kick a soccer ball (or shoot an arrow, fire a missile or throw a stone) it arcs up into the air and comes down again ... ... following the path of a parabola! (Except for how the air affects it.) Try kicking the ball: Definition A parabola is a curve where any point is at an equal distance from: a fixed point (the focus), and a fixed straight line (the directrix) On Paper Get a piece of paper, draw a straight line on it, then make a big dot for the focus (not on the line!). Now play around with some measurements until you have another dot that is exactly the same distance from the focus and the straight line. Keep going until you have lots of little dots, then join the little dots and you will have a parabola! Just like in this interactive (try moving point P): ../sets/images/geom-locus.js?mode=parabola Names Here are the important names: the directrix and focus (explained above) the axis of symmetry (goes through the focus, at right angles to the directrix) the vertex (where the parabola makes its sharpest turn) is halfway between the focus and directrix Reflector And a parabola has this amazing property: Any ray parallel to the axis of symmetry gets reflected off the surface straight to the focus. And that explains why that dot is called the focus ... ... because that's where all the rays get focused! So the parabola can be used for: satellite dishes, radar dishes, concentrating the sun's rays to make a hot spot, the reflector on spotlights and torches, etc We also get a parabola when we slice through a cone (the slice must be parallel to the side of the cone). So the parabola is a conic section (a section of a cone). Equations The simplest equation for a parabola is y = x2 Turned on its side it becomes y2 = x (or y = √x for just the top half) A little more generally: y2 = 4ax where a is the distance from the origin to the focus (and also from the origin to directrix) Example: Find the focus for the equation y2=5x Converting y2 = 5x to y2 = 4ax form, we get y2 = 4 (5/4) x, so a = 5/4, and the focus of y2=5x is: F = (a, 0) = (5/4, 0) The equations of parabolas in different orientations are as follows: y2 = 4ax y2 = −4ax x2 = 4ay x2 = −4ay Measurements for a Parabolic Dish If you want to build a parabolic dish where the focus is 200 mm above the surface, what measurements do you need? To make it easy to build, let's have it pointing upwards, and so we choose the x2 = 4ay equation. And we want "a" to be 200, so the equation becomes: x2 = 4ay = 4 × 200 × y = 800y Rearranging so we can calculate heights: y = x2/800 And here are some height measurements as you run along: \| \| \| \| --- \| \| Distance Along ("x") \| Height ("y") \| \| 0 mm \| 0.0 mm \| \| 100 mm \| 12.5 mm \| \| 200 mm \| 50.0 mm \| \| 300 mm \| 112.5 mm \| \| 400 mm \| 200.0 mm \| \| 500 mm \| 312.5 mm \| \| 600 mm \| 450.0 mm \| \| \| \| Try to build one yourself, it could be fun! Just be careful, a reflective surface can concentrate a lot of heat at the focus. 567,568,833,834, 2088, 2089, 2086, 2087, 3334, 3335 Conic Sections Projectile Animation Geometry Index Copyright © 2025 Rod Pierce
12359	https://www.shmoop.com/common-core-standards/ccss-hs-a-ced-1.html	More on Common Core Standards The Standards About High School: Number and Quantity High School: Algebra High School: Functions High School: Modeling High School: Geometry High School: Statistics and Probability Grade 8 Grade 7 Grade 6 Grade 5 Grade 4 Grade 3 Grade 2 Grade 1 Kindergarten High School: Algebra High School: Algebra Creating Equations HSA-CED.A.1 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Students should be able to interpret word problems and form equations and inequalities in order to solve the problem. That means translating a word problem to an algebraic equation. Let's be real, here. Math is another language, just like Spanish, Japanese, or Icelandic. When you start learning a language, you don't start by translating words like "absquatulate" or "loquacious" or "pneumonoultramicroscopicsilicovolcanoconiosis" (and yes, that is a real word). It's better to start easier, with words like "cat" and "girl" and slowly work your way up. Just the same, if you use simple linear equations that are familiar to students, they can focus on the translation process and it'll all go a lot smoother. Translation is a useful analogy in and of itself because it emphasizes that the algebraic equation is the same as the word problem, just presented in a different way. In addition to helping students to understand the process, the translation analogy can also help reassure struggling learners and encourage practice. After they've gotten a hang of the basics, students can start learning quadratic, rational, and exponential functions to address all aspects of this standard. Once students are familiar with these operations individually, they should be asked to distinguish them from each another. As students gain experience, there are additional strategies that should be introduced. One experienced problem solver strategy is to read the question twice before beginning. It's a useful piece of advice in general, actually. Writing a list of what is known and a list of what needs to be calculated is also an excellent strategy. Such lists are especially useful when sorting out unnecessary information, identifying an appropriate formula to utilize, or constructing a proof. These strategies should be suggested and shown to students after they are proficient with the basic translation process. To start off, the chart below may be presented as a dictionary to support word to symbol translation. Students can also add to the chart as they find other key words or phrases. \| \| \| --- \| \| Algebra Symbol \| Key Words \| \| = equals \| 1. all 2. equals 3. gives 4. is, are, was, were, will be 5. results 6. same 7. yields \| \| < is less than \| 1. below 2. less than \| \| ≤ is less than or equal to \| 1. maximum of 2. not more than \| \| > is greater than \| 1. greater than 2. more than 3. over \| \| ≥ is greater than or equal to \| 1. at least 2. minimum of 3. not less than \| \| + addition \| 1. add 2. and 3. combine 4. increase 5. more 6. plus 7. raise 8. sum 9. together 10. total \| \| – subtraction \| 1. decrease 2. difference 3. fewer 4. less 5. lose 6. minus 7. reduce \| \| × multiplication \| 1. directly proportional 2. double(× 2), triple(× 3), etc. 3. group of 4. linear 5. multiplied 6. product 7. times \| \| / division \| 1. average 2. cut 3. divided by/into 4. each 5. inversely proportional 6. out of 7. per 8. pieces 9. quotient 10. ratio 11. share 12. split \| \| xn power \| 1. power 2. square (n = 2), cube (n = 3), etc. \| \| nx exponential \| 1. decays 2. doubles (n = 2), triples (n = 3), quadruples (n = 4), etc. 3. grows 4. rate of n per x \| If you needed another word problem example or video to show your students, here is one such example: Drills There are 60 students going on a field trip to the chocolate factory. The students are from three different classes. Mrs. Hooper's class has 24 students and Mr. Gomez's class has 18 students. Which of the equalities correctly describes the students and could be used to solve for how many students are from Mr. Anderson's class? (Let A = the number of students in Mr. Anderson's class.) Correct Answer: Answer Explanation: The relation Hooper's class + Gomez's class + Anderson's class = 60 students going on a field trip becomes an equation by changing the written descriptions into numbers and variables. Mrs. Hooper's class has 24 students and Mr. Gomez's class has 18 students giving 24 + 18 + Anderson's class = 60 students going on a field trip. The number of students in Anderson's class is the unknown and must be represented by a variable like A for Anderson. That means 24 + 18 + A = 60. There are 60 students going on a field trip to the chocolate factory. The students are from three different classes. Mrs. Hooper's class has 24 students and Mr. Gomez's class has 18 students. How many students are from Mr. Anderson's class? Correct Answer: Answer Explanation: The equation 24 + 18 + A = 60 (where A = students in Anderson's class) needs to be solved to isolate A. First, simplify 24 + 18 to 42 to get 42 + A = 60. Then, subtract 42 from both sides to get A = 60 – 42. This gives A = 18. There are 18 students from Anderson's class. There are six chaperones going on a field trip. There are two buses for the trip. The chaperones divide so there is the same number of chaperones on each bus. Which of the equations could be utilized to find the number of chaperones on the first bus? (Let c = number of chaperones on first bus.) Correct Answer: Answer Explanation: The relation becomes an equation by changing the written descriptions into numbers and variables. There are six chaperones on the trip giving . There are two buses, giving . The number of chaperones on the first bus is an unknown represented by the variable c. This results in the equation . There are six chaperones going on a field trip. There are two buses for the trip. The chaperones divide so there is the same number of chaperones on each bus. How many chaperones are there on the first bus? Correct Answer: Answer Explanation: The equation = c (where c = number of chaperones on first bus) needs to be solved for the variable c. This gives 3 = c or c = 3. There are 3 chaperones on the first bus. A total of 66 people attended a field trip to a chocolate factory for a tour. A maximum of 15 people are allowed to tour at one time. Which equation correctly describes how many tour groups to organize? (Let g = the number of groups.) Correct Answer: Answer Explanation: The relation becomes an equation by changing the written descriptions into numbers and variables. The 66 and 15 become numbers directly giving . The number of tour groups is an unknown so we need to make a variable g = number of tour groups. The resulting equation is . A total of 66 people attended a field trip to a chocolate factory for a tour. A maximum of 15 people are allowed to tour at one time. What is the minimum number of tour groups that can be formed? Correct Answer: Answer Explanation: The equation (where g = number of tour groups) needs to be solved for the variable g. Multiplying each side by g to remove it from the denominator gives 66 ≤ 15 × g. Dividing each side by 15 then gives which becomes 4.4 ≤ g. Reversing this gives g ≥ 4.4. The actual number of tour groups formed must be a whole number since there cannot be fractions of tours. The smallest whole number that is greater than or equal to 4.4 is 5. A minimum of 5 tour groups must be formed. A heart shaped chocolate box is composed of one square and two half circles. The total number of chocolates in the box is calculated by adding the area of a square given by 4x2 and the area of a circle approximated by 3x2. The company plans to add a small additional box for a promotional campaign containing one row (2x) of chocolates. If the total combined heart shape and small box contain 69 chocolates, which of these equations could be utilized to solve for the number of chocolates in the small box (2x)? Correct Answer: Answer Explanation: Begin with the relation: combine heart shape and small box = 69 chocolates total. The key words "and, "adding," and "add" all indicate summing the three relations for numbers of chocolates. This gives 4x2 + 3x2 + 2x. Combining these gives the equation 4x2 + 3x2 + 2x = 69. A heart shaped chocolate box is composed of one square and two half circles. The total number of chocolates in the box is calculated by adding the area of a square given by 4x2 and the area of a circle approximated by 3x2. The company plans to add a small additional box for a promotional campaign containing one row (2x) of chocolates. If the total combined heart shape and small box contain 69 chocolates, how many chocolates are in the small box (2x)? Correct Answer: Answer Explanation: The equation 4x2 + 3x2 + 2x = 69 can be solved for 2x, the number of chocolates in the small box. This gives 7x2 + 2x – 69 = 0 which is factored to (7x + 23)(x – 3) = 0. The solutions are and x = 3. These give 2x = -6.572 and 2x = 6, respectively. The negative solution does not make sense because you cannot have a negative number of chocolates in a box. So, the correct solution must be 2x = 6. There are 6 chocolates in the small box. On the day of the class field trip, the chocolate factory produced three times as many plain chocolate bars as crispy bars. They produced 50 more nutty bars than crispy bars. The ratio of plain chocolate bars produced to nutty bars produced was 2 to 1. Which of the equations below could be utilized to solve for the number of crispy bars produced on the day of the field trip? Correct Answer: Answer Explanation: The key word "ratio" indicates division such that . Then, you need number of plain chocolate bars in terms of crispy bars, which is 3c from the key word "times." And, you need number of nutty bars in terms of crispy bars, which is c + 50 from the key word "more." Combining these gives the equation . On the day of the class field trip, the chocolate factory produced three times as many plain chocolate bars as crispy bars. They produced 50 more nutty bars than crispy bars. The ratio of plain chocolate bars produced to nutty bars produced is 2 to 1. How many crispy bars were produced? Correct Answer: Answer Explanation: The equation equates the 2 to 1 ratio with 3c (the number of plain chocolate bars) to c + 50 (the number of nutty bars). To solve for c, the number of crispy bars, first multiply both sides by c + 50 to get 3c = 2c + 100. Then, subtract 2c from both sides to get c = 100. So, 100 crispy bars were produced. A large box of 144 chocolates has a width that is three times the height of the box and a length that is twice the width of the boxes. Each chocolate rests in a cube that is 1 in × 1 in × 1 in. Which equation could be utilized to calculate the height of the box in inches? Correct Answer: Answer Explanation: The initial relation for this problem requires knowing that the volume of the box will be length × width × height. Since each chocolate has a volume of 1 in3, the total volume of the box with 144 chocolates will be 144 in3. So, the basic relationship is length × width × height = 144. We do not know length, but we do know that it is twice the width. And, we know the width is three times the height. The key words "twice" and "times" both indicate multiplication so we have width = 3 × height = 3h and length = 2 × width = 2 × 3h. And the height is, of course, just h. So, the overall equation becomes (2 × 3h) × 3h × h = 144. A large box of 144 chocolates has a width that is three times the height of the box and a length that is twice the width of the boxes. Each chocolate rests in a cube that is 1 in × 1 in × 1 in. What is the height of the box in inches? Correct Answer: Answer Explanation: Given the relation length × width × height = volume, the resulting equation is (2 × 3h) × 3h × h = 144. This simplifies to 18h3 = 144. First, each side is divided by 18 to get h3 = 8. Then, taking the cubed root of both sides gives h = 2. The height of the box is 2 in. The candy company's total revenue this year was 1.1 times the revenue last year. If they experience the same growth every year, what equation describes how many years will it take before the revenue is more than double? Correct Answer: Answer Explanation: The basic relation here is for year revenue to be at least initial revenue. That is yearly revenue ≥ 2 × initial revenue. But there is no information given about the total initial revenue or the total year revenue. Instead, we know that the first year the revenue will be 1.1 × initial revenue. The second year, it will be 1.1 × first year revenue, or 1.1 × 1.1 × initial revenue, or 1.12 × initial revenue. For any given year, the revenue is 1.1y × initial revenue, where y is the number of years. Going back into the inequality gives 1.1y × initial revenue ≥ 2 × initial revenue. Canceling the initial revenue from both sides gives the equation 1.1y ≥ 2. The candy company's total revenue this year was 1.1 times the sales last year. If they experience the same growth every year, how many years will it take before the revenue is more than double? Correct Answer: Answer Explanation: For any given year, the revenue will be 1.1y times the initial revenue. More than double means 1.1y ≥ 2. Taking the log10 of both sides gives y × log101.1 ≥ log102. Using a calculator for the base 10 log gives log101.1 = 0.0414 and log102 = 0.301. The equation becomes 0.0414y ≥ 0.301. Dividing both sides by 0.0414 gives y ≥ 7.27. Looking at whole numbers of years, the first year with more than double the current revenue is 8 years. Aligned Resources Video MathShack Teacher Guides More standards from High School: Algebra - Creating Equations Aristotle Albert Einstein Tired of ads? Logging out… Logging out... You've been inactive for a while, logging you out in a few seconds... Why's This Funny?
12360	https://planetmath.org/divisorfunctionismultiplicativethe	divisor function is multiplicative, the divisor function is multiplicative, the Theorem. The divisor function ( is multiplicative. Proof. Let t=m n t=m⁢n with m,n m,ncoprime. Applying the fundamental theorem of arithmetic, we can write m=p a 1 1 p a 2 2⋯p a r r,n=q b 1 1 q b 2 2⋯q b s s,m=p 1 a 1⁢p 2 a 2⁢⋯⁢p r a r,n=q 1 b 1⁢q 2 b 2⁢⋯⁢q s b s, where each p j p j and q i q i are prime. Moreover, since m m and n n are coprime, we conclude that t=p a 1 1 p a 2 2⋯p a r r q b 1 1 q b 2 2⋯q b s s.t=p 1 a 1⁢p 2 a 2⁢⋯⁢p r a r⁢q 1 b 1⁢q 2 b 2⁢⋯⁢q s b s. Now, each divisor of t t is of the form t=p k 1 1 p k 2 2⋯p k r r q h 1 1 q h 2 2⋯q h s s.t=p 1 k 1⁢p 2 k 2⁢⋯⁢p r k r⁢q 1 h 1⁢q 2 h 2⁢⋯⁢q s h s. with 0≤k j≤a j 0≤k j≤a j and 0≤h i≤b i 0≤h i≤b i, and for each such divisor we get a divisor of m m and a divisor of n n, given respectively by u=p k 1 1 p k 2 2⋯p k r r,v=q h 1 1 q h 2 2⋯q h s s.u=p 1 k 1⁢p 2 k 2⁢⋯⁢p r k r,v=q 1 h 1⁢q 2 h 2⁢⋯⁢q s h s. Now, each respective divisor of m m, n n is of the form above, and for each such pair their product is also a divisor of t t. Therefore we get a bijection between the set of positive divisors of t t and the set of pairs of divisors of m m, n n respectively. Such bijection implies that the cardinalities of both sets are the same, and thus d(m n)=d(m)d(n).d⁢(m⁢n)=d⁢(m)⁢d⁢(n). \| Title \| divisor function is multiplicative, the \| \| Canonical name \| DivisorFunctionIsMultiplicativeThe \| \| Date of creation \| 2013-03-22 15:03:47 \| \| Last modified on \| 2013-03-22 15:03:47 \| \| Owner \| yark (2760) \| \| Last modified by \| yark (2760) \| \| Numerical id \| 9 \| \| Author \| yark (2760) \| \| Entry type \| Theorem \| \| Classification \| msc 11A25 \| Generated on Fri Feb 9 15:13:08 2018 by LaTeXML
12361	https://www.pinterest.com/ideas/subtraction-with-borrowing-anchor-chart/957507343783/	Skip to content When autocomplete results are available use up and down arrows to review and enter to select. Touch device users, explore by touch or with swipe gestures. Log in Sign up Explore Education School Supplies Subtraction with borrowing anchor chart Discover Pinterest’s best ideas and inspiration for Subtraction with borrowing anchor chart. Get inspired and try out new things. 69 people searched this · Last updated 3mo Addition Subtract across zeros Key Poem Double digit Subtracting mixed numbers Regrouping Vocabulary 1st grade 3 digit Subtraction Anchor Chart Math Subtraction Math Anchor Charts Math Intervention Math Strategies 2nd Grade Classroom Second Grade Math Third Grade Math Homeschool Math Last Week in Review, Visual Plans (my way), and a BIG FREEBIE ------------------------------------------------------------- ### If you follow me other places like FB or Instagram, then you have already seen this picture, but I wanted to share where I decided to hang the anchor chart. I put it right next to our door so that when we leave the room for specials, lunch, recess, etc that the students see the anchor chart over and over. We also say it as a class as it has a nice sing-songy tune to it! I need a few more days with this topic before I assess my kids (because subtraction with regrouping is HARD stuff) so I put… 5.1k Subtraction Anchor Chart 2nd Grade Addition And Subtraction Anchor Chart First Grade Subtraction Strategy Anchor Chart Subtraction Poem Anchor Chart Subtraction With Borrowing Anchor Chart Math Anchor Charts Second Grade Math Homeschool Math Future Classroom Subtraction Anchor Chart 2nd Grade ---------------------------------- ### A Subtraction Poem: More on top? No need to stop! More on the floor? Go next door and get 10 more! Numbers the same? Zero's the game! 332 3 Digit Subtraction Strategies 3 Digit Subtraction With Regrouping Anchor Chart Colorful Educational Poster Creative Math Classroom Ideas Fun Math Teaching Tool How To Teach Subtraction Effectively Stealing Subtraction Math Poster Subtraction With Borrowing Anchor Chart 3 Digit Subtraction Strategies ------------------------------ ### Regrouping three digit addition. They are "stealing" a ten in order to finish their subtraction. Even though it doesn't say regrouping, the students understand that you have to "take" a ten or "borrow" a ten to add to the ones in order to subtract. Hate me for having fun, but they loved it!! 1.8k Creative Math Teaching Ideas Elementary Math Teaching Tool Fun Math Learning Activities Math Poem Classroom Subtraction Poem For Kids Creative Math Teaching Ideas ---------------------------- ### Two digit and three digit subtraction. Use this poem to help students understand the borrowing rule! 1.4k Subtracting Anchor Chart Third Grade Subtraction Subtraction Charts For Grade 2 Regrouping Anchor Chart 2nd Grade 3 Digit Subtraction Anchor Chart How To Subtract Regrouping Subtraction Anchor Chart Math Regrouping Subtraction Addition Regrouping Anchor Chart The scoop on subtraction ------------------------ ### Subtraction with regrouping 7.4k Regrouping Subtraction Anchor Chart 3 Digit Addition And Subtraction With Regrouping Anchor Chart Adding 3 Digit Numbers With Regrouping Anchor Chart Subtraction With Regrouping Anchor Chart Subtraction With Base Ten Blocks Subtracting 3 Digit Numbers Anchor Chart Subtraction With Regrouping Using Base Ten Blocks Subtraction With Borrowing Anchor Chart Subtraction Anchor Chart How to Teach 2 Digit Subtraction with Regrouping Strategies ----------------------------------------------------------- ### Teaching 2 digit subtraction with regrouping strategies? These anchor charts cover base ten, expanded form, number line, standard algorithm, box it and more. Students will be learning alternatives to borrowing in no time. Your 2nd and 3rd grade students NEED these new common core math aligned strategies. 64 3rd Grade Subtraction St Math Subtraction With Regrouping Anchor Chart Teaching Regrouping Addition Subtraction Strategies 2nd Grade Subtraction With Regrouping Subtraction Strategies For Grade 3 Addition And Subtraction Lesson Plans Subtraction Task Cards Grade 3 3rd Grade Subtraction --------------------- 478 Teaching Regrouping Subtraction Teaching Subtraction With Regrouping Subtract Mixed Numbers With Regrouping Subtraction Poem Regrouping Subtraction With Regrouping Visual Teaching Subtraction Math Subtraction Math Anchor Charts Math Strategies Who are the people in your neighborhood? . . . ---------------------------------------------- ### We had to change things up a bit, since I learned yesterday that P had forgotten subtraction with regrouping. He had a hard time with the curriculum at school, because it had boxes everywhere regro… 485 1st Grade Subtraction Activities Kindergarten Math Anchor Chart Subtracting Anchor Chart Subtraction Activities For Preschool Subtraction Anchor Chart Ideas Subtraction Anchor Chart For Kindergarten Subtraction Centers First Grade Subtraction Charts Subtraction Ideas For Kindergarten Subtraction Anchor Chart ------------------------ ### Kindergarten math anchor chart for subtraction. 3.3k Ice Cream Subtraction Math Poster Addition And Subtraction Anchor Chart 3rd Classroom Decoration Inspiration Colorful Math Classroom Decor Fun Math Teaching Ideas Ice Cream Themed Learning Tool Ordering Numbers From Least To Greatest Anchor Chart Subtraction Anchor Chart 3rd Grade Subtraction Anchor Chart 4th Ice Cream Subtraction Math Poster --------------------------------- 283 Related interests Subtraction Strategies Anchor Chart Addition Anchor Chart Kindergarten Addition Strategies Anchor Chart Addition And Subtraction Practice Addition And Subtraction Anchor Chart Teaching Subtraction Addition With Regrouping Anchor Chart Fun Math Worksheets Subtraction With Regrouping Anchor Chart Subtraction Activities For Grade 2 See more Subtraction with borrowing anchor chart and more Explore related boards Ideas like this Subtraction with Borrowing Anchor Chart ===============
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12363	https://mathresearch.utsa.edu/wiki/index.php?title=Maxima_and_Minima_Problems	Maxima and Minima Problems From Department of Mathematics at UTSA Jump to navigation Jump to search The four types of extrema. Maxima and minima are points where a function reaches a highest or lowest value, respectively. There are two kinds of extrema (a word meaning maximum or minimum): global and local, sometimes referred to as "absolute" and "relative", respectively. A global maximum is a point that takes the largest value on the entire range of the function, while a global minimum is the point that takes the smallest value on the range of the function. On the other hand, local extrema are the largest or smallest values of the function in the immediate vicinity. In many cases, extrema look like the crest of a hill or the bottom of a bowl on a graph of the function. A global extremum is always a local extremum too, because it is the largest or smallest value on the entire range of the function, and therefore also its vicinity. It is also possible to have a function with no extrema, global or local: is a simple example. At any extremum, the slope of the graph is necessarily 0 (or is undefined, as in the case of ), as the graph must stop rising or falling at an extremum, and begin to head in the opposite direction. Because of this, extrema are also commonly called stationary points or turning points. Therefore, the first derivative of a function is equal to 0 at extrema. If the graph has one or more of these stationary points, these may be found by setting the first derivative equal to 0 and finding the roots of the resulting equation. The function , which contains a saddle point at the point . However, a slope of zero does not guarantee a maximum or minimum: there is a third class of stationary point called a saddle point. Consider the function The derivative is The slope at is 0. We have a slope of 0, but while this makes it a stationary point, this doesn't mean that it is a maximum or minimum. Looking at the graph of the function you will see that is neither, it's just a spot at which the function flattens out. True extrema require a sign change in the first derivative. This makes sense - you have to rise (positive slope) to and fall (negative slope) from a maximum. In between rising and falling, on a smooth curve, there will be a point of zero slope - the maximum. A minimum would exhibit similar properties, just in reverse. Good (B and C, green) and bad (D and E, blue) points to check in order to classify the extremum (A, black). The bad points lead to an incorrect classification of A as a minimum. This leads to a simple method to classify a stationary point - plug x values slightly left and right into the derivative of the function. If the results have opposite signs then it is a true maximum/minimum. You can also use these slopes to figure out if it is a maximum or a minimum: the left side slope will be positive for a maximum and negative for a minimum. However, you must exercise caution with this method, as, if you pick a point too far from the extremum, you could take it on the far side of another extremum and incorrectly classify the point. Contents 1 The Extremum Test 2 Critical Points 2.1 Example 1 2.2 Example 2 3 Saddle Point 3.1 Mathematical discussion 3.2 Saddle surface 3.3 Examples 3.4 Other uses 4 Extrema for functions of two variables 4.1 Functions of many variables 5 Resources 6 Licensing The Extremum Test A more rigorous method to classify a stationary point is called the extremum test, or 2nd Derivative Test. As we mentioned before, the sign of the first derivative must change for a stationary point to be a true extremum. Now, the second derivative of the function tells us the rate of change of the first derivative. It therefore follows that if the second derivative is positive at the stationary point, then the gradient is increasing. The fact that it is a stationary point in the first place means that this can only be a minimum. Conversely, if the second derivative is negative at that point, then it is a maximum. Now, if the second derivative is 0, we have a problem. It could be a point of inflexion, or it could still be an extremum. Examples of each of these cases are below - all have a second derivative equal to 0 at the stationary point in question: has a point of inflexion at has a minimum at has a maximum at However, this is not an insoluble problem. What we must do is continue to differentiate until we get, at the th derivative, a non-zero result at the stationary point: If is odd, then the stationary point is a true extremum. If the th derivative is positive, it is a minimum; if the th derivative is negative, it is a maximum. If is even, then the stationary point is a point of inflexion. As an example, let us consider the function We now differentiate until we get a non-zero result at the stationary point at (assume we have already found this point as usual): Therefore, is 4, so is 3. This is odd, and the fourth derivative is negative, so we have a maximum. Note that none of the methods given can tell you if this is a global extremum or just a local one. To do this, you would have to set the function equal to the height of the extremum and look for other roots. Critical Points Critical points are the points where a function's derivative is 0 or not defined. Suppose we are interested in finding the maximum or minimum on given closed interval of a function that is continuous on that interval. The extreme values of the function on that interval will be at one or more of the critical points and/or at one or both of the endpoints. We can prove this by contradiction. Suppose that the function has maximum at a point in the interval where the derivative of the function is defined and not . If the derivative is positive, then values slightly greater than will cause the function to increase. Since is not an endpoint, at least some of these values are in . But this contradicts the assumption that is the maximum of for in . Similarly, if the derivative is negative, then values slightly less than will cause the function to increase. Since is not an endpoint, at least some of these values are in . This contradicts the assumption that is the maximum of for in . A similar argument could be made for the minimum. Example 1 Consider the function on the interval . The unrestricted function has no maximum or minimum. On the interval , however, it is obvious that the minimum will be , which occurs at and the maximum will be , which occurs at . Since there are no critical points ( exists and equals everywhere), the extreme values must occur at the endpoints. Example 2 Find the maximum and minimum of the function on the interval . First start by finding the roots of the function derivative: : Now evaluate the function at all critical points and endpoints to find the extreme values. : From this we can see that the minimum on the interval is -24 when and the maximum on the interval is when Saddle Point A saddle point (in red) on the graph of z=x2−y2 (hyperbolic paraboloid) Saddle point between two hills (the intersection of the figure-eight -contour) In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function has a critical point at that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the -direction. The name derives from the fact that the prototypical example in two dimensions is a surface that curves up in one direction, and curves down in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In terms of contour lines, a saddle point in two dimensions gives rise to a contour graph or trace in which the contour corresponding to the saddle point's value appears to intersect itself. Saddle point on the countour plot is the point where level curves cross Mathematical discussion A simple criterion for checking if a given stationary point of a real-valued function F(x,y) of two real variables is a saddle point is to compute the function's Hessian matrix at that point: if the Hessian is indefinite, then that point is a saddle point. For example, the Hessian matrix of the function at the stationary point is the matrix which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point is a saddle point for the function but the Hessian matrix of this function at the origin is the null matrix, which is not indefinite. In the most general terms, a saddle point for a smooth function (whose graph is a curve, surface or hypersurface) is a stationary point such that the curve/surface/etc. in the neighborhood of that point is not entirely on any side of the tangent space at that point. The plot of y = x3 with a saddle point at 0 In a domain of one dimension, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum. Saddle surface Hyperbolic paraboloid A model of an elliptic hyperboloid of one sheet A monkey saddle A saddle surface is a smooth surface containing one or more saddle points. Classical examples of two-dimensional saddle surfaces in the Euclidean space are second order surfaces, the hyperbolic paraboloid (which is often referred to as "the saddle surface" or "the standard saddle surface") and the hyperboloid of one sheet. The Pringles potato chip or crisp is an everyday example of a hyperbolic paraboloid shape. Saddle surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature. A classical third-order saddle surface is the monkey saddle. Examples In a two-player zero sum game defined on a continuous space, the equilibrium point is a saddle point. For a second-order linear autonomous system, a critical point is a saddle point if the characteristic equation has one positive and one negative real eigenvalue. In optimization subject to equality constraints, the first-order conditions describe a saddle point of the Lagrangian. Other uses In dynamical systems, if the dynamic is given by a differentiable map f then a point is hyperbolic if and only if the differential of ƒ n (where n is the period of the point) has no eigenvalue on the (complex) unit circle when computed at the point. Then a saddle point is a hyperbolic periodic point whose stable and unstable manifolds have a dimension that is not zero. A saddle point of a matrix is an element which is both the largest element in its column and the smallest element in its row. Extrema for functions of two variables Suppose that f(x, y) is a differentiable real function of two variables whose second partial derivatives exist and are continuous. The Hessian matrix H of f is the 2 × 2 matrix of partial derivatives of f: Define D(x, y) to be the determinant of H. Finally, suppose that (a, b) is a critical point of f (that is, fx(a, b) = fy(a, b) = 0). Then the second partial derivative test asserts the following: If D(a, b) > 0 and fxx(a, b) > 0 then (a, b) is a local minimum of f. If D(a, b) > 0 and fxx(a, b) < 0 then (a, b) is a local maximum of f. If D(a, b) < 0 then (a, b) is a saddle point of f. If D(a, b) = 0 then the second derivative test is inconclusive, and the point (a, b) could be any of a minimum, maximum or saddle point. Sometimes other equivalent versions of the test are used. Note that in cases 1 and 2, the requirement that fxx fyy − fxy2 is positive at (x, y) implies that fxx and fyy have the same sign there. Therefore the second condition, that fxx be greater (or less) than zero, could equivalently be that fyy or tr(H) = fxx + fyy be greater (or less) than zero at that point. Functions of many variables For a function f of three or more variables, there is a generalization of the rule above. In this context, instead of examining the determinant of the Hessian matrix, one must look at the eigenvalues of the Hessian matrix at the critical point. The following test can be applied at any critical point a for which the Hessian matrix is invertible: If the Hessian is positive definite (equivalently, has all eigenvalues positive) at a, then f attains a local minimum at a. If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a. If the Hessian has both positive and negative eigenvalues then a is a saddle point for f (and in fact this is true even if a is degenerate). In those cases not listed above, the test is inconclusive. For functions of three or more variables, the determinant of the Hessian does not provide enough information to classify the critical point, because the number of jointly sufficient second-order conditions is equal to the number of variables, and the sign condition on the determinant of the Hessian is only one of the conditions. Note that in the one-variable case, the Hessian condition simply gives the usual second derivative test. In the two variable case, and are the principal minors of the Hessian. The first two conditions listed above on the signs of these minors are the conditions for the positive or negative definiteness of the Hessian. For the general case of an arbitrary number n of variables, there are n sign conditions on the n principal minors of the Hessian matrix that together are equivalent to positive or negative definiteness of the Hessian (Sylvester's criterion): for a local minimum, all the principal minors need to be positive, while for a local maximum, the minors with an odd number of rows and columns need to be negative and the minors with an even number of rows and columns need to be positive. Resources Local Extrema, Critical Points, & Saddle Points of Multivariable Functions - Calculus 3- Video by he Organic Chemistry Tutor 2019 Absolute Maximum and Minimum Values of Multivariable Functions - Calculus 3 Video by The Organic Chemistry Tutor 2019 Licensing Content obtained and/or adapted from: Extrema and Points of Inflection, Wikibooks: Calculus under a CC BY-SA license Second partial derivative test, Wikipedia under a CC BY-SA license Saddle point, Wikipedia under a CC BY-SA license Retrieved from " Navigation menu Personal tools Log in Namespaces Page Discussion Navigation Main page Recent changes Random page Help about MediaWiki Tools What links here Related changes Special pages Printable version Permanent link Page information Cite this page
12364	https://www.gigacalculator.com/calculators/percentage-calculator.php	Percentage Calculator ％ Use this versatile percentage calculator to easily find the percentage difference between two numbers, to calculate percent change (percentage increase, percentage decrease from a baseline), to find out what % is a given number from any other given number, as well as how much is x percent of y. Share calculator: Embed this tool: get code Related calculators Quick navigation: What is a percentage How to calculate percent change How to increase or decrease X by Y percent How to calculate X is what percent of Y How to calculate X percent of Y How to calculate percent difference Percent Points vs Relative Percentages Compounding and averaging percentages Many uses for a percentage calculator What is a percentage Before explaining how to use a percentage calculator to calculate percent change, percent difference or percentage of one number from another, it is useful to examine the basics of the concept of percentages. A percentage is a dimensionless number, represented as a fraction of 100, e.g. 50 out of 100 can be written as 50%, and 1 out of 10 can be written as 10%. A percentage is by definition a ratio. The sign for percent is "%", but the abbreviation "pct" is sometimes used in its place, while in older literature and documents one can encounter "per cent", where "cent" is an abbreviation of the Latin "centum" which literally means "one hundred", so the phrase means "per one hundred" - the literal definition of percentage. Percentages have a wide array of applications in many disciplines and everyday usage. They are common in statistics, social sciences, economics, finance, accounting. In everyday usage we often encounter percent off coupons. Promotions, sales, and various discounts are often expressed as percentage from a previous reference price of an item or service. Percentage calculations can be used when measuring productivity or load of a person or machine, e.g. "he is working at 100%" (at maximum capacity). Percentage increase or decrease are used to describe the relative growth or decline of something, e.g. a population, capital, personal wealth, etc. Differences between any two objects can be expressed as ratios or as percentage difference. The measurement error of a tool or process can be described in terms of percent error and can easily be computed using a percentage calculator. Read more on percent vs percentage. How to calculate percent change This is what most people mean when they want to know "how to calculate percentage", but for other possible percentage calculations see below. Percent change calculators are commonly employed when comparing quantities, business metrics, or other measurements from two time periods, the earlier one serving as a baseline. A percentage change calculation is also useful when comparing a new state of things to an old state of things, e.g. using the census to compare the number of people living in villages in a given municipality before versus after industrialization. Our calculator is of great assistance for calculating percent increase / decrease, but you can also find the percentage change on your own. For example, say you are reviewing the performance of your business on a monthly basis and you see that the past month you had 80 customers while the month before you were able to acquire only 64. To find the growth rate of your business versus prior month's base value you need to calculate percent change using the equation below. Percent Change Formula Percent change = new / old 100 - 100 where new is the newer quantity or measure, and old is the older quantity or measure. In the above example this would be 80 / 64 100 - 100 = 1.25 100 - 100 = 125 - 100 = 25%. Your monthly percentage change (percent growth, percent increase) from 64 to 80 was thus 25 percent versus the baseline from the month prior as you can verify by using the percentage change calculator. In another situation, you might be examining a proposition to increase your salary from $100,000 a year to $120,000 a year to keep you on the payroll and want to find what percent is the new salary versus your old one. If you do the math manually, start by dividing 120,000 by 100,000 to get 1.2. Then multiply by 100 to get 120. Finally subtract 100 which leaves 20%. Therefore, you were offered a 20% increase of your salary and as the new salary is 120% of your current salary. Similarly, you can use the calculator to calculate change in speed of different modes of transportation. If you compare a car or bus moving at 60 miles per hour to a high-speed train moving at 120 miles per hour, you can obtain the percentage change to be 100% meaning that the railway is twice as fast as the vehicle. How to increase or decrease X by Y percent Oftentimes one may want to alter an original number by a percentage of its value. For example, if you want to calculate what a 20% increase to the price of an item due to VAT would result in, you can use the percent increase formula below. Increase by percentage formula Increased value = base + base % increase / 100 For example, if the current price is $100, increasing it by 20% means calculating: $100 + $100 20 / 100 = $100 + $20 = $120 price after the percent increase. Decrease by percentage formula Decreased value = base - base % increase / 100 The formula for decreasing a value by a percentage is almost identical, but the plus has been replaced with a minus. For an example application, say one is calculating a price discount of 50% from an original price of $200, the calculation would be: $200 - $200 50 / 100 = $200 - $100 = $100 discounted price. How to calculate X is what percent of Y Let's say you are a car salesman and you have a car originally priced at $50,000, but you have done some calculation and determined that you can take $5,000 off the price of the car and still be ahead after the sale. How can you determine what percentage is $5,000 from $50,000? Obviously, just plugging in the numbers in the percent calculator above is the fastest way, but to do the math manually use the following formula: X is What Percent of Y Formula x is x / y 100 % of y so in this case that would be 5,000 / 50,000 100 = 0.1 100 = 10%. If you were to offer a $5,000 discount on a $50,000 car, that would be a 10% discount. In another example you might want to find what percentage of your total yearly income you have to pay in taxes. If your yearly income is $80,000 and you have calculated that your total tax amount is $36,000, then your tax rate is 36,000 / 80,000 / 100 = 0.45 100 = 45%, since $36,000 is 45 percent of $80,000. Other applications of this equation can be found in percent error calculations. Say a measurement of width, height or weight is off by a given absolute value, then the absolute error can be translated into percentage error by the X is what percent of Y formula. How to calculate X percent of Y Let's say you are told you are eligible to get a 20% discount on an item costing $500. How can you determine what the discount value is at this percentage? The formula to use is: X Percent of Y Formula x% of y is y (x / 100) In the example above that would be calculated as 500 (20 / 100) = 500 0.2 = 100. If you were to purchase the $500 item with 20% off, you would be getting a discount of $100. Another example of using a percent calculator would be if you wonder how many minutes are 75% of a 60-minute video. The answer is 60 (75 / 100) = 45 minutes as you can verify with our tool. How to calculate percent difference The percentage difference of two numbers (quantities) a and b is the relative difference, expressed as a percent. It should be calculated using the formula: Percentage Difference Formula Percent Difference = \|a - b\| / ((a + b) / 2) 100 percent For example, if one item costs $5 and another costs $6 the percent difference between them is: \|5 - 6\| / ((5 + 6) / 2) 100 = 1 / (11 / 2) 100 = 1 / 5.5 100 = 18.18%. Please, note that this doesn't mean that 5 is 18.18% smaller than 6, or that 6 is 18.18% larger than 5. The correct percentages if you are asking the question of "what percent is a from b" would be 16.66% and 20%, respectively, as explained above. Similarly, a percentage difference calculator might be useful if computing the difference in elevation of two mountains. If one is 6,000 feet high and the other 3,000 feet, the absolute difference is 2000 ft while the percent difference is 40%. Percentage difference is useful in a few situations, so it should be used with care. For example, one should not use percentage difference when comparing time periods, as the first metric is another state of the second metric, so percent change is the appropriate calculation. Similarly, finding a price changed by some percentage should not be done using percentage difference. Percent Points vs Relative Percentages You might have noted that often when newspapers are talking about USA elections for members of parliament, premier, or president of a town, municipality, or a whole country, there would be polls tracking the favorability of each candidate. Differences in sentiment and changes in said polling measures will often be communicated in percent points instead of percentages. A percent point is like absolute difference, but expressed in percentages. Note that such calculations only apply when comparing differences in percentage measures - something not supported by our percentage difference calculator mode. For example, the percent in favor of candidate A might be 40% before a certain political event and only 35% after. The percent change is simply 40% minus 35% which equals 5 p.p. (but is a percentage change of 12.5 percent). Compounding and averaging percentages Percentages should not be added up (compounded) or averaged like simple numbers, as this will result in an incorrect end result. Compounding is often encountered in finance, e.g. when calculating compound interest or multi-year return of a financial portfolio. Averaging percentages is often encountered in business calculations, for example to determine the average growth of a company, but also in finance and banking where average growth of an asset or asset portfolio may be calculated. Here is an example of adding percentages: say you have a $100,000 bank deposit at a 2% interest rate, applied yearly at the end of the year. If you keep it for 5 years, you might think that the way to calculate your deposit's value at the end of the 5-year period is to simply multiply 2% x 5 = 10% (or, equivalently, 2% + 2% + 2% + 2% + 2% = 10%), add 100% and then use our percentage calculator to calculate 110% of $100,000. By this calculation you would expect to have $110,000 at the end of the period (10% of 100,000 is $10,000). However, you will have $110,408, since at the end of each year you will get your interest but then in each of the following years you will accrue interest over the interest from the first year. And so on for the second, third... To average percentage growth a year, it would be incorrect to just sum up the growth % in each year and then divide by the number of years. Let us say you are the founder of an organization which has an asset that grew 5% the first year, 6% the second year, 10% the third year, and then lost 10% the fourth year. The growth of the principal value is not 5% + 6% + 10% - 10% = 11%, but it is instead the geometric mean: 2.4549% times the number of years = 2.4549 x 4 = 9.82%. Again, this is not something you can solve for using the above calculator. The above peculiarities are also the reason why if you lose 25% of some asset, you need to grow it 33.33% to recoup the loss. A quick calculation shows that $10,000 25% = $7,500, while $10,000 of $7,500 = 133.33% (or $10,000 - $7,500 = $2,500 to return to zero, and $2.500 is 33.33% of $7,500). Many uses for a percentage calculator As the numerous examples above demonstrate, a percentage calculator can be useful in so many practical scenarios. Changes in electoral sentiment in democratic processes, industrial production, stocks of material goods in warehouses, albums sold by an artist / musician, publications in a scientific journal, personnel changes in an organization, as well as natural things such as the amount of flora or fauna in given island, etc. can be computed using such a tool. Another application of percentage calculations is for differences in achievements such as records at Olympic games and other tournaments and championships. A specialized calculator can be used to measure the relative difference in physical quantities such as speed, density, luminosity, reflectivity, strength, tensile strength, and so on. However, a calculator is most often used for financial assets and benchmarks and for everyday things such as discounts, commissions, and tipping at a restaurant. Cite this calculator & page If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Percentage Calculator", [online] Available at: URL [Accessed Date: 26 Sep, 2025]. 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12365	https://www.nejm.org/doi/full/10.1056/NEJMoa1501035	Skip to main content Create an E-mail Alert for This Article Perioperative Bridging Anticoagulation in Patients with Atrial Fibrillation Authors: James D. Douketis, M.D., Alex C. Spyropoulos, M.D., Scott Kaatz, D.O., Richard C. Becker, M.D., Joseph A. Caprini, M.D., Andrew S. Dunn, M.D., David A. Garcia, M.D., +7 , Alan Jacobson, M.D., Amir K. Jaffer, M.D., M.B.A., David F. Kong, M.D., Sam Schulman, M.D., Ph.D., Alexander G.G. Turpie, M.B., Vic Hasselblad, Ph.D., and Thomas L. Ortel, M.D., Ph.D., for the BRIDGE Investigators -7Author Info & Affiliations Published August 27, 2015 N Engl J Med 2015;373:823-833 DOI: 10.1056/NEJMoa1501035 VOL. 373 NO. 9 Copyright © 2015 Abstract Background It is uncertain whether bridging anticoagulation is necessary for patients with atrial fibrillation who need an interruption in warfarin treatment for an elective operation or other elective invasive procedure. We hypothesized that forgoing bridging anticoagulation would be noninferior to bridging with low-molecular-weight heparin for the prevention of perioperative arterial thromboembolism and would be superior to bridging with respect to major bleeding. Methods We performed a randomized, double-blind, placebo-controlled trial in which, after perioperative interruption of warfarin therapy, patients were randomly assigned to receive bridging anticoagulation therapy with low-molecular-weight heparin (100 IU of dalteparin per kilogram of body weight) or matching placebo administered subcutaneously twice daily, from 3 days before the procedure until 24 hours before the procedure and then for 5 to 10 days after the procedure. Warfarin treatment was stopped 5 days before the procedure and was resumed within 24 hours after the procedure. Follow-up of patients continued for 30 days after the procedure. The primary outcomes were arterial thromboembolism (stroke, systemic embolism, or transient ischemic attack) and major bleeding. Results In total, 1884 patients were enrolled, with 950 assigned to receive no bridging therapy and 934 assigned to receive bridging therapy. The incidence of arterial thromboembolism was 0.4% in the no-bridging group and 0.3% in the bridging group (risk difference, 0.1 percentage points; 95% confidence interval [CI], −0.6 to 0.8; P=0.01 for noninferiority). The incidence of major bleeding was 1.3% in the no-bridging group and 3.2% in the bridging group (relative risk, 0.41; 95% CI, 0.20 to 0.78; P=0.005 for superiority). Conclusions In patients with atrial fibrillation who had warfarin treatment interrupted for an elective operation or other elective invasive procedure, forgoing bridging anticoagulation was noninferior to perioperative bridging with low-molecular-weight heparin for the prevention of arterial thromboembolism and decreased the risk of major bleeding. (Funded by the National Heart, Lung, and Blood Institute of the National Institutes of Health; BRIDGE ClinicalTrials.gov number, NCT00786474.) Quick Take The BRIDGE Trial 1m 35s For patients with atrial fibrillation who are receiving warfarin and require an elective operation or other elective invasive procedure, the need for bridging anticoagulation during perioperative interruption of warfarin treatment has long been uncertain.1–3 Each year, this common clinical scenario affects approximately one in six warfarin-treated patients with atrial fibrillation.4,5 Warfarin treatment is typically stopped 5 days before an elective procedure to allow its anticoagulant effect to wane; it is resumed after the procedure, when hemostasis is secured, at which point 5 to 10 days of treatment is required to attain therapeutic anticoagulation.6,7 During the interruption of warfarin treatment, bridging anticoagulation therapy, typically with low-molecular-weight heparin, can be given to minimize the time that patients do not have an adequate level of anticoagulation, with the intent of minimizing the risk of perioperative arterial thromboembolism, such as stroke.6 Multiple observational studies have assessed the timing and dosing of perioperative bridging with low-molecular-weight heparin.8–15 However, the fundamental question of whether bridging anticoagulation is necessary during perioperative warfarin interruption has remained unanswered.16–18 Because of the lack of evidence, practice guidelines have provided weak and inconsistent recommendations regarding the need for bridging anticoagulation.19–21 Against this background, the Bridging Anticoagulation in Patients who Require Temporary Interruption of Warfarin Therapy for an Elective Invasive Procedure or Surgery (BRIDGE) trial was designed to address a simple question: in patients with atrial fibrillation, is heparin bridging needed during interruption of warfarin therapy before and after an operation or other invasive procedure? We hypothesized that forgoing bridging altogether would be noninferior to bridging with low-molecular-weight heparin for the prevention of perioperative arterial thromboembolism and would be superior to bridging with regard to the outcome of major bleeding. Methods Study Design and Oversight The BRIDGE trial was a randomized, double-blind, placebo-controlled trial. The protocol (available with the full text of this article at NEJM.org) was designed by the steering committee (see the Supplementary Appendix, available at NEJM.org, for a full list of trial personnel) and approved by the institutional review board at each participating clinical center. The Duke Clinical Research Institute managed the study. The clinical coordinating center was responsible for study coordination, randomization, and distribution of the study drug. The data coordinating center was responsible for maintenance of the study database, data validation, and analyses. Eisai donated the dalteparin, and University of Iowa Pharmaceuticals prepared the matching placebo. Eisai had no role in the design or conduct of the study, the analysis of the data, or the preparation of the manuscript. The steering committee vouches for the completeness and accuracy of the data and analyses and for the fidelity of this report to the trial protocol. Patients Patients were eligible to participate in the trial if they were 18 years of age or older; had chronic (permanent or paroxysmal) atrial fibrillation or flutter, confirmed by means of previous electrocardiography or pacemaker interrogation (patients with atrial fibrillation associated with valvular disease, including mitral valve disease, were eligible); had received warfarin therapy for 3 months or longer, with an international normalized ratio (INR) therapeutic range of 2.0 to 3.0; were undergoing an elective operation or other elective invasive procedure that required interruption of warfarin therapy; and had at least one of the following CHADS2 stroke risk factors: congestive heart failure or left ventricular dysfunction, hypertension, age of 75 years or older, diabetes mellitus, or previous ischemic stroke, systemic embolism, or transient ischemic attack. Patients were not eligible if they had one or more of the following: a mechanical heart valve; stroke, systemic embolism, or transient ischemic attack within the previous 12 weeks; major bleeding within the previous 6 weeks; creatinine clearance of less than 30 ml per minute; platelet count of less than 100×103 per cubic millimeter; or planned cardiac, intracranial, or intraspinal surgery. A complete list of the trial inclusion and exclusion criteria is provided in the Supplementary Appendix. All participants provided written informed consent. Procedures Patients were randomly assigned to receive bridging anticoagulation therapy with dalteparin sodium (100 IU per kilogram of body weight administered subcutaneously twice daily) or to receive no bridging therapy (i.e., a matching subcutaneous placebo) from 3 days before the procedure until 24 hours before the procedure and then for 5 to 10 days after the procedure. Randomization was stratified according to study center either with the use of an interactive voice-response system with a toll-free telephone number and access codes or through the Internet. The study drugs were provided in identical vials. The administration of study drug followed a standardized perioperative management protocol (Figure 1). Warfarin treatment was stopped 5 days before the procedure, and administration of the study drug (dalteparin or matching placebo) was started 3 days before the procedure. The last preprocedure dose of dalteparin or placebo was given in the morning approximately 24 hours before the procedure.22,23 Warfarin treatment was restarted on the evening of or the day after the procedure, at the patient’s usual dose. Administration of dalteparin or placebo was resumed 12 to 24 hours after a minor (or low-bleeding-risk) procedure and 48 to 72 hours after a major (or high-bleeding-risk) procedure.8,10 The designation of a procedure as having a low or high bleeding risk was guided by means of a classification scheme (see Table S1 in the Supplementary Appendix), but the final determination of risk was left to the investigator’s discretion. The patient continued to take the study drug after the procedure until the INR was 2 or higher on one occasion. Patients had follow-up encounters by telephone weekly, with the final encounter 30 to 37 days after the procedure. Perioperative management of antiplatelet therapy was left to the site investigator’s discretion. Figure 1 Study Outcomes All study outcomes were assessed by 37 days after the procedure. The primary efficacy outcome was arterial thromboembolism, including stroke (ischemic or hemorrhagic), transient ischemic attack, and systemic embolism, and the primary safety outcome was major bleeding. The secondary efficacy outcomes were acute myocardial infarction, deep-vein thrombosis, pulmonary embolism, and death, and the secondary safety outcome was minor bleeding. The definitions of the outcomes are provided in the Supplementary Appendix. All study outcomes were independently and blindly adjudicated. Statistical Analysis The primary efficacy outcome was arterial thromboembolism at 30 days. The initial sample-size estimates for arterial thromboembolism were based on the results of contemporaneous cohort studies, which suggested that the rate in the bridging group would be 1.0%.8–10,24,25 We also assumed that the rate in the no-bridging group would be 1.0%. The primary analysis of efficacy was a noninferiority analysis with a one-sided test at the 0.025 level. The noninferiority margin was set at 1.0%. We determined that the hypothesis of inferiority would be rejected if the upper boundary of the 95% confidence interval for the difference in rates would be less than 1.0 percentage point. We prespecified that the 95% confidence interval for the difference in event rates would be calculated with the use of methods based on Barnard’s test,26 because this test permits the calculation of confidence intervals in analyses with small sample sizes. The confidence interval values were calculated with the use of StatXact software, version 9 (Cytel).27 The primary safety outcome was major bleeding at 30 days after the procedure. The null hypothesis of no difference in the incidence of major bleeding was tested with a two-sided test at the 0.05 level. The expected bleeding rates were 1.0% in the no-bridging arm and 3.0% in the bridging arm. The P value was calculated with the use of Fisher’s mid-P test, as implemented in SAS software, version 9.3 (SAS Institute), and the 95% confidence interval was a likelihood-ratio confidence interval calculated with the same version of SAS. We calculated that a sample of 1641 patients per group would give the study 80% power to detect the noninferiority of no bridging therapy, assuming a rate of arterial thromboembolism of 1.0% in each group and a noninferiority margin of 1.0%, at a one-sided alpha level of 0.025 for arterial thromboembolism and a two-sided alpha level of 0.05 for bleeding. With a 10% allowance for patients withdrawing from the study, the required sample size was 1813 per group. We calculated that this sample size would also give the study more than 99% power to detect the expected difference in bleeding rates. After approximately 850 patients had been enrolled, it was clear that the rate of arterial thromboembolism, as assessed by investigators who were unaware of the study-group assignments, was less than 0.5%, and we determined that a revised sample size of 2526 would provide at least 90% power for each primary end point. After 1720 patients were enrolled, the rate of arterial thromboembolism was 0.46%, and the bleeding rate was 2.3% in the entire population. A revised sample size of 1882 was calculated on the basis of the estimate that this would provide nearly 90% power for the two primary end points. Results Patients As shown in Figure 2, we recruited 1884 patients during the period from July 2009 through December 2014 at 108 sites in the United States and Canada; 950 patients were assigned to the placebo (no-bridging) group, and 934 patients were assigned to receive bridging treatment with dalteparin (bridging group). Table 1 shows the characteristics of the patients at baseline. The mean age of the patients was 71.7 years, and 73.4% of patients were male; the mean body weight was 95.8 kg. The mean CHADS2 score (CHADS2 scores range from 1 to 6, with higher scores indicating a greater risk of stroke) was 2.3; 38.3% of patients had a CHADS2 score of 3 or higher. A total of 34.7% of the patients were taking aspirin, and 7.2% were taking another antiplatelet drug. Figure 2 Table 1 Of the 1884 patients enrolled in the trial, 1722 actually underwent the anticipated procedure (as-treated group), and 162 did not. The categories and types of operations and procedures that the participants underwent are shown in Table S2 in the Supplementary Appendix. The most common procedures were gastrointestinal (44.0%), cardiothoracic (17.2%), and orthopedic (9.2%). Overall, 89.4% of patients underwent a procedure that was classified as minor (low bleeding risk) according to the prespecified classification; however, 69.1% were treated as having a low bleeding risk by the site investigator. Perioperative Anticoagulant Management The mean (±SD) number of doses of study drug administered was 5.0±1.1 before the procedure and 16.0±7.9 after the procedure (Table 2). The mean dose of dalteparin administered was 9093±2240 IU subcutaneously twice daily. Adherence to the study-drug protocol, defined as administration of 100% of protocol-specified doses of study drug, was 86.5% before the procedure and 96.5% after the procedure. Table 2 Study Outcomes Of the 1884 patients enrolled in the trial, 71 discontinued participation and did not provide outcome data; therefore, data from 1813 patients were available for the analysis (Figure 2). At 30 days after the procedure, the incidence of arterial thromboembolism was 0.4% (four events among 918 patients) in the no-bridging group and 0.3% (three events among 895 patients) in the bridging group (mean between-group difference, 0.1 percentage points; 95% confidence interval [CI], −0.6 to 0.8; P=0.01 for noninferiority; P=0.73 for superiority) (Table 3). In an as-treated analysis, the rates of arterial thromboembolism were 0.3% (three events among 875 patients) in the no-bridging group and 0.4% (three events among 847 patients) in the bridging group (mean between-group difference, 0.0 percentage points; 95% CI, −0.7 to 0.7; P=0.006 for noninferiority). Patients in whom arterial thromboembolism occurred had a mean CHADS2 score of 2.6 (range, 1 to 4), and five of the seven events occurred after a minor procedure. The median time to an arterial thromboembolism event after the procedure was 19.0 days (interquartile range, 6.0 to 23.0). Table 3 Major bleeding occurred in 1.3% of the patients (12 of 918) in the no-bridging group and in 3.2% (29 of 895) in the bridging group, which indicated that no bridging was superior to bridging with regard to major bleeding (relative risk, 0.41; 95% CI, 0.20 to 0.78; P=0.005). None of the instances of major bleeding were fatal. Forgoing bridging was associated with a risk of minor bleeding that was significantly lower than the risk associated with bridging (12.0% vs. 20.9%, P<0.001). The median time to a major bleeding outcome after the procedure was 7.0 days (interquartile range, 4.0 to 18.0). There was no significant difference between the groups in the rates of acute myocardial infarction, deep-vein thrombosis, pulmonary embolism, or death. Information on the causes of death and times to death is provided in Table S3 in the Supplementary Appendix. Discussion We found that in patients with atrial fibrillation who require perioperative interruption of warfarin treatment for an elective procedure, a strategy of discontinuing warfarin treatment without the use of bridging anticoagulation was noninferior to the use of bridging anticoagulation for the prevention of arterial thromboembolism; in addition, bridging conferred a risk of major bleeding that was nearly triple the risk associated with no bridging. There was also less minor bleeding without bridging than there was with bridging, and there was no significant difference between the groups with regard to myocardial infarction, venous thromboembolism, or death. Taken together, these findings show that there is a net clinical benefit in favor of a strategy of forgoing bridging, as compared with perioperative bridging with low-molecular-weight heparin. The findings in our trial are consistent with those from nonrandomized comparisons of these strategies. A meta-analysis of observational studies involving a total of 12,278 patients with atrial fibrillation or mechanical heart valves who received or did not receive bridging with low-molecular-weight heparin showed no significant difference in the rate of arterial thromboembolism (odds ratio with bridging, 0.80; 95% CI, 0.42 to 1.54) but a higher rate of major bleeding (odds ratio, 3.60; 95% CI, 1.52 to 8.50) in association with bridging.28 In a substudy of the Randomized Evaluation of Long-Term Anticoagulation Therapy (RE-LY),29 in which patients with atrial fibrillation were randomly assigned to receive warfarin or dabigatran in an open-label manner, bridging anticoagulation was associated with a rate of major bleeding that was higher than that associated with no bridging (6.8% vs. 1.6%, P<0.001) among 1424 warfarin-treated patients who had treatment interruption for an elective procedure, and there was no significant effect on arterial thromboembolism (0.5% vs. 0.2%, P=0.32).30 The Outcomes Registry for Better Informed Treatment of Atrial Fibrillation study (ORBIT-AF), which involved 2200 patients with atrial fibrillation who required an elective procedure, also showed a higher rate of bleeding if bridging anticoagulation therapy was used during perioperative interruption of warfarin treatment.31 The rationale for the use of bridging anticoagulation therapy has been anchored on the premise that the associated higher bleeding risk was clinically acceptable because it would be offset by a lower risk of perioperative arterial thromboembolism.32 The findings from the BRIDGE trial as well as from nonrandomized studies suggest that the perioperative risk of arterial thromboembolism in patients with atrial fibrillation during interruption of warfarin treatment may have been overstated and may not be mitigated by bridging anticoagulation. Indeed, the mechanisms of perioperative arterial thromboembolism may be more closely related to factors such as the type of procedure33 and to intraoperative alterations in blood pressure.34 The premise that warfarin interruption leads to rebound hypercoagulability and that the milieu of the procedure confers a prothrombotic state, which in turn leads to arterial thromboembolism, is not supported by the results of this trial.35–37 There are potential limitations of the BRIDGE trial. First, although we aimed to recruit a representative sample of patients with atrial fibrillation for whom bridging anticoagulation is normally considered, certain groups were underrepresented. Few patients had a CHADS2 score of 5 or 6, although the mean score of 2.3 is similar to that among patients with atrial fibrillation who were assessed in recent trials and patient registries, in which the mean scores were between 2.1 and 2.8.29,38–40 Patients undergoing major surgical procedures associated with high rates of arterial thromboembolism and bleeding (e.g., carotid endarterectomy, major cancer surgery, cardiac surgery, or neurosurgery)19,33 were not represented in the trial, although the procedures performed were representative of the most common interventions patients undergo during an interruption of therapeutic anticoagulation, the majority of which are low-risk procedures, such as colonoscopy or ambulatory surgery.4,5,41 In addition, the findings should not be applied to patients with mechanical heart valves, who were specifically not included in the trial. Second, the overall rate of arterial thromboembolism was lower than expected, which potentially affected the power of the trial to detect a benefit associated with bridging. Although we had expected perioperative arterial thromboembolism rates to be approximately 1.0%,8,9,12,24 the observed rate (0.4%) is similar to rates in recent studies involving patients who had perioperative interruption of warfarin treatment.4,5,31,42 In addition, the noninferiority margin we selected turned out to be large in relation to the actual observed event rate; it reflected the original estimate of the event rate as specified in the trial protocol. Third, the observed rate of major bleeding in the bridging group (3.2%, with no instances of fatal bleeding) may be considered to be modest. However, our bridging protocol was designed to minimize bleeding, and the higher rates of bleeding reported in other studies of bridging anticoagulation probably reflect resumption of bridging therapy too soon after operations with a high bleeding risk10,43 or a lack of standardized bridging protocols.28,30 Fourth, the reduction in the study sample size may raise concerns. This reduction was driven by the lower rate of arterial thromboembolism overall, with the proviso that power was maintained to address the primary study hypotheses. Although extending the trial was considered, this was not done because the added statistical power would have been negligible and because recruitment had been challenging throughout the course of the trial. Finally, one may contend that the trial findings have diminished relevance because of the decreasing use of warfarin in the treatment of patients with atrial fibrillation, given the availability of the newer direct oral anticoagulants.6 However, warfarin remains widely used among patients with atrial fibrillation.44–46 Furthermore, the trial findings may also apply to the newer agents. In the substudy of the RE-LY trial discussed above, dabigatran-treated patients who had treatment interruption for an elective procedure had more major bleeding with bridging therapy than without bridging therapy, and there was no significant effect on arterial thromboembolism.30 In conclusion, in the BRIDGE trial, we found that for patients with atrial fibrillation who require temporary interruption of warfarin treatment for an elective operation or other elective invasive procedure, a strategy of forgoing bridging anticoagulation was noninferior to perioperative bridging with low-molecular-weight heparin for the prevention of arterial thromboembolism. The strategy of forgoing bridging treatment also decreased the risk of major bleeding. Notes This article was published on June 22, 2015, at NEJM.org. Supported by grants from the National Heart, Lung, and Blood Institute for the clinical coordinating center (U01HL087229, to Dr. Ortel) and for the data coordinating center (U01HL086755, to Dr. Hasselblad). Eisai supplied the active drug, dalteparin sodium (Fragmin), through an unrestricted investigator-initiated grant to Dr. Ortel. Dr. Douketis reports receiving fees for serving on advisory boards from Biotie, Portola, and the Medicines Company, honoraria from Bristol-Myers Squibb, Pfizer, and Sanofi Aventis, consulting fees from Boehringer Ingelheim, Bayer, Janssen, Bristol-Myers Squibb, Daiichi-Sankyo, and Actelion, and grant support from Boehringer Ingelheim; Dr. Spyropoulos, receiving consulting fees from Janssen, Boehringer Ingelheim, Daiichi-Sankyo, and Pfizer, and grant support from Daiichi-Sankyo; Dr. Kaatz, receiving consulting and lecture fees from Boehringer Ingelheim, Bristol-Myers Squibb/Pfizer Alliance, Janssen, CSL Behring, and Daiichi-Sankyo, and grant support through his institution from Boehringer Ingelheim, Bristol-Myers Squibb/Pfizer Alliance, Janssen, Iverson Genetics Diagnostics/Medicare, and Blue Cross Blue Shield of Michigan; Dr. Becker, receiving fees for serving on scientific advisory boards from Janssen, Portola, and Daiichi-Sankyo; Dr. Caprini, receiving fees for serving on advisory boards from Bristol-Myers Squibb and Pfizer, fees for serving on a steering committee from Janssen, and lecture fees from Sanofi Aventis; Dr. Garcia, receiving consulting fees from Pfizer, Genzyme, Boehringer Ingelheim, Bristol-Myers Squibb, Portola, and Daiichi-Sankyo, and grant support from Bayer; Dr. Jacobson, receiving consulting fees from Bristol-Myers Squibb/Pfizer Alliance, Daiichi-Sankyo, Boehringer Ingelheim, Janssen, Roche Diagnostics, and Alere, and lecture fees from Bristol-Myers Squibb/Pfizer Alliance, Daiichi-Sankyo, Boehringer Ingelheim, and Janssen; Dr. Jaffer, receiving consulting fees from Pfizer, Janssen, Daiichi-Sankyo, Boehringer Ingelheim, Marathon, and Medtronic; and Dr. Ortel, receiving consulting fees from Instrumentation Laboratory, CSL Behring, and Daiichi-Sankyo, and grant support from Instrumentation Laboratory. No other potential conflict of interest relevant to this article was reported. Disclosure forms provided by the authors are available with the full text of this article at NEJM.org. We thank Andrei L. Kindzelski, who served as the National Institutes of Health program official; Jill Lynch, University of Iowa Pharmaceuticals, who was the manufacturer of the matching placebo; and James Bernstein, Live Oak Pharmaceuticals Consulting, for pharmacy consultancy regarding the processes involved in the manufacture of the matching placebo and for packaging the study drug kits for distribution to the trial sites. Supplementary Material Protocol (nejmoa1501035_protocol.pdf) Download 1.05 MB Supplementary Appendix (nejmoa1501035_appendix.pdf) Download 459.49 KB Financial Disclosures (nejmoa1501035_disclosures.pdf) Download 492.27 KB References 1. Kearon, C, Hirsh, J. Management of anticoagulation before and after elective surgery. N Engl J Med 1997;336:1506-1511 Go to Citation Crossref PubMed Web of Science Google Scholar 2. Piazza, G, Goldhaber, SZ. Periprocedural management of the chronically anticoagulated patient: critical pathways for bridging therapy. Crit Pathw Cardiol 2003;2:96-103 Go to Citation Crossref PubMed Google Scholar 3. Gallego, P, Apostolakis, S, Lip, GY. Bridging evidence-based practice and practice-based evidence in periprocedural anticoagulation. Circulation 2012;126:1573-1576 Go to Citation Crossref PubMed Web of Science Google Scholar 4. Healey, JS, Eikelboom, J, Douketis, J, et al. Periprocedural bleeding and thromboembolic events with dabigatran compared with warfarin: results from the Randomized Evaluation of Long-Term Anticoagulation Therapy (RE-LY) randomized trial. Circulation 2012;126:343-348[Erratum, Circulation 2012;126(10):e160.] Crossref PubMed Web of Science Google Scholar a [...] patients with atrial fibrillation. b [...] such as colonoscopy or ambulatory surgery. c [...] interruption of warfarin treatment. 5. Garcia, D, Alexander, JH, Wallentin, L, et al. Management and clinical outcomes in patients treated with apixaban vs warfarin undergoing procedures. Blood 2014;124:3692-3698 Crossref PubMed Web of Science Google Scholar a [...] patients with atrial fibrillation. b [...] such as colonoscopy or ambulatory surgery. c [...] interruption of warfarin treatment. 6. Baron, TH, Kamath, PS, McBane, RD. Management of antithrombotic therapy in patients undergoing invasive procedures. N Engl J Med 2013;368:2113-2124 Crossref PubMed Web of Science Google Scholar a [...] to attain therapeutic anticoagulation. b [...] arterial thromboembolism, such as stroke. c [...] of the newer direct oral anticoagulants. 7. Schulman, S, Hwang, HG, Eikelboom, JW, Kearon, C, Pai, M, Delaney, J. Loading dose vs. maintenance dose of warfarin for reinitiation after invasive procedures: a randomized trial. J Thromb Haemost 2014;12:1254-1259 Go to Citation Crossref PubMed Web of Science Google Scholar 8. Douketis, JD, Johnson, JA, Turpie, AG. Low-molecular-weight heparin as bridging anticoagulation during interruption of warfarin: assessment of a standardized periprocedural anticoagulation regimen. Arch Intern Med 2004;164:1319-1326 Crossref PubMed Web of Science Google Scholar a [...] bridging with low-molecular-weight heparin. b [...] a major (or high-bleeding-risk) procedure. c [...] rate in the bridging group would be 1.0%. d [...] rates to be approximately 1.0%, 9. Kovacs, MJ, Kearon, C, Rodger, M, et al. Single-arm study of bridging therapy with low-molecular-weight heparin for patients at risk of arterial embolism who require temporary interruption of warfarin. 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Garcia, DA, Regan, S, Henault, LE, et al. Risk of thromboembolism with short-term interruption of warfarin therapy. Arch Intern Med 2008;168:63-69 Go to Citation Crossref PubMed Web of Science Google Scholar 26. Barnard, GA. Significance tests for 2 X 2 tables. Biometrika 1947;34:123-138 Go to Citation PubMed Web of Science Google Scholar 27. StatXact Version 9 with Cytel Studio. Cambridge, MA: CYTEL Software, 2010. Go to Citation Google Scholar 28. Siegal, D, Yudin, J, Kaatz, S, Douketis, JD, Lim, W, Spyropoulos, AC. Periprocedural heparin bridging in patients receiving vitamin K antagonists: systematic review and meta-analysis of bleeding and thromboembolic rates. Circulation 2012;126:1630-1639 Crossref PubMed Web of Science Google Scholar a [...] 1.52 to 8.50) in association with bridging. b [...] a lack of standardized bridging protocols. 29. Connolly, SJ, Ezekowitz, MD, Yusuf, S, et al. Dabigatran versus warfarin in patients with atrial fibrillation. 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Patterns of initiation of oral anticoagulants in patients with atrial fibrillation- quality and cost implications. Am J Med 2014;127:1075.e1-1082.e1 Go to Citation Crossref Web of Science Google Scholar 46. Olesen, JB, Sørensen, R, Hansen, ML, et al. Non-vitamin K antagonist oral anticoagulation agents in anticoagulant naïve atrial fibrillation patients: Danish nationwide descriptive data 2011-2013. Europace 2015;17:187-193 Go to Citation Crossref PubMed Web of Science Google Scholar Information & Authors Information Published In New England Journal of Medicine Volume 373 • Number 9 • August 27, 2015 Pages: 823-833 Copyright Copyright © 2015 Massachusetts Medical Society. All rights reserved. For personal use only. Any commercial reuse of NEJM Group content requires permission. History Published ahead of print: June 22, 2015 Published online: August 27, 2015 Published in issue: August 27, 2015 Topics Anticoagulation/Thromboembolism (Cardiology) Anticoagulation/Thromboembolism (Pulmonary/Critical Care) Arrhythmias/Pacemakers/Defibrillators Clinical Medicine General Coagulation Authors Authors James D. Douketis, M.D., Alex C. Spyropoulos, M.D., Scott Kaatz, D.O., Richard C. Becker, M.D., Joseph A. Caprini, M.D., Andrew S. Dunn, M.D., David A. Garcia, M.D., Alan Jacobson, M.D., Amir K. Jaffer, M.D., M.B.A., David F. Kong, M.D., Sam Schulman, M.D., Ph.D., Alexander G.G. Turpie, M.B., Vic Hasselblad, Ph.D., and Thomas L. Ortel, M.D., Ph.D., for the BRIDGE Investigators Affiliations From St. Joseph’s Healthcare Hamilton (J.D.D.) and the Department of Medicine (J.D.D.) and Hamilton Health Science Center (S.S., A.G.G.T.), McMaster University, Hamilton, ON, Canada; Hofstra North Shore–Long Island Jewish School of Medicine, Manhasset (A.C.S.), and Mount Sinai Medical Center, New York (A.S.D.) — both in New York; Hurley Medical Center, Flint, MI (S.K.); University of Cincinnati College of Medicine, Cincinnati (R.C.B.); NorthShore University HealthSystem, Evanston (J.A.C.), and Rush University Medical Center, Chicago (A.K.J.) — both in Illinois; University of Washington Medical Center, Seattle (D.A.G.); Veterans Affairs Loma Linda Healthcare System, Loma Linda, CA (A.J.); and Duke Clinical Research Institute (D.F.K., V.H.) and Department of Medicine (T.L.O.), Duke University Medical Center, Durham, NC. Notes Address reprint requests to Dr. Ortel at Duke University Medical Center, Box 3422, Durham, NC, 27710, or at thomas.ortel@duke.edu. A complete list of investigators in the Bridging Anticoagulation in Patients who Require Temporary Interruption of Warfarin Therapy for an Elective Invasive Procedure or Surgery (BRIDGE) study is provided in the Supplementary Appendix, available at NEJM.org. Metrics & Citations Metrics Altmetrics See more details Picked up by 6 news outlets Blogged by 6 Posted by 800 X users Mentioned by 1 peer review sites On 51 Facebook pages Referenced in 1 Wikipedia pages Mentioned in 7 Google+ posts Highlighted by 1 platforms On 1 videos Referenced in 22 clinical guideline sources 1557 readers on Mendeley 5 readers on CiteULike Citations Export citation Select the format you want to export the citation of this publication. Cited by Ashish Tripathi, Sanjana Arsha, Daniel Casey, Fiyinfoluwa Olasoko, William H. Frishman, Wilbert S. Aronow,Perioperative Anticoagulation Management: Revisiting Bridging Strategies for Mechanical and Nonmechanical Indications, Cardiology in Review, (2025). Crossref 2. Paloma Pellegrino, Thomas Paul Scherer, Marian Severin Wettstein, Melanie Baumgartner, Daniel Eberli, Cédric Poyet, Uwe Bieri,Development of a nomogram for prediction of postoperative bleeding after transurethral resection of bladder tumors, International Urology and Nephrology, (2025). Crossref 3. Liqi Shu, Yasmin N. Aziz, Adam de Havenon, Steven R. Messe, Thanh N. Nguyen, Nicole B. Sur, Lize Xiong, Shadi Yaghi,Perioperative Stroke: Mechanisms, Risk Stratification, and Management, Stroke, 56, 9, (2798-2809), (2025). Crossref 4. Sungmin Suh, Young-Lan Kwak, Jae-Kwang Shim,Response to letter regarding ‘association between ex vivo thrombogenicity and ischemic outcome in off-pump coronary surgery’, Annals of Medicine, 57, 1, (2025). Crossref 5. Filip P.A. Casselman, Marcus D. 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Fellahi,Anestesia-rianimazione per chirurgia carotidea, EMC - Anestesia-Rianimazione, 30, 3, (1-14), (2025). Crossref 8. P. Guerci, J.-L. Fellahi,Anestesia y reanimación para cirugía carotídea, EMC - Anestesia-Reanimación, 51, 3, (1-16), (2025). Crossref 9. Brijesh Sathian, Javed Iqbal, Hanadi Al Hamad,Concerns regarding the predictive use of thromboelastography in coronary bypass surgery, Annals of Medicine, 57, 1, (2025). Crossref 10. Kuo-Feng Hua, Hsin-Chiao Yu, Hsien-Ta Hsu,Spinal surgery in the context of end-stage renal disease: Balancing risks and surgical strategies – A narrative review, Tzu Chi Medical Journal, (2025). Crossref See more Loading... View Options View options PDF View PDF Media Figures BRIDGE Study Design. Screening visits occurred between 30 days and 5 days before the planned procedure, and randomization (R) occurred 5 days before the procedure. Warfarin treatment was discontinued 5 days before the procedure, and administration of the study drug was initiated 3 days before the procedure. It was recommended that the international normalized ratio (INR) be measured 1 day before the procedure; if the INR was greater than 1.8, oral vitamin K (1.0 to 2.5 mg) was recommended; if the INR was 1.5 to 1.8, oral vitamin K was optional. If the procedure or surgery was delayed up to 3 days, administration of the study drug was continued until 24 hours before the procedure. Warfarin treatment was restarted on the evening of or the day after the procedure, and the study drug was restarted 12 to 24 hours after a minor (or low-bleeding-risk) procedure and 48 to 72 hours after a major (or high-bleeding-risk) procedure. Administration of the study drug was continued after the procedure until the INR was 2.0 or higher on one occasion. The final patient follow-up occurred 30 days after the procedure. LMWH denotes low-molecular-weight heparin. Go to Figure Screening, Randomization, and Follow-up. Go to Figure Other Tables Baseline Characteristics of the Patients. Go to Table Perioperative Anticoagulant Management. Go to Table Study Outcomes. Go to Table Share Share CONTENT LINK Copied! Copying failed. Share FacebookX (formerly Twitter)LinkedInemailBluesky References References 1. Kearon, C, Hirsh, J. Management of anticoagulation before and after elective surgery. N Engl J Med 1997;336:1506-1511 Go to Citation Crossref PubMed Web of Science Google Scholar 2. Piazza, G, Goldhaber, SZ. Periprocedural management of the chronically anticoagulated patient: critical pathways for bridging therapy. Crit Pathw Cardiol 2003;2:96-103 Go to Citation Crossref PubMed Google Scholar 3. Gallego, P, Apostolakis, S, Lip, GY. Bridging evidence-based practice and practice-based evidence in periprocedural anticoagulation. Circulation 2012;126:1573-1576 Go to Citation Crossref PubMed Web of Science Google Scholar 4. Healey, JS, Eikelboom, J, Douketis, J, et al. Periprocedural bleeding and thromboembolic events with dabigatran compared with warfarin: results from the Randomized Evaluation of Long-Term Anticoagulation Therapy (RE-LY) randomized trial. Circulation 2012;126:343-348[Erratum, Circulation 2012;126(10):e160.] Crossref PubMed Web of Science Google Scholar a [...] patients with atrial fibrillation. b [...] such as colonoscopy or ambulatory surgery. c [...] interruption of warfarin treatment. 5. Garcia, D, Alexander, JH, Wallentin, L, et al. Management and clinical outcomes in patients treated with apixaban vs warfarin undergoing procedures. Blood 2014;124:3692-3698 Crossref PubMed Web of Science Google Scholar a [...] patients with atrial fibrillation. b [...] such as colonoscopy or ambulatory surgery. c [...] interruption of warfarin treatment. 6. Baron, TH, Kamath, PS, McBane, RD. Management of antithrombotic therapy in patients undergoing invasive procedures. N Engl J Med 2013;368:2113-2124 Crossref PubMed Web of Science Google Scholar a [...] to attain therapeutic anticoagulation. b [...] arterial thromboembolism, such as stroke. c [...] of the newer direct oral anticoagulants. 7. Schulman, S, Hwang, HG, Eikelboom, JW, Kearon, C, Pai, M, Delaney, J. Loading dose vs. maintenance dose of warfarin for reinitiation after invasive procedures: a randomized trial. J Thromb Haemost 2014;12:1254-1259 Go to Citation Crossref PubMed Web of Science Google Scholar 8. Douketis, JD, Johnson, JA, Turpie, AG. Low-molecular-weight heparin as bridging anticoagulation during interruption of warfarin: assessment of a standardized periprocedural anticoagulation regimen. Arch Intern Med 2004;164:1319-1326 Crossref PubMed Web of Science Google Scholar a [...] bridging with low-molecular-weight heparin. b [...] a major (or high-bleeding-risk) procedure. c [...] rate in the bridging group would be 1.0%. d [...] rates to be approximately 1.0%, 9. Kovacs, MJ, Kearon, C, Rodger, M, et al. Single-arm study of bridging therapy with low-molecular-weight heparin for patients at risk of arterial embolism who require temporary interruption of warfarin. Circulation 2004;110:1658-1663 Crossref PubMed Web of Science Google Scholar a [...] bridging with low-molecular-weight heparin. b [...] rate in the bridging group would be 1.0%. c [...] rates to be approximately 1.0%, 10. Dunn, AS, Spyropoulos, AC, Turpie, AG. Bridging therapy in patients on long-term oral anticoagulants who require surgery: the Prospective Peri-operative Enoxaparin Cohort Trial (PROSPECT). J Thromb Haemost 2007;5:2211-2218 Crossref PubMed Web of Science Google Scholar a [...] bridging with low-molecular-weight heparin. b [...] a major (or high-bleeding-risk) procedure. c [...] rate in the bridging group would be 1.0%. d [...] after operations with a high bleeding risk 11. Halbritter, KM, Wawer, A, Beyer, J, Oettler, W, Schellong, SM. Bridging anticoagulation for patients on long-term vitamin-K-antagonists: a prospective 1 year registry of 311 episodes. J Thromb Haemost 2005;3:2823-2825 Go to Citation Crossref PubMed Web of Science Google Scholar 12. Spyropoulos, AC, Turpie, AGG, Dunn, AS, et al. Clinical outcomes with unfractionated heparin or low-molecular-weight heparin as bridging therapy in patients on long-term oral anticoagulants: the REGIMEN registry. J Thromb Haemost 2006;4:1246-1252 Crossref PubMed Web of Science Google Scholar a [...] bridging with low-molecular-weight heparin. b [...] rates to be approximately 1.0%, 13. Wysokinski, WE, McBane, RD, Daniels, PR, et al. Periprocedural anticoagulation management of patients with nonvalvular atrial fibrillation. Mayo Clin Proc 2008;83:639-645 Go to Citation Crossref PubMed Web of Science Google Scholar 14. Pengo, V, Cucchini, U, Denas, G, et al. Standardized low-molecular-weight heparin bridging regimen in outpatients on oral anticoagulants undergoing invasive procedure or surgery: an inception cohort management study. Circulation 2009;119:2920-2927 Go to Citation Crossref PubMed Web of Science Google Scholar 15. Malato, A, Saccullo, G, Lo Coco, L, et al. Patients requiring interruption of long-term oral anticoagulant therapy: the use of fixed sub-therapeutic doses of low-molecular-weight heparin. J Thromb Haemost 2010;8:107-113 Go to Citation Crossref PubMed Web of Science Google Scholar 16. Ansell, JE. The perioperative management of warfarin therapy. Arch Intern Med 2003;163:881-883 Go to Citation Crossref PubMed Web of Science Google Scholar 17. BRIDGE Study Investigators. Bridging anticoagulation: is it needed when warfarin is interrupted around the time of a surgery or procedure? Circulation 2012;125:e496-e498 Go to Citation Crossref PubMed Web of Science Google Scholar 18. Patel, JP, Arya, R. The current status of bridging anticoagulation. Br J Haematol 2014;164:619-629 Go to Citation Crossref PubMed Web of Science Google Scholar 19. Douketis, JD, Berger, PB, Dunn, AS, et al. The perioperative management of antithrombotic therapy: American College of Chest Physicians Evidence-Based Clinical Practice Guidelines (8th Edition). Chest 2008;133:Suppl:299S-339S Crossref PubMed Web of Science Google Scholar a [...] the need for bridging anticoagulation. b [...] surgery, cardiac surgery, or neurosurgery) 20. Fuster, V, Rydén, LE, Cannom, DS, et al. 2011 ACCF/AHA/HRS focused updates incorporated into the ACC/AHA/ESC 2006 Guidelines for the management of patients with atrial fibrillation: a report of the American College of Cardiology Foundation/American Heart Association Task Force on Practice Guidelines developed in partnership with the European Society of Cardiology and in collaboration with the European Heart Rhythm Association and the Heart Rhythm Society. 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Periprocedural heparin bridging in patients receiving vitamin K antagonists: systematic review and meta-analysis of bleeding and thromboembolic rates. Circulation 2012;126:1630-1639 Crossref PubMed Web of Science Google Scholar a [...] 1.52 to 8.50) in association with bridging. b [...] a lack of standardized bridging protocols. 29. Connolly, SJ, Ezekowitz, MD, Yusuf, S, et al. Dabigatran versus warfarin in patients with atrial fibrillation. N Engl J Med 2009;361:1139-1151 Crossref PubMed Web of Science Google Scholar a [...] Long-Term Anticoagulation Therapy (RE-LY), b [...] the mean scores were between 2.1 and 2.8. 30. Douketis, JD, Healey, JS, Brueckmann, M, et al. Perioperative bridging anticoagulation during dabigatran or warfarin interruption among patients who had an elective surgery or procedure: substudy of the RE-LY trial. Thromb Haemost 2015;113:625-632 Crossref PubMed Web of Science Google Scholar a [...] thromboembolism (0.5% vs. 0.2%, P=0.32). b [...] a lack of standardized bridging protocols. c [...] effect on arterial thromboembolism. 31. Steinberg, BA, Peterson, ED, Kim, S, et al. Use and outcomes associated with bridging during anticoagulation interruptions in patients with atrial fibrillation: findings from the Outcomes Registry for Better Informed Treatment of Atrial Fibrillation (ORBIT-AF). Circulation 2015;131:488-494 Crossref PubMed Web of Science Google Scholar a [...] interruption of warfarin treatment. b [...] interruption of warfarin treatment. 32. Spyropoulos, AC. Pro: “Bridging anticoagulation is needed during warfarin interruption in patients who require elective surgery.” Thromb Haemost 2012;108:213-216 Go to Citation Crossref PubMed Web of Science Google Scholar 33. Kaatz, S, Douketis, JD, Zhou, H, Gage, BF, White, RH. Risk of stroke after surgery in patients with and without chronic atrial fibrillation. J Thromb Haemost 2010;8:884-890 Crossref PubMed Web of Science Google Scholar a [...] to factors such as the type of procedure b [...] surgery, cardiac surgery, or neurosurgery) 34. Cheung, CC, Martyn, A, Campbell, N, et al. Predictors of intraoperative hypotension and bradycardia. Am J Med 2015;128:532-538 Go to Citation Crossref PubMed Web of Science Google Scholar 35. Grip, L, Blombäck, M, Schulman, S. Hypercoagulable state and thromboembolism following warfarin withdrawal in post-myocardial-infarction patients. Eur Heart J 1991;12:1225-1233 Go to Citation Crossref PubMed Web of Science Google Scholar 36. Palareti, G, Legnani, C. Warfarin withdrawal: pharmacokinetic-pharmacodynamic considerations. Clin Pharmacokinet 1996;30:300-313 Go to Citation Crossref PubMed Web of Science Google Scholar 37. Kosir, MA, Schmittinger, L, Barno-Winarski, L, et al. Prospective double-arm study of fibrinolysis in surgical patients. J Surg Res 1998;74:96-101 Go to Citation Crossref PubMed Web of Science Google Scholar 38. Granger, CB, Alexander, JH, McMurray, JJ, et al. Apixaban versus warfarin in patients with atrial fibrillation. N Engl J Med 2011;365:981-992 Go to Citation Crossref PubMed Web of Science Google Scholar 39. Giugliano, RP, Ruff, CT, Braunwald, E, et al. Edoxaban versus warfarin in patients with atrial fibrillation. N Engl J Med 2013;369:2093-2104 Go to Citation Crossref PubMed Web of Science Google Scholar 40. Graham, DJ, Reichman, ME, Wernecke, M, et al. Cardiovascular, bleeding, and mortality risks in elderly Medicare patients treated with dabigatran or warfarin for nonvalvular atrial fibrillation. Circulation 2015;131:157-164 Go to Citation Crossref PubMed Web of Science Google Scholar 41. Beyer-Westendorf, J, Gelbricht, V, Förster, K, et al. Peri-interventional management of novel oral anticoagulants in daily care: results from the prospective Dresden NOAC registry. Eur Heart J 2014;35:1888-1896 Go to Citation Crossref PubMed Web of Science Google Scholar 42. Sherwood, MW, Douketis, JD, Patel, MR, et al. Outcomes of temporary interruption of rivaroxaban compared with warfarin in patients with nonvalvular atrial fibrillation: results from the Rivaroxaban Once Daily, Oral, Direct Factor Xa Inhibition Compared With Vitamin K Antagonism for Prevention of Stroke and Embolism Trial in Atrial Fibrillation (ROCKET AF). Circulation 2014;129:1850-1859 Go to Citation Crossref PubMed Web of Science Google Scholar 43. Birnie, DH, Healey, JS, Wells, GA, et al. Pacemaker or defibrillator surgery without interruption of anticoagulation. N Engl J Med 2013;368:2084-2093 Go to Citation Crossref PubMed Web of Science Google Scholar 44. Xu, Y, Holbrook, AM, Simpson, CS, Dowlatshahi, D, Johnson, AP. Prescribing patterns of novel oral anticoagulants following regulatory approval for atrial fibrillation in Ontario, Canada: a population-based descriptive analysis. CMAJ Open 2013;1:E115-9 Go to Citation Crossref PubMed Google Scholar 45. Desai, NR, Krumme, AA, Schneeweiss, S, et al. Patterns of initiation of oral anticoagulants in patients with atrial fibrillation- quality and cost implications. Am J Med 2014;127:1075.e1-1082.e1 Go to Citation Crossref Web of Science Google Scholar 46. Olesen, JB, Sørensen, R, Hansen, ML, et al. Non-vitamin K antagonist oral anticoagulation agents in anticoagulant naïve atrial fibrillation patients: Danish nationwide descriptive data 2011-2013. 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12366	https://hmmt-archive.s3.amazonaws.com/tournaments/2015/feb/alg/solutions.pdf	HMMT February 2015 Saturday 21 February 2015 Algebra 1. Let Q be a polynomial Q(x) = a0 + a1x + · · · + anxn, where a0, . . . , an are nonnegative integers. Given that Q(1) = 4 and Q(5) = 152, ﬁnd Q(6). Answer: 254 Since each ai is a nonnegative integer, 152 = Q(5) ≡a0 (mod 5) and Q(1) = 4 = ⇒ ai ≤4 for each i. Thus, a0 = 2. Also, since 54 > 152 = Q(5), a4, a5, . . . , an = 0. Now we simply need to solve the system of equations 5a1 + 52a2 2 + 53a3 3 = 150 a1 + a2 + a3 = 2 to get a2 + 6a3 = 7. Since a2 and a3 are nonnegative integers, a2 = 1, a3 = 1, and a1 = 0. Therefore, Q(6) = 63 + 62 + 2 = 254. 2. The fraction 1 2015 has a unique “(restricted) partial fraction decomposition” of the form 1 2015 = a 5 + b 13 + c 31, where a, b, c are integers with 0 ≤a < 5 and 0 ≤b < 13. Find a + b. Answer: 14 This is equivalent to 1 = 13 · 31a + 5 · 31b + 5 · 13c.1 Taking modulo 5 gives 1 ≡3 · 1a (mod 5), so a ≡2 (mod 5). Taking modulo 13 gives 1 ≡5 · 5b = 25b ≡−b (mod 13), so b ≡12 (mod 13). The size constraints on a, b give a = 2, b = 12, so a + b = 14. Remark. This problem illustrates the analogy between polynomials and integers, with prime powers (here 51, 131, 311) taking the role of powers of irreducible polynomials (such as (x −1)1 or (x2 + 1)3, when working with polynomials over the real numbers). Remark. The “partial fraction decomposition” needs to be restricted since it’s only unique “modulo 1”. Abstractly, the abelian group (or Z-module) Q/Z has a “prime power direct sum decomposition” (more or less equivalent to Bezout’s identity, or the Chinese remainder theorem), but Q itself (as an abelian group under addition) does not. You may wonder whether there’s a similar “prime power decomposition” of Q that accounts not just for addition, but also for multiplication (i.e. the full ring structure of the rationals). In some sense, the “adeles/ideles” serve this purpose, but it’s not as clean as the partial fraction decomposition (for additive structure alone)—in fact, the subtlety of adeles/ideles reﬂects much of the diﬃculty in number theory! 3. Let p be a real number and c ̸= 0 an integer such that c −0.1 < xp 1 −(1 + x)10 1 + (1 + x)10 < c + 0.1 for all (positive) real numbers x with 0 < x < 10−100. (The exact value 10−100 is not important. You could replace it with any “suﬃciently small number”.) Find the ordered pair (p, c). 1Note that this does actually have integer solutions by Bezout’s identity, as gcd(13 · 31, 5 · 31, 5 · 13) = 1. Algebra Answer: (−1, −5) This is essentially a problem about limits, but phrased concretely in terms of “small numbers” (like 0.1 and 10−100). We are essentially studying the rational function f(x) := 1−(1+x)10 1+(1+x)10 = −10x+O(x2) 2+O(x) , where the “big-O” notation simply make precise the notion of “error terms”.2 Intuitively, f(x) ≈−10x 2 = −5x for “small nonzero x”. (We could easily make this more precise if we wanted to, by specifying the error terms more carefully, but it’s not so important.) So g(x) := xpf(x) ≈−5xp+1 for “small nonzero x”. • If p + 1 > 0, g will approach 0 (“get very small”) as x approaches 0 (often denoted x →0), so there’s no way it can stay above the lower bound c −0.1 for all small nonzero x. • If p + 1 < 0, g will approach −∞(“get very large in the negative direction”) as x →0, so there’s no way it can stay below the upper bound c + 0.1 for all small nonzero x. • If p + 1 = 0, g ≈−5 becomes approximately constant as x →0. Since c is an integer, we must have c = −5 (as −5 is the only integer within 0.1 of −5). Remark. Why does (p, c) = (−1, −5) actually satisfy the inequality? This is where the 10−100 kicks in: for such small values of x, the “error” \|g(x) −(−5)\| of the approximation g ≈−5 does actually lie within the permitted threshold of ±0.1. (You can easily work out the details yourself, if you’re interested. It’s something you might want to work out once or twice in your life, but rational functions are “well-behaved” enough that we can usually rely on our intuition in these kinds of scenarios.) 4. Compute the number of sequences of integers (a1, . . . , a200) such that the following conditions hold. • 0 ≤a1 < a2 < · · · < a200 ≤202. • There exists a positive integer N with the following property: for every index i ∈{1, . . . , 200} there exists an index j ∈{1, . . . , 200} such that ai + aj −N is divisible by 203. Answer: 20503 Let m := 203 be an integer not divisible by 3. We’ll show the answer for general such m is m⌈m−1 2 ⌉. Let x, y, z be the three excluded residues. Then N works if and only if {x, y, z} ≡{N −x, N −y, N −z} (mod m). Since x, y, z (mod m) has opposite orientation as N −x, N −y, N −z (mod m), this is equivalent to x, y, z forming an arithmetic progression (in some order) modulo m centered at one of x, y, z (or algebraically, one of N ≡2x ≡y + z, N ≡2y ≡z + x, N ≡2z ≡x + y holds, respectively). Since 3 ∤m, it’s impossible for more than one of these congruences to hold (or else x, y, z would have to be equally spaced modulo m, i.e. x−y ≡y−z ≡z−x). So the number of distinct 3-sets corresponding to arithmetic progressions is m⌈m−1 2 ⌉(choose a center and a diﬀerence, noting that ±d give the same arithmetic progression). Since our speciﬁc m = 203 is odd this gives m m−1 2 = 203 · 101 = 20503. Remark. This problem is a discrete analog of certain so-called Frieze patterns. (See also Chapter 6, Exercise 5.8 of Artin’s Algebra textbook.) 5. Let a, b, c be positive real numbers such that a+b+c = 10 and ab+bc+ca = 25. Let m = min{ab, bc, ca}. Find the largest possible value of m. Answer: 25 9 Without loss of generality, we assume that c ≥b ≥a. We see that 3c ≥a+b+c = 10. Therefore, c ≥10 3 . 2For instance, the O(x2) refers to a function bounded by C\|x\|2 for some positive constant C, whenever x is close enough to 0 (and as the 10−100 suggests, that’s all we care about). Algebra Since 0 ≤(a −b)2 = (a + b)2 −4ab = (10 −c)2 −4 (25 −c(a + b)) = (10 −c)2 −4 (25 −c(10 −c)) = c(20 −3c), we obtain c ≤20 3 . Consider m = min{ab, bc, ca} = ab, as bc ≥ca ≥ab. We compute ab = 25−c(a+b) = 25 −c(10 −c) = (c −5)2. Since 10 3 ≤c ≤20 3 , we get that ab ≤25 9 . Therefore, m ≤25 9 in all cases and the equality can be obtained when (a, b, c) = ( 5 3, 5 3, 20 3 ). 6. Let a, b, c, d, e be nonnegative integers such that 625a + 250b + 100c + 40d + 16e = 153. What is the maximum possible value of a + b + c + d + e? Answer: 153 The intuition is that as much should be in e as possible. But divisibility obstructions like 16 ∤153 are in our way. However, the way the coeﬃcients 54 > 53 · 2 > · · · are set up, we can at least easily avoid having a, b, c, d too large (speiﬁcally, ≥2). This is formalized below. First, we observe that (a1, a2, a3, a4, a5) = (5, 1, 0, 0, 0) is a solution. Then given a solution, replacing (ai, ai+1) with (ai −2, ai+1 + 5), where 1 ≤i ≤4, also yields a solution. Given a solution, it turns out all solutions can be achieved by some combination of these swaps (or inverses of these swaps). Thus, to optimize the sum, we want (a, b, c, d) ∈{0, 1}4, since in this situation, there would be no way to make swaps to increase the sum. So the sequence of swaps looks like (5, 1, 0, 0, 0) →(1, 11, 0, 0, 0) → (1, 1, 25, 0, 0) →(1, 1, 1, 60, 0) →(1, 1, 1, 0, 150), yielding a sum of 1 + 1 + 1 + 0 + 150 = 153. Why is this optimal? Suppose (a, b, c, d, e) maximizes a + b + c + d + e. Then a, b, c, d ≤1, or else we could use a replacement (ai, ai+1) →(ai −2, ai+1 + 5) to strictly increase the sum. But modulo 2 forces a odd, so a = 1. Subtracting oﬀand continuing in this manner3 shows that we must have b = 1, then c = 1, then d = 0, and ﬁnally e = 150. Remark. The answer is coincidentally obtained by dropping the exponent of 153 into the one’s place. 7. Suppose (a1, a2, a3, a4) is a 4-term sequence of real numbers satisfying the following two conditions: • a3 = a2 + a1 and a4 = a3 + a2; • there exist real numbers a, b, c such that an2 + bn + c = cos(an) for all n ∈{1, 2, 3, 4}. Compute the maximum possible value of cos(a1) −cos(a4) over all such sequences (a1, a2, a3, a4). Answer: −9 + 3 √ 13 Let f(n) = cos an and m = 1. The second (“quadratic interpolation”) condition on f(m), f(m + 1), f(m + 2), f(m + 3) is equivalent to having a vanishing third ﬁnite diﬀerence f(m + 3) −3f(m + 2) + 3f(m + 1) −f(m) = 0. 3This is analogous to the “number theoretic” proof of the uniqueness of the base 2 expansion of a nonnegative integer. Algebra This is equivalent to f(m + 3) −f(m) = 3 [f(m + 2) −f(m + 1)] ⇐ ⇒cos(am+3) −cos(am) = 3 (cos(am+2) −cos(am+1)) = −6 sin am+2 + am+1 2 sin am+2 −am+1 2 = −6 sin am+3 2 sin am 2 . Set x = sin am+3 2 and y = sin am 2 . Then the above rearranges to (1 −2x2) −(1 −2y2) = −6xy ⇐ ⇒x2 −y2 = 3xy. Solving gives y = x −3± √ 13 2 . The expression we are trying to maximize is 2(x2 −y2) = 6xy, so we want x, y to have the same sign; thus y = x −3+ √ 13 2 . Then \|y\| ≤\|x\|, so since \|x\|, \|y\| ≤1, to maximize 6xy we can simply set x = 1, for a maximal value of 6 · −3+ √ 13 2 = −9 + 3 √ 13. 8. Find the number of ordered pairs of integers (a, b) ∈{1, 2, . . . , 35}2 (not necessarily distinct) such that ax + b is a “quadratic residue modulo x2 + 1 and 35”, i.e. there exists a polynomial f(x) with integer coeﬃcients such that either of the following equivalent conditions holds: • there exist polynomials P, Q with integer coeﬃcients such that f(x)2 −(ax + b) = (x2 + 1)P(x) + 35Q(x); • or more conceptually, the remainder when (the polynomial) f(x)2 −(ax + b) is divided by (the polynomial) x2 + 1 is a polynomial with (integer) coeﬃcients all divisible by 35. Answer: 225 By the Chinese remainder theorem, we want the product of the answers modulo 5 and modulo 7 (i.e. when 35 is replaced by 5 and 7, respectively). First we do the modulo 7 case. Since x2 + 1 is irreducible modulo 7 (or more conceptually, in F7[x]), exactly half of the nonzero residues modulo x2 + 1 and 7 (or just modulo x2 + 1 if we’re working in F7[x]) are quadratic residues, i.e. our answer is 1 + 72−1 2 = 25 (where we add back one for the zero polynomial). Now we do the modulo 5 case. Since x2 +1 factors as (x+2)(x−2) modulo 5 (or more conceptually, in F5[x]), by the polynomial Chinese remainder theorem modulo x2 + 1 (working in F5[x]), we want the product of the number of polynomial quadratic residues modulo x ± 2. By centering/evaluating polynomials at ∓2 accordingly, the polynomial squares modulo these linear polynomials are just those reducing to integer squares modulo 5.4 So we have an answer of (1 + 5−1 2 )2 = 9 in this case. Our ﬁnal answer is thus 25 · 9 = 225. Remark. This problem illustrates the analogy between integers and polynomials (speciﬁcally here, polynomials over the ﬁnite ﬁeld of integers modulo 5 or 7), with x2 + 1 (mod 7) or x ± 2 (mod 5) taking the role of a prime number. Indeed, just as in the integer case, we expect exactly half of the (coprime) residues to be (coprime, esp. nonzero) quadratic residues. 9. Let N = 302015. Find the number of ordered 4-tuples of integers (A, B, C, D) ∈{1, 2, . . . , N}4 (not necessarily distinct) such that for every integer n, An3 + Bn2 + 2Cn + D is divisible by N. Answer: 24 Note that n0 = n 0 , n1 = n 1 , n2 = 2 n 2 + n 1 , n3 = 6 n 3 + 6 n 2 + n 1 (generally see Thus the polynomial rewrites as 6A n 3 + (6A + 2B) n 2 + (A + B + 2C) n 1 + D n 0 , 4This is more explicit than necessary. By the same reasoning as in the previous paragraph, we can abstractly count 1+ 51−1 2 quadratic residues modulo x±2 (irreducible polynomials in F5[x]) each (and then multiply/square to get the answer for x2 +1). Algebra which by the classiﬁcation of integer-valued polynomials is divisible by N always if and only if 6A, 6A+ 2B, A + B + 2C, D are always divisible by N. We can eliminate B and (trivially) D from the system: it’s equivalent to the system 6A ≡0 (mod N), 4A −4C ≡0 (mod N), B ≡−A −2C (mod N), D ≡0 (mod N). So we want 12 times the number of (A, C) with A ≡0 (mod N/6), C ≡A (mod N/4). So there are N/(N/6) = 6 choices for A, and then given such a choice of A there are N/(N/4) = 4 choices for C. So we have 6·4·12 = 24 solutions total. 10. Find all ordered 4-tuples of integers (a, b, c, d) (not necessarily distinct) satisfying the following system of equations: a2 −b2 −c2 −d2 = c −b −2 2ab = a −d −32 2ac = 28 −a −d 2ad = b + c + 31. Answer: (5, −3, 2, 3) We ﬁrst give two systematic solutions using standard manipulations and divisibility conditions (with some casework), and then a third solution using quaternionic number theory (not very practical, so mostly for your cultural beneﬁt). Solution 1. Subtract the second equation from the third to get a(c −b + 1) = 30. Add the second and third to get 2a(b + c) = −4 −2d. Substitute into the fourth to get 2a(2ad −31) = −4 −2d ⇐ ⇒a(31 −2ad) = 2 + d ⇐ ⇒d = 31a −2 2a2 + 1, which in particular gives a ̸≡1 (mod 3). Then plugging in a factor of 30 for a gives us the system of equations b + c = 2ad −31 and c −b + 1 = 30/a in b, c. Here, observe that b + c is odd, so c −b + 1 is even. Thus a must be odd (and from earlier a ̸≡1 (mod 3)), so a ∈{−1, ±3, 5, ±15}. Manually checking these, we see that the only possibilities we need to check are (a, d) = (5, 3), (−1, −11), (−3, −5), corresponding to (b, c) = (−3, 2), (11, −20), (5, −6). Then check the three candidates against ﬁrst condition a2 −b2 −c2 −d2 = c −b −2 to ﬁnd our only solution (a, b, c, d) = (5, −3, 2, 3). Solution 2. Here’s an alternative casework solution. From 2ad = b + c + 31, we have that b + c is odd. So, b and c has diﬀerent parity. Thus, b2 + c2 ≡1 (mod 4). Plugging this into the ﬁrst equation, we get that a and d also have the same parity. So, a2 −b2 −c2 −d2 ≡−1 (mod 4). Thus, c −b −2 ≡−1 (mod 4). So, c ≡b + 1 (mod 4). From taking modulo a in the second and third equation, we have a \| d + 32 and a \| 28 −d. So, a \| 60. Now, if a is even, let a = 2k and d = 2m. Plugging this in the second and third equation, we get 2kc = 14 −k −m and 2kb = k −m −16. So, k(c −b) = 15 −k. We can see that k ̸= 0. Therefore, c −b = 15−k k = 15 k −1. But c −b ≡1 (mod 4). So, 15 k −1 ≡1 (mod 4), or 15 k ≡2 (mod 4) which leads to a contradiction. So, a is odd. And we have a \| 60. So, a \| 15. This gives us 8 easy possibilities to check... Solution 3. The left hand sides clue us in to the fact that this problem is secretly about quaternions. Indeed, we see that letting z = a + bi + cj + dk gives (z −i + j)z = −2 −32i + 28j + 31k. Taking norms gives N(z−i+j)N(z) = 22 +322 +282 +312 = 2773 = 47·59. By the triangle inequality, N(z), N(z −i + j) aren’t too far apart, so they must be 47, 59 (in some order). Thus z, z −i + j are Hurwitz primes.5 We rely on the following foundational lemma in quaternion number theory: 5For the purposes of quaternion number theory, it’s simpler to work in the the Hurwitz quaternions H = ⟨i, j, k, 1+i+j+k 2 ⟩Z, which has a left- (or right-) division algorithm, left- (resp. right-) Euclidean algorithm, is a left- (resp. right-) principal ideal domain, etc. There’s no corresponding division algorithms when we’re working with the Lipschitz quaternions, i.e. those with integer coordinates. Algebra Lemma. Let p ∈Z be an integer prime, and A a Hurwitz quaternion. If p \| N(A), then the HA + Hp (a left ideal, hence principal) has all element norms divisible by p, hence is nontrivial. (So it’s either Hp or of the form HP for some Hurwitz prime P.) In our case, it will suﬃce to apply the lemma for A = −2−32i+28j +31k at primes p = 47 and q = 59 to get factorizations (unique up to suitable left/right unit multiplication) A = QP and A = P ′Q′ (respectively), with P, P ′ Hurwitz primes of norm p, and Q, Q′ Hurwitz primes of norm q. Indeed, these factorizations come from HA + Hp = HP and HA + Hq = HQ′. We compute by the Euclidean algorithm: HA + H(47) = H(−2 −32i + 28j + 31k) + H(47) = H(−2 + 15i −19j −16k) + H(47) = [H(47 · 18) + H(47)(−2 −15i + 19j + 16k)]−2 + 15i −19j −16k 47 · 18 = [H18 + H(−2 + 3i + j −2k)]−2 + 15i −19j −16k 18 = H(−2 + 3i + j −2k)−2 + 15i −19j −16k 18 = H−54 −90i + 54j −36k 18 = H(−3 −5i + 3j −2k). Thus6 there’s a unit7 ǫ such that P = ǫ(−3 −5i + 3j −2k). Similarly, to get P ′, we compute AH + 47H = (−2 −32i + 28j + 31k)H + 47H = (−2 + 15i −19j −16k)H + 47H = −2 + 15i −19j −16k 47 · 18 [(47 · 18)H + 47(−2 −15i + 19j + 16k)H] = −2 + 15i −19j −16k 18 [18H + (−2 + 3i + j −2k)H] = −2 + 15i −19j −16k 18 (−2 + 3i + j −2k)H = −54 + 18i + 18j + 108k 18 H = (−3 + i + j + 6k)H, so there’s a unit ǫ′ with P ′ = (−3 + i + j + 6k)ǫ′. Finally, we have either z = ǫ(−3 −5i + 3j −2k) for some ǫ, or z −i + j = (−3 + i + j + 6k)ǫ′ for some ǫ′. Checking the 24 + 24 cases (many of which don’t have integer coeﬃcients, and can be ruled out immediately) gives z = iP = 5 −3i + 2j + 3k as the only possibility. Remark. We have presented the most conceptual proof possible. It’s also possible to directly compute based on the norms only, and do some casework. For example, since 47 ≡3 (mod 4), it’s easy to check the only ways to write it as a sum of four squares are (±5)2 + (±3)2 + (±3)2 + (±2)2 and (±3)2 + (±1)2 + (±1)2 + (±6)2. Remark. For a systematic treatment of quaternions (including the number theory used above), one good resource is On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry by John H. Conway and Derek A. Smith. A more focused treatment is the expository paper Factorization of Hurwitz Quaternions by Boyd Coan and Cherng-tiao Perng. For an example of interesting research in this rather exotic area, see the Metacommutation of Hurwitz primes paper by Henry Cohn and Abhinav Kumar. 6Hidden computations: we’ve used 47 · 18 = 846 = 22 + 152 + 192 + 162, and 18 = N(−2 + 3i + j −2k). 7i.e. one of ±1, ±i, ±j, ±k, ±1±i±j±k 2 Algebra
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CopyConvert another number?Swap conversion: Meters to KilometerWhat is 4 Kilometer in Meters? Unit rate 1 Kilometer = 1,000 Meters Current ratio 1,000 : 1 Scale insight m is 1000.00 times larger than km Significant digits 4 Round based on measurement precision: construction (2-3 decimals), engineering (4-6 decimals) Mental shortcut No common shortcut available Quick mental estimates - Within 5-10% of actual value Proportion 1,000 : 1 100,000% Kilometer A kilometer is a unit of length in the metric system equal to 1,000 meters or approximately 0.621 miles [Source: Wikipedia] Meters A meter is the base unit of length in the International System of Units (SI), equal to 100 centimeters or approximately 39.3701 inches. [Source: Wikipedia] Quick conversion chart of Kilometer to Meters 3 Kilometer To Meters = 3000 Meters 13 Kilometer To Meters = 13000 Meters 23 Kilometer To Meters = 23000 Meters 33 Kilometer To Meters = 33000 Meters 43 Kilometer To Meters = 43000 Meters 53 Kilometer To Meters = 53000 Meters 63 Kilometer To Meters = 63000 Meters 73 Kilometer To Meters = 73000 Meters 83 Kilometer To Meters = 83000 Meters 93 Kilometer To Meters = 93000 Meters Other ways to convert Kilometer What is 3 kilometers in nanometer?What is 3 kilometers in millimeter?What is 3 kilometers in centimeter?What is 3 kilometers in meters?What is 3 kilometers in inches?What is 3 kilometers in feet?What is 3 kilometers in yards?What is 3 kilometers in miles?What is 3 kilometers in nanometer?3km x 3km to meters \| 3x3 km in meters Further calculations of kilometer to Meters \| Kilometer \| Meters \| --- \| \| 2.1 km to m \| 2100 \| \| 2.2 km to m \| 2200 \| \| 2.3 km to m \| 2300 \| \| 2.4 km to m \| 2400 \| \| 2.5 km to m \| 2500 \| \| 2.6 km to m \| 2600 \| \| 2.7 km to m \| 2700 \| \| 2.8 km to m \| 2800 \| \| 2.9 km to m \| 2900 \| \| 3 km to m \| 3000 \| \| 3.1 km to m \| 3100 \| \| 3.2 km to m \| 3200 \| \| 3.3 km to m \| 3300 \| \| 3.4 km to m \| 3400 \| \| 3.5 km to m \| 3500 \| \| 3.6 km to m \| 3600 \| \| 3.7 km to m \| 3700 \| \| 3.8 km to m \| 3800 \| List of 1% to 99% Calculations What is 1% of 3?What is 2% of 3?What is 3% of 3?What is 4% of 3?What is 5% of 3?What is 6% of 3?What is 7% of 3?What is 8% of 3?What is 9% of 3?What is 10% of 3?What is 11% of 3?What is 12% of 3?What is 13% of 3?What is 14% of 3?What is 15% of 3?What is 16% of 3?What is 17% of 3?What is 18% of 3?What is 19% of 3?What is 20% of 3?What is 21% of 3?What is 22% of 3?What is 23% of 3?What is 24% of 3?What is 25% of 3?What is 26% of 3?What is 27% of 3?What is 28% of 3?What is 29% of 3?What is 30% of 3?What is 31% of 3?What is 32% of 3?What is 33% of 3?What is 34% of 3?What is 35% of 3?What is 36% of 3?What is 37% of 3?What is 38% of 3?What is 39% of 3?What is 40% of 3?What is 41% of 3?What is 42% of 3?What is 43% of 3?What is 44% of 3?What is 45% of 3?What is 46% of 3?What is 47% of 3?What is 48% of 3?What is 49% of 3?What is 50% of 3?What is 51% of 3?What is 52% of 3?What is 53% of 3?What is 54% of 3?What is 55% of 3?What is 56% of 3?What is 57% of 3?What is 58% of 3?What is 59% of 3?What is 60% of 3?What is 61% of 3?What is 62% of 3?What is 63% of 3?What is 64% of 3?What is 65% of 3?What is 66% of 3?What is 67% of 3?What is 68% of 3?What is 69% of 3?What is 70% of 3?What is 71% of 3?What is 72% of 3?What is 73% of 3?What is 74% of 3?What is 75% of 3?What is 76% of 3?What is 77% of 3?What is 78% of 3?What is 79% of 3?What is 80% of 3?What is 81% of 3?What is 82% of 3?What is 83% of 3?What is 84% of 3?What is 85% of 3?What is 86% of 3?What is 87% of 3?What is 88% of 3?What is 89% of 3?What is 90% of 3?What is 91% of 3?What is 92% of 3?What is 93% of 3?What is 94% of 3?What is 95% of 3?What is 96% of 3?What is 97% of 3?What is 98% of 3?What is 99% of 3? ↓ Show more ↑ Show less What percentage is this fraction? 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↓ Show more ↑ Show less Kilometer (km)Meters (meters) 3.010 Kilometers 3010 meters) 3.020 Kilometers 3020 meters) 3.030 Kilometers 3030 meters) 3.040 Kilometers 3040 meters) 3.050 Kilometers 3050 meters) 3.060 Kilometers 3060 meters) 3.070 Kilometers 3070 meters) 3.080 Kilometers 3080 meters) 3.090 Kilometers 3090 meters) 3.100 Kilometers 3100 meters) 3.110 Kilometers 3110 meters) 3.120 Kilometers 3120 meters) 3.130 Kilometers 3130 meters) 3.140 Kilometers 3140 meters) 3.150 Kilometers 3150 meters) 3.160 Kilometers 3160 meters) 3.170 Kilometers 3170 meters) 3.180 Kilometers 3180 meters) 3.190 Kilometers 3190 meters) 3.200 Kilometers 3200 meters) 3.210 Kilometers 3210 meters) 3.220 Kilometers 3220 meters) 3.230 Kilometers 3230 meters) 3.240 Kilometers 3240 meters) 3.250 Kilometers 3250 meters) 3.260 Kilometers 3260 meters) 3.270 Kilometers 3270 meters) 3.280 Kilometers 3280 meters) 3.290 Kilometers 3290 meters) 3.300 Kilometers 3300 meters) 3.310 Kilometers 3310 meters) 3.320 Kilometers 3320 meters) 3.330 Kilometers 3330 meters) 3.340 Kilometers 3340 meters) 3.350 Kilometers 3350 meters) 3.360 Kilometers 3360 meters) 3.370 Kilometers 3370 meters) 3.380 Kilometers 3380 meters) 3.390 Kilometers 3390 meters) 3.400 Kilometers 3400 meters) 3.410 Kilometers 3410 meters) 3.420 Kilometers 3420 meters) 3.430 Kilometers 3430 meters) 3.440 Kilometers 3440 meters) 3.450 Kilometers 3450 meters) 3.460 Kilometers 3460 meters) 3.470 Kilometers 3470 meters) 3.480 Kilometers 3480 meters) 3.490 Kilometers 3490 meters) 3.500 Kilometers 3500 meters) Kilometer (km)Meters (meters) 3.500 Kilometers 3500 meters) 3.510 Kilometers 3510 meters) 3.520 Kilometers 3520 meters) 3.530 Kilometers 3530 meters) 3.540 Kilometers 3540 meters) 3.550 Kilometers 3550 meters) 3.560 Kilometers 3560 meters) 3.570 Kilometers 3570 meters) 3.580 Kilometers 3580 meters) 3.590 Kilometers 3590 meters) 3.600 Kilometers 3600 meters) 3.610 Kilometers 3610 meters) 3.620 Kilometers 3620 meters) 3.630 Kilometers 3630 meters) 3.640 Kilometers 3640 meters) 3.650 Kilometers 3650 meters) 3.660 Kilometers 3660 meters) 3.670 Kilometers 3670 meters) 3.680 Kilometers 3680 meters) 3.690 Kilometers 3690 meters) 3.700 Kilometers 3700 meters) 3.710 Kilometers 3710 meters) 3.720 Kilometers 3720 meters) 3.730 Kilometers 3730 meters) 3.740 Kilometers 3740 meters) 3.750 Kilometers 3750 meters) 3.760 Kilometers 3760 meters) 3.770 Kilometers 3770 meters) 3.780 Kilometers 3780 meters) 3.790 Kilometers 3790 meters) 3.800 Kilometers 3800 meters) 3.810 Kilometers 3810 meters) 3.820 Kilometers 3820 meters) 3.830 Kilometers 3830 meters) 3.840 Kilometers 3840 meters) 3.850 Kilometers 3850 meters) 3.860 Kilometers 3860 meters) 3.870 Kilometers 3870 meters) 3.880 Kilometers 3880 meters) 3.890 Kilometers 3890 meters) 3.900 Kilometers 3900 meters) 3.910 Kilometers 3910 meters) 3.920 Kilometers 3920 meters) 3.930 Kilometers 3930 meters) 3.940 Kilometers 3940 meters) 3.950 Kilometers 3950 meters) 3.960 Kilometers 3960 meters) 3.970 Kilometers 3970 meters) 3.980 Kilometers 3980 meters) 3.990 Kilometers 3990 meters) Frequently Asked Questions What is 3 Kilometer in Meters? 3 Kilometer is equal to Meters 3000. How do you convert Kilometer to Meters? Use the appropriate conversion factor. For example, 3 Kilometer equals Meters 3000. Is 3 Kilometer more than one Meters? Yes. 3 Kilometer equals Meters 3000, which is more than one 3000. What is the formula to convert Kilometer to Meters? Value in 3000 = value in Kilometer × conversion factor. Example: 3 Kilometer = Meters 3000. Why is converting Kilometer to Meters useful? It helps to express measurements in the most convenient unit. For long distances, Meters may be easier than Kilometer. How many Meters are in 3 Kilometer? There are Meters 3000 in 3 Kilometer. Can I use this calculator for other values? Yes. Enter any value in Kilometer to get the result in Meters. Where is converting Kilometer to Meters commonly used? This conversion is common in science, engineering, and everyday life. Available in Other Languages العربيةكيلومتر إلى متر AzeriMetrədək kilometr българскиКилометър до метър বাংলাকিলোমিটার থেকে মিটার CatalàQuilòmetre a comptabilitat ČeštinaKilometr do metru DanskKilometer til meter DeutschKilometer zum Meter GreekΧιλιόμετρο για μετρητή EspañolaKilómetro a metro فارسیکیلومتر تا متر SuomiKilometri metriin PhilippinesKilometro hanggang metro FrançaisKilomètre à compteur ગુજરાતીકિલોમીટર સુધી עִבְרִיתקילומטר למטר हिंदीकिलोमीटर से मीटर HrvatskiKilometar do metra HungarianKilométerre a méterre ՀայաստանԿիլոմետր մինչեւ մետր IndonesianKilometer ke Meter ItalianChilometro a metro 日本語キロメートルからメーター Қазақ тіліКилометрді метрге дейін ಕನ್ನಡಕಿಲೋಮೀಟರ್ ಟು ಮೀಟರ್ 한국어킬로미터에서 미터 КыргызМетрге километр LatviešuKilometrs līdz skaitītājam മലയാളംമീറ്ററിൽ കിലോമീറ്റർ नेपालीमिटर किलोमिटर NederlandsKilometer tot meter NorskKilometer til meter PolskiKilometr do metra پښتوکیلو میتر ته کیلومتره PortuguêsQuilômetro a metro RomânăKilometru la contor РусскийКилометр до метра SlovenčinaKilometer na merač Albanian – ShqipKilometër në njehsor CрпскиКилометар до мерача كِسوَحِيلِKilomita kwa mita தமிழ்கிலோமீட்டர் முதல் மீட்டர் வரை తెలుగుకిలోమీటర్ నుండి మీటర్ ไทยกิโลเมตรเป็นเมตร TürkçeKilometre ila metre УкраїнськаКілометр до лічильника اردوکلومیٹر سے میٹر Tiếng ViệtKm đến mét ↓ Show more ↑ Show less Share this tool with your friends Page link HTML link code Related Tools Fahrenheit To Kelvin Ounces To Cups Joule To Calorie Miles to Yards Centimeter to Mile Megawatt to Watt Celsius To Fahrenheit Yards to Centimeters Calorie To Kilojoule Mile To Nm By continuing to use this site you consent to the use of cookies in accordance with our Privacy Policy. 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12368	https://mathoverflow.net/questions/396776/three-squares-in-a-rectangle	reference request - Three squares in a rectangle - MathOverflow Join MathOverflow By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Three squares in a rectangle Ask Question Asked 4 years, 2 months ago Modified4 years, 2 months ago Viewed 984 times This question shows research effort; it is useful and clear 15 Save this question. Show activity on this post. One of my colleagues gave me the following problem about 15 years ago: Given three squares inside a 1 by 2 rectangle, with no two squares overlapping, prove that the sum of side lengths is at most 2. (The sides of the squares and the rectangle need not be parallel to each other.) I couldn't find a solution or even a source for this problem. Does anyone know about it? I was told that this was like an exercise in combinatorial geometry for high school contests such as IMO, but I'm not sure if this problem was listed in such competitions. Has anyone heard of this problem, or does anyone know how to solve it? Any kind of help will be appreciated. reference-request mg.metric-geometry discrete-geometry plane-geometry Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Improve this question Follow Follow this question to receive notifications edited Jul 4, 2021 at 13:42 user44143 asked Jul 4, 2021 at 12:13 udaqueudaque 153 3 3 bronze badges 7 1 So you have three squares of sides a,b,c a,b,c and want to show a+b+c≤2 a+b+c≤2 you know from the conditions that the sum of the areas is at most 2.2. Also, each of a,b,c a,b,c is at most 1.1. work with that.Aaron Meyerowitz –Aaron Meyerowitz 2021-07-04 13:41:50 +00:00 Commented Jul 4, 2021 at 13:41 6 @AaronMeyerowitz, that only proves that a+b+c≤6–√a+b+c≤6; e.g. you can have a=b=c=2/3−−−√a=b=c=2/3.user44143 –user44143 2021-07-04 13:44:04 +00:00 Commented Jul 4, 2021 at 13:44 1 This area approach corresponds to partitioning a circle of radius 2/π−−−√2/π into three congruent circular sectors. Somehow the actual shapes need to be considered. Since two unit squares side by side achieve a+b+c=2 a+b+c=2, maybe condition on whether the central vertical line is crossed by some square.RobPratt –RobPratt 2021-07-04 14:22:10 +00:00 Commented Jul 4, 2021 at 14:22 2 @JuanMoreno you are supposed to sum the three side lengths, not all twelve sides.RobPratt –RobPratt 2021-07-04 21:15:50 +00:00 Commented Jul 4, 2021 at 21:15 5 More generally, what is the maximum sum of side lengths of n n squares in a 1×x 1×x rectangle, for any real number x>1 x>1?Richard Stanley –Richard Stanley 2021-07-04 23:33:12 +00:00 Commented Jul 4, 2021 at 23:33 \|Show 2 more comments 2 Answers 2 Sorted by: Reset to default This answer is useful 5 Save this answer. Show activity on this post. Since the squares are convex, we can draw lines which separate them. In particular, if two separating lines go from (b−a,0)(b−a,0) to (b,1)(b,1) and from (c,1)(c,1) to (c+d,0)(c+d,0), then we can prove the result in terms of those lines and those variables. So: let the rectangle go from (0,0)(0,0) to (2,1)(2,1). Let A A be the leftmost square (or one such square). Let C C be the rightmost square (or one such square). Let B B be the other square. Draw a line separating A A and B B, and let (b,1)(b,1) and (b−a,0)(b−a,0) be its intersections with the lines y=1 y=1 and y=0 y=0. Draw a line separating B B and C C, and let (c,1)(c,1) and (c+d,0)(c+d,0) be its intersections with the lines y=1 y=1 and y=0 y=0. Reasoning as in user21820's answer, we assume wlog that: 0<a 0<a, so A A is left and above the line separating A A and B B; 0<d 0<d, so C C is right and above the line separating B B and C C; 0<b 0<b and c<2 c<2 and a−b+c+d>0 a−b+c+d>0, so B B is below both lines. (Since the lines may leave the rectangle, we do not assume b−a>0 b−a>0 or b<2 b<2 or c>0 c>0 or c+d<2 c+d<2.) Lemma (proved at the end): sidelength of A sidelength of B sidelength of C≤min(b a+1,1)≤min((a−b+c+d)u,1)≤min(2−c d+1,1)sidelength of A≤min(b a+1,1)sidelength of B≤min((a−b+c+d)u,1)sidelength of C≤min(2−c d+1,1) where u=max(1 a+d+1,a 2+1−−−−−√a 2+a+d+1,d 2+1−−−−−√d 2+d+a+1)u=max(1 a+d+1,a 2+1 a 2+a+d+1,d 2+1 d 2+d+a+1) The factor u u satisfies 1/(a+1)>u 1/(a+1)>u and 1/(d+1)>u 1/(d+1)>u so long as a<3.66 a<3.66 and d<3.66 d<3.66 respectively. I will assume those inequalities for now to show that some functions are increasing or decreasing; I don't have a clean proof for those inequalities or without them yet. In the corner case of b=a+1 b=a+1 and c=1−d c=1−d, the side lengths are 1 1, 0 0 and 1 1, and they sum to exactly 2 2. We now use this in analyzing four cases. Case I, b≤a+1 b≤a+1 and c≤1−d c≤1−d: The sum of the sidelengths is at most b a+1+(a−b+c+d)u+1 b a+1+(a−b+c+d)u+1 This is increasing in both b b and c c, so its value is at most the corner value of 2 2. Case II, b≤a+1 b≤a+1 and c≥1−d c≥1−d: The sum of the sidelengths is at most b a+1+(a−b+c+d)u+2−c d+1 b a+1+(a−b+c+d)u+2−c d+1 This is increasing in b b and decreasing in c c, so its value is at most the corner value of 2 2. Case III, b≥a+1 b≥a+1 and c≤1−d c≤1−d: The sum of the sidelengths is at most 1+(a−b+c+d)u+1 1+(a−b+c+d)u+1 This is decreasing in b b and increasing in c c, so its value is at most the corner value of 2 2. Case IV, b≥a+1 b≥a+1 and c≥1−d c≥1−d: The sum of the sidelengths is at most 1+(a−b+c+d)u+2−c d+1 1+(a−b+c+d)u+2−c d+1 This is decreasing in both b b and c c, so its value is at most the corner value of 2 2. So the sum of the sidelengths is at most 2 2 in each case. Proof of Lemma: The sidelength of A A is clearly less than 1, and also clearly less than the maximum sidelength inscribed in the right triangle bounded by x=0 x=0, y=1 y=1, and the separator of A A and B B. We use Polya's formula here to calculate the sidelength in the triangle as the maximum of b/(a+1)b/(a+1) and b a 2+1−−−−−√/(a 2+a+1)b a 2+1/(a 2+a+1); since a>0 a>0, the maximum is just b/(a+1)b/(a+1). A similar use of Polya's result bounds the sidelength of C C. Yet another use of that result, now for a triangle which may be acute or obtuse, bounds the sidelength of B B by \|a−b+c+d\|u\|a−b+c+d\|u. Since we assumed a−b+c+d>0 a−b+c+d>0, we write this bound as (a−b+c+d)u(a−b+c+d)u. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Jul 6, 2021 at 21:23 answered Jul 6, 2021 at 9:26 user44143 user44143 6 2 "Since the squares are convex, we can draw lines which separate them": really?André Henriques –André Henriques 2021-07-06 09:28:06 +00:00 Commented Jul 6, 2021 at 9:28 Yes: this is the hyperplane separation theorem. en.wikipedia.org/wiki/Hyperplane_separation_theoremuser44143 –user44143 2021-07-06 09:31:17 +00:00 Commented Jul 6, 2021 at 9:31 I don't think you understood @AndréHenriques's objection: i.sstatic.net/TzoUd.pnguser21820 –user21820 2021-07-06 10:46:14 +00:00 Commented Jul 6, 2021 at 10:46 1 @user21820, this does not assume that the lines go from top to bottom in the rectangle — that is why a,b,c,d a,b,c,d are defined in terms of intersections with lines instead of line segments, and why I noted that b−a b−a may be negative. The definition also applies to the illustration in your comments, which indeed allows separating A A and B B with one line, and then separating B B and C C with another.user44143 –user44143 2021-07-06 12:20:01 +00:00 Commented Jul 6, 2021 at 12:20 1 @user2182: well, the only case in which a line does not intersect two parallel lines is if it is parallel to both. In such a case, the sum of sidelengths of these two squares is at most 1, and the third has at most length 1 because of height constraints. This degenerate case so doesn't influence much the proof.Andrea Marino –Andrea Marino 2021-07-06 12:27:55 +00:00 Commented Jul 6, 2021 at 12:27 \|Show 1 more comment This answer is useful 1 Save this answer. Show activity on this post. Partial solution Let the rectangle R have the longer side horizontal. It is easy to prove that you can label the squares A,B,C such that A can be shifted (i.e. translated) all the way to the left without overlapping the other squares, and C can be shifted all the way to the right. Shift them as just described. Then we have A entirely in the left half of R and C entirely in the right half of R. Since A and C are in separate halves of R, we can shift each of them to touch two sides of R, because B can block only one direction. Now expand A until it touches B or it cannot be expanded without going beyond R, and do the same for C. As thus constrained, A can be fully described by its size x∈[0,1] and orientation angle r∈[0,π/2] and the vertical position p∈[0,1] of its centre. The set of possible parameters (x,r,p) is compact. Likewise for C. Moreover, the maximum size of B that can fit into R without overlapping A,C is continuous with respect to the parameters specifying A,C and the parameter in [0,π/2] specifying the orientation of B. Thus by EVT (extreme-value theorem) we can assume that A,B,C actually achieve a maximum total size. So thus far, A is in the left half of R and C is in the right half of R, each touching two sides of R, and B must fall into one of the following cases (otherwise it can be expanded contradicting maximality): B has sides s,t,u,v with s,t being opposite sides and can slide towards v. If either of s,t, say s by symmetry, has only a single contact point, that point must be touching a square (not R), and we can expand B while keeping t,u on the same lines and contracting the touching square, yielding a higher total size, contradicting maximality. Thus both s,t have multiple contact points, so B must be flush with A on one side and flush with C on the opposite side. If A is not flush with the side of R, then we can expand B slightly while contracting A by the same amount to maintain total size and create a gap between A and the left side of R, and so after that we can shift A leftward and expand A slightly, contradicting maximality. Therefore A,B,C have sides parallel with R, yielding maximum total size 2. B cannot slide towards any side, so each side is blocked by something. Thus B must touch R, otherwise A,C must block two sides each, which requires an edge of A touching a corner of B and an edge of C touching the opposite corner of B, in which case we can expand B slightly perhaps with some rotation, contradicting maximality. If B touches the left side of R, then B is entirely in the left half of R, so it suffices to show that two non-overlapping squares within a unit square have total size at most 1 (which can be done via similar tricks). Similarly if B touches the right side of R. The remaining case is if B touches the top or bottom side of R, say the top by symmetry. Then the lowest side of B must be blocked, and so A,C must both be touching the bottom side of R and a lower corner of B. Thus B is basically a maximal square inscribed in a triangle, and I believe it is not difficult to finish from here. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications answered Jul 4, 2021 at 21:42 user21820user21820 3,116 1 1 gold badge 20 20 silver badges 38 38 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 reference-request mg.metric-geometry discrete-geometry plane-geometry See similar questions with these tags. 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12370	https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/04%3A_Sets/4.03%3A_Set_Operations	4.3: Set Operations - 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: Sets Gentle Introduction to the Art of Mathematics (Fields) { } { "4.01:_Basic_Notions_of_Set_Theory" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.02:_Containment" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.03:_Set_Operations" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.04:_Venn_Diagrams" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.05:_4.5_Russells_Paradox" : "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:_Introduction_and_Notation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Logic_and_Quantifiers" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Proof_Techniques_I" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Sets" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Proof_Techniques_II-Induction" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Relations_and_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "07:_Proof_Techniques_III-Combinatorics" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "08:_Cardinality" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "09:_Proof_Techniques_IV-_Magic" : "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" } Sun, 05 Sep 2021 07:05:45 GMT 4.3: Set Operations 19383 19383 admin { } Anonymous Anonymous 2 false false [ "article:topic", "showtoc:no", "license:gnufdl", "authorname:joefields", "symmetric difference" ] [ "article:topic", "showtoc:no", "license:gnufdl", "authorname:joefields", "symmetric difference" ] Search site Search Search Go back to previous article Sign in Username Password Sign in Sign in Sign in Forgot password Expand/collapse global hierarchy 1. Home 2. Bookshelves 3. Mathematical Logic and Proofs 4. Gentle Introduction to the Art of Mathematics (Fields) 5. 4: Sets 6. 4.3: Set Operations Expand/collapse global location 4.3: Set Operations Last updated Sep 5, 2021 Save as PDF 4.2: Containment 4.4: Venn Diagrams Page ID 19383 Joseph Fields Southern Connecticut State University ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Practice 2. Practice 3. Practice 4. Practice 5. Practice 6. Exercises: 1. Exercise 4.3.1 2. Exercise 4.3.2 3. Exercise 4.3.3 4. Exercise 4.3.4 5. Exercise 4.3.5 6. Exercise 4.3.6 7. Exercise 4.3.7 8. Exercise 4.3.8 9. Exercise 4.3.9 10. Exercise 4.3.10 11. Exercise 4.3.11 Notes: In this section, we’ll continue to develop the correspondence between Logic and Set theory. The logical connectors ∧ and ∨ correspond to the set-theoretic notions of union (∪) and intersection (∩). The symbols are designed to provide a mnemonic for the correspondence; the Set theory symbols are just rounded versions of those from Logic. Explicitly, if P⁡(x) and Q⁡(x) are open sentences, then the union of the corresponding truth sets S P and S Q is defined by (4.3.1)S P∪S Q={x∈U⁢P⁡(x)∨Q⁡(x)}. Practice Suppose two sets A and B are given. Re-express the previous definition of “union” using their membership criteria, M A⁡(x)=“x∈A” and M B⁡(x)=“x∈B.” The union of more than two sets can be expressed using a big union symbol. For example, consider the family of real intervals defined by I n=(n,n+1].1 There’s an interval for every integer n. Also, every real number is in one of these intervals. The previous sentence can be expressed as (4.3.2)R=⋃n∈Z I n. The intersection of two sets is conceptualized as “what they have in common” but the precise definition is found by considering conjunctions, (4.3.3)A∩B={x∈U⁢x∈A∧x∈B}. Practice With reference to two open sentences P⁡(x) and Q⁡(x), define the intersection of their truth sets, S P∩S Q. There is also a “big” version of the intersection symbol. Using the same family of intervals as before, (4.3.4)∅=⋂n∈Z I n. Of course, the intersection of any distinct pair of these intervals is empty so the statement above isn’t particularly strong. Negation in Logic corresponds to complementation in Set theory. The complement of a set A is usually denoted by A― (although some prefer a superscript c – as in A c), this is the set of all things that aren’t in A. In thinking about complementation one quickly sees why the importance of working within a well-defined universal set is stressed. Consider the set of all math textbooks. Obviously, the complement of this set would contain texts in English, Engineering and Evolution – but that statement is implicitly assuming that the universe of discourse is “textbooks.” It’s equally valid to say that a very long sequence of zeros and ones, a luscious red strawberry, and the number π are not math textbooks and so these things are all elements of the complement of the set of all math textbooks. What is really a concern for us is the issue of whether or not the complement of a set is well-defined, that is, can we tell for sure whether a given item is or is not in the complement of a set. This question is decidable exactly when the membership question for the original set is decidable. Many people think that the main reason for working within a fixed universal set is that we then have well-defined complements. The real reason that we accept this restriction is to ensure that both membership criteria, M A⁡(x) and M A―⁡(x), are decidable open sentences. As an example of the sort of strangeness that can crop up, consider that during the time that I, as the author of this book, was writing the last paragraph, this text was nothing more than a very long sequence of zeros and ones in the memory of my computer. . . Every rule that we learned in Chapter 2 (see Table 2.3⁢.1) has a set-theoretic equivalent. These set-theoretic versions are expressed using equalities (i.e. the symbol = in between two sets) which is actually a little bit funny if you think about it. We normally use = to mean that two numbers or variables have the same numerical magnitude, as in 12 2=144, we are doing something altogether different when we use that symbol between two sets, as in {1,2,3}={1,4,9}, but people seem to be used to this so there’s no sense in quibbling. Practice Develop a useful definition for set equality. In other words, come up with a (quantified) logical statement that means the same thing as “A=B” for two arbitrary sets A and B. Practice What symbol in Logic should go between the membership criteria M A⁡(x) and M B⁡(x) if A and B are equal sets? In Table 4.3⁢.1 the rules governing the interactions between the set-theoretic operations are collected. We are now in a position somewhat similar to when we jumped from proving logical assertions with truth tables to doing two-column proofs. We have two different approaches for showing that two sets are equal. We can do a so-called “element chasing” proof (to show A=B, assume x∈A and prove x∈B and then vice versa). Or, we can construct a proof using the basic set equalities given in Table 4.3⁢.1. Often the latter can take the form of a two-column proof. \| Table 4.3⁢.1: Basic Set Theoretic Equalities \| \| \| ∩ \| ∪ \| \| Commutative Laws \| A∩B=B∩A \| A∪B=B∪A \| \| Associative Laws \| A∩(B∩C)=(A∩B)∩C \| A∪(B∪C)=(A∪B)∪C \| \| Distributive Laws \| A∩(B∪C)=(A∩B)∪(A∩C) \| A∪(B∩C)=(A∪B)∩(A∪C) \| \| DeMorgan's Laws \| A∩B―=A―∪B― \| A∪B―=A―∩B― \| \| Double Complement \| A――=A \| same \| \| Complementarity \| A∩A―=∅ \| A∪A―=U \| \| Identity Laws \| A∩U=A \| A∪∅=A \| \| Domination \| A∩∅=∅ \| A∪U=U \| \| Idempotence \| A∩A=A \| A∪A=A \| \| Absorption \| A∩(A∪B)=A \| A∪(A∩B)=A \| Before we proceed much further in our study of set theory it would be a good idea to give you an example. We’re going to prove the same assertion in two different ways — once via element chasing and once using the basic set-theoretic equalities from Table 4.3⁢.1. The statement we’ll prove is A∪B=A∪(A∩B). First, by chasing elements: Proof: Suppose x is an element of A∪B. By the definition of union we know that x∈A∨x∈B. The conjunctive identity law and the fact that x∈A∨x∉A is a tautology gives us an equivalent logical statement: (x∈A∨x∉A)∧(x∈A∨x∈B). Finally, this last statement is equivalent to x∈A∨(x∉A∧x∈B) which is the definition of x∈A∪(A∩B). On the other hand, if we assume that x∈A∪(A∩B), it follows that x∈A∨(x∉A∧x∈B). Applying the distributive law, disjunctive complementarity and the identity law, in sequence we obtain \begin{array} x ∈ A ∨ (x \notin A ∧ x ∈ B) &\cong (x ∈ A ∨ x \notin A) ∧ (x ∈ A ∨ x ∈ B) \ &\cong t ∧ (x ∈ A ∨ x ∈ B) \ &\cong x ∈ A ∨ x ∈ B \end{array} The last statement in this chain of logical equivalences provides the definition of x∈A∪B. Q.E.D. A two-column proof of the same statement looks like this: Proof: A∪B =U∩(A∪B) =(A∪A)∩(A∪B) =(A∪(A∩B) Q.E.D. There are some notions within Set theory that don’t have any clear parallels in Logic. One of these is essentially a generalization of the concept of “complements.” If you think of the set A as being the difference between the universal set U and the set A you are on the right track. The difference between two sets is written A∖B (sadly, sometimes this is denoted using the ordinary subtraction symbol A−B) and is defined by A∖B=A∩B―. The difference, A∖B, consists of those elements of A that aren’t in B. In some developments of Set theory, the difference of sets is defined first and then complementation is defined by A=U∖A. The difference of sets (like the difference of real numbers) is not a commutative operation. In other words A∖B≠B∖A (in general). It is possible to define an operation that acts somewhat like the difference, but that is commutative. The symmetric difference of two sets is denoted using a triangle (really a capital Greek delta). (4.3.5)A△B=(A∖B)∪(B∖A). Practice Show that A△B=(A∪B)∖(A∩B). Come on! You read right past that exercise without even pausing! What? You say you did try it and it was too hard? Okay, just for you (and this time only) I’ve prepared an aid to help you through. . . On the next page is a two-column proof of the result you need to prove, but the lines of the proof are all scrambled. Make a copy and cut out all the pieces and then glue them together into a valid proof. So, no more excuses, just do it! =(A∩B)∪(B∩A)Identity Law =(A∪B)∩(A∩B)Def. of Relative Difference (A∪B)∖(A∩B)Given =((A∩A)∪(A∩B))∪((B∩A)∪(B∩B))Distributive Law =(A∖B)∪(B∖A)Def. of Relative Difference =(A∩(A∩B))∪(B∩(A∩B))Distributive Law =A△B Def. of Relative Difference =(A∩(A∪B)∪(B∩(A∪B))DeMorgan's Law =(∅∪(A∩B))∪((B∩A)∪∅)Complementarity Exercises: Exercise 4.3.1 Let A={1,2,{1,2},b} and let B={a,b,{1,2}}. Find the following: A∩B A∪B A∖B B∖A A△B Exercise 4.3.2 In a standard deck of playing cards, one can distinguish sets based on face-value and/or suit. Let A,2,...9,10,J,Q and K represent the sets of cards having the various face-values. Also, let ♥,♠,♣ and ♦ be the sets of cards having the possible suits. Find the following A∩♥ A∪♥ J∩(♠∪♥) K∩♥ A∩K A∪K Exercise 4.3.3 The following is a screenshot from the computational geometry program OpenSCAD (very handy for making models for 3-d printing. . . ) In computational geometry, we use the basic set operations together with a few other types of transformations to create interesting models using simple components. Across the top of the image below we see 3 sets of points in R 3, a ball, a sort of 3-dimensional plus sign, and a disk. Let’s call the ball A, the plus sign B and the disk C. The nine shapes shown below them are made from A, B and C using union, intersection and set difference. Identify them! Exercise 4.3.4 Do element-chasing proofs (show that an element is in the left-hand side if and only if it is in the right-hand side) to prove each of the following set equalities. A∩B=A∪B A∪B=A∪(A∩B) A△B=(A∪B)∖(A∩B) (A∪B)∖C=(A∖C)∪(B∖C) Exercise 4.3.5 For each positive integer n, we’ll define an interval I n by I n=[−n,1 n). Find the union and intersection of all the intervals in this infinite family. ⋃n∈Z+I n= ⋂n∈Z+I n= Exercise 4.3.6 There is a set X such that, for all sets A, we have X△A=A. What is X? Exercise 4.3.7 There is a set Y such that, for all sets A, we have Y△A=A. What is Y? Exercise 4.3.8 In proving a set-theoretic identity, we are basically showing that two sets are equal. One reasonable way to proceed is to show that each is contained in the other. Prove that A∩(B∪C)=(A∩B)∪(A∩C) by showing that A∩(B∪C)⊆(A∩B)∪(A∩C)⁢a⁢n⁢d⁢(A∩B)∪(A∩C)⊆A∩(B∪C). Exercise 4.3.9 Prove that A∪(B∩C)=(A∪B)∩(A∪C) by showing that A∪(B∩C)⊆(A∪B)∩(A∪C) and (A∪B)∩(A∪C)⊆A∪(B∩C). Exercise 4.3.10 Prove the set-theoretic versions of DeMorgan’s laws using the technique discussed in the previous problems. Exercise 4.3.11 The previous technique (showing that A=B by arguing that A⊆B∧B⊆A) will have an outline something like Proof: First we will show that A⊆B. Towards that end, suppose x∈A. . . . Thus x∈B. Now, we will show that B⊆A. Suppose that x∈B. . . . Thus x∈A. Therefore A⊆B∧B⊆A so we conclude that A=B. Q.E.D. Formulate a proof that A△B=(A∪B)∖(A∩B) that follows this outline. Notes: 1. The elements of I n can also be distinguished as the solution sets of the inequalities n<x≤n+1. This page titled 4.3: Set Operations is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Joseph Fields. Back to top 4.2: Containment 4.4: Venn Diagrams Was this article helpful? Yes No Recommended articles 4.1: Basic Notions of Set TheoryIn modern mathematics, there is an area called Category theory which studies the relationships between different areas of mathematics. More precisely,... 4.2: ContainmentThere are two notions of being “inside” a set. A thing may be an element of a set, or may be contained as a subset. Distinguishing these two notions o... 4.4: Venn DiagramsHopefully, you’ve seen Venn diagrams before, but possibly you haven’t thought deeply about them. Venn diagrams take advantage of an obvious but import... 4.5: 4.5 Russell’s ParadoxBertrand Russell was one of the twentieth century’s most colorful intellectuals. He was perhaps better known as a philosopher. It’s hard to conceive o... 1: Introduction and Notation Article typeSection or PageAuthorJoe FieldsLicenseGNU FDLShow Page TOCno Tags symmetric difference © 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. 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12371	https://luckytoilet.wordpress.com/2011/10/11/calculating-the-plane-angles-of-platonic-solids/	Calculating the Plane Angles of Platonic Solids – Lucky's Notes Search for: Email Subscription Enter your email address to subscribe to this blog and receive notifications of new posts by email. Email Address: Sign me up! Join 203 other subscribers Categories Languages (8) Machine Learning (14) Mathematics (70) Meta (1) Opinion (19) Programming (47) Research (13) Side Project (4) Travel (6) Uncategorized (14) Recent Posts How to deploy your deep learning side project on a budget Effectively self-studying over the Internet Lessons learned after 6 months of building a language learning startup How I write NLP papers in 8 months: idea to publication Virtual NLP Conferences: The Good and the Bad The Efficient Market Hypothesis in Research How can deep learning be combined with theoretical linguistics? The biggest headache with Chinese NLP: indeterminate word segmentation RSS - Posts RSS - Comments Blog Stats 1,029,662 hits Skip to content Lucky's Notes Notes on math, coding, and other stuff Primary Menu Top Posts Archives About Widgets Posted on October 11, 2011 June 23, 2015 in Mathematics Calculating the Plane Angles of Platonic Solids What is the angle between two planes of a tetrahedron? The angle between any two edges of the tetrahedron is , so it’s easy to (falsely) conclude that the angle between two faces must also be . But this isn’t the case: the plane angle is defined as the angle between two lines on the two planes that are both perpendicular to the edge. None of the edges in a tetrahedron is perpendicular to any other, so the answer of is invalid. We can try to compute the angle using regular Euclidean solid geometry, but things tend to get messy. A different way to approach the problem is by using vector geometry: using vector methods we can easily calculate the plane angle of the tetrahedron (as well as the icosahedron and the dodecahedron). Assume symmetry. We represent three concurrent edges of the polyhedron as three vectors beginning at the same point: , , and ; let and be the angles between the vectors (by symmetry we’re assuming that the two alpha’s are equal): (in case my poor drawing skills does not make this apparent, vectors a and b form one face of the polyhedron and c and b form another face) For simplicity, let’s also say the lengths of each of the three vectors is 1. We want to compute the angle between the plane formed by and , and the plane formed by and . Hence let and be the perpendiculars to and respectively each ending at the same point as their respective vectors: For any two vectors, the dot product is defined with being the angle between the vectors. Given that the lengths of the vectors are all 1, we have: Also, by vector addition: Hence and . We want to find the angle between and — call this angle . Then The dot product of vectors x and y is simply: Additionally . Hence the cosine of the angle is: We can now use this newly derived formula to calculate plane angles! For example… Tetrahedron In the tetrahedron, both and are 60: So and . Icosahedron In the icosahedron, but : The top ‘cap’ is a regular pentagon, which has a vertex angle of 108; each side of the pentagon constitutes a side of an equilateral triangle. Since , , and . Dodecahedron Computing angles for the dodecahedron works a bit differently from the tetrahedron and icosahedron. Instead of using existing edges as vectors, we construct an equilateral triangle by connecting three vertices: So (since it’s part of a regular pentagon) and . Then and . Share this: Click to email a link to a friend (Opens in new window)Email Click to share on Facebook (Opens in new window)Facebook Click to share on LinkedIn (Opens in new window)LinkedIn Click to share on Reddit (Opens in new window)Reddit Click to share on X (Opens in new window)X Like Loading... Related dodecahedrongeometryisosahedronplatonic solidstetrahedronvector geometry Post navigation Varignon’s theorem proved in one line with vectors Solving the AB Game by Brute Force 6 thoughts on “Calculating the Plane Angles of Platonic Solids” Vsevolod Tokarevsays: July 27, 2015 at 1:21 pmReply Briliantly simple! I was calculating dihedral angles of truncated icosahedron using what you refer to “regular Euclidian solid geometry”, and that was messy. May I point out that your sketch of icosahedron looks funny (cap base is a hexagon rather than pentagon.) What was wrong with using the edges of dodecahedron as a, b, and c? LikeLike 2. Pat M says: October 20, 2015 at 6:01 pmReply A couple of typos to keep us on our toes. Very nice work though, appreciate it. LikeLike 3. JN says: October 8, 2018 at 2:23 pmReply Messy? Plane trig works fine with less abstraction. LikeLiked by 1 person 4. haleba says: September 25, 2019 at 2:13 pmReply Your approach looks interesting. Thanks. LikeLike 5. the Rodent of Unusual Size says: November 11, 2019 at 11:25 amReply How about an example applying this to a cube to get the plane angle of 90°? LikeLike 1. luckytoiletsays: November 11, 2019 at 12:10 pmReply Sure, just put alpha = beta = 90 in the formula. LikeLike Leave a comment Cancel reply Δ Create a free website or blog at WordPress.com. Comment Reblog SubscribeSubscribed Lucky's Notes Join 203 other subscribers Sign me up Already have a WordPress.com account? Log in now. 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12372	https://www.youtube.com/watch?v=kXCqOwprmsQ	Expand and Simplify (a + b)(a - b) MagnetsAndMotors (Dr. B's Other Channel) 84500 subscribers 68 likes Description 10068 views Posted: 14 Aug 2023 Here's the template filled in for the expression (a + b)(a - b): Here is how to expand and simplify the expression (a + b)(a - b) using the FOIL method (also known as the distributive property). FOIL stands for First, Outside, Inside, and Last. Step 1: Multiply the terms in the first parentheses with the terms in the second parentheses. Multiply a by a: a a = a^2 Multiply a by -b: a -b = -ab Multiply b by a: b a = ba (which can be simplified to ab, since multiplication is commutative) Multiply b by -b: b -b = -b^2 Step 2: Combine the like terms (terms with the same variable exponent). Combine the terms -ab and ab: -ab + ab = 0 (they cancel out) Combine the terms a^2 and -b^2: a^2 - b^2 So, now we have: a^2 - b^2 The expanded and simplified expression is a^2 - b^2. 5 comments Transcript: let's expand this here we have a plus b times a minus B so the quantity a plus b multiplied by the quantity a minus B we're going to use the foil method to do this so we'll take the first term a times a that'll give us a squared then we'll take the outside so the outside here and here 8 times negative B that gives us negative a be the inside B times a which is really just a b same thing and then the last term B times negative B that gives us negative B squared so at this point we can add up the similar terms here we have a squared and negative a B plus a b these cancel out and we're left with a squared minus B squared this is kind of an important pattern in math this is called the difference of squares this is where you'll see this so when we expand and simplify a plus b times the quantity a minus B we end up with a squared minus B squared that's it this is Dr B and thanks for watching
12373	https://www.raymondgeddes.com/pages/learning-about-money-and-making-change?srsltid=AfmBOorDrWV8AJeZukB5LrVhCa2wbukAZixrrGoVMOfvWa5JkKRI37Va	Lesson 8: Learning About Making Change – Raymond Geddes Skip to content Spend $45.00 to get free shipping Spend $45.00 to get free shipping Spend $45.00 to get free shipping Spend $45.00 to get free shipping Spend $45.00 to get free shipping Spend $45.00 to get free shipping Spend $45.00 to get free shipping Spend $45.00 to get free shipping Menu Shapes Navigation bar Close (esc) Shop Offers $1 and Under Geddes Exclusives Best Sellers Bundles New Items Sale Items Weekly Specials Price Reductions Clearance School Supplies Writing Supplies & Accessories Pens Pencils Pencil Toppers & Grips Pencil Pouches Mechanical Pencils & Lead Refills Highlighters Erasers Pencil Sharpeners Scented School Supplies Novelty Toys & Games Stress Balls Keychains Plush 3D Printed Toys Putty, Slime, and Goo Squishy Toys Tech Gadgets Wearables Classroom Supplies Art Supplies Backpacks Back-to-School Essentials Bookmarks Desk Pets Fidget & Sensory Toys Happy Birthday Incentives & Rewards Paper Products Treasure Boxes Fan-Favorite Brands Pete the Cat Dr. Seuss™ Dog Man™ Diary of a Wimpy Kid Seasonal Valentines Easter/Spring Halloween/Fall Christmas/Winter Grinch School Stores School Kits Sold Individually Goodie Bags Contact Us Log in Create account Search Search Facebook Twitter Pinterest Instagram TikTok YouTube LinkedIn Shop Offers $1 and Under Geddes Exclusives Best Sellers Bundles New Items Sale Items Weekly Specials Price Reductions Clearance School Supplies Writing Supplies & Accessories Pens Pencils Pencil Toppers & Grips Pencil Pouches Mechanical Pencils & Lead Refills Highlighters Erasers Pencil Sharpeners Scented School Supplies Novelty Toys & Games Stress Balls Keychains Plush 3D Printed Toys Putty, Slime, and Goo Squishy Toys Tech Gadgets Wearables Classroom Supplies Art Supplies Backpacks Back-to-School Essentials Bookmarks Desk Pets Fidget & Sensory Toys Happy Birthday Incentives & Rewards Paper Products Treasure Boxes Fan-Favorite Brands Pete the Cat Dr. Seuss™ Dog Man™ Diary of a Wimpy Kid Seasonal Valentines Easter/Spring Halloween/Fall Christmas/Winter Grinch School Stores School Kits Sold Individually Goodie Bags Contact Us Log in Search our site Search Close (esc) Cart items Lesson 8: Learning About Making Change Return to Lesson Plan Home Page \| Download This Lesson \| Next Lesson Lesson Title: Learning About Making Change Grade Band: 3-5 Lesson Length: Approximately 2 days NCTM Standard: Number and Operations Learning Objectives The student will understand the roles of the cashier and customer when making and receiving change. The student will calculate change based on single item purchases paid with a $1.00 and $5.00 bill. Assuming the role of a cashier, the student will calculate total purchase amounts for multi-item sales receipts and the amount of change to give to a customer based on cash paid. The student will compute change, given various item prices and cash paid, in a game of bingo. Connection to Bloom’s Taxonomy Comprehension Application Analysis Synthesis Lesson Materials Bingo Markers Calculators Crayons or Colored Pencils Index Cards Pencils Play or real money (a $1 bill, a $5 bill, and assorted coins) Worksheets for lesson plan 8 (see sidebar) Making Connections What is exact change? Students should understand that this is the same amount as the purchase amount. For example, if you buy a candy bar for $.50 you give the seller exactly $.50 and not a penny less or more. Ask the students, “What if you don’t have the exact amount of money for the candy bar?” The opposite of exact change is simple change back from your purchase. If you purchased the candy bar with a dollar bill, your change would be $.50 in coins. Next, as a class, brainstorm a list of places where you and your family shop or go to for services that require payment. Some possible responses may include the following: Grocery store Gas station Department store Post office Barber shop or hair salon Pet store Convenience store Wholesale shopping club Shoe store Book store Sporting goods store Toy store Ice cream shop Coffee shop Fast food restaurants School store Explain that often, adults may use a credit or debit card to make a purchase. However, many times people pay for products and services with cash. If you don’t have the exact amount of cash (also referred to as exact change), you can give more than the total and receive money back in return. This is called receiving change. Receiving change back may include both coin and paper money. Have students test their shopping knowledge by asking them how much cash in specific denominations is needed to make each assumed purchase on the Going Shopping (PDF) chart. Answers will vary depending on student experiences and money sense. Exploring and Learning Ask students to consider the following: Who is responsible for making sure that you receive the correct change when making a purchase or paying for a service with cash? Students may think it is the responsibility of the cashier. However, the customer (person who is making the purchase) is also responsible and should always be aware that mistakes can often be made by cashiers. Both the customer and cashier should always check to make sure that the correct change is given and/or received. Provide students with practice making change for single item purchases. Pair students together and provide each group with a copy of the Change Chart (PDF). This worksheet is based on customers purchasing each item with $1 and $5 bills. Students may use play money, real money, and/or a calculator to help with calculations if needed. Together as a class, use the Change Chart Answer Key (PDF) to review answers and to check for student progress. Present students with the following scenario: Sniffer is working at the Raymond Geddes Elementary School Store as the school store cashier. RG, Hannie, and several friends are visiting the school store during lunch. Can you help Sniffer calculate the change that should be returned to each customer based on the cash they use to make each purchase? 5. To help complete the scenario, pair students together and provide each group with a copy of Making Change Worksheet (PDF). Explain and list on the board or as a transparency the following instructions with Receipt 1 completes as a sample: Start with Receipt #1 Compute the total prices for each item by multiplying the quantity sold by the item’s retail price. 5 Eraser Grips x $.35 each = $1.75 1 Piranha Sharpener x $.50 each = $.50 Calculate the total purchase amounts. ( $1.75 + $.50 = $2.25 ) If the items are purchased using a $5 bill, how much change should be received? ( $2.75 ) Together as a class, use Making Change Worksheet Key (PDF) to review answers and to check for student progress for all five receipts. Provide students additional practice by play a game of Geddes Bingo - Making Change. Prior to the start of the game you will need to reproduce the 3 different Bingo cards from the Geddes Bingo Game (PDF). You may choose to do this activity individually, in pairs or groups of 3 or more. Explain and list on the board or as a transparency the following game instruction: Each student or group will select a Bingo card. Each student or group needs Bingo markers to cover squares as the game is played or a marker can be used to “X” out the square. Cover the Free Space in the center of their bingo card. Place the item cards from the Geddes Bingo Game (PDF) into a container and select one person to be the “caller”. The caller will draw cards and read the item’s price and the amount paid ($1.00, $2.00, $3.00, $4.00, or $5.00) aloud to the class and it can also be written on the board. Based on the information contained on the card drawn, students must determine the amount of change to be given. For example: The Retro Pencil card is drawn and the caller calls the $1.00 paid column with a price of $.20 for the pencil. The Bingo Columns are the paid amounts of $1.00, $2.00, $3.00, $4.00, and $5.00. These dollar amounts replace the letters B I N G O on a regular bingo game card. Students will determine that $.80 is the change to be given from $1.00 paid. If the Bingo card has a $.80 square in the $1.00 column it can be covered. The first team to cover their bingo card squares to complete a row horizontally, vertically, or diagonally, calls out BINGO. The caller, with help from the teacher, will verify that the team has covered the squares correctly. If the team’s card is covered correctly, they win that round of bingo. Extended Learning and Practice Practice making change by recalculating the amount owed to each school store customer based on each customer paying with a $20 bill. Students can also play a variation of the Making Change Bingo game where the entire board needs to be covered in order to win. Visit the United States Mint H.I.P. Pocket Change website at for additional games and activities. Assessment The lesson objectives can be assessed by evaluating the Change Chart (PDF), Making Change Worksheet (PDF) with the Making Change Worksheet Answer Key (PDF). Use the Assessment of Student Progress (PDF) to assess students’ overall abilities to meet the lesson’s learning objectives, which include understanding the roles of the cashier and customer when making and receiving change; calculating change based on single item purchases paid with a $1.00 and $5.00 bill, assuming the role of a cashier to calculate total purchase amounts for multi-item sales receipts and the amount of change to give to a customer based on cash paid; and computing change, given various item prices and cash paid, in a game of bingo. Closure Provide each student with an index card and have them answer the following questions on one side of the card: What are two new things that you have learned? What else would you like to learn about this topic? On the back side of the index card, instruct the students to draw a picture of something they learned about during this lesson. The index cards can be hole punched and held together with a simple shower curtain ring. Learning Model Making Connections Exploring and Learning Extended Learning and Practice Assessment Closure Lesson Worksheets Bingo Game: Making Change Change Chart Change Chart Answer Key Going Shopping Making Change Worksheet Making Change Worksheet Key Assessment of Student Progress Download all Worksheet PDFs Download all worksheets & the lesson. Teaching Strategies Brainstorming Guided Practice Paired Learning See teaching strategies for all lessons. Literary Connection The Coin Counting Book by Rozanne Lanczak Williams is an excellent introductory book to the world of money. The author cleverly uses quick rhymes and jingles to count and add coins of different denominations. Students have the opportunity to practice using math skills that are needed in order to count and make correct change. Coins are displayed in a photograph form that is easy to see and includes simple mathematics equations. This book is an excellent addition to any elementary media center and will prove to be a teacher favorite. See book recommendations for all lessons. Word Origins Denomination is originally from the Latin word denominationem which means “a naming” and was from the late 1300’s. In the mid 1600’s it was from the Latin word denominare which means “to name completely” and is from de meaning completely and nominare meaning “to name”. Denomination is a naming or designation and can be used to describe or name many things. For example, it is used to name religious groups such as the Catholic, Lutheran, Methodist, and others. In the case of money it is used as a value system. See word origins for all lessons. 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12374	https://www.cuemath.com/algebra/square-1-to-100/	Squares 1 to 100 Squares 1 to 100 is the list of squares of all numbers from 1 to 100. The values of squares from 1 to 100 range from 1 to 10000. Remembering these values will help students to simplify the time-consuming math equations quickly. The square 1 to 100 in the exponential form is expressed as (x)2. Square 1 to 100: Exponent form: (x)2 Highest Value: 1002 = 10000 Lowest Value: 12 = 1 \| \| \| --- \| \| 1. \| Square 1 to 100 \| \| 2. \| Square 1 to 100 PDF \| \| 3. \| How to Calculate Square 1 to 100? \| \| 4. \| FAQs \| Squares 1 to 100 Chart Squares from 1 to 100 Learning squares 1 to 100 can help students to recognize all perfect squares up to 5 digits and approximate a square root by interpolating between known squares. The values of squares 1 to 100 are listed in the table below. \| \| \| \| \| \| --- --- \| List of All Squares from 1 to 100 \| \| \| \| \| \| 12 = 1 \| 22 = 4 \| 32 = 9 \| 42 = 16 \| 52 = 25 \| \| 62 = 36 \| 72 = 49 \| 82 = 64 \| 92 = 81 \| 102 = 100 \| \| 112 = 121 \| 122 = 144 \| 132 = 169 \| 142 = 196 \| 152 = 225 \| \| 162 = 256 \| 172 = 289 \| 182 = 324 \| 192 = 361 \| 202 = 400 \| \| 212 = 441 \| 222 = 484 \| 232 = 529 \| 242 = 576 \| 252 = 625 \| \| 262 = 676 \| 272 = 729 \| 282 = 784 \| 292 = 841 \| 302 = 900 \| \| 312 = 961 \| 322 = 1024 \| 332 = 1089 \| 342 = 1156 \| 352 = 1225 \| \| 362 = 1296 \| 372 = 1369 \| 382 = 1444 \| 392 = 1521 \| 402 = 1600 \| \| 412 = 1681 \| 422 = 1764 \| 432 = 1849 \| 442 = 1936 \| 452 = 2025 \| \| 462 = 2116 \| 472 = 2209 \| 482 = 2304 \| 492 = 2401 \| 502 = 2500 \| \| 512 = 2601 \| 522 = 2704 \| 532 = 2809 \| 542 = 2916 \| 552 = 3025 \| \| 562 = 3136 \| 572 = 3249 \| 582 = 3364 \| 592 = 3481 \| 602 = 3600 \| \| 612 = 3721 \| 622 = 3844 \| 632 = 3969 \| 642 = 4096 \| 652 = 4225 \| \| 662 = 4356 \| 672 = 4489 \| 682 = 4624 \| 692 = 4761 \| 702 = 4900 \| \| 712 = 5041 \| 722 = 5184 \| 732 = 5329 \| 742 = 5476 \| 752 = 5625 \| \| 762 = 5776 \| 772 = 5929 \| 782 = 6084 \| 792 = 6241 \| 802 = 6400 \| \| 812 = 6561 \| 822 = 6724 \| 832 = 6889 \| 842 = 7056 \| 852 = 7225 \| \| 862 = 7396 \| 872 = 7569 \| 882 = 7744 \| 892 = 7921 \| 902 = 8100 \| \| 912 = 8281 \| 922 = 8464 \| 932 = 8649 \| 942 = 8836 \| 952 = 9025 \| \| 962 = 9216 \| 972 = 9409 \| 982 = 9604 \| 992 = 9801 \| 1002 = 10000 \| ☛ Squares 1 to 100 PDF The students are advised to memorize these squares 1 to 100 values thoroughly for faster math calculations. Square 1 to 100 - Even Numbers The table below shows the values of squares from 1 to 100 for even numbers. \| \| \| \| \| \| --- --- \| 22 = 4 \| 42 = 16 \| 62 = 36 \| 82 = 64 \| 102 = 100 \| \| 122 = 144 \| 142 = 196 \| 162 = 256 \| 182 = 324 \| 202 = 400 \| \| 222 = 484 \| 242 = 576 \| 262 = 676 \| 282 = 784 \| 302 = 900 \| \| 322 = 1024 \| 342 = 1156 \| 362 = 1296 \| 382 = 1444 \| 402 = 1600 \| \| 422 = 1764 \| 442 = 1936 \| 462 = 2116 \| 482 = 2304 \| 502 = 2500 \| \| 522 = 2704 \| 542 = 2916 \| 562 = 3136 \| 582 = 3364 \| 602 = 3600 \| \| 622 = 3844 \| 642 = 4096 \| 662 = 4356 \| 682 = 4624 \| 702 = 4900 \| \| 722 = 5184 \| 742 = 5476 \| 762 = 5776 \| 782 = 6084 \| 802 = 6400 \| \| 822 = 6724 \| 842 = 7056 \| 862 = 7396 \| 882 = 7744 \| 902 = 8100 \| \| 922 = 8464 \| 942 = 8836 \| 962 = 9216 \| 982 = 9604 \| 1002 = 10000 \| Square 1 to 100 - Odd Numbers The table below shows the values of squares from 1 to 100 for odd numbers. \| \| \| \| \| \| --- --- \| 12 = 1 \| 32 = 9 \| 52 = 25 \| 72 = 49 \| 92 = 81 \| \| 112 = 121 \| 132 = 169 \| 152 = 225 \| 172 = 289 \| 192 = 361 \| \| 212 = 441 \| 232 = 529 \| 252 = 625 \| 272 = 729 \| 292 = 841 \| \| 312 = 961 \| 332 = 1089 \| 352 = 1225 \| 372 = 1369 \| 392 = 1521 \| \| 412 = 1681 \| 432 = 1849 \| 452 = 2025 \| 472 = 2209 \| 492 = 2401 \| \| 512 = 2601 \| 532 = 2809 \| 552 = 3025 \| 572 = 3249 \| 592 = 3481 \| \| 612 = 3721 \| 632 = 3969 \| 652 = 4225 \| 672 = 4489 \| 692 = 4761 \| \| 712 = 5041 \| 732 = 5329 \| 752 = 5625 \| 772 = 5929 \| 792 = 6241 \| \| 812 = 6561 \| 832 = 6889 \| 852 = 7225 \| 872 = 7569 \| 892 = 7921 \| \| 912 = 8281 \| 932 = 8649 \| 952 = 9025 \| 972 = 9409 \| 992 = 9801 \| How to Calculate the Values of Squares 1 to 100? In order to calculate the squares from 1 to 100, we can use any one of the following methods: Method 1: Multiplication by itself: In this method, the number is multiplied by itself and the resultant product gives us the square of that number. For example, the square of 8 = 8 × 8 = 64. Here, the resultant product “64” gives us the square of the number “8.” This method works well for smaller numbers. Method 2: Using basic algebraic identities: For example, to find the square of 49, we can express 49 as: Option 1: (40 + 9) Option 2: (50 - 1) In the next step, we use the basic algebraic identity formula and get Option 1: [40² + 9² + (2 × 40 × 9)] or Option 2: [50² + 1² - (2 × 50 × 1)]. Solving the expressions further, we get Option 1: (1600 + 81 + 720) = 2401 or Option 2: (2500 + 1 - 100) = 2401. Solved Examples on Square 1 to 100 Example 1: If a circular tabletop has a radius of 50 inches. Find the area of the tabletop in sq. inches? [Use π = 3.14] Solution: Area of circular tabletop = πr2 = π (50)2 Using values from square 1 to 100 chart; i.e. A = 2500π Therefore, the area of the tabletop = 7850 inches2. 2. Example 2: Find the area of a square window whose side length is 34 inches. Solution: Area of square window (A) = Side2 i.e. A = 342 = 1156 Therefore, the area of a square window is 1156 inches2. 3. Example 3: Two square wooden planks have sides 30m and 42m respectively. Find the combined area of both the wooden planks? Solution: Area of wooden plank = (side)2 ⇒ Area of 1st wooden plank = 302 = 900 m2 ⇒ Area of 2nd wooden plank = 422 = 1764 m2 Therefore, the combined area of wooden planks is 900 + 1764 = 2664 m2 4. Example 4: Find the sum of the first 100 odd numbers. Solution: The sum of first n odd numbers is given as n² ⇒ Sum of first 100 odd numbers (n) = 100² Using values from square 1 to 100 chart, the sum of first 100 odd numbers = 100² = 10000 Show Solution > Ready to see the world through math’s eyes? Math is at the core of everything we do. Enjoy solving real-world math problems in live classes and become an expert at everything. Book a Free Trial Class FAQs on Square 1 to 100 What is the Value of Squares 1 to 100? The square 1 to 100 is the list of numbers obtained by multiplying an integer two times (z × z). It will always be a positive number. From 1 to 100, the value of squares of numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98 will be even and the value of squares of numbers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99 will be odd. What are the Methods to Calculate Squares from 1 to 100? We can calculate the square of a number by using the a² + b² + 2ab formula. For example (12)² can be calculated by splitting 12 into 10 and 2. Other methods that can be used to calculate squares from 1 to 100: Finding Square by Column Method Finding Squares by Diagonal Method If you take Squares from 1 to 100, how many of them will be Even Numbers? The even numbers between 1 to 100 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98. Since the squares of even numbers are always even. Therefore, the value of squares of numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98 will be even. Using Squares 1 to 100 Chart, find the value of 100 plus 70 Square plus 50 Square. The value of 70² is 4900 and 50² is 2500. So, 100 + 70² + 50² = 7500. Hence, the 100 plus 70 square plus 50 Square is 7500. How Many Numbers in Squares 1 to 100 are Odd? The odd numbers between 1 to 100 are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99. Since the squares of odd numbers are always odd. Therefore, the value of squares of numbers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99 will be odd. What is the Sum of all Perfect Squares from 1 to 100? The sum of all perfect squares from 1 to 100 is 321 i.e. 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385 What Values of Squares from 1 to 100 are Between 30 and 50? The values of squares 1 to 100 between 30 and 50 are 6² (36) and 7² (49). 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12375	https://www.intlpress.com/site/pub/files/_fulltext/journals/joc/2010/0001/0001/JOC-2010-0001-0001-a001.pdf	Journal of Combinatorics Volume 1, Number 1, 1–27, 2010 A generalization of Apollonian packing of circles Gerhard Guettler and Colin Mallows Three circles touching one another at distinct points form two curvilinear triangles. Into one of these we can pack three new cir-cles, touching each other, with each new circle touching two of the original circles. In such a sextuple of circles there are three pairs of circles, with each of the circles in a pair touching all four circles in the other two pairs. Repeating the construction in each curvilinear triangle that is formed results in a generalized Apollonian pack-ing. We can invert the whole packing in every circle in it, getting a “generalized Apollonian super-packing”. Many of the properties of the Descartes conﬁguration and the standard Apollonian pack-ing carry over to this case. In particular, there is an equation of degree 2 connecting the bends (curvatures) of a sextuple; all the bends can be integers; and if they are, the packing can be placed in the plane so that for each circle with bend b and center (x, y), the quantities bx/ √ 2 and by are integers. The construction provides a generalization of the Farey series and the associated Ford circles. 1. Introduction and summary Recently there has been renewed interest in a very old idea, that of Apol-lonian packing of circles, in which an initial conﬁguration of three mutually tangent circles is augmented by repeatedly drawing new circles in each curvi-linear gap. See for example Mumford et al . We can also study “super-Apollonian” packings which are obtained by repeatedly inverting an Apol-lonian packing in every circle in it. It is a remarkable fact that Apollonian and super-Apollonian packings exist in which all the bends (curvatures) are integers. This property, and the associated group theory, has been studied in detail by Graham et al [4–7]. Also, if all the bends are integers, the super-Apollonian packing can be placed in the plane so that all the “bend times center” quantities are integers. Several extensions of the Apollonian idea have been studied, for example Mauldon studied conﬁgurations in which adjacent circles do not touch but have constant “separation”. Stephenson takes the theory in rather diﬀerent directions. Our own interest lies in extending these ideas in new directions, partic-ularly by packing not one but three circles within each triangular gap, thus 1 2 Gerhard Guettler and Colin Mallows Figure 1: Sextuple conﬁgurations. forming sextuples of circles, and in exploring the degree to which the theory associated with the classical packings can be extended to cover this case. We ﬁnd that all the bends in such a generalized packing can be integers; and there are results relating to the positions of the centers of the circles that directly generalize those found by Lagarias et al in the classical Descartes-Apollonian case. Into either of the curvilinear triangles formed by three mutually tangent circles we can pack three more circles, forming a sextuple. Figure 1 shows the four possible conﬁgurations of a sextuple. There can be zero, one, or two circles with bend zero (i.e. straight lines), and at most one bend can be negative, as in case (a). The circles are labelled with their bends. The bends of the three small circles in Figure 1(b) are 88, 89, and 90. These generalize the classical Descartes conﬁguration, in which just one circle is placed in a curvilinear triangle. Such a sextuple of circles forms an n = 4 example of what we call a “ball-bearing” conﬁguration, in which a ring of n circles (each touching two others) have the property that there are “inner” and “outer” circles that each touch all n circles in the ring. The n = 3 case contains the classical Descartes 4-circle conﬁguration. With A generalization of Apollonian packing of circles 3 Figure 2: A generalized Apollonian packing. n = 4 the circles come in three pairs, with each of the circles in a pair touching all four circles in the other two pairs. The circles of a pair do not touch one another. The sextuple thus has the symmetry of the vertices of an octahedron (or of the faces of a cube), rather than the symmetry of a tetrahedron as in the Descartes case. We describe the bends in a sextuple by a vector of the form (a, a′; b, b′; c, c′). Thus in Figure 1(a) the bends are (−1,7;2,4;2,4) and in Figure 1(c) they are (0,2;0,2;1,1). Repeating the construction in each curvilinear triangle that is formed results in a “generalized Apollonian packing”. See Figures 2 (based on Figure 1(a)) and 3 (based on Figure 1(c)). In these packings, all the bends are integers. In this and subsequent ﬁgures, we have drawn only the circles with bends less than 100. There are also packings that ﬁll either a half-plane, or the whole plane; but their bends cannot all be integral. Many of the properties of the Descartes conﬁguration and the standard Apollonian packing carry over to this case. In later sections we will establish these results: (i) Given three mutually tangent circles, there is a quadratic equation ((5) below) whose coeﬃcients involve the bends of these circles, and whose roots determine the bends of the two sets of three circles that can be in-scribed in the curvilinear triangles that they form, thus making two sex-tuples. Also, given a ring of four circles, formed by two pairs of circles in 4 Gerhard Guettler and Colin Mallows Figure 3: Another generalized Apollonian packing. a sextuple, there is a quadratic equation ((6) below) whose coeﬃcients in-volve the bends of these four circles, the roots of which are the bends of the other pair of circles in the sextuple. These results generalize the classical Descartes equation ((2) below). Replacing each bend by the corresponding bend(complex) center in each of these equations gives results which gener-alize the “Complex Descartes Theorem” of . (ii) There is an analog of “Descartes reﬂection” (see ) in which three circles (one from each pair in a sextuple, these three circles occupying a curvilinear triangle formed by the other three circles of the sextuple) are replaced by three circles occupying the other triangle formed by these three circles, thus forming another sextuple. Given a sextuple, this operation can be performed in eight diﬀerent ways. See Section 4. Iteration of this operation creates a generalized Apollonian packing in which the interiors of all circles are disjoint. A packing is determined by any three touching circles within it. (iii) All six bends of the circles in a sextuple can be integers. See Sec-tion 5. The integrality property is inherited by all derived circles. (iv) if all bends of a sextuple are integers, the sextuple can be placed in the plane so that for each circle with bend b and center (x, y), the “bend times center” quantities (bx, by) are of the form (m √ 2, n) with m, n integers. This property is inherited by the generalized packing based on this sextuple. See Section 7. (v) The construction of the generalized packing can be realised by inte-gral linear operations acting on matrices representing the sextuples. These matrices could be 6 × 4 matrices with each row containing the “abbc” or “augmented bend, bend times center” coordinates introduced in , and de-ﬁned in Section 2 below. However it is convenient to represent a sextuple by a 4 × 4 matrix in which the ﬁrst three rows contain the abbc coordinates of three of the circles in the sextuple (one from each pair) and the fourth row A generalization of Apollonian packing of circles 5 is the average of the two rows that represent a pair of circles in the sextuple (this average is the same for each of the three pairs, see Lemma 4 below). This row does not represent a circle. See Section 3. (vi) There are dual operations acting on the right, which represent Moe-bius transformations. See Section 3. (vii) Among the sextuples with integer bends, there are “root” sextuples (see and ) having the property that any application of the reﬂection operation in (ii) results in circles with larger bends. These root sextuples can be found by applying a reduction algorithm, just as in the Descartes case. Except for the special sextuple with bends (0,2; 0,2; 1,1) (Figure 1(a)), each root sextuple has exactly one circle with negative bend. We have a conjecture (7 below) as to the number of primitive root sextuples with smallest bend −n. See Section 6. (viii) By inverting a generalized Apollonian packing in each circle in the packing, and then again in every circle, and so on, we obtain a “gen-eralized Apollonian super-packing”, directly analogous to the Apollonian super-packing studied in . There is essentially just one super-packing in which all bends are integral. This super-packing can be placed in the plane, in exactly four ways, so that each bx/ √ 2, by is integral. In each version of this super-packing, there is a basic rectangle (0, √ 2) × (0, 1) which repeats by translation and reﬂection to cover the whole plane. See Section 7 and Figure 4. (ix) Each primitive integral sextuple appears exactly once in the (0, √ 2)× (0, 1) rectangle of the super-packing. See Section 7. Computation suggests that there are some symmetries within the basic rectangle, like those shown in . See Figures 5(a)–(d) below. (x) One can consider “ball-bearing” structures of circles, in which a ring of n balls (each touching two neighbors) have the property that there exist “inner” and “outer” circles that each touch each of the “balls” in the ring. The case n = 3 reproduces the Descartes conﬁguration; the case n = 4 gives the sextuples studied in this paper. The bends of all n + 2 circles can be integral only when n = 3, 4, 6. There is a quadratic equation (equation (8) below) whose roots are the bends of the “inner” and “outer” circles, and whose coeﬃcients involve the bends of the circles in the ring. See Section 8. (xi) There is an analog of the Farey series and the associated Ford circles, in which at every stage we insert two new fractions (and two new touching circles) instead of just one, between every existing adjacent pair of fractions. See Section 9 and Figure 6 below. 6 Gerhard Guettler and Colin Mallows Figure 4: A generalized Apollonian super-packing. 2. Conventions and notation We adopt the conventions in [4–8]. A circle C with radius r and center (x, y) is described by its “augmented bend, bendcenter” (abbc) coordinates a(C) = (¯ b, b, bx, by) where b is the bend (i.e. curvature) b = 1/r, and where ¯ b is the bend of the circle that is the inverse of C in the unit circle, namely (1) ¯ b = b(x2 + y2) −1/b Sometimes we will write z = x + iy and work in the complex plane. A straight line is a circle with bend zero, and we need a deﬁnition to replace (1). Consider the line with equation p1x+p2y = h where p = (p1, p2) is a unit vector. This is the limit as λ →∞of a circle with center λp and radius (λ −h), so it makes sense to deﬁne the abbc coordinates of this line as (2h, 0, p1, p2). If the bend of a circle is positive, the “interior” of the circle contains its center. In all cases, the abbc coordinates uniquely determine the circle and its orientation. We say that two circles “touch” or “are tangent” when they have just one point in common and their interiors are disjoint. A generalization of Apollonian packing of circles 7 We are considering sets of six circles such that each touches four others, and the interiors are all disjoint. We can describe a sextuple S by the 6 × 4 matrix that has the abbc coordinates of the circles in its rows, where the circles are taken in what we call standard order, namely rows 1,2 contain the circles in a pair, as do rows 3,4 and rows 5,6. Since we can permute the three pairs, and also can take the circles of a pair in either order, there are 48 diﬀerent standard orders, all representing the same sextuple. We introduce the following matrix W. This is twice the inverse of what is called the “Wilker quadratic form” in , so named in honor of Wilker . W = ⎛ ⎜ ⎜ ⎜ ⎝ 0 −1/2 0 0 −1/2 0 0 0 0 0 1 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠. From and we have Lemma 1. For any two circles C and C′, a(C)Wa(C′)T = 1 if C = C′ = −1 if C and C′ are externally tangent Proof. Simple algebra. We also introduce the matrix of the “Descartes quadratic form” (twice what is deﬁned in ) D = ⎛ ⎜ ⎜ ⎜ ⎝ 1 −1 −1 −1 −1 1 −1 −1 −1 −1 1 −1 −1 −1 −1 1 ⎞ ⎟ ⎟ ⎟ ⎠. Note that D−1 = D/4. We honor Rene Descartes here because in 1643 he derived a result connecting the bends of four mutually tangent circles, equivalent to (2) 2(b2 1 + b2 2 + b2 3 + b2 4) = (b1 + b2 + b3 + b4)2 and this can be written bDbT = 0 where b is the vector (b1, b2, b3, b4). Given a Descartes conﬁguration consisting of four mutually tangent circles 8 Gerhard Guettler and Colin Mallows C1, C2, C3, C4 with disjoint interiors, we deﬁne the 4 × 4 matrix A as con-taining rows a(Cj), j = 1, 2, 3, 4. From Lemma 1 we have AWAT = D. Inverting both sides of this relation we immediately obtain the “Augmented Euclidean Descartes Theorem” of , namely (3) AT DA = 4W−1 The Descartes equation (2) is the second diagonal element of this relation. From the (3,4) coordinates we have the “Complex Descartes equation” of , namely (b ∗z)D(b ∗z)T = 0 where b ∗z = (b1z1, b2z2, b3z3, b4z4). We will ﬁnd generalizations of this and (2) in the sextuple case, but the same argument will not work because we have a 6 × 4 matrix of abbc coordinates, and we get singular matrices. The following Lemma will be very useful. Lemma 2 (, Appendix A). The eﬀect of an arbitrary Moebius transfor-mation on the abbc vector a(C) describing a circle C is to replace a(C) by a(C)m where m is an element of a certain group Moeb of 4 × 4 matrices. All matrices in this group satisfy mWmT = W. The proof is straightforward, using the following description of the ele-ments of Moeb. The group Moeb is generated by matrices of the following kinds: scaling by λ: m is diagonal, with elements (1/λ, λ, 1, 1). rigid rotation about the origin: m is the 4 × 4 unit matrix, with the (3, 4) × (3, 4) submatrix replaced by cos θ sin θ −sin θ cos θ . inversion in the unit circle: m replaces the leading 2×2 submatrix of the 4 × 4 unit matrix by 0 1 1 0 . A generalization of Apollonian packing of circles 9 translation by x, y: m = ⎛ ⎜ ⎜ ⎜ ⎝ 1 0 0 0 x2 + y2 1 x y 2x 0 1 0 2y 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠. 3. Sextuples Lemma 3. If C and C′ are a pair of (non-tangent) circles in a sextuple conﬁguration, then a(C)Wa(C′)T = −3. Proof. The relation is easily veriﬁed for the special sextuple shown in Figure 1(c), with the origin at the contact-point of the two circles with bend 2, for which the abbc coordinates are the rows of ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2 0 0 1 4 2 0 −1 2 0 0 −1 4 2 0 1 1 1 √ 2 0 1 1 − √ 2 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . Any sextuple can be obtained from this one by a Moebius transformation. By Lemma 2, the eﬀect of a Moebius transformation is to replace a(C) and a(C′) by a(C)m and a(C′)m respectively, for some m in Moeb. Each such m satisﬁes mWmT = W and the Lemma follows. This simple result has an important consequence. Lemma 4. In a sextuple S containing circles C1, C′ 1 etc. we have a(C1) + a(C′ 1) = a(C2) + a(C′ 2) = a(C3) + a(C′ 3). Proof. For j = 1, 2, 3 deﬁne wj = (a(Cj)+a(C′ j))/2, and let Fj be the 4×4 matrix whose rows are a(C1), a(C2), a(C3), wj . From Lemmas 1 and 3 we have that w1WwT 1 = (1 −3 −3 + 1)/4 = −1, and w1WwT 2 = (−1 −1 −1 − 1)/4 = −1 also. So (4) F1WFT 1 = K = F2WFT 1 10 Gerhard Guettler and Colin Mallows where K = ⎛ ⎜ ⎜ ⎜ ⎝ 1 −1 −1 −1 −1 1 −1 −1 −1 −1 1 −1 −1 −1 −1 −1 ⎞ ⎟ ⎟ ⎟ ⎠. Since all the matrices in (4) are non-singular, this shows that F1 = F2, so that w1 = w2. Similarly both are equal to w3. This result makes possible a convenient representation of a sextuple. F notation. Given a sextuple S containing six circles C1, C′ 1; C2, C′ 2; C3, C′ 3, an F matrix describing the sextuple is a 4 × 4 matrix whose ﬁrst three rows contain the abbc coordinates of three of the circles (one from each pair) and the fourth row is the average of the abbc coordinates of the two circles in any pair. Given a sextuple, there are 48 such representations, obtained by choosing the pairs in diﬀerent orders, and choosing diﬀerent representatives of each pair. We can move between diﬀerent representations by premultiplying F by matrices like ⎛ ⎜ ⎜ ⎜ ⎝ 0 −1 0 2 0 0 −1 2 1 0 0 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ which replaces C1, C2, C3 by C′ 2, C′ 3, C1. The set of 48 such integral matrices form a group which we call Perm. We can recover all the 48 6 × 4 standard representations as G(S) = EF(S) where E = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 0 0 −1 0 0 2 0 1 0 0 0 −1 0 2 0 0 1 0 0 0 −1 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . Lemma 2 above pointed out that the eﬀect of a Moebius transformation on the abbc coordinates of a circle is right-multiplication by an element m of a certain group. Since such a transformation retains adjacency properties, it follows that the eﬀect on an F matrix describing a sextuple is to replace F by Fm. The next Theorem generalizes the Descartes equation, in two ways. A generalization of Apollonian packing of circles 11 Theorem 1. Given three mutually tangent circles with bends b1, b2, b3, en-closing two triangular gaps, the bends of the two sets of three circles that can be inscribed in these gaps are (2w −b1, 2w −b2, 2w −b3) and (2w′ −b1, 2w′ − b2, 2w′ −b3) where w, w′ are the roots of the equation (5) w2 −2w(b1 + b2 + b3) + b2 1 + b2 2 + b2 3 = 0 A similar result holds for the bendcenters. Also, given the bends b1, b′ 1, b2, b′ 2 of the circles in two pairs of a sextuple, forming a ring, the bends of the other pair, each of which touches all four of these circles, are the roots of the equation (6) 2x2 −xσ + τ −3 8σ2 = 0 where σ = b1 + b′ 1 + b2 + b′ 2, τ = b2 1 + b′ 1 2 + b2 2 + b′ 2 2. The same equation holds when each bend b is replaced by bz, where z is the (complex) center of the circle with bend b. Proof. If w is the (4,2) element of the F matrix that describes a sextuple, so that 2w = b1 + b′ 1 = b2 + b′ 2 = b3 + b′ 3, then inverting both sides of the equation FWFT = K we ﬁnd FT K−1F = W−1 in which K−1 = 1 2 ⎛ ⎜ ⎜ ⎜ ⎝ 1 0 0 −1 0 1 0 −1 0 0 1 −1 −1 −1 −1 1 ⎞ ⎟ ⎟ ⎟ ⎠. The (2,2) element of this equation shows that (5) holds. Also, since b′ 1 = 2w −b1, b′ 2 = 2w −b2, we have σ = 4w, τ = 2(b2 1 + b2 2) −4w(b1 + b2) + 8w2 and (6) follows immediately. The bendcenter results follow by the same argument. This validates claim (i) of the Introduction. Equations (5) and (6) can each be regarded as generalizing the Descartes equation. 12 Gerhard Guettler and Colin Mallows 4. Reﬂection and generalized Apollonian packing In the operation of “Descartes reﬂection” was deﬁned, in which one of the circles in a Descartes quadruple is replaced by the other circle that is tangent to the same three circles as the chosen one. Given three mutually tangent circles C1, C2, C3 with bends b1, b2, b3 and (complex) centers z1, z2, z3, the bends b4, b′ 4 of the two circles that are each tangent to these three circles are the roots of the Descartes equation (2). So “reﬂection” of the b4 circle replaces this circle by one with bend b′ 4 = 2(b1 +b2 +b3)−b4, and (using the “Generalized Descartes equation”) with bendcenter b′ 4z′ 4 = 2(b1z1 + b2z2 + b3z3) −b4z4. This operation can be performed in four ways, taking each of the circles in the Descartes quadruple in turn. Using equation (5), we can perform a similar “reﬂection” operation on the circles in a sextuple. Speciﬁcally, let F be a 4 × 4 matrix describing a sextuple. Then (5) has two roots w, w′ say. One of these gives the bends of the other three circles in the sextuple described by F, namely 2w −b1, 2w − b2, 2w −b3. The other root gives the bends of another set of three circles, namely 2w′ −b1, 2w′ −b2, 2w′ −b3 which also form a sextuple with the three circles in the ﬁrst three rows of F. The (complex) centers of these circles can be found by using (5) with the bends replaced by bendcenters. The new sextuple is the result of inverting the original one in the circle that passes through the three points of contact of the three circles that stay invariant. This validates claim (ii). Since from (5) w + w′ = 2(b1 + b2 + b3), we do not need to solve a quadratic explicitly. Lemma 5. The operation of reﬂection in circles C1, C2, C3 can be performed by premultiplying an F matrix that describes the sextuple, with the abbc coordinates of these three circles in its ﬁrst three rows, by the matrix R, where R = ⎛ ⎜ ⎜ ⎜ ⎝ 1 0 0 0 0 1 0 0 0 0 1 0 2 2 2 −1 ⎞ ⎟ ⎟ ⎟ ⎠. Proof. R replaces w by w′ and w by w′. This validates the “reﬂect” part of claim (v). Note that reﬂection is an involution: R2 = I. R is the same as the matrix S4 of that implements reﬂection of the fourth circle in a Descartes quadruple. To reﬂect in a dif-ferent triad of circles in the sextuple, we use the “permute” group Perm to A generalization of Apollonian packing of circles 13 bring the required triad of circles into rows 1,2,3 of the representation F. So the operation can be performed in eight ways. Note that each of these eight operations puts a new triad of circles into a diﬀerent one of the gaps formed by the original six circles. The generalized Apollonian packing based on a given sextuple S0 is obtained by premultiplying F(S0) by “permute” and “reﬂect” matrices, in all possible ways. Thus a general sextuple in the generalized Apollonian packing based on S0 is represented by an F matrix of the form RPnRPn−1 · · · RP1F(S0). (Since R is an involution, we do not need powers of R higher than the ﬁrst; since the “permute” matrices form a group, we never need more than one between two R’s.) Figures 2 and 3 show generalized Apollonian packings based on the sex-tuples in Figure 1(a) and 1(c) respectively. Note that in such packings, the interiors of all circles are disjoint (a circle with negative bend has the point at inﬁnity in its interior) and no circles intersect (though many touch). As in the Descartes case, there are several ways to regard the generalized Apollonian packing: (i) as a packing of circles in the plane, each touching four others; these four all being touched by one other circle; (ii) as a packing of triads of touching circles, each triad surrounding (and being surrounded by) another triad of circles; (iii) as an arrangement formed by conﬁgurations consisting of sextuples of circles, interlocking by overlapping sets of three tangent circles. The view (iii) gives rise to intriguing group-theory problems, analogous to those studied in Graham et al [4–5]. 5. Integral packings We will consider only primitive sextuples, where the bends have no common factor. Suppose (a, a′; b, b′; c, c′) is a primitive integral sextuple. Since 2w = a + a′ etc., 2w is a positive integer. Since from (5) w = a + b + c ± 2(ab + ac + bc), ab + ac + bc must be twice a square, say 2m2 with m an integer, and w is an integer. 14 Gerhard Guettler and Colin Mallows At least one of a, b, c must be odd, since otherwise all six bends are even. Suppose a is odd. Then (a + b)(a + c) = a2 + 2m2 which is odd, so each of a + b and a + c must be odd, and b and c must be even. Hence w = a + b + c ± 2m is an odd positive integer. We can generate (all) primitive integral sextuples of bends by choosing a odd and m arbitrarily (without loss of generality we can take m non-negative, but a can have either sign). Factorize a2 + 2m2 as fg, with f and g positive, and set b = f −a, c = g −a, w = a+b+c±2m, a′ = 2w −a etc. The sextuple is primitive iﬀa, f, g have no common factor. This settles claim (iii). Lemma 6. If the bends of a sextuple are all integers, the same is true for the bends of all the circles in the generalized packing based on this sextuple. If the abbc coordinates of the circles in a sextuple are of the form (integer, integer, integer √ 2, integer), then the same is true for every circle in the packing. Proof. The matrix R and the matrices in the group Perm have integer en-tries. This lemma is not empty, since we have the examples in ﬁgure 1(a) and 1(c). 6. Reduction algorithm and root sextuples The size of a sextuple is conveniently measured by the value of w. Given the bends of a sextuple, by applying a permutation and then reﬂecting, we may be able to ﬁnd a sextuple with smaller value of w. If this is not possible, we call this a root sextuple. Except for the special sextuple (shown in Figure 1(c)) with bends (0,2;0,2;1,1), a root sextuple has exactly one negative bend. As in the Descartes case (see ), starting from a given sextuple, there is a root sextuple that can be reached by repeated reﬂections. The root sextuple is unique because its three smallest bends are the bends of the three largest circles in the packing, and so are unique. However the packing may contain more than one copy of this sextuple, for example the packing in Figure 2 contains two copies of the root sextuple (−1,7;2,4;2,4) and the packing in Figure 3 contains inﬁnitely many copies of the root sextuple (0,2;0,2;1,1). A generalization of Apollonian packing of circles 15 Lemma 7. The sextuple described by integer bends a ≤b ≤c ≤w (satisfy-ing (5)) is a root sextuple iﬀw ≤a + b + c. Proof. w′ ≥w, and this is true also if we replace any of a, b, c by a′, b′, c′. Computation ﬁnds the numbers of primitive root sextuples with smallest bend −n, for n = 1, . . . , 100, in the following table. The number of root sextuples with smallest bend −n. i 1 2 3 4 5 6 7 8 9 10 n = 0 + i 1 2 1 3 2 4 3 5 2 8 n = 10 + i 3 6 4 10 4 9 5 8 5 14 n = 20 + i 5 12 7 10 8 16 5 18 8 16 n = 30 + i 9 17 6 18 13 14 10 20 8 26 n = 40 + i 11 20 11 22 10 26 13 18 15 32 n = 50 + i 9 30 14 20 16 34 10 32 15 28 n = 60 + i 16 34 13 33 22 24 17 34 13 52 n = 70 + i 19 26 19 40 16 38 21 32 21 50 n = 80 + i 14 42 21 36 25 44 16 42 23 40 n = 90 + i 29 50 17 50 28 34 25 58 16 62 For example, the three primitive root sextuples with smallest bend −4 are: (−4,26; 7,15; 10,12) (−4,30; 6,20; 13,13) (−4,46; 5,37; 20,22). From the counts in the table and several more with larger values of n, we conjecture a formula (having some resemblance to the one proved in for the corresponding Descartes case) for the number of root sextu-ples. Conjecture. The number of primitive root sextuples with smallest bend −n is (7) k(n) = 1 4 ⎛ ⎝n p\|n p + χ(p) p + r(n) ⎞ ⎠ 16 Gerhard Guettler and Colin Mallows where χ(2) = 2, χ(p) = −1, −1, 1, 1 for p = 1, 3, 5, 7 (mod 8),and the re-mainder term depends on the number of diﬀerent odd prime factors of n, which we name ρ(n): if n is odd, ρ(n) = 1, then r(n) = φ(p) where φ(p) = 4, 2, 2, 4 when p = 1, 3, 5, 7 (mod 8). if n is odd and ρ(n) ≥2, r(n) = 2ρ(n) if n is even, r(n) = 4.2ρ(n). This conjecture constitutes claim (vi). 7. Inversion and the generalized super-packing Given a generalized Apollonian packing, we can invert the whole packing in every circle in the packing, and then again in every circle in the resulting packing, and so on. The result is what we call a “generalized super-packing” (compare ). In this packing, every two circles either touch, or are disjoint. See Figure 4. It is convenient to regard the super-packing as being composed of sextu-ples. We can generate the sextuples of the super-packing by adding another generator to the generalized Apollonian group, representing inversion in the ﬁrst circle of a general sextuple. Lemma 8. Given a sextuple S described by an F matrix F(S), the result of inversion in the circle described by the top row of F is a sextuple S′ with F(S′) = VF(S) where V = ⎛ ⎜ ⎜ ⎜ ⎝ −1 0 0 0 2 1 0 0 2 0 1 0 2 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠. Proof. Suppose the rows of F(S) are a, b, c, w. From (3.26) of (or by simple geometry) we have that since each of the b, c circles touches the a circle, their transformed abbc coordinates are given correctly. Since the b′ circle (whose abbc coordinates are 2w −b) also touches the a circle, its transformed abbc coordinates are 2a + 2w −b so that the new fourth row of F is (1/2)(2a + b + 2a + 2w −b) = 2a + w. Note that V is an involu-tion. A generalization of Apollonian packing of circles 17 This validates claim (v). The situation is exactly analogous to that of the standard Apollonian super-packing, see . It is convenient to work with the matrix group B that has eight generators of the form PRPT , with PϵPerm, which reﬂect in the eight diﬀerent triads in a sextuple, and six generators of the form PVPT which each invert with respect to one of the circles in a sextuple. Explicitly, given a sextuple with circles C1, C2, C3, C′ 1, C′ 2, C′ 3 described by an F matrix with the ﬁrst three rows corresponding to the ﬁrst three of these circles, we have R000 = R as given above, which reﬂects leaving the circles C1, C2, C3 invariant, R100 = ⎛ ⎜ ⎜ ⎜ ⎝ −3 4 4 4 0 1 0 0 0 0 1 0 −2 2 2 3 ⎞ ⎟ ⎟ ⎟ ⎠ which reﬂects leaving C′ 1, C2, C3 invariant, and similarly R010 and R001, R110 = ⎛ ⎜ ⎜ ⎜ ⎝ −3 −4 4 12 −4 −3 4 12 0 0 1 0 −2 −2 2 7 ⎞ ⎟ ⎟ ⎟ ⎠ which reﬂects leaving C′ 1, C′ 2, C3 invariant, and similarly R101 and R011, and ﬁnally R111 = ⎛ ⎜ ⎜ ⎜ ⎝ −3 −4 −4 20 −4 −3 −4 20 −4 −4 −3 20 −2 −2 −2 11 ⎞ ⎟ ⎟ ⎟ ⎠ which reﬂects leaving C′ 1, C′ 2, C′ 3 invariant. Also we have V1 = V as above, which inverts with respect to the circle C1, and similarly V2 and V3, and V1′ = ⎛ ⎜ ⎜ ⎜ ⎝ −5 0 0 12 −2 1 0 4 −2 0 1 4 −2 0 0 5 ⎞ ⎟ ⎟ ⎟ ⎠ which inverts with respect to the circle C′ 1, and similarly V2′ and V3′. We can write a word in the super-group B as U = UnUn−1 · · · U1, in which each Ui is one of these generators Rx and Vy. Each such word in B has a normal form, obtained by cancelling all squares, and moving each 18 Gerhard Guettler and Colin Mallows appearance of each inversion Vy as far to the right (i.e. earlier) as possible, so that Vy follows a reﬂection that does not leave the circle Cy invariant. Then the sextuples in the super-packing are in 1-1 correspondence with normal-form words in the super-group. We need to show that there are no identities in this group other than those described above. Theorem 2. The generators R000 etc. and V1 etc described above are in-volutions. Also each Vj commutes with the four Rs that leave Cj invariant. There are no other identities among these generators. Proof. Direct computation veriﬁes the involution and commutativity rela-tions. The hard part is showing that there are no other relations. The fol-lowing argument mimics the key part of the proof of Theorem 3.1 in . We consider the eﬀect of applying an element of the group (i.e. a word in nor-mal form) to an arbitrary basic sextuple S0. There are two cases. First, the word contains no inversions. In this case, a reﬂection R that leaves circles C1, C2, C3 invariant puts three circles inside one of the curvilinear trian-gles that they form. We can get out of this triangle only by repeating this reﬂection, but this is not allowed in a normal-form word until some other reﬂection has intervened, and then not all of C1, C2, C3 are still members of the sextuple. Second, suppose an inversion Vj appears. This puts ﬁve circles inside Cj, and we can only get outside of this circle by another application of Vj. But in a normal form word, Vj cannot reappear until either (i) another V, or (ii) an R that does not leave Cj invariant, has intervened. In either case Cj is no longer a member of the sextuple. Thus we can never get back to the original sextuple. Theorem 3. The integral super-packing can be placed in the plane so that all bendcenters are of the form (m √ 2, n) with m and n integral, in exactly four ways. Proof. The argument exactly parallels that in the Descartes case, see The-orems 6.2 and 6.3 of . Given the bends of any primitive integral sex-tuple S, there is an unique normal-form word U in the generators of the super-Apollonian group that reduces it to the special integral sextuple S0 = (0, 2; 0, 2; 1, 1). This sextuple can be placed in the plane making bx/ √ 2 and by integers for every circle, by making the straight lines parallel to the x-axis, and with the point of contact of the two bend-2 circles at any point m √ 2, n with m and n integral. Application of inversions in the bend-0 cir-cles and reﬂections leaving these circles invariant then ﬁlls the whole plane, and forms just four diﬀerent placements of the whole ﬁgure, namely with the origin at A generalization of Apollonian packing of circles 19 (i) the center of a bend-1 circle, (ii) the point of contact of a bend-1 circle and one of the lines, (iii) the contact-point of the two bend-2 circles, (iv) the point of contact of a bend-2 circle and one of the lines. We get only four placements, not eight as in the Descartes case, because here we cannot reﬂect in the line y = x while keeping the values of bx/ √ 2 and by integral. In each placement, the basic rectangle (0, √ 2)×(0, 1) repeats by reﬂections to cover the whole plane. Now applying the elements of the word U in reverse order to the F matrix describing this placed basic sextuple S0, we ﬁnd exactly one placing of the sextuple S within each version of the basic rectangle. This validates claims (vii) and (viii) of the Introduction. Computation suggests that there are some symmetries within the basic rectangle, as in the Descartes case, namely circles with even bends appear symmetrically about x = √ 2/2 circles with bends = 0 mod 8 appear symmetrically about x = √ 2/4 circles with bends = 2 or = 6 mod 8 appear symmetrically about y = 1/2 a circle with bend = 4 mod 8 and center x, y has a mate at √ 2/2 −x, 1 −y See Figures 5(a)–(d). Each circle in these ﬁgures is the largest circle of a unique root sextuple. For example, the three root sextuples with smallest bend −4, which we listed in section 6 above, lie within the three largest circles in Figure 5(c). We do not see any symmetries for circles with odd bends. Similar symmetries in the Descartes case were conjectured in and proved by S. Northshield . 8. Ball-bearing conﬁgurations We deﬁne a “ball-bearing” structure as consisting of a ring of circles, each touching two neighbors, with the property that there exist “inner” and “outer” circles that each touch each of the circles in the ring. Theorem 4. In an n-ball ball-bearing structure, where the bends of the “ball” circles are (in order) b1, b2, . . . , bn, the bends of the “inner” and “outer” 20 Gerhard Guettler and Colin Mallows Figure 5: (a) Circles with bend = 0 mod 8. Figure 5: (b) Circles with bend = 2 mod 8. A generalization of Apollonian packing of circles 21 Figure 5: (c) Circles with bend = 4 mod 8. Figure 5: (d) Circles with bend = 6 mod 8. 22 Gerhard Guettler and Colin Mallows circles are the roots of the equation (8) Dnx2 −xσ + τ −3 2nσ2 = 0 where σ = bj, τ = (bj)2, and Dn = (n/2) cot(π/n)2. Also, if the cen-ters of the ring circles are the complex numbers z1, ..., zn, the bendcenter coordinates of the inner and outer circles are the roots of the equation (9) DnX2 −XS + T −3 2nS2 = 0 where S = bjzj, T = (bjzj)2. Remark 1. For n = 3, D3 = 1/2 and (8) is the Descartes relation that connects the bends of four mutually tangent circles. Also (9) is then the “Complex Descartes Theorem” of . Remark 2. An entry in MathWorld entitled “Soddy Circles” gives a for-mula that is diﬀerent from ours. However this must be in error, since it is not homogeneous in the bends. If the bends b1, b2, . . . , bn are integers, the roots of (8) can be rational only when n = 3, 4, 6 (Dn = 1/2, 2, 9). Only in these cases can all the bends be integers. Proof of the Theorem. We start by proving the relation (8) for a special conﬁguration, in which the “inner” and “outer” circles are centered at the origin, with the j-th ring circle having center ω2j, where ωn = −1. The radius of each ring circle is s = sin π/n; the radii of the inner and outer circles are 1 −s, −(1 + s). Let a0 contain the abbc coordinates of the “inner” circle, and for j = 1, . . . , n let aj contain the abbc coordinates of the j-th circle in the ring, so that a0 = (s −1, 1/(1 −s), 0, 0) aj = (1/s −s, 1/s, cos(2πj/n)/s, sin(2πj/n)/s). We assemble these vectors in an (n+1)×4 matrix A. Let Q be the n+1×n+1 matrix Q = Dn −1 21T n −1 21n In −3 2n1n1T n where 1n is the n × 1 column vector (1, 1, . . . , 1)T . The equation in the The-orem is bT Qb = 0 where b is the vector of bends bT = (1/(1−s), (1/s)1T n). A generalization of Apollonian packing of circles 23 Straightforward computation shows that AT QA = n s2 W−1 where W was deﬁned in Section 2. Picking oﬀthe (2,2) element proves the ﬁrst claim of the theorem (for this special case). The second claim follows from the (3, 4) × (3, 4) submatrix. Since for any m in Moeb we have mT W−1m = W−1, (8) remains true when A is replaced by Am. Thus the relation holds for every ball-bearing conﬁguration. This validates claim (ix). Remark 3. The bends of the ring circles satisfy n−3 linear identities, as do the “bendcenter” quantities. For n = 4 the identities are b1 + b3 = b2 + b4 and b1z1 + b3z3 = b2z2 + b4z4. For n = 6 they are b1 + b4 = b2 + b5 = b3 + b6, b1 + b3 + b5 = b2 + b4 + b6, etc. For all n ≥3 a ball-bearing conﬁguration is determined by the abbc coordinates of the “inner” (or “outer”) circle and any three consecutive “ring” circles. Thus we can work with 4 × 4 matrices in all cases. Remark 4. Given three mutually tangent circles with integral bends a, b, c, set q = ab + ac + bc. Then the circles can belong to an integral Descartes quadruple iﬀq is a perfect square; can belong to an integral sextuple (in the sense of this paper) iﬀ2q is a perfect square; and can form part of a n = 6 ball-bearing conﬁguration iﬀ3q is a perfect square. Remark 5. In the n = 6 case the basic integral conﬁguration (analogous to the (0,0,1,1) conﬁguration in the n = 3 case, and (0,2;0,2;1,1) in the n = 4 case) is (1,1; 0,3,6,6,3,0), consisting of two parallel lines with two unit circles touching both, and a pile of four circles between them. This conﬁguration can be placed in the plane so that all values of bx/ √ 3 and by are integers. However it is not the case that all derived bends and bendscenters are integral. Remark 6. One can also consider multiple-ring structures, for example a 1-5-5-1 structure. Here, every circle touches ﬁve others, and the circles have the symmetry of the vertices of an icosahedron. This suggests an-other generalization of Apollonian packing, in which nine circles are placed in every curvilinear triangle. In this case the bends cannot all be inte-gers. 24 Gerhard Guettler and Colin Mallows Figure 6: Generalized Ford circles. Remark 7. Similar constructions are possible in three dimensions. Between four spheres, each of which touches the other three, we can (uniquely) pack four more spheres, touching each other and each touching three of the orig-inal spheres. We leave consideration of these conﬁgurations to another oc-casion. 9. Generalized Farey series and Ford circles In the generalized Apollonian packing based on the sextuple (0,2;0,2;1,1) depicted in Figure 3, the circles that touch the lower bend-zero line are analogous to Ford circles, and their points of contact are analogous to terms in the Farey series. See Figure 6. We start with a circle with bend 1 touching the x-axis at x = 0, and another circle with bend 2, touching the ﬁrst one and touching the x-axis at x = √ 2. Packing three circles between these (as in Figure 1(d)) gives two new circles touching the x-axis, namely a circle with bend 8 touching at x = (1/2) √ 2 and another with bend 9 touching at x = (2/3) √ 2. The A generalization of Apollonian packing of circles 25 third circle is ignored. Note that 8 = 2·22, 9 = 32, and 1/2 = (0+1)/(1+1), 2/3 = (0+2·1)/(1+2·1). This pattern continues. In general, in a gap between a circle with bend q2 touching at x = (p/q) √ 2 and another with bend 2s2 touching at x = (r/s) √ 2, where p is even and r is odd, we construct a circle with bend (q +2s)2 touching at x = ((p+2r)/(q +2s)) √ 2 and another with bend 2(q + s)2 touching at x = ((p + r)/(q + s)) √ 2. Two fractions p/q, r/s are neighbors, and the corresponding circles touch each other, if and only if \|ps −qr\| = 1. At every stage, the two types of circles (with bends q2 and 2s2) alternate. See Figure 6. 10. Conclusion, and some open questions We have shown that much of the theory of the classical Apollonian packing and super-packing can be carried over to the sextuple case. We have left open several possible extensions of the theory. For example, what if we mix classical Apollonian insertions (one circle per gap) with the present rule of three circles per gap? Can all the bends be integral? We think not, except in cases where the initial triad of circles has some symmetry. What about placing six or ten circles in each gap? It seems that in all such cases we can use 4×4 matrices to describe the conﬁgurations that arise. We have not explored in depth the group theory associated with our generalized packings and super-packings. Here are a few more questions. Is the conjectured formula for the number of root sextuples cor-rect? Are the conjectured symmetries within the basic cell of the super-packing valid? Do all integers arise as bends of circles in generalized Apollonian packings? (It is known that a positive fraction of possible integer bends does occur in a standard Apollonian packing, see . It may be that all suﬃciently large integers appear in each of our generalized packings.) Is the Hausdorﬀdimension of our generalized packing the same as in the Apollonian case? Acknowledgment Thanks to an anonymous referee for his very careful reading, and useful comments. 26 Gerhard Guettler and Colin Mallows References D. Aharonov and K. Stephenson. Geometric sequences of discs in the Apollonian packing. Algebra i Analiz., 1997. MR1466797 J. Bourgain and E. Fuchs. A proof of the positive density conjecture for integer Apollonian circle packings. Arxiv preprint arXiv:1001.3894, 2010. R. Descartes. Oeuvres de Descartes, Correspondance IV (eds. C. Adam and P. Tannery). Leopold Cerf, Paris, 1901. R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks, and C. Yan. Apollonian circle packings: number theory. J. Number Theory 100 (2003), 1–45. MR1971245 R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks, and C. Yan. Apollonian circle packings: Geometry and Group Theory I. The Apollo-nian Group. Discrete and Computational Geometry 34 (2005), 547–585. MR2173929 R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks, and C. Yan. Apollonian circle packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings. Discrete and Computational Geometry 35 (2006), 1–36. MR2183489 R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks, and C. Yan. Apollonian circle packings: Geometry and Group Theory III. Higher Dimensions. Discrete and Computational Geometry 35 (2006), 37–72. MR2183490 J. C. Lagarias, C. L. Mallows, and A. R. Wilks. Beyond the Descartes circle theorem. Amer. Math. Monthly 109 (2002), 338–361. MR1903421 J. G. Mauldon. Sets if equally inclined spheres. Canadian J. Math 14 (1962), 509–516. MR0142031 D. Mumford, C. Series, and D. Wright. Indra’s Pearls. Cambridge U. P., 2002. MR1913879 S. Northshield. On Apollonian circle packings. Preprint, July 26, 2005. K. Stephenson. Introduction to circle packing. Cambridge U. P., 2005. MR2131318 J. B. Wilker. Inversive Geometry, in The Geometric Vein (eds. C. Davis, B. Grunbaum and F. A. Sherk), Springer-Verlag, New York, 1981, 379– 442. MR0661793 A generalization of Apollonian packing of circles 27 Gerhard Guettler University of Applied Sciences Giessen Friedberg Germany E-mail address: dr.gerhard.guettler@swd-servotech.de Colin Mallows Avaya Labs Basking Ridge, NJ 07920 USA E-mail address: colinm@research.avayalabs.com Received January 27, 2010
12376	https://geometry.stevejtrettel.site/plane/lines/	geometry - 12 Lines The Plane 12 Lines geometry Dedication The Greeks 1 Euclid 2 Parallels 3 Pythagoras 4 Archimedes 5 Modern Axioms Calculus 6 Fundamental Strategy 7 Working Infinitesimally 8 Zooming In 9 Zooming Out The Plane 10 Foundations 11 Isometries 12 Lines 13 Shapes 14 Angles 15 Area 16 π The Sphere 17 Foundations 18 Lines & Circles 19 Curvature 20 Polygons Maps 21 Cartography 22 Examples 23 Mercator 24 Stereographic Hyperbolic Space 25 Discovery 26 Models 27 Geometry 28 Life in Curved Space Lorentzian Geometry 29 A Strange Inner Product 30 Geometry of Minkowski Space 31 Geometry of Spacetime 32 Relativity Assignments Table of contents 12.1 Shortest 12.1.1 Uniqueness of Lines 12.2 Straightest 12.3 Folding 12.4 Distance 12.4.1 Useful Computations The Plane 12 Lines 12 Lines Laying down the foundations at a deeper level than the Greeks, we have some work to do before we can hope to recover the axioms of Euclid. Indeed - no where in our foundations does the term line even appear: we are in the awkward position of being able to work with any curve we like, but we do not know which among them is a straight line! To find the lines among the sea of curves, we need a good and precise definition. Definitions single out an important property characterizing the object being defined, and for that definition to be good, we would like that condition to be checkable within the framework we are building. So - Euclid’s definition of a line as a breadthless length is not going to do much for us here. However, looking to history, we find several good candidate definitions among the properties of lines the ancients took as essential. Definition 12.1 (Essential Properties of Lines) Archimedes used as an axiom of length that the line segment between two points is the shortest among all curves connecting them. This could be turned upside down and directly used as the defining feature of lines: whichever curve is shortest, we call a line. As a followup to the infamously unhelpful breadthless length Euclid states the important feature of a line being that the points lie evenly with themselves. This also requires a bit of translation, but if we can define what it means for a curve to turn, we could then specify straight lines as curves that do not turn. The term line also shows up in phrases such as line of symmetry - for instance in discussing that the human form is left-right-symmetric. The fact that reflections fix a line is foundational to geometric arts like Origami, which is what allows the use of Euclidean geometry to describe the collection of creases made: they arise as lines of symmetry, so they are the lines of Euclid! In fact all three of these things can be made into precise statements in our new geometry, and we can compute exactly what sort of curves satisfy each of them. The main purpose of this section is to do so, and to show that all three of them end up specifying exactly the same class of curves! This is one reason that lines are so important to geometry: they are the single objects sharing all three of these very natural properties! 12.1 Shortest We start first with the insight of archimedes, and attempt to make precise the notion of shortest curve between two points. In doing so, we will actually first define line segment, and then use this to define lines more generally. Definition 12.2 (Line Segment) Given two points p,q∈E 2, a curve γ starting at p and ending at q is called a line segment if it is distance minimizing. That is, for all other curves α from p to q, we have length(γ)≤length(α) Defining a line segment γ as the shortest curve joining its endpoints. This definition seems very powerful: if you know something is a line segment you know a lot about it: you know how its length relates to the length of every single other curve! Theorem 12.1 (Segments of x-axis are Minimizers) Finite segments of the x axis, that is, curves of the form γ(t)=(t,0)a≤t≤b are length minimizers. Proof. First, we compute the length of the x-axis between 0 and L by integrating the infinitesimal lengths of γ: γ(t)=(t,0)⟹γ′(t)=(1,0)⟹∥γ′(t)∥=1 length(γ)=∫0 L∥γ′(t)∥d t=∫0 L d t=L This of course is unsurprising! But its good to know explicitly that we have found a curve of length exactly L. Now, let α(t)=(x(t),y(t)) be any arbitrary (regular)curve connecting (0,0) to (L,0). Our goal is now to show that length(α)≥L, as this would mean no curve can have a length less than L, and our segment of the x-axis above is indeed the shortest curve! The difficulty in doing so is that we know very little about α, and hence very little about its coordinate functions x(t),y(t). If α is defined on the interval [a,b] knowing that it starts and ends at (0,0) and (L,0) implies x(a)=0 x(b)=L y(a)=0 y(b)=0 but this is essentially all we know. Nonetheless, let’s push onwards and see what we can learn about length(α) by writing out its definition. length(α)=∫a b∥α′(t)∥d t=∫a b x′(t)2+y′(t)2 d t Now we do some estimation: we know that whatever y is, y′(t)2 is nonnegative - because its squared, after all! So y′(t)2≥0⟹x′(t)2+y′(t)2≥x′(t)2 We can then take the square root of both sides of this equation (which preserves inequalities) to get x′(t)2+y′(t)2≥x′(t)2=\|x′(t)\|≥x′(t) Igoring all the middle terms in this string of inequalities, (and recalling the left hand side is the norm of α′) we see that ∥α′(t)∥≥x′(t)for all t Thus, as functions of t, we see that the curve x′ lies below the curve ∥α′∥: since the area under the lower curve must be less than or equal to the upper, this inequality is still preserved after we integrate. ∫a b∥α′(t)∥d t≥∫a b x′(t)d t But now we have really made some progress: on the right side here we are integrating a derivative, so we can use the fundamental theorem of calculus! The antiderivative of x′(t) is just x(t) of course, so we evaluate at the endpoints: ∫a b x′(t)d t=x(t)\|a b=x(b)−x(a)=L−0=L And with that, we’ve done it! The integral on the left side was precisely the length of α, so length(α)≥L Now that we have a firm understanding of segments, how can we properly bootstrap this idea to a definition of lines? A line itself has no endpoints, and so is not a distance minimizing curve! However, it has the property that if you cut out any segment from it, that segment is distance minimizing. To say this formally, we need a word for “cut out a segment of a curve” Definition 12.3 (Finite Segment of a Curve) Given a curve γ:R→E 2, a finite segment of γ is the restriction of γ to some finite interval [a,b]⊂R. A segment of a curve is a restriction of that curve to a sub-interval of its domain. This makes the definition for a line completely precise: Definition 12.4 (Line) A curve γ is a line if all of its finite segments are line segments. This sounds pretty useless until we unpack it: since line segments are distance minimizers, this is saying that to be a line, a curve must have the property that it is distance minimizing between any two points it passes through! A strong condition indeed. However, given the work we did above on segments of the x axis, we can now immediately apply this to the entire axis itself. Corollary 12.1 (The x-axis is a line) Every finite segment of the x-axis is a distance minimizing line segment, so the x-axis is a line. Well, after all this theory we have finally managed to track down one line in the plane! How can we find more? One option of course is to mimic the argument given here: with trivial modifications we can similarly prove that the y axis is a line, and that curves of the form x=a or y=b are all lines as well. But it would take a little more work (in the form of a clever u-substitution) to apply this further: we took big advantage of the fact that one of the coordinate derivatives was zero in our proof! Instead, we take this as our first opportunity to use one of the most powerful ideas in modern geometry: symmetry. We proved that isometries preserve the length of all curves, and this has an important consequence: isometries send lines to lines! Theorem 12.2 (Isometries Send Lines to Lines) Let γ:R→E 2 be a line, and ϕ:E 2→E 2 be an isometry. Then ϕ∘γ is also a line. Proof. To argue that ϕ∘γ is a line, we need to show that all of its finite segments are length-minimizing. So, pick some arbitrary interval [a,b]⊂R and look at the restriction of our curve to that segment, which goes from ϕ(γ(a))=p to ϕ(γ(b))=q. A line segment γ and its image under an isometry ϕ. Assume for the sake of contradiction that this is not length minimizing: then there is some other curve α connecting p to q which is of shorter length. A mysterious curve α which is assumed to be shorter than ϕ∘γ. Now, apply the inverse function ϕ−1 to everything: this takes the segment ϕ∘γ back to γ, and takes α to a new curve ϕ−1∘α, starting and ending at the same points as the corresponding segment of γ: Applying ϕ−1 moves α back to share endpoints with the original γ, which we know to be length-minimizing Since isometries preserve length, we know that since α was shorter than ϕ∘γ, we must now have that ϕ−1∘α is shorter than γ! But this is impossible: we assumed that γ itself was a line, so all of its segments are length-minimizing: there are no shorter curves! Thus, its impossible that α exists, so ϕ∘γ must have been the shortest segment between p and q after all. As all segments of this curve are distance minimizers, its a line! This gives us an easy prescription to track down lines: we already know γ(t)=(t,0) is a line - and if we apply any isometry at all to this, we will get another line! Corollary 12.2 (Affine Equations are Lines) Every linear equation f(t)=(a t,b t) describes a line that passes through the origin. Every affine equation of the form ℓ(t)=(a t+c b t+d) is also a Euclidean line. Here we concentrate on the main case where ⟨a,b⟩ is a unit vector. We comment below the proof on the small change needed when it is not. Proof. Then, the rotation ϕ=(a−b b a) taking ⟨1,0⟩∈T(0,0)E 2 to ⟨a,b⟩ is an isometry, so it sends lines to lines. Applying it to the x-axis γ(t)=(t,0), we see ϕ∘γ(t)=(a−b b a)(t 0)=(a t b t) Thus, t↦(a t,b t) is a line! Next, we can use the fact that for fixed c,d the translation ψ(x,y)=(x+c,y+d) is an isometry of E 2, so ψ(a t,b t)=(a t+c,b t+d) is also a line! If v=⟨a,b⟩ is not a unit vector, then we can run the argument above using the unit vector v∥v∥. This gives us that the curve below is a line β(t)=(a a 2+b 2 t,b a 2+b 2 t) Since we know that the length of curves does not depend on their parameterization, we can speed up or slow down β by pre-composing it with another function, and not change the fact that it is distance minimizing! Speeding it up by t↦a 2+b 2 t gives β(a 2+b 2 t)=(a t,b t) Thus t↦(a t,b t) is a line for any a,b∈R! We saw in Theorem 12.2 that any isometry will carry a line to another line. The same is true more generally of similarities: Exercise 12.1 (Similarities Send Lines to Lines) Let γ:R→E 2 be a line, and σ:E 2→E 2 be a similarity. Prove that σ∘γ is also a line. Hint-replicate the proof of Theorem 12.2 as closely as possible, replacing the isometry ϕ with the similarity σ, and keeping track of the scaling factors of σ versus σ−1 (Proposition 11.4). Using these tools, we can already start our process of rebuilding the Elements from below! Theorem 12.3 (Proving Euclid’s Axiom I) Given any two points p,q∈E 2, there is a line segment connecting p to q. Proof. Knowing that line segments are given by affine equations, we need just fine an affine equation γ(t) where γ(0)=p and γ(1)=q. Perhaps the simplest such is γ(t)=p+(q−p)t Theorem 12.4 (Proving Euclid’s Axiom II) Given a line segment between two points of E 2, it can be extended indefinitely in either direction. Proof. Let p,q∈E 2 and define the line segment γ:[0,1]→E 2 by γ(t)=p+(q−p)t as in the previous theorem. To extend this line segment indefinitely, we need only extend the domain from [0,1] to an arbitrary interval [a,b] containing [0,1]. The result is still an affine equation on a closed interval, and so still is a line segment by Corollary 12.2. And, as [a,b] contains [0,1] this new segment contains the original segment from p to q, so it represents an extension of the segment. 12.1.1 Uniqueness of Lines Above we proved the existence of lines, and found that all affine equations describe lines in the plane. But are these all the lines there are to be found? In fact they are - and we can confirm this with very little extra work: we had all the ideas in place already during the proof of Theorem 12.1. Proposition 12.1 Segments of the x-axis are the unique distance minimizers between their endpoints. Proof. Let γ(t)=(t,0) between t=a and t=b, and α(t)=(x(t),y(t)) be a different curve with the same endpoints. Then since α does not just trace the x axis, we must have y(t)≠0 at some point. But y(a)=0 and y(b)=0 at the endpoints, so for y to go from zero to nonzero, it must have nonzero derivative on some interval. Proving that the line segments we already know of are the unique minimizers. But, this means that y′(t)2 is strictly greater than zero on some interval, so ∥α′(t)∥>∥γ′(t)∥, and ∥α′(t)∥−∥γ′(t)∥>0 on some interval inside of [a,b]. Furthermore, since we already knew ∥α′∥≥∥γ′, this quantity is never negative. Thus ∫a b∥α′(t)∥−∥γ′(t)∥d t>0 And, re-arranging the integral, this immediately implies length(α)=∫a b∥α′∥d t>∫a b∥γ′∥d t=length(γ) Applying isometries to this, we can extend this result to any of the segments we already know: Exercise 12.2 Prove that all Euclidean lines are given by affine equations. Hint: we already know that the affine equation γ(t)=(a t+b,c t+d) defines a line. Can you show there is no curve of an equally short length, by using steps similar to the proofs Theorem 12.2 and ?cor-cor-affline-eqns-are-lines to reach a contradiction given that we just proved Proposition 12.1? 12.2 Straightest Another notion of line is “curve that doesn’t turn”. How do we make this precise? The unit tangent vector to a curve gives its direction, so we say a curve “turns” if the tangent changes direction. The derivative of the tangent vector is acceleration, a “straight curve” would have acceleration zero. Definition 12.5 (Straight) A curve γ is called straight if its tangent vector does not change. That is, if its acceleration is zero. Remark 12.1. You might worry what it means to say that tangent vectors are constant, since each one of them technically lives in a different tangent space! This difficulty will be absolutely crucial to deal with later on, when space itself is curved. But here in E 2, we can take advantage of the fact that we can make sense of the basis vectors ⟨1,0⟩ and ⟨0,1⟩ in each tangent space T p E 2: then constant just means that the components of the vectors are constant in time. This curve is not straight: which we can measure by seeing that its tangent vectors are not constant: they change in direction as we move along the curve. We’ve already done all of the hard work above, and we can now quickly confirm that this alternative definition picks out exactly the same class of curves. Theorem 12.5 (Distance Minimizers are Straight) A curve γ is distance minimizing if and only if it is straight. Proof. This is a direct computation, now that we’ve proven that every distance minimizer is given by a linear equation ℓ(t)=p+t v. Differentiating once leaves ℓ′(t)=v, and differentiating twice gives ℓ′′(t)=(0 0) Thus, ℓ is straight. To prove the other direction, we now assume we start with a straight curve γ, and we wish to prove its distance minimizing. If γ is straight, then γ′′=0 and integrating twice we see that γ(t)=(a t+c,b t+d) for some constants a,b,c,d. Thus, γ is an affine curve, and we know affine curves are distance minimizers (Corollary 12.2). So, we are done! This will turn out to be true in general: while we will have to be a little more careful when moving onwards to other geometries, curves that are straight will coincide with curves that minimize distance. 12.3 Folding Finally we come to the third possible definition of line, and show that it also picks out the same collection of curves! Definition 12.6 (Line of Symmetry) A fixed point of an isometry ϕ:E 2→E 2 is a point p with ϕ(p)=p. A curve γ is called a line of symmetry of E 2 if there exists an isometry which fixes γ(t) for all t. This captures the intuitive notion of a crease from folding paper, or reflecting across a line: this swaps the two sides of the plane but leaves What are the fixed sets of reflections? Proposition 12.2 (Reflecting in the x-Axis is an Isometry) The map ϕ(x,y)=(x,−y) is an isometry of E 2. Reflection across the x axis is an isometry of E 2. Proof. First, notice that ϕ is actually a linear map, so we can write it as a matrix: ϕ(x,y)=(1 0 0−1)(x y) Since ϕ is linear, its derivative is constant and also equal to ϕ at every point. Thus to check that it is an isometry, we only need to see that it does not change the length of any vectors. Let v=⟨v 1,v 2⟩p∈T p E 2 be a tangent vector based at some arbitrary point p. Then D ϕ p(v)=(1 0 0−1)(v 1 v 2)=(v 1−v 2) And measuring lengths with the vector norm, ∥D ϕ p(v)∥=v 1 2+(−v 2)2=v 1 2+v 2 2=∥v∥ Thus, ϕ is an isometry. This map fixes the line of points (x,0) as it only negates the y component.Thus, the x axis is a line of symmetry! Similar to before, we can use isometries to prove that every affine curve is the fixed point of some reflection. Exercise 12.3 (Reflections in Any Line) Prove that every affine curve is a line of symmetry. Hint: given an isometry that reflects in the x axis, can you build an isometry that reflects in any other line? Consider moving the line to the x axis, reflecting, and then moving back. The converse is also true: that every line of symmetry is an affine equation: so this characterization of lines exactly agrees with the two previous. To prove this, we will need a bit better understanding of the isometries of Euclidean space, and so will postpone until that chapter, 12.4 Distance So far in our development of Euclidean geometry, we have defined the length of a curve, but we have not defined any notion of distance between two points. This makes some sense, as the distance between two locations depends on how you get from one to the other, and that’s exactly what our definition captures! However, now that we know there is a unique shortest curve between any two points, there’s a natural candidate for distance: the shortest possible path. Definition 12.7 (Distance) The distance between two points p,q∈E 2 is the length of the shortest possible curve starting at p and ending at q. Because of all of our hard work above, we can turn this rather abstract definition into something concrete and practical! Theorem 12.6 (The Euclidean Distance) Let p and q be any two points in the plane. Then the Euclidean distance between them is given by dist(p,q)=∥p−q∥=(p 1−q 1)2+(p 2−q 2)2 Because lines are affine equations (and thus have constant derivative), the infinitesimal pythagorean theorem scales up to the distance function. Proof. We can write down a distance minimizing curve from p to q as an affine equation: γ(t)=p+t(q−p) This is equal to p when t=0 and q when t=1. Thus, its length is given by the integral of γ′ over [0,1]. Computing the derivative is straightforward since γ is affine: γ′(t)=q−p=(q 1−p 1,q 2−p 2), and so the length is dist(p,q)=length(γ)=∫0 1∥γ′∥d t=∫0 1(q 1−p 1)2+(q 2−p 2)2 d t=(q 1−p 1)2+(q 2−p 2)2∫0 1 d t=(q 1−p 1)2+(q 2−p 2)2 Proposition 12.3 (Distance is preserved by isometries) If p and q are any two points in the plane and ϕ is an isometry, then dist(ϕ(p),ϕ(q))=dist(p,q) Distance is invariant under isometry since isometries send lines to lines, and preserve the length of all curves. Proof. First, we start with the isometry case. Given two points p,q we can construct a line segment γ from p to q (Theorem 12.3), and as this segment is the minimizer we know its length accurately measures the distance: $dist(p,q)=length(γ). Applying ϕ we recall that isometries carry lines to lines (Theorem 12.2) to note that ϕ∘γ is a line segment between ϕ(p) and ϕ(q), and as line segments are distance minimizers, we know dist(ϕ(p),ϕ(q))=length(ϕ∘γ) Finally, we recall that isometries don’t change the length of curves (Theorem 11.2) to see length(γ)=length(ϕ∘γ) and stringing all these equalities together gives dist(p,q)=length(γ)=length(ϕ∘γ)=dist(ϕ(p),ϕ(q)) Exercise 12.4 If p,q are any two points in the plane and σ is a similarity with scaling factor k, prove dist(σ(p),σ(q))=k dist(p,q) Hint: follow closely the argument for isometries above, replacing the theorems relating isometries, line segments, and lengths with the corresponding results for similarities. Distance is scaled under similarity since similarities send lines to lines, and linearly scale the length of all curves. It will often be useful to measure distances not just to points, but to more complicated objects in the plane. Remark 12.2. We are avoiding a detail here, that anyone who has seen real analysis may be interested in. Sometimes, the minimum distance between p and a point in R doesn’t exist, but only the infimum of such distances does. However, we will never encounter such cases in this text. Definition 12.8 (Distance to a Set) Let R⊂E 2 be a region in the plane. Then the distance from a point p∈E 2 to R is defined as the shortest line segment connecting p to any point of R, and is denoted dist(p,R). The set of points at constant distance from a set (red region). On the right, a collection of points and the shortest line connecting them to a point of the set. 12.4.1 Useful Computations ' Now that we know exactly what lines are, we can convert elementary geometric problems - such as when they intersect - into algebraic problems, solvable via systems of equations. Here’s an example. Exercise 12.5 (Intersecting Lines) Calculate the point of intersection between two lines a x+b y=c and the the diagonal line x=y. When is there no intersection? With the ability to solve equations (since lines are given by affine equations, which are easy to work with!) we have developed a geometric superpower. To demonstrate this, we can use this to prove Playfair’s axiom (remember, this is equivalent to Euclid’s 5th Postulate!) Proposition 12.4 Given any line L in E 2, and any point p∈E 2 not lying on L, there exists a unique line Λ through p which does not intersect p. Exercise 12.6 Prove Proposition 1. _Hint: use isometries to help you out! First, use an isometry to move L to the x-axis. Then, use another isometry to keep L on the x axis, but to move p to some point along the y axis (and possibly, use a reflection to then insure p has been moved to a point on the positive y axis, if you like!). Then, prove that through any point on the y axis there is a unique line that does not intersect the x-axis._ In addition to algebra, founding our new geometry on calculus makes all of these tools also available to us. As a first example, we will use our knowledge of derivatives to minimize the distance between a point and a line. Minimizing distance turns out to be a pretty common thing one needs to do in applications of geometry, and while straightforward theoretically (take the derivative, set it equal to zero), its annoying in practice because of the square root in the distance formula. But there is a nice trick to get around this: Exercise 12.7 (Minimizing the Square: A Very Useful Trick!) Let f(x) be a differentiable positive function of one variable, and let s(x)=f(x)2 be its square. Show that the minima of s(x) and f(x) occur at the same points, by following the steps below: First, assume x=a is the location of a minimum of f. What does the first and second derivative test tell you about the values f′(a) and f′′(a)? Use this, together with the fact that f(a)>0 to show that x=a is also the location of a minimum of s (using the second derivative test). Conversely, assume x=a is the location of a minimum of s(x). Now, you know information about the derivatives s′(a) and s′′(a). Use this to conclude information about f′(a) and f′′(a) to show that a is a minimum for f as well. This tells us anytime we want to minimize a positive function, we could always choose to find where its square is minimized instead, if that turns out to be easier. The main question is below, where without loss of generality we have taken the point to be the origin (as we could always slide it there via an isometry). Exercise 12.8 (Closest Point on Line) Let L be the line traced by the affine curve γ(t)=(a t+c b t+d), and O be the origin, as usual. Calculate dist(O,L) Hint: use calculus to find the closest point on L to O. Can you minimize the squared distance from γ(t) to (0,0)? 11 Isometries 13 Shapes' class='pagination-link' href='/plane/shapes/'>13 Shapes
12377	https://www.gauthmath.com/solution/5-x-2-12x-36-x-6-2-TRUE-FALSE-1712657325474821	Sign in Homework Assignment Solver Calculator Calculator Resources Resources Blog Blog App App Gauth Unlimited answers Gauth AI Pro Start Free Trial Homework Helper Study Resources Math Equation Questions Question x^2-12x+36=(x-6)^2 TRUE FALSE Expert Verified Solution 93%(705 rated) Answer TRUE Explanation To verify the equation, we can expand $$(x-6)^{2}$$(x−6)2 using the formula $$(x-a)^{2}=x^{2}-2ax+a^{2}$$(x−a)2=x2−2ax+a2 . Expand $$(x-6)^{2}$$(x−6)2 : $$(x-6)^{2} = x^{2} - 2(x)(6) + 6^{2} = x^{2} - 12x + 36$$(x−6)2=x2−2(x)(6)+62=x2−12x+36 . Compare the expanded form with the original equation: $$x^{2} - 12x + 36 = x^{2} - 12x + 36$$x2−12x+36=x2−12x+36 . Since both forms are identical, the original equation $$x^{2}-12x+36=(x-6)^{2}$$x2−12x+36=(x−6)2 is true. Helpful Not Helpful Explain Simplify this solution Related The polynomial awedge 2-ab-3a+3b 1 point can be factored by grouping. false TRUE xwedge 2-12x+36=x-6wedge 2 1 point x2-12x+36=x-62 FALSE TRUE 3xwedge 2+5x+2=3x+2x+1 1 point FALSE TRUE 100% (4 rated) x2-12x a 36; x2-12x+36=x-62 b 144; x2-12x-144=x-122 C 144; x2-12x+144=x-122 d 36; x2-12x-36=x-62 100% (8 rated) Complete the square for the binomial. Then factor the resulting perfect square trinomial. x2-12x a 36; x2-12x-36=x-62 b 144; x2-12x+144=x-122 C 36; x2-12x+36=x-62 d 144 . x2-12x-144=x-122 100% (7 rated) Complete the square for the binomial. Then factor the resulting perfect square trinomial. x2-12x B 36; x2-12x+36=x-62 b 36; x2-12x-36=x-62 C 144; x2-12x+144=x-122 d 144; x2-12x-144=x-122 93% (14 rated) DUTINVENIORS 1. Work each exercise. 2. Find the code letter for the correct answer. 3. Write the code letter in each blank having that exercise number. Exercises Factor. 1. x2-12x+36=x-62 V 13. 12x2-12x+3 100% (4 rated) Tell whether each statement is True or False. _1. The GCF of the integers 12 and 24 is 6 _2. The trinomial 9x2+155x+25 is a perfect square trinomial. _3. The polynomial a2-ab-3a+3b can be factored by grouping. _4 x2-12x+36=x-62 _5. The product of the sum and difference of the same two terms is equal to a difference of two squares. _6. The polynomial 25y+1 is a prime polynomial. _7. The greatest common factor of the terms of the polynomial 8x3y . 12x2y3 is 4x2y. _ _8. The polynomial 3x2+54 cannot be factored. _10. The polynomial a2+25 is a difference of two squares. 100% (5 rated) a i How many even numbers are there from 100 to 1000 inclusive? ii I write down m consecutive odd numbers, the smallest of which is л. Find in terms of m and n, the value of the largest odd number I wrote down. b y is a positive whole number such that y/11 lies between 81/13 and 98/15 . Find all possible values of y. 80% (5 rated) A company is using linear programming to decide how many units of each of its two products to make each week. Weekly production will be x units of Product X and y units of Product Y. At least 50 units of X must be produced each week, and at least twice as many units of Y as of X must be produced each week. Each urt of X requires 30 minutes of labour, and each unit of Y requires two hours of labour. There are 5,000 hours of labour available each week. Which of the following is the correct set of constraints? Submit your answer to view the feedback. 0.5x+2y ≤ 5,000 0.5x+2y ≤ 5,000 x ≥ 50 x ≥ 50 y ≤ 2x y ≥ 100 x+4y ≤ 5,000 0.5x+2y ≤ 5,000 x ≥ 50 x ≥ 50 y ≥ 2x y ≥ 2x 100% (4 rated) Simplify 6+7i/1-4i giving your answer in the form a+bi. 75% (8 rated) x-2x+3-x+1x-4=0 10. 3x+ 2/x =4 11. 1/x + x-2/2 =3 12. 4x+5=2x2 13. 2x/5 - 3/x =1 14. 2/x + x-1/4 = 3x/2 15. 3x+1/2x - 1/3x = 5/6 100% (6 rated)
12378	https://physics.stackexchange.com/questions/647739/null-point-of-an-electric-field	electrostatics - Null point of an electric field - Physics Stack Exchange Join Physics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Null point of an electric field Ask Question Asked 4 years, 3 months ago Modified4 years, 3 months ago Viewed 1k times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. For opposite charges, why is the null point (E net=0) of an electric field closer to the smaller charge? electrostatics electric-fields vector-fields Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Improve this question Follow Follow this question to receive notifications edited Jun 25, 2021 at 14:08 Vincent Thacker 15.5k 16 16 gold badges 44 44 silver badges 63 63 bronze badges asked Jun 25, 2021 at 11:47 KaranKaran 69 9 9 bronze badges Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. It's the place where there is no resultant force. For charges of equal signs: Let's imagine moving on a line from the larger to the smaller charge. The larger charge provides a bigger force, but it falls of as an inverse square law as we go further away from it. The force from the smaller charge gets larger as we come closer to it. Half way between, the force from the larger charge is still larger, but there will be a point past half way, where the force from each charge balances. At that point the electric field is zero. For charges of opposite signs: The resultant force is zero at a point nearer the smaller charge (in magnitude), but at a point not between the charges. That way the distance to the small charge can be smaller than the distance to the big charge and from arguments similar to 1) the forces can balance. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Jun 25, 2021 at 12:32 answered Jun 25, 2021 at 12:20 John HunterJohn Hunter 13.9k 2 2 gold badges 26 26 silver badges 58 58 bronze badges Add a comment\| Your Answer Thanks for contributing an answer to Physics Stack Exchange! Please be sure to answer the question. Provide details and share your research! 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12379	https://en.wikipedia.org/wiki/Okadaic_acid	Okadaic acid - Wikipedia Jump to content [x] Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search [x] Appearance Donate Create account Log in [x] Personal tools Donate Create account Log in Pages for logged out editors learn more Contributions Talk Contents move to sidebar hide (Top) 1 History 2 SynthesisToggle Synthesis subsection 2.1 Derivatives 2.2 Biosynthesis 2.3 Laboratory syntheses 3 BiologyToggle Biology subsection 3.1 Mechanism of action 3.2 Toxicology 3.3 Medical uses 4 See also 5 References 6 External links [x] Toggle the table of contents Okadaic acid [x] 11 languages تۆرکجه Deutsch Español فارسی Français Italiano 日本語 Српски / srpski Srpskohrvatski / српскохрватски Suomi 中文 Edit links Article Talk [x] English Read Edit View history [x] Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikidata item Appearance move to sidebar hide From Wikipedia, the free encyclopedia Okadaic acid Names Preferred IUPAC name (2 R)-2-Hydroxy-2-{[(2 S,5 R,6 R,8 S)-5-hydroxy-8-{(2 R,3 E)-4-[(2 R,4′a R,6′S,8′R,8′a S)-8′-hydroxy-6′-{(1 S,3 S)-1-hydroxy-3-[(2 S,3 R,6 S)-3-methyl-1,7-dioxaspiro[5.5]undecan-2-yl]butyl}-7′-methylidenehexahydro-3′H-spiro[oxolane-2,2′-pyrano[3,2-b]pyran]-5-yl]but-3-en-2-yl}-10-methyl-1,7-dioxaspiro[5.5]undec-10-en-2-yl]methyl}propanoic acid Other names 9,10-Deepithio-9,10-didehydroacanthifolicin Identifiers CAS Number 78111-17-8Y 3D model (JSmol) Interactive image ChEBI CHEBI:CHEBI:7733N ChEMBL ChEMBL280487N ChemSpider 393845Y DrugBank DB02169 ECHA InfoCard100.116.145 EC Number 616-589-8 IUPHAR/BPS 5349 KEGG C01945N MeSHAcid Okadaic Acid PubChemCID 446512 UNII 1W21G5Q4N2Y CompTox Dashboard(EPA) DTXSID60880002 InChI InChI=1S/C44H68O13/c1-25-21-34(55-44(23-25)35(46)12-11-31(54-44)24-41(6,50)40(48)49)26(2)9-10-30-14-18-43(53-30)19-15-33-39(57-43)36(47)29(5)38(52-33)32(45)22-28(4)37-27(3)13-17-42(56-37)16-7-8-20-51-42/h9-10,23,26-28,30-39,45-47,50H,5,7-8,11-22,24H2,1-4,6H3,(H,48,49)/b10-9+/t26-,27-,28+,30+,31+,32+,33-,34+,35-,36-,37+,38+,39-,41-,42+,43-,44-/m1/s1Y Key:QNDVLZJODHBUFM-WFXQOWMNSA-NY InChI=1/C44H68O13/c1-25-21-34(55-44(23-25)35(46)12-11-31(54-44)24-41(6,50)40(48)49)26(2)9-10-30-14-18-43(53-30)19-15-33-39(57-43)36(47)29(5)38(52-33)32(45)22-28(4)37-27(3)13-17-42(56-37)16-7-8-20-51-42/h9-10,23,26-28,30-39,45-47,50H,5,7-8,11-22,24H2,1-4,6H3,(H,48,49)/b10-9+/t26-,27-,28+,30+,31+,32+,33-,34+,35-,36-,37+,38+,39-,41-,42+,43-,44-/m1/s1 Key:QNDVLZJODHBUFM-WFXQOWMNBB InChI=1S/C44H68O13/c1-25-21-34(55-44(23-25)35(46)12-11-31(54-44)24-41(6,50)40(48)49)26(2)9-10-30-14-18-43(53-30)19-15-33-39(57-43)36(47)29(5)38(52-33)32(45)22-28(4)37-27(3)13-17-42(56-37)16-7-8-20-51-42/h9-10,23,26-28,30-39,45-47,50H,5,7-8,11-22,24H2,1-4,6H3,(H,48,49)/b10-9+/t26-,27-,28+,30+,31+,32+,33-,34+,35-,36-,37+,38+,39-,41-,42+,43-,44-/m1/s1 Key:QNDVLZJODHBUFM-WFXQOWMNSA-N SMILES O=C(O)C@@(C)C[C@H]7O[C@]/1(OC@@HC@@HC)C@HCC7 Properties Chemical formulaC 44 H 68 O 13 Molar mass805.015 g·mol−1 Melting point164-166 °C Hazards GHS labelling: Pictograms Signal wordDanger Hazard statementsH301, H311, H315, H331 Precautionary statementsP261, P262, P264, P270, P271, P280, P301+P316, P302+P352, P304+P340, P316, P321, P330, P332+P317, P361+P364, P362+P364, P403+P233, P405, P501 Except where otherwise noted, data are given for materials in their standard state (at 25°C [77°F], 100 kPa). Nverify(what isYN?) Infobox references Chemical compound Okadaic acid, C 44 H 68 O 13, is a toxin produced by several species of dinoflagellates. It is known to accumulate in both marine sponges and shellfish. One of the primary causes of diarrhetic shellfish poisoning, okadaic acid is a potent inhibitor of specific protein phosphatases, and has a variety of negative effects on cells. A polyketide, polyether derivative of a C 38fatty acid, okadaic acid and other members of its family have illuminated many biological processes both with respect to dinoflagellate polyketide synthesis as well as the role of protein phosphatases in cell growth. History [edit] As early as 1961, reports of gastrointestinal disorders following the consumption of cooked mussels appeared in both the Netherlands and Los Lagos. Attempts were made to determine the source of the symptoms; however, they failed to elucidate the true culprit, instead implicating a species of microplanctonicdinoflagellates. In the summers of the late 1970s, a series of food poisoning outbreaks in Japan led to the discovery of a new type of shellfish poisoning. Named for the most prominent symptoms, the new Diarrhetic Shellfish Poisoning (DSP) only affected the northern portion of Honshu during 1976; however, by 1977 large cities such as Tokyo and Yokohama were affected. Research into the shellfish consumed in the affected regions showed that a fat-soluble toxin was responsible for the 164 documented cases, and this toxin was traced to mussels and scallops harvested in the Miyagi prefecture. In northeastern Japan, a legend had existed that during the season of paulownia flowers, shellfish can be poisonous. Studies following this outbreak showed that toxicity of these mussels and scallops appeared and increased during the months of June and July, and all but disappeared between August and October. Elsewhere in Japan, in 1975 Fujisawa pharmaceutical company observed that the extract of a black sponge, Halichondria okadai, was a potent cytotoxin; this was dubbed Halichondrine-A. In 1981, the structure of one such toxin, okadaic acid, was determined after it was extracted from both the black sponge in Japan, Halichondria okadai, for which it was named, and a sponge in the Florida Keys, Halichondria melanodocia. Okadaic acid sparked research both for its cytotoxicicity, and for being the first reported marine ionophore. Several years later, one of the toxins responsible for DSP, dinophysistoxin-1 (DTX-1) (named for one of the organisms implicated in its production, Dinophysis fortii) was compared to and shown to be very chemically similar to okadaic acid; okadaic acid itself was implicated in DSP around the same time. Since its initial discovery, reports of DSP have spread throughout the world, and are especially concentrated in Japan, South America and Europe. Synthesis [edit] Derivatives [edit] Okadaic acid (OA) and its derivatives, the dinophysistoxins (DTX), are members of a group of molecules called polyketides. The complex structure of these molecules include multiple spiroketals, along with fused ether rings. Structures of Okadaic Acid and the Dinophysistoxins Biosynthesis [edit] Being polyketides, the okadaic acid family of molecules are synthesized by dinoflagellates via polyketide synthase (PKS). However, unlike the majority of polyketides, the dinoflagellate group of polyketides undergo a variety of unusual modifications. Okadaic acid and its derivatives are some of the most well studied of these polyketides, and research on these molecules via isotopic labeling has helped to elucidate some of those modifications. Okadaic acid is formed from a starter unit of glycolate, found at carbons 37 and 38, and all subsequent carbons in the chain are derived from acetate. Because polyketide synthesis is similar to fatty acid synthesis, during chain extension the molecule may undergo reduction of the ketone, dehydration, and reduction of the olefin. Failure to perform one of more of these three steps, combined with several unusual reactions is what allows for the formation of the functionality of okadaic acid. Carbon deletion and addition at the alpha and beta position comprise the other transformations present in the okadaic acid biosynthesis. Carbon deletion occurs by way of a Favorskii rearrangement and subsequent decarboxylation. Attack of a ketone in the growing chain by enzyme-bound acetates, and subsequent decarboxylation/dehydration results in an olefin replacing the ketone, in both alpha and beta alkylation. After this the olefin can isomerize to more thermodynamically stable positions, or can be activated for cyclizations, in order to produce the natural product. Laboratory syntheses [edit] Isobe's Synthesis of Okadaic Acid. To date, several studies have been performed toward the synthesis of okadaic acid and its derivatives. Three total syntheses of okadaic acid have been achieved, along with many more formal syntheses and several total syntheses of the other dinophysistoxins. The first total synthesis of okadaic acid was completed in 1986 by Isobe et al., just five years after the molecule's structure was elucidated. The next two were completed in 1997 and 1998 by the Forsyth and Ley groups respectively. In Isobe's synthesis, the molecule was broken into three pieces, along the C14–C15 bonds, and the C27–C28 bonds. This formed fragments A, B, and C, which were all synthesized separately, after which the B and C fragments were combined, and then combined with the A fragment. This synthesis contained 106 steps, with a longest linear sequence of 54 steps. The precursors to all three fragments were all glucose derivatives obtained from the chiral pool. Spiroketals were obtained from precursor ketone diols, and were therefore formed thermally in acid. Forsyth's Synthesis of Okadaic Acid. Similar to Isobe's synthesis, the Forsyth synthesis sought to reduce the number of steps, and to increase potential for designing analogues late in the synthesis. To do this, Forsyth et al. designed the synthesis to allow for structural changes and installation of important functional groups before large pieces were joined. Their resulting synthesis was 3% yielding, with 26 steps in the longest linear sequence. As above, spiroketalization was performed thermodynamically with introduction of acid. Ley's Synthesis of Okadaic Acid. Ley's synthesis of okadaic acid is most unlike its predecessors, although it still contains similar motifs. Like the others, this synthesis divided okadaic acid into three components along the acyclic segments. However, designed to display new techniques developed in their group, Ley's synthesis included forming the spiroketals using (diphenylphosphineoxide)-tetrahydrofuran and (phenylsulfonyl)-tetrahydropyrans, allowing for more mild conditions. Similar to those above, a portion of the stereochemistry in the molecule was set by starting materials obtained from the chiral pool, in this case mannose. Biology [edit] Mechanism of action [edit] Okadaic acid (OA) and its relatives are known to strongly inhibit protein phosphatases, specifically serine/threonine phosphatases. Furthermore, of the four such phosphatases, okadaic acid and its relatives specifically target protein phosphatase 1 (PP1) and protein phosphatase 2A (PP2A), at the exclusion of the other two, with dissociation constants for the two proteins of 150 nM and 30 pM respectively. Because of this, this class of molecules has been used to study the action of these phosphatases in cells. Once OA binds to the phosphatase protein(s), it results in hyperphosphorylation of specific proteins within the afflicted cell, which in turn reduces control over sodium secretion and solute permeability of the cell. Affinity between okadaic acid and its derivatives and PP2A has been tested, and it was shown that the only derivative with a lower dissociation constant, and therefore higher affinity, was DTX1, which has been shown to be 1.6 times stronger. Furthermore, for the purpose of determining the toxicity of mixtures of different okadaic acid derivatives, inhibitory equivalency factors for the relatives of okadaic acid have been studied. In wild type PP2A, the inhibitory equivalency relative to okadaic acid were 0.9 for DTX-1 and 0.6 for DTX-2. Toxicology [edit] The main route of exposure to DSP from okadaic acid and its relatives is through the consumption of shellfish. It was initially shown that the toxic agents responsible for DSP tend to be most concentrated in the hepatopancreas, followed by the gills for certain shellfish. The symptoms for diarrhetic shellfish poisoning include intense diarrhea and severe abdominal pains, and rarely nausea and vomiting, and they tend to occur anytime between 30 minutes and at most 12 hours after consuming toxic shellfish. It has been estimated that it takes roughly 40 μg of okadaic acid to trigger diarrhea in adult humans. Medical uses [edit] Because of its inhibitory effects in phosphatases, okadaic acid has shown promise in the world of medicine for numerous potential uses. During its initial discovery, okadaic acid, specifically the crude source extract, showed potent inhibition of cancer cells, and so initial interest in the family of molecules tended to center around that feature. However, it was shown that the more cytotoxic component of H. okadai was actually a separate family of compounds, the Halichondrines, and as such research into the cytotoxicity of okadaic acid decreased. However, the unique function of okadaic acid upon cells maintained biological interest in the molecule. Okadaic acid has been shown to have neurotoxic, immunotoxic, and embryotoxic effects. Furthermore, in two-stage carcinogenesis of mouse skin, the molecule and its relatives have been shown to have tumor promoting effects. Because of this, the effects of okadaic acid on Alzheimer's, AIDS, diabetes, and other human diseases have been studied. See also [edit] Canadian Reference Materials Brevetoxin Ciguatoxin Domoic acid Saxitoxin Tetrodotoxin Rubratoxin References [edit] ^ abcdReguera, B.; Riobo, P.; Rodriguez, F.; Diaz, P. A.; Pizarro, G.; Paz, B.; Franco, J. M.; Blanco, J. (January 2014). "Dinophysis Toxins: Causative Organisms, Distribution and Fate in Shellfish". Mar. Drugs. 12 (1): 394–461. doi:10.3390/md12010394. PMC3917280. PMID24447996. ^Holmes, C. F. B.; Luu, H. A.; Carrier, F.; Schmitz, F. J. (1990). "Inhibition of protein phosphatases-1 and -2A with acanthifolicin: Comparison with diarrhetic shellfish toxins and identification of a region on okadaic acid important for phosphatase inhibition". FEBS Lett. 270 (1–2): 216–218. Bibcode:1990FEBSL.270..216H. doi:10.1016/0014-5793(90)81271-O. PMID2171991. ^ abcdefValdiglesias, V.; Prego-Faraldo, M. V.; Pasaro, E.; Mendez, J; Laffon, B (2013). "Okadaic Acid: More than a Diarrheic Toxin". Mar. Drugs. 11 (11): 4328–4349. doi:10.3390/md11114328. PMC3853731. PMID24184795. ^ abGarcia, A.; Cayla, X.; Guergnon, J.; Dessauge, F.; Hospital, V.; Rebollo, M. P.; Fleischer, A.; Rebollo, A. (2003). "Serine/threonine protein phosphatases PP1 and PP2A are key players in apoptosis". Biochimie. 85 (8): 721–726. doi:10.1016/j.biochi.2003.09.004. PMID14585537. ^ abcdefghDounay, A. B.; Forsyth, C. J. (2002). "Okadaic acid: The archetypal serine/threonine protein phosphatase inhibitor". Curr. Med. Chem. 9 (22): 1939–1980. doi:10.2174/0929867023368791. PMID12369865. ^ abcdVan Wagoner, R. M.; Satake, M.; Wright, J. L. C. (2014). "Polyketide biosynthesis in dinoftagellates: what makes it different?". Nat. Prod. Rep. 31 (9): 1101–1137. doi:10.1039/c4np00016a. PMID24930430. ^ abcdYasumoto, T.; Oshima, Y.; Yamaguchi, M. (1978). "Occurrence of a new type of shellfish poisoning in the Tohoku district". Bull. Jpn. Soc. Sci. Fish. 44 (11): 1249–1255. doi:10.2331/suisan.44.1249. ^ abcYasumoto, T.; Murata, M.; Oshima, Y.; Sano, M.; Matsumoto, G. K.; Clardy, J. (1985). "Diarrhetic shellfish toxins". Tetrahedron. 41 (6): 1019–1025. doi:10.1016/s0040-4020(01)96469-5. ^ abTachibana, K.; Scheuer, P. J.; Tsukitani, Y.; Kikuchi, H.; Vanengen, D.; Clardy, J.; Gopichand, Y.; Schmitz, F. J. (1981). "Okadaic acid, a cytotoxic polyether from two marine sponges of the genus Halichondria". J. Am. Chem. Soc. 103 (9): 2469–2471. Bibcode:1981JAChS.103.2469T. doi:10.1021/ja00399a082. ^Shibata, S.; Ishida, Y.; Kitano, H.; Ohizumi, Y.; Habon, J.; Tsukitani, Y.; Kikuchi, H. (1982). "Contractile effects of okadaic acid, a novel ionophore-like substance from black sponge, on isolated smooth muscles under the condition of Ca deficiency". J. Pharmacol. Exp. Ther. 223 (1): 135–143. doi:10.1016/S0022-3565(25)33303-3. PMID6214623. ^ abScoging, A.; Bahl, M. (1998). "Diarrhetic shellfish poisoning in the UK". The Lancet. 352 (9122): 117. doi:10.1016/S0140-6736(98)85023-X. PMID9672285. S2CID35211241. ^Weissman, K. J. (2009). "Chapter 1 Introduction to Polyketide Biosynthesis". Complex Enzymes in Microbial Natural Product Biosynthesis, Part B: Polyketides, Aminocoumarins and Carbohydrates. Methods in Enzymology. Vol.459. pp.3–16. doi:10.1016/S0076-6879(09)04601-1. ISBN9780123745910. PMID19362633. ^ abIsobe, M.; Ichikawa, Y.; Goto, T. (1986). "Synthetic studies toward marine toxic polyethers the total synthesis of okadaic acid". Tetrahedron Lett. 27 (8): 963–966. doi:10.1016/S0040-4039(00)84149-0. ^ abForsyth, C. J.; Sabes, S. F.; Urbanek, R. A. (1997). "An Efficient Total Synthesis of Okadaic Acid". J. Am. Chem. Soc. 119 (35): 8381–8382. Bibcode:1997JAChS.119.8381F. doi:10.1021/ja9715206. ^ abLey, S. V.; Humphries, A. C.; Eick, H.; Downham, R.; Ross, A. R.; Boyce, R. J.; Pavey, J. B. J.; Pietruszka, J. (1998). "Total synthesis of the protein phosphatase inhibitor okadaic acid". J. Chem. Soc., Perkin Trans. 1 (23): 3907–3912. doi:10.1039/A807957I. ^ abcdTakai, A.; Murata, M.; Torigoe, K.; Isobe, M.; Mieskes, G.; Yasumoto, T. (1992). "Inhibitory effect of okadaic acid derivatives on protein phosphatases. A study on structure-affinity relationship". Biochem. J. 284 (2): 539–544. doi:10.1042/bj2840539. PMC1132671. PMID1318034. ^Dawson, J. F.; Holmes, C. F. (1999). "Molecular mechanisms underlying inhibition of protein phosphatases by marine toxins". Front. Biosci. 4 (1–3): D646-58. doi:10.2741/Dawson. PMID10502549. ^ abGaribo, D.; de la Iglesia, P.; Diogène, J; Campàs, M. (2013). "Inhibition Equivalency Factors for Dinophysistoxin-1 and Dinophysistoxin-2 in Protein Phosphatase Assays: Applicability to the Analysis of Shellfish Samples and Comparison with LC-MS/MS". J. Agric. Food Chem. 61 (10): 2572–2579. Bibcode:2013JAFC...61.2572G. doi:10.1021/jf305334n. PMID23406170. ^Suzuki, T.; Igarashi, T.; Ichimi, K.; Watai, M.; Suzuki, M.; Ogiso, E.; Yasumoto, T. (2005). "Kinetics of diarrhetic shellfish poisoning toxins, okadaic acid, dinophysistoxin-1, pectenotoxin-6 and yessotoxin in scallops Patinopecten yessoensis". Fisheries Science. 71 (4): 948–955. Bibcode:2005FisSc..71..948S. doi:10.1111/j.1444-2906.2005.01049.x. S2CID2185421. ^ abKamat, P. K.; Rai, S.; Swarnkar, S.; Shukla, R.; Nath, C. (2014). "Molecular and Cellular Mechanism of Okadaic Acid (OKA)-Induced Neurotoxicity: A Novel Tool for Alzheimer's Disease Therapeutic Application". Mol. Neurobiol. 50 (3): 852–865. doi:10.1007/s12035-014-8699-4. PMID24710687. S2CID19032730. External links [edit] Forsyth, C. J.; Sabes, S. F.; Urbanek, R. A. (1997). "An Efficient Total Synthesis of Okadaic Acid". Journal of the American Chemical Society. 119 (35): 8381–8382. Bibcode:1997JAChS.119.8381F. doi:10.1021/ja9715206. \| v t e Xenobiotic-sensing receptormodulators \| \| CARTooltip Constitutive androstane receptor \| Agonists:6,7-Dimethylesculetin Amiodarone Artemisinin Benfuracarb Carbamazepine Carvedilol Chlorpromazine Chrysin CITCO Clotrimazole Cyclophosphamide Cypermethrin DHEA (prasterone) Efavirenz Ellagic acid Griseofulvin Methoxychlor Mifepristone Nefazodone Nevirapine Nicardipine Octicizer Permethrin Phenobarbital Phenytoin Pregnanedione (5β-dihydroprogesterone) Reserpine TCPOBOP Telmisartan Tolnaftate Troglitazone Valproic acid Antagonists:3,17β-Estradiol 3α-Androstanol 3α-Androstenol 3β-Androstanol 17-Androstanol AITC Ethinylestradiol Meclizine Nigramide J Okadaic acid PK-11195 S-07662 T-0901317 \| \| PXRTooltip Pregnane X receptor \| Agonists:17α-Hydroxypregnenolone 17α-Hydroxyprogesterone Δ 4-Androstenedione Δ 5-Androstenediol Δ 5-Androstenedione AA-861 Allopregnanediol Allopregnanedione (5α-dihydroprogesterone) Allopregnanolone (brexanolone) Alpha-Lipoic acid Ambrisentan AMI-193 Amlodipine besylate Antimycotics Artemisinin Aurothioglucose Bile acids Bithionol Bosentan Bumecaine Cafestol Cephaloridine Cephradine Chlorpromazine Ciglitazone Clindamycin Clofenvinfos Chloroxine Clotrimazole Colforsin Corticosterone Cyclophosphamide Cyproterone acetate Demecolcine Dexamethasone DHEA (prasterone) DHEA-S (prasterone sulfate) Dibunate sodium Diclazuril Dicloxacillin Dimercaprol Dinaline Docetaxel Docusate calcium Dodecylbenzenesulfonic acid Dronabinol Droxidopa Eburnamonine Ecopipam Enzacamene Epothilone B Erythromycin Famprofazone Febantel Felodipine Fenbendazole Fentanyl Flucloxacillin Fluorometholone Griseofulvin Guggulsterone Haloprogin Hetacillin potassium Hyperforin Hypericum perforatum (St John's wort) Indinavir sulfate Lasalocid sodium Levothyroxine Linolenic acid: α-Linolenic acid and γ-Linolenic acid LOE-908 Loratadine Lovastatin Meclizine Metacycline Methylprednisolone Metyrapone Mevastatin Mifepristone Nafcillin Nicardipine Nicotine Nifedipine Nilvadipine Nisoldipine Norelgestromin Omeprazole Orlistat Oxatomide Paclitaxel Phenobarbital Piperine Plicamycin Prednisolone Pregnanediol Pregnanedione (5β-dihydroprogesterone) Pregnanolone Pregnenolone Pregnenolone 16α-carbonitrile Proadifen Progesterone Quingestrone Reserpine Reverse triiodothyronine Rifampicin Rifaximin Rimexolone Riodipine Ritonavir Simvastatin Sirolimus Spironolactone Spiroxatrine SR-12813 Suberoylanilide Sulfisoxazole Suramin Tacrolimus Tenylidone Terconazole Testosterone isocaproate Tetracycline Thiamylal sodium Thiothixene Thonzonium bromide Tianeptine Troglitazone Troleandomycin Tropanyl 3,5-dimethulbenzoate Zafirlukast Zeranol Antagonists:Ketoconazole Sesamin \| \| See also Receptor/signaling modulators \| Retrieved from " Categories: Carboxylic acids Laxatives Phycotoxins Polyketides Polyether toxins Spiro compounds Oxygen heterocycles Phosphatase inhibitors Hidden categories: Chemical articles with multiple compound IDs Multiple chemicals in an infobox that need indexing Articles with changed EBI identifier ECHA InfoCard ID from Wikidata Articles with changed KEGG identifier Chembox having GHS data Articles containing unverified chemical infoboxes Chembox image size set Articles with short description Short description matches Wikidata Use dmy dates from April 2020 This page was last edited on 25 August 2025, at 06:05(UTC). 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12380	https://www.krayonnz.com/user/doubts/detail/649c7677097bbb00577882f3/prove-the-sum-to-product-identities-sin-x-sin-y-2sin-x-y-2-cos-x-y-2-and-cos-x-cos-y-2cos-x-y-2-cos-x-y-2	Prove the sum-to-product identities: sin(x) + sin(y) = 2sin((x+y)/2)cos((x-y)/2) and cos(x) + cos(y) = 2cos((x+y)/2)cos((x-y)/2). Klubs Clips Competitions Quizzes Notes Doubts Login/Signup Question Details Anupama+Follow 342 Followers Asked on 28-06-2023 more_vert Q: Prove the sum-to-product identities: sin(x) + sin(y) = 2sin((x+y)/2)cos((x-y)/2) and cos(x) + cos(y) = 2cos((x+y)/2)cos((x-y)/2). Please answer in brief. Mathematics Higher trigonometry 0AnswerShareSave 130 Suggested Notes 7 109 By : Tara Rani Theory X and Y 4 135 By : Krish Sarkar Regression Equation Of X On Y \| Statistics ( Dibrugarh University) 4 188 By : Tashmeen Kaur Theory Of X & Theory Of Y \| Management Studies (Lucknow University) 2 140 By : Devashish Mandal If X Tan 45° Sin 30° = Cos 30° Tan 30°, Then X Is Equal To \| Mathematics (Guru Nanak Dev University) 1 108 By : Ankita Trigonometric Identities - Cos X \| Mathematics (CCSU) Ask Question Questions My Questions My Answers Related Questions Find dy/dx at x = 1, y = π/4 if sin^2 y + cos xy = K? Solve the following differential equation. x dy – y dx = √(x^2+y^2) dx, given that y = 0 when x = 1? Prove that. (Cos x + cos y)²+(sin x + sin y)² = 4cos² (x-y/2) Find the derivative of z(x) = cos(x^2) - sin(x^3). If tan A = 3/4, prove that Sin A Cos A = 12/25. Prove the double angle identities for sine and cosine: sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x). Prove the half-angle identities for sine and cosine: sin(x/2) = ±√[(1 - cos(x))/2] and cos(x/2) = ±√[(1 + cos(x))/2], depending on the quadrant. Find the Intervals in which the function f(x) = sin x + cos x, 0 ≤ x ≤ 2π is strictly increasing and strictly decreasing? Prove : (sin 3x + sin x) sin x + (cos 3x - cos x) cos x = 0. Quiz you might be interested View All Beginner Brainiac Highlights: September Edition (हाइलाइट्स – सितंबर एडिशन) ₹2500 Prizepool 29 Sep, 2025 02:30 PM Join, Get free 20 Koins, Register Beginner Brainiac KO: Guess the Brand (गेस द ब्रांड) ₹800 Prizepool 30 Sep, 2025 02:30 PM Join, Get free 20 Koins, Register My Questions My Answers Ask Question Questions You are not Logged in clear Log in to experience the fastest growing social learning network. Sign in Or Signup Made in India with ♥️ ️ Test Mode Save Item 7 pages 4 pages 2 pages 1 page
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12382	https://www.khanacademy.org/science/biology/membranes-and-transport/active-transport/a/active-transport	Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer. More information Manage Consent Preferences Strictly Necessary Cookies Always Active Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account. For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session. Functional Cookies These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences. Targeting Cookies These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission. We do not use cookies to serve third party ads on our Service. Performance Cookies These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates. For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users. We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails.
12383	https://math.stackexchange.com/questions/637042/calculate-maximum-velocity-given-accel-decel-initial-v-final-position	physics - Calculate maximum velocity given accel, decel, initial v, final position - 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 Calculate maximum velocity given accel, decel, initial v, final position Ask Question Asked 11 years, 8 months ago Modified8 years, 5 months ago Viewed 54k times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. I'm trying to calculate the maximum velocity of an object given: acceleration deceleration starting position (0) ending position initial velocity I've been trying to use formulas from: but I still can't seem to figure it out. The closest I've gotten is: t=V m a x−V 0 a+0−V m a x d t=V m a x−V 0 a+0−V m a x d But even trying to solve that for Vmax leaves an unknown 't t' in the equation, and doesn't take into account the ending position either. A sort of real world example for this, if it helps, is of a car. The car starts at p=0, with a rolling start (in either direction). At the finish line (ending position) is a brick wall. The car must stop with the front bumper touching the brick wall. It must accelerate at a for as long as possible, then decelerate at d, ending at that brick wall, with v=0. (I'm writing an Arduino library for controlling servos, with speed limits, acceleration and deceleration. I need to know whether the max velocity it can reach given the acceleration and deceleration will ever exceed the max velocity the library user specifies, so I can calculate whether there needs to be any time spent "coasting" between acceleration and deceleration.) physics Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited May 23, 2017 at 12:39 CommunityBot 1 asked Jan 13, 2014 at 15:40 Derek LewisDerek Lewis 123 1 1 gold badge 1 1 silver badge 4 4 bronze badges 1 a and d are both constant, and there's a given "ending position" the object must reach, and be at rest, so there must be a maximum velocity, unless I'm missing something.Derek Lewis –Derek Lewis 2014-01-13 16:43:17 +00:00 Commented Jan 13, 2014 at 16:43 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 4 Save this answer. Show activity on this post. We assume the object starts at time t=0 t=0 at position x=0 x=0 and ends at position ℓ ℓ. It accelerates constantly with acceleration a a and then suddenly at time t=t 1 t=t 1 it decelerates constantly with deceleration d d (positive) until stopping at the point ℓ ℓ. Let us call t 2 t 2 the deceleration time. Then v max=v 0+a t 1 v max=v 0+a t 1. To find t 1 t 1, note that v 0+a t 1−d t 2=0;v 0+a t 1−d t 2=0; this is the condition that the speed at x=ℓ x=ℓ is zero. And the distance has to be ℓ ℓ, so v 0 t 1+1 2 a t 2 1+(v 0 t 2+a t 1 t 2−1 2 d t 2 2)=ℓ.v 0 t 1+1 2 a t 1 2+(v 0 t 2+a t 1 t 2−1 2 d t 2 2)=ℓ. The term between brackets comes from the fact that the speed during the deceleration process is v max−d t v max−d t, so the distance covered during the deceleration process is v max t 2−1 2 d t 2 2 v max t 2−1 2 d t 2 2. From the first equation, we get t 2=(v 0+a t 1)/d t 2=(v 0+a t 1)/d. Plugging this into the second equation and solving, t 1=−v 0 a+1 a d v 2 0+2 a ℓ d a+d−−−−−−−−−√.t 1=−v 0 a+1 a d v 0 2+2 a ℓ d a+d. So v max=v 0+a t 1=d v 2 0+2 a ℓ d a+d−−−−−−−−−√.v max=v 0+a t 1=d v 0 2+2 a ℓ d a+d. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Jan 13, 2014 at 18:12 answered Jan 13, 2014 at 17:21 Martin ArgeramiMartin Argerami 219k 17 17 gold badges 161 161 silver badges 299 299 bronze badges 7 This looks really good, but I think it doesn't account for a non-zero initial velocity. It's definitely a step in the right direction though. I have no idea how to work that into your first equation though. (at1−dt2=0) That's where I get stuck every time I try to solve this.Derek Lewis –Derek Lewis 2014-01-13 17:31:52 +00:00 Commented Jan 13, 2014 at 17:31 Sorry, I didn't notice that the initial velocity was non-zero. I'll edit in a few minutes.Martin Argerami –Martin Argerami 2014-01-13 17:49:28 +00:00 Commented Jan 13, 2014 at 17:49 Please see the edit.Martin Argerami –Martin Argerami 2014-01-13 18:13:03 +00:00 Commented Jan 13, 2014 at 18:13 Thanks for the edit. :) I'm not entirely clear on the term in the brackets in the 2nd equation. I think I follow your explanation of it, though I'm confused about the "v0t2 + at1" part of it. I think it's intended to be equal to vmax. If that's the case, should it be something like (v0t1 + at1)? (With the brackets, to multiply all that by t2 when substituting vmax with that.)Derek Lewis –Derek Lewis 2014-01-13 19:01:46 +00:00 Commented Jan 13, 2014 at 19:01 v 0 t 1+a t 1 v 0 t 1+a t 1, makes no sense: v 0 t 1 v 0 t 1 has units of distance, while a t 1 a t 1 has units of speed. When the deceleration starts, the initial speed for this second part of the problem is v 0+a t 1 v 0+a t 1. The distance covered is the integral of the speed, −1 2 d t 2 2−1 2 d t 2 2 plus the initial speed times t 2 t 2; that's precisely the expression in brackets.Martin Argerami –Martin Argerami 2014-01-13 19:22:48 +00:00 Commented Jan 13, 2014 at 19:22 \|Show 2 more comments You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions physics See similar questions with these tags. 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12384	https://www.youtube.com/watch?v=3FxE0qyEu7I	Opposition, Reposition Thumb Movement (Flexion, Abduction) \| Anatomy Body Movement Terms RegisteredNurseRN 4750000 subscribers 1197 likes Description 81687 views Posted: 15 Dec 2020 Opposition and reposition are special body movements of the thumb. Opposition involves abduction and flexion, allowing the thumb to meet (and oppose) the tips of any of the fingers on the same hand. The thumb (metacarpal 1) articulates with the trapezium bone of the carpus via a saddle joint. This joint allows the thumb to perform flexion, extension, circumduction, abduction, adduction, as well as the special movements of opposition and reposition. Opposition is the movement of the thumb so that it touches any of the digits of the hand. Reposition is very simple: It involves returning the thumb/fingers to their original position. This video also includes a couple of memory tricks to help you remember opposition. QUIZ: Anatomy playlist: anatomy opposition Website: More Videos: Nursing Gear: Instagram: Facebook: Twitter: Popular Playlists: NCLEX Reviews: Fluid & Electrolytes: Nursing Skills: 28 comments Transcript: hey everyone this is ben with registerednessrn.com and in this video i'm going to demonstrate opposition and reposition which are special movements involving the thumb now the thumb articulates or forms a joint with the trapezium bone of the wrist via a saddle joint a type of synovial joint featuring interlocking convex and concave surfaces and they call it a saddle joint because well it looks like a saddle saddle up cowboy yeehaw now thanks to this saddle joint the thumb can perform various movements such as circumduction flexion and extension abduction and adduction as well as special movements called opposition and reposition opposition of the thumb occurs when the tip of the thumb comes to meet and oppose the tip of another finger from the same hand now a super simple way to remember this movement is that you've probably heard someone say that humans have opposable thumbs and we do and that's what this movement opposition is all about it's about taking the thumb and moving it around so that it opposes each of the digits or fingers by touching the tip like that also check this out whenever opposition occurs the thumb moves and meets the tip of one of the fingers and when you look at the shape that results it kind of makes the shape of a letter o o for opposition with our opposable thumbs now reposition is super simple because it is the opposite action of opposition during reposition the thumb and finger return to their original position okay so that wraps up this video on opposition and reposition we have a whole anatomy playlist if you click the link in the description below also take a free quiz on our website to test yourself really quickly over opposition and reposition to help lock it in your brain thank you so much for watching and please subscribe
12385	https://oercommons.org/courseware/lesson/737/overview	Math, Grade 6, Rational Numbers, Reflections \| OER Commons Donate to ISKME Discover Resources Collections Providers Hubs Sign in to see your Hubs Login Featured HubsAI & OER Community Hub Open Textbooks #GoOpen Professional Learning Climate Education California Community Colleges Hub See all Hubs Groups Sign in to see your Groups Login Featured GroupsAdult Education Open Community of Resources OpenStax Biology 2e PA STEM Toolkit Pathways Project \| Language Teaching Repository @ Boise State Student Advocacy See all Groups Learn More About Help Center About Hubs Services OER 101 Add OER Open Author Create a standalone learning module, lesson, assignment, assessment or activity Create Resource Submit from Web Submit OER from the web for review by our librarians Add Link Learn more about creating OER Add OER Add Link Create Resource About creating OER Search Advanced Search Notifications Sign In/Register Sign In/Register Discover Resources Collections Providers Hubs Sign in to see your Hubs Login Featured HubsAI & OER Community Hub Open Textbooks #GoOpen Professional Learning Climate Education California Community Colleges Hub See all Hubs Groups Sign in to see your Groups Login Featured GroupsAdult Education Open Community of Resources OpenStax Biology 2e PA STEM Toolkit Pathways Project \| Language Teaching Repository @ Boise State Student Advocacy See all Groups Learn More About Help Center About Hubs Services OER 101 Add OER Open Author Create a standalone learning module, lesson, assignment, assessment or activity Create Resource Submit from Web Submit OER from the web for review by our librarians Add Link Learn more about creating OER Add OER Add Link Create Resource About creating OER Student View Preview Copy Save Please log in to save materials. Log in Report Details Standards Resource Library Subject:Numbers and Operations Material Type:Lesson Plan Level:Middle School Grade:6 Provider:PearsonTags: 6th Grade Mathematics Graphing Log in to add tags to this item. License:Creative Commons Attribution Non-CommercialLanguage:English Media Formats:Downloadable docs, Text/HTML Show MoreShow Less Education Standards 1. 1 2. 2 3. 3 4. 4 5. 5 6. ... 7. 6 8. 7 9. 8 10. 9 11. 10 MCCRS.Math.Content.6.G.A.3 Maryland College and Career Ready Math Standards Grade 6 Learning Domain: Geometry Standard: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. MCCRS.Math.Content.6.NS.C.6 Maryland College and Career Ready Math Standards Grade 6 Learning Domain: The Number System Standard: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. MCCRS.Math.Content.6.NS.C.6b Maryland College and Career Ready Math Standards Grade 6 Learning Domain: The Number System Standard: Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. MCCRS.Math.Content.6.NS.C.6c Maryland College and Career Ready Math Standards Grade 6 Learning Domain: The Number System Standard: Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. MCCRS.Math.Content.6.NS.C.8 Maryland College and Career Ready Math Standards Grade 6 Learning Domain: The Number System Standard: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. CCSS.Math.Content.6.G.A.3 Common Core State Standards Math Grade 6 Cluster: Solve real-world and mathematical problems involving area, surface area, and volume Standard: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. CCSS.Math.Content.6.NS.C.6 Common Core State Standards Math Grade 6 Cluster: Apply and extend previous understandings of numbers to the system of rational numbers Standard: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. CCSS.Math.Content.6.NS.C.6b Common Core State Standards Math Grade 6 Cluster: Apply and extend previous understandings of numbers to the system of rational numbers Standard: Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. CCSS.Math.Content.6.NS.C.6c Common Core State Standards Math Grade 6 Cluster: Apply and extend previous understandings of numbers to the system of rational numbers Standard: Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. CCSS.Math.Content.6.NS.C.8 Common Core State Standards Math Grade 6 Cluster: Apply and extend previous understandings of numbers to the system of rational numbers Standard: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. PDF Check Your Understanding DownloadView Coordinate Plane Plotter View Coordinate Point Plotter View PDF Locate the Vertices DownloadView PDF New Reflections DownloadView Math, Grade 6 Rational Numbers Getting Started Rational Numbers Fractions and Decimals Ratios Expressions Equations and Inequalities Rate Putting Math to Work Distributions and Variability Surface Area and Volume Reflections Above and Below Sea Level Opposite of a Number Using Negative and Absolute Numbers Possible or Impossible? Inequalities Assess and Revise Gallery Problems Coordinate Plane Drawing Figures on the Coordinate Plane Reflections Peer Review and Revise Gallery Problems Reflections Overview Students reflect a figure across one of the axes on the coordinate plane and name the vertices of the reflection. As they are working, students look for and make use of structure to identify a convention for naming the coordinates of the reflected figure. Key Concepts When point (m,n) is reflected across the y-axis, the reflected point is (−m,n). When point (m,n) is reflected across the x-axis, the reflected point is(m,−n). When point (m,n) is reflected across the origin (0,0), the reflected point is (−m,−n). Goals and Learning Objectives Reflect a figure across one of the axes on the coordinate plane. Name the vertices of the reflected figure. Discern a pattern in the coordinates of the reflected figure. Reflecting Lesson Guide Put a line on the floor with masking tape. Have a student stand on one side of the line in an interesting position, such as standing on one foot. Have another student stand on the other side of the line in a mirror image. Now have students look at the images of the cars and the triangles. Discuss how each image has been reflected across the line to get a mirror image. Be sure students understand that each mirror image is the same size as the original and each original point and its mirror image are the same distance from the line of reflection. Mathematical Practices Mathematical Practice 7: Look for and make use of structure. Review the importance of looking for and making use of structure in math. Remind students that they used the structure of a number line to decide if an inequality was true or false. Encourage students to look for structure in the math today. Tell students to find a pattern in the way the math works. Opening Reflecting Read and Discuss You can reflect a figure across a line to get a mirror image of the figure. Every point is the same distance from the line of reflection, and the reflection, or mirror image, is the same size as the original image. Reflect Points Lesson Guide Tell students that they will use the x- and y-axes on the coordinate plane to reflect points across the x- and y-axes. Have pairs work through the problem and then discuss it as a class. Opening Reflect Points You can reflect a point across the x-axis or y-axis on the coordinate plane. In this case, the point (−3,2) has been reflected across each axis. What are the coordinates of the reflection of (−3,2) across the x- axis? The y-axis? Reflect Figures Lesson Guide Have students reflect the figure and locate the vertices of the reflection. Opening Reflect Figures Look for a pattern, or structure, as you reflect the figure in the Coordinate Point Plotter interactive. Locate the vertices of the mirror image, or reflection, of the figure. Use the x- or y-axis as the mirror. Reflect the figure across the x- or y-axis. INTERACTIVE: Coordinate Point Plotter Math Mission Lesson Guide Discuss the Math Mission. Students will find a pattern for naming the vertices of a figure reflected across one of the axes on the coordinate plane. Opening Find a pattern for naming the vertices of a figure reflected across one of the axes on the coordinate plane. Locate the Vertices Lesson Guide Tell students that they will continue solving reflection problems. Monitor and check students' work as they progress through the problems. After students have worked individually on a few problems, have them share their strategy with a partner. SWD: During Partner Work, monitor student discussions and provide guiding questions that help students see the mathematics and find their own way to a solution. If a student is struggling with a particular concept, hold an individual conference. If many students are struggling with a concept, pull a small group to discuss the concept. ELL: Be sure that ELLs are productive during this activity and that they are paired in a way that will foster participation. Sometimes, pair them up with native English speakers so that ELLs can learn from their counterparts’ English language skills. At other times, pair them up with students at the same level of language skills, so ELLs can take a more active role and partners can work things out together. Interventions Student has difficulty getting started. What are you trying to do? What are the coordinates of the vertices of the original figure? What axis are you reflecting the figure across? What will you do first? Second? Student has an incorrect solution. Explain how you reflected the figure across the axis. What does it mean to reflect a point across an axis? How do you know that the point is reflected across the axis correctly? Student has a correct solution. What method did you use to reflect the figure across the axis? Did you reach any conclusions about the coordinates of the reflected figure’s vertices? ELLs: In posing these questions, make sure that if the student involved is an ELL, your pace is adequate and you are providing ample wait time to allow for a thoughtful response. Mathematical Practices Mathematical Practice 1: Make sense of problems and persevere in solving them. Some students may get frustrated if their reflections and the originals do not match up. Look for students who persevere when trying to plot the reflections. Mathematical Practice 6: Attend to precision. Identify students who carefully plot the coordinates of the reflected figures. Compare their methods with students who just “eyeball” the reflections. Mathematical Practice 7: Look for and make use of structure. Watch for students who begin to see a pattern in the coordinates of the reflected vertices and plot the reflections using the pattern. Work Time Locate the Vertices Locate the vertices of each figure reflected across the x- or y-axis. Use the Coordinate Plane Plotter interactive if you find it to be helpful. HANDOUT: Locate the Vertices INTERACTIVE: Coordinate Plane Plotter Start with one vertex and find its reflection. Prepare a Presentation Preparing for Ways of Thinking Have partners prepare a presentation together. Observe how students are locating the reflections and identify different strategies that can be shared in Ways of Thinking. Challenge Problem Answers Answer will vary. Work Time Prepare a Presentation Describe the process you used to find each reflected vertex. Challenge Problem Choose a line of reflection that is not the x- or y-axis. Reflect the figure across the line on the coordinate plane. Use the Coordinate Plane Plotter if you find it to be helpful. HANDOUT: New Reflections INTERACTIVE: Coordinate Plane Plotter Make Connections Lesson Guide Have student pairs give their presentations. Discuss the Work Time activity and students’ presentations. Ask questions such as the following: Compare [Name]’s and [Name]’s strategies. How are they the same? How are they different? Did anyone have trouble reflecting the figures to start? Tell us the different methods you tried. What do you notice about the ordered pairs that name the corresponding vertices of a reflected figure? (Answer: They only differ by their signs.) Have students who did the Challenge Problem share their work. Performance Task Ways of Thinking: Make Connections Take notes about the strategies you can use to reflect figures on a coordinate plane. As students present, ask questions such as: What strategy did you use to reflect the figure? How is your strategy different from the other methods mentioned so far? How is it similar? How do you know that each point is reflected correctly? What did you notice about the coordinates of the reflected figure’s vertices? Describe Reflections A Possible Summary If you have two ordered pairs and the first numbers in the pair are opposite, the points will be located on opposite sides of the y-axis. If you have two ordered pairs and the second numbers in the pair are opposite, the points will be located on opposite sides of the x-axis. Additional Discussion Points When two ordered pairs differ only by their signs, the locations of the points are related by reflections across onme or both axes. When point (m,n) is reflected across the y-axis, the reflected point is (−m,n). When point (m,n) is reflected across the x-axis, the reflected point is(m,−n). When the point (m,n) is reflected across the origin (0,0), the reflected point is (−m,−n). ELL: Make sure all students have this information in their notebook. Provide some students with certain things to listen for during this portion of instruction. Some students may need copies of the notes from this portion of the lesson. Formative Assessment Summary of the Math: Describe Reflections Write a summary describing how to reflect images on a coordinate plane. Check your summary. Do you include what you observed about the coordinates of the reflected points? Check Your Understanding Lesson Guide This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery and which students would benefit from review before the assessment. Have students work on the Self Check individually. Assessment Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches. Do not score students’ work. Share with each student the most appropriate interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows. Interventions Student has difficulty getting started. What do you need to do? Approximately where will this figure be located on the coordinate plane? Can you find one point that meets one of the criteria? Can you find another point that meets a different criterion? Student does not seem to understand what a coordinate (ordered pair) is. Where can you go to review the meaning of coordinates (ordered pairs)? What does the first number in the ordered pair tell you? What does the second number in the ordered pair tell you? Student does not seem to understand the opposite of a number or has drawn a figure that does not meet the criteria for the opposite of a number. Where can you go to review information about the opposite of a number? What is an example of a number and its opposite? How would you write those numbers as an ordered pair? Where would that point be located on the coordinate plane? Student does not seem to understand integers and non-integers or has drawn a figure that does not meet the criteria for non-integers. Where can you go to review integers? What is an example of two numbers that are both non-integers? How would you write those numbers as an ordered pair? Where would that point be located on the coordinate plane? Student does not seem to understand absolute value or has drawn a figure that does not meet the criteria for absolute value. Where can you go to review absolute value? What is an example of two numbers that have equal absolute values? Student produces a correct solution but does not give an explanation of why it is correct. How does your figure or design meet each of the criteria? Student presents the work poorly. Is your work shown clearly? Have you given enough explanation and is it clear? Student presents a correct solution. Can you find a different way to solve the problem? Possible Answers Sample answer: Explanations will vary. Formative Assessment Check Your Understanding Work on this Self Check by yourself. Create a figure or design on the coordinate plane that satisfies the following criteria: At least one vertex has coordinates that include a number and its opposite. At least one vertex has coordinates that are both non-integers. At least one vertex is in each quadrant. At least two vertices have x-coordinates that have equal absolute values. At least two vertices have y-coordinates that have equal absolute values. Label each vertex with its ordered pair, and explain how your figure or design meets the criteria. Use the Coordinate Plane Plotter interactive if you find it to be helpful, but record your work on the handout. HANDOUT: Check Your Understanding INTERACTIVE: Coordinate Plane Plotter Reflect On Your Work Lesson Guide Have each student write a brief reflection before the end of class. Review the reflections to find out what strategies students found useful when reflecting a figure on the coordinate plane. Work Time Reflection Write a reflection about the ideas discussed in class today. Use the sentence starter if you find it to be helpful. A strategy that I found useful in reflecting a figure on the coordinate plane is … Discover Resources Collections Providers Community All Hubs All Groups Create Open Author Submit a Resource Our Services About Hubs About OER Commons OER 101 Help Center My Account My Items My Groups My Hubs Subscribe to OER Newsletter Subscribe Connect with OER CommonsFacebook, Opens in new windowTwitter, Opens in new window Donate to ISKME Powered By Privacy PolicyTerms of ServiceCollection PolicyDMCA © 2007 - 2025, OER Commons A project created by ISKME. 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12386	https://soneven.com/blogs/blogs/how-many-golf-balls-fit-in-a-trash-can?srsltid=AfmBOoruqTNZhAKTXgHefdkS0RgkOt63_puPue-st-SEJ7-KSG57SmjY	How Many Golf Balls Fit in a Trash Can? How Many Golf Balls Fit in a Trash Can? Have you ever wondered how many golf balls can fit in a trash can while playing golf? A standard golf ball is about 1.68 inches in diameter and has a volume of 2.48 cubic inches. A typical 13-gallon trash can holds about 3,003 cubic inches of space. Based on these measurements, a 13-gallon trash can can theoretically hold about 1,100 golf balls, but actual capacity will be affected by factors such as packing efficiency. In this article, we'll explore how these variables come into play and calculate a more accurate estimate. What Are the Dimensions of a Golf Ball? According to the United States Golf Association (USGA), the minimum diameter of a standard golf ball is 1.68 inches (42.67 mm). Therefore, its radius is approximately 0.84 inches (21.34 mm). Using the volume formula for a sphere, the volume of a golf ball is approximately 2.48 cubic inches (40.68 cubic centimeters). Additionally, the circumference of a golf ball is approximately 5.28 inches (134 mm). These consistent dimensions ensure consistent performance on the golf course, allowing players to expect the ball to perform predictably in terms of roll, flight, and spin. What Are the Dimensions of a Standard Trash Can? Trash cans come in a variety of sizes and shapes, each with a different impact on packing efficiency. Here are the most common sizes: 10-Gallon Trash Can: A 10-gallon trash can is a great choice for smaller spaces like a kitchen or bathroom. With a capacity of 10 gallons (2,310 cubic inches), a height of approximately 28 inches, and a diameter of approximately 12 inches, it is compact and easy to fit into tight corners. 13-Gallon Trash Can: A 13-gallon trash can is perfect for a medium-sized kitchen or office. With a capacity of 13 gallons (3,003 cubic inches), a height of approximately 30 inches, and a diameter of approximately 14 inches, it has plenty of space without being too bulky. 30-Gallon Trash Can: For larger spaces like a garage or outdoor area, a 30-gallon trash can is plenty of capacity. With a capacity of 30 gallons (6,930 cubic inches), a height of approximately 36 inches, and a diameter of approximately 16 inches, it is perfect for handling large amounts of waste. How to Calculate the Maximum Number of Golf Balls? Calculating how many golf balls can fit in a trash can requires a few simple math steps. Knowing the dimensions of the golf balls and the trash can, and factoring in packing efficiency, you can come up with a realistic estimate. Determine the Volume of One Golf Ball First, you need to know the volume of a single golf ball. Diameter: A standard golf ball has a diameter of 1.68 inches (42.67 mm). Radius: The radius is half of the diameter, so it’s 0.84 inches (21.335 mm). Volume Formula: The volume VVV of a sphere is calculated using the formula： Calculation: From this, we know that each golf ball has a volume of approximately 2.48 cubic inches (40.68 cm³). Find the Volume of the Trash Can Next, determine the total volume of the trash can you’re using. Capacity: Let’s consider a standard 13-gallon trash can. Conversion: A 13-gallon trash can holds approximately 3,003 cubic inches of space. Shape Considerations: Most trash cans are cylindrical, but some might be rectangular or have tapered sides. For a cylindrical trash can, you can calculate the volume using: where D is the diameter and H is the height. Ensure all measurements are in the same units. Calculate the Theoretical Maximum Number of Golf Balls Now, divide the trash can’s total volume by the volume of one golf ball to find the theoretical maximum number of balls that can fit, assuming perfect packing with no wasted space. So, theoretically, a 13-gallon trash can could hold about 1,211 golf balls if they fit perfectly without any gaps. Adjust for Packing Efficiency In reality, golf balls cannot be packed perfectly due to their spherical shape, which leaves empty spaces between them. This is where packing efficiency comes into play. Packing Efficiency: For spheres, the packing efficiency can range from 60% to 70% depending on how well they are arranged. Realistic Estimate Calculation: Realistic Number=Theoretical Maximum×Packing Efficiency Using 65% packing efficiency as an example: Realistic Number=1,211×0.65≈787 golf balls Thus, considering a packing efficiency of around 65%, a 13-gallon trash can can realistically hold approximately 787 golf balls. What Factors Will Affect the Actual Number? When figuring out how many golf balls can fit into a trash can, it's not just about the total volume. Several real-world factors can change the final number. Let’s look at the main ones: 1. Shape and Design of the Trash Can: The shape of your trash bin plays a big role in how many golf balls you can fit. While most trash bins are round, some have tapered sides or even rectangular shapes, which can make packing more difficult. Additionally, features like lids, handles, or built-in dividers take up extra space, which means you can fit fewer golf balls than the total capacity would suggest. 2. Consistency of Golf Balls: Although golf balls should be the same size, they may have slightly different diameters or shapes. These slight variations can create more gaps when you try to pack them together. Additionally, if the golf balls are worn or have irregular surfaces, they won't stack neatly, reducing the total number you can fit. 3. How You Pack the Balls: The way you put the golf balls in the trash bin can also greatly affect the final number. If you just throw them in randomly, there may be more gaps between the balls. On the other hand, stacking them in an orderly manner or using fillers like foam or paper can help you fit more golf balls by minimizing gaps. However, using fillers can also take up some space, so it's a question of balance. 4. Packing Efficiency: Packing efficiency is how well the golf balls make use of the available space. In theory, you might think you can fit all the golf balls perfectly, but in reality, there will always be some gaps. Therefore, you can expect about 60 to 70 percent packing efficiency, depending on how you arrange the golf balls and the shape of your trash can. 5. Internal Obstacles in the Trash Can: Many trash cans have internal features like handles or retaining rings that can get in the way when you're trying to pack golf balls. These obstructions take up space and make it harder to pack the balls tightly together, which reduces the total number you can store. 6. Environmental Conditions: Factors like temperature and humidity can also affect how well you pack your golf balls. For example, if it's very cold, the golf balls may become less flexible and more rigid, creating more voids. High humidity may make the golf balls a little sticky or soggy, which can affect how they stack. Also, if you move or shake the trash can while it's full of trash, the golf balls may move around and create more voids. 7. Using Fillers: Adding padding like foam padding or paper can help keep the golf balls in place and reduce movement, making for a more stable pack. However, padding also takes up some space, so you need to find a balance between using enough padding to stabilize the golf balls and not using too much padding to reduce capacity loss. What Role Does Golf Clothing Play in the Process? Golf clothing plays a key role in ensuring comfort and performance on the course. The right gear, like women's golf clothes, long sleeve polo, women's leggings, and lady thermal wear, is designed to provide flexibility and breathability, so you can move freely during your swing and other movements. Lightweight, breathable fabrics keep you cool, while thermal tops offer warmth when it's chilly, allowing you to stay comfortable no matter the weather. Whether you are practicing or playing, various types of skirts can help you concentrate and perform at your best. Because the right clothing can not only make you more focused on the game, improve your overall performance, and enjoy your time on the court. In addition, these clothes can also allow you to wear more, so that you can stay optimized in leisure moments, such as how to match a polo shirt with a suit, etc. Conclusion In summary, how many golf balls a trash can can hold depends on the size of the trash can and the efficiency of its packing. While a standard 13-gallon trash can holds approximately 3,003 cubic inches, it can theoretically hold about 1,100 golf balls. However, due to irregular packing and the spaces between the balls, the actual number will be lower. By knowing the dimensions of your golf balls and trash can, you can estimate how many golf balls you can fit, keeping in mind that packing efficiency plays a vital role in the final count. Share Article Tags Latest Products Leave a comment Please note: comments must be approved before they are published. 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12387	https://teachy.ai/en/summaries/high-school/12th-grade/mathematics-en/1x1-determinants-the-foundation-for-practical-mathematical-solutions-4bfa0	Summary of 1x1 Determinants: The Foundation for Practical Mathematical Solutions We use cookies Teachy uses cookies to enhance your browsing experience, analyze site traffic, and improve the overall performance of our website. You can manage your preferences or accept all cookies. Manage preferences Accept all TeachersSchoolsStudents Teaching Materials EN Log In Teachy> Summaries> Mathematics> 12th grade> Determinant: 1x1 Summary of Determinant: 1x1 Lara from Teachy Subject Mathematics Mathematics Source Teachy Original Teachy Original Topic Determinant: 1x1 Determinant: 1x1 1x1 Determinants: The Foundation for Practical Mathematical Solutions Objectives 1. Understand the concept of determinants of a 1x1 matrix. 2. Learn to calculate the determinants of a matrix with one row and one column. 3. Recognize the importance of determinants in practical applications and in the job market. Contextualization Determinants are crucial mathematical tools used to solve systems of linear equations and analyze properties of matrices. Imagine having to decide the most efficient route for a delivery with multiple stops; determinants help simplify complex calculations and find quick and accurate solutions. They are used in various fields such as engineering, physics, and economics, becoming essential for solving practical and real problems. Relevance of the Theme Determinants are fundamental for solving complex problems across various fields of knowledge. In an increasingly data-driven and efficiency-oriented world, understanding and correctly using determinants can enhance decision-making, optimize processes, and significantly contribute to the development of innovative solutions in areas such as engineering, economics, and computer science. Concept of Determinant of a 1x1 Matrix The determinant of a 1x1 matrix is a fundamental concept in linear algebra. A 1x1 matrix consists of only a single element, and its determinant is equal to the value of that element itself. This simple concept is the basis for understanding determinants of larger and more complex matrices. A 1x1 matrix consists of a single element. The determinant of a 1x1 matrix is the value of the unique element. This concept is the foundation for understanding determinants of larger matrices. Fundamental Properties of Determinants Determinants have fundamental properties that facilitate the solving of linear systems and the analysis of matrices. In the case of a 1x1 matrix, these properties are simple, but they become more complex and powerful as we study higher-order matrices. Determinants assist in solving linear systems. They facilitate the analysis of matrices. In larger matrices, determinants have properties that allow for the simplification of complex calculations. Practical Applications of Determinants Determinants are widely used in various fields such as engineering, physics, and economics to solve real problems. For example, they are used to calculate areas and volumes, analyze systems of linear equations, and determine the invertibility of matrices, which is essential in computational algorithms. Used to calculate areas and volumes. Fundamental in the analysis of systems of linear equations. Determine the invertibility of matrices, essential for computational algorithms. Practical Applications In engineering, determinants are used to analyze force systems and determine the stability of structures. In physics, determinants help solve differential equations that model natural phenomena. In finance, determinants are used in risk analysis and the optimization of investment portfolios. Key Terms Determinant: A value that can be calculated from the elements of a square matrix that summarizes various properties of that matrix. 1x1 Matrix: A matrix that consists of a single row and a single column, containing only one element. Linear Algebra: A branch of mathematics that deals with vectors, vector spaces, matrices, and systems of linear equations. Questions How can understanding determinants help in your future career? In what ways are determinants applied in solving complex everyday problems? What other fields of knowledge, beyond those mentioned, can benefit from the use of determinants? Conclusion To Reflect Understanding the determinants of 1x1 matrices is an essential step in the learning journey in mathematics. These concepts are more than just simple numbers; they represent a powerful tool that can be applied in various knowledge fields such as engineering, physics, economics, and computer science. By mastering the fundamentals of determinants, you are preparing to face and solve complex problems efficiently and accurately. Consider how this skill can be utilized in your future career and the numerous possibilities that open up by understanding how determinants work. Practice and application of these concepts are essential for transforming theoretical knowledge into real and innovative solutions. Mini Challenge - Determinant in Practice: Planning an Efficient Route Let's apply the concept of determinants of 1x1 matrices to solve a practical navigation problem, helping to plan the most efficient route between different points. Divide into groups of 3 to 4 students. Draw a simple map on a sheet of grid paper, including 4 to 5 stopping points. Use a 1x1 matrix to represent the distance between two points on the map. Calculate the determinant of that matrix for each pair of adjacent points. Determine the most efficient route based on the calculated determinants. 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12388	https://www.quora.com/How-can-I-prove-simply-that-sin-%CF%80-x-sin-x	Something went wrong. Wait a moment and try again. Sin Function Trigonometric Identities Proofs (mathematics) Sine and Cosine Sine Theta Trigonometric Expressions Sine (math function) 5 How can I prove (simply) that sin (π-x) =sin x? Pankaj Kumar B Tech from National Institute of Technology, Patna (Graduated 2022) · 6y To prove this, use sine Subtraction formula. i.e, sin(a-b)= sin(a)cos(b)-cos(a)sin(b) Here a=π and b=x sin(π-x) = sin(π)cos(x)-cos(π)sin(x) = 0×{cos(x)}-{-1×sin(x)} = sin(x) Hence proved Related questions For all x in [0, π 2 , How do I prove that cos ( sin x ) sin ( cos x ) ? How can I simply prove that sin(π/2-x) =cosx? How do I prove that sin (180-x) =sin x? How does one prove sin ( 2 x ) sin ( x ) − cos ( 2 x ) cos ( x ) = sec ( x ) ? How do I prove s i n ( s i n ( x ) ) + s i n ( s i n ( x 3 ) ) + s i n ( s i n ( x 9 ) ) + . . . + s i n ( s i n ( x 3 n ) ) < 1.5 x ∀ 0 < x < 1 , n ∈ N ? Ashis Ghosh Former Retired officer at State Bank of India (SBI) · Author has 2.6K answers and 2.9M answer views · 6y sin(pi-x)=sin pi.cosx-cos pi.sinx or,sin(pi-x)=2sin pi/2.cos pi/2.cosx-[2(cos pi/2)^2-1].sinx or,sin(pi-x)=2×1×0×cosx-[2×0-1].sinx or,sin(pi-x)=0–(-1)sinx=+(1.sin x)=sin x So,sin(pi-x)=sin x proven S.p B.Sc maths in Mathematics & Physics, SIVETCollege (Graduated 1978) · Author has 4.3K answers and 2.2M answer views · Jul 5 sin(pi-x)=sinpicosx-cospisinx[Since sin(A-B) =sinAcosB -cosAsinB] sinpi=0 and cospi=-1 we have, 0.cosx-(-1)sinx=0+sinx=sinx which is proved.Thanks. Assistant Bot · 1y To prove that sin ( π − x ) = sin x , you can use the properties of the sine function and the unit circle. Here’s a simple step-by-step proof: Understanding the Unit Circle : The sine of an angle in the unit circle is the y-coordinate of the corresponding point on the circle. Finding sin ( π − x ) : When you consider the angle π − x , it represents a point that is in the second quadrant of the unit circle. The coordinates of this point are ( − cos x , sin x ) . 3. Using the Sine Function : The sine function gives the y-coordinate of the point on the unit circle. Thus: \sin(\pi - x) = \text{y-coordin To prove that sin ( π − x ) = sin x , you can use the properties of the sine function and the unit circle. Here’s a simple step-by-step proof: Understanding the Unit Circle : The sine of an angle in the unit circle is the y-coordinate of the corresponding point on the circle. Finding sin ( π − x ) : When you consider the angle π − x , it represents a point that is in the second quadrant of the unit circle. The coordinates of this point are ( − cos x , sin x ) . 3. Using the Sine Function : The sine function gives the y-coordinate of the point on the unit circle. Thus: sin ( π − x ) = y-coordinate of the point at π − x = sin x 4. Conclusion : Therefore, we conclude that: sin ( π − x ) = sin x This proof shows that the sine function is symmetric about the line x = π 2 , confirming that sin ( π − x ) = sin x . Related questions How can I prove ∫ π 0 cos ( sin ( sin x ) e cos x ) e e cos x cos ( sin x ) d x = π e ? How can I prove that x s i n 3 ( x ) = − x ( s i n ( 3 x ) − 3 s i n ( x ) ) 4 ? How to prove 1 + sin x = ( sin x 2 + cos x 2 ) 2 ? How can we prove that sin x + cos x = sin x cos x ? How will you prove 1 − sin x = 2 sin 2 ( π / 4 − x / 2 ) ? Rajeev Raj Studied at Resonance, Kota · Author has 133 answers and 220K answer views · 6y LHS=sin(π-x) Apply sin(A-B)=sinA cosB -cosAsinB LHS=sinπ cosx -cosπ sinx LHS=0×cosx -(-1)sinx [sinπ=0 ,cosπ=-1] LHS=sinx RHS prove Philip Lloyd Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist. · Upvoted by Jeremy Collins , M.A. Mathematics, Trinity College, Cambridge · Author has 6.8K answers and 52.8M answer views · 4y Related How do you prove that -sin(x) = sin(-x)? Very often it is not a PROOF that people want, it is an EXPLANATION so that real understanding can occur! These are the best DEFINITIONS of sine, cosine and tangent. In the diagram below we imagine that OP can rotate about the origin. This is called a “unit circle” because OP = 1 unit. Angles are measured from the positive x axis in an anti-clockwise direction. (Negative angles are measured in a clockwise direction.) We don’t need to refer to SOH CAH TOA for this work. The definitions of sin (θ), cos (θ) and tan (θ) are clearly shown on this diagram. Now let’s just consider an angle of 30 degrees for Very often it is not a PROOF that people want, it is an EXPLANATION so that real understanding can occur! These are the best DEFINITIONS of sine, cosine and tangent. In the diagram below we imagine that OP can rotate about the origin. This is called a “unit circle” because OP = 1 unit. Angles are measured from the positive x axis in an anti-clockwise direction. (Negative angles are measured in a clockwise direction.) We don’t need to refer to SOH CAH TOA for this work. The definitions of sin (θ), cos (θ) and tan (θ) are clearly shown on this diagram. Now let’s just consider an angle of 30 degrees for convenience. This clearly shows WHY sin (–30) = –sin (30) This sort of thing is best explained IN PERSON but as the next best thing I would strongly encourage you to have a look at this short 2 minute video I did: 2019-01-15 1708 Basic Trigometer-1.m4v World's leading screen capture + recorder from Snagit + Screencast by Techsmith. Capture, edit and share professional-quality content seamlessly. Sponsored by Grammarly Is your writing working as hard as your ideas? Grammarly’s AI brings research, clarity, and structure—so your writing gets sharper with every step. Ved Prakash Sharma Former Lecturer at Sbm Inter College, Rishikesh (1971–2007) · Author has 14.3K answers and 16.5M answer views · 6y sin(π-x)=sinx L.H.S. =sinπ.cosx-cosπ.sinx by putting sinπ=0. and cosπ= -1. =0.cosx - (-1).sinx = sin x. Proved. Girija Warrier Studied at Sufficiently Educated · Author has 5.9K answers and 13.9M answer views · Updated 3y Related How can I simply prove that cos(π/2-x)=sinx? I’m proving it geometrically.. In the above image, the initial ray is rotated 90° anti clockwise direction, it takes position OC. Then, x° is rotated in clockwise direction, which is = -x° ⇒ ( 90° - x) = y° . . . . . .(1) AC is drawn // BO Hence, <C becomes 90° ⇒ ABOC is a rectangle. ⇒ AC = BO . . . . . . . .(2) In triangle ABO Cos y = OB/OA But cos y = cos(90° - x) ( by (1) ) ⇒ cos( 90° - x) = OB/OA . . . . . . .(3) In tri ACO Sin x = AC/ OA But, AC = OB ( by (2) ) ⇒ sin x = OB/OA . . . . . . .(4) By comparing (3) & (4) We get, cos ( 90° - x) = sin x (Proved) I’m proving it geometrically.. In the above image, the initial ray is rotated 90° anti clockwise direction, it takes position OC. Then, x° is rotated in clockwise direction, which is = -x° ⇒ ( 90° - x) = y° . . . . . .(1) AC is drawn // BO Hence, <C becomes 90° ⇒ ABOC is a rectangle. ⇒ AC = BO . . . . . . . .(2) In triangle ABO Cos y = OB/OA But cos y = cos(90° - x) ( by (1) ) ⇒ cos( 90° - x) = OB/OA . . . . . . .(3) In tri ACO Sin x = AC/ OA But, AC = OB ( by (2) ) ⇒ sin x = OB/OA . . . . . . .(4) By comparing (3) & (4) We get, cos ( 90° - x) = sin x (Proved) Promoted by The Penny Hoarder Lisa Dawson Finance Writer at The Penny Hoarder · Updated Sep 16 What's some brutally honest advice that everyone should know? Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included. And honestly? Putting them to use was way easier than I expected. I bet you can knock out at least three or four of these right now—yes, even from your phone. Don’t wait like I did. 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WithKashKick, you can actually get paid to play. No weird surveys, no endless ads, just real money for playing games you’d probably be playing anyway. Some people are even making over $1,000 a month just doing this! Oh, and here’s a little pro tip: If you wanna cash out even faster, spending $2 on an in-app purchase to skip levels can help you hit your first $50+ payout way quicker. Once you’ve got $10, you can cash out instantly through PayPal—no waiting around, just straight-up money in your account. Seriously, you’re already playing—might as well make some money while you’re at it.Sign up for KashKick and start earning now! Nikhil Prashant Fulmare Lives in Delhi (2021–present) · Author has 443 answers and 656K answer views · 6y Related How can I prove (simply) that sin (π+x) =-sin x? For the simplest proof, I say you simply need to look at the graphs. Following is the graph of sin x. For graph of sin(π+x), simply shift the graph backward along X-axis by π. Comparing the two, we find that sin(π+x) is the negative of sin x. Note: I was asked to give a simple proof and hence I resorted to use graphs. There are many more, better ways to prove this. For the simplest proof, I say you simply need to look at the graphs. Following is the graph of sin x. For graph of sin(π+x), simply shift the graph backward along X-axis by π. Comparing the two, we find that sin(π+x) is the negative of sin x. Note: I was asked to give a simple proof and hence I resorted to use graphs. There are many more, better ways to prove this. Levi Carton Lives in The Universe · 6y Related How can I prove that sin 3 X = 3 sin x -4 sin³ x? sin3x is same thing as sin(x+2x), sin3x is same thing as sin(x+2x), Sponsored by CDW Corporation Want document workflows to be more productive? The new Acrobat Studio turns documents into dynamic workspaces. Adobe and CDW deliver AI for business. Peter Butcher Former Former Professional Engineer, Teacher, Tutor · Author has 1.4K answers and 4.6M answer views · 2y Related Can you explain why sin (x+Pi/2) = -sin x? No! but I can explain why Sin(x+ Pi/2) DOESN’T = Sin(x) Consider Graph 1 (below) Y = Sin(x) Note that: Sin(x) = 0 when x = 0 Sin(x) = 0 when x = pi Sin(x) = 1 when x = pi/2 Sin(x) =-1 when x = 3pi/2 Now consider Graph 2 (below) where Sin(x)is in Black and Sin(x + pi/2) is in purple. In Graph 2, note that: Sin(x+pi/2) = Sin(0) = 1 Sin(x+pi/2) = Sin(pi/2) = 0 Sin(pi) = Sin(x + pi/2) = Sin(pi) = -1 and most important Sin(x+pi/2) = Cos(x) NOT -Sin(x) Finally consider Graph 3 (below) Sin(x) is drawn in Black Sin(x+pi/2) is drawn in Purple and -Sin(x) is drawn in Acqa. Clearly Sin(x+pi/2) (Purple) is not the same as No! but I can explain why Sin(x+ Pi/2) DOESN’T = Sin(x) Consider Graph 1 (below) Y = Sin(x) Note that: Sin(x) = 0 when x = 0 Sin(x) = 0 when x = pi Sin(x) = 1 when x = pi/2 Sin(x) =-1 when x = 3pi/2 Now consider Graph 2 (below) where Sin(x)is in Black and Sin(x + pi/2) is in purple. In Graph 2, note that: Sin(x+pi/2) = Sin(0) = 1 Sin(x+pi/2) = Sin(pi/2) = 0 Sin(pi) = Sin(x + pi/2) = Sin(pi) = -1 and most important Sin(x+pi/2) = Cos(x) NOT -Sin(x) Finally consider Graph 3 (below) Sin(x) is drawn in Black Sin(x+pi/2) is drawn in Purple and -Sin(x) is drawn in Acqa. Clearly Sin(x+pi/2) (Purple) is not the same as -Sin(x) (Acqa) Sin(x+pi/2) is not the same as -Sin(x) but is the same as Cos(x). Peter Aouad M.S.E.E Engineering, Prague -1972 · Author has 1.8K answers and 1.6M answer views · 3y Related How do you prove that sin(π+x) =sinx? Plot both sides: Both functions are not equal for every x. sin(x)=-sin(pi+x) are equa... Plot both sides: Both functions are not equal for every x. sin(x)=-sin(pi+x) are equa... Philip Lloyd Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist. · Author has 6.8K answers and 52.8M answer views · 3y Related How do you prove that sin(x) + cos(x) = 1? You cannot prove this because it is only true for certain values of x. I will work in degrees so that more people can follow. Sin(0) = 0 and cos(0) = 1 so it is true for x = 0 degrees Sin(90) = 1 and cos(90) = 0 so it is true for x = 90 degrees Sin(360) = 0 and cos(360) = 1 so it is true for x = 360 degrees Sin(450) = 0 and cos(450) = 1 so it is true for x = 450 degrees I hope you get the idea now? Here is a graph of y = sin(x) + cos(x) which shows cases where sin(x) + cos(x) = 1 You cannot prove this because it is only true for certain values of x. I will work in degrees so that more people can follow. Sin(0) = 0 and cos(0) = 1 so it is true for x = 0 degrees Sin(90) = 1 and cos(90) = 0 so it is true for x = 90 degrees Sin(360) = 0 and cos(360) = 1 so it is true for x = 360 degrees Sin(450) = 0 and cos(450) = 1 so it is true for x = 450 degrees etc… I hope you get the idea now? Here is a graph of y = sin(x) + cos(x) which shows cases where sin(x) + cos(x) = 1 Related questions For all x in [0, π 2 , How do I prove that cos ( sin x ) sin ( cos x ) ? How can I simply prove that sin(π/2-x) =cosx? How do I prove that sin (180-x) =sin x? How does one prove sin ( 2 x ) sin ( x ) − cos ( 2 x ) cos ( x ) = sec ( x ) ? How do I prove s i n ( s i n ( x ) ) + s i n ( s i n ( x 3 ) ) + s i n ( s i n ( x 9 ) ) + . . . + s i n ( s i n ( x 3 n ) ) < 1.5 x ∀ 0 < x < 1 , n ∈ N ? How can I prove ∫ π 0 cos ( sin ( sin x ) e cos x ) e e cos x cos ( sin x ) d x = π e ? How can I prove that x s i n 3 ( x ) = − x ( s i n ( 3 x ) − 3 s i n ( x ) ) 4 ? How to prove 1 + sin x = ( sin x 2 + cos x 2 ) 2 ? How can we prove that sin x + cos x = sin x cos x ? How will you prove 1 − sin x = 2 sin 2 ( π / 4 − x / 2 ) ? How do I prove sin 2 x = 2 sin x cos x ? How do you prove that \| sin x \| ≤ \| x \| ? How can we prove x/ ([x/pi]! [-x/pi]!) = sin(x)? How do you prove sin ( 2 x ) sin x = 2 ( sin x tan x − tan x sin 3 x ) ? How do you prove that sin ( x + y ) = sin x + cos y cos x + sin y ? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12389	https://es.scribd.com/document/724885351/Laboratorio-8-Dilatacion-lineal	Laboratorio 8 Dilatación Lineal \| PDF \| Expansión térmica \| Temperatura Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 0 ratings 0% found this document useful (0 votes) 60 views 8 pages Laboratorio 8 Dilatación Lineal Este documento tiene como objetivo principal observar y analizar el comportamiento de los materiales ante el aumento de temperatura y cómo se produce la dilatación lineal. 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INFORME DE LABORATORIO Dilatación lineal Yenifer Julieth Navarrete Tejedor – Cod: 7305982 Lisandro Gómez Anzola – Cod: 7305936 Ronald Steven Yepes Cañas – Cod: 7305450 José Albiro Gamboa Gamboa – Cod: 7305870 Docente: Cristina Díaz González RESUMEN: se tiene como objetivo principal observar y analizar el comportamiento materiales frente al aumento de la temperatura, como se produce una dilatación lineal para esto se toman los datos de videos, longitud inicial, dilatación de las varillas, temperatura inicial, temperatura final , y el ΔT para hallar el α de cada uno de los materiales el resultado que se obtiene se compara con los coeficientes de dilatación lineal teóricos para poder saber que materiales fueron los de las varillas, las conclusiones y el análisis se enfoca en resolver las preguntas de la guía. 1 OBJETIVOS 1.1 OBJETIVO GENERAL Observar que los cuerpos cambian sus dimensiones en función de los cambios en la temperatura. Hallar el“coeficiente de dilatación lineal” y con este resultado identificar el tipo de material al cual pertenecen.Figura: Dilatómetro con el reloj comparador 1.2 OBJETIVOS ESPECIFICOS •Ana liz ar e l f enó men o f ísi co de dil ata ció n l ine al.•Compr ender la re laci ón qu e ex iste entre la es truct ura d el ma teria l y dicho fenóm eno.•Calcu lar el coef icie nte de dilat ación linea l a par tir d e los datos obten idos de la gráfi ca de l experimento.•Com par ar lo s va lor es te óri cos c on lo s va lor es ex per ime nta les. 2 MARCO TEÓRICO La tem per atu ra es una mag nit ud fís ica que se emp lea par a med ir el cal or o el frí o de un cue rpo. Est á directamente relacionada con la energía cinética de las partículas o componentes de un cuerpo o sistema. En otras palabras, cuanto mayor sea la energía cinética, mayor será la temperatura del objeto en cuestión. Los instrumentos utilizados para medir la temperatura son:  Termómetros de líquido.  Mercurio. Pueden medir temperaturas entre -39 °C y 357 °C. Son de fácil transporte y lectura 1 adDownload to read ad-free . .Alcohol coloreado. Pueden medir tem peraturas entre -112 °C y 78 °C. Son de fácil transport e, pero de menor precisión que los termómetros de mercurio.Los termómetros de gas, con su capacidad para medir temperaturas que oscilan entre -27 °C y 1.477 °C,ofrecen una amplia gama y precisión. Sin embargo, su utilización es compleja, lo que los relega principalmente al propósito de calibrar otros termómetros.Los termómetros de resistencia de platino son capaces de detectar temperaturas que oscilan desde -259 °C hasta 1.127 °C, aunque su exactitud se considera fiable solamente hasta los 631 °C.Pilas termoe léct ricas. Est án forma das por dos cables metá lico s unido s, cuyo volt aje varí a depend iendo de la temperatura. Miden temperaturas de entre -248 °C y 1.477 °C y se utilizan principalmente para reaccionar con rapidez a cambios bruscos de temperatura. 2 adDownload to read ad-free . Las escalas de temperatura utilizadas son las siguientes:  La escala Fahrenheit, concebida por Daniel Gabriel Fahrenheit en 1724, difiere completamente de las escalas Celsius y Kelvin. En esta escala, el punto de congelación del agua se sitúa en 32 °F y el punto de ebullición en 212 °F.  El grado Celsius (°C), anteriormente conocido como grado centígrado, establece una diferencia de 100 grados entre el punto de congelación y el de ebullición del agua.  La escala de temperatura Kelvin, creada por William Thomson Kelvin en 1848, se basa en la concepción del calor como movimiento de partículas.  A continuación, se presentan las fórmulas para conv ertir entre las dif erentes escalas de temperatura.Como mencionamos previamente, la temperatura de un objeto está estrechamente relacionada con su energía cinética. Por lo tanto, cuando un objeto recibe calor, su energía cinética aumenta, lo que hace que las partículas se muevan con mayor rapidez. Esta mayor velocidad de movimiento provoca que las partículas tiendan a ocupar más esp aci o, lo que res ult a en un aum ent o del volum en del objet o. Por el con tra rio, si el obj eto pierde calor,experimentará una disminución en su energía cinética y, por ende, en su volumen.La magnitud del cambio de tamaño depende de varios factores, como la naturaleza del objeto, sus dimensiones iniciales y la cantidad de calor recibido o la variación de temperatura experimentada.Como explicamos anteriormente, los objetos, al experimentar un cambio de temperatura, tienden a cambiar su volumen, ya sea expandiéndose o contrayéndose. Para cuantificar estos cambios, se utiliza el coeficiente de dilatación, que es el cociente que mide las variaciones de longitud o volumen de los objetos.Exis ten varias forma s de dilat ación térmic a. La dilat ación lineal ocurre cuand o una dimen sión del objet o predomina sobre las otras dos. Ejemplos de objetos que se dilatan linealmente incluyen varillas, alambres y barras.La expresión matemática para la dilatación lineal es la siguiente: 3 adDownload to read ad-free . La dilatación superficial ocurre cuando se enfatizan dos dimensiones, como una superficie, en comparación con una tercera dimensión. Ejemplos de objetos que experimentan dilatación superficial incluyen láminas y planchas. La fórmula para calcular la dilatación superficial es la siguiente:La dilatación volumétrica ocurre cuando todas las dimensiones del objeto se expanden de manera proporcional.Ejemplos de objetos que experimentan este tipo de dilatación son los dados del parchís y las estatuas en los jardines.La dilatación en los líquidos se manifiesta de manera más notoria que en los sólidos debido a que las moléculas líquidas tienen mayor libertad de movimiento. Esto facilita que el volumen ocupado por cada molécula aumente con la temperatura, lo que a su vez incrementa el volumen total del líquido. Este fenómeno se expresa de manera similar a la dilatación volumétrica observada en los sólidos.La dilatación de los gases es notablemente perceptible y se ve influenciada principalmente por la temperatura y la presión. Esto se debe a que las fuerzas de atracción entre las partículas en los gases son mucho más débiles en comparación con otros estados de la materia. Su comportamiento en términos de cambio de volumen es análogo a la dilatación que experimentan los sólidos.El coeficiente de dilatación lineal α representa el aumento en la longitud por unidad de longitud de un cuerpo cuando su temperatura se incrementa en 1 grado Celsius. Conociendo α y la longitud inicial de un cuerpo, es posible calcular su longitud una temperatura T específica mediante la siguiente ecuación:En un sólido, las moléculas ocupan posiciones relativament e fijas dentro de su estructura. Cada átomo en la red cristalina experimenta vibraciones determinadas por un pozo de potencial, y la amplitud de estas vibraciones 4 adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free Share this document Share on Facebook, opens a new window Share on LinkedIn, opens a new window Share with Email, opens mail client Copy link Millions of documents at your fingertips, ad-free Subscribe with a free trial You might also like Dilatacion Termica No ratings yet Dilatacion Termica 10 pages Informe Dilatación Lineal No ratings yet Informe Dilatación Lineal 4 pages Informe de Dilatación Térmica No ratings yet Informe de Dilatación Térmica 5 pages Termodinámica: Temperatura y Calor No ratings yet Termodinámica: Temperatura y Calor 57 pages Clase N°10 - Dilatación Térmica No ratings yet Clase N°10 - Dilatación Térmica 22 pages 4 Física Plan Común Temperatura y Dilatación 100% (1) 4 Física Plan Común Temperatura y Dilatación 7 pages 5° UA 05 Ses2-2TemperaturaDilataciónActiv. 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12390	https://www.zhihu.com/question/49583170	雨果是一个怎样的存在？ - 知乎关注推荐热榜专栏圈子 New付费咨询知学堂直答切换模式登录/注册雨果是一个怎样的存在？关注问题写回答登录/注册作家文学雨果法国文学西方文学雨果是一个怎样的存在？如题，关注者 334 被浏览 278,198 关注问题写回答邀请回答好问题 5 1 条评论分享 35 个回答默认排序匂宫出梦御建鸣神主尊大御所关注谢邀首先，毫无疑问，雨果无疑是法兰西文学史上的巨擘、最闪耀的明星之一，他的作品至今仍旧让我难以忘怀（当然我看的是中文版，感谢诸位老师的翻译），他才华横溢，并且富有激情，而且善于用激烈的故事情节来打动人心，是我最喜欢的外国文学家之一。不过，除了这些标签式的赞誉之外，雨果另有其作为凡人的一面，他同样在生活当中经历了坎坷，犯下了多种错误，并且也有和每个人一样的人性弱点。他像那时候上流社会的男人一样放荡，对妻子并不忠贞，情妇前后好几个，有一个甚至在身边长留了三十年——也许这种放荡因为文学家的才华而被蒙上了一层脂粉，但是毫无疑问，这种不忠贞深深地伤害了自己的妻子，并且给孩子们树立了并不完美的榜样。（不过说到底，19世纪的文豪们，又有哪个不是那样？只能说，他和其他人一样放纵自己的欲望而已）如果说在当时的环境下，放荡的生活只是道德上的小小瑕疵的话，那么他早年在政治上的前后摇摆，更加让喜欢或者不喜欢他的人都对他颇有微词。这里必须讲一下当时法国的政治背景——自从大革命之后，法兰西一直都陷入到了一种政治上的精神分裂当中，各个政治派别纷争不断，时常冷战偶尔热战，所以政府的垮台频频出现，有的时候甚至会发展成革命，把整个国家倾覆。从1789年到1871年，法国经历了三个王国、三个共和国和两个帝国，可谓是风云激荡。在这种激荡，共和派、帝国派和支持王朝的派别（里面又分支持波旁王家的正统派和支持七月王朝的秩序党）互不相让，彼此攻讦撕咬，互相仇恨不已。可以说当时法国的三家王族总是有一家在台上，有一家被迫流亡，有一家在被监视或坐牢，彼此轮回。而雨果却多次改换了门庭，最初，在极端保王党的妈妈的影响下，雨果是十分拥护波旁复辟王朝的正统派分子，他成名也是在这个阶段，而到了复辟王朝末期，他却成为了一个共和派，呼吁打倒王权。可是在1830年七月王朝建立之后，没多久他又改变了立场，成为了拥护奥尔良王室的秩序党，并且接受了国王提供的贵族院议员职位，成为七月王朝的核心拥护者之一。不管他这么做有多少个人理由，但是在很多人看来，他这种做法无疑就是为了高官厚禄抛弃个人立场，是一个“三姓家奴”，而这在重视立场的政界参与者看来是极不可取的行为，于是有很多朋友就此和他交恶，他在文学界和戏剧界的地位也由此而大大受损。而这一时期，他上映的剧本开始不受捧场，几度失败，以至于被人留下了“雨果已经完蛋啦！”的讥嘲。如果没有后面的风云变幻的话，雨果的文学成就和名声恐怕就会到此为止了，成为“文章憎命达”的俗话的一个新注解。可是最后，命运终究还是再作弄了他，1848年法国再度爆发革命，七月王朝倒台了，而这时候雨果重新成为了共和派（又一次改换门庭），拥护共和国。可是路易-波拿巴很快就借着共和国总统的职位篡夺了国家的权力，并且把共和国变成了帝国，拿破仑帝国派开始得势。这一次，雨果终于不再改变立场了，他极端反感波拿巴政权，并且做出了斗争，然后被波拿巴政权流放——他寓居于比利时，并且在帝国政府多次劝诱下仍旧没有选择归国。而就在这一段时间里面，在困窘的生活当中，他写出了像《悲惨世界》这样的经典作品，并且真正奠定了自己的地位。而这一次他在政治上的坚持，也终于抹清了之前在政治上的种种污点，成为了被法国人广泛认可的人（几乎每个法国人都是天生的反政府主义者，不管台上的是谁）。雨果的作品，激情澎湃，然而同托尔斯泰一样，他喜欢把自己的作品当成自己人生观和世界观的宣讲台，所以会有大段大段的分析和解说（看过《战争与和平》和《悲惨世界》的人肯定会印象很深），有的时候这些评论会让人觉得失之乏味，有的时候却能让人产生共鸣，不管怎么样，雨果对文字的驾驭是超出于常人的，也是值得敬佩的。同时，雨果也是当时欧洲著名作家当中，唯一一个大声疾呼反对欧洲入侵中国的人，更加控诉英法联军抢掠烧毁圆明园的罪行——他这么做，并不仅仅是出于对中国或者中国人的热爱，而是出于朴素的人本主义思想，他认为欧洲人跑到中国去屠杀劫掠，本身就是一种非道义的行为，他的这种表态，也让身为中国人的我对他更加多了一分好感。这种思想，在他的《九三年》这部作品当中也有所体现，他认为“在一切政治纷争当中，唯有人道主义是超越一切的，也是不应该被违背的”。即使看上去这么说有些天真，但是我依旧认为世界上多一些这样的人将会是好事。所以，纵观他的一生，即使有种种问题，雨果依旧是个伟大的文学家，一个至少忠诚于自己的人道主义信念的人，在时间的长河当中，政治上的纷争、国王皇帝们的辉煌、个人的荣辱都会随风而逝，但是他的作品将会永存，他的激情也将会跨越时间的障壁，感染每一个用心去研读他的人。他是一个因为有凡人的种种毛病而并没有显得超脱尘世的人、一个因天才而注定被人永远铭记的人。【写了这么多想来也不会有多少人看，就当是自娱自乐吧。】展开阅读全文赞同 118569 条评论分享收藏喜欢知乎用户 8 人赞同了该回答雨果的五大长篇都读全了。法国最宏伟的作家。何谓“宏伟”呢？这就像你可以说苏州园林很美，云南竹楼很有特点，某些古城很悠远，上海的大楼很魔幻……但是你不能否认中国最“”宏伟“”的建筑物，是天安门和长城。雨果之于法国文学，就像天安门或者长城，之于中国的其他建筑。发布于 2019-08-11 04:31 赞同 83 条评论分享收藏喜欢轩允善说风凉话关注雨果，容我想想。最高票数已经谈了雨果在世俗生活层面的种种切切，我且谈谈雨果的作品。分割线将瓦莱里的几句话作为文献会言之中肯的：“他的作品中没有一首非诗的诗篇，也未有一处形式上的错误……他把词汇的整个领域都踏遍了，他尝试各种文体。” 有关对雨果的评论，我无法援引更多，雨果长期以来被视为“公共建筑”，而后我将会谈到。至今，我尚未在别处谋到对其小说的偏爱的评论（后世的严肃的文学评论）。雨果在法国民族的地位是“法兰西的莎士比亚”，这有可能归功他的政治责任感，雨果当过官，官拜参议员，但文学家参与政治多以失败而告终，在参与革命期间，他被小拿破仑赶出巴黎。而雨果死后安葬于先贤祠，他的葬仪规模超过了巴尔扎克，司汤达，福楼拜。身后哀荣有点近似中国的鲁迅，得利的统治阶级迫切地需要他。我重读了雨果的散文，发现其中多半与政治事件挂钩（政治狂飙之徒）。《拿破仑的葬礼》仅是其中之一。他很喜欢单纯描绘见闻，似乎只做呈现，不做主见，敏锐而讽刺，我承认我更喜欢雨果如诗般的对印象的记录。相比瓦莱里来说，他的政论就不那么具有魔力。我们熟悉他的《巴黎圣母院》及《悲惨世界》等小说，他是中学生必读物（真正去读的人不会很多）。他的小说可视作讽喻小说。美，丑，善，恶，在他笔下的人物上，界限清晰。尼采在讨伐瓦格纳时，曾将雨果与其类比，抱怨他们是“拯救”性质的艺术家，抱怨他们过分参与革命，带有“帝王派气”（站在某一政治革命立场）。雨果与德拉克洛瓦的英雄主义气质，也许都令尼采不舒服。波德莱尔著有《我看德拉克洛瓦》，他将德拉克洛瓦喻为一面镜子，我会抽空推送波德莱尔对绘画的评论集。他甚至准确地预知了马奈。波德莱尔，雨果，德加均毕业于巴黎路易大帝中学。在少年时就一鸣惊人，十五岁就收到法兰西院士的赞扬。他有着和他的年龄不相称的少年老成。纵观其一生，雨果的创作周期异乎寻常的漫长，从青年时期一直到垂暮，他不断在超越自己。大部分是诗人在青年时即夭折，或者苍白无力的吃青年本。这是雨果的一大现象。浪漫派领袖，《克伦威尔》序言发表，就是对古典派的宣战。据说雨果死后50年还在被人争论，也说明了雨果的前程广大。雨果塔的稳固仰仗的是一种语言。关于哲学 “雨果的哲学思想潜藏在每一部作品背后，在大文学家那里，哲学是同文学混合的。”这是史学家的原话，但纪德却不认定雨果有思想。他在别人问雨果是否有杰作时说：我承认对东方集有所偏爱，它是以“色彩”和“音响”著称的诗集。”只字不提雨果的思想，只强调雨果是语言大师。在生活和作品中表现出非常好的仁慈心。他是某种意义上的“自由思想者，比任何都更确信，天主是一个传说。雨果，真的把所有词汇领域全踏遍了，他尝试过各种文体，从戏剧到小说，颂歌到讽刺、评论、雄辩。这个高度，从来没有被掌握或实施过。而他也许到了一种近乎滥用的程度。冗长，是他的一大问题。罗兰·巴特在谈到，夏布里昂多与雨果的时候说：“唯独雨果以其丰厚，得以向古典主义写作施压并使之达到一种分裂的前夜。” 诗歌：《秋叶集》、《暮歌集》、《心声集》、《光影集》都是早期抒情诗。《暮歌集》有政治色彩，《心声集》哲理意味。之后搁笔十年，流亡时作《惩罚集》是讽刺诗，运用一切能够用的诗歌形式，或揭露，或痛斥、嬉笑，怒而不骂冷嘲热讽。，之后《静观集》转向沉思凝郁，更上一层楼，艺术成就高于前面几部抒情诗。《上帝集》和《撒旦的结局》想象恣肆，意象诡谲。《林园集》雨果已是花甲老人。《凶年集》，乔治桑对雨果从敬佩转为不满。《历代传说》神话、史诗。早期把诗歌变成人民斗争的武器，波德莱尔写完《恶之花》雨果大赞：“您创造了新的战栗。”19世纪天才都是大大方方的互相赞赏。不像中国现在，就怕称别人为天才以后自己没位置了。雨果的绘画：写信给波德莱尔，自称是个人消遣娱乐，超现实主义画家们在看到雨果的作品时，惊呼“先于兰波……探求自己的生活潜意识。”雨果的画是一系列幻视的图景，二十世纪被主流绘画界认作是先辈。雨果的画先于印象派，但已有了抽象和超现实的元素。小说：《巴黎圣母院》大家最熟悉，里面那个敲钟人的人格是模糊的。情节就不介绍，可以私下去看看。《悲惨世界》，把小说融入到史诗中去。美国哲学家安兰德说：“浪漫主义的最高目的：映射道德价值观。”道德诸如美与善，对应丑与恶，正义与非正义，从腐朽到生命，有很鲜明的对照。讲白点，在绘画上也有类似的例子，比如西班牙的柯勒惠支，戈雅。这种道德价值观的投射，是二十世纪小说家所不取的。悲惨世界共五部，一千七百多页 19世纪应当说是小说的世纪。小说被提升到拥有和诗歌一样的地位。戏剧赋予人物性格，谈戏剧必谈到性格。基督教要人做到的是抛弃性格，它反对性格，向每个人散播同样的理想。试想一下，一个完美的非性格化的基督徒站在舞台上，他还有什么话要讲？（沉默就好了）中年流亡，晚年雨果成了公众的偶像，获得诗人荣誉，政治地位，像一座高大上的公共建筑。佩吉充满热情的赞赏他，克洛岱尔和纪德对雨果却提出严厉的保留。特别是在心理学方面。“雨果是最有分量的意象之集大成者，音响和节奏操纵自如，是我们法兰西文学中句法和法语形式最好的大师。”用最高级的形容词达到贬人的目的，这是纪德的水平。参与政治的雨果，当过贵族院议员，制宪会议议员，立法会议议员。像雨果这样颇有治国抱负的文学家，类似的，像德国的歌德。纯个人手写，未完，待修改展开阅读全文赞同 10111 条评论分享收藏喜欢查看剩余 32 条回答写回答 4 个回答被折叠（为什么？）下载知乎客户端与世界分享知识、经验和见解相关问题雨果到底是怎么样的人? 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12391	https://math.stackexchange.com/questions/1380300/find-the-area-of-a-triangle-given-the-radius-of-its-incircle-and-a-tangential-po	geometry - Find the area of a triangle given the radius of its incircle and a tangential point - 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 Find the area of a triangle given the radius of its incircle and a tangential point Ask Question Asked 10 years, 2 months ago Modified10 years, 2 months ago Viewed 2k times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. A friend gave me recently the following interesting problem and I would like to share a couple of solutions. Any additional contributions are welcome. A triangle △A B C△A B C is given and we know the radius r r of its incircle(O,r)(O,r). Let D D be the tangential point of the incircle on A C A C which partitions A C A C into A D=α A D=α and D C=β D C=β. Determine the area of △A B C△A B C as a function of α α, β β and r r. geometry trigonometry euclidean-geometry Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Jul 31, 2015 at 18:27 Pantelis SopasakisPantelis Sopasakis asked Jul 31, 2015 at 14:48 Pantelis SopasakisPantelis Sopasakis 5,409 32 32 silver badges 66 66 bronze badges 7 Have you got B B and C C swapped around in your diagram?Rob Arthan –Rob Arthan 2015-07-31 14:54:17 +00:00 Commented Jul 31, 2015 at 14:54 @DavidQuinn The area of A B C A B C is S=r τ S=r τ where τ τ is the semiperimeter of A B C A B C, i.e., τ=(A B+B C+C A)/2 τ=(A B+B C+C A)/2. I updated my answer with the complete formula of S S.Pantelis Sopasakis –Pantelis Sopasakis 2015-07-31 15:04:29 +00:00 Commented Jul 31, 2015 at 15:04 @RobArthan Yes, there was indeed a typo in the question. Thanks for pointing this out.Pantelis Sopasakis –Pantelis Sopasakis 2015-07-31 15:07:59 +00:00 Commented Jul 31, 2015 at 15:07 @Sawarnik What are a a and b b? It should be a=s−α=β+B E a=s−α=β+B E and b=s−β=α+B E b=s−β=α+B E but I can't see how this helps.Pantelis Sopasakis –Pantelis Sopasakis 2015-07-31 15:12:48 +00:00 Commented Jul 31, 2015 at 15:12 cot C=cot 2 C 2−1 2 cot C 2=(s−c)2−r 2 r 2 2 s−c r=β 2−r 2 2 r β cot⁡C=cot 2⁡C 2−1 2 cot⁡C 2=(s−c)2−r 2 r 2 2 s−c r=β 2−r 2 2 r β Similarly, we can get cot B cot⁡B, and use this formula, and done: Δ=(α+β)2 2(cot B+cot C)Δ=(α+β)2 2(cot⁡B+cot⁡C) Sawarnik –Sawarnik 2015-07-31 17:34:39 +00:00 Commented Jul 31, 2015 at 17:34 \|Show 2 more comments 4 Answers 4 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. Let s,S,R s,S,R be semiperimeter, area and B B-exradius of triangle A B C A B C respectively. It is well-known that r R=α β r R=α β and r s=S=R(s−α−β)r s=S=R(s−α−β) (see below). From the latter we get s=R(α+β)R−r s=R(α+β)R−r which leads to S=r s=r R(α+β)R−r=r α β(α+β)r R−r 2=r α β(α+β)α β−r 2.S=r s=r R(α+β)R−r=r α β(α+β)r R−r 2=r α β(α+β)α β−r 2. ~~~~~~~~~~~~~~~~~~ For a proof that r R=α β r R=α β denote the B B-excenter by J J. Let X X be a point of tangency of B B-excircle with side A C A C. It is well-known that A X=C D A X=C D. Note that triangles J A X J A X and C O D C O D are similar as their angles are equal. Thus R β=J X A X=C D O D=α r R β=J X A X=C D O D=α r yielding r R=α β r R=α β. Now we'll prove that S=R(s−α−β)S=R(s−α−β). We'll use a notation [F][F] which denotes area of F F. We have S=[A B C]=[A B J]+[C B J]−[B C J]=A B⋅R 2+B C⋅R 2−A C⋅R 2=A B+B C−A C 2⋅R=(A B+B C+A C 2−A C)⋅R=(s−(α+β))⋅R=R(s−α−β)S=[A B C]=[A B J]+[C B J]−[B C J]=A B⋅R 2+B C⋅R 2−A C⋅R 2=A B+B C−A C 2⋅R=(A B+B C+A C 2−A C)⋅R=(s−(α+β))⋅R=R(s−α−β) Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Aug 1, 2015 at 11:09 answered Jul 31, 2015 at 16:37 timon92timon92 12.4k 1 1 gold badge 23 23 silver badges 44 44 bronze badges 2 Could you provide some references about r R=α β r R=α β and S=R(x−α−β)S=R(x−α−β) (which means that S=R⋅B E S=R⋅B E)?Pantelis Sopasakis –Pantelis Sopasakis 2015-07-31 17:52:43 +00:00 Commented Jul 31, 2015 at 17:52 these are easy facts, I'll sketch proofs of them in a sec timon92 –timon92 2015-07-31 19:37:09 +00:00 Commented Jul 31, 2015 at 19:37 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. Solution 1: Draw A O A O, A O A O and C O C O. We know that these are the bisectors of ∠B A C∠B A C, ∠A B C∠A B C and ∠B C A∠B C A respectively. Let ∠O A C=1 2∠B A C=:ϕ A∠O A C=1 2∠B A C=:ϕ A and ∠O B E=1 2∠A B C=:ϕ B∠O B E=1 2∠A B C=:ϕ B and ∠O C A=1 2∠B C A=:ϕ C∠O C A=1 2∠B C A=:ϕ C. We have that ∠O D A∠O D A is a right angle (and so is ∠O D C∠O D C). Thus tan ϕ A=r α tan⁡ϕ A=r α and tan ϕ C=r β tan⁡ϕ C=r β. Therefore ∠A O D=π 2−ϕ A∠A O D=π 2−ϕ A. The two right-angled triangles △A D O△A D O and △A E O△A E O are equal (indeed, they share a common side A O A O and O E=O D=r O E=O D=r). Thus, ∠E O A=∠D O A=π 2−ϕ A∠E O A=∠D O A=π 2−ϕ A. Similarly, ∠C O D=π 2−ϕ C∠C O D=π 2−ϕ C and since △C O D=△C O F△C O D=△C O F it is ∠C O F=∠C O D=π 2−ϕ C∠C O F=∠C O D=π 2−ϕ C. Using the fact that ∠A O D+∠D O C+∠C O F+∠F O E+∠E O A=2 π,2(π 2−ϕ A)+2(π 2−ϕ C)+∠F O E=2 π,∠F O E=2(ϕ A+ϕ C).∠A O D+∠D O C+∠C O F+∠F O E+∠E O A=2 π,2(π 2−ϕ A)+2(π 2−ϕ C)+∠F O E=2 π,∠F O E=2(ϕ A+ϕ C). Because of the equality of △B E O△B E O and △B F O△B F O we have ∠E O B=∠F O E 2=ϕ A+ϕ C∠E O B=∠F O E 2=ϕ A+ϕ C, so tan∠E O B=B E r,B E=r tan(ϕ A+ϕ C)=r tan ϕ A+tan ϕ C 1−tan ϕ A tan ϕ C==r r α+r β 1−r α r β=r r(α+β)α β α β−r 2 α β=r 2 α+β α β−r 2 tan⁡∠E O B=B E r,B E=r tan⁡(ϕ A+ϕ C)=r tan⁡ϕ A+tan⁡ϕ C 1−tan⁡ϕ A tan⁡ϕ C==r r α+r β 1−r α r β=r r(α+β)α β α β−r 2 α β=r 2 α+β α β−r 2 From the aforementioned triangle equalities we have A E=α A E=α, C F=β C F=β and B F=B E B F=B E, therefore, the semiperimeter of △A B C△A B C is τ=α+β+B E=α+β+r 2 α+β α β−r 2=α β(α+β)α β−r 2,τ=α+β+B E=α+β+r 2 α+β α β−r 2=α β(α+β)α β−r 2, and the area of the triangle is S=r τ=r α β(α+β)α β−r 2 S=r τ=r α β(α+β)α β−r 2 Solution 2: This is a solution without trigonometry and is less elegant. Since ∠B O E=ϕ A+ϕ C∠B O E=ϕ A+ϕ C there is a point M M on B E B E so that ∠M O E=ϕ A∠M O E=ϕ A and ∠M O B=ϕ C∠M O B=ϕ C as in the following figure: It is easy to see that the two triangles △M E O△M E O and △O E A△O E A are similar (they have all their angles equal to one another), so, E M E O=E O E A,E M=r 2 α.E M E O=E O E A,E M=r 2 α. We now need to determine B M B M. What is the same, △B M O△B M O and △B O C△B O C are similar (all angles equal to one another). Note that (Pythagorean theorem on △B E O△B E O) B O=B E 2+r 2−−−−−−−−√=(B M+M E)2+r 2−−−−−−−−−−−−−−−√,B O=B E 2+r 2=(B M+M E)2+r 2, and (Pythagorean theorem on △O C D△O C D) O C=r 2+β 2−−−−−−√.O C=r 2+β 2. We also have M O=r 2+M E 2−−−−−−−−√=r 2+r 4 α 2−−−−−−−√=r 1+(r/a)2−−−−−−−−√M O=r 2+M E 2=r 2+r 4 α 2=r 1+(r/a)2 Using the fact that △B M O△B M O and △B O C△B O C are similar (all angles equal to one another), we arrive at B M B O=M O O C,B M(B M+r 2 α 2)2+r 2−−−−−−−−−−−−−−√=r 1+r 2 α 2−−−−−−√r 2+β 2−−−−−−√,B M 2(B M+r 2 α 2)2+r 2=r 2(1+r 2 α 2)r 2+β 2,B M B O=M O O C,B M(B M+r 2 α 2)2+r 2=r 1+r 2 α 2 r 2+β 2,B M 2(B M+r 2 α 2)2+r 2=r 2(1+r 2 α 2)r 2+β 2, which is a quadratic equation one can solve to derive a formula for B M B M (the non-positive solution should be discarded) in terms of r r, α α and β β. Having determined B M B M and M E M E we can easily determine the semiperimeter of △A B C△A B C, τ=α+β+B M+M E τ=α+β+B M+M E and the area of △A B C△A B C is again S=r τ S=r τ. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Jul 31, 2015 at 17:40 answered Jul 31, 2015 at 14:48 Pantelis SopasakisPantelis Sopasakis 5,409 32 32 silver badges 66 66 bronze badges 2 Have you checked your answer against any examples? For an equilateral with sides of length 2 3–√2 3, you have r=1 r=1 and α=β=3–√α=β=3 and the area should come out as 3 3–√3 3.Rob Arthan –Rob Arthan 2015-07-31 16:00:58 +00:00 Commented Jul 31, 2015 at 16:00 @RobArthan I have made some modifications in my answer and wrote S S in a simpler form which agrees with the answers of other contributors.Pantelis Sopasakis –Pantelis Sopasakis 2015-07-31 17:41:46 +00:00 Commented Jul 31, 2015 at 17:41 Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. If I I is the incenter of A B C A B C and I A,I B,I C I A,I B,I C are the projections of the incenter on the sides B C,A C,A B B C,A C,A B (sorry, but to call the incenter O O gives me some itch), we have: A I B=A I C=b+c 2 B I A=B I C=a+c 2,C I A=C I B=a+b 2 A I B=A I C=b+c 2 B I A=B I C=a+c 2,C I A=C I B=a+b 2 and 2 Δ=r(a+b+c)2 Δ=r(a+b+c), or 2 Δ=r(A I B+B I C+C I A)2 Δ=r(A I B+B I C+C I A), so we just need to find B I C B I C in terms of r,A I B,C I B r,A I B,C I B. Pretty easy: B I C=r tan B ˆ 2=r tan A ˆ+C ˆ 2=r⋅tan A ˆ 2+tan C ˆ 2 1−tan A ˆ 2 tan C ˆ 2 B I C=r tan⁡B^2=r tan⁡A^+C^2=r⋅tan⁡A^2+tan⁡C^2 1−tan⁡A^2 tan⁡C^2 hence: B I C=r 2⋅A I B+C I B A I B⋅C I B−r 2 B I C=r 2⋅A I B+C I B A I B⋅C I B−r 2 and: 2 Δ=(A I B+C I B)A I B C I B r A I B C I B−r 2.2 Δ=(A I B+C I B)A I B C I B r A I B C I B−r 2. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Jul 31, 2015 at 15:36 Jack D'AurizioJack D'Aurizio 372k 42 42 gold badges 419 419 silver badges 886 886 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. I'll use the more-traditional notation with tangent points D D, E E, F F opposite respective vertices A A, B B, C C. I'll also take α α, β β, γ γ (the first two given, the last unknown) to be the lengths of the tangent segments from vertices A A, B B, C C. Finally, I'll reduce some notational clutter by writing A 2 A 2, B 2 B 2, C 2 C 2 for half-angles A/2 A/2, B/2 B/2, C/2 C/2. (So, A 2+B 2+C 2=90∘A 2+B 2+C 2=90∘.) Since α+β+γ α+β+γ is the semi-perimeter, we know the area of the triangle is given by \|△A B C\|=r(α+β+γ)(⋆)(⋆)\|△A B C\|=r(α+β+γ) Our task is to replace γ γ. Clearly, r=α tan A 2=β tan B 2=γ tan C 2 r=α tan⁡A 2=β tan⁡B 2=γ tan⁡C 2 But, tan C 2=tan(90∘−A 2−B 2)=cot(A 2+B 2)=cot A 2 cot B 2−1 cot A 2+cot B 2=α r β r−1 α r+β r=α β−r 2 r(α+β)tan⁡C 2=tan⁡(90∘−A 2−B 2)=cot⁡(A 2+B 2)=cot⁡A 2 cot⁡B 2−1 cot⁡A 2+cot⁡B 2=α r β r−1 α r+β r=α β−r 2 r(α+β) Therefore, γ=r tan C 2=r 2(α+β)α β−r 2 γ=r tan⁡C 2=r 2(α+β)α β−r 2 so that, by (⋆)(⋆), \|△A B C\|=r(α+β+r 2(α+β)α β−r 2)=r α β(α+β)α β−r 2\|△A B C\|=r(α+β+r 2(α+β)α β−r 2)=r α β(α+β)α β−r 2 Observation: Recall that the circle with diameter A B¯¯¯¯¯¯¯¯A B¯ meets the perpendicular at F F in points P P and Q Q such that \|A P¯¯¯¯¯¯¯¯\|2=\|A Q¯¯¯¯¯¯¯¯\|2=\|A F¯¯¯¯¯¯¯¯\|\|B F¯¯¯¯¯¯¯¯\|=α β\|A P¯\|2=\|A Q¯\|2=\|A F¯\|\|B F¯\|=α β. This says exactly that the "power" of point F F with respect to that circle is α β α β. Moreover, since that perpendicular passes through I I (which, we'll say, lies between F F and P P), we have \|I P¯¯¯¯¯¯\|\|I Q¯¯¯¯¯¯¯\|=(\|A P¯¯¯¯¯¯¯¯\|−\|A I¯¯¯¯¯¯\|)(\|A Q¯¯¯¯¯¯¯¯\|+\|A I¯¯¯¯¯¯\|)=(α β−−−√−r)(α β−−−√+r)=α β−r 2\|I P¯\|\|I Q¯\|=(\|A P¯\|−\|A I¯\|)(\|A Q¯\|+\|A I¯\|)=(α β−r)(α β+r)=α β−r 2 Thus, the denominator of the area formula is the power of I I with respect to that circle. This seems significant ... or coincidental. (It may also be worth noting that that circle contains the feet of the altitudes from A A and B B.) Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Jul 31, 2015 at 17:18 answered Jul 31, 2015 at 15:43 BlueBlue 84.4k 15 15 gold badges 128 128 silver badges 266 266 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 geometry trigonometry euclidean-geometry 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 3Triangle and Incircle 0Find the maximum radius of a incircle? 1Relation between the radius and the area of tangential polygon 2Finding the area of a triangle, given the distance between center of incircle and circumscribed circle 0The incircle of a triangle A B C A B C touches A B A B at point P P and has radius equal to 21 21.If A P=23 A P=23 and P B=27 P B=27 1Finding the angles of a right triangle if area of triangle area of incircle=2 3√+3 π area of triangle area of incircle=2 3+3 π 0Two overlapping triangles △A B C△A B C, △D B E△D B E. 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12392	https://wiingy.com/learn/ap-statistics/standard-deviation/	Learn Subjects Music & Languages Find a tutor Singing Guitar Piano Violin Music Spanish English French Hindi Yoga Home Learn AP-STATISTICS How to Calculate Standard Deviation? AP-STATISTICS How to Calculate Standard Deviation? ByShifa Alion Feb 25, 2023 Updated Mar 18, 2025 In this article How to calculate standard deviation? Examples of standard deviation in action Advantages and disadvantages of standard deviation Solved examples Conclusion Frequently asked questions (FAQs) References Looking for a private tutor? Choose your tutor from 40+ subjects Find your private tutor SingingGuitarPianoSpanishEnglishMathBiologyChemistryOther Standard deviation is a measure of how dispersed the data is in relation to the mean. Standard deviation is important because it helps in understanding the measurements when the data is distributed. The more the data is distributed, the greater will be the standard deviation of that data. A low standard deviation means data are clustered around the mean, and a high standard deviation indicates data are more spread out. The standard deviation is fixed and well-defined for a set of data and hence in analysis, it helps to predict performance trends because the level of dispersion would indicate the defined amount of variation or we can say deviation from the normal or mean value. Looking for last minute help for your AP Statistics exam in May? Find an expert 1-on-1 onlineAP Statistics tutorfrom Wiingy and give your exam prep a boost! How to calculate standard deviation? Let us see the step-by-step calculation of the standard deviation Step 1:Find the mean of the data; The mean is calculated by adding all the data points and dividing the obtained total by the number of data points. The mean for grouped data can be calculated by multiplying each data point with its respective frequency and then finding the sum of all of them and dividing the sum by the total number by the data points. Step 2:For each data point we now find the square of the difference from the calculated mean. Step 3:Now sum all the values of squares obtained in step 2. Step 4: now divide the obtained sum by the number of data points. Step 5:Take the square root of the value obtained in step 4 and we get the standard deviation of a given data set. The formula for the standard deviation Standard deviation = The relation between the statistical unit called variance and the standard deviation is Variance which is equal to the square of the standard deviation of a data set. The variables in the above equation are as follows: (1) σ is the standard deviation. (2) ∑ is the summation of the squared terms. (3) x is the data point. (4) µ is the mean of the data set. (5) N is the number of data points in the set. Examples of standard deviation in action Calculating standard deviation for a small data set Example 1:Find the standard deviation for the given data set. 4 18 45 9 30 14 50 37 23 30 Solution 1: The mean of the data set is Which gives us the mean as 26. Now following the steps, we need to find the sum of the squares of the difference of the data points with the mean. Calculating the above we get Which is equal to 2080. Now we have to divide the above obtained sum by the number of data points. Which gives us 208. Now we got to take the square root of the obtained value. We get the standard deviation to be 14.42 Using standard deviation to compare data sets. Example 2: Use standard deviation to compare the given sets of data. Set 1: 45 68 17 34 16 Set 2: 23 20 47 73 25 Solution 2: Following the steps, we need to find the mean For set 1: The mean for set 1 is 36. For set 2: The mean for set 2 is 37.6. We need to find the sum of the squares of the difference of the data points with the mean. For set 1: For set 2: Now we have to divide the obtained sum of squares by their respective number of data points. We get For set 1: For set 2: Now to obtain the standard deviation we need to take the square root of the above-calculated values For set 1: For set 2: Hence the standard deviation for set 1 is 19.33 and set 2 is 20.11 respectively. We can observe from the obtained values the standard deviation of set 2 is larger than the standard deviation of set 1. This indicates that the values in data set one are more varied compared to set two. Advantages and disadvantages of standard deviation The advantages of standard deviation The standard deviation is always fixed and well-defined. It is very sensitive to changes in the data It is based on all the data points in the set. It is less affected by the sampling fluctuation. Because of it being defined, it can be used for the analysis of huge amounts of varied data. The disadvantages of standard deviation Outliers will add a huge value to the numerator when the differences are squared since squaring large values makes them even larger. The standard deviation, therefore, gives extreme values greater weight. As a result, the standard deviation is susceptible to the impact of outliers. Standard deviation assumes a normal distribution, so it may not be appropriate for data sets that are not normally distributed. Solved examples Q 1. A test is conducted for a class of 5 students and the scores out of 10 are as follows. Solution 1: 5. 4 7 10 8 6 Find the standard deviation of the test result. First, we find the mean of the data set. Now we find the sum of the square of the difference of each data point from the mean. The next step is to divide the obtained sum by the total number of data points. Now we take the square root to get the standard deviation which is 2. 2. The average salaries of people working in different fields. Calculate the standard deviation, then interpret what the standard deviation means in terms of each field. Marketing Education Banking Technology Mean salary 60,000 45,000 75,000 15,000 Variance 900,000,000 25,000,000 100,000,000 16,000,000 Solution 2: To find the standard deviation we just need to find the square root of the variance. Therefore the standard deviation of each of the work fields is as follows. Marketing Education Banking Technology Standard deviation 30000 5000 10000 4000 3. Find the mean deviation when the data points and their respective frequencies are given. Xi 10 30 50 70 90 Fi 4 24 28 16 8 Solution 3: First, we need to find the product of XiFi XiFi 40 720 1400 1120 720 Now we find the sum of Fi which is 80. And then find the sum of XiFi is 4000. Now we find the mean of the data points The next step is to find the mod difference of each data point from the mean. \|Xi-50\| 40 20 0 20 40 Now we multiply the respective frequencies with the mod difference. Fi\|Xi-50\| 160 480 0 320 320 Now we need the sum of the calculated Fi\|Xi-50\| which is 1280. Mean(x)= Mean deviation about the mean= 4. Find the standard deviation of the following data and round off to the nearest two decimals. x 1 2 3 4 5 f 3 11 4 9 2 Solution 4: First, we find the square of the data points 1 4 9 16 25 Now we need to calculate the product of f and x fx 3 22 12 36 10 The next step is to calculate the product of f and f 3 44 36 144 50 Finding the sum of f = Finding the sum of f= now we have to find the mean of the data set Which is Now to calculate the standard deviation we use the below equation We get Which is equal to 1.17 Hence the standard deviation of the set is 1.17. Looking for last minute help for your AP Statistics exam in May? Find an expert 1-on-1 onlineAP Statistics tutorfrom Wiingy and give your exam prep a boost! Conclusion Standard deviation is a powerful tool to study the various characteristics of given data, albeit it has some drawbacks particularly when it comes to more advanced situations such as Machine Learning and Regression Analysis. Standard deviation is a sensitive and well-proven tool to understand the behavior of data, however, it is plagued majorly by outliers which can be very frequently seen in real-world data. Hence, filtering said outliers, which can be an arduous task, is essential to apply Standard Deviation in a conceptually sound manner. Frequently asked questions (FAQs) What is a standard error? The Standard error shows how closely any given sample of a population’s mean will likely be to the actual population mean. Any given mean is more likely to be a subpar representation of the true population means as the standard error increases, suggesting that the means are more equally spread. What is the difference between variance and standard deviation? The term “variance” refers to the average squared deviations from the mean, whereas the term “standard deviation” is determined by taking the square root of this number. Despite the fact that both metrics show distributional variability, their units are different. What does standard deviation indicate in normal distribution? A higher standard deviation in normal distributions denotes that the values are further from the mean. A reduced standard deviation indicates that the values are closely clustered around the arithmetic mean value. What is the best measure of dispersion? Standard deviation is the best measure of dispersion followed by variance. Why is standard deviation the best measure of dispersion? It contains information for the entire series because it depends on all values. As a result, the standard deviation can be affected by even a minor change in one variable. References Lee, D. K., In, J., & Lee, S. (2015). Standard deviation and standard error of the mean. Korean journal of anesthesiology, 68(3), 220-223. Altman, D. G., & Bland, J. M. (2005). Standard deviations and standard errors. Bmj, 331(7521), 903. Reviewed by Wiingy Mar 18, 2025 Was this helpful? You might also like ### How to Find Z-Score? Feb 24, 2023 ### Residuals and Correlation Feb 03, 2025 ### Introduction to Random Variables Feb 03, 2025 ### Chi-Square Test Feb 03, 2025 Explore more topics ### Python Tutorials ### C++ Tutorials ### R Studio Tutorials ### Solidworks Tutorials ### AP Statistics Tutorials Wiingy Help Centre About Us Contact Us Newsroom Student Blogs Resources Learn Dashboard Tutor Become a Tutor Online Tutoring Jobs Our Tutors Online Tutors Contact us Live Chat . 2 min Whatsapp . 2 min +1 6184485208 8 The Green, Dover, Delaware, US, 19901 © 2025 Wiingy, Inc • 1. Privacy 2. • 3. Terms 4. • 5. Sitemap English (US) English (CA) English (AU) English (IN) English (US) Standard Deviation: Definition, Calculation, and Solved Examples - Wiingy ===============
12393	https://physics.stackexchange.com/questions/661127/finding-final-equilibrium-temperature	thermodynamics - Finding final equilibrium temperature? - Physics Stack Exchange Join Physics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Physics helpchat Physics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Finding final equilibrium temperature? Ask Question Asked 4 years, 1 month ago Modified4 years, 1 month ago Viewed 147 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. \begingroup "A 100g mass is heated with a strong flame for 10 seconds then placed into a crucible on a heat proof mat. After cooling for 5 minutes the 100g mass is at 67.6C while the table it's resting on is at 24.2C. What will be the final temperature of the mass?" Our teacher assigned us this problem to work on and its one I don't think is possible to solve without some extra information such as the heat capacities of both of the masses. I tried adding both temps together and divided them to get the final temperature/average but I don't think that approach was going to be accurate. It takes different amounts of joules to increase the temperature of dissimilar materials, meaning if one loses a certain amount of joules the other will increase in temp but not by the same amount as the other substance(correct me if I'm wrong). This is why I think the problem might need more information but when confronting my teacher with this she just said this in reply "Basically, the final temp of the mass will end up being the same temperature of the room. It will heat the room by a tiny, tiny, tiny quantity, but essentially end up at room temp." Am I missing something here? This is my first thermodynamics orientated class so I'm not well versed. Thank you in advance. Edit: here is the other information we are given: The initial temp is 25.9C and the sample was at 180.2C after being heat up. The flame temp 190.5C. Conduction, convection, radiation are all involved. thermodynamics temperature equilibrium Share Cite Improve this question Follow Follow this question to receive notifications edited Aug 24, 2021 at 2:49 import_hillimport_hill asked Aug 24, 2021 at 2:34 import_hillimport_hill 329 1 1 gold badge 2 2 silver badges 9 9 bronze badges \endgroup 6 \begingroup You don't know the initial temperature of the sample, and you don't know what form of heat transfer is involved (e.g., conduction, convection, radiation). A bunsen burner can heat the sample up to 1083 deg C, but the problem didn't tell you what type of flame was used. You need more information.\endgroup David White –David White 2021-08-24 02:45:34 +00:00 Commented Aug 24, 2021 at 2:45 \begingroup Sorry, that info is given I just didn't include it. The initial temp is 25.9C and the sample was at 180.2C after being heat up. Conduction, convection, radiation are all involved. Could I solve the problem with this info?\endgroup import_hill –import_hill 2021-08-24 02:49:10 +00:00 Commented Aug 24, 2021 at 2:49 \begingroup The initial temperature of what? The sample before it is heated up? Also note - conduction requires a thermal conductivity. Convection can be natural convection or forced convection, and is VERY dependent on the exact physical setup. Radiation is proportional to the forth power of the temperatures involved. You still need a LOT more data.\endgroup David White –David White 2021-08-24 02:55:25 +00:00 Commented Aug 24, 2021 at 2:55 \begingroup The sample before it is heated up. Conduction because the object is metal but we aren't given what type of metal. Convection from the bunsen burner. Radiation meaning the mass emits radiation after being heat up and dissipating into the environment. But why would you even need the type of heat transfer to calculate final temp, we already know it heat up to 180.2?\endgroup import_hill –import_hill 2021-08-24 02:59:47 +00:00 Commented Aug 24, 2021 at 2:59 \begingroup Are. you asked to find the final temperature, or the temperature after a specified amount of time?\endgroup Chet Miller –Chet Miller 2021-08-24 03:08:39 +00:00 Commented Aug 24, 2021 at 3:08 \|Show 1 more comment 1 Answer 1 Sorted by: Reset to default This answer is useful 0 Save this answer. Show activity on this post. \begingroup the crucial information is "After cooling for 5 minutes the 100g mass is at 67.6C while the table it's resting on is at 24.2C" since the table did noch change temperature in the first 5 minutes, why should it with smaller temperature difference in the next time. So you have to read the question very carefully, to look for the answer. So your teacher was right. Share Cite Improve this answer Follow Follow this answer to receive notifications answered Aug 24, 2021 at 10:52 trulatrula 7,073 1 1 gold badge 10 10 silver badges 13 13 bronze badges \endgroup 3 \begingroup The mass will cool to the temperature of the room, but it may take more than 5 min.\endgroup R.W. Bird –R.W. Bird 2021-08-24 14:44:26 +00:00 Commented Aug 24, 2021 at 14:44 \begingroup@ R-W.Bird: "may take more than 5 minutea" ? It is stated in the text that it takes more time, so what shall this coment say?\endgroup trula –trula 2021-08-24 20:06:33 +00:00 Commented Aug 24, 2021 at 20:06 \begingroup You are right, my mistake.\endgroup R.W. Bird –R.W. Bird 2021-08-25 13:55:39 +00:00 Commented Aug 25, 2021 at 13:55 Add a comment\| Your Answer Thanks for contributing an answer to Physics Stack Exchange! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. 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12394	https://dictionary.cambridge.org/us/dictionary/learner-english/advocate	Definition of advocate – Learner’s Dictionary Your browser doesn't support HTML5 audio Your browser doesn't support HTML5 audio advocacy Your browser doesn't support HTML5 audio Your browser doesn't support HTML5 audio Your browser doesn't support HTML5 audio Your browser doesn't support HTML5 audio advocate noun [C] (SUPPORTER) advocate noun [C] (LAWYER) (Definition of advocate from the Cambridge Learner's Dictionary © Cambridge University Press) Translations of advocate Get a quick, free translation! Browse Word of the Day take something back to admit that something you said was wrong Blog Calm and collected (The language of staying calm in a crisis) New Words vibe coding © Cambridge University Press & Assessment 2025 © Cambridge University Press & Assessment 2025 Learn more with +Plus Learn more with +Plus To add advocate to a word list please sign up or log in. Add advocate to one of your lists below, or create a new one. {{message}} {{message}} Something went wrong. {{message}} {{message}} Something went wrong. {{message}} {{message}} There was a problem sending your report. {{message}} {{message}} There was a problem sending your report.
12395	https://courses.lumenlearning.com/suny-orgbiochemistry/chapter/converting-units/	Chapter 1: Chemistry, Matter, and Measurement 1.7 Converting Units Learning Objective Convert a value reported in one unit to a corresponding value in a different unit. The ability to convert from one unit to another is an important skill. For example, a nurse with 50 mg aspirin tablets who must administer 0.2 g of aspirin to a patient needs to know that 0.2 g equals 200 mg, so 4 tablets are needed. Fortunately, there is a simple way to convert from one unit to another. Conversion Factors If you learned the SI units and prefixes described in Section 1.6 “The International System of Units”, then you know that 1 cm is 1/100th of a meter. [latex]1\text{ cm}=\frac{1}{100\text{ m}}[/latex] or 100 cm = 1 m Suppose we divide both sides of the equation by 1 m (both the number and the unit): [latex]\frac{100\text{ cm}}{1\text{ m}}=\frac{1\text{ m}}{1\text{ m}}[/latex] As long as we perform the same operation on both sides of the equals sign, the expression remains an equality. Look at the right side of the equation; it now has the same quantity in the numerator (the top) as it has in the denominator (the bottom). Any fraction that has the same quantity in the numerator and the denominator has a value of 1: We know that 100 cm is 1 m, so we have the same quantity on the top and the bottom of our fraction, although it is expressed in different units. A fraction that has equivalent quantities in the numerator and the denominator but expressed in different units is called a conversion factor. Here is a simple example. How many centimeters are there in 3.55 m? Perhaps you can determine the answer in your head. If there are 100 cm in every meter, then 3.55 m equals 355 cm. To solve the problem more formally with a conversion factor, we first write the quantity we are given, 3.55 m. Then we multiply this quantity by a conversion factor, which is the same as multiplying it by 1. We can write 1 as 100 cm1 m and multiply: [latex]3.55\text{ m}\times\frac{100\text{ cm}}{1\text{ m}}[/latex] The 3.55 m can be thought of as a fraction with a 1 in the denominator. Because m, the abbreviation for meters, occurs in both the numerator and the denominator of our expression, they cancel out: [latex]\frac{3.55\cancel{\text{ m}}}{1}\times\frac{100\text{ cm}}{1\cancel{\text{ m}}}[/latex] The final step is to perform the calculation that remains once the units have been canceled: [latex]\frac{3.55}{1}\times\frac{100\text{ cm}}{1}=355\text{ cm}[/latex] In the final answer, we omit the 1 in the denominator. Thus, by a more formal procedure, we find that 3.55 m equals 355 cm. A generalized description of this process is as follows: quantity (in old units) × conversion factor = quantity (in new units) You may be wondering why we use a seemingly complicated procedure for a straightforward conversion. In later studies, the conversion problems you will encounter will not always be so simple. If you can master the technique of applying conversion factors, you will be able to solve a large variety of problems. In the previous example, we used the fraction $$\frac{100\text{ cm}}{1\text{ m}}$$ as a conversion factor. Does the conversion factor $$\frac{1\text{ m}}{100\text{ cm}}$$ also equal 1? Yes, it does; it has the same quantity in the numerator as in the denominator (except that they are expressed in different units). Why did we not use that conversion factor? If we had used the second conversion factor, the original unit would not have canceled, and the result would have been meaningless. Here is what we would have gotten: [latex]3.55\text{ m} \times \frac{1\text{ m}}{100\text{ cm}}=0.0355 \frac{\text{m}^2}{\text{cm}}[/latex] For the answer to be meaningful, we have to construct the conversion factor in a form that causes the original unit to cancel out. Figure 1.10 “A Concept Map for Conversions” shows a concept map for constructing a proper conversion. The steps for doing a unit conversion problem are summarized in the margin. Figure 1.10 A Concept Map for Conversions.This is how you construct a conversion factor to convert from one unit to another. Significant Figures in Conversions How do conversion factors affect the determination of significant figures? Numbers in conversion factors based on prefix changes, such as kilograms to grams, are not considered in the determination of significant figures in a calculation because the numbers in such conversion factors are exact. Exact numbersA number that is defined or counted. are defined or counted numbers, not measured numbers, and can be considered as having an infinite number of significant figures. (In other words, 1 kg is exactly 1,000 g, by the definition of kilo-.) Counted numbers are also exact. If there are 16 students in a classroom, the number 16 is exact. In contrast, conversion factors that come from measurements (such as density, as we will see shortly) or are approximations have a limited number of significant figures and should be considered in determining the significant figures of the final answer. Example 12 The average volume of blood in an adult male is 4.7 L. What is this volume in milliliters? A hummingbird can flap its wings once in 18 ms. How many seconds are in 18 ms? Solution Show Answer We start with what we are given, 4.7 L. We want to change the unit from liters to milliliters. There are 1,000 mL in 1 L. From this relationship, we can construct two conversion factors: [latex]\frac{1\text{ L}}{1,000\text{ mL}}\text{ or }\frac{1,000\text{ mL}}{1\text{ L}}[/latex] We use the conversion factor that will cancel out the original unit, liters, and introduce the unit we are converting to, which is milliliters. The conversion factor that does this is the one on the right. [latex]4.7\cancel{\text{ L}} \times \frac{1,000\text{ mL}}{1\cancel{\text{ L}}}=4,700\text{ mL}[/latex] Because the numbers in the conversion factor are exact, we do not consider them when determining the number of significant figures in the final answer. Thus, we report two significant figures in the final answer. We can construct two conversion factors from the relationships between milliseconds and seconds: [latex]\frac{1,000\text{ ms}}{1\text{ s}}\text{ or }\frac{1 \text{s}}{1,000\text{ ms}}[/latex] To convert 18 ms to seconds, we choose the conversion factor that will cancel out milliseconds and introduce seconds. The conversion factor on the right is the appropriate one. We set up the conversion as follows: [latex]18\cancel{\text{ ms}}\times \frac{1\text{ s}}{1,000\cancel{\text{ ms}}}=0.018\text{ s}[/latex] The conversion factor’s numerical values do not affect our determination of the number of significant figures in the final answer. Skill-Building Exercise Perform each conversion 101,000 ns to seconds 32.08 kg to grams Conversion factors can also be constructed for converting between different kinds of units. For example, density can be used to convert between the mass and the volume of a substance. Consider mercury, which is a liquid at room temperature and has a density of 13.6 g/mL. The density tells us that 13.6 g of mercury have a volume of 1 mL. We can write that relationship as follows: 13.6 g mercury = 1 mL mercury This relationship can be used to construct two conversion factors: [latex]\frac{13.6\text{ g}}{1\text{ mL}}\text{ and }\frac{1\text{ mL}}{13.6\text{ g}}[/latex] Which one do we use? It depends, as usual, on the units we need to cancel and introduce. For example, suppose we want to know the mass of 16 mL of mercury. We would use the conversion factor that has milliliters on the bottom (so that the milliliter unit cancels) and grams on top so that our final answer has a unit of mass: [latex]16\cancel{\text{ mL}}\times \frac{13.6\text{ g}}{1\text{ mL}}=217.6\text{ g}=220\text{ g}[/latex] In the last step, we limit our final answer to two significant figures because the volume quantity has only two significant figures; the 1 in the volume unit is considered an exact number, so it does not affect the number of significant figures. The other conversion factor would be useful if we were given a mass and asked to find volume, as the following example illustrates. Note Density can be used as a conversion factor between mass and volume. Example 13 A mercury thermometer for measuring a patient’s temperature contains 0.750 g of mercury. What is the volume of this mass of mercury? Solution Show Answer Because we are starting with grams, we want to use the conversion factor that has grams in the denominator. The gram unit will cancel algebraically, and milliliters will be introduced in the numerator. [latex]0.750\cancel{\text{ g}}\times \frac{1\text{ mL}}{13.6\cancel{\text{ g}}}=0.055147\text{ ... mL}=0.0551\text{ mL}[/latex] We have limited the final answer to three significant figures. Skill-Building Exercise What is the volume of 100.0 g of air if its density is 1.3 g/L? Looking Closer: Density and the Body The densities of many components and products of the body have a bearing on our health. Bones. Bone density is important because bone tissue of lower-than-normal density is mechanically weaker and susceptible to breaking. The density of bone is, in part, related to the amount of calcium in one’s diet; people who have a diet deficient in calcium, which is an important component of bones, tend to have weaker bones. Dietary supplements or adding dairy products to the diet seems to help strengthen bones. As a group, women experience a decrease in bone density as they age. It has been estimated that fully half of women over age 50 suffer from excessive bone loss, a condition known as osteoporosis. Exact bone densities vary within the body, but for a healthy 30-year-old female, it is about 0.95–1.05 g/cm3. Osteoporosis is diagnosed if the bone density is below 0.6–0.7 g/cm3. Urine. The density of urine can be affected by a variety of medical conditions. Sufferers of diabetes insipidus produce an abnormally large volume of urine with a relatively low density. In another form of diabetes, called diabetes mellitus, there is excess glucose dissolved in the urine, so that the density of urine is abnormally high. The density of urine may also be abnormally high because of excess protein in the urine, which can be caused by congestive heart failure or certain renal (kidney) problems. Thus, a urine density test can provide clues to various kinds of health problems. The density of urine is commonly expressed as a specific gravity, which is a unitless quantity defined as $$\frac{\text{density of some material}}{\text{density of water}}$$. Normal values for the specific gravity of urine range from 1.002 to 1.028. Body Fat. The overall density of the body is one indicator of a person’s total body fat. Fat is less dense than muscle and other tissues, so as it accumulates, the overall density of the body decreases. Measurements of a person’s weight and volume provide the overall body density, which can then be correlated to the percentage of body fat. (The body’s volume can be measured by immersion in a large tank of water. The amount of water displaced is equal to the volume of the body.) Sometimes you will have to perform more than one conversion to obtain the desired unit. For example, suppose you want to convert 54.7 km into millimeters. You can either memorize the relationship between kilometers and millimeters, or you can do the conversion in steps. Most people prefer to convert in steps. To do a stepwise conversion, we first convert the given amount to the base unit. In this example, the base unit is meters. We know that there are 1,000 m in 1 km: [latex]54.7\cancel{\text{ km}}\times \frac{1,000\text{ m}}{1\cancel{\text{ km}}}=54,700\text{ m}[/latex] Then we take the result (54,700 m) and convert it to millimeters, remembering that there are 1,000 mm for every 1 m: [latex]54,700\cancel{\text{ m}}\times \frac{1,000\text{ m}}{1\cancel{\text{ m}}}=54,700,000\text{ mm}=5.47\times10^7\text{ mm}[/latex] We have expressed the final answer in scientific notation. As a shortcut, both steps in the conversion can be combined into a single, multistep expression: [latex]54.7\cancel{\text{ km}}\times \frac{1,000\cancel{\text{ m}}}{1\cancel{\text{ km}}}\times\frac{1,000\text{ mm}}{1\cancel{\text{ m}}}\times=54,700,000\text{ mm}=5.47\times10^7\text{ mm}[/latex] Either method—one step at a time or all the steps together—is acceptable. If you do all the steps together, the restriction for the proper number of significant figures should be done after the last step. As long as the math is performed correctly, you should get the same answer no matter which method you use. Example 14 Convert 58.2 ms to megaseconds in one multistep calculation. Solution Show Answer First, convert the given unit to the base unit—in this case, seconds—and then convert seconds to the final unit, megaseconds: [latex]58.2\cancel{\text{ ms}}\times\frac{1\cancel{\text{ s}}}{1,000\cancel{\text{ ms}}}\times\frac{1\cancel{\text{ Ms}}}{1,000,000\cancel{\text{ s}}}=0.0000000582\text{ Ms}=5.82\times10^{-8}\text{ Ms}[/latex] Neither conversion factor affects the number of significant figures in the final answer. Skill-Building Exercise Convert 43.007 ng to kilograms in one multistep calculation. Career Focus: Pharmacist A pharmacist dispenses drugs that have been prescribed by a doctor. Although that may sound straightforward, pharmacists in the United States must hold a doctorate in pharmacy and be licensed by the state in which they work. Most pharmacy programs require four years of education in a specialty pharmacy school. Pharmacists must know a lot of chemistry and biology so they can understand the effects that drugs (which are chemicals, after all) have on the body. Pharmacists can advise physicians on the selection, dosage, interactions, and side effects of drugs. They can also advise patients on the proper use of their medications, including when and how to take specific drugs properly. Pharmacists can be found in drugstores, hospitals, and other medical facilities. Curiously, an outdated name for pharmacist is chemist, which was used when pharmacists formerly did a lot of drug preparation, or compounding. In modern times, pharmacists rarely compound their own drugs, but their knowledge of the sciences, including chemistry, helps them provide valuable services in support of everyone’s health. Concept Review Exercises How do you determine which quantity in a conversion factor goes in the denominator of the fraction? State the guidelines for determining significant figures when using a conversion factor. Answers Show Answer 1.The unit you want to cancel from the numerator goes in the denominator of the conversion factor. 2.Exact numbers that appear in many conversion factors do not affect the number of significant figures; otherwise, the normal rules of multiplication and division for significant figures apply. Key Takeaway A unit can be converted to another unit of the same type with a conversion factor. Exercises Give the two conversion factors you can construct using each pair of units. meters and kilometers liters and microliters seconds and milliseconds Give the two conversion factors you can construct using each pair of units. grams and centigrams millimeters and meters liters and megaliters How many meters are in 56.2 km? How many seconds are in 209.7 ms? How many microliters are in 44.1 L? How many megagrams are in 90.532 g? Convert 109.6 kg into micrograms. Express your final answer in scientific notation. Convert 3.8 × 105 mm into kilometers. Express your final answer in scientific notation. Convert 3.009 × 10−5 ML into centiliters. Express your final answer in scientific notation. Convert 99.04 dm into micrometers. Express your final answer in scientific notation. The density of ethyl alcohol is 0.79 g/mL. What is the mass of 340 mL of ethyl alcohol? The density of a certain fraction of crude oil is 1.209 g/mL. What is the mass of 13,500 mL of this fraction? The density of ethyl alcohol is 0.79 g/mL. What is the volume of 340 g of ethyl alcohol? The density of a certain component of crude oil is 1.209 g/mL. What is the volume of 13,500 g of this component? Vitamin C tablets can come in 500 mg tablets. How many of these tablets are needed to obtain 10 g of vitamin C? A tablet of penicillin contains 250 mg of the antibacterial drug. A prescription contains 44 tablets. What is the total mass of penicillin in the prescription? Answers Show Answer $$\frac{1,000\text{ m}}{1\text{ km}}; \frac{1\text{ km}}{1,000\text{ m}}$$ $$\frac{1,000,000\text{ μL}}{1\text{ L}}; \frac{1\text{ L}}{1,000,000\text{ μL}}$$ $$\frac{1,000\text{ ms}}{1\text{ s}}; \frac{1\text{ s}}{1,000\text{ ms}}$$ 5.62 × 104 m 4.41 × 107 µL 1.096 × 108 µg 3.009 × 103 cL 270 g 430 mL 20 tablets Candela Citations CC licensed content, Shared previously The Basics of General, Organic, and Biological Chemistry v. 1.0. Provided by: Saylor Academy. Located at: License: CC BY-NC: Attribution-NonCommercial. License Terms: This text was adapted by Saylor Academy under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License without attribution as requested by the work's original creator or licensor. Licenses and Attributions CC licensed content, Shared previously The Basics of General, Organic, and Biological Chemistry v. 1.0. Provided by: Saylor Academy. Located at: License: CC BY-NC: Attribution-NonCommercial. License Terms: This text was adapted by Saylor Academy under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License without attribution as requested by the work's original creator or licensor. Privacy Policy
12396	https://math.stackexchange.com/questions/1816137/for-fx-frac-1x-and-gx-sqrtx-4-find-the-domain-of-the-composite	For $f(x)=\frac {1}{x}$ and $g(x)=\sqrt{x-4}$, find the domain of the composite function $g\circ f(x)$. - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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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 For f(x)=1 x f(x)=1 x and g(x)=x−4−−−−√g(x)=x−4, find the domain of the composite function g∘f(x)g∘f(x). Ask Question Asked 9 years, 3 months ago Modified9 years, 3 months ago Viewed 918 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. For f(x)=1 x f(x)=1 x and g(x)=x−4−−−−−√g(x)=x−4, find the domain of the composite function g∘f(x)g∘f(x). My Attempt Here, f(x)=1 x f(x)=1 x g(x)=x−4−−−−−√g(x)=x−4 Now, g∘f(x)=1−4 x x−−−−−−√g∘f(x)=1−4 x x How can I proceed further? functions Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Jun 6, 2016 at 17:20 Em. 16.2k 7 7 gold badges 28 28 silver badges 40 40 bronze badges asked Jun 6, 2016 at 17:16 user335710user335710 350 1 1 silver badge 10 10 bronze badges 5 Hint: what's the domain of g g?lulu –lulu 2016-06-06 17:20:51 +00:00 Commented Jun 6, 2016 at 17:20 x∈(0,0.25]x∈(0,0.25]clueless –clueless 2016-06-06 17:21:05 +00:00 Commented Jun 6, 2016 at 17:21 @lulu, I don't know about that.user335710 –user335710 2016-06-06 17:22:37 +00:00 Commented Jun 6, 2016 at 17:22 Well...what's the domain of h(x)=x−−√h(x)=x?lulu –lulu 2016-06-06 17:23:43 +00:00 Commented Jun 6, 2016 at 17:23 Two hints: what can't you find the square root of, and when is 1−4 x x 1−4 x x undefined? Whatever is outside of these two spaces is your domain.Clarinetist –Clarinetist 2016-06-06 17:24:11 +00:00 Commented Jun 6, 2016 at 17:24 Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 4 Save this answer. Show activity on this post. We have g∘f(x)=f(x)−4−−−−−−−√=1 x−4−−−−−√g∘f(x)=f(x)−4=1 x−4 This function is defined when x≠0 x≠0 because of the 1 x 1 x and when 1 x−4≥0 1 x−4≥0 because of the −−√. Now, if x<0 x<0 then 1 x<0 1 x<0 and so 1 x−4<0 1 x−4<0. Hence, we need x>0 x>0. Finally if x>0 x>0, then 1 x−4≥0⟺1 x≥4⟺1≥4 x⟺1 4≥x.1 x−4≥0⟺1 x≥4⟺1≥4 x⟺1 4≥x. This finally shows that the domain of g∘f g∘f is D=(0,1 4]D=(0,1 4]. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Jun 6, 2016 at 17:36 SurbSurb 57.4k 11 11 gold badges 68 68 silver badges 119 119 bronze badges 2 The answer I have is x=>4 x=>4 user335710 –user335710 2016-06-07 03:02:49 +00:00 Commented Jun 7, 2016 at 3:02 @user335710 How do you get this answer? Try to plug in x=5 x=5 and see what happens.Surb –Surb 2016-06-07 07:06:27 +00:00 Commented Jun 7, 2016 at 7:06 Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. D g o f={x∈D f\|f(x)∈D g}D g o f={x∈D f\|f(x)∈D g} D f=R−{0}D f=R−{0}, D g=[4,∞)D g=[4,∞) and f(x)∈D g f(x)∈D g was interpreted as follow 1 x∈[4,∞)⟹1 x≥4⟹1−4 x x≥0 1 x∈[4,∞)⟹1 x≥4⟹1−4 x x≥0 then D g o f=R−{0}∩(0,1 4]=(0,1 4]D g o f=R−{0}∩(0,1 4]=(0,1 4] Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Jun 6, 2016 at 18:23 Surb 57.4k 11 11 gold badges 68 68 silver badges 119 119 bronze badges answered Jun 6, 2016 at 18:14 Behrouz MalekiBehrouz Maleki 11.4k 3 3 gold badges 29 29 silver badges 49 49 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 functions See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Related 1How do I solve this composite function and find its domain? 0Composite function (State the domain) 1Domain of the definition of a composite function 5Domain and range of composite functions 1Confusion About Domain and Range of Linear Composite Functions 0Domain of a composite function by using intersections 4Domain of composite function (f∘g)(x).(f∘g)(x). 1Finding the implied domain and image of a composite function 0Absolute method to find the domain of a composite function Hot Network Questions My dissertation is wrong, but I already defended. How to remedy? Discussing strategy reduces winning chances of everyone! Gluteus medius inactivity while riding How long would it take for me to get all the items in Bongo Cat? 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12397	https://litfl.com/glasgow-coma-scale-gcs/	Skip to content Glasgow Coma Scale (GCS) Chris Nickson Reviewed and revised 12 October 2019 The GCS is a neurological scoring system used to assess conscious level after head injury Teasdale and Jennet invented the GCS in 1974 It is now usually scored out of 15 and is comprised of 3 categories, best eye response, best vocal response and best motor response (e.g. E4V5M6 = GCS15) CALCULATION OF GCS Eye response (E) No eye opening Eye opening in response to pain stimulus (a peripheral pain stimulus, such as squeezing the lunula area of the patient’s fingernail is more effective than a central stimulus such as a trapezius squeeze, due to a grimacing effect) Eye opening to speech (not to be confused with the awakening of a sleeping person; such patients receive a score of 4, not 3) Eyes opening spontaneously Verbal response (V) No verbal response Incomprehensible sounds (moaning but no words) Inappropriate words (random or exclamatory articulated speech, but no conversational exchange) Confused (the patient responds to questions coherently but there is some disorientation and confusion) Oriented (patient responds coherently and appropriately to questions such as the patient’s name and age, where they are and why, the year, month) Motor response (M) No motor response Extension to pain (extensor posturing: abduction of arm, external rotation of shoulder, supination of forearm, extension of wrist, decerebrate response) Abnormal flexion to pain (flexor posturing: adduction of arm, internal rotation of shoulder, pronation of forearm, flexion of wrist, decorticate response) Flexion/Withdrawal to pain (flexion of elbow, supination of forearm, flexion of wrist when supra-orbital pressure applied ; pulls part of body away when nailbed pinched) Localizes to pain (Purposeful movements towards painful stimuli; e.g., hand crosses mid-line and gets above clavicle when supra-orbital pressure applied) Obeys commands (the patient does simple things as asked, e.g. stick out tongue or move toes) USES categorises severity of TBI into mild (13-15), moderate (9-12) and severe (8 or less) used in BTF guidelines as part of the indications for ICP monitoring (e.g. GCS 8 or less and abnormal CT head) used for determining the need for CT head in TBI by validated tools such as the Canadian CT Head Rule Traditional ATLS mantra is “GCS 8, intubate” used in APACHE II originally described for monitoring depth of coma over time in a neurosurgical unit (never validated) ADVANTAGES most widely recognised of all conscious level scoring systems in the world has face validity (looks like it should work) quick reproducible (this is controversial, in one study 38% of the cases the GCS scores were the same and in 33% of cases the scores varied with more than two points) skewed towards motor score, which is good since this is the most reliable measure of short-term prognosis in TBI the distinction between a motor score of 2, 3 and 4 is a very useful clinical indicator of the severity of TBI, and the area of brain function that has been affected correlates with adverse neurological outcomes such as brain injury, neurosurgical intervention, and mortality DISADVANTAGES Problems with the use of GCS not originally intended to be converted into a single score — the components (E4,V5, M6) are more important than the total score does not incorporate brain-stem reflexes M score does not factor in unilateral pathology unreliable in patients in the middle range of 9-12 The same GCS score will predict different TBI mortality depending on the components— GCS of 4 with the components 1+1+2 (E+V+M) predicts a mortality rate of 48%— GCS of 4 with the components 2+1+1 (E+V+M) predicts a mortality rate of 19% grossly predictive but cannot accurately predict outcomes in individual patients (on par with weather presenters predicting rain or WBC predicting appendicitis!) Problems with performing GCS designed as a tool for repeated bedside assessment of various neurological functions in patients in a neurosurgical ward, not for use in TBI It is difficult for untrained staff to apply properly, especially distinguishing between M= 3,4,5 (even neurosurgeons get it wrong ~50% of the time) Variation in scoring V in intubated patients subject to language barriers cannot be applied to small children may be affected by other factors influencing level of consciousness, e.g. drugs such as alcohol and sedatives GCS is often used in settings such as toxicology where it is unvalidated Debates within the literature as to when GCS can be first applied after TBI, i.e when is the first post-resuscitation GCS applicable Problems with accuracy and validity of GCS Controversy in the literature There is poor inter-observer reliability Reproducibility is poor (only 50% in neurosurgeons!) There is little evidence demonstrating validity and reliability of the GCS Not proven to be better than unstructured clinical judgement There are numerous other neurological scoring systems that have demonstrated similar or greater validity and reliability e.g. the FOUR score, AVPU in children GCS 8 does not reliably correlate with the presence or absence of airway reflexes We have never recommended using the GCS alone, either as a means of monitoring coma, or to assess the severity of brain damage or predict outcome. Teasdale and Jennet in 1978 ALTERNATIVES FOUR score AVPU Simplified Motor Score (aka TROLL: Test Responsiveness: Obeys, Localizes, or Less) unstructured clinical judgement MY APPROACH Due to widespread adoption I still use GCS in TBI in conjunction with other clinical information and investigations in the assessment of TBI severity, to guide monitoring and management and as an aid to prognostication However, because of its limitations, GCS must be used cautiously all staff need to be aware of the same criteria for its use and application and have a standardised approach to its assessment on-going education is needed to make sure that it is used correctly VIDEOS Videos by Jake Timothy (Consultant Neurosurgeon) and Sir Graham Teasdale (professor of Neurosurgery) on the history and use of GCS: References and Links CCC Neurocritical Care Series Emergencies: Brain Herniation, Eclampsia, Elevated ICP, Status Epilepticus, Status Epilepticus in PaedsDDx: Acute Non-Traumatic Weakness, Bulbar Dysfunction, Coma, Coma-like Syndromes, Delayed Awakening, Hearing Loss in ICU, ICU acquired Weakness, Post-Op Confusion, Pseudocoma, Pupillary AbnormalitiesNeurology: Anti-NMDA Encephalitis, Basilar Artery Occlusion, Central Diabetes Insipidus, Cerebral Oedema, Cerebral Venous Sinus Thrombosis, Cervical (Carotid / Vertebral) Artery Dissections, Delirium, GBS vs CIP, GBS vs MG vs MND, Guillain-Barre Syndrome, Horner’s Syndrome, Hypoxic Brain Injury, Intracerebral Haemorrhage (ICH), Myasthenia Gravis, Non-convulsive Status Epilepticus, Post-Hypoxic Myoclonus, PRES, Stroke Thrombolysis, Transverse Myelitis, Watershed Infarcts, Wernicke’s EncephalopathyNeurosurgery: Cerebral Salt Wasting, Decompressive Craniectomy, Decompressive Craniectomy for Malignant MCA Syndrome, Intracerebral Haemorrhage (ICH)— SCI: Anatomy and Syndromes, Acute Traumatic Spinal Cord Injury, C-Spine Assessment, C-Spine Fractures, Spinal Cord Infarction, Syndomes,— SAH: Acute management, Coiling vs Clipping, Complications, Grading Systems, Literature Summaries, ICU Management, Monitoring, Overview, Prognostication, Vasospasm— TBI: Assessment,Base of skull fracture, Brain Impact Apnoea, Cerebral Perfusion Pressure (CPP), DI in TBI, Elevated ICP, Limitations of CT, Lund Concept, Management, Moderate Head Injury, Monitoring, Overview, Paediatric TBI, Polyuria incl. CSW, Prognosis, Seizures, TemperatureID in NeuroCrit. Care: Aseptic Meningitis, Bacterial Meningitis, Botulism, Cryptococcosis, Encephalitis, HSV Encephalitis, Meningococcaemia, Spinal Epidural AbscessEquipment/Investigations: BIS Monitoring, Codman ICP Monitor, Continuous EEG, CSF Analysis, CT Head, CT Head Interpretation, EEG, Extradural ICP Monitors, External Ventricular Drain (EVD), Evoked Potentials, Jugular Bulb Oxygen Saturation, MRI Head, MRI and the Critically Ill, Train of Four (TOF), Transcranial DopplerPharmacology: Desmopressin, Hypertonic Saline, Levetiracetam (Keppra), Mannitol, Midazolam, Sedation in ICU, ThiopentoneMISC: Brainstem Rules of 4, Cognitive Impairment in Critically Ill, Eye Movements in Coma, Examination of the Unconscious Patient, Glasgow Coma Scale (GCS), Hiccoughs, Myopathy vs Neuropathy, Neurology Literature Summaries, NSx Literature Summaries, Occulocephalic and occulovestibular reflexes, Prognosis after Cardiac Arrest, SIADH vs Cerebral Salt Wasting, Sleep in ICU Journal articles Teasdale G, Jennett B. Assessment of coma and impaired consciousness. A practical scale. Lancet. 1974 Jul 13;2(7872):81-4. Teasdale G, Jennett B, Murray L, Murray G. Glasgow coma scale: to sum or not to sum. Lancet. 1983 Sep 17;2(8351):678. Chou R, Totten AM, Carney N, Dandy S, Fu R, Grusing S, Pappas M, Wasson N, Newgard CD. Predictive Utility of the Total Glasgow Coma Scale Versus the Motor Component of the Glasgow Coma Scale for Identification of Patients With Serious Traumatic Injuries. Ann Emerg Med. 2017 Aug;70(2):143-157.e6. Green SM. Cheerio, laddie! Bidding farewell to the Glasgow Coma Scale. Ann Emerg Med. 2011 Nov;58(5):427-30. Zuercher M, Ummenhofer W, Baltussen A, Walder B. The use of Glasgow Coma Scale in injury assessment: a critical review. Brain Inj. 2009 May;23(5):371-84. FOAM and web resources Glasgow Coma Scale website Critical Care Compendium …more CCC Chris Nickson Chris is an Intensivist and ECMO specialist at The Alfred ICU, where he is Deputy Director (Education). He is a Clinical Adjunct Associate Professor at Monash University, the Lead for the Clinician Educator Incubator programme, and a CICM First Part Examiner. He is an internationally recognised Clinician Educator with a passion for helping clinicians learn and for improving the clinical performance of individuals and collectives. He was one of the founders of the FOAM movement (Free Open-Access Medical education)has been recognised for his contributions to education with awards from ANZICS, ANZAHPE, and ACEM. His one great achievement is being the father of three amazing children. On Bluesky, he is @precordialthump.bsky.social and on the site that Elon has screwed up, he is @precordialthump. \| INTENSIVE \| RAGE \| Resuscitology \| SMACC 3 Comments Dear Dr Nickson, Indeed interesting reading! Especially this part:Problems with accuracy and validity of GCS: •There are numerous other neurological scoring systems that have demonstrated greater validity and reliability e.g. the FOUR score, AVPU in Children” I would be very happy if you would inform me on the publications that favours AVPU in a pediatric setting. //Kind regardsDavid BjörnhedenSwedish consultant in pediatric emergeny and pediatric neurology. Hi David As with any tool, the validity of the result depends on who is using it, for what reason, and in which patients. I have revised the sentence to say “similar or greater” in different settings, as the evidence base for the examples given -FOUR score and AVPU ( not to mention other simpler scales) – are not equivalent. This 2018 Study by Nuttall et al supports the use of AVPU in infants/ small children for the initial assessment of head injury, but argues it lacks sensitivity to subtler level of consciousness changes in ongoing assessment in in older children. This follows on from McNarry and Goldhill (2008) who showed the correlation of AVPU with different levels of GCS. I’m not familiar with studies of AVPU that use something other than GCS as the comparison/ gold standard. Cheers Chris 2. Hi there, I think the M2 description for decerebrate posturing is incorrect. It should be internal rotation of the shoulder, extension at elbow, pronation of forearm and flexion of fingers based on my reference to Cheers! Leave a ReplyCancel reply This site uses Akismet to reduce spam. Learn how your comment data is processed.
12398	http://astro1.physics.utoledo.edu/~megeath/ph6820/lecture26_ph6820.pdf	Stellar Nucleosynthesis They Might be Giants Elemental Abundances in the Solar System Synthesized in the Big Bang We need to consider the creation of all the other elements in stars. Elemental Abundances in the Solar System 3 Slide: Stephen Smartt Abundances of nuclei Review: Nuclear Fusion in Stars Review: Binding Energy of Nuclei Review: Mass Excess Atomic Number (Z) Atomic Weight Rate of Reactions The Coloumb Barrier 1 fm = 10-15 m For 12C and α 9 Occurrence of fusion reactions Now will discuss the conditions under which fusion can occur – and whether such conditions exist in stellar interiors • Nuclei interact through four forces of physics – only electromagnetic and strong nuclear important here • Two positively charged nuclei must overcome coulomb barrier (long range force ∝1/r2), to reach separation distances where strong force dominates (10-15 m, typical size of nucleus) Schematic plots • V (potential energy) vs nuclei separation distance • Wave function representing penetration of a potential barrier by nucleus with kinetic energy of approach Ekin (below barrier height). Slide: Stephen Smartt 10 Quantum Tunneling As derived in your Quantum Mechanics courses, there is a finite probability for a particle to penetrate the Coulomb barrier as if “tunnel” existed. Quantum effect discovered by George Gamow (1928) in connection with radioactivity. Penetration probability (calculated by Gamow) is given as: e −πZ1Z 2e 2 ε 0hv Hence this increases with v (particle velocity), but we know v will be Maxwellian distribution for ideal gas. Hence fusion probability is product prob( fusion) ∝e −πZ1Z 2e 2 ε 0hv e −mv 2 2kT Slide: Stephen Smartt 11 The Gamow peak Schematically this is plotted, and the fusion most likely occurs in the energy window defined as the Gamow Peak. The Gamow peak is the product of the Maxwellian distribution and tunnelling probability. The area under the Gamow peak determines the reaction rate. The higher the electric charges of the interacting nuclei, the greater the repulsive force, hence the higher the Ekin and T before reactions occur. Highly charged nuclei are obviously the more massive, so reactions between light elements occur at lower T than reactions between heavy elements. Slide: Stephen Smartt The Gamow Peak Resonances From Clayton: Principles of Stellar Evolution and Nucleosynthesis Calculation of Gamow peak assumes that the process is far from a resonances. Resonances can increase the cross section of a reaction signiﬁcantly and consequently increase the reaction rate. Resonance C12 + P => N13 + γ 14 Hydrogen and helium burning The most important series of fusion reactions are those converting H to He (H-burning). As we shall see this dominates ~90% of lifetime of nearly all stars. • Fusion of 4 protons to give one 4He is completely negligible • Reaction proceeds through steps – involving close encounter of 2 particles • We will consider the main ones: the PP-chain and the CNO cycle The PP Chain The PP chain has three main branches called the PPI, PPII and PPIII chains. PPI Chain 1 p + p → d + e+ + νe 2 d + p → 3He + γ 3 3He + 3He → 4He + 2p PPII Chain 3' 3He + 4He → 7Be + γ 4' 7Be + e− → 7Li + νe 5' 7Li + p → 4He + 4He PPIII Chain 4'' 7Be + p → 8B + γ 5'' 8B → 8Be + e+ + νe 6'' 8Be → 24He Slide: Stephen Smartt 15 Relative importance of PPI and PPII chains (branching ratios) depend on conditions of H-burning (T,ρ , abundances). The transition from PPI to PPII occurs at temperatures in excess of 1.3×107 K. Above 3×107 K the PPIII chain dominates over the other two, but another process takes over in this case. Slide: Stephen Smartt The Three P-P Chains From Clayton: Principles of Stellar Evolution and Nucleosynthesis When does a given P-P chain dominate? FPPIII FPPII FPPI From Clayton: Principles of Stellar Evolution and Nucleosynthesis 18 Energy production and neutrino emission Energy released in the formation of an α particle by fusion of four protons. Is essentially given by the difference of the mass excesses of four protons and one α particle. Qp−p = 4ΔM(1H) −ΔM(4He) [ ]c 2 = 26.7 MeV Since any reaction branch that completes this must turn 2 protons in 2 neutrons, two neutrinos are also emitted, which carry energy away from the reaction site. It is these neutrinos that directly confirm the occurrence of nuclear reactions in the interior of the Sun. No other direct observational test of nuclear reactions is possible. The mean neutrino energy flux is ~0.26MeV for d creation (PPI/II) and ~7.2MeV for B decay (PPIII). But as PPIII is negligible, the energy released for each He nucleus assembled is ~26MeV (or 6 ×1014 JKg-1 ) Slide: Stephen Smartt 19 The CNO Cycle At birth stars contain a small (2%) mix of heavy elements, some of the most abundant of which are carbon, oxygen and nitrogen (CNO). These nuclei may induce a chain of H-burning reactions in which they act as catalysts. The process is known as the CNO Cycle. There are alternative names that you may come across : • The CNO bi-cycle • The CNOF cycle • The CN and NO cycles • The CN and NO bi-cycles In this course we will just refer to it all as the CNO cycle – and discuss the branches, but not specifically label them. Slide: Stephen Smartt 20 The main branch 1 12C + p → 13N + γ 2 13N → 13C + e+ + νe 3 13C + p → 14N + γ 4 14N + p → 15O + γ 5 15O → 15N + e+ + νe 6 15N + p → 12C + 4He In the steady state case, the abundances of isotopes must take values such that the isotopes which react more slowly have higher abundance. The slowest reaction is p capture by 14N . Hence most of 12C is converted to 14N. Slide: Stephen Smartt 21 Temperature dependence of PP chain and CNO Cycle The two processes have very different temperature dependences. The rate of energy production in each: € εPP = ε0ρX H 2 T T0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 4.6 € εCNO = ε0ρX H XCNO fN T 25 ×106 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 16.7 Equating this two gives the T at which they produce the same rate of energy production: € T ≈1.7 ×107 X H 50XCN ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 12.1 K Below this temperature the PP chain is most important, and above it the CNO Cycle dominates. This occurs in stars slightly more massive than the sun e.g. 1.2-1.5M. Slide: Stephen Smartt 22 Slide: Stephen Smartt Changes in Elemental Abundances During CNO Caughlan 1962 Caughlan & Fowler 1962 Changes in Elemental Abundances During CNO 25 Helium Burning: the triple-α reaction. Simplest reaction in a helium gas should be the fusion of two helium nuclei. There is no stable configuration with A=8. For example the beryllium isotope 8Be has a lifetime of only 2.6×10-16 s 4He + 4He → 8Be But a third helium nucleus can be added to 8Be before decay, forming 12C by the “triple-alpha” reaction 4He + 4He → 8Be 8Be + 4He → 12C + γ Slide: Stephen Smartt 26 Helium Burning: the triple-α reaction. Fred Hoyle (1952-54) suggested this small probability of α-capture by short lived 8Be would be greatly enhanced if the C nucleus had an energy level close to the combined energies of the reacting 8Be and 4He nuclei. The reaction would be a faster “resonant” reaction. This resonant energy level of 12C was not experimentally known at the time. Hoyle’s prediction led to nuclear experiment at Caltech, and resonant level discovered. Thus helium burning proceeds in a 2-stage reaction, and energy released is Q3α = 3ΔM(4He) −ΔM(12C) [ ]c 2 = 7.275MeV In terms of energy generated per unit mass ≡ 5.8 ×1013 JKg-1 (I.e. 1/10 of energy generated by H-burning). But the T dependence is astounding: ε3α ∝ρ2T 40 Slide: Stephen Smartt 27 Carbon and oxygen burning Carbon burning (fusion of 2 C nuclei) requires temperatures above 5 ×108 K, and oxygen burning in excess of 109 K. Interactions of C and O nuclei are negligible – as at the intermediate temperatures required by the coulomb barrier the C nuclei are quickly destroyed by interacting with themselves 12C + 12C → 24Mg + γ → 23Mg + n → 23Na + p → 20Ne + α → 16O + 2α 16O + 16O → 32S + γ → 31S + n → 31P + p → 28Si + α → 24Mg + 2α The branching ratios for these reactions are temperature dependent probabilities. 12C + 12C → ~13MeV (~5.2 ×1013 JKg-1) 16O + 16O → ~16MeV (~4.8 ×1013 JKg-1) These reactions produce p, n, α, which are immediately captured by heavy nuclei, thus many isotopes created by secondary reactions. Slide: Stephen Smartt 28 Silicon burning: nuclear statistical equilibrium Two Si nuclei could fuse to create 56Fe – the end of the fusion chain. But now very high Coulomb barrier, at T above O burning, but below that required for Si burning, photodisintegration takes place 16O + α ↔ 20Ne + γ This produces Ne at T~109K but reverses above 1.5 ×109 K. Si disintegration occurs around 3×109 K, and the light particles emitted are recaptured by other Si nuclei. Although the reactions tend to a state of equilibrium, a leakage occurs towards the stable iron group nuclei (Fe, Co, Ni), which resist photodisintegration up to 7×109 K. Slide: Stephen Smartt 29 Major nuclear burning processes Common feature is release of energy by consumption of nuclear fuel. Rates of energy release vary enormously. Nuclear processes can also absorb energy from radiation field, we shall see consequences can by catastrophic. Nuclear Fuel Process Tthreshold 106K Products Energy per nucleon (Mev) H PP ~4 He 6.55 H CNO 15 He 6.25 He 3α 100 C,O 0.61 C C+C 600 O,Ne,Ma,Mg 0.54 O O+O 1000 Mg,S,P,Si ~0.3 Si Nuc eq. 3000 Co,Fe,Ni <0.18 Slide: Stephen Smartt 30 The s-process and r-process Interaction between nuclei and free neutrons (neutron capture) – the neutrons are produced during C, O and Si burning. Neutrons capture by heavy nuclei is not limited by the Coulomb barrier – so could proceed at relatively low temperatures. The obstacle is the scarcity of free neutrons. If enough neutrons available, chain of reactions possible: I(A, Z) + n → I1(A+1, Z) I1(A+1, Z) + n → I2(A+2, Z) I2(A+2, Z) + n → I3(A+3, Z) …etc If a radioactive isotope is formed it will undergo β-decay, creating new element. IN(A+N, Z) → J(A+N, Z+1) + e− + If new element stable, it will resume neutron capture, otherwise my undergo series of β-decays J(A+N, Z+1) → K(A+N,Z+2) + e− + K(A+N, Z+2) → L(A+N, Z+3) + e− + ν ν Slide: Stephen Smartt 31 In the process two types of reactions and two types of nuclei are involved: Neutron captures and β-decays ; stable and unstable nuclei Stable nuclei may undergo only neutron captures, unstable ones my undergo both, with the outcome depending on the timescales for the two processes. What can we say about the timescales of these processes ? Hence neutron capture reactions may proceed more slowly or more rapidly than the competing β-decays. The different chains of reactions and products are called the s-process and r-process. Slide: Stephen Smartt 32 Some basic examples ⇒ Simple explanation of r and s process on: ultraman.ssl.berkeley.edu/nucleosynthesis.html Slide: Stephen Smartt Summary 1.We described the basic accounting for nuclear reactions. This accounting includes: 1.1.the mass excess (binding energy released when creating an element) 1.2.the rate of interactions (depends on the density, mass fractions, and rate coefﬁcient). 1.4.This hides most of the physics. 2.Iron has highest binding energy 3. In the Sun, the P-P chain dominates 4.There are three P-P chains, the other two require He3 5. At higher temperatures (> 106 K), the CNO cycle takes over (using existing C, N and O abundances in star). This rearranges the C, N, and O temperatures. 6. At 100 K, Carbon burning occurs. 7. Triple α requires unstable Be8 as an intermediary state. 8.Silicon Burning (at 109 K) will be described in next lecture 9.We introduced the S and R processes. These work by Neutron capture: S-process (slow process) occurs when Β decay faster than neutron capture, the R-process (rapid process) when neutron capture is much faster than Β decay.
12399	https://physicsexperiments.eu/2047/cooling-mixture-of-water,-ice-and-salt	Cooling Mixture of Water, Ice and Salt — Collection of Experiments Collection of Physics Experiments Physics Physics Thermodynamics Mechanics Thermodynamics Electromagnetism Optics Cooling Mixture of Water, Ice and Salt Experiment number :2047 Goal of experiment The goal of this experiment is to verify that adding kitchen salt into a mixture of water and ice can decrease the melting/freezing point of the mixture. #### Theory Melting points (or freezing points) of crystalline substances generally depend on the surrounding pressure and chemical pureness of the substance – possible impurities can considerably affect the melting/freezing point of the substance. Water can coexist with ice in an equilibrium state under normal pressure at a temperature of 0°C. However, when we mix water, ice and salt (NaCl), this temperature decreases. Figure 1 shows a phase diagram of a mixture of water and salt; the vertical axis shows temperature, the horizontal axis shows the concentration of salt. For each pair temperature-concentration can be found the state of the mixture. Example: For 50% concentration of salt at temperature 15°C the graph shows a green depicted phase “water+NaCl”, which means that some salt does not dissolve (supersaturated solution). We can see from the graph that the lowest temperature, at which a mixture of water, ice and salt can exist, is ca. −21°C and the salt concentration is approximately 23%. Therefore, if we add salt into a mixture of ice and water, the melting/freezing point of the mixture decreases and the ice begins to melt. In order for a phase change to occur, the ice draws the heat of fusion from its surroundings, which allows the temperature to decrease. #### Tools Ice, hammer, beaker, styrofoam block, water, kitchen salt, thermometer (Fig.2). #### Procedure 1. Use the hammer to crash some of the ice cubes and them into a beaker with a small amount of water. Insulate the beaker with a styrofoam block. 2. When the temperature of the mixture stabilizes at approximately 0°C, start adding salt into the mixture while stirring it; use the thermometer to measure the temperature of the mixture. Sample result In the sample experiment we were able to reach the temperature of −15°C without measuring the ratio between water, ice and salt. #### Technical notes When crushing the ice, it is advised to wrap the ice cubes in a towel or a rag (Fig.3). Otherwise you will have pieces of crushed ice flying all around the room. It is convenient to use such a thermometer that can be connected to a computer and plot the time dependence of the temperature. A thermometer with a firm body can also be used as a stirrer. The phenomenon is also observed when no insulation is used. During the experiment, air humidity freezes on the surface of the beaker; the beaker can also freeze to the mat. It is not necessary to use kitchen salt for this experiment. Same effect can be found when using other salts dissoluble in water (for example CaCl 2, FeCl 2, MgCl 2 etc.). Pedagogical notes Practical use of this experiment is, apart from preparing a cooling mixture, also salting frozen roads in the winter. Practice shows that students do not know how the experiment is related to this practical use. (Suggestive question asked by the teacher can be: “I want the ice on the road to melt. How does it help that I decrease its temperature?”) Therefore it is necessary to emphasize, that adding salt primarily decreases the melting/freezing point of the mixture – therefore we can have liquid water even at temperatures well below 0°C. In other words, from a frozen road we can make a wet road. Attentive students can object, that intense stirring can be counterproductive, since mechanical work increases the internal energy of the mixture and the mixture therefore warms up. This reasoning is of course correct and it is good to appreciate it; on the other hand, it is advisable to point out that the temperature increase during stirring is in this case negligible. Complementary experiment A complementary experiment is to show that even dissolving salt in water can cause temperature decrease of the mixture. The physics behind this phenomenon is, however, different from the one used in the previous experiment – dissolving salt is an endothermic process, when the mixture draws heat from its surroundings to break the lattice of sodium chloride (we can talk about so called heat of dissolution), change in state does not occur. Gradually pour some salt (at room temperature) into a glass with water at room temperature, stir the mixture and observe, that the temperature decreases. The video below illustrates the impementation of the experiment using the Vernier system. ×Original source: Kácovský, P. (2016). Experimenty podporující výuku termodynamiky na středoškolské úrovni. (Disertační práce.) Matematicko-fyzikální fakulta UK, Praha. About Show experiment Experiments filter Particulate Nature of Matter (2) Brownian Motion Diameter of oleic acid molecule Change in Internal Energy by Performing Work (5) Change in Internal Energy by Performing Work: Nail Hammering Conversion of Kinetic Energy into Internal Energy: Blow with a Mallet Conversion of Kinetic Energy to Internal Energy: Falling Weight Change in Internal Energy by Performing Work: Drilling in Wood Change in Internal Energy by Performing Work: Dragging a Heavy Object Heat and Specific Heat (4) Experimental determination of Specific Heat of Water Comparing Specific Heat of Water and Vegetable Oil Comparing Specific Heat of Ethanol and Water Temperature Changes of Nostrils During Breathing Thermal Conductivity (10) Thermal Conductivity of Plastic and Metal I. Comparing Thermal Conductivity of Copper, Aluminium and Brass Thermal Conductivity of Plastic and Metal II. 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Choose required ranks and required tasks. The table of contents will list only tasks having one of the required ranks in corresponding rankings and at least one of the required tags (overall). If you wish to filter only according to some rankings or tags, leave the other groups empty. Experiment rankings Type of experiment- [x] Qualitative - [x] Quantitative Difficulty level- [x] From Kindergarten - [x] From Lower secondary level - [x] From Upper secondary level - [x] University level Necessary tools- [x] Easily obtained tools and equipment - [x] Tools and equipment usually found at schools - [x] Specific tools and equipment required Preparation time- [x] Under 3 minutes - [x] 3–10 minutes - [x] Longer than 10 minutes Duration of experiment- [x] Under 3 minutes - [x] 3–10 minutes - [x] Longer than 10 minutes Task tags Attributes- [x] Experiment is video recorded « Updated: Aug. 3rd, 2022

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