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Can you find the missing number in 20 seconds? Only 15% manage to do so without running out of time. Think youâ€™re fast? Prove it! See if you can find the missing number in 20 seconds!Ready to test your speed? Put your brain to the challenge and see if you can find the missing number in 20 seconds! (c) Abmeyerwood Are you up for a challenge? This brain teaser is sure to get your mind working. Look at the picture below and find the missing number in the logical sequence. You have 20 seconds to find the answer. Can you solve it in time? Finding the missing number in a logical sequence in 20 soncds max Finding the missing number in a logical sequence is a challenging task. It requires understanding of both the pattern of the sequence and the logic behind it. To solve this challenge, one must identify the pattern of the numbers and determine how many numbers are part of the sequence. Once this is identified, you can then look for any gaps or missing numbers. If there are more than 20 numbers in the sequence, it may become difficult to identify and find the missing number. It is important to use logical thinking and look for patterns that could indicate the missing number. (c) Abmeyerwood To find a pattern of a logical sequence, it is important to break down the problem into smaller parts. Identify the variables that are needed and determine which of them are related to each other. Look for the relationships between those variables and try to articulate them in a logical manner. It is helpful to draw a diagram or chart that visually represents the relationships between the variables. Use trial and error by changing one variable at a time and observing its impact on the other variables. This can help uncover patterns in the sequence and help you discover the logical order. We sure hope you’ve found the solution to your problem! Ready to check if you’ve succeeded? Click the next page to find out! Explore these related topics	{"url":"https://abmeyerwood.com/can-you-find-the-missing-number-in-20-seconds-only-15-manage-to-do-so-without-running-out-of-time/","timestamp":"2024-11-15T04:46:10Z","content_type":"text/html","content_length":"252286","record_id":"<urn:uuid:3b2b1706-7571-437e-891e-25765a857f2b>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00654.warc.gz"}
GMAT Problems on TrainsProblems on Trains - Learn How to Solve in 1 Minute! \| Leverage Edu If you are practising to ace competitive exams like GRE or GMAT then logical reasoning is one section that you really need to focus on. Logical reasoning is a part of almost every competitive exam and students usually have a hard time solving that section of the question paper but with correct tricks and proper guidance and practice, you can easily solve that section in no time. Problems on trains are among the popular examination questions, every entrance exam usually has a few questions that involve problems related to trains, their speed, direction, and length. This blog will try to simplify and explain this section of the paper in order to give you a basic idea of how to go about problems on trains. Problems on Trains: Concepts and Logics The problems on trains follow a certain format that revolves around some fundamental concepts, these are: Distance and time The question based on trains usually includes concepts like relative motion, speed and time. If you are good in physics or even if you remember the basic concepts of it then you can easily ace this section. There are questions that involve distance and time. One formula that you need to keep in mind is d= st, where d is distance, s is the speed and t stands for time. Let’s look at a question which involves this formula: Question: Train A travels at 50 mph, Train B travels at 70 mph. Taking into account that both trains leave the station at the same time, evaluate how far apart they will be after two hours? Solution: To calculate the distance here you need to add the distance traveled by both the trains together. Distance traveled by train A will be: d=st, the rate at which it travels here is 50mph and the time is 2 hours. So the distance traveled by train A will be 50×2 = 100. Similarly, distance traveled by train B will be 70×2 = 140. Now if we add the distances 140+100= 240, hence the distance at which they travel apart from each other would be 240 miles. If the trains are traveling in the same direction then we would subtract distances traveled by both the trains i.e. 140-100= 40, then the answer would be 40 miles. Length of the train In these type of questions, you need to find out the length of the train with the given information in the question. These questions usually include relative speed and the formula used here is the same d= st. Let’s look at an example in order to understand the problem more clearly. Question: A train is traveling at a speed of 60kmph it then overtakes a bike that is traveling at 30kmph in about 50 seconds. Now calculate the length of the train. Solution: To calculate the length of the train we need to find out the distance traveled by the train, since the bike is also in motion we need to look at the relative speed of the bike and the Since both the objects are moving in the same direction we need to subtract the distance traveled by both the objects i.e. 60-30= 30, so the relative speed is 30kmph. Now let’s find out the distance traveled by train while taking over the bike, applying the formula, d= st, distance will be 30kmphx50 seconds. Now let’s convert the distance into meter per second. 1 kmph = 5/18 m/sec Therefore 30kmph= 30×5/18= 8.33 m/sec Therefore distance traveled will be 8.33×50= 416.5 meters, the length of the train is 416.5 meters. Types of Questions: Problems on Train The number of applicants has increased throughout the years, and each year, candidates observe a new pattern or format in which questions are asked for various topics in the syllabus. In order to minimise the chance of receiving a bad grade, candidates should be aware of the several ways questions may be phrased or asked throughout the exam. As a result, the types of questions that might be posed from train-based challenges are listed below: • Time Taken by Train to Traverse Any Stationary Body or Platform – A question may require the applicant to determine the amount of time it takes a train to cross a particular type of stationary object, such as a pole, a standing person, a platform, or a bridge. • Time it takes for two trains to cross paths – The duration it might take for two trains to cross each other is still another possible inquiry. • Train Equation-Based Problems- The question may present two situations, and the candidates must build equations based on the conditions. Key Formulas For Problems on Train A candidate must memorise the relevant formulas in order to solve any numerical ability question so they can respond quickly and effectively. The following key formulas for train-related questions will aid candidates in responding to questions based on this subject: • Speed of the Train = Total distance covered by the train / Time taken • The time it takes for two trains to cross each other is equal to (a+b) / (x+y) if the lengths of the trains, say a and b, are known and they are going at speeds of x and y, respectively. • When the length of two trains, let’s say a and b, is known and they are travelling at speeds of x and y, respectively, in the same direction, the time it takes for them to cross each other is equal to (a+b) / (x-y). • When two trains begin travelling in the same direction from points x and y and cross each other after travelling in opposite directions for times t1 and t2, respectively, the ratio of the speeds of the two trains is equal to t2:t1. • If two trains depart from stations x and y at times t1 and t2, respectively, and they go at speeds L and M, respectively, then the distance from x at which they will collide is equal to (t2 – t1) (speed product) / (difference in speed). • When a train stops, it travels the same distance at an average speed of y rather than the normal average speed of x. Hourly Rest Time = (Difference in Average Speed) / (Speed without stoppage) • If it takes two trains of similar length and speed t1 and t2 to pass a pole, the time it takes for them to cross each other if the trains are going in the opposite direction is equal to (2t1t2) / • If it takes two trains of equal length and speed t1 and t2 to cross a pole, the time it takes for the trains to cross each other if they are travelling in the same direction is equal to (2t1t2). Things to Keep in Mind While Solving Problems on Trains While solving the section related to problems on trains it is important to keep certain things in mind to ensure efficiency and accuracy. Here are a few things that you should keep in mind: • Make sure all the units are the same, if not don’t forget to change them. • Read the questions carefully and keep in mind the concepts of speed and relative speed. • Remember the basic formulas • Be clear about all the concepts. How do you solve a train problem in math? Always read the question carefully before responding, as train-based topics are frequently given in a convoluted manner. Try to apply a formula after reading the question; this may lead to a quick solution and save you time. What is relative speed in train problems? When two bodies are moving in the same direction, the relative speed is equal to the difference in their speeds. For example, a person in a train travelling at 60 km/hr in the west will perceive the speed of the other train travelling at 40 km/hr as being 20 km/hr (60-40). What is the formula of train? x km/h = x*(5/18) m/s. The amount of time needed for a train of length/meters to pass a pole, a single post, or a standing person is equal to the distance the train must go in/meters. Understanding the concepts behind problems on trains can help you in developing strategies to solve those questions. While we have tried to give you all the important information required to tackle these questions, it is natural to feel stressed about the entrances. The experts at Leverage Edu can help you plan for these exams so that nothing comes between you and your dreams.	{"url":"https://leverageedu.com/blog/problems-on-trains/","timestamp":"2024-11-14T22:05:34Z","content_type":"text/html","content_length":"339105","record_id":"<urn:uuid:eb6fa8a0-c1fc-4bc2-b27b-3169865a121e>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00353.warc.gz"}
ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 17 Visualising Solid Shapes Ex 17.2 - CBSE Tuts ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 17 Visualising Solid Shapes Ex 17.2 Question 1. Can a polyhedron have for its faces (i) 3 triangles? (ii) 4 triangles? (iii) a square and four triangles? Question 2. Which are prisms among the following? Question 3. Verify Euler’s formula for these solids: Question 4. Can a polyhedron have 15 faces, 30 edges and 20 vertices? Question 5. If a polyhedron has 8 faces and 8 vertices, find the number of edges. Question 6. If a polyhedron has 7 faces and 10 vertices, find the number of edges. Question 7. Write the number of faces, vertices and edges in (i) an octagonal prism (ii) decagonal pyramid. Question 8. Using Euler’s formula, complete the following table:	{"url":"https://www.cbsetuts.com/ml-aggarwal-class-8-solutions-for-icse-maths-chapter-17-ex-17-2/","timestamp":"2024-11-09T23:13:15Z","content_type":"text/html","content_length":"65508","record_id":"<urn:uuid:f3b7f3e1-af7b-436f-a2bf-67e35811757e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00218.warc.gz"}
Vedic Mathematics Teacher's Manual 1 Pothi paperback (for India only): Rs.400 Click here for the FRENCH EDITION of MANUAL 1. Click here for the ITALIAN EDITION of MANUAL 1 (goes to another website). This book is designed for teachers of children in grades 3 to 7. It shows how Vedic Mathematics can be used in a school course but does not cover all school topics (see contents). The book can be used for teachers who wish to learn the Vedic system or to teach courses on Vedic mathematics for this level. The Manual contains many topics that are not in the other Manuals that are suitable for this age range and many topics that are also in Manual 2 are covered in greater detail here. 166 + v pages. Size: A4 Paperback. 2009 Author: Kenneth Williams ISBN 978-1-902517-16-2. Please note that these Manuals do not form a sequence: there is some overlap between the three books. "The author should be commended for his thorough grasp of Krishna Tirthaji's book on Vedic Mathematics, his layout of the divisions of mathematics and various exercises provided for the students which enhance the value of these books [Teacher's Manuals]. They are ideal companions for the teacher and adult students alike." - Journal of oriental research, Vol. 78-80, 2006-9. "I am a student of class XI and planning to appear in IIT entrance exam next year and would like to take this opportunity of telling you what a difference your books have made to my life. I hated maths till class x and believed that it is the only subject which was a brain freak and I felt that I could never achieve excellent marks in math’s, as I could in other subjects. I am extremely thankful to you for writing these books. They are great source of encouragement for me and many other students like me. I have read your primary level and dying to read next two levels..." Rajesh Gupta This Manual is the first of three (elementary, intermediate and advanced) Manuals which are designed for adults with a basic understanding of mathematics to learn or teach the Vedic system. So teachers could use it to learn Vedic Mathematics, though it is not suitable as a text for children (for that the Cosmic Calculator Course is recommended). Or it could be used to teach a course on Vedic Mathematics. The sixteen lessons of this course are based on a series of one week summer courses given at Oxford University by the author to Swedish mathematics teachers between 1990 and 1995. Those courses were quite intensive consisting of eighteen, one and a half hour, lessons. All techniques are fully explained and proofs are given where appropriate, the relevant Sutras are indicated throughout (these are listed at the end of this Manual) and, for convenience, answers are given after each exercise. Cross-references are given showing what alternative topics may be continued with at certain points. It should also be noted that the Vedic system encourages mental work so we always encourage students to work mentally as long as it is comfortable. In the Cosmic Calculator Course pupils are given a short mental test at the start of most or all lessons, which makes a good start to the lesson, revises previous work and introduces some of the ideas needed in the current lesson. In the Cosmic Calculator course there are also many games that help to establish and promote confidence in the ideas used here. Some topics will be found to be missing in this text: for example, there is no section on area, only a brief mention. This is because the actual methods are the same as currently taught so that the only difference would be to give the relevant Sutra(s). Vedic Mathematics is an ancient system of mathematics which was rediscovered early last century by Sri Bharati Krsna Tirthaji (henceforth referred to as Bharati Krsna). The Sanskrit word “veda” means “knowledge”. The Vedas are ancient writings whose date is disputed but which date from at least several centuries BC. According to Indian tradition the content of the Vedas was known long before writing was invented and was freely available to everyone. It was passed on by word of mouth. The writings called the Vedas consist of a huge number of documents (there are said to be millions of such documents in India, many of which have not yet been translated) and these have recently been shown to be highly structured, both within themselves and in relation to each other (see Reference 2). Subjects covered in the Vedas include Grammar, Astronomy, Architecture, Psychology, Philosophy, Archery etc., etc. A hundred years ago Sanskrit scholars were translating the Vedic documents and were surprised at the depth and breadth of knowledge contained in them. But some documents headed “Ganita Sutras”, which means mathematics, could not be interpreted by them in terms of mathematics. One verse, for example, said “in the reign of King Kamse famine, pestilence and unsanitary conditions prevailed”. This is not mathematics they said, but nonsense. Bharati Krsna was born in 1884 and died in 1960. He was a brilliant student, obtaining the highest honours in all the subjects he studied, including Sanskrit, Philosophy, English, Mathematics, History and Science. When he heard what the European scholars were saying about the parts of the Vedas which were supposed to contain mathematics he resolved to study the documents and find their meaning. Between 1911 and 1918 he was able to reconstruct the ancient system of mathematics which we now call Vedic Mathematics. He wrote sixteen books expounding this system, but unfortunately these have been lost and when the loss was confirmed in 1958 Bharati Krsna wrote a single introductory book entitled “Vedic Mathematics”. This is currently available and is a best-seller (see Reference 1). There are many special aspects and features of Vedic Mathematics which are better discussed as we go along rather than now because you will need to see the system in action to appreciate it fully. But the main points for now are: 1) The system rediscovered by Bharati Krsna is based on sixteen formulae (or Sutras) and some sub-formulae (sub-Sutras). These Sutras are given in word form: for example By One More than the One Before and Vertically and Crosswise. In this text they are indicated by italics. These Sutras can be related to natural mental functions such as completing a whole, noticing analogies, generalisation and so on. 2) Not only does the system give many striking general and special methods, previously unknown to modern mathematics, but it is far more coherent and integrated as a system. 3) Vedic Mathematics is a system of mental mathematics (though it can also be written down). Many of the Vedic methods are new, simple and striking. They are also beautifully interrelated so that division, for example, can be seen as an easy reversal of the simple multiplication method (similarly with squaring and square roots). This is in complete contrast to the modern system. Because the Vedic methods are so different to the conventional methods, and also to gain familiarity with the Vedic system, it is best to practice the techniques as you go along. LESSON 1 COMPLETING THE WHOLE DEFICIENCY AND COMPLETION TOGETHER COMPLETING THE WHOLE COLUMNS OF FIGURES SUBTRACTING NUMBERS NEAR A BASE LESSON 2 DOUBLING AND HALVING MULTIPLYING BY 4, 8 SPLITTING NUMBERS DIVIDING BY 4, 8 MULTIPLYING BY 5, 50, 25 DIVIDING BY 5, 50, 25 DIVIDING BY 5 DIVIDING BY 50, 25 LESSON 3 DIGIT SUMS MORE DIGIT SUM PUZZLES MULTIPLICATION CHECK NUMBER NINE LESSON 4 LEFT TO RIGHT ADDITION: LEFT TO RIGHT LESSON 5 ALL FROM 9 AND THE LAST FROM 10 ALL FROM 9 AND THE LAST FROM 10 ADDING ZEROS ONE LESS ONE MORE ONE LESS AGAIN LESSON 6 NUMBER SPLITTING LESSON 7 BASE MULTIPLICATION RECURRING DECIMALS NUMBERS CLOSE TO 100 NUMBERS OVER 100 MENTAL MATHS RUSSIAN PEASANT MULTIPLICATION NUMBERS ABOVE THE BASE ANOTHER APPLICATION OF PROPORTIONATELY LESSON 8 CHECKING AND DIVISIBILITY THE FIRST BY THE FIRST AND THE LAST BY THE LAST THE FIRST BY THE FIRST THE LAST BY THE LAST DIVISIBILITY BY 4 DIVISIBILITY BY 11 REMAINDER AFTER DIVISION BY 11 ANOTHER DIGIT SUM CHECK LESSON 9 BAR NUMBERS ALL FROM 9 AND THE LAST FROM 10 MULTIPLICATION BY 11 LONGER NUMBERS BY ONE MORE THAN THE ONE BEFORE THE FIRST BY THE FIRST AND THE LAST BY THE LAST REPEATING NUMBERS LESSON 12 SQUARING SQUARING NUMBERS THAT END IN 5 SQUARING NUMBERS NEAR 50 THE DUPLEX 3 AND 4 FIGURE NUMBERS LESSON 13 EQUATIONS LESSON 14 FRACTIONS LESSON 15 SPECIAL DIVISION DIVISION BY 9 LONGER NUMBERS A SHORT CUT DIVISION BY 8 ETC. DIVISION BY 99, 98 ETC. TWO-FIGURE ANSWERS LESSON 16 THE CROWNING GEM 9-POINT CIRCLES INDEX OF THE VEDIC FORMULAE Back Cover ¯ Vedic Mathematics was reconstructed from ancient Vedic texts early last century by Sri Bharati Krsna Tirthaji (1884-1960). It is a complete system of mathematics which has many surprising properties and applies at all levels and areas of mathematics, pure and applied. ¯ It has a remarkable coherence and simplicity that make it easy to do and easy to understand. Through its amazingly easy methods complex problems can often be solved in one line. ¯ The system is based on sixteen word-formulae (Sutras) that relate to the way in which we use our mind. ¯ The benefits of using Vedic Mathematics include more enjoyment of maths, increased flexibility, creativity and confidence, improved memory, greater mental agility and so on. ¯ This Elementary Manual is the first of three designed for teachers who wish to teach the Vedic system, either to a class or to other adults/teachers. It is also suitable for anyone who would like to teach themselves the basic Vedic methods.	{"url":"https://www.vedicmaths.org/1-vedic-mathematics-teacher-s-manual-1","timestamp":"2024-11-02T03:11:44Z","content_type":"application/xhtml+xml","content_length":"106352","record_id":"<urn:uuid:aabbde2c-a72e-40d8-9d51-e0ea2a54ae71>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00446.warc.gz"}
5 Tips for Getting Over Your Fear of Math Originally published at https://blog.tarynmcmillan.com/5-tips-for-getting-over-your-fear-of-math Do you like math? Maybe you were a star math student in high school or university. Or maybe itâ€™s been years, or even decades since your last math class. If youâ€™re interested in learning how to code, by now youâ€™ve probably realized that you need to be comfortable with math. Being self-taught means teaching yourself everything, and that includes the basic mathematical operations used in programming. Iâ€™ve recently realized that so much of my initial aversion to math really stemmed from fear. If this sounds like you, know that by committing yourself to becoming better at math, youâ€™ll accelerate your learning and gain a newfound sense of confidence. Today Iâ€™m sharing the five strategies I used to get over my fear of math and become a better, more capable coder. 1. Donâ€™t overthink it As someone without much academic experience in math, I used to totally freeze up whenever I saw an equation. Iâ€™d also rack my brain trying to remember minute details about the math I learned in high school. Many of these details, I'd later realize, werenâ€™t actually important in the long run. High school math puts a big emphasis on teaching material that can be easily graded. It puts far less emphasis on abstract thinking or discrete mathematics, both of which are important in coding. But theyâ€™re also harder to grasp, and as a result, can lead to overthinking. Overthinking can trigger the flight-or-fight response in your brain, which leads to a bunch of undesirable symptoms such as increased heart rate and brain fog. It can also lead to what's called "analysis paralysis", meaning you've analyzed a problem so much that you're actually paralyzed from making any decisions. As you can see, overthinking can cause a lot of problems when you are studying math. Instead, try to relax and project confidence when you are learning. Pay attention to your thought patterns, take regular breaks, and don't beat yourself up if you don't understand something on the first pass. As you gain more confidence in math, youâ€™ll become more reliant on your past experiences and realize you probably know a lot more than you think. 2. Investigate your languageâ€™s Math library Something I really recommend doing early on is finding out how your language deals with mathematical operations. You can find this information in your languageâ€™s documentation and match it up with the math that you remember. Some languages, like JavaScript, use a math object while others, like C# use a math library. A library is essentially a database of common math functions, such as square root, rounding, and finding the minimum and maximum between two values. These functions build upon your knowledge of different variable types, such as integer, float, and double. The following list shows some of the common operations youâ€™ll be using as a programmer. The syntax differs between programming languages, but the basic functionality is the same. These operations are a good place to start if you are a beginner. • Round- rounds the value to the nearest integer • Ceiling- rounds the value up to the nearest integer • Floor- rounds the value down to the nearest integer • Random- returns a random number within a range • Max- finds the highest value • Min- finds the lowest value • Abs- returns the absolute value • Sqrt- returns the square root Here are two examples of basic syntax, just to get you started: JavaScript example: Math.sqrt(36); // returns 6 C# (Unity) example: Mathf.Min(1, 3); // returns 1 3. Practice The best way to practice your coding math is to simply code. Try creating a simple app or game around a basic equation, like finding the average between two numbers. You'll be surprised at how much you can do with such a simple operation. There are many online resources for practicing math that are also worth checking out if you want to brush up on your skills. Here are some good ones: • Scripy Lecture Notes (Python specific) If youâ€™re looking for structured courses in math, itâ€™s worth visiting the MIT open courseware site , browsing the Math section on Khan Academy (note that this site starts with very basic math and progresses from there), or checking out some of the math courses on EdX. The YouTube channel by Professor Leonard is also a popular choice for coders brushing up on their math skills. 4. Look for the everyday uses Math seemed far less foreign and intimidating when I considered how often I used it in my everyday life. So much of my fear of math was really fear of the unknown. But then I got to thinking about the measuring I did in my baking, or the budget management, or even home maintenance like hanging shelves in my garage. In case you need a refresher, here are some everyday uses of math: • Exercise: setting target heart rate, counting reps, calculate calories burned • Leisure: calculating a tip to leave at a restaurant, planning and budgeting for a vacation, playing or composing music, gardening and landscaping • Finance: comparing interest rates, calculating car or mortgage payments, creating a grocery budget, managing investments • Cooking: measuring ingredients, converting recipes between two units of measurements (ie. grams to mL) 5. Change your mindset I didn't have a lot of confidence as a coder at first, especially since I didnâ€™t start coding until my thirties . Even when I took my first Udemy course on C# I remember feeling like a total impostor during the math-heavy lectures. Eventually I realized I needed to start seeing myself as a woman in STEM if I wanted to become one professionally. The more active I became in the world of tech, the more comfortable I felt exploring the math Iâ€™d previously been so afraid of. I took active steps to become a member of the online coding community and I suggest you do the same! This could include: • Joining Discord groups • Participating in Twitter chats on coding or tech-related topics • Becoming active in the tech community on Instagram • Joining the Dev.to or Hashnode community (or both!) In your social media bio, you can be honest about what you don't know, but don't sell yourself short! Remember that there is no 'end' to learning and everyone you meet is a beginner in something. I hope these tips will help you on your coding journey. Remember: learning takes time so you shouldnâ€™t expect to master a subject in just a few weeks of work. Experienced programmers have been working with math for years, and they still learn something new all the time. Keep an open mind and always remember to have fun! Top comments (2) Cepearre • Getting over your fear of math starts with shifting your mindset. Instead of seeing math as a daunting challenge, treat it as a puzzle that sharpens your brain. Break problems into smaller steps, practice regularly, and don't be afraid to make mistakesâ€”it's how you learn. Use resources like tutorials, games, or apps that make learning math fun and engaging. One great way to start is by playing interactive games like Hit the Button Maths Game which helps boost your confidence through fast-paced challenges. With consistency and patience, math will soon feel like a manageable and even enjoyable subject. jamesboston816 • Awesome work thanks for sharing this Smooth Maintenance valuable info with us. 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Categorification of pre-Lie Algebras and solutions of 2-graded classical Yang-Baxter equations Categorification of pre-Lie Algebras and solutions of 2-graded classical Yang-Baxter equations Yunhe Sheng In this paper, we introduce the notion of a pre-Lie 2-algebra, which is the categorification of a pre-Lie algebra. We prove that the category of pre-Lie 2-algebras and the category of 2-term pre-Lie$_\infty$-algebras are equivalent. We classify skeletal pre-Lie 2-algebras by the third cohomology group of a pre-Lie algebra. We prove that crossed modules of pre-Lie algebras are in one-to-one correspondence with strict pre-Lie 2-algebras. O-operators on Lie 2-algebras are introduced, which can be used to construct pre-Lie 2-algebras. As an application, we give solutions of 2-graded classical Yang-Baxter equations in some semidirect product Lie 2-algebras. Keywords: 2-algebras, pre-Lie$_\infty$-algebras, Lie 2-algebras, O-operators, 2-graded classical Yang-Baxter equations 2010 MSC: 17B99, 55U15 Theory and Applications of Categories, Vol. 34, 2019, No. 11, pp 269-294. Published 2019-04-05. TAC Home	{"url":"http://www.tac.mta.ca/tac/volumes/34/11/34-11abs.html","timestamp":"2024-11-07T03:56:31Z","content_type":"text/html","content_length":"1956","record_id":"<urn:uuid:c6467bf8-9a5c-47e5-bd91-acfb0de69c0e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00610.warc.gz"}
Quartiles and Interquartile Range - SAT II Math I All SAT II Math I Resources Example Questions Example Question #111 : Data Analysis And Statistics What is the 3rd quartile of this set? Correct answer: This first step is to find the median which is To find the 3rd quartile, you find the middle number of the set of numbers above the median. For this set those numbers would be The middle number for this new set, which is the 3rd quartile, is Example Question #1 : How To Find Interquartile Range Given the following set of data, what is twice the interquartile range? Correct answer: How do you find the interquartile range? We can find the interquartile range or IQR in four simple steps: 1. Order the data from least to greatest 2. Find the median 3. Calculate the median of both the lower and upper half of the data 4. The IQR is the difference between the upper and lower medians Step 1: Order the data In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest). Let's sort an example data set with an odd number of values into ascending order. Now, let's perform this task with another example data set that is comprised of an even number of values. Rearrange into ascending order. Step 2: Calculate the median Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two. First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point The median of the odd valued data set is four. Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values. Find the average of the two centermost values. The median of the even valued set is four. Step 3: Upper and lower medians Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data. Omit the centermost value. Find the median of the lower portion. Calculate the average of the two values. The median of the lower portion is Find the median of the upper portion. Calculate the average of the two values. The median of the upper potion is If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions. Find the median of the lower portion. The median of the lower portion is two. Find the median of the upper portion. The median of the upper portion is eight. Step 4: Calculate the difference Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR. Let's find the IQR of the odd data set. Finally, we will find the IQR of the even data set. In order to better illustrate these values, their positions in a box plot have been labeled in the provided image. Now that we have solved a few examples, let's use this knowledge to solve the given problem. First, we need to put the data in order from smallest to largest. The median of the lower half falls between two values. The median of the upper half falls between two values. The interquartile range is the difference between the third and first quartiles. Multiply by Example Question #1 : How To Find Interquartile Range Determine the interquartile range of the following numbers: 42, 51, 62, 47, 38, 50, 54, 43 Correct answer: How do you find the interquartile range? We can find the interquartile range or IQR in four simple steps: 1. Order the data from least to greatest 2. Find the median 3. Calculate the median of both the lower and upper half of the data 4. The IQR is the difference between the upper and lower medians Step 1: Order the data In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest). Let's sort an example data set with an odd number of values into ascending order. Now, let's perform this task with another example data set that is comprised of an even number of values. Rearrange into ascending order. Step 2: Calculate the median Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two. First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point The median of the odd valued data set is four. Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values. Find the average of the two centermost values. The median of the even valued set is four. Step 3: Upper and lower medians Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data. Omit the centermost value. Find the median of the lower portion. Calculate the average of the two values. The median of the lower portion is Find the median of the upper portion. Calculate the average of the two values. The median of the upper potion is If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions. Find the median of the lower portion. The median of the lower portion is two. Find the median of the upper portion. The median of the upper portion is eight. Step 4: Calculate the difference Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR. Let's find the IQR of the odd data set. Finally, we will find the IQR of the even data set. In order to better illustrate these values, their positions in a box plot have been labeled in the provided image. Now that we have solved a few examples, let's use this knowledge to solve the given problem. First reorder the numbers in ascending order: 38, 42, 43, 47, 50, 51, 54, 62 Then divide the numbers into 2 groups, each containing an equal number of values: (38, 42, 43, 47)(50, 51, 54, 62) Q1 is the median of the group on the left, and Q3 is the median of the group on the right. Because there is an even number in each group, we'll need to find the average of the 2 middle numbers: The interquartile range is the difference between Q3 and Q1: Example Question #1 : How To Find Interquartile Range The interquartile range is the difference in value between the upper quartile and lower quartile. Find the interquartile range for the data set. Correct answer: How do you find the interquartile range? We can find the interquartile range or IQR in four simple steps: 1. Order the data from least to greatest 2. Find the median 3. Calculate the median of both the lower and upper half of the data 4. The IQR is the difference between the upper and lower medians Step 1: Order the data In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest). Let's sort an example data set with an odd number of values into ascending order. Now, let's perform this task with another example data set that is comprised of an even number of values. Rearrange into ascending order. Step 2: Calculate the median Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two. First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point The median of the odd valued data set is four. Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values. Find the average of the two centermost values. The median of the even valued set is four. Step 3: Upper and lower medians Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data. Omit the centermost value. Find the median of the lower portion. Calculate the average of the two values. The median of the lower portion is Find the median of the upper portion. Calculate the average of the two values. The median of the upper potion is If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions. Find the median of the lower portion. The median of the lower portion is two. Find the median of the upper portion. The median of the upper portion is eight. Step 4: Calculate the difference Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR. Let's find the IQR of the odd data set. Finally, we will find the IQR of the even data set. In order to better illustrate these values, their positions in a box plot have been labeled in the provided image. Now that we have solved a few examples, let's use this knowledge to solve the given problem. As always, rearranging the data set helps us immensely: To find To find Lastly, our interquartile range is Example Question #1 : How To Find Interquartile Range The interquartile range is the difference in value between the upper quartile and lower quartile. Find the interquartile range of the following data set: Correct answer: How do you find the interquartile range? We can find the interquartile range or IQR in four simple steps: 1. Order the data from least to greatest 2. Find the median 3. Calculate the median of both the lower and upper half of the data 4. The IQR is the difference between the upper and lower medians Step 1: Order the data In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest). Let's sort an example data set with an odd number of values into ascending order. Now, let's perform this task with another example data set that is comprised of an even number of values. Rearrange into ascending order. Step 2: Calculate the median Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two. First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point The median of the odd valued data set is four. Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values. Find the average of the two centermost values. The median of the even valued set is four. Step 3: Upper and lower medians Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data. Omit the centermost value. Find the median of the lower portion. Calculate the average of the two values. The median of the lower portion is Find the median of the upper portion. Calculate the average of the two values. The median of the upper potion is If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions. Find the median of the lower portion. The median of the lower portion is two. Find the median of the upper portion. The median of the upper portion is eight. Step 4: Calculate the difference Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR. Let's find the IQR of the odd data set. Finally, we will find the IQR of the even data set. In order to better illustrate these values, their positions in a box plot have been labeled in the provided image. Now that we have solved a few examples, let's use this knowledge to solve the given problem. The first step (as with most data set problems) is to rearrange the data set from least to greatest value: To find the lower quartile ( Thus, our lower quartile is at Since our 3rd number is 2, and our 4th number is 3, we need to find 1/4 of the way between 2 and 3. We will use the equation Thus, our We can repeat the process above to find the upper quartile ( So, our The last step is easy by comparison. Subtract Thus, our interquartile range is Example Question #1 : How To Find Interquartile Range Using the data provided above, what is the interquartile range (IQR)? Correct answer: How do you find the interquartile range? We can find the interquartile range or IQR in four simple steps: 1. Order the data from least to greatest 2. Find the median 3. Calculate the median of both the lower and upper half of the data 4. The IQR is the difference between the upper and lower medians Step 1: Order the data In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest). Let's sort an example data set with an odd number of values into ascending order. Now, let's perform this task with another example data set that is comprised of an even number of values. Rearrange into ascending order. Step 2: Calculate the median Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two. First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point The median of the odd valued data set is four. Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values. Find the average of the two centermost values. The median of the even valued set is four. Step 3: Upper and lower medians Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data. Omit the centermost value. Find the median of the lower portion. Calculate the average of the two values. The median of the lower portion is Find the median of the upper portion. Calculate the average of the two values. The median of the upper potion is If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions. Find the median of the lower portion. The median of the lower portion is two. Find the median of the upper portion. The median of the upper portion is eight. Step 4: Calculate the difference Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR. Let's find the IQR of the odd data set. Finally, we will find the IQR of the even data set. In order to better illustrate these values, their positions in a box plot have been labeled in the provided image. Now that we have solved a few examples, let's use this knowledge to solve the given problem. To find the IQR, we first must find the In data sets, and found Our upper half of our data set, the numbers above our median, now consists of We now have our Thus our Example Question #1 : How To Find Interquartile Range Using the data above, what is the interquartile range? Correct answer: How do you find the interquartile range? We can find the interquartile range or IQR in four simple steps: 1. Order the data from least to greatest 2. Find the median 3. Calculate the median of both the lower and upper half of the data 4. The IQR is the difference between the upper and lower medians Step 1: Order the data In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest). Let's sort an example data set with an odd number of values into ascending order. Now, let's perform this task with another example data set that is comprised of an even number of values. Rearrange into ascending order. Step 2: Calculate the median Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two. First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point The median of the odd valued data set is four. Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values. Find the average of the two centermost values. The median of the even valued set is four. Step 3: Upper and lower medians Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data. Omit the centermost value. Find the median of the lower portion. Calculate the average of the two values. The median of the lower portion is Find the median of the upper portion. Calculate the average of the two values. The median of the upper potion is If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions. Find the median of the lower portion. The median of the lower portion is two. Find the median of the upper portion. The median of the upper portion is eight. Step 4: Calculate the difference Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR. Let's find the IQR of the odd data set. Finally, we will find the IQR of the even data set. In order to better illustrate these values, their positions in a box plot have been labeled in the provided image. Now that we have solved a few examples, let's use this knowledge to solve the given problem. To find the IQR, we first must find the In data sets, In a previous problem, we placed the data pieces in numerical order: and found Our upper half of our data set, the numbers above our median, now consists of We now have our Thus our Example Question #2 : How To Find Interquartile Range Using the data provided, find the Interquartile range, IQR. Correct answer: How do you find the interquartile range? We can find the interquartile range or IQR in four simple steps: 1. Order the data from least to greatest 2. Find the median 3. Calculate the median of both the lower and upper half of the data 4. The IQR is the difference between the upper and lower medians Step 1: Order the data In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest). Let's sort an example data set with an odd number of values into ascending order. Now, let's perform this task with another example data set that is comprised of an even number of values. Rearrange into ascending order. Step 2: Calculate the median Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two. First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point The median of the odd valued data set is four. Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values. Find the average of the two centermost values. The median of the even valued set is four. Step 3: Upper and lower medians Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data. Omit the centermost value. Find the median of the lower portion. Calculate the average of the two values. The median of the lower portion is Find the median of the upper portion. Calculate the average of the two values. The median of the upper potion is If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions. Find the median of the lower portion. The median of the lower portion is two. Find the median of the upper portion. The median of the upper portion is eight. Step 4: Calculate the difference Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR. Let's find the IQR of the odd data set. Finally, we will find the IQR of the even data set. In order to better illustrate these values, their positions in a box plot have been labeled in the provided image. Now that we have solved a few examples, let's use this knowledge to solve the given problem. The data set provided is called a five number summary. These data values allow us to find the median, IQR, and range. This question is asking for the IQR which is Example Question #1 : How To Find Interquartile Range Using the data above, find the interquartile range. Correct answer: How do you find the interquartile range? We can find the interquartile range or IQR in four simple steps: 1. Order the data from least to greatest 2. Find the median 3. Calculate the median of both the lower and upper half of the data 4. The IQR is the difference between the upper and lower medians Step 1: Order the data In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest). Let's sort an example data set with an odd number of values into ascending order. Now, let's perform this task with another example data set that is comprised of an even number of values. Rearrange into ascending order. Step 2: Calculate the median Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two. First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point The median of the odd valued data set is four. Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values. Find the average of the two centermost values. The median of the even valued set is four. Step 3: Upper and lower medians Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data. Omit the centermost value. Find the median of the lower portion. Calculate the average of the two values. The median of the lower portion is Find the median of the upper portion. Calculate the average of the two values. The median of the upper potion is If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions. Find the median of the lower portion. The median of the lower portion is two. Find the median of the upper portion. The median of the upper portion is eight. Step 4: Calculate the difference Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR. Let's find the IQR of the odd data set. Finally, we will find the IQR of the even data set. In order to better illustrate these values, their positions in a box plot have been labeled in the provided image. Now that we have solved a few examples, let's use this knowledge to solve the given problem. The Interquartile range, or IQR, is defined as the The first step is the find the median of the data set, which in this case is For the upper quartile, if placed in numerical order we see that there is an even number, thus we must take the center two numbers and find the average to find the true center of this data set, giving us We do the same for the lower quartile, giving us a When we subtract Example Question #1 : How To Find Interquartile Range Using the data above, find the IQR. (interquartile range) Correct answer: How do you find the interquartile range? We can find the interquartile range or IQR in four simple steps: 1. Order the data from least to greatest 2. Find the median 3. Calculate the median of both the lower and upper half of the data 4. The IQR is the difference between the upper and lower medians Step 1: Order the data In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest). Let's sort an example data set with an odd number of values into ascending order. Now, let's perform this task with another example data set that is comprised of an even number of values. Rearrange into ascending order. Step 2: Calculate the median Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two. First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point The median of the odd valued data set is four. Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values. Find the average of the two centermost values. The median of the even valued set is four. Step 3: Upper and lower medians Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data. Omit the centermost value. Find the median of the lower portion. Calculate the average of the two values. The median of the lower portion is Find the median of the upper portion. Calculate the average of the two values. The median of the upper potion is If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions. Find the median of the lower portion. The median of the lower portion is two. Find the median of the upper portion. The median of the upper portion is eight. Step 4: Calculate the difference Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR. Let's find the IQR of the odd data set. Finally, we will find the IQR of the even data set. In order to better illustrate these values, their positions in a box plot have been labeled in the provided image. Now that we have solved a few examples, let's use this knowledge to solve the given problem. To find the IQR, we must first find the thus the thus the Certified Tutor Bank Street College of Education , Master's/Graduate, Early Childhood and Childhood General Education. Certified Tutor Emory University, Bachelor of Science, Mathematics/Economics. Certified Tutor Georgia Southern University, Bachelor of Science, Biochemistry. University of Georgia, Current Grad Student, Pharmacy. All SAT II Math I Resources	{"url":"https://www.varsitytutors.com/sat_ii_math_i-help/quartiles-and-interquartile-range?page=1","timestamp":"2024-11-07T23:44:32Z","content_type":"application/xhtml+xml","content_length":"266576","record_id":"<urn:uuid:f38fe9d8-4294-4665-8371-3e4b12e50b5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00235.warc.gz"}
How to Insert Section from Source Doc into Target Doc 05-03-2024, 08:27 AM #1 VBAX Newbie May 2024 Hello all, I am looking for help inserting a full section from a source doc into a target document. For example: "3. SECTION HEADING EXAMPLE 3.1 Text here 3.2 Text here 3.3 Text here Table here: Table Heading Text Text Text Image Image 3.4 Text here" I need the section from the source document to be pasted into a specific area in the target document, for example, between "2. SECTION HEADING EXAMPLE" and "4. SECTION HEADING EXAMPLE" I know this is possible because I used a code that did this previously but I cannot find it. I have a code that this sub will be inserted into that loops through all documents in a folder and performs a subroutine -- it will open all documents in a folder, perform the action, and close the document. All the documents need this specific section added between existing Section X and Y. Thank you for any help you can provide! Welcome to VBAX iwritedoc62. Until one of the word guru's drops by, perhaps this will get you started in the right direction Sub InsertAfterMethod() Dim MyText As String Dim MyRange As Object Set MyRange = ActiveDocument.Range MyText = "<Replace this with your text>" ' Selection Example: Selection.InsertAfter (MyText) ' Range Example: ' (Inserts text at the current position of the insertion point.) MyRange.InsertAfter (MyText) End Sub Yes, it doesn't loop through your documents, but if you wish to test it and see if it can be modified. Remember To Do the Following.... Use [Code].... [/Code] tags when posting code to the thread. Mark your thread as Solved if satisfied by using the Thread Tools options. If posting the same issue to another forum please show the link Here's a VBA code that should accomplish this: Sub InsertSection() Dim srcDoc As Document Dim tgtDoc As Document Dim srcRng As Range Dim tgtRng As Range Dim shp As Shape Const srcHeading As String = "3. SECTION HEADING EXAMPLE" Const tgtHeading As String = "2. SECTION HEADING EXAMPLE" For Each srcDoc In ActiveDocument.Parent.Documents ' Find the source section based on the heading Set srcRng = srcDoc.Sections(1).Range While srcRng.Find.Execute(srcHeading) Set srcRng = srcRng.Find.Found Exit While End While ' Copy the entire section ' Find the target section in the target document Set tgtDoc = ActiveDocument Set tgtRng = tgtDoc.Sections(1).Range While tgtRng.Find.Execute(tgtHeading) Set tgtRng = tgtRng.Find.Found Exit While End While ' Paste the copied section after the target heading tgtRng.Collapse wdCollapseEnd ' Copy any shapes within the section For Each shp In srcDoc.Shapes If shp.Range.Start >= srcRng.Start And shp.Range.End <= srcRng.End Then End If Next shp Next srcDoc End Sub bro, the post was made 4 months ago.. Remember To Do the Following.... Use [Code].... [/Code] tags when posting code to the thread. Mark your thread as Solved if satisfied by using the Thread Tools options. If posting the same issue to another forum please show the link Here's a VBA code that should accomplish this: Sub InsertSection() Dim srcDoc As Document Dim tgtDoc As Document Dim srcRng As Range Dim tgtRng As Range Dim shp As Shape Const srcHeading As String = "3. SECTION HEADING EXAMPLE" Const tgtHeading As String = "2. SECTION HEADING EXAMPLE" For Each srcDoc In ActiveDocument.Parent.Documents ' Find the source section based on the heading Set srcRng = srcDoc.Sections(1).Range While srcRng.Find.Execute(srcHeading) Set srcRng = srcRng.Find.Found Exit While End While ' Copy the entire section ' Find the target section in the target document Set tgtDoc = ActiveDocument Set tgtRng = tgtDoc.Sections(1).Range While tgtRng.Find.Execute(tgtHeading) Set tgtRng = tgtRng.Find.Found Exit While End While ' Paste the copied section after the target heading tgtRng.Collapse wdCollapseEnd ' Copy any shapes within the section For Each shp In srcDoc.Shapes If shp.Range.Start >= srcRng.Start And shp.Range.End <= srcRng.End Then End If Next shp Next srcDoc End Sub I entered your code but the system still reports an error, why? Remember To Do the Following.... Use [Code].... [/Code] tags when posting code to the thread. Mark your thread as Solved if satisfied by using the Thread Tools options. If posting the same issue to another forum please show the link That code is pretty awful. Try: Sub XferHeadingRange()Application.ScreenUpdating = False Dim DocSrc As Document, RngSrc As Range Dim DocTgt As Document, RngTgt As Range Set DocTgt = ActiveDocument With Application.Dialogs(wdDialogFileOpen) If .Show = -1 Then .AddToMru = False .ReadOnly = True .Visible = False Set DocSrc = ActiveDocument End If End With If DocSrc Is Nothing Then Exit Sub With DocSrc.Range With .Find .Text = "SOURCE SECTION HEADING EXAMPLE" .Replacement.Text = "" .Forward = True .Wrap = wdFindStop .MatchWildcards = False End With If .Find.Found = True Then Set RngSrc = .Paragraphs(1).Range Set RngSrc = RngSrc.GoTo(What:=wdGoToBookmark, Name:="\HeadingLevel") MsgBox "Source Content Not Found!", vbExclamation GoTo CleanUp End If End With With DocTgt.Range With .Find .Text = "TARGET SECTION HEADING EXAMPLE" .Replacement.Text = "" .Forward = True .Wrap = wdFindStop .MatchWildcards = False End With If .Find.Found = True Then Set RngTgt = .Paragraphs(1).Range RngTgt.Collapse wdCollapseStart RngTgt.FormattedText = RngSrc.FormattedText MsgBox "Destination Not Found!", vbExclamation End If End With : CleanUp DocSrc.Close SaveChanges:=False Set RngSrc = Nothing: Set DocSrc = Nothing Set RngTgt = Nothing: Set DocTgt = Nothing Application.ScreenUpdating = True End Sub Note: The code assumes the target document is already open and that Word's Heading Styles are used in the source document to denote the various ranges. When running the macro, you select the source document to open from the dialog box. For the target document, the heading to specify is the one after the location where you want the source content inserted. Paul Edstein [Fmr MS MVP - Word] 05-03-2024, 01:08 PM #2 09-19-2024, 01:07 AM #3 VBAX Newbie Sep 2024 09-19-2024, 02:44 AM #4 VBAX Expert Sep 2019 09-19-2024, 07:27 PM #5 10-01-2024, 09:43 PM #6 VBAX Newbie Oct 2024 10-02-2024, 01:07 AM #7 10-02-2024, 05:45 PM #8 VBAX Guru Jul 2008	{"url":"http://www.vbaexpress.com/forum/showthread.php?71610-How-to-Insert-Section-from-Source-Doc-into-Target-Doc&s=deca2f50a67866b017e70884e29a3ce6","timestamp":"2024-11-13T19:44:30Z","content_type":"application/xhtml+xml","content_length":"79494","record_id":"<urn:uuid:97d16c8f-4001-4fe3-af4e-9f8f73c93c82>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00675.warc.gz"}
A3F-CNS Summer School 2021 In this research, we tested a new idea to measure proton-distribution radii ($r_{\rm{p}}$) by heavy-ion secondary beam experiments. It is important for understanding the structures of nuclei to know the proton- and the neutron-distribution radii independently. From this point of view, we tried to develop a new method to deduce proton-distribution radii ($r_{\rm{p}}$) very efficiently using nuclear collisions . Now, $r_{\rm{p}}$ can be measured by electron scattering and isotope shift measurements. They have high accuracy and precision, but applicable unstable nuclei are rather limited. On the other hand, the present new method could have the same degrees of accuracy and could measure a wide range of unstable nuclei. The experiment was carried out at HIMAC, Heavy Ion Medical Accelerator in Chiba, in Japan. We measured charge changing cross sections ($\sigma_{\rm{cc}}$) for $^{7-12}$Be isotopes on proton, Be, C, and Al targets. Charge changing cross section ($\sigma_{\rm{cc}}$) is the cross section of changing the number of protons in the collision with the target nucleus. We can deduce charge changing cross sections ($\sigma_{\rm{cc}}$) from the number of incident particles $N_1$ and charge changed particles $N_2$: In the zeroth-order approximation, charge changing reaction can be attributed to the abrasion of protons in the incident nucleus by nucleons in the target nucleus. A schematic drawing of this process is shown in figure 1. Thus it is approximated by equation (2). ![charge changing][1] From eq (2), we can derive proton radii if target’s nucleon radius $r_{\rm{T}}$ and $\sigma_{\rm{cc}}$ are known. In practice, we need to use Glauber calculation with more realistic proton and neutron distributions both in the projectile and the target nuclei. Thus, when trying to link the charge change cross-section and the proton distribution radius, the consideration of the proton evaporation process shown in fig. 2 is considered to be very important. In this process, neutrons are firstly abraded, which excites prefragment and results in the evaporation of protons. If this process could be extracted independently, it would be very useful in deriving the proton-distribution radii from the charge change cross sections. ![proton evaporation][2] In the experiment, we used proton, Be, C, and Al targets. Proton target is particularly sensitive to neutrons in the projectile reflecting the isospin asymmetry of the nucleon-nucleon total cross sections, which amplifies neutron abrasion. In short, the proton-evaporation effect has large portion of the charge changing cross section on proton target $\sigma^{\rm{p}}_{cc}$. So, we assumed that $\sigma^{\rm{p}}_{cc}$ multiplied by some value x: $x\sigma^{\rm{p}}_{cc}$ is the cross section of proton evaporation for Be, C, and Al targets. Therefore, adding $x\sigma^{\rm {p}}_{cc}$ to eq (2) would reproduce the experimental results of charge changing cross sections. In practice, we introduced x for each target and a constant parameter Y as the first and second approximation terms: \sigma_{cc} = \sigma_{\rm{Glauber}} + As a result, we figured out that only 4 parameters, x(for 3 targets) and Y could reproduce 15 data of charge changing cross section for Be isotopes very well. It suggests a possibility of this new method for the deduction of proton-distribution radii with high accuracy and efficiency applicable to a wide range of unstable nuclei. ![proton distribution radii][3]	{"url":"https://indico3.cns.s.u-tokyo.ac.jp/event/145/timetable/?print=1&view=standard","timestamp":"2024-11-08T05:42:02Z","content_type":"text/html","content_length":"196506","record_id":"<urn:uuid:897c138d-24fb-432f-bfc9-a1e6bc295b25>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00584.warc.gz"}
Tyler needs at least $205 for a new video game system. He has already saved $30. He earns $7 an hour at his job. Write and solve an inequality to find how many hours he will need to work to buy the system. Interpret the solution._? \| Socratic Tyler needs at least $205 for a new video game system. He has already saved $30. He earns $7 an hour at his job. Write and solve an inequality to find how many hours he will need to work to buy the system. Interpret the solution._? 1 Answer Tyler needs to work at least 25 hours The words "at least" indicate that an "inequality" is needed here to describe Tyler's situation. let's use the letter $T$ (Target) to represent the amount of money that Tyler needs, $x$ for the amount of money he has and $h$ for the number of hours he works. things we are told: $T \ge 205$ $x = 30 + 7 h$ If he reaches his target then $x = T$ so: $30 + 7 h \ge 205$ subtract 30 from both sides $7 h \ge 203 - 30$ $7 h \ge 175$ divide both sides by 7 $h \ge 25$ Impact of this question 4869 views around the world	{"url":"https://socratic.org/questions/tyler-needs-at-least-205-for-a-new-video-game-system-he-has-already-saved-30-he-","timestamp":"2024-11-05T07:12:22Z","content_type":"text/html","content_length":"33034","record_id":"<urn:uuid:ec5be0b2-e702-4eeb-9e3d-bede5ab7f860>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00356.warc.gz"}
2nd Pulley Diameter `D_2 = D_1 (( RPM_1 )/( RPM_2 ))` Enter a value for all fields The 2nd Pulley Diameter equation calculates the diameter of a second pulley based INSTRUCTIONS: Choose units and enter the following: • (RPM[1]) RPMs of Pulley 1 • (D[1]) Diameter of Pulley 1 • (RPM[2]) RPMs of Pulley 2 2nd Pulley Diameter (D[2]): The calculator returns the diameter of pulley 2 in meters. However this can be automatically converted to other length units via the pull-down menu. The Math / Science The formula for the diameter of the second pulley is: `D_2 = D_1((RPM_1)/(RPM_2))` • D[2] = Diameter of second pulley • D[1] = Diameter of first pulley • RPM[1] = Rotation rate of first pulley • RPM[2] = Rotation rate of second pulley Enhance your vCalc experience with a free account Sign Up Now! Sorry, JavaScript must be enabled. Change your browser options, then try again.	{"url":"https://www.vcalc.com/wiki/2nd-pulley-diameter","timestamp":"2024-11-07T07:43:56Z","content_type":"text/html","content_length":"55532","record_id":"<urn:uuid:0fb10871-6b65-4878-bef8-a06d3bf03db5>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00665.warc.gz"}
Cryptographic Hardness of Learning Halfspaces with Massart Noise Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track Ilias Diakonikolas, Daniel Kane, Pasin Manurangsi, Lisheng Ren We study the complexity of PAC learning halfspaces in the presence of Massart noise. In this problem, we are given i.i.d. labeled examples $(\mathbf{x}, y) \in \mathbb{R}^N \times \{ \pm 1\}$, where the distribution of $\mathbf{x}$ is arbitrary and the label $y$ is a Massart corruption of $f(\mathbf{x})$, for an unknown halfspace $f: \mathbb{R}^N \to \{ \pm 1\}$, with flipping probability $\eta (\mathbf{x}) \leq \eta < 1/2$. The goal of the learner is to compute a hypothesis with small 0-1 error. Our main result is the first computational hardness result for this learning problem. Specifically, assuming the (widely believed) subexponential-time hardness of the Learning with Errors (LWE) problem, we show that no polynomial-time Massart halfspace learner can achieve error better than $\Omega(\eta)$, even if the optimal 0-1 error is small, namely $\mathrm{OPT} = 2^{-\log^{c} (N)}$ for any universal constant $c \in (0, 1)$. Prior work had provided qualitatively similar evidence of hardness in the Statistical Query model. Our computational hardness result essentially resolves the polynomial PAC learnability of Massart halfspaces, by showing that known efficient learning algorithms for the problem are nearly best possible.	{"url":"https://proceedings.nips.cc/paper_files/paper/2022/hash/17826a22eb8b58494dfdfca61e772c39-Abstract-Conference.html","timestamp":"2024-11-12T00:55:57Z","content_type":"text/html","content_length":"9226","record_id":"<urn:uuid:c13f7eed-0834-436e-9cc2-7ae28d475faf>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00426.warc.gz"}
Two sides of the coin problem In the coin problem, one is given n independent flips of a coin that has bias β > 0 towards either Head or Tail. The goal is to decide which side the coin is biased towards, with high confidence. An optimal strategy for solving the coin problem is to apply the majority function on the n samples. This simple strategy works as long as β > Ω(1/√n). However, computing majority is an impossible task for several natural computational models, such as bounded width read once branching programs and AC[0] circuits. Brody and Verbin [8] proved that a length n, width w read once branching program cannot solve the coin problem for β < O(1/(log[n])[w]). This result was tightened by Steinberger [20] to O(1/(log n)[w-2]). The coin problem in the model of AC[0] circuits was first studied by Shaltiel and Viola [19], and later by Aaronson [1] who proved that a depth d size s Boolean circuit cannot solve the coin problem for β < O(1/(log[s])[d+2]). This work has two contributions: We strengthen Steinberger result and show that any Santha-Vazirani source with bias β < O(1/(log n)[w-2]) fools length n, width w read once branching programs. In other words, the strong independence assumption in the coin problem is completely redundant in the model of read once branching programs, assuming the bias remains small. That is, the exact same result holds for a much more general class of sources. We tighten Aaronson's result and show that a depth d, size s Boolean circuit cannot solve the coin problem for β < O(1/(log[s])[d-1]). Moreover, our proof technique is different and we believe that it is simpler and more natural. Publication series Name Leibniz International Proceedings in Informatics, LIPIcs Volume 28 Conference 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2014 and the 18th International Workshop on Randomization and Computation, RANDOM 2014 Country/ Spain City Barcelona Period 4/09/14 → 6/09/14 • Bounded depth circuits • Read once branching programs • Santha-Vazirani sources • The coin problem All Science Journal Classification (ASJC) codes Dive into the research topics of 'Two sides of the coin problem'. Together they form a unique fingerprint.	{"url":"https://cris.iucc.ac.il/en/publications/two-sides-of-the-coin-problem","timestamp":"2024-11-02T22:21:58Z","content_type":"text/html","content_length":"46782","record_id":"<urn:uuid:0d753c58-eaf2-4471-b911-1c3a3d97401f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00836.warc.gz"}
Let's Get Rid of Zero! One of my tweeps sent me a link to a delightful pile of rubbish: a self-published “paper” by a gentleman named Robbert van Dalen that purports to solve the “problem” of zero. It’s an amusing pseudo-technical paper that defines a new kind of number which doesn’t work without the old numbers, and which gets rid of zero. Before we start, why does Mr. van Dalen want to get rid of zero? So what is the real problem with zeros? Zeros destroy information. That is why they don’t have a multiplicative inverse: because it is impossible to rebuilt something you have just destroyed. Hopefully this short paper will make the reader consider the author’s firm believe that: One should never destroy anything, if one can help it. We practically abolished zeros. Should we also abolish simplifications? Not if we want to stay practical. There’s nothing I can say to that. So what does he do? He defines a new version of both integers and rational numbers. The new integers are called accounts, and the new rationals are called super-rationals. According to him, these new numbers get rid of that naughty information-destroying zero. (He doesn’t bother to define real numbers in his system; I assume that either he doesn’t know or doesn’t care about them.) Before we can get to his definition of accounts, he starts with something more basic, which he calls “accounting naturals”. He doesn’t bother to actually define them – he handwaves his way through, and sort-of defines addition and multiplication, with: a + b == a concat b a * b = a concat a concat a … (with b repetitions of a) So… a sloppy definition of positive integer addition, and a handwave for multiplication. What can we take from this introduction? Well, our author can’t be bothered to define basic arithmetic properly. What he really wants to say is, roughly, Peano arithmetic, with 0 removed. But my guess is that he has no idea what Peano arithmetic actually is, so he handwaves. The real question is, why did he bother to include this at all? My guess is that he wanted to pretend that he was writing a serious math paper, and he thinks that real math papers define things like this, so he threw it in, even though it’s pointless drivel. With that rubbish out of the way, he defines an “Account” as his new magical integer, as a pair of “account naturals”. The first member of the pair is called a the credit, and the second part is the debit. If the credit is a and the debit is b, then the account is written (a%b). (He used backslash instead of percent; but that caused trouble for my wordpress config, so I switched to a%b ++ c%d = (a+c)%(b+d) a%b ** c%d = ((ac)+(bd))%((ad)+(bc)) – a%b = b%a So… for example, consider 56. We need an “account” for each: We’ll use (7%2) for 5, and (9%3) for 6, just to keep things interesting. That gives us: 56 = (7%2)(9%3) = (63+6)%(21+18) = 69%39, or 30 in regular numbers. Yippee, we’ve just redefined multiplication in a way that makes us use good old natural number multiplication, only now we need to do it four times, plus 2 additions to multiply two numbers! Wow, progress! (Of a sort. I suppose that if you’re a cloud computing provider, where you’re renting CPUs, then this would be progress. Oh, but that’s not all. See, each of these “accounts” isn’t really a number. The numbers are equivalence classes of accounts. So once you get the result, you “simplify” it, to make it easier to work So make that 4 multiplications, 2 additions, and one subtraction. Yeah, this is looking nice, huh? So… what does it give us? As far as I can tell, absolutely nothing. The author promises that we’re getting rid of zero, but it sure likes like this has zeros: 1%1 is zero, isn’t it? (And even if we pretend that there is no zero, Mr. van Dalen specifically doesn’t define division on accounts, we don’t even get anything nice like closure.) But here’s where it gets really rich. See, this is great, cuz there’s no zero. But as I just said, it looks like 1%1 is 0, right? Well it isn’t. Why not? Because he says so, that’s why! Really. Here’s a verbatim quote: An Account is balanced when Debit and Credit are equal. Such a balanced Account can be interpreted as (being in the equivalence class of) a zero but we won’t. But, according to him, we don’t actually get to see these glorious benefits of no zero until we add rationals. But not just any rationals, dum-ta-da-dum-ta-da! super-rationals. Why super-rationals, instead of account rationals? I don’t know. (I’m imagining a fraction with blue tights and a red cape, flying over a building. That would be a lot more fun than this little “paper”.) So let’s look as the glory that is super-rationals. Suppose we have two accounts, e = a%b, and f = c%d. Then a “super-rational” is a ratio like e/f. So… we can now define arithmetic on the super-rationals: e/f +++ g/h = ((eh)++(gf))/(fh); or in other words, pretty much exactly what we normally do to add two fractions. Only now those multiplications are much more laborious. e/f g/h = (eg)/(fh); again, standard rational mechanics. Multiplication Inverse (aka Reciprocal) `e/f = f/e; (he introduces this hideous notation for no apparent reason – backquote is reciprocal. Why? I guess for the same reason that he did ++ and +++ – aka, no particularly good reason. So, how does this actually help anything? It doesn’t. See, zero is now not really properly defined anymore, and that’s what he wants to accomplish. We’ve got the simplified integer 0 (aka “balance”), defined as 1%1. We’ve got a whole universe of rational pseudo-zeros – 0/1, 0/2, 0/3, 0/4, all of which are distinct. In this system, (1%1)/(4%2) (aka 0/2) is not the same thing as (1%1)/(5%2) (aka 0/3)! The “advantage” of this is that if you work through this stupid arithmetic, you essentially get something sort-of close to 0/0 = 0. Kind-of. (There’s no rule for converting a super-rational to an account; assuming that if the denominator is 1, you can eliminate it, you get 1/0 = 0: I’m guessing that he intends identities to apply, so: (4%1)/(1%1) = ((4%1)/(2%1)) * `((2%1)/(1%1)) = ((4%1)/(2%1)) *** ((1%1)/(2%1)) = (1%1)/(2%1). So 1/0 = 0/1 = 0… If you do the same process with 2/0, you end up getting the result being 0/2. And so on. So we’ve gotten closure over division and reciprocal by getting rid of zero, and replacing it with an infinite number of non-equal What’s his answer to that? Of course, more hand-waving! Note that we also can decide to simplify a Super- Rational as we would a Rational by calculating the Greatest Common Divisor (GCD) between Numerator and Denominator (and then divide them by their GCD). There is a catch, but we leave that for further research. The catch that he just waved away? Exactly what I just pointed out – an infinite number of pseudo-0s, unless, of course, you admit that there is a zero, in which case they all collapse down to be zero… in which case this is all pointless. Essentially, this is all a stupidly overcomplicated way of saying something simple, but dumb: “I don’t like the fact that you can’t divide by zero, and so I want to define x/0=0.” Why is that stupid? Because dividing by zero is undefined for a reason: it doesn’t mean anything! The nonsense of it becomes obvious when you really think about identities. If 4/2 = 2, then 22=4; if x/y=z, then x=zy. But mix zero in to that: if 4/0 = 0, then 00=4. That’s nonsense. You can also see it by rephrasing division in english. Asking “what is four divided by two” is asking “If I have 4 apples, and I want to distribute them into 2 equal piles, how many apples will be in each pile?”. If I say that with zero, “I want to distribute 4 apples into 0 piles, how many apples will there be in each pile?”: you’re not distributing the apples into piles. You can’t, because there’s no piles to distribute them to. That’s exactly the point: you can’t divide by zero. If you do as Mr. van Dalen did, and basically define x/0 = 0, you end up with a mess. You can handwave your way around it in a variety of ways – but they all end up breaking things. In the case of this account nonsense, you end up replacing zero with an infinite number of pseudo-zeros which aren’t equal to each other. (Or, if you define the pseudo-zeros as all being equal, then you end up with a different mess, where (2/0)/(4/0) = 2/4, or other weirdness, depending on exactly how you defie things.) The other main approach is another pile of nonsense I wrote about a while ago, called nullity. Zero is an inevitable necessity to make numbers work. You can hate the fact that division by zero is undefined all you want, but the fact is, it’s both necessary and right. Division by zero doesn’t mean anything, so mathematically, division by zero is undefined. 43 thoughts on “Let's Get Rid of Zero!” 1. Simplicissimus Mr van Dalen seems to have rediscovered the construction of the integers from the natural numbers in the most roundabout and notationally grotesque way possible, modulo refusing to identify the equivalence class corresponding to the integer 0 for reasons of pure superstition. Now that is an achievement! 2. Alex R I wouldn’t call this all completely* pointless… In fact, except for the “leaving out zero” part, this is the standard approach for building up the definitions of various sets of numbers from pure set theory: start by defining the natural numbers as Von Neumann ordinals, then define the integers as equivalence classes of ordered pairs of naturals, then define the rationals as equivalence classes of ordered pairs of integers, etc. I’m sure, though, that in order for addition and multiplication on integers to be well defined, his “balanced accounts” will all have to be in the same equivalence class. He can call that class whatever he wants, but I’ll keep calling it zero… 3. Vince Vatter Not that I disagree, but I hope the stroke to your ego was worth the time you put into this. 1. clonus its not about ego. If you derived your self worth from comparing yourself to people like that in terms of math that would be extremely ridiculous. 4. Vince Vatter Let me just add one more thing. I have no idea how old this author is, but from your post it seems you don’t know either. Suppose this is an audacious 14-year-old. You have just replied with “Why is that stupid?”. Why can’t you reply instead (if you really have to reply at all) with “Let me tell you we don’t do this”? Again, I have no idea who you are critiquing (and you have provided no background), but it seems a bit aloof and out-of-touch to call them “stupid” when they might just be playing with ideas at this point in their development. Even if this wasn’t written by a 14-year-old, would you send this same signal to a 14-year-old who thought such thoughts? If so, I think you are slowly destroying mathematics. We ought to give people the space and time to think wrong thoughts, because the right ones don’t come easy. 1. Mikael Vejdemo-Johansson I suppose you are new to this blog? Because for anyone who has paid any sort of attention to Good Math/Bad Math, this is sort of what Mark does: he discusses various degrees of Wrong Thoughts, and points out what the problem with this particular brand of Wrong Thought actually is. And if you seriously believe mathematics can be so easily destroyed that a single dedicated layman can pull the entire edifice down — merely by heckling Things That Are Wrong — then I’m worried about the chances your mathematics stands overall… 2. John Fringe Wow, a 14 year old boy who works at a bank as a developer. That would be something! Do you realize that this post is not criticizing the paper for being wrong, but for being elementarily wrong while erroneously claiming a great accomplishment (abolishing zero) in an absurdly convoluted (yet trivial) way? If the paper were called “playing with the concept of zero” and didn’t claim to have solved anything, nobody would be mocking it. 5. Mikael Vejdemo-Johansson In our discussion on reddit, where Mr. van Dalen shows up and participates, it turns out that _equality_ is defined in the standard way for constructing integers and rationals. How his construction is supposed to be different at this stage is becoming somewhat unclear — although he is invoking “wheels” as the saving algebraic structure he aims for… 1. Mikael Vejdemo-Johansson The discussion is at http://www.reddit.com/r/math/comments/13vvx8/division_by_zero_is_undefined_so_we_have_decided/ btw. 6. John Fringe Even without wanting to listen anyone telling him that his zero is a zero, “his system” “destroys information” in the very same way than zero. If I tell you that I multiplied a number x by zero and it gave me zero, you can’t tell me what my number x was. Which is the way it should be. I assume this is what he calls “destruction of In the same sense, if I tell you I multiplied an “account” (x%y) by (1%1) and the result was (3%3), you still can’t tell me what the account (x%y) was, nor even the class of equivalence. (2%1) (1%1) = (3%3) = (1%2)(1%1). You’ll not only have to erase zero from books, obfuscate computations for no reason and forbid people to speak about multiplicative inverses. You’ll also have to kill anyone asking quantitative questions of this kind. 7. Robbert van Dalen Author here. Thank you for your in-depth review. The article is not an academic paper so I can say whatever I want. It may have helped to google my name, showing that I’m not 14 years old and that I do now what reals are (but I am finitist so I don’t like them). Why not leave a message on my blog? Anyway, I do think you missed a subtle point. This has to do with lazy evaluation (of number expressions) versus strict evaluation (‘simplification’ of number expressions). For example the expression 8%6 is in the same equivalence class as expression 10%8 but that doesn’t make them equal. Same for expression 12%12 and 8%8. Lastly, the system I propose has similarities with the wheels and meadows algebras. Do you also consider those to be the works of pseudo scientists? 1. John Armstrong The system you “propose” is isomorphic to the standard construction for models of the integers and rationals from models of the Peano axioms. Just because you don’t understand what you’ve written doesn’t mean it’s anything new. 1. Robbert van Dalen So you know what I understand and don’t understand; great mind reading!. And do two isomorphic things make them equal? Same fallacy of equality. Also, I’ve already done some (previous) work that might make reconsider your last statement. Check out http://www.enchiladacode.nl and http://github.com/rapido/spread/tree/master/src/ But I guess my work is not interesting for mathematicians: that’s ok – you weren’t the target audience to begin with. 1. Mikael Vejdemo-Johansson There is definitely something interesting about making arithmetic lazy (in the sense of functional programming). If only you had written about that instead… 1. Robbert van Dalen Fair enough. I understand that my article was incomplete. But I do think that it does contain all the goods (implicitly). In my defense: the article was a kind of a (serious) satire that has not been picked up as such. I mean, can you take this statement seriously without LOL? ‘Note that the actual choice of the tally symbol is insignificant: we could even choose to use the zero symbol, but we will not.’ Also the property of (T)-Accounts is that they always should be balanced to be valuable: another jest. Nevertheless, zeros do destroy information. Still, I agree that I’ve may have pushed the research idea of ‘non-destructive’ computation to far without properly introducing the Is that a crime? I guess if you are an academic: yes. Anyway, I do enjoy discussions such as these, even if they appear to be a bit aggressive at the start: it sharpens the mind. I don’t like personal attacks though. 8. Tel You guys are missing a bit of context here. Suppose you want to build a computer that solves Newtonian physics problems. Well, we know that Newtonian physics is completely reversible… so if you simply swap time to work in the opposite direction then you can run any Newtonian physics in reverse to get back where you started. Because it is reversible the information content remains constant at all times. However, Thermodynamics is not reversible, because entropy always increases, and in real physics we have the somewhat strange rule that information CANNOT be destroyed. You might be thinking that your computer contains AND gates and OR gates and those very fundamental binary arithmetic operations are destroying information all the time. Yes, that is correct, but your computer is attached by wires to a massive coal fire and steam turbine that pumps out entropy at such a massive rate, the tiny amount in information destroyed on your desktop is nothing in comparison. By the way, physics does not have a concept of a zero (as far as I’ve ever heard). You can of course plug zero into an equation, but you can’t find it in the real world. That’s the background problem getting people thinking about number systems that can solve problems without destroying information. I agree that van Dalen’s effort probably isn’t going to be the next big breakthrough in this particular reconciliation between maths and physics. 1. MarkCC Post author I think you’re the one that’s missing the point. Zero, in both multiplication and division, does destroy information in some sense. But that’s not a problem, that’s a reality. In physics, when you have a reversible system, you don’t ever multiply or divide by zero. That doesn’t mean that zero doesn’t exist. Rather, it just continues to reinforce the fact that division by zero doesn’t make sense. We don’t make up equations randomly; we write them to fit observations of nature. Since division by zero doesn’t mean anything, then nothing in nature makes our equations divide by zero – because if it did, the equations would be wrong. Nothing in nature makes our equations multiply by 0 in places where it would destroy information, because if it did, the equations would be wrong. None of that says anything about why division by zero is a problem: it isn’t. Division by zero doesn’t work because it’s meaningless. You can spackle over that in all sorts of ways – the infinite zeros of van Dalen, IEEE’s NaN, the despicable nullity, projected geometry, inner infinities – but they’re all limited by the fact that it really doesn’t mean anything. That’s just a 1. Robbert van Dalen I’m not a mathematician, but I’m aware that division by zero is undefined because you want the axioms of a field to hold. But what about the meadow algebra, which is a commutative ring (not a field) extended with a slightly different definition of multiplicative inverse? Do you guys think such algebra is My construction is a variant of the wheel algebra but I’m not sure what kind of algebra class the wheel algebra is in. Do any of you know? I’m asking because I’m designing a total functional programming language and I want to use meadow or wheel numbers. 1. Mikael Vejdemo-Johansson > I’m aware that division by zero is undefined because you want the axioms of a field to hold Noooo, division by zero is undefined because if you defined it you would end up getting an algebraic structure we very seldom have any concrete use for. By all means, you can create a wheel, and that’s one of the sanest possibilities if you insist on having a well-defined division by zero. Not that using a wheel gets you away from any of the special case handling you need otherwise, nor will it make anything we already do any easier to do. But sure, if total definitions of all algebraic operations are THAT important to you, go for it. As for meadows, I hadn’t heard of them until you brought them up here — but I will notice that the original meadows paper itself points out that the authors do not trust the construction works for rationals or reals. Since the construction you propose is (with the equality definitions you gave over on reddit) is the classical construction of the rationals, this is a problem for your approach. If you WANT to complete the rationals to a wheel, why not just do that? Instead of this approximation to the classical constructions? 1. Robbert van Dalen But my construction IS already a wheel, but I did not need the invention of two special symbols to show that. The equality definitions I gave on reddit will tell you that. 2. Paolo G. Giarrusso After agreeing that these proposed systems to avoid zeros make little sense (especially this one since it was a joke). > Division by zero doesn’t work because it’s meaningless. […] That’s just a fact. I think Bertrand Russell explained why there are no “facts” in mathematics, just axioms and implications. I’m used to another interpretation of a/b, that is, how many times does ‘b’ fit into ‘a’. Under that interpretation, it makes intuitively a lot of sense to say that a/0 with a != 0 is infinite. And that works quite well both in physics class and with IEEE doubles (which return signed infinities, not NaN in this case – your reference above seems naive, but I still assume you know better). In most cases you can talk about the limit of a/x for x -> 0 to make these discourses rigorous; but at least at an intuitive level, just taking a/0 = infinity (with a single infinity) is quite simple. I’d dare say that it’s even elegant, since for instance f(x) = a/x is continuous when x = 0 – I think this has the same elegance as projective geometry (where you in fact use no infinitesimals, but have similar intuitions). I’m in fact curious to understand what non-standard analysis would have to say on this fact. What I read is that this discipline justifies the intuitive reasonings with infinitesimals and infinities used by the founders of analysis — where dividing nonzero by zero gives infinity. As I just learned, though, in fact that’s only true for division by infinitesimals (http:// en.wikipedia.org/wiki/Hyperreal_number), so the above is not formally correct in nonstandard analysis because I’d need to distinguish zero from infinitesimals. I wonder where do problems arise with a different approach where zero and infinitesimals are not distinguished. In case you wonder whether I get math at all, I’m not a mathematician, but I was selected to participate at the 2004 International Mathematical Olympiad (http://www.imo-official.org/ 1. MarkCC Post author Yeah, I know that mentioning NaN is a simplification combined with a bit of wishful thinking. I really hate the idea that 1/0 = infinity. There’s some amount of sense to it – but I think that it’s a patch over the real problem – and like most patches, it just makes the problem You can’t divide by zero, because dividing by zero doesn’t work. Saying that the answer is infinity makes it seem like you’ve defined it. But you still haven’t, really, because division is a function from a number to a number, and infinity isn’t a number. So patching division to say that 1/0=infinity doesn’t really fix it. You can patch that, and say, yeah, infinity is a number. But then you lose some essential mathematical identities. For example: 1. 1/0 = infinity 2. 2/0 = infinity 3. That means that 1/0 = 2/0 4. Therefore, 1 == 2 All that “defining” division by 0 does is defer the problem. It doesn’t eliminate it. Division by zero isn’t defined in the standard number field, and you can’t fix that without breaking the field. Breaking the field of real numbers breaks almost everything that we do with numbers. 2. John Fringe Ahem, didn’t we already see that “accounts” destroy information in these sense, too? (x%y) * (1%1) = (3%3) does not let you know what (x%y) are, exactly as it happen with 0. Because (1%1) is a zero! Second, what the hell does “physics does not have a concept of zero” mean? Is that one of those cool sentences nobody knows what it means? When I say a neutrino have zero rest mass,is that not physics? If I say “the net force acting over a body is zero”, is that not a physical statement? “The pressure field at that point is null, so zero pressure”. Hell, even a simple statement as “how many neutrons are in a hydrogen atom? Zero”. What the hell does “physics does not have a concept of zero” mean? Apart, you’d better not mix physics here. Because then you’ll have to tell me how do you measure things in “accounts”. If the net acceleration of a body is zero, I can build a gadget telling you that on a scale. But, what would an “account” gadget return? 1%1? 2%2? 3%3? Because they’re not equal, remember XD 1. John Fringe A: What’s our speed (wrt the floor)? B: Zero A: Arrrghhh, that’s unphysical! 2. Robbert van Dalen Here is a lazy example of not throwing away information as per your example: a = 2%1 * 1%1 => ((21)+(11))((21)+(11)) b = 1%1 * 1%2 => ((11)+(21))((11)+(21)) Note that a and b are not equal (expressions), but are in the same equivalence class. The idea is that you always keep the expressions in the background, not replacing them with their simpler versions (that are in the same equivalence class). Can this be made practical (keeping the expressions)? Yes, I think so. 1. Robbert van Dalen Oh, I hate this formatting stuff. Isn’t there an option to preview what I did? Next try: a = 2%1 * 1%1 => ((21)+(11))%(11)+(21)) b = 1%1 * 2%1 => ((11)+(21))%((11)+(21)) Note that a and b are not equal (expressions), but are in the same equivalence class. The idea is that you always keep the expressions in the background, not replacing them with their simpler versions (that are in the same equivalence class). Can this be made practical (keeping the expressions)? Yes, I think 1. John Fringe I already show you how your “math” loss the same information in the sense that a result does not keep track of the information required to generate it. If you’re now arguing that you can retain information by writing every step in a notebook, you can do that with classic math, too. And writing a lot less, of course XD You’re just trolling, I bet. Last system I work was a system involving 10000×10000 matrices (nothing uncommon these days), which I had to optimize. Try bookkeeping every operation there XD (and finding a reason to do that, and then finding a reason to do that in your add-nothing approach). 9. Nathan And this is what went through my mind halfway through this post: “I can’t blab such blibber blubber! My tongue isn’t make of rubber.” (c) Dr. Seuss by way of Mr Knox. Where do they bring this from? 10. John I’m going to apply the most generous interpretation that I can think of. This is not mathematics, but programming. Mathematicians have the luxury of being able to imagine equivalence classes with infinitely many members, prove that they obey the desired rules, and then work directly with the rules, ignoring the details of any particular representation. Programmers, however, need to find concrete representations. Preferably ones that use little memory and are quick to process. Programmers with little mathematical background don’t care much whether a representation obeys the mathematician’s rules, and don’t worry about the fact that the familiar theorems won’t necessarily apply. If the users don’t complain about too many bugs, it’s good enough. So what we have here is a redundant representation of the integers and rationals. We already have perfectly good representations that are smaller and faster to work with, so this one is going to have to offer something that they don’t. There are infinitely many representations of zero. If we divide a number by one of the zeroes, then multiply the result by the same one, do we get the original number back? Hasn’t everyone wanted to be able to cancel the zeroes in a fraction? Let’s try (using Mark’s % because I don’t know what will happen to a backslash). Say a = 5%3 / 1%0 and b = 1%1 / 1%0 (if I’m doing it right, these are the equivalents of 2 and 0). `b = 1%0 / 1%1 and a * `b = (5%3 1%0) / (1%0 1%1) = (5+0)%(0+3) / (1+0)%(1+0) = 5%3 / Then (5%3 / 1%1) * b = (5%3 1%1) / (1%1 1%0) = (5+3)%(5+3) / (1+0)%(0+1) = 8%8 / 1%1. Oh dear. What if we first multiply by a zero, then divide by it? Sadly, that gives the same result. So what we really have is a redundant representation of the integers and rationals that’s harder to work with and doesn’t present any advantage on the simplest application I could think of. It’s no good defining (by whatever means) what happens when you divide by zero if the result isn’t useful for anything. 1. Robbert van Dalen The outcome of your example equivalent to what happens when you apply the wheels algebra. In the wheels algebra there are two ‘attractors’ from which you cannot escape: 1/0 and 0/0. My construction follows the wheels algebra, except that there are more members of the ‘attractor’ equivalence classes 1/0 and 0/0. Normally, division by zero inescapably throws an error in a regular computing system. Alternatively, when applying the wheels algebra you also cannot escape the equivalent of such error (because of the attractors). So you could argue there is no difference. But there will be a subtle difference when you lazily evaluate (expressions) of numbers. 1. Noah Is the difference you are referring to related to the fact that lazy computations can terminate even if they contain nonterminating expressions (so expressions that would cause errors would not be computed)? I don’t see the value here if these “attractors” absorb in all cases, but perhaps there are ways out of that? 11. K. Curious, “Zeros destroy information.” I thought it was irreversible operators acting on any set of mathematical objects that destroyed information. Or more succintly, information is not conserved for these operators. Take for instance the AND operator. 4 input states -> 2 output states. So it follows, information is “lost” in merging these paths. Or worse yet. Someone hands you the output 10 and says it came from subtracting two numbers, x and y. Now we have an infinite number of input states from which to choose, e.g. any -(x,y) -> 10. 1. Robbert van Dalen You can have gates that are reversible. See reversible computing. It does come at a cost though. 12. Reinier Post I read Van Dalen’s article much more lightly, as a fairly tongue-in-cheek abstraction of the accounting practice of keeping a ‘credit side’ and ‘debit side’ in the general ledger, booking debits and credits on the respective sides and adding up, requiring the totals to be equal when the books close – a practice that does indeed avoid the destruction of information and the need for the number 0 or negative numbers, but clearly produces the same net results. 13. Pete Richter I might be wrong, but I think the “paper” was meant to be a joke, and wasn’t really intended to be serious. 14. John Fringe a = 1 * 0 b = 0 * 1 Notice how a and b are different expressions XD See? If this is what you’re calling “no information loss”, it has no relation to your “accounts” at all. 15. Robbert van DAlen You are correct that using accounts per se does not prevent us from losing information. And yes, bookkeeping any expression in the background would do the same trick, so indeed my argument was bogus. So let’s forget about accounts and concentrate on not destroying information. Here is analogy with spreadsheets: let’s say we have a spreadsheet that holds two 10000×10000 matrices, a and b. You can express the multiplication of a and b in the same spreadsheet. And the matrices a and b will still be there after multiplication: nothing is destroyed. 16. John Fringe Ok, one thing cleared. About the second, why should we concentrate on “not destroying information”? What’s the reason? I mean, when there is a reason, we already have the tool (it’s called “writting what you’re But there is no reason in most cases. Why should we spend resources (memory, etc.) keeping information we don’t need? In any case, all this have nothing to do with zero. 1. John Fringe … nor with math, of course. It’s just a question of bookkeeping. 1. Paolo G. Giarrusso Ignoring the paper, destroying no information is required for quantum computation. Logical gates in quantum computers should not destroy information; so if your gate has two inputs, it will need to have two outputs. Another potentially interesting property is that thermodynamics requires that since information is entropy and entropy cannot decrease, destroying information requires releasing a certain amount of energy to increase entropy elsewhere. If you want your computations to consume less energy than that, you need them to be reversible. Luckily (or sadly) our computations consume much more energy than this theoretical lower bound, so reversible computations won’t help increasing the battery life of your laptop. At least not today, and I guess not for the next 10-20 years. 17. Robbert van DAlen I still do believe there is some merit in my construction (although the paper was meant as a serious joke). Yes, accounts do not prohibit the destruction of information, but they do destroy less information. Also, my “paper” didn’t say anything about equality (which was kind of unfortunate). Equality is more problematic with Rationals that allow 0 as denominators, as you need to distinguish 1/0 and 0/0 as separate equivalent classes. If you don’t have 1/0 and 0/0 separated, everything will equal everything: that is certainly not something I would suggest as being useful. 18. Robbert van Dalen I’m following the wheel algebra where 0x = 0 does not hold in general. This is something I didn’t mention in my paper, but that was my intention. Here are some examples, taken from Jesper Carlstom’s paper ‘Wheels On Division by Zero’: 0x + 0y = 0xy (0/0)x = 0/0 x/x = 1 + 0x/x So with wheels, you cannot replace 0*x with 0. 19. Doggie I SHALL CALL THIS METHMATH	{"url":"http://www.goodmath.org/blog/2012/11/27/lets-get-rid-of-zero/","timestamp":"2024-11-08T21:52:15Z","content_type":"text/html","content_length":"229198","record_id":"<urn:uuid:fd569116-8e66-4df8-8d4a-86e63ff93358>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00255.warc.gz"}
These are medium and long-term projects, regarding scientific topics, that we aim to implement futurely. Space group symmetry from the crystal structure The purpose is to determine the space group symmetry for any crystal structure, including the relevant elements of symmetry. There exists already a free Fortran program that does essentially this, needs to be adapted. Glass construction using geometry and coordination constraints Given geometry constraints as dihedral angle and bond length distributions, and coordination constraints as coordination number distributions, construct glass systems using random number techniques to calculate the coordinates, the rings, etc., this way building atomic structures with order at short distance and disorder at long distance. These algorithms should consider periodic glass systems, where the cell end in a given direction should fit with the cell start in the same direction, including atoms and bonds, which can connect atoms in both ends of the cell.	{"url":"http://www.gamgi.org/project/projects_scientific.html","timestamp":"2024-11-05T02:25:33Z","content_type":"application/xhtml+xml","content_length":"2183","record_id":"<urn:uuid:3da1e0b4-0915-40c6-995a-2d71a3db6ee1>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00605.warc.gz"}
Math Vector Library Machine Learning Extension Programming Reference Manual Version 1.0 DD-00004-110 2.1 Introduction The Math Vector Library Machine Learning Extension is an extension to the Math Vector Library which provides high-performance function library with vectorized versions of standard mathematical functions. The Machine Learning extension builds upon the infrastructure of the Math Vector Library to provide perfomance tuned primitive functions for machine learning applications. The functions can operate both on dense and strided vectors, the latter can be supplied individually for result and operand vectors. Stride only executes the function on every This manual describes the Application Programming Interface (API) of the machine learning extension functions.	{"url":"https://techpubs.adelsbach-research.eu/d/dd-00004-110/html/VecMathRefFtn_PGsu4.html","timestamp":"2024-11-14T11:58:41Z","content_type":"text/html","content_length":"3289","record_id":"<urn:uuid:c3330102-8ab6-4fa2-a692-175c71865cff>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00492.warc.gz"}
MMM Pool Pricing Tl;dr An automated market maker implies that instead of a human determining sell/buy prices, they are determined by various parameters we are going to walk you through below. ๐ Now.. letโ s talk about price calculation! Some definitions Liquidity Pool: simply put, a liquidity pool is a shared pot of digital assets (i.e. tokens) Spot price - s[0]: As defined in the pool Delta - ฮ : As defined in the pool. Will be normalized to decimal value if the pool type is exponential All the below values should be normalized to decimal values for calculations (10% = 0.1, 100bps = 0.01, etc.) Royalty percentage - r: As defined in the pool (for when the pool is buyer) or the input params (for when the pool is seller) Seller fee basis points - f: As defined in metadata Maker fee - m: Maker fee, as defined by our maker/taker fee structure Taker fee - t: Taker fee, as defined by our maker/taker fee structure Effective LP fee - l: conditional function where it is zero if the pool is one sided and taken from the pool data if the pool is two sided Effective price - e: What the person interacting with the pool expects to receive (if pool is buyer) or pay (if pool is seller) Price updates With every trade that occurs, the spot price will update to better reflect the market conditions implied by a trade. After the pool buys a token, the price the pool is willing to pay for another token will be lower; similarly, after the pool sells a token, the price the pool is willing to sell another token will be higher. In the below functions, only discuss how spot price changes with each trade, effective prices will be discussed later. We will define s[n] as the spot price after n trades in a specific direction. If n is positive, it will be after n buy transactions, and if n is negative, it will be after n sell transactions. The graphs shown are also made such that once n becomes negative, it will be red and between parentheses. Linear pools For linear pools, the delta is interpreted as lamports. For ease of representation, we show delta and spot price in SOL s[n]โ =โ s[0]โ โ โ nฮ Exponential pools For exponential pools, the delta is interpreted as basis points. For ease of representation, we show delta in percentage. ${{s}}_{{n}}{\mathrm{ย }}{=}{\mathrm{ย }}{{s}}_{{0}}{\left(}\frac{{1}}{{1}{\mathrm{ย }}{+}{\mathrm{ย }}{\mathrm{ฮ }}}{{\right)}}^{{n}}$ Effective Price To enforce a spread between the buy and sell prices, we calculate effective price for selling with s[โ 1]instead of s[0]. However, the royalty, lp fee, and taker fee also contribute to the spread between the buy and sell prices. Note: The pool will ignore user input r values if the token is a pNFT or has OCP enforced royalties. For pNFT, we will always enforce full royalties, and for OCP tokens, we will either enforce full royalties, or use the dynamic royalty if available. We need to determine if a pool is two-sided to decide whether to charge LP fees. We can determine if a pool is two-sided by looking at how much SOL is deposited in the pool and how many NFTs are deposited into the pool. If the SOL amount deposited is > spot price and the amount of NFTs deposited is > 1, then it is a two-sided pool. Pool Buying (AKA Collection Offers) eโ =โ s[0](1โ rfโ lโ t) Pool Selling (AKA AMM Listings) eย =ย s[โ 1](1+rf+l+t) Example graphs The graphs below assume that the pool is two sided. Given a pool where the curve type is exponential, s[0]ย =ย 1.5, ฮ ย =ย 25%, rโ =โ 50%, fโ =โ 2%, ,tย =ย 1.5%, lย =ย 1%, the initial effective buy and sell prices are shown below. This shows that for example, after selling 2 tokens, the taker price to buy the 3rd taken will be around 3.03 SOL For readability reasons, the red and blue lines do not extend past the 0, even though the pool can still buy NFTs after selling some. However, if the pool was to buy a token, we can simulate this by extending the lines to show what the effective buy/selling prices will be. After buying one token, the pool is willing to sell at approximately 1.5 SOL and buy at 1.15 SOL instead of the previous 1.94 SOL and 1.44 SOL. Below is an example of a linear curve, with the same values as the exponential curve except for delta being 0.1 SOL Buyable/Sellable Tokens To find the number of NFTs to sell, it is just the number of NFTs that are deposited into the pool. The process to find the number of NFTs buyable is a little trickier, but the easiest way to do it is probably iteratively, as seen in the code snippet below. To do it in a non-iterative fashion, we can use the arithmetic and geometric sequence functions to find that if n is the number of NFTs buyable given a spot price and curve delta, and d is the amount deposited, then For linear pools: ${n}{=}\frac{{\left(}{2}{s}{+}{\mathrm{ฮ }}{\right)}{-}\sqrt{{\left(}{2}{s}{+}{\mathrm{ฮ }}{{\right)}}^{{2}}{-}{8}{\mathrm{ฮ }}{d}}}{{2}{\mathrm{ฮ }}}$ This is assuming that the deposited amount d is less than the amount needed to fulfill $\frac{{s}}{{\mathrm{ฮ }}}$ buys. Otherwise, n = d For exponential pools: we can define ${\mathrm{ฮณ}}{=}\frac{{1}}{{1}{\mathrm{ย }}{+}{\mathrm{ย }}{\mathrm{ฮ }}}$, then ${n}{=}{{l}{o}{g}}_{{\mathrm{ฮณ}}}{\left(}\frac{{d}{\left(}{\mathrm{ฮณ}} Maker Fees Most of the above discussion does not involve maker fees. Adding maker fees to the current iteration of MMM will change the behavior in two ways Since the pool maker pays the maker fees from the amount deposited, the above calculations for NFTs buyable will change to replace s with s(1+m) The pool will become unsustainable. Since maker fees are not factored into the spot price, every time a trade occurs, the deposited payment will be lowered by the maker fee amount paid. This amount is not received upon the next trade on the opposite side, as what would usually happen with spot price. We can change this with a new iteration of MMM where LP fees, which are always paid by the taker, can be used to fund maker fees.	{"url":"https://docs.magiceden.io/reference/mmm-pool-pricing","timestamp":"2024-11-03T16:22:48Z","content_type":"text/html","content_length":"1049780","record_id":"<urn:uuid:f8b3ff58-6ed7-40e3-98a4-385fb07cbbe4>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00514.warc.gz"}
Simple proof that the world is not continuous by pashute 2012-11-11 03:45:37 The following is a simple proof that the everything in the physical world is built of discreet that is non-continuous small parts. Space is not continuous. Time is not continuous. Nothing in the physical world is continuous. And the notion of infinity, both large numbers and small numbers is wrong, in the physical world. In math it is possible but in the physical world it is not. And here is why: Lets talk about a small section in space which has a line drawn through it. A location on that line takes up no space (as we are taught in grade school when we learn geometry, about points). Every other location on that line takes up no space. The line is built of the infinite collection of points on the line. But each line has a length of zero. So no matter how many times we multiply zero we should still be left with a length of zero. This proves that the notion of continuity is incorrect. The correct way to understand space is that there is a unit of space, that is the smallest distance. This unit is very small but NOT zero. It is so small, that most of the time we can consider it as insignificant, until there is enough of these to "show on our radar". An extremely large BUT NOT INFINITE number of these units is needed for us to be able to realize its existence, or measure its It is the same with time, mass, energy, frequency and any other measurable quantity in the physical world. Tagged in: You must LOGIN to add comments	{"url":"https://www.hiox.org/36675-simple-proof.php","timestamp":"2024-11-10T02:17:49Z","content_type":"text/html","content_length":"29119","record_id":"<urn:uuid:9c3a9020-08fa-4a72-8f7f-bd830795d5be>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00867.warc.gz"}
Continuous Joint Distribution < Probability distributions list < Continuous joint distribution A continuous joint distribution describes the probability of interaction between two continuous random variables. Its discrete counterpart is the discrete joint distribution which has a countable number of possible outcomes (e.g., 1, 2, 3…). Continuous joint distributions can be described by a non-negative, integrable function [1]. The leap from discrete joint distributions to continuous ones is much like the leap from single variable discrete random variables to continuous ones. However, as continuous joint distributions are two dimensional, double integrals are needed instead of sigma notation (Σ) to solve probability problems. The continuous joint distribution PDF The continuous joint distribution assigns relative likelihoods, via the likelihood function, to combinations of x and y. The probabilities p(x, y) are not traditional probabilities, in the sense that you can get a probability of, say 99% or 50% or 10%; as this is a continuous distribution, the probability of a specific value is always zero. Probabilities for joint continuous distributions are instead treated as volume problems [2], calculate with a double integral ∫∫. Continuous joint distributions are formally described by a joint probability density function, much in the same way that random variables are described by a “single” probability density function (pdf) , defined as follows: Two random variables X and Y are jointly continuous if there exists an integrable, non-negative function (a joint pdf) f[XY]: ℝ^2 → ℝ such that for any set A ∈ ℝ Example 1: Given the following joint PDF, find the constant c: Solution: We are given the x and y bounds (0 to 1), so insert the bounds and the given function (x + cy^2) into the double integral and solve. Watch the following video for a few examples of working with a continuous joint distribution: [1] Liu, M. STA 611: Introduction to Mathematical Statistics; Lecture 4: Random Variables and Distributions. Retrieved November 10, 2021 from: http://www2.stat.duke.edu/courses/Fall18/sta611.01/ [2] Westfall, P. & Henning, K. (2013). Understanding Advanced Statistical Methods. CRC Press. Comments? Need to post a correction? Please Contact Us.	{"url":"https://www.statisticshowto.com/continuous-joint-distribution/","timestamp":"2024-11-04T15:21:37Z","content_type":"text/html","content_length":"70632","record_id":"<urn:uuid:dcd675af-484d-4116-8ca2-d3326f15c7b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00797.warc.gz"}
Comments on Computational Complexity: An interesting serendipitous numberSuddenly invented the following scientific papersâ€™...This reminded me of how, as n -> Inf, (1 + 1/n)...The article is easily found... For instance https:... tag:blogger.com,1999:blog-3722233.post6364597152143042105..comments2024-11-03T21:27:06.200-06:00Lance Fortnowhttp://www.blogger.com/profile/ 06752030912874378610noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-3722233.post-20642217771333867372020-09-20T06:38:27.752-05:002020-09-20T06:38:27.752-05:00Suddenly invented the following <br />scientific papersâ€™ references paradox:<br /><br />https://www.youtube.com/watch?v=CMFcmutIEy4<br /><br />Yuly Shipilevskyhttps://www.blogger.com/profile/ 13699450530150796472noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-90146168831337914532020-09-19T14:00:34.217-05:002020-09-19T14:00:34.217-05:00This reminded me of how, as n -> Inf, (1 + 1/n)^n is about e. So I could imagine that, for "large enough" n, if x_n is "a bit more than" n, then 1/x_n is "a bit less than" 1/n, and x_{n+1} is "somewhat less than" e. Which means that x_{n+2} will have something^{n+2}, which seemed apt to be even larger. So it seemed like it should oscillate, at least sometimes. (Also, induction shows that x_n & gt;= 1 always.)<br /><br />At that point, I basically punted: by writing the five lines of Python to see that this happened (at least sometimes), and looking at the solution in this post :/<br />Josh article is easily found... For instance https://doi.org/10.1007/978-1-4471-0751-4_8 . Of course, it is behind a paywall, but then Sci-hub comes to rescue and you find the complete book: http://	{"url":"https://blog.computationalcomplexity.org/feeds/6364597152143042105/comments/default","timestamp":"2024-11-04T23:15:12Z","content_type":"application/atom+xml","content_length":"7874","record_id":"<urn:uuid:34e829d7-1e43-43cd-b60f-5eccb5a88c80>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00334.warc.gz"}
what is centrifuging in ball mill BALL MILL. Definition: The ball mills are also called grinding machines or fine crushers, date back to 1876 and are characterized by the use of balls (made of iron, steel or tungsten carbide) as grinding mills are horizontal, rotating cylindrical or cylindroconical steel shells, usually working as continuous machines. The size reduction is accomplished by impact of these balls as ... WhatsApp: +86 18838072829 To study the effect of RPM on the power consumption of ball mill. To calculate the critical speed (n c) of ball mill. 3. I NTRODUCTION: Generally the ball mills are known as the secondary size reduction equipment. The ball mill is made in a great many types and sizes and can be used on a greater variety of soft WhatsApp: +86 18838072829 If the mill is operated at very high speeds, the balls are carried right around in contact with the side of the mill, and the mill is said to be centrifugal. The minimum speed at which centrifuging occurs is called the critical speed of the mill, and under these conditions, centrifugal force will be exactly balanced by the weight of the ball. WhatsApp: +86 18838072829 sourcehut Log in — Register. ~changjiangsx/ sbm . summary; tree; log; refs WhatsApp: +86 18838072829 The mill was no longer overloaded after hours. The bottom graph illustrates selected samples of the conductivity probe profiles and clearly indicates the presence of centrifuging when the conductivity probe data shows no variation. The feed end of the mill started centrifuging first, followed later at the middle of the mill. WhatsApp: +86 18838072829 The MLB rumor mill is buzzing ahead of the Winter Meetings, with new reports on the sport's top stars. Who is at the forefront of the most intriguing rumors. WhatsApp: +86 18838072829 The length of the mill is approximately equal to its diameter. Principle of Ball Mill : Ball Mill Diagram. • The balls occupy about 30 to 50 percent of the volume of the mill. The diameter of ball used is/lies in between 12 mm and 125 mm. The optimum diameter is approximately proportional to the square root of the size of the feed. WhatsApp: +86 18838072829 Question. 2 answers. Sep 15, 2023. if a 5gm of copper selenide in grinded to nanoparticles by a planetary ball mill by 550rpm, 250 ml cylinder, 50 balls (10mm diameter), with ball to powder ratio WhatsApp: +86 18838072829 A tumbling mill is a collective name for the generally known ball mills, rod mills, tube mills, pebble mills and autogeneous mills. For all these kinds of mills the mechanics can be dealt with together, there being no substantial difference in the grinding process. There are two kinds of grinding body movements: either they describe an ... WhatsApp: +86 18838072829 A ball mill also known as pebble mill or tumbling mill is a milling machine that consists of a hallow cylinder containing balls; mounted on a metallic frame such that it can be rotated along its longitudinal axis. The balls which could be of different diameter occupy 30 50 % of the mill volume and its size depends on the feed and mill size. ... WhatsApp: +86 18838072829 A centrifuge machine is basically a rotating platform that can hold some tubelike glassware (or plasticware) and spin them in a high velocity so that the liquid inside them presses down hard on the bottom of the tubes. If you're looking for the definition of a centrifuge, here it is: Operating a swingbucket centrifuge. WhatsApp: +86 18838072829 In a ball mill or pebble mill the sizereduction is carried out by impact as the balls or pebbles drop from near the top of the shell. In a large ball mill the diameter of shell may be 3 m and in length. ... Little or no grinding is done when a mill is centrifuging, and operating speeds must be less than the critical. The critical speed ... WhatsApp: +86 18838072829 Although there are instances in which ball mills have been successfully operated at speeds ranging from 60% to 90% of their critical speed, it is a common practice to run the mills at speeds ... WhatsApp: +86 18838072829 CHAPTER 3 CENTRIFUGATION INTRODUCTION General Background: Centrifugal dewatering is widely used method for separating solid liquid or liquidliquid in several industries due to the higher gravitational forces affecting on the WhatsApp: +86 18838072829 Find out the critical speed of ball mill by using the following data: Diameter of the ball mill is 450mm and Diameter of the ball is 25 mm. 2. Calculate the operating speed of the ball mill from the given data below Diameter of the ball mill is 800 mm and Diameter of the ball is 60 mm. if critical speed is 40% less than the operating speed. WhatsApp: +86 18838072829 A cascadecataract charge flow model for power draft of tumbling mills. The existing models of mill power draft are in general not flexible and versatile enough to incorporate the effects of high mill speed and charge loading, liner design and slurry rheology etc., in a phenomenologically consistent manner. A multitorque model is derived in ... WhatsApp: +86 18838072829 What happens when ball mill is centrifuging? The critical speed of a rotating mill is the RPM at which a grinding medium will begin to "centrifuge", namely will start rotating with the mill and therefore cease to carry out useful work. ... Ball mills are used for grinding materials such as mining ores, coal, pigments, and feldspar for ... WhatsApp: +86 18838072829 where d is the maximum size of feed (mm); σ is compression strength (MPa); E is modulus of elasticity (MPa); ρb is density of material of balls (kg/m 3); D is inner diameter of the mill body (m).. Generally, a maximum allowed ball size is situated in the range from D /18 to D/24.. The degree of filling the mill with balls also influences productivity of the mill and milling WhatsApp: +86 18838072829 BeadBlaster™ 96 Ball Mill Homogenizer. Benchmark Scientific. The BeadBlaster™ 96 is an extremely versatile bead mill homogenizer that has applications in a variety of areas, including biological research, environmental testing, and industrial settings. Adapters are available for microplates, microtubes, and 50ml tubes. WhatsApp: +86 18838072829 f BALL MILL WORKING. CONTINUOUS OPERATION. Feed from left through 60° cone; Product discharge from 30° cone to the right. Balls rise and then drop on feed; Solids are ground and reduced in size by impact. When shell rotates, large balls segregate at feed end, small balls near product end. WhatsApp: +86 18838072829 The maintenance cost of the ball mill is reduced as the lifetime of grinding media and partition grates is extended. ... The shell rotates faster than the critical speed which leads to centrifuging of the material. The main feature is the roller inside the shell which is rotated by the material freely on its shaft without a drive. WhatsApp: +86 18838072829 Objectives. At the end of this lesson students should be able to: Explain the role of ball mill in mineral industry and why it is extensively used. Describe different types of ball mill design. Describe the components of ball mill. Explain their understanding of ball mill operation. Explain the role of critical speed and power draw in design ... WhatsApp: +86 18838072829 A Slice Mill is the same diameter as the production mill but shorter in length. Request Price Quote. Click to request a ball mill quote online or call to speak with an expert at Paul O. Abbe® to help you determine which design and size ball mill would be best for your process. See our Size Reduction Options. WhatsApp: +86 18838072829 Discrete Element Methods (DEM) is a numerical tool consolidated to the simulations of collisions in particulate systems. In this paper, the method was used to study the collisions between grinding media and grinding media and walls in ball mills, which is the most used unit operation in clinker grinding, the majority component of the cement. Amongst the variables that affect the dynamics of ... WhatsApp: +86 18838072829 Based on his work, this formula can be derived for ball diameter sizing and selection: Dm <= 6 (log dk) * d^ where D m = the diameter of the singlesized balls in = the diameter of the largest chunks of ore in the mill feed in mm. dk = the P90 or fineness of the finished product in microns (um)with this the finished product is ... WhatsApp: +86 18838072829 A Ball Mill Critical Speed (actually ball, rod, AG or SAG) is the speed at which the centrifugal forces equal gravitational forces at the mill shell's inside surface and no balls will fall from its position onto the shell. The imagery below helps explain what goes on inside a mill as speed varies. Use our online formula. The mill speed is typically defined as the percent of the Theoretical ... WhatsApp: +86 18838072829 Industrial tumbling mills typically operate in the high Froude range (cascading or cataracting) and exhibit a rich coexistence of flow regimes (Mellmann, 2001) that are bounded by nonlinear surfaces as illustrated in Fig. 1. 1 The "rising enmasse" region is dense and follows closely the rotation of the atop the rising enmasse is a gravitydriven layer, termed the ... WhatsApp: +86 18838072829 Scribd is the world's largest social reading and publishing site. WhatsApp: +86 18838072829 Type of ball mill: • There is no fundamental restriction to the type of ball mill used for organic synthesis (planetary ball mill, mixer ball mill, vibration ball mill, .). • The scale of reaction determines the size and the type of ball mill. • Vessels for laboratory vibration ball mills are normally restricted to a volume of 50 cm3. WhatsApp: +86 18838072829 Critical rotation speed of dry ballmill was studied by experiments and by numerical simulation using Discrete Element Method (DEM). The results carried out by both methods showed good agreement. ... The conditions, under which transitions between the flows regimes (rolling, cascading, cataracting, and centrifuging) occurred, were identified ... 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Car Crash Web Quest Car Crash Web Quest Ray Wagner, Pat Suter, Mike LiPani 12/21/11 Physics R (3+4) A traffic accident has just occurred on Millway Street. A 3000 kg Cadillac Escalade SUV has just rear-ended a 2000 kg Subaru Outback at a stop sign. We are trying to determine if the vehicle exceeded the speed limit of 35 km/hr before the time of impact. The results of this investigation will determine the punishment of the SUV driver. We examined the crash site and found 24 meter skid marks from the wagon and 2 meter skid marks from the SUV. The drivers evidently slammed on their brakes after impact. Furthermore, we examined each vehicles braking capabilities and discovered that with the brakes locked, the SUV will accelerate at -2/ms^2 and the wagon will accelerate at -3 m/s^2, meaning the vehicles will slow down at the said rate when the driver floors the brakes. Since we know the acceleration, initial velocity of the Subaru, breaking distance for both vehicles and the final velocity of the SUV, we can use out kinematic equation (vf^2=vi^2+2ad) to individually calculate the final velocity of the Subaru and the initial velocity of the SUV before impact. Our collision experts have determined that the SUV reached a speed of 2.83 m/s immediately following the crash and the wagon reached a speed of 12 m/s. In order to determine if the SUV was speeding before the collision, we had to find the momentum of both vehicles. We multiplied both vehicle’s masses and change in velocities to get the total momentum. We found the momentum of the SUV right after the crash to be 8490 newtons/s and the momentum of the wagon to be 24000 n/s. Since the law of Conservation of Momentum states that the momentum of an explosion is the same before and after, we could combine the each momentum to get a total momentum of 32490 n/s without any extra work. This probably means nothing to you at this point but it’s essential to the last part of our investigation. Momentum equals mass times velocity so we did some simple algebra and divided the total momentum into the mass of the SUV and found its velocity of 10.83 m/s. As you know, speed limits are not measured in meters per second, so we converted meters per second to kilometers per hour. The SUV was traveling 38.953 km/hr, 3.953 km/hr too fast. So the driver was in fact speeding before impact.	{"url":"https://aplusphysics.com/community/index.php?/topic/524-car-crash-web-quest/","timestamp":"2024-11-10T14:33:17Z","content_type":"text/html","content_length":"99421","record_id":"<urn:uuid:021e0fd4-5ba6-4b7a-80a5-c6efd3f1f86e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00109.warc.gz"}
Università degli Studi di Perugia Study-unit Code In all curricula Matteo Duranti Course Regulation Coorte 2018 Learning activities Attività formative affini o integrative Academic discipline Type of study-unit Opzionale (Optional) Type of learning activities Attività formativa monodisciplinare Language of instruction The course is focused on • The Linux operative system, common commands and shell environment; • The C/C++ programming language and the usage of Makefile; • Techniques and algorithms for the simulation of problems in physics (MonteCarlo) • Techniques and algorithms for the resolution of problems in physics (numerical integration, resolution of systems of differential equations); Reference texts • W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Third Edition (Cambridge University Press, 2007, ISBN-10: 0521880688). • E. Gamma, R. Helm, R. Johnson, J. Vlissides, Design Patterns: Elements of Reusable Object-Oriented Software, (Pearson Education, 1994, ISBN: 0201633612, ISBN-13: 9780201633610) Educational objectives Goal of the course is the learning of common computer science techniques and algorithms, applied to the solution of problems common in the physics research. Since the knowledge and competence of the programming and of the computer environment is a fundamental prerequisite for such a goal, the course has, as secondary goal, but preparatory for the main one, the detailed study of the computer science competences. So the acquired knowledges will be: • role and meaning of the various steps of the compilation of a C/C++ program; • the role, the structure and the usage of a Makefile; • the most relevant random number generators; • the MonteCarlo technique and its application; • the most relevant methods for the numerical integration (rectangle, trapeze, Simpson, Gauss, .); • the most relevant methods for the numerica resolution of differential equation systems (Eulero, punto medio, Runge-Kutta, .); The acquired abilities: • compilation of a C/C++ software; • writing of a Makefile; • writing a LCG random number generator and usage of the random number generators; • wiriting of simple MonteCarlo programs; • writing of simple programs for the numerical integration of functions; • writing of simple programs for the numerical resolution of differential equations; To understand and being able to apply the techniques described during the lessons, is mandatory to have passed the Laboratorio di Informatica exam. The examples used during the course and in the final practice, in addition, require the ability to solve simple Mechanics and/or Electromagnetism and/or Statistics problems. Teaching methods • Lessons (theory): 10 lessons of 1 hours each, during the which the techniques will be discussed from a theoretical point of view; • Practical exercitations with the computer: 10 exercitations of 3 hours each, during the which exercises requiring the implementation of the techniques described from the theoretical point of view, within simple programs, will be solved; Other information Is strongly suggested to attend the lessons Learning verification modality The final examination is based on a practical test at the computer and on an oral examination. The practical test foresee the resolution, through the implementation of simple programs, of, at most, 2 simple exercises. The exercises are simple Mechanics and/or Electromagnetism and/or Statistics problems to be solved using the techniques described during the lessons. The oral examination, during at most 60 minutes, is based on the discussion of a written essay about a project (writing of a software to solve different problematics) given at the end of the course. The discussion of the project will be just the starting point to the verification of the knowledges foreseen to be acquired during the course. Extended program The course aim to provide a solid background level to the usage of the computer science in the field of physics and scientific in general research. This background will be provided through the study and implementation (working with the computer) of some examples of algorithms and solutions, common in the routine research in physics. The program of the course foresee the detailed study of: • The Linux operative system, common commands and shell environment; • The C/C++ programming language and the usage of Makefile; • Techniques and algorithms for the simulation of problems in physics (MonteCarlo) • Techniques and algorithms for the resolution of problems in physics (numerical integration, resolution of systems of differential equations);	{"url":"https://www.unipg.it/en/ects/ects-course-catalogue-2021-22?annoregolamento=2021&layout=insegnamento&idcorso=231&idinsegnamento=156796","timestamp":"2024-11-02T10:58:51Z","content_type":"application/xhtml+xml","content_length":"58717","record_id":"<urn:uuid:894d0d37-4cd5-4ffc-805e-f3dd55456926>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00725.warc.gz"}
Quadratic Vector Equations on Complex Upper Half-Planesearch Item Successfully Added to Cart An error was encountered while trying to add the item to the cart. Please try again. Please make all selections above before adding to cart Quadratic Vector Equations on Complex Upper Half-Plane Oskari Ajanki : Institute of Science and Technology, Klosterneuberg, Austria László Erdős : Institute of Science and Technology, Klosterneuberg, Austria Torben Krüger : Institute of Science and Technology, Klosterneuberg, Austria Softcover ISBN: 978-1-4704-3683-4 Product Code: MEMO/261/1261 List Price: $81.00 MAA Member Price: $72.90 AMS Member Price: $48.60 eBook ISBN: 978-1-4704-5414-2 Product Code: MEMO/261/1261.E List Price: $81.00 MAA Member Price: $72.90 AMS Member Price: $48.60 Softcover ISBN: 978-1-4704-3683-4 eBook: ISBN: 978-1-4704-5414-2 Product Code: MEMO/261/1261.B List Price: $162.00 $121.50 MAA Member Price: $145.80 $109.35 AMS Member Price: $97.20 $72.90 Click above image for expanded view Quadratic Vector Equations on Complex Upper Half-Plane Oskari Ajanki : Institute of Science and Technology, Klosterneuberg, Austria László Erdős : Institute of Science and Technology, Klosterneuberg, Austria Torben Krüger : Institute of Science and Technology, Klosterneuberg, Austria Softcover ISBN: 978-1-4704-3683-4 Product Code: MEMO/261/1261 List Price: $81.00 MAA Member Price: $72.90 AMS Member Price: $48.60 eBook ISBN: 978-1-4704-5414-2 Product Code: MEMO/261/1261.E List Price: $81.00 MAA Member Price: $72.90 AMS Member Price: $48.60 Softcover ISBN: 978-1-4704-3683-4 eBook ISBN: 978-1-4704-5414-2 Product Code: MEMO/261/1261.B List Price: $162.00 $121.50 MAA Member Price: $145.80 $109.35 AMS Member Price: $97.20 $72.90 • Memoirs of the American Mathematical Society Volume: 261; 2019; 133 pp MSC: Primary 45; Secondary 46; 60; 15 The authors consider the nonlinear equation $-\frac 1m=z+Sm$ with a parameter $z$ in the complex upper half plane $\mathbb H $, where $S$ is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in $ \mathbb H$ is unique and its $z$-dependence is conveniently described as the Stieltjes transforms of a family of measures \ (v\) on $\mathbb R$. In a previous paper the authors qualitatively identified the possible singular behaviors of $v$: under suitable conditions on $S$ we showed that in the density of $v$ only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any $z\ in \mathbb H$, including the vicinity of the singularities. □ Chapters □ 1. Introduction □ 2. Set-up and main results □ 3. Local laws for large random matrices □ 4. Existence, uniqueness and $\mathrm {L}^{\!2}$-bound □ 5. Properties of solution □ 6. Uniform bounds □ 7. Regularity of solution □ 8. Perturbations when generating density is small □ 9. Behavior of generating density where it is small □ 10. Stability around small minima of generating density □ 11. Examples □ A. Appendix • Permission – for use of book, eBook, or Journal content • Book Details • Table of Contents • Additional Material • Requests Volume: 261; 2019; 133 pp MSC: Primary 45; Secondary 46; 60; 15 The authors consider the nonlinear equation $-\frac 1m=z+Sm$ with a parameter $z$ in the complex upper half plane $\mathbb H $, where $S$ is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in $ \mathbb H$ is unique and its $z$-dependence is conveniently described as the Stieltjes transforms of a family of measures $v$ on $\ mathbb R$. In a previous paper the authors qualitatively identified the possible singular behaviors of $v$: under suitable conditions on $S$ we showed that in the density of $v$ only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any $z\in \mathbb H $, including the vicinity of the singularities. • Chapters • 1. Introduction • 2. Set-up and main results • 3. Local laws for large random matrices • 4. Existence, uniqueness and $\mathrm {L}^{\!2}$-bound • 5. Properties of solution • 6. Uniform bounds • 7. Regularity of solution • 8. Perturbations when generating density is small • 9. Behavior of generating density where it is small • 10. Stability around small minima of generating density • 11. Examples • A. Appendix Permission – for use of book, eBook, or Journal content Please select which format for which you are requesting permissions.	{"url":"https://bookstore.ams.org/MEMO/261/1261","timestamp":"2024-11-12T05:51:38Z","content_type":"text/html","content_length":"96110","record_id":"<urn:uuid:13256088-d8f4-4851-bb3d-8aa10654ec7f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00863.warc.gz"}
Statistics for Data Analyst - Statistics MCQs, Analysis, Software The post is about MCQs correlation and Regression Analysis with Answers. There are 20 multiple-choice questions covering topics related to correlation and regression analysis, coefficient of determination, testing of correlation and regression coefficient, Interpretation of regression coefficients, and the method of least squares, etc. Let us start with MCQS Correlation and Regression Analysis with answers. Online Multiple-Choice Questions about Correlation and Regression Analysis with Answers 1. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is 2. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0: \rho =04 against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is 3. Which one of the following statements is true? 4. The correlation coefficient 5. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is 6. Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$ 7. What do we mean when a simple linear regression model is “statistically” useful? 8. In a simple linear regression problem, $r$ and $\beta_1$ 9. The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by 10. If the coefficient of determination is 0.49, the correlation coefficient may be 11. If the correlation coefficient ($r=1.00$) then 12. The $Y$ intercept ($b_0$) represents the 13. Which one of the following situations is inconsistent? 14. The slope ($b_1$) represents 15. Testing for the existence of correlation is equivalent to 16. Which of the following does the least squares method minimize? 17. If the correlation coefficient $r=1.00$ then 18. The true correlation coefficient $\rho$ will be zero only if 19. If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a 20. The strength of the linear relationship between two numerical variables may be measured by the MCQs Correlation and Regression Analysis • The $Y$ intercept ($b_0$) represents the • The slope ($b_1$) represents • Which of the following does the least squares method minimize? • What do we mean when a simple linear regression model is “statistically” useful? • If the correlation coefficient $r=1.00$ then • If the correlation coefficient ($r=1.00$) then • Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$ • Testing for the existence of correlation is equivalent to • The strength of the linear relationship between two numerical variables may be measured by the • In a simple linear regression problem, $r$ and $\beta_1$ • The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is • The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04 against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is • If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a • The correlation coefficient • If the coefficient of determination is 0.49, the correlation coefficient may be • The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by • Which one of the following situations is inconsistent? • Which one of the following statements is true? • The true correlation coefficient $\rho$ will be zero only if https://rfaqs.com, https://gmstat.com MCQs Probability Quiz The post is about the Online MCQs Probability Quiz. There are 20 multiple-choice questions covering topics related to random experiments, random variables, expectations, rules of probability, events and types of events, and sample space. Let us start with the Probability Quiz. Please go to MCQs Probability Quiz to view the test Online MCQs Probability Quiz with Answers • Consider a dice with the property that the probability of a face with $n$ dots showing up is proportional to $n$. What is the probability of the face showing 4 dots? • Let $X$ be a random variable with a probability distribution function $$f (x) = \begin{cases} 0.2 & \text{forÂ } \|x\|<1 \ 0.1 & \text{for } 1 < \|x\| < 4\ 0 & \text{otherwise} \end{cases}$$ The probability P (0.5 < x < 5) is ————- • Runs scored by batsmen in 5 one day matches are 50, 70, 82, 93, and 20. The standard deviation is ————-. • Find the median and mode of the messages received on 9 consecutive days 15, 11, 9, 5, 18, 4, 15, 13, 17. • $E (XY)=E (X)E (Y)$ if $x$ and $y$ are independent. • Mode is the value of $x$ where $f(x)$ is a maximum if $X$ is continuous. • A coin is tossed up 4 times. The probability that tails turn up in 3 cases is ————–. • If $E$ denotes the expectation the variance of a random variable $X$ is denoted as? • $X$ is a variate between 0 and 3. The value of $E(X^2)$ is ————-. • The random variables $X$ and $Y$ have variances of 0.2 and 0.5, respectively. Let $Z= 5X-2Y$. The variance of $Z$ is? • In a random experiment, observations of a random variable are classified as • A number of individuals arriving at the boarding counter at an airport is an example of • If $A$ and $B$ are independent, $P(A) = 0.45$ and $P(B) = 0.20$ then $P(A \cup B)$ • If a fair dice is rolled twice, the probability of getting doublet is • If a fair coin is tossed 4 times, the probability of getting at least 2 heads is • If $P(B) \ne 0$ then $P(A\|B) = $ • The collection of all possible outcomes of an experiment is called • An event consisting of one sample point is called • An event consisting of more than one sample point is called • When the occurrence of an event does not affect the probability of occurrence of another event, it is called https://rfaqs.com, https://gmstat.com How to Perform Paired Samples t test in SPSS In this post, we will learn about performing paired samples t test in SPSS. The paired samples t-test is a statistical hypothesis testing procedure used to determine whether the mean differences between two sets of observations are zero. In paired samples t-tests (also known as dependent samples) t-tests, each observation in one set is paired with the corresponding observation in another. In this test means/averages of two related groups are compared. By related we mean that the observations in the two groups are paired or matched in some way. Table of Contents Points to Remember The following are points that need to be remembered: A Paired samples t-test can be used when two measurements are taken from the same individuals/objects/respondents or related units. The paired measurements can be: • Before and After Comparisons: A comparison of before and after situations, such as measuring blood pressure before and after taking medication. • Matched Pairs: Used when comparing the test scores of twins or blood relations. • Repeated Measures: When measuring a person’s happiness level at different points in time. A paired samples t-test is also known as a dependent samples t-test, paired samples t-test, or repeated measures t-test. Paired Samples t-test Cannot be used Note that a paired samples t-test can only be used to compare the means for two related (paired) units having a continuous outcome that is normally distributed. This test is not appropriate when • The data is unpaired • There are more than two units/ groups • The continuous outcome is not normally distribution • The outcome is ordinal or ranked Hypothesis for Paired Samples t test The hypotheses for a paired/ dependent samples t-test can be stated as $H_0:\mu_d = 0$ (the difference between the mean of pairs is zero (or equal) ) $H_1: \mu_d \ne$ (the difference between the mean of pairs is not zero (or different) ) $H_1: \mu_d < 0$ (upper tailed test) $H_1: \mu_d > 0$ (lower-tailed test) The test statistics for a paired samples t-test are as follows $$t=\frac{\overline{d} }{\frac{s_d}{\sqrt{n}} }$$ • $\overline{d}$ is the sample mean of the differences • $n$ is the sample size • $s_d$ is the sample standard deviation of the differences Performing Paired Samples t test in SPSS To run a paired samples t test in SPSS, click Analyze > Compare Means > Paired Samples t-test. Paired samples t-test dialog box, the user needs to specify the variables to be used in the analysis. The variables from the left side need to be moved from the paired variables box. A blue button in between both boxes may be used to shift the variables from left to right or right to left side. Note that the variables you specify in paired variables pan need to be in pair form. In the above dialog box, the following are important points to follow: • Pair: The pair row (on the right side pane) represents the number of paired samples t-tests to run. More than one paired samples t-test may be run simultaneously by selecting multiple sets of matched variables. • Variables 1: The first variable represents the first match group (such as the before situation). • Variables 2: The second variable represents the second match group (such as the after situation). • Options: The options button can be used to specify the confidence interval percentage and how the analysis will deal with the missing values. Note that setting the confidence interval percentage does not have any impact on the calculation of the p-value. Paired Samples t test Data Example Consider the following example about 20 students’ academic performance by taking an examination before and after a particular teaching methodology. Student Number Marks before Teaching Methodology Marks After Teaching Methodology Testing the Assumptions of Paired Samples t-test Before performing the Paired Samples t-test, it is better to test the assumptions (or requirements) of the paired samples t-test. • The dependent variable should be continuous (that is measured on interval or ratio level). • The dependent observations (related samples) should have the same subject/ objects, that is, the subjects in the first group are also in the second group. • Sampled data should be random from the respective population. • The differences between the paired values should follow the normal (or approximately) normal distribution • There should be no outliers in the differences between the two related groups. Note that when testing the assumptions (such as normality, and outliers detection) related to paired samples t-test, one must use a variable that represents the differences between the paired values, not the original variables themselves. Also note that when one or more assumptions for a paired samples t-test are not met, you may run the non-parametric test, Wilcoxon Signed Ranks Test. Output: Paired Samples T test The SPSS will result in four tables: 1. Paired Samples Statistics The paired samples statistics table gives univariate descriptive statistics (such as mean, sample, size, standard deviation, and standard error) for each variable entered as paired variables. 2. Paired Samples Correlations The paired samples correlation table gives the bivariate Person correlation coefficient for each pair of variables entered. 3. Paired Samples Test The paired samples test table gives the hypothesis test results with p-value and confidence interval of difference. 4. Paired Samples Effect Sizes The paired sample Effect sizes tables give Cohen’s d and Hedges’ Correction values with confidence interval Interpreting the Paired Samples t test Output From the “Paired Samples Test” the two-tailed p-value (0.121) is greater than 0.05 (level of significance), which means that the null hypothesis is accepted which means that there is no difference between marks before and after the teaching methodology. It means that improvement in marks is due to chance or random variation marely. The “Paired Samples Correlations” Table shows that the paired variables are correlated/ related to each other as the p-value for Pearson’s Correlation is less than 0.05. The marks related to before and after teaching methodology are statistically and significantly related to each other, however, the average difference of marks between before and after teaching methodology is not statistically significant. The differences are due to change or random variation. How to report the Paired Samples t-test Results One might report the statistics in the following format: t(degrees of freedom) = t-value, p = significance level. From the above example, this would be: t(9) = -1.714, p > 0.05. Due to the averages of the two situations and the direction of the t-value, one can conclude that there was a statistically non-significant improvement in marks due to the teaching methodology from 19.9 Â± 2.685 marks to 21.50 Â± 3.922 marks (p > 0.05). So, there is no improvement due to the teaching methodology. https://rfaqs.com, https://gmstat.com Properties of a Good Estimator Introduction (Properties of a Good Estimator) The post is about a comprehensive discussion of the Properties of a Good Estimator. In statistics, an estimator is a function of sample data used to estimate an unknown population parameter. A good estimator is both efficient and unbiased. An estimator is considered as a good estimator if it satisfies the following properties: • Unbiasedness • Consistency • Efficiency • Sufficiency • Invariance Table of Contents Let us discuss these properties of a good estimator one by one. An estimator is said to be an unbiased estimator if its expected value (that is mean of its sampling distribution) is equal to its true population parameter value. Let $\hat{\theta}$ be an unbiased estimator of its true population parameter $\theta$ then $\hat{\theta}$. If $E(\hat{\theta}) = E(\theta)$ the estimator ($\hat{\theta}$) will be unbiased. If $E(\hat{\theta})\ne \theta$, then $\hat{\ theta}$ will be a biased estimator of $\theta$. • If $E(\hat{\theta}) > \theta$, then $\hat{\theta}$ will be positively biased. • If $E(\hat{\theta}) < \theta$, then $\hat{\theta}$ will be negatively biased. Some examples of biased or unbiased estimators are: • $\overline{X}$ is an unbiased estimator of $\mu$, that is, $E(\overline{X}) = \mu$ • $\widetilde{X}$ is also an unbiased estimator when the population is normally distributed, that is, $E(\widetilde{X}) =\mu$ • Sample variance $S^2$ is biased estimator of $\sigma^2$, that is, $E(S^2)\ne \sigma^2$ • $\hat{p} = \frac{x}{n}$ is an unbiased estimator of $E(\hat{p})=p$ It means that if the sampling process is repeated many times and calculations about the estimator for each sample are made, the average of these estimates would be very close to the true population An unbiased estimator does not systematically overestimate or underestimate the true parameter. An estimator is said to be a consistent estimator if the statistic to be used as an estimator approaches the true population parameter value by increasing the sample size. OR An estimator $\hat{\theta}$ is called a consistent estimator of $\theta$ if the probability that $\hat{\theta}$ becomes closer and closer to $\theta$, approaches unity with increasing the sample Symbolically, $\hat{\theta}$ is a consistent estimator of the parameter $\theta$ if for any arbitrary small positive quantity $e$ or $\epsilon$. \lim\limits_{n\rightarrow \infty} P\left[\|\hat{\theta}-\theta\|\le \varepsilon\right] &= 1\\ \lim\limits_{n\rightarrow \infty} P\left[\|\hat{\theta}-\theta\|> \varepsilon\right] &= 0 A consistent estimator may or may not be unbiased. The sample mean $\overline{X}=\frac{\Sigma X_i}{n}$ and sample proportion $\hat{p} = \frac{x}{n}$ are unbiased estimators of $\mu$ and $p$, respectively and are also consistent. It means that as one collects more and more data, the estimator becomes more and more accurate in approximating the true population value. An efficient estimator is less likely to produce extreme values, making it more reliable. An unbiased estimator is said to be efficient if the variance of its sampling distribution is smaller than that of the sampling distribution of any other unbiased estimator of the same parameter. Suppose there are two unbiased estimators $T_1$ and $T_2$ of the sample parameter $\theta$, then $T_1$ will be said to be a more efficient estimator compared to the $T_2$ if $Var(T_1) < Var(T_2)$. The relative efficiency of $T_1$ compared to $T_2$ is given by the ration $$E = \frac{Var(T_2)}{Var(T_1)} > 1$$ Note that when two estimators are biased then MSE is used to compare. A more efficient estimator has a smaller sampling error, meaning it is less likely to deviate significantly from the true population parameter. An efficient estimator is less likely to produce extreme values, making it more reliable. An estimator is said to be sufficient if the statistic used as an estimator utilizes all the information contained in the sample. Any statistic that is not computed from all values in the sample is not a sufficient estimator. The sample mean $\overline{X}=\frac{\Sigma X}{n}$ and sample proportion $\hat{p} = \frac{x}{n}$ are sufficient estimators of the population mean $\mu$ and population proportion $p$, respectively but the median is not a sufficient estimator because it does not use all the information contained in the sample. A sufficient estimator provides us with maximum information as it is close to a population which is why, it also measures variability. A sufficient estimator captures all the useful information from the data without any loss. A sufficient estimator captures all the useful information from the data. Invariance (Property of Love) If the function of the parameter changes, the estimator also changes with some functional applications. This property is known as invariance. E(X-\mu)^2 &= \sigma^2 \\ \text{or } \sqrt{E(X-\mu)^2} &= \sigma\\ \text{or } [E(X-\mu)^2]^2 &= (\sigma^2)^2 The property states that if $\hat{\theta}$ is the MLE of $\theta$ then $\tau(\hat{\theta})$ is the MLE of $\tau(\hat{\theta})$ for any function. The Taw ($\tau$) is the general form of any function. for example $\theta=\overline{X}$, $\theta^2=\overline{X}^2$, and $\sqrt{\theta}=\sqrt{\overline{X}}$. From the above diagrammatic representations, one can visualize the properties of a good estimator as described below. • Unbiasedness: The estimator should be centered around the true value. • Efficiency: The estimator should have a smaller spread (variance) around the true value. • Consistency: As the sample size increases, the estimator should become more accurate. • Sufficiency: The estimator should capture all relevant information from the sample. In summary, regarding the properties of a good estimator, a good estimator is unbiased, efficient, consistent, and ideally sufficient. It should also be robust to outliers and have a low MSE. https://rfaqs.com, https://gmstat.com	{"url":"https://itfeature.com","timestamp":"2024-11-02T20:04:19Z","content_type":"text/html","content_length":"359112","record_id":"<urn:uuid:61fb0c7e-5a87-42f5-862c-61022b3ad211>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00370.warc.gz"}
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Adelic descent theory \| Compositio Mathematica \| Cambridge Core For a smooth, irreducible, complete, algebraic curve $X$ , we denote by $F$ the field of rational functions, by $\mathbb{O}$ the product $\prod _{x\in X_{\text{cl}}}\widehat{{\mathcal{O}}}_{x}$ ranging over closed points, and $$\begin{eqnarray}\mathbb{A}=\mathop{\prod }_{x\in X_{\text{cl}}}^{\!\!\!\prime }\,\widehat{F}_{x}=\{(f_{x})_{x\in X_{\text{cl}}}\|f_{x}\in \widehat{{\mathcal{O}}}_{x}~\text{for almost all}~x\}.\end This object is called the ring of adèles. André Weil was probably the first to appreciate the close connection between adèles and the geometry of curves (see the letter [Reference WeilWei38b] to Hasse where the case of line bundles is discussed, and [Reference WeilWei38a] for the closely related notion of matrix divisors). Theorem 0.1 (Weil). Let $X$ be an algebraic curve, defined over an algebraically closed field $k$ , and let $G$ be a reductive algebraic group. We then have an equivalence between the groupoid of $G$ -torsors on $X$ , $BG(X)$ , and the groupoid defined by the double quotient $[G(F)\setminus G(\mathbb{A})/G(\mathbb{O})]$ . Weil’s theorem is central to the geometric Langlands programme, as it connects the arithmetic conjectures to their geometric counterpart. For a survey of this connection see [Reference FrenkelFre07]. The interplay of Weil’s result with conformal field theory is discussed by Witten [Reference WittenWit88, § V]. In this article we present a generalisation of Weil’s theorem to arbitrary Noetherian schemes. We will deduce it from an adelic descent result for the perfect complexes. The co-simplicial ring $\ mathbb{A}_{X}^{\bullet }$ was introduced by Beilinson in [Reference BeilinsonBei80], as a generalisation of the theory of adèles for curves. A similar construction has also been obtained by Parshin for algebraic surfaces. If $X$ is a curve, the co-simplicial ring is given by the diagram which captures the adelic rings $F$ , $\mathbb{O}_{X}$ and $\mathbb{A}_{X}$ , and the various maps between them, used to formulate Weil’s Theorem 0.1. Theorem 0.2 (Adelic descent). Let $X$ be a Noetherian scheme. We denote by $\mathbb{A}_{X}^{\bullet }$ Beilinson’s co-simplicial ring of adèles (see Definition 1.4). We have an equivalence of symmetric monoidal $\infty$ -categories $\mathsf{Perf}(X)_{\otimes }\simeq \|\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })_{\otimes }\|$ , where the right-hand side denotes the $\infty$ -category of cartesian $\mathbb{A}_{X}^{\bullet }$ -modules. This theorem also holds for almost perfect complexes, as we show in Corollary 3.39. According to Lieblich, the study of perfect complexes is the mother of all moduli problems (see the abstract of [ Reference LieblichLie06]). The Tannakian formalism enables us to make this philosophical principle precise. Using the results of Bhatt [Reference BhattBha16] and Bhatt and Halpern-Leistner [Reference Bhatt and Halpern-LeistnerBHL15], we obtain a descent result for $G$ -torsors (we may replace $BG$ by more general algebraic stacks). Theorem 0.3. Let $X$ be a Noetherian scheme. The geometric realisation of the simplicial affine scheme $\operatorname{Spec}\mathbb{A}_{X}^{\bullet }$ in the category of Noetherian algebraic stacks with quasi-affine diagonal is canonically equivalent to $X$ . In particular, we have $BG(X)\simeq \|BG(\operatorname{Spec}\mathbb{A}_{X}^{\bullet })\|$ , if $G$ is a Noetherian affine algebraic group scheme. Let $G$ be a special group scheme (for example, $G=\operatorname{GL}_{n}$ ). We denote by $G(\mathbb{A}_{X}^{1})^{\text{cocycle}}$ the subset consisting of $\unicode[STIX]{x1D719}\in \mathbb {G}(\mathbb{A}_{X}^{1})$ satisfying the cocycle condition $\unicode[STIX]{x1D719}_{02}=\unicode[STIX]{x1D719}_{01}\circ \unicode[STIX]{x1D719}_{12}$ in $G(\mathbb{A}_{X}^{2})$ . There is an equivalence of groupoids $BG(X)\simeq [G(\mathbb{A}_{X}^{1})^{\text{cocycle}}/G(\mathbb{A}_{X}^{0})]$ . In characteristic 0, the assumption that $G$ be Noetherian can often be dropped. We refer the reader to Corollary 3.35. We refer the reader to § 3.2.4 for a more detailed discussion of the adelic description of $G$ -bundles on Noetherian schemes $X$ . The case of punctured surfaces has also been considered by Garland and Patnaik in [Reference Garland and PatnaikGP]. In [Reference Parshin Par83], Parshin used adelic cocycles for $G$ -bundles as above to obtain formulae for Chern classes in adelic terms. As a further consequence of the adelic descent formalism, we obtain an analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces [Reference Gelfand and NaimarkGN43]. Recall that [Reference Gelfand and NaimarkGN43] shows that a locally compact topological space can be reconstructed from the ring of continuous functions. It is well-known that a similar result cannot hold for non-affine schemes. However, our result implies that a Noetherian scheme $X$ can be reconstructed from the co-simplicial ring of adèles. Theorem 0.4. The functor $\mathbb{A}^{\bullet }:\mathsf{Sch}^{N}{\hookrightarrow}(\mathsf{Rng}^{\unicode[STIX]{x1D6E5}})^{\mathsf{op}}$ from the category of Noetherian schemes to the dual category of co-simplicial commutative rings, has an explicit left-inverse, sending $R^{\bullet }$ to $\|\operatorname{Spec}\,R^{\bullet }\|$ . It is instructive to meditate on the differences and similarities with Gelfand–Naimark’s theorem. While their result copes with plain rings, our Theorem 0.4 requires a diagram of rings (see Corollary 3.33 for a precise statement to which extent the co-simplicial structure is needed). However, the necessary condition of local compactness for topological spaces is not unlike the restriction that the scheme be Noetherian. For a quasi-compact and quasi-separated scheme $X$ we may choose a finite cover by affine open subschemes $\{U_{i}\}_{i=1,\ldots ,n}$ . The coproduct $U=\coprod _{i=1}^{n}U_{i}$ is then still an affine scheme, and we have a map $U\rightarrow X$ . Choosing a finite affine covering for $U\times _{X}U$ , and iterating this procedure, we arrive at a simplicial affine scheme $U_{\bullet }\ rightarrow X$ , which yields a hypercovering of $X$ . The coordinate ring yields a co-simplicial ring $\unicode[STIX]{x1D6E4}(U_{\bullet })$ associated to $X$ . However, this construction is a priori not functorial, since it depends on the chosen coverings. Nonetheless, using the construction $X\mapsto X^{Z}$ introduced by Bhatt and Scholze [Reference Bhatt and ScholzeBS15], one obtains another functor as in Theorem 0.4 (the author thanks Bhatt for bringing this to his attention). Our Theorem 0.2 relies heavily on Beilinson’s [Reference BeilinsonBei80], which constructs a functor, sending a quasi-coherent sheaf ${\mathcal{F}}$ on $X$ to an $\mathbb{A}_{X}^{\bullet }$ -module $ \mathbb{A}_{X}^{\bullet }({\mathcal{F}})$ . Beilinson observes that the latter co-simplicial module gives rise to a chain complex, computing the cohomology of ${\mathcal{F}}$ . This chain complex can be obtained by applying the Dold–Kan correspondence, or taking the alternating sum of the face maps in each degree: $[\mathbb{A}^{0}({\mathcal{F}})\xrightarrow[{}]{\unicode[STIX]{x2202}_{0}-\unicode [STIX]{x2202}_{1}}\mathbb{A}^{1}({\mathcal{F}})\cdots \,]$ . Beilinson’s result can be stated as $$\begin{eqnarray}H^{i}(X,{\mathcal{F}})\simeq H^{i}(\mathsf{DK}(\mathbb{A}_{X}^{\bullet }({\mathcal{F}}))).\end{eqnarray}$$ The reason is that the sheaves $\mathsf{A}_{X}^{k}:U\mapsto \mathbb{A}_{U}^{k}({\mathcal{F}})$ are flasque, and hence it remains to show that the corresponding complex of sheaves defines a flasque resolution of ${\mathcal{F}}$ . The details are explained in [Reference HuberHub91]. Since morphisms in $\mathsf{Perf}(X)$ are closely related to sheaf cohomology, it is not difficult to deduce from Beilinson’s observation the existence of a fully faithful functor $\mathsf{Perf}(X){\hookrightarrow}\|\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })\|$ , amounting to a sort of a cohomological descent result. Our proof of the adelic descent theorem is, hence, mainly concerned with establishing that this functor is essentially surjective, that is, establishing effectivity of descent for objects in those $\infty$ -categories. In heuristic terms, our theorem asserts that a perfect complex $M$ on $X$ can be described by an iterative formal glueing procedure from the adelic parts $\mathbb{A}_{X}^ {\bullet }(M)$ . Our main theorem uses the language of stable $\infty$ -categories. Replacing $\mathsf{Perf}(X)$ and $\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })$ by their homotopy categories would render the result incorrect. However, the theorem could be formulated in more classical language. The $\infty$ -category of cartesian perfect modules over a co-simplicial ring, such as $\mathsf{Perf}(\mathbb{A}_{X}^{\ bullet })$ , has a model, as is discussed in [Reference Toën and VezzosiTV08, 1.2.12] by Toën and Vezzosi. Since the construction of $\mathbb{A}_{X}^{\bullet }$ involves iterative completion and localisation procedures, the formal descent result of Beauville and Laszlo [Reference Beauville and LaszloBL95] and Ben-Bassat and Temkin [Reference Ben-Bassat and TemkinBBT13] (for quasi-coherent sheaves in each case) are closely related. These theorems allow one to glue sheaves on a scheme $X$ with respect to the formal neighbourhood of a closed subvariety $Y$ , and its open complement $X\setminus Y$ . Beauville–Laszlo developed such a descent theory for an affine scheme $X$ , and a closed subvariety $Y$ given by a principal ideal. This result was motivated by the study of conformal blocks [Reference Beauville and LaszloBL94]. The second article does not require the restrictions of $X$ to be affine and $Y$ to be principal, however it utilises the theory of Berkovich spaces to formulate the glueing result. Recently their theory has been extended to treat flags of subvarieties by Hennion, Porta and Vezzosi [Reference Hennion, Porta and VezzosiHPV16]. Our Theorem 0.2 gives a very similar descent theory, but uses all closed subvarieties at once and avoids rigid geometry. One of the key properties of the adèles allowing one to establish effectivity of adelic descent data is a strengthening of the theory of flasque sheaves of algebras. Theorem 0.5. Let $X$ be a quasi-compact topological space and $\mathsf{A}$ a lâche sheaf of algebras (see Definition 2.9), with ring of global sections $R=\unicode[STIX]{x1D6E4}(\mathsf{A})$ . The global section functor $\unicode[STIX]{x1D6E4}:\mathsf{Mod}(\mathsf{A})\rightarrow \mathsf{Mod}(\mathsf{A})$ restricts to a symmetric monoidal equivalence $\mathsf{Perf}(\mathsf{A})_{\otimes }\simeq \mathsf{Perf}(R)_{\otimes }$ . We show in Lemma 1.14 that the adèles $\mathbb{A}_{X}^{k}$ are lâche sheaves of algebras. The derived equivalence underlying our theorems decomposes into two parts: $$\begin{eqnarray}\mathsf{Perf}(X)\simeq \|\mathsf{Perf}(\mathsf{A}_{X}^{\bullet })\|\simeq \|\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })\|.\end{eqnarray}$$ The second equivalence is deduced from Theorem 0.5. The first equivalence can be established by local verifications. 1 A reminder of Beilinson–Parshin adèles In [Reference BeilinsonBei80] Beilinson generalised the notion of adèles to arbitrary Noetherian schemes, and studied the connection adèles bear with coherent cohomology. We briefly review his definition and the main properties of relevance to us. Except for the assertion that adèles are flasque sheaves (Corollary 1.15), we will not provide a proof for those statements and refer the reader instead to Huber [Reference HuberHub91]. Examples can be found in Morrow’s survey article about adèles and their relation to higher local fields [Reference MorrowMor12]. 1.1 Recollection Henceforth we denote by $X$ a Noetherian scheme, with underlying topological space $\|X\|$ and structure sheaf ${\mathcal{O}}_{X}$ . Definition 1.1. Let $X$ be a scheme with underlying topological space $\|X\|$ . For $x,y\in \|X\|$ we write $x\leqslant y$ for $x\in \overline{\{y\}}$ ; this defines a partially ordered set $\|X\|_{{\ leqslant}}$ . We denote the set $\{(x_{0},\ldots ,x_{k})\in \|X\|^{\times k+1}\|x_{0}\leqslant \cdots \leqslant x_{k}\}$ by $\|X\|_{k}$ . One sees that $\|X\|_{k}$ is in fact the set of $k$ -simplices of a simplicial set $\|X\|_{\bullet }$ . This simplicial structure will be reflected in a co-simplicial structure for Beilinson–Parshin Definition 1.2. The simplicial set $\|X\|_{\bullet }:\unicode[STIX]{x1D6E5}^{\mathsf{op}}\rightarrow \operatorname{Set}$ is defined to be the functor, sending $[n]\in \unicode[STIX]{x1D6E5}^{\mathsf {op}}$ to the set of ordered maps $[n]\rightarrow \|X\|_{{\leqslant}}$ , where $\|X\|_{{\leqslant}}$ refers to the partially ordered set defined in Definition 1.1. Following [Reference BeilinsonBei80] we define adèles with respect to a subset $T\subset \|X\|_{k}$ . The case of interest to us will be $T=\|X\|_{k}$ , but the recursive nature of the definition necessitates a definition for general subsets $T\subset \|X\|_{k}$ . We begin with the following preliminary definitions. Definition 1.3. Let $X$ be a Noetherian scheme and $k\in \mathbb{N}$ a non-negative integer: 1. (a) for $x\in \|X\|$ and $T\subset \|X\|_{k}$ we define $\text{}_{x}T=\{(x_{0}\leqslant \cdots \leqslant x_{k-1})\in \|X\|^{\times k}\|(x_{0}\leqslant \cdots \leqslant x_{k-1}\leqslant x)\in T\}$ ; 2. (b) for $x\in \|X\|$ we denote by ${\mathcal{O}}_{x}$ the local ring at $x$ with maximal ideal $\mathfrak{m}_{x}$ ; there is a canonical morphism $j_{rx}:\operatorname{Spec}{\mathcal{O}}_{x}/\ mathfrak{m}_{x}^{r}\rightarrow X$ . It is convenient to define adèles in a higher-dimensional situation as sheaves of ${\mathcal{O}}_{X}$ -modules. Definition 1.4. Let $X$ be a Noetherian scheme. The adèles are the unique family of exact functors $\mathsf{A}_{X,T}(-):\mathsf{QCoh}(X)\rightarrow \mathsf{Mod}({\mathcal{O}}_{X})$ , indexed by subsets $T\subset \|X\|_{k}$ , satisfying the following conditions: 1. (a) the functor $\mathsf{A}_{X,T}(-)$ commutes with directed colimits; 2. (b) for ${\mathcal{F}}\in \mathsf{Coh}(X)$ and $k=0$ we have $\mathsf{A}_{X,T}({\mathcal{F}})=\prod _{x\in T}\mathop{\varprojlim }\nolimits_{r\geqslant 0}(j_{rx})_{\ast }(j_{rx})^{\ast }{\mathcal {F}}$ ; 3. (c) for ${\mathcal{F}}\in \mathsf{Coh}(X)$ and $k\geqslant 1$ we have $\mathsf{A}_{X,T}({\mathcal{F}})=\prod _{x\in \|X\|}\mathop{\varprojlim }\nolimits_{r\geqslant 0}\mathsf{A}_{X,\text{}_{x}T} ((j_{rx})_{\ast }(j_{rx})^{\ast }{\mathcal{F}})$ . We refer the reader to [Reference HuberHub91] for a detailed verification that the above family of functors is well-defined and exact. The ring of adéles with respect to $T\subset \|X\|_{n}$ is defined by taking global sections of the sheaf of rings $\mathsf{A}_{X,T}({\mathcal{O}}_{X})$ . Moreover, it is important to emphasise that the sheaves of ${\mathcal{O}}_{X}$ -modules $\mathsf{A}_{X,T}({\ mathcal{F}})$ are in general not quasi-coherent. Definition 1.5. We denote the abelian group $\unicode[STIX]{x1D6E4}(X,\mathsf{A}_{X,T}({\mathcal{F}}))$ by $\mathbb{A}_{X,T}({\mathcal{F}})$ ; and reserve the notation $\mathbb{A}_{X,T}$ for $\mathbb {A}_{X,T}({\mathcal{O}}_{X})$ . By construction $\mathbb{A}_{X,T}({\mathcal{F}})$ is an $\mathbb{A}_{X,T}$ -module. As we already alluded to, the most interesting case for us is when $T=\|X\|_{k}$ . We reserve a particular notation for this situation. Definition 1.6. We denote the sheaf $\mathsf{A}_{X,\|X\|_{k}}({\mathcal{F}})$ by $\mathsf{A}_{X}^{k}({\mathcal{F}})$ . The abelian group $\unicode[STIX]{x1D6E4}(X,\mathsf{A}_{X}^{k}({\mathcal{F}}))$ will be denoted by $\mathbb{A}_{X}^{k}({\mathcal{F}})$ . As the superscript indicates, these sheaves can be assembled into a co-simplicial object. The proof of this can be found in [Reference HuberHub91, Theorem 2.4.1]. Proposition 1.7. Let $X$ be a Noetherian scheme, and $T_{\bullet }\subset \|X\|_{\bullet }$ a simplicial subset. There is an exact functor $\mathsf{A}_{X,T_{\bullet }}^{\bullet }:\mathsf{QCoh}(X)\ rightarrow \mathsf{Fun}(\unicode[STIX]{x1D6E5},\mathsf{Mod}({\mathcal{O}}_{X}))$ , which commutes with directed colimits, and maps $([k],{\mathcal{F}})$ to $\mathsf{A}_{X,T_{k}}({\mathcal{F}})$ . We denote the functor $\unicode[STIX]{x1D6E4}(X,\mathsf{A}_{X,T_{\bullet }}^{\bullet }(-)):\mathsf{QCoh}(X)\rightarrow \mathsf{Fun}(\unicode[STIX]{x1D6E5},\mathsf{AbGrp})$ by $\mathbb{A}_{X,T_{\bullet }}^{\bullet }(-)$ ; it is exact and commutes with directed colimits. The notation $\mathbb{A}_{X}^{\bullet }(-)$ is reserved for the functor corresponding to the case $T_{\bullet }=\|X\|_{\bullet }$ . We shall write $\mathbb{A}_{X}^{\bullet }$ to denote the co-simplicial ring obtained by applying this functor to the structure sheaf ${\mathcal{O}}_{X}$ . Let $X$ be an irreducible Noetherian scheme of dimension 1, and ${\mathcal{F}}$ a coherent sheaf on $X$ . We will discuss how the definitions above recover the classical theory of adèles for algebraic curves. Following classical conventions, we denote by $$\begin{eqnarray}\mathbb{A}_{X}({\mathcal{F}})=\mathop{\prod }_{x\in \|X\|_{\text{cl}}}^{\!\!\!\!\prime }\,\widehat{{\mathcal{F}}}_{x}\otimes _{\widehat{{\mathcal{O}}}_{x}}\operatorname{Frac}\widehat the restricted product ranging over all closed points $x\in \|X\|_{\text{cl}}$ . We denote by $$\begin{eqnarray}\mathbb{O}_{X}({\mathcal{F}})=\mathop{\prod }_{x\in X_{\text{cl}}}\widehat{{\mathcal{F}}}_{x};\end{eqnarray}$$ and by $F_{X}({\mathcal{F}})$ the ${\mathcal{O}}$ -module ${\mathcal{F}}_{\unicode[STIX]{x1D702}}$ , where $\unicode[STIX]{x1D702}$ is the generic point of $X$ . With respect to this notation we may identify the co-simplicial ${\mathcal{O}}$ -module $\mathbb{A}_{X}^{\bullet }({\mathcal{F}})$ with where $F_{X}({\mathcal{F}})\rightarrow \mathbb{A}_{X}({\mathcal{F}})$ is the diagonal inclusion, and $\mathbb{O}_{X}({\mathcal{F}})\rightarrow \mathbb{A}_{X}({\mathcal{F}})$ the canonical map. Embracing the usual redundancies in co-simplicial objects, that is the continual re-appearance of factors already seen at a lower degree level, we observe that Beilinson’s $\mathbb{A}_{X}^{\bullet }$ captures the classical theory of adèles. It is also helpful to understand the co-simplicial structure in the local case. Let $\unicode[STIX]{x1D70E}:[n]\rightarrow \|X\|_{{\leqslant}}$ be an element of $\|X\|_{n}$ . We denote the ring of adèles $\mathbb{A}_{X,\{\unicode[STIX]{x1D70E}\}}$ corresponding to $T=\{\unicode[STIX]{x1D70E}\}\subset \|X\|_{n}$ by $\mathbb{A}_{X,\unicode[STIX]{x1D70E}}$ . Proposition 1.7 implies that for every map $f: [m]\rightarrow [n]$ in $\unicode[STIX]{x1D6E5}$ we have a ring homomorphism $\mathbb{A}_{X,\unicode[STIX]{x1D70E}\circ f}\rightarrow \mathbb{A}_{X,\unicode[STIX]{x1D70E}}$ . The following assertion is also proven in [Reference HuberHub91, Theorem 2.4.1]. Lemma 1.8. Let $X$ and $T_{\bullet }$ be as in Proposition 1.7. The co-simplicial ring $\mathbb{A}_{X,T_{\bullet }}^{\bullet }$ injects into the co-simplicial ring $$\begin{eqnarray}[n]\mapsto \mathop{\prod }_{\unicode[STIX]{x1D70E}:[n]\rightarrow \|X\|_{{\leqslant}}}\mathbb{A}_{X,\unicode[STIX]{x1D70E}}.\end{eqnarray}$$ We also need the following observation, which is a consequence of the definitions of adèles. Remark 1.9. If ${\mathcal{F}}$ is a quasi-coherent sheaf on $X$ , set-theoretically supported on a finite union of closed points $Z\subset X$ , then we have ${\mathcal{F}}\simeq \mathsf{A}_{X}^{k}({\ mathcal{F}})$ . Another observation which we will need is that for an affine Noetherian scheme $X$ , the functor $$\begin{eqnarray}\mathbb{A}_{X,T}(-):\mathsf{QCoh}(X)\rightarrow \mathsf{Mod}(\unicode[STIX]{x1D6E4}({\mathcal{O}}_{X}))\end{eqnarray}$$ can be expressed as $-\otimes _{\unicode[STIX]{x1D6E4}({\mathcal{O}}_{X})}\mathbb{A}_{X,T}$ . This is the case, since $\mathbb{A}_{X,T}(-)$ , and $-\otimes -$ commute with filtered colimits. Since $\ mathbb{A}_{X,T}(-)$ takes values in flasque sheaves by [Reference HuberHub91, Proposition 2.1.5] (see also Corollary 1.15), we see that $\mathbb{A}_{X,T}(-)$ is an exact functor. Therefore, $\mathbb {A}_{X,T}$ is a flat algebra over $\unicode[STIX]{x1D6E4}({\mathcal{O}}_{X})$ . We record this for later use. Lemma 1.10. Let $X=\operatorname{Spec}R$ be an affine Noetherian scheme. Then $\mathbb{A}_{X,T}$ is a flat $R$ -algebra. 1.2 Functoriality If $f:X\rightarrow Y$ is a morphism of Noetherian schemes, we have an induced map of partially ordered sets $\|X\|_{{\leqslant}}\rightarrow \|Y\|_{{\leqslant}}$ . Indeed, $x\in \overline{\{y\}}$ implies $f(x)\in \overline{\{f(y)\}}$ . In addition, we have an induced morphism of local rings ${\mathcal{O}}_{Y,f(x)}\rightarrow {\mathcal{O}}_{X,x}$ . These observations are the building blocks of a functoriality property satisfied by adèles. To the best of the author’s knowledge, this property has not yet been recorded in the literature. Lemma 1.11. Let $f:X\rightarrow Y$ be a morphism of Noetherian schemes, and ${\mathcal{F}}\in \mathsf{QCoh}(Y)$ a quasi-coherent sheaf on $Y$ . For $T\subset \|X\|_{n}$ and $f(T)\subset T^{\prime }\ subset \|Y\|_{n}$ we have a morphism $\mathsf{A}_{Y,T^{\prime }}({\mathcal{F}})\rightarrow f_{\ast }\mathsf{A}_{X,T}(f^{\ast }{\mathcal{F}})$ in $\mathsf{Mod}({\mathcal{O}}_{Y})$ , fitting into the following commutative diagram. Commutativity of this diagram amounts to the construction inducing a map of augmented co-simplicial objects in $\mathsf{Mod}({\mathcal{O}}_{Y})$ from ${\mathcal{F}}\rightarrow \mathsf{A}_{Y}^{\bullet }({\mathcal{F}})$ to $f_{\ast }f^{\ast }{\mathcal{F}}\rightarrow f_{\ast }\mathsf{A}_{X}^{\bullet }(f^{\ast }{\mathcal{F}})$ . Proof of Lemma 1.11. The morphism $\mathsf{A}_{Y,T^{\prime }}({\mathcal{F}})\rightarrow f_{\ast }\mathsf{A}_{X,T}(f^{\ast }{\mathcal{F}})$ is constructed by induction on $n$ (where $T\subset \|X\|_{n}$ ). For $n=0$ and ${\ mathcal{F}}\in \mathsf{Coh}(Y)$ , we have $$\begin{eqnarray}\mathsf{A}_{X,T}(f^{\ast }{\mathcal{F}})=\mathop{\prod }_{x\in T}\underset{r\geqslant 0}{\mathsf{lim}}(j_{rx})_{\ast }j_{rx}^{\ast }f^{\ast }{\mathcal{F}},\end{eqnarray}$$ and $\mathsf{A}_{Y,T^{\prime }}{\twoheadrightarrow}\mathsf{A}_{Y,f(T)}({\mathcal{F}})=\prod _{x\in f(T)}\mathsf{lim}_{r\geqslant 0}(j_{rx})_{\ast }j_{rx}^{\ast }{\mathcal{F}}$ . We have a map $$\begin{eqnarray}{\mathcal{F}}\otimes _{{\mathcal{O}}_{Y}}{\mathcal{O}}_{Y,f(x)}\rightarrow f^{\ast }{\mathcal{F}}\otimes _{{\mathcal{O}}_{X}}{\mathcal{O}}_{X,x}\end{eqnarray}$$ for each $x\in T$ , which defines the required map for $T\subset \|X\|_{0}$ . Assume that the morphism $\mathsf{A}_{Y,T^{\prime }}({\mathcal{F}})\rightarrow f_{\ast }\mathsf{A}_{X,T}(f^{\ast }{\mathcal{F}})$ has been constructed for all $T\subset \|X\|_{k}$ and $f(T)\subset T^{\ prime }\subset \|Y\|_{k}$ , where $k\leqslant n$ , such that we have the following commutative diagram. Let $T\subset \|X\|_{n+1}$ . For every $x\in X$ , we then have a well-defined map $\mathsf{A}_{Y,\text{}_{f(x)}f(T)}({\mathcal{F}})\rightarrow f_{\ast }\mathsf{A}_{X,\text{}_{x}T}(f^{\ast }{\mathcal {F}})$ , since $f(\text{}_{x}T)\subset \text{}_{f(x)}f(T)\subset \text{}_{f(x)}T^{\prime }$ . This induces a morphism (1) $$\begin{eqnarray}\mathop{\prod }_{x\in \|X\|}\underset{r\geqslant 0}{\mathsf{lim}}\mathsf{A}_{Y,\text{}_{f(x)}f(T)}((j_{rf(x)})_{\ast }j_{rf(x)}^{\ast }f^{\ast }{\mathcal{F}})\rightarrow \mathop{\ prod }_{x\in \|X\|}\underset{r\geqslant 0}{\mathsf{lim}}\mathsf{A}_{X,\text{}_{x}T}((j_{rx})_{\ast }j_{rx}^{\ast }f^{\ast }{\mathcal{F}}).\end{eqnarray}$$ We have a commutative diagram which commutes levelwise (before taking the inverse limits and products) by the induction hypothesis. We precompose the map (1) with (3) $$\begin{eqnarray}\displaystyle \mathop{\prod }_{y\in \|Y\|}\underset{r\geqslant 0}{\mathsf{lim}}\mathsf{A}_{Y,\text{}_{y}T^{\prime }}((j_{ry})_{\ast }j_{ry}^{\ast }f^{\ast }{\mathcal{F}}) & {\ twoheadrightarrow} & \displaystyle \mathop{\prod }_{y\in f(\|X\|)}\underset{r\geqslant 0}{\mathsf{lim}}\mathsf{A}_{Y,\text{}_{y}T^{\prime }}((j_{ry})_{\ast }j_{ry}^{\ast }f^{\ast }{\mathcal{F}})\ nonumber\\ \displaystyle & \rightarrow & \displaystyle \mathop{\prod }_{x\in \|X\|}\underset{r\geqslant 0}{\mathsf{lim}}\mathsf{A}_{Y,\text{}_{f(x)}f(T)}((j_{rf(x)})_{\ast }j_{rf(x)}^{\ast }f^{\ast }{\ to obtain the required morphism $\mathsf{A}_{Y,T^{\prime }}({\mathcal{F}})\rightarrow f_{\ast }\mathsf{A}_{X,T}(f^{\ast }{\mathcal{F}})$ . The required commutativity assumption holds by commutativity of (2).◻ Setting ${\mathcal{F}}={\mathcal{O}}_{Y}$ in the assertion above, we obtain the following result as a consequence. Corollary 1.12. For a morphism of Noetherian schemes, we obtain a map of augmented co-simplicial objects in sheaves of algebras on the topological space $\|Y\|$ . Taking global sections, we obtain a functor from Noetherian schemes to co-simplicial rings. Definition 1.13. We denote the functor $(\mathsf{Sch}^{\mathsf{N}})^{\mathsf{op}}\rightarrow (\mathsf{Rng}^{\unicode[STIX]{x1D6E5}})$ , sending a Noetherian scheme $X$ to the co-simplicial ring $\ mathbb{A}_{X}^{\bullet }({\mathcal{O}}_{X})$ by $\mathbb{A}^{\bullet }$ . As we have alluded to in Theorem 0.4, and will prove as Corollary 3.32, this functor has a left-inverse. 1.3 Taking a closer look at the flasque sheaf of adèles In this subsection we give a close analysis of flasqueness of the sheaf $\mathsf{A}_{X,T}({\mathcal{F}})$ . We show that the restriction map $\mathbb{A}_{X,T}({\mathcal{F}})\rightarrow \mathbb{A}_ {U,T\cap \|U\|_{n}}({\mathcal{F}})$ is not only surjective, but admits an $\mathbb{A}_{X,T}({\mathcal{O}}_{X})$ -linear section. As a consequence we obtain that $\mathsf{A}_{X,T}({\mathcal{O}}_{X})$ is what we call a lâche sheaf of algebras in Definition 2.9 (see also Corollary 2.17). A similar statement is contained in [Reference HuberHub91, Proposition 2.1.5], however the $\mathsf{A}_{X,T}({\ mathcal{O}}_{X})$ -linearity is not investigated in [Reference HuberHub91]. Lemma 1.14. Let $X$ be a Noetherian scheme, $T\subset \|X\|_{n}$ and ${\mathcal{F}}$ a quasi-coherent sheaf on $X$ . For every open subset $U\subset X$ the restriction map $\mathbb{A}_{X,T}({\mathcal {F}})\rightarrow \mathbb{A}_{U,T}({\mathcal{F}})$ has a section, which is moreover $\mathbb{A}_{X,T}({\mathcal{O}}_{X})$ -linear and functorial in ${\mathcal{F}}$ . Proof. We denote the inclusion $U{\hookrightarrow}X$ by $j$ , and will construct a section to the map of sheaves $\mathsf{A}_{X,T}({\mathcal{F}})\rightarrow j_{\ast }\mathsf{A}_{U,T}({\mathcal{F}})$ . Recall that for a coherent sheaf ${\mathcal{F}}$ we have an equivalence $\mathsf{A}_{X,T}({\mathcal{F}})\simeq \prod _{x\in X}\mathop{\varprojlim }\nolimits_{r}\mathsf{A}_{X,\text{}_{x}T}(j_{rx}^{\ ast }{\mathcal{F}}),$ and $j_{\ast }\mathsf{A}_{U,T\cap \|U\|_{n}}({\mathcal{F}})\simeq \prod _{x\in U}\mathop{\varprojlim }\nolimits_{r}\mathsf{A}_{U,\text{}_{x}T\cap \|U\|_{n-1}}(j_{rx}^{\ast }{\ mathcal{F}}).$ Suppose that we have already constructed a section $\unicode[STIX]{x1D719}_{{\mathcal{F}}}$ for $\mathsf{A}_{X,T^{\prime }}({\mathcal{F}})\rightarrow j_{\ast }\mathsf{A}_{U,T^{\prime }}({\mathcal{F}})$ for $T^{\prime }\subset \|X\|_{n-1}$ , such that for each map ${\mathcal{F}}\rightarrow {\mathcal{G}}$ we obtain the following commutative square. We can then map the limit $\prod _{x\in U}\mathop{\varprojlim }\nolimits_{r}\mathsf{A}_{U}(\text{}_{x}T;j_{rx}^{\ast }{\mathcal{F}})$ to $\prod _{x\in X}\mathop{\varprojlim }\nolimits_{r}\mathsf{A}_ {U}(\text{}_{x}T;j_{rx}^{\ast }{\mathcal{F}})$ , by defining the components of the map to be $0$ for $x\in X\setminus U$ , and given by the section $\unicode[STIX]{x1D719}$ otherwise. By induction we see that we may assume that $T\subset \|X\|_{0}$ . We may also assume that ${\mathcal{F}}$ is coherent. Hence, $\mathsf{A}_{X,T}({\mathcal{F}})$ is equal to the product $\prod _{x\in T} \mathop{\varprojlim }\nolimits_{r}j_{rx}^{\ast }{\mathcal{F}}$ , and $\mathsf{A}_{U}(T,{\mathcal{F}})$ to $\prod _{x\in T\cap U}\mathop{\varprojlim }\nolimits_{r}j_{rx}^{\ast }{\mathcal{F}}$ . The natural restriction map is given by the canonical projection. A canonical section with the required properties is given by the identity map for components corresponding to $x\in U\cap T$ , and the map 0 for $x\in T\setminus U$ .◻ As a corollary one obtains the following observation of Beilinson. Corollary 1.15. The sheaves $\mathbb{A}_{X,T}({\mathcal{F}})$ are flasque. In [Reference BeilinsonBei80] Beilinson continues to observe that for any quasi-coherent sheaf ${\mathcal{F}}$ the canonical augmentation ${\mathcal{F}}\rightarrow \mathsf{A}_{X}^{\bullet }({\mathcal {F}})$ induces an equivalence ${\mathcal{F}}\simeq \|\mathsf{A}_{X}^{\bullet }({\mathcal{F}})\|$ . A detailed proof is given by Huber [Reference HuberHub91, Theorem 4.1.1]. Theorem 1.16 (Beilinson). Let $X$ be a Noetherian scheme and ${\mathcal{F}}$ a quasi-coherent sheaf on $X$ . The augmentation ${\mathcal{F}}\rightarrow \mathsf{A}_{X}^{\bullet }({\mathcal{F}})$ defines a co-simplicial resolution of ${\mathcal{F}}$ by flasque ${\mathcal{O}}_{X}$ -modules. Applying the global sections functor $\unicode[STIX]{x1D6E4}(X,-)$ we obtain $R\unicode[STIX]{x1D6E4}(X,{\mathcal{F}})=\|\mathbb {A}_{X}^{\bullet }({\mathcal{F}})\|$ , where the co-simplicial realisation $\|\cdot \|$ is taken in the derived $\infty$ -category $\mathsf{D}(\mathsf{AbGrp})$ of abelian groups. It is instructive to test the general considerations above on the special case of algebraic curves. For the rest of this subsection we will thus assume that $X$ is an algebraic curve. We denote by $\ mathsf{A}_{X}$ the sheaf, assigning to an open subset $U\subset X$ the ring of adèles $\mathbb{A}_{U}$ . Similarly we have sheaves $\mathsf{F}_{X}$ and $\mathsf{O}_{X}$ of rational functions and integral adèles. The sheaves $\mathsf{A}_{X}$ and $\mathsf{O}_{X}$ satisfy the conclusion of Lemma 1.14, because a section over $U\subset X$ can be extended by 0, outside of $U$ . Since $\mathsf{F}_{X}(U)=\mathsf{F} _{X}(X)$ , as long as $U\neq \emptyset$ , we see that the conclusion of Lemma 1.14 is trivially satisfied for $\mathsf{F}$ . Beilinson’s Theorem 1.16 is in the present situation tantamount to the assertion that the complex $$\begin{eqnarray}[{\mathcal{O}}_{X}\rightarrow \mathsf{F}_{X}\oplus \mathsf{O}_{X}\rightarrow \mathsf{A}_{X}]\end{eqnarray}$$ is exact. In other words, we observe that a rational function without any poles on $U\subset X$ , defines a regular function on $U$ . While this is a tautology in the one-dimensional case, the general setting of Noetherian schemes requires more subtle arguments from commutative algebra. We refer the reader to the proof of [Reference HuberHub91, Theorem 4.1.1] for more details. 2 Perfect complexes and lâche sheaves of algebras In this section we introduce the notion of lâche sheaves of algebras and prove Theorem 0.5. 2.1 Lâche sheaves of algebras The main example of a lâche sheaf of algebras $\mathsf{A}$ is Beilinson’s sheaf of adèles. This is the content of Corollary 2.17 below. 2.1.1 Flasque sheaves In this section we record a few well-known lemmas on flasque sheaves for the convenience of the reader. Lemma 2.1. If ${\mathcal{F}}$ is a sheaf on $X$ , such that every point $x\in X$ has an open neighbourhood $U$ with ${\mathcal{F}}\|_{U}$ flasque, then ${\mathcal{F}}$ is a flasque sheaf. Proof. Let $V\subset X$ be an open subset and $s\in {\mathcal{F}}(V)$ a section. We claim that there exists $t\in {\mathcal{F}}(X)$ with $t\|_{V}=s$ . Consider the set $I$ of pairs $(W,t)$ , where $W\ subset X$ is an open subset containing $V$ , and $t\in {\mathcal{F}}(W)$ , such that $t\|_{V}=s$ . Inclusion of open subsets induces a partial ordering on $I$ , where we say that $(W,t)\leqslant (W^ {\prime },t^{\prime })$ if $W\subset W^{\prime }$ , and $t^{\prime }\|_{W}=t$ . Moreover, $I$ is inductively ordered, that is, for every totally ordered subposet $J\subset I$ , there exists a common upper bound $i\in I$ , such that we have $i\geqslant j$ for all $j\in J$ . Indeed, denoting the pair corresponding to $j\in J$ by $(W_{j},t_{j})$ , we have $W_{j}\subset W_{k}$ for $j\leqslant k$ in $J$ , and $t_{k}\|_{W_{j}}=t_{j}$ . If we define $W=\bigcup _{j\in J}W_{j}$ , the fact that ${\mathcal{F}}$ is a sheaf allows us to define a section $t\in {\mathcal{F}}(W)$ with $t\|_{W_{j}}=t_{j}$ . In particular, $(W,t)\in I$ is a common upper bound for the elements of $J$ . Zorn’s lemma implies that the poset $I$ has a maximal element $(W,t)$ . It remains to show that $W=X$ . Assume that there exists $x\in X\setminus W$ . By assumption, $x$ has an open neighbourhood $U$ , such that ${\mathcal{F}}\|_{U}$ is flasque. In particular, there exists a section $r\in {\mathcal{F}}(U)$ , such that $r\|_{U\cap W}=t\|_{U\cap W}$ . By virtue of the sheaf property we obtain a section $t^{\prime }\in {\mathcal{F}}(W\cup U)$ , satisfying $t^{\prime }\|_{W}=t$ , which contradicts maximality of $(W,t)$ .◻ Lemma 2.2. If $X$ is a quasi-compact topological space and $\mathsf{A}$ a sheaf of algebras, then every locally finitely generated $\mathsf{A}$ -module $\mathsf{M}$ which is flasque is globally finitely generated, that is there exists a surjection $\mathsf{A}^{n}\rightarrow \mathsf{M}$ . Proof. For every point $x\in X$ there exists a neighbourhood $U_{x}$ , such that $\mathsf{M}\|_{U_{x}}$ is finitely generated. Since $X$ is quasi-compact, we may choose a finite subcover $X=\bigcup _ {i=1}^{n}U_{i}$ , and generating sections $(s_{ij})_{j=1,\ldots n_{i}}$ . Because $\mathsf{M}$ is assumed to be flasque, we may extend each $s_{ij}$ to a global section $t_{ij}$ , and see that this finite subset of $\unicode[STIX]{x1D6E4}(X,\mathsf{M})$ generates $\mathsf{M}$ .◻ Lemma 2.3. Assume that we have a short exact sequence of $\mathsf{A}$ -modules with $$\begin{eqnarray}0\rightarrow \mathsf{M}_{2}\rightarrow \mathsf{M}_{1}\rightarrow \mathsf{M}_{0}\rightarrow 0,\end{eqnarray}$$ with $\mathsf{M}_{i}$ flasque for $i>0$ , then $\mathsf{M}_{0}$ is flasque as well. Proof. Since flasque sheaves have no higher cohomology, we have $H^{1}(X,\mathsf{M}_{2})=0$ , and therefore the following commutative diagram has exact rows Commutativity of the right-hand square, and the fact that $\mathsf{M}_{1}(X){\twoheadrightarrow}\mathsf{M}_{1}(U){\twoheadrightarrow}\mathsf{M}_{0}(U)$ is surjective, implies surjectivity of $\mathsf {M}_{0}(X){\twoheadrightarrow}\mathsf{M}_{0}(U)$ .◻ Lemma 2.4. Let $\mathsf{A}$ be an arbitrary sheaf of algebras on a topological space $X$ . Consider the abelian category of sheaves of $\mathsf{A}$ -modules. The full subcategory, given by $\mathsf {A}$ -modules $\mathsf{M}$ , such that $\mathsf{M}$ is a flasque sheaf, is extension-closed. Proof. Assume that we have a short exact sequence of $\mathsf{A}$ -modules $\mathsf{M}_{1}{\hookrightarrow}\mathsf{M}_{2}{\twoheadrightarrow}\mathsf{M}_{3}$ , with $\mathsf{M}_{i}$ flasque for $i=1$ and $i=3$ . Since flasque sheaves do not have higher cohomology, we see that for every open subset $U\subset X$ we have a short exact sequence of abelian groups $\mathsf{M}_{1}(U){\hookrightarrow}\ mathsf{M}_{2}(U){\twoheadrightarrow}\mathsf{M}_{3}(U)$ . In particular, we obtain a commutative diagram with exact rows with the left and right vertical arrows being surjective. The snake lemma or a simple diagram chase reveal that the vertical map in the middle also has to be surjective. This proves that $\mathsf{M}_ {2}$ is a flasque sheaf.◻ Definition 2.5. We denote the exact category, given by the extension-closed full subcategory of $\mathsf{Mod}(\mathsf{A})$ consisting of modules whose underlying sheaf is flasque, by $\mathsf{Mod}_{\ mathsf{fl}}(\mathsf{A})$ . We refer the reader to [Reference KellerKel96] and [Reference BühlerBüh10] for the notion of derived categories of exact categories. We also emphasise that we use the notation $\mathsf{D}(-)$ to denote the stable $\infty$ -category obtained by applying the dg-nerve construction of [Reference LurieLur, § 1.3.1] to the dg-category of [Reference KellerKel96]. We will also consider similarly constructed stable $\infty$ -categories $\mathsf{D}^{+}$ , $\mathsf{D}^{-}$ and $\mathsf{D}^{b}$ , corresponding to complexes which are bounded below, bounded above and bounded, respectively. It is important to emphasise that for a substantial part of this text we will not need to delve deeply into the theory of stable $\infty$ -categories. The homotopy category of a stable $\infty$ -category is naturally triangulated. To check that a functor $F:\mathsf{C}\rightarrow \mathsf{D}$ is fully faithful, essentially surjective, or an equivalence, it suffices to prove the same statement for its homotopy category (that is a classical triangulated category). This is essentially a consequence of the Whitehead lemma. Distinguished triangles $X\rightarrow Y\rightarrow Z\rightarrow \ unicode[STIX]{x1D6F4}X$ correspond to so-called bi-cartesian squares that is, commutative diagrams which are cartesian and co-cartesian. A functor $\mathsf{C}\rightarrow \mathsf{D}$ between stable $\infty$ -categories is called exact, if it preserves bi-cartesian squares. In particular, this is the case if and only if the induced functor $\mathsf{Ho}(\mathsf{C})\rightarrow \mathsf{Ho}(\mathsf{D})$ is exact in the sense of triangulated categories. The embedding $\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}){\hookrightarrow}\mathsf{Mod}(\mathsf{A})$ induces an exact functor of derived categories. Lemma 2.6. The canonical functor $\mathsf{D}^{+}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))\rightarrow \mathsf{D}^{+}(\mathsf{A})$ , induced by the exact functor $\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}) {\hookrightarrow}\mathsf{Mod}(\mathsf{A})$ , is fully faithful. Proof. According to a theorem of Keller [Reference KellerKel96, Theorem 12.1] it suffices to check that every short exact sequence of $\mathsf{A}$ -modules $\mathsf{M}_{1}{\hookrightarrow}\mathsf{M}_ {2}{\twoheadrightarrow}\mathsf{M}_{3}$ with $\mathsf{M}_{1}$ flasque fits into a commutative diagram with exact rows with $\mathsf{M}_{2}^{\prime }$ and $\mathsf{M}_{3}^{\prime }$ flasque. To produce this diagram, we recall that every $\mathsf{A}$ -module $\mathsf{M}$ can be embedded into a flasque $\mathsf{A}$ -module. Indeed, the sheaf of discontinuous sections, that is, $\mathsf{M}^{\text{dc}}(U)=\prod _{x\in U}\mathsf{M}_{x}$ provides such an embedding. Let $\mathsf{M}_{2}{\hookrightarrow}\mathsf{M}_{2} ^{\prime }$ be an embedding of $\mathsf{M}_{2}$ into a flasque $\mathsf{A}$ -module. Then, the quotient $\mathsf{M}_{3}^{\prime }=\mathsf{M}_{2}^{\prime }/\mathsf{M}_{1}$ is also flasque by Lemma 2.3 2.1.2 Flasque sheaves of algebras In this subsection we ponder over what can be said about quasi-coherent sheaves of $\mathsf{A}$ -modules, if the sheaves of algebras $\mathsf{A}$ itself is known to be flasque. Recall that an $\ mathsf{A}$ -module $\mathsf{M}$ is quasi-coherent, if every point $x\in X$ has a neighbourhood $U\subset X$ , such that the restriction $\mathsf{M}\|_{U}$ can be represented as a cokernel of a morphism $\mathsf{A}^{\oplus J}\|_{U}\rightarrow \mathsf{A}^{\oplus I}\|_{U}$ of free $\mathsf{A}$ -modules. Remark 2.7. For a general sheaf of algebras $\mathsf{A}$ the category $\mathsf{QCoh}(\mathsf{A})$ of quasi-coherent $\mathsf{A}$ -modules is in general not closed under taking kernels in the abelian category of $\mathsf{A}$ -modules $\mathsf{Mod}(\mathsf{A})$ . In particular, one does not expect $\mathsf{QCoh}(\mathsf{A})$ to be abelian in general. If the restriction maps $\mathsf{A}(V)\ rightarrow \mathsf{A}(U)$ for $U\subset V$ belonging to a specific subbase for the topology are known to be flat, $\mathsf{QCoh}(\mathsf{A})$ can be shown to be abelian. This assumption is too strong for the sheaves of algebras we care about in this article. We see from Lemma 2.1 that every locally finite free or locally finite projective sheaf of $\mathsf{A}$ -modules is flasque (where $\mathsf{A}$ is itself a flasque sheaf of algebras). In general, one cannot expect every quasi-coherent sheaf of $\mathsf{A}$ -modules to be flasque. However, we will see in the next subsection that there are certain flasque sheaves of algebras for which this is true. Lemma 2.8. Let $\mathsf{A}$ be a sheaf of algebras on $X$ , such that every free $\mathsf{A}$ -module is flasque. We denote by $\mathsf{P}(\mathsf{A})$ the exact category given by the idempotent completion of free $\mathsf{A}$ -modules, and refer to its objects as projective $\mathsf{A}$ -modules. The functor $\mathsf{D}^{-}(\mathsf{P}(\mathsf{A})){\hookrightarrow}\mathsf{D}^{-}(\mathsf{Mod} _{\mathsf{fl}}(\mathsf{A}))$ , induced by the inclusion $\mathsf{P}(\mathsf{A}){\hookrightarrow}\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A})$ , is fully faithful. Proof. We will apply the dual of the result in Keller [Reference KellerKel96, Theorem 12.1], by which it suffices to check that every short exact sequence of flasque $\mathsf{A}$ -modules $\mathsf{M} _{1}{\hookrightarrow}\mathsf{M}_{2}{\twoheadrightarrow}\mathsf{M}_{3}$ with $\mathsf{M}_{3}$ projective fits into a commutative diagram with exact rows with $\mathsf{M}_{1}^{\prime }$ and $\mathsf{M}_{2}^{\prime }$ projective. Since $\mathsf{M}_{2}\in \mathsf{Mod}_{\mathsf{fl}}(\mathsf{A})$ is flasque by assumption, there exists a surjection $\ mathsf{A}^{\oplus I}\rightarrow \mathsf{M}_{2}$ for some index set $I$ . Indeed, we can take $I=\{(U,s)\|U\subset X\text{ open, }s\in \mathsf{M}_{2}(U)\}$ . Choosing an extension $t_{(U,s)}\in \ mathsf{M}_{2}(X)$ , satisfying $t_{(U,s)}\|_{U}=s$ for every element of $I$ , we obtain a surjective morphism of $\mathsf{A}$ -module $\mathsf{A}^{\oplus I}\rightarrow \mathsf{M}_{2}$ . Let $\mathsf{M}_{2}^{\prime }=\mathsf{A}^{\oplus I}$ , and define $M_{1}^{\prime }$ to be the kernel of the composition $\mathsf{M}_{2}^{\prime }\rightarrow \mathsf{M}_{2}\rightarrow \mathsf{M}_{3}$ . Since $\mathsf{M}_{3}$ is a direct summand of a free $\mathsf{A}$ -module, and $\mathsf{M}_{2}^{\prime }$ is flasque, there exists a splitting to this surjection. Therefore, $\mathsf{M}_{1}^{\prime }$ belongs to $\mathsf{P}(\mathsf{A})$ , since it is a direct summand of $\mathsf{M}_{2}^{\prime }$ . This concludes the proof.◻ 2.1.3 Definition of lâche sheaves of algebras If $\mathsf{A}$ is a sheaf of algebras on a topological space, then there is a strengthening of the notion of $\mathsf{A}$ being flasque. Definition 2.9. A sheaf of algebras $\mathsf{A}$ on $X$ is called lâche if for every open subset $U\subset X$ and every map of free $\mathsf{A}_{U}$ -modules $\mathsf{A}_{U}^{\oplus J}\xrightarrow [{}]{f}\mathsf{A}_{U}^{\oplus I}$ , the kernel $\ker f$ is a flasque sheaf on $U$ . To see that there are non-trivial lâche sheaves of algebras, we let $X$ be a topological space where every open subset is also closed in the following example. Example 2.10. Let $X$ be a topological space, where every open subset is also closed. Then every sheaf of abelian groups ${\mathcal{F}}$ is flasque. If $U\subset X$ is open, and $s\in {\mathcal{F}} (U)$ , then using the sheaf property of ${\mathcal{F}}$ we see that there is a unique section $t\in {\mathcal{F}}(X)$ , such that $t\|_{U}=s$ and $t\|_{X\setminus U}=0$ . This is possible because $X\ setminus U$ is open by assumption. Hence, every sheaf of algebras on $X$ is lâche. The lemma below implies that for a lâche sheaf of algebras $\mathsf{A}$ , and a morphism $f:\mathsf{A}^{\oplus J}\rightarrow \mathsf{A}^{\oplus I}$ the sheaves $\operatorname{im}f$ and $\operatorname {coker}f$ are flasque as well. Lemma 2.11. Let $V_{1}\xrightarrow[{}]{f}V_{2}$ be a morphism of flasque sheaves, such that $\ker f$ is flasque. Then, the sheaves $\operatorname{im}f$ , and $\operatorname{coker}f$ are flasque. Proof. We have a short exact sequence $\ker f{\hookrightarrow}V_{1}{\twoheadrightarrow}\operatorname{im}f$ , since the first two sheaves are flasque, so is the third (Lemma 2.3). The same argument applies to the short exact sequence $\operatorname{im}f{\hookrightarrow}V_{2}{\twoheadrightarrow}\operatorname{coker}f$ , and implies that $\operatorname{coker}f$ is flasque.◻ We can further generalise the assertion. Lemma 2.12. Let $V_{1}\xrightarrow[{}]{f}V_{2}$ be a morphism of projective quasi-coherent $\mathsf{A}$ -modules (that is, direct summands of free modules), where $\mathsf{A}$ is lâche. Then the sheaves $\ker f$ , $\operatorname{im}f$ and $\operatorname{coker}f$ are flasque. Proof. Since every projective quasi-coherent $\mathsf{A}$ -module is a direct summand of a free $\mathsf{A}$ -module, there exist quasi-coherent $\mathsf{A}$ -modules $W_{1}$ and $W_{2}$ , such that $V_{i}\oplus W_{i}$ are free $\mathsf{A}$ -modules for $i=1,2$ . The induced map $$\begin{eqnarray}f\oplus \operatorname{id}:V_{1}\oplus (W_{1}\oplus W_{2}\oplus V_{1}\oplus V_{2})^{\oplus \mathbb{N}}\rightarrow V_{2}\oplus (W_{1}\oplus W_{2}\oplus V_{1}\oplus V_{2})^{\oplus \ has the same kernel $\ker f\simeq \ker (f\oplus \operatorname{id})$ . However, the Eilenberg swindle $$\begin{eqnarray}(W_{1}\oplus W_{2}\oplus V_{1}\oplus V_{2})^{\oplus \mathbb{N}}\simeq (W_{1})^{i}\oplus (W_{2})^{j}\oplus (W_{1}\oplus W_{2}\oplus V_{1}\oplus V_{2})^{\oplus \mathbb{N}}\end allows us to see that the two sides are, in fact, free $\mathsf{A}$ -modules. Therefore, the defining property of lâche sheaf of algebras implies that $\ker f$ is flasque. Lemma 2.11 yields that $\ operatorname{im}f$ and $\operatorname{coker}f$ are flasque sheaves.◻ The considerations above imply, in particular, that every quasi-coherent $\mathsf{A}$ -module of a lâche sheaf of algebras $\mathsf{A}$ is flasque. However, we have to keep in mind that the category of quasi-coherent sheaves is not abelian in general, as we pointed out in Remark 2.7. We have the following corollary to Lemma 2.12. Corollary 2.13. If $M^{\bullet }\in \mathsf{DMod}(\mathsf{A})$ is a complex of sheaves of $\mathsf{A}$ -modules which is locally quasi-isomorphic to an object of $\mathsf{D}(\mathsf{P}(\mathsf{A}))$ , then its cohomology sheaves ${\mathcal{H}}^{i}(\mathsf{M}^{\bullet })$ are flasque. Proof. We have seen in Lemma 2.1 that a sheaf is flasque if and only if it is locally flasque. Therefore, we may assume $M\in \mathsf{D}(\mathsf{P}(\mathsf{A}))$ . Let us choose an explicit presentation by a complex $(V^{\bullet },d)$ , where each $V^{i}$ is a projective $\mathsf{A}$ -module. We have ${\mathcal{H}}^{i}(M)\simeq (\ker d^{i})/(\operatorname{im}d^{i-1})$ . By Lemma 2.12, $ \ker d^{i}$ and $\operatorname{im}d^{i-1}$ are flasque. By Lemma 2.3, the quotient ${\mathcal{H}}^{i}(M)$ is flasque.◻ 2.1.4 A criterion for being lâche In this subsection we observe that every sheaf of algebras $\mathsf{A}$ , which admits linear sections to the restriction maps $\mathsf{A}(X)\rightarrow \mathsf{A}(U)$ , is in fact lâche. As a consequence, we obtain that the sheaf of adèles on a Noetherian scheme is lâche (Corollary 2.17). Definition 2.14. A sheaf of algebras $\mathsf{A}$ is called very flasque if for every open subset $U\subset X$ there exists an $\mathsf{A}(X)$ -linear section $\unicode[STIX]{x1D719}_{U}$ of the restriction map $r_{U}:\mathsf{A}(X)\rightarrow \mathsf{A}(U)$ . Typically the section $\unicode[STIX]{x1D719}_{U}$ is given by a map which extends $s\in \mathsf{A}(U)$ by 0 outside of $U$ , as in the following example. Example 2.15. Let $X$ be a topological space where every open subset is closed and $\mathsf{A}$ an arbitrary sheaf of algebras, then $\mathsf{A}$ is very flasque. Proof. For an open subset $U\subset X$ we have a map $\unicode[STIX]{x1D719}:\mathsf{A}(U)\rightarrow \mathsf{A}(X)$ which sends $s\in \mathsf{A}(U)$ to the unique section $\widehat{s}\in \mathsf{A} (X)$ , such that $\widehat{s}\|_{U}=s$ , and $\widehat{s}\|_{X\setminus U}=0$ . This definition makes sense because $\mathsf{A}$ is a sheaf, and $X=U\cup X\setminus U$ a disjoint open covering. Since this map is $\mathsf{A}(X)$ -linear, we have shown that $\mathsf{A}$ is very flasque.◻ In hindsight we have shown in Lemma 1.14 that for every quasi-coherent sheaf of algebras ${\mathcal{F}}$ on a Noetherian scheme $X$ , the sheaves of algebras $\mathbb{A}_{X,T}({\mathcal{F}})$ are very flasque. See also Corollary 2.17 below, where an important consequence of this observation is recorded. The next lemma is the aforementioned criterion for a sheaf of algebras being lâche. Lemma 2.16. A very flasque sheaf of algebras $\mathsf{A}$ is lâche. Proof. Let $f:\mathsf{A}_{V}^{\oplus J}\rightarrow \mathsf{A}_{V}^{\oplus I}$ be an $\mathsf{A}_{V}$ -linear map, where $V\subset X$ is open. We have to show that $K=\ker f$ is a flasque sheaf on $V$ . For $U\subset V$ open we have a commutative diagram with exact rows, because taking global sections is a left exact functor. However, $\mathsf{A}(V)$ -linearity of the section $r_{V}\circ \unicode[STIX]{x1D719}_{U}:\mathsf{A}(U)\rightarrow \mathsf{A} (V)$ implies that we have a commutative diagram where the dashed arrow is provided by the universal property of kernels. The dashed arrow is therefore right-inverse to the restriction map $K(V)\rightarrow K(U)$ , and we conclude that $K=\ker f$ is Corollary 2.17. For a Noetherian scheme $X$ and a quasi-coherent sheaf ${\mathcal{F}}$ of algebras, the sheaves of Beilinson–Parshin adèles $\mathsf{A}_{X,T}({\mathcal{F}})$ are lâche sheaves of Proof. Lemma 1.14 asserts that $\mathsf{A}_{X,T}({\mathcal{F}})$ is very flasque. According to Lemma 2.16 this implies that $\mathsf{A}_{X,T}({\mathcal{F}})$ is also lâche.◻ 2.2 Perfect complexes In this subsection we study the $\infty$ -category of perfect complexes of $\mathsf{A}$ -modules. This is necessary since the classical category of quasi-coherent $\mathsf{A}$ -modules is not necessarily abelian (see Remark 2.7). Definition 2.18. Let $\mathsf{P}(\mathsf{A})$ denote the exact category obtained as the idempotent completion of the exact category of free $\mathsf{A}$ -modules. We denote by $\mathsf{D}^{-}(\mathsf {A})$ the $\infty$ -category corresponding to the full subcategory of $\mathsf{D}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))$ given by complexes of flasque $\mathsf{A}$ -modules $\mathsf{M}^{\bullet }$ , which are locally equivalent to objects of $\mathsf{D}^{-}(\mathsf{P}(\mathsf{A}))$ . That is, there exists an open covering $X=\bigcup _{i\in I}U_{i}$ and complexes $\mathsf{N}_{i}^{\bullet }\in \ mathsf{D}^{-}(\mathsf{P}(\mathsf{A}\|_{U_{i}}))$ such that we have equivalences $\mathsf{N}_{i}^{\bullet }\simeq \mathsf{M}^{\bullet }\|_{U_{i}}$ . Recall that every exact functor between exact categories induces a functor between derived $\infty$ -categories. Lemma 2.19. Let $X$ be a quasi-compact topological space and $\mathsf{A}$ a lâche sheaf of algebras on $X$ . We denote by $R=\unicode[STIX]{x1D6E4}(\mathsf{A})$ the ring of global sections of $\ mathsf{A}$ . The global sections functor $\unicode[STIX]{x1D6E4}:\mathsf{D}^{-}(\mathsf{A})\rightarrow \mathsf{D}^{-}(R)$ , induced by the exact functor $\unicode[STIX]{x1D6E4}:\mathsf{Mod}_{\mathsf {fl}}(\mathsf{A})\rightarrow \mathsf{Mod}(R)$ , is conservative. Proof. Pick a complex $\mathsf{M}^{\bullet }=[\cdots \xrightarrow[{}]{d^{i-1}}M^{i}\xrightarrow[{}]{d^{i}}M^{i+1}\xrightarrow[{}]{d^{i+1}}\cdots \,]\in \mathsf{D}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf {A}))$ representing an object of $\mathsf{D}^{-}(\mathsf{A})$ . By definition, we have $\unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet })=[\cdots \xrightarrow[{}]{\unicode[STIX]{x1D6E4}(d^{i-1})}\unicode [STIX]{x1D6E4}(M^{i})\xrightarrow[{}]{\unicode[STIX]{x1D6E4}(d^{i})}\unicode[STIX]{x1D6E4}(M^{i+1})\xrightarrow[{}]{\unicode[STIX]{x1D6E4}(d^{i+1})}\cdots \,]$ . We shall assume that $\unicode[STIX] {x1D6E4}(\mathsf{M}^{\bullet })$ is acyclic, that is quasi-isomorphic to the 0-complex in $D^{-}(R)$ . To establish conservativity of the functor $\unicode[STIX]{x1D6E4}$ , we must show that $\ mathsf{M}^{\bullet }$ is acyclic in $D(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))$ . Since $X$ is assumed to be quasi-compact, and $\mathsf{M}^{\bullet }$ locally quasi-isomorphic to an object of $D^{-}(\mathsf{P}(\mathsf{A}))$ , we see that there exists an $i\in \mathbb{Z}$ , such that the cohomology sheaves ${\mathcal{H}}^{j}(\mathsf{M}^{\bullet })=0$ vanish for $j>i$ . We claim that for such an integer $i$ we have that ${\mathcal{Z}}^{i}=\ker d^{i}$ has no higher cohomology. Indeed, by assumption the stupid truncation $\unicode[STIX]{x1D70E}_{i}\mathsf{M}^{\bullet }=[\cdots \rightarrow 0\rightarrow M^{i}\rightarrow M^{i+1}\rightarrow \cdots \,]$ is a flasque resolution of ${\mathcal{Z}}^{i}[-i]$ , by assumption on the vanishing of ${\mathcal{H}}^{j}(\mathsf{M}^{\bullet })$ for $j>i$ . However, since $\unicode[STIX]{x1D6E4}(M^{\bullet })$ has no cohomology in all degrees, we see that $\unicode[STIX]{x1D70E}_{i}\mathsf{M}^{\bullet }$ has no cohomology in degrees $j>i$ . This shows that ${\mathcal{Z}}^{i}$ has no higher cohomology. Let us denote the image sheaf of $d^{i-1}$ by ${\mathcal{B}}^{i}$ . It fits into a short exact sequence ${\mathcal{B}}^{i}\rightarrow M^{i}{\twoheadrightarrow}{\mathcal{Z}}^{i}$ . Since $M^{i}$ and $ {\mathcal{Z}}^{i}$ are acyclic, one sees from the associated long exact sequence that ${\mathcal{B}}^{i}$ has no higher cohomology if and only if $\unicode[STIX]{x1D6E4}(M^{i})\xrightarrow[{}]{\ unicode[STIX]{x1D6E4}(d^{i})}\unicode[STIX]{x1D6E4}({\mathcal{Z}}^{i})$ is surjective. By definition, the cokernel of this map equals $R^{i}\unicode[STIX]{x1D6E4}(M^{\bullet })=0$ . This is the case because $\unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet })$ is acyclic, and therefore all its cohomology groups vanish. We have a commutative diagram where the lower zigzag is a short exact sequence of sheaves without higher cohomology. Applying the functor $\unicode[STIX]{x1D6E4}$ (the short exact sequence is preserved by virtue of the fact that ${\mathcal{B}}^{i}$ has no higher cohomology) we obtain Using again that $\unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet })$ has no non-zero cohomology groups, we see that $\unicode[STIX]{x1D6E4}(d^{i-1})$ is surjective, and therefore so is the map $\unicode [STIX]{x1D6E4}({\mathcal{B}}^{i})\rightarrow \unicode[STIX]{x1D6E4}({\mathcal{Z}}^{i})$ . This implies $\unicode[STIX]{x1D6E4}({\mathcal{H}}^{i})=0$ . However, we know from Corollary 2.13 that the cohomology sheaves of $\mathsf{M}^{\bullet }$ are locally flasque, and therefore flasque by Lemma 2.1. Since a flasque sheaf without non-zero global sections is the zero sheaf, we obtain ${\mathcal {H}}^{i}(\mathsf{M}^{\bullet })=0$ . Continuing this process by downward induction, we obtain that all cohomology sheaves of $\mathsf{M}^{\bullet }$ vanish. To conclude the proof we need to show that $\mathsf{M}^{\bullet }$ is acyclic in $\mathsf{D}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))$ . By [Reference BühlerBüh10, Definition 10.1 and Remark 10.19] this is equivalent to the assertion that the sheaves ${\mathcal{Z}}^{i}$ and ${\mathcal{B}}^{i}$ are flasque. Since sheaves are flasque if they are locally flasque by Lemma 2.1, it suffices to show that $\mathsf{M}^{\bullet }$ is locally acyclic. By assumption we can cover $X$ by open subsets $U$ , such that $\mathsf{M}^{\bullet }\|_{U}$ is quasi-isomorphic to a complex of projective $\mathsf{A}$ -modules $P^{\bullet }\in \mathsf{D}^{-}(\ mathsf{P}(\mathsf{A}\|_{U}))$ . In particular, we see that $P^{\bullet }$ is a complex of projective sheaves of $\mathsf{A}\|_{U}$ -modules $[\cdots \rightarrow P^{i-1}\rightarrow P^{i}\rightarrow 0\ rightarrow \cdots \,]$ , such that ${\mathcal{H}}^{k}(P^{\bullet })=0$ for all $k\in \mathbb{Z}$ . This implies the existence of a factorisation where the lower zigzags are short exact sequences. Since $P^{i}=Q^{i}$ is projective, we see that the first sequence splits, and therefore $Q^{i-1}$ is projective too. Continuing by downward induction, we see that $Q^{j}$ is projective for all integers $j$ , and therefore $P^{\bullet }\simeq 0$ in $\mathsf{D}(\mathsf{P}(\mathsf{A}\|_{U}))$ . We have seen in Lemma 2.8 that the functor $\ mathsf{D}^{-}(\mathsf{P}(\mathsf{A}\|_{U})){\hookrightarrow}\mathsf{D}^{-}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}\|_{U}))$ is fully faithful. This shows that $\mathsf{M}^{\bullet }\|_{U}\simeq 0$ is acyclic, and therefore that the restriction of the sheaves ${\mathcal{B}}^{i}$ and ${\mathcal{Z}}^{i}$ to $U$ are flasque. As discussed above this concludes the proof that $\mathsf{M}^{\bullet }$ is acyclic in $\mathsf{D}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))$ .◻ We also have a localisation functor. Definition 2.20. For $\mathsf{A}$ a lâche sheaf of algebras on $X$ , we have an exact functor between exact categories $-\otimes _{R}\mathsf{A}:\mathsf{P}(R)\rightarrow \mathsf{P}(\mathsf{A}){\ hookrightarrow}\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A})$ . The induced exact functor between derived $\infty$ -categories will be denoted by $$\begin{eqnarray}\mathsf{Loc}:\mathsf{D}^{-}(R)\rightarrow \mathsf{D}^{-}(\mathsf{A}).\end{eqnarray}$$ Proposition 2.21. If $X$ is quasi-compact and $\mathsf{A}$ is lâche, then $\unicode[STIX]{x1D6E4}:\mathsf{D}^{-}(\mathsf{A})\rightarrow \mathsf{D}^{-}(R)$ is an equivalence of $\infty$ -categories, with inverse equivalence $\mathsf{Loc}$ . Proof. There is a commutative triangle of exact functors, inducing a natural equivalence of functors $\operatorname{id}_{\mathsf{D}^{-}(\mathsf{A})}\simeq \mathsf{Loc}\circ \unicode[STIX]{x1D6E4}$ . We claim that we also have an equivalence $\unicode[STIX]{x1D6E4}\circ \mathsf{Loc}\simeq \operatorname{id}_{\mathsf{D}^{-}(R)}$ . To see this, consider $\mathsf{M}^{\bullet }\in \mathsf{D}^{-}(\mathsf{A})$ . We will show that $\ mathsf{M}^{\bullet }$ belongs to the essential image of $\mathsf{Loc}$ . Let $g:P^{\bullet }\rightarrow \unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet })$ be a projective replacement of $\unicode[STIX] {x1D6E4}(\mathsf{M}^{\bullet })$ , given by an actual morphism between chain complexes in $\mathsf{Mod}(R)$ . By the adjunction between $-\otimes _{R}\,\mathsf{A}$ and $\unicode[STIX]{x1D6E4}$ , this yields a morphism $f:\mathsf{Loc}(P^{\bullet })\simeq \mathsf{Loc}(\unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet }))\rightarrow \mathsf{M}^{\bullet }$ in $\mathsf{D}^{-}(\mathsf{A})$ . Since $\unicode [STIX]{x1D6E4}(f)=g$ is a quasi-isomorphism, and $\unicode[STIX]{x1D6E4}$ is conservative by Lemma 2.19, we see that $f$ is an equivalence. This implies that every $\mathsf{M}^{\bullet }\in \mathsf {D}^{-}(\mathsf{A})$ is in fact equivalent to an object of $\mathsf{D}^{-}(\mathsf{P}(\mathsf{A}))$ . Therefore, we have a natural equivalence $\mathsf{Loc}\circ \unicode[STIX]{x1D6E4}\simeq \ operatorname{id}_{\mathsf{D}^{-}(\mathsf{A})}$ as a consequence of the commutative diagram of exact functors, and Lemma 2.8, which asserted	{"url":"https://core-cms.prod.aop.cambridge.org/core/journals/compositio-mathematica/article/adelic-descent-theory/885CDDA26D395D02DDCAD28DC4805D73","timestamp":"2024-11-02T14:33:37Z","content_type":"text/html","content_length":"1049977","record_id":"<urn:uuid:acd4f419-720a-422b-bf91-03bdf71f863d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00259.warc.gz"}
Taylor Diagram Software 1. Rainfed wheat (Triticum aestivum L.) yield prediction using economical, meteorological, and drought indicators through pooled panel data and statistical downscaling You can access sample input files for download here: DropBox What is Taylor Diagram software tool? When assessing the quality of multiple models compared to observation data and conducting various statistical tests, the process can be complex, especially when simultaneously dealing with several models, scenarios, and parameters. In such cases, using a Taylor diagram software tool proves to be the most efficient approach to achieving accurate results in a shorter time. Agrimetsoft recognized the need for a Taylor diagram software tool and explored existing options, including programming languages like NCL (Ncar Command Language / Taylor diagram NCL), MATLAB (MATLAB Taylor diagram), Python (python Taylor diagram), and various R packages (R Taylor diagram) that enable the creation of Taylor diagrams. However, for those unfamiliar with coding in these software environments, utilizing these options can become a tedious challenge. To address this issue, Agrimetsoft developed its own Taylor diagram software, specifically designed to streamline the process for scholars and scientists. This software provides a user-friendly interface, making it easy and efficient to generate Taylor diagrams. With the Agrimetsoft Taylor diagram software, users can effortlessly create Taylor diagrams with multiple models and input variables, saving valuable time. The software also offers the flexibility to save the diagrams in various file formats, ensuring high-quality outputs. Before ordering the Taylor diagram software, it is recommended to consult us and use the demo version(You can input your data and not draw the chart) and watch the accompanying YouTube video, which provides step-by-step guidance on running the software. By utilizing this tool, researchers can simplify the process of depicting Taylor diagrams, ultimately enhancing their analysis and facilitating their work in scientific research. How to Draw Taylor Diagram \| Windows Software What is Taylor Diagram? Taylor diagrams are a valuable tool for evaluating multiple models and comparing their performance across multiple aspects. They provide a graphical summary of how well a set of models matches observations by considering the standard deviation of model values' time series and the correlation between the model values and observations. Taylor (2001) introduced these diagrams as a comprehensive means of assessing model performance, incorporating information such as root mean square error (RMSE) and correlation coefficients. Taylor diagrams allow for a visual comparison of various variables from one or more test data sets against reference data sets. Typically, the test data sets consist of model experiments(Model/ Predicted data), while the reference data set serves as a control or includes observational data. The plotted values on the diagram are often derived from climatological monthly, seasonal, or annual means. To account for differing numerical values across variables, normalization is applied using the reference variables. The ratio of normalized variances reflects the relative amplitude of variations between the models and observations. In a Taylor diagram, various statistical indicators for each model are combined within a single quadrant. The x-axis represents the RMSE (Root Mean Square Error) or normalized RMSE, while the y-axis represents standard deviation values or their normalized counterparts. The internal semi-circles on the diagram correspond to both RMSE and standard deviation values. Additionally, the arc axis represents the correlation coefficient, and the correlation coefficient values continue as diagonal lines into the graph. These indicators in Taylor Diagram collectively assess the performance of each model in comparison to the observations. Ideally, a strong linear relationship between the model and observations is indicated by a correlation coefficient close to one. Greater accuracy is reflected by smaller RMSE values, and the standard deviation of the model results should closely align with that of the RMSE serves as a measure of the discrepancies between predicted values and observed values, providing an assessment of accuracy. On the other hand, the correlation coefficient quantifies the strength of the linear relationship or pattern similarity between two variables. It is commonly used to gauge the extent of correlation between variables. Another advantage of Taylor diagrams is their ability to plot different parameters together by normalizing the indicators(RMSE, SD), enabling a comprehensive evaluation across multiple dimensions. This second type of Taylor diagram is recommended for How to Draw Taylor Diagram with Negative Correlation Values Taylor Diagram Software How can we compute Taylor Diagram? In essence, the Taylor diagram provides a statistical representation of the relationship between two fields: a "test" field, typically representing a model simulation, and a "reference" field, typically representing observed data. Taylor (2005) discovered that each point in the two-dimensional space of the Taylor diagram can simultaneously depict three key statistics: the centered RMS difference, the correlation coefficient, and the standard deviation. This is achieved through a simple formula that relates these statistics to the Taylor diagram. The construction of the Taylor diagram is based on the similarity between the equation mentioned earlier and the Law of Cosines. In the equation, R represents the correlation coefficient between the test and reference fields, E'2 denotes the centered RMS difference between the fields, and σf^2 and σr^2 represent the variances of the test and reference fields, respectively. By considering the correlation coefficient as the cosine of the azimuthal angle, the Taylor diagram takes shape, leveraging the resemblance to the Law of Cosines. There are several minor variations of the Taylor diagram that have proven to be useful for different purposes. For more in-depth information, I recommend referring to Taylor's original paper from 2001. It provides detailed insights into these variations and their specific applications. • IPCC, 2001: Climate Change 2001: The Scientific Basis, Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change [Houghton, J.T., Y. Ding, D.J. Griggs, M. Noguer, P.J. van der Linden, X. Dai, K. Maskell, and C.A. Johnson (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, 881 pp. • Elvidge, S., Angling, M.J., et al., 2014. On the use of modified Taylor diagrams to compare ionospheric assimilation models, 978-1-4673-5225-3/14/$31.00, 2014 IEEE. • Taylor KE. 2001. Summarizing multiple aspects of model performance in a single diagram. Journal of Geophysical Research 106: 7183-7192. • Taylor Diagram Primer, 2005, Karl E. Taylor. 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How to Implement Machine Learning Algorithms in Python from Scratch • June 23, 2023 • Posted by: Oyesh • Category: Education Online Learning Guide Technology As we navigate the digital age, machine learning has emerged as a vital technology, driving innovation across various sectors. With Python’s simplicity and extensive libraries, it has become the go-to language for implementing machine learning algorithms. In this blog post, we will delve into how you can implement machine learning algorithms in Python from scratch. Whether you’re a seasoned coder or a beginner in the field, you will find this guide helpful and insightful. Understanding the Basics: What is Machine Learning? Machine learning, a branch of artificial intelligence, allows computers to learn from data without being explicitly programmed. It uses algorithms that iteratively learn from the data, allowing the system to independently adapt to new scenarios. Machine learning algorithms in Python are utilized across many industries, aiding in decision making, predictive analysis, and pattern recognition. Why Use Python for Machine Learning? Python’s appeal lies in its simplicity, versatility, and extensive library support. It’s favored by developers and data scientists worldwide for its readability, making it perfect for beginners looking to implement machine learning algorithms. Python’s robust libraries, such as NumPy, Pandas, Matplotlib, and Scikit-learn, simplify the process of creating complex algorithms. Getting Started with Python and Machine Learning Before we delve into the process of implementing machine learning algorithms in Python from scratch, you’ll need to have Python installed on your system. Additionally, you should have a basic understanding of Python syntax, variables, loops, and functions. Key Steps to Implementing Machine Learning Algorithms in Python Implementing machine learning algorithms in Python involves a systematic process. Here are the key steps: Step 1: Importing Libraries Start by importing the necessary Python libraries. The most commonly used libraries for machine learning are NumPy, Pandas, Matplotlib, and Scikit-learn. Step 2: Data Preprocessing Data preprocessing is a crucial step in machine learning. This stage involves cleaning and transforming raw data into a format that the machine learning model can understand. Step 3: Selecting a Machine Learning Algorithm There are various types of machine learning algorithms in Python, each suitable for different kinds of problems. The choice of algorithm depends on the nature of the data and the problem you’re trying to solve. Step 4: Training the Algorithm Once you’ve selected an algorithm, the next step is to train it. This involves providing the algorithm with training data, from which it learns patterns and information. Step 5: Testing the Algorithm After training the algorithm, you need to test it using testing data. This stage is crucial for assessing the model’s performance. Step 6: Evaluation and Optimization Finally, evaluate your machine learning model using appropriate metrics, and optimize the algorithm for better performance if necessary. Implementing Machine Learning Algorithms in Python: A Detailed Walkthrough Having understood the key steps involved, let’s now delve into a detailed guide on how to implement two commonly used machine learning algorithms in Python from scratch: linear regression and k-nearest neighbors. Linear Regression in Python Linear regression is a popular algorithm in machine learning for predicting a continuous outcome variable (Y) based on one or more predictor variables (X). Here’s how to implement it from scratch: 1. Import Necessary Libraries: As always, we begin by importing the necessary libraries. 2. Generate or Load the Dataset: You can either generate a dataset or load one for regression analysis. 3. Implement the Algorithm: Implement the linear regression algorithm, which involves calculating the slope and intercept of the best fit line. 4. Train the Model: Split the data into a training set and a testing set. Use the training set to train the model. 5. Test the Model: Test the model using the testing set and evaluate its performance. K-Nearest Neighbors (K-NN) in Python K-NN is a simple, easy-to-implement supervised machine learning algorithm that can be used for both classification and regression. Here’s how to implement it from scratch: 1. Import Necessary Libraries: Begin by importing the necessary Python libraries. 2. Generate or Load the Dataset: Generate or load a dataset for classification or regression. 3. Implement the Algorithm: Implement the K-NN algorithm, which involves finding the K nearest neighbors and using majority vote or averaging for prediction. 4. Train the Model: Split the data into a training set and a testing set, and then use the training set to train the model. 5. Test the Model: Finally, test the model using the testing set and evaluate its performance. Leveraging Python’s Powerful Libraries While implementing machine learning algorithms in Python from scratch is a fantastic way to learn and understand them, in a real-world setting, you’ll often leverage powerful libraries such as Scikit-learn. These libraries simplify the process, making it easier to implement complex machine learning algorithms. Wrapping Up Implementing machine learning algorithms in Python from scratch might seem daunting, but with practice and persistence, it becomes an achievable task. It offers you a deep understanding of how the algorithms work and allows you to customize them to suit your specific needs. As we approach the conclusion of this article, it’s essential to note that understanding and implementing machine learning algorithms is just the beginning. If you want to delve deeper and explore advanced concepts in data science, consider enrolling in our Advanced Data Science course. Mastering the implementation of machine learning algorithms in Python is a valuable skill, opening a world of opportunities in the tech industry. So, roll up your sleeves, get your Python environment set up, and start coding! Your machine learning journey is just beginning. Frequently Asked Questions Q: What are the prerequisites for implementing machine learning algorithms in Python from scratch? A: You should have a basic understanding of Python programming, including knowledge of variables, loops, and functions. Familiarity with libraries like NumPy, Pandas, and Matplotlib can be beneficial. A basic understanding of machine learning concepts and algorithms is also necessary. Q: Why is Python a popular choice for implementing machine learning algorithms? A: Python is known for its simplicity, readability, and extensive libraries. These characteristics make it perfect for implementing machine learning algorithms. Libraries such as NumPy, Pandas, and Scikit-learn provide powerful tools for data manipulation and analysis, which are crucial in machine learning. Q: What are some of the most commonly used machine learning algorithms in Python? A: Some of the most commonly used machine learning algorithms include linear regression, logistic regression, decision trees, random forest, K-nearest neighbors (K-NN), and support vector machines (SVM). The choice of algorithm depends on the problem at hand. Q: Can I implement machine learning algorithms in Python without using any libraries? A: Yes, it’s possible to implement machine learning algorithms from scratch without using any specific machine learning libraries. However, this is more complex and time-consuming. Python libraries like Scikit-learn simplify the process of implementing and working with machine learning algorithms. Q: What’s the importance of data preprocessing in implementing machine learning algorithms in Python? A: Data preprocessing is a crucial step in machine learning. It involves cleaning the data (handling missing values, removing outliers) and transforming it (normalization, encoding categorical variables) to a format that the machine learning algorithm can work with. Without proper preprocessing, the performance of your machine learning model may be negatively impacted.	{"url":"https://vidalinternational.in/how-to-implement-machine-learning-algorithms-in-python-from-scratch/","timestamp":"2024-11-09T06:09:16Z","content_type":"text/html","content_length":"161786","record_id":"<urn:uuid:7f79be3e-2aac-4044-a162-45584f89c5a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00762.warc.gz"}
Book of Engineering Think about an object floating in a water. There are two forces acting on the object in this case which are upthrust of water and weight of the object. These two forces must be equal so that the object can float in the water. Continue reading BOUYANCY FORMULA FOR FLOATING OR SUSPENDED (BALANCED IN FLUID) OBJECTS Momentum is a classical physics term where it is calculated by the multiplication of the object’s mass and velocity. It is a vector like velocity that’s why it has value and direction. Continue reading WHAT IS MOMENTUM?	{"url":"https://bookofengineering.com/tag/force/","timestamp":"2024-11-11T14:48:31Z","content_type":"text/html","content_length":"57636","record_id":"<urn:uuid:71ebc3cd-318f-4c5d-9031-15a1da9701e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00858.warc.gz"}
attrition ball mill specification I Attrition bead mill The model used for the simulation of attntlon grinding is the basic kinetic model. in where the first order breakage hypothesis for particles is assumed (Arbiter and Brahny (1960). This assumption is accepted by many authors. by example Austin et al. (197In2. 1976). Heiskanen (1978).	{"url":"https://www.ophripes.fr/2021/06/24_7515.html","timestamp":"2024-11-13T07:31:29Z","content_type":"text/html","content_length":"45058","record_id":"<urn:uuid:6e1b84fa-5683-4902-8c26-1b9eef369025>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00021.warc.gz"}
Engora Data Blog Scientific fraud and business manipulation Sadly, there's a long history of scientific fraud and misrepresentation of data. Modern computing technology has provided better tools for those trying to mislead, but the fortunate flip side is, modern tools provide ways of exposing misrepresented data. It turns out, the right tools can indicate what's really going on. In business, companies often say they can increase sales, or reduce costs, or do so some other desirable thing. The evidence is sometimes in the form of summary statistics like means and standard deviations. Do you think you could assess the credibility of evidence based on the mean and standard deviation summary data alone? In this blog post, I'm going to talk about how you can use one tool to investigate the credibility of mean and standard deviation evidence. Discrete quantities Discrete quantities are quantities that can only take discrete values. An example is a count, for example, a count of the number of sales. You can have 0, 1, 2, 3... sales, but you can't have -1 sales or 563.27 sales. Some business quantities are measured on scales of 1 to 5 or 1 to 10, for example, net promoter scores or employee satisfaction scores. These scales are often called Likert scales. For our example, let's imagine a company is selling a product on the internet and asks its customers how likely they are to recommend the product. The recommendation is on a scale of 0 to 10, where 0 is very unlikely to recommend and 10 is very likely to recommend. This is obviously based on the net promoter idea, but I'm simplifying things here. Very unlikely to recommend Very likely to recommend Imagine the salesperson for the company tells you the results of a 500-person study are a mean of 9 and a standard deviation of 2.5. They tell you that customers love the product, but obviously, there's some variation. The standard deviation shows you that not everyone's satisfied and that the numbers are therefore credible. But are these numbers really credible? Stop for a second and think about it. It's quite possible that their customers love the product. A mean of 9 on a scale of 10 isn't perfection, and the standard deviation of 2.5 suggests there is some variation, which you would expect. Would you believe these numbers? Investigating credibility We have three numbers; a mean, a standard deviation, and a sample size. Lots of different distributions could have given rise to these numbers, how can we backtrack to the original data? The answer is, we can't fully backtrack, but we can investigate possibilities. In 2018, a group of academic researchers in The Netherlands and the US released software you can use to backtrack to possible distributions from mean and standard deviation data. Their goal was to provide a tool to help investigate academic fraud. They wrote up how their software works and published it online, you can read their writeup here. They called their software SPRITE (Sample Parameter Reconstruction via Iterative TEchniques) and made it open-source, even going so far as to make a version of it available online. The software will show you the possible distributions that could give rise to the summary statistics you have. One of the online versions is here. Let's plug in the salesperson's numbers to see if they're credible. If you go to the SPRITE site, you'll see a menu on the left-hand side. In my screenshot, I've plugged in the numbers we have: • Our scale goes from 0 to 10, • Our mean is 9, • Our standard deviation is 2.5, • The number of samples is 500. • We'll choose 2 decimal places for now • We'll just see the top 9 possible distributions. Here are the top 9 results. Something doesn't smell right. I would expect the data to show some form of more even distribution about the mean. For a mean of 9, I would expect there to be a number of 10s and a number of 8s too. These estimated distributions suggest that almost everyone is deliriously happy, with just a small handful of people unhappy. Is this credible in the real world? Probably not. I don't have outright evidence of wrongdoing, but I'm now suspicious of the data. A good next step would be to ask for the underlying data. At the very least, I should view any other data the salesperson provides with suspicion. To be fair to the salesperson, they were probably provided with the data by someone else. What if the salesperson had given me different numbers, for example, a mean of 8.5, a standard deviation of 1.2, and 100 samples? Looking at the results from SPRITE, the possible distributions seem much more likely. Yes, misrepresentation is still possible, but on the face of it, the data is credible. Did you spot the other problem? There's another, more obvious problem with the data. The scale is from 0 to 10, but the results are a mean of 9 and a standard deviation of 2.5, which implies a confidence interval of 6.5 to 11.5. To state the obvious, the maximum score is 10 but the upper range of the confidence interval is 11.5. This type of mistake is very common and doesn't of itself indicate fraud. I'll blog more about this type of mistake later. What does this mean? Due diligence is about checking claims for veracity before spending money. If there's a lot of money involved, it behooves the person doing the due diligence to check the consistency of the numbers they've been given. Tools like SPRITE are very helpful for sniffing out areas to check in more detail. However, just because a tool like SPRITE flags something up it doesn't mean to say there's fraud; people make mistakes with statistics all the time. However, if something is flagged up, you need to get to the bottom of it. Other ways of detecting dodgy numbers Finding out more The echoes of history Sometimes, there are weird connections between very different cultural areas and we see the echoes of history playing out. I'm going to tell you how pulsars, Nobel Prizes, an iconic album cover, Nazi atrocities, and software chart plotting all came to be connected. The discovery of pulsars In 1967, Jocelyn Bell was working on her Ph.D. as a post-graduate researcher at the Mullard Radio Astronomy Observatory, near Cambridge in the UK. She had helped build a new radio telescope and now she was operating it. On November 28, 1967, she saw a strikingly unusual and regular signal, which the team nicknamed "little green men". The signal turned out to be a pulsar, a type of star new to This was an outstanding discovery that shook up astronomy. The team published a paper in Nature, but that wasn't the end of it. In 1974, the Nobel committee awarded the Nobel Physics Prize to the team. To everyone on the team except Jocelyn Bell. Over the years, there's been a lot of controversy over the decision, with many people thinking she was robbed of her share of the prize, either because she was a Ph.D. student or because she was a woman. Bell herself has been very gracious about the whole thing; she is indeed a very classy lady. The pulsar and early computer graphics In the late 1960s, a group of Ph.D. students from Cornell University were analyzing data from the pulsar Bell discovered. Among them was Harold Craft, who used early computer systems to visualize the results. Here's what he said to the Scientific American in 2015: "I found that it was just too confusing. So then, I wrote the program so that I would block out when a hill here was high enough, then the stuff behind it would stay hidden." Here are three pages from Craft's Ph.D. thesis. Take a close look at the center plot. If Craft had made every line visible, it would have been very difficult to see what was going on. Craft re-imagined the data as if he were looking at it at an angle, for example, as if it were a mountain range ridgeline he was looking down on. With a mountain ridgeline, the taller peaks hide what's behind them. It was a simple idea, but very effective. (Credit: JEN CHRISTIANSEN/HAROLD D. CRAFT) The center plot is very striking. So striking in fact, that it found its way into the Cambridge Encyclopaedia of Astronomy (1977 edition): (Cambridge Encyclopedia of Astronomy, 1977 edition, via Tim O'Riley) Joy Division England in the 1970s was not a happy place, especially in the de-industrialized north. Four young men in Manchester had formed a band and recorded an album. The story goes that one of them, Bernard Sumner, was working in central Manchester and took a break in the city library. He came across the pulsar image in the encyclopedia and liked it a lot. The band needed an image for their debut album, so they selected this one. They gave it to a recently graduated designer called Peter Saville, with the instructions it was to be a black-on-white image. Saville felt the image would look better white-on-black, so he designed this cover. This is the iconic Unknown Pleasures album from Joy Division. The starkness of the cover, without the band's name or the album's name, set it apart. The album itself was critically acclaimed, but it never rose high in the charts at the time. However, over time, the iconic status of the band and the album cover grew. In 1980, the lead singer, Ian Curtis, committed suicide. The remaining band members formed a new band, New Order, that went on to massive international fame. By the 21st century, versions of the album cover were on beach towels, shoes, and tattoos. Joy plots In 2017, Claus Wilke created a new charting library for R, ggjoy. His package enabled developers to create plots like the famous Unknown Pleasures album cover. In honor of the album cover, he called these plots joy plots. Ridgeline plots This story has a final twist to it. Although joy plots sound great, there's a problem. Joy Division took their name from a real Nazi atrocity fictionalized in a book called House of Dolls. In some of their concentration camps, the Nazis forced women into prostitution. The camp brothels were called Joy Division. The name joy plots was meant to be fun and a callback to an iconic data visualization, but there's little joy in evil. Given this history, Wilke renamed his package ggridges and the plots ridgeline Here's an example of the great visualizations you can produce with it. If you search around online, you can find people who've re-created the pulsar image using ggridges. (Andrew B. Collier via Twitter) It's not just R programmers who are playing with Unknown Pleasures, Python programmers have got into the act too. Nicolas P. Rougier created a great animation based on the pulsar data set using the venerable Matplotlib plotting package - you can see the animation here. If you liked this post You might like these ones: What is truth? Statistical testing is ultimately all about probabilities and thresholds for believing an effect is there or not. These thresholds and associated ideas are crucial to the decision-making process but are widely misunderstood and misapplied. In this blog post, I'm going to talk about three testing concepts: confidence, significance, and p-values; I'll deal with the hugely important topic of statistical power in a later post. (peas - not p-values. Author: Malyadri, Source: Wikimedia Commons, License: Creative Commons) Types of error To simplify, there are two kinds of errors in statistical testing: • Type I - false positive. You say there's an effect when there isn't. This is decided by a threshold $\alpha$, usually set to 5%. $\alpha$ is called significance. • Type II - false negative. You say there isn't an effect but there is. This is decided by a threshold $\beta$ but is usually expressed as the statistical power which is $1 - \beta$. In this blog post, I'm going to talk about the first kind of error, Type I. Distribution and significance Let's imagine you're running an A/B test on a website and you're measuring conversion on the A branch ($c_A$) and on the B branch ($c_B$). The null hypothesis is that there's no effect, which we can write as: \[H_0: c_A - c_B = 0\] This next piece is a little technical but bear with me. Tests of conversion are usually large tests (mostly, > 10,000 samples in practice). The conversion rate is the mean conversion for all website visitors. Because there are a large number of samples, and we're measuring a mean, the Central Limit Theorem (CLT) applies, which means the mean conversion rates will be normally distributed. By extension from the CLT, the quantity $ c_A - c_B$ will also be normally distributed. If we could take many measurements of $ c_A - c_B$ and the null hypothesis was true, we would theoretically expect the results to look like something like this. Look closely at the chart. Although I've cut off the x-axis, the values go off to $\pm \infty$. If all values of $ c_A - c_B$ are possible, how can we reject the null hypothesis and say there's an effect? Significance - $\alpha$ To decide if there's an effect there, we use a threshold. This threshold is referred to as the level of significance and is called $\alpha$. It's usually set at the 0.05 or 5% level. Confusingly, sometimes people refer to confidence instead, which is 1 - significance, so a 5% significance level corresponds to a 95% confidence level. In the chart below, I've colored the 95% region around the mean value blue and the 5% region (2.5% at each end) red. The blue region is called the acceptance region and the red region is called the rejection region. What we do is compare the measurement we actually make with the chart. If our measurement lands in the red zone, we decide there's an effect there (reject the null), if our measurement lands in the blue zone, we'll decide there isn't an effect there (fail to reject the null or 'accept' the null). One-sided or two-sided tests On the chart with the blue and the red region, there are two rejection (red) regions. This means we'll reject the null hypothesis if we get a value that's more than a threshold above or below our null value. In most tests, this is what we want; we're trying to detect if there's an effect there or not and the effect can be positive or negative. This is called a two-sided test because we can reject the null in two ways (too negative, too positive). But sometimes, we only want to detect if the treatment effect is bigger than the control. This is called a one-sided test. Technically, the null hypothesis in this case is: \[H_0: c_A - c_B \leq 0\] Graphically, it looks like this: So we'll reject the null hypothesis only if our measured value lands in the red region on the right. Because there's only one rejection region and it's on one side of the chart, we call this a one-sided test. I've very blithely talked about measured values landing in the rejection region or not. In practice, that's not what we do; in practice, we use p-values. Let's say we measured some value x. What's the probability we would measure this value if the null hypothesis were true (in other words, if there were no effect)? Technically, zero because the distribution is continuous, but that isn't helpful. Let's try a more helpful form of words. Assuming the null hypothesis is true, what's the probability we would see a value of x or a more extreme value? Graphically, this looks something like the green area on the chart below. Let me say again what the p-value is. If there's no effect at all, it's the probability we would see the result (or a more extreme result) that we measured, or how likely is it that our measurement could have been due to chance alone? The chart below shows the probability the null hypothesis is true; it shows the acceptance region (blue), the rejection region (red), and the measurement p-value (green). The p-value is in the rejection region, so we'll reject the null hypothesis in this case. If the green overlapped with the blue region, we would accept (fail to reject) the null hypothesis. There are some common misunderstandings around testing that can have quite bad commercial effects. • 95% confidence is too high a bar - we should drop the threshold to 90%. In effect, this means you'll accept a lot of changes that have no effect. This will reduce the overall effectiveness of your testing program (see this prior blog post for an explanation). • One-sided tests are OK and give a smaller sample size, so we should use them. This is true, but it's often important to determine if a change is having a negative effect. In general, hypothesis testing tests a single hypothesis, but sadly, people try and read more into test results than they should and want to answer several questions with a single question. • p-values represent the probability of an effect being present. This is just not true at all. • A small p-value indicates a big effect. p-values do not make any indication about the size of an effect; a low p-value does not mean there's a big effect. Practical tensions In practice, there can be considerable tension between business and technical people over statistical tests. A lot of statistical practices (e.g. 5% significance levels, two-sided testing) are based on experience built up over a long time. Unfortunately, this all sounds very academic to the business person who needs results now and wants to take shortcuts. Sadly, in the long run, shortcuts always catch you up. There's an old saying that's very true: "there ain't no such thing as a free lunch." Promotion paths for technical people I’ve worked in technology companies and I’ve seen the same question arise several times: what to do with technical people who don’t want to be managers? What do you promote them to? (Image credit: Louis-Henri de Rudder, source: Old Book Illustrations) The traditional engineering career ladder emphasizes management as the desired end-goal and devalues anyone not in a management position. Not everyone wants to be a manager and not everyone is good at management. Some people are extremely technically competent and want to stay technical. What are they offered? Separate, but not equal Most companies deal with the problem by creating a parallel career path for engineers who don’t want to be managers. This is supposedly separate but equal, but it always ends up being very unequal in the management branch’s favor. The inequality is reflected in job titles. Director is a senior position in most companies and it comes with management responsibility. The equivalent technical role might be ‘Fellow’, which has overtones of putting someone out to grass. A popular alternative is ‘Technical Director’, but note the management equivalent is Director - the engineers get a qualifying word the managers don’t, it’s letting people know the person isn’t a real Director (they're technically a Director, but...). Until you get to VP or C-level, the engineering titles are always worse. The management and technical tracks have power differences too. The managers get to decide pay raises and promotions and hiring and firing, the technical people don’t. Part of this is obviously why people choose management (and why some people don’t choose management), but often the technical path people aren’t even given a seat at the table. When there are business decisions to be made, the technical people are usually frozen out, even when the business decisions aren't people ones. Sometimes this is legitimate, but most of the time it’s a power thing. The message is clear: if you want the power to change things, you need to be a manager. A way forward Here’s what I suggest. The managerial/technical divide is a real one. Not everyone wants to be a manager and there should be a career path upward for them. I suggest having the same job titles for the managerial path and the technical path. There should be no Technical Directors and Directors, just Directors. People on the technical path should be given a seat at the power table and should be equals when it comes to making business decisions. This means managers will have to give up power and it will mean a cultural shift, but if we’re to give meaningful advancement to the engineering track, this is the way it has to be. All the executives laughed A few years ago, I was at an industry event. The speaker was an executive talking about his A/B testing program. He joked that vendors and his team were unreliable because the overall result was less than the sum of the individual tests. Everyone laughed knowingly. But we shouldn't have laughed. The statistics are clear and he should have known better. By the rules of the statistical game, the benefits of an A/B program will be less than the sum of the parts and I'm going to tell you why. Thresholds and testing An individual A/B test is a null hypothesis test with thresholds that decide the result of the test. We don't know whether there is an effect or not, we're making a decision based on probability. There are two important threshold numbers: • $\alpha$ - also known as significance and usually set around 5%. If there really is no effect, $\alpha$ is the probability we will say there is an effect. In other words, it's the false positive rate (Type I errors). • $\beta$ - is usually set around 20%. If there really is an effect, $\beta$ is the probability we will say there is no effect. In other words, it's the false negative rate (Type II errors). In practice, power is used instead of $\beta$, power is $1-\beta$, so it's usual to set the power to 80%. Standard statistical practice focuses on just a single test, but an organization's choice of $\alpha$ and $\beta$ affect the entire test program. $\alpha$, $\beta$ and the test program To see how the choice of $\alpha$ and $\beta$ affect the entire test program, let's run a simplified thought experiment. Imagine we choose $\alpha = 5\%$ and $\beta = 20\%$, which are standard settings in most organizations. Now imagine we run 1,000 tests, in 100 of them there's a real effect and in 900 of them there's no effect. Of course, we don't know which tests have an effect and which don't. Take a second to think about these questions before moving on: • How many many positive test results will we measure? • How many false positives will we see? • How many true positives will we see? At this stage, you should have numbers in mind. I'm asking you to do this so you understand the importance of what happens next. The logic to answer these questions is straightforward. In the picture below, I've shown how it works, but I'll talk you through it so you can understand it in more detail. Of the 1,000 tests, 100 have a real effect. These are the tests that $\beta$ applies to and $\beta=20\%$, so we'll end up with: • 20 false negatives, 80 true positives Of the 1,000 tests, 900 have no effect. These are the tests that $\alpha$ applies to and $\alpha=5\%$, so we'll end up with: • 855 true negatives, 45 false positives Overall we'll measure: • 125 positives made up of • 80 true positives • 45 false positives Crucially, we won't know which of the 125 positives are true and which are false. Because this is so important, I'm going to lay it out again: in this example, 36% of all test results we thought were positive are wrong, but we don't know which ones they are. They will dilute the overall results of the overall program. The overall results of the test program will be less than the sum of the individual test results. What happens in reality In reality, you don't know what proportion of test results are 'true'. It might be 10%, or 20%, or even 5%. Of course, the reason for the test is that you don't know the result. What this means is, it's hard to do this calculation on real data, but the fact that you can't easily do the calculation doesn't mean the limits don't apply. Can you make things better? To get a higher proportion of true positives, you can do at least three things. • Run fewer tests - selecting only tests where you have a good reason to believe there is a real effect. This would certainly work, but you would forgo a lot of the benefits of a testing program. • Run with a lower $\alpha$ value. There's a huge debate in the scientific community about significance levels. Many authors are pushing for a 0.5% level instead of a 5% level. So why don't you just lower $\alpha$? Because the sample size will increase greatly. • Run with a higher power (lower $\beta$). Using a power of 80% is "industry standard", but it shouldn't be - in another blog post I'll explain why. The reason people don't do it is because of test duration - increasing the power increases the sample size. Are there other ways to get results? Maybe, but none that are simple. Everything I've spoken about so far uses a frequentist approach. Bayesian testing offers the possibility of smaller test sizes, meaning you could increase power and reduce $\alpha$ while still maintaining workable sample sizes. Of course, A/B testing isn't the only testing method available and other methods offer higher power with lower sample sizes. No such thing as a free lunch Like any discipline, statistical testing comes with its own rules and logic. There are trade-offs to be made and everything comes with a price. Yes, you can get great results from A/B testing programs, and yes companies have increased conversion, etc. using them, but all of them invested in the right people and technical resources to get there and all of them know the trade-offs. There's no such thing as a free lunch in statistical testing. Why use the Poisson distribution? Because it has properties that make it great to work with, data scientists use the Poisson distribution to model different kinds of counting data. But these properties can be seductive, and sometimes people model data using the Poisson distribution when they shouldn't. In this blog post, I'll explain why the Poisson distribution is so popular and why you should think twice before using it. (SimÃ©on-Denis Poisson by E. Marcellot, Public domain, via Wikimedia Commons) Poisson processes The Poisson distribution is a discrete event probability distribution used to model events created using a Poisson process. Drilling down a level, a Poisson process is a series of events that have these properties: • They occur at random but at a constant mean rate, • They are independent of one another, • Two (or more) events can't occur at the same time Good examples of Poisson processes are website visits, radioactive decay, and calls to a help center. The properties of a Poisson distribution Mathematically, the Poisson probability mass function looks like this: \[ P_r (X=k) = \frac{\lambda^k e^{- \lambda}}{k!} \] • k is the number of events (always an integer) • $\lambda$ is the mean value (or expected rate) It's a discrete distribution, so it's only defined for integer values of $k$. Graphically, it looks like this for $\lambda=6$. Note that it isn't symmetrical and it stops at 0, you can't have -1 events. (Let's imagine we were modeling calls per hour in a call center. In this case, $k$ is the measured calls per hour, $P$ is their frequency of occurrence, and $\lambda$ is the mean number of calls per hour). Here are some of the Poisson distribution's properties: • Mean: $\lambda$ • Variance: $\lambda$ • Mode: floor($\lambda$) The fact that some of the key properties are given by $\lambda$ alone makes using it easy. If your data follows a Poisson distribution, once you know the mean value, you've got the variance (and standard deviation), and the mode too. In fact, you've pretty much got a full description of your data's distribution with just a single number. When to use it and when not to use it Because you can describe the entire distribution with just a single number, it's very tempting to assume that any data that involves counting follows a Poisson distribution because it makes analysis easier. Sadly, not all counts follow a Poisson distribution. In the list below, which counts do you think might follow a Poisson distribution and which might not? • The number of goals in English Premier League soccer matches. • The number of earthquakes of at least a given size per year around the world. • Bus arrivals. • The number of web pages a person visits before they make a purchase. Bus arrivals are not well modeled by a Poisson distribution because in practice they're not independent of one another and don't occur at a constant rate. Bus operators change bus frequencies throughout the day, with more buses scheduled at busy times; they may also hold buses at stops to even out arrival times. Interestingly, bus arrivals are one of the textbook examples of a Poisson process, which shows that you need to think before applying a model. The number of web pages a person visits before they make a purchase is better modeled using a negative binomial distribution. Earthquakes are well-modeled by a Poisson distribution. Earthquakes in different parts of the world are independent of one another and geological forces are relatively constant, giving a constant mean rate for quakes. It's possible that two earthquakes could happen simultaneously in different parts of the world, which shows that even if one of the criteria might not apply, data can still be well-modeled by Poisson. What about soccer matches? We know two goals can't happen at the same time. The length of matches is fixed and soccer is a low-scoring game, so the assumption of a constant rate for goals is probably OK. But what about independence? If you've watched enough soccer, you know that the energy level in a game steps up as soon as a goal is scored. Is this enough to violate the independence requirement? Apparently not, scores in soccer matches are well-modeled by a Poisson distribution. What should a data scientist do? Just because the data you're modeling is a count doesn't mean it follows a Poisson distribution. More generally, you should be wary of making choices motivated by convenience. If you have count data, look at the properties of your data before deciding on a distribution to model it with. If you liked this blog post you might like The moon and the cold war In 1959, Cold War rivalries were intense and drove geopolitics; the protagonists had already fought several proxy wars and the nuclear arms race was well underway. The Soviet Union put the first satellite into space in 1957, which was a wake-up call to the United States; if the Soviet Union could put a satellite in orbit, they could put a missile in orbit. By extension, if the Soviet Union got to the moon first, they could build a lunar military base and dominate the moon. The Soviet Union had announced plans to celebrate its 50th anniversary (1967) with a lunar landing. The race was on with a clear deadline. (Phadke09, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons) In response to the perceived threat, the US Army developed an audacious plan to set up a military base on the moon by 1965 and beat the Soviets. The plan, Project Horizon, was 'published' in 1959 but only declassified in 2014. The plan is very extensive and covers living arrangements, spacesuits, power, and transport, it was published in two volumes with illustrations (Volume I, Volume II). In some alternative history, something like this could have happened. The ideas in it still have some relevance, so let's dive in and take a look. Getting there In 1959, the enormous Saturn V rockets didn't exist and the army planners knew that heavy-lift rockets would take years to develop. To meet the 1965 deadline, they needed to get going with current and near-future technology, which meant Saturn I and Saturn II rockets. The plan called for 61 Saturn I rockets and 88 Saturn IIs, with a launch rate of 5.3 per month. In reality, only 19 Saturn Is were ever launched (the Apollo project used Saturn Vs, of which 13 were launched). To state the obvious, smaller rockets carry smaller payloads. To maximize payloads to orbit, you need to think about launch sites; the closer you are to the equator the bigger boost you get from the Earth's rotation. Project Horizon considered several launch sites on the equator, none of which were in US territory. The map below comes from the report and I've indicated prospective launch sites in red. • Somalia. Rejected because of remoteness. • Manus Island. Rejected because of remoteness. • Christmas Island. Remote, but given serious consideration because of logistics. • Brazil. Closer to the US and given serious consideration. The report doesn't decide between Christmas Island and Brazil but makes a good case for both. The launch site would obviously be huge and would have several gantries for multiple launches to hit the 5.3 launches per month target. The next question is: how do you get to the moon? Do you go directly or do you attempt equatorial orbit refueling? In 1969, Apollo 11 went directly to the moon, but it was launched by the far larger Saturn V rocket. With just Saturn I and Saturn IIs, the team needed to take a different approach. They settled on orbital refueling from a 10-person space station followed by a direct approach once more powerful rockets became available. Landing on the moon The direct and indirect methods of getting to the moon led to two different lander designs, one of which I've shown below. The obvious question is, why is it streamlined? The upper stage is the return-to-earth vehicle so it's shaped for re-entry. In the Apollo missions, the reentry vehicle was part of the command module that stayed in lunar orbit, so the lunar lander could be any shape. The plan was for an initial landing by two astronauts followed by construction teams to build the base. The role of the first astronauts was scouting and investigating possible base sites, and they were to stay on the moon for 30-90 days, living in their lander. The construction crews would build the base in 18 months but the maximum tour of duty for a construction crew member was 12 months. By late 1965, the base would be staffed by a team of 12 (all men, of course, this was planned in 1959 after all). The moon base The moon crew was to have two different sorts of rides: a lunar bulldozer for construction and a rover for exploration (and 'surveillance'). The bulldozer was to dig trenches and drop the living quarter into them (the trenches would also be partially excavated by high explosives). The living quarters themselves were 10ft x 20ft cylinders. Burying living quarters helps with temperature regulation and provides protection against radiation. The cylinders themselves were to be giant double-walled thermos flasks (vacuum insulated) for thermal stability. The finished base was to be l-shaped. Lunar living The initial construction camp looks spartan at best and things only improve marginally when the l-shaped base is completed. In the finished base, toilets were to be 'bucket-type' with activated charcoal for odor control; urine was to be stored on the moon surface for later recycling. The men were to be rationed to 6lb of water (2.7 liters) and 4lb of food per day - not starvation or dehydration, but not generous either. The initial plan was for all meals to be pre-cooked, but the soldiers would later set up a hydroponics farm for fresh fruit and vegetables. Curiously, there isn't as much as you'd expect about spacesuits, only a few pages. They knew that spacesuits would be restrictive and went so far as defining the body measurements of a standard man, including details like palm length. The idea seems straightforward, if technology restricts your design flexibility, then select your crew to fit what you can build. Perhaps unsurprisingly for the 1950s, power for the base was to come from two nuclear reactors. both of which needed to be a safe distance from the base and recessed into the regolith in case of accidents. It seems like the lunar bulldozer was going to be very busy. Soldiers mean guns or at least weapons. The report is surprisingly coy about weapons; it alludes to R&D work necessary to develop lunar weapons, but that's about it. $6 billion total in 1959 dollars. Back then, this was an awful lot of money. The real Apollo program cost $25.4 billion and it's highly likely $6 billion was a substantial underestimate, probably by an order of magnitude. Project Horizon's impact As far as I can tell, very little. The plan was put to Eisenhower, who rejected it. Instead, NASA was created and the race to the moon as we know it started. But maybe some of the Project Horizon ideas might come back. Burying habitats in the lunar regolith is an idea the Soviets used in their lunar base plans and has been used several times in science fiction. It's a compelling idea because it insulates the base from temperature extremes and from radiation. However, we now know lunar regolith is a difficult substance to work with. Nuclear power makes sense but has obvious problems, and transporting nuclear power systems to orbit has risks. The 1970s British TV science fiction series "Space:1999" had a nuclear reactor explosion knocking the moon out of orbit, which is far-fetched, but a nuclear problem on the moon would be severe. The ideas of in-flight re-fueling and lunar waystations have come up again in NASA's future lunar exploration plans. What may have dealt a project like Project Horizon a final death blow is the 1967 Outer Space Treaty which bans weapons of mass destruction and the militarization of space. Project Horizon is a footnote in the history of space exploration but an interesting one. It gives insight into the mind of the military planners of the time and provides a glimpse into one alternative path the world might have taken. If you liked this blog post You might like these other ones:	{"url":"https://blog.engora.com/search?updated-max=2021-07-12T07:49:00-04:00&max-results=7&reverse-paginate=true&start=7&by-date=false","timestamp":"2024-11-04T07:17:07Z","content_type":"text/html","content_length":"167436","record_id":"<urn:uuid:fc04b9ca-223b-47ac-8d84-3b118b74021e>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00464.warc.gz"}
NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.1 NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.1 Triangles in Hindi and English Medium prepared for academic session 2024-25. Class 10 Maths Exercise 6.1 contains questions based on proportionality theorem. Class 10 Maths Exercise 6.1 Solutions Class 10 Maths Exercise 6.1 in Hindi Class 10th Maths NCERT Book Download Class 10 Maths Chapter 6 Solutions Class 10 Maths NCERT Solutions Class 10 all Subjects NCERT Solutions NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.1 To effectively prepare for Class 10 Maths Exercise 6.1, start by mastering the basic concepts of similarity and congruence. Class 10 Maths Exercise 6.1 requires a solid understanding of geometric figures, so revise the definitions of similar and congruent figures and how these concepts apply to polygons, particularly triangles. Practice identifying similar triangles based on the criteria of corresponding angles and proportional sides. Make sure you are familiar with common geometric properties, such as the congruency of angles in similar shapes, to answer the fill-in-the-blank questions and state similarities between figures confidently. Now, focus on solving examples that require recognizing and comparing similar and non-similar figures. Review previous chapters on triangles and polygons to strengthen your understanding of the geometrical properties used in Class 10 Maths Exercise 6.1. Practice questions that involve the identification of similar shapes based on their corresponding angles and sides. Pay close attention to how the ratios of sides relate to similarity, as this will be crucial for solving problems that ask you to determine if two figures are similar. Class: 10 Mathematics Chapter 6: Exercise 6.1 Content: NCERT Question Answers Content Type: Text and Videos format Academic Session: Year 2024-25 Medium: English and Hindi Medium Class X Math Ex. 6.1 solutions are free to download in PDF or View in Video Format along with Offline Apps updated for 2024-25. Download UP Board Solutions and NCERT Solutions 2024-25 of other subjects for CBSE, Uttarakhand, Bihar and UP Board students, who are using NCERT Books 2024-25 based on updated CBSE Syllabus for the session 2024-25. Join the Discussion Forum to share your knowledge and ask your doubts. 10 Maths Chapter 6 Exercise 6.1 Solutions NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.1 Triangles English medium use online or download in PDF or View in Video Format. All the solutions are updated for academic session 2024-25. About 10 Maths Exercise 6.1 In 10th Math Exercise 6.1, the questions are based on simple concepts of SIMILAR TRIANGLES, in which the answers may be differ from person to person. In fill in the blanks options, we can fill it according to knowledge of similarity. Important Questions on Similar Triangles 1. Is the triangle with sides 12cm, and 18 cm a right triangle? Give reason. 2. It is given that triangle DEF ~ triangle RPQ. Is it true to say that angle D = angle R and angle F = angle P? 3. If the corresponding Medians of two similar triangles are in the ratio 5:7, then find the ratio of their sides. 4. A right angled triangle has its area numerically equal to its perimeter. The length of each side is an even number and the hypotenuse is 10cm. What is the perimeter of the triangle? 5. An aeroplane leaves an airport and flies due west at a speed of 2100 km/hr. At the same time, another aeroplane leaves the same place at airport and flies due south at a speed of 2000 km/hr. How far apart will be the two planes after 1 hour? Ask your doubts related to NIOS or CBSE Board and share your knowledge with your friends and other users through Discussion Forum. How do we differentiate similar figures and congruent figures in Exercise 6.1 of 10th Maths? Two figures having the same shape (and not necessarily the same size) are called similar figures. Two figures are said to be congruent, if they have the same shape and the same size. How many questions are there in exercise 6.1 of class 10th mathematics chapter 6? There are 3 questions in exercise 6.1 of class 10th mathematics chapter 6 (Triangles). All questions are different from each other but based on same concept. MCQ can come from this exercise. What are the examples of Similar Figures in exercise 6.1 of Class 10 Maths? Examples of Similar Figures are: 1) All equilateral triangles are similar. 2) All circles are similar. 3) All squares are similar Is exercise 6.1 of class 10th mathematics chapter 6 (Triangles) easy? Yes, exercise 6.1 of class 10th mathematics chapter 6 is very easy exercise. This exercise contains only 3 questions and all 3 questions are very easy. In Q1 students have to fill the blanks only, In Q 2 students have to give examples of pair of similar and non similar figures and In Q 3 students have to tell whether the given quadrilaterals are similar or not. Last Edited: October 8, 2024 Content Reviewed: October 8, 2024	{"url":"https://www.tiwariacademy.com/ncert-solutions/class-10/maths/chapter-6/exercise-6-1/","timestamp":"2024-11-04T11:02:59Z","content_type":"text/html","content_length":"241676","record_id":"<urn:uuid:ec37ec33-f6b5-4347-8f53-0118bbc07c52>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00787.warc.gz"}
Accelerate Your Understanding: How to Find the Acceleration from a Position-Time Graph - The Techy Life Accelerate Your Understanding: How to Find the Acceleration from a Position-Time Graph In the world of physics, understanding the concept of acceleration is crucial in comprehending the motion of objects. Whether it’s a speeding car on the highway or a ball falling from a height, analyzing the acceleration can provide valuable insights into the change in velocity over time. One tool that helps visualize this relationship is a position-time graph, which plots the position of an object over time. By examining the shape of this graph, one can determine the acceleration and gain a deeper understanding of the object’s motion. A position-time graph is a graphic representation of an object’s displacement at various points in time. It provides a visual representation of how the position of the object changes over time. This graph typically consists of a horizontal x-axis representing time and a vertical y-axis representing position. While a position-time graph may seem straightforward, it holds a wealth of information regarding the object’s velocity and acceleration, making it a valuable tool in the field of physics. By examining the slope and shape of the graph, we can unravel the mystery of acceleration and gain a comprehensive understanding of an object’s motion. Understanding Position-Time Graphs A. Definition and purpose of a position-time graph A position-time graph is a visual representation of the relationship between an object’s position and the time it takes to reach that position. It provides valuable information about an object’s motion, including its direction, speed, and changes in velocity. By analyzing position-time graphs, you can gain a better understanding of how an object moves and calculate important values such as The purpose of a position-time graph is to simplify the interpretation and analysis of an object’s motion. It allows you to visualize the position of an object at different points in time and observe patterns or trends in its movement. By plotting a series of points on the graph, you can gather information about the object’s velocity and acceleration, which are crucial in understanding its overall motion. B. Explanation of how position and time are represented on the graph In a position-time graph, the horizontal axis represents time, while the vertical axis represents position. The chosen units for time and position will determine the scale used on the graph. As time progresses, the position of the object is recorded and plotted as a point on the graph. By joining these points with a line, the motion of the object can be visualized. By analyzing the steepness or slope of the line on the graph, you can determine the object’s velocity. A steeper slope indicates a higher velocity, while a flatter slope corresponds to a slower velocity. The slope can also indicate the direction of the object’s motion: if the line slopes upwards, it indicates a positive velocity, whereas a downward slope represents a negative velocity or motion in the opposite direction. C. Illustration of different types of motion represented on the graph Position-time graphs can depict various types of motion. For example, a straight line with a positive slope indicates constant velocity, meaning the object is moving at a consistent speed in a straight line. If the line has a negative slope, it represents motion in the opposite direction. Curved lines on a position-time graph indicate changes in velocity. A steeply curving line represents an accelerated motion, suggesting that the object is eTher speeding up or slowing down. Conversely, a gently curving line indicates a deceleration or a gradual change in velocity. Understanding the different types of motion represented on a position-time graph is essential for determining acceleration accurately. By analyzing the shape of the graph, you can gain insights into how an object’s motion changes over time and calculate its acceleration accordingly. In the next section, we will define acceleration and explore how it can be calculated from a position-time graph. Understanding acceleration is crucial in comprehending an object’s motion and using position-time graphs effectively. Definition of Acceleration A. Clear explanation of acceleration as a measure of change in velocity Acceleration is a fundamental concept in physics that measures the rate at which an object’s velocity changes over time. It provides valuable information about how an object is moving and how quickly its velocity is changing. Acceleration can eTher be positive, indicating an increase in velocity, or negative, indicating a decrease in velocity. In simple terms, acceleration measures how quickly an object is speeding up or slowing down. For example, if a car starts from rest and gradually increases its velocity to 60 miles per hour in 10 seconds, it experiences positive acceleration. On the other hand, if a car is traveling at 60 miles per hour and comes to a stop in 5 seconds, it experiences negative acceleration. B. Formula for calculating acceleration Acceleration can be calculated using a simple formula: acceleration (a) = (final velocity – initial velocity) / time This formula takes into account the change in velocity and the time interval over which the change occurs. The units of acceleration are typically meters per second squared (m/s^2) in the International System of Units (SI). For example, if an object initially has a velocity of 10 m/s and its velocity increases to 30 m/s in 5 seconds, the acceleration can be calculated as: acceleration = (30 m/s – 10 m/s) / 5 s = 4 m/s^2 This means that the object is accelerating at a rate of 4 meters per second squared. Understanding the formula for calculating acceleration is crucial for analyzing motion and interpreting position-time graphs. It allows us to quantify how quickly an object’s velocity is changing and provides a basis for further analysis of its motion. By using the formula, we can determine the acceleration of an object at any given moment as long as we have the necessary data: initial and final velocities, and the time interval over which the change in velocity occurs. In the next section, we will explore the relationship between velocity and acceleration and how changes in velocity are reflected in position-time graphs. IRelationship between Velocity and Acceleration A. Explanation of how acceleration affects velocity Velocity and acceleration are closely related concepts in the study of motion. Acceleration refers to the rate at which an object’s velocity is changing, whether that be speeding up, slowing down, or changing direction. It is important to understand how acceleration affects velocity as they both play a crucial role in analyzing motion. When an object experiences acceleration, its velocity is impacted. If an object is accelerating in the same direction as its initial velocity, the object will increase its speed. For example, if a car is initially moving at a speed of 30 mph and experiences a positive acceleration of 5 mph/s, its velocity will increase by 5 mph every second, resulting in a faster speed. On the other hand, if an object’s acceleration is in the opposite direction to its initial velocity, its speed will decrease. This is referred to as negative acceleration or deceleration. If the same car mentioned earlier experiences a negative acceleration of 5 mph/s, its velocity will decrease by 5 mph every second, eventually causing it to come to a stop and potentially reverse direction. B. Demonstration of how changes in velocity are reflected in the position-time graph Changes in velocity can be observed and interpreted by analyzing a position-time graph. In a position-time graph, the object’s velocity is represented by the slope of the graph at any given point. If an object is moving with a constant velocity, the position-time graph will be a straight line with a constant slope. This indicates that the object is not experiencing any acceleration, as its velocity remains constant. When an object is accelerating, the position-time graph will show a changing slope. If the slope of the graph is increasing, it means that the object’s velocity is increasing, indicating positive acceleration. Conversely, if the slope is decreasing, it means that the object’s velocity is decreasing, indicating negative acceleration. By analyzing the position-time graph, it becomes evident how changes in velocity are reflected and can be deduced from the graph itself. This understanding of the relationship between velocity and acceleration allows for the accurate interpretation of motion using position-time graphs. Overall, the relationship between velocity and acceleration is crucial in understanding how an object’s motion changes over time. Acceleration directly influences the velocity of an object, eTher by increasing or decreasing its speed. By analyzing the changes in velocity observed in a position-time graph, one can gain valuable insights into the motion and acceleration of an object. Finding Average Acceleration from a Position-Time Graph Step-by-step guide to finding average acceleration In order to find the average acceleration from a position-time graph, follow these steps: 1. Identify two points on the graph: Start by selecting two points on the position-time graph that correspond to the time interval over which you want to calculate the average acceleration. 2. Determine the change in velocity: Find the change in velocity between the two points by subtracting the initial velocity from the final velocity. The initial velocity can be determined by finding the position at the starting point and the final velocity can be determined by finding the position at the ending point. 3. Calculate the change in time: Find the change in time between the two points by subtracting the initial time from the final time. 4. Use the formula for average acceleration: Divide the change in velocity by the change in time to find the average acceleration. The formula for average acceleration is: average acceleration = (change in velocity) / (change in time) 5. Units and interpretation: Make sure to include the appropriate units for velocity, time, and acceleration in your calculation. For example, if velocity is measured in meters per second and time is measured in seconds, then the units for acceleration would be meters per second squared (m/s^2). Finally, interpret your result by considering whether the object is speeding up or slowing down during the given time interval. Example calculations using real-life scenarios and position-time graphs To further illustrate how to find average acceleration from a position-time graph, let’s consider a couple of real-life scenarios: Scenario 1: An object is moving in a straight line and its position-time graph shows a constant positive slope. Let’s say the initial position is 2 meters and the final position is 12 meters, with a time interval of 4 seconds between the two points. By using the formula for average acceleration, we find: average acceleration = (12 m – 2 m) / (4 s) = 10 m/4 s = 2.5 m/s^2 Therefore, the object is accelerating at an average rate of 2.5 meters per second squared. Scenario 2: An object is moving in a straight line and its position-time graph shows a constant negative slope. Let’s say the initial position is 10 meters and the final position is 4 meters, with a time interval of 2 seconds between the two points. By using the formula for average acceleration, we find: average acceleration = (4 m – 10 m) / (2 s) = -6 m/2 s = -3 m/s^2 In this case, the object is decelerating (or slowing down) at an average rate of 3 meters per second squared. By following these steps and applying the formula for average acceleration, you can confidently analyze position-time graphs to determine average acceleration and gain a deeper understanding of an object’s motion. ## Finding Instantaneous Acceleration from a Position-Time Graph ### A. Definition and Significance of Instantaneous Acceleration Instantaneous acceleration is a measure of how velocity is changing at a specific moment in time. Unlike average acceleration, which provides an overall change in velocity over a given interval, instantaneous acceleration focuses on the precise rate of change at an exact point. This measurement is crucial in understanding the dynamics of an object’s motion. When studying motion, it is often necessary to analyze how an object’s acceleration varies continuously throughout its path. Instantaneous acceleration helps capture these variations, providing insights into the intricate details of an object’s movement. By determining the instantaneous acceleration, it becomes possible to anticipate changes in velocity and predict future positions ### B. Techniques for Approximating Instantaneous Acceleration using Position-Time Graphs To approximate instantaneous acceleration using a position-time graph, one can employ two main techniques: tangent lines and secant lines. Tangent lines involve drawing a straight line that touches the curve of the position-time graph at a specific point. The slope of this tangent line represents the instantaneous acceleration at that particular moment. By choosing a point very close to the desired moment and ensuring the tangent line aligns closely with the curve, a reasonably accurate estimation of the instantaneous acceleration can be obtained. Secant lines, on the other hand, involve drawing a line that connects two points on the position-time graph. The slope of this line represents the average acceleration between those two points. By reducing the interval between these points to a infinitesimally small value (i.e., reducing the interval to zero), the secant line essentially becomes a tangent line. When using eTher the tangent or secant line methods, it is important to choose points that are as close as possible to the desired moment for the most accurate approximation of instantaneous acceleration. Additionally, the precision of the estimate can be improved by using more refined measurement tools, such as digital graphing software or advanced calculators. In conclusion, understanding how to find the instantaneous acceleration from a position-time graph is essential for analyzing the intricate details of an object’s motion. By utilizing tangent and secant lines, it is possible to estimate the precise rate of change of velocity at any given point along the object’s path. These techniques provide valuable insights into the dynamics of motion and aid in predicting future positions accurately. Interpreting Negative Acceleration on a Position-Time Graph A. Explanation of negative acceleration and its meaning in motion In physics, acceleration is defined as the rate at which velocity changes over time. It can be positive or negative, depending on the direction of the change in velocity. Positive acceleration occurs when an object’s velocity increases, while negative acceleration (also known as deceleration or retardation) happens when the velocity decreases. Negative acceleration does not necessarily mean that an object is moving in the opposite direction. Instead, it indicates that the object is slowing down. For example, if a car is traveling forward and its velocity decreases, the acceleration will be negative. This could happen when the car applies brakes or encounters resistance. B. Examples showcasing negative acceleration on position-time graphs Position-time graphs are useful tools for understanding how acceleration affects an object’s motion. When negative acceleration is present, the graph will exhibit specific characteristics. One example is a car coming to a stop. As the car decelerates, its velocity, represented by the slope of the graph, decreases. Consequently, the graph’s slope becomes less steep, indicating negative acceleration. The exact value of the negative acceleration can be calculated by finding the slope of the graph during the deceleration phase. Another example is an object thrown upwards. Due to the force of gravity, the object’s velocity decreases until it reaches its peak height and starts descending. On a position-time graph, this would be represented by a positive slope during the ascent, representing negative acceleration, and a negative slope during the descent, indicating positive acceleration. Understanding negative acceleration on a position-time graph is crucial for analyzing motion accurately. It allows us to determine when an object is slowing down, changing direction, or coming to a stop. By interpreting the graph’s slope and understanding its relationship with negative acceleration, we can gain valuable insights into the underlying physics of an object’s motion. Overall, recognizing negative acceleration on a position-time graph provides essential information about an object’s behavior and helps us comprehend the dynamic nature of motion. Acceleration and Slope of the Position-Time Graph A. Explanation of how the slope of a position-time graph represents acceleration In physics, acceleration refers to the rate at which an object’s velocity changes over time. When analyzing motion using a position-time graph, the slope of the graph represents the object’s acceleration. This relationship can be understood by considering the definition of slope and how it relates to change in position and change in time. The slope of a line is calculated by determining the ratio of the vertical change (change in position) to the horizontal change (change in time) between two points on the graph. In the context of a position-time graph, the vertical change represents the change in position of the object and the horizontal change represents the change in time. Therefore, the slope of the graph represents the ratio of the change in position to the change in time, which is equivalent to the object’s velocity. When analyzing the slope of a position-time graph, a steeper slope indicates a greater change in position over a given time interval. This corresponds to a larger change in velocity and thus a higher acceleration. Conversely, a shallower slope indicates a smaller change in position over the same time interval, meaning a smaller change in velocity and a lower acceleration. Therefore, the slope of a position-time graph directly represents the object’s acceleration. B. Calculating acceleration using the slope of the graph To calculate the acceleration using the slope of the position-time graph, one can determine the slope between two points on the graph that represent the positions of the object at different times. The slope is calculated by dividing the change in position (vertical change) by the change in time (horizontal change) between the two points. For example, if the position-time graph has a slope of 2, this means that for every unit increase in time, the object’s position increases by 2 units. This corresponds to an acceleration of 2 units per unit of time, indicating that the object is accelerating at a constant rate. It is important to note that this method of calculating acceleration assumes that the motion represented by the position-time graph is uniform. If the motion is not uniform, such as in cases of changing velocities or non-linear motion, this method may provide an average or approximate value of acceleration. By understanding the relationship between the slope of a position-time graph and acceleration, one can effectively analyze and interpret the motion of objects. This method provides a graphical approach to calculate acceleration and enables a deeper understanding of the fundamental concepts of physics. Graphical Methods for Calculating Acceleration A. Overview of other graphical methods to calculate acceleration In addition to using the slope of a position-time graph, there are other graphical methods that can be employed to calculate acceleration. These methods offer alternative approaches and can be particularly useful in certain scenarios. One graphical method to determine acceleration involves analyzing the shape of the position-time graph. By examining the curvature of the graph, it is possible to infer information about the object’s acceleration. If the graph exhibits a convex shape, it indicates positive acceleration, while a concave shape suggests negative acceleration. This method is based on the fact that acceleration can be interpreted as the rate at which an object’s velocity is changing, and the shape of the graph reflects this change. Another graphical technique for calculating acceleration involves constructing velocity-time graphs. By finding the slope of the velocity-time graph at a specific point, the instantaneous acceleration at that point can be determined. This method is particularly useful when the position-time graph is not available or difficult to interpret. It provides a direct visualization of the object’s velocity and allows for precise calculations of acceleration. B. Comparison of different graphical techniques and their advantages Each graphical method for calculating acceleration has its own advantages and limitations. The method using the slope of the position-time graph is straightforward and can provide both average and instantaneous acceleration values. It is especially useful when dealing with uniform acceleration, as the slope remains constant throughout. On the other hand, analyzing the shape of the position-time graph offers a qualitative understanding of acceleration. It allows for quick identification of positive or negative acceleration and can provide insights into the object’s changing velocity. However, it may not provide precise numerical values for acceleration. Constructing velocity-time graphs offers a more direct approach to determining instantaneous acceleration. It allows for accurate calculations at specific points in time, even when the position-time graph is unavailable. However, the process of constructing velocity-time graphs can be more time-consuming and may require additional data. In certain situations, a combination of these graphical techniques may be necessary to obtain a comprehensive understanding of the object’s acceleration. By leveraging their respective advantages, researchers, scientists, and students can ensure accurate calculations and interpretations of acceleration from position-time graphs. Overall, these graphical methods for calculating acceleration offer valuable tools for analyzing motion. They provide a visual representation of an object’s velocity and allow for precise determination of acceleration. Understanding and utilizing these techniques can greatly enhance one’s ability to interpret and analyze position-time graphs, leading to a deeper understanding of acceleration and its relationship with motion. Common Errors and Tips for Accuracy A. Common Mistakes when Finding Acceleration from Position-Time Graphs When trying to find acceleration from a position-time graph, there are a few common mistakes that can easily be made. One of the most common errors is misinterpreting the slope of the graph. The slope of the position-time graph represents velocity, not acceleration. It is important to remember that acceleration is the rate of change of velocity, not position. Therefore, simply looking at the slope of the graph will not give an accurate measure of acceleration. Another mistake to watch out for is assuming that a straight line on the position-time graph indicates constant acceleration. While a straight line does indicate constant velocity, it does not necessarily mean constant acceleration. The acceleration could be changing as long as the velocity remains constant. To accurately find the acceleration, it is necessary to calculate the change in velocity over a specific time interval. Additionally, it is important to be mindful of units when calculating acceleration. Make sure that the units for time and displacement are consistent and match the units in the acceleration formula. Mixing up units can lead to incorrect results. B. Tips and Tricks for Achieving Accurate Results To ensure accuracy when finding acceleration from a position-time graph, there are a few tips and tricks that can be helpful. One of the most effective strategies is to zoom in on specific sections of the graph to calculate average acceleration over smaller intervals. This can help reduce errors caused by variations in velocity over larger time intervals. Another useful tip is to use multiple position-time graphs to cross-reference and verify the calculated values of acceleration. Different position-time graphs covering the same motion can provide additional confirmation and increase the reliability of the calculated acceleration. When using real-life scenarios to calculate acceleration, it is important to consider the context and any external factors that may affect the motion. This includes taking into account factors such as friction, air resistance, or gravitational forces, depending on the specific scenario. Failure to account for these factors can lead to inaccurate results. Lastly, double-checking calculations and ensuring that all mathematical operations are performed accurately is crucial. Simple arithmetic errors can lead to significant discrepancies in the calculated acceleration. By avoiding common mistakes and implementing these tips and tricks, accurate results can be achieved when finding acceleration from position-time graphs, providing a solid foundation for understanding the motion of objects and their changing velocities. Accelerate Your Understanding: How to Find the Acceleration from a Position-Time Graph In conclusion, understanding acceleration and its relationship with position-time graphs is essential for analyzing and interpreting the motion of objects. Through the analysis of position-time graphs, we can accurately determine an object’s acceleration, both average and instantaneous. Throughout this article, we have covered various key points. First, we introduced acceleration as a measure of change in velocity and highlighted its importance in studying motion. We then discussed the definition and purpose of position-time graphs, explaining how they represent an object’s position over time. Additionally, we illustrated different types of motion that can be represented on the Furthermore, we provided the formula for calculating acceleration, emphasizing its role in determining how acceleration affects velocity. We demonstrated how changes in velocity are reflected in the position-time graph, providing a visual representation of an object’s acceleration. Moving on, we offered a step-by-step guide for finding average acceleration from a position-time graph, including example calculations using real-life scenarios. We also delved into the topic of instantaneous acceleration and discussed techniques for approximating it using position-time graphs. Moreover, we explored the concept of negative acceleration and its meaning in motion, providing examples showcasing negative acceleration on position-time graphs. We then explained how the slope of a position-time graph represents acceleration, allowing us to calculate it accurately. Furthermore, we provided an overview of other graphical methods for calculating acceleration, comparing different techniques and discussing their advantages. To ensure accuracy, we addressed common mistakes that may arise when finding acceleration from position-time graphs and offered valuable tips and tricks. In conclusion, by understanding acceleration and its relationship with position-time graphs, we gain valuable insights into the motion of objects. Through the analysis of position-time graphs, we can determine an object’s acceleration, enabling us to study and interpret its motion effectively. It is crucial to grasp these concepts to unravel the intricacies of the physical world and make precise predictions about the behavior of objects in motion. Leave a Comment	{"url":"https://thetechy.life/how-to-find-the-acceleration-from-a-position-time-graph/","timestamp":"2024-11-05T20:03:11Z","content_type":"text/html","content_length":"104790","record_id":"<urn:uuid:b8645ed0-f795-4a08-8c65-f4e0ff4866a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00785.warc.gz"}
Bank-switched Farrow resampler - Markus Nentwig Bank-switched Farrow resampler Bank-switched Farrow resampler A modification of the Farrow structure with reduced computational complexity. Compared to a conventional design, the impulse response is broken into a higher number of segments. Interpolation accuracy is achieved with a lower polynomial order, requiring fewer multiplications per output sample at the expense of a higher overall number of coefficients. Example code This code snippet provides a Matlab / Octave implementation. And here is the same in C, with explicit sample-per-sample calculation that allows varying the output rate even during operation (the Matlab version is vectorized). A Farrow structure resamples a signal at an arbitrary, possibly time-varying sampling rate. It utilizes a piecewise polynomial approximation of a continuous-time impulse response. The proposed variant uses multiple segments per tap instead of one. This may allow to achieve the same accuracy with a lower order polynomial. The performance of a Farrow resampler depends mainly on • The ideal impulse response itself (filter prototype, cutoff frequency, steepness, far-off rejection, limitations from finite duration / limited number of taps) • The accuracy of the polynomial approximation to the ideal impulse response The proposed bank switching can improve the approximation accuracy, but the approximation can obviously not outperform the ideal impulse response. In other words, any attempt to improve performance by better approximation accuracy of the impulse response is futile, when the ideal impulse response itself is the limiting factor. "Bank-switching" breaks the interpolation segments into smaller pieces. Fig. 1 shows an example for an impulse response implemented in a conventional Farrow resampler. Each segment corresponds to one sample of the input signal at a time. The resampler interpolates the impulse response at the sample duration, multiplies with the sample sample and sums the resulting product over all taps at the output. To calculate the next output sample, the input stream is shifted, for example by 1/3 sample for an output rate that is 3x higher than the input rate. Either a sample stays within its impulse response segment, or it moves to the next tap (segment) by shifting the delay line. Since the shape of an impulse response can be rather complex over the duration of one segment, a high-order polynomial is required to approximate it with the required accuracy. The arrows in Fig. 1 point out an example segment. ^Fig.1: Resampler impulse response segments (a) and input signal (b) The proposed structure breaks each segment of the impulse response into n sub-segments. This is shown in Fig. 2, with n=2. Since segments become shorter, an accurate approximation can be achieved with a lower-order polynomial. ^Fig.2: Resampler impulse response broken into sub-segments (a) and input signal (b) Only every second impulse response segment contributes to the output signal at any given time. Therefore, the number of taps remains unchanged. The interpolator can use bank-switching between alternative polynomial coefficient sets in each tap. Similarity with polyphase FIR upsampler The bank-switching concept can be understood in terms of a conventional polyphase FIR interpolator, which is recalled below: ^Fig.3: Interpolation using FIR filter As shown in Fig. 3, interpolation by an integer factor n can be performed by inserting (n-1) zero samples and subsequent FIR filtering. Zero samples do not contribute to the output of the FIR filter and their location in time is always known. Therefore, the structure can be rearranged into the polyphase arrangement in Fig. 4. The term "polyphase" highlights that each coefficient bank generates a different "phase" of the signal, that is, a replica that is delayed by a varying amount. ^Fig.4: Polyphase FIR interpolator Instead of fixed FIR coefficients in Fig. 4, the bank-switched Farrow interpolator uses polynomial interpolation (Fig. 5). ^Fig.5: Bank-switched Farrow structure The time instant of each output sample is split into: • A first integer part that corresponds to the position in the input stream. A change results in shifting the delay line and processing one (or more) new input samples. • A second integer part that determines the coefficient bank • A fractional part that interpolates each bank's impulse response segment. The calculation is explicitly written out in the C implementation. Efficient use A Farrow interpolator that requires 3^rd or higher approximation order may be a good candidate for bank switching. Reducing 2^nd order to linear is usually not practical, since the number of sub-segments needed to maintain accuracy grows too large. These numbers should be taken with caution, since accuracy requirements may differ by several orders of magnitudes, depending on the In any case, some trial-and-error with alternative designs may be useful to find the best solution. Your mileage may vary. [ - ] Comment by ●April 1, 2021 what can be the possible choice of the bank index? can it be the interpolation factor? [ - ] Comment by ●April 2, 2021 edit: thinking about your question again: It's equivalent to interpolation by an integer factor (zero sample insertion) before the structure, if that's what you meant. Now if the interpolation factor implemented by the structure (!) is an integer, the problem reduces to a plain polyphase filter with fixed coefficients (e.g. a FIR structure with as many "banks" of coefficients as there are "phases" of each input sample = interpolation factor). In that sense I'm not sure I understand the intent behind the question. But let me take back a step from this 10-year old article. To recap, • Arbitrary resampling is implemented by a FIR structure with coefficients that aren't constant but computed as a function of the output sample's fractional phase • A "conventional" Farrow structure computes each FIR coefficient using a polynomial of said fractional phase • My proposal from the article is to break that polynomial into multiple segments ("banks") to give a better approximation of the function it is trying to approximate. The mathematical concept behind this is "piecewise polynomial approximation": I can use higher-order polynomials to approximate some function more accurately, or I can chop it into segments which I approximate with lower-order polynomials. There is usually a fairly obvious "sweet spot" in the trade-off - too high polynomial order becomes computationally brittle, and increasing the number of segments improves performance only slowly, which won't help if I need order-of-magnitude improvement. One practical approach to piecewise poly approximation is "splines", which turns out to work very well with regard to the spectrum of the approximation error. An alternative approach is discontinuous segments: This gives lowest RMS error but the discontinuities cause an error term that is white across the spectrum, so the filter will have a near-constant alias noise floor. This may or may not be what makes sense in a specific design problem. So what I'd do is • check whether a plain polynomial of acceptable order gives sufficient performance. Hint, the filtering implemented by the Farrow structure is very expensive compared to a conventional rational (/ integer) resampler. I might upsample (with filtering) the signal before the fractional stage to make the arbitrary resampling problem easier (meaning a less complex impulse response function to approximate), but that scales the computational load of the resampler with the higher sample rate. There are a few options to explore in this regard. • Try different approximations for the "continuous-time" impulse response by suitable means e.g. nth-order polynomial vs e.g. k-th order spline with continuity in k derivatives vs piecewise linear 1st order vs plain table lookup (0th order) • while talking about performance evaluation: I'd double-check that my simulation stresses the problematic areas in the approximation. Ideally this needs an infinite sample rate to see all the • ... and one "trick" I found for testbenches is to use an arbitrary test signal that can be considered cyclic in nature (e.g. sufficient zero-padding or true cyclic content). Resample to the output rate by using discrete Fourier transform ("FFT") then evaluate the Fourier-series representation in continuous-time (replace the discrete bin index in the "IFFT" equation with a time-continuous variable). Then compute the vector error at the intended fractional resampling rate by subtracting the actual output signal of the resampler from the above resampled reference signal. This way, all the aliases show up folded back to their final frequencies in the output signal. To post reply to a comment, click on the 'reply' button attached to each comment. To post a new comment (not a reply to a comment) check out the 'Write a Comment' tab at the top of the comments. Please login (on the right) if you already have an account on this platform. 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High order geometric methods with splines: fast solution with explicit time-stepping for Maxwell equations Modern manufacturing engineering is based on a ``design-through-analysis'' workflow. According to this paradigm, a prototype is first designed with Computer-aided-design (CAD) software and then finalized by simulating its physical behavior, which usually i ...	{"url":"https://graphsearch.epfl.ch/en/publication/300346","timestamp":"2024-11-14T00:41:17Z","content_type":"text/html","content_length":"108600","record_id":"<urn:uuid:c47bdca7-6c41-4419-840f-af7363f55327>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00782.warc.gz"}
Find Perimeter & Area of Parallelogram \| Formulas of Parallelogram Area of Parallelogram The parallelogram is a geometrical figure that is formed by the pair of parallel sides having opposite sides of equal length and the opposite angles of equal measure. The height and base of the parallelogram should be perpendicular to each other. The area of a parallelogram is equal to the magnitude of cross-vector products for two adjacent sides. This is possible to create the area of a parallelogram by using any of its diagonals. The leaning rectangular box is a perfect example of the parallelogram. Proof Area of Parallelogram Forluma According to the picture, Area of Parallelogram = Area of Triangle 1 + Area of Rectangle + Area of Triangle 2 => Area of Parallelogram = $ \frac{1}{2} \times Height \times Base $ + $ Height \times Base $ + $ \frac{1}{2} \times Height \times Base $ => Area of Parallelogram = $ \frac{1}{2} \times h\times b_1$ + $ h\times b_3$ + $ \frac{1}{2} \times h\times b_2 $ => Area of Parallelogram = $ h ( \frac{1}{2} \times b_1 + b_3 + \frac{1}{2} \times b_2 $ => Area of Parallelogram = $ h { \frac{1}{2} (b_1 + b_2) } + b_3 $ According to SAS, $ b_1 = b_2 $ => Area of Parallelogram = $ h { \frac{1}{2} (b_1 + b_1) } + b_3 $ => Area of Parallelogram = $ h { \frac{1}{2} \times 2 b_1 }+ b_3 $ => Area of Parallelogram = $ h ( b_1 + b_3 $ According to picture $ b_1+ b_3 = Base $ => Area of Parallelogram = $ height \times Base $ Parallelogram Formula A parallelogram is a four-sided polygon bounded by four infinite segments and it makes a closed figure that is referred to as the quadrilateral. Or we can say that the parallelogram is a special case of quadrilateral where opposite angles are equal and perpendicular to each other. With the help of a basic list of parallelogram formulas, you can calculate the area and the perimeter by putting the values and derive the final output. In the next section, we will discuss popular properties of the parallelogram for quick identification of shape. Perimeter of Parallelogram $ = 2(a+b) $ Diagonal of Parallelogram => $ p=\sqrt{a^{2}+b^{2}-2ab\cos (A)}=\sqrt{a^{2}+b^{2}+2ab\cos (B)} $ => $ q=\sqrt{a^{2}+b^{2}-2ab\cos (A)}=\sqrt{a^{2}+b^{2}-2ab\cos (B)} $ => $ p^{2}+q^{2}=2(a^{2}+b^{2}) $ p,q are the diagonals a,b are the parallel sides What is a Parallelogram? In Euclidean Geometry, the parallelogram is the simplest form of a quadrilateral having two sides parallel to each other. The opposite sides of a parallelogram are equal in length and angles are also the same. The congruent sides are the direct consequence and it can be proved quickly with the help of equivalent formulations. If only two sides are parallel then it is named as the Trapezoid and the three-dimensional counterpart is taken as the parallelepiped. A different type of quadrilateral on the basis of symmetry is defined as the given below – • Rectangle – This is a parallelogram with four equal angles and the opposite sides are also equal. • Rhombus – This is a parallelogram with four sides of equal length. • Rhomboid – This is a parallelogram whose opposite sides are parallel and adjacent or equal but its angles are not right-angled. • Square –This is a parallelogram with four equal sides and angles of the same size. Properties of Parallelogram • Two pairs of the opposite sides are equal in length and angles are of equal measure. • The diagonals of a parallelogram bisect each other. • The pair of opposite sides are equal and they are equal in length. • The adjacent angles of the parallelogram are supplementary. • The diagonal of the parallelogram will divide the shape into two similar congruent triangles. • The shape has the rotational symmetry of the order two. • Based on parallelogram law, the sum of the square of sides is equal to the sum of the square of the diagonals. • The sum of the distances from any given point towards sides is equivalent to the location of the point. • If there is point A in the plane of the quadrilateral then based on the property, every single line will divide the quadrilateral into two equal shapes. Hence, a parallelogram could have all properties listed above if any of the statements become true then this is a parallelogram. Example1: If the base of a parallelogram is equal to 6cm and the height is 4cm, the find its area. Base = 6 cm and height = 4 cm. Use the formula of Area of Parallelogram. => Area of Paralelogram = $ Height \times Base $ [Put the Value of Heigh and Base] => Area of Paralelogram = $ 4\times 6$ => Area of Paralelogram = $ 24 cm^2$ Example 2: The base of the parallelogram is thrice its height. If the area is 192 cm^2, find the base and height. Given, Area of Parallelogram = 192 cm^2 Suppose Height =h and Base = 3h according to question. Area of Parallelogram = $ Height \times base $ => $ 192 = h\times 3h$ => $ 192 = 3h^2$ => $ h^2 = \frac{192}{3}$ => $ h^2 = 64 $ => $ h^2 = 8^2 $ => $ h = 8 $ Height =8 and Base = 3h= 24	{"url":"https://www.andlearning.org/parallelogram-formula/","timestamp":"2024-11-05T00:47:29Z","content_type":"text/html","content_length":"77615","record_id":"<urn:uuid:73ade762-816a-4259-9fc0-f13c068d1003>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00758.warc.gz"}
6.46 ounces to grams Convert 6.46 Ounces to Grams (oz to gm) with our conversion calculator. 6.46 ounces to grams equals 183.137929509474 oz. Enter ounces to convert to grams. Formula for Converting Ounces to Grams (Oz to Gm): grams = ounces * 28.3495 By multiplying the number of grams by 28.3495, you can easily obtain the equivalent weight in grams from ounces. Understanding the Conversion from Ounces to Grams When it comes to converting ounces to grams, it’s essential to know the conversion factor that bridges the gap between these two units of measurement. One ounce is equivalent to approximately 28.3495 grams. This means that to convert ounces to grams, you simply multiply the number of ounces by this conversion factor. The Formula for Converting Ounces to Grams The formula to convert ounces (oz) to grams (g) is straightforward: Grams = Ounces × 28.3495 Step-by-Step Calculation: Converting 6.46 Ounces to Grams Let’s take a closer look at how to convert 6.46 ounces to grams using the formula provided: 1. Start with the number of ounces: 6.46 ounces. 2. Use the conversion factor: 28.3495 grams per ounce. 3. Multiply the ounces by the conversion factor: 6.46 × 28.3495. 4. Perform the calculation: 6.46 × 28.3495 = 183.12 grams. Thus, 6.46 ounces is equal to 183.12 grams when rounded to two decimal places. The Importance of Ounce to Gram Conversion Understanding how to convert ounces to grams is crucial, especially in a world where both the metric and imperial systems are used. This conversion is particularly important in various fields such as cooking, scientific research, and everyday measurements. For instance, many recipes, especially those from different countries, may list ingredients in grams, while you might be more familiar with ounces. Accurately converting these measurements ensures that your dishes turn out perfectly every time. Practical Examples of Ounce to Gram Conversion Here are a few scenarios where converting ounces to grams can be particularly useful: • Cooking: When following a recipe that lists ingredients in grams, knowing how to convert ounces can help you measure accurately, ensuring the right balance of flavors. • Scientific Measurements: In laboratories, precise measurements are crucial. Converting ounces to grams can help scientists and researchers maintain accuracy in their experiments. • Everyday Use: Whether you’re tracking your food intake or measuring out ingredients for a DIY project, being able to convert ounces to grams can simplify your tasks and improve your results. In conclusion, converting 6.46 ounces to grams is a simple yet essential skill that can enhance your cooking, scientific endeavors, and daily life. With the conversion factor of 28.3495, you can easily navigate between these two measurement systems and ensure accuracy in your tasks. Here are 10 items that weigh close to 6.46 ounces to grams – • Standard Baseball Shape: Spherical Dimensions: 9 inches in circumference Usage: Used in the sport of baseball for pitching, hitting, and fielding. Fact: A baseball is made of a cork core wrapped in layers of yarn and covered with leather. • Medium-Sized Avocado Shape: Oval Dimensions: Approximately 4-5 inches long Usage: Commonly used in salads, spreads, and guacamole. Fact: Avocados are technically a fruit and are known for their healthy fats. • Small Pineapple Shape: Cylindrical with a crown Dimensions: About 6-8 inches tall Usage: Eaten fresh, juiced, or used in cooking and baking. Fact: Pineapples take about two years to grow and are a symbol of hospitality. • Glass Paperweight Shape: Spherical or dome-shaped Dimensions: Typically 3-4 inches in diameter Usage: Used to hold down papers on a desk or as a decorative item. Fact: Many glass paperweights are handcrafted and can be quite collectible. • Small Bag of Flour Shape: Rectangular Dimensions: About 5×7 inches Usage: Used in baking and cooking as a primary ingredient. Fact: Flour is made by grinding grains, and different types are used for various recipes. • Medium-Sized Candle Shape: Cylindrical Dimensions: Approximately 3 inches in diameter and 4 inches tall Usage: Used for lighting, decoration, and creating ambiance. Fact: Candles have been used for thousands of years, originally made from tallow or beeswax. • Small Potted Plant Shape: Round (pot) with foliage Dimensions: Pot diameter of about 4-5 inches Usage: Used for decoration and improving indoor air quality. Fact: Many houseplants can help reduce stress and improve mood. • Standard Deck of Playing Cards Shape: Rectangular Dimensions: 2.5 x 3.5 inches per card Usage: Used for various card games and magic tricks. Fact: A standard deck contains 52 cards, plus jokers, and has been around for centuries. • Small Laptop Charger Shape: Rectangular Dimensions: Approximately 4×6 inches Usage: Used to charge laptops and provide power. Fact: Laptop chargers convert AC power from the wall into DC power for the laptop. • Medium-Sized Water Bottle Shape: Cylindrical Dimensions: About 8 inches tall and 3 inches in diameter Usage: Used for carrying water or other beverages. Fact: Staying hydrated is essential for maintaining good health and energy levels. Other Oz <-> Gm Conversions –	{"url":"https://www.gptpromptshub.com/grams-ounce-converter/6-46-ounces-to-grams","timestamp":"2024-11-09T04:42:16Z","content_type":"text/html","content_length":"186888","record_id":"<urn:uuid:cf74b1d7-79e7-44a1-ad80-0444a9325e51>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00503.warc.gz"}
NIST Cybersecurity Framework Informative Reference for 800-171 Rev. 1National Institute of Standards and TechnologyCyberESI Consulting Group, IncorporatedCyberESI Consulting Group, Incorporated 2023-09-15T01:33:17-07:00 0.0.1 1.1.0 OSCAL NIST Team oscal@nist.gov National Institute of Standards and Technology Attn: Computer Security Division Information Technology Laboratory 100 Bureau Drive (Mail Stop 8930) Gaithersburg MD 20899-8930 CyberESI Consulting Group, Incorporated info@cyberesi-cg.com 4109213864 c9c64948-2544-4efc-b94e-193b85ec4551 3c6c85b0-5968-4e9f-8efe-f67ec4186502	{"url":"https://cyberesi-cg.com/oscal_mapping_1c/OSCAL_Mapping_sp_800_171_1_0_0-csf_1_1_0_230915133317.xml","timestamp":"2024-11-08T10:48:59Z","content_type":"application/xml","content_length":"53876","record_id":"<urn:uuid:ec0265ec-1601-415c-8d54-d533953a1a39>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00334.warc.gz"}
Tsing Ch’a Sessions 清茶会 To promote interdisciplinary interaction between different faculty members and students on the campus, a weekly meeting has been organized by our postdoc Jialiang Yan since September 2023, called Tsing Ch’a Sessions (清茶会). Its slogan “know thyself and let others know you better.” ■Schedule for 2024-2025 academic year (Fall) Date Speaker 2024 SEP 12 赵博囡 Bonan Zhao 2024 SEP 19 John Lindqvist, 李文娟 2024 SEP 26 杜鹏昊 Penghao Du 2024 OCT 10 罗昊轩 Haoxuan Luo，欧阳文飞 Wenfei Ouyang 2024 OCT 24 何清瑜 Qingyu He 2024 OCT 31 杨曦 Xi Yang 2024 NOV 07 杨思思 Sisi Yang 2024 NOV 14 王威 Wei Wang，张力竹 Lizhu Zhang 2024 NOV 21 郑文龙 Wenlong Zheng，赵奕辰 Yichen Zhao 2024 NOV 28 Working group 2024 DEC 05 Working group 2024 DEC 12 Working group ■Current Sessions 2024 October 31 14:00-15:30 Xi Yang (杨曦 Tsinghua University) Games for Quantifier Numbers Multi-Structural (MS) games and their variants are designed to capture the number of quantifiers needed to express first-order properties. Recently, Carmosino et al. [1] employed MS games to establish tight bounds on the the number of quantifiers needed to define specific properties of ordered structures. In this talk, I will use the techniques and findings from [1] to show that, on the class of finite linear orders, any first-order sentence with a quantifier depth of n is equivalent to a sentence with approximately 2n quantifiers. Furthermore, I will discuss some potential applications of these games in investigating the succinctness of the finite-variable fragments of first-order logic on linear orders. 1. Carmosino, M., Fagin, R., Immerman, N., Kolaitis, P., Lenchner, J., & Sengupta, R. (2024). On the number of quantifiers needed to define boolean functions. arXiv preprint arXiv:2407.00688. 2024 October 24 14:00-15:30 Qingyu He (何清瑜 Tsinghua University) Boolean dependence logic Baltag and van Benthem[1] introduced a logic of functional dependence (LFD) with local dependence formulas and dependence quantifiers, which can be seen both as a first-order and a modal logic. In the relational semantics of LFD, the dependence quantifiers become modalities and local dependence formulas are treated as special atoms. In particular, the modalities involving multiple variables correspond to intersections of relations. This leads to the study on the interaction between LFD and Boolean Modal Logic [2] (BML)—a poly-modal logic where families of binary relations are closed under the Boolean operations of union, intersection, and complement.In this talk, I will present a BML version of LFD, which can express additional notions of dependence. I will provide an axiomatization, including details about its completeness proof. Furthermore, I will extend the framework by introducing conditional independence atoms, and propose an axiomatization for the extended This is joint work with Chenwei Shi and Qian Chen. [1] Baltag, Alexandru, and Johan van Benthem. “A simple logic of functional dependence.” Journal of Philosophical Logic 50 (2021): 939-1005. [2] Gargov, George, and Solomon Passy. “A note on Boolean modal logic.” Mathematical logic. Boston, MA: Springer US, 1990. 299-309. 2024 October 10 16:00-17:30 Wenfei Ouyang (欧阳文飞 Tsinghua University) Understanding Dependence Relation and its Representation Theorem n Baltag and van Benthem’s paper [1], three representation theorems are proved for the functional dependence relation (Proposition 2.6). In this talk, we will simplify the construction which is key to the proof. Based on this simplification, we give a more detailed characterization of the construction. We will also disscuss other representation theorems in other related works. [1] Baltag, A., van Benthem, J. A Simple Logic of Functional Dependence. J Philos Logic50, 939–1005 (2021). 2024 October 10 14:00-15:30 Haoxuan Luo (罗昊轩 Tsinghua University) A Semantic Model Based on Inconsistent Sets and Its Corresponding Frame Properties In paraconsistent logic, the Principle of Explosion (ECQ) does not hold, meaning that both a proposition A and its negation ¬A can be true at the same time. In this talk, I will briefly introduce Non-adjunctive Discursive Logic and Paraconsistent Logic with Preservationism, which introduce the concept of sets of inconsistent formulas. I will then build a model composed of inconsistent sets, where these inconsistent formulas can be derived. I will also discuss the relationship between this model and the Kripke model. Furthermore, I will provide a characterization of the semantics for some classes of frames and explore more results on compactness. This talk is based on my master’s thesis. 2024 September 26 14:00-15:30 Penghao Du (杜鹏昊 Tsinghua University) Modal logics for the poison game: axiomatization and undecidability Poison modal logic and poison sabotage modal logic have been studied in the existing literature to capture the so-called poison game, which was originally conceived as a paradigm for reasoning about graphical concepts in graph theory and has recently been shown to have significant applications in the theory of abstract argumentation. In this work, we further explore the technical aspects of these two logics and extend existing results by addressing the open questions identified in [Grossi and Rey, 2019, Blando et al., 2020]. Specifically, we show that poison sabotage modal logic has an undecidable satisfiability problem, and we provide both Hilbert-style calculus and tableau calculus for these logics. This is a joint work with Fenrong Liu and Dazhu Li. 2024 September 19 16:00-17:30 John Lindqvist (University of Bergen) Distributed belief – aggregating potentially conflicting information In epistemic logic, the knowledge distributed among a group of agents, or the knowledge possible given the information distributed in the group, can be formalized using the intersection modality. Distributed knowledge can potentially be resolved if the information possessed by the group is shared among its members. However, when we consider belief rather than knowledge, the picture is more complicated. The cumulative information possessed by the agents can be contradictory. In such cases, the distributed belief of the group explodes: the group ends up with distributed belief in everything. Similarly, in such cases, resolving using the intersection operation makes the agents inconsistent.We consider non-explosive alternative definitions of distributed belief, both static and dynamic. For the static case, we offer non-explosive alternative definitions for distributed belief that make use of maximal consistent subgroups. For the dynamic case, we discuss ways of preserving belief properties of individual agents. 2024 September 19 14:00-15:30 Wenjuan Li (李文娟 Beijing Institute of Mathematical Sciences and Applications) Determinacy of omega-languages: from non-determinism to probability I will talk about the interface of the determinacy of Gale-Stewart games and automata on infinite words. The Gale-Stewart game is a two-player turn-based game with perfect information. Given a winning set X, determinacy of X asserts that one of the two players has a winning strategy. The winning set X can also be defined by variants of finite automata as a set of infinite words accepted by such automata. I will review several variants of finite automata on infinite words and the determinacy studies along this topic, then introduce our studies on determinacy strength of infinite games with winning sets defined by pushdown and probabilistic automata with various acceptance conditions. 2024 September 12 13:30-15:30 Bonan Zhao (赵博囡 Princeton University) Computational models of causal generalization Computational Cognitive Science is an interdisciplinary field seeking to understand human cognition and intelligence through computational principles. Logic has played fundamental roles in the early development of cognitive science, and keeps influencing today’s most cutting-edge research in the field. In this talk, I will briefly introduce the historical connection between logic and cognitive science, and share some of my work combining formal representations, probabilistic inference, and behavioral experiments, to account for how people synthesize concepts from very few data and generalize to novel situations. ■Past Sessions Click HERE to check the past sessions.	{"url":"http://tsinghualogic.net/JRC/tsingcha/","timestamp":"2024-11-09T19:28:35Z","content_type":"text/html","content_length":"94535","record_id":"<urn:uuid:ddf586a2-cbdb-40dd-a1d0-f1ea0d5dacb1>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00854.warc.gz"}
Winter School on Strongly Correlated Quantum Matter Oxide perovskites have received widespread attention ever since their discovery due to the multiple physical properties they exhibit, including ferroelectricity, multiferroicity, and superconductivity. One prominent absence in this list of properties that oxide perovskites exhibit is electronic topological order. This is a consequence of the large band gaps of oxide perovskites, which make the band inversions necessary for topology impossible. We find that topological phonons – nodal rings, nodal lines, and Weyl points – are ubiquitous in oxide perovskites in terms of structures (tetragonal, orthorhombic, and rhombohedral), compounds (BaTiO$_3$, PbTiO$_3$, and SrTiO$_3$), and external conditions (photoexcitation, strain, and temperature). In particular, in the tetragonal phase of these compounds all types of topological phonons can simultaneously emerge when stabilized by photoexcitation, whereas the tetragonal phase stabilized by thermal fluctuations only hosts a more limited set of topological phonon states. In addition, we find that the photoexcited carrier density can be used to control the emergent topological states, for example driving the creation/annihilation of Weyl points and switching between nodal lines and nodal rings. Overall, we propose oxide perovskites as a versatile platform in which to study topological phonons and their manipulation with light [1]. Reference: [1] Bo Peng, Yuchen Hu, Shuichi Murakami, Tiantian Zhang, Bartomeu Monserrat. Topological phonons in oxide perovskites controlled by light. Science Advances 6, eabd1618 (2020).	{"url":"https://www.pks.mpg.de/de/scqm20/poster-contributions","timestamp":"2024-11-03T07:07:00Z","content_type":"text/html","content_length":"170083","record_id":"<urn:uuid:82981d39-2388-45d9-833e-36508de2a3ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00425.warc.gz"}
Time Dilation and Relativity: Understanding the Cosmic Clock In the vast landscape of modern physics, few concepts are as intriguing and mind-bending as time dilation, a phenomenon rooted in Einstein’s theory of relativity. While our everyday experience of time seems straightforward—seconds tick away uniformly—relativity reveals a more complex reality. This blog explores the concept of time dilation, its implications, and how it shapes our understanding of the universe. What is Time Dilation? Time dilation refers to the difference in the elapsed time as measured by two observers, due to relative velocity or gravitational fields. Essentially, time can pass at different rates depending on the observer’s speed or proximity to massive objects. This is a radical departure from our intuitive understanding of time as a constant, universal measure. Special Relativity and Velocity The first key insight into time dilation comes from Einstein’s theory of special relativity, formulated in 1905. This theory posits that the laws of physics are the same for all observers, regardless of their relative motion. One of its most famous implications is that as an object moves closer to the speed of light, time slows down for that object relative to a stationary observer. For example, consider a scenario involving two twins—often referred to as the “twin paradox.” One twin stays on Earth while the other travels on a spaceship at near-light speed. Upon returning, the traveling twin finds they have aged less than their Earth-bound sibling. This counterintuitive result underscores the reality of time dilation due to high velocities. The mathematical foundation for this effect can be found in the Lorentz transformation equations, which describe how measurements of time and space change for observers in different inertial frames. As the speed of an object approaches the speed of light, the time experienced by that object dilates compared to a stationary observer. General Relativity and Gravity While special relativity explains time dilation due to velocity, Einstein’s later theory of general relativity (published in 1915) expands the concept to include gravity. According to general relativity, massive objects warp the fabric of spacetime. This curvature affects how time flows in the presence of gravity, leading to the phenomenon known as gravitational time dilation. To illustrate, consider two clocks: one on the surface of the Earth and another in a satellite orbiting the planet. The clock on the satellite, which is farther from Earth’s gravitational influence, ticks slightly faster than the clock on the ground. This effect, while minuscule, has real-world implications, especially for technologies like GPS, which require precise timing to provide accurate positioning data. Experimental Evidence of Time Dilation The principles of time dilation are not merely theoretical; they have been confirmed through numerous experiments. One notable example involved atomic clocks flown on airplanes. When compared with identical clocks on the ground, the airborne clocks were found to have experienced less elapsed time, consistent with both special and general relativity predictions. Another significant experiment involved the observation of muons—subatomic particles that decay over time. When created in the upper atmosphere, muons travel towards the Earth at speeds close to that of light. Due to time dilation, they exist long enough to be detected at the surface, despite having a short half-life. This observation supports the reality of time dilation as a fundamental aspect of our universe. Implications of Time Dilation Time dilation challenges our intuitive grasp of time and raises profound questions about the nature of reality. It suggests that time is not an absolute quantity but rather a relative experience shaped by motion and gravity. This has far-reaching implications for various fields, from astrophysics to philosophy. Astrophysical Significance In astrophysics, time dilation plays a crucial role in our understanding of phenomena like black holes. Near a black hole’s event horizon, time slows dramatically due to immense gravitational forces. For an observer far away, it appears as though objects approaching the black hole take an eternity to cross the event horizon, leading to fascinating discussions about the nature of time and space. Philosophical Considerations Philosophically, time dilation raises questions about the nature of time itself. Is time an inherent part of the universe, or is it a construct shaped by our perceptions and experiences? The relative nature of time challenges our understanding of past, present, and future, prompting debates that span both science and philosophy. Time dilation and relativity fundamentally reshape our understanding of time, challenging our intuitive notions and revealing a universe far more intricate than we can perceive. As we continue to explore the cosmos and develop new technologies, the implications of time dilation will remain at the forefront of scientific inquiry. From the twin paradox to gravitational effects near black holes, the interplay of time, space, and gravity reveals a tapestry of interconnected phenomena that beckon us to delve deeper into the mysteries of the universe. As we ponder these complexities, we realize that time is not just a ticking clock; it is a dynamic, fluid aspect of our existence—an integral part of the cosmic dance that shapes our reality. Leave a Comment	{"url":"https://socialzoe.com/time-dilation-and-relativity-understanding-the-cosmic-clock/","timestamp":"2024-11-02T15:52:59Z","content_type":"text/html","content_length":"129361","record_id":"<urn:uuid:0429f9b0-70e7-4a0f-97ff-2d3566f1554e>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00167.warc.gz"}
An upper bound on the convergence time for quantized consensus We analyze a class of distributed quantized consensus algorithms for arbitrary networks. In the initial setting, each node in the network has an integer value. Nodes exchange their current estimate of the mean value in the network, and then update their estimate by communicating with their neighbors in a limited capacity channel in an asynchronous clock setting. Eventually, all nodes reach consensus with quantized precision. We start the analysis with a special case of a distributed binary voting algorithm, then proceed to the expected convergence time for the general quantized consensus algorithm proposed by Kashyap et al. We use the theory of electric networks, random walks, and couplings of Markov chains to derive an O(N^3 log N) upper bound for the expected convergence time on an arbitrary graph of size N, improving on the state of art bound of O(N^4 log N) for binary consensus and O(N ^5) for quantized consensus algorithms. Our result is not dependent on the graph topology. Simulations are performed to validate the analysis. Publication series Name Proceedings - IEEE INFOCOM ISSN (Print) 0743-166X Other 32nd IEEE Conference on Computer Communications, IEEE INFOCOM 2013 Country/Territory Italy City Turin Period 4/14/13 → 4/19/13 All Science Journal Classification (ASJC) codes • General Computer Science • Electrical and Electronic Engineering • Distributed quantized consensus • convergence time • gossip Dive into the research topics of 'An upper bound on the convergence time for quantized consensus'. Together they form a unique fingerprint.	{"url":"https://collaborate.princeton.edu/en/publications/an-upper-bound-on-the-convergence-time-for-quantized-consensus","timestamp":"2024-11-14T20:10:27Z","content_type":"text/html","content_length":"52365","record_id":"<urn:uuid:ca467aff-435d-403f-a79b-2698a05276a5>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00372.warc.gz"}
Variance of purified MPS state Hi there, I am currently using TenPy to try to calculate thermodynamic observables for bigger yet finite systems. For the first test I wanted to do some smaller ones to test all tools. There I noticed that the MPO.variance method does not work for "non standard MPS" I checked the code and saw the exception case. My question is now , is there a workaround that can handle that , just as for example constructing a H*2 and calculate it with MPO.expectation or similiar without requring to rewrite complete bases of classes? Already thanks in advance for any help that might come! Re: Variance of purified MPS state It's not hard to generalize and adjust the source of that variance function to work with purificationMPS - all you need to do is to initially check whether you have the additional q leg (at the top of the function), and if so, in addition contract the additional ['q'] with ['q'] leg in the tensordot where you "close" one column, i.e. the tensordots of contr with th.conj() and B.conj(), Re: Variance of purified MPS state Ah okay so for example for the 0 site i would add Python: Select all contr = npc.tensordot(contr, contr, axes=[ 'q', 'q' ]) in addition to the previos p leg contraction and same further on for B ? Since the contraciton with the W's only happens on the physical legs p and all that gets added new are the virtual legs q and q on top and bottom of the B - MPS right? Or is it smarter to do it together with Python: Select all contr = npc.tensordot(contr, B.conj(), axes=[['vR', 'p' , 'q'], ['vL', 'p, 'q']]) not sure how much of a different in calculation time it would be	{"url":"https://tenpy.johannes-hauschild.de/viewtopic.php?t=723","timestamp":"2024-11-07T16:28:37Z","content_type":"text/html","content_length":"27838","record_id":"<urn:uuid:510fa352-70ac-4007-946a-f9388d326522>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00420.warc.gz"}
ChatGPT - I asked some questions - You should too! I had the idea of asking ChatGPT all about Electricity. I've been find it very useful for explaining how it works, I highly Recommend beginners to ask your Questions to ChatGPT, it never gets offended and you can Asked it again and again to rephrase it for you. I have asked it to explain Voltage and Current, Electron Drift and Maxwell's Equations, does Electron Spin create electricity, Poynting Vectors... all those gritty details that you will be hard to find elsewhere with 1 Exception The Science Asylum https://www.youtube.com/playlist?list=PLOVL_fPox2K9MtRv68T_cmWwQUbg9YR4F It's only been about 6 months since I discovered Veritasium's Video and his Follow up. In following the trail, I then found Rick Hartley (PCB designer), followed by Eric Bogatin, and Others: Most useful of these the Altium Designer videos (PCB software company) has been posting their keynote speakers and I've learnt so much. Rick Hartley at Altium I just want to say, that Steady State DC works great with Lumped Models, until ... we add variation (Transient) into our circuits, as in anything that Switches, a 555 timer, a Microcontroller, a The Reality is that we will not get to talk to EE's that understand Maxwell's equations or Poynting vectors. I would like to think that this information can get some hobbyists mind out of the Water Model and into the Electromagnetic Field Model. Some might argue that I'm creating division or confusion, I would say I'm bridging the baby steps we have been taught towards better understanding of the Universe (The Electromagnetic Universe) What's the difference between Light (you see with your eyes) and energy in a circuit? Do you have Questions? Ask GPT Answer: frequency Think about this then, how do we create Radio Waves from electronic parts to be transmitted into 'thin air'	{"url":"https://www.eevblog.com/forum/chatgptai/chatgpt-i-asked-some-questions-you-should-too!/","timestamp":"2024-11-10T04:54:56Z","content_type":"application/xhtml+xml","content_length":"140871","record_id":"<urn:uuid:a73cea2a-2378-43a8-8877-2152a53b0a4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00491.warc.gz"}
User-defined kernels Size parameter in a binomial distribution Imagine that we are interested in learning the size of a population given an observed proportion. In this case, we already know about the prevalence of a disease. Furthermore, assume that 20% of the individuals acquire this disease, and we have a random sample from the population $y \sim \mbox{Binomial}(0.2, N)$. We don’t know $N$. Such a scenario, while perhaps a bit uncommon, needs special treatment in a Bayesian/MCMC framework. The parameter to estimate is not continuous, so we would like to draw samples from a discrete distribution. Using the “normal” (pun intended) transition kernel may still be able to estimate something but does not provide us with the correct posterior distribution. In this case, a transition kernel that makes discrete proposals would be desired. Let’s simulate some data, say, 300 observations from this Binomial random variable with parameters $p = .2$ and $N = 500$: set.seed(1) # Always set the seed!!!! # Population parameters p <- .2 N <- 500 y <- rbinom(300, size = N, prob = p) Our goal is to be able to estimate the parameter $N$. As in any MCMC function, we need to define the log-likelihood function: Now comes the kernel object. In order to create an fmcmc_kernel, we can use the helper function kernel_new as follows: kernel_unif_int <- kernel_new( proposal = function(env) env$theta0 + sample(-3:3, 1), logratio = function(env) env$f1 - env$f0 # We could have skipped this Here, the kernel is in the form of $\theta_1 = \theta_0 + R, R\sim \mbox{U}\{-3, ..., 3\}$, this is, proposals are done by adding a number $R$ drawn from a discrete uniform distribution with values between -3 and 3. While in this example, we could have skipped the logratio function (as this transition kernel is symmetric), but we defined it so that the user can see an example of it. Let’s take a look at the object: #> An environment of class fmcmc_kernel: #> logratio : function (env) #> proposal : function (env) The object itself is an R environment. If we added more parameters to kernel_new, we would have seen those as well. Now that we have our transition kernel, let’s give it a first try with the MCMC ans <- MCMC( ll, # The log-likleihood function initial = max(y), # A fair initial guess kernel = kernel_unif_int, # Our new kernel function nsteps = 1000, # 1,000 MCMC draws thin = 10, # We will sample every 10 p. = p # Passing extra parameters to be used by `ll`. Notice that for the initial guess, we are using the max of y', which is a reasonable starting point (the $N$ parameter MUST be at least the max ofy’). Since the returning object is an object of class mcmc from the coda R package, we can use any available method. Let’s start by plotting the chain: As you can see, the trace of the parameter started to go up right away, and then stayed around 500, the actual population parameter $N$. As the first part of the chain is useless (we are essentially moving away from the starting point); it is wise (if not necessary) to start the MCMC chain from the last point of ans. We can easily do so by just passing ans as a starting point, since MCMC will automatically take the last value of the chain as the starting point of this new one. This time, let’s increase the sample size as well: ans <- MCMC( initial = ans, # MCMC will use tail(ans, 0) automatically kernel = kernel_unif_int, # same as before nsteps = 10000, # More steps this time thin = 10, # same as before p. = p # same as before Let’s take a look at the posterior distribution: #> Iterations = 10:10000 #> Thinning interval = 10 #> Number of chains = 1 #> Sample size per chain = 1000 #> 1. Empirical mean and standard deviation for each variable, #> plus standard error of the mean: #> Mean SD Naive SE Time-series SE #> 504.30500 2.67902 0.08472 0.09960 #> 2. Quantiles for each variable: #> 2.5% 25% 50% 75% 97.5% #> 499 503 504 506 510 #> ans #> 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 #> 3 9 25 44 69 89 144 150 143 127 82 57 28 17 8 4 1 A very lovely mixing (at least visually) and a posterior distribution from which we can safely sample parameters.	{"url":"https://cran.usk.ac.id/web/packages/fmcmc/vignettes/user-defined-kernels.html","timestamp":"2024-11-03T13:48:43Z","content_type":"text/html","content_length":"81404","record_id":"<urn:uuid:f60c65c9-f693-46db-9401-2d9b0589a661>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00086.warc.gz"}
Introducing Multiplication and The Two Words You Should Always Use - Thrifty in Third Grade There are two words you should always include when you first introduce multiplication. Those words are: groups of (or rows of). When students first learn the concept of multiplication, it’s really important for them to understand what it means to multiply, and how this is similar to, but different from adding. Using visuals, and the words groups of (and also rows of) will help your students understand the topic. This concept should be taught and instilled in your students before you introduce the properties of multiplication (specifically the commutative property, identity property, and zero property). Teaching these students the concept of multiplication before the properties will allow them to develop a foundation of understanding what it truly means to multiply. When they are later faced with equations they don’t have memorized, they will know how to solve for the answer. Don’t get me wrong–the properties of multiplication are important and need to be taught! Just not yet! How to Teach “Groups of” With Hands-On Practice You can use any manipulatives as you teach groups. I recommend giving your students twelve buttons (cubes, paperclips, erasers, etc…) to use as their manipulatives. The reason I like to use twelve is that it can be used to make several equations. Using the manipulatives on their desk, you can ask your students to make: • One group of twelve • Two groups of six • Three groups of four • Six groups of two • Twelve groups of one With each equation, I would have my students write the equation to represent the groups of. You can also teach your students repeated addition using this hands-on activity. As you do this activity, your students might begin to naturally notice the commutative property in action! (But remember–you haven’t taught this property explicitly yet.) Add or take away more manipulatives to do other equations. Once your students have enough practice, you can have them draw pictures to represent groups of. You can show real-world pictures to your students and have them write equations as well. How to Teach “Rows of” With Hands-On Practice Just like the groups of activities, you can use manipulatives to practice building arrays and creating equal rows. You can also use a carton of eggs as an example when you are teaching arrays. Once students have learned the concepts of groups of and rows of, we say this whenever we write out a multiplication equation. For example, if I wrote the equation 3×5 on the board, my students would say: • Three groups of five • Three rows of five • Three times five By saying it this way, when students come to an equation they don’t have memorized, they will instinctively be telling themselves what to draw. Want these and other real-world multiplication photos? Subscribe to my email list and I’ll send them right over! I hope these tips can help you as you introduce multiplication to your students! Let me help you teach multiplication! Check out these third grade favorites: For when you are first introducing multiplication in third grade: For going deeper with third grade multiplication skills: Check out our next post and learn more about building your students’ multiplication fluency skills.	{"url":"https://thriftyinthirdgrade.com/introducing-multiplication-and-the-two-words-you-should-always-use/","timestamp":"2024-11-02T20:19:58Z","content_type":"text/html","content_length":"176683","record_id":"<urn:uuid:58139b3b-5673-4d9e-8ff7-f8af60b78fd8>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00276.warc.gz"}
Pyramids Vs. Prisms: Understanding The Key Differences » Differencess Hey there! Have you ever wondered about the difference between pyramids and prisms? Well, you’re in the right place! In this article, I’ll break down the key distinctions between these two geometric shapes. So, whether you’re a math enthusiast or just curious about the world around you, keep reading to discover the unique characteristics that set pyramids and prisms apart. First things first, let’s talk about pyramids. These three-dimensional figures have a polygonal base and triangular faces that converge at a single point called the apex. Pyramids come in various forms, such as square pyramids, triangular pyramids, and even pentagonal pyramids. They are known for their towering structures and iconic appearances in ancient civilizations, like the famous Egyptian pyramids. On the other hand, prisms have two parallel polygonal bases connected by rectangular faces. Unlike pyramids, prisms don’t have an apex and are known for their solid, block-like What are Pyramids? Pyramids are fascinating structures that have captured the imaginations of people throughout history. As an expert blogger, I’m here to shed some light on what makes pyramids unique and special. Pyramids can be defined as solid geometric shapes with a polygonal base and triangular faces that converge at a single point called the apex. The base can be any polygon, such as a square, triangular, or pentagonal shape, but it must be a flat, closed figure. This base is connected to the apex by triangular faces, creating a distinctive and iconic design. One of the most well-known pyramids is the Great Pyramid of Giza in Egypt. Impressive in both size and precision, this pyramid is a testament to the engineering skills of the ancient Egyptians. It stands as a symbol of their advanced civilization and serves as a reminder of their ingenuity and architectural prowess. But pyramids are not limited to ancient civilizations alone. They can also be found in various forms and shapes across different cultures and time periods. From the Mayan pyramids of Central America to the Step Pyramid of Djoser in Egypt, these structures have left their mark on the world and continue to awe and inspire. The tall, towering structure of a pyramid is one of its defining features. These structures were often built as tombs for pharaohs and rulers, serving as monumental reminders of their power and importance. The pyramids have withstood the test of time, enduring for thousands of years and serving as a testament to human history and achievement. Pyramids are remarkable structures with a polygonal base and triangular faces that converge at a single apex. They come in various forms and have played significant roles in ancient civilizations. Their towering presence and cultural significance make them an enduring symbol of human achievement. So now that we have a better understanding of pyramids, let’s delve into prisms in the next Characteristics of Pyramids Pyramids are fascinating structures with distinct characteristics that set them apart from other geometric shapes. Here are some key features of pyramids: 1. Triangular Faces: Pyramids have triangular faces that converge at a single point called the apex. These triangular faces give pyramids their iconic shape, with a wide polygonal base and sloping sides that come together to form a point at the top. 2. Polygonal Base: Unlike prisms, which have a rectangular or polygonal base, pyramids have a polygonal base with straight sides. The base can be of different shapes, including square, triangular, pentagonal, or even more complex polygons. 3. Single Apex: Pyramids have a central point, known as the apex, where all the triangular faces converge. This apex is the highest point of the pyramid and is often considered symbolic of power and 4. Solid Structure: Unlike the hollow structure of prisms, pyramids are solid geometric shapes. They are made up of faces, edges, and vertices, and the entire volume within the pyramid is filled. 5. Historical Significance: Pyramids have great cultural and historical significance. Throughout history, they were often built as grand tombs for pharaohs and rulers, serving as monumental reminders of their power and importance. The most famous example is the Great Pyramid of Giza in Egypt, which is one of the Seven Wonders of the Ancient World. 6. Architectural Marvels: Pyramids are remarkable engineering achievements. The construction of pyramids required meticulous planning, precision, and labor. The ability of ancient civilizations to erect such massive structures with limited technological resources is a testament to their ingenuity and skill. Pyramids are not just architectural wonders, but they also hold immense cultural value. The enduring legacy of pyramids continues to captivate and inspire people worldwide. Understandably, it’s no wonder that these impressive structures continue to be a subject of fascination and exploration even to this day. Different Types of Pyramids When it comes to pyramids, there’s more than meets the eye. Pyramids aren’t just a single type of structure; they come in various shapes and sizes, each with its own unique characteristics. In this section, I’ll delve into the different types of pyramids and what sets them apart. 1. Egyptian Pyramids: The Egyptian pyramids are the most iconic and well-known type of pyramid. These massive structures, built thousands of years ago by ancient Egyptians, served as grand tombs for their pharaohs. The Great Pyramid of Giza, built for Pharaoh Khufu, is a shining example of the architectural marvels that dominated the Egyptian landscape. 2. Step Pyramids: Step pyramids are another intriguing variation of pyramids. These structures, found in ancient Mesopotamia and Mesoamerica, feature a series of rectangular or square tiers stacked one on top of the other. The most famous example is the Pyramid of Djoser in Saqqara, Egypt. 3. Nubian Pyramids: Moving south to the African continent, we find the Nubian pyramids, primarily located in modern-day Sudan. These pyramids are characterized by their steep and narrow sides, reminiscent of the early Egyptian pyramids. They were built as burial places for the kings and queens of ancient Nubia. 4. Mayan Pyramids: Heading over to Central America, we encounter the awe-inspiring Mayan pyramids. These pyramids, built by the ancient Maya civilization, showcase sophisticated architectural techniques and are often adorned with intricate carvings and hieroglyphs. The Pyramid of Kukulkan in Chichen Itza, Mexico, is a prime example of Mayan pyramid construction. 5. Giza Solar Boat Museum: The Giza Solar Boat Museum, located near the Great Pyramid of Giza, is a testament to the ingenuity of the ancient Egyptians. It houses the reconstructed Khufu ship, a boat that was buried alongside Pharaoh Khufu’s pyramid to transport him to the afterlife. Each type of pyramid holds its own piece of history and cultural significance. These architectural wonders continue to amaze and inspire us with their grandeur and precision. Stay tuned as we explore more fascinating aspects of pyramids in the upcoming sections. Historical and Cultural Significance of Pyramids Pyramids hold a significant place in history and culture, serving as magnificent structures that continue to captivate us. Let’s explore the rich historical and cultural significance of pyramids. • Monuments of Power and Leadership: Many pyramids were built as grand tombs for pharaohs, rulers, and prominent figures. These larger-than-life structures were meant to showcase their power, authority, and importance. The towering heights and impressive architecture of pyramids spoke volumes about the might and influence of those who commissioned their construction. • Symbol of Immortality and Afterlife: Pyramids were intricately linked to the belief in the afterlife in ancient civilizations. The designs and inscriptions inside pyramids often depicted the journey of the deceased into the next realm. They were believed to be passageways to the afterlife, with the pyramid shape symbolizing the ascending pathway to immortality. • Religious and Spiritual Significance: Pyramids held immense religious and spiritual significance in various ancient cultures. These structures were considered sacred spaces for performing rituals, ceremonies, and offerings to the gods. The pyramid’s shape was believed to connect the earthly realm with the divine, making it a conduit for communication with higher powers. • Engineering Marvels: As magnificent architectural achievements, pyramids showcased the advanced engineering skills of ancient civilizations. The meticulous planning, precision, and labor required to construct these massive structures without modern technology are truly awe-inspiring. The alignment of the pyramids with heavenly bodies such as the sun and stars further showcases the incredible mathematical and astronomical knowledge of these cultures. • Cultural Identity and Heritage: Pyramids have become iconic symbols of their respective cultures. Egyptian pyramids represent the grandeur and power of the pharaohs, while Mayan pyramids stand as testament to the Maya civilization’s advanced knowledge and spiritual practices. Pyramids have become emblems of cultural identity and heritage, preserving the memory and legacy of these ancient Throughout history, pyramids have stood the test of time, remaining as lasting legacies of human achievement and cultural significance. They continue to intrigue and inspire us with their majesty and the mysteries they hold. As we delve deeper into the world of pyramids, we will explore the different types and their unique characteristics. What are Prisms? Prisms are another type of three-dimensional geometric figure that differ from pyramids in several ways. While pyramids have a base and triangular faces that converge at a single point called the apex, prisms have two parallel bases connected by rectangular or polygonal faces. These connecting faces are called lateral faces. Prisms come in various shapes and sizes, including triangular, rectangular, pentagonal, hexagonal, and so on, depending on the number of sides in their base. They can be classified as right prisms or oblique prisms, depending on whether the lateral faces are perpendicular to the bases or at an angle. One of the distinguishing features of prisms is that their bases are congruent, meaning they have the same shape and size. This property gives prisms a uniform cross-section throughout their length, which allows them to have a consistent volume. In addition to their geometric properties, prisms can be made from different materials, such as glass, plastic, or crystal, and can be used in various applications, including optics, architecture, and even in the creative arts. Prisms have historically fascinated scientists, mathematicians, and artists alike due to their unique properties and visual appeal. Even though they may not have the cultural or historical significance of pyramids, prisms are still worthy of examination and study for their contributions to our understanding of geometry and their practical applications in various fields. In the next sections, I’ll delve deeper into the specific characteristics of different types of prisms and explore their significance in different areas of study and human endeavors. So, join me as we continue our exploration of the fascinating world of prisms. Characteristics of Prisms Prisms, like pyramids, are three-dimensional geometric figures. However, they exhibit different characteristics that set them apart. Here are some key characteristics of prisms: 1. Shape and Structure: Prisms have two parallel bases that are connected by rectangular or polygonal faces. The bases are congruent and lie on parallel planes, while the lateral faces are parallelograms. This unique structure gives prisms their distinctive look. 2. Number of Faces: A prism has three pairs of congruent faces: the two bases and the lateral faces. The number of faces varies depending on the type of prism. For example, a rectangular prism has six faces, while a triangular prism has five faces. 3. Edges: Prisms have a fixed number of edges. The number of edges is equal to the sum of the number of edges in the bases plus the number of edges on the lateral faces. For instance, a rectangular prism has 12 edges, while a triangular prism has nine edges. 4. Vertices: The number of vertices of a prism can be calculated by adding the number of vertices in the bases to two times the number of sides of the base shape. So, a rectangular prism has eight vertices, while a triangular prism has six vertices. 5. Volume: The volume of a prism can be calculated by multiplying the area of the base by the height. The formula for calculating the volume of a prism is V = Bh, where V represents volume, B represents the area of the base, and h represents the height. Prisms are not only fascinating geometric figures, but they also have practical applications in various fields. They are used in architecture and construction to create structures such as buildings and bridges. Prisms also play a crucial role in optics, where they are employed to manipulate light, such as in cameras and binoculars. Prisms differ from pyramids in their shape, structure, and characteristics. Understanding the unique properties of prisms contributes to a deeper understanding of geometry and their applications in various fields. Comparison between Pyramids and Prisms Pyramids and prisms are two distinct types of three-dimensional geometric figures that have their own unique characteristics. In this section, I will compare the main differences between pyramids and Shape and Structure A pyramid is a solid shape that has a polygonal base and triangular faces that meet at a single point called the apex. On the other hand, a prism has two parallel bases that are congruent and connected by rectangular or polygonal faces. Faces, Edges, and Vertices Pyramids have a varying number of faces, edges, and vertices depending on the shape of their base. For example, a pyramid with a triangular base will have four faces, six edges, and four vertices. As for prisms, they always have two bases, several lateral faces (often rectangular), and they have the same number of edges and vertices as their bases. Volume Calculation Calculating the volume of pyramids and prisms also differs. The volume of a pyramid can be calculated by multiplying the area of its base by its height and dividing the result by 3. On the other hand, the volume of a prism is calculated by multiplying the area of its base by its height. It’s worth noting that the height in both cases is measured perpendicular to the bases. Practical Applications Both pyramids and prisms have practical applications in various fields. Pyramids are commonly used in architecture for monumental structures such as the Pyramids of Giza. Prisms, on the other hand, are frequently used in architecture, construction, and optics. The rectangular shape of prisms makes them useful for building and designing structures, while the triangular prism is often used in optics for light refraction. Understanding the differences between pyramids and prisms is essential for a deeper understanding of geometry and their applications in different fields. By recognizing their unique characteristics and properties, we can appreciate the role they play in our daily lives. Stay tuned for the next section, where I will delve into the practical applications of prisms in more detail. Understanding the differences between pyramids and prisms is crucial for gaining a deeper understanding of geometry and their applications in various fields. Pyramids are three-dimensional geometric figures with a polygonal base and triangular faces that converge to a single point called the apex. On the other hand, prisms have two parallel bases connected by rectangular or polygonal faces. They both have unique characteristics that set them apart. Pyramids and prisms differ in their shape and structure, the number of faces, edges, and vertices they possess, as well as how to calculate their volume. Pyramids have a single base and a pointed apex, while prisms have two parallel bases. Pyramids have fewer faces, edges, and vertices compared to prisms. Calculating the volume of pyramids and prisms involves different formulas. Both pyramids and prisms find practical applications in various fields, including architecture, construction, and optics. Architects use pyramids and prisms to create visually appealing structures, while construction professionals rely on their solid shapes for stability. In optics, prisms are used to manipulate light and create stunning visual effects. Understanding the differences between pyramids and prisms enhances our knowledge of geometry and allows us to appreciate their significance in different industries. Frequently Asked Questions 1. What are prisms in geometry? Prisms are three-dimensional geometric figures that have two parallel bases connected by rectangular or polygonal faces. They have a defined shape and structure, which distinguishes them from other geometric figures. 2. How many faces, edges, and vertices does a prism have? A prism has two parallel bases and several rectangular or polygonal faces connecting them. The number of faces depends on the shape of the bases and the number of rectangular or polygonal faces. It has as many edges as the sum of the edges of the bases and the connecting faces. The number of vertices is equal to the number of corners where the edges meet. 3. How do you calculate the volume of a prism? To calculate the volume of a prism, multiply the area of the base by the height of the prism. The formula for calculating the volume of a prism is V = Bh, where V is the volume, B is the area of the base, and h is the height. 4. What are the practical applications of prisms? Prisms have several practical applications in various fields. In architecture and construction, prisms are used to create solid structures and design buildings. In optics, prisms are used to refract and disperse light. They are also used in photography, telescopes, and other optical instruments to manipulate light and create different effects. 5. What are the main differences between pyramids and prisms? Pyramids have a single base and triangular faces that converge at a vertex, while prisms have two parallel bases and rectangular or polygonal faces that connect the bases. Pyramids have fewer faces, edges, and vertices compared to prisms. The calculation of the volume is also different, with pyramids using a different formula. Additionally, the practical applications of pyramids and prisms differ, with pyramids commonly used in ancient architecture and prisms widely used in modern architecture and optics.	{"url":"https://differencess.com/pyramids-vs-prisms-understanding-the-key-differences/","timestamp":"2024-11-14T08:30:55Z","content_type":"text/html","content_length":"95233","record_id":"<urn:uuid:19b0a5c6-f8a4-40c5-8026-b33e7752f0da>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00518.warc.gz"}
Reliable Fault Diagnosis with Few Tests for Combinatorics Probability and Computing Combinatorics Probability and Computing Reliable Fault Diagnosis with Few Tests View publication We consider the problem of fault diagnosis in multiprocessor systems. Processors perform tests on one another: fault-free testers correctly identify the fault status of tested processors, while faulty testers can give arbitrary test results. Processors fail independently with constant probability p < 1/2 and the goal is to identify correctly the status of all processors, based on the set of test results. For 0 < q < 1, q-diagnosis is a fault diagnosis algorithm whose probability of error does not exceed q. We show that the minimum number of tests to perform q-diagnosis for n processors is Θ(n log 1/q) in the nonadaptive case and n + Θ(log 1/q) in the adaptive case. We also investigate q-diagnosis algorithms that minimize the maximum number of tests performed by, and performed on, processors in the system, constructing testing schemes in which each processor is involved in very few tests. Our results demonstrate that the flexibility yielded by adaptive testing permits a significant saving in the number of tests for the same reliability of diagnosis.	{"url":"https://research.ibm.com/publications/reliable-fault-diagnosis-with-few-tests","timestamp":"2024-11-03T14:31:31Z","content_type":"text/html","content_length":"68648","record_id":"<urn:uuid:bea77910-af2d-4511-a3b7-72facdcd7b89>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00110.warc.gz"}
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You landed on this page because you entered a search term similar to this: writing linear equations powerpoint presentation . We have an extensive database of resources on writing linear equations powerpoint presentation . Below is one of them. If you need further help, please take a look at our software , a software program that can solve any algebra problem you enter! Q. What is Holistic Numerical Methods Institute (HNMI)? In 1985, while pursuing his doctorate in Clemson University, Autar Kaw revised a WPAFB Apple IIe based BASIC program for laminate analysis of composite materials. Since migration to PCs was not an easy task then (1988), he and a few hardworking independent study students wrote a completely new laminate analysis program called PROMAL. This tool was then used in teaching graduate level and senior elective course in Advanced Composite Materials course. Since 1988, PROMAL, which is now written in Visual Basic for Windows, has evolved into a product that is used in over 40 universities worldwide, and accompanies the Mechanics of Composite Materials textbook (1997). Naturally, the success of this idea of developing computational tools was extended in 1990 to a course in Numerical Methods. At that time, we started developing simulations for Numerical Methods using Microsoft Quickbasic3.0, and then in Visual Basic for Windows. In recent years with advances in mathematical packages and the web browsers/web development, and with funding from National Science Foundation, the idea was transformed to what it is now - extension to several mathematical packages and engineering majors. It is free of charge to anyone in the world. To quote the OCW initiative at MIT, we are strong believers in "having open dissemination of educational materials, philosophy, and modes of thought, that will help lead to fundamental changes in the way colleges and universities utilize the Web as a vehicle for education." We are continually looking for self-sustaining avenues of dissemination, and we have been fortunate to find sponsors such as Maple and Mathcad to keep it free. Q. What does it mean that this is a developing website? The core of undergraduate numerical methods consists mainly of eight topics/mathematical procedures, namely, 1) approximations, errors & modeling 2) nonlinear equations, 3) simultaneous linear equations including eigenvalues/eigenvectors, 4) interpolation, 5) regression, 6) differentiation, 7) integration, 8) differential equations Under the NSF funding for the prototype, we developed modules for approved in February 2004, we are developing four more modules. The timeline is as follows Integration - December 2004 Simultaneous Linear Equations - December 2005 Ordinary Differential Equations - June 2006 Regression - June 2006 Intermediate versions of modules are accessible as developed. Q. Will all topics of numerical methods be included in the future and when? To complete a typical Numerical Methods course, we will seek funding for two more modules Fundamentals of Scientific Computing Do you think there are other important mathematical procedures that should be covered in an undergraduate Numerical Methods course? Drop us an Q. How do I register to use the course materials? There is no registration needed to use the course material. I want to keep it purely open access without any hassles or obstacles such as payment of use, registration, downloading, buying expensive software, etc. But drop me a note to tell me how you are using the resources. However, I am requiring faculty members who use any of the the course material to send me brief info and put link(s) to HNMI on their web site. Q. How can a faculty member use the course materials? Faculty members can use the materials to enhance their classroom lecture by using the power point presentations and simulations. They can ask students to quickly assess their knowledge by taking the online assessment of multiple choice questions. They can ask students to pre-study the topics so that class time is used for discussion purposes. However, I am requiring faculty members who use any of the the course material to send me brief info and put link(s) to HNMI on their web site. This will help us to keep this site unrestricted and at the same time show where it is being used. Q. How can a student use the course material? A student can use it to review background information on a topic, perform their own simulations, review course material, go for self-assessment of knowledge, learn how other engineering majors use numerical methods, have seven different examples to illustrate each method. Q. How does courseware here differ from others? This courseware would only be possible with the web. We have taken a holistic approach where users can review the background information as well as see the higher level application of what they have learned. We have also taken a customized approach because had the contents been written in a a book form, one would have to write 28 versions of the book. But with this courseware, a student has 24/ 7 access and can work at his own pace with help from text book notes, simulations and assessment. Q. What intellectual property policies govern the materials? The materials given on the web site are mostly original and written by the faculty and students at the University of South Florida. Any other material is either in public domain, or permission has been given for its use and is acknowledged. If you have any questions about the ownership of the materials, please contact us. Q. How do we define non-commercial use? The material on the web site or its derivatives can only be used for nonprofit purposes in educational institutions of any grade level. Providing direct links on user's website are critical in fair use of the materials. Any use of the material on the website should be acknowledged to the Holistic Numerical Methods Institute, University of South Florida, National Science Foundation and the original authors of the material. Q. How was the courseware developed? The course was developed using Microsoft FrontPage and JavaScript for the web pages and assessment tools; Microsoft Office for the text notes and lecture presentations; Mathcad, Maple, Mathematica and Matlab for the simulations; Acrobat for making of the PDF files; Adobe Photoshop for editing images; Flash V for drawing sketches; Microsoft Publisher for advertisements. Q. What are the system and technical requirements for the course materials? Read all the system and technical requirements. Q. How do I search the course material I am looking for? • If you know a topic and have a language of choice, the best page to find the course material is the resource page. The same resources are also given as text links if you face any technical trouble with the GO buttons or images. • You can also use google search that searches the whole website. • You can also , but be aware that it takes some time for it load as it has all the links to the resources. The current number of links in the site index is more than 2000. Q. How is HNMI supported? The Holistic Numerical Methods Institute is currently funded Foundation through their CCLI-EMD program. Support also comes from • the Mechanical Engineering Department at USF via faculty release time and conference support, • the College of Engineering at USF via undergraduate students through their REU program, • the Engineering Computing at USF via software support and web site maintenance, • the University of South Florida via offering the PI a sabbatical in Fall 2002 to develop the web site and form a basis for a full development proposal to be submitted to NSF in June 2003, and • Academic Computing at USF via support of software training and Blackboard access. • Maple and Mathcad • Wright State University • Florida A&m University Q. What are the long-term goals of HNMI? The long term goals of HNMI are to develop course materials for all the main topics taught in a course in Numerical Methods. This will depend on continued funding of the project. Q. How does a faculty member contact a real person from HNMI? Contact the Principal Investigator - Autar Kaw via telephone, e-mail, fax or mail. Any inquiry, especially from instructors of Numerical Methods, will be answered. We welcome your questions, comments, and suggestions. We would like to help you in incorporating the contents of the website in your course. Q. Why are the simulations, especially ones written in Matlab not modular? The simulations, especially those written in Matlab, at first might appear not well written; they do not take advantage of modular programming. However, in our website modules, we wanted to keep all subroutines and functions within a single script file. This might not seem logical at first glance. But, it is important to note the overall goal of the simulations is to provide the student only one file. The content is in one file for simplicity. Otherwise, there would be dozens of files that the student would have to manage and understand-- not just one. Also we want to show a numerical method worked out step-by-step as if the student was working it out by hand. This is why many of the simulations show each iteration separately as opposed to in a We hope that you would ask your students to write procedures (subroutines, functions, etc) and use modular programming techniques as part of the learning process of Numerical Methods.	{"url":"https://softmath.com/tutorials/writing-linear-equations-powerpoint-presentation.html","timestamp":"2024-11-12T22:39:12Z","content_type":"text/html","content_length":"62359","record_id":"<urn:uuid:e4359e54-9855-4c86-befa-6adfd98a74d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00622.warc.gz"}
Is math 110 pre calc? Math 110 - Pre Calculus (Self-Paced) Credit: 2 Credits. Is math 110 a precalculus? Trigonometric and inverse trigonometric functions, equations, and graphs, fundamental trigonometric identities, and the polar coordinate system. What type of math is math 110? College Algebra develops the concepts of graphing functions, polynomial and rational functions, exponential and logarithmic functions, conic sections, solving systems of equations, the binomial theorem, permutations, combinations, and probability. What level of math is pre calc? AP Precalculus is for any student seeking a third- or fourth-year mathematics course following completion of both Geometry and Algebra 2, or Integrated Math 3. Students who've taken these courses at any level have covered all the content necessary for AP Precalculus. Is math 111 precalculus? MATH 111 - Precalculus Trigonometry. PreCalculus Full Course For Beginners What is math 111 in college? This course prepares science majors for the calculus sequence and algebra based physics emphasizing basic concepts of algebra and is also suitable as a general education elective for non-science Is math 107 precalculus? About Precalculus Online Course. As the name indicates, Math 107: Precalculus serves as a stepping-stone to the study of calculus. In fact, it will provide you with skills that are indispensable for success in calculus—algebra and trigonometry. Many calculus students do not consider calculus itself difficult. Is math 102 pre calc? There are two possible prerequisite paths to Calculus and Linear Algebra as shown in the diagram below. Students may take either MATH 101(i) College Algebra and MATH 101T, or MATH 102 Precalculus. What is the hardest math? This blog is about the five most difficult topics in mathematics that students fear. What is the lowest math class in college? Short answer: Algebra II. Theses classes are the “lowest” math classes that you can receive credit for at most colleges, and these two classes are required for almost every major. What does math 110 mean? MATH 110 is an entry-level course in mathematics that introduces the student to skills associated with the application of calculus techniques to business and social science applications. Is math 108 precalculus? Topics include polynomial functions; factor and remainder theorems; complex roots; exponential, logarithmic, and trigonometric functions; and coordinate geometry. What is the difference between math 115 and math 110? Math 110 is a condensed, half-term version of Math 105 designed specifically to prepare students for Math 115. It is open only to students who have enrolled in Math 115 and whose performance on the first uniform examination indicates that they will have difficulty completing that course successfully. Is math 110 algebra? This course investigates the concepts of college algebra. The course covers the concepts of algebra, graphing and solution of linear and quadratic equations, inequalities and the solution of systems of linear equations. What math is before pre-calculus? In college, the following courses come before Pre-Calculus: Pre-Algebra, Introductory Algebra, Intermediate Algebra, and College Algebra. Is math 115 a pre-calculus? Additional Course Description: Math 115 - Precalculus, is an in-depth study of functions. Several function classes are studied, including linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions. What is the 1 hardest math problem? Today's mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. Is pre calc hard? Why is precalculus hard? Precalculus, which is a combination of trigonometry and math analysis, bridges the gap to calculus, but it can feel like a potpourri of concepts at times. Students are suddenly required to memorize a lot of material as well as recall various concepts from their previous math courses. What is the easiest math? Basic Math and Consumer Math are typically considered the easiest math classes in high school because they focus on practical, real-world math skills. Is math 112 precal? MTH 112: Precalculus Algebra. Is math 105 precalculus? MATH 105: Precalculus Mathematics. Is pre calc 12th grade math? Grade 12 Pre-Calculus Mathematics (40S) is designed for students who intend to study calculus and related mathematics as part of post-secondary education. Is math 114 calculus? Functions, limits, continuity, derivatives of all functions including trig, exponential, log, inverse trig and implicit functions. Is math 109 a calculus? Limits, continuity, derivatives, integrals, and their applications. Requisites: Prerequisite: MATH 101T or MATH 102 or ALEKS PPL assessment score 78-100 or appropriate high school coursework. Is math 101 a precalculus? Math 101 is the first semester of a the two-semester of M101-102 Precalculus sequence. This first course is a review of Intermediate algebra with an introduction to functions. Math 101 alone does not satisfy the R1 general education requirement for mathematics.	{"url":"https://www.spainexchange.com/faq/is-math-110-pre-calc","timestamp":"2024-11-02T14:15:02Z","content_type":"text/html","content_length":"346944","record_id":"<urn:uuid:0962412b-8609-48d9-bda2-db03724341db>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00210.warc.gz"}
GRE Math Section: An Overview of Topics & Question Types - InspiricaPros What’s covered in the GRE math section is no secret, but that doesn’t make it easier to tackle. Many students applying to graduate programs haven’t taken a math class in a while; in fact, unless you’re going to school for math, it’s probably been years since you did math in an academic setting. And even if you are pursuing a graduate degree in math, you probably haven’t touched the topics covered on this test for years! The material covered in the GRE math section is a strange blend of algebra, geometry, number theory, and advanced counting (yes, that’s actually a real thing). In this post, we’ll cover what’s on the test, what to study, and how to prep. GRE Math Topics and Types The material in the GRE math section can generally be divided into four distinct groups: arithmetic, algebra, geometry, and data analysis. □ Arithmetic: This category consists primarily of questions covering number properties, fractions/rates/percentages, divisibility and primes, arithmetic operations, and exponents/roots. Many students find the easy questions in this area very easy and the hard questions extremely difficultโ sometimes among the most difficult on the test. Prime number theory in particular can cause problems for students. □ Algebra: These problems cover algebraic properties, functions, rates and ratios, single- and double-variable word problems, and sequences. Most students are familiar with the basics of this section, but they often need a refresher on the advanced content. Transforming word problems into equations is a particular focus of questions in this category. □ Geometry: This topic includes two- and three-dimensional geometry, coordinate geometry, and mixed geometry. It focuses primarily on triangles, then circles, then other polygons, and questions in this area often include algebra as a secondary topic. Most students learned this material early in high school, so it might be a challenge to recall all of it. □ Data Analysis: Questions in this category cover a fairly wide rangeโ no pun intendedโ of concepts. The one-off problems typically deal with mean/median/mode, probability, combinatorics, and some slightly more advanced statistics concepts, such as normal distributions. Each Quant section also contains 4-5 problems that ask students to analyze a given chart or graph and sometimes to do math with data pulled from that figure; percent change in particular is a favorite topic of these questions. If you donโ t recognize many of these topics, it might be worth doing some intensive review even before you take a practice or diagnostic test. Take a moment to go through the list above and make a list of topics where you think you need a refresh. Many of them are easy to review using online or free resources, such as Khan Academy or YouTube. For more in-depth instruction, our excellent GRE tutors can help. If you canโ t easily identify your weaknesses by looking at the list of concepts, work through a couple of practice sections to get a clearer picture of where you need targeted review. GRE Math Question Types Each of the math topics mentioned above can be tested in several ways. The GRE math section has three primary question-types that testers should be aware of: • Multiple Choice: These are your standard, boring test questions. Students will choose from five options, and questions can cover any of the topics laid out earlier. Your usual elimination and plug-in tactics work well here. To spice things up, each Quant section will include 1-2 MC questions that allow for more than one answer selection, with students being instructed to select all choices that are correct. This subtype feels scarier, but don’t worryโ most of the usual approaches still work just fine! • Numeric Entry: These questions have no answer choices; instead, students will bubble the answer they calculate directly into a blank box. As you might expect, these problems can be tough: plug-ins typically donโ t work as well as they do on MC questions, and your options for checking your answer are limited. You’ll need to be confident in your calculation or find a clever way to check your answer to be sure. • Quantitative Comparison: These are the strangest questions on the test. Instead of simply asking you to evaluate an equation or complete a series of concrete calculations, these problems ask whether you have enough information to determine which of two given quantities is greater. This is the question-type that most students are the least familiar with, so we spend a good deal of time on these questions in session. The question-types listed above may seem very different, but youโ ll be surprised by how many techniques they have in common. In order to ace the GRE, testers need to know not only what’s covered in the GRE math section but also how to tackle each problem-type. Get Started Today with Online GRE Prep If you’re uneasy about the GRE Quantitative section or the GRE in general, you’re not alone. That’s where our test prep experts come in. If youโ re looking for customized one-on-one prep thatโ s 100% tailored to your unique needs, Inspirica Pros has dozens of expert GRE tutors with decades of combined experience. Find an online tutor now to get started! Related GRE Resources:	{"url":"https://inspirica.com/blog/standardized-tests/gre/whats-covered-in-the-gre-math-section/","timestamp":"2024-11-10T09:33:36Z","content_type":"text/html","content_length":"313285","record_id":"<urn:uuid:e031fe06-fb91-470f-8807-d90b940af419>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00271.warc.gz"}
Vlookup #no match error - search value is the output of another formula My vlookup formula is not working as the search value cell is the output of a separate formula. The Vlookup works once I type the numerical value into the search value. Is there any way around this? • Hey @Jon B I'm guessing that smartsheet is interpreting your search value output as a text string rather than a number. Without seeing the both formulas, it's harder to suggest which one of these to try Try wrapping your search value within the Vlookup with the VALUE() function. If that doesn't work, wrap the VALUE around your entire Output function. =VALUE(entire formula, including parentheses) If that doesn't work, let me know. At that point, if you could also provide the formulas you are using, it would be easier to troubleshoot. • Hi Kelly, The first one worked! I wrapped my search value with the VALUE() function. Thanks very much I have another question. I am trying to separate numbers from a text that changes format slightly eg. LINE ColumnName Outcome that I want 1 ABCD-CH111 111 2 ABCD-PMP123 123 3 ABCD-VF313 313 If tried using several formulas. I am struggling as 8th digit can change from a letter to a number and the 8/9th number can be from 0 - 9. This formula below returns "P12" for line 2 instead of "123" =IF(MID(ColumnName@row, 8, 1) >= 1, MID(ColumnName@row, 8, 3), MID(ColumnName@row, 9, 2)) I also tried these formulas to no avail. IF(ISTEXT(MID(ColumnName@row, 8, 1)) = 1, MID(ColumnName@row, 8, 3)) IF(ISTEXT(MID(ColumnName@row, 8, 1)) = 1, MID(ColumnName@row, 8, 3),MID(ColumnName@row9,3) IF(ISNUMBER(MID(ColumnName@row, 8, 1)) = 1, MID(ColumnName@row, 8, 3)) I can create a new question also if that is easier. • Hey Jon Happy to try to help. If I understand correctly, the data does not always present itself so that it is the last three characters that need to be extracted?. Is this correct? If it is always the last three digits then we could use RIGHT instead of MID. To take care of a leading zero, we might have to add +"" to the end. This will force the value to be a textstring - this would actually do the opposite of what we did in your first question. If RIGHT() works then: =RIGHT(ColumnName@row, 3)+"" If the RIGHT function won't work, can you give me a screenshot of some actual data, as much as possible? It is hard to get a feel in how much and exactly how the data varies from one row to another, from the few lines above. You'll find you will almost always get more complete from the community when screenshots are provided. Please advise • Hi Kelly, Sorry the format changed of the data that I entered so it made it unclear. The right function won't work unfortunately as I want to pull numbers from the middle of the data. E.g. "ABCD-CH111 -1-345". Desired outcome "111". E.g "FD-PMP251-56-5757" Desired outcome "251" • Is the data to be pulled always in front of the second hyphen, and once found, it's always 3 characters? • Yes its always 3 characters before the second hyphen. • Hey Jon I think this will work for you. Awhile back in my real work I had tried using the same approach as you started but I couldn't get reliable results. Thankfully you had special characters to work off of as markers. The heavy lifting for the formula was posted earlier this year by @Leibel Shuchat. I had been waiting for an opportunity to try it out, it's such a clever use of SUBSTITUTION- and for your example, I took some liberties with his formula. I like to use the ASCII/HTML characters for special characters as sometimes smartsheet can become confused - especially if the character needed is a comma or parenthesis. Although not a problem in your case with the hyphens, since I keep this link handy, it's always quick for me to look them up. CHAR(45) equals your hyphen. Using the CHAR (45) in the formula also made it easier as I worked on the formula to distinguish your hyphen from the SUBSTITUTION placeholder "~". As Leibel pointed out in his post, the SUBSTITUTE allows you to call out which instance of Search_Text you are substituting. This makes it straight-forward to direct the Find function to the appropriate position. By subtracting position #1 from position #2, this gives you the number of characters needed for the last term in the MID function. Since , in your case, you always need the last three characters in that parsed text, I wrapped the MID function with RIGHT(). Thankfully the number of characters you needed were constant or we would have been forced to try to make the ISTEXT or ISNUMBER work. =RIGHT(MID(ColumnName@row, FIND("~", SUBSTITUTE(ColumnName@row, CHAR(45), "~", 1)) + 1, FIND("~", SUBSTITUTE(ColumnName@row, CHAR(45), "~", 2)) - FIND("~", SUBSTITUTE(ColumnName@row, CHAR(45), "~", 1)) - 1), 3) Help Article Resources	{"url":"https://community.smartsheet.com/discussion/89872/vlookup-no-match-error-search-value-is-the-output-of-another-formula","timestamp":"2024-11-12T11:00:26Z","content_type":"text/html","content_length":"423833","record_id":"<urn:uuid:223e7715-5779-49cf-8f8d-619d50d2caf8>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00827.warc.gz"}
Adventures in Pedagogy: Conversations My 11 year old was asked to describe what he could about the relationships between 1/a and 1/b if a < b (a, b are Natural numbers) and a/b and b/a. He explained to me that 1/a > 1/b since it had a smaller denominator. Well because if the denominator is larger, it's broken up into smaller pieces. OK, so what about a/b and b/a? a/b is a proper fraction and b/a is an improper fraction. Then which one's bigger? thinks for a moment Because b/a will have a whole number and a/b won't. Then what can you tell me about all improper fractions? looks confused I get up and motion for Dawson to do the same. If I'm 0 and you're 1, where would the proper fractions be? points between us Then where are the improper fractions? points away from me Then what can you tell me about all improper fractions? They're all bigger than 1. Then what can you tell me about all proper... They're all between 0 and 1. There you go. So do I have to do this problem? Didn't you just do it? Well yeah, but I didn't write it down. Will writing it down help you understand it better? Not really. Looking my son in the eyes 6 comments: What about having him explain it to someone else? Do you think that would serve to cement his understanding of it even further? Okay, you just cemented the need for at least one round of conferring as assessment in my classroom this year. I know you made the right call on writing it down, because if he had seemed a little shaky you totally would have made him write it down. In the process of writing it down, he would have had to think through his reasoning again. I suppose the more he explained it, the better. Problem would be finding someone who'd understand his explanation. I did much more in the way of conversing with my students on an indicidual basis this last year and found it to be very telling. Kids can learn to manipulate the symbols but had to really dig to express their understanding verbally. What I find very impressive about this story is the relationship of trust you have with your kid. It would seem to take a lot to be able to talk your kid about something such as math, to reach a point where they are confused, and to have them not want to walk away in the it's-my-prerogative-because-I-am-a-kid-and-I-am-impatient sort of way. I wonder: Is this relationship what enables you to homeschool, or is this relationship a result of homeschooling? This comment has been removed by the author. What a great question. At this point I'd have to say the conversation was a result of homeschooling. My son(s) is/are very stubborn and when he was in school, the last thing he wanted to do was talk about math. He'd take the I already have a teacher attitude very quickly. Now that I'm his math teacher, that's changing. We talk a lot in our family, so that helps. But talking about math is a new thing for us.	{"url":"https://coxmath.blogspot.com/2010/07/adventures-in-pedagogy-conversations.html","timestamp":"2024-11-06T07:52:07Z","content_type":"text/html","content_length":"75998","record_id":"<urn:uuid:f1c1495d-dee4-4df8-8b43-ab738af14d10>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00284.warc.gz"}
Analogmachine Blog Here is an example of runnign a simulation and plotting the results as an animation. The model is a 20 step linear pathway where at time zero, all concentrations are zero. The boundary species Xo is set to 10 and as the simualtion runs the species slowly fill up to their steady-state levels. We can capture the data and turn it intot an animation using the matplotlib.animation from matlpotlib. At the end we convert the frames into a mp4 file. I generated the model using hte buildnetworks class in teUtils import tellurium as te import matplotlib.pyplot as plt from matplotlib.animation import FuncAnimation, FFMpegFileWriter r = te.loada(''' J1: $Xo -> S1; e1(k10Xo - k11S1); J2: S1 -> S2; e2(k20S1 - k21S2); J3: S2 -> S3; e3(k30S2 - k31S3); J4: S3 -> S4; e4(k40S3 - k41S4); J5: S4 -> S5; e5(k50S4 - k51S5); J6: S5 -> S6; e6(k60S5 - k61S6); J7: S6 -> S7; e7(k70S6 - k71S7); J8: S7 -> S8; e8(k80S7 - k81S8); J9: S8 -> S9; e9(k90S8 - k91S9); J10: S9 -> S10; e10(k100S9 - k101S10); J11: S10 -> S11; e11(k110S10 - k111S11); J12: S11 -> S12; e12(k120S11 - k121S12); J13: S12 -> S13; e13(k130S12 - k131S13); J14: S13 -> S14; e14(k140S13 - k141S14); J15: S14 -> S15; e15(k150S14 - k151S15); J16: S15 -> S16; e16(k160S15 - k161S16); J17: S16 -> S17; e17(k170S16 - k171S17); J18: S17 -> S18; e18(k180S17 - k181S18); J19: S18 -> S19; e19(k190S18 - k191S19); J20: S19 -> $X1; e20(k200S19 - k201X1); k10 = 0.86; k11 = 0.24 e1= 1; k20 = 2.24; k21 = 0.51 e2= 1; k30 = 2.00; k31 = 0.49 e3= 1; k40 = 2.54; k41 = 0.83 e4= 1; k50 = 3.05; k51 = 0.97 e5= 1; k60 = 1.29; k61 = 0.10 e6= 1; k70 = 1.94; k71 = 0.20 e7= 1; k80 = 3.01; k81 = 0.17 e8= 1; k90 = 0.64; k91 = 0.19 e9= 1; k100 = 1.96; k101 = 0.42 e10= 1; k110 = 2.47; k111 = 0.63 e11= 1; k120 = 4.95; k121 = 0.51 e12= 1; k130 = 4.78; k131 = 0.70 e13= 1; k140 = 3.77; k141 = 0.78 e14= 1; k150 = 1.93; k151 = 0.37 e15= 1; k160 = 3.87; k161 = 0.96 e16= 1; k170 = 0.83; k171 = 0.37 e17= 1; k180 = 1.82; k181 = 0.69 e18= 1; k190 = 2.62; k191 = 0.83 e19= 1; k200 = 4.85; k201 = 0.67 e20= 1; Xo = 10.00 X1 = 0 S1 = 0; S2 = 0; S3 = 0; S4 = 0; S5 = 0; S6 = 0; S7 = 0; S8 = 0; S9 = 0; S10 = 0; S11 = 0; S12 = 0; S13 = 0; S14 = 0; S15 = 0; S16 = 0; S17 = 0; S18 = 0; S19 = 0; # I'm running this on windows so I had to install the ffmpeg binary # in order to save the video as an mp4 file # for more info go tot his page: # https://suryadayn.medium.com/error-requested-moviewriter-ffmpeg-not-available-easy-fix-9d1890a487d3 plt.rcParams['animation.ffmpeg_path'] ="C:\\ffmpeg\\bin\\ffmpeg.exe" endTime = 18 fig, ax = plt.subplots(figsize=(8, 5)) ax.set(xlim=(-1, r.getNumIndFloatingSpecies()), ylim=(0, 18)) m = r.simulate(0, endTime, 100) label = ax.text(15.6, 19, 'Time=', ha='center', va='center', fontsize=18, color="Black") bars = ax.bar(r.getFloatingSpeciesIds(), m[0,1:], color='b', alpha = 0.5) def barAnimate(i): label.set_text('Time = ' + f'{m[i,0]:.2f}') for bar, h in zip(bars, m[i,1:]): bar.set_color ('r') anim = FuncAnimation(fig, barAnimate, interval=50, frames=100, repeat=False) # You can save themp4 file anywhere you want anim.save('c:\\tmp\\animation.mp4', fps=30) This is the resulting video: I just updated teUtils to version 2.9 UPDATE: The documentation was broken, now fixed. This is a set of utilities that can be used with our simulation environment Tellurium The update includes some updates to the build synthetic networks functionality. The new version adds 'ei(' terms to mass actions rate laws, eg S1 -> S2; e1(k1S1 - k2S2) This makes it easier to compute control coefficients with repect to 'e' I also added a new mass-action rate of the form: v = k1A(1 - (B/A)/Keq1) These can be generated using the call: model = teUtils.buildNetworks.getLinearChain(10, rateLawType="ModifiedMassAction") Useful if you want to more easily control the equilbrium constant for a reaction. Here is an example of a three step random linear chain: model = teUtils.buildNetworks.getLinearChain(10, rateLawType="ModifiedMassAction") print (model) J1: $Xo -> S1; k1Xo(1 - (S1/Xo)/Keq1); J2: S1 -> S2; k2S1(1 - (S2/S1)/Keq2); J3: S2 -> $X1; k3S2(1 - (X1/S2)/Keq3); k1 = 1.42; Keq1 = 3.61 e1 = 1; k2 = 1.42; Keq2 = 7.91 e2 = 1; k3 = 3.50; Keq3 = 5.64 e3 = 1; Xo = 5.00 X1 = 0 S1 = 1E-6; S2 = 1E-6; I needed to be able to generate simulation plots for a sphinx document but I didn't want to have to generate the plots separately and then include them manually. I wanted the simulations done from within sphinx so that they would be automatically included when building the document. The key to this is to add the following to the list of extensions in the sphinx conf.py file: extensions.append ('matplotlib.sphinxext.plot_directive') To use it in a sphinx document use the plot directive, for example: .. plot:: import tellurium as te r = te.loada ('''A -> B; k1A; k1=0.1; A = 10''') m = r.simulate (0, 40, 100) It assumes you have matplotlib installed in your python setup (I am using Windows 10, Python 3.11). Further information can be found here: This is a note for myself but others might find useful. I needed to execute python in a sphinx document at build time. The package that comes up in a search doesn't work with Python 3.x. Instead I found which does work with Python 3.11. (I'm running Windows 10) Install it by typing the following on the command line pip install sphinx-execute-code-python3 To use it don't forget to add this line to your sphinx conf.py file: Here is an example in a rst file: .. execute_code:: a = [1,2,3] b = [4,5,6] print ('Printing a list: ', a + b) The docs have various options you can use. Analysis of a simple pathway: This model is model a) from the paper by Tyson et al (2003) "Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell". I changed their R to P however so I could use the symbol R for response. Assume $ v_1 = k_1 S $ and $ v_2 = k_2 P $ Then $\frac{dP}{dt} = k_1 S - k_2 P $. At steady-state this equals 0 so that $$ P = \ frac{k_1 S}{k_2} $$ In other words P is a linear function of S. In terms of logarithmic senstivities or the reponse coefficient: $$ R^P_S = \frac{dP}{dS} \frac{S}{P} = \frac{k_1}{k_2} \frac{k_2}{k_1} = 1 $$ That is a 1% change in S will lead to a 1% change in the steady-state level of P. This is important because in their pathway (d, shown below) they use this property to ensure that when S also regulates a secondary pathway that in turn up regulates the consumption step v2, the level of P remains unchanged. This property is not very robust however and any change to the rate law on $v_1$ will result in the pathway failing to maintain P constant. For example if $v_1 = k_o + k_1 S$, where we've added a basal rate of $k_o$, then the response becomes: $$ R^P_S = = \frac{k_1 S}{k_o + k_1 S} $$ that is the response is no longer proportional. Change S in this case will result in P changing. The code below will model this system. One important point to make is that this isn't an example of integral control. import tellurium as te import matplotlib.pyplot as plt r1 = te.loada(""" -> P; k0 + k1S P ->; k2XP -> X; k3S X ->; k4X k1 = 1; k2 = 1 k3 = 1; k4 = 1 # Set basal rate to zero k0 = 0; S = 1 # Change signal, P won't change at time > 10: S = S2 # Change basal rate and set S back to what it was at time > 25: k0 = 0.3, S = 1; # Change signal, this time P will change at time > 40: S = S2 m = r1.simulate(0, 60, 200, ['time', 'P', 'S']) plt.plot (m['time'], m['P'], label='P') plt.plot (m['time'], m['S'], label='S') plt.text(2, 0.75, "Basal = 0") plt.text(9, 0.9, "Change S") plt.text(14, 1.2, "P restored") plt.text(20, 0.75, "Set basal > 0") plt.text(35, 0.9, "Change S, P not restored") It's always surprising to find students who don't quite get what a steady-state is. Here is a simulation of a three step pathway that might help. You'll need to install tellurium for this to work. $X_o \stackrel{v_1}{\rightarrow} S_1 \stackrel{v_2}{\rightarrow} S_2 \stackrel{v_3}{\rightarrow}$ we assume that $X_o$ is fixed and does not change in time. I don't care where $S_3$ goes. If we start with zero concentrations for $S_1$ and $S_2$ we get the following plots. The left plot shows the change in concentration and the right plot the reaction rates. Note that all three reaction rates approach the same rate since at steady-state all rates must be equal. Here is the python code that will generate that plot. To get the reaction rate symbols above the reaction arrows I used a bit of a hack by printing two text strings, one above the other. For some reason I couldn't get matplotlib to recognise the LaTeX command stackrel. import tellurium as te import matplotlib.pyplot as plt r = te.loada(""" J1: $Xo -> S1; k1Xo - k2S1 J2: S1 -> S2; k3S1 - k4S2 J3: S2 -> ; k5S2 k1 = 0.1; k2 = 0.04 k3 = 0.14; k4 = 0.09 k5 = 0.16 Xo = 10 m = r.simulate (0, 60, 100) plt.subplot (1,2, 1) plt.plot(m['time'], m['[S1]'], label='S1') plt.plot(m['time'], m['[S2]'], label='S2') plt.ylim((0, 9)) plt.text(5, 8.4, 'Approach to steady-state') plt.text(5, 7.8, '$X_o$ is fixed') plt.text(14, 1.5, r'$\quad\ v_1 \quad v_2 \quad v_3$', fontsize = 12) plt.text(14, 1, r'$X_o \rightarrow S_1 \rightarrow S_2 \rightarrow$', fontsize = 12) plt.legend(bbox_to_anchor=(0.4, 0.5)) # Next generate the reaction rates m = r.simulate (0, 40, 100, ['time', 'J1', 'J2', 'J3']) plt.subplot (1,2, 2) plt.plot(m['time'], m['J1'], label='$v_1$') plt.plot(m['time'], m['J2'], label='$v_2$') plt.plot(m['time'], m['J3'], label='$v_3$') plt.text(14, 0.2, r'Reaction rates', fontsize = 12) Nothing to do with cells or modeling, but I recently purchased a 1885 copy of John Casey's rendition of Euclid's Elements. John Casey was born in Limerick, Ireland, and became a lecturer in mathematics at University College Dublin. He wrote one of the more well-known editions of Euclid's elements. In his preface he states: "This edition of the Elements of Euclid, undertaken at the request of the principals of some of the leading Colleges and Schools of Ireland, is intended to supply a want much felt by teachers at the present day—the production of a work which, while giving the unrivalled [sic] original in all its integrity, would also contain the modern conceptions and developments of the portion of Geometry over which the Elements extend." You can find a copy at Project Gutenberg (https://www.gutenberg.org/ebooks/21076) if you are curious. What's interesting about the book I received, is not so much the geometry, but what I found inside. Inside was a copy of a math Pass Examination paper from 1888, presumably from University College Dublin but at least somewhere in Ireland (I purchased the book from Dublin). Update: There are two inscriptions in the book. The first is a simple "W.H Dunlop May 1886". The second is more interesting and written in a very elegant cursive style where the book was transferred two years later to: Michael J Buckley. Catholic University Dublin. 19.1.'88. This is the name of the first university dedicated to accepting Catholics from Ireland (Trinity College was the Anglican University founded by Elizabeth I, so you can imagine the problem). However, the university didn't do too well financially and had problems awarding degrees because it didn't have a royal charter. In 1908/1909, it became the University College Dublin with its own charter. The exam had 10 questions, one involving reaping a field, another about selling a horse, and another about carpeting a room. The remaining 7 are pure math questions, with a number of them being numerical estimation questions, eg, roots, squares etc. I think modern students could do these questions, though they might struggle with the numeric computations unless they have a calculator at hand. I took a photograph of the exam for all to see: Here is an interesting metabolic question to ask. Consider a pathway with two steps: $$\large X_0 \stackrel{e_1}{\rightarrow} S \stackrel{e_2}{\rightarrow} $$ where the first step is catalyzed by enzyme $e_1$ and the second step by $e_2$. We can assume that a bacterial cell has a finite volume, meaning there is only so much space to fit all the proteins necessary for the bacterium to live. If the cell make more of one protein, it presumably needs to find the extra space by down expressing other proteins. If the two step pathway is our cell, this translates to there being a fixed amount of protein that can be distributed betwen the two, that is: $$ e_1 + e_2 = e_t $$ Presumably evolutionary pressure will maximize the flux through the pathway per unit protein. This means that given a fixed total amount of protein $e_t$ there must be a particular distribution of protein between $e_1$ and $e_2$ that maximizes flux. For example, if most of the protein were allocated to $e_1$ and very little to $e_2$ then the flux would be quite low. Likewise if most of the protein were allocated to $e_2$ and very little to $e_1$ then again the flux would be low. Between these two extremes there must be a maximum flux. To show this is the case we can do a simple simulation shown in the python code below. This code varies the levels of $e_1$ and $e_2$ in a loop such that their total is always 1 unit. Each time we change $e_1$ we must change $e_2$ by the same amount in the opposite direction in order to keep the constant fixed. As we do this we collect the steady-state flux and the current level of $e_1$. Finally we plot the flux versus $e_1$. import tellurium as te import roadrunner import matplotlib.pyplot as plt r = te.loada(""" J1: $Xo -> S; e1(k1Xo - k2S) J2: S ->; e2k3S Xo = 1 k1 = 0.5; k2 = 0.2 k3 = 0.45 e1 = 0.01; e2 = 0.99 x = []; y = []; for i in range (49): r.e1 = r.e1 + 0.02 r.e2 = 1 - r.e1 # Total assumed to be one x.append (r.e1) y.append (r.J1) plt.grid(b=True, which='major', color='#666666', linestyle='-') plt.grid(b=True, which='minor', color='#999999', linestyle='-', alpha=0.2) plt.plot (x, y, linewidth = 2) plt.xlabel('$e_1$', fontsize=36) plt.ylabel('Flux', fontsize=36) The following graph show the results from the simulation (the font sizes might be on the big size if you have a low res monitor): You can see the flux reaching a maximum at around $e_1 = 0.6$, meaning also that $e_2 = 0.4$ in order to keep the total fixed. The actual position of the peak wil depend on the rate constants in the rate expressions. Let's assume we are at the maxium. The slope, $dJ/de_1$, at the maxium is obviously zero. That means if we were to move a small amount of protein from $e_1$ to $e_2$ the flux won't change. We can write this experiment in terms of the two flux control coefficients: $$ \frac{\delta J}{J} = 0 = C^J_{e_1} \frac{\delta e_1}{e_1} + C^J_{e_2} \frac{\delta e_2}{e_2} $$ However, we know that in this particlar experiment the change in $e_1$ is the same but oppisite to the change in $e_2$. That is $\delta e_1 + \delta e_2 = 0$ or $$ \delta e_1 = -\delta e_2$$ Replacing $e_2$ with $-\delta e_1$ gives: $$ \frac{\delta J}{J} = 0 = C^J_{e_1} \frac {\delta e_1}{e_1} - C^J_{e_2} \frac{\delta e_1}{e_2} $$ $$ \frac{\delta J}{J} = 0 = C^J_{e_1} \frac{1}{e_1} - C^J_{e_2} \frac{1}{e_2} $$ $$ C^J_{e_1} \frac{1}{e_1} = C^J_{e_2} \frac{1}{e_2} $$ Giving the final result: $$ \frac{C^J_{e_1}}{C^J_{e_2}} = \frac{e_1}{e_2} $$ That is, when the protein distribution is optimized to maximize the flux, the flux control coefficients are in the same ratio as the ratio of enzyme amounts. This generalizes to any size pathway with multiple enzymes. This gives us a tantgilizing suggestion that we can obtain the flux control coeficients just by measuring the protein levels. There is obviously a lot more one can write there and maybe I do that in future blogs but for now you can get further information: S. Waley, “A note on the kinetics of multi-enzyme systems,” Biochemical Journal, vol. 91, no. 3, p. 514, 1964. J Burns: “Studies on complex enzyme system.” https://era.ed.ac.uk/handle/1842/13276, 1971 (page 141-) I have a LaTeX version at: https://github.com/hsauro/JumBurnsThesis Guy Brown, Total cell protein concentration as an evolutionary constraint on the metabolic control distribution in cells,” Journal of theoretical biology, vol. 153, no. 2, pp. 195–203, 1991. E. Klipp and R. Heinrich, “Competition for enzymes in metabolic pathways:: Implications for optimal distributions of enzyme concentrations and for the distribution of flux control,” Biosystems, vol. 54, no. 1-2, pp. 1–14, 1999 Sauro HM Systems Biology: An Introduction to Metabolic Control Analysis, 2018 To get hold of me it is best to contact me via my email address at: hsauro@uw.edu Someone asked me the other day what the relationship was between the steady-state fluxe through a reaction and the coresponding level of enzyme. Someone else suggested that there would be a linear, or proportional relatinship between a flux and the enzyme level. However, this can’t be true, at least at steady. Considder a 10 step linear pathway. At steady-state each step in the pathway will, by defintion, carry the same flux. This is true even if each step has a different enzyme level. Hence the relationship is so simple. In fact the flux a given step carries is a systemic properties, dependent on all steps in the pathway. As an experiment I decided to do a simulation on some synthetic netowrks with random parameters and enzyme levels. For this exmaple I just used a simple rate law of the form: $$ v = e_i (k_1 A - k_2 B) $$ For a bibi reaction, A + B -> C + D, the coresponding rate law would be: $$ v = e_i (k_1 A B - k_2 C D) $$ a similar picture would be seen for the unibi and biuni reactions. Using our teUtils package I generated random networks with 60 species and 150 reactions. The reactions allowed are uiui-uinbi, biui or bibi. I then randomized the values for the enzymne levels $e_i$ and computed the steady-state flux. I used the following code to do the analysis. I have a small loop that generates 5 random models but obviously this number can be changed. I generate a random model, load the model into roadrunner, randomize the values for the $e_i$ parameters between 0 and 10, compute the steady-state (I do a presimulation to help things along) and collect the corresponding $e_i$ and flux values. Finally I plot each pair in a scatter plot. import tellurium as te import roadrunner import teUtils as tu import matplotlib.pyplot as plt import random for i in range (5): J = []; E = [] antStr = tu.buildNetworks.getRandomNetwork(60, 150, isReversible=True) r = te.loada(antStr) n = r.getNumReactions() for i in range (n): r.setValue ('E' + str (i), random.random()10) m = r.simulate(0, 200, 300) for i in range (n): J.append (abs (r.getValue ('J' + str (i)))) E.append (r.getValue ('E' + str (i))) plt.figure(figsize=(12, 8)) plt.plot(E, J, '.') print ('Error: bad model') The results for five random networks is shown below. Note the x axis is the enzyme level and the y axis the corresponding steady-state flux through that enzyme. It's intersting to see that there is a rough correlation between enzyme amount and the corresponding flux, but its not very strong. Many of the points are just scattered randomly with some showing a definite correlation. The short answer is the realtinship is not so simple. I recently came across one of the ten simple rules aricles in PLoS Comp Bio: "Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students" by Korryn Bodner et al One thing that struck me was Rule 5 on coding best practices with commenting being one of the discussion points. What struck me was their screen shot of a documented function shown below (in R): My take on commenting is that it should be used to add human readable metadata on elements of a program that are not immediately obvious. Most of the time, code should be sufficiently readable to indicate what it's doing. Obviously some languages are better than others when desribing an algorithm but it is also dependent on the programmer. I've seen code written in clear languages that are unintelliglbe, but I've also seen code written in poorly expressible languages that are easily readable. Although the programming language itself can influence code reability I think the programmer has much more influence. But back to Rule 5. In the example you'll see something like: # calculate the mean of the data u <- mean (x) This is completely redundant, as the coding states what it is going to do. In fact the authors comment every line like this. If anythng, I think the extent of comments actually hinders the reabilty of the code. The code itself is mostly clear as to what it is doing. There may be a justification to include a comment on next line that computes the standard error because the variables names are so badly chosen, e.g what does the following line do: s <- sd(x) sd might stand for standard deviation but the rest of the line offers no clue. If it had been written as: standardDeviation <- sd(x) It would have been much clearer, instead the authors add a comment to make up for poor choice of variable names. They also give the function itself a nondescriptive name, in this case ci. It would have been better to write the function using getConfidenceInterval or similar: getConfidenceInterval <- f (data) { Nothing to do with cells and networks, but I have an interest in Euclid's Elements and decided to publish a new edition of Book I. The difference with previous editions is that this one is in color and also has a chapter on the history of the elements, together with commentaries on each proposition. For those unfamiliar with Euclid's Elements, it's a series of books (more like chapters in today's language) laying the foundation for geometry and number theory. Book I focuses on the foundation of geometry culminating in a proof for Pythagoras' theorem and other important but less known results. The key innovation is that it describes a deductive approach to geometry. It starts with definitions and axioms from which all results are derived using logical proofs. It occurred to me that something similar could be done with deriving the properties of biochemical networks. For example, we might define the following three primitives: I. Species II. Reaction III. Steady-state We might then define the following axioms: I. A species has associated with it a value called the concentration, x_i. II. All concentrations are positive. III. A reaction has a value associated with it called the reaction rate, v_i. IV. Reaction rates can be negative, zero, or positive. V. A reaction transforms one or more species (reactants) into one or more other species (products). VI. The reaction rate is a continuous function of the reactants and products. VII. The rate of change of a species can be described using a differential equation, dx/dt VIII. All steps are reversible unless otherwise stated (may this can be derived?) Given these axioms, we could build a series of propositions. This might be an interesting exercise to do. Some of the more obvious propositions would be the results from metabolic control analysis, such as the summation and connectivity theorems.	{"url":"https://blog.hsauro.org/2023/","timestamp":"2024-11-06T17:21:05Z","content_type":"application/xhtml+xml","content_length":"149446","record_id":"<urn:uuid:87772ee6-05c0-4cbe-ab4c-b0b05d55c7b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00631.warc.gz"}
Consumer arithmetic-online maths test consumer arithmetic-online maths test Related topics: ks3 maths practice tests online gcf worksheets how to do distributive property with fractions algebra 1 cheats multiplying and dividing rational expressions free calculator free calculator algebra fraction from least to the greatest expand brackets in algebra worksheets college algebra solvers How To Do Multiple Variable Equations strategies for teaching multiplying and dividing integers ordering decimals from least to greatest calculators Author Message Imterloj Posted: Friday 17th of Jul 12:29 Hi everyone, I heard that there are various programs that can help with us studying ,like a tutor substitute. Is this really true? Is there a program that can aid me with algebra ? I have never tried one thus far , but they are probably not hard to use I assume. If anyone tried such a software , I would really appreciate some more detail about it. I'm in Algebra 2 now, so I've been studying things like consumer arithmetic-online maths test and it's not easy at all. From: London, UK Back to top oc_rana Posted: Saturday 18th of Jul 08:52 Don’t fear, Algebrator is here ! I was in a same situation sometime back, when my friend advised that I should try Algebrator. And I didn’t just clear my test; I went on to score really well in it . Algebrator has a really simple to use GUI but it can help you crack the most difficult of the problems that you might face in math at school. Just try it and I’m sure you’ll do well in your test. Back to top daujk_vv7 Posted: Sunday 19th of Jul 07:07 Hello, I am a mathematics professor . I use this software whenever I get stuck at any equation . Algebrator is undoubtedly a very handy software. From: I dunno, I've lost it. Back to top thicxolmed01 Posted: Sunday 19th of Jul 11:47 Algebrator is a very incredible product and is definitely worth a try. You will find several interesting stuff there. I use it as reference software for my math problems and can swear that it has made learning math much more fun . From: Welly, NZ Back to top C-3PO Posted: Tuesday 21st of Jul 07:17 It’s amazing that a program can perform that. I didn’t expect something like that could help in algebra. I’m used to be taught by a teacher but this really sounds cool. Do you have any links for this software ? From: Belgium Back to top MoonBuggy Posted: Wednesday 22nd of Jul 08:54 Here you go, click on this link – https://softmath.com/algebra-policy.html. I personally think it’s quite good for a math software and the fact that they even offer an unconstrained money back guarantee makes it a deal, you can’t miss. From: Leeds, UK Back to top	{"url":"https://softmath.com/algebra-software/exponential-equations/consumer-arithmetic-online.html","timestamp":"2024-11-10T17:28:57Z","content_type":"text/html","content_length":"43094","record_id":"<urn:uuid:80381619-1338-4f3b-a189-a10883c63578>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00545.warc.gz"}
next → ← prev Teradata Hashing Algorithm The hashing algorithm is a piece of code that acts as a translation table. A row is assigned to a particular AMP based on the primary index value. Teradata uses a hashing algorithm to determine which AMP gets the row. The Teradata Database hashing algorithms are proprietary mathematical functions that transform an input data value of any length into a 32-bit value referred to as a rowhash, which is used to assign a row to an AMP. Whether the input is a combination of different column values, comes from differently-sized values from a variable-length column, or is composed of different data types, the hashing algorithm's output will always be a fixed size and format. It dealing with uniformly-sized row identifiers simplifies the database's effort during storing, aggregating, sorting, and joining rows. Following is a high-level diagram on the hashing algorithm. Here is the explanation of the above diagram, how to insert the data: • First, the client submits a query. • Then, the parser receives the query and passes the PI value of the record to the hashing algorithm. • The hashing algorithm hashes the primary index value and returns a 32-bit number, called Row Hash. • The higher-order bits of the row hash (first 16 bits) are used to identify the hash map entry. The hash map contains one AMP #. Hash map is an array of buckets that contain specific AMP #. • BYNET sends the data to the identified AMP. • AMP uses the 32 bit Row hash to locate the row within its disk. • If there is any record with the same row hash, it increments the uniqueness ID, a 32-bit number. For new row hash, uniqueness ID is assigned as 1 and incremented whenever a record with the same row hash is inserted. • The combination of Row hash and Uniqueness ID is called Row ID. • Row ID prefixes each record in the disk. • Each table row in the AMP is logically sorted by their Row IDs. Hashing Functions Teradata SQL provides several functions that can be used to analyze the hashing characteristics of the existing indexes and candidate indexes. These functions are documented fully in SQL Functions, Operators, Expressions, and Predicates. There are four types of hashing functions available in the Teradata. 1. HASHROW: It describes hexadecimal rowhash value for an expression. The Query would give the same results if we ran again and again. The HASHROW function produces the 32-bit binary Row Hash that is stored as part of the data row. It returns maximum of 4,294,967,295 unique values. 2. HASHAMP: The HASHAMP function returns the identification number of the primary AMP for any Hash Bucket number. When no value is passed through the HASHAMP function, it returns a number less than the number of AMPs in the current system configuration. 3. HASHBUCKET: The HASHBUCKET function produces 16bit binary Hash Bucket used with the Hash Map to determine the AMP that store and retrieve the data row. The values range from 0 to 1,048,575, not counting the NULL as a possible result. 4. HASHBAKAMP: The HASHBAKAMP function returns the identification number of the Fallback AMP for any Hash Bucket number. Hash Collisions The Hash Collision is a situation in which the rowhash value for different rows is identical, making it difficult for a system to discriminate among the hash synonyms when one unique row is requested for retrieval from a set of hash synonyms. Minimizing Hash Collisions To minimize the Hash Collision problem, Teradata Database defines 4.2 billion hash values. The AMP software adds a system-generated 32-bit uniqueness value to the rowhash value. The resulting 64-bit value prefixed with an internal partition number is called the rowID. This value uniquely identifies each row in a system, making a scan to retrieve a particular row among several having the same rowhash a trivial task. A scan must check each of the rows to determine if it has the searched-for value and not another value with the same rowhash value. Hash Maps A hash map is a mechanism for determining which AMP a row is stored on, or, in the case of the Open PDE hash map, the destination AMP for a message sent by the Parser subsystem. Each cell in the map array corresponds to a hash bucket, and each hash bucket is assigned to an AMP. Hash map entries are maintained by the BYNET. ← prev next →	{"url":"https://www.javatpoint.com/teradata-hashing-algorithm","timestamp":"2024-11-07T06:56:26Z","content_type":"text/html","content_length":"39118","record_id":"<urn:uuid:32f34f7a-ee07-48ab-92a5-8a2d85ba51f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00670.warc.gz"}
Guggemos, Fabien Voici les Ã©lÃ©ments 1 - 1 sur 1 • Publication MÃ©tadonnÃ©es seulement Penalized Calibration in Survey Sampling: Design-Based Estimation Assisted by Mixed Models Calibration techniques in survey sampling, such as generalized regression estimation (GREG), were formalized in the 1990s to produce efficient estimators of linear combinations of study variables, such as totals or means. They implicitly lie on the assumption of a linear regression model between the variable of interest and some auxiliary variables in order to yield estimates with lower variance if the model is true and remaining approximately design-unbiased even if the model does not hold. We propose a new class of model-assisted estimators obtained by releasing a few calibration constraints and replacing them with a penalty term. This penalization is added to the distance criterion to minimize. By introducing the concept of penalizedcalibration, combining usual calibration and this â€˜relaxedâ€™ calibration, we are able to adjust the weight given to the available auxiliary information. We obtain a more flexible estimation procedure giving better estimates particularly when the auxiliary information is overly abundant or not fully appropriate to be completely used. Such an approach can also be seen as a design-based alternative to the estimation procedures based on the more general class of mixedmodels, presenting new prospects in some scopes of application such as inference on small domains.	{"url":"https://libra.unine.ch/entities/person/96472f02-f2d1-4e53-b6ef-b775ea63f0df/details?view=list&spc.page=1&f.author=e5769349-b8c2-4b2f-9261-40da256e00d8,authority&f.author=96472f02-f2d1-4e53-b6ef-b775ea63f0df,authority&f.has_content_in_original_bundle=false,equals&f.organization=8df2631a-a18e-4a52-b984-47dbca6b664b,authority&f.itemtype=journal%20article,equals","timestamp":"2024-11-02T07:32:19Z","content_type":"text/html","content_length":"985908","record_id":"<urn:uuid:788b2e24-44a8-4cdd-ad03-bc938eabeb8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00418.warc.gz"}
Numbers and percents Percents are a way of expressing a number as a fraction of 100. The word "percent" means "per hundred", so when you see a percent sign (%) it represents a number out of 100. For example, 50% is equivalent to the fraction 50/100, which simplifies to 1/2. To convert a percent to a decimal, you divide by 100. For example, 25% is equal to 0.25 as a decimal. To convert a fraction to a percent, you can divide the numerator by the denominator and then multiply by 100. For example, to convert 1/4 to a percent, you would divide 1 by 4 to get 0.25, and then multiply by 100 to get 25%. To calculate a percentage of a number, you can use the formula: Percentage = (Part/Whole) x 100 For example, if you want to find 20% of 150, you would use the formula: (20/100) x 150 = 0.20 x 150 = 30 So, 20% of 150 is 30. Percent Increase and Decrease To calculate a percent increase or decrease, you can use the formula: Percent Change = ((New Value - Original Value) / Original Value) x 100 If the result is positive, it's a percent increase. If it's negative, it's a percent decrease. Study Guide 1. Understand the concept of percents as a way of expressing a number out of 100. 2. Practice converting between percents, decimals, and fractions. 3. Learn how to calculate percentages using the formula (Part/Whole) x 100. 4. Practice calculating percent increase and decrease using the formula ((New Value - Original Value) / Original Value) x 100. It's important to practice various types of problems related to percents and to understand how they are used in real-life situations such as discounts, taxes, and interest rates.	{"url":"https://newpathworksheets.com/math/grade-8/numbers-and-percents","timestamp":"2024-11-02T21:40:09Z","content_type":"text/html","content_length":"44547","record_id":"<urn:uuid:8ca4c960-1b7c-4c59-9d2a-c193a6131710>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00681.warc.gz"}
Learning SciPy for Numerical and Scientific Computing, 2nd Edition Learning SciPy for Numerical and Scientific Computing, 2nd Edition Quick solutions to complex numerical problems in physics, applied mathematics, and science with SciPy SciPy is an open source Python library used to perform scientific computing. The SciPy (Scientific Python) package extends the functionality of NumPy with a substantial collection of useful algorithms. The book starts with a brief description of the SciPy libraries, followed by a chapter that is a fun and fast-paced primer on array creation, manipulation, and problem-solving. You will also learn how to use SciPy in linear algebra, which includes topics such as computation of eigenvalues and eigenvectors. Furthermore, the book is based on interesting subjects such as definition and manipulation of functions, computation of derivatives, integration, interpolation, and regression. You will also learn how to use SciPy in signal processing and how applications of SciPy can be used to collect, organize, analyze, and interpret data. Herausgeber Packt Publishing Autor(en) Sergio J. Rojas G., Erik A Christensen, Francisco J. Blanco-Silva ISBN 978-1-78398-770-2 veröffentlicht 2015 Seiten 188 Sprache English Ähnliche Bücher	{"url":"http://www.englische-fachbuecher.de/books/88498719-9781783987702-learning-scipy-for-numerical-and-scientific-computing-2nd-edition","timestamp":"2024-11-12T07:18:28Z","content_type":"text/html","content_length":"14546","record_id":"<urn:uuid:c512dcb4-dae0-4b51-91c4-22e65a1123dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00391.warc.gz"}
Tax Calculator Indiana Salary Tax Calculator for the Tax Year 2024/25 You are able to use our Indiana State Tax Calculator to calculate your total tax costs in the tax year 2024/25. Our calculator has recently been updated to include both the latest Federal Tax Rates, along with the latest State Tax Rates. It has been specially developed to provide users not only with the amount of tax they will be paying, but also with a breakdown of all the tax costs that will be incurred, taking into consideration any deductions they may be eligible to receive. Our Indiana Salary Tax Calculator has only one goal, to provide you with a transparent financial situation. By seeing how all of your taxes are split up and where each of them go, you have a better understanding of why you pay the tax you do, where the money goes, and why each tax has a purpose. Note: Keep in mind that when you are filing both your State and Federal tax returns, you should always consult with a professional. Failure to do so could result in filing these taxes wrongly, and thus landing you in trouble. While Indiana collects income tax, there are certain states that do not collect income tax. These states include: South Dakota, New Hampshire, Nevada, Washington, Florida, Alaska, Wyoming, Texas, Note: New Hampshire only taxes interest and dividend income. Washington taxes the capital gains income of high-earners. Enter your Gross Salary and click "Calculate" to see how much Tax you'll need to Pay Calculation Results: $ $ $ % Gross Yearly Income Yearly Taxes Yearly Take Home Effective Tax Rate Yearly Monthly Weekly Daily Gross Income Taxable Income Federal Tax Indiana State Tax Social Security Total FICA You Take Home Compare with previous year Using our Indiana Salary Tax Calculator To use our Indiana Salary Tax Calculator, all you need to do is enter the necessary details and click on the "Calculate" button. After a few seconds, you will be provided with a full breakdown of the tax you are paying. This breakdown will include how much income tax you are paying, state taxes, federal taxes, and many other costs. To determine the amount of tax you'll need to pay, make sure to include any deductions you are eligible for. You can choose another state to calculate both state and federal income tax here. Your deductions play a key part in filing your taxes, and while some people find it easier to not file their deductions and "just pay the extra tax", it makes a difference in the long run. For example, if you are running a small business, there are many simple things that you are able to have tax deducted from. Perhaps office supplies, repairs on your office computers, building repairs, or even purchasing your workers' suitable clothing for their job (e.g. a plumber should be wearing a boiler suit). While they are little things that not many people think about, your business thrives on these tiny details and therefore you are able to have tax deductions on them. IN Tax Calculations Click to view a salary illustration and print as required. IN Tax Calculations Click to view a salary illustration and print as required.	{"url":"https://goodcalculators.com/us-salary-tax-calculator/indiana/","timestamp":"2024-11-01T22:59:10Z","content_type":"application/xhtml+xml","content_length":"152627","record_id":"<urn:uuid:700cbb5d-10b9-4c6c-9ab6-3cf41e854e12>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00148.warc.gz"}
11,313 research outputs found This paper considers analytical issues associated with the notion of the energy release rate in quasi-static elastic crack propagation This paper describes some recent theoretical results pertaining to the experimentally-observed relation between the speed of a shock wave in a solid and the particle velocity immediately behind the shock. The new feature in the present analysis is the assumption that compressive strains are limited by a materially-determined critical value, and that the internal energy density characterizing the material is unbounded as this critical strain is approached. It is shown that, with this assumption in force, the theoretical relation between shock speed and particle velocity is consistent with many experimental observations in the sense that it is asymptotically linear for strong shocks of the kind often arising in the laboratory Rectangular nozzles are increasingly used for modern military aircraft propulsion installations, including the roll nozzles on the F-35B vertical/short take-off and landing strike fighter. A peculiar phenomenon known as axis-switching is generally observed in such non-axisymmetric nozzle flows during which the jet spreads faster along the minor axis compared to the major axis. This might affect the under-wing stores and aircraft structure. A computational fluid dynamics study was performed to understand the effects of changing the upstream nozzle geometry on a rectangular free jet. A method is proposed, involving the formulation of an equation based upon a statistical model for a rectangular nozzle with an exit aspect ratio (ARe) of 4; the variables under consideration (for a constant nozzle pressure ratio (NPR)) being inlet aspect ratio (ARi) and length of the contraction section. The jet development was characterised using two parameters: location of the cross-over point (Xc) and the difference in the jet half-velocity widths along the major and minor axes (Î B30). Based on the observed results, two statistical models were formulated for the prediction of axis-switching; the first model gives the location of the cross-over point, while the second model indicates the occurrence of axis-switching for the given configuration The wings of many insect species including crane flies and damselflies are petiolate (on stalks), with the wing planform beginning some distance away from the wing hinge, rather than at the hinge. The aerodynamic impact of flapping petiolate wings is relatively unknown, particularly on the formation of the lift-augmenting leading-edge vortex (LEV): a key flow structure exploited by many insects, birds and bats to enhance their lift coefficient. We investigated the aerodynamic implications of petiolation P using particle image velocimetry flow field measurements on an array of rectangular wings of aspect ratio 3 and petiolation values of P = 1â 3. The wings were driven using a mechanical device, the â Flapperatusâ , to produce highly repeatable insect-like kinematics. The wings maintained a constant Reynolds number of 1400 and dimensionless stroke amplitude Î * (number of chords traversed by the wingtip) of 6.5 across all test cases. Our results showed that for more petiolate wings the LEV is generally larger, stronger in circulation, and covers a greater area of the wing surface, particularly at the mid-span and inboard locations early in the wing stroke cycle. In each case, the LEV was initially arch-like in form with its outboard end terminating in a focus-sink on the wing surface, before transitioning to become continuous with the tip vortex thereafter. In the second half of the wing stroke, more petiolate wings exhibit a more detached LEV, with detachment initiating at approximately 70% and 50% span for P = 1 and 3, respectively. As a consequence, lift coefficients based on the LEV are higher in the first half of the wing stroke for petiolate wings, but more comparable in the second half. Time-averaged LEV lift coefficients show a general rise with petiolation over the range tested.This work was supported by an EPSRC Career Acceleration Fellowship to R.J.B. (EP/H004025/1) Insect wing shapes are diverse and a renowned source of inspiration for the new generation of autonomous flapping vehicles, yet the aerodynamic consequences of varying geometry is not well understood. One of the most defining and aerodynamically significant measures of wing shape is the aspect ratio, defined as the ratio of wing length (R) to mean wing chord ($\bar{c}$). We investigated the impact of aspect ratio, AR, on the induced flow field around a flapping wing using a robotic device. Rigid rectangular wings ranging from AR = 1.5 to 7.5 were flapped with insect-like kinematics in air with a constant Reynolds number (Re) of 1400, and a dimensionless stroke amplitude of $6.5\bar{c}$ (number of chords traversed by the wingtip). Pseudo-volumetric, ensemble-averaged, flow fields around the wings were captured using particle image velocimetry at 11 instances throughout simulated downstrokes. Results confirmed the presence of a high-lift, separated flow field with a leading-edge vortex (LEV), and revealed that the conical, primary LEV grows in size and strength with increasing AR. In each case, the LEV had an arch-shaped axis with its outboard end originating from a focus-sink singularity on the wing surface near the tip. LEV detachment was observed for $\mathrm{AR}\gt 1.5$ around mid-stroke at $\sim 70\%$ span, and initiated sooner over higher aspect ratio wings. At $\mathrm{AR}\gt 3$ the larger, stronger vortex persisted under the wing surface well into the next half-stroke leading to a reduction in lift. Circulatory lift attributable to the LEV increased with AR up to AR = 6. Higher aspect ratios generated proportionally less lift distally because of LEV breakdown, and also less lift closer to the wing root due to the previous LEV's continuing presence under the wing. In nature, insect wings go no higher than $\mathrm{AR}\sim 5,$ likely in part due to architectural and physiological constraints but also because of the reducing aerodynamic benefits of high AR wings The fountain flow created by two underexpanded axisymmetric, turbulent jets impinging on a ground plane was studied through the use of laser-based experimental techniques. Velocity and turbulence data were acquired in the jet and fountain flow regions using laser doppler velocimetry and particle image velocimetry. Profiles of mean and rms velocities along the jet centreline are presented for nozzle pressure ratios of two, three and four. The unsteady nature of the fountain flow was examined and the presence of large-scale coherent structures identified. A spectral analysis of the fountain flow data was performed using the Welch method. The results have relevance to ongoing studies of the fountain flow using large eddy simulation techniques Eli Sternberg, perhaps the best known scholar in the field of elasticity during most of the past half-century, died suddenly in Pasadena, California, on October 8, 1988, shortly before his seventy-first birthday A new energy estimate is given for a boundary value problem for the biharmonic equation. The result is applied to the estimation of stresses in a plane elasticity problem The structure of harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space can be described by proceeding from the following assumptions: (1) the plane boundary is free of surface traction; (2) the LaimĂ© potentials, and consequently all physical quantities, decay exponentially with distance away from the boundary. In the absence of further a priori assumptions, the resulting surface waves need be neither plane nor axially symmetric, and thus the derivation sketched here constitutes a generalization of the ones usually given in the textbook literature [e.g., Love, 1944; Ewing et al., 1957]. With reference to Cartesian coordinates Ï _1, Ï _2, Ï _3, the half-space under consideration occupies the region x_3â „0. The displacement vector u of a typical point has Cartesian components u_j, and the associated components of stress are denoted by Ï _(jk). The summation convention is used, Latin and Greek subscripts have the respective ranges 1, 2, 3 and 1, 2, and a subscript preceded by a comma indicates differentiation with respect to the corresponding coordinate	{"url":"https://core.ac.uk/search/?q=author%3A(Knowles%2C%20K.)","timestamp":"2024-11-02T05:46:00Z","content_type":"text/html","content_length":"147935","record_id":"<urn:uuid:a94d7afa-949f-4b61-8636-e07f93934bcc>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00714.warc.gz"}
A Multi-stage Representation of Cell Proliferation as a Markov Process Yates, Christian A. and Ford, Matthew J. and Mort, Richard L. (2017) A Multi-stage Representation of Cell Proliferation as a Markov Process. Bulletin of Mathematical Biology, 79 (12). pp. 2905-2928. ISSN 0092-8240 Full text not available from this repository. The stochastic simulation algorithm commonly known as Gillespie's algorithm (originally derived for modelling well-mixed systems of chemical reactions) is now used ubiquitously in the modelling of biological processes in which stochastic effects play an important role. In well-mixed scenarios at the sub-cellular level it is often reasonable to assume that times between successive reaction/ interaction events are exponentially distributed and can be appropriately modelled as a Markov process and hence simulated by the Gillespie algorithm. However, Gillespie's algorithm is routinely applied to model biological systems for which it was never intended. In particular, processes in which cell proliferation is important (e.g. embryonic development, cancer formation) should not be simulated naively using the Gillespie algorithm since the history-dependent nature of the cell cycle breaks the Markov process. The variance in experimentally measured cell cycle times is far less than in an exponential cell cycle time distribution with the same mean.Here we suggest a method of modelling the cell cycle that restores the memoryless property to the system and is therefore consistent with simulation via the Gillespie algorithm. By breaking the cell cycle into a number of independent exponentially distributed stages, we can restore the Markov property at the same time as more accurately approximating the appropriate cell cycle time distributions. The consequences of our revised mathematical model are explored analytically as far as possible. We demonstrate the importance of employing the correct cell cycle time distribution by recapitulating the results from two models incorporating cellular proliferation (one spatial and one non-spatial) and demonstrating that changing the cell cycle time distribution makes quantitative and qualitative differences to the outcome of the models. Our adaptation will allow modellers and experimentalists alike to appropriately represent cellular proliferation-vital to the accurate modelling of many biological processes-whilst still being able to take advantage of the power and efficiency of the popular Gillespie algorithm. Item Type: Journal Article Journal or Publication Title: Bulletin of Mathematical Biology Uncontrolled Keywords: ?? cell cyclemarkovian representationstochastic simulation gillespie algorithm exponentially modified erlang general agricultural and biological sciencesgeneral neurosciencegeneral biochemistry,genetics and molecular biologygeneral environmental scienceimmun ?? Deposited On: 07 Dec 2017 15:24 Last Modified: 16 Jul 2024 10:34	{"url":"https://eprints.lancs.ac.uk/id/eprint/88839/?template=browse","timestamp":"2024-11-02T21:56:37Z","content_type":"application/xhtml+xml","content_length":"25741","record_id":"<urn:uuid:63ddce53-1b0c-4431-bd30-3610ae436a02>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00147.warc.gz"}
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Design and Application of Fuzzy Logic Based Fire Monitoring and Warning Systems for Smart Buildings Department of Computer Science & IT, The Islamia University Bahawalpur, Bahawalpur 63100, Pakistan Department of Computer Science, Govt. Sadiq College Women University, Bahawalpur 63100, Pakistan Author to whom correspondence should be addressed. Submission received: 8 September 2018 / Revised: 29 October 2018 / Accepted: 31 October 2018 / Published: 9 November 2018 Typical fire monitoring and warning systems use a single smoke detector that is connected to a fire management system to give early warnings before the fire spreads out up to a damaging level. However, it is found that only smoke detector-based fire monitoring systems are not efficient and intelligent since they generate false warnings in case of a person is smoking, etc. There is need of a multi-sensor based intelligent and smart fire monitoring system that employs various parameters, such as presence of flame, temperature of the room, smoke, etc. To achieve such a smart solution, a multi-sensor solution is required that can intelligently use the data of sensors and generate true warnings for further fire control and management. This paper presents an intelligent Fire Monitoring and Warning System (FMWS) that is based on Fuzzy Logic to identify the true existence of dangerous fire and send alert to Fire Management System (FMS). This paper discusses design and application of a Fuzzy Logic Fire Monitoring and Warning System that also sends an alert message using Global System for Mobile Communication (GSM) technology. The system is based on tiny, low cost, and very small in size sensors to ensure that the solution is reproduceable. Simulation work is done in MATLAB ver. 7.1 (The MathWorks, Natick, MA, USA) and the results of the experiments are satisfactory. 1. Introduction Fire is very useful, it serves a lot of purpose for us as long as it is under control, but when it goes out of control, it can be the cause of a big disasters. Since, for household activities, like cooking, heating, or lighting, fire is a common solution however such fire-based activities result in a high mortality rate and damage of property of almost $8.6 billion per year. The available solutions of home based Fire Monitoring and Warning System (FMWS) are of two types; smoke alarm and vision based Camera systems. The first one, smoke alarm is not reliable while the second one, vision based camera systems are highly expensive. For developing countries, there is need of cheap and reproduceable solutions that can also be provided on commercial basis. There is an urgent need to design and develop an intelligent and smart Fire Monitoring and Warning System (FMES) using Fuzzy Logic [ ] that plays an important role in achieving the safe environments and buildings. To achieve the above-mentioned goals, a home based Fire Monitoring and Warning System (FMWS) is introduced in this paper that uses Arduino Uno R3 (Arduino, Somerville, TX, USA) that is economical and affordable and a set of easy-to-buy sensors. The proposed system is efficient in detecting fire during the presence of smoke and flame at a level with the particular increase in the room temperature. The proposed systems uses a fuzzy logic based system to identify the true incidents of fire that can result in a critical and dangerous situation and send alerts via GSM to the concerning person. The proposed system is efficient in detecting false-fire incidents and report only the true-fire incidents as previous solutions [ ] do not identify false-fire incidents. Here, the false-fire incidents are those incidents that is not actual fire but a fir-detecting system gives an alarm for true-fire. In the literature, researchers proposed different methods [ ] to detect and control fire, whereas these methods are fire alarm system, autonomous fire fighting Robots, Fire Monitoring and Warning System using wireless sensor networks (WSN) and Global System for Mobile Communication (GSM) technology, and videos and images based fire detection systems. Many controllers have good effects as well as negative like false alarm which can be due to any environmental changes, smoking, etc. and such false alarms result in consumption of lots of energy. Due to these reasons, a system is much needed that is less expensive and can detect fire at early stages, uses multiple sensors to detect true incidents of fire, reduce false alarm rate and reliability, which can be used in houses, buildings, and offices. The proposed system also sends alerts to the concerning people for better and in-time control and management of fire incidents. In this study, the novel idea is the use of Fuzzy Logic method for identification of true incidents of fire. Fuzzy Logic was originally proposed by Dr. Lotfi Zadeh of the University of California at Berkeley in 1960s. He gave idea of membership function and in 1964 which had become the backbone of fuzzy set theory. In 1965, [ ], the idea of fuzzy set theory and fuzzy logic technology was introduced to define membership function and rule set, which plays an important role in simulation work to construct a FMWS. The previous studies show that the existing fire detection and warning systems are mostly uni-sensor systems that only decide on the base of presence of smoke. However, smoke cannot always be a reason of a true fire incident. There is need to consider the other parameters of fire in design of an intelligent and fully automatic system for detecting incidents of true fire. To fill this research gap, a multi-sensor system is presented in this paper that uses the decision making power of fuzzy logic techniques for the purpose of decision making and implementing the system in MATLAB. The experiments show that the proposed system is accurate and reliable in identification of incidents of true fire. The rest of the paper is structured, as follows: Section 2 discusses the outcomes of the detailed literature review carried out during the research and a course of related work; Section 3 describes the materials and methods of the presented research along the used fuzzy logic system for the proposed Fire Monitoring and Warning System; Section 4 provides details of the experiments their results and discussions to show the performance testing and outcomes of the presented approach; and, Section 5 presents a conclusion of the presented research. Related Work Several years before, fuzzy control became one of the most beneficial area where the focus of the research was applications of fuzzy set theory. There were many technologies which were proposed for early fire detection for safety, such as Neural Networks, Image Processing, Video-based techniques, Fuzzy Logic, etc. Fire detection has become an important research topic for the safety of human life, many other technologies are used for the surveillance of early fire detection. Faisal et al. designed and evaluated wireless sensor network using multiple sensors for early fire detection of a house fire. The author also used GSM to avoid false alarms. The system was tested in a smart home [ ]. Hamdy et al. build “Smart Forest Fire Early Detection Sensory System (SFFEDSS)”, by combining wireless sensor networks with artificial neural networks (ANNs). In this system, low-cost sensor nodes, temperature, light, and smoke data spread out on the forest in order to get information, and this information is taken as input to ANN models that is converted into intelligence. Without any human involvement, this SFFEDSS system monitors the forest [ ]. Giovanni Laneve et al. discussed that in the Mediterranean regions, to detect fire at the early stage, author analysis the application of Spinning Enhanced Visible and Infrared Imager SEVIRI images. In this techniques, two or more images are compared with each other after the interval (15-min) [ Digvijay Singh et al. proposed a fire alarm system, by using less expensive instruments, connectivity, and wireless communication. It is real-time monitoring system, in which system detects the presence of fire as well as captures images by using the camera and display images on the screen. In this system two controllers are used, Controller 1 sends signals to GSM and Controller 2, and Controller 2 will turn ON the screen. When it finds that temperature, humidity, CO2, and fire are increased above the threshold, these outputs are taken from sensors. One or more Arduino is connected and trigger automatically [ ]. Md Iftekharul Mobin et al. proposed a System Safe from Fire (SFF) by using multi-sensor, actuators, and operated by micro-controller unit (MCU) is an intelligent self-controlled smart fire extinguisher system. Sensors placed in different areas for monitoring purpose and input signals are taken from that sensors and combine integrated fuzzy logic to detect fire breakout location and severity and discard false fire situation, such as cigarette, smoke, welding etc. SFF notifies fire services and other by text messages and telephone calls when the fire is detected [ Navyanth et al. presented a system design for fire-fighting robots, which detect the fire and reach the target area without hitting any obstacle for preventing damage lives and property using the Fuzzy Logic techniques. In another contribution, there are many ultrasonic sensors mounted on the robot, which sensed turn angle between the robot head and the target, distance of the obstacles around the robot (front, left, and right, including other mobile robots) and used as input fuzzy members. The objective of designing the fire-fighting robots is to reach at fire area zone without hitting any obstacle to prevent any damage in the unknown environment [ ]. Liu et al. describe a smart bushfire monitoring system that alerts through warning message in case of a bushfire. Global system for mobile (GSM) communication technology is used as a communication medium for sending short message service (SMS). The temperature and humidity sensors are connected to a microcontroller composed of a device. To report module position, Microcontroller interfaces with a GPS as well as GSM module to communicate the sensory information. Sent message through GSM is received by other mobile phones for further processing [ Vikshant et al. presented a work in the detection of forest fire using wireless sensor networks (WSNs) [ ] with the idea of implementing Fuzzy Logic. Multiple sensors are used for detecting the probability of fire and fire direction as well. Information that is collected from different sensors will be passed on the cluster head using Event Detection Mechanism. Multiple sensors temperature, Humidity, light, and CO density are used to detect fire probability and fire direction and these all sensors are mounted in each nod in order to improve accuracy and reduce false alarm rate as well [ ]. Muralidharan et al. develop a simple way of detecting fire using multi-sensor instead of the single sensor with the use of Artificial Intelligence technology Fuzzy Logic. Results were presented in MATLAB. The developed system was simpler work in a human way of thinking and closely related to the real model concept [ ]. Mirjana et al. developed a system with the internet of things (IoT) concept for making right decision according to the situation for monitoring and determining fire confidence and reduce the number of rules by doing so sensor activities also reduced and extend battery life time as well as improve efficiency [ Robert et al. describe a system for detecting fire in automobiles while using Arduino microcontroller and Artificial Intelligence technology Fuzzy Logic in order to avoid any damages in an automobile due to fire. Temperature sensors, flame sensors, and smoke sensors are used, when the fire is detected then sound alarm and provide carbon dioxide to location. A system is implemented and tested on a medium sized physical car and 2 kg Cylinder is mounted behind the rear seats of the passenger [ ]. Hongliang et al. proposed a multi-features fusion (MFF) algorithm with self-adaptive, self-learning, and fault tolerance, the probability of burning fire is detected by taking flame’s twinkling frequency and its dynamic contour into account. For detecting fire areas, first dynamic video of fire picked up, detect fire area, and build a quantified distinguishing rule using disperse to the eigenvector. In complex situations, the MFF algorithm has an ability to detect white spots from moving and static objects, with a spotlight, automobile light, and illumination changing [ Harjinder et al. describe a solution of an early detection of the forest fire with the wireless sensors. Data is collected from sensors using Arduino development board and it is sent to the base station wirelessly after detecting fire also alert message is sent using GSM module [ ]. Turgay et al. present a model for detecting fire and smoke without using any sensor, this approach is based on image processing. The system is designed in which color information with motion analysis are combined by using the extracted model [ Wang et al. describe the idea of automatic fire alarm and fire control linkage system in the intelligent building. The system predicts fire intelligently, it controls gas, automatic fire alarm, and linkage function as well [ ]. Viktor et al. describes the architecture for fire detection using social media networks. Author said that as social network and services are increasing there has been large amount of information sharing on the internet. Author proposed architecture of wildfire social sensor (WSS) platform. In result of this work, a human-centric sensor network can be provided by social media for the early detection of disasters like fire [ 2. Materials and Methods In this paper, the fuzzy control algorithm is used to detect fire in the residential area which is placed where it is needed and used to detect any abnormal behavior. Whenever any abnormal behavior is detected, FMWS are used to set an alarm and other control mechanisms. Rule-based fuzzy logic is implemented in this approach on data that is collected from different sensors. In FMWS, MATLAB Fuzzy Logic Toolbox is used for simulation work with more accuracy, flexibility, and scalability with other systems. To process the Fuzzy Logic in this research Fuzzy Rule, Fuzzy Inference System, and Defuzzification process are involved. 2.1. Nomenclature of Proposed Fuzzy Logic Control In the implementation of fuzzy logic control, the fuzzification component, the inference rules component and the defuzzification component are implemented and each component have different methods, such as singleton fuzzification, min-max inference, and centroid defuzzification methods with different characteristics. In general, fuzzy logic control method of each components can be represented by the number of operations and memory requirements. In Fuzzy Logic Control, program is implemented in the C language in Arduino IDE (Integrated Development Environment), which is integrated with Arduino UNO microcontroller and model is ATmega328P. Basic operation that is, addition, subtraction, and division is used to count number of operations. Table 1 presents the symbols which is used in this paper. 2.2. Architecture of the Proposed Fuzzy Logic System The general architecture of a proposed FLS is shown in Figure 1 . In a broader sense, the fuzzy logic refers to a fuzzy set [ ]. Typically, a fuzzy set is a pair ( ) where universe of discourse is a set and characterized by a membership function where takes the value in the interval [0, 1], i.e., → [0, 1]. Whereas, the Universe of Discourse and the membership function plays a vital role in this approach. The fuzzy set can be represented as a set of ordered pairs of elements and its degree of membership function, as described in [ ] and the same is shown in Equation (1): The probability that belongs to is the membership function ). Here, three parameters (three fuzzy subset of S (A, B, and C)) are used in the proposed approach. In Equations (2) and (3), the m (x) is the degree of membership of x in A, mB (x), the degree of membership function of x in B and m (x) is the degree of membership of x in C. Here, a variable A represents , variable B represents and Variable C represents . Considering this, the fuzzy union set and fuzzy intersection set are defined as: A ∪ B ∪ C = { x, max(m[A](x), m[B](x), m[C](x))\|x is an element of S} A ∩ B ∩ C = { x, min(m[A](x), m[B](x), m[C](x))\|x is an element of S} The Equations (2) and (3) are implemented in this approach to compute rules strength. In Equation (2), the union of three fuzzy subsets is computed by using fuzzy operator “or” by combining the fuzzified inputs using fuzzy combination. Similarly, in Equation (3), to compute the intersection of three subsets, the fuzzy “and” operator is used by clipping the output membership function at the rule strength. According to the proposed work, Equation (3) is implemented with three inputs because in this approach, we need to obtain rule strength by combining the fuzzified inputs, and it clips the output membership function at the rule strength. Typically, a fuzzy set describes the values of linguistic variables as quantitatively and qualitatively. A membership functions defines that each point is mapped to a membership value between 0 and 1 in the defined Universe of Discourse. In the proposed FMWS, the fuzzy rules are implemented using if-then rules, which are used to express a piece of knowledge. The fuzzy rules are most widely and commonly used for interpretation also used in this approach, like “if x is A and y is B then z is C” For the collection of fuzzy rules, “if x is A[i] and y is B[i] then z is C[i]”, where i = 1....n, The proposed fuzzy logic expert system in this paper consists of the four basic components: fuzzification, inference, knowledge base, rules, and defuzzification. Every component is involved in the decision making process by the proposed FMWS. Fuzzy logic system with its components is shown in Figure 1 and working of the proposed fuzzy logic system is described in Algorithm 1 that is based on [ ]. This algorithm implements the fuzzy logic system in the proposed Fire Management and Warning System (FMWS). Algorithm 1 Proposed Algorithm for proposed Fuzzy Logic System 1. Define: Linguistic variables and terms (initialization) 2. Construct: Membership Functions (initialization) 3. Construct: Rule base (initialization) 4. Construct: Crisp input data to fuzzy values using Membership Function (fuzzification) 5. Evaluate: Rules in the rule base (inference) 6. Combine: Results of each rule (inference) 7. Convert: Output data to non-fuzzy values (defuzzification) According to the Algorithm 1, a set of input data was collected from sensors and using data variables, fuzzy linguistic terms, and membership function set is converted to a fuzzy set and this process is called as fuzzification. Afterwards, using a set of rules an inference is made. In the last, step which is called as defuzzifictaion, fuzzy output is mapped to an output using the membership functions. The description of each components of the proposed fuzzy logic systems is given below: 2.2.1. Used Fuzzification Method The fuzzification is the first phase in the proposed fuzzy logic based FMWS. In this fuzzification process, the crisp values are converted into fuzzy set using a set of membership functions that are described in next section. Each membership function represents quality of the sensor variables being fuzzified, e.g., the membership function of a variable “Change rate of Temperature” and “Change rate of Humidity” are valued as “Low”, “Mid”, and “High”, whereas the membership function of variable “Time” is “short” and “long”, The fuzzified set can be described, as shown in Equation (4): Ã = µ[1] K(x[1]) + µ[2] K(x[2]) +...... + µ[n] K(x[n]) In this relation, the fuzzy set K(xi) is called as kernel of fuzzification. To implement this method, µi constant and xi is being transformed to a fuzzy set K(xi). In the proposed work, the Equation (4) is used in the process of fuzzification where the universe of discourse and the membership functions are implemented. 2.2.2. Used Membership Function In the proposed fuzzy logic based FMWS, a membership function is defined as µ :X [0, 1] for a fuzzy set A and X is a universe of discourse, and each element of X is mapped to a degree of membership function between 0 to 1. In the proposed approach, a fuzzy set can be represented graphically using a membership function. The -axis represents universe of discourse and -axis represents the degree of membership function from 0 to 1. In the proposed approach, the triangular membership function is used, and the details of the triangular membership function. The reason of using the triangular membership function in the proposed approach is that it is simple to implement in MATLAB and it provides accurate results. It is available with the function name “trimf” in MATLAB. The triangular membership function [ ] can be defined by a lower limit a, an upper limit b and a value mean m, where a < m < b, as shown in Figure 2 and Equation (5). $μ A ( X ) = { 0 , ( x − a ) / ( m − a ) , ( b − x ) / ( b − m ) , 0 , x ≤ a a < x ≤ m m < x < b x ≥ b$ There are three parameters in -axis (a, m, b), as shown in Figure 2 that represents lower boundary, peak, and upper boundary of a membership function, -axis represents the degree of membership function. Alternative expression for the preceding equation can be written using min and max operations, as shown in Equation (6): $triangle ( x ; a , b , c ) = max ( min ( x − a b − c , c − x c − b ) )$ The Equation (6) shows that there are three parameters (a, b, c) with (a < b < c) that determines coordinates of the three corners of the underlying triangular membership function. The Equations (5) and (6) are used to make membership functions for all four variables (three inputs and one output) of the used fuzzy logic system. The membership function is designed in this approach using above equations for each input linguistic variables, which are “change rate of temperature”, “change rate of humidity”, “time” and output linguistic variable “fire-chances”. The used membership functions are defined in MATLAB Fuzzy Logic toolbox, as explained in Section 3 2.2.3. Used Fuzzy Inference System In the architecture of the proposed FMWS, a fuzzy inference system is used that is based on a set of if-then rules, a set of membership functions and fuzzy logic operators, such as “and”, “or”. In the proposed approach, the fuzzy inference is mapped from an input to an output using fuzzy logic. The fuzzy inference has three types, such as Mamdani fuzzy inference system, Sugeno fuzzy inference system, and Tsukamoto fuzzy inference system. In the proposed approach, Mamdani and Sugeno fuzzy inference system [ ] is used that was originally introduced by WeEbrahim Mamdani in 1975. The used Mamdani fuzzy inference system is implemented in the following six steps: • Fuzzification of input using membership function • According to fuzzy set theory the fuzzified inputs are combines • Built fuzzy rules • Finding outcomes of the rules by combining rule strength and output membership function • Get an output distribution by combining the outcomes • Defuzzification of the output membership function IF (X[1] is A^1[1] AND y[1] is A^1[2] AND z[1] is A^1[3]) THEN (w is B^1) IF (X[2] is A^2[1] AND y[2] is A^2[2] AND z[2] is A^2[3]) THEN (w is B^2) Figure 3 , a detailed process of Mamdani FIS [ ] using three inputs and two rules is illustrated and defined in Equations (7) and (8), where the input x , y , and z is fuzzy, not crisp. The three inputs are fuzzified by applying intersection on crisp input value with the input membership function. For combining, the three fuzzified inputs to obtain rule strength by using operator “and”. The used Mamdani inference system uses a membership function for each rule and then according to the condition of the rule reaches to a conclusion. 2.2.4. Used Defuzzification Method This is a last step in the implementation of FMWS whereas the output is evaluated from a rule set that is written in the form of if-then statements and is saved in a knowledge base of the system. Here, a scalar value of fuzzy system is fuzzified, rules are applied, and each rule generates a fuzzy output and converts it to a scalar quantity. The two typically used defuzzification methods are the Centroid Defuzzification method and Weighted Average Defuzzification method This step is done by MATLAB fuzzy logic tool box and centroid method is implemented in this research and its details are described below: Centroid Defuzzification Method: In this approach centroid defuzzification method is used because it gives an accurate result which is originally developed by Sugeno in 1985. This technique is expressed mathematically in Equation (9) as $x * = ∫ µ i ( x ) dx ∫ µ i ( x ) dx$ where x* is the defuzzified output, µ (x) aggregated membership function, and is the output variable in Equation (9). It is defined graphically in Figure 4 -axis represents the output variable x, -axis represents µ (x) aggregated membership function and x* is the defuzzified output. The shape of output is obtained by applying fuzzy operator “and” by clipping the output membership function according to the rule 3. Implementation of Fire Monitoring and Warning System (FMWS) The proposed FMWS system is implemented with less-expensive and tiny sensors to detect true fire incidents. Input data is taken from different sensors, like temperature, humidity, and flame sensors. After collecting data from sensors, the first step is to check flame’s presence using flame sensor and if there is a flame found, then the decision systems goes to next step for further processing otherwise the system goes back to initial phase. In next step, a rule-based logic is applied on data to detect fire incidents and check its intensity. If the fire intensity is low then just an alert is generated to the people or management in order to take a particular action to control this respective kind of situation. However, if fire intensity is medium, then a fire control-mechanism is applied but the systems outputs a ‘medium’ signal, alarm sounds and turn off gas and turns on the water shower. However, if fire intensity is high then a solution similar to ‘medium’ mechanism is applied, but the level of application will be in high mode. The workflow of the proposed FMWS is shown in Figure 5 Figure 5 , the incidents of true fire are identified by using a set of variables, such as temperature, humidity values, time-span, fire-presence, etc. and with respect to the identification, a respective action is taken. However, when situation become under control then system stops the control mechanism and moves back to the normal behaviour. The following sections explain the implementation with used hardware and software: 3.1. Used Hardware in Proposed FMWS (Fire Monitoring and Warning System) In proposed FMWS, a temperature sensor DHT22 (Aosong Electronics, Shenzhen, China) is used that performs dual action by measuring both temperature and humidity, while a flame sensor is used to detect flame presence. These two sensors are attached with Arduino UNO microcontroller. The properties and working of sensors and microcontroller are discussed below: Arduino UNO microcontroller: To implements the proposed FMWS, an ATmega328P microcontroller (see Figure 6 ) is used due to its special features, such as analog input, PWM output, etc. To access sensors data, the sensors were attached with microcontroller using USB cable. Table 2 Shows the key properties of microcontroller Ardunio UNO. The microcontroller consists of six analog input pins from A0 to A5 and 14 digital pins, out of which six provides PWM output. When the microcontroller is connected to computer using USB port, it appears on PC as virtual com port to software. In our implementation, the textual data is sent to and from Arduino board using serial monitor. Arduino IDE is used in this approach to program the Arduino UNO board. Temperature and Humidity Sensor (DHT22): To detect presence of fire in the surroundings, the DHT22 (see Figure 7 ) was used as one of the sensors that gives two measurements, such as temperature and humidity, and these two values play an important role in the implementation of FMWS. After digitization, through serial port data is transferred to microcontroller. DHT22 temperature sensor gives temperature value in °C and humidity value in %. The reason of using DHT22 in this approach is because of measuring two inputs using single sensor. DHT22 is attached with microcontroller with three pins, which are +ve, data and −ve pins. Table 3 , the properties of DHT22 sensor are described. DHT22 measures temperature between −40 °C to 80 °C and 0 to 100% humidity. The model AM2302 is suitable for this type of systems, as it has small size, low energy-consumption, and long transformation distance (100 m). In the proposed work, DHT22 sensor is configured by using jumper wires with Arduino microcontroller. To code and configure DHT22 with Arduino IDE, the DHT22 library is used. Flame sensor: In the implementation of FMWS, a flame sensor is also used as shown in Figure 8 . The purpose of using flame sensor in proposed work, to check the presence of flame. It detects fire within short range, its detection range is 3 feet and angle is 60 degree. In the implementation, the flame sensor is attached to microcontroller. The flame sensor VCC pin is connected to the Ardunio 5 V pin, GND pin of module is connected to Arduino GND pin, and module output pin is connected to A0 pin of Arduino using jumper wires. After configuring the flame sensor with Arduino, a program is written in Arduino IDE by altering the code of online code available for the flame-sensor. The properties of flame sensor are given in Table 4 3.2. Coding in Arduino IDE (Integrated Development Environment) Once the hardware is configured with Arduino UNO microcontroller, the code is written on Arduino IDE that is available. However, the code was customized a bit to get desired format of the sensor’s data. Once the sensors’ data is obtained and the Universe of Discourse is defined that is necessary to perform simulation work in MATLAB. In implementation, only one DHT22 library is used that is also available online. The program is written using C language in Arduino IDE and then uploaded on board. The used IDE is shown in Figure 9 In the implementation, Arduino 1.8.5 version is used for hardware configuration and programming. Once the code is written in Arduino IDE using C language, it is uploaded into Arduino IDE and output is achieved on serial monitor, as shown in Figure 9 . Here, the data is collected from each sensor in an Excel sheet, and for integration between Arduino IDE and Excel sheet, another piece of code is written in Arduino IDE to access data from sensors. 3.3. Recording Sensor’s Data in Experiments The simulation work is done in MATLAB using Fuzzy Logic toolbox is used for proposed FMWS to record readings of the sensors for real time situations. To test the proposed FMWS, a set of different scenarios were chosen, such as scenarios with no fire, scenarios with medium fire, and scenarios with high fire. Here, the actual data is gathered from sensors (Temperature, Humidity, and Flame) to find Universe of Discourse for simulation. For all different scenarios, the data is collected in an Excel sheet directly from sensors when fire is detected. This data helps in finding the Universe of Discourse. An extract of the collected data is shown in Table 5 Table 5 , the shown data is collected from sensors to find universe of discourse for simulation work in MATLAB for the proposed FMWS. The shown data is taken by two experiments. Here, the first column represents time interval of fire to occur and second column represent the change rate of temperature (C-R-Temp) that shows how much the temperature is affected and when the incident of true fire is detected. Column three represent the change rate of humidity (C-R-Humidity) that shows how much the humidity is affected when the fire is detected. The proposed system was tested with more than fifty (50) experiments with at least 500 values, however two of them are described in Table 5 . In Experiment 1, first iteration shows that the change rate of temperature and the change rate of humidity is 0 and at this time no fire is detected, but in second iteration, the change rate of temperature is 3 which means that temperature is changed by 3 °C and humidity is changed by 3% within 2 min. In the third iteration, the value of change rate of temperature is 3.5, which means that temperature is changed by 3.5 °C and humidity is changed by −2.9% within 3.4 min. In 4th iteration, the value of change rate of temperature is 6.4, which means that temperature is changed by 36.4 °C and humidity is changed by −10.6% within 4 min. In experiment 2, in first iteration, the value of change rate of temperature is 0.7, which means temperature is changed by 0.7 °C and humidity is changed by 0.6% within 1.5 min. The value of change rate of temperature and change rate of humidity is changed according to the fire intensity. So, we conclude that the value of change rate of temperature and change rate of humidity is changed around 2 °C within 2 to 3 min, and 3 °C to 5 °C within 4 min, etc. So, according to these experiments, we find the Universe of Discourse illustrated Table 6 Table 6 , the first column represents the linguistic variable of each membership function, the second column represents the Universe of Discourse for input variable C-R-Temp for each variable. For example, if C-R-Temp is 4 it means change rate of temp lies between range of 2–5 and the change rate is medium and if C-R-humidity is 10 then it lies between range of 5–12 and C-R-humidity is also medium and if time interval of fire detection is 2 min, which is short time it lies in 0–4 and after that rules are applied then computed fire chances. The third column represents the Universe of Discourse for input variable C-R-humidity. The fourth column represents the Universe of Discourse for output Fire-chances, which represents the chances of fire as output, and according to it, fire-control mechanism will be activated. The fifth column represents the Universe of Discourse for Time, which represents the time interval at which the fire is detected. After data collection and the universe of discourse selection, the next step is to configuration of MATLAB for the simulation of the proposed FMWS. 3.4. Configuring MATLAB Implementation with Universe of Discourse In the proposed FMWS, once the MATLAB simulation is ready, it is configured with the Universe of Discourse. In MATLAB, a fuzzy controller block is available to simulate a fuzzy logic control system [ ]. Here, the values in the input vector and its respective fuzzy rules are defined, the values are assigned to the output vector that are interpreted using fuzzy inference method. A set of editors and viewers are used for simulation work of proposed FMWS in which a rules set is built, the membership functions are defined and then analyzed the behavior of a fuzzy inference system in fuzzy logic toolbox, as shown in Figure 10 . All of these editors and viewers are implemented in MATLAB for FMWS and simulation work of this approach is described in this section. We used MATLAB 2014b x64 version in this approach for simulation work. In MATLAB Fuzzy Logic Toolbox is opened by writing “fuzzy” in command window. Afterwards, we entered the names of input and output variables. Figure 10 , three inputs C-R-Temp, C-R-humidity, and Time, which are shown on left side and only one output Fire-chances on right side are added for simulation work of Fire Monitoring and Warning System 3.4.1. Implementing Fuzzification in MATLAB In fuzzification, a crisp value is converted into fuzzy linguistic variables using membership functions. In this paper, four inputs are used the change rate of temperature, change rate of humidity, time, and flame, and one output is evaluated, which is fire-chances. Since the Flame- presence variable remains constant because we just use it to check flame is present or not, then the rules are applied after confirming flame’s presence. Further, three inputs, such as the change rate of temp and humidity and time are used in simulation work for the evaluation of fire-chances to be detected by the FMWS. Fuzzification of the input and output variables are presented below: Membership Function of Change Rate of Temp (C-R-temp): Figure 11 , three linguistic variables of C-R-Temp Low, Mid, and High are used. -axis represents the value of change rate of temp from 0 to 10 °C and -axis represents the degree of membership from 0 to 1. The range of linguistic variables Low is from 0 to 2 °C, Mid is from 2 to 5 °C, and High is from 5 to 10 °C according to our defined Universe of Discourse of each variable in Table 5 Membership Function of Change Rate of humidity (C-R-humidity): Figure 12 , three linguistic variables of C-R-humidity is Low, Mid and High are used and -axis represent the value of change rate of humidity from 0 to 20% and -axis represent the degree of membership from 0 to 1. The range of linguistic variable Low is from 0 to 5%, Mid is from 5 to 12%, and High is from 12 to 20% according to the Universe of Discourse of each variable, which we defined in Table 5 Membership Function of Time: Figure 13 three linguistic variables for Time is Short and Long are used and -axis represent the value of Time from 0 to 10 min and -axis represent the degree of membership from 0 to 1. The range of linguistic variable Time Short is from 0 to 4 and Long is from 4 to 10 min according to our defined Universe of Discourse of each variable in Table 5 Membership Function of Output Fire-chances: Figure 14 three linguistic variables of output Fire-chances Low, Mid and High are used and axis represent the value of Fire-chances from 0 to 100% and axis represent the degree of membership from 0 to 1. The range of linguistic variable Low is from 0 to 30%, Mid is from 30 to 60%, and High is from 60 to 100%, according to our defined universe of discourse of each variable in Table 5 3.4.2. Defining Fuzzy Rules in MATLAB Rules Editor Once the membership functions are designed, the fuzzy rules are defined using simple IF-Then rules, which are used to express piece of knowledge in MATLAB rules editor. The fuzzy rules are most widely and commonly used for interpretation. Rules for FMWS (according to flame presence): The defined rules are applied when fire is detected otherwise FMWS again start sensing surroundings until the fire is not detected. Table 7 flame is used as for Boolean value (Yes, No) to check the flame presence. It is because when the fire is detected, it means there must be flame, second column is “Go to Start” when flame is present it will go back until flame remains present, in column three “Go to next step” means when flame is present it will go to next step for further procedure for evaluating fire chances by using other inputs variables, otherwise will never go to next step until flame is not present. In the proposed work, we applied three and two linguistic variables consisting of three linguistic values each thereby resulting in 3 × 3 × 2 = 18 rules. Using these rules a controller system was designed. Rules for FMWS (when input time is short): Table 8 , the rule set is formulated for short time according to two inputs C-R-Temp and C-R-Humidity by using variables Low, Mid, and High. Which means when C-R-Temp and C-R-Humidity values changed in short Rules for FMWS (when input time is long): Table 9 is formulated similar to Table 8 , but the only difference between them that in Table 9 , rule set is applied when time is Long. Apply solution mechanisms according to outcomes: Table 10 , represents the solution mechanism according to outcome (Fire chances) for FMWS. If the outcome is Low, then FMWS alerts the people, if outcome is Mid then alarm is activated, water showers are turned ON but with High intensity Gas is turned Off, if outcome is High then alarm is activated, water showers are turned ON and Gas will turned Off until situation turns to under control. Add rules in MATLAB using Rules Editor: Figure 15 represents MATLAB rule editor in which rule sets are added for FMWS. First box represents the rule set, second left box shows C-R-Temp variables, third represents C-R-humidity variables, and fourth box represents time variables, these sides of box represents the input variables and in right side box represents Fire-chances, which we used for output. 4. Results and Discussion A set of experiments were conducted to test the performance of the proposed FMWS. The performance of the FMWS is defined with respect to the accuracy of flame detection. Here, the accuracy of flame detection is represented by the Fire-chances in the experiments discussed below. To test the performance of proposed FMWS, a set of twelve experiments was performed, as shown in Table 11 . Out of these twelve experiments, the visual representation of three experiment cases is given in Figure 16 Figure 17 Figure 18 as experiments with medium, lowest, and highest chances of fire, respectively. In the experiments discussed below, the fire-chance is calculated from 1 to 100 in terms of ratio to chances of true fire. The score of fire-chances in our experiments had a range from 8 to 83. At this point, FMWS is ready to accept input values and generate output value according to rule set. 4.1. Results of MATLAB Implementation Figure 16 , the result of experiment 1 are shown in rule viewer. First column of rule viewer represents the change rate of temperature (C-R-Temp), second column represents the change rate of humidity (C-R-humidity), and third column represents the time, these three columns represented input values. The fourth column is for output, which represents the chances of Fire (how much chances of fire to occur). In Figure 16 , the value of C-R-Temp is 5, C-R-humidity is 10 and Time is 5. The output of this experiment is 50, which is shown in the last column that means according to input values the fire-chances is 50% o, according to rules there is urgent need of activating fire control mechanisms discussed above. Figure 17 , the results of experiment 2 in proposed work is shown in rule viewer. First column of rule viewer represents the change rate of temperature (C-R-Temp), second column represents the change rate of humidity (C-R-humidity) and third column represents the time, these three columns represented input values. The fourth column is for output which represents the chances of Fire (how much chances of fire to occur). In Figure 17 the value of C-R-Temp is 1.52, C-R-humidity is 7.83 and Time is 2.53. The output of this experiment is 13 which are shown in last column which means according to input values the fire- chances is 13, there are 13% chances of fire to occur, situation is normal and according to rules there is need to alert people using alarm or SMS. Figure 18 represents the experiment result in rules viewer for FMWS, in which the first three columns represent input values and fourth represents output values, which is evaluated by input values and according to rule set. First column represents C-R-Temp, which means how much temperature is changed and value is set by 6.33. Second column represents C-R-humidity, which means how much humidity is changed within defined time interval and in value is set by 13.9. Third column show time which represents time interval and the value in this is 3.07. Fourth column represents output value of Fire-chances which is evaluated by input values. The input values in rule viewer can set manually in fuzzy logic toolbox to check rule set is working according to requirements and system accuracy. At this point, the construction of the FMWS is complete, because inference and defuzzification are built in function in the software, which is done by MATLAB. FMWS is ready to accept input values; we entered input values (6.33, 13.9, 3.03), as shown in Figure 18 the input values are related to fuzzy sets and decision rules are applied. The fuzzy results of the output variable are composed and defuzzified using Centroid Defuzzification method which are done by MATLAB. In Figure 18 the output is 82.4 which is High chances of fire. All of these steps are implemented by using MATLAB fuzzy logic toolbox based on fuzzy logic rules and operations. Figure 19 represents FMWS results graphically in surface viewer. -axis represents input value of C-R-Temp, -axis represents input value of C-R-humidity, and -axis represents the output value of Fire-chances. It is clearly shown in Figure 19 that C-R-temp and C-R-humidity gets higher value over fire regions and lower values over non-fire region. After experiments, all the results are collected, now in this section results are discussed, and all experiment results are illustrated in Table 11 Results are shown in Table 11 . In experiment 1, the change rate of temp is 1.51, change rate of humidity is 7.83 and time is 2.53, and the chance of fire is 8.4%, which is low and actual case also Low, which means that the accuracy is 100%. In experiment 2, change rate of temp is 5, change rate of humidity is 10 and time is 5 and the chance of fire is 50% which mean there are 50% chances of fire to occur and actual case also Mid which means the accuracy is 100%. In experiment 3, change rate of temp is 3.67, change rate of humidity is 9.28, and time is 2.53, and chance of fire is 40 and actual case also Mid, which means in this case that accuracy is 100%. In experiment 4, change rate of temp is 7.41, change rate of humidity is 12.9, and time is 1.99 and chance of fire is 61.2, which is high and actual case also High which means the accuracy is 100%. But in exp 5, change rate of temp is 7.89, change rate of humidity is 14.3 and time is 8.3 and the chance of fire is 82.9, which is high chances of fire and actual case also High, which means that the accuracy is 100%. In experiment 6 to 12 few of them are High and few are Mid chances of fire. As it can be observed from Table 11 the accuracy of proposed FMWS in most of experiments is 100%, which means that the system is working according to our defined rules for FMWS. The overall FMWS is calculated below: $FMWS accuracy = ∑ µ ( ai ) n$ In Equation (10), we calculated the FMWS accuracy where µ(ai) represents the accuracy percentage of each experiment and n represents the total number of experiments. According to experiments we achieved accuracy 95.83%. 4.2. Computational Complexity of Proposed Approach The computational complexity of the proposed system is examined when the control enters in loop and run in controller hardware then inference speed of fuzzy logic control can be observed. In Table 12 , the clock speed of CPU is described, which is observed when controller is running and instruction cycles represent the number of instruction which is executed. 4.3. Contribution to Knowledge The comparison between existing work and proposed work is described in Table 13 that three existing works are compared to our proposed work. The feature, which is the base of this comparison, are multiple-sensors, alarm, decision on two authentications, and false alarm. The proposed work is novel and important in multiple ways. A comparison between the presented work and previous work is shown in Table 13 . First novelty is that existing work of Tan et al., Yunus et al. and Son B. et al., have many short comings in their work, such as the use of one or two sensors, which gives less accuracy, while the proposed work employs a multi sensor solution and it uses four inputs to provide more accurate and reliable results. Second, there was no alarm system to alert people but in proposed work alarm activation is used. Third, in existing work, the decision is not verified by two authentications but in proposed work multiple sensors are used as the flame is detected initially and then change rate of temp and humidity is calculated. Fourth, the major shortcoming in the previous work was false alarm, sometimes temperature is changed according to environmental changes and the system notifies fire detection, which consume lot of energy consumption and create hustle and disturbance, but in proposed work, it never produces false alarm. The results of the experiments are accurate up to 95.83%, as shown in Table 11 , which outperforms the similar work in the domain. 4.4. Limitation of the Proposed Approach The proposed system is designed to detect fire at early stage, it is very effective for this purpose, which is found in some previous solution, and it also has many advantages as well as few disadvantages. The advantage and disadvantages of proposed solution with traditional solution are described in Table 14 Table 15 5. Conclusions In this research paper, the fuzzy logic based Fire Monitoring and Warning System (FMWS) is presented to save lives and property damages. The objective of this paper is to detect true-fire incidents at early stages, and alert people and extinguish fire as soon as possible. Researchers proposed different methods to monitor and detect fire incidents. In this paper, fuzzy logic is used as one of the latest technology in which it supported execution requirements very easily. Multiple sensors (Temperature sensor, flame sensor) are used to get the accurate results to reduce false alarm rate. Four parameters are used as input, such as the change rate of temperature, the change rate of humidity, presence of flame, and time. The purpose of using change rate is to make the system more general rather than specific because different countries have different temperatures, increase in temperature in short time shows that there is something wrong in environment. The system will alert people if any unwanted situation occurs anywhere. The output ‘chances of fire’ is achieved after applying rules when a fire is detected somewhere, then, according to situation alternative control mechanism, can be activated like water showers, etc. The proposed work also discards the false alarm rate. Simulation work is done in MATLAB Fuzzy Logic toolbox and satisfactory results are discussed in this paper as well. Author Contributions B.S. is the main author of this paper and she has contributed in investigation of the problem, research design, experiments design and writing the original draft. I.S.B. has supervised this research work and contributed in writing, review and editing this paper. S.R. has contributed in experiments and editing of this paper. B.R. contributed in implementation and coding of this research. M.K. contributed in MATLAB Coding and experiments of this research. This research received no external funding. Conflicts of Interest The authors declare no conflict of interest. Symbol Description K Kernal of Fuzzification x The number of input fuzzy sets µ The number of inputs R The number of rules X The number of output fuzzy sets x* Defuzzified output C-R-Temp The change rate of temperature C-R-humidity The change rate of humidity Model ATmega328P Operating Voltage 5 V Input Voltage 6–20 V Analog Input Pins 6 (A0–A5) Digital I/O Pins 14 (out of which 6 provide PWM output) DC Current on I/O Pins 40 mA DC Current on 3.3 V Pin 50 mA Flash memory 32 KB SRAM 2 KB EEPROM 1 KB Clock Speed 16 MHz Model AM2302 Power supply 3.3–6 V DC Output Signal Digital Signal via 1-wire bus Sensing element Polymer capacitor Operating range Humidity 0–100%RH, temperature −40~80 Celsius Accuracy Humidity ±2% RH (Max ±5% RH), Temperature < ±0.5 Celsius Resolution of sensitivity Humidity 0.1% RH; Temperature 0.1 Celsius Repeatability Humidity ±1% RH; Temperature ±0.2 Celsius Humidity hystersis ±0.3% RH Long-term stability ±0.5% RH/year Sensing period Average: 2 s Interchange ability Fully interchangeable Dimensions Small size 14,185.5 mm, Big size 22,285 mm Positive supply +5 V Output Digital and Analog Flame wavelength 760 nm to 1100 nm Power supply 3.3–5.5 V DC, DO, high/low electric level signal output Detection angle range about 60 degrees Sr. No. Experiment 1 Experiment 2 Time Interval (minutes) C-R-Temp (°C) C-R-Humidity (%) Time Interval (minutes) C-R-Temp (°C) C-R-Humidity (%) 1 2 0 0 1.5 0.7 0.6 2 2 3 −2.4 1.10 2 1.8 3 3.4 3.5 −2.9 39 s 1 2 4 4 6.4 −10.6 2.38 2 1.5 5 2 3 6.1 2.12 2 3 6 2.3 3.6 9 1.56 3 5 7 1.2 2 5 2 3 8 8 2.35 4 6 2 3.2 6 Variable Universe of Discourse for C-R-Temp (°C) Universe of Discourse for C-R-Humidity Universe of Discourse for Fire-Chances Universe of Discourse for Time Low 0–2 0–7 0–30 - Mid 2–5 5–12 30–60 - High 5–10 12–20 60–100 - Short (Time) - - - 0–4 Long (Time) - - - 4–10 Flame Go to Start Go to Next Step Flame is present No Yes Flame is not present Yes No C-R-Humidity C-R-Temp Low Mid High Low Low Mid High Mid Low Mid High High Mid Mid High C-R-Humidity C-R-Temp Low Mid High Low Low Low Mid Mid Low Low Mid High Low High High Outcome Solution Low Alert Mid Alarm/Water Shower(Low)/Off Gas High Alarm/Water Shower(High)/Off Gas Sr. No. C-R-Temp (°C) Presence of Flame C-R-Humidity (% age) Time (minutes) Chances of True Fire Test Case Actual Case Accuracy (% age) (% age) 1 1.51 1 7.83 2.53 8.4 Low Low 100% 2 5 1 10 5 50 Mid Mid 100% 3 3.67 1 9.28 2.53 40 Mid Mid 100% 4 7.41 1 12.9 1.99 61.2 High High 100% 5 7.89 1 14.3 8.3 82.9 High High 100% 6 1.87 1 3.01 1.39 14.3 Low Low 100% 7 2.71 1 5.18 2.47 45 Mid Mid 100% 8 7.29 1 7.35 1.39 67.7 High High 100% 9 6.69 1 16.3 2.35 83.4 High High 100% 10 1.27 1 15.8 2.35 45 Mid Low 50% 11 5.48 1 11 1.39 65.9 High High 100% 12 9.1 1 15.3 4.04 83.9 High High 100% CPU (Clock Speed) Instruction Cycle (Clocks) Inference Speed Add. Com. Sub. Mul. Div. Arduino UNO (16 MHz) 1 1 1 1 1 3 s AM2302 (0.5 Hz) 1 16 2 1 1 250 ms Flame Sensor 1 2 1 1 1 2 s Features Tan et al. [12] Yunus et al. [13] Son B. et al. [14] Proposed Method Multiple-sensors No No No Yes Alarm No No No Yes Decision on two Authentications No No No Yes False Alarm Detection No No No Yes Proposed Solution Traditional Solutions • Proposed system is less expensive. • It reduce false alarm rate. • Image processing and video based systems are good for fire detection on large scale but are expensive. • It detects true fire at early stage. • In WSN, number of clusters are sensing environment and send information when fire is detected. • It can be used in residential areas. • It detect fire automatically without any supervision. Proposed Solution Traditional Solution • Some traditional solution can produce false alarm. • Some system cannot work automatically. • It is used only in small area such as rooms. Accuracy may differ in big halls. • Single sensor is not enough to detect true fire. • Flame sensor also detect sun light instead of flame or fire. The systems • Some systems only suitable to detect fire in large scale like wildfire and not suitable for residential area. is effective for indoor solutions. • It can be very expensive. • In WSN, if any cluster is not able to send information to cluster head due to any reason then it cannot detect fire until cluster head got information from that cluster. © 2018 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 (http:/ Share and Cite MDPI and ACS Style Sarwar, B.; Bajwa, I.S.; Ramzan, S.; Ramzan, B.; Kausar, M. Design and Application of Fuzzy Logic Based Fire Monitoring and Warning Systems for Smart Buildings. Symmetry 2018, 10, 615. https:// AMA Style Sarwar B, Bajwa IS, Ramzan S, Ramzan B, Kausar M. Design and Application of Fuzzy Logic Based Fire Monitoring and Warning Systems for Smart Buildings. Symmetry. 2018; 10(11):615. https://doi.org/ Chicago/Turabian Style Sarwar, Barera, Imran Sarwar Bajwa, Shabana Ramzan, Bushra Ramzan, and Mubeen Kausar. 2018. "Design and Application of Fuzzy Logic Based Fire Monitoring and Warning Systems for Smart Buildings" Symmetry 10, no. 11: 615. https://doi.org/10.3390/sym10110615 Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details Article Metrics	{"url":"https://www.mdpi.com/2073-8994/10/11/615","timestamp":"2024-11-14T21:07:26Z","content_type":"text/html","content_length":"507745","record_id":"<urn:uuid:b661afa8-ea06-4dd7-914a-0b24a3f0a846>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00306.warc.gz"}
Multiplication Chart 1-42 2024 - Multiplication Chart Printable Multiplication Chart 1-42 Multiplication Chart 1-42 – You can get a blank Multiplication Chart if you are looking for a fun way to teach your child the multiplication facts. This may let your child to fill in the details independently. You can get empty multiplication charts for various merchandise varieties, which includes 1-9, 10-12, and 15 goods. If you want to make your chart more exciting, you can add a Game to it. Here are some tips to get the little one started: Multiplication Chart 1-42. Multiplication Charts You may use multiplication charts in your child’s college student binder to assist them to remember math details. Even though many kids can commit to memory their math details by natural means, it will take lots of others time to do this. Multiplication maps are an ideal way to reinforce their learning and boost their self-confidence. In addition to being educative, these graphs could be laminated for additional durability. The following are some helpful methods to use multiplication graphs. Also you can have a look at these web sites for useful multiplication fact sources. This lesson addresses the essentials in the multiplication kitchen table. Together with discovering the guidelines for multiplying, college students will understand the idea of aspects and patterning. By understanding how the factors work, students will be able to recall basic facts like five times four. They can also be able to use the property of one and zero to fix more advanced merchandise. Students should be able to recognize patterns in multiplication chart 1, by the end of the lesson. In addition to the standard multiplication graph, individuals should create a graph or chart with additional variables or fewer elements. To generate a multiplication graph or chart with additional factors, college students should produce 12 furniture, every with 12 rows and 3 columns. All 12 tables should match on one page of papers. Facial lines must be driven with a ruler. Graph document is right for this undertaking. If graph paper is not an option, students can use spreadsheet programs to make their own tables. Game suggestions Whether you are instructing a newcomer multiplication course or focusing on the mastery of your multiplication kitchen table, you may come up with entertaining and interesting activity tips for Multiplication Graph 1. A number of entertaining concepts are listed below. This game necessitates the students to remain pairs and work on the very same difficulty. Then, they will likely all last their charge cards and go over the best solution to get a minute. If they get it right, they win! When you’re teaching youngsters about multiplication, one of the best resources it is possible to let them have is a printable multiplication graph. These computer sheets arrive in a number of models and may be printed using one web page or numerous. Children can find out their multiplication specifics by copying them through the chart and memorizing them. A multiplication chart will be helpful for a lot of good reasons, from aiding them learn their math concepts facts to teaching them the way you use a calculator. Gallery of Multiplication Chart 1-42 Taylormath Multiplication Prime Multiplication Sequence Multiplication Chart Multiplication Table Of 42 Learn 42 Times Table Multiplication Table Of 42 Leave a Comment	{"url":"https://www.multiplicationchartprintable.com/multiplication-chart-1-42-2/","timestamp":"2024-11-13T22:13:04Z","content_type":"text/html","content_length":"52923","record_id":"<urn:uuid:411d634d-1eb4-4e7e-a222-1eeac5f7a804>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00688.warc.gz"}
1090 - And Now, a Remainder from Our Sponsor IBM has decided that all messages sent to and from teams competing in the ACM programming contest should be encoded. They have decided that instead of sending the letters of a message, they will transmit their remainders relative to some secret keys which are four, two-digit integers that are pairwise relatively prime. For example, consider the message "THE CAT IN THE HAT". The letters of this message are first converted into numeric equivalents, where A=01, B=02, ..., Z=26 and a blank=27. Each group of 3 letters is then combined to create a 6 digit number. (If the last group does not contain 3 letters it is padded on the right with blanks and then transformed into a 6 digit number.) For example THE CAT IN THE HAT ! 200805 270301 202709 142720 080527 080120 Each six-digit integer is then encoded by replacing it with the remainders modulo the secret keys as follows: Each remainder should be padded with leading 0’s, if necessary, to make it two digits long. After this, the remainders are concatenated together and then any leading 0’s are removed. For example, if the secret keys are 34, 81, 65, and 43, then the first integer 200805 would have remainders 1, 6, 20 and 38. Following the rules above, these combine to get the encoding 1062038. The entire sample message above would be encoded as The input consists of multiple test cases. The first line of input consists of a single positive integer n indicating the number of test cases. The next 2n lines of the input consist of the test cases. The first line of each test case contains a positive integer (< 50) giving the number of groups in the encoded message. The second line of each test case consists of the four keys followed by the encoded message. Each message group is separated with a space. For each test case write the decoded message. You should not print any trailing blanks. sample input sample output THE CAT IN THE HAT THE END The 2007 ACM East Central North America	{"url":"http://hustoj.org/problem/1090","timestamp":"2024-11-13T15:01:36Z","content_type":"text/html","content_length":"9431","record_id":"<urn:uuid:bd326a9a-2a02-4ee6-bf8c-b4e69b6c0f87>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00601.warc.gz"}
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The formulas and usage of those statistical procedures are explained as well. Any serious analysis starts with data preparation. The course introduces some common data preparation tasks and shows how to implement them in T-SQL. No analysis is good without good data quality. The course introduces data quality issues, and shows how you can check for completeness and accuracy with T-SQL, and how to measure improvements of data quality over time. And since the talk is already about the time, the course shows how you can optimize queries with temporal data, for example when you search for overlapping intervals. More advanced time-oriented information includes hazard and survival analysis. Then the course switches to the currently most fashionable topic, the data science. Some of quite advanced algorithms can also be implemented in T-SQL. The reader learns about the market basket analysis with association rules using different measures like support and confidence, and even sequential market basket analysis, when there is a sequence in the basket. Then the course shows how to develop predictive models with a mixture of k-nearest neighbor and decision trees algorithms and with Bayesian inference analysis. Analyzing text, or text mining, is another modern topic. However, many people do not realize that you can do really a lot of text mining also in pure T-SQL. SQL Server can also become a text mining engine. The course shows how to analyze text in multiple natural languages with pure T-SQL, using also features from the full-text search (FTS). In short, this course teaches you how to use T-SQL for: • Statistical analysis • Data science methods • Text mining. What Attendees Will Learn? • Describe the distribution of a variable with statistical measures. • Find associations between pairs of variables. • Evaluate the quality of the data. • Analyze data over time. • Do the market basket analysis. • Predict outcome of a target variable by using few input variables. • Extract the key words from text data in order to categorize the texts. Advanced Analytics with Transact-SQL is a course for database developers and database administrators who want to take their T-SQL programming skills to the max. It is for the attendees who want to analyze huge amounts of data in an efficient way by using their existing knowledge of the T-SQL language. It is also for the attendees who want to improve their querying by teaching them new and original optimization techniques. 1. Descriptive Statistics 2. Associations between Pairs of Variables 3. Data Preparation 4. Data Quality 5. Time-Oriented Data 6. Time-Oriented Analyses 7. Data Mining 8. Text Mining EmbRace R (E-R) Delivery format Instructor-led seminar in class with discussions. There are no guided labs; however, the attendees get all of the code and access to the virtual machines, so they can test the code in the evenings. Short Description R is the most popular environment and language for statistical analyses, data mining, and machine learning. Managed and scalable version of R runs in SQL Server and Azure ML. Learn how to program in R to squeeze the information from your data. This course is both, an independent seminar, and a complement to the Python for SQL Server Specialists seminar, meaning that the two courses overlap only partially; in each course, some different algorithms and techniques are introduced. Target Audience The target audience is everybody that wants to start analyzing data with R. Database developers that deal with SQL Server and code in T-SQL and want to move more to advanced analytics can get the most of this course. Acquired Skills After completing the course, the delegates are able to start analyzing their data with statistical and machine learning methods immediately. In addition, they can also prepare the data accordingly for the target analysis, and deploy the solution in SQL Server. Besides practical skills, the delegates also learn the basics of the mathematics behind the algorithms. As being an open-source development, R is the most popular analytical engine and programming language for data scientists worldwide. The number of libraries with new analytical functions is enormous and continuously growing. However, there are also some drawbacks. R is a programming language, so you have to learn it to use it. Open-source development also means less control over code. Finally, the free R engine is not scalable. Microsoft added support for R code in SQL Server 2016 and, Azure Machine Learning, or Azure ML, and in Power BI. A parallelized highly scalable execution engine is used to execute the R scripts. In addition, not every library is allowed in these two environments. Attendees of this seminar learn to program with R from the scratch. Basic R code is introduced using the free R engine and RStudio IDE. Then the seminar shows some more advanced data manipulations, matrix calculations and statistical analysis together with graphing options. The mathematics behind is briefly explained as well. Then the seminar switches more advanced data mining and machine learning analyses. Attendees also learn how to use the R code in SQL Server. 1. Introduction to R 2. Data overview and manipulation 3. Basic and advanced visualizations 4. Data mining and machine learning methods 5. Scalable R in SQL Server Python for Data Analysis (P-DA) Delivery format Instructor-led seminar in class with discussions. There are no guided labs; however, the attendees get all of the code and access to the virtual machines, so they can test the code in the evenings. Short Description Although R is the most popular environment and language for statistical analyses, data mining, and machine learning, Python as a more general language might be even more popular. Lately, Python is more widely used for data science as well. SQL Server 2017 adds support for running Python code inside the Database Engine. This course is both, an independent seminar, and a complement to the EmbRaceR seminar, meaning that the two courses overlap only partially; in each course, some different algorithms and techniques are introduced. Target Audience The target audience is everybody that wants to start developing with Python and use the language for machine learning. However, the course is focused on data science, not general development. Database developers that deal with SQL Server and code in T-SQL and want to move more to advanced analytics can get the most of this course. Acquired Skills After completing the course, the delegates are able to start analyzing their data with statistical and machine learning methods immediately. In addition, they can also prepare the data accordingly for the target analysis, and deploy the solution in SQL Server. Besides practical skills, the delegates also learn the basics of the mathematics behind the algorithms. Python is more organized language than R. In last years, many data analytics libraries for Python evolved, and thus Python is catching up with R even in the data science area. Microsoft added support for Python code in SQL Server in version 2017. Now you can use either R or Python inside the Database Engine for advanced tasks like predictive analytics. Therefore, you can use the language that suits you better. Statisticians and mathematicians might prefer R, while developers tend to be more Python oriented. Python has also become overwhelming analytical language in the Azure cloud. Attendees of this seminar learn to program with Python from the scratch. Basic Python code is introduced using the Python engine installed with SQL Server and Visual Studio. The seminar shows some more advanced data manipulations, matrix calculations and statistical analysis together with graphing options. The mathematics behind is briefly explained as well. Then the seminar switches to more advanced data mining and machine learning analyses. Finally, the seminar introduces how you can use Python in SQL Server and in Azure. 1. Introduction to Python 2. Data overview and manipulation 3. Basic and advanced visualizations 4. Data mining and machine learning methods 5. Scalable Python in SQL Server, Power BI, and Azure ML Naročite se na Xnet novice in ostanite na tekočem glede novih tečajev, seminarjev, možnosti pridobitve novih certificiranj in akcijskih cen.	{"url":"https://www.kompas-xnet.si/izobrazevanja/koledar-tecajev-t/BundleSQL3","timestamp":"2024-11-09T14:28:47Z","content_type":"text/html","content_length":"151355","record_id":"<urn:uuid:ff850de9-0980-4f59-9bce-150dc47447b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00380.warc.gz"}
Number 89 Interesting facts about the number 89 More photos ... • (89) Julia is asteroid number 89. It was discovered by E. J. M. Stephan from Marseille Observatory on 8/6/1866. Areas, mountains and surfaces • The total area of Anticosti Island is 3,066 square miles (7,941 square km). Country Canada (Quebec). 89th largest island in the world. • There is a 89 miles (142 km) direct distance between Bucheon-si (South Korea) and Daejeon (South Korea). • There is a 89 miles (143 km) direct distance between Daejeon (South Korea) and Seoul (South Korea). • There is a 89 miles (142 km) direct distance between Hangzhou (China) and Wuxi (China). • There is a 89 miles (142 km) direct distance between Hefei (China) and Nanjing (China). • More distances ... • There is a 89 miles (142 km) direct distance between Ibadan (Nigeria) and Ilorin (Nigeria). • Pokémon Muk (Betobeton, Betbeton) is a Pokémon number # 089 of the National Pokédex. Muk is Poison-type Pokémon in the first generation. It is an amorphous egg group Pokémon. Muk's other indexes are Hoenn index 107 , Johto index 117 , Teselia index 065 . History and politics • United Nations Security Council Resolution number 89, adopted 17 November 1950. Complaints regarding expulsion of Palestinian people. Resolution text. In other fields • The lowest natural temperature ever directly recorded at ground level on Earth is −89 °C (−128.6 °F; 184.0 K) at the then-Soviet Vostok Station in Antarctica on 21 July 1983 by ground • The ISBN Group Identifier for books published in South Korea. Example book: ISBN 89-04-02003-4 존 칼빈, 기독교강요 (상), 생명의말씀사 (1988) • 89 is the smallest number for which the positive sum of all odd primes less than or equal to it is a square • 89 is the smallest prime that is a concatenation of p^q and q^p where p and q are prime • 89 is the smallest prime whose digits are composites • 89 = 8^1 + 9^2 • Actinium is the chemical element in the periodic table that has the symbol Ac and atomic number 89. • The best athletes to wear number 89 Mike Ditka, NFL; Gino Marchetti, NFL; Mark Bavaro, NFL; Clyde Lovellette, NBA Tv, movies and cinematography • 89 (Official Trailer 2017) What is 89 in other units The decimal (Arabic) number converted to a Roman number Roman and decimal number conversions The number 89 converted to a Mayan number is Decimal and Mayan number conversions. Weight conversion 89 kilograms (kg) = 196.2 pounds (lbs) 89 pounds (lbs) = 40.4 kilograms (kg) Length conversion 89 kilometers (km) equals to 55.302 miles (mi). 89 miles (mi) equals to 143.232 kilometers (km). 89 meters (m) equals to 143.232 feet (ft). 89 feet (ft) equals 27.128 meters (m). 89 centimeters (cm) equals to 35.0 inches (in). 89 inches (in) equals to 226.1 centimeters (cm). Temperature conversion 89° Fahrenheit (°F) equals to 31.7° Celsius (°C) 89° Celsius (°C) equals to 192.2° Fahrenheit (°F) Power conversion 89 Horsepower (hp) equals to 65.45 kilowatts (kW) 89 kilowatts (kW) equals to 121.02 horsepower (hp) Time conversion (hours, minutes, seconds, days, weeks) 89 seconds equals to 1 minute, 29 seconds 89 minutes equals to 1 hour, 29 minutes Number 89 morse code: ---.. ----. Sign language for number 89: Number 89 in braille: Year 89 AD • Pliny the Younger becomes urban quaestor. • Aquincum (modern Budapest) was founded. • Han Chinese Emperor Chang Ti dies, and was succeeded by Ho Ti Gregorian, Hebrew, Islamic, Persian and Buddhist Year (Calendar) Gregorian year 89 is Buddhist year 632. Buddhist year 89 is Gregorian year 454 a. C. Gregorian year 89 is Islamic year -549 or -548. Islamic year 89 is Gregorian year 707 or 708. Gregorian year 89 is Persian year -534 or -533. Persian year 89 is Gregorian 710 or 711. Gregorian year 89 is Hebrew year 3849 or 3850. Hebrew year 89 is Gregorian year 3671 a. C. The Buddhist calendar is used in Sri Lanka, Cambodia, Laos, Thailand, and Burma. The Persian calendar is the official calendar in Iran and Afghanistan. Share in social networks Advanced math operations Is Prime? The number 89 is a prime number . The closest prime numbers are The 89th prime number in order is Factorization and factors (dividers) The prime factors of 89 Prime numbers have no prime factors smaller than themselves. The factors of 89 are , 89. Total factors 2. Sum of factors 90 (1). Prime factor tree 89 is a prime number. The second power of 89 is 7.921. The third power of 89 is 704.969. The square root √ is 9,433981. The cube root of is 4,464745. The natural logarithm of No. ln 89 = log 89 = 4,488636. The logarithm to base 10 of No. log 89 = 1,94939. The Napierian logarithm of No. log 89 = -4,488636. Trigonometric functions The cosine of 89 is 0,510177. The sine of 89 is 0,860069. The tangent of 89 is 1,685825. Number 89 in Computer Science Code type Code value ASCII Html Y Uppercase Y Y (ISO 8859-1 Characters) Unix time Unix time 89 is equal to Thursday Jan. 1, 1970, 12:01:29 a.m. GMT IPv4, IPv6 Number 89 internet address in dotted format v4 0.0.0.89, v6 ::59 89 Decimal = 1011001 Binary 89 Decimal = 10022 Ternary 89 Decimal = 131 Octal 89 Decimal = 59 Hexadecimal (0x59 hex) 89 BASE64 ODk= 89 MD5 7647966b7343c29048673252e490f736 89 SHA1 16b06bd9b738835e2d134fe8d596e9ab0086a985 89 SHA224 32580a22449ef2e5b12f1e5a794cc0f90f1f1463fd15036f2b00ed67 89 SHA256 cd70bea023f752a0564abb6ed08d42c1440f2e33e29914e55e0be1595e24f45a 89 SHA384 18b62dcda274df291192c2b36caa24be6d3f9c30c68666b53813efd366a9b0b19c9e2ea4ac4abcce391b89f8c019d91a More SHA codes related to the number 89 ... If you know something interesting about the 89 number that you did not find on this page, do not hesitate to write us here. Numerology 89 The meaning of the number 9 (nine), numerology 9 Character frequency 9: 1 The number 9 (nine) is the sign of ideals, Universal interest and the spirit of combat for humanitarian purposes. It symbolizes the inner Light, prioritizing ideals and dreams, experienced through emotions and intuition. It represents the ascension to a higher degree of consciousness and the ability to display love for others. He/she is creative, idealistic, original and caring. More about the the number 9 (nine), numerology 9 ... The meaning of the number 8 (eight), numerology 8 Character frequency 8: 1 The number eight (8) is the sign of organization, perseverance and control of energy to produce material and spiritual achievements. It represents the power of realization, abundance in the spiritual and material world. Sometimes it denotes a tendency to sacrifice but also to be unscrupulous. More about the the number 8 (eight), numerology 8 ... № 89 in other languages How to say or write the number eighty-nine in Spanish, German, French and other languages. Spanish: 🔊 (número 89) ochenta y nueve German: 🔊 (Nummer 89) neunundachtzig French: 🔊 (nombre 89) quatre-vingt-neuf Portuguese: 🔊 (número 89) oitenta e nove Hindi: 🔊 (संख्या 89) नवासी Chinese: 🔊 (数 89) 八十九 Arabian: 🔊 (عدد 89) تسعة و ثمانون Czech: 🔊 (číslo 89) osmdesát devět Korean: 🔊 (번호 89) 팔십구 Danish: 🔊 (nummer 89) niogfirs Hebrew: (מספר 89) שמונים ותשע Dutch: 🔊 (nummer 89) negenentachtig Japanese: 🔊 (数 89) 八十九 Indonesian: 🔊 (jumlah 89) delapan puluh sembilan Italian: 🔊 (numero 89) ottantanove Norwegian: 🔊 (nummer 89) åtti-ni Polish: 🔊 (liczba 89) osiemdziesiąt dziewięć Russian: 🔊 (номер 89) восемьдесят девять Turkish: 🔊 (numara 89) seksendokuz Thai: 🔊 (จำนวน 89) แปดสิบเก้า Ukrainian: 🔊 (номер 89) вісімдесят дев'ять Vietnamese: 🔊 (con số 89) tám mươi chín Other languages ... News to email If you know something interesting about the number 89 or any other natural number (positive integer), please write to us here or on Facebook. Legal Notices & Terms of Use The content of the comments is the opinion of the users and not of number.academy. It is not allowed to pour comments contrary to the laws, insulting, illegal or harmful to third parties. Number.academy reserves the right to remove or not publish any inappropriate comment. It also reserves the right to publish a comment on another topic. Privacy Policy Frequently asked questions about the number 89 • Is 89 prime and why? The number 89 is a prime number because its divisors are: 1, 89. • How do you write the number 89 in words? 89 can be written as "eighty-nine". Number 89 in News NASA: NASA to Provide Coverage of Progress 89 Launch, Space Station Docking NASA will provide live launch and docking coverage of a Roscosmos cargo spacecraft delivering nearly three tons of food, fuel, and supplies to the Expedition 71 crew aboard the International Space Station. The unpiloted Progress 89 spacecraft is scheduled to launch at 11:20 p.m. EDT, Wednesday, Aug. 14 (8:20 a.m. … Aug. 12, 2024 BBC: Maui fire: 89 killed as governor warns of 'significant' death toll rise Eighty-nine people have been killed and hundreds are unaccounted for days after fires broke out in Hawaii. Aug. 13, 2023 What is your opinion?	{"url":"https://number.academy/89","timestamp":"2024-11-07T15:57:47Z","content_type":"text/html","content_length":"43491","record_id":"<urn:uuid:a579c9f6-d870-4e88-9ccb-14328816090e>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00493.warc.gz"}
Biophysics Problem 37 A cell's oxygen requirement is proportional to its mass, but its oxygen intake is proportional to its surface area. Show that a cell cannot grow indefinitely and survive. Since the oxygen required is proportional to mass and mass itself is proportional to volume $(M = \rho V),$ the oxygen required scales as $L^3.$ Thus the oxygen required will equal some constant multiplied by $L^3,$ i.e., $O_{required} = K_1 L^3,$ where $K_1$ is this constant. Since the oxygen supplied is proportional to \emph{surface area}, it similarly scales as some constant times $L^2,$ i.e., $O_{supplied} = K_2 L^2,$ where $K_2$ is a different constant than $K_1.$ In the limiting case, the required and supplied oxygen will be equal. Therefore, $K_1 L^3 = K_2 L^2\\ L = \frac{K_1}{K_2}$ When $(K_1/K_2) < L,$ the above equation will not be balanced and the cell's oxygen requirements cannot be satisfied.	{"url":"https://www.physics.uoguelph.ca/biophysics-problem-37","timestamp":"2024-11-05T12:17:52Z","content_type":"text/html","content_length":"56959","record_id":"<urn:uuid:fdf2a1a0-0e1e-48bc-aa9d-2fffee037b37>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00852.warc.gz"}
Elephant Learning Being asked, “What is 5 times 4?” or “Why does 5 times 4 equal 20?” There’s a reason teachers go through extensive educational training before they attempt to teach math to others. But what if you could do it yourself – at home? Searching for the answer to that question is what brought Raelyn, mother of first-grader Ellis, to Elephant Learning. Ellis’ teachers had already identified his limited attention span as a potential roadblock to learning. Raelyn explains, “He struggles with focusing in a classroom setting, but can be easily redirected. In talking to his teacher, it’s topics he may not have that much knowledge about or it’s information that doesn’t catch his attention.” When Raelyn found Elephant Learning, she discovered an app filled with engaging math games that would likely hold Ellis’ attention long enough to learn something new. Raelyn wants Ellis to love learning math as much as he loves reading, and the two skills are in fact tied to each other. Research shows that children who do more math are better readers, writers, speakers, and problem solvers. Teaching math to kids these days is a daunting task for any parent, and it doesn’t matter how confident parents are in their own math skills. It can be hard to put into words the how and why of math concepts for an adult audience, and trying to make these concepts make sense to a child is even harder. Adding to that dilemma is the trend of constantly evolving math curriculum. New insights into teaching math can impact students in a positive way. The rapid development of various math problem-solving procedures can keep parents and students in a vicious cycle of re-learning, rather than building on concept mastery and moving forward. For example, if a student has already mastered addition, they might still be expected to learn a new series of steps for addition, even though it’s a concept they’ve already mastered. This trend is evident in many educational settings: learning math concepts has shifted to learning multiple math procedures, with students expected to learn several ways to solve the same problem. Suddenly, the way many adults were taught math in elementary and middle school is no longer the standard approach. In fact, it can feel like there’s no standard approach at all. For many parents, their math knowledge is considered obsolete when it comes time to help their kids with math homework. Any confidence a parent may have in helping their child answer a math problem is often met with their child grumbling at them, “That’s not how the teacher wants us to do it” — even if the answer is Suddenly the roles are reversed, and now it’s your child's job to teach you the various ways they’re supposed to do the math, even as they themselves are struggling to understand it, let alone put it into words. This role reversal would be a small miracle for a younger child who is still learning how to communicate in general. In this familiar scenario, feelings of frustration in both parents and their children can leave everyone feeling helpless and discouraged. Raelyn shares this frustration with her son Ellis. “The strategies they teach [in school] involve using more than one way to find the answer which is challenging. He forgets steps in between and it crushes him. He’s frustrated and I’m frustrated.” Imagine Raelyn’s relief when she found Elephant Learning, a math program designed to empower children with their math progress. What makes Elephant Learning so effective is that it teaches math concepts, not procedures. That means your child is learning how to problem-solve rather than memorizing a series of steps for a designated problem. In other words, your child builds their own toolbox of diverse problem-solving skills. And like a toolbox, they can use those skills in a variety of contexts and at any grade level. Their confidence in their math abilities isn’t tied to a specific style of solving math problems. That’s how Elephant Learning is 100 percent compatible with all math standards and curriculums. When your child learns the universal language of mathematics, it means they can more easily adapt to their rapidly changing world — new teachers, new schools, new curriculum, or real-world The result is a student who experiences increased understanding, increased learning, and increased confidence. For kids like Ellis, that means their confidence in their math abilities isn’t tied to how well they know a specific type of problem. They can rely on their trained, mathematical intuition to tackle a problem regardless of the context. And thank goodness for that, because the math curriculum continues to evolve at every grade level. Even if Ellis masters his first-grade math curriculum, he will likely face a new curriculum with a new methodology in the future, leaving him and his mom back at square one to learn an entirely new method for solving the same problem. But Elephant Learning removes parents and students from that vicious cycle of re-learning how to do math every year. Learning math isn’t automatically more fun just because it’s on an app instead of in a classroom. What makes Elephant Learning so effective is that it turns learning math into a game. For example, your child might be presented with a screen full of pandas and asked to make eight equal groups of pandas. Or, they might be given an empty pattern and asked to fill in a fraction of the Don’t let the fun graphics and animations in Elephant Learning mislead you: learning math concepts through games is a research-based approach to ensure engagement and information retention. Elephant Learning math games are designed by early-age education researchers who have studied successful math gamification models. For a kid like Ellis, this fun, game-like approach to doing math is exactly what he needs to hold his attention. As Raelyn says, “I just want him to have fun,” and with Elephant Learning it’s hard not to, with a variety of games at Ellis’ fingertips, whether he’s using the mobile app or a computer. As Ellis plays his math games, the program adapts to his learning progress. He can’t possibly get bored, because once he’s mastered a concept the program introduces more challenging material. And if he does struggle, the program adjusts accordingly to prevent unnecessary frustration. This model is paying off for Ellis already. When he began Elephant Learning he was doing math below his age level. After six months, he’s mastered over four years of material and is now ahead of Elephant Learning makes sure parents stay in the loop when it comes to their child’s progress too. The detailed progress report lets parents see exactly which areas their kids need help in, and which areas show progress. That gives parents the freedom to focus their time and attention on where they think it matters. Regardless of where kids start in their math journey, Elephant Learning meets them where they are and builds their confidence. A confident learner is a happy and motivated learner, and motivation is what keeps kids actively learning as they grow. Imagine the frustration melting away as your child masters a year’s worth of math in just three months, after playing with the Elephant Learning app for 30 minutes a week. You’ve finally escaped that vicious cycle once and for all. Related: The Critical Differences Between Elephant Learning and Other Math Apps “I want Ellis to feel confident about learning” - Mom, Raelyn Your child will learn at least 1 year of mathematics over the course of the next 3 months using our system just 10 minutes/day, 3 days per week or we will provide you a full refund.	{"url":"https://www.elephantlearning.com/case-studies/how-ellis-escaped-the-vicious-cycle-of-re-learning-math-concepts-every-year","timestamp":"2024-11-08T14:24:49Z","content_type":"text/html","content_length":"51918","record_id":"<urn:uuid:7466c844-f3d3-4dfa-bff3-d893fec7cb3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00691.warc.gz"}
How to Build Wooden Steps for Mobile Homes You can learn to build stairs for a mobile home or trailer in few steps. Why hire an expensive handyman or professional carpenter when you can tackle the project yourself? Most trailers and mobile homes are sitting on a concrete base on the ground, with a slab for the stairs already in place. If there is no slab present, you can easily construct one to accommodate a set of stairs. Take measurements for the stairs: First, measure the height of the stairs; you will want to keep them a couple of inches below the door opening: this measurement will be the total rise. Next, measure the length of the stairs; most stairs are built with a 30 to 35 degree angle slope. The length is otherwise knows as the total run of the stairs. • You can learn to build stairs for a mobile home or trailer in few steps. • Take measurements for the stairs: First, measure the height of the stairs; you will want to keep them a couple of inches below the door opening: this measurement will be the total rise. Measure the riser height: First, write down the total rise; then convert the total rise to inches. Divide the total rise by 7 (7 is used because this is the recommended height in inches of all stair risers). Take this number and divide it into the total rise. Take the numbers to the right of the decimal and convert these to inches by dividing them by 16. (You use 16ths because that is how many fractions of an inch there are on a tape measure.) • Lay out and cut the stair stringers: The stringer will be either 2x10 or 2x12 pieces of lumber. • Place on the stringer and trace out each step, sliding the framing square to the right so that the tread measurement lines up with the last riser line. • ( • Measure the riser height: First, write down the total rise; then convert the total rise to inches. Example of how to get unit rise: 4'6" total rise 4' multiplied by 12"=48"+6"=54" total rise in inches 54" divided by 7"=7.714" So there will be 7 total risers for the stairs 54" divided by 7=7.714; Convert .714 to inches .714 multiplied by 16=11.424 or 11/16 of an inch. According to these measurements, the unit rise will be 7-11/16 inches and the stair set will contain seven steps. Measure the tread width: Convert the total run to inches by multiplying the feet by 12 and then add any leftover inches. Divide this number by 6. Minus 1 from the total number of steps needed for the stairs; you must do this because the concrete slab is counted as a tread. Convert the decimals into inches, and then multiply the three numbers to the right of the decimal by 16. • Measure the tread width: Convert the total run to inches by multiplying the feet by 12 and then add any leftover inches. Example to get tread width: 5'3" total run 5' multiplied by 12"=60"+3"=63" total run in inches 63" divided 6=10.5 .5 multiplied by 16=8 or 8/16 or ½ inch. Tread width will be 10½ inches The tread is 10½ inches and the riser is 7-11/16 inches. According to the measurements, there are seven steps and six treads. Lay out and cut the stair stringers: The stringer will be either 2x10 or 2x12 pieces of lumber. The stairs will need only two stringers, but you can build with more if you prefer. Take the framing square and tighten the square gauges on the outer edge at the unit riser height and tread width. The tread will be on the tongue of the framing square and the tread will be on the blade of the framing square. (The tongue is the shorter end of the framing square while the blade is the longer end.) Place on the stringer and trace out each step, sliding the framing square to the right so that the tread measurement lines up with the last riser line. (Your last tread mark will be part of the landing.) Reverse the framing square and mark the bottom of the first step. Move the square to the top end of the stringer. Align the tongue with the last riser line and mark the top cut. Cut the stringers, stopping the saw at the end of each line and using a cut out saw to finish the cut. Cut out a notch at the bottom of the stringers to accommodate the bottom plate that you will attach to the concrete slab in step five. • Lay out and cut the stair stringers: The stringer will be either 2x10 or 2x12 pieces of lumber. • Place on the stringer and trace out each step, sliding the framing square to the right so that the tread measurement lines up with the last riser line. • ( Construct the stairs: Fasten a bottom plate, usually a 2x4, to the concrete slab using a powder actuated nail gun or self tapping concrete screws. This plate will be located where the bottom face of the stairs will be located. There are several ways to attach the stringers to the house; you can nail directly into the joist of the house, nail a ledger plate to the joist to set the stringers on, or use brackets to hold the stringers to attach to the house. Cut the treads out of lumber best suited for outdoor environments. Use split treads, two treads per step, to allow water to drain. Glue with wood glue and screw in place.	{"url":"https://www.ehow.co.uk/how_4865829_build-wooden-steps-mobile-homes.html","timestamp":"2024-11-07T13:56:59Z","content_type":"text/html","content_length":"121892","record_id":"<urn:uuid:f9bfda5f-e3a6-40df-b425-61de6e7504bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00871.warc.gz"}
WAEC Marking Scheme For Mathematics 2022/2023WAEC Marking Scheme For Mathematics 2022/2023 WAEC Marking Scheme For Mathematics 2022/2023 April 9, 2024 WAEC No Comments How Is Waec Percentage Calculated For Mathematics 2022/2023 Academic Session? Waec marking scheme for mathematics: If you are a candidate who is asking about how WAEC Mathematics examinations are marked and graded? See the latest WAEC marking scheme here for Mathematics. Many Candidates are scared while some are not, If you are a student and you are thinking of how to score A1, and B2 or probably escape by passing Mathematics, then you need to understand how WAEC examiners mark their papers. The fact is that as a candidate who is writing the 2022 Waec Exam, you are to be aware of the Waec marking scheme for mathematics in other for you to be prepared on how to tackle and answer their questions. In case you are looking for the 2022 Waec Mathematics syllabus kindly check here. The fact is that marks are subdivided into three categories which are • Method (M) • Accuracy (A) • Independent accuracy marks not preceded by M mark (B) The M mark is given for a particular stage if the method used is correct. That is if correctly done without numerical errors yielding the right answer. M marks are not subdivided and unless the M marks for a stage have been awarded no A marks can be gained for that stage • Examiners will deduct 2 marks if you misread and present the data given falsely. • Examiners will deduct 1 mark for an answer not given to the accuracy asked for. • Deductions under iii and iv above shall not be more than once on each occasion in one question • 1 mark is deductible for premature approximations that do not considerably simplify the subsequent work. • Deductions can only be made from A or B marks and not from M marks. • Examiners will give zero for results obtained from work they cannot decipher or which is wholly suppressed and 1 mark is deductible for the omission of an essential working. • Once a correct answer is stated examiners will ignore any further workings beyond that stage. • For geometric proofs, 1 mark is deductible for the omission of an essential reason or for a wrong reason given. But this should not be more than once in a question. • 1 mark is to be deducted for the omission of units or for wrong units. But this should not be more than once in a question. • If a single question is attempted on more than one occasion the mark for the best attempt is to be awarded to the student.SOS (see other solutions) shall be written against the others. • If more questions are attempted then the rubric allows the lowest marks per those extra questions will be deleted. MQA ANSWERED will be indicated by the examiner to indicate he was aware while • Unless otherwise stated, equivalent methods not specified in the will be accepted and given appropriate marks. The final total is 100 marks and will be rounded upwards to the nearest whole number. Actually, early preparation is the key to success, Easy step-by-step on how to pass your Waec is already published here. Let continue with How do Waec mark mathematics? • Do you know that to get an A in WAEC 2022 Mathematics, you need to score above 75% in the Exam? • Do you know that if you are given 50 questions in your mathematics subject and you answer 37 questions you answered so well? • Do you know WAEC mathematics comprises 50 objectives and 13 theory questions? Related Article For You: How is the 2022 Waec Mathematics Percentage Calculated? For you to have A, B, or C, in your upcoming coming Waec 2021 then you need to follow these simple steps, Wassce mathematics comprises 50 objective questions and 13 theory questions. The theory questions are divided into parts I and Part II. You are to answer ten questions in mathematics theory; all the questions in Part I and five questions in Part II. The objective questions take 40% of the total mark while the theory makes the remaining 60%. Therefore, for you to score A, you must answer 40 correct questions in the objective and at least 8 in Calculate how many marks you are very sure of, then divide by the total marks the question carry and multiply by 100% For Example, you are required to answer five questions in the theory questions, Each question carries 20mark Let say, for instance, you are able to answer 5 questions and you are sure of 18 marks out of a total of 20 marks for each question The Total marks you are sure of is 18 x 5 = 90 The total number of marks you can get is 20 x 5 = 100 Your Percentage score is (90/100) x 100% = 90% = A1 This is to inform all Waec candidates about How Is Waec Percentage Calculated For Mathematics? and the Waec Marking Scheme For Mathematics If you have any questions to ask based on this Waec marking scheme for mathematics, How do Waec mark mathematics? Waec Marking Scheme For Mathematics PDF kindly asks us by dropping a comment in our comment box below… If actually, this information on “Waec marking scheme for mathematics” is awesome and useful to you please kindly share using via Facebook, WhatsApp, Twitter and Google+	{"url":"https://funloaded.org.ng/waec-marking-scheme-for-mathematics/","timestamp":"2024-11-02T18:21:06Z","content_type":"text/html","content_length":"176875","record_id":"<urn:uuid:4986d14e-2eb0-43f7-b255-5a0bdc86fe51>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00402.warc.gz"}
Adapted By Darlene Young Introductory Statistics Module 8: Confidence Intervals Introduction: Confidence Intervals Barbara Illowsky & OpenStax et al. Have you ever wondered what the average number of M&Ms in a bag at the grocery store is? You can use confidence intervals to answer this question. (credit: comedy_nose/flickr) Learning Objectives By the end of this chapter, the student should be able to: • Calculate and interpret confidence intervals for estimating a population mean and a population proportion. • Interpret the Student’s t probability distribution as the sample size changes. • Discriminate between problems applying the normal and the Student’s t distributions. • Calculate the sample size required to estimate a population mean and a population proportion given a desired confidence level and margin of error. Suppose you were trying to determine the mean rent of a two-bedroom apartment in your town. You might look in the classified section of the newspaper, write down several rents listed, and average them together. You would have obtained a point estimate of the true mean. If you are trying to determine the percentage of times you make a basket when shooting a basketball, you might count the number of shots you make and divide that by the number of shots you attempted. In this case, you would have obtained a point estimate for the true proportion. We use sample data to make generalizations about an unknown population. This part of statistics is called inferential statistics. The sample data help us to make an estimate of a population parameter . We realize that the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals. In this chapter, you will learn to construct and interpret confidence intervals. You will also learn a new distribution, the Student’s-t, and how it is used with these intervals. Throughout the chapter, it is important to keep in mind that the confidence interval is a random variable. It is the population parameter that is fixed. If you worked in the marketing department of an entertainment company, you might be interested in the mean number of songs a consumer downloads a month from iTunes. If so, you could conduct a survey and calculate the sample mean, [latex]displaystyleoverline{x}[/latex], and the sample standard deviation, s. You would use [latex]displaystyleoverline{x}[/latex] to estimate the population mean and s to estimate the population standard deviation. The sample mean, [latex]displaystyleoverline{x}[/latex], is the point estimate for the population mean, μ. The sample standard deviation, s, is the point estimate for the population standard deviation, σ. Each of [latex]displaystyleoverline{x}[/latex] and s is called a statistic. A confidence interval is another type of estimate but, instead of being just one number, it is an interval of numbers. The interval of numbers is a range of values calculated from a given set of sample data. The confidence interval is likely to include an unknown population parameter. Suppose, for the iTunes example, we do not know the population mean μ, but we do know that the population standard deviation is σ = 1 and our sample size is 100. Then, by the central limit theorem, the standard deviation for the sample mean is [latex]displaystylefrac{{sigma}}{{sqrt{n}}} = frac{{1}}{{sqrt{100}}}=0.1[/latex] The empirical rule, which applies to bell-shaped distributions, says that in approximately 95% of the samples, the sample mean, [latex]displaystyleoverline{x}[/latex] , will be within two standard deviations of the population mean μ. For our iTunes example, two standard deviations is (2)(0.1) = 0.2. The sample mean [latex]displaystyleoverline{x}[/latex] =0.1 is likely to be within 0.2 units of Because [latex]displaystyleoverline{x}[/latex] is within 0.2 units of μ, which is unknown, then μ is likely to be within 0.2 units of [latex]displaystyleoverline{x}[/latex] in 95% of the samples. The population mean μ is contained in an interval whose lower number is calculated by taking the sample mean and subtracting two standard deviations (2)(0.1) and whose upper number is calculated by taking the sample mean and adding two standard deviations. In other words, μ is between [latex]displaystyleoverline{x}[/latex]− 0.2 and [latex]displaystyleoverline{x}[/latex] + 0.2 in 95% of all the For the iTunes example, suppose that a sample produced a sample mean [latex]displaystyleoverline{x}[/latex]= 2. Then the unknown population mean μ is between [latex]displaystyleoverline{x}[/latex]− 0.2=2−0.2=1.8 and [latex]displaystyleoverline{x}[/latex] +0.2=2+0.2=2.2 We say that we are 95% confident that the unknown population mean number of songs downloaded from iTunes per month is between 1.8 and 2.2. The 95% confidence interval is (1.8, 2.2). The 95% confidence interval implies two possibilities. Either the interval (1.8, 2.2) contains the true mean μ or our sample produced an [latex]displaystyleoverline{x}[/latex] that is not within 0.2 units of the true mean μ. The second possibility happens for only 5% of all the samples (95–100%). Remember that a confidence interval is created for an unknown population parameter like the population mean, μ. Confidence intervals for some parameters have the form: (point estimate – margin of error, point estimate + margin of error) The margin of error depends on the confidence level or percentage of confidence and the standard error of the mean. When you read newspapers and journals, some reports will use the phrase “margin of error.” Other reports will not use that phrase, but include a confidence interval as the point estimate plus or minus the margin of error. These are two ways of expressing the same concept. Although the text only covers symmetrical confidence intervals, there are non-symmetrical confidence intervals (for example, a confidence interval for the standard deviation). Have your instructor record the number of meals each student in your class eats out in a week. Assume that the standard deviation is known to be three meals. Construct an approximate 95% confidence interval for the true mean number of meals students eat out each week. 1. Calculate the sample mean. 2. Let σ = 3 and n = the number of students surveyed. 3. Construct the interval (([latex]displaystyleoverline{x}[/latex]−2)([latex]displaystylefrac{{sigma}}{{sqrt{n}}}[/latex]), ([latex]displaystyleoverline{x}[/latex]+ 2)([latex]displaystylefrac We say we are approximately 95% confident that the true mean number of meals that students eat out in a week is between __________ and ___________.	{"url":"https://psu.pb.unizin.org/introductorystatyoungsu18/chapter/introduction-confidence-intervals/","timestamp":"2024-11-12T10:25:35Z","content_type":"text/html","content_length":"104853","record_id":"<urn:uuid:b72b924d-17bb-4e0b-ae68-d219415cd02c>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00583.warc.gz"}
Coffee Time Challenge A short little puzzle to solve over your next morning coffee: • 1, x, y are three numbers in geometric progression. • x, 6, y are in arithmetic progression. What are the values of x, y ? Easy Solution Without even breaking out math, you probably got the solution x = 3, y = 9 This gives the geometric progression of: 1, 3, 9 and the arithmetic progression as: 3, 6, 9. There is, however, another solution. Can you find it?	{"url":"https://datagenetics.com/blog/april42021/index.html","timestamp":"2024-11-08T10:31:39Z","content_type":"application/xhtml+xml","content_length":"6949","record_id":"<urn:uuid:6eb4c365-c698-4281-be83-edf77b9d24ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00238.warc.gz"}
How to plot piecemeal equations 06-24-2022, 01:29 PM Post: #1 Nick1941 Posts: 8 Junior Member Joined: Jun 2022 How to plot piecemeal equations On the HP Prime Graphing Calculator, is there a way to plot piecemeal equations such as the one shown in the attachment below? 06-25-2022, 05:01 AM Post: #2 Didier Lachieze Posts: 1,658 Senior Member Joined: Dec 2013 RE: How to plot piecemeal equations Yes, with the PIECEWISE function. In Algebraic mode enter PIECEWISE(X<0,X+2,X≥0,1-X) In Textbook mode just enter it as in your example. 06-25-2022, 05:07 AM Post: #3 Mark Power Posts: 108 Member Joined: Dec 2013 RE: How to plot piecemeal equations Or use IFTE(X<0,X+2,1-X) 06-25-2022, 01:01 PM Post: #4 jonmoore Posts: 313 Senior Member Joined: Apr 2020 RE: How to plot piecemeal equations There's also a nifty template option on the third key from the left of the top row. You should recognise the temple from literature, it's the second from the right of the top row. 06-25-2022, 02:59 PM Post: #5 Nick1941 Posts: 8 Junior Member Joined: Jun 2022 RE: How to plot piecemeal equations Thanks to all of you for your help. I especially liked using jonmoore's method in the Advanced Graphing App. User(s) browsing this thread: 1 Guest(s)	{"url":"https://hpmuseum.org/forum/thread-18505-post-161649.html#pid161649","timestamp":"2024-11-09T19:36:22Z","content_type":"application/xhtml+xml","content_length":"25957","record_id":"<urn:uuid:8fa4d75a-2530-4a89-8c14-4a96d17a17f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00176.warc.gz"}
Lesson 12 Prisms and Pyramids 12.1: The Faces of Geometry (5 minutes) In this warm-up, students practice visualizing and drawing the faces of several solids. This will be helpful in upcoming activities as they categorize solids based on features of their choosing, and as they build solids from nets as a foundation for developing the formula for the volume of a pyramid. Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner. Student Facing Three solids are shown. Draw all the surfaces of each solid. Anticipated Misconceptions Students may struggle to draw the cone surface that’s in the shape of a sector of a circle. Ask them to consider snipping the cone in a straight line along this face and unrolling it. Activity Synthesis Here are questions for discussion. • “What are the names of these solids?” (rectangular pyramid, triangular prism, cone) • “What is the same and different about the surfaces of the prism and the pyramid?” (Each of these solids has 5 faces, and the faces are all triangles and rectangles. The prism has a triangle for a base and rectangles for the other faces, while the pyramid has a rectangle for a base and triangles for the other faces.) • “Which is the only surface that’s not a polygon?” (The cone has one “face” shaped like a wedge from a circle.) 12.2: Card Sort: Sorting Shapes (10 minutes) A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). In this task, students sort solids based on features of their choosing. The structures students identify will allow them to extend the adjectives right and oblique to pyramids and cones. Monitor for different ways groups choose to categorize the solids, but especially for categories that distinguish between right and oblique solids, and between solids that have an apex (pyramids and cones) and those that do not (prisms and cylinders). As students work, encourage them to refine their descriptions of the solids using more precise language and mathematical terms (MP6). Arrange students in groups of 2 and distribute pre-cut slips. Tell students that in this activity, they will sort some cards into categories of their choosing. When they sort the solids, they should work with their partner to come up with categories. Conversing: MLR2 Collect and Display. As students work on this activity, listen for and collect the language students use to distinguish between right and oblique solids. Also, collect the language students use to distinguish between solids with an apex and solids with two congruent bases. Write the students’ words and phrases on a visual display. As students review the visual display, create bridges between current student language and new terminology. For example, “the tip of the cone” is the apex of the cone. The phrase “the pyramid is slanted” can be rephrased as “the altitude of the pyramid does not pass through the center of the base.” This will help students use the mathematical language necessary to precisely describe the differences between categories of solids. Design Principle(s): Optimize output (for comparison); Maximize meta-awareness Student Facing Your teacher will give you a set of cards that show geometric solids. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories. Then, sort the cards into 2 categories in a different way. Be prepared to explain the meaning of your new categories. Student Facing Are you ready for more? The platonic solids are a special group of solids with some specific criteria: • The faces are all congruent and are all the same regular polygon. • They are convex, meaning that the faces only meet at their edges. • The same number of faces meet at every vertex. 1. Draw a platonic solid constructed with faces that are squares. 2. Draw a platonic solid constructed with faces that are triangles. 3. Draw a different platonic solid constructed with faces that are triangles. Activity Synthesis Select groups to share their categories and how they sorted their solids. Choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between right and oblique solids, and another distinguishes between solids with an apex versus those without. Attend to the language that students use to describe their categories, giving them opportunities to describe their solids more precisely. Highlight the use of terms like hexagonal, perpendicular, and circular. If students use phrasing such as: “The pyramids and cones get smaller while the cylinders and prisms do not,” encourage them to use the language of cross sections. A sample response might be: “Cross sections taken parallel to the base of prisms and cylinders are congruent throughout the solid. On the other hand, cross sections taken parallel to pyramid and cone bases are similar to each other, but are not congruent.” Tell students that we can use the categories they created to define some characteristics of solids. A pyramid is a solid with one face (called the base) that’s a polygon. All the other faces are triangles that all meet at a single vertex, called the apex. A cone also has a base and an apex, but its base is a circle and its other surface is curved. Just like prisms and cylinders can be right and oblique, so can cones and some pyramids. For a cone, imagine dropping a line from the cone’s apex straight down at a right angle to the base. If this line goes through the center of the base, then the cone is right. Otherwise, the cone is oblique. Pyramids with bases that have a center, like a square, a a pentagon, or an equilateral triangle, can also be considered right or oblique in the same way as cones. For example, the cone on Card E from this activity is a right cone because its apex is centered directly over the center of its base. However, the pyramid on Card G has its center shifted; if we drop a height line straight down at right angles to the plane of the base, the line doesn’t hit the center of the pyramid’s base. This pyramid is oblique. Point out that some mathematicians consider a cone to be a “circular pyramid,” others consider pyramids to be “polygonal cones,” and still others classify them in totally separate categories. Regardless of what we call them, the two kinds of solids share certain properties that will be explored in upcoming activities. 12.3: Building a Prism from Pyramids (15 minutes) In this activity, students build a triangular prism out of 3 pyramids and make a conjecture about the volume of one of the pyramids. This activity creates a foundation for upcoming activities in which students will derive the formula for the volume of a pyramid. Arrange students in groups of 3. Provide each group with scissors, tape, and 1 set of nets. Tell students that they’ll be building pyramids, and that they should save the pyramids when they’re done for use in an upcoming activity. Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one step at a time and monitor students to ensure they are making progress throughout the activity. Supports accessibility for: Organization; Attention Student Facing Your teacher will give your group 3 nets. Each student should select 1 of the 3 nets. 1. Cut out your net and assemble a pyramid. The printed side of the net should face outward. 2. Assemble your group’s 3 pyramids into a triangular prism. Each pair of triangles with a matching pattern will come together to form one of the rectangular faces of the prism. You will need to disassemble the prism in a later activity, so use only a small amount of tape (or no tape at all if possible). 3. Make a conjecture about the relationship between the volume of the pyramid marked P1 and the volume of the prism. 4. What information would you need to verify that your conjecture is true? Don’t throw away your pyramids! You’ll use them again. Activity Synthesis The goal of the discussion is to make observations about the relationships between the 3 pyramids and the prism. Here are some questions for discussion. It’s okay for the answers about triangle congruence to be informal. • “Take a look at the pyramid marked P3. Which face would you consider its base? Is there only one possibility?” (Any face of this pyramid could be considered the base. No matter which face we choose to call the base, the remaining faces are all triangles. This is actually true for all 3 pyramids.) • “Which faces of the prism are its bases?” (The faces marked P1 and P3 are the prism’s bases.) • “Do the pyramids marked P1 and P3 have any congruent faces? If so, which are they, and how do you know?” (The faces marked P1 and P3 are congruent because they are the prism’s bases. The faces with the lines are also congruent, because together, they form a rectangle.) • “Do the pyramids marked P2 and P3 have any congruent faces? If so, which are they, and how do you know?” (The gray-colored faces are congruent because together, they form a rectangle. The two faces that are unmarked are congruent to each other. When assembled into the prism, each line segment that forms the sides of the triangles is shared between the two shapes.) To ensure the pyramids are available for an upcoming activity, collect the assembled pyramids or direct students to place them in a safe storage area. Speaking: MLR8 Discussion Supports. As students share the congruent faces between the pyramids marked P1, P2, and P3, press for details by asking how they know that the faces are congruent. Invite students to repeat their reasoning using mathematical language relevant to the lesson such as prism, base, and rectangle. For example, ask, "Can you say that again, using the term 'base'?" Consider inviting the remaining students to repeat these phrases to provide additional opportunities for all students to produce this language. This will help students justify their reasoning for why certain faces of the pyramids are congruent. Design Principle(s): Support sense-making; Optimize output (for justification) Lesson Synthesis The goal of the discussion is to consider what information would be needed to show that the volumes of the 3 pyramids are equal. Display this image for all to see. Ask students, “How do the volumes of these 2 pyramids compare? How do you know?” Challenge them to use the language of cross sections. The bases of the pyramids have equal area, and the pyramids have the same height. Even though one pyramid’s apex is shifted compared to the other pyramid, the cross sections of the two pyramids have the same area at all heights. Therefore, the pyramids have the same volume. Choose 2 of the 3 assembled pyramids and display them for all to see. Ask students, “What would we need to know in order to verify the volumes of these two pyramids are equal?” We would need to know the pyramids have bases with equal area, and that the heights of the pyramids are equal. Then, the pyramids’ cross sections would have equal area at all heights, and the pyramids would have equal 12.4: Cool-down - How Many Faces? (5 minutes) Student Facing Pyramids and cones are different from prisms and cylinders in that they have just one base and an apex, or a single point at which the other faces of the solid meet. Cones are like cylinders and prisms in that they can be oblique or right. If a line dropped from the cone’s apex at right angles to the base goes through the center of the base, then the cone is right. Otherwise, the cone is oblique. Pyramids that have a clear center in their bases can also be considered right or oblique. We can use relationships between pyramids and prisms to build a formula for the volume of a pyramid. The image shows 3 square pyramids assembled into a cube. We’ll use similar thinking, but with triangular pyramids and prisms, to create a pyramid volume formula in an upcoming lesson.	{"url":"https://curriculum.illustrativemathematics.org/HS/teachers/2/5/12/index.html","timestamp":"2024-11-07T23:15:12Z","content_type":"text/html","content_length":"110275","record_id":"<urn:uuid:fd650a1c-6e80-4c58-a4ec-f957281078fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00434.warc.gz"}
Mitchell Feigenbaum (1944–2019), 4.66920160910299067185320382…—Stephen Wolfram Writings Behind the Feigenbaum Constant It’s called the Feigenbaum constant, and it’s about 4.6692016. And it shows up, quite universally, in certain kinds of mathematical—and physical—systems that can exhibit chaotic behavior. Mitchell Feigenbaum, who died on June 30 at the age of 74, was the person who discovered it—back in 1975, by doing experimental mathematics on a pocket calculator. It became a defining discovery in the history of chaos theory. But when it was first discovered, it was a surprising, almost bizarre result, that didn’t really connect with anything that had been studied before. Somehow, though, it’s fitting that it should have been Mitchell Feigenbaum—who I knew for nearly 40 years—who would discover it. Trained in theoretical physics, and a connoisseur of its mathematical traditions, Mitchell always seemed to see himself as an outsider. He looked a bit like Beethoven—and projected a certain stylish sense of intellectual mystery. He would often make strong assertions, usually with a conspiratorial air, a twinkle in his eye, and a glass of wine or a cigarette in his hand. He would talk in long, flowing sentences which exuded a certain erudite intelligence. But ideas would jump around. Sometimes detailed and technical. Sometimes leaps of intuition that I, for one, could not follow. He was always calculating, staying up until 5 or 6 am, filling yellow pads with formulas and stressing Mathematica with elaborate algebraic computations that might run for hours. He published very little, and what he did publish he was often disappointed wasn’t widely understood. When he died, he had been working for years on the optics of perception, and on questions like why the Moon appears larger when it’s close to the horizon. But he never got to the point of publishing anything on any of this. For more than 30 years, Mitchell’s official position (obtained essentially on the basis of his Feigenbaum constant result) was as a professor at the Rockefeller University in New York City. (To fit with Rockefeller’s biological research mission, he was themed as the Head of the “Laboratory of Mathematical Physics”.) But he dabbled elsewhere, lending his name to a financial computation startup, and becoming deeply involved in inventing new cartographic methods for the Hammond World Atlas. What Mitchell Discovered The basic idea is quite simple. Take a value x between 0 and 1. Then iteratively replace x by a x (1 – x). Let’s say one starts from x = , and takes a = 3.2. Then here’s what one gets for the successive values of x: ListLinePlot[NestList[Compile[x, 3.2 x (1 - x)], N[1/3], 50], Mesh -> All, PlotRange -> {0, 1}, Frame -> True] After a little transient, the values of x are periodic, with period 2. But what happens with other values of a? Here are a few results for this so-called “logistic map”: ListLinePlot[NestList[Compile[x, a x (1 - x)], N[1/3], 50], Mesh -> All, PlotRange -> {0, 1}, Frame -> True, FrameTicks -> None]], StringTemplate["a = ``"][a]], {a, 2.75, 4, .25}], 3], Spacings -> {.1, -.1}] For small a, the values of x quickly go to a fixed point. For larger a they become periodic, first with period 2, then 4. And finally, for larger a, the values start bouncing around seemingly One can summarize this by plotting the values of x (here, 300, after dropping the first 50 to avoid transients) reached as a function of the value of a: Table[{a, #} & /@ Drop[NestList[Compile[x, a x (1 - x)], N[1/3], 300], 50], {a, 0, 4, .01}], 1], Frame -> True, FrameLabel -> {"a", "x"}] As a increases, one sees a cascade of “period doublings”. In this case, they’re at a = 3, a ≃ 3.449, a ≃ 3.544090, a ≃ 3.5644072. What Mitchell noticed is that these successive values approach a limit (here a[∞] ≃ 3.569946) in a geometric sequence, with a[∞] – a[n] ~ δ^–n and δ ≃ 4.669. That’s a nice little result. But here’s what makes it much more significant: it isn’t just true about the specific iterated map x ⟶ a x (1 – x); it’s true about any map like that. Here, for example, is the “bifurcation diagram” for x ⟶ a sin(π): Table[{a, #} & /@ Drop[NestList[Compile[x, a Sin[Pi Sqrt@x]], N[1/3], 300], 50], {a, 0, 1, .002}], 1], Frame -> True, FrameLabel -> {"a", "x"}] The details are different. But what Mitchell noticed is that the positions of the period doublings again form a geometric sequence, with the exact same base: δ ≃ 4.669. It’s not just that different iterated maps give qualitatively similar results; when one measures the convergence rate this turns out be exactly and quantitatively the same—always δ ≃ 4.669. And this was Mitchell’s big discovery: a quantitatively universal feature of the approach to chaos in a class of systems. The Scientific Backstory The basic idea behind iterated maps has a long history, stretching all the way back to antiquity. Early versions arose in connection with finding successive approximations, say to square roots. For example, using Newton’s method from the late 1600s, can be obtained by iterating x ⟶ (here starting from x = 1): NestList[Function[x, 1/x + x/2], N[1, 8], 6] The notion of iterating an arbitrary function seems to have first been formalized in an 1870 paper by Ernst Schröder (who was notable for his work in formalizing things from powers to Boolean algebra ), although most of the discussion that arose was around solving functional equations, not actually doing iterations. (An exception was the investigation of regions of convergence for Newton’s approximation by Arthur Cayley in 1879.) In 1918 Gaston Julia made a fairly extensive study of iterated rational functions in the complex plane—inventing, if not drawing, Julia sets. But until fractals in the late 1970s (which soon led to the Mandelbrot set), this area of mathematics basically languished. But quite independent of any pure mathematical developments, iterated maps with forms similar to x ⟶ a x (1 – x) started appearing in the 1930s as possible practical models in fields like population biology and business cycle theory—usually arising as discrete annualized versions of continuous equations like the Verhulst logistic differential equation from the mid-1800s. Oscillatory behavior was often seen—and in 1954 William Ricker (one of the founders of fisheries science) also found more complex behavior when he iterated some empirical fish reproduction curves. Back in pure mathematics, versions of iterated maps had also shown up from time to time in number theory. In 1799 Carl Friedrich Gauss effectively studied the map x → FractionalPart[] in connection with continued fractions. And starting in the late 1800s there was interest in studying maps like x ⟶ FractionalPart[a x] and their connections to the properties of the number a. Particularly following Henri Poincaré’s work on celestial mechanics around 1900, the idea of sensitive dependence on initial conditions arose, and it was eventually noted that iterated maps could effectively “excavate digits” in their initial conditions. For example, iterating x ⟶ FractionalPart[10 x], starting with the digits of π, gives (effectively just shifting the sequence of digits one place to the left at each step): N[NestList[Function[x, FractionalPart[10 x]], N[Pi, 100], 5], 10] Rest@N[NestList[Function[x, FractionalPart[10 x]], N[Pi, 100], 50], 40], Mesh -> All] (Confusingly enough, with typical “machine precision” computer arithmetic, this doesn’t work correctly, because even though one “runs out of precision”, the IEEE Floating Point standard says to keep on delivering digits, even though they are completely wrong. Arbitrary precision in the Wolfram Language gets it right.) Maps like x ⟶ a x(1 – x) show similar kinds of “digit excavation” behavior (for example, replacing x by sin[π u]^2, x ⟶ 4 x(1 – x) becomes exactly u ⟶ FractionalPart[u, 2]—and this was already known by the 1940s, and, for example, commented on by John von Neumann in connection with his 1949 iterative “middle-square” method for generating pseudorandom numbers by computer. But what about doing experimental math on iterated maps? There wasn’t too much experimental math at all on early digital computers (after all, most computer time was expensive). But in the aftermath of the Manhattan Project, Los Alamos had built its own computer (named MANIAC), that ended up being used for a whole series of experimental math studies. And in 1964 Paul Stein and Stan Ulam wrote a report entitled “Non-linear Transformation Studies on Electronic Computers” that included photographs of oscilloscope-like MANIAC screens displaying output from some fairly elaborate iterated maps. In 1971, another “just out of curiosity” report from Los Alamos (this time by Nick Metropolis [leader of the MANIAC project, and developer of the Monte Carlo method], Paul Stein and his brother Myron Stein) started to give more specific computer results for the behavior logistic maps, and noted the basic phenomenon of period doubling (which they called the “U-sequence”), as well as its qualitative robustness under changes in the underlying map. But quite separately from all of this, there were other developments in physics and mathematics. In 1964 Ed Lorenz (a meteorologist at MIT) introduced and simulated his “naturally occurring” Lorenz differential equations, that showed sensitive dependence on initial conditions. Starting in the 1940s (but following on from Poincaré’s work around 1900) there’d been a steady stream of developments in mathematics in so-called dynamical systems theory—particularly investigating global properties of the solutions to differential equations. Usually there’d be simple fixed points observed; sometimes “limit cycles”. But by the 1970s, particularly after the arrival of early computer simulations (like Lorenz’s), it was clear that for nonlinear equations something else could happen: a so-called “strange attractor”. And in studying so-called “return maps” for strange attractors, iterated maps like the logistic map again appeared. But it was in 1975 that various threads of development around iterated maps somehow converged. On the mathematical side, dynamical systems theorist Jim Yorke and his student Tien-Yien Li at the University of Maryland published their paper “Period Three Implies Chaos”, showing that in an iterated map with a particular parameter value, if there’s ever an initial condition that leads to a cycle of length 3, there must be other initial conditions that don’t lead to cycles at all—or, as they put it, show chaos. (As it turned out, Aleksandr Sarkovskii—who was part of a Ukrainian school of dynamical systems research—had already in 1962 proved the slightly weaker result that a cycle of period 3 implies cycles of all periods.) But meanwhile there had also been growing interest in things like the logistic maps among mathematically oriented population biologists, leading to the rather readable review (published in mid-1976) entitled “Simple Mathematical Models with Very Complicated Dynamics” by physics-trained Australian Robert May, who was then a biology professor at Princeton (and would subsequently become science advisor to the UK government, and is now “Baron May of Oxford”). But even though things like sketches of bifurcation diagrams existed, the discovery of their quantitatively universal properties had to await Mitchell Feigenbaum and his discovery. Mitchell’s Journey Mitchell Feigenbaum grew up in Brooklyn, New York. His father was an analytical chemist, and his mother was a public-school teacher. Mitchell was unenthusiastic about school, though did well on math and science tests, and managed to teach himself calculus and piano. In 1960, at age 16, as something of a prodigy, he enrolled in the City College of New York, officially studying electrical engineering, but also taking physics and math classes. After graduating in 1964, he went to MIT. Initially he was going to do a PhD in electrical engineering, but he quickly switched to physics. But although he was enamored of classic mathematical physics (as represented, for example, in the books of Landau and Lifshiftz), he ended up writing his thesis on a topic set by his advisor about particle physics, and specifically about evaluating a class of Feynman diagrams for the scattering of photons by scalar particles (with lots of integrals, if not special functions). It wasn’t a terribly exciting thesis, but in 1970 he was duly dispatched to Cornell for a postdoc position. Mitchell struggled with motivation, preferring to hang out in coffee shops doing the New York Times crossword (at which he was apparently very fast) to doing physics. But at Cornell, Mitchell made several friends who were to be important to him. One was Predrag Cvitanović, a star graduate student from what is now Croatia, who was studying quantum electrodynamics, and with whom he shared an interest in German literature. Another was a young poet named Kathleen Doorish (later, Kathy Hammond), who was a friend of Predrag’s. And another was a rising-star physics professor named Pete Carruthers, with whom he shared an interest in classical music. In the early 1970s quantum field theory was entering a golden age. But despite the topic of his thesis, Mitchell didn’t get involved, and in the end, during his two years at Cornell, he produced no visible output at all. Still, he had managed to impress Hans Bethe enough to be dispatched for another postdoc position, though now at a place lower in the pecking order of physics, Virginia Polytechnic Institute, in rural Virginia. At Virginia Tech, Mitchell did even less well than at Cornell. He didn’t interact much with people, and he produced only one three-page paper: “The Relationship between the Normalization Coefficient and Dispersion Function for the Multigroup Transport Equation”. As its title might suggest, the paper was quite technical and quite unexciting. As Mitchell’s two years at Virginia Tech drew to a close it wasn’t clear what was going to happen. But luck intervened. Mitchell’s friend from Cornell, Pete Carruthers, had just been hired to build up the theory division (“T Division”) at Los Alamos, and given carte blanche to hire several bright young physicists. Pete would later tell me with pride (as part of his advice to me about general scientific management) that he had a gut feeling that Mitchell could do something great, and that despite other people’s input—and the evidence—he decided to bet on Mitchell. Having brought Mitchell to Los Alamos, Pete set about suggesting projects for him. At first, it was following up on some of Pete’s own work, and trying to compute bulk collective (“transport”) properties of quantum field theories as a way to understand high-energy particle collisions—a kind of foreshadowing of investigations of quark-gluon plasma. But soon Pete suggested that Mitchell try looking at fluid turbulence, and in particular on seeing whether renormalization group methods might help in understanding it. Whenever a fluid—like water—flows sufficiently rapidly it forms lots of little eddies and behaves in a complex and seemingly random way. But even though this qualitative phenomenon had been discussed for centuries (with, for example, Leonardo da Vinci making nice pictures of it), physics had had remarkably little to say about it—though in the 1940s Andrei Kolmogorov had given a simple argument that the eddies should form a cascade with a k distribution of energies. At Los Alamos, though, with its focus on nuclear weapons development (inevitably involving violent fluid phenomena), turbulence was a very important thing to understand—even if it wasn’t obvious how to approach it. But in 1974, there was news that Ken Wilson from Cornell had just “solved the Kondo problem” using a technique called the renormalization group. And Pete Carruthers suggested that Mitchell should try to apply this technique to turbulence. The renormalization group is about seeing how changes of scale (or other parameters) affect descriptions (and behavior) of systems. And as it happened, it was Mitchell’s thesis advisor at MIT, Francis Low, who, along with Murray Gell-Mann, had introduced it back in 1954 in the context of quantum electrodynamics. The idea had lain dormant for many years, but in the early 1970s it came back to life with dramatic—though quite different—applications in both particle physics (specifically, QCD) and condensed matter physics. In a piece of iron at room temperature, you can basically get all electron spins associated with each atom lined up, so the iron is magnetized. But if you heat the iron up, there start to be fluctuations, and suddenly—above the so-called Curie temperature (770°C for iron)—there’s effectively so much randomness that the magnetization disappears. And in fact there are lots of situations (think, for example, melting or boiling—or, for that matter, the formation of traffic jams) where this kind of sudden so-called phase transition occurs. But what is actually going on in a phase transition? I think the clearest way to see this is by looking at an analog in cellular automata. With the particular rule shown below, if there aren’t very many initial black cells, the whole system will soon be white. But if you increase the number of initial black cells (as a kind of analog of increasing the temperature in a magnetic system), then suddenly, in this case at 50% black, there’s a sharp transition, and now the whole system eventually becomes black. (For phase transition experts: yes, this is a phase transition in a 1D system; one only needs 2D if the system is required to be microscopically reversible.) "RuleNumber" -> 294869764523995749814890097794812493824, "Colors" -> 4\|>, 3 Boole[Thread[RandomReal[{0, 1}, 2000] < rho]], {500, {-300, 300}}], FrameLabel -> {None, Round[100 rho], "% black"}]}], {rho, {0.4, 0.45, 0.55, 0.6}}], -30] But what does the system do near 50% black? In effect, it can’t decide whether to finally become black or white. And so it ends up showing a whole hierarchy of “fluctuations” from the smallest scales to the largest. And what became clear by the 1960s is that the “critical exponents” characterizing the power laws describing these fluctuations are universal across many different systems. But how can one compute these critical exponents? In a few toy cases, analytical methods were known. But mostly, something else was needed. And in the late 1960s Ken Wilson realized that one could use the renormalization group, and computers. One might have a model for how individual spins interact. But the renormalization group gives a procedure for “scaling up” to the interactions of larger and larger blocks of spins. And by studying that on a computer, Ken Wilson was able to start computing critical exponents. At first, the physics world didn’t pay much attention, not least because they weren’t used to computers being so intimately in the loop in theoretical physics. But then there was the Kondo problem (and, yes, so far as I know, it has no relation to modern Kondoing—though it does relate to modern quantum dot cellular automata). In most materials, electrical resistivity decreases as the temperature decreases (going to zero for superconductors even above absolute zero). But back in the 1930s, measurements on gold had shown instead an increase of resistivity at low temperatures. By the 1960s, it was believed that this was due to the scattering of electrons from magnetic impurities—but calculations ran into trouble, generating infinite results. But then, in 1975, Ken Wilson applied his renormalization group methods—and correctly managed to compute the effect. There was still a certain mystery about the whole thing (and it probably didn’t help that—at least when I knew him in the 1980s and beyond—I often found Ken Wilson’s explanations quite hard to understand). But the idea that the renormalization group could be important was So how might it apply to fluid turbulence? Kolmogorov’s power law seemed suggestive. But could one take the Navier–Stokes equations which govern idealized fluid flow and actually derive something like this? This was the project on which Mitchell Feigenbaum embarked. The Big Discovery The Navier–Stokes equations are very hard to work with. In fact, to this day it’s still not clear how even the most obvious feature of turbulence—its apparent randomness—arises from these equations. (It could be that the equations aren’t a full or consistent mathematical description, and one’s actually seeing amplified microscopic molecular motions. It could be that—as in chaos theory and the Lorenz equations—it’s due to amplification of randomness in the initial conditions. But my own belief, based on work I did in the 1980s, is that it’s actually an intrinsic computational phenomenon —analogous to the randomness one sees in my rule 30 cellular automaton.) So how did Mitchell approach the problem? He tried simplifying it—first by going from equations depending on both space and time to ones depending only on time, and then by effectively making time discrete, and looking at iterated maps. Through Paul Stein, Mitchell knew about the (not widely known) previous work at Los Alamos on iterated maps. But Mitchell didn’t quite know where to go with it, though having just got a swank new HP-65 programmable calculator, he decided to program iterated maps on it. Then in July 1975, Mitchell went (as I also did a few times in the early 1980s) to the summer physics hang-out-together event in Aspen, CO. There he ran into Steve Smale—a well-known mathematician who’d been studying dynamical systems—and was surprised to find Smale talking about iterated maps. Smale mentioned that someone had asked him if the limit of the period-doubling cascade a[∞] ≃ 3.56995 could be expressed in terms of standard constants like π and . Smale related that he’d said he didn’t know. But Mitchell’s interest was piqued, and he set about trying to figure it out. He didn’t have his HP-65 with him, but he dove into the problem using the standard tools of a well-educated mathematical physicist, and had soon turned it into something about poles of functions in the complex plane—about which he couldn’t really say anything. Back at Los Alamos in August, though, he had his HP-65, and he set about programming it to find the bifurcation points a[n]. The iterative procedure ran pretty fast for small n. But by n = 5 it was taking 30 seconds. And for n = 6 it took minutes. While it was computing, however, Mitchell decided to look at the a[n] values he had so far—and noticed something: they seemed to be converging geometrically to a final value. At first, he just used this fact to estimate a[∞], which he tried—unsuccessfully—to express in terms of standard constants. But soon he began to think that actually the convergence exponent δ was more significant than a[∞]—since its value stayed the same under simple changes of variables in the map. For perhaps a month Mitchell tried to express δ in terms of standard constants. But then, in early October 1975, he remembered that Paul Stein had said period doubling seemed to look the same not just for logistic maps but for any iterated map with a single hump. Reunited with his HP-65 after a trip to Caltech, Mitchell immediately tried the map x ⟶ sin(x)—and discovered that, at least to 3-digit precision, the exponent δ was exactly the same. He was immediately convinced that he’d discovered something great. But Stein told him he needed more digits to really conclude much. Los Alamos had plenty of powerful computers—so the next day Mitchell got someone to show him how to write a program in FORTRAN on one of them to go further—and by the end of the day he had managed to compute that in both cases δ was about 4.6692. The computer he used was a typical workhorse US scientific computer of the day: a CDC 6000 series machine (of the same type I used when I first moved to the US in 1978). It had been designed by Seymour Cray, and by default it used 60-bit floating-point numbers. But at this precision (about 14 decimal digits), 4.6692 was as far as Mitchell could compute. Fortunately, however, Pete’s wife Lucy Carruthers was a programmer at Los Alamos, and she showed Mitchell how to use double precision—with the result that he was able to compute δ to 11-digit precision, and determine that the values for his two different iterated maps agreed. Within a few weeks, Mitchell had found that δ seemed to be universal whenever the iterated map had a single quadratic maximum. But he didn’t know why this was, or have any particular framework for thinking about it. But still, finally, at the age of 30, Mitchell had discovered something that he thought was really interesting. On Mitchell’s birthday, December 19, he saw his friend Predrag, and told him about his result. But at the time, Predrag was working hard on mainstream particle physics, and didn’t pay too much Mitchell continued working, and within a few months he was convinced that not only was the exponent δ universal—the appropriately scaled, limiting, infinitely wiggly, actual iteration of the map was too. In April 1976 Mitchell wrote a report announcing his results. Then on May 2, 1976, he gave a talk about them at the Institute for Advanced Study in Princeton. Predrag was there, and now he got interested in what Mitchell was doing. As so often, however, it was hard to understand just what Mitchell was talking about. But by the next day, Predrag had successfully simplified things, and come up with a single, explicit, functional equation for the limiting form of the scaled iterated map: g(g(x)) = , with α ≃ 2.50290—implying that for any iterated map of the appropriate type, the limiting form would always look like an even wigglier version of: fUD[z_] = 1. - 1.5276329970363323 z^2 + 0.1048151947874277 z^4 + 0.026705670524930787 z^6 - 0.003527409660464297 z^8 + 0.00008160096594827505 z^10 + 0.000025285084886512315 z^12 - 2.5563177536625283^-6 z^14 - 9.65122702290271^-8 z^16 + 2.8193175723520713^-8 z^18 - 2.771441260107602^-10 z^20 - 3.0292086423142963^-10 z^22 + 2.6739057855563045^-11 z^24 + 9.838888060875235^-13 z^26 - 3.5838769501333333^-13 z^28 + 2.063994985307743*^-14 z^30; fCF = Compile[{z}, Module[{\[Alpha] = -2.5029078750959130867, n, \[Zeta]}, n = If[Abs[z] <= 1., 0, Ceiling[Log[-\[Alpha], Abs[z]]]]; \[Zeta] = z/\[Alpha]^n; Do[\[Zeta] = #, {2^n}]; \[Alpha]^n \[Zeta]]] &[fUD[\[Zeta]]]; Plot[fCF[x], {x, -100, 100}, MaxRecursion -> 5, PlotRange -> All] How It Developed The whole area of iterated maps got a boost on June 10, 1976, with the publication in Nature of Robert May’s survey about them, written independent of Mitchell and (of course) not mentioning his results. But in the months that followed, Mitchell traveled around and gave talks about his results. The reactions were mixed. Physicists wondered how the results related to physics. Mathematicians wondered about their status, given that they came from experimental mathematics, without any formal mathematical proof. And—as always—people found Mitchell’s explanations hard to understand. In the fall of 1976, Predrag went as a postdoc to Oxford—and on the very first day that I showed up as 17-year-old particle-physics-paper-writing undergraduate, I ran into him. We talked mostly about his elegant “bird tracks” method for doing group theory (about which he finally published a book 32 years later). But he also tried to explain iterated maps. And I still remember him talking about an idealized model for fish populations in the Adriatic Sea (only years later did I make the connection that Predrag was from what is now Croatia). At the time I didn’t pay much attention, but somehow the idea of iterated maps lodged in my consciousness, soon mixed together with the notion of fractals that I learned from Benoit Mandelbrot’s book . And when I began to concentrate on issues of complexity a couple of years later, these ideas helped guide me towards systems like cellular automata. But back in 1976, Mitchell (who I wouldn’t meet for several more years) was off giving lots of talks about his results. He also submitted a paper to the prestigious academic journal Advances in Mathematics. For 6 months he heard nothing. But eventually the paper was rejected. He tried again with another paper, now sending it to the SIAM Journal of Applied Mathematics. Same result. I have to say I’m not surprised this happened. In my own experience of academic publishing (now long in the past), if one was reporting progress within an established area it wasn’t too hard to get a paper published. But anything genuinely new or original one could pretty much count on getting rejected by the peer review process, either through intellectual shortsightedness or through academic corruption. And for Mitchell there was the additional problem that his explanations weren’t easy to understand. But finally, in late 1977, Joel Lebowitz, editor of the Journal of Statistical Physics, agreed to publish Mitchell’s paper—essentially on the basis of knowing Mitchell, even though he admitted he didn’t really understand the paper. And so it was that early in 1978 “Quantitative Universality for a Class of Nonlinear Transformations”—reporting Mitchell’s big result—officially appeared. (For purposes of academic priority, Mitchell would sometimes quote a summary of a talk he gave on August 26, 1976, that was published in the Los Alamos Theoretical Division Annual Report 1975–1976. Mitchell was quite affected by the rejection of his papers, and for years kept the rejection letters in his desk drawer.) Mitchell continued to travel the world talking about his results. There was interest, but also confusion. But in the summer of 1979, something exciting happened: Albert Libchaber in Paris reported results on a physical experiment on the transition to turbulence in convection in liquid helium—where he saw period doubling, with exactly the exponent δ that Mitchell had calculated. Mitchell’s δ apparently wasn’t just universal to a class of mathematical systems—it also showed up in real, physical systems. Pretty much immediately, Mitchell was famous. Connections to the renormalization group had been made, and his work was becoming fashionable among both physicists and mathematicians. Mitchell himself was still traveling around, but now he was regularly hobnobbing with the top physicists and mathematicians. I remember him coming to Caltech, perhaps in the fall of 1979. There was a certain rock-star character to the whole thing. Mitchell showed up, gave a stylish but somewhat mysterious talk, and was then whisked away to talk privately with Richard Feynman and Murray Gell-Mann. Soon Mitchell was being offered all sorts of high-level jobs, and in 1982 he triumphantly returned to Cornell as a full professor of physics. There was an air of Nobel Prize–worthiness, and by June 1984 he was appearing in the New York Times magazine, in full Beethoven mode, in front of a Cornell waterfall: Still, the mathematicians weren’t satisfied. As with Benoit Mandelbrot’s work, they tended to see Mitchell’s results as mere “numerical conjectures”, not proven and not always even quite worth citing. But top mathematicians (who Mitchell had befriended) were soon working on the problem, and results began to appear—though it took a decade for there to be a full, final proof of the universality of δ. Where the Science Went So what happened to Mitchell’s big discovery? It was famous, for sure. And, yes, period-doubling cascades with his universal features were seen in a whole sequence of systems—in fluids, optics and more. But how general was it, really? And could it, for example, be extended to the full problem of fluid turbulence? Mitchell and others studied systems other than iterated maps, and found some related phenomena. But none were quite as striking as Mitchell’s original discovery. In a sense, my own efforts on cellular automata and the behavior of simple programs, beginning around 1981, have tried to address some of the same bigger questions as Mitchell’s work might have led to. But the methods and results have been very different. Mitchell always tried to stay close to the kinds of things that traditional mathematical physics can address, while I unabashedly struck out into the computational universe, investigating the phenomena that occur there. I tried to see how Mitchell’s work might relate to mine—and even in my very first paper on cellular automata in 1981 I noted for example that the average density of black cells on successive steps of a cellular automaton’s evolution can be approximated (in “mean field theory”) by an iterated map. I also noted that mathematically the whole evolution of a cellular automaton can be viewed as an iterated map—though on the Cantor set, rather than on ordinary real numbers. In my first paper, I even plotted the analog of Mitchell’s smooth mappings, but now they were wild and discontinuous: Table[FromDigits[CellularAutomaton[#, IntegerDigits[n, 2, 12]], 2], {n, 0, 2^12 - 1}], Sequence[ AspectRatio -> 1, Frame -> True, FrameTicks -> None]], Text[StringTemplate["rule ``"][#]]] & /@ {22, 42, 90, 110}] But try as I might, I could never find any strong connection with Mitchell’s work. I looked for analogs of things like period doubling, and Sarkovskii’s theorem, but didn’t find much. In my computational framework, even thinking about real numbers, with their infinite sequence of digits, was a bit unnatural. Years later, in A New Kind of Science, I had a note entitled “Smooth iterated maps”. I showed their digit sequences, and observed, rather undramatically, that Mitchell’s discovery implied an unusual nested structure at the beginning of the sequences: FractionalDigits[x_, digs_Integer] := NestList[{Mod[2 First[#], 1], Floor[2 First[#]]} &, {x, 0}, digs][[ 2 ;;, -1]]; FractionalDigits[#, 40] & /@ NestList[a # (1 - #) &, N[1/8, 80], 80]]] /@ {2.5, 3.3, 3.4, 3.5, 3.6, 4}] The Rest of the Story (Photograph by Predrag Cvitanović) So what became of Mitchell? After four years at Cornell, he moved to the Rockefeller University in New York, and for the next 30 years settled into a somewhat Bohemian existence, spending most of his time at his apartment on the Upper East Side of Manhattan. While he was still at Los Alamos, Mitchell had married a woman from Germany named Cornelia, who was the sister of the wife of physicist (and longtime friend of mine) David Campbell, who had started the Center for Nonlinear Studies at Los Alamos, and would later go on to be provost at Boston University. But after not too long, Cornelia left Mitchell, taking up instead with none other than Pete Carruthers. (Pete—who struggled with alcoholism and other issues—later reunited with Lucy, but died in 1997 at the age of 61.) When he was back at Cornell, Mitchell met a woman named Gunilla, who had run away from her life as a pastor’s daughter in a small town in northern Sweden at the age of 14, had ended up as a model for Salvador Dalí, and then in 1966 had been brought to New York as a fashion model. Gunilla had been a journalist, video maker, playwright and painter. Mitchell and she married in 1986, and remained married for 26 years, during which time Gunilla developed quite a career as a figurative painter. Mitchell’s last solo academic paper was published in 1987. He did publish a handful of other papers with various collaborators, though none were terribly remarkable. Most were extensions of his earlier work, or attempts to apply traditional methods of mathematical physics to various complex fluid-like phenomena. Mitchell liked interacting with the upper echelons of academia. He received all sorts of honors and recognition (though never a Nobel Prize). But to the end he viewed himself as something of an outsider—a Renaissance man who happened to have focused on physics, but didn’t really buy into all its institutions or practices. From the early 1980s on, I used to see Mitchell fairly regularly, in New York or elsewhere. He became a daily user of Mathematica, singing its praises and often telling me about elaborate calculations he had done with it. Like many mathematical physicists, Mitchell was a connoisseur of special functions, and would regularly talk to me about more and more exotic functions he thought we should add. Mitchell had two major excursions outside of academia. By the mid-1980s, the young poetess—now named Kathy Hammond—that Mitchell had known at Cornell had been an advertising manager for the New York Times and had then married into the family that owned the Hammond World Atlas. And through this connection, Mitchell was pulled into a completely new field for him: cartography. I talked to him about it many times. He was very proud of figuring out how to use the Riemann mapping theorem to produce custom local projections for maps. He described (though I never fully understood it) a very physics-based algorithm for placing labels on maps. And he was very pleased when finally an entirely new edition of the Hammond World Atlas (that he would refer to as “my atlas”) came out. Starting in the 1980s, there’d been an increasing trend for physics ideas to be applied to quantitative finance, and for physicists to become Wall Street quants. And with people in finance continually looking for a unique edge, there was always an interest in new methods. I was certainly contacted a lot about this—but with the success of James Gleick’s 1987 book Chaos (for which I did a long interview, though was only mentioned, misspelled, in a list of scientists who’d been helpful), there was a whole new set of people looking to see how “chaos” could help them in finance. One of those was a certain Michael Goodkin. When he was in college back in the early 1960s, Goodkin had started a company that marketed the legal research services of law students. A few years later, he enlisted several Nobel Prize–winning economists and started what may have been the first hedge fund to do computerized arbitrage trading. Goodkin had always been a high-rolling, globetrotting gambler and backgammon player, and he made and lost a lot of money. And, down on his luck, he was looking for the next big thing—and found chaos theory, and Mitchell Feigenbaum. For a few years he cultivated various physicists, then in 1995 he found a team to start a company called Numerix to commercialize the use of physics-like methods in computations for increasingly exotic financial instruments. Mitchell Feigenbaum was the marquee name, though the heavy lifting was mostly done by my longtime friend Nigel Goldenfeld, and a younger colleague of his named Sasha At the beginning there was lots of mathematical-physics-like work, and Mitchell was quite involved. (He was an enthusiast of Itô calculus, gave lectures about it, and was proud of having found 1000 speed-ups of stochastic integrations.) But what the company actually did was to write C++ libraries for banks to integrate into their systems. It wasn’t something Mitchell wanted to do long term. And after a number of years, Mitchell’s active involvement in the company declined. (I’d met Michael Goodkin back in 1998, and 14 years later—having recently written his autobiography The Wrong Answer Faster: The Inside Story of Making the Machine That Trades Trillions—he suddenly contacted me again, pitching my involvement in a rather undefined new venture. Mitchell still spoke highly of Michael, though when the discussion rather bizarrely pivoted to me basically starting and CEOing a new company, I quickly dropped it.) I had many interactions with Mitchell over the years, though they’re not as well archived as they might be, because they tended to be verbal rather than written, since, as Mitchell told me (in email): “I dislike corresponding by email. I still prefer to hear an actual voice and interact…” There are fragments in my archive, though. There’s correspondence, for example, about Mitchell’s 2004 60th-birthday event, that I couldn’t attend because it conflicted with a significant birthday for one of my children. In lieu of attending, I commissioned the creation of a “Feigenbaum–Cvitanović Crystal”—a 3D rendering in glass of the limiting function g(z) in the complex plane. It was a little complex to solve the functional equation, and the laser manufacturing method initially shattered a few blocks of glass, but eventually the object was duly made, and sent—and I was pleased many years later to see it nicely displayed in Mitchell’s apartment: Sometimes my archives record mentions of Mitchell by others, usually Predrag. In 2007, Predrag reported (with characteristic wit): “Other news: just saw Mitchell, he is dating Odyssey. No, no, it’s not a high-level Washington type escort service—he is dating Homer’s Odyssey, by computing the positions of low stars as function of the 26000 year precession—says Hiparcus [sic] had it all figured out, but Catholic church succeeded in destroying every single copy of his tables.” Living up to the Renaissance man tradition, Mitchell always had a serious interest in history. In 2013, responding to a piece of mine about Leibniz, Mitchell said he’d been a Leibniz enthusiast since he was a teenager, then explained: “The Newton hagiographer (literally) Voltaire had no idea of the substance of the Monadology, so could only spoof ‘the best of all possible worlds’. Long ago I’ve published this as a verbal means of explaining 2^n universality. Leibniz’s second published paper at age 19, ‘On the Method of Inverse Tangents’, or something like that, is actually the invention of the method of isoclines to solve ODEs, quite contrary to the extant scholarly commentary. Both Leibniz and Newton start with differential equations, already having received the diff. calculus. This is quite an intriguing story.” But the mainstay of Mitchell’s intellectual life was always mathematical physics, though done more as a personal matter than as part of institutional academic work. At some point he was asked by his then-young goddaughter (he never had children of his own) why the Moon looks larger when it’s close to the horizon. He wrote back an explanation (a bit in the style of Euler’s Letters to a German Princess), then realized he wasn’t sure of the answer, and got launched into many years of investigation of optics and image formation. (He’d actually been generally interested in the retina since he was at MIT, influenced by Jerry Lettvin of “What the Frog’s Eye Tells the Frog’s Brain” fame.) He would tell me about it, explaining that the usual theory of image formation was wrong, and he had a better one. He always used the size of the Moon as an example, but I was never quite clear whether the issue was one of optics or perception. He never published anything about what he did, though with luck his manuscripts (rumored to have the makings of a book) will eventually see the light of day—assuming others can understand them. When I would visit Mitchell (and Gunilla), their apartment had a distinctly Bohemian feel, with books, papers, paintings and various devices strewn around. And then there was The Bird. It was a cockatoo, and it was loud. I’m not sure who got it or why. But it was a handful. Mitchell and Gunilla nearly got ejected from their apartment because of noise complaints from neighbors, and they ended up having to take The Bird to therapy. (As I learned in a slightly bizarre—and never executed—plan to make videogames for “they-are-alien-intelligences-right-here-on-this-planet” pets, cockatoos are social and, as pets, arguably really need a “Twitter for Cockatoos”.) (Photograph by Predrag Cvitanović) In the end, though, it was Gunilla who left, with the rumor being that she’d been driven away by The Bird. The last time I saw Mitchell in person was a few years ago. My son Christopher and I visited him at his apartment—and he was in full Mitchell form, with eyes bright, talking rapidly and just a little conspiratorially about the mathematical physics of image formation. “Bird eyes are overrated”, he said, even as his cockatoo squawked in the next room. “Eagles have very small foveas, you know. Their eyes are like telescopes.” “Fish have the best eyes”, he said, explaining that all eyes evolved underwater—and that the architecture hadn’t really changed since. “Fish keep their entire field of view in focus, not like us”, he said. It was charming, eccentric, and very Mitchell. For years, we had talked from time to time on the phone, usually late at night. I saw Predrag a few months ago, saying that I was surprised not to have heard from Mitchell. He explained that Mitchell was sick, but was being very private about it. Then, a few weeks ago, just after midnight, Predrag sent me an email with the subject line “Mitchell is dead”, explaining that Mitchell had died at around 8 pm, and attaching a quintessential Mitchell-in-New-York picture: (Photograph by Predrag Cvitanović) It’s kind of a ritual I’ve developed when I hear that someone I know has died: I immediately search my archives. And this time I was surprised to find that a few years ago Mitchell had successfully reached voicemail I didn’t know I had. So now we can give Mitchell the last word: And, of course, the last number too: 4.66920160910299067185320382… 13 comments 1. Makes me think someone should compile a “Lives of the Mathematicians” and encourage apt young folks to read it, though it might be hard to find others this good to include. Thank you. 2. Thanks for that obituary. It complements well the ones from the Washington post and the New York Times. Universality was one of the great discoveries of the last century. It will certainly have more ramifications in the future. Also nice is the visualization of the math which Feigenbaum discovered experimentally. It illustrates how important experiments have become in mathematics (which contrasts in an astounding way with an increasing dispatch of modern theoretical physics with experiments). Feigenbaum, who was somewhere between mathematics and physics illustrates the fields. I also appreciate the information about the person which one can not find any where else. The story, including of course the episode about `the bird’ could produce material for a movie. It is actually quite surprising that the `person` Mitchell Feigenbaum is hard found on the web (like talks, lectures or interviews). The preference of personal communication rather than through electronic means explains a bit this mystery. 3. Stephen, I thank you for a fascinating article remembering Mitchell. As a long time admirer of your development of Mathematica and the Wolfram Language I wish for your continued success in developing computational knowledge and extending its availability to a broad world-wide user base. Best wishes… Syd Geraghty 4. To Vincent DiCarlo A classic example of what you would like is: Men of Mathematics E.T. Bell Simon and Schuster, Oct 15, 1986 – Biography & Autobiography – 590 pages 7 Reviews From one of the greatest minds in contemporary mathematics, Professor E.T. Bell, comes a witty, accessible, and fascinating look at the beautiful craft and enthralling history of mathematics. Men of Mathematics provides a rich account of major mathematical milestones, from the geometry of the Greeks through Newton’s calculus, and on to the laws of probability, symbolic logic, and the fourth dimension. Bell breaks down this majestic history of ideas into a series of engrossing biographies of the great mathematicians who made progress possible—and who also led intriguing, complicated, and often surprisingly entertaining lives. Never pedantic or dense, Bell writes with clarity and simplicity to distill great mathematical concepts into their most understandable forms for the curious everyday reader. Anyone with an interest in math may learn from these rich lessons, an advanced degree or extensive research is never necessary. 5. Thanks for this great article. I found smooth, analytical function, which give us the same results, as iterations from scenario of professor Feigenbaum. It is easy to find Feigenbaum constant from this new function. It is pity, that he never see it. I am living near Montreal and I thought that he will live much longer. 6. Always a big fan of your memoir writing, detailing both personal and technical aspects in a natural and highly interesting manner. Thank you. 7. I’m wondering if you could possibly tell me (or write a full-blown post on it), how the Ricci Flow relates to renormalization group and universality! I read in Grisha Perelman’s biography that he wasn’t really interested in having closed the Poincare conjecture, rather, he was fascinated by the connections he saw between Ricci Flow and renormalization. Could you possibly write about 8. This is the best obituary I’ve ever read. It not only brings this unusual person alive (after all, the real purpose of an obituary!), but invokes and explains and contextualizas his work. I knew almost nothing about Feigenbaum, and now he shines for me. What a gift; many thanks. 9. It’s so great somebody wrote Mitchell Feigenbaum’s story. I’m still trying to grasp his theories and so glad I found this article, and to hear his voice. Thank You. 10. Thank you for writing a beautiful précis of this talented man’s life. Being the same age roughly, and remembering back to the 60’s and 70’s, it is illustrative of the challenges faced in academia by those with a lot to offer who didn’t exactly fit into a particular pigeonhole. Multidisciplinarity, if there is such a word, is thankfully becoming more tolerated, at least conceptually. Young academics would do well to read what you have written here. 11. I am not a mathematician, squeeked by in calculus. I want to know if I understand correctly the meaning of this number. I think it means in any randomness [or chaos] there is an eventual order that repeats at a dependable frequency. Essentially a sort of mandelbrat evolves from any chaos. By extension of that thought Deep Time, could explain the accidental Big Bang. By further consideration this number could answer, “Who made God?” And physical reality reveal fractals and apparently the Feigenbaums constant, perhaps as cause or fundamentals to the existence of the same? 12. I was pleasantly surprised to find this today. Thank you for sharing! 13. Answering myself, from elswhere I learned at Los Alamos, Mitchell was tasked with answering why atomic explosions stop. With a mathematical model of steady fluid dynamics he introduced, at various mathematical points, frequencies to produce chaos and accidently discovered what appeared to be a constant rate of increase IN CHAOS. With supercomputing he discovered various frequencies produced dependable fractal bifurcations at the constant rate named for him. I had been reading on Mandelbrot fractals and assumed these Feigenbaum fractals reiterated infinitely; were similar to Mandelbrots. But I think I understand that Feigenbaum’s constant is more in line with ‘cooling universe’ : order descends into chaos, showing a dependable rate for that. (I guess that answers why an atomic blast fizzles out.) For an abstract-number-universe the Mandelbrots stem from random interations of seed conditions and suggested structure appears at a constant rate from chaos: Fractals. I thought perhaps Feigenbaum’s constant showed something similar AND was applicable to non-abstract, physically-real systems. I wasn’t thinking of increasing chaos – I was thinking increasing complexity. Hence my wonder over the idea of a seed for either accidental big bang or creation by an abstract complexity approaching sentience. Not being versed in the terminology I misunderstood. But: Feigenbaum fired my imagination and I want to share this: I can’t help but feel that mathematicians could between these constants and the fine particle constant mathematically link abstract Mandelbrot structure with string or brane theory formation of non abstract physical laws. Reciprocal of Feigenbaum for creation? (Big-bang or God. 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Higher Math Class Course Content Prepare ahead of everyone with online courses in all subjects of SSC, HSC, and university admission exams at the lowest cost! Copyright © For example, I served as an Associate Instructor at Indiana University in Bloomington for the courses. M – Finite Mathematics,; M – Brief Survey of. First Year Students: I want to try to place into a higher math course for fall. What are my options? · UArizona New Start Summer Program, which begins June. Higher Mathematics · Course Specification (23/05/) · Past Papers and Marking Instructions · Additional question papers resources · Understanding Standards . Course general description: Math covers topics in ordinary differential equations including: first-order, second- order and higher-order differential. Emphasis will be placed on algebraic thinking and its teaching in high school. Forty hours of secondary classroom observations will be a required activity in. 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Identifying gender bias in candidate lists in proportional representation elections The Norwegian parliamentary elections uses a system of proportional representation. Each county has a number of seats in parliament (based on number of inhabitants and area), and the number of seats given to each party almost proportional to the number of votes the party receives on that county. Since each party can win more than one seat the parties has to prepare a ranked list of people to be elected, where the top name is given the first seat, the second name given the second seat etc. Proportional representation systems like the Norwegian one has been show to be associated with greater gender balance in parliaments than other systems (see table 1 in this paper). Also, the proportion of women in the Norwegian Storting has also increased the last 30 years: Data source: Statistics Norway, table 08219. At the 1981 election, 26% of the elected representatives where women. At the 2013 election, the proportion was almost 40%. One mechanism that can explain this persistent female underrepresentation is that men are overrepresented at the top of the electoral lists. Inspired by a bioinformatics method called Gene Set Enrichment (GSEA) I am going to put this hypothesis to the test. The method is rather simple. Explained in general terms, this is how it works: First you need to calculate a score witch represents the degree of overrepresentation of a category near the top of the list. Each time you encounter an instance belonging to the category your testing you increase the score, otherwise you decrease it. To make the score be a measure of overrepsentation at the top of the list the increase and decrease must be weighted accordingly. The maximum score of this ‘running sum’ is the test statistic. Here I have chosen the function $\frac{1}{\sqrt(i)}$ where i is the number the candidate is on the list (number 1 is the top candidate). To calculate the p-value the same thing is done again repeatedly with different random permutations of the list. The proportion of times the score from these randomizations are greater or equal to the observed score is then the p-value. I am going to use this method on the election lists from Hordaland county from the 1981 and 2013 election. Hordaland had 15 seats in 1981, and 16 seats in 2013. 3 (20 %) women were elected in 1981 and 5 (31.3 %) in 2013. The election lists are available from the Norwegian Social Science Data Services and the National Library of Norway. Here are the results for each party at the two elections: Party 2013 1981 Ap 1 (0.43) 3.58 (0.49) Frp 3.28 (0.195) 3.56 (0.49) H 1.018 (0.66) 3.17 (0.35) Krf 1.24 (0.43) 2.32(0.138) Sp 2.86 (0.49) 2.86 (0.48) Sv 1 (0.24) 0.29 (0.72) V 1.49 (0.59) 1.37 (0.29) The number shown is the score, while the p-value is in parenthesis. A higher score means a higher over representation of men at the top of the list. Even if we ignore problems with multiple testing, none of the parties have a significant over representation of men at the top if the traditional significance threshold of $p \le 0.05$ is used. This is perhaps unexpected, as at least the gender balance in the elected candidates after the 1981 election is significantly biased (p = 0.018, one sided exact binomial test). This really tells us that this method is not really powerful enough to make inferences about this kinds of data. I think one possible improvement would be to somehow score all lists in combination to find an overall gender bias. One could also try a different null model. The one I have used here has randomly shuffled the list in question, maintaining the bias in gender ratio (if any). Instead a the observed score could be compared to random samplings where each gender were sampled with equal probabilities. My final thought is that this whole significance testing approach is inappropriate. Even if the bias is statistical insignificant, it is still there to influence the gender ratio of the elected members of parliament. From looking at some of the lists and their scores, I will say that all scores greater than 1 at least indicate a positive bias towards having more men at the top.	{"url":"https://opisthokonta.net/?p=750","timestamp":"2024-11-05T21:31:28Z","content_type":"text/html","content_length":"33535","record_id":"<urn:uuid:d57acbee-ed11-470c-b983-c9467e77727a>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00707.warc.gz"}
Unit converters - Energy \| energyfaculty.com On this page you will find nine types of energy related unit converters, those are unit converters for: Here you can make conversions of the most commonly used units between the metric and the imperial systems. The energy converter, converts to and from: Joule, Kilojoule, Megajoule, Gigajoule, Petajoule and Exajoule, Watt – hour, Kilowatt – hour and Megawatt – hour, Ampere – hour, Newton – metre, Kilopond metre, Calorie and Kilocalorie and Btu. Calculate CO[2] emissions from various fuels, get the chemical formula for common fuels, generate combustion equations. Try our energy calculators The power converter, converts to and from: Watt, Kilowatt, Horsepower and Newtom metre. The length converter, converts to and from: Ångstrøm, Nanometre, Micrometre, Millimetre, Centimetre, Decimetre Metre and Kilometre, Inch, Foot, Yard and Mile. The area converter, converts to and from: Square Millimetre, Square Centimetre, Square Decimetre, Square metre and Square Kilometre, Square Inch, Square Foot, Square yard and Acre. The volume converter, converts to and from: Cubic Millimetre, Cubic Centimetre, Litre, Cubic metre, Cubic Inch and Cubic Foot, Gallon and Barrel. The weight and mass converter, converts to and from: Picogram, nanogram, Microgram, Milligram, Gram Hectogram, Kilogram and Ton, Pound and Ounce. The temperature converter, converts to and from: Celsius, Kelvin and Fahrenheit. The pressure converter, converts to and from: Millibar, Bar and Atmosphere, Pound, Pascal, Kilopascal and Newton. The speed converter, converts to and from: Metre per second, Kilometres per hour, Miles per hour, Speed of light, Mach (Speed of Sound).	{"url":"https://energyfaculty.com/unit-converters/","timestamp":"2024-11-12T10:04:01Z","content_type":"text/html","content_length":"186378","record_id":"<urn:uuid:963655a2-8e62-459b-9cc7-b0efb861ac15>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00751.warc.gz"}
Computer Science Archives - Quentin Santos This year’s Advent of Code has been brutal (compare the stats of 2023 with that of 2022, especially day 1 part 1 vs. day 1 part 2). It included a problem to solve with dynamic programming as soon as day 12, which discouraged some people I know. This specific problem was particularly gnarly for Advent of Code, with multiple special cases to take into account, making it basically intractable if you are not already familiar with dynamic programming. However, dynamic programming itself is mostly natural when you understand what it does. And many common algorithms are actually just the application of dynamic programming to specific problems, including omnipresent path-finding algorithms such as Dijkstra’s algorithm. This motivated me to write a gentler introduction and a detailed explanation of solving Day 12. Let me start with a rant. You can skip to the following section if you want to get to the meat of the article. The Rant Software engineering is terrible at naming things. • “Bootstrap” is an imaged expression to point to the absurdity and impossibility of a task, but it has become synonymous with “start” without providing any additional information, such as “boot a computer”. The illusion that it is actually meaningful had lead to an absurd level of polysemy. • “Daemon”, for a process that is detached from your terminal • “Cascading Style Sheets”, just to mean that properties can be overridden • “Cookie”, for a piece of data stored on the Web browser, which is automatically sent to the server • “Artificial Intelligence” which is so vague it refers just as well to if-conditions, or to AGI Now, let’s take a look at “dynamic programming”. What can we learn from the name? “Programming” must refer to a style of programming, such as “functional programming”, or maybe “test-driven development”. Then, “dynamic” could mean: • like in dynamic typing, maybe it could refer to the more general idea of handling objects of arbitrary types, with techniques such as virtual classes, trait objects, or a base Object type. • maybe it could be the opposite of preferring immutable state • if it’s not about data, maybe it could be about dynamic code, such as self-modifying code, or JIT • maybe it could be yet another framework such as Agile, SCRUM, XP, V-model, RTE, RAD • maybe it could be referring to using practices from competitive programming? Yes, it’s a stretch, but that might make sense Guess what. It means nothing of that, and it has nothing to do with being “dynamic”. It is an idea that you can use to design an algorithm, so there is a link to “programming”; I will grant it that. Edit: There is a reason why it is named this way. When you look at the historical meaning of “programming”, the expression made sense. See niconiconi’s comment. So, what is it? Basic Caching Let’s say we want to solve a problem by splitting it in smaller similar problems. Basically, a recursive function. Often, we end-up having to solve the same smaller problems many times. The typical example is Fibonacci, where you want to evaluate f(n), which is defined as f(n - 1) + f(n - 2). If we implement it naively, we will end up evaluating f(1) many times: Call tree for evaluating f(6), the 6-th Fibonacci number. To evaluate, f(6), we need to evaluate both f(5) and f(4). To evaluate f(5), we will need f(4) and f(3). Already, we see that we are going to need f(4) in two places. If we go further, we see that we will need f(1) 8 eight times, which happens to be f(6). In fact, in this case, since only f(1) adds anything to the overall result, the number of times we will need f(1) is equal to f(n). And f(n) grows very fast as n grows. Of course, we can avoid doing this. We can just cache the results (or memoize f, in terrible academic vernacular). In the example, once we have evaluated f(4) once, there is no need to evaluate it again, saving 3 evaluations of f(1). By doing the same for f(3) and f(2), we get down to 2 evaluations of f(1). In total, f(…) is evaluated 7 times (f(0), f(1), f(2), f(3), f(4), f(5), f(6)), which is just f(n) + 1. This is theoretically (asymptotically) optimal. But we can look at this in a different way. Optimized Caching With memoization, we keep the recursion: “to solve f(6), I need f(5), which will itself need f(4) […] and f(3) […], and f(4), which will itself need f(3) […] and f(2) […].”. Basically, we figure out what we need just when we need them. Instead, we can make the simple observation that we will need f(0) and f(1) for all other evaluations of f(…). Once we have them, we can evaluate f(2), which will need for all other evaluations of f You can think of it as plucking the leaves (the nodes without descendants) from the call tree we saw before, and repeat until there are no more nodes. In other words, perform a topological sort. With the example, if we have some array F where we can store our partial results: • F[0] = f(0) = 0 • F[1] = f(1) = 1 • F[2] = f(2) = f(1) + f(0) = F[1] + F[0] = 1 + 0 = 1 • F[3] = f(3) = f(2) + f(1) = F[2] + F[1] = 1 + 1 = 2 • F[4] = f(4) = f(3) + f(2) = F[3] + F[2] = 2 + 1 = 3 • F[5] = f(5) = f(4) + f(3) = F[4] + F[3] = 3 + 2 = 5 • F[6] = f(6) = f(5) + f(4) = F[5] + F[4] = 5 + 3 = 8 With this approach, we do not have any recursive call anymore. And that is dynamic programming. It also forces us to think clearly about what information we will be storing. In fact, in the case of Fibonacci we can notice that we only need the two last previous values. In other words: • F[0] = f(0) = 0 • F[1] = f(1) = 1 • F[2] = f(2) = previous + previous_previous = 1 + 0 = 1 • F[3] = f(3) = previous + previous_previous = 1 + 1 = 2 • F[4] = f(4) = previous + previous_previous = 2 + 1 = 3 • F[5] = f(5) = previous + previous_previous = 3 + 2 = 5 • F[6] = f(6) = previous + previous_previous = 5 + 3 = 8 So, we can discard other values and just keep two of them. Doing this in Python, we get: def fibo(n): if n == 0: return 0 previous_previous = 0 previous = 1 for _ in range(n - 1): current = previous_previous + previous (previous, previous_previous) = (current, previous) return previous I like that this gives us a natural and systematic progression from the mathematical definition of the Fibonacci function, to the iterative implementation (not the optimal one, though). Now, Fibonacci is more of a toy example. Let’s have a look at Edit Distance The edit distance between two strings is the smallest number of edits needed to transform one string into the other one. There are actually several versions, depending on what you count as an “edit”. For instance, if you only allow replacing a character by another, you get Hamming distance; evaluating the Hamming distance between two strings is algorithmically very simple. Things become more interesting if you allow insertion and deletion of characters as well. This is the Levenstein distance. Considering this title of the present article, this is of course something that can be solved efficiently using ✨ dynamic programming ✨. To do that, we’ll need to find how we can derive a full solution from solutions to smaller-problems. Let’s say we have two strings: A and B. We’ll note d(X, Y) the edit distance between strings X and Y, and we’ll note x the length of string X. We need to formulate d(A, B) from any combination of d(X, Y) where X is a substring of A and Y a substring of B^1. We’re going to look at a single character. We’ll use the last one. The first one would work just as well but using a middle one would not be as convenient. So, let’s look at A[a – 1] and B[b – 1] (using zero-indexing). We have four cases: • A[a - 1] == B[b - 1], then we can ignore that character and look at the rest, so d(A, B) = d(A[0..a - 1], B[0..b - 1]) • A[a - 1] != B[b - 1], then we could apply any of the three rules. Since we want the smallest number of edits, we’ll need to select the smallest value given by applying each rule: □ substitute the last character of A by that of B, in which case d(A, B) = d(A[0..a - 1], B[0..b - 1]) + 1 □ delete the last character of A, in which case d(A, B) = d(A[0..a - 1], B) + 1 □ insert the last character of B, in which case d(A, B) = d(A, B[0..b - 1]) + 1 • A is actually empty (a = 0), then we need to insert all characters from B^2, so d(A, B) = b • B is actually empty (b = 0), then we need to delete all characters from A, so d(A, B) = a By translating this directly to Python, we get: def levenstein(A: str, B: str) -> int: a = len(A) b = len(B) if a == 0: return b elif b == 0: return a elif A[a - 1] == B[b - 1]: return levenstein(A[:a - 1], B[:b - 1]) return min([ levenstein(A[:a - 1], B[:b - 1]) + 1, levenstein(A[:a - 1], B) + 1, levenstein(A, B[:b - 1]) + 1, assert levenstein("", "puppy") == 5 assert levenstein("kitten", "sitting") == 3 assert levenstein("uninformed", "uniformed") == 1 # way too slow! # assert levenstein("pneumonoultramicroscopicsilicovolcanoconiosis", "sisoinoconaclovociliscipocsorcimartluonomuenp") == 36 As hinted by the last test, this version becomes very slow when comparing long strings with lots of differences. In Fibonnacci, we were doubling the number of instances for each level in the call tree; here, we are tripling it! In Python, we can easily apply memoization: from functools import cache def levenstein(A: str, B: str) -> int: a = len(A) b = len(B) if a == 0: return b elif b == 0: return a elif A[a - 1] == B[b - 1]: return levenstein(A[:a - 1], B[:b - 1]) return min([ levenstein(A[:a - 1], B[:b - 1]) + 1, levenstein(A[:a - 1], B) + 1, levenstein(A, B[:b - 1]) + 1, assert levenstein("", "puppy") == 5 assert levenstein("kitten", "sitting") == 3 assert levenstein("uninformed", "uniformed") == 1 # instantaneous! assert levenstein("pneumonoultramicroscopicsilicovolcanoconiosis", "sisoinoconaclovociliscipocsorcimartluonomuenp") == 36 Now, there is something that makes the code nicer, and more performant, but it is not technically necessary. The trick is that we do not actually need to create new strings in our recursive functions. We can just pass arounds the lengths of the substrings, and always refer to the original strings A and B. Then, our code becomes: from functools import cache def levenstein(A: str, B: str) -> int: def aux(a: int, b: int) -> int: if a == 0: return b elif b == 0: return a elif A[a - 1] == B[b - 1]: return aux(a - 1, b - 1) return min([ aux(a - 1, b - 1) + 1, aux(a - 1, b) + 1, aux(a, b - 1) + 1, return aux(len(A), len(B)) assert levenstein("", "puppy") == 5 assert levenstein("kitten", "sitting") == 3 assert levenstein("uninformed", "uniformed") == 1 # instantaneous! assert levenstein("pneumonoultramicroscopicsilicovolcanoconiosis", "sisoinoconaclovociliscipocsorcimartluonomuenp") == 36 The next step is to build the cache ourselves: def levenstein(A: str, B: str) -> int: # cache[a][b] = levenstein(A[:a], B[:b]) # note the + 1 so that we can actually do cache[len(A)][len(B)] # the list comprehension ensures we create independent rows, not references to the same one cache = [[None] * (len(B) + 1) for _ in range(len(A) + 1)] def aux(a: int, b: int) -> int: if cache[a][b] == None: if a == 0: cache[a][b] = b elif b == 0: cache[a][b] = a elif A[a - 1] == B[b - 1]: cache[a][b] = aux(a - 1, b - 1) cache[a][b] = min([ aux(a - 1, b - 1) + 1, aux(a - 1, b) + 1, aux(a, b - 1) + 1, return cache[a][b] return aux(len(A), len(B)) assert levenstein("", "puppy") == 5 assert levenstein("kitten", "sitting") == 3 assert levenstein("uninformed", "uniformed") == 1 # instantaneous! assert levenstein("pneumonoultramicroscopicsilicovolcanoconiosis", "sisoinoconaclovociliscipocsorcimartluonomuenp") == 36 The last thing we need to do is to replace the recursion with iterations. The important thing is to make sure we do that in the right order^3: def levenstein(A: str, B: str) -> int: # cache[a][b] = levenstein(A[:a], B[:b]) # note the + 1 so that we can actually do cache[len(A)][len(B)] # the list comprehension ensures we create independent rows, not references to the same one cache = [[None] * (len(B) + 1) for _ in range(len(A) + 1)] for a in range(0, len(A) + 1): for b in range(0, len(B) + 1): if a == 0: cache[a][b] = b elif b == 0: cache[a][b] = a elif A[a - 1] == B[b - 1]: # since we are at row a, we have already filled in row a - 1 cache[a][b] = cache[a - 1][b - 1] cache[a][b] = min([ # since we are at row a, we have already filled in row a - 1 cache[a - 1][b - 1] + 1, # since we are at row a, we have already filled in row a - 1 cache[a - 1][b] + 1, # since we are at column b, we have already filled column b - 1 cache[a][b - 1] + 1, return cache[len(A)][len(B)] assert levenstein("", "puppy") == 5 assert levenstein("kitten", "sitting") == 3 assert levenstein("uninformed", "uniformed") == 1 # instantaneous! assert levenstein("pneumonoultramicroscopicsilicovolcanoconiosis", "sisoinoconaclovociliscipocsorcimartluonomuenp") == 36 Now, if you really want to grok dynamic programming, I invite you to try it yourself on the following problems, preferrably in this order: Once you are comfortable with dynamic programming, Day 12 should become much less daunting! Advent of Code, Day 12 In the Advent of Code of December 12th, 2023, you have to solve 1D nonograms. Rather than rephrasing the problem, I will let you read the official description. .??..??...?##. 1,1,3 This can be solved by brute-force. The proper technique for that is backtracking, another terrible name. But the asymptotic complexity is exponential (for n question marks, we have to evaluate 2^n potential solutions). Let’s see how it goes with this example: • .??..??...?##. 1,1,3 the first question mark could be either a . or a #; in the second case, we “consume” the first group of size 1, and the second question mark has to be a . 1. ..?..??...?##. 1,1,3 the next question mark could be either a . or a #; in the second case, we “consume” the first group of size 1, and the next character has to be a ., which is the case 1. .....??...?##. 1,1,3 the backtracking algorithm will continue to explore the 8 cases, but none of them is a valid solution 2. ..#..??...?##. (1),1,3 2. .#...??...?##. (1),1,3 There are 32 candidates, which would make 63 list items. I’ll spare you that. Instead, I want to draw your attention to the items 2.2 and 2: • 2.2. ..#..??...?##. (1),1,3 • 2. .#...??...?##. (1),1,3 They are extremely similar. In fact, if we discard the part that has already been accounted for, they are more like: • 2.2. .??...?##. 1,3 • 2. ..??...?##. 1,3 There is an extra . on the second one, but we can clearly see that it is actually the same problem, and has the same solutions. In other words, just like with Fibonacci, the total number of cases is huge, but many of them will just be repeats of other ones. So we are going to apply memoization. And then, dynamic programming. When we implement the “backtracking” algorithm we’ve overviewed above, we get something like this (not my code): def count_arrangements(conditions, rules): if not rules: return 0 if "#" in conditions else 1 if not conditions: return 1 if not rules else 0 result = 0 if conditions[0] in ".?": result += count_arrangements(conditions[1:], rules) if conditions[0] in "#?": if ( rules[0] <= len(conditions) and "." not in conditions[: rules[0]] and (rules[0] == len(conditions) or conditions[rules[0]] != "#") result += count_arrangements(conditions[rules[0] + 1 :], rules[1:]) return result Note the program above handles ? by treating it as both . and #. The first case is easy, but the second case need to check that it matches the next rules; and for that, it needs to check that there is a separator afterwards, or the end of the string. Since it’s Python, to memoize, we just need to add @cache. To make it dynamic programing, we use the same trick as in the example of the edit distance: we pass the offset in the string, and the offset in the rules as parameters in the recursion. This def count_arrangements(conditions, rules): def aux(i, j): if not rules[j:]: return 0 if "#" in conditions[i:] else 1 if not conditions[i:]: return 1 if not rules[j:] else 0 result = 0 if conditions[i] in ".?": result += aux(i + 1, j) if conditions[i] in "#?": if ( rules[j] <= len(conditions[i:]) and "." not in conditions[i:i + rules[j]] and (rules[j] == len(conditions[i:]) or conditions[i + rules[j]] != "#") result += aux(i + rules[j] + 1, j + 1) return result return aux(0, 0) Then, we implement our own cache and fill it in the right order: def count_arrangements(conditions, rules): cache = [[0] * (len(rules) + 1) for _ in range(len(conditions) + 1)] # note that we are in the indices in reverse order here for i in reversed(range(0, len(conditions) + 1)): for j in reversed(range(0, len(rules) + 1)): if not rules[j:]: result = 0 if "#" in conditions[i:] else 1 elif not conditions[i:]: result = 1 if not rules[j:] else 0 result = 0 if conditions[i] in ".?": # since we are at row i, we already filled in row i + 1 result += cache[i + 1][j] if conditions[i] in "#?": if ( rules[j] <= len(conditions[i:]) and "." not in conditions[i:i + rules[j]] if rules[j] == len(conditions[i:]): # since we are at row i, we already filled in row i + rules[j] > i result += cache[i + rules[j]][j + 1] elif conditions[i + rules[j]] != "#": # since we are at row i, we already filled in row i + rules[j] + 1 > i result += cache[i + rules[j] + 1][j + 1] cache[i][j] = result return cache[0][0] And, voilà! You can also have a look at a Rust implementation (my code, this time). Note: In this case, it looks like the dynamic programming version is slower than the memoized one. But that’s probably due to it being written in unoptimized Python. Note: Independently from using a faster language and micro-optimizations, the dynamic programming version allows us to see that we only need the previous column. Thus, we could replace the 2D array by two 1D arrays (one for the previous column, and one for the column being filled). I’ll concede that dynamic programming is not trivial. But it is far from being unreachable for most programmers. Being able to understand how to split a problem in smaller problems will enable you to use memoization in various contexts, which is already a huge improvement above a naive implementation. However, mastering dynamic programming will let us understand a whole class of algorithms, better understand trade-offs, and make other optimizations possible. So, if you have not already done them, I strongly encourage you to practice on these problems: And don’t forget to benchmark and profile your code! 1. Excluding, of course, d(A, B) itself ↩︎ 2. B could be empty as well, in which case we need to insert 0 characters ↩︎ 3. Note that we could permute the inner and outer loops as shown below. In this case, it works just as well: for b in range(0, len(B) + 1): for a in range(0, len(A) + 1):	{"url":"https://qsantos.fr/category/computer-science/","timestamp":"2024-11-08T05:46:44Z","content_type":"text/html","content_length":"65746","record_id":"<urn:uuid:c5159131-da4f-4ddf-a7d8-bd4a6abcc08b>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00726.warc.gz"}
Geometric Sequences and Series - A Plus Topper Geometric Sequences and Series A sequence is an ordered list of numbers. The sum of the terms of a sequence is called a series. Two such sequences are the arithmetic and geometric sequences. Let’s investigate the geometric sequence. If a sequence of values follows a pattern of multiplying a fixed amount (not zero) times each term to arrive at the following term, it is referred to as a geometric sequence. The number multiplied each time is constant (always the same). The fixed amount multiplied is called the common ratio, r, referring to the fact that the ratio (fraction) of the second term to the first term yields this common multiple. To find the common ratio, divide the second term by the first term. Notice the non-linear nature of the scatter plot of the terms of a geometric sequence. The domain consists of the counting numbers 1, 2, 3, 4, … and the range consists of the terms of the sequence. While the x value increases by a constant value of one, the y value increases by multiples of two (for this graph). Formulas used with geometric sequences and geometric series:	{"url":"https://www.aplustopper.com/geometric-sequences-series/","timestamp":"2024-11-04T15:38:39Z","content_type":"text/html","content_length":"44419","record_id":"<urn:uuid:842440ef-f955-4f56-b68c-8295cc242bcd>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00885.warc.gz"}
IB Biology Mock Exams & Developing Exam Writing Skills Mock exams are happening right now at my school here in Morocco... and what a whirlwind experience they are for our Seniors. They are such an important tool in the students' preparation for the IB exams in May and if you do not do them in your school then I highly recommend you do so! It is an excellent way for students to get a feel for the format, duration and stress of IB exams in a much lower stakes situation. It is also an excellent guide for focusing future revision and determining predicted grades. In this blog post I will describe the three papers of the IB Biology exam and which skills students need for each as well as how I maximize my students learning from our Mocks. The first paper of the IB Biology exam is designed to test the breadth of the content covered in the course, it is the shortest component of the IB exam, taking only 45 minutes for SL students and 1 hour for HL students. It is worth 20% of the overall IB Diploma score for both SL & HL students, the same value as the IA, but less than Paper 2. The tables below list the approximate number of questions from each topic for SL & HL students; this can be a useful tool for students and teachers as they prepare for exams. Paper 1 questions always have four answers and are testing Objectives 1 (Knowledge & Understanding), 2 (Application) & 3 (Formulation, Analysis & Evaluation) with a ratio of 1+2:3, so Objective 3 is much more important on this Paper. Students need to demonstrate higher order thinking skills as well as memorization of facts to be successful on this paper. The Importance of Past Papers One of the best ways to prepare is to practice with past papers, luckily the current syllabus is quite old so there are many exams available. As long as the exam date is after May 2016 you are good to go! This usually means 3 exams are available per year, with Time Zones 1 & 2 (TZ1 & TZ2) in May and one exam in November. You can purchase these exams from the IBO directly or use software such as IB Questionbank, an excellent use of your Science budget in my opinion. If your school has been an IB school for a while you should have paper copies of old exams around, the IBO always sends some extras every year just in case and you get to keep them! Multiple Choice Exam Writing Skills It is always a good idea to spend some time with your students going through different strategies for each of the papers. Here are some of my favourite tips I give students for Paper 1 questions: • Write in the question booklet, the IBO only checks your answer sheet! • Eliminate answers whenever you can, cross them out right on the paper • Circle or underline key words, this is especially important for Objective 3 questions • When in doubt go with your gut, only change your first answer if you are 100% sure • Skip hard questions on the first go through, put a star beside them and come back later • Pay careful attention to any included diagrams, they are there for a reason! Paper 2: Data Analysis & Connecting Topics Paper two is the most important paper of the IB Biology exam, it is worth 40% of the students grade in SL Biology & 36% of the grade in HL Biology. This is also the longest paper of the exam, it is 1 hour and 15 minutes in length for SL students and 2 hours and 15 minutes in length for HL students. When preparing your students for this paper it is wise to divide it into three sections: Section A (34 marks SL & 40 marks HL) includes: • Question 1 a data-based questions (12 marks SL & 15 - 18 marks HL) • Remaining questions are short answer, structured questions which often connect multiple subtopics and may include drawings or calculations Section B (16 marks SL & 32 marks HL) • Students always have a choice in Section B • SL students much choose one from two groups of questions, each group containing three questions, one of which is worth 7 or 8 marks • HL students much choose two from three groups of questions, each group containing three questions, one of which is worth 7 or 8 marks • Each group of questions is worth 15 points with an additional point for the quality of the writing, if an examiner only has to read your response once then you will most likely get than extra The very first question of Section A is a single, long question which includes a large number of graphs and diagrams for students to interpret. These questions usually connect to several different course topics, although they focus much more on analysis and evaluation skills than knowledge and understanding. In preparing for this question it is good to spend some time with students reviewing how to interpret different types of graphs, the importance of slowing down and really examining the axes, the key, the title and any background information provided. Oftentimes the answers to these questions require that students slow down and really read with care. It is also very important to include units and quantities in answers whenever possible. I highly recommend incorporating these types of Past Paper questions into your course assessments and review tools, although perhaps more in Year 2 once students are more familiar with the majority of the course content. Section B includes short answer questions which the IBO calls "essays". One of the first things that students will need to do is choose which questions they want to tackle. Each cluster of questions is worth the same point total - 15 points per question group.You may recall that SL need only choose one cluster while HL students need to choose two. Students should keep in mind the number of points they are confident that they can earn when making their choice... if they are unsure about the 7 or 8 point question then they need to be very confident about the other two questions (usually worth 3 - 5 points each). Once students make their choice they should take a minute to brainstorm the components of their answer, they can even write a short outline (to be crossed out later) to help them keep their ideas organized. It is always a good idea to be sure to define any key terms in the question as this is often a point in the markscheme. For any biological processes students should write in a sequential manner, being sure to describe each step with precision and using the correct vocabulary. Students can often get points for annotated diagrams in these questions as well, although they must be relevant to the question. Paper 3: Laboratory Skills & Options The final paper of the IB Biology exam is designed to test students knowledge of the required skills and laboratory Practicals in Section A (usually 3 questions) and then the chosen Option in Section B (usually 4 or 5 short answer questions per option). Students need only complete one of the four option sections, based on which option they studied. Paper 3 is middling in length, it is allocated 1 hour for SL students and 1 hour and 15 minutes for HL students. It is worth 24% of HL students final score and 20% for SL students. Unfortunately, this paper is often neglected by both students and teachers as it focuses on Options and lab skills rather than the entirety of the course. It is equal in value for SL students and for HL students it is worth 4% more than both the IA & Paper 1, so be smart and prepare well for it! Like every paper of this exam using past papers and grading your answers with the markschemes is excellent practice. The Importance of Command Terms Command terms are the verbs used to declare which skill the IBO wants students to demonstrate with a particular question. The complete list can be found on pages 166 & 167 of the IB Biology subject guide or in the document below. Students often struggle to gain the points in the markscheme simply because they did not follow the command term correctly. For this reason a couple of years into teaching IBDP Biology I started allowing students to use the command term glossary during test and quizzes in Year 1. This allows them to practice interpreting Paper 2 and 3 questions correctly and gradually learn all of the command terms. I simply printed and laminated the document below and pull it out for tests, quizzes and practice questions. This allows students to familiarize themselves with the command terms over multiple exposures and understand how to use them with real IB questions. Once it's time for your exam review, for both Mocks and the real deal in May (or November) then I recommend taking some time to do some activities in class. I suggest you look at how they are broken down into each Objective and remind your students that Objective 3 is the most important one for all three papers! One fun activity and brain break is to post a paper in 3 corners of your classroom, one for "Objective 1", "Objective 2" and "Objective 3" and then reading out a command term and having students move to the objective where they think it belongs. There are also free Quizlets online that you can have students use to practice, here is a good one. As you may know I am a big fan of sort activities, they get students using their hands and collaborating as they sort terms, images &/or concepts. Here is a fun sort activity you can do to see if students really remember those command terms. If you incorporate practice of the command terms into your course I am certain that your students' will improve on the IB exam! I have also put together a few more Command Term Activities including a domino loop game and a crossword puzzle. I have found that it is critical to review Command Terms regularly and often throughout the course, I am for at least once a semester, so a variety of activities are necessary to keep students engaged. Making a mock exam can be a time consuming task for teachers. In order to ensure that it follows the correct ratio of questions per topic you have a few choices... you can build it from scratch using the question ratios included in the tables earlier in this post. You can also choose one IB exam and give it to your students to complete, you will need to make sure that there are no missing diagrams (sometimes copyrighted images are removed after the exam) and that you have covered all of the content in the course. The last option is to combine two exams, so that any questions you have to remove (because you haven't covered it yet or there is a missing diagram) can easily be taken from the other exam. I used to do this with a pair of scissors and a glue stick, but now my school has IB Questionbank, which makes everything much easier. Whichever format you choose, try to ensure that the exam mimics a real IB exam as closely as possible to ensure your students get good practice. My Mock Exam Feedback Form After the exam is over I grade them using the IB markschemes and then fill out this form for my students and their parents. This helps them to understand the trends in their performance and guides their revision. I like to scan these reports for my own files as I make predicted grades and track my students performance on the IB exam. I hope that this post was helpful and for those who are making, grading or writing IB exams: good luck! Thanks for reading teachers, travelers & curious souls of all kinds.	{"url":"https://www.theroamingscientist.com/post/ib-biology-mock-exams-developing-exam-writing-skills","timestamp":"2024-11-12T20:03:32Z","content_type":"text/html","content_length":"1050616","record_id":"<urn:uuid:1a8cdcb7-0d71-4b0a-9d99-da279653954f>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00073.warc.gz"}
Graph Theory Graph Theory. Instructor: Dr. L. Sunil Chandran, Department of Computer Science and Automation, IISc Bangalore. In computer science, graph theory is used extensively. The aim of this course is to introduce the subject of graph theory to computer science students in a thorough way. While the course will cover all elementary concepts such as coloring, covering, hamiltonicity, planarity, connectivity and so on, it will also introduce the students to some advanced concepts. (from nptel.ac.in) Covering Problems Lecture 01 - Introduction: Vertex Cover and Independent Set Lecture 02 - Matchings: Konig's Theorem and Hall's Theorem Lecture 03 - More on Hall's Theorem and Some Applications Lecture 04 - Tutte's Theorem on Existence of a Perfect Matching Lecture 05 - More on Tutte's Theorem Lecture 06 - More on Matchings Lecture 07 - Dominating Set, Path Cover Lecture 08 - Gallai-Milgram Theorem, Dilworth's Theorem Lecture 09 - Connectivity: 2-Connected and 3-Connected Graphs Lecture 10 - Menger's Theorem Lecture 11 - More on Connectivity: K-Linkedness Lecture 12 - Minors, Topological Minors and More on K-Linkedness Lecture 13 - Vertex Coloring: Brooks' Theorem Lecture 14 - More on Vertex Coloring Lecture 15 - Edge Coloring: Vizing's Theorem Lecture 16 - Proof of Vizing's Theorem, Introduction to Planarity Lecture 17 - 5-Coloring Planar Graphs, Kuratowski's Theorem Lecture 18 - Proof of Kuratowski's Theorem, List Coloring Lecture 19 - List Chromatic Index Lecture 20 - Adjacency Polynomial of a Graph and Combinatorial Nullstellensatz Lecture 21 - Chromatic Polynomial, K-Critical Graphs Lecture 22 - Gallai-Roy Theorem, Acyclic Coloring, Hadwiger Conjecture Special Classes of Graphs Lecture 23 - Perfect Graphs: Examples Lecture 24 - Interval Graphs, Chordal Graphs Lecture 25 - Proof of Weak Perfect Graph Theorem (WPGT) Lecture 26 - Second Proof of WPGT, Some Non-perfect Graph Classes Lecture 27 - More Special Classes of Graphs Lecture 28 - Boxicity, Sphericity, Hamiltonian Circuits Lecture 29 - More on Hamiltonicity: Chvatal's Theorem Lecture 30 - Chvatal's Theorem, Toughness, Hamiltonicity and 4-Color Conjecture Network Flows Lecture 31 - Network Flows: Max-Flow Min-Cut Theorem Lecture 32 - More on Network Flows: Circulations Lecture 33 - Circulations and Tensions Lecture 34 - More on Circulations and Tensions, Flow Number and Tutte's Flow Conjectures Random Graphs and Probabilistic Method Lecture 35 - Random Graphs and Probabilistic Method: Preliminaries Lecture 36 - Probabilistic Method: Markov's Inequality, Ramsey Number Lecture 37 - Probabilistic Method: Graphs of High Girth and High Chromatic Number Lecture 38 - Probabilistic Method: Second Moment Method, Lovasz Local Lemma Graph Minors Lecture 39 - Graph Minors and Hadwiger Conjecture Lecture 40 - More on Graph Minors, Tree Decompositions Graph Theory Instructor: Dr. L. Sunil Chandran, Department of Computer Science and Automation, IISc Bangalore. An introduction to the subject of graph theory.	{"url":"http://www.infocobuild.com/education/audio-video-courses/computer-science/graph-theory-iisc-bangalore.html","timestamp":"2024-11-06T20:43:00Z","content_type":"text/html","content_length":"15824","record_id":"<urn:uuid:e0665f30-349d-4671-a24b-8f322f6db83b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00768.warc.gz"}
Nmanhattan gmat geometry pdf It covers not only fundamental geometric principles and techniques but also nuanced strategies for tackling tricky questions involving polygons, the. However,if thereissomethingabouttheproblemthatleadsyoutothinka. Difference between manhatten gmat 5th and 6th edition. Pdf gmat geometry rules quant all geometry usman khan. May 31st, 2011 dear student, thank you for picking up a copy of advanced gmat math. There are many more rules you must memorize area, perimeter, triangle ratios, etc. From 660 to 760 a brief message from rich before you take your 5th cat 5 mins quant. Well there are just two people who can help me out at this point in time, either it has to be some math guru or it has to be the almighty himself. In fact, if you diligently work through the guides guide 1. Y ou may get three to five questions from geometry and solid geometry in the gmat quant section in both variants viz. At manhattan gmat, we continually aspire to provide the best instructors and resources possible. Additional practice each of the above video lessons features a reinforcement activities box containing practice questions that are specifically related to the particular concepts covered in that lesson e. As per my experience with the manhattan gmat, there wont be any b. For a question which asks what is the distance between 2 points on a coordinate plane this question is really asking you to find the hypotenuse. Question in the figure given below, abc and cde are two identical semicircles of radius 2 units. These chapters will help you interpret your practice test results and chart your game plan. Manhattan geometry gmat strategy guide pdf download 4. Gmat rate questions get tricky when the test combines multiple rates. By statement 2 alone, since alternating interior angles are congruent, but no conclusion can be drawn about the relationship of, since the actual measures of the angles are not given. Gmat word problems manhattan prep gmat strategy guides. Manhattan geometry gmat strategy guide pdf download. Foundations of gmat math by manhattan gmat nook book. This session will demonstrate that by following our methodical approach, you can easily disseminate the seemingly difficult questions. For example, 23 2 2 2, or 2 multiplied by itself three times. Every gmat geometry formula you need to know prepscholar. If a square has an area of 49 ft2, what is the length of one of its sides. Ive heard that manhatten gmat is the best study guide set to follow in conjunction with the official guide for a total gmat study package. Even the most prepared testtakers can feel a lot of anxiety on test day. Designed to be userfriendly for all students, this book provides easytofollow explanations of fundamental math concepts and stepbystep application of these concepts to example problems. If youve signed up for a manhattan gmat course, read before your course starts but after you take a manhattan gmat practice exam. Unlike other guides that attempt to convey everything in a single tome, the gmat word problems strategy guide is designed to provide deep, focused coverage of one specialized area tested on the gmat. Get your kindle here, or download a free kindle reading app. Gmat overview the graduate management admission test gmat is indeed a difficult examination, and as such, it was required that media be chosen by which intellectual ability could be measured. Youre given a couple of pieces of info to start and you have to figure out the 4 or 5 steps that will get you over to the answer, or what youre trying to prove. Written by manhattan preps highcaliber gre instructors, the geometry gre strategy guide equips you with powerful tools to comprehend and solve every geometry problem on the gre. The concepts tested include lines, angles, triangles, quadrilaterals, circles, coordinate geometry, solid geometry, finding the area, perimeter of two dimensional geometric shapes and surface area, volume, longest diagnal of solids. That doesnt mean you can treat geometry as completely separate from the rest of gmat math, but there are a few issues to focus on. We hope that you will find our commitment manifest in this book. Refresh your knowledge of shapes, planes, lines, angles, objects, and more. Where can i download the latest edition of gmat manhattan. Manhattan gmat s foundations of gmat math book gives a refresher of the important math concepts examined on the gmat. The exponent indicates how many times to multiple the base, b, by itself. Before getting to work problems, well take a look at a couple of others way in which the gmat will combine rates. A salesmans income consists of commission and base salary. Manhattan prep gmat blog heres why youre getting geometry problems on gmat data did you know that. We are then told that 24 of the boys are juniors, and that they represent a proportion of total boys equal to the proportion of sophomore girls to total girls. Illuminates the two initial sections of the gmat teaches effective strategies for the new ir section enables optimal performance on the argument essay. Designed to be userfriendly for all students, this book provides easytofollow explanations of fundamental math concepts and stepbystep application of these concepts to example. These chapters are designed to guide you through our course or through nine weeks of self. Download gmatclub guide from below url and go through geometry. It covers not only fundamental geometric principles and techniques but also nuanced strategies for tackling tricky questions involving polygons, the coordinate plane, and many other topics. We cover exactly what you need to know for the gmat quantitative section. I just started gmat journey from manhattan 6th edition. T his gmat quant practice question is a problem solving question in geometry. The gmat geometry strategy guide equips you with powerful tools to grasp and solve every geometry problem tested on the gmat. Triangle and semicircle geometry problem solving question 8 2. Today we will discuss a pretty advanced gmat question, because we can still use our basic gmat concepts to find the answer. Considered the goldstandard in gmat test prep, manhattan gmats ten strategy guides are the first books on the market to be aligned with the th edition gmac official guide. Many geometry problems allow you to eliminate some answer choices using. Relearn the basics through easytofollow explanations of concepts and reinforce learning with hundreds of practice problems. Manhattan gmats foundations of math book 520 pages provides a refresher of the basic math concepts tested on the gmat. In fact, the question will look familiar at first, but will present. I solved only a couple of questions from this but its good for those. This question seems convoluted but is actually more simple than it seems. The base is the number that we multiply by itself n times. Manhattan gmats foundations of math book provides a refresher of the basic math concepts tested on the gmat. Download foundations of gmat math, 5th edition pdf ebook. Gmat geometry manhattan prep gmat strategy guides manhattan prep on. Free gmat prep access free practice tests, flashcards. Two ways to solve surface area problems in gmat quant. Im looking to take the gmat about 6 months to a year from now. Geometry on the gmat can be a bit like the proofs that we learned to do in high school. Used by itself or with other manhattan prep strategy guides, the gmat geometry strategy guide will help students develop all the knowledge, skills, and strategic thinking necessary for success on the gmat. Statement 1 alone establishes by definition that, but does not establish any relationship between and. Two sides of a triangle are 7 and ind the third side. In fact, one type of question, sometimes called work or simultaneous rate, merits the entire next chapter. It may seem like we will need trigonometry to handle this question, but that is not so. Designed to be shoppernice for all school college students, this book gives simpletoadjust to explanations of elementary math concepts and stepbystep software of these concepts to occasion points. Adapting to the everchanging gmat exam, manhattan preps 6th edition. Updated for the revised gre, the geometry guide illustrates every geometric principle, formula, and problem type tested on the gre to help you understand and master the intricacies of shapes, planes, lines, angles. I have a method to solve this type of question, but not sure if its exactly valid. If you have any questions or comments, please email me at. I have the full set of manhatten gmat 5th edition currently and was wondering if i should spend the money to get the full 6th edition set. Geometry has a number of concepts that can be tested in an even higher number of combinations. This comprehensive guide illustrates every geometric principle, formula, and problem type tested on the. As a result, students benefit from thorough and comprehensive subject material, clear explanations of fundamental principles, and stepbystep. In the past fewyears, short passages have been more common on the gmat than tong passages. You have worked through the 5 mathfocused manhattan gmat strategy guides. The geometry guide equips you with powerful tools to grasp and solve every geometry problem tested on the gmat. Properties of polygons quadrilaterals in specific and their shapes in geometry and elementary concepts in coordinate geometry finding length of a line segment, if the coordinates of its end points are known. I stopped thinking about the difficulty of the question and instead, worked through it. But i am pretty sure that you can get all manhattan study guides th edition in the internet. Purchase of this book includes one year of access to manhattan preps geometry question bank. Foundations of gmat math is a stressfree guide to mastering the fundamental math topics tested on the gmat, including algebra, number properties, inequalities, exponents, fractions, divisibility, geometry, and more. I am not sure whether you will get the latest editions of manhattan in the internet.	{"url":"https://sariphatreea.web.app/200.html","timestamp":"2024-11-12T00:44:23Z","content_type":"text/html","content_length":"15297","record_id":"<urn:uuid:72525e76-ef10-4e6b-9f75-6fa2be3545bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00562.warc.gz"}
Rotate, Reverse, Block Swap, Repeat. I was reading through a paper about block merge sort and in the article was a list of helper functions that the algorithm utilizes during its execution. This list was kind of a "who's who" of array manipulation algorithms. These algorithms are useful for a whole lot more than just block merge sort, and so I though It would be fun to implement different array manipulation algorithms and see what can be done with them, besides their obvious intended use. Amongst the algorithms listed as helper functions were some you would expect to find in an optimized sorting algorithm, such as swap(), insertion_sort(), and even binary_search(). What really piqued my interest though were some of less common algorithms: rotate(), reverse(), floorPowerOf2(). And these got me thinking about what some other useful array manipulation algorithms, such as block_swap (), partition(), and rotate_left(). So without further, ado lets play with some arrays. The Humble Swap The act of exchanging the values of two variables is fundamental to a great many algorithms, and as such it has gotten its share of attention. Interestingly, the most straight forward "naïve" approach to swapping to values, also happens to be the best way to accomplish this task, an attribute shared by very few algorithms. //the humble swap algorithm template <class T> void exch(T a[], int l, int r) { T tmp = a[l]; a[l] = a[r]; a[r] = tmp; Many people write the algorithm using references instead of passing the array as an argument, extending the algorithm for use on any two values, not just two in the same array: template <class T> void exch(T& l, T& r) { T tmp = l; l = r; r = tmp; //or using pointers template <class T> void exch(T* a, T* b) { T tmp = a; a = b; b = tmp; And of course, there is std::swap. Of interest, but not of practical use, is swapping without the use of extra storage, which can be done using bitwise operations such as xor swapping, or even good old fashioned addition and subtraction: //using xor to swap two values x = x ^ y; y = y ^ x; x = x ^ y; //using addition and subtraction: a = a + b; b = b - a; a = a - b; The last two methods will only work when the values aren't equal. It is of some debate whether it is worthwhile to check for equality before swapping using the traditional methods as well, as it may be more costly to check for equality than to swap unneccisarily. Block Swap and Reverse Building directly from swap comes our next two algorithms, block swap, which is used to swap ranges of equal size, and reverse which reverse the order of values in an array. template <class T> void block_swap(T a[], int l1, int r1, int l1, int r2) { for (int i = l1, j = l2; i < r1 && j < r2; i++, j++) exch(a, i, j); Block swap was the impetus that launched on a rabbit hole of O(log N) array reverse algorithms, but I will get into that in more detail below. For now when it comes to reversing arrays, the following O(N) solution will suffice: template <class T> void reverse(T a[], int l, int r) { while (r > l) exch(a, l++, --r); Now, I said it will have to suffice, because our next algorithm makes clever use of reverse() to accomplish it's task. Rotate & Rotate Middle Left A rotation in an array, preserves the overall ordering of the elements but shifts their position N steps to the right or left. It sounds straight forward, but without reading any further, I implore you to try implementing an array rotation algorithm that utilizes O(1) extra space. It's not as easy a task it sounds, which is part of what makes the following algorithm interesting: template <class T> void rotate(T a[], int l, int r, int k) { reverse(a, l, r); reverse(a, l, l+k); reverse(a, l+k, r); Rotation through reversal works by making three calls to reverse(), the first call reverses the entire array, the next two following calls use the reversal amount - k - similar to the pivot in quicksort, to "un-reverse" the values, while placing them in their shifted position. It then follows that if we can come up with a O(log N) reverse() algorithm, then we can implement an O(log N) rotate algorithm as well. The above algorithm rotates the elements of an array K positions to the right. std::rotate from the C++ standard library functions much differently, and in my opinion, is poorly named. While std::rotate does perform a rotation, it performs a very specific rotation: It rotates a range (first, last) to the left, so that the value that was in (first+last)/2 is now in the first position and all other values have been re-positioned accordingly. //produces same ordering as std::rotate template <class T> void stl_rotate(T a[], int l, int r) { int m = (l+r)/2; int n = m; while (l != n) { exch(a, l++, n++); if (n == r) n = m; else if (l == m) m = n;	{"url":"http://maxgcoding.com/rotate-reverse-block-swap-repeat","timestamp":"2024-11-03T06:48:24Z","content_type":"text/html","content_length":"9265","record_id":"<urn:uuid:1a4d7637-2dbb-4f3e-9c40-41853bf54958>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00774.warc.gz"}
Pivot table functionality Cell/sample selection and metadata quantification The UniApp provides flexible data analysis by allowing to subset your data in a virtually unlimited manner. This offers great flexibility in your analysis and experimental design, and can be accomplished in the Pivot table module. 1.1 Creating a new plot To begin, you should assign a name, write a description for your plot and choose "Pivot table" as your analysis algorithm. You can find these on the Create Plot field. Then, you should choose "Pivot table" from the Choose algorithm to run your analysis dropdown menu. Once you input all desired information, you can click on Select algorithm button to finish this stage. 1.2 Select data In this step, you need to present the right data to UniApp to be processed. By clicking on Choose track element dropdown menu, you can select the input data. 1.3 Selecting parameters The spatial expression visualization algorithm does not have any available parameters at the moment. You can proceed to run the analysis by clicking the Run button. 2. Spatial expression visualization interactive plot page In the pivot table interactive plot page you can interactively explore your data. 2.1 Pivot table layout The pivot table is partitioned in six panels: • Panel 1 lists all metadata variables. From here you can drag and drop variables to Panel 2 and Panel 3 to determine which variables make up the rows and columns of the pivot table, respectively. By default, this panel is empty until the user selects the measurements to work with (see below). • Panel 2 is the area where you can drag and drop metadata variables to be summarized and visualized in Panel 4. This will correspond to the y-axis of the plots visualized in Panel 4. • Panel 3 is the second area where you can drag and drop metadata variables to further details the plots and tables displayed in Panel 4. • Panel 4 displays the resulting pivot table or visualization. By default is empy until the user configure a visualization. • Panel 5 presents different drop-down menus for further customize the pivot table. Some are available by default (see list below), other appears depending on the pivot table configuration □ "Select metadata measurements" allows to decide which metadata variables are displayed in Panel 1. □ "Specify metadata type": here the users can specify whether the selected metadata measurements are categorical or numerical □ "Select molecular dataset": the same molecular dataset can be present with different preprocessing: normalized, scaled, log-transformed, etc. This menu allows to decide which one to use. □ "Select molecular measurements" allows to select from the chosen molecular dataset one or more genes / metabolites / proteins (depending on the nature of your omics dataset). The seleted variable available for exploration with the pivot table. □ "Visualization type" specifies how the data are visualized in the Pivot Table (Panel 4). Particularly, it allows to switch between a tabular visualization of the data and different graphical representations (plots). □ "Summary type" defines how your data should be aggregated. For example, Here you can decide to visualize the average values rather than the maximum values for a continuous variable. 2.2 Selecting measurements Each project can involve a large number of measurements (a.k.a. variables), recorded either as metadata (e.g., phenotype information) or molecular measurments (e.g., gene expression levels, protein abundance levels). The user is first asked to select the subset of variables that should be analyzed within the pivot table. This can be done through the buttons in Panel 5. The Select metadata measurements button provides a searchable menu where the user can select the metadata variables to be listed in Panel 1 and being included in the analysis. The Specify metadata type button instead allows the user to specify whether each selected metadata variable is numerical or categorical (see next subsection for more details on this topic). The next button, Select molecular dataset, can be used for selecting a specific molecular dataset from which extraxcting molecular measurements to analyze. If several version of the same dataset (e.g, normalized, scaled, etc.) are present, they will all appear in this menu. With the Select molecular measurements button, the user can finally spaecify which molecular variables should be listed in Panel 1. 2.2.1 Variables types Metadata can be represented by different types of variables. The Pivot Table will present different visualizations depending pn which types of variables are selected. UniApp distinguishes between two main types: 1. Numerical variables: measurements like age, height, number of cells are all numerical variables. Further subcategories are possible as well: 1. Count values: quantities like number of cells or number of patients are inherently discrete, meaning they can only assume values that are integer: 0, 1, 2, etc. 2. Continuous values: these variables can assume any real numerical values, e.g., 3.8, 4.1, etc. 2. Categorical variables: these measurements contain categories like colors or locations that cannot usually be described as numbers. These variables can be further classified as: 1. Binary: containing only two values, e.g., Yes or No, True or False. 2. Nominal: they contain more than two values, e.g., red, yellow, blue, or east, west, north and south 3. Ordinal: these variables have a few number of categories (less than 10), and an order can be established among the catergories, for example short, medium, tall. Numerical variables are marked violet, while categorical variables in green. 2.3 Selecting subsets of cells To subset you data, in the bottom left field where all metadata variables (i.e. columns) are listed, click on the downward pointing arrow to right of a metadata variable. The subsetting mechanism is different for numerical or categorical variables. Selecting cells in the Data pretreatment module will perform a "hard" subset. This means that only the cells in the cell subset used in data pretreatment will be available in the downstream analyses. 2.3.1 Selecting subsets of cells on the basis of a categorical variable If you select a categorical variable, then you perform the subsetting by selecting the categories to keep / discard. For example, in the metadata "ClusterID" we can deselect all cells corresponding to clusters 6 and 7. This means that these cells will not be retained for the following visualizations / analyses. Click on the Apply button to save your selection. 2.3.1 Selecting subsets of cells on the basis of a numerical variable If you select a numerical variable, then you must specify an interval of suitable values that corresponds to cells / samples / observations that must be retained. Subsetting can be performed on multiple different variables at same time. For example, we may select on the basis of both "ClusterID" and "percent_mito". 2.4 Visualizing single variables In order to visualize the properties of a numerical variable you need to drag the variable from Panel 1 to Panel 2 or, equivalently, Panel 3. Depending on whether the selected variable is categorical or numerical, the type of information that can be visualized changes. Particularly, the options that are available in the two dropdown menus "Visualization types" and "Summary Type" of Panel 5 will change depending on the type of seleted variables. 2.4.1 Visualizing a single numerical variable As soon as a single numerical variable is dropped in Panel 2 (or Panel 3), the following visualization types will become available in the dropdown menu of Panel 5: 1. Statistics 2. Histogram 3. Density plot The "Summary type" dropdown menu in Panel 2 will be disabled, since for numerical variables we only visualize their original distribution. The following plot presents the "Statistics: summary" visualization for single numerical variables Important: any time a table or a plot is visualized in the Pivot Table, Panel five is modified in two ways: 1) The button "Save settings" appears below the table/plot. This button allows to open a new view where the table or plot can be saved within a Track. 2) A number of menus appear under the title "Interactive view parameters". These menus allow to interactively modify the table / plot contained in Panel 3) The button "Enlarge view" appears as well. This button can be used for enlarging Panel 4, while Panel 1, 2 and 3 are folded away. Choosing the histogram as visualization will instead lead to the following: Pressing the "Enlarge view" button will magnify Panel 4: Finally, selecting the density plot will land this visualization: Important: once a numerical variable is dropped in Panel 2 or 3, no other variable can be dropped in the same panel. 2.4.2 Visualizing a single categorical variable When a categorical variable is dropped in Panel 3 or Panel 5, there will be two different visualization options, while no aggregation method will be available: 1. Statistics 2. Bar plot Choosing the Statistics visualization will display a table of statistics, which includes counts and frequency for each category. The least frequent and the most frequent (mode) category are also The bar plot visualization is as follow, where the summary is replaced by a graph: Contrarily to what happens with numerical variables, it is possible to drop more than a categorical variable in Panel 2 or 3. See section "7 Combining several categorical variables in a single axis" for details. Categorical variables with more than 100 values by default will generate a warning and the user will be asked to confirm that they want to proceed with the visualization. This is because a large number of categories usually leads to plots difficult to interpret. 2.5 Visualizing variables pairs In order to visualize the joint distribution of two variables, one of them must be dropped from Panel 1 to Panel 2, and one of them from Panel 1 to Panel 3. The available visualizations and data aggregation approaches depend upon the chosen variables types. The possible alternatives are described below. 2.5.1 Visualizing two numerical variables Dropping one numerical variable in Panel 2 and one in Panel 3 will allow to investigate their joint distribution. The "Summary type" dropdown menu will be disabled, while the available visualization types will be: • Statistics: correlation analysis • Statistics: linear model • Scatterplot • Density plot The "Statistics: correlation analysis" visualization is reported below. The pivot table reports the Pearson, Spearman and Tau correlation coefficients, along with their corresponding p-values. both Chosing "Statistics: linear model" will fit a linear model between the two variables.Particulary, the variable dropped in Panel 3 acts as independent variable (x-axis) and the variable dropped in Panel 2 as dependent variable (y-axis). Passing to the scatterplot visualization will produce the following: The density visualization simply replaces the scatter plot with a 2D density plot: 2.5.2 Visualizing two categorical variables Numerous visualization types are available for representing the joint distribution of two categorical variables 1. Statistics: contingency table 2. Statistics: association test 3. Bar plots 4. Heatmap 5. Sankey plot 6. Mosaic plot Furthermore, two summary types are also available: 1. Counts 2. Frequencies Let's start with visualizing descriptive statistics, i.e., contingency tables. This generates a table with row corresponding to the values of the categorical variable dropped in Panel2, while the column correspond to the values of the variable in Panel 3. If summary type is set to count, each cell of the table reports the number of samples corresponding to the respective row and column. If the summary type is set to frequencies, than each cell reports the proportion of samples falling in its combination of values. Marginal counts and frequencies are also displayed. The next visualization type presents the results of the chi squared test assessing whether the there is any association between the two chosen variables. For 2 x 2 tables the Fisher exact test is used instead of chi squared. The Bar plot visualization provides a quick overview of the data. Depending on the choice in the summary type dropdown meny, this height of the bars will correspond either to the counts or to the frequencies of each category combination. Note that the variable dropped in Panel 3 will be placed on the x-axis of the plot, while the variable in Panel2 will indicate the color of the bars. Heatmaps are direct translation of table into figures; a heatmap is organized as a table, with colors indicating how many samples (counts) or what proportion (frequencies) of samples are present in each heatmap cell. Sankey plots are one additiona tool to graphically represent how samples are distributed across two variables. Only counts can be visualized in Sankey plots, not frequencies. Finally, Mosaic plots are a graphically appealing variation with respect to the classical barplots. Only frequencies can be represented in this type of plots. 2.5.3 Visualizing one categorical variable and a numerical variable When a numerical variable is selected along side a categorical one, the available sample is divided in subgroups according to the categorical variable and the distribution of the numerical variable can be studied within each subgroup. The available visualization types for this case are the following: 1. Statistics: descriptive 2. Statistics: anova test 3. Bar plot 4. Heatmap 5. Box plot The descriptive statistics include: 1. Average 2. 25% Quantile 3. Median 4. 75% Quantile 5. Maximum 6. Standard deviation 7. Interquantile range 8. Sum 9. Number of missing values The summary type menu is disabled when the "Statistics: descriptive" visualization type is active. The "Statistics: anova test" visualization reports the results of an ANOVA test, or a t-test if the categorical variable is binary: Barplots provides one more opportunity for exploring the distribution of the numerical variable within each category of the categorical variable. When the users select the bar plot visualization type, they must also select one of the options out of the Summary types. The eight of the bars then reflect the chosen summary type. For example, this is the plot resulting from selecting bar plot as visualization type and the average as summary type: The following plot was instead obtained by selecting "standard deviation" in the summary type dropdown menu: Heatmaps work in a way that is identical to the bar plots approach, using shade of color rather than height for comparing different summary statistics across categories. More interestingly, box plots and violin plots provide a bird-eye view on several aspects of the distribution of the continuous variable. The selection of the summary type is deactivated when box plots are shown. Violin plots can be constructed using the interactive view parameters. 2.6 Visualizing two categorical and one numerical variables When two categorical variables are selected (see section 5.2 above), a fourth drop down menu appear in Panel 5. This menu allows to select one numerical variable among the ones contained in Panel 1. Once this variable is selected, the cell of the pivot table will start visualizing summary statistics computed on the selected variable. For example, in the figure below, "Percent_mito" was selected as additional variable, "Average" was selected as Summary type, and thus the cells of the Pivot Table report the average of Percent_mito within each subgroup defined by the two chosen categorical variables. Summary statistics available for the additional continuous variable are the same used in section "5.3 Visualizing one categorical variable and a numerical variable". Also the available visualization types are the same: 1. Statistics: descriptive 2. Statistics: anova test 3. Bar plot 4. Heatmap 5. Box plot The Anova and Heatmap visualizations operates as in section 5.3. Bar plots and box plots needs to be slightly modified in order to represent all three variables. For bar plots, one categorical variable dictates the x-axis, the other regulate the color of the bars, and the chosen summary statistic from the numerical variable decides the height of the bars. Box plots works in a similar way, however no summary statistics needs to be selected: 2.7 Combining several categorical variables in a single axis Two or more categorical variable can be simultaneosuly being included in a single axis, by dragging both of them either in Panel 2 or Panel 3. In this case, the categories of these variables are combined so that to create a single list of categories. For example, ClusterID and Orig_ident are combined together in Panel 2: This Pivot Table can now be analyzed as if a single categorical variable was in Panel 2. Other variables can be added to Panel 3 as well for more in depth explorations. 2.8 Visualizing date variables in a Gantt chart The Pivot table in the Management analytics screen is able to visulize date variables as a Gantt chart. A Gantt chart is a special type of bar chart that illustrates a project schedule. The tasks of project are listed on the y-axis while the start and expected due date of the task are plotted on the x-axis. To create a Gantt chart select the "Gantt chart" display option. Then drag the task variable in the y-axis and the start and due dates of the task in the y-axis. Addiotionally, dragging in addiotional categorical variables will color-code the bars of the Gannt chart. For example, adding the "Assignee" variable will show which task is assigned to which person. 3. Saving plots From Save output tab you can save the plot you have currently set up as a new track element. Frst add the plot name and desctiption, then click on the Save button. The plot should now appear in the analysis track. 4. Reopening pivot table plots From the plot details page of an exported pivot table plot, you can reopen the saved plot in the pivot table. First, go to the plot details page of an exported pivot table plot. Now click the "Open pivot table" button. This will load the original pivot table setup that was used to create the original plot. From here you can modify the original plot and save it as a new plot. Was this article helpful? That’s Great! Thank you for your feedback Sorry! We couldn't be helpful Thank you for your feedback Feedback sent We appreciate your effort and will try to fix the article	{"url":"https://support.unicle.life/support/solutions/articles/101000451086-super-pivot-table-functionality","timestamp":"2024-11-03T21:38:41Z","content_type":"text/html","content_length":"407501","record_id":"<urn:uuid:f269962a-5f56-48c8-9424-e5e69010f936>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00799.warc.gz"}
Standard Form Standard Form Standard Form is another way to write the equation of a line. The Standard Form equation is Ax+By=C. A, B, and C all represent coefficients. It's similar to Slope-Intercept Form, however, x and y are on the same side, unlike the prior. The slope in Standard Form is represented by the "A" and the "C" represents the y-intercept. To turn Standard Form into Slope-Intercept Form, you add or subtract (depending on whether A is positive or negative) the "Ax" to the other side, so you get By=-Ax+C. Then, you divide "Ax+C" by B, and you get your equation into Slope-Intercept Form. To turn Slope-Intercept into Standard Form, you add or subtract (depending if m is positive or negative) the "mx" to the other side so that you get "Ax+By=C." Another helpful tip to remember when learning Standard Form is that "A" should NOT be negative. Try this practice problem below! Try finding this in Slope-Intercept Form, and transfer it into Standard Form! First, find the Slope-Intercept Form equation of the line. Remember, to find the slope, do y[2]-y[1]/x[2]-x[1]. When you do this, you should get 2-5/-1-2, which equals a slope of -3/-3, or 1. Then, you need to find the y-intercept. This is the point where the line hits the y-axis. On this problem, that point is at (0,3). Now, you have a Slope-Intercept equation of y=1x+3. Then, follow the steps that I talked about above to transfer this equation into Standard Form. This is pretty easy to remember, because you just need to subtract 1x from each side. You know have an equation of -1x+y=3. Also, since A cannot be negative, you need to divide the whole equation then by -1 (just for this case) so that your A value can be positive. After doing this, you have your Standard Form equation of	{"url":"https://www.geogebra.org/m/JMtvfFtX","timestamp":"2024-11-10T06:02:39Z","content_type":"text/html","content_length":"91809","record_id":"<urn:uuid:3b343104-ec9e-4d2c-8d62-4442be2f7be6>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00899.warc.gz"}
13+ Eli5 Volts Amps - collegebeautybuff.com13+ Eli5 Volts Amps 13+ Eli5 Volts Amps 13+ Eli5 Volts Amps. Today’s top 29 volt jobs in moreno valley, california, united states. And the more conductive, the more amps you will have. ELI5 What's the difference between Amps, Watts , and Volts? explainlikeimfive from www.reddit.com Eli5 amps volts [ eli5 volts amps ] volts, amps, and watts explained in detail. Features 1,150 square feet of office, 16 auto stalls, 18'. Leverage your professional network, and get hired. You Can Calculate It With Ohm's Law, V= I (Amps) X R (Resistance). Every time i come i'm in a chair getting my cut in under 2 minutes. Details you can watch and learn the content, video and information about this eli5 volts amps from the video below. In other words if you look at the battery, it has a potential of 12 volts. Volts To Amps Calculation The Current I In Amps (A) Is Equal To The Power P In Watts (W), Divided By The Voltage V In Volts (V): And the more conductive, the more amps you will have. Today’s top 29 volt jobs in moreno valley, california, united states. So a device that uses 2 amps when plugged into a 120 volt electrical outlet uses 240 watts of power. Available For Lease 7,180 Industrial Warehouse With Secured Yard. In order to find the solution, simply select the type (watts or ohms) and then enter the value of current in. Eli5 amps volts [ eli5 volts amps ] volts, amps, and watts explained in detail. Posted on june 5, 2022 june 5, 2022 by blog author. Amps To Volts Calculation The Voltage V In Volts (V) Is Equal To The Power P In Watts (W), Divided By The Current I In Amps (A): 7 reviews of volt barbers i've come here a few times and the service is excellent, it's very clean in here and has good vibes. Leverage your professional network, and get hired. I(a) = p(w) / v(v) the current i in amps (a) is equal to the. Features 1,150 Square Feet Of Office, 16 Auto Stalls, 18'. You can reorganize the equation to work out the other values. 13.1 v to amps converter converts 13.1 volts into amperes. Whether you need a residential commercial or industrial circuit breaker challenger breakers span a wide variety of applications.	{"url":"https://collegebeautybuff.com/blog/2022/12/04/13-eli5-volts-amps/","timestamp":"2024-11-10T22:13:45Z","content_type":"text/html","content_length":"58042","record_id":"<urn:uuid:f0bed735-5f1c-4ca7-bfb0-641cb8bfdebc>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00859.warc.gz"}
Lagrange's Theorem \documentclass[12pt]{article} \usepackage{amsmath} \usepackage{amsthm} \begin{document} \begin{center} Senior Seminar: Project 1 \\Shateil French \\Math 4991; Dr. Yi Jiang \\February 01, 2016 \end {center} \subsection{Lagrange's Theorem (Group Theory)} For any finite group \emph{G}, the order (number of elements) of every subgroup \emph{H} of \emph{G} divides the order of \emph{G}. \ subsection{Theorem} The order of a subgroup \emph{H} of group \emph{G} divides the order of \emph{G}. group \emph{G}, a subgroup of \emph{H} of \emph{G}, and a subgroup \emph{K} of \emph{H}, (\emph {G:K)}=\emph{(G:H)(H:K)}. Proof: For any element \textbf{x} of \emph{G}, \{\emph{H}\textbf{x} = {\textbf{h $\cdot$ x} $\mid$ \textbf{h} is in \emph{H}}\} defines a right coset of \emph{H}. By the cancellation law each \textbf{h} in \emph{H} will give a different product when multiplied on the left onto \textbf{x}. Thus \emph{H}\textbf{x} will have the same number of elements as \emph{H}. Lemma: Two right cosets of a subgroup \emph{H} of a group \emph{G} are either identical or disjoint. Proof: Suppose $\emph{H}\textbf{x}$ and $\emph{H}\textbf{y}$ have an element in common. Then for some elements $\textbf{h}_1$ and $\textbf{h}_2$ of $\emph{H}$ $$\textbf{h}_1 \cdot \textbf{x} = \textbf{h}_2 \cdot \textbf{y}$$ Since \emph{H} is closed this means there is some element $\textbf{h} _3$ of \emph{H} such that $\textbf{x} = \textbf{h}_3 \cdot y$. This means that every element of \emph{H}\textbf{x} can be written as an element of \emph{H}\textbf{y} by the correspondence $$\textbf {h} \cdot \textbf{x} = (\textbf{h} \cdot \textbf{h}_3) \cdot \textbf{y}$$ for every \textbf{h} in \emph{H}. We have shown that if \emph{H}\textbf{x} and \emph{H}\textbf{y} have a single element in common then every element of \emph{H}\textbf{x} is in \emph{H}\textbf{y}. By a symmetrical argument it follows that every element of \emph{H}\textbf{y} is in \emph{H}\textbf{x} and therefore the "two" cosets must be the same coset. Since every element \textbf{g} of \emph{G} is in some coset the elements of \emph{G} can be distributed among \textbf{H} and its right cosets without duplication. If \emph{k} is the number of right cosets and \emph{n} is the number of elements in each coset then $\mid$\emph{G}$\mid$ = \emph{kn}. \end{document}	{"url":"https://www.overleaf.com/articles/lagranges-theorem/kgdjqdwxsqgv","timestamp":"2024-11-09T22:26:57Z","content_type":"text/html","content_length":"37261","record_id":"<urn:uuid:87382da6-789e-460a-ab47-b9164a601f6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00366.warc.gz"}
NA Digest Sunday, February 4, 1990 Volume 90 : Issue 05 NA Digest Sunday, February 4, 1990 Volume 90 : Issue 05 Today's Editor: Cleve Moler Today's Topics: From: T. J. Garratt <tjg%maths.bath.ac.uk@nsfnet-relay.ac.uk> Date: Mon, 29 Jan 90 15:30:32 GMT Subject: Roommate Needed at Copper Mountain Conference WANTED: Person to share room for conference: "ITERATIVE METHODS", Copper Mountain, Colorado, 1st - 5th April, 1990. I am a male postgraduate studying for my PhD in Numerical Analysis at Bath University, and will be attending the above conference. To help with the costs of accommodation, I am looking for someone to share a lodge room or deluxe studio. Perhaps a student in a similar situation might be interested. If you are interested or know someone who may be, then please contact: Tony Garratt, School of Mathematical Sciences, Univeristy of Bath, Claverton Down, Bath. AVON. BA2 7AY. United Kindgom. E-mail: tjg@uk.ac.bath.baths (OR na.spence@edu.stanford.na-net) From: Bob Ward <ward@rcwsun.EPM.ORNL.GOV> Date: Tue, 30 Jan 90 10:42:07 EST Subject: Liz Jessup Wins Householder Fellowship at Oak Ridge Elizabeth R. Jessup has been selected as the winner of the first Householder Fellowship at the Oak Ridge National Laboratory (ORNL). Dr. Jessup, who received her doctorate degree in Computer Science in 1989 from Yale University, is currently an Assistant Professor of Computer Science at the University of Colorado at Boulder. Her research interests are in parallel computing and numerical linear Dr. Jessup will be collaborating with the researchers in ORNL's Mathematical Sciences Section and with applied computational scientists in various divisions at ORNL on scientific problems involving high performance computing. Her primary interest will be on parallel algorithms for solving large-scale eigenproblems on a distributed-memory MIMD multiprocessor. Her fellowship appointment will begin this summer. Alston S. Householder was the organizer and founding Director of the Mathematics Division (precursor of the current Mathematical Sciences Section) at ORNL. In recognition of the seminal research contributions of Dr. Householder to the fields of numerical analysis and scientific computing, a distinguished postdoctoral fellowship program was established and named in his honor. Householder Fellows will be appointed annually for a term of one year, renewable for a second The Householder Fellowship Program is supported by the Applied Mathematical Sciences Subprogram of the U.S. Department of Energy. From: Jorge More <more@antares.mcs.anl.gov> Date: Wed, 31 Jan 90 09:21:30 CST Subject: Barry Smith Wins Wilkinson Fellowship at Argonne We are pleased to announce that Barry Smith from the Courant Institute of Mathematical Sciences is the 1990 Wilkinson fellow. Barry is a student of Olof Widlund working on domain decomposition algorithms for the partial differential equations of linear elasticity. In addition to Courant, he has worked at the IBM T. J. Watson Research Center, Los Alamos National Laboratory, and at the University of Bergen. He will join the Mathematics and Computer Science Division of Argonne National Laboratory in the summer. From: Henry Wolkowicz <hwolkocz@orion.waterloo.edu> Date: Mon, 29 Jan 90 15:48:23 EST Subject: Distance of a Matrix to a Subspace How would one find (numerically) the distance between a given real n by n matrix A and the given subspace S, where S is the subspace of upper triangular matrices which are themselves made up of k by k upper triangular blocks ? The distance is the inf of spectral norms (largest singular value). Henry Wolkowicz; Department of Combinatorics and Optimization; Faculty of Mathematics; University of Waterloo; Waterloo, Ontario, Canada N2L 3G1 (519-888-4597 office; 746-6592 FAX) {hwolkowicz@water.bitnet; na.wolkowicz@na-net.stanford.edu} {hwolkowicz@water.uwaterloo.ca; usersunn@ualtamts.bitnet } From: Ben Lotto <ben@cps3xx.egr.msu.edu> Date: 1 Feb 90 20:10:23 GMT Subject: Numerical Integration Program Wanted I would like a numerical integration program that will handle a Cauchy principal value integral of the following form: \lim_{\epsilon\to 0} \int_{\epsilon}^{\pi} (f(\theta - t) - f(\theta + t)) / tan(t/2) dt (this computes the conjugate function of f) where f is a function that has a a couple of jump discontinuities (I could probably fudge things and get rid of this) and a log x-type singularity. In particular, I would like the algorithm to work for the function f(x) = log \|x\|, if \|x\| < \pi / 2 0, if \|x\| >= \pi / 2 Reply by e-mail, please, as I don't read this newsgroup regularly. Thanks in advance. -B. A. Lotto (ben@nsf1.mth.msu.edu) Department of Mathematics/Michigan State University/East Lansing, MI 48824 From: Bill Anderson <XB.N64@Forsythe.Stanford.EDU> Date: Thu, 1 Feb 90 20:41:31 PST Subject: Summer Programs for Undergraduates Last week's NA Digest included an announcement of a Summer program for undergraduates at CNSF at Cornell. Are there additional Summer programs to which I could encourage two highly qualified undergraduates to apply? One is a math major, the other CS. Thanks in advance! Bill Anderson email: xa.e71@forsythe.stanford.edu From: G. W. Stewart <stewart@cs.UMD.EDU> Date: Fri, 2 Feb 90 07:47:33 -0500 Subject: Nominations Sought for Fifth Householder Prize Alston S. Householder Award V (1990) (Second Posting) In recognition of the outstanding services of Alston Householder, former Director of the Mathematics Division of the Oak Ridge National Laboratory and Professor at the University of Tennessee, to numerical analysis and linear algebra, it was decided at the Fourth Gatlinburg Symposium (now renamed the Householder Symposium) in 1969 to establish the Householder Award. This award is in the area in which Professor Householder has worked and its natural developments, as exemplified by the international Gatlinburg Symposia [see A. S. Householder, The Gatlinburgs, SIAM Review 16:340-343 (1974)]. Recent recipients of the award include James Demmel (Berkeley), Ralph Byers (Cornell), and Nicholas Higham (Manchester). The Householder Prize V (1990) will be awarded to the author of the best thesis in Numerical Algebra. The term Numerical Algebra is intended to describe those parts of mathematical research which have both algebraic aspects and numerical content or implications. Thus the term covers, for example, linear algebra that has numerical applications or the algebraic aspects of ordinary differential, partial differential, integral, and nonlinear equations. The thesis will be assessed by an international committee consisting of Chandler Davis (Toronto), Beresford Parlett (Berkeley), Axel Ruhe (Gothenburg), Pete Stewart (Maryland), and Paul Van Dooren (Phillips, To qualify, the thesis must be for a degree at the level of an American Ph.D. awarded between 1 January 1987 and 31 December 1989. An equivalent piece of work will be acceptable from those countries where no formal thesis is normally written at that level. The candidate's sponsor (e.g., supervisor of his research) should submit five copies of the thesis (or equivalent) together with an appraisal Professor G. W. Stewart Department of Computer Science University of Maryland College Park, MD 20742 by 28 February 1990. The award will be announced at the Householder XI meeting and the candidates on the short list will receive invitations to that meeting. From: Michael Mascagni <mascagni@ncifcrf.gov> Date: Fri, 2 Feb 90 13:04:41 EST Subject: Washington, DC Area E-mailing List I am happy to announce a newly formed mailing list. The list's purpose is to distribute information on scholarly talks, meetings, and other events of interest to the "greater" Washington, DC area community involved in applied mathematics, computer science, numerical analysis, high performance computing, and scientific computing. We have identified people at several sites in the area who have agreed to serve as site contributors. We are quite biased, and have no doubt left out several sites, group, etc. Our purpose was not to offend, but to get things going ASAP. So if you wish to be a site contributor, please send in a request. If you wish to be placed on the mailing list also send us e-mail. DO NOT E-MAIL TO MY NA-NET ADDRESS. Instead, send mail to mascagni@jvncf.csc.org with your request. As soon as we have a reasonable number of announcements, the first mailing will go out. Until then, spread the word, and please communicate with mascagni@jvncf.csc.org!! Thanks for your help in this.--Michael Mascagni (na.mascagni, but mascagni@jvncf.csc.org for this) From: Jerzy Wasniewski <mfci!wasniews@uunet.UU.NET> Date: Mon, 29 Jan 90 07:38:27 EST Subject: Dr. Zahari Zlatev Visiting Multiflow Computer, Inc. Dr. Zahari Zlatev National Environmental Research Institute, Division for Emissions and Air Pollution, Frederiksborgvej 399, 4000 Roskilde, Denmark visiting Multiflow Computer, Inc. Feb 14 - 16, 1990. Dr. Zlatev will present two lectures. 1) Thursday, February 15th, 1990 - 12:00 a.m. Multiflow Computer, Inc. 31 Business Park Drive Branford, CT 06405 Tel: (203) 488-6090 A b s t r a c t The long-range transport of air pollutants ( LRTAP ) over Europe is studied, at the Air Pollution Laboratory of the Danish Agency of Environmental Protection, by a mathematical model based on a system of partial differential equations ( PDE's ) . Four different physical processes, advection, diffusion, deposition and chemical reactions (together with emission sources), are the main components of the LRTAP . These four processes are described by different terms in the model (the system of PDE's). Since the space domain is very large (including the whole of Europe together with parts of the Atlantic Ocean, Asia and Africa), the discretization of the system of PDE's leads to huge systems of linear algebraic equations ( LAE's ) . In the three dimensional case on a 32 x 32 x 9 grid the number of LAE's that are to be solved at each time-step is more than 106 when 29 chemical species are involved in the model. Even if the model is considered as a two-dimensional model, the number of LAE's is still very large; more than 105 . This explains why one should make some simplifications in the model description (which are not always very well justified physically, but lead to a model that can be handled on the computer used) and/or one should use high-speed computers. In the latter case, high performance can be achieved by efficiently implementing certain kernels which perform the bulk of the computational work. Fortunately, regular grids are to be used during the discretization of the LRTAP model. This leads to the solution of LAE's whose coefficient matrices are banded and whose solution dominates the computational load. Several such kernels for solving banded systems of LAE's will be described. Experimental results obtained on AMDAHL VP1100, CRAY X-MP and ALLIANT will be presented and discussed. 2) Friday, February 16, 1990 - 11:00 a.m. Yale University - Numerical Analysis A. K. Watson Hall - 51 Prospect Street - room 200 New Haven, CT 06520 A b s t r a c t Consider the system ! Ax = b !. Assume that !A! is a large and sparse, but neither any special property of this matrix (such as symmetry and/or positive definiteness) nor any structure of its non-zero elements (such as bandedness) can be exploited. For such systems direct methods may be both time and storage consuming, while iterative methods may not converge. A hybrid method, which attempts to avoid the drawbacks of both direct methods and iterative methods, is proposed. We start with some factors !L! and !U! obtained by removing "small" non-zero elements during Gaussian elimination and use them to precondition the system. Then one of three conjugate gradients-type methods (ORTHOMIN, GMRES and CGS) can be used. If the iterative process does not converge, then the criterion used in the decision whether a non-zero element is small or not is made more stringent and new factors are calculated and used to precondition the system. This process can, if necessary, be repeated several times. If after a prescribed number of trials the iterative method is still not convergent, then a switch is made to Gaussian elimination. Thus, with regard to the accuracy requirements the hybrid method is not worse than Gaussian elimination. However, even more important is the fact that the method is often less time and storage consuming than Gaussian elimination. This is demonstrated by many numerical examples (including the well-known Boeing-Harwell test-matrices). From: Mikko Tarkiainen <mcsun!sunic!tut!tukki!tarkiain@uunet.uu.net> Date: 29 Jan 90 16:41:45 GMT Subject: Conference on Numerical Methods for Free Boundary Problems Second Announcement of the July 23-27, 1990 in Jyvaskyla, Finland TOPICS OF THE MEETING. The topics covered at the conference will be: Free boundary problems in fluid mechanics, in hydrodynamics, in mechanics, in ground freezing and in optimal shape design, capillary free boundaries, shape memory problems, inverse and identification problems, control of phase transition, solidification process, etc. PARTICIPANTS. So far, among others, the following persons are intending to attend: Barbu, V. (Romania), Bossavit, A. (France), Chizikalov, V.A. (USSR), Cuvelier, C. (The Netherlands), Fage, D. (USSR), Fasano, A. (Italy), Gets, I. (USSR), Grossman, Ch. (DDR), Haslinger, J. (Czechoslovakia), Hoffmann, K-H (BRD), Kaliev, I. (USSR), Kenmochi, N. (Japan), Khludnev, A.M. (USSR), Knabner, P. (BRD), Kurtze, D.A. (USA), Magenes, E. (Italy), Maximov, A. (USSR), Meirmanov, A. (USSR), Mittelmann, H. (USA), Myslinski, A. (Poland), Niezgodka, M. (Poland), O'Carrol, M.J. (USA), Paolini, M. (Italy), Primicerio, M. (Italy), Rivkind, V. (USSR), Rogers, J.C.W. (USA), Sahm, P.R. (BRD), Schulkes, R.M.S.M.(The Netherlands), Shemetov, N. (USSR), Shopov, P.J. (Bulgaria), Verdi, C. (Italy). REGISTRATION. Registration forms can be ordered from the address below. Notice that the registration must be done before March 31, 1990. A detailed program and abstracts of the lectures will be issued to those attending. Registration forms should be sent to Professor Pekka Neittaanm{ki. You may contact us also by email. CONFERENCE FEE. The conference fee, which includes attendance at the conference, conference material, refreshments during breaks, ship cruise on Lake P{ij{nne and conference dinner, will be $ 100. Participants especially from East and Southeast Europe may be given some support for the conference fee and local expenses (travel in Finland, living costs in Finland). Please inform us about required financial support in the registration form. ACCOMMODATION. Accommodation for the conference is available at the Hotel Alba on the University campus. Also, student hotels are available (2 km from the University). Please make the reservation for the accommodation, including the dates, on the accommodation registration form. If you want another hotel please inform us. If you want to stay longer in Finland before or after the conference we can help you to make reservations (hotels, summer houses, camping places, THIRD ANNOUNCEMENT including a preliminary conference program, information on preparing the paper for the conference proceedings, travel connections in Finland, etc., will be sent at the end of April Prof. Pekka Neittaanmaki University of Jyvaskyla Department of Mathematics Seminaarinkatu 15 SF-40100 Jyvaskyla, Finland email: Neittaanmaki@finjyu.bitnet tel.: (+358 41)602733 telefax: 358-41602701 telex: 28219 JYK SF Mikko Tarkiainen e-mail: mtt@jylk.jyu.fi Department of Mathematics tarkiain@tukki.jyu.fi University of Jyvaskyla, Finland phone: +358 41 292715 From: Germund Dahlquist <dahlquis@nada.kth.se> Date: Fri, 2 Feb 90 12:56:51 +0100 Subject: SIAM Nordic Section meeting, June 1990 Third Annual Meeting of June 26-27 1990 Stockholm, Sweden SIAM Nordic Section was founded in 1987. The objectives of the section are within the Nordic countries - to further the application of mathematics to industry and science - to promote basic research in mathematics leading to new methods and techniques useful to industry and science - to unite the community of researchers and graduate students in applied - to provide media for the exchange of information and ideas between mathematicians and other technical and scientific personnel. The first annual meeting was held in 1988 in Bergen, Norway, the second one in Espoo, Finland. All kinds of contributions of 25 minutes duration (including discussion) are welcome, but presentations from doctoral students and nonacademic organisations are especially invited. Please send a title of your talk and an abstract (at most one page long) before April 18, 1990. At the SIAM Nordic Section Meeting The GOLUB PRIZE will be awarded for the best contributed paper presented at the Section Meeting by a student who is from a Nordic country and has not yet finished PhD. The second Golub Prize was given to Rune Karlsson from Linkoping at the 1989 meeting in Helsinki. In addition to the contributed talks, there will be a number of talks by leading researchers from the Nordic countries. There will be a registration fee of 200 Sw.Cr. For members of the SIAM Nordic Section, 150 Sw.Cr. only. Membership can be arranged at the meeting. There will be no registration fee for graduate students from the Nordic countries. There will be a "Wine & Carrots" -party on Tuesday, June 26, at 5 p.m. The local organizer of the meeting is the Department of Numerical Analysis and Computing Science (NADA) at the Royal Institute of Housing has been arranged at a tourist class hotel, Hostel Frescati, located at the University campus, about 5 km north of Stockholm centre, while the meeting takes place at the Royal Institute of You can either have a nice (?) walk (less than 3 kms) or go by bus and subway. The same bus can also bring you downtown in about 10 minutes. Rates per night are 170 Sw Crs (about US$ 27) for a single room, 130 Sw Crs per person in a double room. The reception of the hotel is open all the time There is an extra cost (30 Sw Crs) for linen unless you bring linen yourself. Breakfast is not included but is served in a Campus restaurant. If you want us to book a room for you on Hostel Frescati, please send in the enclosed registration form as soon as possible. Hotel prices in Stockholm are high, about 1000 Sw Crs for a single room. For more information and questions, please contact: Berit Gudmundson Germund Dahlquist K T H K T H S-100 44 Stockholm S-100 44 Stockholm Sweden Sweden Tel. +46 (8) 790 8077 +46 (8) 790 7142 Email: dahlquis@nada.kth.se We like to mention that during the week June 18-22 there are two Applied Mathematics meetings in the Nordic countries: 1) The 1990 Conference on Solution of Ordinary Differential Equations, Helsinki, Finland (Register before April 30,1990) Information from Prof Olavi Nevanlinna, Institute of Mathematics, Helsinki University of Technology, 02150 Espoo 15, Finland Email: mat-on@finhut.bitnet 2) The Householder Symposium XI Meeting in Numerical Algebra, Tylosand,Sweden. (Deadline was Novenber 1, 1989) Information from Prof Ake Bjorck, Dept of Mathematics, Linkoping Univ, S-581 83 Linkoping, Sweden So, if you decide to participate in one of the above meetings, you are encouraged to extend your visit to the Nordic countries by attending to the SIAM Nordic Section meeting. In between there is the famous Nordic Midsummer Weekend, with midnight sun and all that + a Monday for recovery. By the way, there is also a great meeting in the week June 11-15: 3) 3rd International Conference on Hyperbolic Problems, Uppsala, Sweden Information from Lena Jutestal, Dept of Scientific Computing, Uppsala Univ, Stureg 4B, S-752 23 Uppsala, Sweden, Email: lena@tdb.uu.se From: Sven Hammarling <NAGSVEN%vax.oxford.ac.uk@nsfnet-relay.ac.uk> Date: Mon, 29 Jan 90 18:04 GMT Subject: NAG Floating-point Test Package FPV is a program which attempts to test the floating-point operations + - * / sqrt, and comparisons .LT. .GT. etc., on a systematically chosen set of operands. The code is written with all floating-point operations in loops that will vectorise easily. It can test that the arithmetic is rounded according to a number of rounding rules, including all the IEEE rules. There are currently Fortran-77 and ISO standard Pascal versions of FPV. Unlike Paranoia though, FPV is a commercial product. Anyone interested in receiving more information should contact The Numerical Algorithms Group. Sven Hammarling. From: Andy Sherman <cs.yale.edu!topcat!sherman-andy@CS.YALE.EDU> Date: 30 Jan 90 20:56:06 GMT Subject: PCGPAK2 for Solving Sparse Linear Equations SCIENTIFIC Computing Associates, Inc. is pleased to announce the availability of PCGPAK2, its new package of subroutines for the iterative solution of large, sparse systems of linear equations. PCGPAK2 offers a choice of solution methods based on a collection of preprocessing, preconditioning, and iterative techniques that includes some of the most robust and efficient methods known. The entire package is written in portable Fortran 77, so it can be easily merged with the large amount of existing scientific and engineering software that depends on solving sparse linear systems. Four basic iterative methods are available in PCGPAK2: --- the conjugate gradient method (CG); --- the generalized minimal residual method (GMRES(k)); --- ORTHOMIN(k); --- the restarted generalized conjugate residual method (GCR(k)). All of these are Krylov subspace methods that minimize a norm of the residual error at each step. CG is applicable only to symmetric, positive definite systems; the others are general methods designed mainly for systems having nonsymmetric or non-positive-definite symmetric coefficient matrices. PCGPAK2 includes several options that can enhance the performance of the basic iterative methods. Among these are: 1. Incomplete factorization preconditioning -- The system is preconditioned with an approximate factorization of the coefficient matrix generated with sparse Gaussian elimination, ignoring some or all of the fill-in. A levelparameter is used to control the amount of fill-in that is neglected, and a relaxation parameter is available to fully or partially preserve the matrix row sums. 2. Reduced system preprocessing -- A preprocessing step generates a smaller, denser system that is solved using one of the preconditioned basic iterative methods.The solution to the full system is recovered by postprocessing the solution to the smaller reduced system. 3. Block iteration -- All of the methods in PCGPAK2 can exploit general block structure in the coefficient matrix. This leads to iterative methods that are extremely robust and natural for problems with underlying block structure arising from geometric or modeling considerations. Both constant and variable blocksizes are supported. PCGPAK2 is applicable to a wide range of engineering and scientific problems that depend on the solution of large sparse systems of linear equations. Examples of application areas include structural engineering analysis, aerodynamic and hydrodynamic modeling, oil reservoir simulation, ocean acoustics, simulation of VLSI circuit designs and combustion physics. For many problems, PCGPAK2 is substantially faster and uses far less storage than alternative banded or sparse Gaussian elimination methods. For example, on one relatively-small nonsymmetric system of order 3969 arising from a nine-point discretization of an elliptic partial differential equation on the unit square, PCGPAK2 required less than one-fourth of the time and less than one-fifth of the storage required by the band Gaussian elimination routines from LINPACK. For larger two-dimensional and three-dimensional partial differential equations, the savings are far greater. The standard Fortran version of PCGPAK2 will run on essentially any computer. Optimized versions of PCGPAK2 are available for a number of vector machines, including the Cray 1, Cray XMP, Cray YMP, Cray 2, IBM 3090, Convex C-1, Convex C-2, and DEC VAX 9000. For further information, contact SCIENTIFIC at SCIENTIFIC Computing Associates, Inc. 246 Church Street, Suite 307 New Haven, CT 06510 Tel.: (203) 777-7442 FAX: (203) 776-4074 Email: sca@yale.edu or yale!sca PCGPAK2 is a registered trademark of SCIENTIFIC Computing Associates, Inc. Computers mentioned may be trademarks of their respective manufacturers. From: Iain Duff <duff@antares.mcs.anl.gov> Date: Sun, 28 Jan 90 16:32:16 CST Subject: IMA Journal of Numerical Analysis Contents The contents of the current issue of the IMA Journal of Numerical Analysis are given below. IMA Journal of Numerical Analysis - Volume 10, Number 1 A Iserles Stability and dynamics of numerical methods for non-linear ordinary differential M Z Liu and M N Spijker The stability of the i-methods in the numerical solution of delay differential J Gilbert and W A Light Envelope solutions for implicit ordinary differential equations D Funaro Convergence analysis for pseudospectral multidomain approximations of linear advection equations J Solar Vortex filament method A Bellen, A Jackiewicz, Stability analysis of Runge-Kutta methods R Vermiglio and for Volterra integral equations of the M Zennaro second kind R Coquereaux, A Grossmann Iterative method for calculation of the and B E Lautrup Weierstrass elliptic function H Brass Optimal estimation rules for functions of high smoothness N Dyn, D Levin and Data dependent triangulations for piecewise S Rippa linear interpolation The annual subscription rate for IMAJNA is $216 (120 pounds outside North America and 92 pounds in UK), with a reduced rate for members of the IMA of 38.50 pounds. There are four issues (each of approximately 150 pages) each year. Note that it is now possible to pay for IMA journals and IMA membership using major credit cards. From: Bob Plemmons <plemmons%matple@ncsuvx.ncsu.edu> Date: Wed, 31 Jan 90 14:35:45 EST Subject: SIMAX April Contents Table of Contents SIAM J. on Matrix Analysis and Applications April 1990, Vol. 11 no. 2. 1. On Perhermitian Matrices Richard D. Hill, Ronald G. Bates, and Steven R. Waters 2. A Matrix Approach to the Design of Low-Order Regulators L.H. Keel and S.P. Bhattacharyya 3. Some 0-1 Solutions to the Matrix Equation A(m) - A(n) = I Chi Fai Ho 4. Sets of Positive Operators with Suprema W.N. Anderson, Jr., T.D. Morley, and G.E. Trapp 5. Algebraic Polar Decomposition Irving Kaplansky 6. The Laplacian Spectrum of a Graph Robert Grone, Russell Merris, and V.S. Sunder 7. Robust Stability and Performance Analysis for State Space Systems via Quadratic Lyapunov Bounds Dennis S. Bernstein and Wassim M. Haddad 8. On the Singular Values of a Product of Operators Rajendra Bhatia and Fuad Kittaneh 9. Points of Continuity of the Kronecker Canonical Form Immaculada de Hoyas 10. On Rutishauser's Approach to Self-Similar Flows D.S. Watkins and L. Elsner 11. Incremental Condition Estimation Christian Bischof 12. A New Algorithm for Finding a Pseudoperipheral Node in a Graph Roger G. Grimes, Daniel J. Pierce, and Horst D. Simon ***Looking ahead - The July and October issues will contain, in part, invited papers from the Salishan, Oregon, Sparse Matrix Symposium held in 1989. From: K. McKinnon <EFTM11%emas-a.edinburgh.ac.uk@nsfnet-relay.ac.uk> Date: 01 Feb 90 10:07:38 gmt Subject: Lectureship in Mathematics at Edinburgh University Lectureship in Mathematics Particulars of Appointment Applications are invited for a LECTURESHIP IN MATHEMATICS tenable in the above Department. The appointment will commence on 1 October 1989 or at a date to be decided between the department and the successful candidate. The Department wishes to appoint an applied mathematician with strong research interests. The ideal candidate will work in optimization theory or numerical analysis, but strong candidates in other areas of applied mathematics will be considered seriously. The successful candidate will have the opportunity to interact fruitfully with the research groups in the Department, and with other departments in the University. There are three established chairs. The chair in Applied Mathematics is held by D.F.Parker, whose interests include nonlinear wave propagation in solids and optics. The other two are held by T.J.Lyons (currently Head of Department) whose interests relate to probability theory, particularly in analysis and geometry; and E.G.Rees whose interests are in topology and geometry. There are five Readers, twenty four other teaching staff, two computing officers and a number of other research workers. The interests of the other teaching staff include optimization, numerical analysis, dynamical systems, differential equations, analysis, probability, algebra, topology and The Department is responsible for teaching and research in Pure and Applied Mathematics, and also runs (jointly with Heriot-Watt University) an MSc course in Nonlinear Mathematics, supported by the SERC. There are separate departments of Chemical, Electrical and Mechanical Engineering, Computer Science, Statistics, Artificial Intelligence, Geology and Geophysics, as well as a large Theoretical Physics group within the Department of Physics. The Mathematics Department has strong links with the new Edinburgh-based SERC-funded programme for the development of new techniques for design, optimisation and control in the process engineering industries. The Department is housed in the James Clerk Maxwell Building on the King's Buildings site of the University, together with the combined mathematics libraries of the University and of the Edinburgh Mathematical Society. There are excellent computing facilities, including a 400-transputer parallel processing facility and two Distributed Array Processors (DAPs), in the same building. Edinburgh is an internationally recognised centre for parallel In addition to research, duties would involve lecturing in Mathematics to Honours and Ordinary Degree students and to postgraduate students, preparing and attending tutorials, supervising undergraduates, examining, supervising postgraduate students and assisting generally in the work of the Department. The appointee is expected to join the Universities Superannuation Scheme (USS), and to contribute 6.35% of annual salary, in which case the University will contribute an additional sum equal to 18.55% of annual salary. The current salary scales for lecturers A and B are 10,458 to 20,469 pounds. The University is prepared to contribute towards removal expenses of staff coming from other parts of the United Kingdom to Edinburgh on a first appointment to an established post within the University, the full cost of any reasonable vouched expenditure on removal of furniture and effects, including insurance thereon, and the cost of fares of bringing the family to Edinburgh. Claims in respect of travel etc from overseas will be considered on their Applications (7 copies), including curriculum vitae and the names and addresses of three referees, should be sent to Professor T.J.Lyons, Department of Mathematics, Room 5320, JCMB, The King's Buildings, Edinburgh EH9 3JZ, not later than 2nd March 1990. In the case of overseas candidates, later applications may be considered. Such candidates need supply only one copy of their application. PLEASE QUOTE REFERENCE NUMBER 1486 From: Bo Kagstrom <BOKG%SEUMDC51.BITNET@Forsythe.Stanford.EDU> Date: Thu, 1 Feb 90 13:09 EDT Subject: Chair in Scientific Computing at Umea Announcement of SWEDEN's first chair as Professor in Computer Graphics and Visualization in Scientific Computing at the University of Umea, Sweden (Reference number: Dnr 321-189-90) Umea university is a young university that lies at the mouth of the river Ume, equidistant from both the capital, Stockholm, and Sweden+s most northerly town Kiruna. Today the campus has some 3 000 employees and 11 000 students. The university has achieved prominence in many fields, of which bio-technology, environmental ecology and information technology are some of those in which now intensive activity is taking Expertise in the field of information technology in its broadest sense is rapidily growing and in certain areas such as Scientific computing great progress has already been made, and international collaboration established, primarily with European and American researchers. A couple of years ago a special action program for Information technology - Scientific Computing was established at the faculty of Mathematical and Natural Sciences. The program aims towards development of advanced methods, algorithms and software in Scientific computing for different parallel computer architectures. The university is together with the Technical University of Lulea, the Institute of Space Physics i Kiruna and the Industrial Development Center in Skelleftea, founder of Supercomputer Center North (SDCN). SDCN is one of two national centers for supercomputing in Sweden and is connected to all swedish universities through the Swedish University Network (SUNET). Thereby scientists have access to an IBM 3090-600 E/VF, placed in Skelleftea, soon to be upgraded to a 600 J-model. At the university we have a distributed-memory multiprocessor-system Intel iPSC/2 hypercube with 64 nodes of which 16 nodes have a vector facility and are about to aquire a shared-memory multiprocessor-system with both high-performance computing power and advanced graphic facilities for visualization. Due to the partnership in SDCN Sweden+s first chair as professor in Computer Graphics and Visualization in Scientific Computing is now established at the university. The field is very wide and interdisciplinary to its nature and candidates for the chair can have different scientific profiles ranging from research in tools and methods for Computer Graphics and Visualization in Scientific Computing to graphics computing and visualization in Scientific Computing with an emphasis on applications from biology, biotechnology, chemistry, physics and medicine. At the university we have applications/possible applications in for instance biotechnology - molecular biology, chemometry, environmental chemistry, geographical information systems, industrial design, medicine, physical chemistry, psychology, theoretical physics and space physics. The professorship is placed at the department of Computing Science. At the department there are professor chairs in numerical analysis, computer science, and numerical analysis and parallel computing. Since a couple of years there has been an intense development of knowledge in the fields of parallel computing and environments and tools for parallel computer architectures. The university now announces a professorship in Computer Graphics and Visualization in Scientific Computing as vacant, reference number Dnr 321-189-90. Notice that the reference number must be mentioned on the application! To get started in this field as soon as possible the position can also be a visiting professorship. Send the application to Rektorsambetet, University of Umea, S-901 87 UMEA, Sweden before the 30th of March 1990. Enclosed to the application should be curriculum vitae, short summary of scientific and educational work, and publications and ev. interest of a visiting professorship. Questions will be answered by Professor Bo Kagstrom, Dept of Computing Science, Umea University, S-901 87 Umea, phone +46-90165419, email: bokg@biovax.umdc.umu.se (or na.kagstrom@na-net.stanford.edu) or by Project coordinator Torbjorn Johansson, Supercomputer Center North, Umea University, S-901 87 Umea, phone +46-90166585, email: From: David Womble <dewombl@sandia.gov> Date: 2 Feb 90 13:32:00 MST Subject: Fellowship at Sandia National Labs (Please distribute this announcement to colleagues and students who do not receive the NANET distributions.) Mathematics and Computational Science Department Sandia National Laboratories Sandia National Laboratories is seeking outstanding candidates in the areas of numerical analysis, scientific computing, or symbolic computing to fill its 1990 Applied Mathematical Sciences Research Fellowship. The Fellowship is supported by a special grant from the Applied Mathematical Sciences Research Program at the U.S. Department of Energy. The Fellowship is intended to provide an exceptional opportunity for young researchers. Sandia's Mathematics and Computational Science Department maintains strong programs in theoretical computer science, analytical and computational mathematics, computational physics and engineering, advanced computational approaches for parallel computers, graphics, and architectures and languages. Sandia provides a unique parallel computing environment, including a 1024-processor NCUBE 3200 hypercube, a 1024-processor NCUBE 6400 hypercube, a Connection Machine-2, and several large Cray supercomputers. The successful candidate must be a U.S. citizen, must have earned a Ph.D. degree or the equivalent, and should have a strong interest in advanced computing research. The fellowship appointment is for a period of one year, and may be renewed for a second year. It includes a highly competitive salary, moving expenses, and a generous professional travel allowance. Applications from qualified candidates, or nominations for the Fellowship, should be addressed to Robert H. Banks, Division 3531-24B, Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185. Applications should include a resume, a statement of research goals, and the names of three references. The closing date for applications is April 30, 1990. The position will commence during 1990. Further inquiries can be made by calling (505) 844-2248 or by sending E-mail to Equal Opportunity Employer M/F/V/H U.S. Citizenship is Required End of NA Digest	{"url":"https://www.netlib.org/na-digest-html/90/v90n05.html","timestamp":"2024-11-11T16:32:36Z","content_type":"text/html","content_length":"47367","record_id":"<urn:uuid:e048808b-44e9-48d7-8405-998f0920a6c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00710.warc.gz"}
How To Calculate Probability Of Ignition - Calculatorey How To Calculate Probability Of Ignition Understanding the Probability of Ignition When it comes to fire safety, one important factor to consider is the probability of ignition. This refers to the likelihood that a fire will start in a given area or under certain conditions. Understanding this probability can help individuals and organizations take appropriate measures to prevent fires and protect property and lives. Factors Affecting Probability of Ignition Several factors can affect the probability of ignition in a specific setting. These factors include the presence of flammable materials, the temperature and humidity of the environment, the presence of ignition sources such as open flames or electrical sparks, and the amount of oxygen available. Understanding how these factors interact can help determine the likelihood of a fire starting. Calculating the Probability of Ignition There are various methods for calculating the probability of ignition in a given scenario. One common approach is to use mathematical models that take into account the factors mentioned above. These models assign probabilities to each factor and then combine them to determine the overall probability of ignition. Another method is to rely on empirical data and statistical analysis to estimate the likelihood of a fire starting under certain conditions. By studying past fires and their causes, researchers can develop models that predict the probability of ignition in different scenarios. Using the Fire Triangle The fire triangle is a useful tool for understanding the elements necessary for a fire to start. These elements are fuel, oxygen, and heat. By assessing the availability of these elements in a given setting, individuals can determine the likelihood of a fire starting. For example, a room filled with flammable materials, poor ventilation, and high temperatures would have a higher probability of ignition than a well-ventilated room with minimal combustible materials. Reducing the Probability of Ignition There are several measures that can be taken to reduce the probability of ignition in a specific setting. These measures include proper storage and handling of flammable materials, regular maintenance of electrical equipment to prevent sparks, and enforcing strict no-smoking policies in areas with combustible materials. By implementing these measures, individuals and organizations can decrease the likelihood of a fire starting. Understanding and calculating the probability of ignition is crucial for fire safety. By considering factors such as fuel, oxygen, heat, and ignition sources, individuals can assess the likelihood of a fire starting in a given scenario. By taking proactive measures to reduce this probability, individuals and organizations can protect property and lives from the devastating effects of fires.	{"url":"https://calculatorey.com/how-to-calculate-probability-of-ignition/","timestamp":"2024-11-05T19:25:21Z","content_type":"text/html","content_length":"75668","record_id":"<urn:uuid:c0dc8e06-64e2-4db4-a2f5-d3e858d25029>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00390.warc.gz"}
Blog - Academia Essay Writers by Carl Martins Assignment: Contingency Tables And Odds In Excel Assignment: Contingency Tables And Odds In Excel Assignment: Contingency Tables And Odds In Excel In the prerequisite course, Quantitative Reasoning and Analysis, you constructed basic contingency (crosstab) tables. You might be surprised to learn that you can estimate a simple logistic regression model, with a categorical predictor, using the descriptive values presented in the crosstab table. In this assignment, you use Microsoft Excel to construct a specialized tool that creates basic logistic regression models given a crosstab/contingency table. As if that were not useful enough, this Excel tool is not specialized—you can use it given any crosstab/contingency tables you encounter in research. In the field of statistical research, this is just about as exciting as you can get! To prepare • Review the sections in the Osborne text that present a template for constructing an Excel worksheet. • Review the video in the Learning Resources, in which Dr. Matt Jones explains how to harness the power of Excel using contingency tables. • Think about the types of variables that are useful for cross tab tables. Using one of the datasets provided, select two variables that allow you to construct a 2×2 contingency table. Use SPSS to run the initial crosstab table, using any two variables that you think are appropriate. Then, use Excel to construct a table in which you report: • Conditional probabilities • Conditional odds • Logits • Odds ratios • Relative risk • Slope Be sure to apply the template from the Osborne text. Note that page 42 has a completed example that should help you determine these values. Be sure to use formulas and cell references in Excel so that the spreadsheet you create can be used as a tool for calculating similar values for other datasets. Once you have created the tool, write a 1- to 2-paragraph summary in APA format interpreting your results. Submit both your Excel file and your summary to complete this assignment. You must proofread your paper. But do not strictly rely on your computer’s spell-checker and grammar-checker; failure to do so indicates a lack of effort on your part and you can expect your grade to suffer accordingly. Papers with numerous misspelled words and grammatical mistakes will be penalized. Read over your paper – in silence and then aloud – before handing it in and make corrections as necessary. Often it is advantageous to have a friend proofread your paper for obvious errors. Handwritten corrections are preferable to uncorrected mistakes. Use a standard 10 to 12 point (10 to 12 characters per inch) typeface. Smaller or compressed type and papers with small margins or single-spacing are hard to read. It is better to let your essay run over the recommended number of pages than to try to compress it into fewer pages. Likewise, large type, large margins, large indentations, triple-spacing, increased leading (space between lines), increased kerning (space between letters), and any other such attempts at “padding” to increase the length of a paper are unacceptable, wasteful of trees, and will not fool your professor. The paper must be neatly formatted, double-spaced with a one-inch margin on the top, bottom, and sides of each page. When submitting hard copy, be sure to use white paper and print out using dark ink. If it is hard to read your essay, it will also be hard to follow your argument. Psychosocial Development Case Study Assessment Psychosocial Development Case Study Assessment Psychosocial Development Case Study Assessment The purpose of this assignment is to use what you have learned about lifespan theories, models of resilience, and psychosocial development to assess how well individuals and families are functioning in relation to all three of these areas. To address realistic situations without violating personal rights to privacy, you will view the movie My Big Fat Greek Wedding, or one of the other approved films for this assignment, in order to provide a case scenario. Psychosocial Development Case Study Assessment If you have not yet viewed the movie, you need to do so to complete this assignment. Each movie represents characters in multiple life stages. Identify three characters moving through a different life stage and focus on details that will make your analysis applicable to your specialization. Psychosocial Development Case Study Assessment For this assignment, imagine you are a counselor working with each of these three characters. Now, conduct the following analysis for each character: · Identify the life stage he or she is in, along with the psychological crisis each is experiencing. · Apply psychosocial developmental theory to the situation presented, from the perspective of a counselor: . Conceptualize your ideas for the developmental tasks of each character selected, grounding your conceptualization in your own area of specialization. . Include a discussion of the character’s life and factors that might affect behaviors, including cultural and other influences related to the stage of development assessed. To the extent that it is relevant for each character, include an analysis of interrelationships among work, family, and other life roles. Include an analysis of the impact of cultural influences, as well. Refer to specific actions and words of the characters in the movie as evidence for your analysis. . Support your ideas with specific lifespan theories discussed in this course, citing and referencing your sources. From a clinical perspective, assess how these three characters function as a family unit: · Examine their functioning in relation to a model of resilience appropriate to your specialization, and evaluate their challenges and strengths related to wellness and resilience. As each character transitions to his or her next developmental stage, how will the transition impact the functioning of the family unit? · Support your ideas with appropriate sources on the model of resilience you chose. Use the document Unit 8 Assignment Template (given in the resources) to prepare and submit your report. Do not include a synopsis of the movie in your paper. Instead, follow the template. Focus on the three selected characters, and the realities of their functioning as a family unit and transitioning within the family unit, as described in the template. Then, provide your assessment of how well they are functioning in relation to your choice of a model of wellness and resilience, as described in the template. Your summary, at the end of the template, should provide a brief, focused review of the key insights in your assessment. Review the scoring guide given in the resources to make sure you understand how this assignment will be graded. Other Requirements Your paper must meet the following requirements: · Resources: Cite and reference at least three resources from the professional literature that you use as the basis of your ideas related to life span theory and resilience models. · APA formatting: Resources and citations must be formatted according to current APA style. · Font and Font size: Times New Roman, 12 point. · Length of Paper: Doing a thorough job on this assignment is likely to require approximately 7–10 typed, double-spaced pages. · Turnitin: You are required to submit your final version of this paper to Turnitin to generate a final report prior to submitting the assignment for grading. From the Turnitin tool, first submit to the draft link to check your work for any necessary edits. Once the paper is finalized and all edits have been made, submit your final paper to the Final report option for the assignment. Please be advised it can take up to a day to obtain the percentage from Turnitin. When your paper is downloaded and viewable in Turnitin, save the originality report. Refer to the Turnitin Tutorial: Viewing an Originality Report (linked in the Resources) for guidance. 1. Submit your assignment using the following file naming format: Your Name_AssignmentNumber_Assignment Title (example: Ima_Learner_u08a1_CaseStudyAssessment). 2. In the comment section, provide the percentage from the final Turnitin report (example: Final Turnitin percentage = 4%). Please be prepared to provide your faculty member with a copy of the Turnitin report should this be requested of you. · Psychosocial Development Case Study Assessment Scoring Guide . · Unit 8 Assignment Template [DOCX] . Feminist, Solution-Focused, And Narrative Theory Application Feminist, Solution-Focused, And Narrative Theory Application Read the “Case Study Analysis.” Select one of the following theories that you feel best applies to treating the client in the case study: Write a 750-1,000-word analysis of the case study using the theory you chose. Include the following in your analysis. 1.What concepts of the theory make it the most appropriate for the client in the case study? 2.Why did you choose this theory over the others? 3.What will be the goals of counseling and what intervention strategies are used to accomplish those goals? 4.Is the theory designed for short- or long-term counseling? 5.What will be the counselor’s role with this client? 6.What is the client’s role in counseling? 7.For what population(s) is this theory most appropriate? How does this theory address the social and cultural needs of the client? 8.What additional information might be helpful to know about this case? 9.What may be a risk in using this approach? Include at least three scholarly references in your paper. APA form Case Study Analysis Client Name: Ana Client age: 24 Gender: F Presenting Problem Client states, “I recently lost my job and feel hopeless. I can’t sleep and don’t feel like eating.” Client also reports she has lost 10 pounds during the last two months. Client states that she is a solo parent and is worried about becoming homeless. Client states, “I worry all the time. I can’t get my brain to shut off. My husband is in the military and currently serving in an overseas combat zone for the next eight months. I worry about him all the time.” Behavioral Observations Client arrived 30 minutes early for her appointment. Client stated that she had never been in counseling before. Client depressed and anxious, as evidenced by shaking hands and tearfulness as she filled out her intake paperwork. Ana made little eye contact as she described what brought her into treatment. Client speech was halting. Client affect flat. Client appeared willing to commit to eight sessions of treatment authorized by her insurance company. General Background Client is a 24-year-old first-generation immigrant from Guatemala. Ana was furloughed from her job as a loan officer at local bank three months ago. Client reported that she was from a wealthy family in Guatemala, but does not want to ask for help. Client speaks fluent Spanish. Client has completed one year of college with a major in business. Client states that she left college after her son was born as she found it difficult to manage a baby, college, and a full-time job. Family Background Client is the middle of four siblings. Client has two older brothers and one younger sister. Client’s parents have been married for 27 years. Client states that she has had a “close” relationship with her family, although she states that her father is a “heavy drinker.” Client states that all her brothers and sisters have graduated from college and have professional careers. Client states that her father is a banker and her mother is an educator. Client states that she has not seen her family for 1 year. Client has a 1-year-old son and states that she is sometimes “overwhelmed” by raising him alone. Major Stressors •Lack of family and supportive friends •Financial problems due to job loss •Husband deployed overseas •Raising a baby by herself Statistics Project, Part 4: Correlation Statistics Project, Part 4: Correlation USE ATTACHMENTS AS GUIDE One week away from the final project, and now you calculate your first correlational stats on your data set. You will calculate the Pearson product-moment correlations between at least two sets of variables in your data set. Do one correlation between two independent variables such as age and education. Do the second correlation on an independent variable (such as age) and the dependent variable (such as score). Remember that most people never see the actual output or data; they read the results statements by the researcher, so your summary must be accurate. Calculate the Pearson product-moment correlations between at least 2 sets of variables in your data set. Do one correlation between two independent variables such as age and education. Do the second correlation on an independent variable (such as age) and the dependent variable (such as score). Summarize the results of the calculation in 45 to 90 words. • Frequencies Table College no collegesome college CaffeineCaffeine yesnoyesno CountCountCountCountGenderMaleTest Prepno preparation3030 moderate preparation2011 high preparation0001 FemaleTest Prepno preparation3111 moderate preparation1041 high preparation0010 College associate’s degreebachelor’s degree CaffeineCaffeine yesnoyesno CountCountCountCountGenderMaleTest Prepno preparation0000 moderate preparation5121 high preparation1210 FemaleTest Prepno preparation1010 moderate preparation6020 high preparation1020The number of males that had no preparation in the test took caffeine. Those who had moderate preparation only three did not take caffeine, which were one from some college, one from associate’s degree and one from bachelor’s degree. Majority of males had moderate preparation were under associates degree and bachelor’s degree and most took caffeine. There is a low number of males who had high preparations and majority of them took caffeine. Majority of females took caffeine and were highly prepared. Identify Your Sources Of Statistics Anxiety Identify Your Sources Of Statistics Anxiety Discussion 1 – Statistics Anxiety Identify your sources of statistics anxiety as you begin this course. Reflect on your thoughts, feelings, and behaviors related to statistics. How can you minimize statistics anxiety moving forward? 1.1 Introduction This textbook assumes that students have taken basic courses in statistics and research methods. A typical first course in statistics includes methods to describe the distribution of scores on a single variable (such as frequency distribution tables, medians, means, variances, and standard deviations) and a few widely used bivariate statistics . Bivariate statistics (such as the Pearson correlation, the independent samples t test, and one-way analysis of variance or ANOVA) assess how pairs of variables are related. This textbook is intended for use in a second course in statistics; the presentation in this textbook assumes that students recognize the names of statistics such as t test and Pearson correlation but may not yet fully understand how to apply and interpret these analyses. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details The first goal in this course is for students to develop a better understanding of these basic bivariate statistics and the problems that arise when these analytic methods are applied to real-life research problems. Chapters 1 through 3 deal with basic concepts that are often a source of confusion because actual researcher behaviors often differ from the recommended methods described in basic textbooks; this includes issues such as sampling and statistical significance testing. Chapter 4 discusses methods for preliminary data screening; before doing any statistical analysis, it is important to remove errors from data, assess whether the data violate assumptions for the statistical procedures, and decide how to handle any violations of assumptions that are detected. Chapters 5 through 9 review familiar bivariate statistics that can be used to assess how scores on one X predictor variable are related to scores on one Y outcome variable (such as the Pearson correlation and the independent samples t test). Chapters 10 through 13 discuss the questions that arise when a third variable is added to the analysis. Later chapters discuss analyses that include multiple predictor and/or multiple outcome variables. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details When students begin to read journal articles or conduct their own research, it is a challenge to understand how textbook knowledge is applied in real-life research situations. This textbook provides guidance on dealing with the problems that arise when researchers apply statistical methods to actual data. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details 1.2 A Simple Example of a Research Problem Suppose that a student wants to do a simple experiment to assess the effect of caffeine on anxiety. (This study would not yield new information because there has already been substantial research on the effects of caffeine; however, this is a simple research question that does not require a complicated background story about the nature of the variables.) In the United States, before researchers can collect data, they must have the proposed methods for the study reviewed and approved by an institutional review board (IRB) . If the research poses unacceptable risks to participants, the IRB may require modification of procedures prior to approval. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details To run a simple experiment to assess the effects of caffeine on anxiety, the researcher would obtain IRB approval for the procedure, recruit a sample of participants, divide the participants into two groups, give one group a beverage that contains some fixed dosage level of caffeine, give the other group a beverage that does not contain caffeine, wait for the caffeine to take effect, and measure each person’s anxiety, perhaps by using a self-report measure. Next, the researcher would decide what statistical analysis to apply to the data to evaluate whether anxiety differs between the group that received caffeine and the group that did not receive caffeine. After conducting an appropriate data analysis, the researcher would write up an interpretation of the results that takes the design and the limitations of the study into account. The researcher might find that participants who consumed caffeine have higher self-reported anxiety than participants who did not consume caffeine. Researchers generally hope that they can generalize the results obtained from a sample to make inferences about outcomes that might conceivably occur in some larger population. If caffeine increases anxiety for the participants in the study, the researcher may want to argue that caffeine would have similar effects on other people who were not actuallyincluded in the study. This simple experiment will be used to illustrate several basic problems that arise in actual research: 1. Selection of a sample from a population 2. Evaluating whether a sample is representative of a population 3. Descriptive versus inferential applications of statistics 4. Levels of measurement and types of variables 5. Selection of a statistical analysis that is appropriate for the type of data 6. Experimental design versus nonexperimental design The following discussion focuses on the problems that arise when these concepts are applied in actual research situations and comments on the connections between research methods and statistical analyses. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details 1.3 Discrepancies Between Real and Ideal Research Situations Terms that appear simple (such as sample vs. population) can be a source of confusion because the actual behaviors of researchers often differ from the idealized research process described in introductory textbooks. Researchers need to understand how compromises that are often made in actual research (such as the use of convenience samples) affect the interpretability of research results. Each of the following sections describes common practices in actual research in contrast to idealized textbook approaches. Unfortunately, because of limitations in time and money, researchers often cannot afford to conduct studies in the most ideal manner. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details 1.4 Samples and Populations A sample is a subset of members of a population. 1 Usually, it is too costly and time-consuming to collect data for all members of an actual population of interest (such as all registered voters in the United States), and therefore researchers usually collect data for a relatively small sample and use the results from that sample to make inferences about behavior or attitudes in larger populations. In the ideal research situation described in research methods and statistics textbooks, there is an actual population of interest. All members of that population of interest should be identifiable; for example, the researcher should have a list of names for all members of the population of interest. Next, the researcher selects a sample from that population using either simple random sampling or other sampling methods (Cozby, 2004). In a simple random sample, sample members are selected from the population using methods that should give every member of the population an equal chance of being included in the sample. Random sampling can be done in a variety of ways. For a small population, the researcher can put each participant’s name on a slip of paper, mix up the slips of paper in a jar, and draw names from the jar. For a larger population, if the names of participants are listed in a spreadsheet such as Excel, the researcher can generate a column of random numbers next to the names and make decisions about which individuals to include in the sample based on those random numbers. For instance, if the researcher wants to select of the members of the population at random, the researcher may decide to include each participant whose name is next to a random number that ends in one arbitrarily chosen value (such as 3). Identify Your Sources Of Statistics Anxiety In theory, if a sample is chosen randomly from a population, that simple random sample should be representative of the population from which it is drawn. A sample is representative if it has characteristics similar to those of the population. Suppose that the population of interest to a researcher is all the 500 students at Corinth College in the United States. Suppose the researcher randomly chooses a sample of 50 students from this population by using one of the methods just described. The researcher can evaluate whether this random sample is representative of the entire population of all Corinth College students by comparing the characteristics of the sample with the characteristics of the entire population. For example, if the entire population of Corinth College students has a mean age of 19.5 years and is 60% female and 40% male, the sample would be representative of the population with respect to age and gender composition if the sample had a mean age close to 19.5 years and a gender composition of about 60% female and 40% male. Representativeness of a sample can be assessed for many other characteristics, of course. Some characteristics may be particularly relevant to a research question; for example, if a researcher were primarily interested in the political attitudes of the population of students at Corinth, it would be important to evaluate whether the composition of the sample was similar to that of the overall Corinth College population in terms of political party preference. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details Random selection may be combined with systematic sampling methods such as stratification. A stratified random sample is obtained when the researcher divides the population into “strata,” or groups (such as Buddhist/Christian/Hindu/Islamic/Jewish/other religion or male/female), and then draws a random sample from each stratum or group. Stratified sampling can be used to ensure equal representation of groups (such as 50% women and 50% men in the sample) or that the proportional representation of groups in the sample is the same as in the population (if the entire population of students at Corinth College consists of 60% women and 40% men, the researcher might want the sample to contain the same proportion of women and men). 2 Basic sampling methods are reviewed in Cozby (2004); more complex survey sampling methods are discussed by Kalton (1983).U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details In some research domains (such as the public opinion polls done by the Gallup and Harris organizations), sophisticated sampling methods are used, and great care is taken to ensure that the sample is representative of the population of interest. In contrast, many behavioral and social science studies do not use such rigorous sampling procedures. Researchers in education, psychology, medicine, and many other disciplines often use accidental or convenience samples (instead of random samples). An accidental or convenience sample is not drawn randomly from a well-defined population of interest. Instead, a convenience sample consists of participants who are readily available to the researcher. For example, a teacher might use his class of students or a physician might use her current group of patients. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details A systematic difference between the characteristics of a sample and a population can be termed bias . For example, if 25% of Corinth College students are in each of the 4 years of the program, but 80% of the members of the convenience sample obtained through the subject pool are first-year students, this convenience sample is biased (it includes more first-year students, and fewer second-, third-, and fourth-year students, than the population). U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details The widespread use of convenience samples in disciplines such as psychology leads to underrepresentation of many types of people. Convenience samples that consist primarily of first-year North American college students typically underrepresent many kinds of people, such as persons younger than 17 and older than 30 years, persons with serious physical health problems, people who are not interested in or eligible for a college education, persons living in poverty, and persons from cultural backgrounds that are not numerically well represented in North America. For many kinds of research, it would be highly desirable for researchers to obtain samples from more diverse populations, particularly when the outcome variables of interest are likely to differ across age and cultural background. The main reason for the use of convenience samples is the low cost. The extensive use of college students as research participants limits the potential generalizability of results (Sears, 1986); this limitation should be explicitly acknowledged when researchers report and interpret research results. 1.5 Descriptive Versus Inferential Uses of Statistics Statistics that are used only to summarize information about a sample are called descriptive statistics. One common situation where statistics are used only as descriptive information occurs when teachers compute summary statistics, such as a mean for exam scores for students in a class. A teacher at Corinth College would typically use a mean exam score only to describe the performance of that specific classroom of students and not to make inferences about some broader population (such as the population of all students at Corinth College or all college students in North America). Identify Your Sources Of Statistics Anxiety Researchers in the behavioral and social sciences almost always want to make inferences beyond their samples; they hope that the attitudes or behaviors that they find in the small groups of college students who actually participate in their studies will provide evidence about attitudes or behaviors in broader populations in the world outside the laboratory. Thus, almost all the statistics reported in journal articles are inferential statistics. Researchers may want to estimate a population mean from a sample mean or a population correlation from a sample correlation. When means or correlations based on samples of scores are used to make inferences about (i.e., estimates of) the means or correlations for broader populations, they are called inferential statistics. If a researcher finds a strong correlation between self-esteem and popularity in a convenience sample of Corinth College students, the researcher typically hopes that these variables are similarly related in broader populations, such as all North American college students.U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details In some applications of statistics, such as political opinion polling, researchers often obtain representative samples from actual, well-defined populations by using well-thought-out sampling procedures (such as a combination of stratified and random sampling). When good sampling methods are used to obtain representative samples, it increases researcher confidence that the results from a sample (such as the stated intention to vote for one specific candidate in an election) will provide a good basis for making inferences about outcomes in the broader population of interest. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details However, in many types of research (such as experiments and small-scale surveys in psychology, education, and medicine), it is not practical to obtain random samples from the entire population of a country. Instead, researchers in these disciplines often use convenience samples when they conduct small-scale studies. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details Consider the example introduced earlier: A researcher wants to run an experiment to assess whether caffeine increases anxiety. It would not be reasonable to try to obtain a sample of participants from the entire adult population of the United States (consider the logistics involved in travel, for example). In practice, studies similar to this are usually conducted using convenience samples. At most colleges or universities in the United States, convenience samples primarily include persons between 18 and 22 years of age. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See When researchers obtain information about behavior from convenience samples, they cannot confidently use their results to make inferences about the responses of an actual, well-defined population. For example, if the researcher shows that a convenience sample of Corinth College students scores higher on anxiety after consuming a dose of caffeine, it would not be safe to assume that this result is generalizable to all adults or to all college students in the United States. Why not? For example, the effects of caffeine might be quite different for adults older than 70 than for 20-year-olds. The effects of caffeine might differ for people who regularly consume large amounts of caffeine than for people who never use caffeine. The effects of caffeine might depend on physical health. Although this is rarely explicitly discussed, most researchers implicitly rely on a principle that Campbell (cited in Trochim, 2001) has called “proximal similarity” when they evaluate the potential generalizability of research results based on convenience samples. It is possible to imagine a hypothetical population —that is, a larger group of people that is similar in many ways to the participants who were included in the convenience sample—and to make cautious inferences about this hypothetical population based on the responses of the sample. Campbell suggested that researchers evaluate the degree of similarity between a sample and hypothetical populations of interest and limit generalizations to hypothetical populations that are similar to the sample of participants actually included in the study. If the convenience sample consists of 50 Corinth College students who are between the ages of 18 and 22 and mostly of Northern European family background, it might be reasonable to argue (cautiously, of course) that the results of this study potentially apply to a hypothetical broader population of 18- to 22-year-old U.S. college students who come from similar ethnic or cultural backgrounds. This hypothetical population—all U.S. college students between 18 and 22 years from a Northern European family background—has a composition fairly similar to the composition of the convenience sample. It would be questionable to generalize about response to caffeine for populations that have drastically different characteristics from the members of the sample (such as persons who are older than age 50 or who have health problems that members of the convenience sample do not have). Generalization of results beyond a sample to make inferences about a broader population is always risky, so researchers should be cautious in making generalizations. An example involving research on drugs highlights the potential problems that can arise when researchers are too quick to assume that results from convenience samples provide accurate information about the effects of a treatment on a broader population. For example, suppose that a researcher conducts a series of studies to evaluate the effects of a new antidepressant drug on depression. Suppose that the participants are a convenience sample of depressed young adults between the ages of 18 and 22. If the researcher uses appropriate experimental designs and finds that the new drug significantly reduces depression in these studies, the researcher might tentatively say that this drug may be effective for other depressed young adults in this age range. It could be misleading, however, to generalize the results of the study to children or to older adults. A drug that appears to be safe and effective for a convenience sample of young adults might not be safe or effective in patients who are younger or older. To summarize, when a study uses data from a convenience sample, the researcher should clearly state that the nature of the sample limits the potential generalizability of the results. Of course, inferences about hypothetical or real populations based on data from a single study are never conclusive, even when random selection procedures are used to obtain the sample. An individual study may yield incorrect or misleading results for many reasons. Replication across many samples and studies is required before researchers can begin to feel confident about their conclusions. 1.6 Levels of Measurement and Types of Variables A controversial issue introduced early in statistics courses involves types of measurement for variables. Many introductory textbooks list the classic levels of measurement defined by S. Stevens (1946): nominal , ordinal , interval , and ratio (see Table 1.1 for a summary and Note 3 for a more detailed review of these levels of measurement). 3 Strict adherents to the Stevens theory of measurement argue that the level of measurement of a variable limits the set of logical and arithmetic operations that can appropriately be applied to scores. That, in turn, limits the choice of statistics. For example, if scores are nominal or categorical level of measurement, then according to Stevens, the only things we can legitimately do with the scores are count how many persons belong to each group (and compute proportions or percentages of persons in each group); we can also note whether two persons have equal or unequal scores. It would be nonsense to add up scores for a nominal variable such as eye color (coded 1 = blue, 2 = green, 3 = brown, 4 = hazel, 5 = other) and calculate a “mean eye color” based on a sum of these scores. 4 Table 1.1 Levels of Measurement, Arithmetic Operations, and Types of Statistics a. Jaccard and Becker (2002). b. Many variables that are widely used in the social and behavioral sciences, such as 5-point ratings for attitude and personality measurement, probably fall short of satisfying the requirement that equal differences between scores represent exactly equal changes in the amount of the underlying characteristics being measured. However, most authors (such as Harris, 2001) argue that application of parametric statistics to scores that fall somewhat short of the requirements for interval level of measurement does not necessarily lead to problems. In recent years, many statisticians have argued for less strict application of level of measurement requirements. In practice, many common types of variables (such as 5-point ratings of degree of agreement with an attitude statement) probably fall short of meeting the strict requirements for equal interval level of measurement. A strict enforcement of the level of measurement requirements outlined in many introductory textbooks creates a problem: Can researchers legitimately compute statistics (such as mean, t test, and correlation) for scores such as 5-point ratings when the differences between these scores may not represent exactly equal amounts of change in the underlying variable that the researcher wants to measure (in this case, strength of agreement)? Many researchers implicitly assume that the answer to this question is yes. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details The variables that are presented as examples when ordinal, interval, and ratio levels of measurement are defined in introductory textbooks are generally classic examples that are easy to classify. In actual practice, however, it is often difficult to decide whether scores on a variable meet the requirements for interval and ratio levels of measurement. The scores on many types of variables (such as 5-point ratings) probably fall into a fuzzy region somewhere between the ordinal and interval levels of measurement. How crucial is it that scores meet the strict requirements for interval level of measurement? U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details Many statisticians have commented on this problem, noting that there are strong differences of opinion among researchers. Vogt (1999) noted that there is considerable controversy about the need for a true interval level of measurement as a condition for the use of statistics such as mean, variance, and Pearson’s r , stating that “as with constitutional law, there are in statistics strict and loose constructionists in the interpretation of adherence to assumptions” (p. 158). Although some statisticians adhere closely to Stevens’s recommendations, many authors argue that it is not necessary to have data that satisfy the strict requirements for interval level of measurement to obtain interpretable and useful results for statistics such as mean and Pearson’s r. Identify Your Sources Of Statistics Anxiety Howell (1992) reviewed the arguments and concluded that the underlying level of measurement is not crucial in the choice of a statistic: U1D1-64 – Identify Your Sources Of Statistics Anxiety – See The validity of statements about the objects or events that we think we are measuring hinges primarily on our knowledge of those objects or events, not on the measurement scale. We do our best to ensure that our measures relate as closely as possible to what we want to measure, but our results are ultimately only the numbers we obtain and our faith in the relationship between those numbers and the underlying objects or events … the underlying measurement scale is not crucial in our choice of statistical techniques … a certain amount of common sense is required in interpreting the results of these statistical manipulations. (pp. 8–9) Harris (2001) says, I do not accept Stevens’s position on the relationship between strength [level] of measurement and “permissible” statistical procedures … the most fundamental reason for [my] willingness to apply multivariate statistical techniques to such data, despite the warnings of Stevens and his associates, is the fact that the validity of statistical conclusions depends only on whether the numbers to which they are applied meet the distributional assumptions … used to derive them, and not on the scaling procedures used to obtain the numbers. (pp. 444–445) Gaito (1980) reviewed these issues and concluded that “scale properties do not enter into any of the mathematical requirements” for various statistical procedures, such as ANOVA. Tabachnick and Fidell (2007) addressed this issue in their multivariate textbook: “The property of variables that is crucial to application of multivariate procedures is not type of measurement so much as the shape of the distribution” (p. 6). Zumbo and Zimmerman (1993) used computer simulations to demonstrate that varying the level of measurement for an underlying empirical structure (between ordinal and interval) did not lead to problems when several widely used statistics were applied. Based on these arguments, it seems reasonable to apply statistics (such as the sample mean, Pearson’s r, and ANOVA) to scores that do not satisfy the strict requirements for interval level of measurement. (Some teachers and journal reviewers continue to prefer the more conservative statistical practices advocated by Stevens; they may advise you to avoid the computation of means, variances, and Pearson correlations for data that aren’t clearly interval/ratio level of measurement.) When making decisions about the type of statistical analysis to apply, it is useful to make a simpler distinction between two types of variables: categorical versus quantitative (Jaccard & Becker, 2002). For a categorical or nominal variable, each number is merely a label for group membership. A categorical variable may represent naturally occurring groups or categories (the categorical variable gender can be coded 1 = male, 2 = female). Alternatively, a categorical variable can identify groups that receive different treatments in an experiment. In the hypothetical study described in this chapter, the categorical variable treatment can be coded 1 for participants who did not receive caffeine and 2 for participants who received 150 mg of caffeine. It is possible that the outcome variable for this imaginary study, anxiety, could also be a categorical or nominal variable; that is, a researcher could classify each participant as either 1 = anxious or 0 = not anxious, based on observations of behaviors such as speech rate or fidgeting. Quantitative variables have scores that provide information about the magnitude of differences between participants in terms of the amount of some characteristic (such as anxiety in this example). The outcome variable, anxiety, can be measured in several different ways. An observer who does not know whether each person had caffeine could observe behaviors such as speech rate and fidgeting and make a judgment about each individual’s anxiety level. An observer could rank order the participants in order of anxiety: 1 = most anxious, 2 = second most anxious, and so forth. (Note that ranking can be quite time-consuming if the total number of persons in the study is large.) A more typical measurement method for this type of research situation would be self-report of anxiety, perhaps using a 5-point rating similar to the one below. Each participant would be asked to choose a number from 1 to 5 in response to a statement such as “I am very anxious.” Conventionally, 5-point ratings (where the five response alternatives correspond to “degrees of agreement” with a statement about an attitude, a belief, or a behavior) are called Likert scales . However, questions can have any number of response alternatives, and the response alternatives may have different labels—for example, reports of the frequency of a behavior. (See Chapter 21 for further discussion of self-report questions and response alternatives.) What level of measurement does a 5-point rating similar to the one above provide? The answer is that we really don’t know. Scores on 5-point ratings probably do not have true equal interval measurement properties; we cannot demonstrate that the increase in the underlying amount of anxiety represented by a difference between 4 points and 3 points corresponds exactly to the increase in the amount of anxiety represented by the difference between 5 points and 4 points. None of the responses may represent a true 0 point. Five-point ratings similar to the example above probably fall into a fuzzy category somewhere between ordinal and interval levels of measurement. In practice, many researchers apply statistics such as means and standard deviations to this kind of data despite the fact that these ratings may fall short of the strict requirements for equal interval level of measurement; the arguments made by Harris and others above suggest that this common practice is not necessarily problematic. Usually researchers sum or average scores across a number of Likert items to obtain total scores for scales (as discussed in Chapter 21 ). These total scale scores are often nearly normally distributed; Carifio and Perla (2008) review evidence that application of parametric statistics to these scale scores produces meaningful results. Tabachnick and Fidell (2007) and the other authors cited above have argued that it is more important to consider the distribution shapes for scores on quantitative variables (rather than their levels of measurement). Many of the statistical tests covered in introductory statistics books were developed based on assumptions that scores on quantitative variables are normally distributed. To evaluate whether a batch of scores in a sample has a nearly normal distribution shape, we need to know what an ideal normal distribution looks like. The next section reviews the characteristics of the standard normal distribution. 1.7 The Normal Distribution Introductory statistics books typically present both empirical and theoretical distributions. An empirical distribution is based on frequencies of scores from a sample, while a theoretical distribution is defined by a mathematical function or equation. A description of an empirical distribution can be presented as a table of frequencies or in a graph such as a histogram . Sometimes it is helpful to group scores to obtain a more compact view of the distribution. SPSS® makes reasonable default decisions about grouping scores and the number of intervals and interval widths to use; these decisions can be modified by the user. Details about the decisions involved in grouping scores are provided in most introductory statistics textbooks and will not be discussed here. Thus, each bar in an SPSS histogram may correspond to an interval that contains a group of scores (rather than a single score). An example of an empirical distribution appears in Figure 1.1 . This shows a distribution of measurements of women’s heights (in inches). The height of each bar is proportional to the number of cases; for example, the tallest bar in the histogram corresponds to the number of women whose height is 64 in. For this empirical sample distribution, the mean female height M = 64 in., and the standard deviation for female height (denoted by s or SD) is 2.56 in. Note that if these heights were transformed into centimeters (M = 162.56 cm, s = 6.50 cm), the shape of the distribution would be identical; the labels of values on the X axis are the only feature of the graph that would change. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details Figure 1.1 A Histogram Showing an Empirical Distribution of Scores That Is Nearly Normal in Shape The smooth curve superimposed on the histogram is a plot of the mathematical (i.e., theoretical) function for an ideal normal distribution with a population mean μ = 64 and a population standard deviation σ = 2.56. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details Empirical distributions can have many different shapes. For example, the distribution of number of births across days of the week in the bar chart in Figure 1.2 is approximately uniform; that is, approximately one seventh of the births take place on each of the 7 days of the week (see Figure 1.2 ). Some empirical distributions have shapes that can be closely approximated by mathematical functions, and it is often convenient to use that mathematical function and a few parameters (such as mean and standard deviation) as a compact and convenient way to summarize information about the distribution of scores on a variable. The proportion of area that falls within a slice of an empirical distribution (in a bar chart or histogram) can be interpreted as a probability. Thus, based on the bar chart in Figure 1.2 , we can say descriptively that “about one seventh of the births occurred on Monday”; we can also say that if we draw an individual case from the distribution at random, there is approximately a one-seventh probability that a birth occurred on a Monday. When a relatively uncommon behavior (such as crying) is assessed through self-report, the distribution of frequencies is often a J-shaped, or roughly exponential, curve, as in Figure 1.3 , which shows responses to the question, “How many times did you cry last week?” (data from Brackett, Mayer, & Warner, 2004). Most people reported crying 0 times per week, a few reported crying 1 to 2 times a week, and very few reported crying more than 11 times per week. (When variables are frequency counts of behaviors, distributions are often skewed; in some cases, 0 is the most common behavior frequency. Statistics reviewed in this book assume normal distribution shapes; researchers whose data are extremely skewed and/or include a large number of 0s may need to consider alternative methods based on Poisson or negative binomial distributions, as discussed by Atkins & Gallop, 2007.). Identify Your Sources Of Statistics Anxiety Figure 1.2 A Bar Chart Showing a Fairly Uniform Distribution for Number of Births (Y Axis) by Day of the Week (X Axis) A theoretical distribution shape that is of particular interest in statistics is the normal (or Gaussian) distribution illustrated in Figure 1.4 . Students should be familiar with the shape of this distribution from introductory statistics. The curve is symmetrical, with a peak in the middle and tails that fall off gradually on both sides. The normal curve is often described as a bell-shaped curve. A precise mathematical definition of the theoretical normal distribution is given by the following equations: Figure 1.3 Bar Chart Showing a J-Shaped or Exponential Distribution Figure 1.4 A Standard Normal Distribution, Showing the Correspondence Between Distance From the Mean (Given as Number of σ Units or z Scores) and Proportion of Area Under the Curve π is a mathematical constant, approximate value 3.1416 … e is a mathematical constant, approximate value 2.7183 … μ is the mean—that is, the center of the distribution. σ is the standard deviation; that is, it corresponds to the dispersion of the distribution. In Figure 1.4 , the X value is mapped on the horizontal axis, and the Y height of the curve is mapped on the vertical axis. For the normal distribution curve defined by this mathematical function, there is a fixed relationship between the distance from the center of the distribution and the area under the curve, as shown in Figure 1.4 . The Y value (the height) of the normal curve asymptotically approaches 0 as the X distance from the mean increases; thus, the curve theoretically has a range of X from −∞ to +∞. Despite this infinite range of X values, the area under the normal curve is finite. The total area under the normal curve is set equal to 1.0 so that the proportions of this area can be interpreted as probabilities. The standard normal distribution is defined by Equations 1.1 and 1.2 with μ set equal to 0 and σ set equal to 1. In Figure 1.4 , distances from the mean are marked in numbers of standard deviations—for example, +1σ, +2σ, and so forth. From Figure 1.4 , one can see that the proportion of area under the curve that lies between 0 and +1σ is about .3413; the proportion of area that lies above +3σ is .0013. In other words, about 1 of 1,000 cases lies more than 3σ above the mean, or, to state this another way, the probability that a randomly sampled individual from this population will have a score that is more than 3σ above the mean is about .0013. A “family” of normal distributions (with different means and standard deviations) can be created by substituting in any specific values for the population parameters μ and σ. The term parameter is (unfortunately) used to mean many different things in various contexts. Within statistics, the term generally refers to a characteristic of a population distribution that can be estimated by using a corresponding sample statistic. The parameters that are most often discussed in introductory statistics are the population mean μ (estimated by a sample mean M) and the population standard deviation σ (estimated by the sample standard deviation , usually denoted by either s or SD). For example, assuming that the empirical distribution of women’s heights has a nearly normal shape with a mean of 64 in. and a standard deviation of 2.56 in. (as in Figure 1.1 ), a theoretical normal distribution with μ = 64 and σ = 2.56 will approximately match the location and shape of the empirical distribution of heights. It is possible to generate a family of normal distributions by using different values for μ and σ. For example, intelligence quotient (IQ) scores are normally distributed with μ = 100 and σ = 15; heart rate for a population of healthy young adults might have a mean of μ = 70 beats per minute (bpm) and a standard deviation σ of 11 bpm. When a population has a known shape (such as “normal”) and known parameters (such as μ = 70 and σ = 11), this is sufficient information to draw a curve that represents the shape of the distribution. If a population has an unknown distribution shape, we can still compute a mean and standard deviation, but that information is not sufficient to draw a sketch of the distribution. The standard normal distribution is the distribution generated by Equations 1.1 and 1.2 for the specific values μ = 0 and σ = 1.0 (i.e., population mean of 0 and population standard deviation of 1). When normally distributed X scores are rescaled so that they have a mean of 0 and a standard deviation of 1, they are called standard scores or z scores. Figure 1.4 shows the standard normal distribution for z. There is a fixed relationship between distance from the mean and area, as shown in Figure 1.4 . For example, the proportion of area under the curve that lies between z = 0 and z = +1 under the standard normal curve is always .3413 (34.13% of the area). Recall that a proportion of area under the uniform distribution can be interpreted as a probability; similarly, a proportion of area under a section of the normal curve can also be interpreted as a probability. Because the normal distribution is widely used, it is useful for students to remember some of the areas that correspond to z scores. For example, the bottom 2.5% and top 2.5% of the area of a standard normal distribution lie below z = −1.96 and above z = +1.96, respectively. That is, 5% of the scores in a normally distributed population lie more than 1.96 standard deviations above or below the mean. We will want to know when scores or test statistics have extreme or unusual values relative to some distribution of possible values. In most situations, the outcomes that correspond to the most extreme 5% of a distribution are the outcomes that are considered extreme or unlikely. The statistics covered in introductory textbooks, such as the t test, ANOVA, and Pearson’s r, were developed based on the assumption that scores on quantitative variables are normally distributed in the population. Thus, distribution shape is one factor that is taken into account when deciding what type of statistical analysis to use. Students should be aware that although normal distribution shapes are relatively common, many variables are not normally distributed. For example, income tends to have a distribution that is asymmetric; it has a lower limit of 0, but there is typically a long tail on the upper end of the distribution. Relatively uncommon behaviors, such as crying, often have a J-shaped distribution (as shown in Figure 1.3 ). Thus, it is important to examine distribution shapes for quantitative variables before applying statistical analyses such as ANOVA or Pearson’s r. Methods for assessing whether variables are normally distributed are described in Chapter 4 . Identify Your Sources Of Statistics Anxiety 1.8 Research Design Up to this point, the discussion has touched on two important issues that should be taken into account when deciding what statistical analysis to use: the types of variables involved (the level of measurement and whether the variables are categorical or quantitative) and the distribution shapes of scores on quantitative variables. We now turn from a discussion of individual variables (e.g., categorical vs. quantitative types of variables and the shapes of distributions of scores on variables) to a brief consideration of research design. Identify Your Sources Of Statistics Anxiety It is extremely important for students to recognize that a researcher’s ability to draw causal inferences is based on the nature of the research design (i.e., whether the study is an experiment) rather than the type of analysis (such as correlation vs. ANOVA). This section briefly reviews basic research design terminology. Readers who have not taken a course in research methods may want to consult a basic research methods textbook (such as Cozby, 2004) for a more thorough discussion of these issues. Identify Your Sources Of Statistics Anxiety 1.8.1 Experimental Design In behavioral and social sciences, an experiment typically includes the following elements: 1. Random assignment of participants to groups or treatment levels (or other methods of assignment of participants to treatments, such as matched samples or repeated measures , that ensure equivalence of participant characteristics across treatments). 2. Two or more researcher-administered treatments, dosage levels, or interventions. 3. Experimental control of other “nuisance” or “error” variables that might influence the outcome: The goal is to avoid confounding other variables with the different treatments and to minimize random variations in response due to other variables that might influence participant behavior; the researcher wants to make certain that no other variable is confounded with treatment dosage level. If the caffeine group is tested before a midterm exam and the no-caffeine group is tested on a Friday afternoon before a holiday weekend, there would be a confound between the effects of the exam and those of the caffeine; to avoid a confound, both groups should be tested at the same time under similar circumstances. The researcher also wants to make sure that random variation of scores within each treatment condition due to nuisance variables is not too great; for example, testing persons in the caffeine group at many different times of the day and days of the week could lead to substantial variability in anxiety scores within this group. One simple way to ensure that a nuisance variance is neither confounded with treatment nor a source of variability of scores within groups is to “hold the variable constant”; for example, to avoid any potential confound of the effects of cigarette smoking with the effects of caffeine and to minimize the variability of anxiety within groups that might be associated with smoking, the researcher could “hold the variable smoking constant” by including only those participants who are not smokers. 4. Assessment of an outcome variable after the treatment has been administered. 5. Comparison of scores on outcome variables across people who have received different treatments, interventions, or dosage levels: Statistical analyses are used to compare group means and assess how strongly scores on the outcome variable are associated with scores on the treatment variable. In contrast, nonexperimental studies usually lack a researcher-administered intervention, experimenter control of nuisance or error variables, and random assignment of participants to groups; they typically involve measuring or observing several variables in naturally occurring situations. The goal of experimental design is to create a situation in which it is possible to make a causal inference about the effect of a manipulated treatment variable (X) on a measured outcome variable (Y ). A study that satisfies the conditions for causal inferences is said to have internal validity . The conditions required for causal inference (a claim of the form “X causes Y”) (from Cozby, 2004) include the following: 1. The X and Y variables that represent the “cause” and the “effect” must be systematically associated in the study. That is, it only makes sense to theorize that X might cause Y if X and Y covary (i.e., if X and Y are statistically related). Covariation between variables can be assessed using statistical methods such as ANOVA and Pearson’s r. Covariation of X and Y is a necessary, but not sufficient, condition for causal inference. In practice, we do not require perfect covariation between X and Y before we are willing to consider causal theories. However, we look for evidence that X and Y covary “significantly” (i.e., we use statistical significance tests to try to rule out chance as an explanation for the obtained pattern of results). 2. The cause, X, must precede the effect, Y, in time. In an experiment, this requirement of temporal precedence is met by manipulating the treatment variable X prior to measuring or observing the outcome variable Y. 3. There must not be any other variable confounded with (or systematically associated with) the X treatment variable. If there is a confound between X and some other variable, the confounded variable is a rival explanation for any observed differences between groups. Random assignment of participants to treatment groups is supposed to ensure equivalence in the kinds of participants in the treatment groups and to prevent a confound between individual difference variables and treatment condition. Holding other situational factors constant for groups that receive different treatments should avoid confounding treatment with situational variables, such as day of the week or setting. 4. There should be some reasonable theory that would predict or explain a cause-and-effect relationship between the variables. Of course, the results of a single study are never sufficient to prove causality. However, if a relationship between variables is replicated many times across well-designed experiments, belief that there could be a potential causal connection tends to increase as the amount of supporting evidence increases. Compared with quasi-experimental or nonexperimental designs, experimental designs provide relatively stronger evidence for causality, but causality cannot be proved conclusively by a single study even if it has a well-controlled experimental design. There is no necessary connection between the type of design (experimental vs. non-experimental) and the type of statistic applied to the data (such as t test vs. Pearson’s r) (see Table 1.2 ). Because experiments often (but not always) involve comparison of group means, ANOVA and t tests are often applied to experimental data. However, the choice of a statistic depends on the type of variables in the dataset rather than on experimental design. An experiment does not have to compare a small number of groups. For example, participants in a drug study might be given 20 different dosage levels of a drug (the independent variable X could be the amount of drug administered), and a response to the drug (such as self-reported pain, Y) could be measured. Pairs of scores (for X = drug dosage and Y = reported pain) could be analyzed using methods such as Pearson correlation, although this type of analysis is uncommon in experiments. Internal validity is the degree to which the results of a study can be used to make causal inferences; internal validity increases with greater experimental control of extraneous variables. On the other hand, external validity is the degree to which the results of a study can be generalized to groups of people, settings, and events that occur in the real world. A well-controlled experimental situation should have good internal validity. Table 1.2 Statistical Analysis and Research Designs External validity is the degree to which the results of a study can be generalized (beyond the specific participants, setting, and materials involved in the study) to apply to real-world situations. Some well-controlled experiments involve such artificial situations that it is unclear whether results are generalizable. For example, an experiment that involves systematically presenting different schedules of reinforcement or reward to a rat in a Skinner box, holding all other variables constant, typically has high internal validity (if the rat’s behavior changes, the researcher can be reasonably certain that the changes in behavior are caused by the changes in reward schedule, because all other variables are held constant). However, this type of research may have lower external validity; it is not clear whether the results of a study of rats isolated in Skinner boxes can be generalized to populations of children in school classrooms, because the situation of children in a classroom is quite different from the situation of rats in a Skinner box. External validity is better when the research situation is closely analogous to, or resembles, the real-world situations that the researcher wants to learn about. Some experimental situations achieve strong internal validity at the cost of external validity. However, it is possible to conduct experiments in field settings or to create extremely lifelike and involving situations in laboratory settings, and this can improve the external validity or generalizability of research results. The strength of internal and external validity depends on the nature of the research situation; it is not determined by the type of statistical analysis that happens to be applied to the data. 1.8.2 Quasi-Experimental Design Quasi-experimental designs typically include some, but not all, of the features of a true experiment. Often they involve comparison of groups that have received different treatments and/or comparison of groups before versus after an intervention program. Often they are conducted in field rather than in laboratory settings. Usually, the groups in quasi-experimental designs are not formed by random assignment, and thus, the assumption of equivalence of participant characteristics across treatment conditions is not satisfied. Often the intervention in a quasi experiment is not completely controlled by the researcher, or the researcher is unable to hold other variables constant. To the extent that a quasi experiment lacks the controls that define a well-designed experiment, a quasi-experimental design provides much weaker evidence about possible causality (and, thus, weaker internal validity). Because quasi experiments often focus on interventions that take place in real-world settings (such as schools), they may have stronger external validity than laboratory-based studies. Campbell and Stanley (1966) provide an extensive outline of threats to internal and external validity in quasi-experimental designs that is still valuable. Shadish, Cook, and Campbell (2001) provide further information about issues in the design and analysis of quasi-experimental 1.8.3 Nonexperimental Research Design Many studies do not involve any manipulated treatment variable. Instead, the researcher measures a number of variables that are believed to be meaningfully related. Variables may be measured at one point in time or, sometimes, at multiple points in time. Then, statistical analyses are done to see whether the variables are related in ways that are consistent with the researcher’s expectations. The problem with nonexperimental research design is that any potential independent variable is usually correlated or confounded with other possible independent variables; therefore, it is not possible to determine which, if any, of the variables have a causal impact on the dependent variable . In some nonexperimental studies, researchers make distinctions between independent and dependent variables (based on implicit theories about possible causal connections). However, in some nonexperimental studies, there may be little or no basis to make such a distinction. Nonexperimental research is sometimes called “correlational” research. This use of terminology is unfortunate because it can confuse beginning students. It is helpful to refer to studies that do not involve interventions as “nonexperimental” (rather than correlational) to avoid possible confusion between the Pearson’s r correlation statistic and nonexperimental design. As shown in Table 1.2 , a Pearson correlation can be performed on data that come from experimental designs, although it is much more often encountered in reports of non-experimental data. A t test or ANOVA is often used to analyze data from experiments, but these tests are also used to compare means between naturally occurring groups in non-experimental studies. (In other words, to judge whether a study is experimental or non-experimental, it is not useful to ask whether the reported statistics were ANOVAs or correlations. We have to look at the way the study was conducted, that is, whether it has the features typically found in experiments that were listed earlier.) The degree to which research results can be interpreted as evidence of possible causality depends on the nature of the design (experimental vs. nonexperimental), not on the type of statistic that happens to be applied to the data (Pearson’s r vs. t test or ANOVA). While experiments often involve comparison of groups, group comparisons are not necessarily experimental. A nonexperimental study usually has weak internal validity; that is, merely observing that two variables are correlated is not a sufficient basis for causal inferences. If a researcher finds a strong correlation between an X and Y variable in a nonexperimental study, the researcher typically cannot rule out rival explanations (e.g., changes in Y might be caused by some other variable that is confounded with X rather than by X). On the other hand, some nonexperimental studies (particularly those that take place in field settings) may have good external validity; that is, they may examine naturally occurring events and behaviors. 1.8.4 Between-Subjects Versus Within-Subjects or Repeated Measures When an experiment involves comparisons of groups, there are many different ways in which participants or cases can be placed in these groups. Despite the fact that most writers now prefer to use the term participant rather than subject to refer to a person who contributes data in a study, the letter S is still widely used to stand for “subjects” when certain types of research designs are When a study involves a categorical or group membership variable, we need to pay attention to the composition of the groups when we decide how to analyze the data. One common type of group composition is called between-subjects (between-S) or independent groups. In a between-S or independent groups study, each participant is a member of one and only one group. A second common type of group composition is called within-subjects (within-S) or repeated measures. In a repeated measures study, each participant is a member of every group; if the study includes several different treatments, each participant is tested under every treatment condition. For example, consider the caffeine/anxiety study. This study could be done using either a between-S or a within-S design . In a between-S version of this study, a sample of 30 participants could be divided randomly into two groups of 15 each. Each group would be given only one treatment (Group 1 would receive a beverage that contains no caffeine; Group 2 would receive a beverage that contains 150 mg caffeine). In a within-S or repeated measures version of this study, each of the 30 participants would be observed twice: once after drinking a beverage that does not contain caffeine and once after drinking a beverage that contains 150 mg caffeine. Another possible variation of design would be to use both within-S and between-S comparisons. For example, the researcher could randomly assign 15 people to each of the two groups, caffeine versus no caffeine, and then assess each person’s anxiety level at two points in time, before and after consuming a beverage. This design has both a between-S comparison (caffeine vs. no caffeine) and a within-S comparison (anxiety before vs. after drinking a beverage that may or may not contain caffeine). Within-S or repeated measures designs raise special problems, such as the need to control for order and carryover effects (discussed in basic research methods textbooks such as Shaughnessy, Zechmeister, & Zechmeister, 2003). In addition, different statistical tests are used to compare group means for within-S versus between-S designs. For a between-S design, one-way ANOVA for independent samples is used; for a within-S design, repeated measures ANOVA is used. A thorough discussion of repeated measures analyses is provided by Keppel (1991), and a brief introduction to repeated measures ANOVA is provided in Chapter 22 of this textbook. Thus, a researcher has to know whether the composition of groups in a study is between-S or within-S in order to choose an appropriate statistical analysis. In nonexperimental studies, the groups are almost always between-S because they are usually based on previously existing participant characteristics (e.g., whether each participant is male or female, a smoker or a nonsmoker). Generally, when we talk about groups based on naturally occurring participant characteristics, group memberships are mutually exclusive; for example, a person cannot be classified as both a smoker and a nonsmoker. (We could, of course, create a larger number of groups such as nonsmoker, occasional smoker, and heavy smoker if the simple distinction between smokers and nonsmokers does not provide a good description of smoking behavior.) In experiments, researchers can choose to use either within-S or between-S designs. An experimenter typically assigns participants to treatment groups in ways that are intended to make the groups equivalent prior to treatment. For example, in a study that examines three different types of stress, each participant may be randomly assigned to one and only one treatment group or type of stress. In this textbook, all group comparisons are assumed to be between-S unless otherwise specified. Chapter 22 deals specifically with repeated measures ANOVA. 1.9 Combinations of These Design Elements For clarity and simplicity, each design element (e.g., comparison of groups formed by an experimenter vs. naturally occurring group; between-and within-S design) has been discussed separately. This book covers most of the “building blocks” that can be included in more complex designs. These design elements can be combined in many ways. Within an experiment, some treatments may be administered using a between-S and others using a within-S or repeated measures design. A factorial study may include a factor that corresponds to an experimental manipulation and also a factor that corresponds to naturally occurring group memberships (as discussed in Chapter 13 on factorial ANOVA). Later chapters in this book include some examples of more complex designs. For example, a study may include predictor variables that represent group memberships and also quantitative predictor variables called covariates (as in analysis of covariance [ANCOVA], Chapter 17 ). For complex experiments (e.g., experiments that include both between-S and within-S factors), researchers should consult books that deal with these in greater detail (e.g., Keppel & Zedeck, 1989; Myers & Well, 1995). 1.10 Parametric Versus Nonparametric Statistics Another issue that should be considered when choosing a statistical method is whether the data satisfy the assumptions for parametric statistical methods. Definitions of the term parametric statistics vary across textbooks. When a variable has a known distribution shape (such as normal), we can draw a sketch of the entire distribution of scores based on just two pieces of information: the shape of the distribution (such as normal) and a small number of population parameters for that distribution (for normally distributed scores, we need to know only two parameters, the population mean μ and the population standard deviation σ, in order to draw a picture of the entire distribution). Parametric statistics involve obtaining sample estimates of these population parameters (e.g., the sample mean M is used to estimate the population mean μ; the sample standard deviation s is used to estimate the population standard deviation σ). Most authors include the following points in their discussion of parametric statistics: 1. Parametric statistics include the analysis of means, variances, and sums of squares . For example, t test, ANOVA, Pearson’s r, and regression are examples of parametric statistics. 2. Parametric statistics require quantitative dependent variables that are at least approximately interval/ratio level of measurement. In practice, as noted in Section 1.6, this requirement is often not strictly observed. For example, parametric statistics (such as mean and correlation) are often applied to scores from 5-point ratings, and these scores may fall short of satisfying the strict requirements for interval level of measurement. 3. The parametric statistics included in this book and in most introductory texts assume that scores on quantitative variables are normally distributed. (This assumption is violated when scores have a uniform or J-shaped distribution, as shown in Figures 1.2 and 1.3 , or when there are extreme outliers .) 4. For analyses that involve comparisons of group means, the variances of dependent variable scores are assumed to be equal across the populations that correspond to the groups in the study. 5. Parametric analyses often have additional assumptions about the distributions of scores on variables (e.g., we need to assume that X and Y are linearly related to use Pearson’s r). It is unfortunate that some students receive little or no education on nonparametric statistics . Most introductory statistics textbooks include one or two chapters on non-parametric methods; however, these are often at the end of the book, and instructors rarely have enough time to cover this material in a one-semester course. Sometimes nonparametric statistics do not necessarily involve estimation of population parameters; they often rely on quite different approaches to sample data—for example, comparing the sum of ranks across groups in the Wilcoxon rank sum test . There is no universally agreed-on definition for nonparametric statistics, but most discussions of nonparametric statistics include the following: 1. Nonparametric statistics include the median, the chi-square (χ2) test of association between categorical variables, the Wilcoxon rank sum test, the sign test, and the Friedman one-way ANOVA by ranks. Many nonparametric methods involve counting frequencies or finding medians. 2. The dependent variables for nonparametric tests may be either nominal or ordinal level of measurement. (Scores may be obtained as ranks initially, or raw scores may be converted into ranks as one of the steps involved in performing a nonparametric analysis.) 3. Nonparametric statistics do not require scores on the outcome variable to be normally distributed. 4. Nonparametric statistics do not typically require an assumption that variances are equal across groups. 5. Outliers are not usually a problem in nonparametric analyses; these are unlikely to arise in ordinal (rank) or nominal (categorical) data. Researchers should consider the use of nonparametric statistics when their data fail to meet some or all of the requirements for parametric statistics. 5 The issues outlined above are summarized in Table 1.3 . Jaccard and Becker (2002) pointed out that there is disagreement among behavioral scientists about when to use parametric versus nonparametric analyses. Some conservative statisticians argue that parametric analyses should be used only when all the assumptions listed in the discussion of parametric statistics above are met (i.e., only when scores on the dependent variable are quantitative, interval/ratio level of measurement, and normally distributed and meet all other assumptions for the use of a specific statistical test). On the other hand, Bohrnstedt and Carter (1971) have advocated a very liberal position; they argued that many parametric techniques are fairly robust 6 to violations of assumptions and concluded that even for variables measured at an ordinal level, “parametric analyses not only can be, but should be, applied.” Identify Your Sources Of Statistics Anxiety The recommendation made here is a compromise between the conservative and liberal positions. It is useful to review all the factors summarized in Table 1.3 when making the choice between parametric and nonparametric tests. A researcher can safely use parametric statistics when all the requirements listed for parametric tests are met. That is, if scores on the dependent variable are quantitative and normally distributed, scores are interval/ratio level of measurement and have equal variances across groups, and there is a minimum N per group of at least 20 or 30, parametric statistics may be used.U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details When only one or two of the requirements for a parametric statistic are violated, or if the violations are not severe (e.g., the distribution shape for scores on the outcome variable is only slightly different from normal), then it may still be reasonable to use a parametric statistic. When in doubt about whether to choose parametric or nonparametric statistics, many researchers lean toward choosing parametric statistics. There are several reasons for this preference. First, the parametric tests are more familiar to most students, researchers, and journal editors. Second, it is widely thought that parametric tests have better statistical power ; that is, they give the researcher a better chance of obtaining a statistically significant outcome (however, this is not necessarily always the case). An additional issue becomes relevant when researchers begin to work with more than one predictor and/or more than one outcome variable. For some combinations of predictor and outcome variables, a parametric analysis exists, but there is no analogous nonparametric test. Thus, researchers who use only nonparametric analyses may be limited to working with fewer variables. (This is not necessarily a bad thing.) When violations of the assumptions for the use of parametric statistics are severe, it is more appropriate to use nonparametric analyses. Violations of assumptions (such as the assumption that scores are distributed normally in the population) become much more problematic when they are accompanied by small (and particularly small and unequal) group sizes. Table 1.3 Parametric and Nonparametric Statistics Parametric Tests (Such as M, t, F, Pearson’s r) Are More Appropriate When Nonparametric Tests (Such as Median, Wilcoxon Rank Sum Test, Friedman One-Way ANOVA by Ranks, Spearman r) Are More Appropriate When The outcome variable Y is interval/ratio level of measurement a The outcome variable Y is nominal or rank ordinal level of measurement (data may be collected as ranks or converted to ranks) Scores on Y are approximately normally distributed b Scores on Y are not necessarily normally distributed There are no extreme outlier values of Y c There can be extreme outlier Y scores Variances of Y scores are approximately equal across populations that correspond to Variances of scores are not necessarily equal across groups groups in the study d The N of cases in each group is “large” e The N of cases in each group can be “small” a. Many variables that are widely used in psychology (such as 5-point or 7-point attitude ratings, personality test scores, and so forth) have scores that probably do not have true equal interval-level measurement properties. For example, consider 5-point degree of agreement ratings: The difference between a score of 4 and 5 and the difference between a score of 1 and 2 probably do not correspond to the same increase of change in agreement. Thus, 5-point ratings probably do not have true equal-interval measurement properties. On the basis of the arguments reported in Section 1.5, many researchers go ahead and apply parametric statistics (such as Pearson’s r and t test) to data from 5-point ratings and personality tests and other measures that probably fall short of satisfying the requirements for a true interval level of measurement as defined by S. Stevens (1946). U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details b. Chapter 4 discusses how to assess this by looking at histograms of Y scores to see if the shape resembles the bell curve shown in Figure 1.4 . c. Chapter 4 also discusses identification and treatment of outliers or extreme scores. d. Parametric statistics such as the t test and ANOVA were developed based on assumption that the Y scores have equal variances in the populations that correspond to the samples in the study. Data that violate the assumption of equal variances can, in theory, lead to misleading results (an increased risk of Type I error, discussed in Chapter 3 ). In practice, however, the t test and ANOVA can yield fairly accurate results even when the equal variance assumption is violated, unless the Ns of cases within groups are small and/or unequal across groups. Also, there is a modified version of the t test (usually called “ separate variances t test ” or “equal variances not assumed t test”) that takes violations of the equal variance assumption into account and corrects for this problem. e. There is no agreed-on standard about an absolute minimum sample size required in the use of parametric statistics. The suggested guideline given here is as follows: Consider nonparametric tests when N is less than 20, and definitely use nonparametric tests when N is less than 10 per group; but this is arbitrary. Smaller Ns are most problematic when there are other problems with the data, such as outliers. In practice, it is useful to consider this entire set of criteria. If the data fail to meet just one criterion for the use of parametric tests (for example, if scores on Y do not quite satisfy the requirements for interval/ratio level of measurement), researchers often go ahead and use parametric tests, as long as there are no other serious problems. However, the larger the number of problems with the data, the stronger the case becomes for the use of nonparametric tests. If the data are clearly ordinal or if Y scores have a drastically nonnormal shape and if, in addition, the Ns within groups are small, a nonparametric test would be strongly preferred. Group Ns that are unequal can make other problems (such as unequal variances across groups) more serious. Almost all the statistics reported in this textbook are parametric. (The only nonparametric statistics reported are the χ2 test of association in Chapter 8 and the binary logistic regression in Chapter 23 .) If a student or researcher anticipates that his or her data will usually require nonparametric analysis, that student should take a course or at least buy a good reference book on nonparametric methods. Special statistical methods have been developed to handle ordinal data—that is, scores that are obtained as ranks or that are converted into ranks during preliminary data handling. Strict adherence to the Stevens theory would lead us to use medians instead of means for ordinal data (because finding a median involves only rank ordering and counting scores, not summing them). The choice between parametric and nonparametric statistics is often difficult because there are no generally agreed-on decision standards. In research methods and statistics, generally, it is more useful to ask, “What are the advantages and disadvantages of each approach to the problem?” than to ask, “Which is the right and which is the wrong answer?” Parametric and nonparametric statistics each have strengths and limitations. Experimental and nonexperimental designs each have advantages and problems. Self-report and behavioral observations each have advantages and disadvantages. In the discussion section of a research report, the author should point out the advantages and also acknowledge the limitations of the choices that he or she has made in research design and data analysis. The limited coverage of nonparametric techniques in this book should not be interpreted as a negative judgment about their value. There are situations (particularly designs with small Ns and severely nonnormal distributions of scores on the outcome variables) where nonparametric analyses are preferable. In particular, when data come in the form of ranks, nonparametric procedures developed specifically for the analysis of ranks may be preferable. For a thorough treatment of nonparametric statistics, see Siegel and Castellan (1988). 1.11 Additional Implicit Assumptions Some additional assumptions are so basic that they generally are not even mentioned, but these are important considerations whether a researcher uses parametric or nonparametric statistics: 1. Scores on the outcome variable are assumed to be independent of each other (except in repeated measures data, where correlations among scores are expected and the pattern of dependence among scores has a relatively simple pattern). It is easier to explain the circumstances that lead to “nonindependent” scores than to define independence of observations formally. Suppose a teacher gives an examination in a crowded classroom, and students in the class talk about the questions and exchange information. The scores of students who communicate with each other will not be independent; that is, if Bob and Jan jointly decide on the same answer to several questions on the exam, they are likely to have similar (statistically related or dependent) scores. Nonindependence among scores can arise due to many kinds of interactions among participants, apart from sharing information or cheating; nonindependence can arise from persuasion, competition, or other kinds of social influence. This problem is not limited to human research subjects. For example, if a researcher measures the heights of trees in a grove, the heights of neighboring trees are not independent; a tree that is surrounded by other tall trees has to grow taller to get exposure to sunlight. For both parametric and nonparametric statistics, different types of analysis are used when the data involve repeated measures than when they involve independent outcome scores. 2. The number of cases in each group included in the analysis should be reasonably large. Parametric tests typically require larger numbers per group than nonparametric tests to yield reasonable results. However, even for nonparametric analyses, extremely small Ns are undesirable. (A researcher does not want to be in a situation where a 1- or 2-point change in score for one participant would completely change the nature of the outcome, and very small Ns sometimes lead to this kind of instability.) There is no agreed-on absolute minimum N for each group. For each analysis presented in this textbook, a discussion about sample size requirements is included. 3. The analysis will yield meaningful and interpretable information about the relations between variables only if we have a “ correctly specified model ”; that is, we have included all the variables that should be included in the analysis, and we have not included any irrelevant or inappropriate variables. In other words, we need a theory that correctly identifies which variables should be included and which variables should be excluded. Unfortunately, we can never be certain that we have a correctly specified model. It is always possible that adding or dropping a variable in the statistical analysis might change the outcome of the analysis substantially. Our inability to be certain about whether we have a correctly specified model is one of the many reasons why we can never take the results of a single study as proof that a theory is correct (or incorrect). U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details 1.12 Selection of an Appropriate Bivariate Analysis Bivariate statistics assess the relation between a pair of variables. Often, one variable is designated as the independent variable and the other as dependent. When one of the variables is manipulated by the researcher, that variable is designated as the independent variable. In nonexperimental research situations, the decision regarding which variable to treat as an independent variable may be arbitrary. In some research situations, it may be preferable not to make a distinction between independent and dependent variables; instead, the researcher may merely report that two variables are correlated without identifying one as the predictor of the other. When a researcher has a theory that X might cause or influence Y, the researcher generally uses scores on X as predictors of Y even when the study is nonexperimental. However, the results of a nonexperimental study cannot be used to make a causal inference, and researchers need to be careful to avoid causal language when they interpret results from nonexperimental studies. During introductory statistics courses, the choice of an appropriate statistic for various types of data is not always explicitly addressed. Aron and Aron (2002) and Jaccard and Becker (2002) provide good guidelines for the choice of bivariate analyses. The last part of this chapter summarizes the issues that have been discussed up to this point and shows how consideration of these issues influences the choice of an appropriate bivariate statistical analysis. Similar issues continue to be important when the analyses include more than one predictor and/or more than one outcome The choice of an appropriate bivariate analysis to assess the relation between two variables is often based, in practice, on the types of variables involved: categorical versus quantitative. The following guidelines for the choice of statistic are based on a discussion in Jaccard and Becker (2002). Suppose that a researcher has a pair of variables X and Y. There are three possible combinations of types of variables (see Table 1.4 ): Case I: Both X and Y are categorical. Case II: X is categorical, and Y is quantitative (or Y is categorical, and X is quantitative). Case III: Both X and Y are quantitative. Consider Case I: The X and Y variables are both categorical; the data are usually summarized in a contingency table that summarizes the numbers of scores in each X, Y group. The chi-square test of association (or one of many other contingency table statistics) can be used to assess whether X and Y are significantly related. This will be discussed in Chapter 8 . There are many other types of statistics for contingency tables (Everitt, 1977). Consider Case III: The X and Y variables are both quantitative variables. If X and Y are linearly related (and if other assumptions required for the use of Pearson’s r are reasonably well satisfied), a researcher is likely to choose Pearson’s r to assess the relation between the X and Y variables. Other types of correlation (such as Spearman r ) may be preferred when the assumptions for Pearson’s r are violated; Spearman r is an appropriate analysis when the X and Y scores consist of ranks (or are converted to ranks to get rid of problems such as extreme outliers). Now consider Case II: One variable (usually the X or independent variable) is categorical, and the other variable (usually the Y or dependent variable) is quantitative. In this situation, the analysis involves comparing means, medians, or sums of ranks on the Y variable across the groups that correspond to scores on the X variable. The choice of an appropriate statistic in this situation depends on several factors; the following list is adapted from Jaccard and Becker (2002): 1. Whether scores on Y satisfy the assumptions for parametric analyses or violate these assumptions badly enough so that nonparametric analyses should be used 2. The number of groups that are compared (i.e., the number of levels of the X variable) 3. Whether the groups correspond to a between-S design (i.e., there are different participants in each group) or to a within-S or repeated measures design One cell in Table 1.4 includes a decision tree for Case II from Jaccard and Becker (2002); this decision tree maps out choices among several common bivariate statistical methods based on the answers to these questions. For example, if the scores meet the assumptions for a parametric analysis, two groups are compared, and the design is between-S, the independent samples t test is a likely choice. Note that although this decision tree leads to just one analysis for each situation, sometimes other analyses could be used. This textbook covers only parametric statistics (i.e., statistics in the parametric branch of the decision tree for Case II in Table 1.4 ). In some situations, however, nonparametric statistics may be preferable (see Siegell & Castellan, 1988, for a thorough presentation of nonparametric methods). Table 1.4 Selecting an Appropriate Bivariate Statistic Based on Type of Independent Variable (IV) and Dependent Variable (DV) SOURCE: Decision tree adapted from Jaccard and Becker (2002). 1.13 Summary Reconsider the hypothetical experiment to assess the effects of caffeine on anxiety. Designing a study and choosing an appropriate analysis raises a large number of questions even for this very simple research question. A nonexperimental study could be done; that is, instead of administering caffeine, a researcher could ask participants to self-report the amount of caffeine consumed within the past 3 hours and then self-report anxiety. A researcher could do an experimental study (i.e., administer caffeine to one group and no caffeine to a comparison group under controlled conditions and subsequently measure anxiety). If the study is conducted as an experiment, it could be done using a between-S design (each participant is tested under only one condition, either with caffeine or without caffeine), or it could be done as a within-S or repeated measures study (each participant is tested under both conditions, with and without caffeine). Let’s assume that the study is conducted as a simple experiment with a between-S design. The outcome measure of anxiety could be a categorical variable (i.e., each participant is identified by an observer as a member of the “anxious” or “nonanxious” group). In this case, a table could be set up to report how many of the persons who consumed caffeine were classified as anxious versus nonanxious, as well as how many of those who did not receive caffeine were classified as anxious versus nonanxious, and a chi-square test could be performed to assess whether people who received caffeine were more likely to be classified as anxious than people who did not receive caffeine. If the outcome variable, anxiety, is assessed by having people self-report their level of anxiety using a 5-point ratings, an independent samples t test could be used to compare mean anxiety scores between the caffeine and no-caffeine groups. If examination of the data indicated serious violations of the assumptions for this parametric test (such as nonnormally distributed scores or unequal group variances, along with very small numbers of participants in the groups), the researcher might choose to use the Wilcoxon rank sum test to analyze the data from this study. This chapter reviewed issues that generally are covered in early chapters of introductory statistics and research methods textbooks. On the basis of this material, the reader should be equipped to think about the following issues, both when reading published research articles and when planning a study: 1. Evaluate whether the sample is a convenience sample or a random sample from a well-defined population, and recognize how the composition of the sample in the study may limit the ability to generalize results to broader populations. 2. Understand that the ability to make inferences about a population from a sample requires that the sample be reasonably representative of or similar to the population. 3. For each variable in a study, understand whether it is categorical or quantitative. 4. Recognize the differences between experimental, nonexperimental, and quasi-experimental designs. 5. Understand the difference between between-S (independent groups) and within-S (repeated measures) designs. 6. Recognize that research designs differ in internal validity (the degree to which they satisfy the conditions necessary to make a causal inference) and external validity (the degree to which results are generalizable to participants, settings, and materials different from those used in the study). 7. Understand why experiments typically have stronger internal validity and why experiments may have weaker external validity compared with nonexperimental studies. 8. Understand the issues involved in making a choice between parametric and nonparametric statistical methods. 9. Be able to identify an appropriate statistical analysis to describe whether scores on two variables are related, taking into account whether the data meet the assumptions for parametric tests; the type(s) of variables, categorical versus quantitative; whether the design is between-S or within-S; and the number of groups that are compared. The decision tree in Table 1.4 identifies the most widely used statistical procedure for each of these situations. 10. Most important, readers should remember that whatever choices researchers make, each choice typically has both advantages and disadvantages. The discussion section of a research report can point out the advantages and strengths of the approach used in the study, but it should also acknowledge potential weaknesses and limitations. If the study was not an experiment, the researcher must avoid using language that implies that the results of the study are proof of causal connections. Even if the study is a well-designed experiment, the researcher should keep in mind that no single study provides definitive proof for any claim. If the sample is not representative of any well-defined, real population of interest, limitations in the generalizability of the results should be acknowledged (e.g., if a study that assesses the safety and effectiveness of a drug is performed on a sample of persons 18 to 22 years old, the results may not be generalizable to younger and older persons). If the data violate many of the assumptions for the statistical tests that were performed, this may invalidate the results. Table 1.5 provides an outline of the process involved in doing research. Some issues that are included (such as the IRB review) apply only to research that involves human participants as subjects, but most of the issues are applicable to research projects in many different disciplines. It is helpful to think about the entire process and anticipate later steps when making early decisions. For example, it is useful to consider what types of variables you will have and what statistical analyses you will apply to those variables at an early stage in planning. It is essential to keep in mind how the planned statistical analyses are related to the primary research questions. This can help researchers avoid collecting data that are difficult or impossible to analyze. Table 1.5 Preview of a Typical Research Process for an Honors Thesis, Master’s Thesis, or Dissertation 1. Examples in this textbook assume that researchers are dealing with human populations; however, similar issues arise when samples are obtained from populations of animals, plants, geographic locations, or other entities. In fact, many of these statistics were originally developed for use in industrial quality control and agriculture, where the units of analysis were manufactured products and plots of ground that received different treatments. 2. Systematic differences in the composition of the sample, compared with the population, can be corrected for by using case-weighting procedures. If the population includes 500 men and 500 women, but the sample includes 25 men and 50 women, case weights could be used so that, in effect, each of the 25 scores from men would be counted twice in computing summary statistics. 3. The four levels of measurement are called nominal, ordinal, interval, and ratio. In nominal level of measurement, each number code serves only as a label for group membership. For example, the nominal variable gender might be coded 1 = male, 2 = female, and the nominal variable religion might be coded 1 = Buddhist, 2 = Christian, 3 = Hindu, 4 = Islamic, 5 = Jewish, 6 = other religion. The sizes of the numbers associated with groups do not imply any rank ordering among groups. Because these numbers serve only as labels, Stevens argued that the only logical operations that could appropriately be applied to the scores are = and ≠. That is, persons with scores of 2 and 3 on religion could be labeled as “the same” or “not the same” on religion. In ordinal measurement, numbers represent ranks, but the differences between scores do not necessarily correspond to equal intervals with respect to any underlying characteristic. The runners in a race can be ranked in terms of speed (runners are tagged 1, 2, and 3 as they cross the finish line, with 1 representing the fastest time). These scores supply information about rank (1 is faster than 2), but the numbers do not necessarily represent equal intervals. The difference in speed between Runners 1 and 2 (i.e., 2 − 1) might be much larger or smaller than the difference in speed between Runners 2 and 3 (i.e., 3 − 2), despite the difference in scores in both cases being one unit. For ordinal scores, the operations > and < would be meaningful (in addition to = and ≠). However, according to Stevens, addition or subtraction would not produce meaningful results with ordinal measures (because a one-unit difference does not correspond to the same “amount of speed” for all pairs of scores). Scores that have interval level of measurement qualities supply ordinal information and, in addition, represent equally spaced intervals. That is, no matter which pair of scores is considered (such as 3 − 2 or 7 − 6), a one-unit difference in scores should correspond to the same amount of the thing that is being measured. Interval level of measurement does not necessarily have a true 0 point. The centigrade temperature scale is a good example of interval level of measurement: The 10-point difference between 40°C and 50°C is equivalent to the 10-point difference between 50°C and 60°C (in each case, 10 represents the same number of degrees of change in temperature). However, because 0°C does not correspond to a complete absence of any heat, it does not make sense to look at a ratio of two temperatures. For example, it would be incorrect to say that 40°C is “twice as hot” as 20°C. Based on this reasoning, it makes sense to apply the plus and minus operations to interval scores (as well as the equality and inequality operators). However, by this reasoning, it would be inappropriate to apply multiplication and division to numbers that do not have a true 0 point. Ratio-level measurements are interval-level scores that also have a true 0 point. A clear example of a ratio-level measurement is height. It is meaningful to say that a person who is 6 ft tall is twice as tall as a person 3 ft tall because there is a true 0 point for height measurements. The narrowest interpretation of this reasoning would suggest that ratio level is the only type of measurement for which multiplication and division would yield meaningful results. Thus, strict adherence to Stevens’s measurement theory would imply that statistics that involve addition, subtraction, multiplication, and division (such as mean, Pearson’s r, t test, analysis of variance, and all other multivariate techniques covered later in this textbook) can legitimately be applied only to data that are at least interval (and preferably ratio) level of measurement. 4. There is one exception. When a nominal variable has only two categories and the codes assigned to these categories are 0 and 1 (e.g., the nominal variable gender could be coded 0 = male, 1 = female), the mean of these scores represents the proportion of persons who are female. 5. Violations of the assumptions for parametric statistics create more serious problems when they are accompanied by small Ns in the groups (and/or unequal Ns in the groups). Sometimes, just having very small Ns is taken as sufficient reason to prefer nonparametric statistics. When Ns are very small, it becomes quite difficult to evaluate whether the assumptions for parametric statistics are satisfied (such as normally distributed scores on quantitative variables). 6. A nontechnical definition of robust is provided at this point. A statistic is robust if it provides “accurate” results even when one or more of its assumptions are violated. A more precise definition of this term will be provided in Chapter 2. Comprehension Questions 1. Chapter 1 distinguished between two different kinds of samples: a. Random samples (selected randomly from a clearly defined population) b. Accidental or convenience samples Which type of sample (a or b) is 1.1. more commonly reported in journal Which type of sample (a or b) is 1.2. more likely to be representative of a clearly defined What does it mean to say that a 1.3. sample is “representative” of a population? Suppose that a researcher tests the safety and effectiveness of a new drug on a convenience sample of male medical students between the ages of 24 and 30. If 2. the drug appears to be effective and safe for this convenience sample, can the researcher safely conclude that the drug would be safe for women, children, and persons older than 70 years of age? Give reasons for your answer. 3. Given below are two applications of statistics. Identify which one of these is descriptive and which is inferential and explain why. Case I: An administrator at Corinth College looks at the verbal Scholastic Aptitude Test (SAT) scores for the entire class of students admitted in the fall of 2005 (mean = 660) and the verbal SAT scores for the entire class admitted in the fall of 2004 (mean = 540) and concludes that the class of students admitted to Corinth in 2005 had higher verbal scores than the class of students admitted in 2004. Case II: An administrator takes a random sample of 45 Corinth College students in the fall of 2005 and asks them to self-report how often they engage in binge drinking. Members of the sample report an average of 2.1 binge drinking episodes per week. The administrator writes a report that says, “The average level of binge drinking among all Corinth College students is about 2.1 episodes per week.” We will distinguish between two types of variables: categorical and quantitative 4. . For your answers to this question, do not use any of the variables used in the textbook or in class as examples; think up your own example. Give an example of a specific variable that is clearly a categorical type of measurement a. (e.g., gender coded 1 = female and 2 = male is a categorical variable). Include the groups or levels (in the example given in the text, when gender is the categorical variable, the groups are female and male). b. Give an example of a specific variable that is clearly a quantitative type of measurement (e.g., IQ scores). If you have scores on a categorical variable (e.g., religion coded 1 = Catholic, 2 = Buddhist, c. 3 = Protestant, 4 = Islamic, 5 = Jewish, 6 = other religion), would it make sense to use these numbers to compute a mean? Give reasons for your answer. Using the guidelines given in this chapter (and Table 1.4), name the most appropriate statistical analysis for each of the following imaginary studies. Also, state which variable is the predictor or independent variable and which 5. variable is the outcome or dependent variable in each situation. Unless otherwise stated, assume that any categorical independent variable is between-S (rather than within-S), and assume that the conditions required for the use of parametric statistics are satisfied. Case I: A researcher measures core body temperature for participants who are either male or female (i.e., gender is a variable in this study). What statistics could the researcher use to see if mean body temperature differs between women and men? Case II: A researcher who is doing a study in the year 2000 obtains yearbook photographs of women who graduated from college in 1970. For each woman, there are two variables. A rating is made of the “happiness” of her facial expression in the yearbook photograph. Each woman is contacted and asked to fill out a questionnaire that yields a score (ranging from 5 to 35) that measures her “life satisfaction” in the year 2000. The researcher wants to know whether “life satisfaction” in 2000 can be predicted from the “happiness” in the college yearbook photo back in 1970. Both variables are quantitative, and the researcher expects them to be linearly related, such that higher levels of happiness in the 1970 photograph will be associated with higher scores on life satisfaction in 2000. (Research on these variables has actually been done; see Harker & Keltner, 2001.) Case III: A researcher wants to know whether preferred type of tobacco use (coded 1 = no tobacco use, 2 = cigarettes, 3 = pipe, 4 = chewing tobacco) is related to gender (coded 1 = male, 2 = female). Case IV: A researcher wants to know which of five drug treatments (Group 1 = placebo, Group 2 = Prozac, Group 3 = Zoloft, Group 4 = Effexor, Group 5 = Celexa) is associated with the lowest mean score on a quantitative measure of depression (the Hamilton Depression Rating Scale). 6. Draw a sketch of the standard normal distribution. What characteristics does this function have? 7. Look at the standard normal distribution in Figure 1.4 to answer the following a. Approximately what proportion (or percentage) of scores in a normal distribution lie within a range from −2σ below the mean to +1σ above the mean? b. Approximately what percentage of scores lie above +3σ? What percentage of scores lie inside the range −2σ below the mean to +2σ above the mean? What c. percentage of scores lie outside this range? U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details 8. For what types of data would you use nonparametric versus parametric statistics? 9. What features of an experiment help us to meet the conditions for causal 10. Briefly, what is the difference between internal and external validity? For each of the following sampling methods, indicate whether it is random (i.e., whether it gives each member of a specific well-defined population an equal chance of being included) or accidental/convenience sampling. Does it involve 11. stratification; that is, does it involve sampling from members of subgroups, such as males and females or members of various political parties? Is a sample obtained in this manner likely to be representative of some well-defined larger population? Or is it likely to be a biased sample , a sample that does not correspond to any clearly defined population? A teacher administers a survey on math anxiety to the 25 members of her introductory a. statistics class. The teacher would like to use the results to make inferences about the average levels of math anxiety for all first-year college students in the United States. A student gives out surveys on eating disorder symptoms to her teammates in gymnastics. She b. wants to be able to use her results to describe the correlates of eating disorders in all college women. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details A researcher sends out 100,000 surveys on problems in personal relationships to mass mailing lists of magazine subscribers. She gets back about 5,000 surveys and writes a book in which c. she argues that the information provided by respondents is informative about the relationship problems of all American women. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See The Nielsen television ratings organization selects a set of telephone area codes to make sure that its sample will include people taken from every geographical region that has its own area code; it then uses a random number generator to get an additional seven-digit telephone d. number. It calls every telephone number generated using this combination of methods (all area codes included and random dialing within an area code). If the person who answers the telephone indicates that this is a residence (not a business), that household is recruited into the sample, and the family members are mailed a survey booklet about their television-viewing habits. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details Give an example of a specific sampling strategy that would yield a random sample 12. of 100 students taken from the incoming freshman class of a state university. You may assume that you have the names of all 1,000 incoming freshmen on a spreadsheet. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details Give an example of a specific sampling strategy that would yield a random sample 13. of 50 households from the population of 2,000 households in that particular town. You may assume that the telephone directory for the town provides a nearly complete list of the households in the town. 14. Is each of the following variables categorical or quantitative? (Note that some variables could be treated as either categorical or quantitative.) Number of children in a family. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Type of pet owned: 1 = none, 2 = dog, 3 = cat, 4 = other animal IQ score Personality type (Type A, coronary prone; Type B, not coronary prone) Do most researchers still insist on at least interval level of measurement as a 15. condition for the use of parametric statistics? U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details 16. How do categorical and quantitative variables differ? How do between-S versus within-S designs differ? Make up a list of names for 17. imaginary participants, and use these names to show an example of between-S groups and within-S groups. When a researcher has an accidental or convenience sample, what kind of 18. population can he or she try to make inferences about? U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details Describe the guidelines for selecting an appropriate bivariate statistic. That 19. is, what do you need to ask about the nature of the data and the research design in order to choose an appropriate analysis? U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details (Warner 30-40) Warner, Rebecca (Becky) (Margaret). Applied Statistics: From Bivariate Through Multivariate Techniques, 2nd Edition. SAGE Publications, Inc, 04/2012. VitalBook file. The citation provided is a guideline. Please check each citation for accuracy before use. U1D1-64 – Identify Your Sources Of Statistics Anxiety – See Details Power is an individual’s ability to influence environments and people within environments. There are different methods in which leaders gain power, which can be highly organized and/or extremely multifaceted. With power, a leader may motivate a group by providing authority to enact change within an organization toward a directed goal. For some leaders, having a variety of ways to access power may produce greater effectiveness for the organization. Just as leaders may work hard to gain power, leaders also must work hard to use power appropriately in order to maintain it and avoid losing it within organizations. For this Discussion, review this week’s Learning Resources and the case study “Organizational Power and Politics” on page 174 of Lussier & Achua (2015). Be sure to consider sources of power in your An analysis of the sources of power that Helen has. What types of power is he using? What influencing tactics is Helen using during the meeting? Are negotiations and exchange tactics appropriate in this situation? Include in your Discussion sources of power and influencing tactics that relate to this case study. Be sure to support your postings and responses with specific references to the current literature and Learning Resources You must proofread your paper. But do not strictly rely on your computer’s spell-checker and grammar-checker; failure to do so indicates a lack of effort on your part and you can expect your grade to suffer accordingly. Papers with numerous misspelled words and grammatical mistakes will be penalized. Read over your paper – in silence and then aloud – before handing it in and make corrections as necessary. Often it is advantageous to have a friend proofread your paper for obvious errors. Handwritten corrections are preferable to uncorrected mistakes. Use a standard 10 to 12 point (10 to 12 characters per inch) typeface. Smaller or compressed type and papers with small margins or single-spacing are hard to read. It is better to let your essay run over the recommended number of pages than to try to compress it into fewer pages. Likewise, large type, large margins, large indentations, triple-spacing, increased leading (space between lines), increased kerning (space between letters), and any other such attempts at “padding” to increase the length of a paper are unacceptable, wasteful of trees, and will not fool your professor. The paper must be neatly formatted, double-spaced with a one-inch margin on the top, bottom, and sides of each page. When submitting hard copy, be sure to use white paper and print out using dark ink. If it is hard to read your essay, it will also be hard to follow your argument. An Accurate Assessment Is Essential For Effective Intervention: The ABC Model Of Crisis Intervention. An Accurate Assessment Is Essential For Effective Intervention: The ABC Model Of Crisis Intervention. Topic: An accurate assessment is essential for effective intervention: The ABC Model of Crisis Intervention. It is often said that Psychology is the science of understanding human behavior, and Counseling is the art of helping others understand their behavior. A person’s response to a crisis situation is idiosyncratic—not all people have the same response. The response is unique to the individual with how they perceived their crisis situation. Two sisters can have completely different responses to the loss of a brother in active duty while completing a tour in a combat zone. It is the therapist’s responsibility to try to see the crisis situation from the client’s perspective. Although this can be challenging, the success of such a goal is largely influenced by an effective assessment. The ABC Model (Kanel, 2015) identifies 3 areas of assessment that are essential when completing an initial interview and providing ongoing treatment/interventions. What are these 3 categories? Define each category. Which category is the most important? You can only pick 1 and you must support your argument with professional sources. Post should be at least 250 words required to use at least 2 peer-reviewed sources for post. Current APA formatting is required for all citations. Textbook Chp 3 Kanel, K. (2015). A guide to crisis intervention (5th ed.). New York, NY: Cengage Learning. ISBN: 9781285739908. You must proofread your paper. But do not strictly rely on your computer’s spell-checker and grammar-checker; failure to do so indicates a lack of effort on your part and you can expect your grade to suffer accordingly. Papers with numerous misspelled words and grammatical mistakes will be penalized. Read over your paper – in silence and then aloud – before handing it in and make corrections as necessary. Often it is advantageous to have a friend proofread your paper for obvious errors. Handwritten corrections are preferable to uncorrected mistakes. Use a standard 10 to 12 point (10 to 12 characters per inch) typeface. Smaller or compressed type and papers with small margins or single-spacing are hard to read. It is better to let your essay run over the recommended number of pages than to try to compress it into fewer pages. Likewise, large type, large margins, large indentations, triple-spacing, increased leading (space between lines), increased kerning (space between letters), and any other such attempts at “padding” to increase the length of a paper are unacceptable, wasteful of trees, and will not fool your professor. The paper must be neatly formatted, double-spaced with a one-inch margin on the top, bottom, and sides of each page. When submitting hard copy, be sure to use white paper and print out using dark ink. If it is hard to read your essay, it will also be hard to follow your argument. Research On Intimate Partner Violence And The Duty To Protect Research On Intimate Partner Violence And The Duty To Protect Use the Research on Intimate Partner Violence and the Duty to Protect attachment to complete the questions on the attachment. Case 4. Research on Intimate Partner Violence and the Duty to Protect Dr. Daniela Yeung, a community psychologist, has been conducting a federally funded ethnographic study of men’s attitudes toward intimate partner violence following conviction and release from prison for spousal abuse. Over the course of a year, she has had individual monthly interviews with 25 participants while they were in jail and following their release. Aiden, a 35-year-old male parolee convicted of seriously injuring his wife, has been interviewed by Dr. Yeung on eight occasions. The interviews have covered a range of personal topics including Aiden’s problem drinking, which is marked by blackouts and threatening phone calls made to his parents and girlfriend when he becomes drunk, usually in the evening. To her knowledge, Aiden has never followed through on these threats. It is clear that Aiden feels very comfortable discussing his life with Dr. Yeung. One evening Dr. Yeung checks her answering machine and finds a message from Aiden. His words are slurred and angry: “Now that you know the truth about what I am you know that there is nothing you can do to help the evil inside me. The bottle is my savior and I will end this with them tonight.” Each time she calls Aiden’s home phone she gets a busy signal. Ethical Dilemma Dr. Yeung has Aiden’s address, and after 2 hours, she is considering whether or not to contact emergency services to go to Aiden’s home or to the homes of his parents and girlfriend. Copyright © 2013 by SAGE Publications, Inc. 360——DECODING THE ETHICS CODE Discussion Questions 1. Why is this an ethical dilemma? Which APA Ethical Principles help frame the nature of the dilemma? 2. Who are the stakeholders and how will they be affected by how Dr. Yeung resolves this dilemma? 3. Does this situation meet the standards set by the Tarasoff decision’s “duty to protect” statute (see Chapter 7)? How might whether or not Dr. Yeung’s state includes researchers under such a statute influence Dr. Yeung’s ethical decision making? How might the fact that Dr. Yeung is a research psychologist without training or licensure in clinical practice influence the ethical decision? 4. In addressing this dilemma, should Dr. Yeung consider how her decision may affect the completion of her research (e.g., the confidentiality concerns of other participants)? 5. How are APA Ethical Standards 2.01f, 3.04, 3.06, 4.01, 4.02, 4.05, and 8.01 relevant to this case? Which other standards might apply? 6. What are Dr. Yeung’s ethical alternatives for resolving this dilemma? Which alternative best reflects the Ethics Code aspirational principles and enforceable standards, legal standards, and obligations to stakeholders? Can you identify the ethical theory (discussed in Chapter 3) guiding your decision? 7. What steps should Dr. Yeung take to implement her decision and monitor its effect? Suggested Readings Appelbaum, P., & Rosenbaum, A. (1989). Tarasoff and the researcher: Does the duty to protect apply in the research setting? American Psychologist, 44(6), 885–894. Fisher, C. B., Oransky, M., Mahadevan, M., Singer, M., Mirhej, G., & Hodge, G. D. (2009). Do drug abuse researchers have a duty to protect third parties from HIV transmission? Moral perspectives of street drug users. In D. Buchanan, C. B. Fisher, & L. Gable (Eds.), Research with high-risk populations: Balancing science, ethics, and law (pp. 189–206). Washington, DC: American Psychological Association. Gable, L. (2009). Legal challenges raised by non-intervention research conducted under high-risk circumstances. In D. Buchanan, C. B. Fisher, & L. Gable (Eds.). Research with high-risk populations: Balancing science, ethics, and law (pp. 47–74). Washington, DC: American Psychological Association. Jordan, C. E., Campbell, R., & Follingstad, D. (2010). Violence and women’s mental health: The impact of physical, sexual, & psychological aggression. Annual Review of Clinical Psychology, 6, 607–628. Draw parallels between the characters in The Outsider & the lead character inI am Jazz. Draw parallels between the characters in The Outsider & the lead character inI am Jazz. Task 1 Watch the two videos “The Outsider” and “I am Jazz”, and write about the following: – Draw parallels between the characters in The Outsider & the lead character inI am Jazz. In what ways are their identity crises similar? In what ways are their identity crises different? Can you relate to either one of their experiences from an identity perspective? If so, how? (NOTE: I am not saying that you need to be Amish and/or have a gender identity crisis to relate to these movies; I am rather asking you to compare your identity experiences either the characters in these movies) It should be about 300 – 350 words Don’t forget to cite from this week’s articles, and the videos Video links: “The Outsiders: Amish Teens” (Note: You must watch all 7 parts on YouTube). This week’s articles: Attached Task 2 This week, you read Why are all of the Black Kids Sitting Together in the Cafeteria? (Tatum, 1999), and Feminine Ideals (Lamb, 2001). I am asking that you choose only one of these articles and answer the following questions: Q1: What struck you most about this article? Why? Q2: Can you personally relate to the discussion in this article? If so, how? If you cannot, why not? Q3: How does this article contribute to your overall understanding of adolescent identity development? A conclusion (Total of 4 paragraphs, each about 200 words) Don’t forget to cite from the readings, you might also relate to the previous readings and draw from them You must proofread your paper. But do not strictly rely on your computer’s spell-checker and grammar-checker; failure to do so indicates a lack of effort on your part and you can expect your grade to suffer accordingly. Papers with numerous misspelled words and grammatical mistakes will be penalized. Read over your paper – in silence and then aloud – before handing it in and make corrections as necessary. Often it is advantageous to have a friend proofread your paper for obvious errors. Handwritten corrections are preferable to uncorrected mistakes. Use a standard 10 to 12 point (10 to 12 characters per inch) typeface. Smaller or compressed type and papers with small margins or single-spacing are hard to read. It is better to let your essay run over the recommended number of pages than to try to compress it into fewer pages. Likewise, large type, large margins, large indentations, triple-spacing, increased leading (space between lines), increased kerning (space between letters), and any other such attempts at “padding” to increase the length of a paper are unacceptable, wasteful of trees, and will not fool your professor. The paper must be neatly formatted, double-spaced with a one-inch margin on the top, bottom, and sides of each page. When submitting hard copy, be sure to use white paper and print out using dark ink. If it is hard to read your essay, it will also be hard to follow your argument. McMinn Discussed Guidelines When Confronting Sin During A Counseling Experience And The Lectures Reviewed Some Factors As Well. McMinn Discussed Guidelines When Confronting Sin During A Counseling Experience And The Lectures Reviewed Some Factors As Well. Discussion Question: McMinn discussed guidelines when confronting sin during a counseling experience and the lectures reviewed some factors as well. Your thread needs to be answered in two parts: First, what would be the challenges (based on the lectures) of confronting clearly wrong behavior/ “sin” in the life of your client if you were working in a secular human services setting? Draw in concepts from the lecture to support your position. How might the approach from psychology make it difficult to confront clearly wrong behavior (worldview and perspective on attribution, for Second, assume that you counseled in a human services setting in which you could integrate spirituality and a Christian worldview. Review the following brief “case” and answer the following 1. Based on the lectures and McMinn, why can’t a sensitive Christian counselor just automatically and quickly confront obvious sin in the life of the counselee? 2. Of the cautions mentioned by the course materials, which ones do you think counselors most often overlook? 3. From what you learned from the lectures/McMinn, how would you best address the clearly sinful behavior of this client? Case Study Jim is a client in your counseling center, who you have seen for about eight months. He has been cycled through several other counselors and one described him as a “basket case.” Jim has several children, each with a different mother. He casually mentions that he rarely sees them, and since he can’t hold down a job, he provides no financial support. Some of his children are now in foster care. He engages in unprotected sex on a weekly basis. Typical of many of your clients, Jim drinks heavily and abuses street drugs. He comes to counseling only because it is required for him to receive the tangible support services of your agency. You are at the point in your counseling with Jim that you’d like to “let him have it” but your counseling training did not include that as a valid counseling technique. There is obviously much more to Jim’s story but suffice it to say that he is repeating many of the behaviors he learned from his parents’ dysfunctional parenting. While you are sharing opinion here, you must demonstrate informed opinion by supporting your points with references to the course materials. by Carl Martins Psychosocial Development Case Study Assessment by Carl Martins Feminist, Solution-Focused, And Narrative Theory Application by Carl Martins Statistics Project, Part 4: Correlation by Carl Martins Identify Your Sources Of Statistics Anxiety by Carl Martins by Carl Martins An Accurate Assessment Is Essential For Effective Intervention: The ABC Model Of Crisis Intervention. by Carl Martins Research On Intimate Partner Violence And The Duty To Protect by Carl Martins Draw parallels between the characters in The Outsider & the lead character inI am Jazz. by Carl Martins McMinn Discussed Guidelines When Confronting Sin During A Counseling Experience And The Lectures Reviewed Some Factors As Well.	{"url":"https://academiaessaywriters.com/blog/","timestamp":"2024-11-09T16:47:27Z","content_type":"text/html","content_length":"258971","record_id":"<urn:uuid:ca8f2eaf-3bc8-4b0a-9a12-40110a02630a>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00797.warc.gz"}
11-14-2021 04:14 AM 11-10-2023 06:39 AM 06-07-2024 04:56 AM 06-07-2024 04:56 AM 06-07-2024 04:56 AM How do I retrieve index names of a table and the columns used for those indexes. on SAS Programming. 11-10-2023 05:36 AM Re: How do I retrieve index names of a table and the columns used for those indexes. on SAS Programming. 11-10-2023 06:39 AM Update Table A (that has more rows per key) with table B that has only ONE row per key. on SAS Programming. 02-28-2024 06:05 AM Re: Update Table A (that has more rows per key) with table B that has only ONE row per key. on SAS Programming. 02-28-2024 07:21 AM Counting decimals in numeric fields. on SAS Programming. 06-06-2024 09:28 AM Re: Counting decimals in numeric fields. on SAS Programming. 06-07-2024 04:41 AM Re: Counting decimals in numeric fields. on SAS Programming. 06-07-2024 04:44 AM Re: Counting decimals in numeric fields. on SAS Programming. 06-07-2024 04:49 AM Re: Counting decimals in numeric fields. on SAS Programming. 06-07-2024 04:50 AM Re: Counting decimals in numeric fields. on SAS Programming. 06-07-2024 04:56 AM Update HUGE indexed table with SQL (in SPD library) seems not to run at all. on SAS Programming. 08-29-2024 03:05 PM Re: Update HUGE indexed table with SQL (in SPD library) seems not to run at all. on SAS Programming. 08-29-2024 04:58 PM Re: Update HUGE indexed table with SQL (in SPD library) seems not to run at all. on SAS Programming. 08-29-2024 05:50 PM It's almost midnight and nobody is on the system ... so I decided to run on the REAL DEAL. The big table in the SPD library. It ran through it all in less than 10 minutes. Yes ... and the updates are correct. What a relieve. Thanx for the feedback. Yes ... I can retrieve a data from a view ... but not update it. "Of course" I think now. 😉 Duh! 😅 ... View more	{"url":"https://communities.sas.com/t5/user/viewprofilepage/user-id/18088","timestamp":"2024-11-10T04:13:47Z","content_type":"text/html","content_length":"279764","record_id":"<urn:uuid:abb3e826-055c-46e7-b066-70aa25320244>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00783.warc.gz"}
Colloquium and Seminars - Kerala School of Mathematics (KSoM) Kerala School of Mathematics frequently conducts regular colloquiums and seminars by eminent and established mathematicians and upcoming researchers. The details of the various colloquiums and seminars at KSoM can be found below Augustin Cauchy Colloquium Title: Imbedding of critical Fractional Sobolev Spaces Speaker: Adi Adimurthi Affiliation: TIFR CAM Venue: Seminar Hall, KSoM Date and Time: August 2, 2024 at 3:30 PM Abstract: It is shown by Dyda that for $sp>1$ or $sp <1$, fractional Sobolev spaces are embedded in Weighted Lebesgue spaces with weight $d(x,\partial \omega)^{-sp}$. He also showed that it is optimal and it will not be true for $sp =1$, the critical case. In this talk I will address this problem and show what is the correct weight to be taken in the imbedding. This is a joint work with Prosenjit Roy, Purbita Jana and Prosenjit Roy, Vivek Sahu. Alan Turing Colloquium. Title: Do stochastic parrots pass the Turing test? Speaker: Madhavan Mukund Affiliation: Chennai Mathematical Institute Venue: Seminar Hall, KSoM Date and Time: June 14, 2024 at 3:30 PM Abstract: Alan Turing not only laid the foundations of modern computing science, he was also one of the pioneers of artificial intelligence. He formulated the imitation game, more commonly called the Turing test, to establish whether a computing system exhibits intelligence. Large language models such as ChatGPT have been all over the news recently. Some people claim these systems exhibit evidence of “emergent intelligence”. We describe how such systems are built and address some of the philosophical questions surrounding their capabilities. KSoM Colloquium talk Title: Finite and Infinite dimensional Lie algebras Speaker: Punita Batra Affiliation: Harish-Chandra Research Institute, Prayagraj Venue: Seminar Hall, KSoM Date and Time: June 11, 2024 [Tuesday] at 03:30 PM Abstract: I will discuss about finite dimensional simple Lie algebras with examples, semisimple Lie algebras, define some basic terms and give a realisation of affine Kac-Moody Lie algebras and discuss toroidal Lie algebras. KSoM Colloquium talk Title: On the trace of powers of Algebraic integers Speaker: R. Thangadurai Affiliation: Harish-Chandra Research Institute, Prayagraj Venue: Seminar Hall, KSoM Date and Time: June 10, 2024 [Monday] at 11:00 AM Abstract: Let $\alpha$ be a non-zero algebraic integer. In this lecture, we prove an interesting characterisation for $\alpha$ to be a root of unity, which is an extension of a classical theorem of Kronecker. Indeed, we prove that {\it if a non-zero algebraic integer $\alpha$ is a root of unity if and only if the sequence $\left(\mathrm{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\alpha^n)\right)_{n\ geq 1}$ is bounded. Moreover, if the sequence $\left(\mathrm{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\alpha^n)\right)_{n\geq 1}$ is bounded, then it is periodic.} Thus, if a non-zero algebraic integer $\alpha$ is not a root of unity, it is clear that the sequence $\left(\mathrm{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\alpha^n)\right)_{n\geq 1}$ is unbounded and hence we study the growth in the next result. Also, we introduce a problem of Polya and its extensions. KSoM Research Talk Speaker: Viswanathan S Affiliation: ICTS Bangalore Venue: Seminar Hall, KSoM Date and Time: June 03, 2024 at 3:30 PM Title: Introduction to Teichmüller Spaces Abstract: Teichmüller spaces have proven to be effective tools to study moduli spaces of Riemann surfaces. It has also found natural applications in dynamics, geometry, and topology. The purpose of this talk would be to introduce these spaces from several perspectives to hammer home the point that the theory at heart brings together algebra, geometry, and analysis. The Henri Poincare Colloquium. Title: Bargmann and Fock meet Hardy: An uncertainty principle for operators. Speaker: S. Thangavelu Affiliation: Indian Institute of Science, Bangalore Venue: Seminar Hall, KSoM Date and Time: April 5, 2024 at 3:30 PM Abstract: Heisenberg uncertainty principle in Quantum Mechanics states that the position and momentum of a moving particle cannot both be measured simultaneously. Viewed as an interesting property of the Fourier transform, this principle has several different manifestations in the realm of Fourier Analysis, including a theorem of Hardy studied extensively by the Indian harmonic analysts. Quite unexpectedly, this principle of Hardy shows its ugly head (or charming face) in connection with the boundedness of certain convolution type operators on the Fock space. The key player is Bargmann who connects Schrodinger with Fock and as expected, Hermite has his own role to play. We plan to describe this story with as little details as possible. One day session on “Engaging Liminal Spaces in Mathematics with Social Sciences” Liminal space refers to a place the person is in during a transition phase. Exactly where, aspiring mathematicians are till they become mathematicians (and thereafter!!!). We work on various problems: real, imaginary and imagined (infinite) juggled around with logic in our heads to come up with propositions and theorems-approximating, affirming and at times even vanishing. Well managed! Then there are other non-mathematical problems. Again, real, imaginary and imagined, but related to – non, in, extra, and ultra-variants of – human. With these problems in our head, we head nowhere. Ill managed! Mind matters, and minds at large matter too! Social sciences concern human worlds and society; how individuals and groups interact and what drives human behavior. Let us gather on 13 March, 2024, for an engaging one day session, by (in order) a real, imaginative and hyperbolic social scientist. Dr. Neetulal (Dept. of Psychology, University of Calicut) Dr. Kiran S (Dept. of Psychology, University of Calicut) Prof. A. F. Mathew (Humanities & Liberal Arts in Management, IIMK) to help us make the ill managed, well managed and to connect better with this not too familiar entity called ‘society’ when we are in our liminal spaces. Ramshida A. P. (Dept. of Psychology, University of Calicut) Pranav Haridas (KSoM) Radhika M. M. (KSoM) The Alexander Grothendieck Colloquium Title: Integrals over the $SU(2)$ character variety and lattice gauge theory. Speaker: T R Ramadas Affiliation: Chennai Mathematical Institute Venue: Seminar Hall, KSoM Date and Time: March 1, 2024 at 3:30 PM Abstract: Given a genus-$g$ Riemann surface $\Sigma$, the moduli space of rank two vector bundles with trivial determinant is, by the Narasimhan-Seshadri Theorem, in bijection with the space of (equivalence classes of) representations in $SU(2)$ of the fundamental group of the surface . In the latter avatar, this space has a symplectic structure and a corresponding finite measure, the Liouville measure. Normalising to total mass one gives a probability measure. There is a natural class of real-valued functions parameterised by isotopy classes of loops on the surface. These are called Wilson loop functions by physicists and Goldman functions by mathematicians. I present a simple scheme to compute joint distributions of these functions for families of loops. This is possible because of the miracle of symplectic geometry called the Duistermaat-Heckman formalism (whose applicability in this context is due to L. Jeffrey and J. Weitsman), and a continuous analogue of the Verlinde algebra. The large $g$ asymptotics can be easily read off. This leads to a highly speculative approach to rigorous analysis of (lattice) gauge theories, as a preliminary step to quantum-field-theoretic constructions. David Hilbert Colloquium Title: Free groups Speaker: Parameswaran Sankaran Affiliation: Chennai Mathematical Institute Venue: Seminar Hall, KSoM Date and Time: January 19, 2024 at 3:30 PM Abstract: Free groups are free objects in the category of groups and as such can be defined in terms of a certain universal property. They arise in geometry, topology, besides group theory itself. After discussing some of their basic properties, we will show that many familiar groups contain (non-abelian) free groups. The talk will be accessible to graduate students and non-specialists. Title: Multiplication operators on Hardy space Speaker: Aneesh M Affiliation: Indian Institute of Technology, Bhubaneswar Venue: Seminar Hall, KSoM Date and Time: December 22, 2023 at 3:00 PM Abstract: This is a short introduction to the subject Dynamics of Linear Operators which is at the cross-road of functional analysis, complex analysis, and dynamical systems. We review a few well known dynamical properties (such as the hypercyclicity and chaos) of multiplication operators on the Hardy-Hilbert space $H^2(\mathbb{D})$ which consists of all analytic functions on the unit disc $\mathbb{D}$, whose Taylor coefficients belong to $l^2$. A multiplication operator is given by $M_\phi (f) = \phi f$ on $H^2(\mathbb{D})$ , where $\phi$ is necessarily a bounded analytic function. When $M_\phi$ is a bounded operator, the closure of the range $\phi(\mathbb{D})$ is the spectrum of $M_\ phi$, and the range itself is the point spectrum of the adjoint $M^_\phi$, and the eigenvectors are nothing but the reproducing kernels at each point of $\mathbb{D}$. These facts along with certain classical results about chaotic operators yield the following: the adjoint $M^_\phi$ is chaotic on $H^2(\mathbb{D})$ if and only if $\phi$ is non-constant and $\phi(\mathbb{D})$ intersects the unit circle $T$. This result is not true for some other Hilbert spaces of analytic functions, such as the Dirichlet space. We will also see a few open problems that are fairly simple to state, yet unsolved so far. National Mathematics Day Colloquium Title: The Problem of Structural Stability Speaker: Shiva Shankar Venue: Seminar Hall, KSoM Date and Time: December 22, 2023 at 12 PM Abstract: The development of a scientific discipline usually follows the pattern of experimental observations, construction of a theoretical model, and an interpretation of its predictions. In order to be useful, the model’s predictions must not be sensitive to the values of the various parameters that appear in its description. This is the criterion of `Stability’. I will explain this in the context of ‘structural stability’ of dynamical systems, first enunciated by Andronov and Pontriagin, and founded on ideas going back to Newton’s treatment of the 3-body problem. John Von Neumann Colloquium Title : What is involved in Quantum Mechanics? Speaker: Krishna Maddaly Affiliation: Ashoka University Venue: Seminar Hall, KSoM Date and Time: December 08, 2023 at 3:30 PM Abstract: Quantum Mechanics emerged as a need to explain many natural phenomena that occur at very small scales and took a while to formulate precisely. In this journey John von Neumann played a crucial role to give a firm Mathematical foundation to the theory. In this talk I will explain the theory and the role of von Neumann. Title: കേരളത്തിന്റെ പ്രാദേശിക ഗണിത ശാസ്ത്രവ്യവഹാരങ്ങൾ :താളിയോല ഗ്രന്ഥങ്ങളിലൂടെ (Regional Mathematical Discourses of Kerala: Through Palm Leaf Texts.) Speaker: Dr. Manju M. P. Affiliation: Department of Malayalam and Kerala Studies, University of Calicut Venue: Seminar Hall, KSoM Date and Time: November 01, 2023 at 11:30 AM Abstract: ഈ പ്രബന്ധത്തിൽ കേരളത്തിന്റെ പ്രാദേശിക ഗണിതവ്യവഹാരങ്ങളെ കുറിച്ചാണ് പറയാൻ ശ്രമിക്കുന്നത്. അതിനു വേണ്ടി കേരളത്തിലെ, താളിയോല ഗ്രന്ഥശേഖരങ്ങളിലെ (പ്രധാനമായും കാലിക്കറ്റ് സർവ്വകലാശാല തുഞ്ചൻ മാനുസ്ക്രിപ്റ്റ് റപോസിറ്ററിയിലെ )ഗണിത സംബന്ധിയായ ഗ്രന്ഥങ്ങളെ പഠനവിഷയമാക്കുന്നു. ഗ്രന്ഥശേഖരത്തിലെ പ്രധാന ഗണിതഗ്രന്ഥങ്ങളെ പരിചയ പെടുത്തുക എന്നതാണ് പ്രബന്ധത്തിന്റെ പ്രധാന ലക്ഷ്യം. അതിനോടൊപ്പം ഇതിൽ നിന്ന് ഒരു താളിയോലഗ്രന്ഥത്തെ പ്രധാന പഠനവസ്തുവായി എടുത്ത് പഠിക്കുകയും ചെയ്യുന്നു. ഇത് വഴി കേരളത്തിന്റെ ജ്ഞാനസമ്പത്ത് , അതിന്റെ കൈവഴികൾ എന്തെല്ലാം തുടങ്ങിയ കാര്യങ്ങളും പറയുന്നു പ്രബന്ധത്തിനോട് അനുബന്ധമായി ഹസ്തലിഖിത വിജ്ഞാനീയത്തിലെ പ്രാചീന എഴുത്തുപാരമ്പര്യത്തെയും പരിചയപ്പെടുത്തുന്നു. Title: Prime numbers Speaker: Kalyan Chakraborty Affiliation: Harish-Chandra Research Institute, Prayagraj Venue: Seminar Hall, KSoM Date and Time: September 01, 2023 at 3:30 PM Abstract: Several historical questions regarding prime numbers are still unsolved. These include Goldbach’s conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes that differ by two. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. Title: p-numerical semigroups and their fundamental properties Speaker: Takao Komatsu Affiliation: Zhejiang Sci-Tech University Venue: Seminar Hall, KSoM Date and Time: August 05, 2023 at 3:30 PM Abstract: Consider the linear Diophantine equation $a_1 x_1+a_2 x_2+...+a_k x_k=n$, where $a_1,a_2,\ldots,a_k$ are positive integers and $n$ is a non-negative integer. When we want to know the non-negative solutions, the number of solutions is finite if and only if $\gcd(a_1,a_2,\ldots,a_k)=1$. For a given non-negative integer $p$, there exists the largest integer $n$ such that the number of solutions is at most $p$. We call this largest integer the $p$-Frobenius number. When $p=0$, the $0$-Frobenius number is the classical Frobenius number, which is the central role of the famous linear Diophantine problem of Frobenius. In this talk, we introduce the basic concept of the $p$-numerical semigroup and mention the fundamental properties. We describe how the $p$-Frobenius number and related values are obtained. Title: Group actions in non-commutative geometry. Speaker: Safdar Quddus Affiliation: Indian Institute of Science, Bangalore Venue: Seminar Hall, KSoM Date and Time: July 31, 2023 at 12:00 PM Abstract: Historically, it is well known that group action changes the homological, differential, and topological properties of manifolds. The aim of this talk is to discuss the action of finite groups on some non-commutative differential and algebraic spaces. We shall first discuss the (co)homological properties of non-commutative and quantum torus then observe how the quotient spaces resulting from the actions of discrete subgroups of $SL(2, \mathbb Z)$ behave, and then shall discuss the flip actions on the non-commutative sphere. We shall see how the tools are developed in non-commutative geometry to understand and tackle the group action in general non-commutative spaces. Title: Rational points on hyperelliptic curves. Speaker: Om Prakash Affiliation: Kerala School of Mathematics Venue: Seminar Hall, KSoM Date and Time: July 28, 2023 at 2:30 PM Abstract: Hyperelliptic curves which can be viewed as generalisation of elliptic curves are a special class of algebraic curves. The theory of elliptic curves is sort of well understood and has a vast literature. However, the theory of hyperelliptic curves has not received as much attention by the research community; and most results concerning hyperelliptic curves which appear in the literature on algebraic geometry are couched in very general terms. In this talk we give an overview of results concerning rational points on hyperelliptic curves and present our recent joint work with Kalyan Chakraborty. Title: Hyperbolicity of the Curve Graph Speaker: George Shaji Affiliation: Department of Mathematics, The University of Utah Venue: Seminar Hall, KSoM Date and Time: July 26, 2023 at 11:00 AM Abstract: We will briefly explore the “Guessing Geodesics Lemma”, and see its immense usefulness by applying it to the curve graph of a surface. Specifically, we shall show that the Curve Graph of a finite type surface (of sufficient complexity) is hyperbolic using “Unicorn Paths” or “Bicorn Paths”. This method provides an extremely beautiful, diversely applicable and significantly simpler alternative to the Masur-Minsky machinery used to prove the same fact much earlier. Title: M-ideals in the Algebra of bounded analytic functions Speaker: Sreejith Siju Affiliation: Kerala School of Mathematics Venue: Seminar Hall, KSoM Date and Time: July 25, 2023 at 3:00 PM Abstract: The distinguished class of subspaces of a Banach space known as M -ideals is a significant tool in the study of the geometry of Banach spaces and approximation theory. The M-ideals in the continuous functions space and the disk algebra are fairly understood. In this talk we will discuss some basic facts regarding M-ideals and some recent results on M-ideals in the algebra of bounded analytic functions. This talk is based on a joint work with Deepak and Jaydeb. Title: On a class of Lebesgue-Ramanujan-Nagell equations Speaker: Azizul Hoque Affiliation: Department of Mathematics, Rangapara College Venue: Seminar Hall, KSoM Date and Time: July 21, 2023 at 11:00 AM Abstract: Finding the solutions of an exponential Diophantine equation is a classical problem and even now it is a very active area of research. One of the most important equations of this type is so-called Lebesgue-Ramanujan-Nagell equation, $x^2+D=\lambda y^n, (x,y,n\in\mathbb{N}, \gcd(x,y)=1, \lambda=1,2,4),$ where $D$ is a fixed positive integer. In this talk, we will discuss some interesting results concerning the integer solutions of certain generalizations of the above equation. This talk is based on some joint works with K. Chakraborty and K. Srinivas. Title: Selmer group associated to the group of degree zero cycles on abelian varieties Speaker: Kalyan Banerjee Affiliation: Vellore Institute of Technology, Chennai Venue: Seminar Hall, KSoM Date and Time: July 11, 2023 at 3:00 PM Abstract: In this talk we will look into a construction of the Selmer group of degree zero cycles on abelian varieties and will prove that it is finite. Title: Genuinely ramified maps and the Galois group of generic projection Speaker: Parameswaran A J Affiliation: TIFR, Mumbai Venue: Seminar Hall, KSoM Date and Time: April 13, 2023 at 12:00 PM Abstract: We show that the Galois closure of a degree d genuinely ramified covering $f : Y \to X$ of smooth projective irreducible curves over an algebraically closed field is the symmetric group $S_d$ if $f$ is a Morse map. Title: Jordan derivations and $n^{\text{th}}$-power property in rings Speaker: Shakir Ali Affiliation: Aligarh Muslim University Venue: Seminar Hall, KSoM Date and Time: February 08, 2023 at 12:00 PM Abstract: Let $R$ be an associative ring(algebra) with center $Z(R)$. Every associative ring $R$ can be turned into a Lie ring(algebra) by introducing a new product $[x, y] = xy - yx$, known as the Lie product. So we may regard $R$ simultaneously as an associative ring(algebra) and as a Lie ring(algebra). An additive mapping $d : R\rightarrow R$ is said to be a derivation on $R$ if $d(xy) = d (x)y + xd(y)$ holds for all $x, y \in R$. An additive mapping $d : R \rightarrow R$ is said to be a Jordan derivation if $d(x^2) = d(x)x+xd(x)$ holds for all $x \in R$. A function $f : R \rightarrow R$ is called a centralizing on $R$ if $[f(x), x)] \in Z(R)$ holds for all $x \in R$. In the special case where $[f(x), x)] = 0$ for all $x \in R$, $f$ is said to be commuting on $R$. The study of such mappings was initiated by E.C. Posner [ Proc. Amer. Math. Soc. 8(1957), 1093-1100]. In 1957, he proved that if a prime ring $R$ has a nonzero commuting derivation on $R$, then $R$ is commutative. An analogous result for centralizing automorphisms on prime rings was obtained by J.H. Mayne [Canad. J. Math. 19 (1976), 113-115]. In this talk, we will discuss the recent progress made on the topic and related areas. Moreover, some examples and counterexamples will be discussed for questions raised naturally. Title: Inversion formulas for the $j$-function around elliptic points. Speaker: Badri Vishal Pandey Affiliation: Universität zu Köln, Germany Venue: Seminar Hall, KSoM Date and Time: February 02, 2023 at 12:00 PM Abstract: Recently, Hong, Mertens, Ono, and Zhang proved a conjecture of Căldăraru, He, and Huang that expresses the Taylor series of the modular $j$-function around the elliptic points $i$ and $\rho = e^{\frac{\pi i}{3}}$ as rational functions arising from the signature $2$ and $3$ cases of Ramanujan’s theory of elliptic functions to alternative bases. We extend these results and give inversion formulas for the $j$-function around $i$ and $\rho$ arising from Gauss’ hypergeometric functions and Ramanujan’s theory in signatures $4$ and $6$. Title: Non-archimedean dynamics in dimension one Speaker: Niladri Patra Affiliation: Tata Institute of Fundamental Research, Mumbai Date and Time: December 09, 2022 at 03:30 PM Venue: Seminar Hall Abstract: In the beginning of the twentieth century, complex analysis gave rise to complex dynamics, which is the study of self-iterations of rational maps defined over complex numbers. Often the study of complex dynamical objects boil down to questions that are of arithmetic nature. This generally motivates the study of arithmetic dynamics. Arithmetic dynamics is roughly divided into two parts, dynamics over global fields and dynamics over local fields. Dynamics over local fields, which is also called non-archimedean dynamics or p-adic dynamics in the literature, has another motivation coming from p-adic analysis. The motivation is to build a theory parallel to complex dynamics with field of complex numbers replaced with $\mathbb{C}_{p}$. The theory that follows is called classical p-adic dynamics. One finds certain anomalies between the theories of complex dynamics and classical p-adic dynamics. These anomalies mostly arise from the fact that unlike $\mathbb {C}$, the topological field $\mathbb{C}_{p}$ is totally disconnected and not locally compact. To rectify these issues, one replaces $\mathbb{C}_{p}$ (or, $\mathbf{P}^{1}(\mathbb{C}_{p})$) with the Berkovich projective line, which is compact, Hausdorff, path connected space and in which $\mathbf{P}^{1}(\mathbb{C}_{p})$ embeds. In this talk, we will introduce discrete, complex, classical p-adic dynamics, dynamics on the Berkovich projective line and mention some of the parallels between complex dynamics and dynamics on the Berkovich projective line. Tittle : RESULTS AND MODELS. AN ILLUSTRATION THROUGH SUMS OF CUBES Speaker : Jean-Marc Deshouillers Affiliation : University of Bordeaux, France. Date and Time : 24-09-2022 at 03;30 PM Venue : Seminar hall, Kerala School of Mathematics Abstract : RESULTS AND MODELS. AN ILLUSTRATION THROUGH SUMS OF CUBES Number theory has the knack for phrasing easily understandable statements which are hard to prove. An archetype is Goldbach’s problem (1742), which is still unsolved : every even integer larger than 4 is a sum of two primes. The root of the talk is Waring’s problem (1770), which states – for cubes – that every integer is a sum of at most 9 cubes. It has been proved in 1909-1912, but we expect much more, namely every sufficiently large integer is a sum of 4 cubes. In the frame of this understandable question, we shall illustrate how mathematicians prove weaker statements, lead computation, build models to comfort their belief in statements they cannot About the speaker Professor Jean-Marc Deshouillers, a number theorist who has authored more than hundred research papers and advised seventeen PhD students and is currently Professor at Institut mathématique de Bordeaux, Bordeaux, France. Jean-Marc Deshouillers received his PhD in 1972 at the University Paris VI In 1985 he showed with Ramachandran Balasubramanian and Francois Dress that, in the case of the fourth powers of Waring’s problem, the least number of fourth powers that is necessary to express any positive integer as a sum of fourth powers is 19. Title: Zero free regions of spectral averages $L$-functions Speaker: Sandeep E. M. Affiliation: Indian Statistical Institute, North East center. Date and Time: September 12, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of MathematicsAbstract: In this talk, I would describe a recent result on the zero-free region of a weighted average of $L$-functions of integral weight Hecke eigenforms and if time permits, some remarks on the same for those corresponding to weight zero Hecke-Maass cusp forms (level $1$). The first is a joint work with Satadal Ganguly. Title: Families of quadratic fields with $3$-rank at least $3$. Speaker: Azizul Hoque Affiliation: Rangapara College Date and Time: August 2, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of MathematicsAbstract: Constructing number fields with large $n$-rank has proved to be a challenging practical problem, due in part to the fact that we believe such examples to be very rare. There is a conjecture (folklore) that the $n$-rank of $k$ is unbounded when $k$ runs through the quadratic fields. It was Quer who constructed $3$ imaginary quadratic fields with $3$-rank equal to $6$, and this result still stands as the current record. We will discuss two methods for constructing quadratic fields with large $n$-rank. We will show that for every large positive real number $x$, there exists a sufficiently large positive constant $c$ such that the number of quadratic fields with $3$-rank at least $3$ and absolute discriminant $\leq x$ is $>cx^ {\frac{1}{3}}$. If time permits, we will construct a parametric family of real (resp. imaginary) quadratic fields with $n$-rank at least $2$. Title: On a Conjecture of Iizuka Speaker: Azizul Hoque Affiliation: Rangapara College Date and Time: August 1, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of MathematicsAbstract: For any odd prime number $p$, we will construct an infinite family of quadruples of imaginary quadratic fields of the form $\mathbb{Q}(\sqrt {d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$ and $\mathbb{Q}(\sqrt{d+4p^2})$ whose class numbers are all divisible by a given odd integer $n\geq 3$. This provides a proof of a particular case of Iizuka’s conjecture (in fact a generalization of it). Title: Snippets from the history of Science in Ancient India. Speaker: T. R. Govindarajan Affiliation: The Institute of Mathematical Sciences Date and Time: July 14, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics Abstract: There are several achievements and there are failures too. Will sketch a few of these in Astronomy, Mathematics and Architecture. Will present some unsolved questions in our understanding too. Will end up with some discussions on what we gave the outside world and what we learnt. Title : Physics and Maths of Braids and Knots: Speaker: T. R. Govindarajan Affiliation: The Institute of Mathematical Sciences Date and Time: July 13, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics Abstract: Braids and Knots have fascinated both Physicists and mathematicians for more than a century. I will present several developments in both the areas as well as new biological questions and end with unanswered questions. Title : $S$-units in Recurrence Sequences Speaker: Sudhansu Sekhar Rout Affiliation: Institute of Mathematics & Applications, Bhubaneswar Date and Time: June 30, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics Abstract: In this talk, we give various finiteness results concerning terms of recurrence sequences $U_n$ representable as a sum of $S$-units with a fixed number of terms. We prove that under certain (necessary) conditions, the number of indices n for which $U_n$ allows such a representation is finite, and can be bounded in terms of the parameters involved. In this generality, our result is ineffective, i.e. we cannot bound the size of the exceptional indices. We also give an effective result, under some stronger assumptions. Title : Some geometric structures on principal Lie 2-group bundles over Lie groupoids. Speaker: Praphulla Koushik Affiliation: IISER Pune Date and Time: June 24, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics Abstract : In this talk we introduce the notion of principal Lie 2-group bundle over a Lie groupoid; as a generalisation of the notion of classical principal (Lie group) bundle over a smooth manifold. Motivated by the idea of Atiyah sequence for G-bundles over manifolds, we introduce the notion of Atiyah sequence for Lie 2-group bundle over Lie groupoids. We then see some notion of (strict) connection and semi-strict connections on such principal bundles. This is a joint work with Saikat Chatterjee and Adittya Chaudhuri Title: On Partial Differential Equations and Diffusion Processes. Speaker: Rajeeva Karandikar Affiliation: Chennai Mathematical Institute. Date and Time: May 26, 2022 at 11:00 AM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: In this talk we will describe connections between second order partial differential equations and the associated Markov processes. This connection has been an active area of research for several decades. Title: Iwasawa theory and Galois groups Speaker: Sujatha Ramdorai Affiliation: University of British Columbia. Date and Time: May 26, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: Understanding the Galois group of the field of rational numbers and of its finite extensions is one of the central problems in Number Theory. Over the last five centuries, this has led to the development of several areas within Pure Mathematics. In this talk, we shall discuss how Iwasawa theory has contributed to studying this problem. Title: Black holes through different windows Speaker: Ajith Parameswaran Affiliation: International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore Date and Time: May 25, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: Made entirely of curved spacetime, black holes are among the most enigmatic objects in the Universe. Although early theoretical ideas on objects like black holes go back to the eighteenth century, the first rigorous mathematical formulation of a black hole was made by Karl Schwarzschild in 1915, based on Einstein’s General Theory of Relativity. A variety of astronomical observations made in the last century confirmed that black holes are not just theoretical constructs — the Universe is littered with them. Recently, a variety of astronomical observations have started to probe the detailed nature of black holes. This talk will provide a summary of this exciting journey. Title: Factorization of Integers using Rational Sieve Speaker:Manika Gupta Affiliation: Kerala School of Mathematics Date and Time: April 20, 2022, 11.00 AM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: We present a discussion on different algorithms for the factorization of integers, focussing on the rational sieve. The rational sieve uses sieve theoretic methods to find the prime factors of a number. It is a special case of the General Number Field Sieve (GNFS) which is the most efficient classical algorithm for factoring ‘large’ integers. We also present a brief overview of the quadratic sieve and GNFS. Title: A lower bound on number of representation of even numbers as sum of an odd prime and product of at most two primes Speaker: Om Prakash Affiliation: Kerala School of Mathematics Date and Time: April 19, 2022, 03.00 PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: This talk is about proving a lower bound on the number of representation of large enough even numbers as sum of an odd prime and a product of at most two primes. Title: TBA Speaker:Pole Vennela Affiliation: Kerala School of Mathematics Date and Time: April 19, 2022, 11.00 AM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: TBA Title: Erdös Distance Problem Speaker:Viswanathan S Affiliation: Kerala School of Mathematics Date and Time:April 18, 2022, 04.00 PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: In this talk we shall introduce Erdös Distance Conjecture. This is now a glowing hot topic in mathematics. Starting with the basic sketch of the problem we shall explore certain partial bounds for the conjecture culminating in Moser’s construction. This problem has now invited tools from several fields of mathematics, some include – Fourier analysis, graph theory, and even information theory Title: Bounded Gaps between Primes Speaker : Viswanathan S Affiliation: Kerala School of Mathematics Date and Time: April 18, 2022, 11.00 AM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: In this talk we shall sketch the idea of Goldston, Pintz, and Yildirim that paved a way to attack the twin prime conjecture by establishing a connecting between equidistribution of primes in certain congruence classes and the gaps between primes (via sieve methods). Their idea later inspired Yitang Zhang and a chain of other mathematicians including Terrence Tao and James Maynard to reduce the infinitely occurring gap between successive primes which led to major breakthroughs in that field. Title: Integers free of large prime factors Speaker: Pole Vennela Affiliation: Kerala School of Mathematics Date and Time:April 17, 2022, 04.15 PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: In this talk we are going to discuss about the number \psi(x,y) of integers < x and free of prime factors > y has been given satisfactory estimates in the regions y < (log X)3/4-6 and y > exp{(log log X)^5/3+6}. In the intermediate range, only very crude estimates have been obtained so far. We close this “gap” and give an expression which approximates \psi(x, y) uniformly for x > y > 2 within a factor 1 + O((log y)/(log x) + (log y)/y). As an application, we derive a simple formula for \psi(cx, y)/\psi(x, y), where 1 < c < y. We also prove a short interval estimate for \psi(x, y). Title: The Paley-Weiner Theorem Speaker: Manika Gupta Affiliation: Kerala School of Mathematics Date and Time: April 17, 2022, 02.30 PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: In this talk, a proof of the Paley-Weiner theorem is presented. The theorem relates decay properties of a function at infinity with analyticity of its Fourier transform. The theorem plays a crucial role in the proof of the main large sieve inequality. We shall also discuss the background and other applications of this result. Title: A lower bound on number of representation of even numbers as sum of an odd prime and product of at most two primes. Speaker:Om Prakash Affiliation: Kerala School of Mathematics Date and Time: April 16, 2022, 11.00 AM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: This talk is about proving a lower bound on the number of representation of large enough even numbers as sum of an odd prime and a product of at most two primes. Title: Orthogonality of invariant vectors Speaker: U. K. Anandavardhanan Affiliation:Indian Institute of Technology Bombay Date and Time: March 21, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: This talk is about finite groups and their representation theory. Given a group $G$ and two Gelfand subgroups $H$ and $K$ of $G$, associated to an irreducible representation $\pi$ of $G$, there is a notion of $H$ and $K$ being correlated with respect to $\pi$ in $G$. This notion was defined by Benedict Gross in 1991. The talk will not assume much background material. Towards the end of the talk, we’ll present some recent results regarding this theme (which are joint with Arindam Jana). Title: Nice ideals, their role in ideal convergence and some thoughts Speaker: Pratulananda Das Affiliation: Jadavpur University Date and Time: March 04, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: In this talk we will primarily talk about the so called “nice ideals” in the realm of set theory. We will in particular describe the very basic notion of ideal convergence to see how these ideals have been found to be very useful in summability theory before moving to more deeper observations Title: Nice ideals, their role in ideal convergence and some thoughts Speaker: Pratulananda Das Affiliation: Jadavpur University Date and Time: March 03, 2022 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: In this talk we will primarily talk about the so called “nice ideals” in the realm of set theory. We will in particular describe the very basic notion of ideal convergence to see how these ideals have been found to be very useful in summability theory before moving to more deeper observations Title:On a family of elliptic curves of rank at least $2$. Speaker: Richa Sharma Affiliation: PDF, Kerala School of Mathematics Date and Time: 04 February 2022 at 3.30 PM Venue: Seminar Hall, KSoM Abstract: Let $C_{m} : y^{2} = x^{3} - m^{2}x +p^{2}q^{2}$ be a family of elliptic curves over $\mathbb{Q}$, where $m$ is a positive integer and $p, q$ are distinct odd primes. We prove that the torsion subgroup of $C_{m}(\mathbb{Q})$ is trivial and the $\mathbb{Q}$-rank of this family is at least $2$, whenever $m \equiv 2 \pmod {64}$ with neither $p$ nor $q$ divides $m$. Title: Set partitions, tableaux, and subspace profiles under regular split semisimple matrices Speaker: Amritanshu Prasad Affiliation: The Institute of Mathematical Sciences Date and Time: December 10, 2021 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics Abstract: We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. At $1$, they count set partitions with specified block sizes. At $0$, they count standard tableaux of specified shape. At $-1$ they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express $q$-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions. In a special case, these polynomials coincide with those defined by Touchard in his study of crossings of chord diagrams. Title:Lie algebras associated to closed curves on surfaces. Speaker: Arpan Kabiraj Affiliation: IIT Palakkad Date and Time: November 26, 2021 at 03:30PM Venue: Seminar Hall, Kerala School of Mathematics Abstract:We will discuss various Lie algebras associated to closed curves on orientable surfaces (possibly with boundary and punctures) introduced by Goldman and Wolpert in the 80’s. If time permits, we will discuss a relation between these Lie algebras and skein algebras of three-manifolds. Speaker: Aman Singh Affiliation: Kerala School of Mathematics Date and Time: November 19, 2021 at 03:30 PM Venue: Seminar Hall, Kerala School of Mathematics Title: Dirichlet’s Prime Distribution Theorem – II Abstract: We shall see a complete proof of the fact that there are infinitely many primes in any given arithmetic progression $(a+nd)$, where $\gcd(a,d) = 1$. Speaker: Aman Singh Affiliation: Kerala School of Mathematics Date and Time: November 17, 2021 at 04:00 PM Venue: Seminar Hall, Kerala School of Mathematics Title: Dirichlet’s Prime Distribution Theorem – I Abstract: We will introduce the group of characters associated with a given finite abelian group and prove some orthogonality relations. We shall then see the Dirichlet characters and some of its properties leading up to the proof of non-vanishing of $L(1, \chi)$ for a non-principal character $\chi$. Title: Introduction to Hilbert modular forms and its determination by square-free Fourier coefficients. Speaker: Rishabh Agnihotri Affiliation: HRI, Prayagraj (Allahabad) Date and Time: September 10, 2021 at 03:30PM Venue: Seminar Hall, Kerala School of Mathematics Abstract: We introduce two notions of Hilbert modular forms namely classical and adelic. After that we see the relation between them. We also talk about the determination of adelic hilbert modular forms. More concretely we discuss the following result. Let $\mathbf{f}$ be as above with $C_{\mathbf{f}}(\mathfrak{m})$ denote its Fourier coefficients. Then there exists a square-free ideal $\mathfrak{m}$ with $N(\mathfrak{m})\ll k_0^{3n+\epsilon}N(\ mathfrak{n})^{\frac{6n^2+1}{2}+\epsilon}$ such that $C_{\mathbf{f}}(\mathfrak{m})eq 0$. The implied constant depends only on $\epsilon, F$. Title: Class group of real cyclotomic fields. Speaker: Mohit Mishra Affiliation: HRI, Prayagraj (Allahabad) Date and Time: July 21, 2021 at 03:30PM Venue: Seminar Hall, Kerala School of Mathematics Abstract: For every finite extension of rational numbers, there is a group associated to it called the “Class Group”. Class group is a very mysterious object and there is no (infinite) family known with a prescribed class group. In 1979, G. Cornell proved that every finite abelian group $G$ can be realized as a subgroup of a class group of infinitely many cyclotomic (totally imaginary) fields. In this talk, we will prove the analogue of this result for real cyclotomic fields. This is a joint work with L.C. Washington and R. Schoof. Title: Obstruction Theory in Algebra and Topology. Speaker : Bibekananda Mishra Date and time: 10.00 am to 10.45 am, 17 th June 2021. Online platform: Zoom Abstract: Let P be a projective module of finite rank on a ring A. What is the precise obstruction for P to have a free component (i.e. P \cong Q \oplus A)? This question, very much of algebraic flavour, is intricately related to the question of whether the vector bundles over smooth manifolds have non-vanishing sections on them. We will see in this talk certain invariants, called Nori homotopy groups, both in the algebraic as well as topological context, which gives us an effective description of the obstructions involved. Title: A degeneration of the Compactified Jacobian of irreducible nodal curves. Speaker: Subham Sarkar Date and time: 12.00 pm to 12.45 pm, 17 th June 2021 Online platform: Zoom Abstract: For each k ≥ 1, We construct an algebraic degeneration of the compactified Jacobian of a nodal curve X_k with k-nodes, over a suitable dense subset of the k-fold product of the normalisation X_0 of X_K. A special fiber is isomorphic to the Jacobian of X_0 and k-fold product of a rational nodal curve. We prove that the total space is a quasi-projective variety with k-th product of normal crossing singularity. As an application we computed the mised Hodge number of the cohomogy of compactified Jacobian. Title: Shifted convolution sums and sign changes of Fourier coefficients of certain automorphic forms. Speaker: Lalit Vaishya Date and time: 3.00 pm to 3.45 pm, 17 th June 2021 Online platform: Zoom Abstract: Briefly, we present some of our work which deal with some problems in the theory of automorphic forms. In the first part, we discuss some problems on shifted convolution sums associated to Hecke Maass cusp forms (non holomorphic cusp forms), holomorphic Hecke eigen (cusp) forms and obtain the estimates. In the second part, we prove a quantitative result about sign changes of Fourier coefficients of Hecke eigenform supported at positive integers represented by a primitive integral positive binary quadratic from of negative discriminant having class number 1. We also study the average behavior of Fourier coefficients of Hecke eigenforms supported at positive integers represented by a primitive integral positive definite binary quadratic form of negative discriminant having class number 1. As a consequence, we prove that there are infinitely many sign change of sequence of Fourier coefficients supported at positive integers represented by these binary quadratic forms. On the Topology of Complex Projective Varieties NimaRose Manjila IISER Pune Date and Time: April 09, 2021 at 02:00PM Seminar Hall, Kerala School of Mathematics Abstract: We use Morse Theory and Lefschetz Pencil to find the Topology of Complex Projective Curves and generalise this idea to prove Lefschetz Theorem. Other results include a proof of Poincare Duality and Riemann Hurwitz theorem for Ramified Covers of Curves. Title: Real Unipotent Elements in Classical Lie Groups. Speaker: Krishnendu Gongopadhyay Affiliation: IISER Mohali Date and Time: April 08, 2021 at 03:30PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: Real elements are those elements in a group which are conjugate to their own inverses. Real elements appear naturally at different branches of mathematics. These elements are also known as `reversible’ elements in the literature. These elements are closely related to the so-called strongly real elements in a group which are products of two involutions. After giving a brief exposition on real elements in groups, I shall discuss classification of real unipotent elements in classical Lie groups which is part of a joint work with Chandan Maity. Title: A generalized modified Bessel function and explicit transformations of certain Lambert series Speaker:Rahul kumar Date and Time: March 26, 2021 at 04:00PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: An exact transformation, which we call a master identity, is obtained for the series P∞n=1 σa(n)e−ny for a ∈ C and Re(y) > 0. As corollaries when a is an odd integer, we derive the well-known transformations of the Eisenstein series on SL2 (Z), that of the Dedekind eta function as well as Ramanujan’s famous formula for ζ(2m + 1). Corresponding new transformations when a is a non-zero even integer are also obtained as special cases of the master identity. These include a novel companion to Ramanujan’s formula for ζ(2m+ 1).Although not modular, it is surprising that such explicit transformations exist. The Wigert-Bellman identity arising from the a = 0 case of the master identity is derived too. The latter identity itself is derived using Guinand’s version of the Vorono ̈ı summation formula and an integral evaluation of N. S. Koshliakov involving a generalization of the modified Bessel function Kν(z). Koshliakov’s integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new transformation involving the sums-of-squares function rk(n). This is joint work with Atul Dixit and Aashita Kesarwani. Title: Weak Mordell-Weil Theorem for Chow groups over global function fields Speaker:Kalyan Banerjee Date and Time: March 26, 2021 at 03:00PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: The classical weak Mordell-Weil theorem for an abelian variety A over a number field K says that A(K)/nA(K) is finite for any integer n bigger than 1. This has further consequence that the group A(K) of K-rational points on A is finitely generated. In this talk we are going to consider a variety X defined over the algebraic closure of a function field of a smooth projective curve and consider the group of degree zero cycles modulo rational equivalence on this variety denoted by A_0(X). We are going to consider the question analogous to the weak Mordell-Weil theorem for the Galois invariants of A_0(X), that is whether the group A_0(X)^G/nA_0(X)^G is finite, where G is the absolute Galois group of the function field and n is an integer bigger than 1. We are going to prove this analogue under some assumption on the variety X. Title: Representation theory of finite groups – an Introduction – II. Speaker: Hassain M Affiliation: Kerala School of Mathematics Date and Time: March 17, 2021 at 02:00PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: Let $G$ be a finite group. A $n$-dimensional representation of $G$ is a homomorphism from $G$ to the group $\mathrm{GL}_n(\mathbb C)$ of $n\times n$ invertible matrices over $\mathbb C.$ In this talk, I will discuss some interesting examples and basic results in representation theory of finite groups. Title: Arakelov Geometry of Modular Curves $X_0(p^2)$. Speaker: Chitrabhanu Chaudhuri, NISER Bhubaneshwar Date and Time: March 12, 2021 at 03:30PM Venue: Seminar Hall, Kerala School of Mathematics (online lecture) Abstract: I shall outline the construction of a semisimple and minimal regular model for $X_0(p^2)$ over an appropriate number field. This will be a regular scheme over the spectrum of the ring of integers of that number field, such that the fibres are complete curves with at worst nodal singularities and satisfying certain stability conditions. The generic fibre of the model is isomorphic to $X_0(p^2)$. The purpose of this construction is to use the theory developed by Shou-Wu Zhang, using Arakelov theory, for proving an effective version of a conjecture by Bogomolov in this special case of modular curves $X_0(p^2)$. Title: Representation theory of finite groups – an Introduction – I. Speaker: Hassain M Affiliation: Kerala School of Mathematics Date and Time: March 10, 2021 at 02:00PM Venue: Seminar Hall, Kerala School of Mathematics. Abstract: Let $G$ be a finite group. A $n$-dimensional representation of $G$ is a homomorphism from $G$ to the group $\mathrm{GL}_n(\mathbb C)$ of $n\times n$ invertible matrices over $\mathbb C.$ In this talk, I will discuss some interesting examples and basic results in representation theory of finite groups. Title: Noncommutative Korovkin Theory. Speaker: Arunkumar C. S. Affiliation: Kerala School of Mathematics Date and Time: February 26, 2021 at 03:00PM Venue: Seminar Hall, Kerala School of Mathematics Abstract: In this talk, we will introduce the Hyper Rigidity of operator systems in $C^$-algebras as a non commutative analogue of Korovkin sets in the space of continuous functions, $C[0,1]$. Also, we point out one of our recent results, and a couple of open questions along this direction. Title: A certain kernel function for -values of half-integral weight Hecke eigenforms. Speaker: Sreejith M. M. Affiliation: Kerala School of Mathematics Date and Time : February 05, 2021 at 03:00 p.m. Venue : Seminar Hall, Kerala School of Mathematics Abstract: In this talk we will derive a non-cusp form of weight $k+1/2$ ($k\geq2$, even) for $\Gamma_0 (4)$ in the Kohnen plus space whose Petersson scalar product with a cuspidal Hecke eigenform $f$ is equal to a constant times the $L$ value $L(f,k-1/2).$ We also prove that for such a form $f$ and the associated form $F$ under the $D^{\text{th}}$ Shimura-Kohnen lift the quantity $\frac{a_f(D)L (F,2k-1)}{\pi^{k-1}\langle f,f\rangle L(D,k)}$ is algebraic. Title: Characterization of linear maps preserving unitary conjugation. Speaker: Dr. Shankar P. Affiliation: Indian Statistical Institute, Bangalore Date and Time: January 22, 2021 at 03:00PM Venue: Seminar Hall, Kerala School of Mathematics Abstract: Let $H$ be a complex separable Hilbert space and let $B(H)$ be the algebra of all bounded linear operators on $H$. In this talk, we discuss about what are the linear maps $\alpha:B(H) \ rightarrow B(H)$ which satisfy $\alpha(UXU^)=U\alpha(X) U^~~\forall~~ X\in B(H),$ for every unitary $U$ on $H$. Title: Sign Changes in restricted coefficients of Hilbert Modular forms Speaker: Krishnarjun K Affiliation: Harish Chandra Research Institute, Prayagraj(Allahabad) Date and Time: January 08, 2021 at 03:00PM Venue: Seminar Hall, Kerala School of Mathematics Abstract: Let $\textbf{f}$ be an adelic Hilbert cusp form of weight $k$ and level $\mathfrak{n}$ over a totally real number field $F$. In this talk, we study the sign changes in the Fourier coefficients of $\textbf{f}$ when restricted to square-free integral ideals and integral ideals in an “arithmetic progression”. In both cases we obtain qualitative results and in the former case we obtain quantitative results as well. Our results are general in the sense that we do not impose any restriction of the number field $F$, the weight $k$ or the level $\mathfrak{n}$. Title : Some notions of non-commutative convexity Speaker: Syamkrishnan M. S. Affiliation: Kerala School of Mathematics Date and Time: December 04, 2020 at 03:00PM Venue: Seminar Hall, Kerala School of Mathematics Abstract: In this talk, we shall introduce two non-commutative versions of the classical convexity, namely the $C^$-convexity and matricial convexity in the setting of $C^*$-algebras. We shall discuss the similarities as well as dissimilarities between the convex sets in the classical setting with the convex sets in the non commutative case. Also, we will be discussing its connections with other areas of operator algebras. Title: On non-vanishing of modular L functions inside the critical strip Speaker : Sandeep E. M. Affiliation : Kerala School of Mathematics Date and Time : November 20, 2020 at 03:00 p.m. Venue : Seminar Hall, Kerala School of Mathematics Abstract : The $L$-series associated to a classical modular form $f$ (of weight $k$ and level $1$) denoted by $L(f,s) := \sum_{n\geq 1} \frac{a_f(n)}{n^s}$ where $a_f(n)$ denotes the $n^{\text{th}}$ Fourier coefficient of $f$ (in its $q$-series expansion around $q=0$) is an analytic function on the right half plane $\{\Re(s)>\frac{k+1}{2}\}$ and can be analytically continued to the whole $\mathbb{C}$. The non-trivial zeros of this function lie inside the critical strip $(k-1)/2 < \Re(s) < (k+1)/2$. The analogue (GRH) of the Riemann Hypothesis in this context states that they all lie on the critical line $\Re(s) = k/2$ itself. The following region $\sigma \geq 1-\dfrac{c}{\log (k+\|t\|+3)}$ where $c>0$ is an absolute constant, is currently known to be a zero-free region for $L(f,s)$. Some aspects of this non-vanishing related to my work will be discussed in this talk. This is a joint work with Prof M Manickam and Prof V Kumar Murty Title : Elliptic Curves: Introduction and An Application Speaker: Kalyan Chakraborty Affiliation: Kerala School of Mathematics Date and Time: November 06, 2020 at 03:00 p.m. Venue: Seminar Hall, Kerala School of Mathematics Abstract: This talk will begin with an introduction to elliptic curves. We shall then progress into the BSD conjecture and finally look into the idea of the proof of Fermat’s last theorem. Title: An invitation to the theory of L functions Speaker : Krishnarjun K Affiliation : Harish Chandra Research Institute, Prayagraj(Allahabad) Date and Time : October 23, 2020 at 03:00 p.m. Venue : Seminar Hall, Kerala School of Mathematics Abstract : The aim of this talk is to introduce the notion of an $L$ function and to describe a few basic properties. We shall also prove two classical theorems, one of Riemann and Dirichlet and demonstrate how techniques from complex analysis can be used to prove arithmetic results. We shall briefly touch upon current research topics of interest in the subject, if time permits. Here is a list of the previous colloquiums and seminars	{"url":"https://ksom.res.in/colloquium-and-seminars/","timestamp":"2024-11-06T23:19:36Z","content_type":"text/html","content_length":"249338","record_id":"<urn:uuid:3530c845-dc35-47b0-9787-4265b531b9d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00280.warc.gz"}
Time series lazy evaluation Lazy evaluation is an evaluation strategy that delays the evaluation of an expression until its value is needed. When combined with memoization, lazy evaluation strategy avoids repeated evaluations and can reduce the running time of certain functions by a significant factor. The time series library uses lazy evaluation to process data. Notionally an execution graph is constructed on time series data whose evaluation is triggered only when its output is materialized. Assuming an object is moving in a one dimensional space, whose location is captured by x(t). You can determine the harsh acceleration/braking (h(t)) of this object by using its velocity (v(t)) and acceleration (a(t)) time series as follows: # 1d location timeseries x(t) = input location timeseries # velocity - first derivative of x(t) v(t) = x(t) - x(t-1) # acceleration - second derivative of x(t) a(t) = v(t) - v(t-1) # harsh acceleration/braking using thresholds on acceleration h(t) = +1 if a(t) > threshold_acceleration = -1 if a(t) < threshold_deceleration = 0 otherwise This results in a simple execution graph of the form: x(t) --> v(t) --> a(t) --> h(t) Evaluations are triggered only when an action is performed, such as compute h(5...10), i.e. compute h(5), ..., h(10). The library captures narrow temporal dependencies between time series. In this example, h(5...10) requires a(5...10), which in turn requires v(4...10), which then requires x(3...10). Only the relevant portions of a(t), v(t) and x(t) are evaluated. h(5...10) <-- a(5...10) <-- v(4...10) <-- x(3...10) Furthermore, evaluations are memoized and can thus be reused in subsequent actions on h. For example, when a request for h(7...12) follows a request for h(5...10), the memoized values h(7...10) would be leveraged; further, h(11...12) would be evaluated using a(11...12), v(10...12) and x(9...12), which would in turn leverage v(10) and x(9...10) memoized from the prior computation. In a more general example, you could define a smoothened velocity timeseries as follows: # 1d location timeseries x(t) = input location timeseries # velocity - first derivative of x(t) v(t) = x(t) - x(t-1) # smoothened velocity # alpha is the smoothing factor # n is a smoothing history v_smooth(t) = (v(t)1.0 + v(t-1)alpha + ... + v(t-n)*alpha^n) / (1 + alpha + ... + alpha^n) # acceleration - second derivative of x(t) a(t) = v_smooth(t) - v_smooth(t-1) In this example h(l...u) has the following temporal dependency. Evaluation of h(l...u) would strictly adhere to this temporal dependency with memoization. h(l...u) <-- a(l...u) <-- v_smooth(l-1...u) <-- v(l-n-1...u) <-- x(l-n-2...u) An Example The following example shows a python code snippet that implements harsh acceleration on a simple in-memory time series. The library includes several built-in transforms. In this example the difference transform is applied twice to the location time series to compute acceleration time series. A map operation is applied to the acceleration time series using a harsh lambda function, which is defined after the code sample, that maps acceleration to either +1 (harsh acceleration), -1 (harsh braking) and 0 (otherwise). The filter operation selects only instances wherein either harsh acceleration or harsh braking is observed. Prior to calling get_values, an execution graph is created, but no computations are performed. On calling get_values(5, 10), the evaluation is performed with memoization on the narrowest possible temporal dependency in the execution graph. import tspy from tspy.builders.functions import transformers x = tspy.time_series([1.0, 2.0, 4.0, 7.0, 11.0, 16.0, 22.0, 29.0, 28.0, 30.0, 29.0, 30.0, 30.0]) v = x.transform(transformers.difference()) a = v.transform(transformers.difference()) h = a.map(harsh).filter(lambda h: h != 0) print(h[5, 10]) The harsh lambda is defined as follows: def harsh(a): threshold_acceleration = 2.0 threshold_braking = -2.0 if (a > threshold_acceleration): return +1 elif (a < threshold_braking): return -1 return 0	{"url":"https://www.ibm.com/docs/en/watsonx/w-and-w/1.1.x?topic=analysis-time-series-lazy-evaluation","timestamp":"2024-11-07T11:21:33Z","content_type":"text/html","content_length":"12353","record_id":"<urn:uuid:92eec5af-cdfe-490e-9d24-36d036354ebc>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00081.warc.gz"}
Introduction: Distribution of Differences in Sample Proportions What you’ll learn to do: Recognize when to use a two population proportion hypothesis test to compare two populations/treatment groups. • Recognize when to use a hypothesis test or a confidence interval to compare two population proportions or to investigate a treatment effect for a categorical variable. • Determine if a study involving two proportions is an experiment or an observational study. • Describe the sampling distribution of the difference between two proportions. • Draw conclusions about a difference in population proportions from a simulation.	{"url":"https://courses.lumenlearning.com/atd-herkimer-statisticssocsci/chapter/introduction-distribution-of-differences-in-sample-proportions/","timestamp":"2024-11-10T13:03:52Z","content_type":"text/html","content_length":"46999","record_id":"<urn:uuid:b3fcd657-68c8-4001-b75f-5997b945b440>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00215.warc.gz"}
Interface SingularValueDecomposition<T extends Matrix> All Superinterfaces: All Known Subinterfaces: SingularValueDecomposition_F32<T>, SingularValueDecomposition_F64<T> All Known Implementing Classes: SafeSvd_DDRM, SafeSvd_FDRM, SvdImplicitQrDecompose_DDRM, SvdImplicitQrDecompose_FDRM, SvdImplicitQrDecompose_MT_DDRM, SvdImplicitQrDecompose_MT_FDRM This is an abstract class for computing the singular value decomposition (SVD) of a matrix, which is defined as: A = U * W * V ^T where A is m by n, and U and V are orthogonal matrices, and W is a diagonal matrix. The dimension of U,W,V depends if it is a compact SVD or not. If not compact then U is m by m, W is m by n, V is n by n. If compact then let s be the number of singular values, U is m by s, W is s by s, and V is n by s. Accessor functions for decomposed matrices can return an internally constructed matrix if null is passed in for the optional storage parameter. The exact behavior is implementation specific. If an internally maintained matrix is returned then on the next call to decompose the matrix will be modified. The advantage of this approach is reduced memory overhead. To create a new instance of SingularValueDecomposition see DecompositionFactory_DDRM and SingularOps_DDRM contains additional helpful SVD related functions. Note that the ordering of singular values is not guaranteed, unless done so by a specific implementation. The singular values can be put into descending order while adjusting U and V using • Method Summary Modifier and Type getU(T U, boolean transposed) Returns the orthogonal 'U' matrix. getV(T V, boolean transposed) Returns the orthogonal 'V' matrix. Returns a diagonal matrix with the singular values. If true then compact matrices are returned. The number of singular values in the matrix. Number of columns in the decomposed matrix. Number of rows in the decomposed matrix. • Method Details □ numberOfSingularValues int numberOfSingularValues() The number of singular values in the matrix. This is equal to the length of the smallest side. Number of singular values in the matrix. □ isCompact boolean isCompact() If true then compact matrices are returned. true if results use compact notation. □ getU T getU(@Nullable T U, boolean transposed) Returns the orthogonal 'U' matrix. Internally the SVD algorithm might compute U transposed or it might not. To avoid an unnecessary double transpose the option is provided to select if the transpose is returned. U - Optional storage for U. If null a new instance or internally maintained matrix is returned. Modified. transposed - If the returned U is transposed. An orthogonal matrix. □ getV T getV(@Nullable T V, boolean transposed) Returns the orthogonal 'V' matrix. Internally the SVD algorithm might compute V transposed or it might not. To avoid an unnecessary double transpose the option is provided to select if the transpose is returned. V - Optional storage for v. If null a new instance or internally maintained matrix is returned. Modified. transposed - If the returned V is transposed. An orthogonal matrix. □ getW Returns a diagonal matrix with the singular values. Order of the singular values is not guaranteed. W - Optional storage for W. If null a new instance or internally maintained matrix is returned. Modified. Diagonal matrix with singular values along the diagonal. □ numRows int numRows() Number of rows in the decomposed matrix. Number of rows in the decomposed matrix. □ numCols int numCols() Number of columns in the decomposed matrix. Number of columns in the decomposed matrix.	{"url":"https://ejml.org/javadoc/org/ejml/interfaces/decomposition/SingularValueDecomposition.html","timestamp":"2024-11-09T06:36:16Z","content_type":"text/html","content_length":"18887","record_id":"<urn:uuid:3a3aaa3e-9c73-412b-90f6-ead41b8c434a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00069.warc.gz"}
gg_parallel_slopes: Plot parallel slopes model in moderndive: Tidyverse-Friendly Introductory Linear Regression NOTE: This function is deprecated; please use geom_parallel_slopes() instead. Output a visualization of linear regression when you have one numerical and one categorical explanatory/predictor variable: a separate colored regression line for each level of the categorical variable y Character string of outcome variable in data num_x Character string of numerical explanatory/predictor variable in data cat_x Character string of categorical explanatory/predictor variable in data an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from data environment(formula), typically the environment from which lm is called. alpha Transparency of points an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which lm is called. ## Not run: library(ggplot2) library(dplyr) library(moderndive) # log10() transformations house_prices <- house_prices %>% mutate( log10_price = log10(price), log10_size = log10(sqft_living) ) # Output parallel slopes model plot: gg_parallel_slopes( y = "log10_price", num_x = "log10_size", cat_x = "condition", data = house_prices, alpha = 0.1 ) + labs( x = "log10 square feet living space", y = "log10 price in USD", title = "House prices in Seattle: Parallel slopes model" ) # Compare with interaction model plot: ggplot(house_prices, aes(x = log10_size, y = log10_price, col = condition)) + geom_point(alpha = 0.1) + geom_smooth(method = "lm", se = FALSE, size = 1) + labs( x = "log10 square feet living space", y = "log10 price in USD", title = "House prices in Seattle: Interaction model" ) ## End(Not run) For more information on customizing the embed code, read Embedding Snippets.	{"url":"https://rdrr.io/cran/moderndive/man/gg_parallel_slopes.html","timestamp":"2024-11-11T09:37:31Z","content_type":"text/html","content_length":"36241","record_id":"<urn:uuid:373579c3-15c7-4cad-af95-0d0a62c3e108>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00757.warc.gz"}