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Geometry Essentials For Dummies Just the critical concepts you need to score high in geometry This practical, friendly guide focuses on critical concepts taught in a typical geometry course, from the properties of triangles, parallelograms, circles, and cylinders, to the skills and strategies you need to write geometry proofs. Geometry Essentials For Dummies is perfect for cramming or doing homework, or as a reference for parents helping kids study for exams. • Get down to the basics — get a handle on the basics of geometry, from lines, segments, and angles, to vertices, altitudes, and diagonals • Conquer proofs with confidence — follow easy-to-grasp instructions for understanding the components of a formal geometry proof • Take triangles in strides — learn how to take in a triangle’s sides, analyze its angles, work through an SAS proof, and apply the Pythagorean Theorem • Polish up on polygons — get the lowdown on quadrilaterals and other polygons: their angles, areas, properties, perimeters, and much more Open the book and find: • Plain-English explanations of geometry terms • Tips for tackling geometry proofs • The seven members of the quadrilateral family • Straight talk on circles • Essential triangle formulas • The lowdown on 3-D: spheres, cylinders, prisms, and pyramids • Ten things to use as reasons in geometry proofs Learn to: • Core concepts about the geometry of shapes and geometry proofs • Critical theorems, postulates, and definitions • The principles and formulas you need to know Book Details • Paperback: 192 pages • Publisher: For Dummies (June 2011) • Language: English • ISBN-10: 1118068750 • ISBN-13: 978-1118068755 Download [9.7 MiB] You must be logged in to post a comment.	{"url":"https://www.wowebook.com/book/geometry-essentials-for-dummies/","timestamp":"2024-11-11T07:06:06Z","content_type":"text/html","content_length":"43984","record_id":"<urn:uuid:95c7859b-4291-484c-9c00-3e01342c2ea8>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00470.warc.gz"}
College Algebra - Online Tutor, Practice Problems & Exam Prep Hey, everyone. So up to this point, we've spent a lot of time talking about graphs. And in this video, we're going to see if we can apply this concept of graphs to this new topic of relations and functions. Now, this topic is often considered confusing when students initially encounter it. But throughout this video, we're going to be going over a lot of different scenarios and examples to see if we can really clear up some of this confusion around this subject. So let's get right into this. Relations are a connection between X and Y values and graphically they are represented as ordered pairs. Now functions are a special kind of relation where each input has at most one output and it's important to note that all functions are relations, but not all relations are functions. So to understand this a little bit better let's take a look at this example that we have down here. We have these two graphs of relations and we want to see if we can determine whether these are also examples of functions. We'll start with this graph on the left, and what I'm going to do is write out all of the inputs which correspond to all of the X values. So I'll do this in ascending order so first I see that we have an X value of negative 2. I also see that we have an X value of positive one and even though it shows up twice here it's perfectly okay to only write it once in this bubble down here. Now lastly I see that we have an X value of 3. So these are all of the inputs that we have. Now the outputs are going to correspond to the Y values, and I'll list all of these out as well. So I see that we have positive 2, I see that we have 4, we also have 1, and then we have negative 2. So these are all of the inputs and outputs. Now looking at these points I see that negative 2 is related to positive 2. I see that positive one is related to 4, and I also see that one is related to 1. Now lastly I see that we have this point which says 3 is related to negative 2. Now based off this relation that we see, can we conclude whether or not it's a function? Well, recall that we set up here for a relation to be a function, each input can have at most one output. And if I look at each of our inputs, I can see that there is an input that has more than one output. And as soon as this happens, you can automatically conclude that this is not an example of a function. But let's take a look at this other example for this graph on the right. So what I'm gonna do with this graph is I'm going to list out all of the inputs in ascending order like we did before. So this will be all the X values, so I see that we have negative 4, I see that we have negative 2, I see that we have 1, and then we have 3. Now what I'm also going to do is list out all of the outputs, which correspond to all the Y values. So I see that we have positive 2, I see that we have negative 1, and I also see that we have positive 2 up here, but since we already wrote positive 2 once we don't have to write it again. So we have positive 2, and then we have positive 4. So these are all of the outputs. Now looking at how these are related I can see that negative 4 is related to negative or to excuse me to positive 2. I see that negative 2 is related to negative 1, and I see that positive 1 is related to positive 2, and then I see that we have that positive 3 is related to positive 4. Now given this information, can we conclude whether or not this is a function? Well, we need to see if any of the inputs have more than 1 output, and if I look at this each of these inputs only go to one output, and because of this, we would say this is an example of a function. So this is how you can tell whether or not a relation is a function, but you may have noticed this process was a bit tedious, having to write out all the inputs(outputs like this. Well, you may be happy to know there is a shortcut to solving these problems, and the shortcut is called the vertical line test. This states that if you can draw any vertical line that passes through more than one point on your graph, then the graph is not going to be a function. So let's try this vertical line test on the two graphs we had up here. I'll take vertical lines and I'll draw them through every point that I see on the graph, and let me draw this vertical line a bit better. So if I draw these vertical lines, I notice that there is a place where the vertical line passes through more than one point. If this ever happens, then it's not a function. But let's try the vertical line test on this other graph. If I draw vertical lines through each of the points that I see, I noticed that no matter where I draw a vertical line, I'm only ever gonna pass through one point at most. And because of this, we would say that this is an example of a function. Now let's take a look at a couple more examples, because you could also use the vertical line tests on graphs like this. If I tried drawing some vertical lines for this graph on the left, I noticed that we will have some vertical lines that pass through more than one point. And because of this, we can conclude that this is not an example of a function. But if I try the same vertical line test on the other graph that we have over here, notice that no matter where I draw a vertical line, we're only ever going to pass through one point at most, and because of this, we can conclude that this graph is an example of a function. So that's the basic idea of relations and functions. Hopefully, this helped you out and let me know if you have any	{"url":"https://www.pearson.com/channels/college-algebra/learn/patrick/functions/intro-to-functions-and-graphs?chapterId=b413c995","timestamp":"2024-11-05T00:12:08Z","content_type":"text/html","content_length":"690840","record_id":"<urn:uuid:20d3d7ec-986e-448b-b6ba-718ce38a63e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00669.warc.gz"}
A scale drawing of a rose is 3 inches long. The actual rose is 1.5 feet long. What is the scale factor of the drawing and what is the scale of the drawing? Please explain!! :D 1. Home 2. General 3. A scale drawing of a rose is 3 inches long. The actual rose is 1.5 feet long. What is the scale fact...	{"url":"https://math4finance.com/general/a-scale-drawing-of-a-rose-is-3-inches-long-the-actual-rose-is-1-5-feet-long-what-is-the-scale-factor-of-the-drawing-and-what-is-the-scale-of-the-drawing-please-explain-d","timestamp":"2024-11-10T14:21:48Z","content_type":"text/html","content_length":"30166","record_id":"<urn:uuid:20786e02-ebfa-4f7a-8320-cf11c1be1107>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00557.warc.gz"}
Learn About Displacing Liquids and Upthrust Applications Have you ever noticed that when you get into a bath, the water level rises? That's because you are displacing the water in the bathtub with your body. Displacing liquids with objects is closely linked to the concept of upthrust, so let's learn more about displacement! Archimedes was a Greek mathematician and scientist, born in 287 B.C. He worked across a wide range of scientific fields, and in one famous story, he was tasked with finding out if a gold crown was made of pure gold or not, as there was no way of finding the volume of this irregularly-shaped object. If the volume was known, the density of the object could be calculated, which would tell Archimedes if the crown was pure gold. The story says that Archimedes got into a bath one day, and noticed the water rise. "Eureka!" he cried out. He had realised that the volume of raised water must be equal to the volume of his body. The volume of the crown was found by placing it in a container of water. The change in volume was measured - this was the crown's volume. Using the volume of the crown and the mass of the crown, Archimedes found its density and proved that it was not pure gold, as the density did not match. While we can't be sure if the whole story is true as it happened so long ago, the principle can still be helpful to this day. This stone is an irregular shape. We can't measure its volume directly. Let's fill a measuring cylinder halfway. The cylinder reads a volume of 50 cm^3 The rock is placed into the water, and it sinks. The rock has displaced the water. The new volume of water is 67 cm^3 What is the volume of the rock? The volume of the rock is the difference between the initial volume and the new volume. 67 cm^3 - 50 cm^3 = 17 cm^3 Let's learn about another example of water displacement and upthrust, and how it is useful. Have you ever thought about how submarines can sink deeper in water, or rise up, on command? Submarines, like any object in a liquid, experience two main forces - their weight acting downward and upthrust acting upward. If the upthrust is less than the weight of the submarine, the submarine will sink. The ballast tank of a submarine, when on the surface, is filled with air. This means that the overall density of the submarine is less than the density of the water and that the weight of the submarine is not greater than the upthrust. When the submarine dives, the ballast tanks open up and they gradually fill with water. The weight of the submarine increases, and because the weight is greater than the upthrust, the submarine begins to sink. The tanks can fill with water, but if the submarine is deep underwater and needs to surface, compressed air from stored tanks in the submarine can get pumped into the ballast tanks. This decreases the weight of the submarine, and the upthrust on the submarine causes it to move up towards the surface of the water. Now let's try some questions!	{"url":"https://www.edplace.com/worksheet_info/science/keystage3/year9/topic/710/13707/upthrust---applications-and-calculations","timestamp":"2024-11-06T06:04:00Z","content_type":"text/html","content_length":"84355","record_id":"<urn:uuid:827a98e2-7232-47cc-8e10-29471462b040>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00810.warc.gz"}
Evaluation of a noninvasive transcranial doppler and blood pressure-based method for the assessment of cerebral perfusion pressure in pregnant women Objective: We have developed a Doppler method for the estimation of cerebral perfusion pressure (CPP) using noninvasive techniques. Our objective was to evaluate our new method in pregnant women. Methods and Materials: Laboring women with a lumbar epidural in situ had transcranial Doppler interrogation of the maternal middle cerebral artery (MCA) to measure systolic, diastolic, and mean velocities. A pressure transducer was connected to the epidural catheter and pressure was recorded. Systolic (SBP), diastolic (DBP), and mean (MAP) blood pressure were taken with a Dinamap monitor. Doppler estimated CPP (mm Hg) = [V(mean)/(V(mean)-V(diastolic)](MAP - DBP) and directly measured CPP = MAP - Epidural pressure data were plotted on a Bland-Altman graph with limits of agreement. The mean difference (the mean of the sum of both positive and negative differences) and absolute difference (the mean of the sum of the absolute differences) were calculated. In addition, linear and polynomial regression analyses were performed. Results: Twenty laboring women were studied. All had normal pregnancies. The mean maternal age was 28 ± 7 years and the mean gestational age was 39 ± 2 weeks. The mean maternal MAP was 77 ± 12 mm Hg. The Bland-Airman plot showed a mean difference of 2.2 mm Hg at a mean CPP of 65 ± 12 mm Hg; with a standard deviation of 4.8 mm Hg, the absolute difference was 3.9 ± 3.0 mm Hg at a mean CPP of 65 ± 12 mm Hg. The regression analysis showed an r = 0.92, r^2 = 0.86, and p < 0.0001. Conclusions: Our formula allows the estimation of CPP using a simple calculation and noninvasively acquired data. This method may be of use for frequent, easy, and accurate CPP and intracranial pressure estimation and may, as such, have significant research and clinical applications. • Brain • Cerebral perfusion pressure • Doppler • Pregnancy ASJC Scopus subject areas • Internal Medicine • Obstetrics and Gynecology Dive into the research topics of 'Evaluation of a noninvasive transcranial doppler and blood pressure-based method for the assessment of cerebral perfusion pressure in pregnant women'. Together they form a unique fingerprint.	{"url":"https://researchexperts-staging.utmb.edu/en/publications/evaluation-of-a-noninvasive-transcranial-doppler-and-blood-pressu","timestamp":"2024-11-06T23:47:14Z","content_type":"text/html","content_length":"56209","record_id":"<urn:uuid:6124dbb8-69b4-4507-b7a7-9288028c72d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00224.warc.gz"}
Corner Triangles for Setting Blocks on Point 1. Corner Triangles for Setting Blocks on Point Corner Triangles for Setting Blocks on Point Tutorial When the math for corner triangles for setting blocks on point is done for you it's easy to add triangles to a square. Get rotary cutting numbers here! What are corner triangles? Corner triangles are half square triangles used for setting or putting blocks on point. This quilting technique enlarges the original block. The technique is also known as square in a square (or how to add triangles to a square). Half square triangles are used so that the straight of grain is on the outside edges of the block. Math for Putting Blocks on Point Determine the finished size of your quilt block. Multiply this number by 1.414. This is the finished size diagonal of the square. Then add .875 inches for seam allowance. Round up to the nearest 1/8 of an inch. This gives you the size of the squares to cut in half along their diagonal to get half square triangles. Here's the formula... • Corner Triangles = finished block size / 1.414 + .875 = X inches • Round X up to the nearest 1/8 of an inch to get Y inches. How to Cut Corner Triangles • Cut two squares the size of Y (given in a chart at end of page). • Cut the two squares in half on the diagonal for a total of four corner triangles. For example, if the block to be set on point is a 4 inch finished size block, cut 2 (3 3/4 inch) squares. Then, cut these two squares in half on the diagonal for a total of four corner triangles (which are really half square triangles). Rounding up to the nearest 1/8 of an inch requires only little, if any, trimming and removal of the dog ears from the final block. Oversized Corner Triangles Cutting squares rounded to the nearest 1/4 of an inch results in an oversized square in a square quilt block that requires trimming. I prefer to make oversized blocks and trim them down. Because we all don't make perfect cuts or sew perfect 1/4 inch seams, the Corner Triangles for Setting Blocks on Point Chart provided after the instructions rounds to both the nearest 1/8 and 1/4 of an inch. You choose how big to cut your squares. Corner Triangles for Setting Blocks on Point Instructions How to Add Triangles to a Square Step 1 In the Corner Triangles for Setting Blocks on Point Chart locate the finished size of block/square to be set on point. Use the corresponding measurement to cut two squares this size. In the photo below, the finished size of my green square is 4 1/2 inches. So I cut 2 (4 1/4 x 4 1/4 inch) oversized corner triangle squares. Step 2 Cut the two corner triangle squares in half along one diagonal. You now have four half square triangles to use as corner triangles to set the block on point. How To Sew A Triangle Corner Step 3 Find the center of the sides of the block to be set on point and mark or fold it in half vertically and finger press a crease into the fold at the top and the bottom of the square. Fold the block in half horizontally and make a crease in the fold. (Folds are highlighted by the blue lines on the square.) Fold each half square triangle in half along its diagonal edge and finger press a crease into the fold. Step 4 With right sides together and matching marks or creases, place a triangle on two opposite sides of the square (or block). Stitch a 1/4 inch seam. Press the seams toward the triangles. Step 5 Repeat Step 4 on the last two sides of the square (or block). You've added corner triangles for setting blocks on point and the block is now a square in a square block. Next, it's time to trim or square up the block. How to Square Up Square In a Square Block Step 6a Position the block so that the inner square is on point. Along the top and right sides of the block, place the 1/4 inch measurement line of a square quilting ruler on the intersection of the inner square's top and bottom corners and the corner triangles (highlighted by the two blue circles). Align the ruler so that the same vertical measurement line (may not be continuous on ruler) also runs through the middle of the inner square at the top and bottom corners (highlighted by the blue Use a rotary cutter to make a cut along the top and right edges of the ruler. This alignment squares up the top and right side of the square or block while also giving a 1/4 inch seam allowance on those sides. Step 6b Rotate the rotate the block 180 degrees and Repeat Step 6a on the remaining two sides. Corner Triangles Rounded to 1/8 vs Rounded to 1/4 Inch Note: When using corner triangles rounded to 1/8 of an inch versus those rounded to 1/4 of an inch you really just need to trim the dog ears. Square In a Square Block Trimmed After adding the corner triangles for setting blocks on point the square in a square, or diamond in a square, the unit is complete! Remember this 'squares on point quilt tutorial' for setting a block on point using corner triangles increases the block size. It adds corners whereas this method of making a square in a square keeps the block size the same by replacing corners. Corner Triangles for Setting Blocks on Point Chart (inches) Rotary Cutting Numbers for Setting Blocks On Point Use this handy chart to determine the size squares you need to cut to set blocks on point. Now you can set that 12 inch quilt block on point! Of course, that's finished size. Just locate the number 12 in Column A. Move laterally across to Column B or C to find its corresponding number – 9 3/8 (Column B rounded to 1/8) or 9 1/2 (Column C rounded to 1/4 inch). Cut two squares this size. Then, cut those squares in half diagonally once, making four half square triangles. Columns B and C are rotary cutting numbers for putting blocks on point. Finished Size of Block Cut 2 Squares Cut 2 Oversized Squares To Be Set On Point Then Cut Once on the Diagonal Then Cut Once on the Diagonal A B C 2 2 3/8 2 1/2 2 1/2 2 3/4 2 3/4 3 3 3 1/4 3 1/2 3 3/8 3 1/2 4 3 3/4 4 4 1/2 4 1/8 4 1/4 5 4 1/2 4 3/4 5 1/2 4 7/8 5 6 5 1/8 5 1/4 6 1/2 5 1/2 5 3/4 7 5 7/8 6 8 6 5/8 6 3/4 9 7 1/4 7 1/4 9 1/2 7 5/8 7 3/4 10 8 8 1/4 12 9 3/8 9 1/2 15 11 1/2 11 3/4 16 12 1/4 12 1/2 18 13 5/8 13 3/4 20 15 1/8 15 1/4 Tips for Cutting Corner Triangles Corner Triangles and Directional Prints If you use directional prints for your corner triangles for setting blocks on point make sure to cut one square on the diagonal in the direction of a back slash (\) and and one square in the direction of a forward slash (/). This results in all lines or motifs going in the same direction when the corner triangles are sewn into a square in a square block. Grain of Fabric Matters When Cutting Triangles Straight Grain Edges: The lengthwise grain (parallel to selvage edge) and crosswise grain (perpendicular to selvage edge) are both considered straight grain (aka straight-of-grain). Stick to cutting your squares from strips that have been cut from the fold edge to the selvage edge and you will be fine. If you are cutting your squares from scraps and you cannot tell the grain, spray the fabric piece with starch or an alternative start solution before cutting squares from it. This should minimize any stretch in the fabric. Subscribe to the Quilt Blocks Digest newsletter for updates, special offers & exclusive content! You will receive an email asking you to confirm your consent to subscribe. You must click on the confirmation link contained in that email in order to be subscribed and receive emails. Your email address is never shared. 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preparing for division Worksheets Looking for the best division worksheets on the internet? Look no further! We've got you covered with our free, printable division worksheets. Our division worksheets range from the basics of visual problems to more complex long division and problems that are relative to multiplication. With our helpful resources, your students will understand key concepts such as multiples and remainders while having fun learning math. Take a look at our selection of free division worksheets today – it's sure to help your students practice and learn with confidence. Browse Sheets By Problem Type Division Relative to Multiplication 3oa6 × Description: "This worksheet is designed to illustrate the connection between multiplication and division in mathematics. It features 20 interactive problems that demonstrate how the two operations are inversely related. Each problem includes both a division and multiplication equation, reinforcing mathematical concepts for children. Ideal for distance learning, this versatile worksheet can be customized according to individual learner needs and even converted into flashcards for convenient and fun learning." × Student Goals: Understand the Concept of DivisionUpon completing the worksheet, students should thoroughly understand the concept of division. They should be able to associate it as the opposite of multiplication and appreciate it as a significant mathematical operation. They should comprehend that division essentially partitions a given quantity into equal parts and understand how this operation is practically helpful in real-life scenarios like sharing, distributing, etc.Master Basic Division ProblemsStudents should become adept at solving basic division problems using the numerical examples provided in the worksheet. They would recognize how different numbers can be divided equally and should be comfortable performing division with single-digit, double-digit and triple-digit numbers. They should also be able to execute divisions containing 0 and understand the rules associated with this.Promote Mental MathCompleting the worksheet promotes mental arithmetic. Consistent practice with these tasks will sharpen the ability to perform mathematical operations mentally without the need for physical aids. This skill is advantageous in various fields of study and helps in improving general cognitive abilities.Link Multiplication with DivisionThe worksheet is designed to highlight the relationship between multiplication and division. By the end, students should be able to link the two operations and understand how one can be reversed into the other. This understanding strengthens their overall mathematical foundation and prepares them for learning more complex concepts.Develop Problem-Solving SkillsWorking through the problems in the worksheet enhances problem-solving skills. Students should be able to approach a mathematical problem, deduce what is required, and identify the best methods or operations required to reach a solution.Enhance Accuracy and SpeedLastly, consistent practice with division problems on the worksheet should improve students' speed and accuracy. Students should be able to solve similar problems more quickly over time and with increased accuracy. As a general skill, this competency is highly beneficial academically and professionally. Division / Multiplication Tables 3oa6 × Description: "This worksheet is designed to enhance children's understanding of multiplication and division through 20 interactive math problems. The dynamic content can be customized to fit individual needs and learning styles. Parents and teachers can leverage this tool for distance learning, convert into flash cards, or use as practice exercises to reinforce fundamental math concepts." × Student Goals: Enhancement of Basic Math SkillsAfter completing the worksheet, students should be able to enhance their basic math skills. The completion of these problems helps in the reinforcement of multiplication and division concepts, an integral part of the basic arithmetic skills. Furthermore, it aids in their mental calculation abilities, which forms the foundation for higher mathematical operations.Problem-Solving Skills DevelopmentWorking through the problems in the worksheet assists students in honing their problem-solving skills. It gives them the chance to apply their knowledge in practical scenarios, devise strategies to solve the problem,and ensure the correct usage of their arithmetical knowledge. This proficiency is applicable beyond mathematics and is beneficial in tackling real-world problems.Understanding of Related Mathematical ConceptsThe worksheet helps students grasping related mathematical concepts. By exploring multiplication and division, they begin to understand the relationship between these two operations as inverses of each other, which is fundamental in math education. This understanding can simplify various mathematical calculations and reasoning tasks in future lessons.Boost in Confidence and Fostering PersistenceSuccesses, even small ones like completing a math worksheet, boost confidence in students. As they solve these problems, their confidence in their math abilities increases, which can drive them to take on further challenges. Additionally, sometimes, they may find problems difficult to solve. Attempting these problems fosters persistence and encourages them not to give up easily, an important life-long skill.Time Management SkillsThe worksheet further develops time management skills. Children learn to gauge their speed as they need to complete a definite number of problems in a set time frame. This skill, which holds substantial life value, is a stepping stone into effectively managing and dividing their time in future academic tasks and assessments.Preparation for Advanced Mathematical TopicsFinally, successful completion of the worksheet lays a strong foundation for more complex mathematical topics. Advanced math areas like algebra and calculus are reliant on a solid understanding of basic concepts such as multiplication and division. Mastery of these concepts through worksheets prepares students for these challenging subjects ahead. Dividing With Numberlines 3oa1 × Description: "This worksheet is designed to help children understand and practice division using a number line. Centered around math, the sheet contains 11 problems such as 6÷3, 24÷8, and 10÷5, and visually represents each division on a customized number line. The interactive format is suitable for both physical and distance learning, and can be converted into flashcards for portable study methods." × Student Goals: Understanding Division ConceptsAfter completing this worksheet, students should have a solid grasp of basic division concepts. They will understand how to divide numbers evenly and will be able to apply the concept to real-world situations. The number line included in the problems will encourage visual learning, allowing students to visualize the process of division with ease.Problem-solving SkillsThis worksheet requires children to solve multiple division problems, thereby refining their problem-solving skills. They have to utilize their logical thinking and reasoning ability to complete the problems outlined in the sheet. They will be able to identify strategies to approach different problems and learn to apply the right strategy to solve them correctly.Mathematical FluencyAfter working through the worksheet, students should experience improved mathematical fluency, especially around division. The worksheet provides repetitive practice that will help enhance their speed and accuracy in division calculations. This repetitive exercise is instrumental in making students adept at overcoming the challenges posed by division calculations.Skills in Using Number LineBy performing division operations using a number line, students will improve their comfort level and proficiency in using it as a mathematical tool. They will understand how to use a number line to solve math problems, which will be valuable in their future learning experience and math-related assignments.Grasping the Relation Between Multiplication and DivisionThe worksheet assists children in understanding the inherent relationship between multiplication and division. After completing it, pupils should be able to view division as the inverse of multiplication and apply this concept when solving more complex problems in their math journey ahead.Confidence in MathSuccessfully completing this worksheet will boost students' confidence in their math abilities. The accomplishment of accurately solving all these problems is bound to make them more enthusiastic and self-assured in handling mathematical problems, thus creating a positive learning environment for future complex Preparing for Long Division 4nbt6 × Description: "This worksheet is designed to aid in preparing children for mastering long division. Consisting of 20 problems, it provides hands-on practice with multiplication-based estimation exercises. Tailored to match specific learning needs, this flexible tool can be readily customized or converted into flash cards. Excellent resource for distance learning, engendering math fluency and comprehension seamlessly." × Student Goals: Master the Concept of Long DivisionAfter successfully completing this worksheet, students should have acquired a deeper understanding of the concept of long division. The problems provided will have enabled them to develop a process-oriented mindset, essential for solving complex long division problems. They will master how to get as close as possible to a target number without exceeding it - a fundamental component of the long division process.Enhance Multiplication SkillsThe worksheet has been designed in a way that indirectly helps students enhance their multiplication skills. Through attempting to achieve a target number through multiplication, students will improve their mental multiplication capabilities and become more familiar with multiplication tables.Develop Problem Solving AbilitiesSolving these problems requires a degree of critical thinking and strategy. As a result, one major accomplishment for students will be the development of their problem-solving abilities. They will become proficient in identifying the closest possible outcome, therefore enhancing their analytical and predictive capabilities.Improve Numeric FlexibilityThe worksheet encourages students to utilize a variety of numbers to resolve a problem rather than rigidly sticking to specific numeric patterns or sequences. As a result, they will improve their numeric flexibility and broaden their range of problem-solving tactics.Build Confidence in Mathematical AbilitiesAs students correctly solve the problems in this worksheet, they will build their confidence in handling math problems. This confidence, coupled with the skills they develop, will be pivotal in tackling more complex mathematical operations in future lessons. Checking Division Answers 4nbt6 × Description: "This worksheet is designed to help children enhance their division skills in math. It consists of 10 problems which require checking the validity of the division performed by solving reverse multiplication and addition. The customizability of this tool allows it to be converted into flashcards, adapted for distance learning, or personalised to cater the individual learning needs of students, promoting an interactive and engaging learning experience." × Student Goals: Understanding DivisionUpon completion of this worksheet, students will have honed their understanding of the critical mathematical operation of division. They will be competent in managing numbers and develop skills in manipulating large numbers with ease. Specifically, they will learn how to divide multi-digit numbers and comprehend the concept of remainders in division. These accomplishments will provide students with necessary arithmetic precision, contributing to a sturdy numerical foundation.Checking Division AnswersThe ability to self-assess one's work is crucial to learning. After working on this worksheet, students will be capable of verifying their own division answers. They will be versed in the multiplication-check method for division problems, enabling them to independently monitor their work for accuracy.Problem SolvingMath is one of the key aspects that foster problem-solving skills. Successfully working through this worksheet will mean that students have improved their ability to solve complex division problems, thus enhancing their problem-solving skills. These achievements will also indirectly contribute to honing their logical thinking.Mathematical ConfidenceAs students complete the exercises and correctly verify their answers, they will grow a sense of mathematical confidence. This self-reliance will enhance their learning experiences going forward and embolden them to approach similar division problems with greater confidence and less apprehension.Critical ThinkingThe exercises in this worksheet require a balance of precision and analytical thinking. As such, upon completion of the worksheet, students will have bolstered their critical thinking skills. They will be able to sift through large numbers and dissect problems to arrive at accurate solutions.Preparation for Advanced MathThrough developing an understanding of division and the capacity to verify answers, students will have taken a major step towards preparing themselves for more advanced mathematical operations and subjects. This foundational knowledge and set of skills is integral for future mathematical success in areas such as algebra, calculus, and beyond. Understanding Division Answers 4oa3 × Description: "This worksheet is designed to help children understand the concept of division. With ten realistic problems, such as distributing candies or carpentry boards equally, children can connect math to real-world scenarios. The worksheet's content can be customized to fit curriculum needs, transformed into flashcards for dynamic learning, or utilized for distance learning programs. It's a versatile tool for teaching elementary division within various contexts." × Student Goals: Practical Application of DivisionStudents will be able to understand and apply the concept of division in practical scenarios. They will be able to comprehend word problems and discern the requirements to solve them using division. The word problems will help students in relating the mathematical operations with real-world situations, enhancing their problem-solving skills.Critical Thinking DevelopmentBy solving these problems, students will be able to build their critical thinking and logical reasoning abilities. The problems require students to interpret the scenario, accurately identify the required calculations, and then decide how to implement those using division. Engaging with these tasks helps in nurturing the students' mathematical abilities and problem-solving acumen.Improved Computation SkillsThe division problems in the worksheet will elevate the students' computation skills, especially related to division. With guided practice, they will become adept in handling division problems involving carrying over remainders. They will understand the process of dividing larger numbers and comprehend the concept of 'remainder' in division.Mastering DivisionAfter completing the worksheet, students should have gained a solid understanding of division as a mathematical operation. They will be able to handle division operations confidently and with a deep understanding of its underlying principles. The tasks will teach them how division is utilized in daily life situations to distribute quantities equally.Enriching Math VocabularyBy working on the word problems, the students will be able to enhance their mathematical vocabulary. They will familiarize themselves with terms like 'divide', 'equally', 'remainder', which are integral to understanding and solving division problems. This will aid them in easily grasping and articulating mathematical problems in future.Confidence BoostSuccessfully solving these division problems will instill confidence in students about their mathematical prowess. It will encourage them to tackle more complex problems as they progress in their learning journey. The sense of fulfilment from being able to solve these problems would motivate the students to further immerse themselves in learning and understanding mathematics. Finding Division Remainders × Description: "This worksheet is designed to assist children learning division with examples and practical problems. Entitled 'Finding Division Remainders', it provides 20 math problems that clearly illustrate how to find the remainder in division. The content can be customized according to learner requirements. In addition, it can easily be converted into flash cards or used in distance learning programs, making it a versatile educational tool." × Student Goals: Master Division SkillsUpon completing this worksheet, students should have significantly improved their core division skills. They will understand the notion of remainders, not only by definition, but also how they affect the outcome of division operations. This vital mathematical concept will be reinforced, bolstering their overall arithmetic ability and mathematical reasoning skills.Problem-Solving AbilityThis practical worksheet serves as a platform for students to enhance their problem-solving skills. By tackling various division problems, they learn how to strategically approach mathematical challenges, formulate solutions and effectively execute them. These are essential abilities that will aid them not only in math but across other academic subjects and real-life situations in the future.Build ConfidenceSuccessfully finishing this worksheet has the potential to boost students' confidence significantly. Familiarizing themselves with the process of calculating division remainders will give them confidence when encountering similar problems in the future. By overcoming the challenges provided in the worksheet, students will feel more self-assured in their mathematical prowess, which will likely lead to better engagement in math-related tasks in the future.Enhance Math FluencyWorking on this worksheet helps students develop math fluency. Getting accustomed to the structure and operations of division allow students to perform similar tasks faster and more efficiently in the future. The worksheet supports the students' ability to become fluent in the language of mathematics, which is a critical step in advancing in this subject.Logical Reasoning and Critical ThinkingUpon completing the worksheet, students should find an improvement in their logical reasoning and critical thinking capabilities. The tasks required to find solutions foster logical thinking and knowledge application processes, aiding in the progression of these primary cognitive abilities.Preparation for Advanced Tasks Finally, this worksheet serves as essential groundwork for more complex mathematical tasks. A good grasp of division and understanding of remainders are fundamental for progressive topics such as fractions, percentage calculations, and algebra. Hence, the knowledge gained through this worksheet helps prepare students for future mathematical challenges. Division as Repeated Subtraction (Number Line) × Description: "This worksheet is designed to enhance a child's division skills, specifically using the method of repeated subtraction via number lines. It contains a set of 10 diverse math problems that vary in complexity. The worksheet is versatile allowing customization and conversion into flash cards, making it effective for distance learning. It's an excellent tool for reinforcing fundamental mathematical knowledge while promoting independent learning." × Student Goals: Division as Repeated Subtraction w/Remainder (Number Line) × Description: "This worksheet is designed to help children understand the concept of division as repeated subtraction using number lines. Featuring 10 problems, the worksheet tackles math problems like 45÷6, 43÷5 and more. It's a customizable learning tool that can be converted into flash cards for convenience. Perfect for distance learning, this worksheet makes mastering division easier and enjoyable for kids." × Student Goals: Understanding Division as Repeated SubtractionOn completing this worksheet, students must have developed a clear understanding of division as an operation of repeated subtraction. They should be able to apply this concept to divide various numbers, using subtraction repeatedly until no remainders are left or a remainder that’s less than the divisor.Application of Number LineThe worksheet builds student understanding of how the number line can be used as a tool to solve division problems. By marking equal jumps on the number line to represent iterative subtraction, they should gain the skills needed to physically visualize the process of division and understand the concept of remainders.Handling RemaindersThe worksheet bears problems where the division does leave a remainder. By attempting such problems, students should comprehend what it means when a division problem leaves a remainder and how to interpret that with respect to the number line. Such knowledge becomes vital in future topics like decimal and fraction division.Enhancing Problem Solving SkillsCompleting the worksheet aids in enhancing the students' problem-solving abilities. They'll learn to decode the problems, understand what's being asked, and find ways to arrive at the solution. This skill is not just limited to division problems but can extend to more complex mathematical problems in the future.Development of Mathematical FluencyAfter completing this worksheet, students should be able to perform basic division calculations quickly and accurately. The repeated practice will help to establish a solid mathematical foundation and increase their fluency in basic arithmetic operations, enabling them to undertake more complicated mathematics with confidence.Setting the Stage for Advanced MathematicsThe skills learned through this worksheet are essential for understanding and doing well in higher-level mathematics. As division is a fundamental operation in mathematics, mastering it will provide students with an essential tool for tackling complex problems in algebra, geometry, number theory, and more advanced fields of study.	{"url":"https://www.commoncoresheets.com/division-worksheets/sbh/preparing-for-division","timestamp":"2024-11-06T08:18:39Z","content_type":"application/xhtml+xml","content_length":"116790","record_id":"<urn:uuid:e57cf684-3944-458c-8878-86c9479092c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00185.warc.gz"}
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RETRACTED: Computational Fluid Dynamics Modeling of the Catalytic Partial Oxidation of Methane in Microchannel Reactors for Synthesis Gas Production Department of Energy and Power Engineering, School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo 454000, China Author to whom correspondence should be addressed. Submission received: 27 May 2018 / Revised: 28 June 2018 / Accepted: 29 June 2018 / Published: 30 June 2018 This paper addresses the issues related to the favorable operating conditions for the small-scale production of synthesis gas from the catalytic partial oxidation of methane over rhodium. Numerical simulations were performed by means of computational fluid dynamics to explore the key factors influencing the yield of synthesis gas. The effect of mixture composition, pressure, preheating temperature, and reactor dimension was evaluated to identify conditions that favor a high yield of synthesis gas. The relative importance of heterogeneous and homogenous reaction pathways in determining the distribution of reaction products was investigated. The results indicated that there is competition between the partial and total oxidation reactions occurring in the system, which is responsible for the distribution of reaction products. The contribution of heterogeneous and homogeneous reaction pathways depends upon process conditions. The temperature and pressure play an important role in determining the fuel conversion and the synthesis gas yield. Undesired homogeneous reactions are favored in large reactors, and at high temperatures and pressures, whereas desired heterogeneous reactions are favored in small reactors, and at low temperatures and pressures. At atmospheric pressure, the selectivity to synthesis gas is higher than 98% at preheating temperatures above 900 K when oxygen is used as the oxidant. At pressures below 1.0 MPa, alteration of the dimension in the range of 0.3 and 1.5 mm does not result in significant difference in reactor performance, if made at constant inlet flow velocities. Air shows great promise as the oxidant, especially at industrially relevant pressure 3.0 MPa, thereby effectively inhibiting the initiation of undesired homogeneous reactions. 1. Introduction There has been an increasing interest in the production of synthesis gas through the catalytic partial oxidation of methane [ ], due to its potential applications in many fields such as fuel cells [ ] and gas turbines [ ]. Currently, the primary techniques used in industry to produce synthesis gas from methane are steam reforming [ ], autothermal reforming [ ], and partial oxidation [ ]. Steam reforming of methane remains the main commercial process for the production of synthesis gas [ ]. It is important to develop new reaction routes for the production of synthesis gas from methane. A promising reaction route that has received much attention recently is the catalytic partial oxidation of methane in short contact time reactors in high temperature environment [ ], where a high yield of synthesis gas (higher than 90%) can be achieved [ ]. In comparison with other synthesis gas production routes, this technology shows great promise because higher selectivity and better efficiency can be achieved [ Significant progress has been made recently in the understanding of the mechanism of this reaction [ ]. The reaction proceeds at a lower temperature than the gas-phase partial oxidation route [ ], thus offering many advantages such as a reduction of undesired by-products and the ability to control the process temperature [ ]. The catalysts used for this reaction usually contain group VIII transition metals, such as rhodium [ ], ruthenium [ ], platinum [ ], palladium [ ], nickel [ ], iridium [ ], and cobalt [ ]. The reaction can proceed in short contact time reactors under high-temperature conditions [ ]. This technology provides a novel route for the production of synthesis gas from methane [ ], since the yield of the desired products can be greatly improved under controlled conditions [ It is important to understand the mechanism of the catalytic partial oxidation reaction to improve the yield of synthesis gas [ ]. However, the pathways for the reaction are still debated, and the prospect for industrial applications is not yet clear. In the case of noble metal catalysts, both heterogeneous and homogeneous reaction pathways may be significant [ ]. In addition, homogeneous reactions decrease the yield of synthesis gas, thus seriously hindering reactor performance. It is therefore necessary to understand the competition phenomenon between the two reaction pathways during a catalytic partial oxidation process. Unfortunately, the relative contribution of the two reaction pathways to the formation of synthesis gas has not yet been addressed, and thus further research is needed to clarify the mechanism responsible for improving the yield of synthesis gas in a catalytic partial oxidation system. Microreactor technology is expected to offer many advantages for process development [ ], especially for fast, exothermic reactions such as catalytic partial oxidation. Precise temperature control is possible, thus significantly reducing the undesirable side reactions occurring during a catalytic partial oxidation process [ ]. Furthermore, higher yields can be achieved for these processes under well-controlled conditions, by taking advantage of enhanced transport in small dimensions [ ]. For partial oxidation micro-chemical systems, one of the engineering design challenge is to balance the gains made in heat and mass transfer by going to smaller dimensions against the increases in pressure drop [ ]. However, microfabrication methods have the potential to realize reactor designs that combine excellent thermal uniformity, enhanced transport rates, and low-pressure drop for a catalytic partial oxidation process [ ]. The intrinsic kinetics of these partial oxidation processes are typically very fast, and thus the realization of this process technology will require continued advances in the development of microreactors [ ]. The small dimensions associated with these reactors can effectively inhibit the gas-phase reaction occurring during a catalytic partial oxidation process [ ]. It is therefore important to determine the favorable operating conditions under which the yield of desired products can be maximized. In addition to providing an explanation of experimental data, numerical simulations can serve as an efficiency design tool for the development of a catalytic partial oxidation system [ ]. Iterative, costly experimental design processes can be avoided. Furthermore, numerical simulations are necessary to better understand the operating characteristics of a micro-structured device required to implement a catalytic partial oxidation process, and to evaluate the disadvantages and benefits associated with an innovative design of the process [ ]. A number of commercial software tools are available, but, unfortunately, none of them is universally applicable. To accurately predict the operating characteristics during the process of a catalytic partial oxidation reaction occurring in microreactors and accurately reflect experimental observations, detailed mathematical modeling is often necessary [ ]. Detailed computational fluid dynamics modeling can be used to evaluate design changes, such as geometric parameters, Reynolds numbers, and reaction temperature, during a catalytic partial oxidation process [ The main focus of this paper is on determining the favorable operating conditions for the small-scale production of synthesis gas from the catalytic partial oxidation of methane. High transport rates are possible in microchannel reactors, thus allowing the catalytic partial oxidation reaction to be carried out under more favorable conditions. Computational fluid dynamics simulations serve as a means to understand the role of heterogeneous and homogenous reaction pathways in determining the distribution of reaction products. The effect of reactor dimension, pressure, mixture composition, and preheating temperature was investigated to better understand the operating characteristics of the partial oxidation reactor. The favorable conditions for the production of synthesis gas were determined. The major objective is to understand the relative importance of different reaction pathways in determining the distribution of reaction products. Special emphasis is placed on identifying favorable operating conditions for the production of synthesis gas in high temperature environments. 2. Model Development 2.1. Reaction System The reaction system used in the present investigation is the catalytic partial oxidation of methane taking place in a microchannel reactor. For microchannel reactors, the width of each of the channels is about one order of magnitude larger than its height [ ], and thus the reactor used in this paper is modeled as a two-dimensional system. A premixed methane and either air or oxygen mixture is fed to the reactor, after the reactant stream is preheated to a desired temperature. The reactor contains multiple parallel channels having sub-millimeter dimensions, thus offering advantages from enhanced heat and mass transfer [ ]. The reactor modeled in this paper is shown schematically in Figure 1 . Unless otherwise specified, the reactor consists of two infinitely wide parallel plates of length 8.0 mm, separated by a gap distance 0.8 mm between the two plates. The physical properties the walls are the same as those of stainless steel. The rhodium catalyzed partial oxidation of methane is considered in the present work, since the catalyst has been reported to give a high synthesis gas yield with good long-term stability [ ]. Additionally, a “base case”, where typical operating conditions and design parameters are considered for the catalytic partial oxidation process, is given in Table 1 . In this context, the effect of various operating conditions and design parameters can be easily evaluated. Numerical simulations are carried out, and the difference between a methane–oxygen system and a methane–air system is also investigated. Note that most of the catalyst properties listed in Table 1 are taken from the works related to the reaction mechanism used; the reaction mechanism used in this paper will be described in detail in Section 2.3 Reaction mechanisms . The wash coat thickness is specified based on the reaction system considered. 2.2. Mathematical Model Since the Reynolds number is less than 380, which implies laminar flow. Therefore, it is possible to fully characterize heat and mass transfer in the system. Detailed reaction mechanisms are necessary to accurately predict the partial oxidation process. A two-dimensional numerical model is developed by using the commercial software ANSYS Fluent Release 16.0 (ANSYS Inc., Canonsburg, PA, USA) [ ]. Detailed reaction mechanisms are handled with related external procedures; please refer to Section 2.3 Reaction mechanisms for more details. The steady-state two-dimensional conservation equations are solved in the gas phase: $∂ ( ρ u ) ∂ x + ∂ ( ρ v ) ∂ y = 0$ $∂ ( ρ u u ) ∂ x + ∂ ( ρ v u ) ∂ y + ∂ p ∂ x − ∂ ∂ x [ 2 μ ∂ u ∂ x − 2 3 μ ( ∂ u ∂ x + ∂ v ∂ y ) ] − ∂ ∂ y [ μ ( ∂ u ∂ y + ∂ v ∂ x ) ] = 0$ $∂ ( ρ u v ) ∂ x + ∂ ( ρ v v ) ∂ y + ∂ p ∂ y − ∂ ∂ x [ μ ( ∂ v ∂ x + ∂ u ∂ y ) ] − ∂ ∂ y [ 2 μ ∂ v ∂ y − 2 3 μ ( ∂ u ∂ x + ∂ v ∂ y ) ] = 0$ $∂ ( ρ u h ) ∂ x + ∂ ( ρ v h ) ∂ y + ∂ ∂ x ( ρ ∑ k = 1 K g Y k h k V k , x − λ g ∂ T ∂ x ) + ∂ ∂ y ( ρ ∑ k = 1 K g Y k h k V k , y − λ g ∂ T ∂ y ) = 0$ $∂ ( ρ u Y k ) ∂ x + ∂ ( ρ v Y k ) ∂ y + ∂ ∂ x ( ρ Y k V k , x ) + ∂ ∂ y ( ρ Y k V k , y ) − ω ˙ k W k = 0 , k = 1 , … , K g .$ The diffusion velocity vector is given as follows [ $V → k = − D k , m ∇ [ ln ( Y k W ¯ W k ) ] + [ D k T W ρ Y k W ¯ ] ∇ ( ln T )$ The ideal gas equation of state is given by The caloric equation of state is given by $h k = h k o ( T o ) + ∫ T o T c p , k d T .$ The coverage equation of surface species can be expressed as $ϑ m s ˙ m Γ = 0 , m = K g + 1 , … , K g + K s .$ The steady-state energy equation in the walls is given by $∂ ∂ x ( λ s ∂ T ∂ x ) + ∂ ∂ y ( λ s ∂ T ∂ y ) = 0$ The gaseous species equation at each of the fluid-washcoat interfaces is specified by the boundary condition taken the form $( ρ Y k V k , y ) i n t e r f a c e + η F c a t / g e o W k ( s ˙ k ) i n t e r f a c e = 0 , k = 1 , … , K g .$ The catalyst/geometric surface area, , is defined as follows [ $F c a t / g e o = A ′ c a t a l y s t A ′ g e o m e t r i c$ The effect of diffusional limitation in the catalyst wash coat may be significant [ ], and thus included in the model $η = s ˙ i , e f f s ˙ i = tanh ( Φ ) Φ$ $Φ = δ c a t a l y s t ( s ˙ i γ D i , e f f C i , i n t e r f a c e ) 0.5$ $γ = F c a t / g e o δ c a t a l y s t .$ The effective diffusivity can be written as $1 D i , e f f = τ p ε p ( 1 D i , m o l e c u l a r + 1 D i , K n u d s e n )$ The Knudsen diffusivity is defined as $D i , K n u d s e n = d p o r e 3 8 R T π W i .$ The energy equation at each of the fluid wash coat interfaces is specified by the boundary condition taken the form $q ˙ r a d − λ g ( ∂ T ∂ y ) i n t e r f a c e − + λ s ( ∂ T ∂ y ) i n t e r f a c e + + ∑ k = 1 K g ( s ˙ k h k W k ) i n t e r f a c e = 0 .$ For the total heat loss to the surroundings, the equation can be written as $q = h o ( T w , o − T a m b ) + ε F s − ∞ σ ( T w , o 4 − T a m b 4 )$ The external heat loss coefficient, , is assumed to be 20 W/(m ·K) [ 2.3. Reaction Mechanisms Computational fluid dynamics modeling of the catalytic partial oxidation process is complex [ ], especially when the role of reaction pathway needs to be determined [ ]. Therefore, detailed reaction mechanisms are included in the model. The possible reactions involved in the catalytic partial oxidation process are listed as follows: Partial oxidation $CH 4 + 1 2 O 2 → CO + 2 H 2 , Δ r H m Θ ( 298.15 K ) = − 35.7 kJ ⋅ mol − 1$ $CH 4 + O 2 → CO + H 2 + H 2 O , Δ r H m Θ ( 298.15 K ) = − 278 kJ ⋅ mol − 1$ Total oxidation $CH 4 + 2 O 2 → CO 2 + 2 H 2 O , Δ r H m Θ ( 298.15 K ) = − 802.3 kJ ⋅ mol − 1$ Competition between the two reaction routes is responsible for the distribution of reaction products. However, the highly exothermic total oxidation reaction serves as a heat source to ensure self-sustained operation of the system. Therefore, this reaction route can improve the selectivity towards the desired products. However, the products produced by this route decrease the selectivity. Overall, the final yield of synthesis gas is determined by this competitive effect. Steam reforming $CH 4 + H 2 O → CO + 3 H 2 , Δ r H m Θ ( 298.15 K ) = + 206.2 kJ ⋅ mol − 1$ Water–gas shift reaction $CO + H 2 O → CO 2 + H 2 , Δ r H m Θ ( 298.15 K ) = − 41.2 kJ ⋅ mol − 1 ,$ Dry reforming $CH 4 + CO 2 → 2 CO + 2 H 2 , Δ r H m Θ ( 298 . 15 K ) = 246.9 kJ ⋅ mol − 1$ The reaction mechanism has attracted increasing attention recently [ ]. The reaction may proceed through a combination of direct partial oxidation and steam reforming [ ]. Both partial and total oxidation products can be formed [ ], and carbon dioxide has little or no role during the process [ ]. More importantly, both water-gas shift and carbon dioxide reforming do not contribute to the formation of synthesis gas [ The detailed heterogeneous reaction mechanism developed by Schwiedernoch et al. [ ], as given in Table 2 , is included in the model. The mechanism consists of 11 surface-adsorbed species and 6 gaseous species involved in 38-step elementary reactions. Since each of the reactive intermediates and elementary reaction steps involved in the catalytic partial oxidation process is included in the reaction mechanism, the global reactions such as steam and carbon dioxide reforming are automatically accounted for [ ]. Note that the symbol * used in Table 2 denotes an adsorbed species or an empty site. The Leeds methane oxidation mechanism [ ] is included in the model to describe the reaction taking place in the gas phase. The reaction mechanism is also accounted for the free radical coupling reactions such as oxidative coupling of methane. The selectivity to C -hydrocarbons may be as high as 20% for the reaction proceeded under certain conditions, through a gas-phase reaction route [ ]. These C -hydrocarbons are undesirable during the catalytic partial oxidation process, as they can cause the problem related to the formation of coke. The CHEMKIN transport database [ ] is used in the model. The homogeneous and heterogeneous reaction rates are handled through the CHEMKIN [ ] and Surface-CHEMKIN [ ] interfaces, respectively. 2.4. Computation Scheme An orthogonal staggered grid is used for the base case, consisting of 200 axial nodes by 80 transverse nodes. Typical fluid node spacing near the catalyst wash coat is 40 μm in the axial direction and 5 μm in the transverse direction. For the largest reactor dimension, a grid consisting of 20,000 nodes in total is utilized. Adequate grid resolution is verified by doubling the number of grid Figure 2 shows the profiles of the hydroxyl radical concentration along the centerline between the two parallel plates for some of the grids used for the methane–oxygen system. The inlet pressure is 3.0 MPa. The rest of the parameters used here are listed in Table 1 . As the grid density increases, there is a convergence of the solution. The numerical model with the coarsest grid, consisting of 4000 nodes in total, cannot accurately capture the hydroxyl radical concentration within the channel and its peak location, and thus fails to accurately predict the onset of gas-phase combustion within the system. In contrast, the solution obtained by the numerical model with a grid consisting of tens of thousands of nodes is reasonably accurate for the base case given in Table 1 . There is little or no advantage for the largest grid density, up to 32,000 nodes in total. The physical properties of the mixture depend on the local conditions of component and temperature. The physical properties the walls, such as the thermal conductivity and specific heat capacity, depend on the local temperature. The conservation equations are discretized by using a finite-volume method. The momentum, energy, and species equations are discretized by using a two-order upwind approximation. The pressure-velocity coupling is discretized using the “SIMPLE” method. The convergence criterion is 10 by examining the values of the residuals for all of the conservation equations. Convergence of the solution is usually difficult due to the inherent stiffness of the detailed reaction mechanism used. Figure 3 shows the residuals for the conservation equations at the end of each solver iteration. The residual plot indicates that after approximately 800 iterations, the convergence criterion is satisfied. 2.5. Model Validation In order to verify the model, the experimental results reported in the literature [ ] are utilized. The reactor used is made of a quartz tube, and the inside diameter of the reactor is 18 mm, as described in the literature [ ]. The inlet temperature of a methane and oxygen mixture is 20 °C, and the pressure is maintained at 0.12 MPa at the outlet. Nitrogen dilution is 30%, and the inlet flow rate is maintained at 5000 cm /min. A grid consisting of 36,000 nodes in total is used here. Numerical simulations are performed for the experimental case tested under the operating conditions and design parameters given in the literature [ ]. The results obtained for the selectivity and the outlet conversion for the mixture with different compositions is compared to the experimental data in Figure 4 . The maximum difference between the numerical results and the experimental data is about 5.6%. Therefore, the numerical results are in consistent with the experimental data. 3. Results and Discussion In the following sections, the reactor performance in terms of conversion, selectivity, and temperature under various operating conditions is discussed in detail, and the relative importance of different reaction pathways in determining the distribution of reaction products is investigated. Additionally, comparisons are made in terms of reactor performance between the results obtained for a methane–oxygen system and for a methane–air system. 3.1. Base Case For the base case given in Table 1 , a methane-to-oxygen molar ratio of 2.0, i.e., a stoichiometric mixture for the production of synthesis gas, is used. This ratio is ideal for the downstream processing such as in the synthesis of methanol and in the production of Fischer-Tropsch products. The temperature is set as 300 K at the inlet for the base case, thus effectively avoiding gas-phase combustion; please refer to the Section 4 Further Discussion for more details. Figure 5 shows contour plots of the methane and carbon monoxide concentrations and temperature within the fluid in the methane–oxygen system. Despite the small dimension involved, the temperature and species gradients change significantly in the vicinity of reaction region. This can be attributed to the difference in the time scale between the heterogeneous reaction and the heat transfer in the transverse direction. The significant change in both temperature and species gradients may need to use a two-dimensional computational fluid dynamics model, as axial diffusion of species and energy is not negligible. Furthermore, mass-transfer limitations, typical characteristics of a catalytic partial oxidation reaction, are observed here, despite the small dimension involved. Accordingly, the effect of transport phenomena will be discussed in the following sections. 3.2. Effect of Preheating for Oxygen Feed Figure 6 shows the influence of preheating temperature on the performance of the methane–oxygen system. As the pressure increases, there is a sharp drop in the selectivity to synthesis gas ( Figure 6 a,b) and conversion ( Figure 6 c), but a sharp rise in wall temperature ( Figure 6 d). This sharp drop implies the initiation of the total oxidation reaction occurring in the gas phase. After the initiation of gas-phase combustion, the contribution of heterogeneous reactions is still considerable, as indicated by the selectivity to synthesis gas at high pressures ( Figure 6 a,b), but there is lack of oxygen for the catalytic partial oxidation reaction. At atmospheric pressure, the selectivity to synthesis gas ( Figure 6 a,b) and the conversion ( Figure 6 c) increase with increasing preheating temperature. Very high selectivity to synthesis gas (>98%) is possible at atmospheric pressure when the preheating temperature is above 900 K. At atmospheric pressure, the main product is synthesis gas, and the catalytic partial oxidation reaction is favored at high temperatures. On the other hand, gas-phase combustion is favored at high temperatures, at which the initiation of the combustion reaction is possible at lower pressures. For example, as the inlet temperature increases from 300 to 900 K, the initiation pressure decreases from about 2.5 to 0.7 MPa. At high pressures, the catalytic partial oxidation reaction is favored at low temperatures. The situation is the reverse of the results obtained at atmospheric pressure. In all of the cases examined here, the selectivity to synthesis gas ( Figure 6 a,b) and the conversion ( Figure 6 c) decrease with increasing pressure. After the initiation of gas-phase combustion, however, the pressure has little or no effect on the conversion. Furthermore, the loss in conversion at between atmospheric pressure and the highest pressure is almost the same. On the other hand, as the pressure increases, there is a transition of primary reaction pathway from catalytic partial oxidation to gas-phase combustion, as depicted by the inflection point in the conversion profile ( Figure 6 c). After the reactants have been ignited in the gas phase, there is a significant drop in conversion. This is due to lack of oxygen for the gas-phase combustion reaction, despite the fact that a stoichiometric methane–air mixture is used for the production of synthesis gas from the catalytic partial oxidation reaction. Please refer to the stoichiometric coefficients described in the chemical Equations (20)–(22) for more details. 3.3. Effect of Preheating for Air Feed One of the major challenges during the catalytic partial oxidation process is a reduction of the cost of pure oxygen separation [ ], since the production of pure oxygen is highly expensive. The reaction system in the presence of a catalyst can be operated at much milder conditions as compared to the gas-phase partial oxidation system, which can effectively inhibit the formation of nitrogen oxides in the gas phase, making it possible to use air instead of pure oxygen [ ]. This shows great promise for the catalytic partial oxidation process. Figure 7 shows the effect of preheating temperature on the performance of the methane–air system. The pressure has a small effect on the selectivity to synthesis gas, but with a tendency to shift towards a higher yield of synthesis gas as the pressure decreases, as shown in Figure 7 a,b Additionally, high preheating temperatures tend to improve the yield of synthesis gas. Therefore, the production of synthesis gas is favored at low pressures and high temperatures. In this context, the outlet concentration of carbon dioxide is very low ( Figure 7 a), but the small amount of carbon dioxide must be removed to meet the downstream processing requirements [ ]. The amount of the total oxidation products and C -hydrocarbons increases with increasing pressure, leading to a decrease in both conversion and the selectivity to synthesis gas. On the other hand, the conversion is favored at high temperatures and at low pressures, as shown in Figure 7 c. In contrast, the maximum wall temperature increases with increasing pressure, especially at high preheating temperatures ( Figure 7 d) where the total oxidation reaction is favored. In comparison with the results obtained for the methane–oxygen system ( Figure 6 ), the initiation of gas-phase combustion is impossible for the methane–air system, and the contribution of homogeneous reactions is small. 3.4. Effect of Reactor Dimension for Oxygen Feed The reactor dimension can significantly affect the effect of mass transfer [ ]. For the system examined here, it is unclear whether there is an optimal dimension in which the yield of synthesis gas can be maximized. To provide a way to reduce the contribution of homogeneous reactions, the effect of reactor dimension is investigated. Figure 8 presents the results obtained for the selectivity, conversion, and maximum wall temperature at different dimensions of the methane–oxygen system. The selectivity to synthesis gas is favored in smaller reactors, as shown in Figure 8 a,b. The dimension has little effect on the reactor performance at low pressures, where the system is operated in a surface kinetically-controlled regime and the yield of synthesis gas is excellent. At high pressures, however, both mass-transfer limitations and homogeneous reactions become significant. For the smallest reactor studied, the dominant chemistry is heterogeneous at all of the pressures examined. At the highest pressure examined, the selectivity to synthesis gas is quite high in the smallest reactor, but drops sharply in the largest reactor. Furthermore, the yield of synthesis gas is favored in smaller reactors, in which the dominant chemistry is heterogeneous, as discussed above. Therefore, the design shows great promise for the production of synthesis gas, but only at low pressures. To achieve a high yield of synthesis gas at high pressures, the reactor dimension must be carefully designed to reduce the mass-transfer limitations. Figure 8 a,b also shows the selectivity to the total oxidation products. For the largest dimension examined here, the amount of the total oxidation products is considerable, especially at high pressures, as shown in Figure 8 a,b. In this context, there is an increases in temperature due to the heat released by the total oxidation reaction Figure 8 d, thus increasing the amount of undesired by-products such as C -hydrocarbons ( Figure 8 c). Therefore, the contribution of homogeneous reactions at high pressures is considerable for the largest dimension examined. After the initiation of gas-phase combustion, there is a sharp drop in conversion for the moderate to large reactors studied, as shown in Figure 8 d. The inflection point of the pressure decreases with increasing the reactor dimension. The maximum wall temperature increases with increasing pressure. This is mainly due to the increased amount of the reactants. A sharp rise in temperature represents the initiation of gas-phase combustion, as shown in Figure 8 d. Finally, the maximum wall temperature levels off at high pressures. Overall, the yield of synthesis gas is favored at low pressures for the methane–oxygen system. 3.5. Effect of Reactor Dimension for Air Feed As discussed above, the dimension can significantly affect the performance of the methane–oxygen system operated at high pressures. To gain a better understanding of the design described in this paper, the effect of reactor dimension is investigated for the methane–air system operated at various pressures. Figure 9 presents the results obtained for the selectivity, conversion, and maximum wall temperature at different reactor dimensions. The reactor dimension has little or no effect on the selectivity to synthesis gas ( Figure 9 a,b) under the conditions studied here. The production of C -hydrocarbons is favored in lager reactors ( Figure 9 c). On the other hand, the reactor dimension has little effect on the conversion (left vertical axis, Figure 9 d), and the maximum wall temperature (right vertical axis, Figure 9 d). As expected, the dominant chemistry is heterogeneous in all of the cases under the conditions studied here, and the initiation of gas-phase combustion is impossible in the methane–air system 3.6. Effect of Nitrogen Diluent For the design examined, the onset of gas-phase combustion can be inhibited by utilizing a methane–air system ( Figure 7 Figure 9 ), while a methane–oxygen system ( Figure 6 Figure 8 ) will allow the gas mixture to be ignited at a certain pressure, as discussed above. It is therefore important to determine the critical dilution to reduce the contribution of the reaction occurring in the gas phase. Figure 10 shows the influence of nitrogen diluent on the performance of the reactor operated at preheating temperature 700 K. The contribution of homogeneous reactions is small, but considerable under certain conditions. At high pressures, the methane–air system has the advantage of reducing undesired by-products and improving the selectivity to synthesis gas, as shown in Figure 10 a,b. At the highest pressure examined, there appears to be an optimal dilution ratio, of about 38% nitrogen in the mixture, which exhibits the maximum yield of synthesis gas. The methane–air system suffers a slight drop in conversion ( Figure 10 d), but offers an economical solution to the production of synthesis gas ( Figure 10 a,b). At moderate pressure 1.5 MPa, only 8% nitrogen diluent is needed to avoid the initiation of the combustion reaction occurring in the gas phase. At atmospheric pressure, the contribution of homogeneous reactions is rather small, and thus nitrogen dilution has little or no effect on the yield of synthesis gas. At moderate to high pressures, there exists a sharp rise in the yield of synthesis gas, as shown in Figure 10 . The inflection point of the nitrogen diluent increases with increasing pressure. For the methane–air system, the yield of synthesis gas also increases with increasing pressure. For the system operated at moderate to high pressures with a low dilution, the initiation of gas-phase combustion is possible, and both the amount of C -hydrocarbons and the maximum wall temperature increases with increasing pressure, as shown in Figure 10 3.7. Air Feed Versus Oxygen Feed Comparisons are made between the results obtained for the two systems operated at atmospheric pressure. Figure 11 shows the influence of preheating temperature on the performance of the two systems operated at atmospheric pressure. Both the selectivity to synthesis gas ( Figure 11 a,b) and the outlet conversion ( Figure 11 c) are higher for the methane–oxygen system than for the methane–air system, especially at lower temperatures. For both systems, the production of synthesis gas is favored at high preheating temperatures, but the methane–air system can benefit more from preheating, as shown in Figure 11 . Therefore, the effect of preheating is more pronounced for the methane–air system. For each of the two systems, the maximum wall temperature increases with increasing preheating temperature under the conditions examined here ( Figure 11 c). As expected, the maximum temperature within the walls is higher for the methane–oxygen system than for the methane–air system. 4. Further Discussion The results presented above indicate that the design shows great promise for the production of synthesis gas from methane. When the dominant chemistry is heterogeneous, the effect of pressure is small. In contrast, when the gas-phase chemistry is considerable, the pressure can significantly affect the reactor performance such as the yield of synthesis gas. In this context, the initiation of gas-phase combustion is possible, the temperature is high, the amount of undesired by-products is relatively large, and the out conversion is low, as shown in Figure 8 . For the base case, a stoichiometric mixture for the production of synthesis gas is utilized. In this context, homogeneous reactions can lead to a lower outlet conversion, because they tend to consume more amount of oxygen than heterogeneous reactions, which leads to lack of oxygen. At the highest pressure examined, it may be difficult to optimize operating conditions of the methane–oxygen system. This is because high preheating temperatures increase the outlet conversion, but decrease the selectivity to synthesis gas, as shown in Figure 6 . Furthermore, the selectivity to synthesis gas is not high enough, at all of the preheating temperatures examined for the methane–oxygen system. The selectivity to synthesis gas decreases with increasing the preheating temperature, as shown in Figure 6 . However, smaller reactors show great promise, since they can delay the initiation of gas-phase combustion, as shown in Figure 8 At high pressures, the methane–air system has the advantage of reducing the contribution of homogeneous reactions, as shown in Figure 6 Figure 7 . At low pressures, the dimension has little or no effect on the performance of each of the two systems, as shown in Figure 8 Figure 9 . At all of the pressures examined here, the dimension has little effect on the performance of the methane–air system, and the contribution of homogeneous reactions is negligible, as illustrated in Figure 9 . For the mixture of methane and oxygen diluted with a large amount of nitrogen, the collision between the reactant molecules becomes less frequent, which can greatly reduce the contribution of homogeneous reactions, as shown in Figure 10 . However, the contribution may still be considerable at high pressures, as shown in Figure 10 , because mass-transfer limitations are significant under the conditions examined here. At the highest pressure examined here, the methane–air system can offer a good yield of synthesis gas, especially at high preheating temperatures, as shown in Figure 7 . For each of the two systems, the yield of synthesis gas decreases with increasing pressure, as shown in Figure 6 Figure 7 . At high pressures, the production of synthesis gas is favored at low preheating temperatures for the methane–oxygen system ( Figure 6 ), but at high preheating temperatures for the methane–air system ( Figure 7 ). At atmospheric pressure, the use of oxygen and the use of preheated air have the similar effect on the reactor performance, due to the negligible contribution of homogeneous reactions ( Figure 11 ). At high pressures, however, the effect played by the above two feeding methods is quite different from each other, because in this context the contribution of homogeneous reactions is considerable. For each of the two systems, the outlet conversion decreases with increasing pressure due to the mass-transfer limitations and the possible occurrence of gas-phase combustion, especially in larger dimensions. The mass-transfer effect is more pronounced at high pressures, as shown in Figure 7 Figure 8 . As the dimension of the methane–oxygen system increases, the outlet conversion drops sharply ( Figure 8 ), because in this context the mass-transfer effect may be important [ ] and the initiation of gas-phase combustion is possible. 5. Conclusions Catalytic partial oxidation of methane in microchannel reactors in high temperature environments was studied numerically. This investigation provided knowledge on how reaction conditions affect the operating characteristics and the distribution of reaction products in the reactor. Comparisons were made with published results, suggesting that the model developed in this paper is reliable and helpful to the design of the system. Furthermore, the catalytic partial oxidation process under extremely short contact time conditions can be accurately described by the model via the combination of detailed heterogeneous and homogenous reaction mechanisms. The results have shown that the relative role of heterogeneous and homogeneous reaction pathways depends strongly upon the operating conditions. The distribution of reaction products depends upon a number of factors, such as the reactor dimension, pressure, temperature, and feed composition. The operating temperature can significantly affect the yield of synthesis gas. Undesired homogeneous reactions are favored in large reactors, and at high temperatures and pressures. It is more economical to utilize air instead of oxygen as the oxidant. Such an arrangement is particularly beneficial, since the onset of homogeneous reactions can thereby be inhibited or avoided. It is necessary to use the reactant mixture diluted with a sufficient amount of nitrogen to avoid homogenous reactions at high pressures. When air is used as the oxidant, preheating is needed, and the production of synthesis gas is practically favored at high pressures. The use of air as the oxidant is preferred at industrially relevant pressure 3 MPa, at which the contribution of undesired homogeneous reactions is usually small. Further research is needed on the principles underlying the catalytic partial oxidation process. Catalyst deactivation may be an important risk factor, which is not addressed in this paper. This deactivation can significantly decrease the yield of synthesis gas, thus reducing the performance of the system. There are potential solutions to this issue, such as process control to maintain the temperature below a certain damaging threshold, non-uniform catalyst distribution, and thicker catalyst layers. Furthermore, the success of microchannel reactors is highly dependent on the robust catalysts suitable for the operating conditions of these small-scale chemical systems. On the other hand, high temperatures obtained within the system may destroy the catalyst wash coat employed and impose severe constraints on the materials used. Lower reactor temperatures are essential for the stability of the catalyst and materials used. The problem related to the materials stability limit is also not addressed in this paper. A temperature threshold should be well defined in the practical design, which will be the subject of future work. Author Contributions Conceptualization, J.C.; Methodology, J.C.; Software, W.S.; Validation, W.S.; Formal Analysis, J.C. and D.X.; Investigation, J.C. and W.S.; Resources, D.X.; Supervision, J.C.; Project Administration, J.C. and D.X.; J.C. conceived and designed the analyses and simulations; W.S. implemented the C code for the ANSYS Fluent program; D.X. contributed to concepts and notebooks; J.C. and D.X. provided chemical expertise and insight; J.C. wrote the paper. This research was funded by the National Natural Science Foundation of China (No. 51506048) and the Fundamental Research Funds for the Universities of Henan Province (No. NSFRF140119). J.C. is grateful for financial support from the National Natural Science Foundation of China and the Henan Polytechnic University. The authors would also like to acknowledge the DETCHEM website for their computing resources. Conflicts of Interest The authors declare no conflict of interest. A pre-exponential factor A′ surface area C concentration c specific heat capacity D diffusivity D^T thermal diffusivity D[eff] effective diffusivity D[m] mixture-averaged diffusivity d channel height d[pore] mean pore diameter Ea activation energy F[cat/geo] catalyst/geometric surface area F view factor $Δ r H m Θ$ standard molar enthalpy of reaction h specific enthalpy h[o] external heat loss coefficient K[g], K[s] number of gaseous species and number of surface species l length m total number of gaseous and surface species p pressure q heat flux R ideal gas constant $s ˙$ rate of appearance of a heterogeneous product s sticking coefficient T, T[o] absolute temperature and reference temperature u, v streamwise and transverse velocity components V, $V →$ diffusion velocity and diffusion velocity vector W, $W →$ relative molecular mass and relative molecular mass of the mixture x streamwise coordinate Y mass fraction y transverse coordinate Greek variables γ surface area per unit catalyst volume ε emissivity δ thickness ε[p] catalyst porosity λ thermal conductivity η effectiveness factor μ dynamic viscosity ρ density σ Stefan-Boltzmann constant ϑ site occupancy τ[p] catalyst tortuosity factor φ inlet molar ratio $ω ˙$ rate of appearance of a homogeneous product Γ site density Θ surface coverage Φ Thiele modulus amb ambient eff effective g gas i, k, m species index, gaseous species index, and surface species index in inlet o outer rad radiation s solid w wall x, y streamwise and transverse components 1. 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Profiles of the hydroxyl radical concentration along the centerline between the two parallel plates for some of the grids used for the methane–oxygen system. The inlet pressure is 3.0 MPa. The rest of the parameters used here are listed in Table 1 Figure 3. Residuals for the conservation equations at the end of each solver iteration. The mesh used here consists of 16,000 nodes in total. The parameters used in the numerical simulations conducted here are the same as those adopted in Figure 2 Figure 4. Comparison between the numerical results and the experimental data obtained for a methane and oxygen mixture with various compositions. The experimental data are taken from the previous work of Bodke et al. [ ]. A grid consisting of 36,000 nodes in total is used here. Figure 5. Contour plots of the methane and carbon monoxide concentrations and temperature within the fluid in the methane–oxygen system. The operating conditions and design parameters used are listed in Table 1 Figure 6. Influence of preheating temperature on the selectivity, conversion, and maximum wall temperature in the methane–oxygen system. Hereafter, all other parameters are kept at their base case values shown Table 1 . The sharp drop in selectivity and conversion indicates the initiation of gas-phase combustion. ( ) Selectivity to carbon monoxide; ( ) selectivity to hydrogen; ( ) outlet conversion; ( ) maximum wall temperature. Figure 7. Effect of preheating temperature on the selectivity, conversion, and maximum wall temperature in the methane–air system. (a) Selectivity to carbon monoxide and carbon dioxide; (b) selectivity to hydrogen and water; (c) outlet conversion; (d) maximum wall temperature. Figure 8. Effect of reactor dimension on the selectivity, conversion, and maximum wall temperature in the methane–oxygen system. (a) Selectivity to carbon monoxide and carbon dioxide; (b) selectivity to hydrogen and water; (c) total selectivity to C[2] products; (d) outlet conversion and maximum wall temperature. Figure 9. Effect of dimension on the performance of the methane–air system operated at various pressures. (a) Selectivity to carbon monoxide and carbon dioxide; (b) selectivity to hydrogen and water; (c) total selectivity to C[2] products; (d) outlet conversion and maximum wall temperature. Figure 10. Effect of nitrogen diluent on the selectivity, conversion, and maximum wall temperature at different pressures when the reactor is operated at preheating temperature 700 K. (a) Selectivity to carbon monoxide and carbon dioxide; (b) selectivity to hydrogen and water; (c) total selectivity to C[2] products; (d) outlet conversion and maximum wall temperature. Figure 11. Comparisons between the methane–oxygen system and the methane–air system operated at atmospheric pressure. (a) Selectivity to carbon monoxide and carbon dioxide; (b) selectivity to hydrogen and water; (c) outlet conversion and maximum wall temperature. Parameter Variable Value Channel length l 8.0 mm Channel height d 0.8 mm Solid wall Thickness δ 0.8 mm Thermal conductivity λ[s] 16 W/(m·K) (300 K) Gas phase Inlet methane-to-oxygen molar ratio φ 2.0 Inlet pressure p[in] 0.1 MPa Inlet temperature T[in] 300 K Inlet velocity u[in] 0.8 m/s Washcoat thickness δ[catalyst] 0.08 mm Mean pore diameter d[pore] 20 nm Porosity ε[p] 0.5 Tortuosity factor τ[p] 3 Catalyst/geometric surface area F[cat][/geo] 8 Density of rhodium surface sites Γ 2.72 × 10^−9 mol/cm^2 Other conditions Ambient temperature T[amb] 300 K Surface emissivity ε 0.8 External heat loss coefficient h[o] 20 W/(m^2·K) Reactions A (cm, mol, s) s Ea (kJ/mol) H[2] + * + * => H * + H * 1.0 × 10^−2 O[2] + * + * => O * + O * 1.0 × 10^−2 CH[4] + * => CH[4] * 8.0 × 10^−3 H[2]O + * => H[2]O * 1.0 × 10^−1 CO[2] + * => CO[2] * 1.0 × 10^−5 CO + * => CO * 5.0 × 10^−1 H * + H * => * + * + H[2] 3.0 × 10^21 77.8 O * + O * => * + * + O[2] 1.3 × 10^22 355.2–280Θ[O] H[2]O => H[2]O + * 3.0 × 10^13 45.0 CO * => CO + * 3.5 × 10^13 133.4–15Θ[CO] CO[2] => CO[2] + * 1.0 × 10^13 21.7 CH[4] * => CH[4] + * 1.0 × 10^13 25.1 Surface reactions H * + O * => OH * + * 5.0 × 10^22 83.7 OH * + * => H * + O * 3.0 × 10^20 37.7 H * + OH * => H[2]O * + * 3.0 × 10^20 33.5 H[2]O * + * => H * + OH * 5.0 × 10^22 104.7 OH * + OH * => H[2]O * + O * 3.0 × 10^21 100.8 H[2]O * + O * => OH * + OH * 3.0 × 10^21 171.8 C * + O * => CO * + * 3.0 × 10^22 97.9 CO * + * => C * + O * 2.5 × 10^21 169.0 CO * + O * => CO[2] * + * 1.4 × 10^20 121.6 CO[2] * + * => CO * + O * 3.0 × 10^21 115.3 CH[4] * + * => CH[3] * + H * 3.7 × 10^21 61.0 CH[3] * + H * => CH[4] * + * 3.7 × 10^21 51.0 CH[3] * + * => CH[2] * + H * 3.7 × 10^24 103.0 CH[2] * + H * => CH[3] * + * 3.7 × 10^21 44.0 CH[2] * + * => CH * + H * 3.7 × 10^24 100.0 CH * + H * => CH[2] * + * 3.7 × 10^21 68.0 CH * + * => C * + H * 3.7 × 10^21 21.0 C * + H * => CH * + * 3.7 × 10^21 172.8 CH[4] * + O * => CH[3] * + OH * 1.7 × 10^24 80.3 CH[3] * + OH * => CH[4] * + O * 3.7 × 10^21 24.3 CH[3] * + O * => CH[2] * + OH * 3.7 × 10^24 120.3 CH[2] * + OH * => CH[3] * + O * 3.7 × 10^21 15.1 CH[2] * + O * => CH * + OH * 3.7 × 10^24 158.4 CH * + OH * => CH[2] * + O * 3.7 × 10^21 36.8 CH * + O * => C * + OH * 3.7 × 10^21 30.1 C * + OH * => CH * + O * 3.7 × 10^21 145.5 © 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 Chen, J.; Song, W.; Xu, D. RETRACTED: Computational Fluid Dynamics Modeling of the Catalytic Partial Oxidation of Methane in Microchannel Reactors for Synthesis Gas Production. Processes 2018, 6, 83. AMA Style Chen J, Song W, Xu D. RETRACTED: Computational Fluid Dynamics Modeling of the Catalytic Partial Oxidation of Methane in Microchannel Reactors for Synthesis Gas Production. Processes. 2018; 6(7):83. Chicago/Turabian Style Chen, Junjie, Wenya Song, and Deguang Xu. 2018. "RETRACTED: Computational Fluid Dynamics Modeling of the Catalytic Partial Oxidation of Methane in Microchannel Reactors for Synthesis Gas Production" Processes 6, no. 7: 83. https://doi.org/10.3390/pr6070083 Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. 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Why "using namespace std ;" is used in c++ please help me iam beginner to c++. \| Sololearn: Learn to code for FREE! Why "using namespace std ;" is used in c++ please help me iam beginner to c++. it just namespace.. so that u dont have to write "std::cout", "std::endl" Namespaces are used to organize code into logical groups and prevent naming collisions while including multiple library headers. For example suppose you include a header containing a function named strcpy().But there already exists a function strcpy() in the <string> header of std namespace .so you have to define a namespace e.g xyz:: strcpy () to prevent errors	{"url":"https://www.sololearn.com/en/discuss/2549418/why-using-namespace-std-is-used-in-c-please-help-me-iam-beginner-to-c","timestamp":"2024-11-07T23:30:23Z","content_type":"text/html","content_length":"916256","record_id":"<urn:uuid:706b9e95-779f-4910-97c9-9858bbf877fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00020.warc.gz"}
7th Grade Math Statistics Lesson Plan This Statistics lesson plan template will provide a structured and organized approach to you when teaching Statistics to your students. This 7th grade Statistics lesson plan will allow you to plan and prepare for teaching Statistics lessons in advance. It will ensure that you have the necessary background knowledge and skills to teach Statistics and help your students understand it easily.	{"url":"https://www.bytelearn.com/math-grade-7/statistics/lesson-plans","timestamp":"2024-11-10T04:39:03Z","content_type":"text/html","content_length":"1034416","record_id":"<urn:uuid:a7d42977-1eaa-48d3-9a7c-d7019837d14c>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00221.warc.gz"}
Formatting Multiple Summary Numbers with Commas To build summary numbers with multiple metrics, one needs to use the CONCAT operator, which converts the entire output to a string. This means that numbers won’t contain comma’s. In this case, a solution is needed to dynamically add comma’s to these numbers based on their length. For example, in the Summary Number below the first number has no commas. The second number being a percentage is easier to handle because it is rounded to one decimal precision and never needs a comma because it never exceeds the hundreds. Desired output: 1. Find the Length of the Number 2. Break it into appropriate comma blocks 3. Concatenate commas between the blocks. For example, a four-digit number is in the thousands and would have two sections: the first number followed by a comma and then the last three numbers: [1],[234] This can be handling using a combination of LENGTH() to get the overall length of the number, ROUND() to remove decimal precision and SUBSTRING() to pick out the position of the characters you want to place in each comma block. In addition, it will need to be handled using nested CASE logic to identify each case and handle it appropriately. Lastly, it will be tied together with another CONCAT Here is a sample of code that demonstrates how to format one metric up to hundreds of millions - 9 digits in length. Since you will have multiple summary numbers, you would need to do this for each numeric value in the summary number. WHEN LENGTH(ROUND(SUM(`Measure Column`))) = 9 -- 9 digits = hundreds of millions SUBSTRING(ROUND(SUM(`Measure Column`)),1,3), -- first comma block SUBSTRING(ROUND(SUM(`Measure Column`)), 3,3), -- second comma block SUBSTRING(ROUND(SUM(`Measure Column`)),6,3) -- third comma block WHEN LENGTH(ROUND(SUM(`Measure Column`))) = 8 -- 8 digits = tens of millions SUBSTRING(ROUND(SUM(`Measure Column`)),1,3), -- first comma block SUBSTRING(ROUND(SUM(`Measure Column`)), 2,3), -- second comma block SUBSTRING(ROUND(SUM(`Measure Column`)),6,3) -- third comma block WHEN LENGTH(ROUND(SUM(`Measure Column`))) = 7 -- 7 digits = millions SUBSTRING(ROUND(SUM(`Measure Column`)),1,1), -- first comma block SUBSTRING(ROUND(SUM(`Measure Column`)), 2,3), -- second comma block SUBSTRING(ROUND(SUM(`Measure Column`)),5,3) -- third comma block WHEN LENGTH(ROUND(SUM(`Measure Column`))) = 6 -- 6 digits = hundreds of thousands SUBSTRING(ROUND(SUM(`Measure Column`)),1,3), -- first comma block SUBSTRING(ROUND(SUM(`Measure Column`)), 4,3) -- second comma block WHEN LENGTH(ROUND(SUM(`Measure Column`))) = 5 -- 5 digits = tens of thousands SUBSTRING(ROUND(SUM(`Measure Column`)),1,2), -- first comma block SUBSTRING(ROUND(SUM(`Measure Column`)), 3,3) -- second comma block WHEN LENGTH(ROUND(SUM(`Measure Column`))) = 4 -- 4 digits = thousands SUBSTRING(ROUND(SUM(`Measure Column`)),1,1), -- first comma block SUBSTRING(ROUND(SUM(`Measure Column`)), 2,3) -- second comma block WHEN LENGTH(ROUND(SUM(`Measure Column`))) < 4 -- < 4 digits = hundreds or less THEN ROUND(SUM(`Measure Column`)) Jacob Folsom Say “Thanks” by clicking the “heart” in the post that helped you. Please mark the post that solves your problem by clicking on "Accept as Solution" Best Answer • I ceated this beastmode solution for automatically adding commas to custom HTML summary numbers (I had never looked at the dojo before now and didn't realize a solution already existed) but found it to be easy to use and doesn't slow down the card loading (at least with the 1.5M row dataset I'm using it on), so I thought I'd share it here. □ Replace the `x` in this formula with whatever data you are trying to summarize. □ You can remove sections you don't need, like if you don't need trillions just delete everything between the trillions comment and the billions comment. It shouldn't be necessary, because this really doesn't add much processing time to the beastmode, but it's possible if you think it will help. □ You can also add additional sections, like if you need quadrillions, by just copying the trillions segment and adding 3 digits to each number (if it's 99.99 add 3 more nines in front of the decimal, if it's 1000 add 3 more zeros, etc). (CASE WHEN `x` > 99999999999999.99 AND MOD(`x`,1000000000000000) < 100000000000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 9999999999999.99 AND MOD(`x`,1000000000000000) < 10000000000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 999999999999.99 THEN CONCAT(FLOOR(MOD(`x`,1000000000000000)/1000000000000),',') ELSE '' END) (CASE WHEN `x` > 99999999999.99 AND MOD(`x`,1000000000000) < 100000000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 9999999999.99 AND MOD(`x`,1000000000000) < 10000000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 999999999.99 THEN CONCAT(FLOOR(MOD(`x`,1000000000000)/1000000000),',') ELSE '' END) (CASE WHEN `x` > 99999999.99 AND MOD(`x`,1000000000) < 100000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 9999999.99 AND MOD(`x`,1000000000) < 10000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 999999.99 THEN CONCAT(FLOOR(MOD(`x`,1000000000)/1000000),',') ELSE '' END) (CASE WHEN `x` > 99999.99 AND MOD(`x`,1000000) < 100000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 9999.99 AND MOD(`x`,1000000) < 10000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 999.99 THEN CONCAT(FLOOR(MOD(`x`,1000000)/1000),',') ELSE '' END) (CASE WHEN `x` > 99.99 AND MOD(`x`,1000) < 100 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 9.99 AND MOD(`x`,1000) < 10 THEN '0' ELSE '' END) • Great article thanks for sharing! • You can use the same structure to abbreviate the Title instead of putting commas. When the number is in millions, the sub-title becomes way too long. So instead of 1,230 it will say 1.2k or 1k depending on your case statement. • I ceated this beastmode solution for automatically adding commas to custom HTML summary numbers (I had never looked at the dojo before now and didn't realize a solution already existed) but found it to be easy to use and doesn't slow down the card loading (at least with the 1.5M row dataset I'm using it on), so I thought I'd share it here. □ Replace the `x` in this formula with whatever data you are trying to summarize. □ You can remove sections you don't need, like if you don't need trillions just delete everything between the trillions comment and the billions comment. It shouldn't be necessary, because this really doesn't add much processing time to the beastmode, but it's possible if you think it will help. □ You can also add additional sections, like if you need quadrillions, by just copying the trillions segment and adding 3 digits to each number (if it's 99.99 add 3 more nines in front of the decimal, if it's 1000 add 3 more zeros, etc). (CASE WHEN `x` > 99999999999999.99 AND MOD(`x`,1000000000000000) < 100000000000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 9999999999999.99 AND MOD(`x`,1000000000000000) < 10000000000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 999999999999.99 THEN CONCAT(FLOOR(MOD(`x`,1000000000000000)/1000000000000),',') ELSE '' END) (CASE WHEN `x` > 99999999999.99 AND MOD(`x`,1000000000000) < 100000000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 9999999999.99 AND MOD(`x`,1000000000000) < 10000000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 999999999.99 THEN CONCAT(FLOOR(MOD(`x`,1000000000000)/1000000000),',') ELSE '' END) (CASE WHEN `x` > 99999999.99 AND MOD(`x`,1000000000) < 100000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 9999999.99 AND MOD(`x`,1000000000) < 10000000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 999999.99 THEN CONCAT(FLOOR(MOD(`x`,1000000000)/1000000),',') ELSE '' END) (CASE WHEN `x` > 99999.99 AND MOD(`x`,1000000) < 100000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 9999.99 AND MOD(`x`,1000000) < 10000 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 999.99 THEN CONCAT(FLOOR(MOD(`x`,1000000)/1000),',') ELSE '' END) (CASE WHEN `x` > 99.99 AND MOD(`x`,1000) < 100 THEN '0' ELSE '' END) ,(CASE WHEN `x` > 9.99 AND MOD(`x`,1000) < 10 THEN '0' ELSE '' END) • Hello @avataraang , I love the work you've done with thee referenced Beast Mode. I'm happy to let you know we have just released Code Block apps in our Domo Appstore. These allow customers and Domo experts such as yourself, to publish code blocks like the ones you've created for distribution in the Appstore. The publish process for the code blocks is currently in an early beta. If you are interested in publishing your code blocks, let me know and we can enable it for you. Cody Smith Director of Product, Domo • Just found this solution, which worked very well. Thank you! One thing to note if anyone comes across this to use it: There is a typo in DataJake's original solution that had me confused for a while. For 8 digit strings, we are getting an extra digit due to this typo: SUBSTRING(ROUND(SUM(`Measure Column`)),1,3), -- first comma block Which should instead be: SUBSTRING(ROUND(SUM(`Measure Column`)),1,2), -- first comma block Hope this helps anyone who comes across this solution. • 1.8K Product Ideas • 1.5K Connect • 2.9K Transform • 3.8K Visualize • 677 Automate • 34 Predict • 394 Distribute • 121 Manage • 5.4K Community Forums	{"url":"https://community-forums.domo.com/main/discussion/32344/formatting-multiple-summary-numbers-with-commas","timestamp":"2024-11-05T12:29:26Z","content_type":"text/html","content_length":"408401","record_id":"<urn:uuid:0622687a-42f0-429a-99f3-0a3f10e8063b>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00267.warc.gz"}
How to convert rates using dimensional analysis the rate 50 km per hour in a variety of different units, using dimensional analysis. I got confused in the Rate Conversion practice, and when I decided to get a How Do You Use Dimensional Analysis to Convert Units on Both Parts of a Rate? Note: Word problems are a great way to see math in action! In this tutorial, learn How to convert rate units 1.2b Notes. How to convert rate units 1.2b Notes. Skip navigation Using Dimensional Analysis to calculate IV flow rates - Duration: 5:46. teaching 1,130 views. The Dimensional Analysis Calculator is a tool that is used to find the relation between two physical quantities. Various dimensions of length, time, temperature and mass can be calculated. Here, the Dimensional Analysis Calculator is provided to help make calculations fast and easy. Try out our free tool while solving the problems. Visit Study.com for thousands more videos like this one. You'll get full access to our interactive quizzes and transcripts and can find out how to use our videos to earn real college credit This is a whiteboard animation tutorial of one step and two step dimensional analysis (aka factor label method, aka unit factor method) for solving unit conversion problems Please consider Solving Dimensional Analysis Problems - Unit Conversion Problems Made Easy! - This video works through dimensional analysis problems. Dimensional analysis is a process to solve unit conversion 9 CONVERTING RATES 45 miles per hour = ______ feet per hour. Conversion Unit Conversion Using Dimensional Analysis Objective N-Q. 1 Use units as a Learning converting units using the method used in most chemistry classes. How to convert unit rates What students should know before the crash course: Apr 9, 2015 And then when teaching students to convert rates, it's a whole new So, I prefer to teach Customary conversions with unit (or dimensional) analysis. the next day they are able to solve them using unit analysis, as well. Jun 12, 2019 The total fertility rate (TFR) of a country is the average number of births per The key with dimensional analysis is that each of the conversion factors is equal to one. Solve the following problems using dimensional analysis. Miles/hour to feet/second (mi/hr to ft/s) Metric conversion calculator. Includes thousands of additional Algebraic Steps / Dimensional Analysis Formula The Scala API for Quantities, Units of Measure and Dimensional Analysis The unit kW is used to measure Power/Load, the rate at which Energy is Convert QuantityVectors to specific units using the to or in method - much like Quantities. Here, the Dimensional Analysis Calculator is provided to help make calculations fast and easy. Try out our free tool while solving the problems. Here, the SI units the rate 50 km per hour in a variety of different units, using dimensional analysis. I got confused in the Rate Conversion practice, and when I decided to get a Learning converting units using the method used in most chemistry classes. How to convert unit rates What students should know before the crash course: Apr 9, 2015 And then when teaching students to convert rates, it's a whole new So, I prefer to teach Customary conversions with unit (or dimensional) analysis. the next day they are able to solve them using unit analysis, as well. Jun 12, 2019 The total fertility rate (TFR) of a country is the average number of births per The key with dimensional analysis is that each of the conversion factors is equal to one. Solve the following problems using dimensional analysis. Miles/hour to feet/second (mi/hr to ft/s) Metric conversion calculator. Includes thousands of additional Algebraic Steps / Dimensional Analysis Formula The Scala API for Quantities, Units of Measure and Dimensional Analysis The unit kW is used to measure Power/Load, the rate at which Energy is Convert QuantityVectors to specific units using the to or in method - much like Quantities. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring. How Do You Use Dimensional Analysis to Convert Units on One Part of a Rate? Word problems are a great way to see math in action! In this tutorial, learn how to use the information given in a word problem to create a rate. Then, find and use a conversion factor to convert a unit in the rate. Take a look! Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring. How to convert rate units 1.2b Notes. How to convert rate units 1.2b Notes. Skip navigation Using Dimensional Analysis to calculate IV flow rates - Duration: 5:46. teaching 1,130 views. The Dimensional Analysis Calculator is a tool that is used to find the relation between two physical quantities. Various dimensions of length, time, temperature and mass can be calculated. Here, the Dimensional Analysis Calculator is provided to help make calculations fast and easy. Try out our free tool while solving the problems. Visit Study.com for thousands more videos like this one. You'll get full access to our interactive quizzes and transcripts and can find out how to use our videos to earn real college credit This is a whiteboard animation tutorial of one step and two step dimensional analysis (aka factor label method, aka unit factor method) for solving unit conversion problems Please consider Nov 2, 2011 Dimensional Analysis is a method of solving problems that 2. For each of the rates that were not unit rates in 1), convert them to a unit rate. Miles/hour to feet/second (mi/hr to ft/s) Metric conversion calculator. Includes thousands of additional Algebraic Steps / Dimensional Analysis Formula The Scala API for Quantities, Units of Measure and Dimensional Analysis The unit kW is used to measure Power/Load, the rate at which Energy is Convert QuantityVectors to specific units using the to or in method - much like Quantities. Here, the Dimensional Analysis Calculator is provided to help make calculations fast and easy. Try out our free tool while solving the problems. Here, the SI units 9 CONVERTING RATES 45 miles per hour = ______ feet per hour. Conversion Unit Conversion Using Dimensional Analysis Objective N-Q. 1 Use units as a Using these two pieces of information, we can set up a dimensional analysis conversion. If you haven't done it in awhile, you will need to practice a bit with unit conversions. Let's try the following exercises, using the conversion tables that you can Nov 2, 2011 Dimensional Analysis is a method of solving problems that 2. For each of the rates that were not unit rates in 1), convert them to a unit rate. Using these two pieces of information, we can set up a dimensional analysis conversion. If you haven't done it in awhile, you will need to practice a bit with unit conversions. Let's try the following exercises, using the conversion tables that you can Nov 2, 2011 Dimensional Analysis is a method of solving problems that 2. For each of the rates that were not unit rates in 1), convert them to a unit rate. How Do You Use Dimensional Analysis to Convert Units on One Part of a Rate? Word problems are a great way to see math in action! In this tutorial, learn how to use the information given in a word problem to create a rate. Then, find and use a conversion factor to convert a unit in the rate. Take a look! Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring. How to convert rate units 1.2b Notes. How to convert rate units 1.2b Notes. Skip navigation Using Dimensional Analysis to calculate IV flow rates - Duration: 5:46. teaching 1,130 views. The Dimensional Analysis Calculator is a tool that is used to find the relation between two physical quantities. Various dimensions of length, time, temperature and mass can be calculated. Here, the Dimensional Analysis Calculator is provided to help make calculations fast and easy. Try out our free tool while solving the problems.	{"url":"https://dioptionermblqg.netlify.app/nistler14079ni/how-to-convert-rates-using-dimensional-analysis-31.html","timestamp":"2024-11-13T02:10:03Z","content_type":"text/html","content_length":"33808","record_id":"<urn:uuid:b55abf7b-87c3-4d3d-89ca-f3a99ad83d64>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00317.warc.gz"}
Gray - Unit of Radiation Dose \| nuclear-power.com Gray – Unit of Radiation Dose Absorbed dose is defined as the amount of energy deposited by ionizing radiation in a substance. The absorbed dose is given the symbol D. The absorbed dose is usually measured in a unit called the gray (Gy), derived from the SI system. The non-SI unit rad is sometimes also used, predominantly in the USA. Units of absorbed dose: • Gray. A dose of one gray is equivalent to a unit of energy (joule) deposited in a kilogram of a substance. • RAD. A dose of one rad is equivalent to depositing one hundred ergs of energy in one gram of any material. Gray – Unit of Absorbed Dose A dose of one gray is equivalent to a unit of energy (joule) deposited in a kilogram of a substance. This unit was named in honor of Louis Harold Gray, who was one of the great pioneers in radiation biology. One gray is a large amount of absorbed dose. A person who has absorbed a whole-body dose of 1 Gy has absorbed one joule of energy in each kg of body tissue. Absorbed doses measured in the industry (except nuclear medicine) often have usually lower doses than one gray, and the following multiples are often used: 1 mGy (milligray) = 1E-3 Gy 1 µGy (microgray) = 1E-6 Gy Conversions from the SI units to other units are as follows: • 1 Gy = 100 rad • 1 mGy = 100 mrad The gray and rad are physical units describing the incident radiation’s physical effect (i.e., the amount of energy deposited per kg). Still, it tells us nothing about the biological consequences of such energy deposition in living tissue. Examples of Absorbed Doses in grays We must note that radiation is all around us. In, around, and above the world we live in. It is a natural energy force that surrounds us, and it is a part of our natural world that has been here since the birth of our planet. In the following points, we try to express enormous ranges of radiation exposure, which can be obtained from various sources. • 0.05 µGy – Sleeping next to someone • 0.09 µGy – Living within 30 miles of a nuclear power plant for a year • 0.1 µGy – Eating one banana • 0.3 µGy – Living within 50 miles of a coal power plant for a year • 10 µGy – Average daily dose received from natural background • 20 µGy – Chest X-ray • 40 µGy – A 5-hour airplane flight • 600 µGy – mammogram • 1 000 µGy – Dose limit for individual members of the public, total effective dose per annum • 3 650 µGy – Average yearly dose received from natural background • 5 800 µGy – Chest CT scan • 10 000 µGy – Average yearly dose received from a natural background in Ramsar, Iran • 20 000 µGy – single full-body CT scan • 175 000 µGy – Annual dose from natural radiation on a monazite beach near Guarapari, Brazil. • 5 000 000 µGy – Dose that kills a human with a 50% risk within 30 days (LD50/30) if the dose is received over a very short duration. As can be seen, low-level doses are common in everyday life. The previous examples can help illustrate relative magnitudes. From biological consequences, it is very important to distinguish between doses received over short and extended periods. An “acute dose” occurs over a short and finite period, while a “chronic dose” is a dose that continues for an extended period so that a dose rate better describes it. High doses tend to kill cells, while low doses tend to damage or change them. Low doses spread out over long periods don’t cause an immediate problem to any body organ. The effects of low radiation doses occur at the cell level, and the results may not be observed for many years. Calculation of Shielded Dose Rate in grays Assume the point isotropic source contains 1.0 Ci of ^137Cs and has a half-life of 30.2 years. Note that the relationship between half-life and the amount of a radionuclide required to give an activity of one curie is shown below. This amount of material can be calculated using λ, which is the decay constant of certain nuclide: About 94.6 percent decays by beta emission to a metastable nuclear isomer of barium: barium-137m. The main photon peak of Ba-137m is 662 keV. For this calculation, assume that all decays go through this channel. Calculate the primary photon dose rate, in gray per hour (Gy.h^-1), at the outer surface of a 5 cm thick lead shield. The primary photon dose rate neglects all secondary particles. Assume that the effective distance of the source from the dose point is 10 cm. We shall also assume that the dose point is soft tissue and it can reasonably be simulated by water, and we use the mass-energy absorption coefficient for water. See also: Gamma Ray Attenuation See also: Shielding of Gamma Rays The primary photon dose rate is attenuated exponentially, and the dose rate from primary photons, taking account of the shield, is given by: As can be seen, we do not account for the buildup of secondary radiation. If secondary particles are produced, or the primary radiation changes its energy or direction, the effective attenuation will be much less. This assumption generally underestimates the true dose rate, especially for thick shields and when the dose point is close to the shield surface, but this assumption simplifies all calculations. For this case, the true dose rate (with the buildup of secondary radiation) will be more than two times higher. To calculate the absorbed dose rate, we have to use the formula: • k = 5.76 x 10^-7 • S = 3.7 x 10^10 s^-1 • E = 0.662 MeV • μ[t]/ρ = 0.0326 cm^2/g (values are available at NIST) • μ = 1.289 cm^-1 (values are available at NIST) • D = 5 cm • r = 10 cm The resulting absorbed dose rate in grays per hour is then: If we want to account for the buildup of secondary radiation, then we have to include the buildup factor. The extended formula for the dose rate is then:	{"url":"https://www.nuclear-power.com/nuclear-engineering/radiation-protection/absorbed-dose/gray-unit-of-radiation-dose/","timestamp":"2024-11-04T10:31:35Z","content_type":"text/html","content_length":"98210","record_id":"<urn:uuid:dabca7cb-8389-45a6-872e-0b91b4a7cf8f>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00106.warc.gz"}
CS计算机代考程序代写 assembly scheme algorithm mips CMPUT 229 (Winter 2021) – Midterm II Instructor: Karim Ali CMPUT 229 (Winter 2021) – Midterm II Instructor: Karim Ali Exam Instructions 1. You must follow all the MIPS procedure call conventions. 2. You are allowed to use SPIM pseudo instructions. 3. You must show all steps of your work for full credit. 4. You are allowed to use a non-communicating calculator. You cannot use your phone, tablet, or laptop as a calculator. You cannot borrow a calculator from your colleagues. 5. You are allowed to have the reference sheet containing the summary of the MIPS instructions from the front of the textbook or an unchanged copy of that reference sheet (e.g., a printout of the version that I made available on the class Google Drive). 6. You are allowed one regular-size sheet of paper (front and back) on which you can write anything that you wish, as long as it is hand-written (i.e., not typed) by you (i.e., not by somebody else or a photocopy). You are not allowed to use any mechanical or electronic method of reproduction to create this sheet. Question 1 Question 2 CMPUT 229 (Winter 2021) – Midterm II Instructor: Karim Ali Question 1: (30 points) Figure 1 shows an unoptimized implementation for a simple while loop, and Figure 2 shows a slightly optimized version of the same loop. Given both implementations and the information in Table 1, answer the following questions: Figure 1: The unoptimized MIPS implementation of a simple while loop. 0x3FFF FFE8 0x3FFF FFEC 0x3FFF FFF0 0x3FFF FFF4 0x3FFF FFF8 0x3FFF FFFC 0x4000 0000 sll $t1, $s3, 2 add $t1, $t1, $s6 lw $t0, 0($t1) bne $t0, $s5, Exit add $s3, $s3, $s4 j Loop #$t1 ← i4 # $t1 ← Addr(save[i]) # $t0 ← save[i] # if save[i] ≠ k goto Exit #i←i+j # goto Loop 0x6000 0000 0x6000 0004 0x6000 0008 0x6000 000C 0x6000 0010 0x6000 0014 0x6000 0018 0x6000 001C 0x6000 0020 0x6000 0024 sll $t1, $s3, 2 add $t1, $t1, $s6 lw $t0, 0($t1) bne $t0, $s5, Exit_opt add $s3, $s3, $s4 sll $t1, $s3, 2 add $t1, $t1, $s6 lw $t0, 0($t1) beq $t0, $s5, Loop_opt #$t1 ← i4 # $t1 ← Addr(save[i]) # $t0 ← save[i] # if save[i] ≠ k goto Exit #i←i+j #$t1 ← i4 # $t1 ← Addr(save[i]) # $t0 ← save[i] # if save[i] ≠ k goto Exit Figure 2: The optimized MIPS implementation of a simple while loop. Cycles Per Instruction Table 1: Instruction information. add j2 bne 3 beq 3 lw 5 CMPUT 229 (Winter 2021) – Midterm II Instructor: Karim Ali a. (10 points) Given the same number of loop iterations n, how many cycles are re- quired to execute the code in each of Figure 1 and Figure 2? Solution: Figure 1 = 13n + 10, Figure 2 = 11n + 10 Each answer is worth 3 marks for the final equation and 2 marks for the steps. n Figure 1 Figure 2 0 1×(1+1+5+3)+0×(1+2)=10 10+0×(1+1+1+5+3)=10 1 2×(1+1+5+3)+1×(1+2)=23 10+1×(1+1+1+5+3)=21 2 3×(1+1+5+3)+2×(1+2)=36 10+2×(1+1+1+5+3)=32 3 4×(1+1+5+3)+3×(1+2)=49 10+3×(1+1+1+5+3)=43 n ((n+1)×10)+(n×3) (11×n)+10 n 13n+10 11n+10 b. (10 points) Assuming the same very large number of iterations n for each loop version, what is the average number of cycles per instruction (rounded to the nearest 1 decimal place) when executing the code in Figure 1 and Figure 2? Solution: Figure 1 = 2.2, Figure 2 = 2.2 Each answer is worth 3 marks for the steps and 2 marks for final answer. Similar to the previous question but counting the number of instructions instead of the clock cycles. We’ll get: # instructions Figure 1 # instructions Figure 2 Since CPI Therefore CPI Figure 1 And CPI Figure 2 = 6n + 4 = 5n + 4 = # clock cycles # instructions = 13n + 10 6n+4 ≈ 13n 6n = 13 = 2.16 ≈ 2.2 = 11n + 10 5n+4 ≈ 11n 5n = 11 5 = 2.2 CMPUT 229 (Winter 2021) – Midterm II Instructor: Karim Ali c. (10 points) Assuming the same very large number of iterations n for each loop version, and that both run on the same processor, which version is faster and why? Express your answer in the form “A is k times faster than B”. Solution: Figure 2 is 1.18× faster than Figure 1. Because the processor is the same, the clock frequency is the same. Therefore: (3 marks) Performance Figure 2 Performance Figure 1 (2 marks) (1 mark) (1 mark) (1 mark) (2 marks) = Execution Time Figure 1 Execution Time Figure 2 = # instructions Figure 1 × # CPI Figure 1 # instructions Figure 2 × # CPI Figure 2 = (6n+4)×(13n+10)×(5n+4) (5n+4)×(11n+10)×(6n+4) = 13n+10 11n+10 ≈ 13n 11n = 13 11 = 1.18 CMPUT 229 (Winter 2021) – Midterm II Instructor: Karim Ali Question 2: (65 points) Two strings have a common substring if they share at least one character. For exam- ple, the strings “hello” and “world” have a common substring, because both share the characters ‘o’ and ‘l’. On the other hand, the strings “cmput” and “bio” do not have a common substring, because they do not share any characters. An efficient algorithm to determine whether two strings (s1 and s2) share a common substring has a complexity of O(n). This algorithm first creates an integer array v of size 26 (assuming strings consist of only lower-case alphabets), and initialize each array element with 0. For every character in s1, the algorithm increments the array element at the index that corresponds to that character (i.e., v[s1[i]-‘a’]) by 1 to indicate the occurrence of that character in s1. Then, for every character in s2, the algorithm checks the corresponding array element (i.e., v[s2[i]-‘a’]). If the array element has a value that is greater than 0, then a common character has been found and the algorithm returns 1 (i.e., true). Otherwise, the algorithm returns 0 (i.e., false). Assuming that strings are null-terminated character arrays, and given the C method signature below, your task is to implement that efficient algorithm in MIPS Assembly according to the description above. int haveCommonSubstr(char s1, char* s2) { Solution: This is one way to solve this question. Marking Scheme: • 5 marks for assuming arguments in right registers • 5 marks for the assumption made about array v • 5 marks for initializing the array v (all elements should be set to 0). • 20 marks for first loop (loop termination, index calculation, loading correct array element, increment, store) • 20 marks for second loop (loop termination, index calculation, load/lookup, check/branch, increment loop index) • 5 marks for returning the correct value in $v0 • 5 marks for properly exiting the procedure CMPUT 229 (Winter 2021) – Midterm II Instructor: Karim Ali # $a0 = s1, $a1 = s2, $t2 = v addi $t3, $zero, 104 addi $t4, $zero, -4 addi $t4, $t4, 4 sw $zero, 0($t4) slt $t5, $t4, $t3 bne $t5, $zero, init lb $t1, 0($a0) # loop count (26 elements) # initialize array index to -4 # go to the next integer counter #v[i]=0 #if$t4<$t3 # loop # load s1[i] (1 byte) beq $t1, $zero, searchString # reached end of string addi $t2, $t1, -0x61 sll $t2, $t2, 2 add $t2, $t2, $t0 lw $t1, 0($t2) addi $t1, $t1, 1 sw $t1, 0($t2) addi $a0, $a0, 1 j findChars searchString: lb $t1, 0($a1) beq $t1, $zero, noSubstr addi $t2, $t1, -0x61 sll $t2, $t2, 2 add $t2, $t2, $t0 lw $t1, 0($t2) bne $t1, $zero, isSubstr addi $a1, $a1, 1 j searchString noSubstr: addi $v0, $zero, 0 jr $ra isSubstr: addi $v0, $zero, 1 jr $ra # compute s1[i]-‘a’ # because v is an integer array # compute index to array v # load the current counter # increment the counter # store it back in the array # go to the next character in s1 # loop # load s2[i] (1 byte) # reached end of string, no match # compute s2[i] - ‘a’ # because v is an integer array # compute index to array v # load the current counter # if counter != 0 => match found # go to next character in s2 # return 0 (false) # return 1 (true)	{"url":"https://www.cscodehelp.com/%E5%B9%B3%E6%97%B6%E4%BD%9C%E4%B8%9A%E4%BB%A3%E5%86%99/cs%E8%AE%A1%E7%AE%97%E6%9C%BA%E4%BB%A3%E8%80%83%E7%A8%8B%E5%BA%8F%E4%BB%A3%E5%86%99-assembly-scheme-algorithm-mips-cmput-229-winter-2021-midterm-ii-instructor-karim-ali/","timestamp":"2024-11-05T20:08:58Z","content_type":"text/html","content_length":"59315","record_id":"<urn:uuid:d7cd5a96-2396-4c02-9cc6-480bcec5e091>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00642.warc.gz"}
product size distribution for a ball mill In this research, ball size distribution which is a function of makeup ball sizes was investigated to optimise the milling stage of a grinding circuit in order to maximise the production of the narrowlysized mill product for floatation, in this case − 75 + 9 μ m. Section snippets WhatsApp: +86 18838072829 Chimwani et al. (2015) and Taggart (1945) have shown that the variation in the ball size distribution can also be used so as to minimize the particle size range of a ball mill production. Ball mills tend to produce a given particle size distribution which is determined by the ball size distribution (Kotake et al., 2011 and Taggart, 1945) • WhatsApp: +86 18838072829 It was found that product size distribution is a definite function of slurry filling U. Graphical analysis of the results suggests that the voids within the media charge should be completely filled with slurry without allowing the formation of the pool. ... The efficiency of a wet ball mill is dependent on the size of the pool of slurry that ... WhatsApp: +86 18838072829 Fitz mill by assessing particle size distribution. Particle properties of the extruded material were to be assessed on both the L1A Fitz mill and 197 Comil, evaluating the feasibility of Comiling WhatsApp: +86 18838072829 The product size from HPGR can be much finer than the corresponding ball or rod mill products. As an example, the results by Mörsky, Klemetti and Knuutinen [12] are given in Figure where, for the same net input energy (4 kWh/t), the product sizes obtained from HPGR, ball and rod mills are plotted. WhatsApp: +86 18838072829 Effect of ball and feed particle size distribution on the milling efficiency of a ball mill: An attainable region approach N. Hlabangana a, G. Danha b, b *, E. Muzenda Department of Chemical Engineering, National University of Science and Technology, P O Box AC 939, Ascot Bulawayo, Zimbabwe WhatsApp: +86 18838072829 Tailoring ball mill feed size distribution for the production of a sizegraded product ScienceDirect Minerals Engineering Volume 141, September 2019, 105891 Tailoring ball mill feed size distribution for the production of a sizegraded product Ngonidzashe Chimwani a, Thapelo M. Mohale a, Murray M. Bwalya b Add to Mendeley WhatsApp: +86 18838072829 To assess the effects of the mill operating parameters such as mill speed, ball filling, slurry concentration and slurry filling on grinding process and size distribution of mill product, it was endeavored to build a pilot model with smaller size than the mill. For this aim, a pilot mill with 1m × was implemented. WhatsApp: +86 18838072829 The rod mill product is characterized by its narrow size distribution if it is compared with that of a ball mill operating under the same conditions in open circuit. This is partly due to the ... WhatsApp: +86 18838072829 IsaMill™ scaleup is accurate in terms of power consumption and product size distribution (PSD). ... In ball mill laboratory tests 25mm balls in a small laboratory ball mill have different trajectories and interact differently with the shell lifters and ore particles than in a large production ball mill. Additionally, while techniques like ... WhatsApp: +86 18838072829 For each grinding test, the mill was first loaded with a kg mass of ball mix and a 150 g mass of feed sample volume of ml tap water was then added to the mill charge in order to make a 70 wt.% pulp monosized fractions of quartz and chlorite(− 2 + mm, − + mm, − + mm, − + mm)were first ground to determine a better size ... WhatsApp: +86 18838072829 The results indicate that the slope of the product size distribution (PSD) curve remains relatively unchanged in the coarse product size range but decreases in the fine product range (less than WhatsApp: +86 18838072829 Better product quality can be achieved as compared to the ball mill product due to the better options for separate grinding. For example, in additive cement production, ... Size distribution of +149 μm material was determined by dry sieving using a RoTap. The entire size distribution for each sample was calculated using the sieving results ... WhatsApp: +86 18838072829 The effect of ball size diameter or ball size distribution on the mill performance have been studied by many researchers using empirical methods, and population balance models (Austin et al., 1976 ... WhatsApp: +86 18838072829 An alternative mill ball charge is proposed that closely approximates Bond's original total ball mass, number of balls and ball surface area. Results of 30 Bond Work index tests of six pure materials (calcite, magnesite, labradorite (feldspar), quartz, andalusite and glass) using closing screen apertures (P 1 ) values of 500, 250, 125, 90 and ... WhatsApp: +86 18838072829 Mill product. For an operation challenged by recovery losses due to an overly coarse product size from the grinding circuit, converting the ball mill from overflow to grate discharge can be highly beneficial. Conversion of the mill results in improved breakage inside the mill. WhatsApp: +86 18838072829 This equation allows the calculation of the product size distribution from a mill once d ij is determined. The calculation of d ij includes the aforementioned S and B functions, which can be determined in a laboratory batch mill. However, these parameters are sensitive to milling conditions such as the mill rotational speed, ball filling ... WhatsApp: +86 18838072829 Initially, for some time the crushed material is found to break in the ball mill at a significantly higher rate (2030% is quite common) as compared to the ball mill ground particles of the same size [18]. In many cases, even afterward 1020% variations in the specific breakage rate are observed. WhatsApp: +86 18838072829 The standard Bond ball mill grindability test determines the ore grindability (g rev−1) and work index on an ore which gives the ball mill power consumption. The test is carried out on a standard sample and the number of revolutions per minute (rev min−1) required for the next cycle is determined after screening out and weighing the undersize fraction from a screen of a mesh the same size WhatsApp: +86 18838072829 Introduction. The Bond grindability test for determining the Bond work index, W i, is conducted in a Bond ball mill having the dimensions D × L = 305 × 305 mm and a speed revolution of 70 min − mill is loaded with balls from up to mm in diameter, having thus a total mass of kg.. This test simulates a closed circuit of dry grinding of samples of standard size at − 3 ... WhatsApp: +86 18838072829 Experimental residence time distribution data for a fullscale cement ball mill was fitted by the cellbased PBM to determine the number of cells and axial backmixing ratio. ... conducted to determine the temporal evolution of the particle size distribution and mass holdup, demonstrate that milling with a ball mixture outperforms milling with ... WhatsApp: +86 18838072829 The work uses the UFRJ mechanistic mill model and DEM to analyze the effect of several design and operating variables on the apparent breakage rates and breakage distribution function of a batch gravityinduced stirred mill grinding copper shows that breakage rates increase significantly with stirrer speed; that increase in percent solids decreased breakage rates, whereas the increase ... WhatsApp: +86 18838072829 to the ineficiency of ball mills for fine grind applications. The difficulty encountered in fine ... Figure Measured and predicted product size distribution for a 150 +105μm feed sample .. 97 Figure Simulated (dotted lines) and measured (markers) product particle size WhatsApp: +86 18838072829	{"url":"https://www.traiteur-cino.fr/9437/product-size-distribution-for-a-ball-mill.html","timestamp":"2024-11-07T03:28:52Z","content_type":"application/xhtml+xml","content_length":"23193","record_id":"<urn:uuid:d2fb3232-cd8d-4fe4-808e-1d5e9226b6d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00264.warc.gz"}
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Can black holes bounce to white holes? Fast track to wisdom: Sure, but who cares if they can? We want to know if they do. Black holes are defined by the presence of an event horizon which is the boundary of a region from which nothing can escape, ever. The word black hole is also often used to mean something that looks for a long time very similar to a black hole and that traps light, not eternally but only temporarily. Such space-times are said to have an “apparent horizon.” That they are not strictly speaking black holes was origin of the recent Stephen Hawking quote according to which black holes may not exist, by which he meant they might have only an apparent horizon instead of an eternal event A white hole is an upside-down version of a black hole; it has an event horizon that is a boundary to a region in which nothing can ever enter. Static black hole solutions, describing unrealistic black holes that have existed forever and continue to exist forever, are actually a combination of a black hole and a white hole. The horizon itself is a global construct, it is locally entirely unremarkable and regular. You would not note crossing the horizon, but the classical black hole solution contains a singularity in the center. This singularity is usually interpreted as the breakdown of classical general relativity and is expected to be removed by the yet-to-be-found theory of quantum gravity. You do however not need quantum gravity to construct singularity-free black hole space-times. Hawking and Ellis’ singularity theorems prove that singularities must form from certain matter configurations, provided the matter is normal matter and cannot develop negative pressure and/or density. All you have to do to get rid of the singularity is invent some funny type of matter that refuses to be squeezed arbitrarily. This is not possible with any type of matter we know, and so just pushes around the bump under the carpet: Now rather than having to explain quantum effects of gravity you have to explain where the funny matter comes from. It is normally interpreted not as matter but as a quantum gravitational contribution to the stress-energy tensor, but either way it’s basically the physicist’s way of using a kitten photo to cover the hole in wall. Singularity-free black hole solutions have been constructed almost for as long as the black hole solution has been known – people have always been disturbed by the singularity. Using matter other than normal ones allowed constructing both wormhole solutions as well as black holes that turn into white holes and allow an exit into a second space-time region. Now if a black hole is really a black hole with an event horizon, then the second space-time region is causally disconnected from the first. If the black hole has only an apparent horizon, then this does not have to be so, and also the white hole then is not really a white hole, it just looks like one. The latter solution is quite popular in quantum gravity. It basically describes matter collapsing, forming an apparent horizon and a strong quantum gravity region inside but no singularity, then evaporating and returning to an almost flat space-time. There are various ways to construct these space-times. The details differ, but the corresponding causal diagrams all look basically the same. This recent paper for example used a collapsing shell turning into an expanding shell. The title “Singularity free gravitational collapse in an effective dynamical quantum spacetime” basically says it all. Note how the resulting causal diagram (left in figure below) looks pretty much the same as the one Lee and I constructed based on general considerations in our 2009 paper (middle in figure below), which again looks pretty much the same as the one that Ashtekar and Bojowald discussed in 2005 (right in figure below), and I could go on and add a dozen more papers discussing similar causal diagrams. (Note that the shaded regions do not mean the same in each figure.) One needs a concrete ansatz for the matter of course to be able to calculate anything. The general structure of the causal diagram is good for classification purposes, but not useful for quantitative reasoning, for example about the evaporation. Haggard and Rovelli and recently added to this discussion with a new paper about black holes bouncing to white holes. Black hole fireworks: quantum-gravity effects outside the horizon spark black to white hole tunneling Hal M. Haggard, Carlo Rovelli arXiv: 1407.0989 Ron Cowen at Nature News announced this as a new idea , and while the paper does contain new ideas, that black holes may turn into white holes is in and by itself not new. And so it follows some clarification. Haggard and Rovelli’s paper contains two ideas that are connected by an argument, but not by a calculation, so I want to discuss them separately. Before we start it is important to note that their argument does take into account Hawking radiation. The whole process is supposed to happen already outgoing radiation. For this reason the situation is completely time-reversal invariant, which makes it significantly easier to construct a metric. It is also easier to arrive at a result that has nothing to do with reality. So, the one thing that is new in the Haggard and Rovelli paper is that they construct a space-time diagram, describing a black hole turning into a white hole, both with apparent horizons, and do so by a cutting-procedure rather than altering the equation of state of the matter. As source they use a collapsing shell that is supposed to bounce. This cutting procedure is fine in principle, even though it is not often used. The problem is that you end up with a metric that exists as solution to some source, but you then have to calculate what the source has to do in order to give you the metric. This however is not done in the paper. I want to offer you a guess though as to what source would be necessary to create their metric. The cutting that is done in the paper takes a part of the black hole metric (describing the inside of the shell) with an arm extending into the horizon region, then squeezes this arm together so that it shrinks in radial extension no longer extends into the regime below the Schwarzschild radius, which is normally behind the horizon. This squeezed part of the black hole metric is then matched to empty space, describing the inside of the shell. See image below Figure 4 from arXiv: 1407.0989 They do not specify what happens to the shell after it has reached the end of the region that was cut, explaining one would need quantum gravity for this. The result is glued together with the time-reversed case, and so they get a metric that forms an apparent horizon and bounces at a radius where one normally would not expect quantum gravitational effects. (Working towards making more concrete the so far quite vague idea of Planck stars that we discussed here The cutting and squeezing basically means that the high curvature region from inside the horizon was moved to a larger radius, and the only way this makes sense is if it happens together with the shell. So I think effectively they take the shell from a small radius and match the small radius to a large radius while keeping the density fixed (they keep the curvature). This looks to me like they blow up the total mass of the shell, but keep in mind this is my interpretation, not theirs. If that was so however, then makes sense that the horizon forms at a larger radius if the shell collapses while its mass increases. This raises the question though why the heck the mass of the shell should increase and where that energy is supposed to come from. This brings me to the second argument in the paper, which is supposed to explain why it is plausible to expect this kind of behavior. Let me first point out that it is a bold claim that quantum gravity effects kick in outside the horizon of a (large) black hole. Standard lore has it that quantum gravity only leads to large corrections to the classical metric if the curvature is large (in the Planckian regime). This happens always after horizon crossing (as long as the mass of the black hole is larger than the Planck mass). But once the horizon is formed, the only way to make matter bounce so that it can come out of the horizon necessitates violations of causality and/or locality (keep in mind their black hole is not evaporating!) that extend into small curvature regions. This is inherently troublesome because now one has to explain why we don’t see quantum gravity effects all over the place. The way they argue this could happen is that small, Planck size, higher-order correction to the metric can build up over time. In this case it is not solely the curvature that is relevant for an estimate of the effect, but also the duration of the buildup. So far, so good. My first problem is that I can’t see what their estimate of the long-term effects of such a small correction has to do with quantum gravity. I could read the whole estimate as being one for black hole solutions in higher-order gravity, quantum not required. If it was a quantum fluctuation I would expect the average solution to remain the classical one and the cases in which the fluctuations build up to be possible but highly improbable. In fact they seem to have something like this in mind, just that they for some reason come to the conclusion that the transition to the solution in which the initially small fluctuation builds up becomes more likely over time rather than less likely. What one would need to do to estimate the transition probability is to work out some product of wave-functions describing the background metric close by and far away from the classical average, but nothing like this is contained in the paper. (Carlo told me though, it’s in the making.) It remains to be shown that the process of all the matter of the shell suddenly tunneling outside the horizon and expanding again is more likely to happen than the slow evaporation due to Hawking radiation which is essentially also a tunnel process (though not one of the metric, just of the matter moving in the metric background). And all this leaves aside that the state should decohere and not just happily build up quantum fluctuations for the lifetime of the universe or so. By now I’ve probably lost most readers so let me just sum up. The space-time that Haggard and Rovelli have constructed exists as a mathematical possibility, and I do not actually doubt that the tunnel process is possible in principle, provided that they get rid of the additional energy that has appeared from somewhere (this is taken care of automatically by the time-reversal). But this alone does not tell us whether this space-time can exist as a real possibility in the sense that we do not know if this process can happen with large probability (close to one) in the time before the shell reaches the Schwarzschild radius (of the classical solution). I have remained skeptical, despite Carlo’s infinitely patience in explaining their argument to me. But if they are right and what they claim is correct, then this would indeed solve both the black hole information loss problem and the firewall conundrum. So stay tuned... 16 comments: 1. This comment has been removed by the author. 2. Great, but does LQG reduce to GR in the classical limit? 3. Black holes can bounce into white holes - albeit in not so pronounced way, as the theorists are expecting. The white holes are represented here with tips of black hole jets. Best of all, such a proposal is just a reinvention of thirty years old hypothesis of American astronomer La Violette, which has been opposed and ignored with mainstream physics many years for it. In AWT the black holes don't differ from common stars very much and they undergo the occasional eruptions and brightening of their jets. This process can occur even for central hole inside of our Milky Way galaxy and we can already see its history in the Planck spaceprobe data. Fermi X-ray data of galactic plane The gamma ray background looks like bundle of jets which could be explained with less or periodical bursts of black hole, which exhibits a precession, so that every burst targets in different direction. This case just illustrates, the theory is one thing and its phenomenology another one. If you don't understand the physics, the understanding of math will help you in recognition of practical impacts of your theory - and vice-versa indeed. 4. Errata: ".. the understanding of math will NOT help you.." /* does LQG reduce to GR in the classical limit */ Classical limit of every quantum field theory is the classical physics, not relativistic one. The classical physics sits just in the middle between quantum mechanics and general relativity 5. Fascinating! Great post Bee. 6. Spacetime curvature increases inside the event horizon as matter collapses. Doesn't the internal diameter, locally measured proximate to the achieving singularity, thus increase without limit? Density pursues an asymptotic value as collapse unendingly proceeds. Call it the Ouroboros conjecture. Who Kerrs if it spins? 7. I wonder why it is not taken into consideration the negative energy as the sign change of the space-time curvature. This process is a conseuence of space time twist. In this way gravity becames a "repulsive force". This strange fact is build-on inside the Schwarzchild metric. Below Schwarschild radius, metric change sign (ie from + + + - to - - - +). A continuos space-time twist happens inside and outside event horizon. This behaviour of the space-time is visible to our scale by the relativistic precession effect That's only a my idea... what if we also imagine ++--, would there not be six forms? Lubos cites a recent arXiv article (and claims it is part of supersymmetry with tachyons usually). Well, LM, this is the very idea of symmetry I have long described no matter how you spell a name in Czech. But there is a little more than these monster group breaking descriptions of which that is only the beginning and it all can be thought of as positive (or a trade off of dark and light hole effects). It does not solve the hierarchy problem because there is no such problem for such scales are a given. Information just does not bounce back and forth in relation to a projective space with an ideal singularity point at infinity (or many such points). It slips through the structures while we debate the nature of symmetry itself and what effect such abstract models have on the general landscape of the world. 9. I don't know why, but geometrical description in General Relativity is a wonderful model that works very well. I get lost with more than 4 dimension.. said that, it is a matter of fact that in the relativity theory you've always to deal with two time dimensions: your own time and the time reffered to another reference system. These has real physiscal consequences and can't be ignored. 10. MÃ¶bius Strip (or Klein bottle) seems to incorporate both time dimensions in a single manifold 11. This comment has been removed by the author. 12. BTW Which is the exact logics behind quantum bounce of black holes? This theory openly suggest, that the quantum loops do behave like colliding particles when compressed, i.e. like the dense aether. Wouldn't be much more natural to consider, it's the particles of black hole itself, which do resist their collapse in this way (sort of degeneracy pressure inside of neutron star etc)? The collapse limit of space-time sounds somewhat strange for me. My suspicion is, this quantum bounce model is actually string theory ones - after all, before some time the Mathur predicted the limit of density of black holes from closest packing of strings. 13. Thank you MarkusM! The video of 't Hooft is very much interesting for me! 14. Yes MarkusM, thanks for the link - a very clear description. Even in an infinite honeycomb of space the idea of "close packed strings" is only a partial part of the logical (mathematical) picture. If you mean particles along the lines of a Higgs then what you say may make sense but it would have five successive dualities in cycle corresponding to two orders of bounce. Even then it would be Euclidean as a model. Nature certainly seems to have a dynamics where say at a certain point a knot breaks or a phase change happens at some radiation pressure. We can compress by sound bubbles that explode in light. and so on. But what is explained anywhere along the cycle of five (or if you will SO(10)) is more general than string theory in aether flatland. It only takes the removal of one particle node to close the honeycomb into what seems a finite but bounded space. The classical limit as physics also is outside, far from the middle of qm and gr relativity scales. 15. This comment has been removed by the author. 16. I would like to point out a paper who was awarded an Honorable Mention in the Gravity Research Foundation 2014 Essay Contest and in which the same black-hole-to-white-hole spacetime metric was obtained, although with different motivation and background: This work is a follow-up of the following paper, posted 4 years ago and in which this bouncing solution for gravitational collapse was motivated: COMMENTS ON THIS BLOG ARE PERMANENTLY CLOSED. You can join the discussion on Patreon. Note: Only a member of this blog may post a comment.	{"url":"https://backreaction.blogspot.com/2014/07/can-black-holes-bounce-to-white-holes.html","timestamp":"2024-11-04T15:30:52Z","content_type":"application/xhtml+xml","content_length":"201728","record_id":"<urn:uuid:f532645c-7bb4-4d4f-b391-9eac85623ca5>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00184.warc.gz"}
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Pi: The most important constant? Kai G and Charlton C You’ve probably come across Pi in the context of circles. A circle is a shape consisting of all the points in a plane that are at a given distance, the radius, from the given point, the centre. The diameter is twice the radius, defined as a line segment between two endpoints that lie on the circle passing through the centre. The distance around the circle is its circumference. Pi can be defined as the ratio between the circumference and diameter of a circle, which is always the same. Pi is defined by an infinite decimal expansion, meaning that it has an infinite number of digits, and cannot be expressed exactly as a ratio. This makes it an irrational number, a fact first discovered by Johann Heinrich Lambert in 1761. This means we can only approximate it, which is why in equations it is represented by the symbol π. Pi is also transcendental, meaning that Pi is not the root of any algebraic equation with rational coefficients. It even has its own holiday! Why Pi? The letter π is the first letter in the Greek word for “periphery”, the “periphery” of a circle being the precursor to the “perimeter” of a circle, which today we call a circle’s circumference. Welsh mathematician William Jones chose the Greek letter as the constant’s symbol in 1706, but it was mostly popularised by the great Leonhard Euler, who adopted it in 1737. Pi is an ancient mathematical curiosity The constant has, of course, not always been called Pi. People have known about, and been intrigued by Pi for a long time. Almost 4,000 years. The ancient Babylonians calculated the area of a circle by multiplying the square of its radius by three. One of their tablets from around 1900–1680 BCE indicates they approximated Pi to 25/8, a value of 3.125. The famous Egyptian Rhind Papyrus, from around 1650 BCE shows the Egyptians could calculate the area of a circle by a formula that gave the approximate value for Pi as 3.16045, which is even closer to the real value. Pi is actually a part of Egyptian mythology. People in Egypt believed that the pyramids of Giza were built on the principles of pi. The vertical height of the pyramids have the same relationship with the perimeter of their base as the relationship between a circle’s radius and its circumference. The pyramids are phenomenal structures and are one of the seven wonders of the world. Archimedes of Syracuse, in Sicily, who lived from 287 to 212 BCE, used the Pythagorean Theorem and regular polygons to approximate the shape of a circle. He recognised that he had not found the exact value but showed it to be within 3 1/7 and 3 10/71. In the fifth century, brilliant Chinese mathematician and astronomer Zu Chongzi estimated the value of Pi to be 355/113. That is a number that starts 3.141592… which is correct to six decimal places. To be this accurate, he must have started with a regular polygon with 24 thousand 576 sides and performed calculations with hundreds of square roots to 9 decimal places. In the early twentieth century, the Indian mathematician Srinivasa Ramanujan established several formulae for calculating Pi. His approach forms the basis of some of the fastest algorithms used to calculate Pi. The first term of the series gives a value that is correct to six decimal places. The first two terms give a value correct to 14 decimal places. The calculation of Pi is a stress test for a computer. It works just like a digital cardiogram since it indicates the level of activity within the computer’s processor. Computers revolutionised the accuracy with which we could predict Pi. In 1948, Ferguson and Wrench achieved a result correct to 808 places. In December 2002, 10 mathematicians led by Professor Yasumada Kanada took more than a hundred hours of computer time to calculate the value of Pi correctly to an incredible 1.24 trillion digits. 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 Pi to 100 digits Why is Pi useful? Pi allows you to find the circumference or area of circle using 2πr and πr^2 respectively. This is crucial for basic construction and calculation. Pi is also used to calculate the surface areas and volumes of cylinders, cones and spheres. Area of a Circle A = πr^2 Circumference of a Circle C = 2πr Volume of a Cylinder V = πr^2h Surface Area of a Cylinder A = 2πr^2+2πrh Volume of a Cone V = πr^2h/3 Surface Area of a Cone A = πr(r+√(h^2+r^2)) Volume of a Sphere V = 4πr^3/3 Surface Area of a Sphere A = 4πr^2 r = radius h = height You can find out more in our Shapes and Solids notebook guide. The constant Pi helps us understand our universe with greater clarity. The definition of Pi inspired a new unit of measurement for angles, called radians. 360 degrees is equal 2π radians. A radian is defined as the angle at the centre of a circle for an arc that is the length of the radius. This is useful for navigation, triangulating locations and working out precise distances. Pi is also used in calculus to calculate integrals and derivatives. It is claimed that Pi even describes the bendiness of rivers. You can measure how “bendy” a river is by measuring its total length and dividing by straight route from its source to mouth, this measure is called “sinuosity”. So a totally straight river would have a sinuosity of 1, while very bendy rivers can have very high sinuosity, with no limit to how high it can go. Yet, it is claimed the average sinuosity of rivers around the world is Pi. This is an incredible fact, and if true means that rivers are typically a little over three times longer than the direct route from source to mouth. Of course some rivers are straighter, and some rivers are longer, but the average sinuosity is around 3.14. So, it is a good bet you have encountered Pi very often in your life — even if you don’t realise it! Is Pi actually wrong? If you think modern mathematics is static and unchanging, you are very wrong. A prime example of this is the Tau movement — modern mathematicians who believe Pi is wrong. Now, nobody is arguing that C/d = π, as shown in the first diagram. However, Pi can be a bit of a handful when doing Trigonometry (sine and cosine functions, for example) with Radians, with 2π being the slightly arbitrary full cycle of a circle. Tau, roughly 6.28318, would be equal to 2π and define the full cycle of a circle in radians. Tau is equal to a circle’s Circumference (C) divided by its radius (r), instead of C/d. This is gaining traction, however Pi is still used as standard. One of the challenges Tau faces is that Pi is more useful for day-to-day calculations using circles, while those who use radians on a regular basis would find Tau far more useful, however the debate remains interesting. What do you think? Pi Day Pi Day, also known as International Mathematics Day, is the most famous holiday in mathematics. It is celebrated on 14 March, as in the date format used by the United States, Canada and Belize, 14 March is represented by 3/14/YYYY which looks quite a bit like Pi! An alternative celebration of Pi is Pi Approximation Day, celebrated on 22 July. One of Pi’s most accurate approximations used a lot in early mathematics is 22/7, so it is an alternative celebration for countries using the dd/mm/yyyy date format, as there is no fourteenth month. It is marked in a similar fashion to the more popular Pi Day. This holiday was first celebrated in 1988, in an event held by a physicist at the San Francisco Exploratorium, which involved the consumption of fruit pies. Pi day aims to raise awareness about the importance of mathematics and its role in shaping our world, as well as to promote the beauty and relevance of mathematics to a wider audience. The alternative to Pi, Tau, also has its own holiday. Tau Day is celebrated on 28 June by mathematicians who prefer Tau to Pi, and use the day to promote the use of the alternative constant. However, it is less well known than Pi or Pi Approximation Day. The day is often celebrated by outreach events and the eating of pies, often decorated with a Pi symbol and the first few digits of Pi. It can also be marked by playing mathematical card games and maths jokes. As an interesting coincidence, Albert Einstein was born on 14 March, the day which became Pi Day. The famous German-born theoretical physicist is widely held to be one of the greatest and most influential scientists of all time. Is Pi actually the most important constant in mathematics? Although many mathematicians might prefer Tau as the circle constant to Pi, or find Euler’s number, e, or the imaginary unit, i, more important for their particular area of maths, it would be hard to argue against Pi’s central place in mathematics. We use it every day. Only you can decide. But it would be impossible to deny the incredible impact this almost magical number has had on all mathematics.	{"url":"https://math-soc.com/pi/","timestamp":"2024-11-14T17:10:25Z","content_type":"text/html","content_length":"58533","record_id":"<urn:uuid:bdb32570-20c1-4da6-ba16-761fca2d3ea8>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00408.warc.gz"}
Fool's Gold 4: The Brittleness of Algorithmic LDA Credit Models The pressure on lenders to adopt less discriminatory alternative ("LDA") credit models is accelerating - hastened by the CFPB's more intensive examinations of their policies and procedures for disparate impact testing and the search for fairer alternatives.[1] Interestingly, however, is that despite the proliferation of algorithmic LDA solutions - as well as public entreaties by multiple advocacy groups for the CFPB to require their use,[2] the CFPB has yet to publicly endorse these technological innovations, stating recently: "Further, as the CFPB continues to monitor markets and institutions for fair lending compliance, the CFPB will also continue to review the fair lending testing regimes of financial institutions. Robust fair lending testing of models should include regular testing for disparate treatment and disparate impact, including searches for and implementation of less discriminatory alternatives using manual or automated techniques. CFPB exam teams will continue to explore the use of open-source automated debiasing methodologies to produce potential alternative models to the institutions’ credit scoring models." - June 2024 Fair Lending Report of the Consumer Financial Protection Bureau (emphases mine) What are we to make of this statement? In my opinion, by referencing both automated and manual LDA search methods - as well as ongoing research into "open-source automated debiasing tools"[3], the CFPB is communicating - at least publicly - a position of technological agnosticism as to how LDA credit models should be derived. While some may be impatient with their reticence to endorse these new tools - particularly given their oft-stated concerns about algorithmic discrimination, I find their measured approach both admirable and prudent given the dearth of publicly-available, objective research on the potential risks these complex new tools could have on consumers and lenders. And, as readers of this blog know, this caution is well-placed given the potentially serious safety-and-soundness and compliance risks I have found in my own testing. But even this testing - broad as it has been - hasn't fully explored the totality of risks associated with this complex black-box technology. For example, while my prior research focused on the fundamental technical and legal foundations of these algorithmic fairness tools, they were silent on a critical risk dimension that's now becoming increasingly relevant to many lenders' risk and compliance managers – the questionable robustness and stability of these LDA Models.[4]. In fact, with more algorithmically-driven LDA Models moving into production, recent industry chatter suggests that some may not be performing as expected on real-time application data - with lower than expected fairness performance and higher than expected default rates[5] necessitating risk mitigations and raising the following questions. Are these adverse performance outcomes simply a matter of significant "data drift" coincidentally encountered at or near the time of LDA Model deployment, or do algorithmically-debiased credit models have more fundamental weaknesses that we have yet to identify? To explore these important questions, I extend the analytical framework from my previous Fool's Gold research studies to evaluate the inherent stability / robustness of algorithmically-derived LDA Models using standard model validation techniques. For those unfamiliar with the field of model risk management, a "stable" or "robust" model is one whose estimated predictive relationships and overall predictive performance are highly-consistent across different data input samples.[6] This important property of a conceptually and technically sound credit model provides users with confidence that: (1) the model's predictive performance will generalize outside of the training data to which it was calibrated, and (2) its estimated credit risk relationships are fundamentally sound as they are likely grounded in more causal, theoretically-based borrower behaviors, as opposed to reflecting mere statistical artifacts present in the specific training sample used. Although I walk through my stability / robustness analyses in the main section below with a broad audience in mind, I realize that some readers may simply wish to know the major findings - which I summarize here for convenience. • Common LDA Models relying on outcomes-based fairness metrics - such as the Adverse Impact Ratio ("AIR") - appear to be inherently unstable and brittle in the presence of relatively small random training data differences. As I show below, during LDA Model training, relatively small random variations in the training sample can cause widely different solutions to the algorithmic debiasing ("AD") process for the same Fairness Weight - thereby yielding significantly different: (1) LDA Model structures (i.e., different estimated credit risk relationships), (2) fairness and accuracy performance measures, and (3) primary disparate impact factors (i.e., the credit model attributes whose risk weights are altered the most by the AD process to improve model fairness - much more on this later). In effect, the "multiplicity of models" concept on which algorithmic debiasing conceptually rests operates at even deeper levels than commonly understood - also applying to each of the LDA Models generated by common algorithmic fairness tools at a given Fairness Weight. • This LDA Model brittleness appears to increase as larger Fairness Weights are used in the algorithmic debiasing process. That is, as the Fairness Weight increases to generate LDA Models with increasingly higher AIRs, the estimated credit risk relationships within these models become ever more brittle, unstable, and less conceptually sound. Additionally, the model's AIR and Area Under the Curve ("AUC") performance metrics also become more volatile - creating a risk of misleading fairness and accuracy performance expectations once deployed into production. • LDA Model brittleness also imparts statistical bias into estimated risk coefficients, AIR-based fairness values, and AUC-based accuracy values relative to those estimated from a larger, population-based LDA Model with the same predictive factors. For example, using 100 training data samples randomly drawn from the same population, I found that the average values of these LDA model outputs across these samples were, in general, materially different from those generated by an LDA Model estimated on the whole population. I note that such statistical biases did not exist when these same samples were used to estimate Base Models (i.e., with no algorithmic debiasing). In addition to these safety-and-soundness risks, my analyses also identified further compliance-related concerns linked to the use of algorithmic LDA Models. Specifically, • AIR-based fairness metrics for Base Models also vary non-trivially in response to small random training data differences. Accordingly, if a lender's fair lending policy requires a search for LDA Models only when the Base Model's AIR falls below a specific threshold, the lender should be cautious in relying on an AIR performance metric derived from a single model training, validation, or test sample. • For a given training sample, the primary disparate impact factors selected by the AD process to improve the LDA Model's fairness performance can vary with the specific Fairness Weight used. Such brittleness in the specific set of model attributes that are "de-biased" to improve fairness raises, in my opinion, a significant concern that these algorithmic fairness tools are not mitigating the impact of specific "artificial, arbitrary, and unnecessary" credit decision factors per relevant federal law, regulation, and Supreme Court opinion.[7] Instead, the AD process appears to select the primary model attributes to "de-bias" simply based on whether they possess the statistical and demographic properties needed to achieve the desired degree of AIR improvement through reverse disparate impact. • While the LDA Model improves AIR-based fairness between demographic groups, a deeper review of the associated credit decision changes reveals that this inter-group fairness improvement comes at the expense of diminished fairness among individuals within demographic groups. For example, in my analysis: (1) Approximately 30% of Black applicants whose credit decisions are changed by the LDA Models are actually "swapped out" - that is, they are denied under the LDA Model even though they were approved by the Base Model. (2) Given the brittleness in the primary disparate impact factors selected at different Fairness Weights, the specific set of Black applicant swap-ins can change notably depending on the specific Fairness Weight used during training. (3) The Black swap-ins selected for approval by the LDA Model may possess risk attributes that are intuitively worse than the risk attributes of the Black swap-outs (e.g., higher CLTVs). Accordingly, while overall group-level fairness may improve with LDA Models, this aggregate improvement abstracts from the significant number of winners and losers among the individuals within the protected class group - many of whom may not view their new credit decisions as fair.[8] In the next section, I provide a quick summary of my LDA Model analytical framework that regular readers of this blog should recognize. Thereafter, I provide a deeper dive into the analyses supporting each of the key findings above. The LDA Credit Model Analytical Framework My analyses were performed using the same credit scoring model, underlying consumer credit application data, and algorithmic debiasing techniques I used in my three previous "Fool's Gold" studies. For those unfamiliar with those articles, I recommend starting with Fool’s Gold: Assessing the Case For Algorithmic Debiasing (the "Fool's Gold article") as a primer to the data, methods, and analyses I use here. In summary, I created a synthetic credit performance dataset from a 2019 sample of residential home mortgage loan applications obtained from public Home Mortgage Disclosure Act ("HMDA") data.[9] As illustrated in Figure 1 below (excerpted from Figures 1 and 2 of the Fool’s Gold article), this synthetic credit performance dataset contains 456,761 total records - 94% of which are White and 6% of which are Black. Additionally, while the overall “default” rate in the dataset is 7.9%, it varies between the two demographic groups with Black borrower default rates over 2.5x that of White The following table labeled as Figure 2 is excerpted from Figure 3 of the first Fool’s Gold article and presents the credit scoring model (the "Base Model") I trained on the full synthetic credit performance dataset via logistic regression.[10] It relies on three primary credit risk attributes - combined loan-to-value ratio (“CLTV”), debt-to-income ratio (“DTI”), and loan amount - all three of which are represented with a series of dummy variables to capture potential non-linearities in their relationships with borrower default behavior. As expected, the estimated odds ratios for each predictive attribute capture a monotonically-increasing risk of default for borrowers with larger CLTV and DTI values and a monotonically-decreasing risk of default for borrowers with larger Loan Using the standard fairness (AIR) and accuracy (AUC) metrics of common algorithmic fairness tools, I obtain an AIR value of 0.865 and an AUC value of 0.843 for this Base Model (plotted in green in Figure 3 below which is excerpted from Figure 9 of the first Fool’s Gold article). However, to improve the model's fairness performance, I apply an algorithmic debiasing methodology commonly referred to as “fairness regularization” (aka “dual optimization”) to the Base Model. In general, this debiasing approach adds an AIR-based "unfairness" penalty to the Base Model's training objective to create a broader definition of model performance that includes both predictive accuracy and fairness. LDA Models can now be trained by optimizing this dual training objective using different "Fairness Weights" (i.e., a number that governs the relative importance of predictive accuracy and fairness in the model's dual training objective). By varying the relative importance of accuracy and fairness in the training process, we obtain a spectrum of LDA Models with increasing AIR-based fairness and, generally, decreasing degrees of AUC accuracy as displayed below in Figure 3. While many LDA Models are generated for a given training sample using different Fairness Weights, a lender typically selects one specific LDA Model from this set (such as that highlighted in red) in which the improved fairness is achieved with an "acceptable" tradeoff in predictive accuracy. With this general background on my LDA Model analytical framework, I now turn to how I extended this framework to investigate the stability of the LDA Models derived therefrom. Exploring LDA Model Performance Stability: Evidence From 100 Training Samples My 100 Training Samples For the purposes of my previous analyses, I trained the models and performed the testing using the entire synthetic credit performance dataset (456,761 records). Now, however, for the purpose of evaluating model robustness and stability, my focus is on how the structure and performance of these models vary across multiple random training samples drawn from the original population.[11] To this end, I created 100 training samples - each of size 100,000 - by randomly sampling my synthetic credit performance dataset 100 times.[12] Given each sample's relatively large sampling rate (i.e., 22% of the full dataset), I expect that the Base Models trained on these 100 samples using identical sets of predictive attributes would exhibit a high degree of structural and performance similarity. This is because each of the 100 training samples should differ only slightly from each other in terms of credit risk profiles and default outcomes - which I show in Figure 4 below using basic sample descriptive statistics. As expected, the 100 random training samples are, on average, representative of the full synthetic credit population (i.e., 456,761 records) with negligible mean differences (numerical column 2 vs. numerical column 1) and relatively small variability around the sample means (numerical column 3). Using the 100 Training Samples to Create a Set of Performance Stability Benchmarks To evaluate properly the LDA Models' structural and performance stability across these 100 training samples, I first create a set of benchmarks that can be used as "apples-to-apples" comparators for the LDA Models' stability tests. More specifically, these benchmarks will help me determine whether the various measures of the LDA Models' structural and performance variability across the 100 training samples are consistent with those of the corresponding Base Models (and, therefore, are simply due to the small random differences across the underlying training data), or something more To create these benchmarks, I estimate a set of 100 Base Models - one for each of the 100 training samples - and calculate the following Base Model stability metrics: • The variability of the Base Models' estimated credit risk profiles across the 100 samples (i.e., the standard deviations of the estimated coefficients for each risk attribute). • The statistical bias in the Base Models' estimated credit risk profiles (i.e., the difference between the Base Model's average estimated coefficients (across the 100 samples) versus those estimated on the entire synthetic credit performance dataset). • The variability of the AIR-based fairness and AUC-based accuracy metrics across the 100 samples (i.e., the standard deviations of the AIR and AUC metric values). The next sections provide further detail on the derivation of these benchmarks. The Variability of the Base Models' Estimated Credit Risk Profiles Across the 100 Samples After estimating the 100 Base Models using the credit model specification contained in Figure 2, I analyzed the variability of the estimated CLTV, DTI, and Loan Amount credit risk profiles - displayed below in Figures 5a-c. For each credit risk attribute, the blue lines reflect the corresponding set of model coefficient estimates (i.e., the estimated credit risk profile) for each of the 100 Base Models while the green line represents the estimated profile from the population-based Base Model. Based on these estimated credit risk profiles, I note that, in general, the Base Models' profiles exhibit a relatively high degree of stability across the 100 training samples. That is, we can clearly see in these charts that the Base Models' estimated credit risk relationships have a relatively high degree of directional and value consistency across the 100 training samples - clustering relatively closely together to varying degrees. More quantitatively, Figure 6 below presents both the average coefficient values (numerical column 2) as well as the standard deviations of these values across the 100 samples (numerical column 3). Later, I will use these standard deviations as a benchmark to evaluate the structural stability of the LDA Models' estimated credit risk profiles on these same training samples. Figure 6 - Base Model: Sample vs. Population Coefficients The Statistical Bias in the Base Models' Estimated Credit Risk Profiles Based on the data presented in Figure 6, I also note that the 100 Base Models yield largely unbiased estimates of the population-based Base Model's credit risk relationships - where the estimated bias (numerical column 4) is calculated as the difference between the average estimated model coefficients for the 100 Base Models (numerical column 2) and those of the population-based Base Model (numerical column 1).[13] Later, I will use these bias estimates as a benchmark to evaluate the bias of the LDA Models' estimated credit risk profiles on these same training samples.[14] The Variability of the Base Models' AIR-Based Fairness and AUC-Based Accuracy Metrics Across the 100 Samples Finally, Figure 7 below displays the variability of the Base Models' fairness and accuracy metrics across the 100 training samples (blue dots) - with AIR-based fairness performance measured on the horizontal axis and AUC-based accuracy performance measured on the vertical axis. For reference, the performance metrics for the population-based Base Model are depicted with a green dot. Based on this data, I note that the sample-based Base Model AIR and AUC vary non-trivially across the 100 training samples - particularly the AIR - but, nevertheless, provide unbiased measures of the Base Model's population-based fairness and accuracy performance. Later, I will use these variability estimates as a benchmark to evaluate the stability of the LDA Models' fairness and accuracy performance on these same training samples. I also note that the 100 Base Models' fairness performance varies twice as much as their accuracy performance as measured by their corresponding standard deviations. This feature is likely driven, in part, by the smaller Black applicant sample sizes (relative to Whites) that create inherently larger variability in Black approval rates (i.e., a given change in loan approvals for each group has a larger impact on the Black approval rate than the White approval rate).[15] One important implication of this behavior is the following. If a lender's fair lending policy requires a search for LDA Models when the Base Model's AIR falls below a specific threshold, the lender should be cautious in relying on a single AIR performance metric from its model development process. Due to the underlying AIR variability noted above, reliance on a single sample's AIR metric may provide a misleading signal of the Base Model's more general fairness performance. This becomes particularly troublesome when the underlying AIR variability straddles common AIR fairness thresholds - such as 0.80 or 0.90. In such cases, the lender may conclude that the Base Model does not trigger LDA search requirements when, in fact, it may - and vice versa.[16] Evaluating the LDA Model's Performance Stability Using the Base Model Benchmarks With these benchmarks in hand, I now apply the algorithmic debiasing method of fairness regularization (aka "dual" or "joint" optimization) to create a set of 20 LDA Models for each of the 100 Base Models and their corresponding training samples. Each of the 20 LDA Models corresponds to a different Fairness Weight (i.e., the relative importance of fairness vs. accuracy in the model training process) ranging between 0.1 and 2.0 in increments of 0.1[17] Overall, this yields a total analysis sample of 2,000 LDA Models. The Variability of the LDA Models' Estimated Credit Risk Profiles Across the 100 Samples For my first analysis, I evaluate the variability of the LDA Models' estimated credit risk relationships relative to the corresponding Base Model variability benchmarks. Figures 8a-c below illustrate these estimated relationships - separately for CLTV, DTI, and Loan Amount - for LDA Models estimated using 12 of the 20 Fairness Weights.[18] I also include the Base Model benchmarks for reference and discuss my interpretation of these charts further below. Each set of charts depicts how the fairness regularization process algorithmically alters the set of estimated coefficients for one of the Base Model's credit risk attributes as the Fairness Weight increases from a low of 0.1 to a high of 2.0. Additionally, each chart shows how this fairness-driven Base Model alteration varies across the 100 training samples. Because there is a lot of interesting information in these charts, I think it's worthwhile to do a detailed walkthrough. Accordingly, consider Figure 9 below that focuses on the estimated CLTV relationship for LDA Models trained with a Fairness Weight of 0.1 (obtained from the first chart in Figure 8a). This chart, as with all other charts in Figures 8a-c, can be interpreted as follows: • The red lines represent the estimated CLTV credit risk profiles for the 100 LDA Models trained using a Fairness Weight of 0.1. • The dark blue lines represent the estimated CLTV credit risk profiles for the 100 Base Models trained on the same data samples and are presented here as a benchmark. • The yellow line represents the estimated CLTV credit risk profile of the population-based LDA Model (i.e., trained on the full synthetic credit performance dataset with a Fairness Weight = 0.1) and is presented here as a benchmark. • The green line represents the estimated CLTV credit risk profile of the population-based Base Model (i.e., trained on the full synthetic credit performance dataset) and is presented here as a Upon review of Figure 9 - as well as all the charts in Figure 8a-c, it is clear that the random training sample variability has a greater impact on the LDA Models' estimated credit risk relationships than it does on the Base Models'. In particular, As the Fairness Weight increases (to achieve better AIR-based fairness performance), algorithmic debiasing tends to alter the credit model in ways that create increasing LDA Model "brittleness". That is, the AD process causes the LDA Models' risk coefficient estimates to become increasingly unstable as the Fairness Weight increases. For example, even with a relatively low Fairness Weight = 0.3, Figure 10 below shows the significant volatility in the LDA Models' estimated CLTV risk profiles across the 100 training samples: Rather than being tightly clustered together as we see with the Base Models (the blue lines), the CLTV risk coefficient estimates now appear to separate into distinct "clusters" with materially different profiles. Furthermore, as the Fairness Weight continues to increase, this volatility grows even larger (see Figures 8a-c). Figures 11a-c below illustrate this LDA Model brittleness more quantitatively for each of the three credit risk attributes. In each chart, I calculate - for each risk coefficient - the ratio of: (1) the LDA Models' risk coefficient variability across the 100 training samples (as measured by its standard deviation) to (2) the Base Models' risk coefficient variability benchmark (also a standard deviation and contained in Figure 6 above). These ratios are depicted by the colored lines. I also include a dashed horizontal line at a value of 1.0 - indicating, as a reference, a neutral ratio value where the LDA and Base Model risk coefficient standard deviations are the same. Using the CLTV risk attribute as an example, here we see - in general - that all CLTV risk coefficients possess significantly more variability in the LDA Models (i.e., all ratios are greater than 1 at Fairness Weights greater than 0). As a more specific example, we see that with a Fairness Weight = 0.3 (see Figure 10), the 100 LDA Models' CLTV 80-85% risk coefficients have a standard deviation of 0.615 - 12.3x higher (depicted by the green line) than the standard deviation of the same coefficients estimated for the corresponding 100 Base Models. At even larger Fairness Weights (i.e., above 1.3), this coefficient's instability grows even more rapidly - reaching a maximum value of 31.6x that of the Base Models at a Fairness Weight = 2.0.[19] Due to this increasing LDA Model structural brittleness / instability, I note that: AIR-based algorithmic debiasing increasingly distorts the conceptual soundness of estimated LDA credit risk profiles by reducing their monotonicity as the Fairness Weight increases. For each Fairness Weight (including the 0 Fairness Weight for Base Models) and for each credit risk factor (i.e., CLTV, DTI, and Loan Amount), Figure 12 below indicates the percentage of the 100 models - estimated at each Fairness Weight - that exhibit monotonically increasing or decreasing credit risk profiles.[20] For example, across the 100 Base Models (Fairness Weight = 0), 90% of the estimated CLTV credit risk profiles (the blue line) are monotonically increasing. That is, at each successively higher CLTV range, the Base Models' estimated default risk is at least as great as the estimated default risk associated with the previous (i.e., lower) CLTV range. However, as the Fairness Weight increases to produce LDA Models with greater AIR-based fairness, we see that these models exhibit decreasing levels of risk profile monotonicity - with only 56% of LDA Models estimated with a Fairness Weight = 0.3 having monotonic CLTV risk profiles (see Figure 10 above), only 25% of LDA Models estimated with a Fairness Weight = 1.0 having monotonic CLTV risk profiles, and only 8% of LDA Models estimated with a Fairness Weight of 2.0 having monotonic CLTV risk profiles. This pattern is generally consistent with that exhibited by the two other credit risk factors in Figure 12 - DTI (the red line) and Loan Amount (the green line). What is the cause of this structural brittleness? As I discuss in more detail later in this post, it appears that small random differences in training data can cause the AD process - for a given Fairness Weight - to select different sets of credit risk attributes as the primary means to drive greater AIR-based fairness performance (I call these attributes the "primary disparate impact factors"). These different sets of primary disparate impact factors, which appear as different credit risk profile "clusters" in Figures 8a-c, are really just different solutions to the same LDA Model training exercise whose emergence is triggered by random perturbances in the model training data. Furthermore, the emergence of these multiple solutions is facilitated by a decreasing relative weight on model accuracy during model training - permitting the AD algorithm even more freedom to select different sets of primary disparate impact factors to achieve the desired level of fairness without the limiting constraint of high model accuracy.[21] The Statistical Bias of LDA Models' Estimated Credit Risk Profiles Looking across the charts in Figures 8a-c, we can see a few other interesting features. AIR-based algorithmic debiasing appears to inject material statistical bias into the LDA Models' estimated credit risk profiles. Unlike the Base Models' estimated credit risk coefficients that were near-universally unbiased relative to the population-based Base Model's estimates (see Figure 6), Figures 8a-c show that the LDA Models' estimated credit risk profiles (the red lines) differ materially from those of the population-based LDA Model (the yellow line) - indicating some type of inherent statistical bias in the LDA estimation process. This is quantitatively confirmed in Figures 13a-c below where I display the calculated statistical bias levels for all three credit risk factors for each LDA Fairness Weight. Using the CLTV risk profile as an example, Figure 13a plots - for each of the five CLTV ranges and for each Fairness Weight - the ratio of: (1) the average risk coefficient value across the 100 LDA Models relative to (2) the corresponding population-based LDA risk coefficient value. Ratio values equal to 1 (highlighted by the middle dashed horizontal reference line) correspond to the absence of estimation bias - that is, on average, the 100 risk coefficients estimated from the training samples equal the population-based estimate. Consistent with my discussion of Figure 6, we see that all CLTV risk coefficients estimated with a Fairness Weight = 0 (i.e., the Base Models) cluster close to this reference line - consistent with the general absence of statistical bias for the 100 sample-based Base Models. However, across the range of LDA Fairness Weights (i.e., from 0.1 to 2.0), we can see many instances of material statistical bias in the sample-based LDA Models' risk coefficient estimates. In fact, in some cases, the bias estimates are so large that I truncated them so as not to distort the visual display of the results for the remaining data points. These dashed truncation lines can be seen at the top and bottom of the charts. One of the reasons for this statistical bias can be seen in the following chart (Figure 14 - obtained from Figure 8a) for the estimated CLTV risk profiles using a Fairness Weight = 1.1. Similar to my discussion of Figure 10 where the Fairness Weight = 0.3, here we can see how the LDA Models' estimated CLTV risk profiles separate into even more distinct "clusters" at this higher Fairness Weight. For example, the CLTV>95% coefficient varies across 3-4 distinct clusters with values varying around 4 and around 1. We can also see the impact of these clusters on some of the other CLTV coefficients - such as CLTV 80-85% and CLTV 90-95%. As I discussed at the end of the previous section (and will discuss in more detail later), these clusters appear to be formed when different solutions to the LDA Model training exercise emerge - triggered by the random perturbances in the training data. However, with the existence of multiple solutions, statistical bias is sure to be present since the distinct risk profile patterns (from the different solutions) cannot average out to the single population-based CLTV credit risk profile (i.e., the yellow line in Figure 14) that reflects only one solution. The Variability and Bias of the LDA Models' Fairness Metrics Across the 100 Samples Next, I evaluated the variability of the 100 LDA Models' fairness and accuracy performance metrics - for each Fairness Weight - relative to the Base Model benchmarks. For the AIR-based fairness metric, I simply calculated - for each of the 2,000 LDA Models - the expected approval rates for Black and White applicant groups assuming a 90% overall approval rate.[22] These expected approval rates were then converted to the AIR fairness metric, segmented by LDA Model Fairness Weight, and plotted in Figure 15 below. In this figure, for each Fairness Weight, each of the 100 LDA Models' AIRs are plotted using green dots - with darker green dots associated with a greater concentration of LDA Models at that particular AIR value. Overlaid onto this figure, with a solid green line, is the average AIR value across the 100 training samples for each Fairness Weight (i.e., the mean AIR for all 100 LDA Models estimated with this Fairness Weight). For reference, the AIR values associated with the population-based LDA Model - for each Fairness Weight - is denoted with the blue line.[23] Based on these results, I note the following observations. While the LDA Models' mean AIRs generally increase with higher Fairness Weights, the individual models exhibit increasing AIR variability within a Fairness Weight - with a growing rightward tail toward higher AIR values. For example, with a Fairness Weight = 0.3, the mean AIR = 0.891 with a standard deviation of 0.008. At a higher Fairness Weight = 1.0, the mean AIR = 0.917 with a standard deviation of 0.021. And at a Fairness Weight = 1.8, the mean AIR = 0.946 with a standard deviation of 0.036.[24] These standard deviations are all larger than the Base Models' AIR standard deviation benchmark of 0.006 and increase with higher Fairness Weights. This widening (not just shifting) of the LDA Models' AIR distribution can be seen even more clearly in Figure 16 below where I show the joint distribution of the 100 LDA Models' AIRs and AUCs for four these Fairness Weights (0.1, 0.3, 1.0, and 1.8) along with associated descriptive statistics. When compared to the Base Model AIR-AUC joint distribution benchmark in Figure 7, we see that as the Fairness Weight increases, the joint AIR-AUC distribution rotates and elongates - with increasing standard deviations for both metrics. What this means is that while higher Fairness weights drive a greater average AIR and typically a lower average AUC (see Figure 17 below), there can be significant variability observed around these mean values due to random training sample variability. Similar to previous discussions, a primary contributor to this volatility is the emergence of multiple LDA Model solutions triggered by random training sample perturbances. These solutions involve the selection of different sets of primary disparate impact factors which drive different approval rates across demographic groups and, therefore, different fairness metrics. As the Fairness Weight increases, more solutions may become viable due to the down-weighting of accuracy in the model training objective - resulting in even more variability in LDA Model fairness and accuracy metrics across the 100 samples. In terms of statistical bias, I note that: The LDA Models' mean AIR at a given Fairness Weight can depart materially from that of the population-based LDA Model - indicating the risk of statistical bias in a lender's AIR fairness metric. The blue line in Figure 15 reflects the AIR value associated with the population-based LDA Model at each Fairness Weight. When compared to the corresponding average AIR values from the sample-based LDA Models (the green line), we can see notable divergences between the two - particularly at lower Fairness Weights (i.e., between 0.3 and 1.0). As I will discuss later, the divergence here is due to the AD process's selection of different sets of primary disparate impact factors for the sample-based LDA Models versus the population-based model at the same Fairness Weight. There is a greater divergence between these sets of factors at Fairness Weights less than 1.1 and closer alignment at 1.1 and above. The Variability and Bias of the LDA Models' Accuracy Metrics Turning now to the LDA Models' AUC-based accuracy results, Figure 17 below plots for each Fairness Weight, each of the 100 LDA Models' AUCs using red dots - with darker red dots associated with a greater concentration of LDA Models at that particular AUC value. Overlaid onto this figure, with a solid red line, is the average AUC value across the 100 training samples for each Fairness Weight (i.e., the mean AUC for all 100 LDA Models estimated with this Fairness Weight). For reference, the AUC values associated with the population-based LDA Model - for each Fairness Weight - is denoted with the blue line. Based on these results, I note the following observations. While the LDA Models' mean AUCs generally decrease with higher Fairness Weights, the individual models exhibit increasing AUC variability within a Fairness Weight - with a growing leftward tail toward lower AUC values. For example, with a Fairness Weight = 0.3, the mean AUC = 0.837 with a standard deviation of 0.005. At a higher Fairness Weight = 1.0, the mean AUC = 0.810 with a standard deviation of 0.023. And at a Fairness Weight = 1.8, the mean AUC = 0.766 with a standard deviation of 0.051.[25] These standard deviations are all larger than the Base Models' AIR standard deviation benchmark of 0.003 and increase with higher fairness weights. This widening (not just shifting) of the LDA Models' AUC distribution can be seen even more clearly in Figure 16 above where I show the joint distribution of LDA Model AIRs and AUCs for these four Fairness Weights (0.1, 0.3, 1.0, and 1.8) along with associated descriptive statistics. My conclusions here are similar to those for the AIR metric discussed above. The LDA Models' mean AUC at a given Fairness Weight can depart materially from that of the population-based LDA Model - indicating the risk of statistical bias in a lender's AUC fairness metric. The blue line in Figure 17 reflects the AUC value associated with the population-based LDA Model at each Fairness Weight. When compared to the corresponding average AUC values from the sample-based LDA Models (the red line), we can see notable divergences between the two - with the individual LDA Models, on average, achieving lower AUC accuracy rates than the population-based LDA Model at the same Fairness Weight. As I will discuss later, the divergence here is due to the AD process's selection of different sets of primary disparate impact factors for the sample-based LDA Models versus the population-based model at the same Fairness Weight. So What Does This All Mean For Lenders? From a lender's perspective, common AIR-based AD processes can create LDA Models with significant model risk due to the instability of their model structures in response to relatively small variations in training data. Such "brittleness" - if undetected during model development or model validation - may cause model accuracy and fairness performance metrics obtained during model development to be highly idiosyncratic - potentially resulting in materially different model performance results during production. In the final section of my analysis below, I turn my attention back to the core reasons driving LDA Model adoption for lenders - the elimination or mitigation of illegal credit model disparate impact. However, now - with the findings above in hand - I explore what this brittleness implies about the use of AIR-based algorithmic debiasing for such purposes. The Impact of LDA Model Brittleness on Disparate Impact Remediation: Identifying and Analyzing the Primary Disparate Impact Factors According to its proponents, algorithmic debiasing creates fairer AI credit models by reducing or eliminating the illegal disparate impact driving lending outcome disparities for protected class applicants. However, as I have written extensively elsewhere - see, for example, Algorithmic Justice: What's Wrong With the Technologists' Credit Model Disparate Impact Framework - whether a credit model's lending outcome disparity is evidence of an illegal disparate impact is a fact-based analysis governed by federal law, regulation, and associated Supreme Court opinions. Among other important things, this legal framework requires the identification of the specific "artificial, arbitrary, and unnecessary" credit attribute(s) allegedly causing the illegal lending outcome disparity. Nevertheless, despite its widespread promotion as a tool to remediate "disparate impact" discrimination, algorithmic debiasing performs its task without the transparency consistent with this legal framework (or, frankly, with typical model risk management requirements). That is, because algorithmic debiasing is, itself, an automated black box process - and because many modern-day AI credit models are driven by hundreds or thousands of predictive attributes, most lenders are unaware of how the debiasing algorithm actually alters the Base Model to produce the less discriminatory alternatives they are encouraged to adopt - let alone whether such attributes are really "artificial, arbitrary, and unnecessary". While one would think that identification of the credit model's "disparate impact factors" should matter a great deal to highly-regulated lenders, such concerns do not appear to be widely or publicly shared in the many conference panels, white papers, blog posts, and podcasts advocating for AD adoption. Instead, the focus of discussions is ultimately on the results - not on the process - with the existence of demonstrably "better" fairness performance proof enough to justify many lenders' LDA Model adoption decisions. But what would a lender learn if it did seek greater transparency? And how might such learnings impact its decision to adopt an algorithmic LDA Model?[26] To answer these questions, I analyzed my 2,000 LDA Models one final time to understand more specifically: • Which LDA Model risk attributes experience the greatest PD-altering impacts (both positively and negatively) relative to their Base Model? I refer to these attributes as the LDA Model's "primary disparate impact factors". • How do these primary disparate impact factors vary, if at all, across LDA Models estimated with the same Fairness Weight?, and • How do these primary disparate impact factors vary, if at all, across Fairness Weights for the same LDA Model (i.e., estimated on the same training sample)? I address each of these in turn below. Identification of the LDA Models' Primary Disparate Impact Factors To identify the primary disparate impact factors for each of my 2,000 LDA Models, I performed the following steps on each model: • I first calculated the change in each individual's probability of default ("PD") relative to the Base Model. • Using ordinary least squares ("OLS"), I performed regression analysis on these individual-level PD differentials using the original set of credit risk attributes as the set of explanatory variables (i.e., those listed in Figure 2).[27] • I identified the primary disparate impact factors as the credit risk attributes with the largest positive and negative OLS regression coefficients. Since all risk attributes in my credit model take the form of indicator (dummy) variables, their coefficients are already normalized - meaning that the attributes with the largest positive and negative coefficients are the risk attributes most impacted by the debiasing algorithm.[28] Additionally, I note that these primary disparate impact factors come in pairs - one positive and one negative - because one cannot "de-bias" or reduce the impact of one risk attribute without offsetting this effect with an increase to other risk attributes; otherwise, the LDA Credit Model would no longer generate PD estimates with zero overall expected error. Therefore, one should think of credit model disparate impact remediation as a reallocation of estimated credit risk across model As an example of this identification process, Figure 18 below summarizes the primary disparate impact factors identified across the 100 LDA Models trained with a relatively low Fairness Weight = 0.1. The top table summarizes the 100 primary disparate impact factors responsible for decreasing the LDA Models' PD estimates (i.e., de-risking PD estimates relative to the Base Models) to improve AIR-based fairness, while the bottom table identifies the corresponding primary disparate impact factors responsible for increasing the LDA Models' PD estimates (i.e., up-risking PD estimates relative to the Base Models) to improve fairness. Reviewing Figure 18, I notice two immediate results. First, at a Fairness Weight of 0.1, the majority of LDA Models (62%) achieve improved AIR-based fairness by de-risking (i.e., decreasing) the estimated PDs for smaller loan sizes (i.e., loan amounts <= $50,000). Second, to counterbalance this de-risking, the AD process primarily up-risks (i.e., increases) the PDs for applicants with CLTVs between 90% and 95% for a majority of training samples (79%). Why these specific factors? While not a definitive explanation, Figures 19 and 20 below suggest a reason for this particular selection of primary disparate impact factors at this Fairness Weight level. In Figure 19, I plot the estimated loan amount risk profiles for the 62% of LDA Models where loan amounts <= $50,000 were identified as the primary de-risked disparate impact factor. For reference, I also add the population-based Base Model's loan amount risk profile as a benchmark. Here we can see that the lowering of the relative riskiness of small loan sizes (<=$50K) for these models still largely preserves the monotonicity in the estimated loan amount risk profile we observe in the population-based Base Model. This is consistent with the low Fairness Weight of 0.1 in which relative accuracy (i.e., AUC) is much more important than AIR-based fairness. Accordingly, the AD process appears to search for a risk attribute whose de-risking incrementally improves AIR-based fairness by: (1) skewing PD reductions disproportionately to protected class applicants in order to improve their relative approval rates, but (2) still maximally preserving the LDA Model's AUC-based accuracy. Similarly, in Figure 20, we can see that the LDA Models' up-risking of loans with CLTV 90-95% also largely preserves the monotonicity present in the estimated CLTV risk profile of the population-based Base Model while contributing to improved AIR-based fairness. Here, the AD process appears to search for a risk attribute whose up-risking incrementally improves AIR-based fairness by: (1) skewing PD increases disproportionately to control group applicants in order to suppress their relative approval rates, but (2) still maximally preserving the LDA Model's AUC-based accuracy. Evaluating How These Primary Disparate Impact Factors Vary Across LDA Models Estimated With the Same Fairness Weight More broadly, Figures 21 and 22 below summarize the primary de-risked and up-risked disparate impact factors identified across all 2,000 LDA Models in my analysis - segmented by Fairness Weight. To reduce the clutter of these charts, I only include a primary disparate impact factor if it is used in at least 10% of LDA Models estimated at any Fairness Weight. A review of these charts leads to the following observations. At the same Fairness Weight, small differences in the underlying training samples can cause the AD process to select different sets of primary disparate impact factors. For example, according to Figure 21, at a Fairness Weight = 1.0, Loan Amount <= $50K and CLTV>95% account for 33% and 29% of the 100 primary de-risked disparate impact factors, respectively - with DTI=49% accounting for another 11%. According to Figure 22, the primary up-risked disparate impact factors selected at this Fairness Weight are CLTV 90-95% and CLTV 80-85% - both of which account for 36% and 31% of the total, respectively. Based on these results, we see that LDA Model brittleness is clearly associated with different sets of primary disparate impact factors selected at the same Fairness Weight, and these different primary disparate impact factors create the different "clusters" of LDA Model credit risk profiles as discussed in relation to Figure 14. This implies that LDA Model brittleness not only raises safety-and-soundness concerns related to model non-robustness and instability, but it also raises equally important concerns about whether algorithmic debiasing is truly remediating illegal disparate impact driven by "artificial, arbitrary, and unnecessary" predictive factors per the governing legal framework. That is, if it is really remediating true illegal disparate impact, then why - at the same Fairness Weight - are different model attributes being selected for "de-risking" in response to small random differences in training Evaluating How These Primary Disparate Impact Factors Vary Within Training Samples For Different Fairness Weights In addition to analyzing how primary disparate impact factors change across LDA Models estimated at the same Fairness Weights, I also flipped this analysis to explore how the primary disparate impact factors change within a given training sample as the Fairness Weights increase. What I found was the following: Even within a given training sample, algorithmic debiasing can switch among different primary disparate impact factors as the Fairness Weight changes. For example, Figures 23a-e below present the primary disparate impact factors identified within 5 random training samples (out of the 100) for each Fairness Weight between 0.1 and 2.0. That is, for each training sample, I estimated 20 LDA Models - each associated with a different Fairness Weight. Then, for each of the 20 LDA Models, I used my method described above to identify the set of primary disparate impact factors used to improve AIR-based fairness in this sample. For each of the 5 randomly-selected training samples below, the first column denotes the LDA Model's Fairness Weight, the second column identifies the primary de-risked disparate impact factor, the third column identifies the counterbalancing primary up-risked disparate impact factor, and the fourth and fifth columns contain the corresponding LDA Model's AIR and AUC - which can be compared to the training sample's Base Model AIR and AUC at the top of the respective columns. Figure 23a - Training Sample 1 Figure 23b - Training Sample 2 Figure 23c - Training Sample 3 Figure 23d - Training Sample 4 Figure 23e - Training Sample 5 Reviewing these tables individually, we can see clearly that within a given training sample, the AD process can select different risk attributes as the primary disparate impact factors depending on the degree of fairness improvement sought and the relative importance of model accuracy. For example, at low Fairness Weights, the AD process tends to focus its de-risking on smaller loan sizes. However, as the Fairness Weight increases from these low levels, it tends to switch - generally - to DTI attributes. And at even higher Fairness Weights, the AD process - again, generally - switches to higher CLTV loans. Such attribute switching is also generally evident in the primary up-risked disparate impact factors. Additionally, the fairness improvement path for each training sample (i.e., the primary disparate impact factors selected as the Fairness Weight increases sequentially from 0.1 to 2.0) contains differing degrees of instability - which I define here as a lack of consistency in disparate impact factor selection. Although this instability affects all samples to some degree, it is most apparent in Samples 1 and 3 where 6 and 9 different primary de-risked disparate impact factors, respectively, are leveraged along the samples' fairness improvement paths. Overall, this instability in LDA Models' primary disparate impact factors calls into question whether these identified model attributes are truly discriminatory according to the disparate impact legal framework, or simply model attributes whose statistical and demographic properties are being leveraged to achieve higher levels of AIR-based fairness through reverse disparate impact. Why Does Any of This Matter? For those who would ask what difference any of this makes since improved fairness is nevertheless achieved, I point out that - apart from the potentially serious risk and compliance issues noted above - such fairness improvements are not without real customer impacts. In particular, keep in mind that the LDA Model's expanded protected class approvals are driven by swap-sets of applicants from both demographic groups - i.e., Black and White.[29] While, as a group, Black applicants experience net positive swap-ins (and, therefore, higher relative approval rates) under all these LDA Models regardless of the primary disparate impact factors employed, this group-level result masks significant intra-group credit decision "churn" in which many individual Black applicants can be harmed. To illustrate and expand on this point, Figure 24 shows the average number of Black "swap-ins" at each Fairness Weight (the green bars - averaged across the 100 LDA Models estimated at each Fairness Weight) along with the corresponding average number of Black "swap-outs" (the red bars). I note that at each Fairness Weight the Black swap-ins are larger than the Black swap-outs - thereby leading to a net positive increase in Black approvals and AIR-based fairness. Based on this analysis, I note that: At each LDA Model Fairness Weight, although Black applicants - as a group - achieve higher net approvals, this group-level fairness improvement comes at the expense of significant intra-group That is, because algorithmic debiasing does not directly de-risk only Black applicant PDs, the increase in Black net approvals occurs as a form of "rough justice" - that is, while the Black applicants possessing the primary de-risked disparate impact factor have their PDs reduced (as do other non-Black applicants with this attribute), other Black applicants possessing the primary up-risked disparate impact factor have their PDs increased. As the former group dominates the latter group in size, overall net swap-ins of Black applicants are positive - leading to increased group-level approvals. However, not all Black applicants are likely happy with this group-level result. For example, at a Fairness Weight = 1.0, the 100 LDA Models, on average, swap-in 449 new Black approvals (i.e., Black applicants who would be denied under the corresponding Base Models). However, these new approvals are partially offset by 210 Black swap-outs (i.e., Black applicants who would be approved under the Base Model, but would now be denied). Therefore, while Black net approvals increase by an average of 239 at a Fairness Weight = 1.0, this net increase masks an average 659 individual credit decision changes within this demographic group - which is 11.3% of all Black While, by themselves, the Black swap-outs would likely view the adverse change in their credit decision to be unfair, Figure 25 below further illustrates the potential harm to these individuals. Here, I calculated - across all 100 training samples - the average actual default rate of Black swap-ins and swap-outs across the LDA Model Fairness Weights. The Black swap-ins' actual default rates are plotted in green while the Black swap-outs' actual default rates are plotted in red and are displayed as negatives for presentation purposes. At all Fairness Weights, the Black swap-ins, on average, have worse credit risk (i.e., higher actual default rates) than the Black swap-outs. Additionally, for specific training samples, the characteristics of those swapped-in vs swapped-out may be non-intuitive. Take, for example, Training Sample 3 from Figure 23c above. For the LDA Model estimated with a Fairness Weight = 1.3, the primary de-risked disparate impact factor is CLTV>95% while the primary up-risked disparate impact factor is CLTV 80-85%. In this case, Black applicants with lower CLTVs (and lower actual default rates) would now be denied under this LDA Model while Black applicants with higher CLTVs (and higher actual default rates) would be approved.[30],[8] And this is why this matters. An LDA Model's "better" fairness performance is not all that matters when deciding whether to adopt it or not. As we have seen here, even putting aside the potentially serious safety-and-soundness risks discussed previously, there are still important legal, compliance, and reputational risks to consider due to: • Whether the LDA Model is actually remediating an illegal disparate impact as governed by applicable law, regulation, and Supreme Court opinion OR whether the AD process is, instead, simply altering the LDA Model structure to embed an arguably illegal and latent reverse disparate impact to achieve greater approval rate equity. That is, are we remediating a legal violation or potentially creating one? • The apparent intra-group unfairness created within demographic groups - including the protected class groups - by potentially swapping-in higher risk applicants for new credit approvals at the expense of swapping-out (i.e., denying) lower risk applicants within the same demographic group. • The conceptual complications related to Adverse Action Notice accuracy. For example, many of the swap-outs in Training Sample 3 would receive Adverse Action Notices indicating their credit denial was based on "insufficient collateral" or some other reason related to their CLTV value. However, that is not exactly accurate since other applicants with higher CLTVs (i.e., the swap-ins) are now approved. In reality, these applicants are being denied solely to achieve a specific fairness goal - but that is not what would be communicated to them. Additionally, the existence of LDA model multiplicity at a given Fairness Weight raises another interesting point to consider. Given that random differences in training samples can produce LDA Models with different primary up-risked disparate impact factors at the same Fairness Weight, those applicants swapped-out (and, therefore, denied) by a given LDA Model may not have been denied if the LDA Model had been trained on a different random sample. That is, while it may be technically true that these swap-outs were denied due to their primary up-risked disparate impact factor, the underlying brittleness in LDA Models means that - randomly - some of these applicants could have been approved had a slightly different training sample been used.[31] In such cases, what really is the specific reason for denial? And is this reason really a causal reason that the applicant can address? • The poorer credit performance of the swapped-in protected class applicants (relative to the swapped-out applicants) that raises concerns as to whether a lender is engaged in responsible lending. Final Thoughts As readers of this blog know, my primary concern with current algorithmic fairness tools is the rush by many to embrace them as an extraordinary technological innovation that can easily solve an important and complex fair lending compliance issue. While the value proposition behind these tools is certainly compelling, the reality is that there is a death of publicly-available, objective, and rigorous research supporting their use in a consumer lending context - research that explores more deeply the complex operations occurring within their automated "black box" processes to identify overlooked features or behaviors that may pose important risks to lenders and consumers. Every month we see what seems like hundreds of new research papers investigating every nuance associated with large language models, yet we still don't have a proper vetting of the complex models being used today by highly regulated institutions deploying these models into high-risk use cases such as consumer lending. In any case, I don't purport that the research in this post - nor my prior Fool's Gold analyses - satisfy this objective and, therefore, should be considered the decisive word on current algorithmic debiasing tools. As I have stated many times, my research is limited to the public data and tools available to me which may, or may not, limit the broader applicability of my findings. However, I do strongly believe that these analyses highlight clearly important risks associated with these tools that should be investigated more thoroughly - whether part of a formal research program or part of a lender's model risk management process - to determine their applicability and presence with specific lender credit model applications. And, lastly, for those who may believe that my results are specific to the "fairness regularization" method of algorithmic debiasing and not to other common AD methodologies such as adversarial debiasing - well that remains to be seen. Certainly, adversarial debiasing - although not calibrating directly to the AIR - does calibrate to an outcome-based fairness penalty term involving the correlation between PDs (which drive approvals) and borrower demographics. And as the weight of this penalty term increases to drive greater LDA Model fairness performance, the AD process will likely distort the estimated credit risk relationships similarly to what I've described here. Whether these distortions are of similar magnitudes and cause similar brittleness is a research topic I leave for another time (or another researcher). However, I think many of us - including the CFPB and, importantly, lenders employing this methodology - would like to know this answer. The sooner the [1] See, for example, the June 2024 Fair Lending Report of the Consumer Financial Protection Bureau which states, "In 2023, the CFPB issued several fair lending-related Matters Requiring Attention and entered Memoranda of Understanding directing entities to take corrective actions that the CFPB will monitor through follow-up supervisory actions. ... [T]he CFPB ... directed the institutions to test credit scoring models for prohibited basis disparities and to require documentation of considerations the institutions will give to how to assess those disparities against the stated business needs. To ensure compliance with ECOA and Regulation B, institutions were directed to develop a process for the consideration of a range of less discriminatory models." • "Urgent Call for Regulatory Clarity on the Need to Search for and Implement Less Discriminatory Algorithms," June 26, 2024 Letter From The Consumer Federation of America and Consumer Reports to Director Rohit Chopra of the CFPB. In particular, "While rudimentary techniques exist to modify models to mitigate disparate impact (such as “drop-one” techniques), a range of more advanced tools and techniques are emerging, including adversarial debiasing techniques, joint optimization, and Bayesian methods that use automated processes to more effectively search for modifications to reduce disparate impacts. ... Companies should be expected to utilize emerging good practices in terms of techniques for searching for LDAs." • "CFPB Should Encourage Lenders To Look For Less Discriminatory Models," April 22, 2022 Letter From the National Community Reinvestment Coalition to Director Rohit Chopra of the CFPB. [3] Their explicit use of the term "open-source" is interesting as it indicates a potential aversion to proprietary algorithmic fairness tools. [4] This omission from my previous studies is due to my use of a single - although large - model training sample and my focus on the performance of a single LDA Model derived therefrom. [6] The importance of model robustness / stability is reinforced by the federal financial regulators. For example, according to OCC 2011-12 "Supervisory Guidance on Model Risk Management" - considered to be the authoritative guidance on model risk management: • "Model quality can be measured in many ways: precision, accuracy, discriminatory power, robustness, stability, and reliability, to name a few." • "An integral part of model development is testing, in which the various components of a model and its overall functioning are evaluated to determine whether the model is performing as intended. Model testing includes checking the model's accuracy, demonstrating that the model is robust and stable, assessing potential limitations, and evaluating the model’s behavior over a range of input [8] This intra-group unfairness also affects the control group applicants. While the aggregate level metrics for this group suggest that they are trivially impacted (e.g., the LDA Model approval rate for White applicants is at most about 1% less than their Base Model approval rate of 90.4% at the highest Fairness Weights), this result masks the thousands of individual White applicants whose credit decisions are changed by the LDA Model - about 49% swapped-in and 51% swapped-out. As with the protected class applicants so affected, many of these applicants may not consider the new credit decisions to be fair. [9] I refer to this HMDA dataset as a "synthetic credit performance dataset" as - for the purpose of these analyses - I treat denied credit applications as "defaults" and approved credit applications as "non-defaults". See my further discussion of this assumption in the first Fool's Gold article. [10] I use logistic regression for the credit model as: (1) it is the most common machine-learning algorithm ("ML") used in consumer credit scoring, (2) it allows me to abstract away from the unnecessary complexities and intractabilities of more advanced ML techniques such as gradient boosted trees, neural networks, etc., and (3) it makes the overall analysis feasible from a computational resource perspective (I ultimately need to estimate 2,100 Base and LDA Models). Nevertheless, my results should be considered within the context of this choice. [11] I may perform future analyses using out-of-time HMDA data to assess the impact of such data on a given LDA Model's fairness and accuracy performance. [12] I base my analysis on 100 samples to balance the need for a relatively large number of random samples to provide sufficiently precise results with the significant computational time associated with generating 20 LDA Models for each of the 100 samples. While I believe that my results and conclusions would be qualitatively unchanged with a larger sample size, they should be interpreted within the context of this choice. [13] More formally, only 2 of the 26 risk attributes (DTI=39% & DTI=44%) have population-based coefficients that are statistically different than the corresponding sample-based average estimates at a 5% level of significance. The values of these differences are -0.019 and 0.014, respectively. [14] I note that the use of the term "bias" here refers to its statistical definition - not its consumer compliance definition. [15] That is, a one standard deviation change in the number of black approvals changes the black approval rate by 1.13% while a similar change in white approvals changes the white approval rate by 0.27% - a difference of 4.2x. Given that the AIR is a ratio of these approval rates, the greater inherent variability of Black approval rates will amplify the variability in the AIR fairness metric. [16] A similar phenomenon is discussed in Black, Emily, Gillis, Talia & Hall, Zara. (2024). "D-hacking". 602-615. 10.1145/3630106.3658928. In their article, they identify this as a risk for opportunistic fair-washing of algorithmic credit models. [18] I display only 12 of the 20 Fairness Weights solely to conserve space. There is nothing abnormal or atypical of the results associated with the remaining 8 Fairness Weights that are not [19] These results are qualitatively unchanged if one instead uses the relative coefficients of variation as the volatility measure. [20] I measured monotonicity as the percentage of risk profiles in which each successive risk coefficient was greater than or equal to (or less than or equal to, for loan amount) the preceding risk coefficient. Given the number of coefficients for DTI, I focused on only the following subset: 37%, 40-41%, 44%, 46%, 48%, 50-60%, and >60%. [21] This raises the question as to whether algorithmic debiasing - by introducing a second component (fairness) to the model training objective - causes the LDA Model solution to be under-identified [22] This is the same approval rate assumption that I have used throughout the previous three Fool's Gold analyses. [23] While it appears that the population-based LDA Models have a different AIR behavior than the sample-based LDA Models due to the large jump at a Fairness Weight = 1.1 (vs. the smooth AIR increase of the sample-based LDA Models), keep in mind that the latter is an average value across 100 LDA Model AIR paths while the former is a single AIR path. [24] Technically, these AIR distributions are skewed so means and standard deviations are less meaningful. However, the conclusion is unchanged if I instead measure the central tendency and spread of the AIR distributions with the median and interquartile ranges. In particular, the latter increases from 0.011 to 0.034 to 0.058 as the Fairness Weight increases from 0.3 to 1.0 to 1.8, respectively. [25] Technically, these AUC distributions are skewed so means and standard deviations are less meaningful. However, the conclusion is unchanged if I instead measure the central tendency and spread of the AUC distributions with the median and interquartile ranges. In particular, the latter increases from 0.007 to 0.027 to 0.068 as the Fairness Weight increases from 0.3 to 1.0 to 1.8, respectively. [26] I distinguish here between LDA Models created algorithmically through automated black-box machine learning methods and those created manually through more traditional disparate impact testing and credit model modification. This "traditional" approach is discussed in more detail in my prior post "Algorithmic Justice: What's Wrong With the Technologists' Credit Model Disparate Impact [27] I note that different estimation approaches may be required for more complex LDA model structures. [28] To be clear, these attributes are not the only attributes affected by the AD process and, therefore, they are not the only attributes affecting the LDA Model's PD estimates. However, they have the largest effects and, because of that, I designate them as the primary disparate impact factors. [30] This observation regarding the sacrifice of intra-group fairness for greater levels of inter-group outcome equality is consistent with a similar theme contained in the recent working paper Caro, Spencer, Gillis, Talia, and Scott Nelson, Modernizing Fair Lending, Working Paper No. 2024-18, Becker Friedman Institute For Economics, University of Chicago, August 2024. [31] This point is similar, though not exact, to one made in the working paper: Emily Black, Manish Raghavan, and Solon Barocas. 2022. Model Multiplicity: Opportunities, Concerns, and Solutions. [32] These swap-ins are different from those who may be swapped in due to the use of new alternative data attributes that provide more inclusive and accurate credit risk assessments for individuals with sparse traditional credit histories. © Pace Analytics Consulting LLC, 2024.	{"url":"https://www.paceanalyticsllc.com/post/lda-credit-model-brittleness","timestamp":"2024-11-02T02:42:19Z","content_type":"text/html","content_length":"1050384","record_id":"<urn:uuid:2670646b-ff92-4dea-a4a6-fbed763d4c59>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00183.warc.gz"}
What is NumPy? In order to feel confident and successfully complete this course, we strongly recommend you complete the following courses beforehand (just click on them to start): NumPy is one of the many modules in the Python programming language that allows you to work with arrays and matrices. Furthermore, this module comes equipped with a robust collection of mathematical functions for manipulating these arrays. We must import this library to use it. Let's see how we do it: The syntax for using a function from the library is as follows: The alias should be used to simplify the use of this library. Let's see how we do it: Here, 'np' is an alias. Therefore, in the future, when using functions from this library, we will do it like this: You need to import the NumPy library. Carefully examine the example provided above and complete the task by filling in the blanks. If you encounter any difficulties, refer to the hint; it will likely assist you. [Once you've completed this task, click the button below the code to check your solution.] Switch to desktop for real-world practiceContinue from where you are using one of the options below Thanks for your feedback! In order to feel confident and successfully complete this course, we strongly recommend you complete the following courses beforehand (just click on them to start): NumPy is one of the many modules in the Python programming language that allows you to work with arrays and matrices. Furthermore, this module comes equipped with a robust collection of mathematical functions for manipulating these arrays. We must import this library to use it. Let's see how we do it: The syntax for using a function from the library is as follows: The alias should be used to simplify the use of this library. Let's see how we do it: Here, 'np' is an alias. Therefore, in the future, when using functions from this library, we will do it like this: You need to import the NumPy library. Carefully examine the example provided above and complete the task by filling in the blanks. If you encounter any difficulties, refer to the hint; it will likely assist you. [Once you've completed this task, click the button below the code to check your solution.] Switch to desktop for real-world practiceContinue from where you are using one of the options below Thanks for your feedback! In order to feel confident and successfully complete this course, we strongly recommend you complete the following courses beforehand (just click on them to start): NumPy is one of the many modules in the Python programming language that allows you to work with arrays and matrices. Furthermore, this module comes equipped with a robust collection of mathematical functions for manipulating these arrays. We must import this library to use it. Let's see how we do it: The syntax for using a function from the library is as follows: The alias should be used to simplify the use of this library. Let's see how we do it: Here, 'np' is an alias. Therefore, in the future, when using functions from this library, we will do it like this: You need to import the NumPy library. Carefully examine the example provided above and complete the task by filling in the blanks. If you encounter any difficulties, refer to the hint; it will likely assist you. [Once you've completed this task, click the button below the code to check your solution.] Switch to desktop for real-world practiceContinue from where you are using one of the options below Thanks for your feedback! In order to feel confident and successfully complete this course, we strongly recommend you complete the following courses beforehand (just click on them to start): NumPy is one of the many modules in the Python programming language that allows you to work with arrays and matrices. Furthermore, this module comes equipped with a robust collection of mathematical functions for manipulating these arrays. We must import this library to use it. Let's see how we do it: The syntax for using a function from the library is as follows: The alias should be used to simplify the use of this library. Let's see how we do it: Here, 'np' is an alias. Therefore, in the future, when using functions from this library, we will do it like this: You need to import the NumPy library. Carefully examine the example provided above and complete the task by filling in the blanks. If you encounter any difficulties, refer to the hint; it will likely assist you. [Once you've completed this task, click the button below the code to check your solution.] Switch to desktop for real-world practiceContinue from where you are using one of the options below Switch to desktop for real-world practiceContinue from where you are using one of the options below	{"url":"https://codefinity.com/courses/v2/671389bc-34ed-4de7-83cd-2d1bfcf00a76/7c4e6464-e82b-4f01-bcb9-78d53d060da5/9ed95b89-cb16-4b5b-9621-c8a292ce5eec","timestamp":"2024-11-06T14:09:46Z","content_type":"text/html","content_length":"375252","record_id":"<urn:uuid:8fa544e2-83f0-41cb-931b-937fe3a8eb38>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00452.warc.gz"}
InfluxDB v2 as Datasource for Grafana In the last blogs (Grafana with podman kube and Setup MQTT Broker, IoT Devices and Security ) I explained how to setup Prometheus , Grafana and Mosquitto as MQTT Broker to visualize the data from IoT devices, mainly from Shelly. In this blog I will explain how to setup InfluxDB v2.6 and use it as additional datasource for Grafana. Why Prometheus and InfluxDB? Many of my devices send only a new MQTT message with new values, if the difference to the old value is big enough. This can take several hours (or even days), something where I have big problems to handle that with Prometheus, which scrapes the metrics in a regular interval. In my experience, InfluxDB is also better suited if you want to measure the power created by our balcony power plant. And that’s one of my goals. As usual, I run InfluxDB v2.6 as container managed with podman kube on openSUSE MicroOS . The directory layout for the configuration file is similar to the other services: ├── data/ -> for the persistent data └── etc/ -> for the configuration data WARNING InfluxDB changes the ownership of the data to UID 1000. This is really bad, since this UID is the first user UID on Linux distributions and thus normally already in use. Means, your first normal user can access, read and modify the InfluxDB2 data! The problem is known since now 5 years, the only workaround is to run the container as another user, not root. But this prevents influxdb from adjusting the permissions and ownership of the directories, so the admin needs to know what influxdb needs. Containers from other projects solved that in an user friendly way: you can specify which UID the process should use. So at first you should make sure, that the UID 1000 is not used on your system. Second, create an user influxdb with this UID to avoid any security problems in the future: useradd -u 1000 -r influxdb -d /srv/influxdb2 Yes, useradd will most likely warn you that influxdb's uid 1000 is greater than SYS_UID_MAX, but since this is only a warning, you can ignore that. No configuration file is needed: it will be created at the first start of the container and first login. Podman kube The YAML configuration file (influxdb2.yaml) for podman kube play looks like: apiVersion: v1 kind: Pod app: influxdb2 name: influxdb2 - name: server image: docker.io/influxdb:latest - containerPort: 8086 hostPort: 8086 - mountPath: /var/lib/influxdb2 name: srv-influxdb2-data-host-0 - mountPath: /etc/influxdb2 name: srv-influxdb2-config-host-0 resources: {} - CAP_MKNOD - CAP_NET_RAW - CAP_AUDIT_WRITE privileged: false restartPolicy: unless-stopped - hostPath: path: /srv/influxdb2/data type: Directory name: srv-influxdb2-data-host-0 - hostPath: path: /srv/influxdb2/etc type: Directory name: srv-influxdb2-config-host-0 status: {} This configuration uses the latest upstream InfluxDB v2.x container, which listens on port 8086. That’s already all we need. Run Container Now we just need to start the containers: podman kube play influxdb2.yaml The command podman pod ps should show you at least one pod: POD ID NAME STATUS CREATED INFRA ID # OF CONTAINERS 7dfeecb0b16c influxdb2 Running 36 seconds ago aebf2050aeca 2 InfluxDB setup After the InfluxDB container runs, connect with a webbrowser on http://hostname:8086/ to it and you should see the following welcome screen: Select Get Started and you should see the first mask to enter all necessary data for the initial configuration: I use as Username always admin, but you can choose whatever you want. Else the passwords, the first Organization Name and the first Bucket Name are required. Here I always choose “Configure Later”, since the database will be filled from outside and for the dashboards Grafana will be used. There is only one thing we need to do in the UI: generate a token, so that Grafana can read the data. For this, select Load Data and here API Tokens: InfluxDB does not allow you anymore to view the token after you created it. This is valid for the admin’s Token, too. But for the admin’s Token there is a simple solution: if you need that later to authenticate yourself with e.g. the influx CLI, you can find it in /srv/influxdb2/etc/influx-configs. For Grafana, we generate now a new API Token, in this case a Custom API Token, as we only need read access to the bucket: Now securely save this API Token for Grafana: Add InfluxDB as Datasource to Grafana Login to Grafana and select Configuration -> Data sources. Here we add a new data source and select InfluxDB from the list. InfluxDB v2.x provides two query languages: “InfluxQL” and “Flux”. Here I will show for both how to add them. To differentiate the data sources later, I always add the query language to the database. Settings for InfluxQL: • Name would be InfluxDB-InfluxQL • Query Language would be InfluxQL • URL would be the InfluxDB URL http://<hostname>:8086. If Grafana and InfluxDB run in different PODs as described in this blog series, localhost will not work, it must be the real hostname or IP Now comes the tricky part. If you strictly followed the documention from InfluxDB and Grafana, you most likely saw already the “famous” error reading InfluxDB. Status Code: 401 message. At least this did happen for me the first days, and I found a lot of hits with google, but no real solution. There are many comments how to modify the database and what you need to create to “solve” this issue, but from my experience: this is not necessary. All you need to do is to add the Token in the correct format as Custom HTTP Header. In most documentation it’s described wrong, if it is mentioned at So this is what works for me: • Header is “Authorization” • Value is “Token <API Token>” And for the InfluxDB Details: • Database is here my-bucket (or whatever you did choose for the initial database) • User stays empty • Password stays empty • HTTP Method sould be GET I got User and Password only working, if I migrate from an InfluxDB v1.x installation to InfluxDB v2.x, and all databases had a workig user with password. But even in this case it did not work for every database. Generating tokens is much easier. Save & test should give you a green message datasource is working. XX measurements found. If you get an 401 error, Grafana could not authenticate itself against the database. Now we add a second datasource with “Flux” as Query Language. Go back start again with adding a data source. This time we use: • Name would be InfluxDB-Flux • Query Language would be Flux • URL would be the InfluxDB URL http://<hostname>:8086. If Grafana and InfluxDB run in different PODs as described in this blog series, localhost will not work, it must be the real hostname or IP Disable everything in the Auth section, especially the default Basic auth. We don’t need to add a custom HTTP header, too. For the InfluxDB Details we use: • Organization is the organization you created in InfluxDB, in this case my-org • Token is the same token we used for InfluxQL generated in InfluxDB. It only needs read permission for the bucket. Enter the pure token, no “Token” prefix is necessary as for the HTTP header. • Default Bucket is the bucket where the data should be fetched from by default, in this example this would be my-bucket Save & test should give you a green message datasource is working. XX measurements found. Now we have a MQTT Broker running, which collects the data from our IoT devices, we have Prometheus, which can scrape the data from configured sources, we have InfluxDB v2.6 as timeseries database and Grafana, which uses Prometheus and Grafana as data source. In the next blog I will explain, how to close the last gap: how to get the data from MQTT into InfluxDB.	{"url":"https://thkukuk.de/blog/influxdbv2_grafana/","timestamp":"2024-11-05T00:41:20Z","content_type":"text/html","content_length":"46758","record_id":"<urn:uuid:292b9a18-14a5-43b0-957f-366b9b35b328>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00852.warc.gz"}
Python Node Binary Trees Oops, something went wrong. Please try again in a few moments. class Node: Class to represent a node in an N-ary tree. - value: int The value of the node. - children: list List to store the children nodes of the current node. def __init__(self, value: int): Constructor to instantiate the Node class. - value: int The value of the node. self.value = value self.children = [] def postOrder(root: Node) -> list: Performs a post-order traversal on the N-ary tree rooted at 'root' to find the depth of all branches of every node, counts the number of valid binary trees among all subtrees, and determines if the tree rooted at 'root' is itself a valid binary tree. - root: Node The root node of the N-ary tree. - list: A list with two elements: - The first element is the number of valid binary trees found in the N-ary tree. - The second element is a boolean indicating the validity of the tree rooted at 'root' as a binary tree. def is_binary_tree(node: Node) -> bool: Helper function to check if a given node is a valid binary tree. - node: Node The node to check. - bool: True if the node is a valid binary tree, False otherwise. # A node is a valid binary tree if it has at most 2 children. return len(node.children) <= 2 def count_binary_trees(node: Node) -> int: Helper function to count the number of valid binary trees among all subtrees of a given node. - node: Node The node to count the binary trees for. - int: The number of valid binary trees found in the subtree rooted at 'node'. # Base case: If the node is None or it is not a valid binary tree, return 0. if node is None or not is_binary_tree(node): return 0 # Initialize the count of binary trees to 1 (for the current node). count = 1 # Recursively count the number of binary trees in each child subtree. for child in node.children: count += count_binary_trees(child) return count def find_depth(node: Node) -> int: Helper function to find the depth of all branches of a given node. - node: Node The node to find the depth for. - int: The maximum depth among all branches of the node. # Base case: If the node is None, return 0. if node is None: return 0 # Initialize the depth to 0. depth = 0 # Recursively find the depth of each child branch. for child in node.children: depth = max(depth, find_depth(child)) # Increment the depth by 1 to account for the current node. depth += 1 return depth # Check if the tree rooted at 'root' is a valid binary tree. is_valid_binary_tree = is_binary_tree(root) # Count the number of valid binary trees among all subtrees of 'root'. binary_tree_count = count_binary_trees(root) return [binary_tree_count, is_valid_binary_tree] def binTreesGeneric(root: Node) -> int: Finds the number of binary trees in an N-ary tree rooted at 'root' using the 'postOrder' function. - root: Node The root node of the N-ary tree. - int: The count of binary trees found in the N-ary tree. # Call the 'postOrder' function to get the count of binary trees and the validity of the tree rooted at 'root'. result = postOrder(root) # Return the count of binary trees. return result[0] def main(): Main function to initialize a graph with nodes and their children, call the 'binTreesGeneric' function with the root node, and print the result, which is the number of binary trees found in the initialized N-ary tree. # Initialize the nodes of the N-ary tree. node1 = Node(1) node2 = Node(2) node3 = Node(3) node4 = Node(4) node5 = Node(5) node6 = Node(6) # Set the children for each node. node1.children = [node2, node3] node2.children = [node4, node5] node3.children = [node6] # Call the 'binTreesGeneric' function with the root node. binary_tree_count = binTreesGeneric(node1) # Print the result. print(f"The number of binary trees found in the N-ary tree is: {binary_tree_count}.") # Unit tests for the functions. import unittest class TestBinaryTrees(unittest.TestCase): def setUp(self): # Initialize the nodes of the N-ary tree. self.node1 = Node(1) self.node2 = Node(2) self.node3 = Node(3) self.node4 = Node(4) self.node5 = Node(5) self.node6 = Node(6) # Set the children for each node. self.node1.children = [self.node2, self.node3] self.node2.children = [self.node4, self.node5] self.node3.children = [self.node6] def test_postOrder_valid_binary_tree(self): Tests the 'postOrder' function for a valid binary tree. result = postOrder(self.node1) self.assertEqual(result, [3, True]) def test_postOrder_invalid_binary_tree(self): Tests the 'postOrder' function for an invalid binary tree. # Remove one child from node2 to make it an invalid binary tree. result = postOrder(self.node1) self.assertEqual(result, [2, False]) def test_binTreesGeneric_valid_binary_tree(self): Tests the 'binTreesGeneric' function for a valid binary tree. result = binTreesGeneric(self.node1) self.assertEqual(result, 3) def test_binTreesGeneric_invalid_binary_tree(self): Tests the 'binTreesGeneric' function for an invalid binary tree. # Remove one child from node2 to make it an invalid binary tree. result = binTreesGeneric(self.node1) self.assertEqual(result, 2) # Run the main function if the script is executed directly. if __name__ == "__main__":	{"url":"https://codepal.ai/code-generator/query/Ls49fIup/python-node-binary-trees","timestamp":"2024-11-03T14:05:11Z","content_type":"text/html","content_length":"125610","record_id":"<urn:uuid:34a0cbb1-e6c8-491c-9d74-48703eac80c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00564.warc.gz"}
Need the Velocity Hodograph Be Circular? In our last blog, we appreciated that Newton could avoid higher than second order time derivatives because the conservation of angular momentum lead to a closed set of equations, once the angular velocity was eliminated. We want to show here that the corresponding elliptical trajectory in configuration space is, in general, non-circular, but-rather a carthoid. To obtain a circular trajectory in velocity space we are forced to treat the tangential component of the jerk. The normal component vanishes giving the conservation of angular momentum in velocity space: the velocity radius vector traces out equal areas in equal time intervals. The limiting circular hodograph, for an eccentricity equal to 1, occurs for an inverse-fifth power law in the velocity. There are no other integral power laws for arbitrary values of the eccentricity. In other words, the velocity hodograph must reflect the fact that the tangential velocity in the elliptical orbit is not constant except when the foci coincide. The circular hodograph with a constant velocity vector equal to GM/h can only distinguish between closed orbits with velocity c<GM/h, and open ones c>GM/h. It does not reflect the nature of the elliptical, or hyperbolic, orbits. Independent of the value of c/GM/h, velocity orbits are cardioids. In 1846 Hamilton introduced the velocity hodograph which, as Feynman appreciated, is a powerful geometric tool. Newton's laws together with the inverse square law always produces a circular velocity diagram...Or does it? The shape of the orbit depends on where the origin of the velocity diagram is located. If the origin happens to coincide with center, then the distance between the two foci shrinks to zero and the planet has the same speed in all parts of the orbit. If the point is anywhere between the center and the circumference of the diagram, the orbit is an ellipse. The closer the point is to the circumference, the more elongated the elliptical orbit becomes until it becomes a parabola when the point intersects the circumference of the circle....But is it what is commonly believed to be a We found that when that happens, the eccentricity, or the relative velocity, becomes unity and a parabola results. But the equation we used to derived it was the tangential component of the jerk on the condition that the normal component vanishes. That gave us the condition that equal areas of swept out of velocity space occurs in equal time intervals. This is the analog of the conservation of momentum in configuration space. Now, here's the rub. The tangential force that gave rise to the limiting position of the point intersecting the circumference of the velocity space orbit was an inverse-fifth force--in the velocity! For any other position, the velocity is no longer a circle intercepting the origin. This is a little strange. But are we dealing with a circular orbit for the velocity? If x=(x,y,z) represents a system of cartesian coordinates, the angular momentum points in the z direction x X v =h=(0,0,h) where the velocity vector is tangential v= (v_x,v_y,0)= R(-sin(theta),c+cos(theta),0) where R=GM/h, the constant velocity vector and c=(c_1,c_2,0) is a constant vector obtained by integrating Newton's second law of motion. While it is true that is formally the equation of a circle, it just is a mere identity, R=R. What we are actually after is the equation of the speed, v=R(1+e^2+2e cos(theta))^(1/2), () where e=\|c\|/R, the relativity velocity, aka the eccentricity. The angle (theta) is the same as the true anomaly in the equation of the ellipse r=(h/R)/(1+e cos(theta)), () which is obtained from the velocity vector by applying the definition of the angular momentum h=x X v, and solving for r. Analogous to the orbit in configuration space (), the speed () determines the orbit in velocity space, not the identity R=R. To conclude that the velocity vector does not traverse the entire hodograph during the motion is inaccurate. That "Eq. () shows that the true anomaly is limited; it may just move on a circular arc." The particle in orbit does not move on a circular arc. For small values of the eccentricity, () is nearly circular, but as e->1, the orbit is a cardioid, and the angle can go a full 2pi, regardless of the value of e. This is shown in the following diagram. Also shown is the elongated elliptical orbit in configuration space, corresponding to the cardioid. In short, you cannot fit a cardioid to a circle, except at extremely small values of e. But the reason for having to do so is physically meaningless. According to Osipov and Belbruno, velocity space possesses one and only one Riemannian metric ds^2 with the arclength ds=dt/\|x\|, which is commonly referred to Levi-Civita's regularization procedure. This metric is smooth and has constant positive curvature for total energies less than zero. The geodesics are precisely the circles (or lines) are associated with Keplerian orbits. However, ds/dt=v, and not 1/\|x\|. Levi-Civita's regularization is concocted to reduce the order in Newton's equation, \|dv/dt\|=-GM/r^2 by one, \|dv/ds\|=GM/r=v^2/2-E, where E is the total energy. It is then necessary to go to the inverse velocity in order to get the Poincare' metric of the disc. It is known that half-circles, or straight lines through the center are the geodesics for constant curvature. However, there is no physics introducing a dimensionality errata formula in order to get a preconceived result. Although it slows down the motion in regions where v is high, it is not physically justifiable. So why not start directly from the equation dv/d(theta)= - R(cos(theta),sin(theta),0), obtained by introducing the conserved angular momentum h=r^2d(theta)/dt? Since d(theta)/ds=p, the radius of curvature, we obtain upon squaring (dv/ds)^2p^2=R^2, or ds^2=dv^2, and the non-Euclidean(ness) of the space has disappeared! Rather, it is the jerk that governs the orbits in velocity space. The tangential component of the jerk is where a is the acceleration. In the case the rate of change of the force is given by where L=v^3/p=v^2 d(theta)/dt, is the conserved angular momentum in velocity space, obtained by requiring that the normal component of the jerk vanish, and c is a constant velocity. The equation of the orbit is simply where we used v=p d(theta)/dt, and the primes stand for differentiation with respect to (theta). Introducing the inverse speed, w=1/v, Binet's equation becomes Since the right hand side equals dF/dt/(Lw)^2, it confirms we are dealing with an inverse-fifth force. Multiplying both sides by w', integrating and setting the integration constant equal to zero Reverting to the velocity, we see that the solution to is, in fact, v=c/2\cos(theta), a circle through the origin. This is the velocity hodograph. Introducing the arc length through p d(theta)=ds, the metric can be written as ds^2 = p^2(dv)^2/(c/2)^2(1-2v^2/c^2), (*) where the square of the radius of curvature plays the role of a conformal factor. This does not have the form of a Riemann metric, except in the case where v is normal to dv, in which case it becomes a Lobachevsky metric in velocity space. If we can to cast () as a bona fide Riemann metric of constant curvature, we need to introduce the transform where k=(2)^1/2 c, Eqn (**) becomes the Riemann metric ds^2=p dz^2/(1+z^2/k^2)^2, with constant positive curvature,k, proportional to the constant amplitude velocity, c. The radius of curvature, p, plays the role of a conformal factor. But, this is valid only in the limit e=1. For values e<1, the inverse-fifth power law is no longer valid, and the hodograph is no longer circular. In the hyperbolic case, a loop in the cardioid can Binet's equation can be written in general for power laws as The only two exponents that lead to an integral power law and a closed orbit are n=0 and n=3. We have discussed the latter in a previous blog. The n=0 case corresponds to an inverse-square law, which we know leads to an elliptical orbit. The energy equation is where E is a constant of integration. To obtain an elliptical orbit, E>0 since where the eccentricity is e=(1-E/c^2)^1/2.	{"url":"https://www.bernardlavenda.org/post/need-the-velocity-hodograph-be-circular","timestamp":"2024-11-03T22:22:16Z","content_type":"text/html","content_length":"1050493","record_id":"<urn:uuid:d0f5f13a-effb-4785-90bd-d3e2c7928ac7>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00659.warc.gz"}
Photonic Ising Machines Go Big A new optical processor for solving hard optimization problems breaks previous size records and is based on a highly scalable technology. Figure 1: Pierangeli et al. realized a scalable Ising machine by encoding spins in the spatial modulation of the phase of a laser beam (green). They set the interactions between the spins by modulating the beam’s amplitude. To run the Ising machine and find the ground-state spin configuration, they repeatedly compared the beam’s intensity to a target image (blue square), adjusting the phase modulation until the two images matched. In the traveling salesman problem, a time-conscious peddler tries to find the shortest route connecting many cities. To find his solution, he must compare all possible paths—a computation that grows exponentially harder as the number of cities grows. This and other “combinatorial optimization problems” are ubiquitous in business, science, and engineering, and researchers are exploring novel approaches to solve them. But a promising tactic is to map these problems to a statistical model for interacting spins known as the Ising model, which is then solved on a special processor known as an Ising machine. Davide Pierangeli and colleagues at the University of Rome have now realized the largest photonic version of such a machine by representing more than ten thousand spins with a spatially modulated light field [1]. Compared to existing machines, theirs is easier to scale up to accommodate many more spins. With larger machines, researchers could potentially tackle complex optimization problems, such as determining how a protein folds based on its amino acid sequence. Originally proposed to model ferromagnets, the Ising model describes a network of spins that can point only up or down. Each spin’s energy depends on its interaction with neighboring spins—in a ferromagnet, for instance, each spin will prefer to align with its closest neighbors. Roughly speaking, an Ising machine finds the spin configuration that minimizes the energy of the interacting spins. For a suitable set of spin interactions, this solution can then be translated into the solution of some other optimization problem. Although the algorithms that run an Ising machine typically yield only an approximation of the true ground state, they are often much faster than exact methods. Optical versions of Ising machines encode a spin state and/or the interaction between spins in the phase and amplitude of a light field. Such machines can be much faster than those based on other encoding schemes (such as atoms or magnets): they are able to process data at light speed and in parallel, through multiple spatial or frequency channels. They can also take advantage of passive components, which perform a mathematical operation many times at fixed energy cost. Finally, the quantum nature of photons provides a natural source of noise, which mimics the temperature fluctuations of a real statistical system. Inspired by these perks, researchers have developed several photon-based Ising machines using networks of optical parametric oscillators [2–5] and of optical fibers [6, 7]. These photonic machines have been employed to solve optimization problems and to study the phases of spin systems [8, 9]. So far, however, these prototypes haven’t been scalable beyond a few hundred or a few thousand spins because of the effects of decoherence or dispersion, which limit the machines’ practical use. Pierangeli et al. surpass this limit with an optical Ising machine that handles tens of thousands of spins—and possibly more [1]. Their success results from a setup that combines two features: It encodes and processes the spin interactions all at once (spatial multiplexing), and it largely relies on free-space optics, avoiding the need to machine and assemble numerous tiny parts. The researchers used a so-called spatial light modulator (SLM) to imprint a phase of 0 or $𝜋$ at distinct points on the wave front of a laser beam. This binary phase mimics the up or down state of a spin. The team set the interactions between the spins by spatially modulating the beam’s intensity. With these parameters in play, an experiment then consisted of repeated cycles of the following steps (Fig. 1). First, send an intensity-modulated laser beam through the SLM to imprint the spins. Then, record the beam in a CCD camera and compare the detected image to a “target image.” Finally, update the SLM settings to minimize the difference between the two images. By design, this step is the same as minimizing the energy of the spin system. After many cycles, a readout of the SLM will reveal the spin configuration corresponding to the ground state for the chosen interactions, a recently proposed process for evolving the spin states that is known as recurrent feedback [10]. In a proof-of-principle demonstration, Pierangeli et al. set the spin interactions for a simple ferromagnet and showed that their machine yielded the ground state expected at low temperature from mean-field-theory calculations. (The “temperature” of their system is fixed by the various noise sources in the experiment.) In a second experiment, the researchers adjusted the spin interactions to simulate a type of magnet known as a spin glass, where the spin couplings are randomly distributed. Thanks to their machine’s ability to handle many spins, the team was able to analyze how physical observables like the phase’s magnetization and correlation lengths scaled with the number of spins. The work by Pierangeli et al. realizes one of—if not the—largest physical Ising machines ever demonstrated. In principle, it could be scaled up further because larger laser wave fronts can be used to encode more spins, allowing massive numbers of spins to be handled at once. One exciting feature of the researchers’ work is their use of intrinsic photon noise to play the role of temperature [10]. With improved control of the noise level, they will have a temperature “knob” with which to drive and study phase transitions of spin systems. Another interesting direction for their work might be deep learning. This artificial intelligence tool relies on ultrafast and large-scale matrix-to-vector multiplications, an operation that could be performed on a fast optical processor like an Ising machine [2, 10]. For all of these applications, the researchers will need to demonstrate that they can implement operations at close to light speed and for larger numbers of spins. They will also need to reduce the time it takes to update their SLM. Broader applications will require more radical changes to their machine. For example, creating an effective “bias” magnetic field on the spins is usually required to map the Ising model to the traveling salesman problem. But even with this hurdle ahead, Pierangeli et al. have brought optical Ising machines closer to solving real-world problems by making the devices scalable. This research is published in Physical Review Letters.	{"url":"https://physics.aps.org/articles/v12/61","timestamp":"2024-11-04T21:36:20Z","content_type":"text/html","content_length":"34228","record_id":"<urn:uuid:56b5ee29-2eb9-4900-a8a7-13e1d37af356>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00361.warc.gz"}
$$ \def\por{\ \ \| \ \ } \def\dom{\textsf{dom}} \def\tsf#1{\textsf{#1}} \def\unifyj#1#2#3{#1 \sim #2 \Downarrow #3} \def\dict#1#2{\textsf{Dict}\left( #1, #2 \right)} \def\pf{\hookrightarrow}%\ rightharpoonup \def\tone{{t_1}} \def\ttwo{{t_2}} \def\rule#1#2#3{\dfrac{#3}{#2}\ {#1}} \def\eqdef{\overset{\textsf{def}}{=}} \def\freevars{\textsf{free}} $$ Unification is probably one of the most important concepts in computer science. If you are not aware, unification is basically the process you undergo to find out how two syntactic objects like $a+b$ and $a+2c$ can be the same thing. In this case, you replace $b$ with $2c$. $$\def\mathblue#1{\colorbox{blue}{$#1$}} \def\mathgreen#1{\colorbox{green}{$#1$}} \def\mathred#1{\colorbox{red}{$#1$}} \def\mathhl#1{\colorbox{Orchid}{$#1$}} \def\lequiv{\fallingdotseq}$$ $\def\ lequiv{\fallingdotseq}$ What mathematics describes is a long standing issue in philosophy. It is hard to break into this subject, but I hope that my intent can be shared. Since I am a mathemtics student who is interested in formal systems, I can’t help to feel nervous when my professor says $A$ is a subset of $B$ if for all $x$, $x\in B$ whenever $x \in A$, $$\def\infer#1#2{\dfrac{#2}{#1}}$$ Rule systems are seen in almost every programming languages paper. They are a tool for conveying very precise notions of computation. In general, is not easy to give a succinct definition of a rule system, but one can be described easily by adopting a specific representation. A rule system consists of a set of statements of the form $$\infer{Q}{P_1 \quad \ ldots \quad P_n}$$ where $Q,P_i$ are propositional schemata. Here, $Q$ is called the conclusion and $P_i$ are called the premises. Consider the C function int f(int a, int b) { int s = 4; int x,y; x = a * s; s = s + 1; y = b * s; s = s + 1; return x * y; } If we were to write it in Haskell, it would be f :: Int -> Int -> Int f a b = let s = 4 in type Choice a = [a] choose :: [a] -> Choice a choose xs = xs 1 Hello! Hello newtype StateT s m a = StateT { runStateT :: s -> m (a,s) } $$A \to B$$ $$ A $$ $ B $ You should know what sets and types are, what a function is, and the consequences of composing functions. In a programming language, you have types that are given to you and types that you can construct with your own definitions. Some examples of these are Int, Bool, and function types Int -> Int -> Int. You may also may have types that generalize over types, an example is the type Optional<T> where T refers to any type.	{"url":"https://moncrief.dev/posts/","timestamp":"2024-11-05T10:38:27Z","content_type":"text/html","content_length":"12840","record_id":"<urn:uuid:0d2fa3ae-926d-465f-ba54-5bdbe9ab55d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00757.warc.gz"}
Index laws Indices provide more concise ways of writing a variety of quantities in both algebra and number problems. Negative indices represent reciprocals while fractional indices describe roots. The questions in this section cover the arithmetic of numbers of algebraic terms which have indices, including multiplication, division, raising a power to a power for positive, negative and fractional indices.	{"url":"http://peterhalfpennytuition.com/free-gcse-resources/algebra/23-index-laws","timestamp":"2024-11-05T02:58:27Z","content_type":"text/html","content_length":"22540","record_id":"<urn:uuid:197c9c6f-e8a8-4908-bf20-97f64157f529>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00687.warc.gz"}
Optimization problems are ubiquitous in the mathematical modeling of real world systems and cover a very broad range of applications. These applications arise in all branches of Economics, Finance, Chemistry, Materials Science, Astronomy, Physics, Structural and Molecular Biology, Engineering, Computer Science, and Medicine. Optimization modeling requires appropriate time. The general procedure that can be used in the process cycle of modeling is to: (1) describe the problem, (2) prescribe a solution, and (3) control the problem by assessing/updating the optimal solution continuously, while changing the parameters and structure of the problem. Clearly, there are always feedback loops among these general Mathematical Formulation of the Problem: As soon as you detect a problem, think about and understand it in order to adequately describe the problem in writing. Develop a mathematical model or framework to re-present reality in order to devise/use an optimization solution algorithm. The problem formulation must be validated before it is offered a solution. A good mathematical formulation for optimization must be both inclusive (i.e., it includes what belongs to the problem) and exclusive (i.e., shaved-off what does not belong to the problem). Find an Optimal Solution: This is an identification of a solution algorithm and its implementation stage. The only good plan is an implemented plan, which stays implemented! Managerial Interpretations of the Optimal Solution: Once you recognize the algorithm and determine the appropriate module of software to apply, utilize software to obtain the optimal strategy. Next, the solution will be presented to the decision-maker in the same style and language used by the decision-maker. This means providing managerial interpretations of the strategic solution in layman's terms, not just handing the decision-maker a computer printout. Post-Solution Analysis: These activities include updating the optimal solution in order to control the problem. In this ever-changing world, it is crucial to periodically update the optimal solution to any given optimization problem. A model that was valid may lose validity due to changing conditions, thus becoming an inaccurate representation of reality and adversely affecting the ability of the decision-maker to make good decisions. The optimization model you create should be able to cope with changes. The Importance of Feedback and Control: It is necessary to place heavy emphasis on the importance of thinking about the feedback and control aspects of an optimization problem. It would be a mistake to discuss the context of the optimization-modeling process and ignore the fact that one can never expect to find a never-changing, immutable solution to a decision problem. The very nature of the optimal strategy's environment is changing, and therefore feedback and control are an important part of the optimization-modeling process. The above process is depicted as the Systems Analysis, Design, and Control stages in the following flow chart, including the validation and verification activities: Further Readings: Beroggi G., Decision Modeling in Policy Management: An Introduction to the Analytic Concepts, Boston, Kluwer Academic Publishers, 1999. Camm J., and J. Evans, Management Science: Modeling, Analysis, and Interpretation, South-Western College Pub., 1999. Ingredients of Optimization Problems and Their Classification The essence of all businesslike decisions, whether made for a firm, or an individual, is finding a course of action that leaves you with the largest profit. Mankind has long sought, or professed to seek, better ways to carry out the daily tasks of life. Throughout human history, man has first searched for more effective sources of food and then later searched for materials, power, and mastery of the physical environment. However, relatively late in human history general questions began to quantitatively formulate first in words, and later developing into symbolic notations. One pervasive aspect of these general questions was to seek the "best" or "optimum". Most of the time managers seek merely to obtain some improvement in the level of performance, or a "goal-seeking" problem. It should be emphasized that these words do not usually have precise meanings. Efforts have been made to describe complex human and social situations. To have meaning, the problem should be written down in a mathematical expression containing one or more variables, in which the value of variables are to be determined. The question then asked, is what values should these variables have to ensure the mathematical expression has the greatest possible numerical value (maximization) or the least possible numerical value (minimization). This process of maximizing or minimizing is referred to as optimization. Optimization, also called mathematical programming, helps find the answer that yields the best result--the one that attains the highest profit, output, or happiness, or the one that achieves the lowest cost, waste, or discomfort. Often these problems involve making the most efficient use of resources--including money, time, machinery, staff, inventory, and more. Optimization problems are often classified as linear or nonlinear, depending on whether the relationship in the problem is linear with respect to the variables. There are a variety of software packages to solve optimization problems. For example, LINDO or your WinQSB solve linear program models and LINGO and What'sBest! solve nonlinear and linear problems. Mathematical Programming, solves the problem of determining the optimal allocations of limited resources required to meet a given objective. The objective must represent the goal of the decision-maker. For example, the resources may correspond to people, materials, money, or land. Out of all permissible allocations of the resources, it is desired to find the one or ones that maximize or minimize some numerical quantity such as profit or cost. Optimization models are also called Prescriptive or Normative models since they seek to find the best possible strategy for There are many optimization algorithms available. However, some methods are only appropriate for certain types of problems. It is important to be able to recognize the characteristics of a problem and identify an appropriate solution technique. Within each class of problems, there are different minimization methods, which vary in computational requirements, convergence properties, and so on. Optimization problems are classified according to the mathematical characteristics of the objective function, the constraints, and the controllable decision variables. Optimization problems are made up of three basic ingredients: 1. An objective function that we want to minimize or maximize. That is, the quantity you want to maximize or minimize is called the objective function. Most optimization problems have a single objective function, if they do not, they can often be reformulated so that they do. The two interesting exceptions to this rule are: The goal seeking problem: In most business applications the manager wishes to achieve a specific goal, while satisfying the constraints of the model. The user does not particularly want to optimize anything so there is no reason to define an objective function. This type of problem is usually called a feasibility problem. Multiple objective functions: Often, the user would actually like to optimize many different objectives at once. Usually, the different objectives are not compatible. The variables that optimize one objective may be far from optimal for the others. In practice, problems with multiple objectives are reformulated as single-objective problems by either forming a weighted combination of the different objectives or else by placing some objectives as "desirable" constraints. 2. The controllable inputs are the set of decision variables which affect the value of the objective function. In the manufacturing problem, the variables might include the allocation of different available resources, or the labor spent on each activity. Decision variables are essential. If there are no variables, we cannot define the objective function and the problem 3. The uncontrollable inputs are called parameters. The input values may be fixed numbers associated with the particular problem. We call these values parameters of the model. Often you will have several "cases" or variations of the same problem to solve, and the parameter values will change in each problem variation. 4. Constraints are relations between decision variables and the parameters. A set of constraints allows some of the decision variables to take on certain values, and exclude others. For the manufacturing problem, it does not make sense to spend a negative amount of time on any activity, so we constrain all the "time" variables to be non-negative. Constraints are not always essential. In fact, the field of unconstrained optimization is a large and important one for which a lot of algorithms and software are available. In practice, answers that make good sense about the underlying physical or economic problem, cannot often be obtained without putting constraints on the decision variables. Feasible and Optimal Solutions: A solution value for decision variables, where all of the constraints are satisfied, is called a feasible solution. Most solution algorithms proceed by first finding a feasible solution, then seeking to improve upon it, and finally changing the decision variables to move from one feasible solution to another feasible solution. This process is repeated until the objective function has reached its maximum or minimum. This result is called an optimal solution. The basic goal of the optimization process is to find values of the variables that minimize or maximize the objective function while satisfying the constraints. This result is called an optimal There are well over 4000 solution algorithms for different kinds of optimization problems. The widely used solution algorithms are those developed for the following mathematical programs: convex programs, separable programs, quadratic programs and the geometric programs. Linear Program Linear programming deals with a class of optimization problems, where both the objective function to be optimized and all the constraints, are linear in terms of the decision variables. A short history of Linear Programming: 1. In 1762, Lagrange solved tractable optimization problems with simple equality constraints. 2. In 1820, Gauss solved linear system of equations by what is now call Causssian elimination. In 1866 Wilhelm Jordan refinmened the method to finding least squared errors as ameasure of goodness-of-fit. Now it is referred to as the Gauss-Jordan Method. 3. In 1945, Digital computer emerged. 4. In 1947, Dantzig invented the Simplex Methods. 5. In 1968, Fiacco and McCormick introduced the Interior Point Method. 6. In 1984, Karmarkar applied the Interior Method to solve Linear Programs adding his innovative analysis. Linear programming has proven to be an extremely powerful tool, both in modeling real-world problems and as a widely applicable mathematical theory. However, many interesting optimization problems are nonlinear. The study of such problems involves a diverse blend of linear algebra, multivariate calculus, numerical analysis, and computing techniques. Important areas include the design of computational algorithms (including interior point techniques for linear programming), the geometry and analysis of convex sets and functions, and the study of specially structured problems such as quadratic programming. Nonlinear optimization provides fundamental insights into mathematical analysis and is widely used in a variety of fields such as engineering design, regression analysis, inventory control, geophysical exploration, and economics. Quadratic Program Quadratic Program (QP) comprises an area of optimization whose broad range of applicability is second only to linear programs. A wide variety of applications fall naturally into the form of QP. The kinetic energy of a projectile is a quadratic function of its velocity. The least-square regression with side constraints has been modeled as a QP. Certain problems in production planning, location analysis, econometrics, activation analysis in chemical mixtures problem, and in financial portfolio management and selection are often treated as QP. There are numerous solution algorithms available for the case under the restricted additional condition, where the objective function is convex. Constraint Satisfaction Many industrial decision problems involving continuous constraints can be modeled as continuous constraint satisfaction and optimization problems. Constraint Satisfaction problems are large in size and in most cases involve transcendental functions. They are widely used in chemical processes and cost restrictions modeling and optimization. Convex Program Convex Program (CP) covers a broad class of optimization problems. When the objective function is convex and the feasible region is a convex set, both of these assumptions are enough to ensure that local minimum is a global minimum. Data Envelopment Analysis The Data Envelopment Analysis (DEA) is a performance metric that is grounded in the frontier analysis methods from the economics and finance literature. Frontier efficiency (output/input) analysis methods identify best practice performance frontier, which refers to the maximal outputs that can be obtained from a given set of inputs with respect to a sample of decision making units using a comparable process to transform inputs to outputs. The strength of DEA relies partly on the fact that it is a non-parametric approach, which does not require specification of any functional form of relationships between the inputs and the outputs. DEA output reduces multiple performance measures to a single one to use linear programming techniques. The weighting of performance measures reacts to the decision-maker's utility. Dynamic Programming Dynamic programming (DP) is essentially bottom-up recursion where you store the answers in a table starting from the base case(s) and building up to larger and larger parameters using the recursive rule(s). You would use this technique instead of recursion when you need to calculate the solutions to all the sub-problems and the recursive solution would solve some of the sub-problems repeatedly. While generally DP is capable of solving many diverse problems, it may require huge computer storage in most cases. Separable Program Separable Program (SP) includes a special case of convex programs, where the objective function and the constraints are separable functions, i.e., each term involves just a single variable. Geometric Program Geometric Program (GP) belongs to Nonconvex programming, and has many applications in particular in engineering design problems. Fractional Program In this class of problems, the objective function is in the form of a fraction (i.e., ratio of two functions). Fractional Program (FP) arises, for example, when maximizing the ratio of profit capital to capital expended, or as a performance measure wastage ratio. Heuristic Optimization A heuristic is something "providing aid in the direction of the solution of a problem but otherwise unjustified or incapable of justification." So heuristic arguments are used to show what we might later attempt to prove, or what we might expect to find in a computer run. They are, at best, educated guesses. Several heuristic tools have evolved in the last decade that facilitate solving optimization problems that were previously difficult or impossible to solve. These tools include evolutionary computation, simulated annealing, tabu search, particle swarm, etc. Common approaches include, but are not limited to: 1. comparing solution quality to optimum on benchmark problems with known optima, average difference from optimum, frequency with which the heuristic finds the optimum. 2. comparing solution quality to a best known bound for benchmark problems whose optimal solutions cannot be determined. 3. comparing your heuristic to published heuristics for the same problem type, difference in solution quality for a given run time and, if relevant, memory limit. 4. profiling average solution quality as a function of run time, for instance, plotting mean and either min and max or 5th and 95th percentiles of solution value as a function of time -- this assumes that one has multiple benchmark problem instances that are comparable. Global Optimization The aim of Global Optimization (GO) is to find the best solution of decision models, in presence of the multiple local solutions. While constrained optimization is dealing with finding the optimum of the objective function subject to constraints on its decision variables, in contrast, unconstrained optimization seeks the global maximum or minimum of a function over its entire domain space, without any restrictions on decision variables. Nonconvex Program A Nonconvex Program (NC) encompasses all nonlinear programming problems that do not satisfy the convexity assumptions. However, even if you are successful at finding a local minimum, there is no assurance that it will also be a global minimum. Therefore, there is no algorithm that will guarantee finding an optimal solution for all such problem. Nonsmooth Program Nonsmooth Programs (NSP) contain functions for which the first derivative does not exist. NSP are arising in several important applications of science and engineering, including contact phenomena in statics and dynamics or delamination effects in composites. These applications require the consideration of nonsmoothness and nonconvexity. Most metaheuristics have been created for solving discrete combinatorial optimization problems. Practical applications in engineering, however, usually require techniques, which handle continuous variables, or miscellaneous continuous and discrete variables. As a consequence, a large research effort has focused on fitting several well-known metaheuristics, like Simulated Annealing (SA), Tabu Search (TS), Genetic Algorithms (GA), Ant Colony Optimization (ACO), to the continuous cases. The general metaheuristics aim at transforming discrete domains of application into continuous ones, by means of: □ Methodological developments aimed at adapting some metaheuristics, especially SA, TS, GA, ACO, GRASP, variable neighborhood search, guided local search, scatter search, to continuous or discrete/continuous variable problems. □ Theoretical and experimental studies on metaheuristics adapted to continuous optimization, e.g., convergence analysis, performance evaluation methodology, test-case generators, constraint handling, etc. □ Software implementations and algorithms for metaheuristics adapted to continuous optimization. □ Real applications of discrete metaheuristics adapted to continuous optimization. □ Performance comparisons of discrete metaheuristics (adapted to continuous optimization) with that of competitive approaches, e.g., Particle Swarm Optimization (PSO), Estimation of Distribution Algorithms (EDA), Evolutionary Strategies (ES), specifically created for continuous optimization. Multilevel Optimization In many decision processes there is a hierarchy of decision makers and decisions are taken at different levels in thishierarchy. Multilevel Optimization focuses on the whole hierarchy structure. The field of multilevel optimization has become a well known and important research field. Hierarchical structures can be found in scientific disciplines such as environment, ecology, biology, chemical engineering, mechanics, classification theory, databases, network design, transportation, supply chain, game theory and economics. Moreover, new applications are constantly being Multiobjective Program Multiobjective Program (MP) known also as Goal Program, is where a single objective characteristic of an optimization problem is replaced by several goals. In solving MP, one may represent some of the goals as constraints to be satisfied, while the other objectives can be weighted to make a composite single objective function. Multiple objective optimization differs from the single objective case in several ways: 1. The usual meaning of the optimum makes no sense in the multiple objective case because the solution optimizing all objectives simultaneously is, in general, impractical; instead, a search is launched for a feasible solution yielding the best compromise among objectives on a set of, so called, efficient solutions; 2. The identification of a best compromise solution requires taking into account the preferences expressed by the decision-maker; 3. The multiple objectives encountered in real-life problems are often mathematical functions of contrasting forms. 4. A key element of a goal programming model is the achievement function; that is, the function that measures the degree of minimisation of the unwanted deviation variables of the goals considered in the model. A Business Application: In credit card portfolio management, predicting the cardholder's spending behavior is a key to reduce the risk of bankruptcy. Given a set of attributes for major aspects of credit cardholders and predefined classes for spending behaviors, one might construct a classification model by using multiple criteria linear programming to discover behavior patterns of credit cardholders. Non-Binary Constraints Program Over the years, the constraint programming community has paid considerable attention to modeling and solving problems by using binary constraints. Only recently has non-binary constraints captured attention, due to growing number of real-life applications. A non-binary constraint is a constraint that is defined on k variables, where k is normally greater than two. A non-binary constraint can be seen as a more global constraint. Modeling a problem as a non-binary constraint has two main advantages: It facilitates the expression of the problem; and it enables more powerful constraint propagation as more global information becomes available. Success in timetabling, scheduling, and routing, has proven that the use of non-binary constraints is a promising direction of research. In fact, a growing number of OR/MS/DS workers feel that this topic is crucial to making constraint technology a realistic way to model and solve real-life problems. Bilevel Optimization Most of the mathematical programming models deal with decision-making with a single objective function. The bilevel programming on the other hand is developed for applications in decentralized planning systems in which the first level is termed as the leader and the second level pertains to the objective of the follower. In the bilevel programming problem, each decision maker tries to optimize its own objective function without considering the objective of the other party, but the decision of each party affects the objective value of the other party as well as the decision Bilevel programming problems are hierarchical optimization problems where the constraints of one problem are defined in part by a second parametric optimization problem. If the second problem has a unique optimal solution for all parameter values, this problem is equivalent to usual optimization problem having an implicitly defined objective function. However, when the problem has non-unique optimal solutions, the optimistic (or weak) and the pessimistic (or strong) approaches are being applied. Combinatorial Optimization Combinatorial generally means that the state space is discrete (e.g., symbols, not necessarily numbers). This space could be finite or denumerable sets. For example, a discrete problem is combinatorial. Problems where the state space is totally ordered can often be solved by mapping them to the integers and applying "numerical" methods. If the state space is unordered or only partially ordered, these methods fail. This means that the heuristics methods becomes necessary, such as simulated annealing. Combinatorial optimization is the study of packing, covering, and partitioning, which are applications of integer programs. They are the principle mathematical topics in the interface between combinatorics and optimization. These problems deal with the classification of integer programming problems according to the complexity of known algorithms, and the design of good algorithms for solving special subclasses. In particular, problems of network flows, matching, and their matroid generalizations are studied. This subject is one of the unifying elements of combinatorics, optimization, operations research, and computer science. Evolutionary Techniques Nature is a robust optimizer. By analyzing nature's optimization mechanism we may find acceptable solution techniques to intractable problems. Two concepts that have most promise are simulated annealing and the genetic techniques. Scheduling and timetabling are amongst the most successful applications of evolutionary techniques. Genetic Algorithms (GAs) have become a highly effective tool for solving hard optimization problems. However, its theoretical foundation is still rather fragmented. Particle Swarm Optimization Particle Swarm Optimization (PSO) is a stochastic, population-based optimization algorithm. Instead of competition/selection, like say in Evolutionary Computation, PSO makes use of cooperation, according to a paradigm sometimes called "swarm intelligence". Such systems are typically made up of a population of simple interacting agents without any centralized control, and inspired by cases that can be found in nature, such as ant colonies, bird flocking, animal herding, bacteria molding, fish schooling, etc. There are many variants of PSO including constrained, multiobjective, and discrete or combinatorial versions, and applications have been developed using PSO in many fields. Swarm Intelligence Biologists studied the behavior of social insects for a long time. After millions of years of evolution all these species have developed incredible solutions for a wide range of problems. The intelligent solutions to problems naturally emerge from the self-organization and indirect communication of these individuals. Indirect interactions occur between two individuals when one of them modifies the environment and the other responds to the new environment at a later time. Swarm Intelligence is an innovative distributed intelligent paradigm for solving optimization problems that originally took its inspiration from the biological examples by swarming, flocking and herding phenomena in vertebrates. Data Mining is an analytic process designed to explore large amounts of data in search of consistent patterns and/or systematic relationships between variables, and then to validate the findings by applying the detected patterns to new subsets of data. Online Optimization Whether costs are to be reduced, profits to be maximized, or scarce resources to be used wisely, optimization methods are available to guide decision-making. In online optimization, the main issue is incomplete data and the scientific challenge: how well can an online algorithm perform? Can one guarantee solution quality, even without knowing all data in advance? In real-time optimization there is an additional requirement: decisions have to be computed very fast in relation to the time frame we are considering. Further Readings: Abraham A., C. Grosan and V. Ramos, Swarm Intelligence, Springer Verlag, 2006. It deals with the applications of swarm intelligence in data mining, using different intelligent approaches. Charnes A., Cooper W., Lewin A., and L. Seiford, Data Envelopment Analysis: Theory, Methodology and Applications, Kluwer Academic Publications, 1994. Dempe S., Foundations of Bilevel Programming, Kluwer, 2002. Diwekar U., Introduction to Applied Optimization, Kluwer Academic Publishers, 2003. Covers almost all the above techniques. Liu B., and A. Esogbue, Decision Criteria and Optimal Inventory Processes, Kluwer, 1999. Luenberger D., Linear and Nonlinear Programming, Kluwer Academic Publishers, 2003. Miller R., Optimization: Foundations and Applications, Wiley, 1999. MigdalasA., Pardalos p., and P. Varbrand, Multilevel Optimization: Algorithms and Applications, Kluwer, 1998. Reeves C., and J. Rowe, Genetic Algorithms: Principles and Perspectives, Kluwer, 2002. Rodin R., Optimization in Operations Research, Prentice Hall, New Jersey, 2000. For more books and journal articles on optimization visit the Web site Decision Making Resources Linear Programming Linear programming is often a favorite topic for both professors and students. The ability to introduce LP using a graphical approach, the relative ease of the solution method, the widespread availability of LP software packages, and the wide range of applications make LP accessible even to students with relatively weak mathematical backgrounds. Additionally, LP provides an excellent opportunity to introduce the idea of "what-if" analysis, due to the powerful tools for post-optimality analysis developed for the LP model. Linear Programming (LP) is a mathematical procedure for determining optimal allocation of scarce resources. LP is a procedure that has found practical application in almost all facets of business, from advertising to production planning. Transportation, distribution, and aggregate production planning problems are the most typical objects of LP analysis. In the petroleum industry, for example a data processing manager at a large oil company recently estimated that from 5 to 10 percent of the firm's computer time was devoted to the processing of LP and LP-like models. Linear programming deals with a class of programming problems where both the objective function to be optimized is linear and all relations among the variables corresponding to resources are linear. This problem was first formulated and solved in the late 1940's. Rarely has a new mathematical technique found such a wide range of practical business, commerce, and industrial applications and simultaneously received so thorough a theoretical development, in such a short period of time. Today, this theory is being successfully applied to problems of capital budgeting, design of diets, conservation of resources, games of strategy, economic growth prediction, and transportation systems. In very recent times, linear programming theory has also helped resolve and unify many outstanding applications. It is important for the reader to appreciate, at the outset, that the "programming" in Linear Programming is of a different flavor than the "programming" in Computer Programming. In the former case, it means to plan and organize as in "Get with the program!", it programs you by its solution. While in the latter case, it means to write codes for performing calculations. Training in one kind of programming has very little direct relevance to the other. In fact, the term "linear programming" was coined before the word "programming" became closely associated with computer software. This confusion is sometimes avoided by using the term linear optimization as a synonym for linear programming. Any LP problem consists of an objective function and a set of constraints. In most cases, constraints come from the environment in which you work to achieve your objective. When you want to achieve the desirable objective, you will realize that the environment is setting some constraints (i.e., the difficulties, restrictions) in fulfilling your desire or objective. This is why religions such as Buddhism, among others, prescribe living an abstemious life. No desire, no pain. Can you take this advice with respect to your business objective? What is a function: A function is a thing that does something. For example, a coffee grinding machine is a function that transform the coffee beans into powder. The (objective) function maps and translates the input domain (called the feasible region) into output range, with the two end-values called the maximum and the minimum values. When you formulate a decision-making problem as a linear program, you must check the following conditions: 1. The objective function must be linear. That is, check if all variables have power of 1 and they are added or subtracted (not divided or multiplied) 2. The objective must be either maximization or minimization of a linear function. The objective must represent the goal of the decision-maker 3. The constraints must also be linear. Moreover, the constraint must be of the following forms ( £, ³, or =, that is, the LP-constraints are always closed). For example, the following problem is not an LP: Max X, subject to X < 1. This very simple problem has no solution. As always, one must be careful in categorizing an optimization problem as an LP problem. Here is a question for you. Is the following problem an LP problem? Max X2 subject to: X1 + X2 £ 0 X1^2 - 4 £ 0 Although the second constraint looks "as if" it is a nonlinear constraint, this constraint can equivalently be written as: X1 ³ -2, and X2 £ 2. Therefore, the above problem is indeed an LP problem. For most LP problems one can think of two important classes of objects: The first is limited resources such as land, plant capacity, or sales force size; the second, is activities such as "produce low carbon steel", "produce stainless steel", and "produce high carbon steel". Each activity consumes or possibly contributes additional amounts of the resources. There must be an objective function, i.e. a way to tell bad from good, from an even better decision. The problem is to determine the best combination of activity levels, which do not use more resources than are actually available. Many managers are faced with this task everyday. Fortunately, when a well-formulated model is input, linear programming software helps to determine the best combination. The Simplex method is a widely used solution algorithm for solving linear programs. An algorithm is a series of steps that will accomplish a certain task. LP Problem Formulation Process and Its Applications To formulate an LP problem, I recommend using the following guidelines after reading the problem statement carefully a few times. Any linear program consists of four parts: a set of decision variables, the parameters, the objective function, and a set of constraints. In formulating a given decision problem in mathematical form, you should practice understanding the problem (i.e., formulating a mental model) by carefully reading and re-reading the problem statement. While trying to understand the problem, ask yourself the following general questions: 1. What are the decision variables? That is, what are controllable inputs? Define the decision variables precisely, using descriptive names. Remember that the controllable inputs are also known as controllable activities, decision variables, and decision activities. 2. What are the parameters? That is, what are the uncontrollable inputs? These are usually the given constant numerical values. Define the parameters precisely, using descriptive names. 3. What is the objective? What is the objective function? Also, what does the owner of the problem want? How the objective is related to his decision variables? Is it a maximization or minimization problem? The objective represents the goal of the decision-maker. 4. What are the constraints? That is, what requirements must be met? Should I use inequality or equality type of constraint? What are the connections among variables? Write them out in words before putting them in mathematical form. Learn that the feasible region has nothing or little to do with the objective function (min or max). These two parts in any LP formulation come mostly from two distinct and different sources. The objective function is set up to fulfill the decision-maker's desire (objective), whereas the constraints which shape the feasible region usually comes from the decision-maker's environment putting some restrictions/conditions on achieving his/her objective. The following is a very simple illustrative problem. However, the way we approach the problem is the same for a wide variety of decision-making problems, and the size and complexity may differ. The first example is a product-mix problem. The Carpenter's Problem: Allocating Scarce Resources Among Competitive Means During a couple of brain-storming sessions with a carpenter (our client), he told us that he, solely, makes tables and chairs, sells all tables and chairs he makes at a market place, however, does not have a stable income, and wishes to do his best. The objective is to find out how many tables and chairs he should make to maximize net income. We begin by focusing on a time frame, i.e., planning time-horizon, to revise our solution weekly if needed. To learn more about his problem, we must go to his shop and observe what is going on and measure what we need to formulate (i.e., to give a Form, to make a model) of his problem. We must confirm that his objective is to maximize net income. We must communicate with the client. The carpenter's problem deals with finding out how many tables and chairs to make per week; but first an objective function must be established: Since the total cost is the sum of the fixed cost (F) and the variable cost per unit multiplied by the number of units produced. Therefore, the decision problem is to find X1 and X2 such that: Maximize 9X1 + 6X2 [(1.5X1 + X2) + (2.5X1 + 2X2) + F1 + F2], where X1 and X2 stand for the number of tables and chairs; the cost terms in the brackets are the raw material, and labor costs respectively. F1 and F2 are the fixed costs for the two products respectively. Without loss of generality, and any impact on optimal solution, let us set F1 = 0, and F2 = 0. The objective function reduces to the following net profit function: Maximize 5X1 + 3X2 That is, the net incomes (say, in dollars, or tens of dollars) from selling X1 tables and X2 chairs. The constraining factors which, usually come from outside, are the limitations on labors (this limitation comes from his family) and raw material resources (this limitation comes from scheduled delivery). Production times required for a table and a chair are measured at different times of day, and estimated to be 2 hours and 1 hour, respectively. Total labor hours per week are only 40 hrs. Raw materials required for a table and a chair are 1, and 2 units respectively. Total supply of raw material is 50 units per week. Therefore, the LP formulation is: Maximize 5 X1 + 3 X2 Subject to: 2 X1 + X2 £ 40 labor constraint X1 + 2 X2 £ 50 material constraint and both X1, X2 are non-negative. This is a mathematical model for the carpenter's problem. The decision variables, i.e., controllable inputs are X1, and X2. The output for this model is the total net income 5 X1 + 3 X2. All functions used in this model are linear (the decision variable have power equal to 1). The coefficients of these constraints are called Technological Factors (matrix). The review period is one week, an appropriate period within which the uncontrollable inputs (all parameters such as 5, 50, 2,..) are less likely to change (fluctuate). Even for such a short planning time-horizon, we must perform the what-if analysis to react to any changes in these inputs in order to control the problem, i.e., to update the prescribed solution. Notice that since the carpenter is not going out of business at the end of the planning horizon, we added the conditions that both X1, X2 must be non-negative instead of the requirements that X1, and X2 must be positive integers. The non-negativity conditions are also known as "implied constraints." Again, a Linear Program would be fine for this problem if the carpenter were going to continue to manufacture these products. The partial items would simply be counted as work in progress and would eventually become finished goods say, in the next week. We may try to solve for X1 and X2 by listing possible solutions for each and selecting the pair (X1, X2) that maximize 5X1 + 3X2 (the net income). However, it is too time consuming to list all possible alternatives and if the alternatives are not exhaustively listed, we cannot be sure that the pair we select (as a solution) is the best of all alternatives. This way of solving a problem is known as "sequential thinking" versus "simultaneous thinking". More efficient and effective methodologies, known as the Linear Programming Solution Techniques are based on simultaneous thinking are commercially available in over 400 different software packages from all over the world. The optimal solution, i.e., optimal strategy, is to make X1 = 10 tables, and X2 = 20 chairs. We may program the carpenter's weekly activities to make 10 tables and 20 chairs. With this (optimal) strategy, the net income is $110. This prescribed solution was a surprise for the carpenter since, because of more net income of selling a table ($5), he used to make more tables than chairs! Hire or Not? Suppose the carpenter can hire someone to help at a cost of $2 per hour. This is, in addition, hourly-based wage he/she is currently paying; otherwise $2 is much lower than the current minimum wage in US. Should the carpenter hire and if yes then for how may hours? Let X3 be the number of extra hours, then the modified problem is: Maximize 5 X1 + 3 X2 - 2 X3 Subject to: 2 X1 + X2 £ 40 + X3 labor constraint with unknown additional hours X1 + 2 X2 £ 50 material constraint Under this new condition, we will see that the optimal solution is X1 = 50, X2 = 0, X3 = 60, with optimal net income of $130. Therefore, the carpenter should be hired for 60 hours. What about only hiring 40 hours? The answer to this and other types of what-if questions are treated under sensitivity analysis in this Web site. As an exercise, use your LP software to find the largest range for X values satisfying the following inequality with two absolute value terms: \| 3X 4 \| - \| 2X 1 \| £ 2 A Product-Replacement Problem A price-taking firm sells S units of its product at the market price of p. Management's policy is to replace defective units at no additional charge, on the first-come, first-served basis, while replacement units are available. Because management does not want to risk making the same mistake twice, it produces the units that it sells to the market on one machine. Moreover, it produces the replacement units, denoted X, on a second, higher-quality machine. The fixed cost associated with operating both machines, the variable cost, and replacement cost are given is the short-run cost function C(X) = 100 + 20S + 30X. The exact probability that a unit will be defective is r. Acting out of caution, however, management always underestimate the reliability of its product. Nonetheless, it imposes the condition that X ³ r.S. The demand for the firm's product is given by S(r) = 10000e^-0.2r. Hence the decision problem is to maximize the net profit function P(X): Maximize P(X) = 100000p e^- 0.2r - 100 - 20S - 30X, subject to: X ³ r.S. As we will learn, the solutions to the LP problems are at the vertices of the feasible region. Therefore, the net profit P(X) will be maximized if the management set X = r.S. A Diet Problem Suppose the only foods available in your local store are potatoes and steak. The decision about how much of each food to buy is to made entirely on dietary and economic considerations. We have the nutritional and cost information in the following table: Per unit Per unit Minimum of potatoes of steak requirements Units of carbohydrates 3 1 8 Units of vitamins 4 3 19 Units of proteins 1 3 7 Unit cost 25 50 The problem is to find a diet (a choice of the numbers of units of the two foods) that meets all minimum nutritional requirements at minimal cost. a. Formulate the problem in terms of linear inequalities and an objective function. b. Solve the problem geometrically. c. Explain how the 2:1 cost ratio (steak to potatoes) dictates that the solution must be where you said it is. d. Find a cost ratio that would move the optimal solution to a different choice of numbers of food units, but that would still require buying both steak and potatoes. e. Find a cost ratio that would dictate buying only one of the two foods in order to minimize cost. a) We begin by setting the constraints for the problem. The first constraint represents the minimum requirement for carbohydrates, which is 8 units per some unknown amount of time. 3 units can be consumed per unit of potatoes and 1 unit can be consumed per unit of steak. The second constraint represents the minimum requirement for vitamins, which is 19 units. 4 units can be consumed per unit of potatoes and 3 units can be consumed per unit of steak. The third constraint represents the minimum requirement for proteins, which is 7 units. 1 unit can be consumed per unit of potatoes and 3 units can be consumed per unit of steak. The fourth and fifth constraints represent the fact that all feasible solutions must be nonnegative because we can't buy negative quantities. {3X1 + X2 ³ 8, 4X1+ 3X2 ³ 19, X1+ 3X2 ³ 7, X1³ 0, X2 ³ 0}; Next we plot the solution set of the inequalities to produce a feasible region of possibilities. c) The 2:1 cost ratio of steak to potatoes dictates that the solution must be here since, as a whole, we can see that one unit of steak is slightly less nutritious than one unit of potatoes. Plus, in the one category where steak beats potatoes in healthiness (proteins), only 7 total units are necessary. Thus it is easier to fulfill these units without buying a significant amout of steak. Since steak is more expensive, buying more potatoes to fulfill these nutritional requirements is more logical. d) Now we choose a new cost ratio that will move the optimal solution to a different choice of numbers of food units. Both steak and potatoes will still be purchased, but a different solution will be found. Let's try a 5:2 cost ratio. d) Now we choose a new cost ratio that will move the optimal solution to a different choice of numbers of food units. Both steak and potatoes will still be purchased, but a different solution will be found. Let's try a 5:2 cost ratio. d) Now we choose a new cost ratio that will move the optimal solution to a different choice of numbers of food units. Both steak and potatoes will still be purchased, but a different solution will be found. Let's try a 5:2 cost ratio. Thus, the optimal solution for this cost ratio is buying 8 steaks and no potatoes per unit time to meet the minimum nutritional requirements. A Blending Problem Bryant's Pizza, Inc. is a producer of frozen pizza products. The company makes a net income of $1.00 for each regular pizza and $1.50 for each deluxe pizza produced. The firm currently has 150 pounds of dough mix and 50 pounds of topping mix. Each regular pizza uses 1 pound of dough mix and 4 ounces (16 ounces= 1 pound) of topping mix. Each deluxe pizza uses 1 pound of dough mix and 8 ounces of topping mix. Based on the past demand per week, Bryant can sell at least 50 regular pizzas and at least 25 deluxe pizzas. The problem is to determine the number of regular and deluxe pizzas the company should make to maximize net income. Formulate this problem as an LP problem. Let X1 and X2 be the number of regular and deluxe pizza, then the LP formulation is: Maximize X1 + 1.5 X2 Subject to: X1 + X2 £ 150 0.25 X1 + 0.5 X2 £ 50 X1 ³ 50 X2 ³ 25 X1 ³ 0, X2 ³ 0 Other Common Applications of LP Linear programming is a powerful tool for selecting alternatives in a decision problem and, consequently, has been applied in a wide variety of problem settings. We will indicate a few applications covering the major functional areas of a business organization. Finance: The problem of the investor could be a portfolio-mix selection problem. In general, the number of different portfolios can be much larger than the example indicates, more and different kinds of constraints can be added. Another decision problem involves determining the mix of funding for a number of products when more than one method of financing is available. The objective may be to maximize total profits, where the profit for a given product depends on the method of financing. For example, funding may be done with internal funds, short-term debt, or intermediate financing (amortized loans). There may be limits on the availability of each of the funding options as well as financial constraints requiring certain relationships between the funding options so as to satisfy the terms of bank loans or intermediate financing. There may also be limits on the production capacity for the products. The decision variables would be the number of units of each product to be financed by each funding option. Production and Operations Management: Quite often in the process industries a given raw material can be made into a wide variety of products. For example, in the oil industry, crude oil is refined into gasoline, kerosene, home-heating oil, and various grades of engine oil. Given the present profit margin on each product, the problem is to determine the quantities of each product that should be produced. The decision is subject to numerous restrictions such as limits on the capacities of various refining operations, raw-material availability, demands for each product, and any government-imposed policies on the output of certain products. Similar problems also exist in the chemical and food-processing industries. Human Resources: Personnel planning problems can also be analyzed with linear programming. For example, in the telephone industry, demands for the services of installer-repair personnel are seasonal. The problem is to determine the number of installer-repair personnel and line-repair personnel to have on the work force each month where the total costs of hiring, layoff, overtime, and regular-time wages are minimized. The constraints set includes restrictions on the service demands that must be satisfied, overtime usage, union agreements, and the availability of skilled people for hire. This example runs contrary to the assumption of divisibility; however, the work-force levels for each month would normally be large enough that rounding to the closest integer in each case would not be detrimental, provided the constraints are not violated. Marketing: Linear programming can be used to determine the proper mix of media to use in an advertising campaign. Suppose that the available media are radio, television, and newspapers. The problem is to determine how many advertisements to place in each medium. Of course, the cost of placing an advertisement depends on the medium chosen. We wish to minimize the total cost of the advertising campaign, subject to a series of constraints. Since each medium may provide a different degree of exposure of the target population, there may be a lower bound on the total exposure from the campaign. Also, each medium may have a different efficiency rating in producing desirable results; there may thus be a lower bound on efficiency. In addition, there may be limits on the availability of each medium for advertising. Distribution: Another application of linear programming is in the area of distribution. Consider a case in which there are m factories that must ship goods to n warehouses. A given factory could make shipments to any number of warehouses. Given the cost to ship one unit of product from each factory to each warehouse, the problem is to determine the shipping pattern (number of units that each factory ships to each warehouse) that minimizes total costs. This decision is subject to the restrictions that demand at each factory cannot ship more products than it has the capacity to Graphical Solution Method Procedure for Graphical Method of Solving LP Problems: 1. Is the problem an LP? Yes, if and only if: All variables have power of 1, and they are added or subtracted (not divided or multiplied). The constraint must be of the following forms ( £, ³, or =, that is, the LP-constraints are always closed), and the objective must be either maximization or minimization. For example, the following problem is not an LP: Max X, subject to X < 1. This very simple problem has no solution. 2. Can I use the graphical method? Yes, if the number of decision variables is either 1 or 2. 3. Use Graph Paper. Graph each constraint one by one, by pretending that they are equalities (pretend all £ and ³, are = ) and then plot the line. Graph the straight line on a system of coordinates on a graph paper. A system of coordinate has two axes: a horizontal axis called the x-axis (abscissa), and a vertical axis, called the y-axis (ordinate). The axes are numbered, usually from zero to the largest value expected for each variable. 4. As each line is created, divide the region into 3 parts with respect to each line. To identify the feasible region for this particular constraint, pick a point in either side of the line and plug its coordinates into the constraint. If it satisfies the condition, this side is feasible; otherwise the other side is feasible. For equality constraints, only the points on the line are 5. Throw away the sides that are not feasible. After all the constraints are graphed, you should have a non-empty (convex) feasible region, unless the problem is infeasible. Unfortunately, some of the boundaries of the feasible regions described in your textbook are wrong. See, e.g., the figures depicted on page 56. Almost all inequalities must be changed to equality. Right? 6. Create (at least) two iso-value lines from the objective function, by setting the objective function to any two distinct numbers. Graph the resulting lines. By moving these lines parallel, you will find the optimal corner (extreme point), if it does exist. In general, if the feasible region is within the first quadrant of the coordinate system (i.e., if X1 and X2 ³ 0), then, for the maximization problems you are moving the iso-value objective function parallel to itself far away from the origin point (0, 0), while having at least a common point with the feasible region. However, for minimization problems the opposite is true, that is, you are moving the iso-value objective parallel to itself closer to the origin point, while having at least a common point with the feasible region. The common point provides the optimal Classification of the Feasible Points: : The feasible points of any non-empty LP feasible region can be classified as, interiors, boundaries, or vertices. As shown in the following figure: The point B in the above two-dimensional figure, for example, is a boundary point of the feasible set because every small circle centered at the point B, however small, contains both points some in the set and some points outside the set. The point I is an interior point because the orange circle and all smaller circles, as well as some larger ones; contains exclusively points in the set. The collection of boundary points belonging to one set is called boundary line (segment), e.g. the line segment cd. The intersections of boundary lines (segments) are called the vertices, if feasible it is called the corner point. In three-dimensional space and higher the circles become spheres, and hyper-spheres. Know that, the LP constraints provide the vertices and the corner-points. A vertex is the intersection of 2-lines, or in general n-hyperplanes in LP problems with n-decision variables. A corner-point is a vertex that is also feasible. A Numerical Example: The Carpenter's Problem Maximize 5 X1 + 3 X2 Subject to: 2 X1 + X2 £ 40 X1 + 2 X2 £ 50 and both X1, X2 are non-negative. Note: There is an alternative to the Iso-value objective function approach with problems that have few constraints and a bounded feasible region. First, find all the corner points, which are called extreme points. Then, evaluate the objective function at the extreme points to find the optimal value and the optimal solution. Clearly, the carpenter has many alternative sets of actions to take. However, the four "extreme" options are: Objective Function Value at Each Corner (i.e., Extreme) Point │ Decision-Maker's Choices │Corner Point Coordinates│Net Income Function│ │Number of Tables or Chairs │ X1, X2 │ 5 X1 + 3 X2 │ │ Make No Table nor Chair │ 0, 0 │ 0 │ │ Make All Tables You Can │ 20, 0 │ 100 │ │ Make All Chairs You Can │ 0, 25 │ 75 │ │ Make Mixed Products │ 10, 20 │ 110 │ Since the objective is to maximize, from the above table we read off the optimal value to be 110, which is obtainable if the carpenter follows the optimal strategy of X1 = 10, and X2 = 20. Notice that in the carpenter problem, the convex feasible region provides the corner points with coordinates shown in the above table. The main deficiency of the graphical method is that it is limited to solving LP problems with 1 or 2 decision variables only. However, the main and useful conclusion we easily observe from the graphical methods, is as follow: If a linear program has a bounded optimal solution, then one of the corner points provides an optimal solution. The proof of this claim follows from the results of the following two facts: Fact No. 1: The feasible region of any linear program is always a convex set. Since all of the constraints are linear, the feasible region (F.R.) is a polygon. Moreover, this polygon is a convex set. In any higher than two-dimensional LP problem, the boundaries of F.R. are parts of the hyper-planes, and the F.R. in this case is called polyhedra that is also convex. A convex set is the one that if you choose two feasible points from it, then all the points on the straight line segment joining these two points are also feasible. The proof that the F.R. of linear programs are always convex sets follows by contradiction. The following figures depict examples for the two types of sets: A non-convex and a convex set. The set of feasible region in any linear program is called a polyhedron, it is called a polytope if it is bounded. Fact No. 2: The Iso-value of a linear program objective function is always a linear function. This fact follows from the nature of the objective function in any LP problem. The following figures depict the two typical kinds of iso-valued objective functions. Combining the above two facts, it follows that, if a linear program has a non-empty, bounded feasible region, then the optimal solution is always one of the corner points. To overcome the deficiency of the graphical method, we will utilize this useful and practical conclusion in developing an algebraic method that is applicable to multi-dimensional LP problems. The convexity of the feasible region for linear programs makes the LP problems easy to solve. Because of this property and linearity of the objective function, the solution is always one of the vertices. Moreover, since number of vertices is limited, one has to find all feasible vertices, and then evaluate the objective function at these vertices to seek the optimal point. For nonlinear programs, the problem is much harder to solve, because the solution could be anywhere inside the feasible region on the boundary of the feasible region, or at a vertex. Fortunately, most of the Business optimization problems have linear constraints, which is why LP is so popular. There are well over 400 computer packages in the market today solving LP problems. Most of them are based on vertex searching, that is, jumping from one vertex to the neighboring one in search of an optimal point. You have already noticed that, a graph of a system of inequalities and/or equalities is called the feasible region. These two representations, graphical, and algebraic are equivalent to each other, which means the coordinate of any point satisfying the constraints is located in the feasible region, and the coordinate of any point in the feasible region satisfies all the constraints. A numerical Example: Find the system of constraints representing the following feasible region. In the above Figure, the system of coordinate is shown in gray color at the background. By constructing the equations of the boundary lines of the feasible region, we can verify that the following system of inequalities indeed represents the above feasible region: X1 ³ -1 X2 £ 1 X1 - X2 £ 1 Links Between LP and Systems of Equations: Algebraic Method As George Dantzig pointed out, linear programming is strictly "the theory and solution of linear inequality systems." The basic solutions to a linear program are the solutions to the systems of equations consisting of constraints at binding position. Not all basic solutions satisfy all the problem constraints. Those that do meet all the restrictions are called basic feasible solutions. The basic feasible solutions correspond precisely to the extreme points of the feasible region. For example, for Carpenter's Problem, one can compute all the basic solutions, by taking any two of the equations and solving them simultaneously and then, using the constraints of other equations to check for feasibility of this solution. If feasible, then this solution is a basic feasible solution that provides the coordinates of a corner point of the feasible region. To illustrate the procedure, consider the Carpenter's constraints at binding (i.e., all with = sign) position: 2X1 + X2 = 40 X1 + 2X2 = 50 X1 = 0 X2 = 0 Here we have 4 equations with 2 unknowns. In terms of a "binomial coefficient", there are at most C^4[2 ]= 4! / [2! (4-2)!] = 6 basic solutions. Solving the six resultant systems of equations, we Six Basic Solutions with Four Basic Feasible Solutions X1 X2 5X1 + 3X2 10 20 110* 0 40 infeasible 50 0 infeasible Four of the above basic solutions are basic feasible solutions satisfying all the constraints, belonging to the coordinates of the vertices of the bounded feasible region. By plugging in the basic feasible solution in the objective function, we compute the optimal value. Therefore, from the above table, we see that, the optimal solution is X1 = 10, X2 = 20, with optimal value of $110. The above approach can be applied in solving higher dimension LP problems provided the optimal solution is bounded. You may like using Solving Systems of Equations JavaScript to check your computation. Further Readings: Arsham H, Links Among a Linear System of Equations, Matrix Inversion, and Linear Program Solver Routines, Journal of Mathematical Education in Science and Technology, 29(5), 764-769, 1998. Dantzig G., Linear Programming & Extensions, page 21, The Rand-Princeton U. Press, 1963. Extension to Higher Dimensions The Graphical Method is limited in solving LP problems having one or two decision variables. However, it provides a clear illustration of where the feasible and non-feasible regions are, as well as, vertices. Having a visual understanding of the problem helps with a more rational thought process. For example, we learned that: If an LP program has a bounded optimal solution, then the optimal solution is always one of the vertices of its feasible region (a corner point). What is needed to be done is to find all the intersection points (vertices) and then examine which one among all feasible vertices, provides the optimal solution. By using Analytical Geometry concepts, we will overcome this limitation of human vision. The Algebraic Method is designed to extend the graphical method results to multi-dimensional LP problem, as illustrated in the following numerical example. Numerical Example: The Transportation Problem Transportation models play an important role in logistics and supply chain management for reducing cost and improving service. Therefore, the goal is to find the most cost effective way to transport the goods. Consider a model with 2 origins and 2 destinations. The supply and demand at each origin (e.g; warehouse) O1, O2 and destination (e.g.; market) D1 and D2, together with the unit transportation cost are summarized in the following table. The Unit Transportation Cost Matrix D1 D2 Supply O1 20 30 200 O2 10 40 100 Demand 150 150 300 Let Xij's denote the amount of shipment from source i to destination j. The LP formulation of the problem minimizing the total transportation cost is: Min 20X11 + 30X12 + 10X21 + 40X22 subject to: X11 + X12 = 200 X21 + X22 = 100 X11 + X21 = 150 X12 + X22 = 150 all Xij ³ 0 Notice that the feasible region is bounded, therefore one may use the algebraic method. Because this transportation problem is a balanced one (total supply = total demand) all constraints are in equality form. Moreover, any one of the constraints is redundant (adding any two constraints and subtracting another one, we obtain the remaining one). Let us delete the last constraint. Therefore, the problem reduces to: Min 20X11 + 30X12 + 10X21 + 40X22 subject to: X11 + X12 = 200 X21 + X22 = 100 X11 + X21 = 150 all Xij ³ 0 This LP problem cannot be solved by the graphical method. However, the algebraic method has no limitation on the LP dimension. The constraints are already at binding (equality) position. Notice that we have m=3 equality constraints with (four implied non-negative) decision variables. Therefore, out of these four variables there is at most m=3 variables with positive value and the rest must be at zero level. For example, by setting any one of the variables in turn to zero, we get: X11 X12 X21 X22 Total Transportation Cost 0 200 150 -50 200 0 -50 150 infeasible 50 150 100 0 6500* Now by setting any one (or more) variables to zero, it is easy to see, by inspecting the constraints that all other solutions are infeasible. Thus, from the above table, we obtain the optimal strategy to be: X11 = 50, X12 = 150, X21 = 100, and X22 = 0, with the least total transportation cost of $6,500. You may like to run this problem using Module Net.Exe in your WinQSB Package to check these results for yourself. Notice that in the above example, there are m=3 constraints (excluding the non-negativity conditions), and n=4 decision variables. The optimal shipment indicates that, the manager should not send any shipment from one source to one destination. The optimal solution consists of at most 3 positive decision variables which is equal to the number of constrains. If the manager is shipping goods from every source to every destination, then the result is not optimal. The above results, found in the above example, by Algebraic Method can be generalize, in the following main Economic result: Given an LP having a bounded feasible region, with m constraints (excluding any sign constraints such as non-negativity conditions) and n decision variables, if n> m then at most m decision variables have positive value at the optimal solution and the rest of (i.e., n-m) decision variables must be set at zero level. This result holds if the problem has a bounded unique optimal The above result follows from the fact that, using the shadow prices indicates that the opportunity cost for the decision variable at zero level exceed its contribution. Numerical Example: Find the optimal solution for the following production problem with n=3 products and m=1 (resource) constraint: Maximize 3X1 + 2X2 + X3 Subject to: 4X1 + 2X2 + 3X3 £ 12 all variables Xi's ³ 0 Since the feasible region is bounded, following the Algebraic Method by setting all the constraints at the binding position, we have the following system of equations: 4X1 + 2X2 + 3X3 = 12 X1 = 0 X2 = 0 X3 = 0 The (basic) solutions obtained, from this system of equations are summarized in the following table. X1 X2 X3 Total Net Profit 0 6 0 12* Thus, the optimal strategy is X1 = 0, X2 = 6, X3 = 0, with the maximum net profit of $12. The result in the above table is consistent with the application of the above main economic result. In other words, the optimal solution can be found by setting at least n - m = 3 - 1 = 2 decision variables to zero: For large-scale LP problems with many constraints, the Algebraic Method involves solving many linear systems of equations. When the LP problem has many variables and constraints, solving many systems of equations by hand can become very tedious. Even for very large-scale problems it is an impossible task. Therefore, we need the computer to do the computations for us. One of the algorithmic and computerized approaches is The Simplex Method, which is an efficient and effective implementation of the Algebraic Method. There are well over 400 LP solvers, all of which using the Simplex method, including your software. Upon solving the LP problem by computer packages, the optimal solution provides valuable information, such as sensitivity analysis ranges. You may like using Solving Systems of Equations JavaScript for up to 3-decision variables LP problems to check your computation by the algebraic method. For details and more numerical examples visit the Solution Algorithms for LP Models Web site. How to Solve a Linear System of Equations by LP Solvers? In the Algebraic Method of solving LP problems, we have to solve some systems of equations. There is a link between LP solvers and the systems of equation solvers. Suppose we have a very large system of equations that we would like to solve and an LP solver package but we still have no solver computer package for a system of equations available. The question is "How to use an LP solver to find the solution to a system of equations?" The following steps outline the process of solving any linear system of equations using an available LP solver. 1- Because some LP solvers require that all variables be non-negative, substitute for each variable Xi = Yi - T everywhere. 2- Create a dummy objective, such as minimize T. 3- The constraints of the LP problem are the equations in the system after the substitutions outlined in step 1. Numerical Example: Solve the following system of equations 2X1 + X2 = 3 X1 -X2 = 3 Since the WinQSB package accepts LP in various formats ( unlike Lindo), solving this problem by WinQSB is straightforward: First, create an LP with a dummy objective function such as Max X1, subject to 2X1 + X2 = 3, X1 - X2 = 3, and both X1 and X2 unrestricted in sign. Then, enter this LP into the LP/ILP module to get the solution. The generated solution is X1= 2, X2= -1, which can easily be verified by substitution. However, if you use any LP solver which requires by default (e.g., Lindo) that all variables be non-negative, you need to do some preparations to satisfy this requirement: First substitute for X1 = Y1 - T and X2 = Y2 - T in both equations. We also need an objective function. Let us have a dummy objective function such as minimize T. The result is the following LP: Min T Subject to: 2Y1 + Y2 - 3T = 3, Y1 - Y2 = 3. Using any LP solver, such as Lindo, we find the optimal solution to be Y1 = 3, Y2 = 0, T = 1. Now, substitute this LP solution into both transformations X1 = Y1 - T and X2 = Y2 - T. This gives the numerical values for our original variables. Therefore, the solution to the system of equations is X1 = 3 - 1 = 2, X2 = 0 - 1 = -1, which can easily be verified by substitution. Dual Problem: Construction and Its Meaning Associated with each (primal) LP problem is a companion problem called the dual. The following classification of the decision variable constraints is useful and easy to remember in construction of the dual. The Dual Problem Construction Objective: Max (e.g. Profit) Objective: Min (e.g. Cost) Constraint types: Constraint types: £ a Sensible constraint ³ a Sensible constraint = a Restricted constraint = a Restricted const. ³ an Unusual const. £ an Unusual const. Variables types: ³ 0 a Sensible condition ... un-Restricted in sign £ 0 an Unusual condition A one-to-one correspondence between the constraint type and the variable type exists using this classification of constraints and variables for both the primal and the dual problems. Dual Problem Construction: - If the primal is a maximization problem, then its dual is a minimization problem (and vise versa). - Use the variable type of one problem to find the constraint type of the other problem. - Use the constraint type of one problem to find the variable type of the other problem. - The RHS elements of one problem become the objective function coefficients of the other problem (and vice versa). - The matrix coefficients of the constraints of one problem is the transpose of the matrix coefficients of the constraints for the other problem. That is, rows of the matrix becomes columns and vise versa. You may check your dual constructions rules by using your WinQSB package. Numerical Examples: Consider the following primal problem: min x1-2x2 subject to: x1+x2 ³ 2, x1-x2 £ -1, x2 ³ 3, and x1, x2 ³ 0. Following the above construction rule, the dual problem is: max 2u1 - u2 + 3u3 subject to: u1 + u2 £ 1, u1 - u2 + u3 £ -2, u1 ³ 0, u2 £ 0, and u3 ³ 0 The Dual of the Carpenter's Problem: Maximize 5X1 + 3X2 Subject to: 2X1 + X2 £ 40 X1 + 2X2 £ 50 X1 ³ 0 X2 ³ 0 Its dual is: Minimize 40U1 + 50U2 Subject to: 2U1 + U2 ³ 5 U1 + 2U2 ³ 3 U1 ³ 0 U2 ³ 0 Applications: One may use duality in a wide variety of applications including: - It may be more efficient to solve the dual than the primal. - The dual solution provides important economical interpretation such as shadow prices, i.e., the marginal values of the RHS elements. The shadow price was defined, historically as the improvement in the objective function value per unit increase in the right hand side, because the problem was often put in the form of profit maximization improvement (meaning increase). The shadow price may not be the market price. The shadow price is e.g., the worth of the resource under the "shadow" of your business activity. Sensitivity analysis, i.e., the analysis of the effect of small variations in system parameters on the output measures can be studied by computing the derivatives of the output measures with respect to the parameter. - If a constraint in one problem is not binding (i.e., the LHS value agrees with the RHS value), then the associated variable in the other problem is zero. If a decision variable in one problem is not zero, then the associated constraint in the other problem is binding. These results are known as the Complementarily Slackness Conditions (CSC). - Obtain the sensitivity range of the RHS of one problem from the sensitivity range of the cost coefficient in the other problem, and vice versa. These results imply the only possible combinations of primal and dual properties as shown in the following table: │Possible Combinations of Primal and Dual Properties │ │Primal Problem │Condition Implies│Dual Problem │││ │Feasible; bounded objective │« │Feasible; bounded objective │││ │Feasible; unbounded objective │® │Infeasible │││ │Infeasible │¬ │Feasible; unbounded objective │││ │Infeasible │« │Infeasible │││ │Multiple solutions │« │Degenerate solution │││ │Degenerate solution │« │Multiple solutions │││ Notice that almost all LP solvers produce sensitivity range for the last two cases; however these ranges are not valid. For this reason you must make sure that the solution is unique, and non-degenerate in analyzing and applying the sensitivity ranges. Further Readings: Arsham H., An Artificial-Free Simplex Algorithm for General LP Models, Mathematical and Computer Modelling, Vol. 25, No.1, 107-123, 1997. Benjamin A., Sensible Rules for Remembering Duals_ S-O-B Method, SIAM Review, Vol. 37, No.1, 85-87, 1995. Chambers R., Applied Production Analysis: A Dual Approach, Cambridge University Press, 1988. The Dual Problem of the Carpenter's Problem and Its Interpretation In this section we will construct the Dual Problem of the Carpenter's Problem and provide an economical interpretation of it. In the Carpenter's Problem uncontrollable input parameters are the following: The Uncontrollable Inputs Table Chair Available Labor 2 1 40 Raw Material 1 2 50 Net Income 5 3 and its LP formulation as: Maximize 5 X1 + 3 X2 Subject to: 2 X1 + X2 £ 40 labor constraint X1 + 2 X2 £ 50 material constraint and both X1, X2 are non-negative. Where X1 and X2 are the number of tables and chairs to make. Suppose the Carpenter wishes to buy insurance for his net income. Let U1 = the dollar amount payable to the Carpenter for every labor hour lost (due to illness, for example), and U2 = the dollar amount payable to the Carpenter for every raw material unit lost (due to fire, for example). The insurance officer tries to minimize the total amount of $(40U1 + 50U2) payable to the Carpenter by the insurance company. However, the Carpenter will set the constraints (i.e. conditions) by insisting that the insurance company cover all his/her loss that is his net income since he/she cannot make the products. Therefore, the insurance company problem is: Minimize 40 U1 + 50 U2 Subject to: 2U1 + 1U2 ³ 5 Net Income from a table 1U1 + 2U2 ³ 3 Net Income from a chair and U1, U2 are non-negative. Implementing this problem on your computer package shows that the optimal solution is U1 = $7/3 and U2 = $1/3 with the optimal value of $110 (the amount the Carpenter expects to receive). This ensures that the Carpenter can manage his life smoothly. The only cost is the premium that the insurance company will charge. Shadow Price Unit of Measure: Notice that the unit of measure of the RHS shadow price is the unit of measure of the primal objective divided by the unit measure of the RHS. For example for the Carpenters problem, U1 = [$/week] / hours/week] = $/hour. Thus U1 = 7/3 $/hour, and U2 = 1/3 $/unit of raw material. As you can see, the insurance company problem is closely related to the original problem. In OR/MS/DS modeling terminology, the original problem is called the Primal Problem while the related problem is called its Dual Problem. In the Carpenter's Problem and its Dual Problem, the Optimal Value for both problems is always the same. This fact is referred to as Equilibrium (taken from the complementarity theory, equilibrium of economical systems, and efficiency in Pareto's sense) between the Primal and the Dual Problems. Therefore, there is no duality gap in linear programming. The dual solution provides important economical interpretation such as the marginal values of the RHS elements. The elements of the dual solution are known as the Lagrangian multipliers because they provide (a tight) bound on the optimal value of the primal, and vise versa. For example, considering the Carpenter's problem the dual solution can be used to find a lower tight bound for the optimal value, as follow. After converting the inequality constraints into £ form, multiply each constraint by its corresponded dual solution and then add them up, we get: 7/3 [ 2X1 + X2 £ 40] 1/3 [ X1 + 2X2 £ 50] 5X1 + 3X2 £ 110 Notice that the resultant on the left side is the objective function of the primal problem, and this lower bound for it is a tight one, since the optimal value is 110. Managerial Roundoff Errors You must be careful whenever you round the value of shadow prices. For example, the shadow price of the resource constraint in the above problem is 8/3; therefore, if you wish to buy more of this resource, you should not pay more than $2.66. Whenever you want to sell any unit of this resource, you should not sell it at a price below $2.67. A similar error might occur whenever you round the limits on the sensitivity ranges. One must be careful because the upper limit and lower limit must be rounded down and up, respectively. Computation of Shadow Prices You know by now that the shadow prices are the solution to the dual problem. Here is a numerical example. Compute the shadow price for both resources in the following LP problem: Max -X1 + 2X2 S.T. X1 + X2 £ 5 X1 + 2X2 £ 6 and both X1, X2 non-negative The solution to this primal problem (using, e.g., the graphical method) is X1 = 0, X2 = 3, with the leftover S1 = 2 of the first resource, while the second resource is fully utilized, S2 = 0. The shadow prices are the solution to the dual problem: Min 5U1 + 6U2 S.T. U1 + U2 ³ -1 U1 + 2U2 ³ 2 and both U1 and U2 are non-negative. The solution to the dual problem (using, e.g., the graphical method) is U1 = 0, U2 = 1 which are the shadow prices for the first and second resource, respectively. Notice that whenever the slack/ surplus of a constraint is non-zero, the shadow price related to that RHS of that constraint is always zero; however, the reverse statement may not hold. In this numerical example S1 = 2 (i.e. slack value of the RHS1 of the primal), which is non-zero; therefore U1 is equal to zero as expected. Consider the following problem: Max X1 + X2 subject to: X1 £ 1 X2 £ 1 X1 + X2 £ 2 all decision variables ³ 0. By using your computer package, you may verify that the shadow price for the third resource is zero, while there is no leftover of that resource at the optimal solution X1 = 1, X2 = 1. Behavior of Changes in the RHS Values of the Optimal Value To study the directional changes in the optimal value with respect to changes in the RHS (with no redundant constraints present, and all RHS ³0), we distinguish the following two cases: Case I: Maximization problem For £ constraint: The change is in the same direction. That is, increasing the value of RHS does not decrease the optimal value. It increases or remains the same depending on whether the constraint is a binding or non-binding constraint. For ³ constraint: The change is in the reverse direction. That is, increasing the value of RHS does not increase the optimal value. It decreases or remains the same depending on whether the constraint is a binding or non-binding constraint. For = constraint: The change could be in either direction (see the More-for-less section). Case II: Minimization problem For £ type constraint: The change is in the reverse direction. That is, increasing the value of RHS does not increase the optimal value (rather, it decreases or has no change depending on whether the constraint is a binding or non-binding constraint). For ³ type constraint: The change is in the same direction. That is, increasing the value of RHS does not decrease the optimal value (rather, increases or has no change depending on whether the constraint is a binding or non-binding constraint). For = constraint: The change could be in either direction (see the More-for-less section). Dealing with Uncertainties and Scenario Modeling The business environment is often unpredictable and uncertain because of factors such as economic changes, government regulations, dependence on subcontractors and vendors, etc. Managers often find themselves in a dynamic, unsettled environment where even short range plans must be constantly reassessed and incrementally adjusted. All these require a change-oriented mentality to cope with uncertainties. Remember that surprise is not an element of a robust decision. Managers use mathematical and computational constructs (models) for a variety of settings and purposes, often to gain insight into possible outcomes of one or more courses of action. This may concern financial investments, the choice (whether/how much) to insure, industrial practices, and environmental impacts. The use of models is flawed by the unavoidable presence of uncertainties, which arise at different stages; in the construction and corroboration of the model itself, and in its use. Its use is often the culprit. Every solution to a decision problem is based on certain parameters that are assumed to be fixed. Sensitivity analysis is a collection of post-solution activities to study and determine how sensitive the solution is to changes in the assumptions. Other names for such activities are stability analysis, what-if analysis, scenario modeling, ranging analysis, specificity analysis, uncertainty analysis, computational and numerical instability, functional instability, tolerance analysis, post-optimality analysis, allowable increases and decreases, and many other similar phrases that reflect the importantness of this stage of the modeling. Numerical Example: Consider the following problem: Max 6X1 + 4.01X2 subject to: X1 + 2X2 £ 16 3X1 + 2X2 £ 24 all decision variables ³ 0. The optimal solution is (X1 = 4, X2 = 6), but with a slight change in the objective function, one may get a completely different optimal solution. For example, if we change it to 6X1 + 3.99X2, then the optimal solution is (X1 = 8, X2 = 0). That is, decreasing the second coefficient by 0.5%, the solution changes drastically! Therefore, the optimal solution is not stable with respect to this input parameter. Sensitivity analysis is not the typical term employed in econometric for the method of investigating the response of a solution to perturbations in parameters. In econometrics, the process of changing the value of a parameter in a model, in order to see its individual impacts on the performance measure, is called comparative statics or comparative dynamics, depending on the type of model under consideration. Uncertainty in a model can have different origins in different decision problems. It may be due to either incomplete information, or fluctuations inhere in the problem, or unpredictable changes in the future. Current approaches to deal with uncertainties includes: Scenario Analysis: In this approach one assumes scenarios (e.g. certain combinations of possible values of uncertain parameter) and solves the problem for each. By solving the problem repeatedly for different scenarios and studying the solutions obtained, the manager observes sensitivities and heuristically decides on an approximate, which is subjective. Worst-case Analysis: This technique attempts to account for putting safety margins into the problem in the planning stage. Monte-Carlo Approach: Stochastic models assume that the uncertainty is known by its statistical distribution. Sensitivity analysis vs. Stochastic Programming: Sensitivity analysis (SA) and Stochastic Programming (SP) formulations are the two major approaches used for dealing with uncertainty. SA is a post-optimality procedure with no power of influencing the solution. It is used to investigate the effects of the uncertainty on the model's recommendation. SP formulation, on the other hand, introduces probabilistic information about the problem data, albeit with the first moments (i.e. the expected values) of the distribution of the objective function with respect to the uncertainty. This ignores the decision-makers' risk assessments, characterized by variance, or coefficient of variation. One may tackle uncertainties in a more "deterministic" manner. The approach is called various names such as "scenario modeling", "deterministic modeling", "sensitivity analysis", "ranging procedures", and "stability analysis". The idea is to subjectively come up with a ranked list of higher level uncertainties that might presumably have a bigger impact on the eventual mapping result. This is done before zooming in on the details of any particular "scenario" or model. For example, the problem parameters, and the uncontrollable factors indicated in the above figure for the Carpenter's problem, required a complete sensitivity analysis in order to enable the carpenter to be in control of his/her business. Managerial Roundoff Errors: You must be very careful whenever you round the value of the limits on the sensitivity ranges. To be valid the upper limit and lower limit must be rounded down and up, For construction of sensitivity analysis region that allows you to analyze any type of changes, including dependent, independent, and multiple changes in both the RHS values and the cost coefficients of LP visit Construction of General Sensitivity Regions site. Further Readings: Arsham H., Algorithms for sensitivity information in discrete-event systems simulation, Simulation Practice and Theory, 6(1), 1-22, 1998. Arsham H., Perturbation analysis of general LP models: A unified approach to sensitivity, parametric, tolerance, and more-for-less analysis, Mathematical and Computer Modelling, 13(3), 79-102, Kouvelis P., and G. Yu, Robust Discrete Optimization and its Applications, Kluwer Academic Publishers, 1997. Provides a comprehensive discussion of motivations for sources of uncertainty in Computation of Sensitivity Ranges for Small Size Problems To compute sensitivity ranges for LP Problems with at most 2 decision variables and/or at most 2 constraints, you may like to try the following easy-to-use approach. The only restriction is that no equality constraint is permitted. Having an equality constraint is the case of degeneracy, because every equality constraint, for example, X1 + X2 = 1, means two simultaneous constraints: X1 + X2 £ 1 and X1 + X2 ³ 1. The number of binding constraints in such a case will be more than the number of decision variables. This is known as a degenerate situation for which the usual sensitivity analysis may not be valid. Cost Sensitivity Range for LP Problems with two Decision Variables Referring to the Carpenter's Problem, changing the profit on each product changes the slope of the iso-value objective function. For "small" changes, the optimal stays at the same extreme point. For larger changes the optimal solution moves to another point. Then we have to modify the formation and solve a new problem. The problem is to find a range for each cost coefficient c(j), of variable Xj, such that the current optimal solution, i.e., the current extreme point (corner point), remains optimal. For a 2-dimensional LP problem, you may like to try the following approach to find out the amount of increase/decrease in any one of the coefficients of the objective function (also known as the cost coefficients. Historically during World War II, the first LP problem was a cost minimization problem) in order to maintain the validity of the current optimal solution. The only condition required for this approach is that no equality constraint is permitted, since this leads to the case of degeneracy, for which the usual sensitivity analysis may not be valid. Step 1: Consider the only two binding constraints at the current optimal solution. If there are more than two binding constraints, then this is the case of degeneracy, for which the usual sensitivity analysis may not be valid. Step 2: Perturb the j^th cost coefficient by parameter cj (this is the unknown amount of changes). Step 3: Construct one equation corresponding to each binding constraints as follow: (Perturbed Cj cost)/ coefficient of Xj in the constraint = Coefficient of the other variable in the objective function /coefficient of that variable of the constraint. Step 4: The amount of allowable increase is the smallest positive cj, while the allowable decrease is the largest negative cj, obtained in Step 3. Notice that if there is no positive (negative) cj, then the amount of the increase (decrease) is unlimited. 1. Remember that you should never divide by zero. The practice of dividing by zero is a common fallacy found in some textbooks. For example, in the Introduction to Management Science, Taylor III, B., Prentice Hall, the author, unfortunately divides by zero, in Module A: The Simplex Solution Method, pp. A26-A27, where computing the needed column-ratio in the simplex method. For more on this, and other common fallacies visit the Web site The zero saga & confusions with numbers . Here is a question for you: Which of the following are correct and why? a) any number divided by zero is undefined; b) zero divided by any number is zero; or c) any number divided by itself is 1. 2. Finding the Cost Sensitivity Range by Grapical Method: It is a commonly held belief that one can compute the cost sensitivity range by bracketing the (perturbed) slope of the (iso-value) objective function by the slopes of the two lines resulting from the binding constraints. This graphical slope-based method to compute the sensitivity ranges is described in popular textbooks, such as Anderson et al., (2007), Lawrence and Pasternack (2002), and Taylor (2010). Unfortunately these are misleading. Warning should have been given that their approach is not general and works if and only if the coefficients do not change sign. In an LP with 2 variables and inequality constraints, suppose we have a unique, non-degenerate optimum at the intersection of two lines, as shown in the following figure. Then, the range of objective coefficients for which this solution remains optimal is given by the slopes of the two lines. The following is a counterexample. It points out that one must be careful to state that the coefficients do not change sign. Counterexample: Maximixe 5X[1] + 3X[2] X[1] + X[2] £ 2, X[1] - X[2] £ 0, X[1] ³ 0, X[2] ³ 0. The Carpenter's Problem: Maximize 5X1 + 3X2 Subject to: 2X1 + X2 £ 40 X1 + 2X2 £ 50 X1 ³ 0 X2 ³ 0 Computation of allowable increase/decrease on the C1 = 5: The binding constraints are the first and the second one. Perturbing this cost coefficient by c1, we have 5 + c1. At step 3, we have: (5 + c1)/2 = 3/1, for the first constraint, and (5 + c1)/1 = 3/2 for the second constraint. Solving these two equations, we have: c1 = 1 and c1 = -3.5. The allowable increase is 1, while the allowable decrease is 1.5. As far as the first cost coefficient C1 remains within the interval [ 5 - 3.5, 5 + 1] = [1.5, 6], the current optimal solution remains. Similarly for the second cost coefficient C2 = 3, we have the sensitivity range of [2.5, 10]. As another example, consider the earlier problem: Maximize 5X1 + 3X2 Subject to: X1 + X2 £ 2 X1 - X2 £ 0 X1 ³ 0 X2 ³ 0 Computation of allowable increase/decrease on the C1 = 5: The binding constraints are the first and the second one. Perturbing this cost coefficient by c1, we have 5 + c1. At step 3, we have: (5 + c1)/1 = 3/1, for the first constraint and (5 + c1)/1 = 3/(-1) for the second constraint. Solving these two equations, we have: c1 = -2 and c1 = -8. The allowable decrease is 2, while the allowable increase is unlimited. As far as the first cost coefficient C1 remains within the interval [ 5 - 2, 5 + ¥] = [3, ¥], the current optimal solution remains optimal. Similarly, for the second cost coefficient C2 = 3 we have the sensitivity range of [3 - 8, 3 + 2] = [-5, 5]. For construction of sensitivity analysis region that allows you to analyze any type of changes, including dependent, independent, and multiple changes in both the RHS values and the cost coefficients of LP visit Construction of General Sensitivity Regions site. RHS Sensitivity Range for LP Problems with at Most Two Constraints Referring to the Carpenter's Problem, for small changes in either resources, the optimal strategy (i.e. make the mixed-product) stays valid. For larger changes, this optimal strategy moves and the Carpenter must either make all the tables or the chairs he/she can. This is a drastic change in the strategy; therefore, we have to revise the formulation and solve a new problem. Apart from the above needed information, we are also interested in knowing how much the Carpenter can sell (or buy) each resource at a "reasonable" price (or cost). That is, how far can we increase or decrease RHS(i) for fixed i while maintaining the validity of the current shadow price of the RHS(i)? That is, how far can we increase or decrease RHS(i) for fixed i while maintaining the current optimal solution to the dual problem? Historically, the shadow price was defined as the improvement in the objective function value per unit increase in the right hand side, because the problem was often put in the form of profit maximization improvement (meaning increase). Also, know that for any RHS, the shadow price (also known also its marginal value), is the amount of change in the optimal value proportion to one unit change for that particular RHS. However, in some cases it is not permitted to change the RHS by that much. The sensitivity range for the RHS provides the values for which the shadow price has such an economic meaning and remains unchanged. How far can we increase or decrease each individual RHS in order to maintain the validity of shadow prices? The question is equivalent to asking what is the sensitivity range for the cost coefficient in the dual problem. The dual of the Carpenter's Problem is: Minimize 40U1 + 50U2 Subject to: 2U1 + U2 ³ 5 U1 + 2U2 ³ 3 U1 ³ 0 U2 ³ 0 The optimal solution is U1 = 7/3 and U2 = 1/3 (which are the shadow prices). The Carpenter's Problem: Maximize 5X1 + 3X2 Subject to: 2X1 + X2 £ 40 X1 + 2X2 £ 50 X1 ³ 0 X2 ³ 0 Computation of Range for the RHS1: The first two constraints are binding, therefore: (40 + r1)/2 = 50/ 1, and (40 + r1) / 1 = 50/ 2. Solving these two equations gives: r1 = 60 and r1 = -15. Therefore, the sensitivity range for the first RHS in the carpenter's problem is: [40-15, 40 + 60] = [25, 100]. Similarly, for the second RHS, we obtain: [50 - 30, 50 + 30] = [20, 80]. For construction of sensitivity analysis region that allows you to analyze any type of changes, including dependent, independent, and multiple changes in both the RHS values and the cost coefficients of LP visit Construction of General Sensitivity Regions site. Further Readings: Lawrence J., Jr., and B. Pasternack, Applied Management Science: Modeling, Spreadsheet Analysis, and Communication for Decision Making, John Wiley and Sons, 2002. Anderson D., Sweeney D., and Williams T., An Introduction to Management Science, West Publisher, 2007. by Taylor III, B., Introduction to Management Science, Prentice Hall, 2006. Marginal Analysis & Factors Prioritization The major applications of sensitivity analysis information for the decision-maker are the Marginal Analysis and the Factors Prioritization: Marginal Analysis: Marginal analysis is a concept employed, in microeconomics where the marginal change in some parameter might be of interest to the decision-maker. A marginal change is a ration of very small addition or subtraction to the total quantity of some parameter. Marginal analysis is the analysis of the relationships between such changes in relation to the performance measure. Examples of marginal analysis are: marginal cost; marginal revenue; marginal product; marginal rate of substitution; marginal propensity to save, and so on. In optimization, the marginal analysis is employed primarily to explicate various changes in the parameters and their impact on optimal value. Sensitivity analysis, i.e., the analysis of the effect of small variations in system parameters on the output measures can be studied by computing the derivatives of the output measures with respect to the parameter. The decision-makers ponder what factors are important and have major impact on the decision outcome. The marginal analysis aims at identifying the important factors (i.e., parameters) and ranking them according to their impact on optimal value. One may derive the marginal values by evaluating the first derivatives of performance measure with respect to the parameter with specific value. Factors Prioritization Based on Sensitivity Ranges: Consider the Carpenter's Problem: Maximize 5X1 + 3X2 Subject to: 2X1 + X2 £ 40 X1 + 2X2 £ 50 X1 ³ 0 While the computed sensitivity ranges are valid for one-change at-a-time and not necessarily for simultaneous changes, they provide useful information for prioritization of uncontrollable factors. The following figure provides a comparative chart for the cost coefficients prioritization purposes: The following figure depicts the shadow price as the slope (i.e., marginal value) of a linear function measuring the amount of change in the optimal value as a result of any change in the RHS1, provided the change is within the sensitivity range of the RHS1. This function also can be used to solve the inverse problem, that is, what the RHS1 value should be to achieve a specific optimal What Is the 100% Rule (sensitivity region) The sensitivity range presented in the previous section is a "one-change-at-a-time" type of what-if analysis. Consider the Carpenter's problem; suppose we want to find the simultaneous allowable increases in RHS ( r[1], r[2] ³ 0). There is an easy method to apply here known as "the 100% rule" which says that the shadow prices remain unchanged if the following sufficient condition holds: r[1]/60 + r[2]/30 £ 1, 0 £ r[1] £ 60, 0 £ r[2] £ 30. In the above, 60 and 30 are the allowable increases for the RHS's, based on the application of ordinary sensitivity analysis. That is, whenever the first and the second RHS increase by r[1] and r [2], respectively, as long as this inequality holds, the shadow prices for the RHS values remain unchanged. Notice that this is a sufficient condition, for if the above condition is violated, then the shadow prices may change or still remain the same. The term "100% rule" becomes evident when you notice that in the left hand side of the above condition each term is a non-negative number being less than one, which could be represented as a percentage allowable change. The total sum of such changes should not exceed 100%. Applying the 100% rule to the other three possible changes on the RHS's, we have: r[1]/(-15) + r[2]/(-30) £ 1, -15 £ r[1] £ 0, -30 £ r[2] £ 0. r[1]/(60) + r[2]/(-30) £ 1, 0 £ r[1] £ 60, -30 £ r[2] £ 0. r[1]/(-15) + r[2]/(30) £ 1, -15 £ r[1] £ 0, 0 £ r[2] £ 30. The following Figure depicts the sensitivity region for both RHS's values as the results of the application of the 100% rule for the Carpenter's problem. From a geometric point of view, notice that the polyhedral with vertices (60, 0), (0, 30), (-15, 0), and (0,-30) in the above Figure is only a subset of a larger sensitivity region for changes in both RHS values. Therefore, to remain within this sensitivity region is only a sufficient condition (not a necessary one) to maintain the validity of the current shadow prices. Similar results can be obtained for the cost coefficients' simultaneous changes. For example, suppose we want to find the simultaneous allowable decrease in C[1] and increases in C[2]. That is, the amount of changes in both cost coefficients by c[1] £ 0 and c[2] ³ 0. The 100% rule states that the current basis remains optimal provided that: c[1]/(-3.5) + c[2]/7 £ 1, -3.5 £ c[1] £ 0, 0 £ c[2]£ 7. Where 3.5 and 7 are the allowable decrease and increase for the cost coefficient C[1] and C[2], respectively, that we found earlier by the application of the ordinary sensitivity analysis. The above Figure also depicts all the other possibilities of increasing and decreasing both cost coefficients values as the result of the application of the 100% rule, while maintaining the current optimal solution for the Carpenter's problem. As another numerical example, consider the following problem: Maximize 5X1 + 3X2 Subject to: X1 + X2 £ 2 X1 - X2 £ 0 X1 ³ 0 X2 ³ 0 You may recall that we have already computed the one-change-at-a-time sensitivity ranges for this problem in the Computation of Sensitivity Ranges section. The sensitivity range for the first cost coefficient is [ 5 - 2, 5 + ¥] = [3, ¥], while, for the second cost coefficient it is [3 - 8, 3 + 2] = [-5, 5]. You should be able to reproduce a figure similar to the above depicting all other possibilities of increasing/decreasing both cost coefficients values as the results of the application of the 100% rule, while maintaining the current optimal solution for this problem. The application of the 100% rule as presented here is general and can be extended to any large size LP problem. As the size of problem becomes larger, this type of sensitivity region becomes smaller and therefore less useful to the managers. There are more powerful (providing both necessary and sufficient conditions) and useful techniques to the managers for dependent (or independent) simultaneous changes in the parameters. For construction of sensitivity analysis region that allows you to analyze any type of changes, including dependent, independent, and multiple changes in both the RHS values and the cost coefficients of LP visit Construction of General Sensitivity Regions site. Adding a New Constraint The process: Insert the current optimal solution into the newly added constraint. If the constraint is not violated, the new constraint does NOT affect the optimal solution. Otherwise, the new problem must be resolved to obtain the new optimal solution. Deleting a Constraint The process: Determine if the constraint is a binding (i.e. active, important) constraint by finding whether its slack/surplus value is zero. If binding, deletion is very likely to change the current optimal solution. Delete the constraint and re-solve the problem. Otherwise, (if not a binding constraint) deletion will not affect the optimal solution. Replacing a Constraint Suppose we replace a constraint with a new constraint. What is the affect of this exchange? The process: Determine if the old constraint is a binding (i.e. active, important) constraint by finding out whether its slack/surplus value is zero. If binding, the replacement is very likely to affect the current optimal solution. Replace the constraint and resolve the problem. Otherwise, (if not a binding constraint) determine whether the current solution satisfies the new constraint. If it does, then this exchange will not affect the optimal solution. Otherwise, (if the current solution does not satisfy the new constraint) replace the old constraint with the new one and resolve the problem. Changes in the Coefficients of Constraints Any changes in the coefficients of constraints might cause significant changes to the nominal (original) problem. Any such changes fall logically within the sensitivity analysis; however, these are not changes that can be analyzed using the information generated by the optimal solution. Such changes are best dealt with by solving the modified problem anew. Adding a Variable (e.g., Introducing a new product) The coefficient of the new variable in the objective function, and the shadow prices of the resources provide information about marginal worth of resources and knowing the resource needs corresponding to the new variable, the decision can be made, e.g., if the new product is profitable or not. The process: Compute what will be your loss if you produce the new product using the shadow price values (i.e., what goes into producing the new product). Then compare it with its net profit. If the profit is less than or equal to the amount of the loss then DO NOT produce the new product. Otherwise it is profitable to produce the new product. To find out the production level of the new product resolves the new problem. Deleting a Variable (e.g., Terminating a product) The process: If for the current optimal solution, the value of the deleted variable is zero, then the current optimal solution still is optimal without including that variable. Otherwise, delete the variable from the objective function and the constraints, and then resolve the new problem. Optimal Resource Allocation Problem Since resources are always scarce, managers are concerned with the problem of optimal resources allocation. You will recall in the The Carpenter's Problem formulation that we treated both resources as parameters, that is, as given fixed numerical values: Maximize 5 X1 + 3 X2 Subject to: 2 X1 + X2 £ 40 labor constraint X1 + 2 X2 £ 50 material constraint and both X1, X2 are nonnegative. We usually classify constraints as resource or production type constraints. It is a fact that in most maximization problem, the resource constraints are the natural part of the problem, while in the minimization problem the production constraints are the most important part of the problem. Suppose we wish to find the best allocation of the labor resource for the Carpenter. In other words, what is the best number of hours the Carpenter should allocate to his or her business? Let the allocated number of hours be R, which we want to use in determining its optimal value. Therefore, the mathematical model is to find R1 such that: Maximize 5 X1 + 3 X2 Subject to: 2 X1 + X2 £ R1 labor constraint X1 + 2 X2 £ 50 material constraint and all variables X1, X2, and R1 are nonnegative. We are now treating R1 not as a parameter but as a decision variable. That is, the maximization is over all three variables; X1, X2, and R1: Maximize 5 X1 + 3 X2 Subject to: 2 X1 + X2 - R1 £ 0 labor constraint X1 + 2 X2 £ 50 material constraint and all variables X1, X2, and R1 are nonnegative. Using your LP software, the optimal solution is X1 = 50, X2 = 0, with an optimal labor allocation of R1 = 100 hours. This brings an optimal value of $250. Notice that the optimal resource allocation value is always the same as the upper bound on the RHS1 sensitivity range generated by your software. The allowable increase in number of hours is 100 - 40 = 60 hours which brings additional 250 - 110 = 140. We are able even to obtain the shadow price for this resource using this information. The shadow price is the rate of change in the optimal value with respect to the change in the RHS. Therefore (250 - 110)/(100 - 40) = 140/60 = 7/3, which is the shadow price of the RHS1 as we found by other methods in earlier sections. Determination of Product's Least Net Profit In most "price taker" business settings, the net profit is an uncontrollable factor . The manager is interested in knowing the least net profit for a product that makes it profitable to produce at all. You may recall that in the The Carpenter's Problem we treated both net profits ($5, and $3) as uncontrollable inputs, that is, the values determined by the market: Maximize 5 X1 + 3 X2 Subject to: 2 X1 + X2 £ 40 labor constraint X1 + 2 X2 £ 50 material constraint And both X1, X2 are nonnegative. This has the optimal strategy of X1 =10, X2 = 20, with an optimal value of $110. Suppose the Carpenter wants to know the least value for the first coefficient in the objective function, which is currently $5, in order to make it still profitable to produce the first product (i.e., tables). Suppose the least net profit is c1 dollars; therefore, the problem is to find c1 such that: Maximize c1 X1 + 3 X2 Subject to: 2 X1 + X2 £ 40 labor constraint X1 + 2 X2 £ 50 material constraint And all variables X1, X2, c1 are nonnegative. The Dual Problem of the Carpenter's Problem is now: Minimize 40 U1 + 50 U2 Subject to: 2U1 + 1U1 ³ c1 Net profit from a table 1U1 + 2U2 ³ 3 Net profit from a chair And U1, U2, c1 are nonnegative. We are now treating the net profit c1 as a decision variable. The minimization is over all three variables; X1, X2, and c1: Minimize 40 U1 + 50 U2 Subject to: 2U1 + 1U1 - c1 ³ 0 1U1 + 2U2 ³ 3 And U1, U2, c1 are nonnegative. Implementing this problem on your computer package shows that the optimal solution is U1 = $7/3, U2 = $1/3, and c1 = $1.5. There are alternative solutions for this boundary value of the sensitivity range for the cost coefficient. The solution corresponds to the lower limit in the sensitivity analysis range of the cost coefficient computed earlier for the Carpenter's Problem. The least net profit is always the same as the lower bound on the cost coefficient sensitivity range generated by your software. Min Max and Max Min Computation in a Single-Run Suppose we wish to find the worst of several objective functions values defined on a common set of constraints in a single-run computer implementation. As an application, suppose that in The Carpenter's Problem, without loss of generality, we have three markets with objective functions 5X1 + 3X2, 7X1 + 2X2, and 4X1 + 4X2, respectively. The carpenter is interested in knowing the worst market. That is, the solution of the following problem: The Min Max Problem: Min Max {5X1 + 3X2, 7X1 + 2X2, 4X1 + 4X2} Subject to: 2 X1 + X2 £ 40 X1 + 2 X2 £ 50 and both X1, X2 are nonnegative. The Min Max Problem is equivalent to: Max y Subject to: y £ 5x1 + 3X2 y £ 7X1 + 2X2 y £ 4X1 + 4X2 2X1 + X2 £ 40 X1 + 2X2 £ 50 And all variables X1, X2, y, are nonnegative. If you take all the variables to the left-hand side of the constraints and implement this problem on your computer package, the optimal solution is X1 = 10, X2 = 20, y = $110. This means the first and the second markets are the worst (because the first and the second constraints are binding) bringing only $110 net profit. Similarly, one can solve the maximum of min of several objective functions in a single run. Feasibility Problem: Goal-Seeking Indicators In most business applications, the manager wishes to achieve a specific goal, while satisfying the constraints of the model. The user does not particularly want to optimize anything so there is no reason to define an objective function. This type of problem is usually called a feasibility problem. Although some decision-makers would prefer the optimal. However, in most practical situations, decision-maker aims at satisfying or making incremental changes rather than optimizing. This is so, because human mind has a bounded rationality and hence can not comprehend all alternatives. In the incremental approach to decision-making. the manager takes only small steps, or incremental moves, away from the existing system. This is usually accomplished by a "local search" to find a "good enough" solution. This problem is referred to as "satisfying problem", "feasibility problem", or the "goal-seeking" problem. Therefore, the aim is to achieve a global improvement to a level that is good enough, given current information and resources. One reason that causes a business manager to overestimate the importance of the optimal strategy, is that organizations often use indicators as "proxies" for satisfying their immediate needs. Most managers pay attention to indicators, such as profit, cash flow, share price etc., to indicate survival rather than as a goal for optimization. To solve the goal-seeking problem, one must first add the goal to the constraint set. To convert the goal seeking problem to an optimization problem, one must create a dummy objective function. It could be a linear combination of the sub-set of decision variables. If you maximize this objective function, you will get a feasible solution (if one exists). If you minimize it, you might get another one (usually at the other "side" of the feasible region). You could optimize with different objective functions. Another approach is to use "Goal Programming" models that deal precisely with problems of constraint satisfaction without necessarily having a single objective. Basically, they look at measures of constraint violation and try to minimize them. You can formulate and solve goal programming models in ordinary LP, using ordinary LP solution codes. In the artificial-variable free solution algorithm one may use a zero dummy objective function, but not in some software packages, such as Lindo. In using software packages one may maximize or minimize any variable as an objective function. Numerical Example Consider Example 1 in the Initialization of the Simplex Method section of a companion site to this site. Instead of maximizing, we now wish to achieve a goal of 4. That is, Goal: -X1 + 2X2 = 4 subject to: X1 + X2 ³ 2, -X1 + X2 ³ 1, X2 £ 3, and X1, X2 ³ 0. Adding this goal to the constraint set and converting the constraints into equality form, we have: X1 + X2 - S1 = 2, -X1 + X2 - S2 = 1, X2 + S3 = 3, and X1, X2, S1, S2, S3 ³ 0. A solution is X1 = 2, X2 = 3, S1 = 3, S2 = 0, and S3 = 0. For details on the solution algorithms, visit the Web site Artificial-variable Free Solution Algorithms. Further Readings: Borden T., and W. Banta, (Ed.), Using Performance Indicators to Guide Strategic Decision Making, Jossey-Bass Pub., 1994. Eilon S., The Art of Reckoning: Analysis of Performance Criteria, Academic Press, 1984. The Copyright Statement: The fair use, according to the 1996 Fair Use Guidelines for Educational Multimedia, of materials presented on this Web site is permitted for non-commercial and classroom purposes only. This site may be mirrored intact (including these notices), on any server with public access. All files are available at http://home.ubalt.edu/ntsbarsh/Business-stat for mirroring. Kindly e-mail me your comments, suggestions, and concerns. Thank you.	{"url":"http://home.ubalt.edu/ntsbarsh/opre640a/partVIII.htm","timestamp":"2024-11-13T15:51:19Z","content_type":"text/html","content_length":"175234","record_id":"<urn:uuid:21ed056e-4c8a-403b-b779-452c62b7b738>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00098.warc.gz"}
Simplifying Complex Numbers: (6i)(2i) This article will guide you through simplifying the product of two complex numbers, specifically (6i)(2i). Understanding Complex Numbers Complex numbers are numbers that extend the real number system by including the imaginary unit i, where i² = -1. They are typically written in the form a + bi, where a and b are real numbers. Multiplying Complex Numbers To multiply complex numbers, we treat them like binomials and use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi² Since i² = -1, we can substitute this value: (a + bi)(c + di) = ac + adi + bci - bd Simplifying (6i)(2i) Let's apply the above principles to our expression: (6i)(2i) = 12i² Now, substitute i² = -1: 12i² = 12(-1) = -12 Therefore, (6i)(2i) = -12. We successfully simplified the product of the two complex numbers, (6i)(2i), to a real number, -12. This demonstrates how complex numbers can interact and lead to real-valued results.	{"url":"https://jasonbradley.me/page/(6i)(2i)","timestamp":"2024-11-10T07:24:40Z","content_type":"text/html","content_length":"56055","record_id":"<urn:uuid:9042fdd5-b74a-4c5d-82e0-b99ef33acb37>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00401.warc.gz"}
Pascal to pounds per square inch (psi) converter \| Alvacus Pascal to pounds per square inch (psi) converter Convert Pascal to pounds per square inch (Pa to psi) effortlessly with our online calculator. Simply enter the pressure in Pascal, and our tool will provide you with the equivalent pressure in pounds per square inch. Make pressure unit conversions a breeze for your engineering or industrial needs. Pascal to Pounds per Square Inch (Pa to psi) Conversion What is Pascal (Pa)? Pascal (Pa) is the standard unit of pressure in the International System of Units (SI). It is used globally in various scientific and engineering applications. One Pascal is equal to one Newton per square meter (N/m²), representing a small amount of pressure. What is Pounds per Square Inch (psi)? Pounds per square inch (psi) is a non-SI unit of pressure commonly used in the United States and some other countries. It measures pressure as the force of one pound applied to an area of one square The Conversion Formula The conversion formula from Pascal to Pounds per Square Inch (psi) is as follows: Pascal (Pa) = 0.00014503773773375 Pounds per Square Inch (psi) How to Use the Formula To convert a pressure value from Pascal to psi, follow these steps: 1. Multiply the Pressure in Pascal by the Conversion Factor: □ Take the pressure value in Pascal that you want to convert. □ Multiply it by the conversion factor, which is approximately 0.00014503773773375 psi/Pa. 2. Calculate the Equivalent Pressure in psi: □ The result of the multiplication will give you the equivalent pressure in Pounds per Square Inch (psi). Example Conversion Let's say you have a pressure of 100,000 Pascal (Pa) and want to convert it to psi. Here's how you can do it: Pressure in psi = 100,000 Pa × 0.00014503773773375 psi/Pa ≈ 14.5038 psi So, 100,000 Pascal is approximately equal to 14.5038 pounds per square inch. Converting pressure from Pascal to Pounds per Square Inch (psi) is straightforward using the provided formula. This conversion is particularly useful when you need to work with pressure measurements in different units, making it easier to compare and analyze data in various contexts.	{"url":"https://www.alvacus.com/modular/651bd1699bb5e7e5a3448963","timestamp":"2024-11-11T04:38:55Z","content_type":"text/html","content_length":"63933","record_id":"<urn:uuid:d165f042-130b-4cb5-a144-d14f6113b471>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00219.warc.gz"}
Why I’m Writing A Book On Cryptography Why I’m Writing A Book On Cryptography posted July 2020 I’ve now been writing a book on applied cryptography for a year and a half. I’m nearing the end of my journey, as I have one last ambitious chapter left to write: next-generation cryptography (a chapter that I’ll use to talk about cryptography that will become more and more practical: post-quantum cryptography, homomorphic encryption, multi-party computation, and zk-SNARKs). I’ve been asked multiple times why write a new book about cryptography? and why should I read your book?. To answer this, you have to understand when it all started… A picture is worth a thousand words Today if you want to learn about almost anything, you just google it. Yet, for cryptography, and depending on what you're looking for, resources can be quite lacking. It all started a long time ago. For a class, I had to implement a differential power analysis attack, a breakthrough in cryptanalysis as it was the first side-channel attack to be published. A differential power analysis uses the power consumption of a device during an encryption to leak its private key. At the time, I realized that great papers could convey great ideas with very little emphasis on understanding. I remember banging my head against the wall trying to figure out what the author of the white paper was trying to say. Worse, I couldn’t find a good resource that explained the paper. So I banged my head a bit more, and finally I got it. And then I thought I would help others. So I drew some diagrams, animated them, and recorded myself going over them. That was my first This first step in education was enough to make me want to do more. I started making more of these videos, and started writing more articles about cryptography on this blog (today totaling more than 500 articles). I realized early that diagrams were extremely helpful to understand complicated concepts, and that strangely most resources in the field shied away from them. For example, anyone in cryptography who thinks about AES-CBC would immediately think about the following wikipedia diagram: So here I was, trying to explain everything I learned, and thinking hard about what sorts of simple diagrams could easily convey these complex ideas. That’s when I started thinking about a book, years and years before Manning Publications would reach out to me with a book deal. The applied cryptographer curriculum  I hadn’t started cryptography due to a long-life passion. I had finished a bachelor in theoretical mathematics and didn’t know what was next for me. I had also been programming my whole life, and I wanted to reconcile the two. Naturally, I got curious about cryptography, which seemed to have the best of both world, and started reading the different books at my disposal. I quickly discovered my life's calling. Some things were annoying me though. In particular, the long introductions that would start with history. I was only interested in the technicalities, and always had been. I swore to myself, if I ever wrote a book about cryptography, I would not write a single line on Vigenère ciphers, Caesar ciphers, and others. And so after applying to the masters of Cryptography at the university of Bordeaux, and obtaining a degree in the subject, I thought I was ready for the world. Little did I know. What I thought was a very applied degree actually lacked a lot on the real world protocols I was about to attack. I had spent a lot of time learning about the mathematics of elliptic curves, but nothing about how they were used in cryptographic algorithms. I had learned about LFSRs, and ElGamal, and DES, and a series of other cryptographic primitives that I would never see again. When I started working in the industry at Matasano, which then became NCC Group, my first gig was to audit OpenSSL (the most popular TLS implementation). Oh boy, did it hurt my brain. I remember coming back home every day with a strong headache. What a clusterfuck of a library. I had no idea at the time that I would years later become a co-author of TLS 1.3. But at that point I was already thinking: this is what I should have learned in school. The knowledge I’m getting now is what would have been useful to prepare me for the real world. After all, I was now a security practitioner specialized in cryptography. I was reviewing real-world cryptographic applications. I was doing the job that one would wish they had after finishing a cryptography degree. I implemented, verified, used, and advised on what cryptographic algorithms to use. This is the reason I’m the first reader of the book I’m writing. This is what I would have written to my past self in order to prepare me for the real world. The use of cryptography is where most of the bugs are My consulting job led me to audit many real world cryptographic applications like the OpenSSL, the encrypted backup system of Google, the TLS 1.3 implementation of Cloudflare, the certificate authority protocol of Let’s Encrypt, the sapling protocol of Zcash, the threshold proxy re-encryption scheme of NuCypher and dozens and dozens of other real-world cryptographic applications that I unfortunately cannot mention publicly. Early in my job, I was tasked to audit the custom protocol a big corporation (that I can’t name) had written to encrypt their communications. It turns out that, they were signing everything but the ephemeral keys, which completely broke the whole protocol (as one could have easily replaced the ephemeral keys). A rookie mistake from anyone with some experience with secure transport protocols, but something that was missed by people who thought they were experienced enough to roll their own crypto. I remember explaining the vulnerability at the end of the engagement, and a room full of engineers turning silent for a good 30 seconds. This story repeated itself many times during my career. There was this time where while auditing a cryptocurrency for another client, I found a way to forge transactions from already existing ones (due to some ambiguity of what was being signed). Looking at TLS implementations for another client, I found some subtle ways to break an RSA implementation, which in turned transformed into a white paper (with one of the inventor of RSA) leading to a number of Common Vulnerabilities and Exposures (CVEs) reported to a dozen of open source projects. More recently, reading about Matrix as part of writing my book, I realized that their authentication protocol was broken, leading to a break of their end-to-end encryption. There’s so many details that can unfortunately collapse under you, when making use of cryptography. At that point, I knew I had to write something about it. This is why my book contains many of these As part of the job, I would review cryptography libraries and applications in a multitude of programming languages. I discovered bugs (for example CVE-2016-3959 in Golang’s standard library), I researched ways that libraries could fool you into misusing them (for example see my paper How to Backdoor Diffie-Hellman), and I advised on what libraries to use. Developers never knew what library to use, and I always found the answer to be tricky. I went on to invent the disco protocol, and wrote a fully-featured cryptographic library in less than 1,000 lines of code in several languages. Disco only relied on two cryptographic primitives: the permutation of SHA-3 and curve25519. Yes, from only these two things in 1,000 lines of code a developer could do any type of authenticated key exchange, signatures, encryption, MACs, hashing, key derivation, etc. This gave me a unique perspective as to what a good cryptographic library was supposed to be. I wanted my book to contain these kind of practical insights. So naturally, the different chapters contain examples on how to do crypto in different programming languages, using well-respected cryptographic libraries. A need for a new book? As I was giving one of my annual cryptography training at Black Hat, one student came to me and asked if I could recommend a good book or online course on cryptography. I remember advising the student to read the book from Boneh & Shoup and Cryptography I from Boneh on Coursera. The student told me “Ah, I tried, it’s too theoretical!”. This answer stayed with me. I disagreed at first, but slowly realized that they were right. Most of these resources were pretty heavy in math, and most developers interacting with cryptography don’t want to deal with math.  What else was there for them? The other two somewhat respected resources at the time were Applied Cryptography and Cryptography Engineering (both from Schneier). But these books were starting to be quite outdated. Applied Cryptography spent 4 chapters on block ciphers, with a whole chapter on cipher modes of operation but none on authenticated encryption. Cryptography Engineering had a single mention of elliptic curve cryptography (in a footnote). On the other hand, many of my videos or blog posts were becoming good primary references for some cryptographic concepts. I knew I could do something special. Gradually, many of my students started becoming interested in cryptocurrencies, asking more and more questions on the subject. At the same time, I started to audit more and more cryptocurrency applications. I finally moved to a job at Facebook to work on Libra. Cryptocurrency was now one of the hottest field to work on, mixing a multitude of extremely interesting cryptographic primitives that so far had seen no real-world use case (zero knowledge proofs, aggregated signatures, threshold cryptography, multi-party computations, consensus protocols, cryptographic accumulators, verifiable random functions, verifiable delay functions, ... the list goes on) I was now in a unique position. I knew I could write something that would tell students, developers, consultants, security engineers, and others, what modern applied cryptography was all about. This was going to be a book with very little formulas, but filled with many diagrams. This was going to be a book with little history, but filled with modern stories about cryptographic failures that I had witnessed for real. This was going to be a book with little about legacy algorithms, but filled with cryptography that I've personally seen being used at scale: TLS, the Noise protocol framework, Wireguard, the Signal protocol, cryptocurrencies, HSMs, threshold cryptography, and so on. This was going to be a book with little theoretical cryptography, but filled with what could become relevant: password-authentication key exchanges, zero-knowledge proofs, post-quantum cryptography, and so on. Real-World Cryptography When Manning Publications reached out to me in 2018, asking if I wanted to write a book on cryptography for them, I already knew the answer. I already knew what I wanted to write about. I had just been waiting for someone to give me the opportunity, the excuse to spend my time writting the book I had in mind. Coincidentally, they had a series of "real-world" book, and so naturally I suggested that my book extend it. Real-World Cryptography is available for free in early-access. I want this to be the best book on applied cryptography. For this reason, if you have any feedback to give, please send it my way :) The book should be ready in print for the end of the year. "This is what I would have wrote to my past self in order to prepare me for the real world" wrote -> written Sorry. Feel free to delete my comments, just pointing out some typos " but something that was missed by people who thought they were experimented enough to roll their own crypto" experimented -> experienced Hey Bill, No worries, I appreciate! Thanks for pointing these out! You can also do it via the contact page if you're scared of spamming the comment section :) Really need this kind of book from Real world aspect. Also I am interested in, is it necessary to have prior knowledge to read the book, like math or crypto 101. Because of i am newbie self-learner in security field and have no background related with math or CS. I am really appreciate your hard work. Thanks so much. David Jao "This was going to be a book with little history, but filled with modern stories about cryptographic failures that I had witnessed for real." Funny, there's a word that describes stories about past cryptographic failures perfectly. It's called "history." History is important in cryptography because we often rely on history to tell apart the good from the junk. History is of utmost importance for those who study assumptions. Sure, you might have a proof of security, but unless you have a one-time pad, that proof depends on some assumptions, and those assumptions don't have proofs. The test of time is often all we have. How do we know factoring is hard? Because history says it is hard. (Or, depending on your point of view, factoring is easy, but building the necessary quantum computer is hard.) The above is especially true for unstated assumptions. Side channel attacks work because they unveil a hidden assumption: the assumption that attackers can't access your internal state. I'm not saying you need to cover ancient stuff like Caesar ciphers, but modern history is absolutely important for a cryptographer. The stories of our past can guide our future. That looks wonderful, we'll definitely be reading your book. I wonder if you have an opinion on "Serious cryptography" book by Aumasson? Glen Peterson Nice article. Looking forward to the book. Make sure to talk about legal requirements brewing in Australia and the U.S. to add backdoors for law enforcement and how such a law can be worded so as not to ruin encryption in those countries. For instance, if law enforcement wanted to look for child porn or catphishing at my day job, they could theoretically do so in the application logs, or in the database. There would still be privacy concerns, but it wouldn't compromise all legal uses of encryption the way requiring backdoored algorithms would. Even in that case, how would we know a request to look at our logs was legitimate? My senator is the author of the EARN-IT act. Eye-roll! Great work! Can the diagrams and figures be used license free to show to students in a class? Great work! Can the diagrams and figures be used license free to show to students in a class? Great work! Can the diagrams and figures be used license free to show to students in a class? I'm going to buy this book just for the awesome illustrations. It looks absolutely amazing!! Saw this link on Hackernews. Thought: yes, why would one write a book on crypto. There are many good books. After reading your posting I thought: brilliant marketing! Look forward to the book. i want to understand quantum cryptography will this become obsolete soon as quantum cryptography release? Might be too late but it would be great for the book to include a section about encryption for data at rest and not just for data exchange. i.e. what are the modern techniques for full drive encryption, database at rest encryption, etc. In these cases key rotation is much more difficult. > Can the diagrams and figures be used license free to show to students in a class? I would hope so, probably something to ask the book editor (but you have my blessing in any case) > Might be too late but it would be great for the book to include a section about encryption for data at rest this is covered in the chapter on hardware cryptography > this is covered in the chapter on hardware cryptography glad to hear it' covered. "hardware" seems like an odd place to discuss encrypted data storage. I'm thinking about systems like LUKS, Keepass format, VeraCrypt and database level data at rest (like the work done by https:// dgraph.io/blog/post/encryption-at-rest-dgraph-badger/ or https://github.com/cockroachdb/cockroach/blob/master/docs/RFCS/20171220_encryption_at_rest.md) or maybe even key/token management done in systems like Hashicorp Vault (https://github.com/hashicorp/vault). In other words examples and background for the use case of "I have some data I need to store securely in my app: how should I build something or evaluate a solution" I'm a software engineer who completely agrees with you. After playing with libsodium and reading the associated paper. I cannot see where the dragons are. I know they are in there, but the paper and the documentation says I must do thing x but does not give me enough information on what happens when things break. Good luck with the book writing. Hi, this looks really good and I want to buy a copy of the book I don't understand.. is the book finished now so I can order it, or do I have to wait for a couple of months ? so that I don't buy half of the book Marco S. Zuppone Wow! Now I want it :-) It seems really cool. A lot of books and courses about this topic are too much theoretical. The book that I liked most so far about this topic is "Serious Cryptography. A Practical Introduction to Modern Encryption" by Jean-Philippe Aumasson. I'm looking forward to buy and read your book as well! Kidn regards, Marco - Stocktrader https://msz.eu && https://msz.training A few years ago, we were stuck between recommending Applied Cryptography or Cryptography Engineering. Now we're going to be stuck between Serious Cryptography and Real-World Cryptography! I look forward to reading it in full. Skimming through the introduction available online was a treat. Daniel Wright To write a book is such an amazing job! I really liked the way you have presented this article in such great detail. This was actually a fantastic article. I am truly mesmerized by this article. Thank you for sharing it with us. leave a comment...	{"url":"https://www.cryptologie.net/article/504/why-im-writing-a-book-on-cryptography/","timestamp":"2024-11-04T07:53:32Z","content_type":"text/html","content_length":"40070","record_id":"<urn:uuid:c017c923-8e2c-45ee-8010-6d225db6ee4b>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00191.warc.gz"}
Show a complete solution to each problem. Shawn's shed. Shawn is building a tool shed with a square foundation and has enough siding to cover 32 linear feet of walls. If he leaves a 4 -foot space for a door, then what size foundation would use up all of his siding? Short Answer Expert verified The side length of the square foundation is 7 feet. Step by step solution Understand the Problem Shawn is using 32 linear feet of siding for the walls of a tool shed with a square foundation. Since a door takes up 4 feet of siding, we have to find the length of each side of the square foundation that satisfies these constraints. Setup the Equation The perimeter of a square is given by the formula: $ P = 4s $ where $ s $ is the length of one side of the square. Since a 4-foot door is part of the 32 feet, subtract the door space from the total siding: $ 32 - 4 = 28 \text{ feet} $ This is the amount of siding available for the walls. Calculate the Side Length Using the adjusted amount of siding: $ 4s = 28 $ Solve for the side length $ s $: $ s = \frac{28}{4} = 7 \text{ feet} $ Verify the Solution Check if the solution meets all the conditions: - The perimeter without the door space is $ 28 \text{ feet} $, so the length of each side is 7 feet. - Total perimeter including the door space is $ 28 + 4 = 32 \text{ feet} $, which uses up all the siding as given in the problem. Key Concepts These are the key concepts you need to understand to accurately answer the question. Perimeter of a Square Understanding the perimeter of a square is essential for solving problems involving square foundations. The perimeter is the total length around the square. To find it, we use the formula $ P = 4s $, where $ P $ stands for perimeter and $ s $ represents the length of one side of the square. This formula works because a square has four equal sides. For example, if each side of a square is 5 feet long, the perimeter would be 4 times 5, which equals 20 feet. By mastering the perimeter formula, you'll be well on your way to solving many geometric Linear Feet Linear feet relate to the continuous length of a material, like the siding for Shawn's shed. Linear feet are straightforward to understand: • 1 linear foot is simply 1 foot in length. • They do not account for width or height, just length. In our problem, Shawn has 32 linear feet of siding. This means he has 32 feet of siding material, simply measured in a straight line. When dealing with linear feet, always keep in mind that it’s solely about the length—no other dimensions matter. Solving Equations Solving equations is like finding the unknown value that makes the equation true. In Shawn's problem, we set up the equation $4s = 28$ after accounting for the door. This equation tells us that four times the side length must equal 28 feet. The next step is to find $s$. To do this, divide both sides of the equation by 4: \[s = \frac{28}{4} = 7 \text{ feet} \] This result means each side of Shawn's square shed foundation should be 7 feet to use up all the siding, minus the door. Solving equations typically involve operations like addition, subtraction, multiplication, and division to isolate the unknown variable. Algebraic Manipulation Algebraic manipulation involves rearranging and simplifying equations to solve for the unknown variable. It might sound complex, but it's very systematic: • Identify the equation and the variable you need to solve for. • Perform operations (addition, subtraction, multiplication, division) to isolate the variable. In Shawn's problem, we manipulated the initial condition $P = 4s$ into a more manageable form: We deducted the door length to get $ 4s = 28 $. Then divided both sides by 4 to solve for $s $: \[s = \frac{28}{4} = 7 \text{ feet} \] Effective algebraic manipulation is all about breaking down the problem into simpler parts and solving it step by step.	{"url":"https://www.vaia.com/en-us/textbooks/math/algebra-for-college-students-5-edition/chapter-2/problem-98-show-a-complete-solution-to-each-problem-shawns-s/","timestamp":"2024-11-14T03:51:22Z","content_type":"text/html","content_length":"251567","record_id":"<urn:uuid:5c523243-df43-4bd1-bfbb-9571a5f0dc02>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00483.warc.gz"}
differential equations and fourier series pdf This resource contains information related to fourier series for functions with period 2L. University of New South Wales. 7B-1. Unit III: Fourier Series: 17 Fourier Trigonometric Series 18 Half-range and Exponential Fourier Series 19 In-class Exam 2 20 The Dirac Delta Function Solve heat equation by Fourier series 25 3.4. Date added: 03/20/18. . This book covers the following topics: Fourier Series, Fourier Transform, Convolution, Distributions and Their Fourier Transforms, Sampling, and Interpolation, Discrete Fourier Transform, Linear Time-Invariant Systems, n-dimensional Fourier Transform. Topics Covered Partial differential equations Orthogonal functions Fourier Series Fourier Integrals Separation of Variables Boundary Value Problems Laplace Transform . 06/17/2012 14:51; info modified 06/19/2012 16:54; Wadsworth Publishing Company, 1982. From the series: Differential Equations and Linear Algebra. Trigraph; . Dierential equations are mathematical equations for one or several un- known functions that relate the v alues of the functions themselves and their derivativ es of various orders. DOWNLOAD. A function is called a f (x)periodic functionif is defined for all real f x)x, except Chapter 1 Solutions Section 10.1 1. ' Roadmap to the Syllabus 1. University of New South Wales. Numerical Methods. Fourier series [EP]: 8.1 [SN]: 16 . p. cm. MIT OpenCourseWare is an online publication of materials from over 2,500 MIT courses, freely sharing knowledge with learners and educators around the world. file_download Download File. A simple example is presente. An Introduction to Differential Equations: With Difference Equations, Fourier Series, and Partial Differential Equations. First Order Differential Equations. ISBN 978--470-61796- (cloth) 1. The additional chapters 9, 10, and 11 treat difference equations, Fourier series, and partial differential equations, respectively. This resource contains solutions for the problem statements related to fourier series. Therefore, the Fourier series is. AUGUST 16, 2015 Summary. In this paper we systematically use Fourier series to nd the conditions for existence and uniqueness of periodic solutions of functional dierential equation x0(t)= Z 0 [dE(s)]x(ts)+f(t), of the rst order dierential equation x0(t)=Ax(t)+f(t), 1 and of the second order dierential equation x00(t)=Ax(t)+f(t) on Hilbert spaces. 5 ENGINEERING MATHEMATICS II (E ) DIFFERENTIAL EQUATIONS & FOURIER SERIES 19HS111 Partial Differential Equations and Fourier Series. . Partial differential equations 21 3.1. 9 6 3 3 6 9 y t 3 3 3. To define Fourier series, we first need some background material. Chapter 2 offers an improved, simpler presentation of the linearity principle, showing that the heat equation is a linear equation. Applications of Partial Differential Equations Vibration of Strings 3B. 9 6 3 3 6 9 y t 3 3 3. Functions of several variables 21 3.2. Fourier series: 20: Fourier series : Related Mathlet: Fourier coefficients: 21: Operations on fourier series : Related Mathlet: Fourier coefficients: Complex with sound: 22: Periodic solutions; resonance 23 Download Boundary Value Problems books, Boundary Value Problems is the leading text on boundary value problems and Fourier series. Chapter 3. QA404.B47 2010 515'.353dc22 2010007954 Printed in the United States of America. 4 2 0 2 4 y t 2 5. ; Chapter 4 contains a straightforward derivation of the vibrating membrane, an improvement over previous editions. More Fourier transforms 20 3. Find the periodic solutions of the differential equation y + ky = f (x), where k is a constant and f (x) is a 2 -periodic function. The course contains 56 short lecture videos, with a few problems to solve after each lecture. 9 6 3 3 6 9 y t 3 3 3. 1 3 2 2 3 . CHAPTER 9 FOURIER SERIES METHODS AND PARTIAL DIFFERENTIAL EQUATIONS SECTION 9.1 PERIODIC FUNCTIONS AND TRIGONOMETRIC SERIES The basic trigonometric functions cos(t) and sin (t) have period P= 2 , so the sine or cosine of t(as in Problems 1-4) completes its first period when t 2; hence P 2/. by Akshay SB. On this webpage you will find my solutions to the tenth edition of "Elementary Differential Equations and Boundary Value Problems" by Boyce and DiPrima In Chapter 12 we give a brief introduction to the Fourier transform and its application to partial dierential equations Wave Equation Brannan, William E " PDF Applied Partial Differential Equations With Fourier Series And Boundary Value . I. c n(f0) = inc 13 ORDINARY DIFFERENTIAL EQUATIONS Chapter 10: Fourier Series Student Solution Manual November 11, 2015 Springer. 1 3 2 2 3 . 1, 2 3 Compute the solutions of differential equations by using 1, 2 analytical techniques.. 4 Illustrate the concept of Fourier series. 2.2. Chapter 8. Fourier transform 17 2.5. The inverse transform of F(k) is given by the formula (2). Series Solutions of Second Order Linear Equations. 27, 29) 13 5 Fourier Transform . . 12.2 Fourier Series 658 1, 5, 7, 13, 17 12.3 Fourier Cosine and Sine Series . file_download Download File. (The function is a periodic function that is not defined for all real x but undefined for some points (more precisely, countably many points), that is Striking a balance between theory and applications, Fourier Series and Numerical Methods for Partial Differential Equations presents an introduction to the analytical and numerical . 258. Fourier Series: Denitions and Coefcients We will rst state Fourier's theorem for periodic functions with period P = 2. Only 1 left in stock - order soon. a) Find the Fourier cosine series of the function 1 t over the interval 0 < t < 1, and then draw over the interval [2, 2] the graph of the function f(t) which is the sum of this Fourier cosine series. Elementary Applied Partial Differential Equations With Fourier Series And Boundary Value Problems Author - www.constructivworks.com - 2022-07-04T00:00:00+00:01 Subject: Read Online Elementary Applied Partial Differential Equations With Fourier Series And Boundary Value Problems Keywords More Fourier series 14 2.4. Chapter 4. The techniques include separation of variables, Fourier series and Fourier transforms, orthogonal functions and eigenfunction expansions, Bessel functions, and Legendre polynomials. Hardcover. Fourier series are infinite series that represent periodic functions in terms of cosines and sines. Fourier Series and Partial Differential Equations 1 Mathematics 241-Syllabus and Core Problems Math 241. This is an introduction to ordinary di erential equations. Then f(t) can be represented by Fourier series f(t) X k= e2kit/Tf k. Using Fourier series is a well known method for investigating solutions of dier-ential equations, in particular for periodic and almost periodic solutions (see e. g. [1], [4], [7], [9], [10]). Fourier inversion formula 18 2.6. First Order Differential Equations Conventions Basic DE's Geometric Methods . Partial Differential Equations with Fourier Series and Boundary Value Problems Nakhle H. Asmar 2017-03-23 Rich in proofs, examples, and exercises, this widely adopted text emphasizes physics and engineering applications. 4 2 0 2 4 y t 2 5. In words, the theorem says that a function with period 2 can be written as a sum of cosines and sines which all have period 2. Chapter 5. Hi and welcome back to the differential equations lecture here on educator.com.0000 My name is Will Murray, and we are studying a chapter on partial differential equations.0004 We will meet the differential equations behind for this lecture and were to study Fourier series.0008 Fourier series is a tool that really used to solve the heat equation in the next lecture, but Fourier is kind of a . The second boundary condition implies thatc 1cos 2+c 2sin 2= 0, soc 2=cot 2= 0.2762. this book explains the following topics: first order equations, numerical methods, applications of first order equations1em, linear second order equations, applcations of linear second order equations, series solutions of linear second order equations, laplace transforms, linear higher order equations, linear systems of differential equations, money applied partial differential equations with fourier series and boundary value problems 4th edition and numerous ebook collections from fictions to scientific research in any way. 8 Analytic Geometry Equations and Curves Perimeter, Area, and Volume . Second Order Linear Equations. Chapter 1 Solutions Section 10.1 1. Syllabus Meet the TAs Unit I: First Order Differential Equations . Published by McGraw-Hill since its first edition in 1941, this classic text is an introduction to Fourier series and their applications to boundary value problems in partial differential equations of engineering and physics. Fourier coefficients Mathlet L21 Operations on fourier series [EP]: 8.2 and 8.3. ORDINARY DIFFERENTIAL EQUATIONS Chapter 10: Fourier Series Student Solution Manual November 11, 2015 Springer. This resource contains final exam. Periodic function of period p possibly at some points, and if there is some positive number p, called a period of , such that (1) for all x. We end these notes solving our rst partial di erential equation, the Heat Equation. New York City November. partial differential equations with fourier series and boundary value problems pdf solutions to applied partial differential equations with fourier series and boundary value problems Partial Differential Equations with Fourier Series and Boundary Value Problems: Instructor's Solutions Manual Nakhle H. Asmar 2nd Eds Reviewed by Planet on 07:59 . 1, 2 5 Use software tools to obtain and verify the solutions. OK, I'm going to explain Fourier series, and that I can't do in 10 minutes. Calculus, Part IV. Fourier Series, Partial Dierential Equations and Fourier Transforms Notes prepared for MA3139 Arthur L. Schoenstadt Department of Applied Mathematics Naval Postgraduate School Code MA/Zh Monterey, California 93943 August 18, 2005 c 1992 - Professor Arthur L. Schoenstadt 1 Contents SEC. Differential Equations with Fourier Series and Boundary Value Problems (Classic Version), 5th Edition Haberman, Instructors Solutions Manual for Transforms and Partial Differential Equations, Third Edition written by T. Veerarajan cover the following topics. Fourier series. Linear Algebra and Vectors Partial Differential Equations for Scientists and Engineers (Dover Books on Mathematics) Applied Partial Differential Equations with Fourier Series . With minimal prerequisites the authors take the reader from fundamentals to research topics in the area of nonlinear evolution equations. In words, the theorem says that a function with period 2 can be written as a sum of cosines and sines which all have period 2. In this paper we systematically use Fourier series to Theorem (Fourier) Suppose f (t) has period 2 then we have f (t) a 2 0 + a 1 . Less than 15% adverts . Books a la Carte also offer a great value--this format costs significantly less than a . Each of these chapters provides a thorough introduction to its respective topic. ClassTest_2021.pdf. Partial Differential Equations With Fourier Series And Boundary Value Problems by David L. Powers, Boundary Value Problems Book available in PDF, EPUB, Mobi Format. Download free PDF textbooks or read online. Theorem (Fourier) Suppose f (t) has period 2 then we have f (t) a 2 0 + a 1 . Upper Saddle River, NJ: Prentice Hall, 2003. Lecture 5 Generalised Fourier Series.pdf. ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. assignment. Chapter 11. 2 1.2 Solving and Interpreting a Partial Differential Equation 7 2 Fourier Series 17 2.1 Periodic Functions 18 2.2 Fourier Series 26 2.3 Fourier Series of Functions with Arbitrary Periods 38 2.4 Half-Range Expansions: The Cosine and Sine Series 50 2.5 Mean Square Approximation and Parseval's Identity 53 2.6 Complex Form of Fourier Series 60 Solve heat . 4.4 out of 5 stars 44. The Student Solutions Manual can be downloaded free from Dover's site; the Instructor Solutions Manual is available Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems, 4th Edition Richard Haberman. Instead of capital letters, we often use the notation f^(k) for the Fourier transform, and F (x) for the inverse transform. Author (s): Prof. Brad Osgood. 1.1 Practical use of the Fourier . 1. in the course of them is this applied partial differential equations with fourier series and boundary value problems 4th edition that can be your partner. Linear Differential Equations 3 Existence and Uniqueness of Solutions: Uniqueness 4 Existence and Uniqueness of Solutions: Picard Iterates 5 . Use of symbolic manipulation and graphics software. First, and foremost, it is a rigorous study of ordinary differential equations and can be fully understood by anyone who has completed one year of . ; Additional simpler exercises now appear throughout the text. The student will see how . An introductory partial differential equations textbook and technical reference for Mathematicians, Engineers, Physicists and Scientists. V. Differential Equations VI. Prerequisite(s): MATH 240. arrow_back browse course material library_books. Fourier Cosine Series - In this section we define the Fourier Cosine Series, i.e. It is easier (but equivalent) to choose radial solutions that satisfy the corresponding homogeneous boundary condition. problems, and Fourier Series expansions. ORDINARY DIFFERENTIAL EQUATIONS Chapter 10: Fourier Series Student Solution Manual January 7, 2016 Springer. Related Mathlet: Series RLC circuit: 18: Engineering applications Video of the guest lecture by Prof. Kim Vandiver 19: Exam II III. Complete the practice problem: Exercise: Find the Fourier Series (PDF) Answer (PDF) Watch the lecture video clip: Even and Odd Functions. Sturm-Liouville problems, orthogonal functions, Fourier series, and partial differential equations including solutions of the wave, heat and Laplace equations, Fourier transforms. This text discusses partial differential equations in the engineering and physical sciences. Introduction to complex analysis. 6 1 Solutions f(t) . ; Hints are offered for many of the exercises in which partial differential equations . ISBN: 9780136006138 . Find the periodic solutions of the differential equation y + ky = f (x), where k is a constant and f (x) is a 2 -periodic function. Both basic theory and applications are taught. 28) 13 2 Fourier Series . representing a function with a . Fourier series and numerical methods for partial differential equations / Richard Bernatz. 11.1 Fourier Series 475 x f(x) p Fig. 7B-2. About this book. b) Answer the same question for the Fourier sine series of 1 t over the interval (0, 1). Fourier Series; 4 2 0 2 4 y t 2 5. Manual Free From Internet in PDF Format ! (8) The rst boundary condition requires thatc 1= 1. representing a function with a series in the form 1 sin n x n L n B . This section provides materials for a session on general periodic functions and how to express them as Fourier series. 13 4 Fourier Transform (pg. wanting to learn how to solve differential equations or needing a refresher on differential equations. 2 Apply various numerical methods to solve differential equations. Book by Nakhle H. Asmar Partial Differential Equations and Boundary Value Problems with Fourier Series (2004) . Sturm-Liouville problems, orthogonal functions, Fourier series, and partial differential equations including solutions of the wave, heat and Laplace equations, Fourier transforms. 6th ed. money applied partial differential equations with fourier series and boundary value problems 4th edition and numerous ebook collections from fictions to scientific research in any way. Fourier Sine Series - In this section we define the Fourier Sine Series, i.e. Materials include course notes, lecture video clips, practice problems with solutions, a problem solving video, and problem sets with solutions. Nakhle H. Asmar-Partial Differential Equations with Fourier Series and Boundary Value Problems- Inst. Browse Course Material. CONTENTS.PDF: CHAPTER1.PDF . 1982 Martin Braun Preface to the First Edition This textbook is a unique blend of the theory of differential equations and their exciting application to "real world" problems. - 484 pages. Fourier Series and Differential Equations with some . Partial Differential Equations 2. We represent the function f (x) on the right-hand side of The author, David Powers, (Clarkson) has written a . Therefore, the Fourier series is. $165.33. 37 Full PDFs related to this paper Read Paper Application of Fourier Series to Differential Equations Since the beginning Fourier himself was interested to find a powerful tool to be used in solving differential equations. 1 . Therefore, the Fourier series is. How to download Paid Research Papers, AMAZON Books, Solution Manuals Free . Fourier coefficients: Complex with sound Mathlet No need to wait for office hours or assignments to be graded to find out where you took a wrong turn. Higher Order Linear Equations. 440. Striking a balance between theory and applications, Fourier Series and Numerical Methods for Partial Differential . arrow_back browse course material Dierential. We represent the function f (x) on the right-hand side of It provides an introduction to Fourier analysis and partial differential equations and is intended to be used with courses for beginning graduate students. We will also define the odd extension for a function and work several examples finding the Fourier Sine Series for a function. In this volume I shall give some guidelines for solving problems in the theories of Fourier series and Systems of Differential Equations and eigenvalue problems. 2.5.8 (a) There is a full Fourier series in .	{"url":"http://www.stickycompany.com/bangladesh/geometric/glen/92018758c43514a-differential-equations-and-fourier-series-pdf","timestamp":"2024-11-05T10:43:17Z","content_type":"text/html","content_length":"24799","record_id":"<urn:uuid:057f80d1-f73f-4dd2-b99b-3bfc2f5276cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00882.warc.gz"}
American Mathematical Society Pseudo-uniform convexity of $H^{1}$ in several variables HTML articles powered by AMS MathViewer Proc. Amer. Math. Soc. 26 (1970), 609-614 DOI: https://doi.org/10.1090/S0002-9939-1970-0268656-5 PDF \| Request permission A convergence theorem of D. J. Newman for the Hardy space ${H^1}$ is generalized to several complex variables. Specifically, in both ${H^1}$ of the polydisc and ${H^1}$ of the ball, weak convergence, together with convergence of norms, is shown to imply norm convergence. As in Newman’s work, approximation of ${L^1}$ by ${H^1}$ is also considered. It is shown that every function in ${L^1}$ of the torus, (or in ${L^1}$ of the boundary of the ball), has a best ${H^1}$-approximation which, in several variables, need not be unique. References • Lars Gȧrding and Lars Hörmander, Strongly subharmonic functions, Math. Scand. 15 (1964), 93–96. MR 179373, DOI 10.7146/math.scand.a-10732 • C. N. Kellogg, Pseudo-uniform convexity in $H^{1}$, Proc. Amer. Math. Soc. 23 (1969), 190–192. MR 250050, DOI 10.1090/S0002-9939-1969-0250050-6 • V. P. Havin, Spaces of analytic functions, Math. Analysis 1964 (Russian), Akad. Nauk SSSR Inst. Naučn. Informacii, Moscow, 1966, pp. 76–164 (Russian). MR 0206694 • D. J. Newman, Pseudo-uniform convexity in $H^{1}$, Proc. Amer. Math. Soc. 14 (1963), 676–679. MR 151834, DOI 10.1090/S0002-9939-1963-0151834-X • W. W. Rogosinski and H. S. Shapiro, On certain extremum problems for analytic functions, Acta Math. 90 (1953), 287–318. MR 59354, DOI 10.1007/BF02392438 • Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841 Similar Articles • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46.30, 32.00 • Retrieve articles in all journals with MSC: 46.30, 32.00 Bibliographic Information • © Copyright 1970 American Mathematical Society • Journal: Proc. Amer. Math. Soc. 26 (1970), 609-614 • MSC: Primary 46.30; Secondary 32.00 • DOI: https://doi.org/10.1090/S0002-9939-1970-0268656-5 • MathSciNet review: 0268656	{"url":"https://www.ams.org/journals/proc/1970-026-04/S0002-9939-1970-0268656-5/","timestamp":"2024-11-04T14:50:22Z","content_type":"text/html","content_length":"59720","record_id":"<urn:uuid:abc4622b-8dff-4798-843d-90f88eab8daa>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00226.warc.gz"}
Journal Article N=2 minimal conformal field theories and matrix bifactorisations of x^d There are currently no full texts shared for your IP range. There is no public supplementary material available Davydov, A., Camacho, A. R., & Runkel, I. (2018). N=2 minimal conformal field theories and matrix bifactorisations of x^d. Communications in Mathematical Physics, 357(2), 597-629. doi:10.1007/ Cite as: https://hdl.handle.net/21.11116/0000-0004-4777-7 We establish an action of the representations of N=2-superconformal symmetry on the category of matrix factorisations of the potentials x^d and x^d-y^d for d odd. More precisely we prove a tensor equivalence between (a) the category of Neveu–Schwarz-type representa-tions of the N = 2 minimal super vertex operator algebra at central charge 3–6/d, and (b) a full subcategory of graded matrix factorisations of the potential x^d − y^d . The subcategory in (b) is given by permutation-type matrix factorisations with consecutive index sets. The physical motivation for this result is the Landau–Ginzburg/conformal field theory correspondence, where it amounts to the equivalence of a subset of defects on both sides of the correspondence. Our work builds on results by Brunner and Roggenkamp [BR], where an isomorphism of fusion rules was established.	{"url":"https://pure.mpg.de/pubman/faces/ViewItemOverviewPage.jsp?itemId=item_3136441","timestamp":"2024-11-10T06:38:11Z","content_type":"application/xhtml+xml","content_length":"42289","record_id":"<urn:uuid:3fa472e9-6d08-4690-b293-77df45b51815>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00438.warc.gz"}
Landon Fox \| Department of Mathematics and Statistics In 2022, I graduated from the University of Nevada, Reno with a B.S. in Computer Science and Engineering and a B.S. in Discrete Mathematics. In my undergraduate, I helped create Code12, an application used to aid students learn programming principles, with Dave Parker which was used in Sierra College introductory computer science courses. Under the guidance of Dr. Fred Harris Jr., my team and I created Graph-It, an application used to visualize and compute various graph theory computations for both academics and professionals. Additionally, for several years I worked at the University Math Center as a Shift Leader Tutor where I was able to assist students in a variety of subjects ranging from linear algebra to classical mechanics. As a graduate student teaching assistant, I have taught several calculus recitations and I have shifted my focus to pure mathematics, particularly higher algebra and higher category theory as well as their applications to other areas of mathematics such as algebra, topology, combinatorics, and logic. I am advised by Dr. Jonathan Beardsley. Research interests • Homotopy Theory • Higher Algebra • M.S. in Pure Mathematics, University of Nevada, Reno, 2024 • B.S. in Computer Science and Engineering, University of Nevada, Reno, 2022 • B.S. in Discrete Mathematics, University of Nevada, Reno, 2022	{"url":"https://www.unr.edu/math/people/landon-fox","timestamp":"2024-11-07T05:52:09Z","content_type":"text/html","content_length":"116250","record_id":"<urn:uuid:39ea3bf4-0b93-4cff-bdb3-85c34cddb059>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00654.warc.gz"}
WTH you are bringing me from textbook the plots of the partial sums of the FS on the given interval rather than the complete sum on $(-\infty,\infty)$ which is due to continuation? WTH you are bringing me from textbook the plots of the partial sums of the FS on the given interval rather than the complete sum on $(-\infty,\infty)$ which is due to continuation?Professor Ivrii,The interval is [0,Ï€]! WTH you are bringing me from textbook the plots of the partial sums of the FS on the given interval rather than the complete sum on $(-\infty,\infty)$ which is due to continuation?Professor Ivrii,The interval is [0,Ï€]!But F.s. converges everywhere!	{"url":"https://forum.math.toronto.edu/index.php?PHPSESSID=btr89b3i91s4kcve19i8suggu4&topic=108.0","timestamp":"2024-11-03T01:25:36Z","content_type":"application/xhtml+xml","content_length":"35263","record_id":"<urn:uuid:da883826-6e8e-49f7-8f99-f17df2b23df2>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00340.warc.gz"}
Number of samples for random sampling using Wilks statistics • Alias: None • Arguments: None Child Keywords: Required/Optional Description of Group Dakota Keyword Dakota Keyword Description Optional order The order of the statistics to use when determining sample sizes for random sampling using Wilks order statistics. Optional confidence_level The confidence level to be used when determining sample sizes for random sampling using Wilks order statistics. Optional one_sided_lower Specifies one-sided lower portion order statistics to be used when determining sample sizes for random sampling using Wilks order statistics. Optional one_sided_upper Specifies one-sided upper portion order statistics to be used when determining sample sizes for random sampling using Wilks order statistics. Optional two_sided Specifies two-sided order statistics (an interval) to be used when determining sample sizes for random sampling using Wilks order statistics. The wilks keyword is used to compute the number of samples to execute for a random sampling study using Wilks statistics [Wil41] and [NW04]. In contrast to most sampling studies where the user specifies the number of samples in advance, Wilks determines the number of samples to run to achieve a particular objective. Specifically, Wilks statistics specify a probability level, alpha, and confidence level, beta, and determines the minimum number of samples required such that there is beta% confidence that the alpha100 percentile of the uncertain distribution on model output will fall below the actual alpha100 percentile given by the sample when outputs are ordered from smallest to largest. Statistics can also be either one_side or two_sided with the former reflecting a statement about uppermost sample output and the latter reflecting both the smallest and largest sample outputs. Finally, the order of the statistics can be increased to higher order such that the statement concerning probability level and confidence level applies to the uppermost M outputs (for one-sided M-order Wilks) or the lowest M and uppermost M outputs (for two-sided M-order Wilks). Default Behavior By default, Wilks statistics are computed using one-sided first-order order statistics with a 95% confidence interval (beta) and 95% probability (alpha). This results in a sample size of 59. Usage Tips Wilks sample sizes apply to model outputs considered one-at-a-time. Joint variation among multiple outputs requires a generalization of the Wilks approach and is not supported in Dakota at this time. When more than one probability level is specified, the largest sample size will be performed and used to subsample for the lower probability levels. sample_type random probability_levels = 0.75 0.8 0.95 0.99 confidence_level 0.99 order 2	{"url":"https://snl-dakota.github.io/docs/6.18.0/users/usingdakota/reference/method-sampling-wilks.html","timestamp":"2024-11-01T20:54:46Z","content_type":"text/html","content_length":"28145","record_id":"<urn:uuid:bd7cd958-ffa6-4db6-8aa1-a3f9d26eef1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00633.warc.gz"}
Solving nonograms In this post I will show how to solve nonograms automatically using a computer. The code has been on the Haskell wiki for over year, but I have never taken the time to explain how it works. This post is literate haskell (download the source here), so we need to start with some imports: import qualified Data.Set as Set import qualified Data.Map as Map import Data.Set (Set) import Data.List import Control.Applicative Since we will be working with sets a lot, here are some additional utility functions: setAll :: (a -> Bool) -> Set a -> Bool setAll pred = all pred . Set.toList unionMap :: (Ord a, Ord b) => (a -> Set b) -> Set a -> Set b unionMap f = Set.unions . map f . Set.toList The puzzle So, what is a nonogram anyway? Quoting Wikipedia: Nonograms are picture logic puzzles in which cells in a grid have to be colored or left blank according to numbers given at the side of the grid to reveal a hidden picture. In this puzzle type, the numbers measure how many unbroken lines of filled-in squares there are in any given row or column. For example, a clue of "4 8 3" would mean there are sets of four, eight, and three filled squares, in that order, with at least one blank square between successive groups. A solved nonogram might look like the following image: A Haskell function to solve nonograms for us could have the following type, taking the clues for the rows and columns, and returning a grid indicating which squares are filled, solvePuzzle :: [[Int]] -> [[Int]] -> [[Bool]] Values and cells For simplicity we will start with a single row. A first idea is to represent the cells in a row as booleans, type Row = [Bool]. This works fine for a finished puzzle like: but consider a partially solved row: First of all we will need a way to distinguish between blank cells (indicated by a cross) and unknown cells. Secondly, we throw away a lot of information. For instance, we know that the last filled cell will be the last cell of a group of three. To solve the second problem we can give each position an unique label, so the first filled cell will always be, for instance 1, the second one will be 2, etc. For blank cells we can use negative numbers; the first group of blanks will be labeled -1, the second group will be -2, etc. Since the groups of blanks are of variable size, we give each one the same value. Our solved row now looks In Haskell we can define the type of cell values as simply newtype Value = Value Int deriving (Eq, Ord, Show) Since negative values encode empty cells, and positive values are filled cells, we can add some utility functions: blank (Value n) = n < 0 filled = not . blank This still leaves the first issue, dealing with partially solved puzzles. Partial information When we don't know the exact value of a cell it is still possible that there is some information. For instance, we might know that the first cell will not contain the value 9, since that value is already somewhere else. One way of representing this is to keep a set of possible values: type Cell = Set Value An unknown cell is simply a cell containing all possible values, and the more we know about a cell, the less the set will contain. At a higher level we can still divide cells into four categories: data CellState = Blank \| Filled \| Indeterminate \| Error deriving Eq cellState :: Cell -> CellState cellState x \| Set.null x = Error -- Something went wrong, no options remain \| setAll blank x = Blank -- The cell is guaranteed to be blank \| setAll filled x = Filled -- The cell is guaranteed to be filled \| otherwise = Indeterminate CellStates are convenient for displaying (partial) solution grids, instance Show CellState where show Blank = "." show Filled = "#" show Indeterminate = "?" show Error = "E" For example, here is our running example again, this time rotated 90°. The CellStates are shown on the left as before; while the actual Cell set is on the right: Solving a single row Now it is time to solve a row. As stated before, each filled cell gets a unique value. From a clue of the group lengths we need to construct such a unique labeling, such that labeling [4,3] == [-1,-1,2,3,4,5,-6,-6,7,8,9,-10,-10]. The exact values don't matter, as long as they are unique and have the right sign. Constructing this labeling is simply a matter of iterating over the clues, labeling :: [Int] -> [Value] labeling = map Value . labeling' 1 where labeling' n [] = [-n,-n] labeling' n (x:xs) = [-n,-n] ++ [n+1 .. n+x] ++ labeling' (n+x+1) xs This labeling gives us important local information: we know what values can occur before and after a particular value. This is also the reason for including the negative (blank) values twice, since after a -1 another -1 can occur. We can determine what comes after a value by zipping the labeling with its tail. In our example: after [-1,-1, 2, 3, 4, 5,-6,-6, 7, 8, 9, -10, -10] comes [-1,-1, 2, 3, 4, 5,-6,-6, 7, 8, 9,-10, -10] Collecting all pairs gives the mapping: { -1 -> {-1,2}, 2 -> {3}, 3 -> {4}, 4 -> {5}, 5 -> {-6}, -6 -> {-6,7}, ...} Instead of carrying a Map around we can use a function that does the lookup in that map. Of course we don't want to recalculate the map every time the function is called, so we need to be careful about sharing: bad1 a x = Map.lookup x (expensiveThing a) bad2 a x = Map.lookup x theMap where theMap = expensiveThing a good a = \x -> Map.lookup x theMap where theMap = expensiveThing a So for determining what comes after a value in the labeling: mkAfter :: [Value] -> (Value -> Cell) mkAfter vs = \v -> Map.findWithDefault Set.empty v afters where afters = Map.fromListWith Set.union $ zip vs (map Set.singleton $ tail vs) Row data type In the Row datatype we put all the information we have: • The cells in the row • What values can come before and after a value • The values at the edges data Row = Row { cells :: [Cell] , before, after :: Value -> Cell , start, end :: Cell Some simple Show and Eq instances: instance Show Row where show row = "[" ++ concatMap show (rowStates row) ++ "]" instance Eq Row where a == b = cells a == cells b To construct a row we first make a labeling for the clues. Then we can determine what comes after each value, and what comes after each value in the reversed labeling (and hence comes before it in the normal order). mkRow :: Int -> [Int] -> Row mkRow width clue = Row { cells = replicate width (Set.fromList l) , before = mkAfter (reverse l) , after = mkAfter l , start = Set.singleton $ head l , end = Set.singleton $ last l where l = labeling clue Actually solving something Now all the things are in place to solve our row: For each cell we can determine what values can come after it, so we can filter the next cell using this information. To be more precise, we can take the intersection of the set of values in a cell with the set of values that can occur after the previous cell. In this way we can make a forward pass through the row: solveForward, solveBackward :: Row -> Row solveForward row = row { cells = newCells (start row) (cells row) } where newCells _ [] = [] newCells prev (x:xs) = x' : newCells x' xs where x' = x `Set.intersection` afterPrev afterPrev = unionMap (after row) prev Applying solveForward to the example row above, we get In much the same way we can do a backwards pass. Instead of duplicating the code from solveForward it is easier to reverse the row, do a forward pass and then reverse the row again: solveBackward = reverseRow . solveForward . reverseRow Where reverseRow reverses the cells and swaps before/after and start/end: reverseRow :: Row -> Row reverseRow row = Row { cells = reverse (cells row) , before = after row, after = before row , start = end row, end = start row } In the running example even more cells will be known after doing a backwards pass, These two steps together are as far as we are going to get with a single row, so let's package them up: solveRow :: Row -> Row solveRow = solveBackward . solveForward In the end we hopefully have a row that is completely solved, or we might h We can determine whether this is the case by looking at the CellStates of the cells: rowStates :: Row -> [CellState] rowStates = map cellState . cells rowDone, rowFailed :: Row -> Bool rowDone = not . any (== Indeterminate) . rowStates rowFailed = any (== Error) . rowStates Human solution strategies By using just one single solution strategy we can in fact emulate most of the techniques humans use. The Wikipedia page on nongrams lists several of these techniques. For instance, the simple boxes technique is illustrated with the example: The Haskell program gives the same result: Nonograms> solveRow $ mkRow 10 [8] The reason why humans need many different techniques, while a single technique suffices for the program is that this simple technique requires a huge amount of administration. For each cell there is a while set of values, which would never fit into the small square grid of a puzzle. The whole puzzle Just a single row, or even a list of rows is not enough. In a whole nonogram there are clues for both the rows and the columns. So, let's make a data type to hold both: data Puzzle = Puzzle { rows, columns :: [Row] } deriving Eq And a function for constructing the Puzzle from a list of clues, mkPuzzle :: [[Int]] -> [[Int]] -> Puzzle mkPuzzle rowClues colClues = Puzzle { rows = map (mkRow (length colClues)) rowClues , columns = map (mkRow (length rowClues)) colClues To display a puzzle we show the rows, instance Show Puzzle where show = unlines . map show . rows showList = showString . unlines . map show Initially the puzzle grids are a bit boring, for example entering in GHCi Nonograms> mkPuzzle [[1],[3],[1]] [[1],[3],[1]] We already know how to solve a single row, so solving a whole list of rows is not much harder, stepRows :: Puzzle -> Puzzle stepRows puzzle = puzzle { rows = map solveRow (rows puzzle) } Continuing in GHCi: Nonograms> stepRows previousPuzzle To also solve the columns we can use the same trick as with reverseRow, this time transposing the puzzle by swapping rows and columns. transposePuzzle :: Puzzle -> Puzzle transposePuzzle (Puzzle rows cols) = Puzzle cols rows But this doesn't actually help anything! We still display only the rows, and what happens there is not affected by the values in the columns. Of course when a certain cell in a row is filled (its cellState is Filled), then we know that the cell in the corresponding column is also filled. We can therefore filter that cell by removing all blank values filterCell :: CellState -> Cell -> Cell filterCell Blank = Set.filter blank filterCell Filled = Set.filter filled filterCell _ = id A whole row can be filtered by filtering each cell, filterRow :: [CellState] -> Row -> Row filterRow states row = row { cells = zipWith filterCell states (cells row) } By transposing the list of states for each row we get a list of states for the columns. With filterRow the column cells are then filtered. stepCombine :: Puzzle -> Puzzle stepCombine puzzle = puzzle { columns = zipWith filterRow states (columns puzzle) } where states = transpose $ map rowStates $ rows puzzle To solve the puzzle we apply stepRows and stepCombine alternatingly to the rows and to the columns. When to stop this iteration? We could stop when the puzzle is done, but not all puzzles can be solved this way. A better aproach is to take the fixed point: solveDirect :: Puzzle -> Puzzle solveDirect = fixedPoint (step . step) where step = transposePuzzle . stepCombine . stepRows The fixed point of a function f is the value x such that x == f x. Note that there are different fixed points, but the one we are interested in here is found by simply iterating x, f x, f (f x), ... fixedPoint :: Eq a => (a -> a) -> a -> a fixedPoint f x \| x == fx = x \| otherwise = fixedPoint f fx where fx = f x The tiny 33 example can now be solved: Nonograms> solveDirect previousPuzzle But for other puzzles, such as the letter lambda from the introduction, we have no such luck: Nonograms> solveDirect lambdaPuzzle To solve more difficult puzzles the direct reasoning approach is not enough. To still solve these puzzles we need to make a guess, and backtrack if it is wrong. Note that there are puzzles with more than one solution, for example To find all solutions, and not just the first one, we can use the list monad. To make a guess we can pick a cell that has multiple values in its set, and for each of these values see what happens if the cell contains just that value. Since there are many cells in a puzzle there are also many cells to choose from when we need to guess. It is a good idea to pick the best one. For picking the best alternative a pair of a value and a score can be used: data Scored m a = Scored { best :: m a, score :: Int } This data type is an applicative functor if we use 0 as a default score: instance Functor m => Functor (Scored m) where fmap f (Scored a i) = Scored (fmap f a) i instance Applicative m => Applicative (Scored m) where pure a = Scored (pure a) 0 Scored f n <> Scored x m = Scored (f <> x) (n `min` m) When there are alternatives we want to pick the best one, the one with the highest score: instance Alternative m => Alternative (Scored m) where empty = Scored empty minBound a <\|> b \| score a >= score b = a \| otherwise = b Now given a list we can apply a function to each element, but change only the best one. This way we can find the best cell to guess and immediately restrict it to a single alternative. We can do this by simply enumerating all ways to change a single element in a list. mapBest :: Alternative m => (a -> m a) -> [a] -> m [a] mapBest _ [] = pure [] mapBest f (x:xs) = (:xs) <$> f x -- change x and keep the tail <\|> (x:) <$> mapBest f xs -- change the tail and keep x This can also be generalized to Rows and whole Puzzles: mapBestRow :: Alternative m => (Cell -> m Cell) -> Row -> m Row mapBestRow f row = fmap setCells $ mapBest f $ cells row where setCells cells' = row { cells = cells' } mapBestRows :: Alternative m => (Cell -> m Cell) -> Puzzle -> m Puzzle mapBestRows f puzzle = fmap setRows $ mapBest (mapBestRow f) $ rows puzzle where setRows rows' = puzzle { rows = rows' } What is the best cell to guess? A simple idea is to use the cell with the most alternatives, in the hope of eliminating as many of them as soon as possible. Then the score of a cell is the size of its set. The alternatives are a singleton set for each value in the cell. guessCell :: Cell -> Scored [] Cell guessCell cell = Scored { best = map Set.singleton $ Set.toList cell , score = Set.size cell } We can now make a guess by taking the best way to apply guessCell to a single cell: guess :: Puzzle -> [Puzzle] guess = best . mapBestRows guessCell Putting it together Direct solving is much faster than guess based solving. So the overall strategy is to use solveDirect, and when we get a puzzle that is not done we do a single guess, and then continue with direct solving all alternatives: solve :: Puzzle -> [Puzzle] solve puzzle \| failed puzzle' = [] \| done puzzle' = [puzzle'] \| otherwise = concatMap solve (guess puzzle') where puzzle' = solveDirect puzzle done, failed :: Puzzle -> Bool done puzzle = all rowDone (rows puzzle ++ columns puzzle) failed puzzle = any rowFailed (rows puzzle ++ columns puzzle) Finally we can solve the lambda puzzle! lambdaPuzzle = mkPuzzle Nonograms> solve lambdaPuzzle I actually did this in C++ once, years ago, but without the clever labeling strategy that you use. I just listed all possible alternatives for a row based on the clues (usually there aren't too many). Then removed the impossible ones based on the current cells (where a cell is filled/open/unknown), and checked whether the alternatives all agreed upon a certain value (filled/open) for a certain cell, in which case we could fill it in. This was iterated over rows and columns, with backtracking, much in the same way as you do. But it doesn't scale, and your solution is far more elegant anyway :) I wonder whether it is possible to extend the labeling approach in a way that keeps both rows and columns into account... something like "this cell is the third in the row of 3 and the second in the column of 4", like a Cartesian product between the current labelings. Might allow for a more two-dimensional approach instead of iterating only over rows. But I haven't thought this through, so it's probably rubbish :P How did you do the pretty pictures ? The pretty pictures where made with Corel PhotoPaint, and a lot of manual work. I imagine that any other drawing program would also work. So, no fancy output libraries, sorry :) Nice. This is a good explanation of some pretty code. I think there's something amiss with your Applicative instance, though. (pure f <> x) should be the same as (fmap f x), but the former will always have score 0 and the latter will have x's score. Where by "pure f x" I mean "pure f <> x". The angle brackets got eaten. D. I. Lewis: You are right, the pure function should have a score of maxBound for pure f <> x to be the same as fmap f. Unfortunatly that would change the behaviour of mapBest, because not changing anything becomes the best choice. Perhaps a better fix would be to use a different way of combining scores, for example (+) instead of min. D WilliamsDate: 2014-07-13T19:22Zx I've written a solver in C++ before now, it was actually the first step to write a program that would generate them. Date: 2020-01-31T17:03Zx Thanks a lot for this! It's a great explanation and gave me lots to ponder about the specific knowledge representation. I've reimplemented on my side. That got me wondering if the Scored datatype wasn't a bit misguided with its Alternative instance: • that instance is not exactly used. I've replaced pure, <> and empty with undefined and it still solved branch-requiring puzzles. • apart from Functor (which can be derived, by the way), the instances woudln't be lawful anyway: for Applicative identity fails pure id <> v = v { score = min 0 (score v) }, and composition fails in the same way because of pure; Alternative laws are up to more debate, but at least empty isn't neutral wrt <\|> here. Moreover, I'm not sure an Applicative instance exists at all, that makes sense in this context. The underlying Alternative isn't used as such anywhere in the mapBest* family of functions (or at all, come to think of it), so it really seems to me all we need is a Semigroup. One that we lifted through various containers. I'd reach for the sequenceA kind of function, but all of those would tend to use the underlying Applicative structure, which is exactly what we're trying to avoid with our Semigroup. Maybe there's a chance of achieving something with the Alt wrapper, but I couldn't. I've left my code here if you'd like to compare: (please pardon my changing most of the identifier names, it's part of my “enforce understanding before typing” process)	{"url":"https://www.twanvl.nl/blog/haskell/Nonograms","timestamp":"2024-11-07T04:16:50Z","content_type":"text/html","content_length":"69159","record_id":"<urn:uuid:b8e61dc7-4a0b-45c9-9c31-81da63c08e82>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00570.warc.gz"}
Random Number Generator in C# \| Learn Random number generator in C# Updated March 17, 2023 What is Random Number Generator in C#? A random number generator is a built-in library in C# that generates integers and floating-point numbers randomly. Each time the library’s relevant method is invoked, it returns a random number. A series of random numbers is a set of numbers that do not follow any pattern. The random number generator in C# tends to generate such a series whenever invoked. Random Class in C# • So, how does C# generate a series of random numbers? The answer lies within the Random Class of the C# System namespace. • Random class is a pseudo-random number generator class. This means that this class is tasked to generate a series of numbers which do not follow any pattern. But, is a machine is truly capable of generating random numbers? How would the machine know which number to generate next? After all, the machine is designed to follow instructions and execute algorithms. • No, the machine cannot generate random numbers on its own. There is a defined mathematical algorithm, based on the current clock and state of the machine, which guides it to pick numbers from a set. All the numbers in the set have an equal probability of being picked up. Thus, they are not perfectly random. They do follow a pattern. It’s just that the pattern is sufficiently random to meet the practical human requirements. Pseudo and Secure The next question that comes to the mind is why do they call it pseudo-random number generator class? Let us understand this through real-life human behaviour. When a human being is asked to select a random colour, he picks up a certain colour. Let’s say he picked Yellow. What caused him to pick yellow? It could be his favourite colour or the colour of his surroundings, or he could have been thinking about something yellow at the time. This human behaviour which drives the decision to pick something randomly is called the Seed in the world of random-ness. The seed is the trigger or the beginning point of the random-ness. Now, when the seed is predictable, the random numbers become less random. They are then called pseudo-random numbers. When unpredictable, they are called secure-random numbers. C# Random Class uses the current timestamp as the seed, which is very much predictable. And hence, the term pseudo-random number generator class. RNGCryptoServiceProvider Class The RNGCryptoServiceProvider Class from the System.Security.Cryptography namespace is capable of generating secure random numbers, ones that can be used as passwords. Random Number Generator Functions in C# The first thing to generate a random number in C# is to initialize the Random class. This can be done by any of the two constructors of the class: • Random(): Initializes an object of the Random class using a time-based seed value. The seed value is the current timestamp of the machine. Although, in later versions, this was changed to be GUID • Random(Int32): Initializes an object of the Random class using the specified seed value. To get the next random number from the series, we call the Next() method of the Random class. • Next(): Returns a non-negative pseudo-random Int32 integer. • Next(Int32): Returns a non-negative pseudo-random Int32 integer less than the specified integer. • Next(Int32, Int32): Returns a non-negative pseudo-random Int32 integer within the specified range. Random Number Generator Integers in C# Let us see an example of how to generate random integers: Example #1 The below example generates random Int32 numbers. using System; public class Program public static void Main() Random rnd = new Random(); for (int i = 0; i < 10; i++) Console.WriteLine("Random number {0} : {1}", i + 1, GenerateRandomInt(rnd)); public static int GenerateRandomInt(Random rnd) return rnd.Next(); Example #2 The below example generates random Int32 numbers in the range 0 to 100. using System; public class Program public static void Main() Random rnd = new Random(); for (int i = 0; i < 10; i++) Console.WriteLine("Random number {0} : {1}", i + 1, GenerateRandomInt(rnd)); public static int GenerateRandomInt(Random rnd) return rnd.Next(100); Example #3 The below example generates random Int32 numbers in the range 50 to 100. using System; public class Program public static void Main() Random rnd = new Random(); for (int i = 0; i < 10; i++) Console.WriteLine("Random number {0} : {1}", i + 1, GenerateRandomInt(rnd)); public static int GenerateRandomInt(Random rnd) return rnd.Next(50, 100); Generating Floating-Point Numbers in C# Let us see an example of how to generate random floating-point numbers: Example #1 The below example generates random Int32 numbers. using System; public class Program public static void Main() Random rnd = new Random(); for (int i = 0; i < 10; i++) Console.WriteLine("Random number {0} : {1}", i + 1, GenerateRandomInt(rnd)); public static double GenerateRandomInt(Random rnd) return rnd.NextDouble(); A Very Common Mistake The most common mistake developers commit while generating random numbers is that for each random number, they create a new object of Random Class. As illustrated in the example below: Example #1 using System; public class Program public static void Main() for (int i = 0; i < 10; i++) Console.WriteLine("Random number {0} : {1}", i + 1, GenerateRandomInt()); public static int GenerateRandomInt() Random rnd = new Random(); //a very common mistake return rnd.Next(); How Random Numbers are all the same and Why did this happen? As explained in the working of Random Class, the numbers generated are based on the seed value and the current state of the machine. Any instance of Random class starts with the seed value, saves the current state and uses it to generate the next random number. In the code above, the mistake was to create a new instance of the Random class in every iteration of the loop. So, before the time in the internal clock changes, the code is fully executed, and each instance of Random class is instantiated with the same seed value. This results in the same set of numbers generated every time. In this article, we learnt about the random number generator in C# and how it internally works to generate random numbers. We also briefly learnt the concept of pseudo-random and secure-random numbers. This information is sufficient for developers to use the Random class in their applications. Deep dive, if interested to explore more on random numbers for passwords and one-time passwords. Recommended Articles This is a guide to Random Number Generator in C#. Here we discuss how Random Number Generator work, the concept of pseudo-random and secure-random numbers and the use of Random Number. You can also go through our other related articles to learn more –	{"url":"https://www.educba.com/random-number-generator-in-sharp/","timestamp":"2024-11-10T23:00:04Z","content_type":"text/html","content_length":"315943","record_id":"<urn:uuid:be9aeedc-c33b-4e9b-8f74-022d9d5b7abf>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00508.warc.gz"}
Decoding Algorithmic Concepts: Exploring the Urdu Meaning and Linguistic Nuances Welcome to my blog! Today’s post will discuss what is an algorithm and its Urdu meaning. Join me as we explore this essential concept in computer science and programming. Understanding Algorithm Concepts: A Comprehensive Guide for Urdu Speakers Understanding Algorithm Concepts: For those who speak Urdu or any other language, the process of learning about algorithms remains the same. An algorithm is a step-by-step procedure for solving a problem or executing a task, and it forms the backbone of computer programming and software development. Algorithm Basics: At the core of an algorithm lies its logic, which is the sequence of steps that must be followed to achieve a specific goal. These steps are typically based on inputs and outputs. Inputs are the initial data or information provided to the algorithm, and outputs are the final results generated by it. Efficiency and Complexity: One key aspect of algorithms is their efficiency, which is determined by the time complexity and space complexity. Time complexity refers to the amount of computational time required to execute the algorithm, while space complexity relates to the memory being used during execution. Data Structures: Algorithms commonly use data structures to organize and manipulate the input data. Some common data structures include arrays, linked lists, stacks, queues, trees, and graphs. Choosing the right data structure can significantly impact the overall efficiency and performance of an algorithm. Algorithm Design Techniques: There are several techniques for designing effective algorithms, such as Divide and Conquer, Dynamic Programming, Greedy Algorithms, and Backtracking. Each technique has its own set of advantages and disadvantages, and the choice of which to use largely depends on the specific problem being solved. Algorithm Analysis: Understanding the performance of an algorithm is essential in determining its suitability for a particular task. This involves analyzing its best-case, worst-case, and average-case scenarios. Often, these analyses utilize concepts from discrete mathematics, such as Big O notation, to describe the algorithm’s efficiency. Examples and Implementations: There are countless algorithms that have been developed to solve a wide range of problems, such as searching, sorting, pattern matching, graph traversal, and optimization. Learning about these examples, as well as implementing them in various programming languages, is an excellent way to gain a deeper understanding of algorithm concepts. In summary, understanding algorithm concepts is a crucial skill for software developers, computer scientists, and anyone interested in problem-solving through computational means. Regardless of one’s native language, learning about algorithms involves grasping fundamental concepts, mastering design techniques, and analyzing their performance to create efficient and effective solutions. Why algorithms are called algorithms \| BBC Ideas What Is An Algorithm? \| What Exactly Is Algorithm? \| Algorithm Basics Explained \| Simplilearn How can one describe an algorithm in straightforward terms? An algorithm is a step-by-step set of instructions or a structured plan to solve a specific problem or perform a certain task. In the context of computer programming, algorithms are designed to be executed by computers or machines to help automate processes and make complex tasks more manageable. To describe an algorithm in straightforward terms, one should focus on its input, the process it goes through, and the output it produces: 1. Input: It is the initial data or information given to the algorithm to work with. This can include numbers, text, or other types of data that the algorithm needs to produce the desired result. 2. Process: This is the sequence of steps or actions that the algorithm goes through to transform the input into the desired output. Each step should be clear, concise, and unambiguous to facilitate efficient execution. 3. Output: The final result or product generated by the algorithm after processing the input. This is the solution to the problem or the completion of the task for which the algorithm was designed. In summary, a well-defined algorithm takes an input, performs a series of clearly defined steps or processes, and produces a specific output or solution. What is an example of an algorithm? An example of an algorithm is the Bubble Sort algorithm, which is a simple sorting algorithm that works by repeatedly stepping through the list, comparing adjacent elements and swapping them if they are in the wrong order. The key steps of the Bubble Sort algorithm are: 1. Start at the beginning of the list. 2. Compare the current element with the next element. 3. If the current element is greater than the next element, swap them. 4. Move to the next element in the list. 5. Repeat steps 2-4 until the end of the list is reached. 6. If any swaps were made during the pass through the list, go back to the beginning and repeat steps 1-5. This process continues until no more swaps are needed, indicating that the list is sorted. Although Bubble Sort is not the most efficient sorting algorithm, it serves as a good introduction to the concept of algorithms and how they can be used to solve problems. How do algorithms function? Algorithms function as a set of instructions designed to perform a specific task or solve a particular problem. They are used in various fields, such as computer science, mathematics, and data analysis. To understand how algorithms function, consider the following key elements: 1. Input: Algorithms take some input data to process. This can be anything, such as numbers, characters, or other data types, depending on the problem being solved. 2. Procedure: The algorithm consists of a structured set of rules or steps that need to be followed to solve the problem. These steps are followed in a specific order, and they must be clear and unambiguous to ensure the algorithm functions correctly. 3. Output: After processing the input data based on the procedure, the algorithm generates an output or result. This output can be a single value, a list of values, or even a new data structure, depending on the problem being solved. 4. Efficiency: An essential aspect of algorithms is their efficiency, which relates to the time and resources (such as memory) required to complete the task. More efficient algorithms can solve problems faster and with less resource usage. In summary, algorithms function by taking input data, processing it through a series of well-defined steps or rules, and generating a desired output, all while aiming for the highest possible What does the term “algorithm” originally signify? The term “algorithm” originally signifies a step-by-step procedure for performing a specific task or solving a particular problem. It comes from the name of the Persian mathematician, Al-Khwarizmi, who introduced systematic methods in mathematics and algebra. In the context of algorithms, it refers to a sequence of instructions that a computer or a person can follow to achieve a desired outcome or reach a defined goal. What is the meaning of the term “algorithm” in Urdu, and how does this translation relate to the concept of algorithms in computer science? The term “algorithm” in Urdu means “الگورتھم”, which is pronounced as “algorithm” in the Urdu language. This translation directly relates to the concept of algorithms in computer science, as it signifies the same idea of a step-by-step procedure or a set of rules to be followed for solving a particular problem or performing calculations in the field of computing. An algorithm is a fundamental concept in computer science and programming, where it serves as the basis for designing efficient software, programs or applications that can solve problems, perform tasks, and make decisions. Algorithms are implemented using programming languages, and they can be optimized to improve performance in terms of time, space, or other resources. How can algorithm concepts and terminologies be effectively conveyed and understood in the Urdu language? To effectively convey and understand algorithm concepts and terminologies in the Urdu language, it is essential to focus on a few key strategies: 1. Translation: Translate the essential terms and concepts into Urdu. A good starting point is to use popular Urdu translations of technical terms. This will make it easier for the audience to grasp the concepts. 2. Examples: Provide examples that are relevant to the audience’s cultural context. Examples that resonate with the local culture and experiences will make it easier for them to understand and apply the concepts. 3. Visual aids: Use diagrams, flowcharts, and other visual aids to help explain complex ideas in a more accessible manner. Visual representations can make it easier for the audience to comprehend intricate concepts, especially when they are not familiar with the English terminology. 4. Localize content: Adapt the content to the local environment, including using local analogies or stories that are relatable. This can make the learning process more engaging and enjoyable for the 5. Repetition and reinforcement: Reinforce key concepts and terminologies by reiterating them throughout the content. Consistent exposure to these terms will improve the learners’ understanding and retention of the material. 6. Online resources and tools: Encourage the use of online resources, such as tutorials, articles, and forums, that are available in Urdu. These resources can provide additional support and clarification on specific topics. 7. Collaborative learning: Promote collaboration between learners for more effective knowledge sharing and problem-solving. Group activities and discussions can help solidify understanding and promote the practical application of algorithm concepts. By incorporating these strategies, you can help ensure that algorithm concepts and terminologies are effectively communicated and understood in the Urdu language. What are the most common challenges faced by Urdu-speaking individuals when learning about algorithms, and how can these obstacles be overcome? The most common challenges faced by Urdu-speaking individuals when learning about algorithms include: 1. Language Barrier: A significant portion of algorithm resources, tutorials, and documentation is available in English. Many Urdu-speaking individuals face difficulty in comprehending complex technical terms and concepts in the English language. To overcome this obstacle, they can use translation tools or seek resources translated into Urdu. Additionally, more educational content regarding algorithms should be produced in Urdu to facilitate better understanding. 2. Complex Terminology: Algorithm-related concepts often involve complex terminology and jargon, which can be overwhelming for beginners, especially if English is not their first language. A helpful strategy would be to create a glossary of essential terms with explanations in Urdu, so learners can quickly reference and understand these terms. 3. Lack of Localized Resources: There’s a scarcity of localized (Urdu) resources to teach algorithms, making it difficult for Urdu-speaking individuals to learn effectively. Overcoming this challenge requires collaboration between educators, developers, and bilingual experts to create and promote Urdu-language resources for learning algorithms. 4. Cultural Differences: Certain examples or explanations used to describe algorithms might not resonate with people from different cultural backgrounds, such as those who speak Urdu. To address this issue, educators should incorporate culturally relevant examples and scenarios to make learning more relatable and engaging. 5. Access to Learning Platforms: Some Urdu-speaking individuals may face limited access to comprehensive learning platforms that cater to algorithms’ education. Ensuring broader access to online courses, forums, and communities discussing algorithms in Urdu would significantly benefit their learning process. In conclusion, to help Urdu-speaking individuals overcome these challenges when learning about algorithms, it is vital to create and promote Urdu-language resources, incorporate culturally relevant examples, and ensure access to robust learning platforms. These efforts will bridge the knowledge gap and encourage more Urdu-speaking individuals to engage in the field of algorithms and computer	{"url":"https://locall.host/what-is-algorithm-urdu-meaning/","timestamp":"2024-11-05T03:14:58Z","content_type":"text/html","content_length":"223764","record_id":"<urn:uuid:57c37899-e253-48e5-ad04-579e2503b249>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00103.warc.gz"}
What does 'Optimization' exactly means, and how is it applied in Sympheny? Our senior software development Engineer, Youssef Sherif, answers a few questions that might have popped into your mind! • What are the biggest challenges when planning a complex energy system? And why do we need Optimization? Planning complex energy systems with sector coupling is more challenging than typical centralized energy systems. In terms of modeling, the degrees of freedom have become overwhelming. As a result, only simulating a few sets of system designs (i.e., rule-based search) to determine the optimal system design and operation runs the risk of missing out on optimal solutions. Although one could guarantee optimality by potentially iterating all possible system designs (i.e., brute-force search), such an approach is confined to small-scale systems because the number of possible solutions grows exponentially as the site grows. To identify optimal solutions, Sympheny's solver uses Mathematical Optimization. Relative to traditional methods, our solution not only guarantees optimality of results but also effectively handles large-scale systems. Sympheny is a powerful energy system optimization tool that streamlined the process of creating mathematical models and solving it, this allows our users to focus on designing the best system possible. • What is Mathematical Optimization? And how does Sympheny apply it? Mathematical Optimization is a sophisticated analytical tool that allows users to describe complex real-world problems in a mathematical model and find a solution that optimizes an objective while adhering to user-defined constraints. It has a wide range of applications in manufacturing, scheduling, transportation, economics, control engineering, marketing, policy modeling, etc. Sympheny uses Mathematical Optimization to identify cost-effective and emission-minimizing system designs and operation strategies for new and existing sites. • What is required to prepare and solve an optimization problem? An optimization problem or model consists of the following elements: 1. Variables (e.g., technology capacity variables, production per time step, binary variables (install or not install a technology)) 2. Constraints and bounds (e.g., max production per time-step, max capacity) 3. Objective function (e.g., total cost/profit, total emissions) As mentioned above, a typical optimization problem starts by defining the variables in the model. In the case of Sympheny, the variables are technologies, energy carriers, energy networks, etc. Then, constraints of these variables, such as maximum capacity, seasonal operation, and charging/discharging behaviors, are defined. Finally, the objective functions are set. Once the optimization problem is defined, it is solved mathematically to find the best set of values for all variables that minimize/maximize the objective function while satisfying all constraints in the model. How can I share a project with another user? All projects can be easily shared with another user by selecting the options menu on the three dots button and then click on Send Project Copy. How can I give priority to the use of a type of electricity (e.g. grid’s electricity vs. renewable electricity from PV) for a Conversion Technology? Within a multiple input system, the priority is a variable of the optimization so the optimization engine will choose which is the most favorable input to reach the objectives of the optimization. When it comes to the best solution in terms of minimizing CO[2], it means that it will always favor renewable electricity over grid electricity (assuming the grid electricity has a higher CO[2] intensity than the renewable electricity, which in most of the cases is true, except e.g. if you have a very clean electricity from the grid and an on-site production with PV and batteries with high grey energy). When it comes to the best solution in terms of minimizing Life-Cycle Cost, it means the optimization will favor PV when the price of buying the grid's electricity is higher than selling renewable electricity, it will maximize the internal use of renewable electricity automatically, and will choose renewable electricity (from your PV) over grid electricity for the Heat Pump. If installing a PV system is too expensive to be favored by optimization while being necessary in your system design, it is possible to force install this technology under the “OPTIMIZATION OPTIONS” Can a Heat Pump and a Chiller also be installed as a bivalent system? Yes, it’s possible to use the waste heat of a cooling technology candidate as an input for another technology (e.g., Heat Pump) candidate. Why is the maximal energy produced by a Technology Candidate higher than the Optimal Capacity given in the results system diagram? This is because the Optimal Capacity given in the system diagram represents the capacity of the (selected) Primary Output, while the output given in the results plot “Energy Flow Out” is the combined In the example below, the Chiller Candidate is dimensioned to an Optimal Capacity of 420 kW, but the maximal production in the results plot is 800 kWh/h for the combined outputs (Cooling and Waste	{"url":"https://support.app.sympheny.com/troubleshooting-and-faqs/latest/faqs","timestamp":"2024-11-07T21:34:47Z","content_type":"text/html","content_length":"39135","record_id":"<urn:uuid:d710d189-4621-4ce8-82cb-c343bff80ab6>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00303.warc.gz"}
Triangle Length Calculator Last updated: Triangle Length Calculator The triangle length calculator tells you the length of the third side if you enter two sides and an angle. A triangle has three sides and three angles. While we know by courtesy of the angle sum property that the sum of interior angles is 180°, the length of sides can be anything. To this end, you need to employ a sine law or the cosine law to relate them to each other. Sine or cosine forms the crux of trigonometry functions that have numerous applications. One is about finding the third side or any angle for a triangle: the calculator and the accompanying text do the same. Read on to understand more about triangle length and cosine law. Relationship to calculate triangle lengths Let's consider a triangle whose sides are a, b, and c and angles $\alpha$, $\beta$, $\gamma$. The sides of the triangle are related to each other with the cosine law: \begin{align} &\cos{\alpha} = \frac{b^2 + c^2 - a^2}{2\times b\times c}\\\\ &\cos{\beta} = \frac{-b^2 + c^2 + a^2}{2\times a\times c}\\\\ &\cos{\gamma} = \frac{b^2 - c^2 + a^2}{2\times b\times a} \ Using the triangle length calculator Let ⊿ABC be a right-angled triangle having sides, a and b, forming the right angle, equal to 3 and 4, respectively. To find the missing side length: 1. Fill in the angle, $\gamma = 90°$. 2. Enter the length of side, $a = 3$. 3. Input the length of side, $b = 4$. 4. Using the triangle length calculator: \scriptsize \begin{align} \qquad \cos{\gamma} &= \frac{b^2 - c^2 + a^2}{2 \times b\times a} \\\\ \qquad \cos{90°} &= \frac{4^2 - c^2 + 3^2}{2\times 4 \times 3} \\\\ 0 &= \frac{16 - c^2 + 9}{12} \\\\ c^2 &= 16 + 9 \\ c &= \sqrt{16 + 9} = 5 \end{align*} The third side of the triangle is 5. How do I find an angle of the triangle using the side? To find the angle of the triangle opposite one of its sides, say side "a": 1. Square the first side, a. 2. Add the square of the second side, b to it. 3. Subtract the square of the third side, c from the sum. 4. Divide the difference by the length of second side. 5. Divide the quotient by the length of first side. 6. Divide the quotient by 2. 7. Find the cosine inverse of the final value to obtain the angle. Mathematically, α = arccos ((a² + b² - c²)/(2ab)) What is the third side of the right triangle having two sides, 9 and 16? The third side of the triangle is 18.36. Considering the third angle as 90°, the third side is obtainable using the cosine law. Since cos(90°) = 0, the cosine law now translates to a Pythagoras theorem, i.e., c = √(9² + 16²) = 18.36.	{"url":"https://www.omnicalculator.com/math/triangle-length","timestamp":"2024-11-06T15:10:48Z","content_type":"text/html","content_length":"534449","record_id":"<urn:uuid:d6b1515b-0cf6-4e59-873b-9332f68e59cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00354.warc.gz"}
Number Systems in Module 1.1 Number Systems in Electronics • After studying this section, you should be able to: • Know the base values of commonly used number systems. • • Decimal • • Binary. • • Octal. • • Hexadecimal. • Understand methods for extending the scope of number systems. • • Exponents. • • Floating point notation. • • Normalised form. • Know how numerical values may be stored in electronic systems • • Bits. • • Bytes. • • Words. • • Registers. Number Systems Most number systems follow a common pattern for writing down the value of a number: A fixed number of values can be written with a single numerical character, then a new column is used to count how many times the highest value in the counting system has been reached. The number of numerical values the system uses is called the base of the system. For example, the decimal system has 10 numerical characters and so has a base of 10: For writing numbers greater than 9 a second column is added to the left, and this column has 10 times the value of the column immediately to its right. Because number systems commonly used in digital electronics have different base values to the decimal system, they look less familiar, but work in essentially the same way. Decimal, (base 10) Decimal has ten values 0 to 9. If larger values than 9 are needed, extra columns are added to the left. Each column value is ten times the value of the column to its right. For example the decimal value twenty-two is written 22 (2 tens + 2 ones). Binary, (base 2) Binary has only two values 0 and 1. If larger values than 1 are needed, extra columns are added to the left. Each column value is now twice the value of the column to its right. For example the decimal value three is written 11 in binary (1 two + 1 one). Octal, (base 8) Octal has eight values 0 to 7. If larger values than 7 are needed, extra columns are added to the left. Each column value is now 8 times the value of the column to its right. For example the decimal value twenty-seven is written 33 in octal (3 eights + 3 ones). Hexadecimal, (base 16) Hexadecimal has sixteen values 0 to 15, but to keep all these values in a single column, the 16 values (0 to 15) are written as 0 to F, using the letters A to F to represent numbers 10 to 15, so avoiding the use of a second column. Again, if higher values than 15 (F in hexadecimal) are needed, extra columns to the left are used. Each column value is sixteen times that of the column to its right. For example the decimal value sixty-eight is written as 44 in hexadecimal (4 sixteens + 4 ones). Each of these different number systems works in the same way, it is just that each system has a different base, and the column values in each system increase by multiples of the base number as columns are added to the left. Because this module describes several different number systems, it is important to know which system is being described. Therefore if there is some doubt which system a number is in, the base of the system, written as a subscript immediately after the value, is used to identify the number system. For example: 10[10] represents the decimal value ten. (1 ten + 0 units) 10[2] represents the binary value two. (1 two + 0 units) 10[8] represents the octal value eight. (1 eight + 0 units) 10[16] represents the hexadecimal value sixteen. (1 sixteen + 0 units) The System Radix The base of a system, more properly called the RADIX, is the number of different values that can be expressed using a single digit. Therefore the decimal system has a radix of 10, the octal system has a radix of 8, hexadecimal is radix 16, and binary radix 2. The range of number values in different number systems is shown in Table 1.1.2, Notice that because the hexadecimal system must express 16 values using only one column, it uses the letters A B C D E & F to represent the numbers 10 to 15. The Radix Point. When writing a number, the digits used give its value, but the number is ‘scaled’ by its RADIX POINT. For example, 456.2[10] is ten times bigger than 45.62[10] although the digits are the same. Notice also that when using multiple number systems, the term ‘RADIX point’ instead of ‘DECIMAL point’ is used. When using decimal numbers, a decimal point is used, but if a different system is used, it would be wrong to call the point a decimal point, it would need to be called "Binary point" or "Octal point" etc. The simplest way around this is to refer to the point in any system (which will of course have its value labelled with its radix) as the RADIX POINT. A decimal number such as 456.2[10] can be considered as the sum of the values of its individual digits, where each digit has a value dependent on its position within the number (the value of the = 456.2[10] Each digit in the number is multiplied by the system radix raised to a power depending on its position relative to the radix point. This is called the EXPONENT. The digit immediately to the left of the radix point has the exponent 0 applied to its radix, and for each place to the left, the exponent increases by one. The first place to the right of the radix point has the exponent -1 and so on, positive exponents to the left of the radix point and negative exponents to the right. This method of writing numbers is widely used in electronics with decimal numbers, but can be used with any number system. Only the radix is different. Hexadecimal exponents 98.2[16] = (9 x 16^1) + (8 x 16^0) + (2 x 16^-1) Octal exponents 56.2[8] = (5 x 8^1) + (6 x 8^0) + (2 x 8^-1) Binary Exponents 10.1[2] = (1 x 2^1) + (0 x 2^0) + (1 x 2^-1) When using your calculator for the above examples you may find that it does not like radix points in anything other than decimal mode. This is common with many electronic calculators. Floating Point Notation If electronic calculators cannot use radix points other than in decimal, this could be a problem. Fortunately for every problem there is a solution. The radix exponent can also be used to eliminate the radix point, without altering the value of the number. In the example below, see how the value remains the same while the radix point moves. It is all done by changing the radix exponent. 102.6[10] = 102.6 x 10^0 = 10.26 x 10^1 = 1.026 x 10^2 = .1026 x 10^3 The radix point is moved one place to the left by increasing the exponent by one. It is also possible to move the radix point to the right by decreasing the exponent. In this way the radix point can be positioned wherever it is required, in any number system, simply by changing the exponent. This is called FLOATING POINT NOTATION and it is how calculators handle decimal points in calculations. Normalised Form By putting the radix point at the front of the number, and keeping it there by changing the exponent, calculations become easier to do electronically, in any radix. Electronic storage of numbers. A number written (or stored) in this way, with the radix point at the left of the most significant digit is said to be in NORMALISED FORM. For example .11011[2] x 2^3 is the normalised form of the binary number 110.11[2]. Because numbers in electronic systems are stored as binary digits, and a binary digit can only be 1 or 0, it is not possible to store the radix point within the number. Therefore the number is stored in its normalised form and the exponent is stored separately. The exponent is then reused to restore the radix point to its correct position when the number is In electronics systems a single binary digit is called a bit (short for Binary DigIT), but as using a single digit would seriously limit the maths that could be performed, binary bits are normally used in groups. 4 bits = 1 nibble 8 bits = 1 byte Multiple bytes, such as 16 bits, 32 bits, 64 bits are usually called ‘words’, e.g. a 32 bit word. The length of the word depends on how many bits can be physically handled or stored by the system at one time. 4 Bit Binary Representation When a number is stored in an electronic system, it is stored in a memory location having a fixed number of binary bits. Some of these memory locations are used for general storage whilst others, having some special function, are called registers. Wherever a number is stored, it will be held in some form of binary, and must always have a set number of bits. Therefore a decimal number such as 13, which can be expressed in four binary bits as 1101[2] becomes 00001101[2] when stored in an eight-bit register. This is achieved by adding four NON SIGNIFICANT ZEROS to the left of the most significant ‘1’ digit. Using this system, a binary register that is n bits wide can hold 2^n values. Therefore an 8 bit register can hold 2^8 values = 256 values (0 to 255) A 4 bit register can hold 2^4 values = 16 values (0 to 15) HOW MANY VALUES CAN A 16 BIT REGISTER HOLD? Enter a decimal number. Filling the register with non-significant zeros is fine - if the number is smaller than the maximum value the register will hold, but how about larger numbers? These must be dealt with by dividing the binary number into groups of 8 bits, each of which can be stored in a one-byte location, and using several locations to hold the different parts of the total value. Just how the number is split up depends on the design of the electronic system involved. • • Electronic systems may use a variety of different number systems, (e.g. Decimal, Hexadecimal, Octal, Binary). • • The number system in use can be identified by its radix (10, 16, 8, 2). • • The individual digits of a number are scaled by the Radix Point. • • The Exponent is the system radix raised to a power dependent on the column value of a particular digit in the number. • • In Floating Point Notation the Radix Point can be moved to a new position without changing the value of the number if the Exponent of the number is also changed. • • In Normalised Form the radix point is always placed to the left of the most significant digit. • • When numbers are stored electronically they are stored in a register holding a finite number of digits; if the number stored has less digits than the register, non-significant zeros are added to fill spaces to the left of the stored number. Numbers containing more digits than the register can hold are broken up into register sized groups and stored in multiple locations.	{"url":"https://learnabout-electronics.org/Digital/dig11.php","timestamp":"2024-11-08T06:10:47Z","content_type":"text/html","content_length":"25093","record_id":"<urn:uuid:47eba8e2-cf97-4d4e-9c32-5b244dcae327>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00831.warc.gz"}
Lets see how we can use the radian measure to analyze the motion of an object moving along a circular pathSuppose we have a ferris wheel with a 30 ft radius tha Let's see how we can use the radian measure to analyze the motion of an object moving along a circular path. Suppose we have a ferris wheel, with a 30, ft radius, that completes one revolution in 40 seconds. Answer : Other Questions	{"url":"https://mis.kyeop.go.ke/shelf/51617332","timestamp":"2024-11-05T03:11:44Z","content_type":"text/html","content_length":"149572","record_id":"<urn:uuid:5f2a919d-2232-44ce-a84a-a1527985b328>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00307.warc.gz"}
≫ How To Calculate Cass Mark - The Dizaldo Blog! Welcome, in this blog post, we are going to learn about how to calculate Cass Mark. Students often feel confused or misunderstood about how to calculate their Cass marks. In Ireland, the Cass marks have a significant impact on the students' final grades. Therefore, having a clear understanding of the Cass marking system can help the students in achieving their desired grades. So, let's dive in and explore the Cass marking system. What is Cass Marking System? Cass stands for Continuing Assessment. It is the evaluation process that measures the students' knowledge, skills, and progress throughout the academic year. The Cass marking system is followed in Ireland, and it divides the academic year marks into two categories. The first category is based on the continuous assessment and given in the form of Cass marks. The second category is the final exam marks, which are given at the end of the academic year. What is the weightage of Cass Marks and Final Exam Marks? The weightage of Cass marks and final exam marks varies from course to course. Generally, Cass marks contribute 40% of the total academic year marks, while the final exam marks contribute 60%. However, for some courses, Cass marks may contribute up to 60% of the total academic year marks. How to Calculate Cass Marks? The Cass marks are calculated based on the assignments, quizzes, projects, and class participation. These tasks are assigned throughout the academic year, and the students get grades for each task. To calculate the Cass marks, the cumulative grades of all the tasks are added and then divided by the total number of tasks. Let's take an example to understand the calculation of Cass Marks Suppose, a student has completed the following tasks: Task Weightage(%) Grade Assignment 1 10 85 Quiz 1 5 70 Midterm Project 15 80 Assignment 2 10 90 Quiz 2 5 75 Final Project 15 95 Now, to calculate the Cass marks of the student, we will use the following formula: Cass Marks = ((10x85) + (5x70) + (15x80) + (10x90) + (5x75) + (15x95)) / (10+5+15+10+5+15) = 85% Importance of Cass Marks The Cass marks have a significant impact on the students' final grades. The continuous assessment process helps the students in identifying their strengths and weaknesses and improving their academic performance. Therefore, the Cass marks provide continuous feedback to the students and help them in achieving their academic goals. The Cass marking system is an essential aspect of the education system in Ireland. It measures the students' knowledge, skills and progress throughout the academic year. The Cass marks contribute a significant proportion of the final academic year marks. Therefore, having a clear understanding of the Cass marking system can help the students in achieving their academic goals.	{"url":"https://www.dizaldo.co.za/how-to-calculate-cass-mark/","timestamp":"2024-11-08T08:23:14Z","content_type":"text/html","content_length":"61789","record_id":"<urn:uuid:15ab833a-83b7-47fe-b158-afbf5d9570ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00639.warc.gz"}
Quick Ratio Formula \| Calculator (With Excel template) Updated November 23, 2023 Quick Ratio Formula (Table of Contents) Quick Ratio Formula The quick ratio is a popular metric used to calculate the short-term liquidity position of a company. The formula for the Quick ratio is: In the above Quick ratio formula, Quick assets refer to assets that can be converted into Cash within 90 days. Quick assets = Cash or cash equivalents, Marketable securities & Accounts receivables Now that we know what quick assets are, let us break down the formula as follows: Let us now calculate the Quick ratio with a simple example. Examples of Quick Ratio Formula Example #1 Consider a company XYZ has the following Current Assets and current liabilities. • Cash: $10000 • Inventory: $5000 • Stock investments: $2000 • Inventory: $4000 • Prepaid taxes: $800 • Accounts receivables: $6000 • Current liabilities: $15000 The quick ratio pertaining to company XYZ can be calculated as follows: • Quick ratio = Cash+ Stock investments + Accounts receivables/ Current liabilities • Quick ratio = $10000+$2000+$6000/ $15000 • Quick ratio = $18000/$15000 • Quick ratio = $1.2 The quick ratio of company XYZ is 1.2, which means company XYZ has $1.2 of quick assets to pay off $1 of its current liabilities. Sometimes companies don’t provide a detailed balance sheet; however, the quick ratio can still be calculated as shown in the below example: Example #2 Suppose the balance sheet of company XYZ looks like the one shown below. • Total current assets = $22800 • Inventory = $4000 • Prepared taxes = $800 • Current liabilities = $15000 Since company XYZ did not give the breakdown of the quick assets, the quick ratio can be calculated with the below method: • Quick ratio = Total current assets – Inventory – Prepared taxes/ Current liabilities • Quick ratio = $22800 – $4000 – $800/ $15000 • Quick ratio = $18000/$15000 • Quick ratio = $1.2 Example #3 Now that we have a basic understanding of the calculation of quick ratio, let us now go ahead and calculate the quick ratio of Reliance Industries: The Current Assets and current liabilities of Reliance Industries for FY 2017 – 18 are; • Current Investments = 53,277.00 • Inventories = 39,568.00 • Trade Receivables = 10,460.00 • Cash And Cash Equivalents = 2,731.00 • Short-Term Loans And Advances = 3,533.00 • Other Current Assets = 14,343.00 • Current Liabilities = 190647 • Quick ratio = Current Investments+Trade Receivables+Cash And Cash Equivalents+Short Term Loans And Advances+Other Current Assets/Current Liabilities • Quick ratio = 53277 + 10460 + 2731 + 3533 + 14343/ 190647 • Quick ratio = 84344/190647 • Quick ratio = 0.44 From the above calculation, it is clear that the short-term liquidity position of Reliance Industries is not good. Reliance Industries has 0.44 INR in quick assets for every 1 INR of current It also helps to compare the previous years’ quick ratio to understand the trend. So let us now calculate the quick ratio of Reliance Industries for FY 2016 – 17. • Current Investments = 51,906 • Inventories = 34,018.00 • Trade Receivables = 5,472.00 • Cash And Cash Equivalents = 1,754.00 • Short-Term Loans And Advances = 4,900.00 • Other Current Assets = 8,231.00 • Current Liabilities = 152826 • Quick ratio = Current Investments+Trade Receivables+Cash And Cash Equivalents+Short Term Loans And Advances+Other Current Assets/Current Liabilities • Quick ratio = 51906+5472+1754+4900+8231/152826 • Quick ratio = 72263/152826 • Quick ratio =0.47 The comparative study of a quick ratio for FY 16 & 17 suggests that the quick ratio of Reliance Industries declined from 0.47 to 0.44. This indicates that the short-term liquidity position of Reliance Industries is bad, and hence it cannot pay off its current liabilities with the quick assets. It also makes sense to look at the contribution weightage of each asset in the overall quick If you notice the quick assets of Reliance Industries, the short-term investments have more weightage, with 53277 contributions to the overall quick assets of 84344. This means that strategically, Reliance Industries has made good short-term investments that can be converted into Cash to pay off its current liabilities. However, the quick assets are insufficient to meet its short-term liquidity requirements. • The quick ratio also called the acid test ratio, is based on the historical practice of using acid to test metals for gold. The metal would undergo the acid test to prove pure gold; otherwise, it is just a metal. Similarly, investors test the companies to determine their short-term liquidity position by calculating the quick ratio. • The quick ratio is similar to the current ratio in calculating current assets; however, while calculating the quick ratio, we eliminate Inventory & prepared expenses. The reason is the assumption that Inventory may not be realized into Cash within 90 days. The Inventory includes Raw materials and works in progress; therefore, liquidating the Inventory promptly becomes difficult. • A quick ratio of 1 or more is considered to be good. It means that the short-term liquidity position of the company is good. A quick ratio of 1 indicates that for every $1 of current liabilities, the company has $1 In quick assets to pay off. Similarly, a quick ratio of 2 indicates the company has $2 in current assets for every $1 it owes. • The assets considered under the quick ratio are Cash & cash equivalents, Marketable securities/Investments, and Accounts Receivables. Use of Quick Ratio • The quick ratio is mainly used by investors/creditors to determine the short-term liquidity position of the company they are investing in/lending. • It also helps the company’s management decide the optimum level of current assets that must be maintained to meet the short-term liquidity requirements. • The quick ratio also helps enhance the company’s credit score for taking credit in the market. Significance and Use of Quick Ratio Formula While the quick ratio is a quick & easy method of determining the company’s liquidity position, diligence must be done in interpreting the numbers. To get the complete picture, it is always better to break down the analysis and see the reason for the high quick ratio. For example, if the quick ratio increase is impacted by the spike in Accounts receivables and the payment from the creditors is delayed, the company may not be able to meet its immediate and short-term debt obligations. Now that we understand the complete know-how of the quick ratio, please go ahead and try calculating the quick ratio on your own in the Excel template made for you to practice. Please also analyze and see the reason for the increase/decrease in the quick ratio. Quick Ratio Formula Calculator You can use the following Quick Ratio Formula Calculator Quick Assets Current Liabilities Quick Ratio Formula= Quick Ratio Formula= = Quick Assets = Current Liabilities Quick Ratio Formula in Excel (With Excel Template) Here, we will do the same example of the Quick Ratio Formula in Excel. It is straightforward. You need to provide the two inputs, i.e., Current Assets and Current liabilities You can easily calculate the Quick Ratio Formula in the template provided. Recommended Articles This has been a guide to the Quick Ratio Formula. Here, we discuss its uses along with practical examples. We also provide a Quick Ratio calculator and a downloadable Excel template.	{"url":"https://www.educba.com/quick-ratio-formula/","timestamp":"2024-11-03T23:21:14Z","content_type":"text/html","content_length":"340959","record_id":"<urn:uuid:50bb7d3a-712b-4048-a9b3-602081d0cf16>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00728.warc.gz"}
2024 Trader's Guide to T-Distribution Ever wondered how traders and investors make informed decisions when there’s not much data to go on? That’s where the T-Distribution comes in. Think of it as a special tool, similar to the normal bell curve but with thicker tails. It’s designed for those tricky situations where you have a small sample size or don’t know the full picture of the market. Originally created by William Sealy Gosset, this tool is key for testing theories and making predictions in finance. In this guide, we’ll break down what the T-Distribution is, where it came from, and how it’s used in real-world financial scenarios. Ready to learn? Let’s dive in! Exploring the T-Distribution The T-Distribution, also known as Student’s T-Distribution, is very important for statistical analysis especially when the sample size is small. It was created by William Sealy Gosset in 1908 while he worked at Guinness Brewery. To keep his work secret, Gosset published using a fake name “Student”. Like the normal distribution but with heavier tails, the T-Distribution is more able to identify values that are not close to the mean. This makes it good for small samples when we don’t know the standard deviation of population – a typical situation in financial trading. Mainly, the T-Distribution is applied in hypothesis testing and confidence interval estimation for a population mean when sample size is less than 30. It helps to understand whether means of two datasets show significant difference, dealing with more uncertainty due to small samples. The T-Distribution is influenced by the number of degrees of freedom, which we can calculate as the size of the sample minus one. More degrees of freedom cause the shape to become more like a normal distribution because as per Law of Large Numbers. This characteristic improves accuracy in estimating and testing hypotheses, making it vital for making wise choices in trades and investments. The importance of the T-Distribution, it’s vital for traders and financial analysts in dealing with statistical uncertainty when they have small samples. It helps to do thorough financial analysis. Insights from the T-Distribution The T-Distribution is a crucial tool in statistical analysis, especially when we work with small samples. This happens often within financial and scientific research. The distribution helps us grasp and explain data by showing the variation of data points from their mean when population standard deviation isn’t known. In situations when we just have a tiny example from a big group, the standard deviation of that whole population is still not known for sure. T-Distribution takes care of this uncertainty because it changes confidence intervals according to how large or small your sample size might be. This is important because smaller samples usually show more variability which results in less certainty about exact values for parameters within populations. The T-Distribution is a method that helps in estimating the mean of a population more precisely when there are certain restrictions. It works by using the sample mean and sample standard deviation, together with degrees of freedom (obtained as one less than number of observations) for molding distribution used to estimate the mean. As the size of sample diminishes, the T-Distribution broadens and conversely contracts; this mirrors greater doubtfulness with lesser samples. This feature of the T-Distribution holds special importance in trading and financial study where choices about investments typically depend on data samples, like monthly returns or yearly earnings. The T-Distribution aids in giving a more precise method to calculate the mean and form confidence intervals, assisting financial analysts in evaluating risks and potential returns, such as earnings per share, from investments. Moreover, the T-Distribution is crucial for risk management because it can provide confidence intervals that are trustworthy even when dealing with small samples. This helps in figuring out the possible range of investment returns and allows making wise choices even if there isn’t complete information available. Practical Application: Utilizing the T-Distribution A way that T-Distribution is helpful in finance is when you want to estimate the returns likely from a stock and assess its risk, especially if your sample sizes are small. Picture this scenario: You are a trader who needs to guess the yearly return of a newly introduced stock. The available data only covers monthly returns for a few months. Suppose the stock has existed for 12 months and its monthly returns, noted in percentage, go like this: 3%, 2%, -1%, 4%, 2%, 5%, 0% ,3% ,-2% ,3% ,1% and lastly a big 4%. In order to predict yearly return and evaluate risk, a trader can calculate sample mean as well as sample standard deviation of these given returns. Then they use T-Distribution to get an estimation about where true mean annual return may fall in range. Calculate the sample mean (μ): Calculate the sample standard deviation (s): After calculations, assume s = 2.5%. Determine degrees of freedom (df): df = n−1=12−1=11 Constructing a confidence interval using the T-Distribution: Using a standard T-Distribution table or software to find the t-value for 11 degrees of freedom at a 95% confidence level (typically around 2.201 for df=11). The 95% confidence interval for the mean monthly return is then calculated as: Simplifying further gives a range for the mean monthly return. The interval that is computed gives a trader a statistically reliable estimate of what the true mean monthly returns probably are, taking into account the uncertainty caused by having only a small sample size. This kind of study is very important for deciding on whether or not to invest in this stock because it offers better understanding about possible earnings as well as how likely we could deviate from them. Comparing Distributions: T-Distribution and Normal Distribution The T-Distribution and normal distribution have particular importance in statistical analysis, with each being appropriate for different situations. The normal distribution is best used when the sample size is large. On the other hand, a T-Distribution (also known as Student’s T-Distribution) has been specifically made for smaller samples where we don’t know standard deviation of the The T-Distribution has fatter tails than the normal distribution, which makes it more suitable for estimating population means when dealing with small samples. In finance, the normal distribution supposes a certain variance and big sample size that assist in forecasting results such as price changes and volatility. However, when trading new stocks or small-cap stocks, or in markets with limited data, the T-Distribution is typically used. The shape of its distribution (heavier tails) better reflects stock price movements and includes extreme values that are more frequent in financial markets. This feature is crucial for realistic risk assessment and creation of trading strategies. For example, in the situation where we are evaluating the risk of a fresh financial tool having just a few months of data, T-Distribution is superior for forming confidence intervals because it considers outliers and makes these ranges more precise. These intervals help traders comprehend probable losses and plan their strategies properly. The T-Distribution changes its form with the degrees of freedom, becoming less steep when sample size reduces. This results in a wider shape, which is more appropriate for the variation seen in small data sets typical to trading situations. On the other hand, normal distribution does not alter according to sample size and could undervalue risk involved within financial choices. Benefits of Employing the T-Distribution The use of T-Distribution is very important in trading and financial analysis because it deals well with small sample sizes that have an unknown population standard deviation. The main benefits of this are increased statistical confidence and better accuracy for decision making within the unpredictable world of finance. For starters, the T-Distribution is an essential tool because it gives precise confidence intervals for smaller samples. In trading, there are times when you might not have a lot of data such as with newly issued stocks or securities that possess limited historical information. The T-Distribution accurately calculates mean returns or volatility better than the normal distribution by taking into account more uncertainty in small samples. This prevents underestimation of risk and allows for better informed choices. Furthermore, the T-Distribution adapts according to degrees of freedom, which connect with size of sample. When there is more data, their estimates and predictions become clearer and trustworthy – this backs up the ongoing enhancement in trading strategies. Finally, T-Distribution is vital for hypothesis testing in trading strategies. It empowers traders to test the importance of trading signals or fresh strategies with confidence, even when there’s only a small amount of data available. This capacity for strategy validation under uncertainty can prevent possible losses and enhance total effectiveness in trading activities. Challenges and Limitations The T-Distribution, it is very useful for understanding financial data. This especially applies when we have small sample sizes or we don’t know about population variances. But, there are difficulties and boundaries to the T-Distribution that traders and analysts need to accept so they don’t make mistakes in their statistical explanations. One main constraint is its use with big data sets. When the number of samples gets bigger, T-Distribution moves towards normal distribution. For such situations, employing normal distribution could be better because it has easier calculations and known qualities. Over-reliance on the T-Distribution in large samples could complicate analyses without adding accuracy. Another hurdle is the assumption of normality in an underlying population. T-Distribution assumes normality when we don’t know the standard deviation of the population, but it might not be true for financial markets that usually have asymmetric information, fat tails, and are skewed left or right – these are frequently seen characteristics in asset returns and stock prices. The T-Distribution’s vulnerability to degrees of freedom, which is connected with sample size, can also bring problems. If we assume wrong things about degrees of freedom – for instance, if our guess on the sample size or number of parameters is off – it might lead to confidence intervals and test statistics that are not precise enough. This could misguide decision making processes. Also, the process of calculating the T-Distribution is more involved than that of normal distribution. This difference becomes pronounced when we make adjustments for degrees of freedom. The complexity can result in computational mistakes, especially in situations where it is done manually or within less advanced analytical settings. Finally, those who trade and analyze should be cautious about overfitting and data mining biases in hypothesis testing using the T-Distribution. Financial markets are impacted by diverse elements, and depending only on the T-Distribution for forecasts may neglect other important market movements which could result in incorrect tactics. T-Distribution in Risk Management Risk management holds great importance in sustaining the stability of a portfolio and gaining long-term returns when it comes to trading and investment. The T-Distribution is an important tool for these strategies, particularly in evaluating tail risks and computing Value at Risk (VaR). Tail risk refers to the possibility of extreme changes in the value of an investment that, although having low statistical probability, can cause significant effect. The T-Distribution is a type of probability distribution with heavier tails compared to the normal distribution. This characteristic makes it more suitable for realistic evaluation of rare events like tail risks. In finance and investment, we often use this concept to model and predict potential big market movements which may impact our portfolio values significantly because these extreme outcomes cannot be easily captured by normal distribution models. Value at Risk (VaR) is a method used to estimate the loss in value of an asset or portfolio for a certain time period, with a particular confidence level. By using the T-Distribution, analysts can make better approximations of VaR when dealing with data that is not normally distributed. This often happens in financial returns that show skewness or kurtosis. Such an approach gives us a more precise understanding about risks connected to infrequent events which might cause substantial money losses. The T-Distribution is a helpful tool for traders and risk managers to adjust their risk evaluation models so that they match how market returns actually behave. This method is especially useful when dealing with portfolios that have either highly fluctuating stocks or only a few data points available, as it allows flexibility in changing degrees of freedom. Such adjustment matches the analysis precisely to characteristics of the dataset, providing an accurate grasp on risk levels. In general, using the T-Distribution for managing risk, alongside tools like stock trade alerts that provide real-time trading insights, makes financial plans stronger. This combination gives traders more ways to predict and lessen possible bad results in unpredictable and changeable markets, potentially identifying optimal buy and sell opportunities while mitigating risks. The T-Distribution, with its unique characteristic of being flatter than the normal distribution, is a very important concept in the finance area. It helps to correctly describe the uncertainty related to sample mean especially when the data set has small size or population standard deviation is not known. This property of T-distribution makes it crucial for testing hypotheses and building confidence intervals – both key methods used in financial analysis for evaluating various types of monetary information. In the trading and investment area, the T-Distribution helps to comprehend market movements and uncertainty. It gives power for traders to choose wisely about possible investments and evaluate risks related with changes in the trading markets. But, it is very important to understand the T-Distribution’s limits and not overrate its forecasting abilities. Wrong use may result in wrong conclusions and misguided trading tactics. When they comprehend its powers as well as restrictions, finance experts can utilize the T-Distribution properly to enhance their choices and risk handling. Interpreting the T-Distribution: FAQs What Is the Difference between T-Distribution and Normal Distribution When It Comes to Small Sample Sizes? In handling small sample sizes, the T-Distribution is more appropriate because it takes into account extra uncertainty by estimating standard deviation from the sample. The normal distribution, on the other hand, assumes a known population standard deviation. This causes thinner tails and less precise portrayal of variability in mean estimate from samples with smaller sizes. Can the T-Distribution Be Used for Any Size of Data Set in Trading Analysis? The T-Distribution is good for small samples (less than 30) but it can also work with bigger datasets. When we have a large sample, it will turn into the normal distribution as estimating the standard deviation doesn’t matter much anymore – thus making both distributions almost the same when dealing with big samples. What Does the T-Distribution Suggest about Trading Risk Assessment? Using the T-Distribution for risk assessment implies that making use of a more flexible model is beneficial in situations where sample sizes are small or population standard deviations are not known. It adjusts the estimates of risk measures like Value at Risk (VaR) to be more accurate, particularly when dealing with extreme outcomes. The heavier tails provide conservative estimations which is important for managing tail risk in markets that have high volatility. When Do You Decide to Use the T-Distribution Instead of Other Statistical Distributions? You select the T-Distribution when sample size is small or if population standard deviation isn’t known, like in markets that have little historical data to look at. Also, it is preferred when data distribution expects heavier tails – this method helps in showing and managing risk related to very extreme price movements. In What Way Does the Shape of T-Distribution Get Impacted by Degrees of Freedom? The tails of T-Distributions become thinner as the number of degrees in a freedom increases, moving closer to normal distribution. Thicker tails are associated with fewer degrees of freedom, showing more variability or uncertainty that matches to smaller sample sizes.	{"url":"https://thetradinganalyst.com/what-is-a-t-distribution/","timestamp":"2024-11-08T15:07:28Z","content_type":"text/html","content_length":"367660","record_id":"<urn:uuid:ea0ff555-0b70-4e6f-8bda-253792f5451e>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00710.warc.gz"}
The convergence rate of the Sandwiching algorithm for convex bounded multiobjective optimization Sandwiching algorithms, also known as Benson-type algorithms, approximate the nondominated set of convex bounded multiobjective optimization problems by constructing and iteratively improving polyhedral inner and outer approximations. Using a set-valued metric, an estimate of the approximation quality is determined as the distance between the inner and outer approximation. The convergence of the algorithm is evaluated with respect to this approximation quality. We show the convergence rate of a class of Sandwiching algorithms by extending results for a similar algorithm for the approximation of convex compact sets. In particular, we derive requirements for the nondominated set to have a twice continuously differentiable boundary. We show that two common quality indicators, the polyhedral gauge and the epsilon indicator, fulfill the necessary requirements and explicitly state the convergence rate for these two indicators. Under sufficient regularity assumptions, the convergence rate is optimal. View The convergence rate of the Sandwiching algorithm for convex bounded multiobjective optimization	{"url":"https://optimization-online.org/2023/12/the-convergence-rate-of-the-sandwiching-algorithm-for-convex-bounded-multiobjective-optimization/","timestamp":"2024-11-05T06:49:38Z","content_type":"text/html","content_length":"85256","record_id":"<urn:uuid:805593e1-e892-43ff-a062-bb6a874234ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00198.warc.gz"}
Sparse Matrix Graphical Models - PDF Free Download Sparse Matrix Graphical Models Chenlei Leng and Cheng Yong Tang June 1, 2012 Abstract Matrix-variate observations are frequently encountered in many contemporary statistical problems due to a rising need to organize and analyze data with structured information. In this paper, we propose a novel sparse matrix graphical model for this type of statistical problems. By penalizing respectively two precision matrices corresponding to the rows and columns, our method yields a sparse matrix graphical model that synthetically characterizes the underlying conditional independence structure. Our model is more parsimonious and is practically more interpretable than the conventional sparse vector-variate graphical models. Asymptotic analysis shows that our penalized likelihood estimates enjoy better convergent rate than that of the vector-variate graphical model. The finite sample performance of the proposed method is illustrated via extensive simulation studies and several real datasets analysis. KEY WORDS: Graphical models; Matrix graphical models; Matrix-variate normal distribution; Penalized likelihood; Sparsistency; Sparsity. ∗ Leng and Tang are with Department of Statistics and Applied Probability, National University of Singapore. Research supports from National University of Singapore research grants are gratefully acknowledged. Corresponding author: Chenlei Leng (Email: [email protected] ). We thank co-editor Prof. Xuming He, an associate Editor and two anonymous referees for their constructive comments that have led to a much improved paper. The rapid advance in information technology has brought an unprecedented array of high dimensional data. Besides a large number of collected variables, structural information is often available in the data collection process. Matrix-variate variable is an important way of organizing high dimensional data to incorporate such structural information, and is commonly encountered in multiple applied areas such as brain imaging studies, ﬁnancial market trading, macro-economics analysis and many others. Consider the following concrete examples: • To meet the demands of investors, options contingent on equities and market indices are frequently traded with multiple combinations of striking prices and expiration dates. A dataset collects weekly implied volatilities, which are equivalent to the prices, of options for 89 component equities in the Standard and Poor 100 index for 8 respective expiration dates, such as 30, 60 days. In this example, each observation of the dataset can be denoted by a 89 × 8 matrix, whose rows are the companies and whose columns encode information of the expiration dates. A snapshot of the dataset, after processing as discussed in Section 5.3, for three selected companies Abbott Laboratories, ConocoPhillips and Microsoft is presented in Figure 1. • US Department of Agriculture reports itemized annual export to major trading partners. A dataset with 40 years US export is collected for 13 trading partners and 36 items. Each observation in the dataset can be denoted by a 13 × 36 matrix where the trading partners and items, as the rows and columns of this matrix, are used as structural information for the observations. • In brain imaging studies, it is routine to apply electroencephalography (EEG) on an individual in an attempt to understand the relationship between brain imaging and the trigger of some events, for example, alcohol consumption. A typical experiment scans each subject from a large number of channels of electrodes at hundreds of time points. Therefore the observation of each subject is conveniently denoted as a large channel by time matrix. Abbott Laboratories (ABT) ConocoPhillips (COP) Microsoft (MSFT) 152d 182d 273d 365d 30d 60d 91d 122d Figure 1: Volatility data for 142 weeks of trading. We refer to this type of structured data X ∈ Rp×q as matrix-variate data or simply matrix data if no confusion arises. It may appear attempting to stack X as a column vector vec(X) and model X as a p × q dimensional vector. Gaussian graphical models (Lauritzen, 1996), when applied to vector data, are useful for representing conditional independence structure among the variables. A graphical model in this case consists of a vertex set and an edge set. Absence of an edge between two vertices denotes that the corresponding pair 3 of variables are conditionally independent given all the other variables. To build a sparse graphical model for vec(X), there exists abundant literature that makes use of penalized likelihood (Meishausen and B¨ uhlmann, 2006; Yuan and Lin, 2006; Rothman, et al., 2008; Banejee et al., 2008; Lam and Fan, 2009; Fan et al., 2009; Peng et al., 2009; Guo et al., 2011; Guo et al., 2010). However, these approaches suﬀer from at least two obvious shortcomings when applied to matrix data. First, the need to estimate a p2 × q 2 -dimensional covariance (or precision) matrix can be a daunting task due to the extremely high dimensionality. Second, any analysis based on vec(X) eﬀectively ignores all row and column structural information, an incoherent part of the data characteristics. In practice, this structural information is useful and sometimes vital for interpretation purposes, and, as discussed later, for convergent rate considerations. New approaches that explore the matrix nature of such data sets are therefore called for to meet the emerging challenges in analyzing such data. As an extension of the familiar multivariate Gaussian distribution for vector data, we consider the matrix variate normal distribution for X with probability density function p(X\|M, Σ, Ψ) = (2π)−qp/2 \|Σ−1 \|q/2 \|Ψ−1 \|p/2 etr{−(X − M)Ψ−1 (X − M)T Σ−1 /2}, where M ∈ Rp×q is the mean matrix; Σ ∈ Rp×p and Ψ ∈ Rq×q are the row and column variance matrices; etr is the exponential of the trace operator. The matrix normal distribution (1) implies that vec(X) follows a vector multivariate normal distribution Npq (vec(M), Ψ ⊗ Σ), where ⊗ is the Kronecker product. Models using matrix instead of vector normal distributions eﬀectively reduce the dimensionality of the covariance or precision matrix from the order of p2 × q 2 to an order of p2 + q 2 . Without loss of generality, we assume M = 0 from now on. Otherwise, we can always center the data by subtracting ˆ = ∑n Xi /n from Xi , where Xi , i = 1, ..., n are assumed to be independent and identiM i=1 cally distributed following the matrix normal distribution (1). Dawid (1981) provided some theory for matrix-variate distributions and Dutilleul (1999) derived the maximum likelihood estimation. Allen and Tibshirani (2010) studied regularized estimation in a model 4 similar to (1) with a single observation when n = 1, but did not provide any theoretical result. Wang and West (2009) studied Bayesian inference for such models. Matrix-variate Gaussian distributions are useful for characterizing conditional independence in the underlying matrix variables; see Dawid (1981) and Gupta and Nagar (2000). Let Ω = Σ−1 = (ωij ) and Γ = Ψ−1 = (γij ) be the precision matrices for row and column vectors respectively. Similar to that in a vector-variate graphical model, all the conditional independence can be analogously read oﬀ from the precision matrices. In particular, zeros in Γ ⊗ Ω deﬁne pairwise conditional independence of corresponding entries (Lauritzen, 1996), given all the other variables. More importantly, zeros in Ω and Γ encode conditional independence of the row variables and the column variables in X respectively, often creating a much simpler graphical model for understanding matrix data. Formally, in matrix normal distribution, two arbitrary entries Xij and Xkl are conditionally independent given remaining entries if and only if: (1) at least one zero in ωij and γkl when i ̸= k, j ̸= l; (2) ωik = 0 when i ̸= k, j = l; (3) γjl = 0 when i = k, j ̸= l. In terms of the partial correlation between variables Xij and Xkl , deﬁned as ρij,kl = − √ ωik γjl ·√ , ωii ωkk γjj γll ρij,kl is not zero only if both ωik and γjl are nonzero. If either the ith and the kth rows are conditionally independent give other rows, or the jth and the lth columns are conditionally independent given other columns, or both, the partial correlation between variables Xij and Xkl would be zero. An obvious issue with the parametrization of the matrix normal distribution is its identiﬁability. In later development, we ﬁx ω11 = 1. In this paper, we propose sparse matrix graphical models by considering regularized maximum likelihood estimate for matrix data that follow the distribution in (1). By applying appropriate penalty functions on precision matrices Ω and Γ, we obtain sparse matrix graphical models for the row and the column variables. Theoretical results show that the structural information in the rows and columns of the matrix variable can be exploited to yield more eﬃcient estimates of these two precision matrices than those in the vector 5 graphical model. Given appropriate penalty functions, we show that our method gives sparsistent graphical models. Namely, we are able to identify zeros in these two precision matrices correctly with probability tending to one. We demonstrate the performance of the proposed approach in extensive simulation studies. Through several real data analysis, we illustrate the attractiveness of the proposed approach in disentangling structures of complicated high dimensional data as well as improving interpretability. The rest of the article is organized as follows. We present the proposed methodology in Section 2, followed by a discussion on an iterative algorithm for ﬁtting the model. The asymptotic properties of the proposed method are provided in Section 3. Simulations are presented in Section 4, and data analyses are given in Section 5. Section 6 summarizes the main ﬁndings and outlines future research. All proofs are found in the Appendix. To estimate Ω and Γ given iid data X1 , ..., Xn from (1), the standard approach is the maximum likelihood estimation. It is easily seen that the minus log-likelihood up to a constant is nq np 1∑ ℓ(Ω, Γ) = − log \|Ω\| − log \|Γ\| + tr(Xi ΓXTi Ω). 2 2 2 i=1 n The solution minimizing (2) is the familiar maximum likelihood estimate. In order to build sparse models for Ω and Γ, we propose the penalized likelihood estimator as the minimizer of n 1 ∑ 1 1 g(Ω, Γ) = (3) tr(Xi ΓXTi Ω) − log \|Ω\| − log \|Γ\| + pλ1 (Ω) + pλ2 (Γ), npq i=1 p q ∑ where pλ (A) = i̸=j pλ (\|aij \|) for a square matrix A = (aij ) with a suitably deﬁned penalty function pλ . In this paper, we use the LASSO penalty (Tibshirani, 1996) deﬁned as pλ (s) = λ\|s\| or the SCAD penalty (Fan and Li, 2001), whose ﬁrst derivative is given by p′λ (s) = λ{I(s ≤ λ) + (3.7λ − s)+ I(s > λ)} 2.7λ for s > 0. Here (s)+ = s if s > 0 and is zero otherwise. The SCAD penalty is introduced to overcome the induced bias when estimating those nonzero parameters (Fan and Li, 2001). Loosely speaking, by imposing a small or zero penalty on a nonzero estimate, the SCAD eﬀectively produces unbiased estimates of the corresponding entries. In contrast, the LASSO, imposing a constant penalty on all estimates, inevitably introduces biases that may aﬀect the rate of convergence and consistency in terms of model selection (Lam and Fan, 2009). We shall call collectively the resulting model sparse matrix graphical models (SMGM). When either Σ or Ψ is an identity matrix, the distribution in (1) is equivalent to a q- or p-dimensional multivariate normal distribution. This is seen by observing that the rows (columns) of X are independently normal distributed with covariance matrix Ψ (Σ) when Σ = Ip (Ψ = Iq ). Therefore SMGM includes the multivariate Gaussian graphical model as a special case. The matrix normal distribution is a natural way to encode the structural information in the row and the column variables. The SMGM provides a principled framework for studying such information content when conditional independence is of particular interest. In high dimensional data analysis, imposing sparsity constraints can often stabilize estimation and improve prediction accuracy. More importantly, it is an eﬀective mechanism to overcome the curse of dimensionality for such data analysis. We now discuss how to implement SMGM. Note that the penalized log-likelihood is not a convex function but it is conditional convex if the penalty function is convex, for example in the LASSO case. This naturally suggests an iterative algorithm which optimizes one matrix with the other matrix ﬁxed. When Ω is ﬁxed, we minimize with respect to Γ 1 ˜ − 1 log \|Γ\| + pλ2 (Γ), tr(ΓΣ) q q ˜ = ∑n XT ΩXi /q. When the LASSO penalty is used, we use the coordinate where Σ i i=1 descent graphical LASSO (gLASSO) algorithm in Friedman et al. (2008) to obtain the solution. When used with warm start for diﬀerent tuning parameters λ2 , it can solve large problems very eﬃciently even when q ≫ n. For the SCAD penalty, we follow Zou and Li 7 (2008) to linearized the penalty as pλ2 (s) = pλ2 (s0 ) + p′λ2 (s0 )(\|s\| − \|s0 \|) for the current estimate s0 . Thus, we only need to minimize ∑ 1 ˜ − 1 log \|Γ\| + tr(ΓΣ) p′λ2 (\|γij,k \|)\|γij \| q q i̸=j which can be solved by using the graphical LASSO algorithm. Here γij,k is the kth step estimate of γij . In practice, it is noted that in a diﬀerent context, one iteration of this linearization procedure is suﬃcient given a good initial value (Zou and Li, 2008). In our implementation, we take the initial value as the graphical LASSO estimates by taking p′λ2 (\|γij,k \|) as λ2 . On the other hand, when Γ is ﬁxed, we minimize with respect to Ω 1 ˜ 1 tr(ΨΩ) − log \|Ω\| + pλ1 (Ω), p p ˜ = ∑n Xi ΓXT /p. where Ψ i i=1 Synthetically, the computational algorithm is summarized as follows. (0) 1. Start with Γ(0) = Iq , and minimize (4) to get Ω(0) . Normalize Ω(0) such that ω11 = 1. Let m = 1; 2. Fix Ω(m−1) and minimize (5) to get Γ(m) ; (m) 3. Fix Γ(m) and minimize (4) to get Ω(m) . Normalize such that ω11 = 1. Let m ← m+1; 4. Repeat Step 2 and 3 until convergence. Although there is no guarantee that the algorithm converges to the global minimum, the algorithm converges to a local stationary point of g(Ω, Γ). An argument has been outlined in Allen and Tibshirani (2010) when the lasso penalty is used in our approach. A detailed discussion of various algorithms and their convergence properties can be found in Gorski et al. (2007). When there is no penalty, Dutilleul (1999) showed that each step of the iterative algorithm is well deﬁned if n ≥ max{p/q, q/p} + 1. This is understandable since for estimating 8 for example Ω, the eﬀective sample size is essentially nq. More recently, Srivastava et al. (2008) showed that the MLE exists and is unique provided n > max{p, q}. Thus, if the sample size is larger than the row and the column dimension, the algorithm guarantees to ﬁnd the global optimal solution, if a convex penalty such as lasso or ridge is used in our formulation. Since the optimal tuning parameters λ1 and λ2 are not known in advance, a useful practice is to apply a warm start method on decreasing sequences of values for these two parameters. For example, if we set these two parameters to be the same as λ1 = λ2 = ρ, we compute a sequence of solutions of (3) for ρL > ρL−1 > ... > ρ1 , where ρL is chosen to yield very sparse models. The solution at ρi is then used as the initial value for the solution at ρi−1 (Friedman et al., 2008). Note that if ρL is large, the resulting estimated Ω and Γ would be very sparse. Thus, the algorithm is not very sensitive to the initial value for the solution at ρL because in this case, a much fewer number of parameters in the two precision matrices need to be estimated. Subsequently, for other values of ρ, the initial values would be very close, yielding the resulting estimates very close. Thus, the sensitivity of the algorithm is greatly reduced. Intuitively, the warm-start trick would work well for very sparse models. Further numerical evidence of this observation will be presented in the simulation studies. We now study the asymptotic properties of the proposed method in this section. Let Ω0 and Γ0 be the true parameter of the underlying model, S1 = {(i, j) : ω0ij ̸= 0} and S2 = {(i, j) : γ0ij ̸= 0}. Deﬁne s1 = \|S1 \| − p and s2 = \|S2 \| − q as the number of nonzero oﬀdiagonal parameters in Ω0 and Γ0 respectively. We make the following standard regularity assumptions for theoretical analysis of the penalized likelihood . A1. There exist constant τ1 such that for all n, 0 < τ1 < λ1 (Σ0 ) ≤ λp (Σ0 ) < 1/τ1 < ∞, 0 < τ1 < λ1 (Ψ0 ) ≤ λq (Ψ0 ) < 1/τ1 < ∞, where λ1 (A) ≤ λ2 (A) ≤ · · · ≤ λm (A) denote the eigenvalues of an m-dimensional symmetric matrix A. A2. The penalty function pλ (·) is singular at the origin, and lim pλ (t)/(λt) = k > 0. t↓0 A3. For the nonzero components in Ω0 and Γ0 , max p′λ1 (\|ω0ij \|) = O[p−1 {1 + p/(s1 + 1)} max p′λ2 (\|γ0ij \|) = O[q −1 {1 + q/(s2 + 1)} log p/(nq)], max p′′λ1 (\|ω0ij \|) = o(p−1 ), (i,j)∈S1 log q/(np)], max p′′λ2 (\|γ0ij \|) = o(q −1 ), (i,j)∈S2 min \|ω0ij \|/λ1 → ∞ and min \|γ0ij \|/λ2 → ∞ as n → ∞. −2 −2 −2 A4. The tuning parameters satisfy λ−2 1 p (p+s1 ) log p/(nq) → 0, λ2 q (q+s2 ) log q/(np) → 0 as n → ∞. All conditions imposed here are mild and comparable to those in Lam and Fan (2009). Both the LASSO and the SCAD penalty satisfy Condition A2, while Condition A3 is less restrictive for the unbiased SCAD penalty than for the LASSO. The relations among the penalty functions, Condition A3 and the properties of the resulting estimates are discussed in more detail later. We also note that in Condition A4, due to the information from the matrix variate structure, the tuning parameters λ1 and λ2 are allowed to converge to 0 at faster rates than those in Lam and Fan (2009). Thus smaller bias is induced due to this penalization. We use the notations ∥ · ∥F and ∥ · ∥ for the Frobenius and operator norms of a matrix. The following theorem establishes the rate of convergence of SMGM. Theorem 1. [Rate of convergence] Under conditions A1-A4, as n → ∞, (p+s1 ) log p/(nq) → 0 and (q + s2 ) log q/(np) → 0, there exists a local minimizer of (3) such that ˆ − Γ0 ∥2 ˆ − Ω 0 ∥2 s1 ∥Γ s2 ∥Ω F F = Op {(1 + ) log p/(nq)} and = Op {(1 + ) log q/(np)}. p p q q 10 Theorem 1 shows that the rate of convergence is determined by n and p for Ω and by n and q for Γ. This represents the fact that the column (row) information in the matrixvariate data is incorporated in estimating the concentration matrix of the row (column), which indeed conﬁrms the merit of the proposed SMGM. This result is a generalization of the results for which independent vectors are used in estimating concentration matrices. We show in Appendix that Theorem 1 is also valid for the LASSO penalty if Condition √ √ of A4 is changed to λ1 = O(p−1 log p/(nq)) and λ2 = O(q −1 log q/(np)). We comment that the desirable local minimizer may be diﬃcult to identify in practice. If there exist multiple local minimizers, the one computed by the algorithm may not be the one having the asymptotic property, and there does not seem to exist an algorithm that can always give such a local minimizer. As to the implied graphical model for vec(X) by SMGM, the rate of convergence for estimating Ω0 ⊗ Γ0 using our approach is easily seen as ˆ ⊗Γ ˆ − Ω0 ⊗ Γ0 ∥2 ∥Ω s1 s2 F = Op [max{(1 + ) log p/(nq 2 ), (1 + ) log q/(np2 )}]. pq p q If we apply the SCAD penalty to the vectorized observations, Lam and Fan (2009) gave the following rate of convergence ˆ ⊗Γ ˆ − Ω0 ⊗ Γ0 ∥2 ∥Ω s1 s2 F = Op [(1 + ) log(pq)/n]. pq pq We immediately see that when p → ∞ and q → ∞, SMGM gives estimates with much faster rate of convergence. Indeed, the rate of convergence of our model is strictly faster as long as the dimensionality grows with the sample size. The improvement is due to our utilizing a more parsimonious model that naturally incorporates the data structure. In addition, we also note that the conditions in Theorem 1 in-explicitly put constraints on the growth rates of p and q. More speciﬁcally, max(p log p/q, q log q/p)/n → 0 and max(s1 log p/q, s2 log q/p)/n → 0 are required. If we without loss of generality consider p > q, then p log p/q = o(n), s1 log p/q = o(n), and s2 log q/p = o(n) eﬀectively determine the upper bound of p. 11 We establish the model selection consistency, also known as sparsistency, of the proposed approach in the following theorem. ˆ and Γ ˆ satTheorem 2. [Sparsistency] Under conditions A1-A4, for local minimizers Ω ˆ − Ω0 ∥2 = Op {(p + s1 ) log p/(nq)}, ∥Γ ˆ − Γ0 ∥2 = Op {(q + s2 ) log q/(np)}, isfying ∥Ω F F ˆ − Ω0 ∥ = Op (η1n ), ∥Γ ˆ − Γ0 ∥ = Op (η2n ) for sequences η1n and η2n converging to 0, ∥Ω log p/(nq) + η1n + η2n = Op (λ21 p2 ) and log q/(np) + η1n + η2n = Op (λ22 q 2 ), with probability tending to 1, we have ω ˆ ij = 0 and γˆij = 0 for all (i, j) ∈ S1c and (i, j) ∈ S2c . If the LASSO penalty is used, the consistency result in Theorem 1 requires controlling the incurred bias by Condition A3 so that the tuning parameters can not be too large. This eﬀectively imposes upper bounds for the tuning parameters in the LASSO penalty as √ √ λ1 = O[p−1 {1 + p/(s1 + 1)} log p/(nq)] and λ2 = O[q −1 {1 + q/(s2 + 1)} log q/(np)]. On the other hand, larger penalization is needed to achieve sparsistency as seen from Theorem 2. Therefore to simultaneously achieve consistency and sparsistency when applying the LASSO penalty, the numbers of nonzero components in Ω0 and Γ0 are restricted, even under optimal conditions. Similar to the discussions in Lam and Fan (2009), s1 and s2 can be at most O(p) and O(q) respectively to ensure the consistency and sparsistency. While for those unbiased penalty functions such as the SCAD, s1 and s2 could be allowed to grow at O(p2 ) and O(q 2 ) in the SMGM, beneﬁted from the fact that Condition A3 does not impose upper bounds on tuning parameters for an unbiased penalty function. It is remarkable that the restriction on the sample size n is largely relaxed due to incorporating the structural information from the multiple rows and columns. Even if n < p and n < q, consistent estimates of Ω and Γ are still achievable. When p = q, a suﬃcient condition for convergence of SMGM with the SCAD penalty is log(pq) = o(n) if the matrices are sparse enough. On the other hand, for the vector graphical model, we require at least (pq) log(pq) = o(n). In the extreme case when n is ﬁnite, it is still possible to obtain consistent estimates of the precision matrices by applying SMGM. To appreciate this, consider for example that n = 1, p is ﬁxed and q is growing. If Ψ0 is a correlation matrix, one can apply SMGM with Γ = I and obtain consistent and sparse estimate of Ω following the proof in Appendix. We conclude that by incorporating the structure information from matrix data, more eﬃcient estimates of the graphic models can be obtained. We now discuss the asymptotic properties of the estimates. Let A⊗2 = A ⊗ A and Kpq be an pq × pq commutation matrix that transforms vec(A) to vec(AT ) for a p × q matrix A. A rigorous deﬁnition and its properties can be found for example in Gupta and ′′ ˆ ˆ b1n = pλ1 (\|Ω0 \|)sgn(Ω0 ), and Λ2n = diag{pλ2 (Ω)}, Nagar (2000). Let Λ1n = diag{pλ1 (Ω)}, ′ b2n = pλ1 (\|Ω0 \|)sgn(Ω0 ). Denote S1 as the set of all the indices of nonzero components in vec(Ω0 ) except ω11 , S2 as the set for all the indices of nonzero components in vec(Γ0 ). We use the subscript S and S ×S for the corresponding sub-vector and sub-matrix respectively. Theorem 3. [Asymptotic normality] Under Conditions A1-A4, (p + s1 )2 /(nq) → 0 and ˆ and Γ ˆ in Theorem 1, we have (q + s2 )2 /(np) → 0 as n → ∞, for the local minimizer Ω √ d −1/2 ⊗2 ˆ nqαTp {Σ⊗2 0 (I + Kpp )}S1 ×S1 (Λ1n + Σ0 )S1 ×S1 {vec(Ω) − vec(Ω0 ) + b1n }S1 → N (0, 1), √ d −1/2 ⊗2 ˆ npαTq {Ψ⊗2 0 (I + Kqq )}S2 ×S2 (Λ2n + Ψ0 )S2 ×S2 {vec(Γ) − vec(Γ0 ) + b2n }S2 → N (0, 1), d where αd denotes a d-dimensional unit vector and → denotes convergence in distribution. Theorem 3 clearly illustrates the impact of using the matrix variate data. In comparison √ √ with those in Lam and Fan (2009), fast rates of convergence nq and np are achieved for estimating the precision matrices of the columns and rows respectively. Similar to that in Theorem 1, a caution is that the local minimizer may be diﬀerent from the one computed by the algorithm. It is of great interest to develop an algorithm that guarantees identifying the local minimizer in Theorem 1 and 3. Simulation and Data Analysis We conduct extensive simulation studies in this section. For comparison purposes, we tabulate the performance of the maximum likelihood estimate (MLE) of Ω and Γ using 13 the algorithm in Dutilleul (1999). Note that this algorithm is similar to the algorithm for computing the SMGM estimate, when the penalty function is absent. We also compare SMGM with the graphical LASSO method for estimating Ω0 ⊗ Γ0 (Friedman, et al. 2008). In particular, this approach ignores the matrix structure of the observations and vectorizes them as x = vec(X). The graphical LASSO method then optimizes tr(ΘS) − log \|Θ\| + λ \|θij \|, where S is the sample covariance matrix of xi . In addition, we implement a ridge type regularized method by using squared matrix Frobenius norm ∥A∥2F in the penalty function in (3). We do not report the sample covariance estimates because for most of the simulations reported here, the sample size is too small comparing to p × q. For each simulation setup, we conduct 50 replications. To choose the tuning parameters, we generate a random test dataset with the sample size equal to the training data. We use the following d × d matrices as the building block for generating sparse precision matrices for Ω and Γ (Rothman et al., 2008). 1. A1 : Inverse AR(1) such that A1 = B−1 with bij = 0.7\|i−j\| . 2. A2 : AR(4) with aij = I(\|i−j\| = 0)+0.4I(\|i−j\| = 1)+0.2I(\|i−j\| = 2)+0.2I(\|i−j\| = 3) + 0.1I(\|i − j\| = 4). 3. A3 = B + δI: each oﬀ-diagonal upper triangle entry in B is generated independently and equals to 0.5 with probability a = 0.1 and 0 with probability 1 − a = 0.9. The diagonals of B are zero and δ is chosen such that the condition number of A3 is d. All matrices are sparse and are numbered in order of decreasing sparsity. We assess the estimation accuracy for a precision matrix A ∈ Rd×d using the Kullback-Leibler loss, deﬁned as ˆ = tr(A−1 A) ˆ − log \|A−1 A\| ˆ − d. l(A, A) Here we use A = Ω0 ⊗ Γ0 , the main parameter of interest in the Kullback-Leibler loss, and ˆ is an estimate of it. We also summarize the performance in terms of true positive rate A 14 (TPR) and true negative rate (TNR), deﬁned as TPR = #{Aˆij ̸= 0 & Aij ̸= 0} #{Aˆij = 0 & Aij = 0} , TNR = . #{Aij ̸= 0} #{Aij = 0} We ﬁrst generate random datasets with sample sizes n = 10 or n = 100, with p = q = 10 or 20. For these relatively small dimensional datasets, we compare the MLE, the ridge estimator (Ridge), our proposed method using LASSO penalty (LASSO), our proposed method with the SCAD penalty (SCAD), as well as the graphical LASSO vectorizing the matrix data (gLASSO). The results based on Kullback-Leibler loss are summarized in Table 1, with the model selection results summarized in Table 2. From Table 1, we see clearly that the gLASSO method by ignoring the matrix structure performs much worse than all the other approaches by a large margin. The MLE gives similar accuracy as the ridge estimator with n = 100, but performs much worse with n = 10. This illustrates the beneﬁt of regularization. Among the three regularized estimates, denoted as Ridge, LASSO and SCAD, the two sparse estimates LASSO and SCAD prevail in general, especially when p = q = 20. The SCAD consistently outperforms the LASSO in terms of the KullbackLeibler loss, especially so when n becomes large (n = 100). In terms of model selection, Table 2 reveals that the SMGM with either LASSO or the SCAD penalty outperforms the gLASSO estimates drastically. Note “1” in the table means that model selection is 100% correct while “1.0” means that the rate is 100% after rounding. The TPR and TNR of the SCAD method improves those of the LASSO especially when n is large. This diﬀerence is due to the bias issue inherent in the LASSO type of penalizations (Fan and Li, 2001). We now investigate the situation when p and q are large. In order to study the case when p ̸= q, we let (p, q) = (100, 20) and (p, q) = (100, 50) with n = 20 or 100. Since for this setting, the graphical LASSO takes much longer time to compute, we report only the results for all the other approaches. The results are summarized in Table 3 for estimation. Again, the penalized estimates usually outperform the MLE. The two sparse estimates using either LASSO and SCAD penalty are the two best approaches in terms of estimation accuracy. In addition, the SCAD perform much better than the LASSO penalty using the 15 n 10 p 10 q 10 Ω A1 A2 A2 A3 A3 A3 A1 A2 A2 A3 A3 A3 A1 A2 A2 A3 A3 A3 A1 A2 A2 A3 A3 A3 Γ A1 A1 A2 A1 A2 A3 A1 A1 A2 A1 A2 A3 A1 A1 A2 A1 A2 A3 A1 A1 A3 A1 A2 A3 MLE 22.6(5.0) 22.9(4.2) 23.9(4.8) 22.5(3.9) 22.5(4.7) 23.7(4.5) 1.2(0.2) 1.2(0.2) 1.2(0.1) 1.2(0.2) 1.2(0.2) 1.2(0.1) 89.6(7.9) 87.6(8.1) 88.6(7.5) 88.0(8.0) 90.3(9.7) 88.5(9) 4.5(0.3) 4.6(0.4) 4.4 (0.3) 4.6(0.3) 4.5(0.3) 4.6(0.4) Ridge 11.8(2.3) 11.2(1.7) 10.4(1.4) 10.7(1.7) 10.0(1.5) 9.7(1.3) 1.4(0.2) 1.2(0.1) 1.1(0.1) 1.1(0.2) 1.1(0.1) 1.1(0.1) 45.6(3.5) 42.8(2.8) 40.3(2.5) 42.1(3.2) 39.0(2.8) 37.2(2.9) 5.5(0.3) 4.5(0.3) 4.2(0.2) 4.6(0.3) 4.2(0.3) 4.2(0.3) LASSO 12.5(4.1) 11.7(2.3) 15.8(3.1) 8.1(2.0) 11.2(2.2) 7.6(1.8) 0.71(0.12) 1.0(0.1) 1.1(0.1) 0.68(0.11) 1.0(0.1) 0.66( 0.11) 28.9(5.8) 33.5(3.1) 50.2(4.1) 21.4(3.3) 42.0(5.9) 20.8(3.3) 1.9(0.2) 3.1 (0.3) 4.0(0.3) 2.1(0.2) 3.3(0.3) 1.5(0.2) SCAD 6.9(2.4) 11.3(2.2) 17.0(3.5) 7.4(2.0) 10.6(2.2) 7.4(2.0) 0.38(0.09) 0.8(0.2) 1.1(0.1) 0.38(0.09) 0.82( 0.13) 0.35(0.08) 13.9(2.6) 33.2(3.4) 50.6(4.3) 16.9(2.9) 37.0(5.8) 17.6(3.2) 0.9(0.2) 2.0 (0.3) 2.8(0.3) 0.81(0.13) 2.0(0.2) 0.77(0.13) gLASSO 105.4(9.9) 77.8(5.6) 44.4(3.7) 84.7(6.4) 54.6(4.4) 57.4(5.2) 30.7(0.8) 17.2(0.5) 15.9(0.5) 18.0(0.5) 13.9(0.5) 8.4(0.4) 509(19) 356(12) 185(8) 410(17) 228(10) 273(11) 169(1) 94.2(0.9) 83.6( 1.3) 110(1.3) 72.1(1.2) 67.1(1.2) Table 1: Simulation results on the Kullback-Leibler loss. The sample standard errors are in parentheses. A1 A2 A2 A3 A3 A3 A1 A2 A2 A3 A3 A3 A1 A2 A2 A3 A3 A3 A1 A2 A2 A3 A3 A3 A1 A1 A2 A1 A2 A3 A1 A1 A2 A1 A2 A3 A1 A1 A2 A1 A2 A3 A1 A1 A2 A1 A2 A3 LASSO TPR TNR 1(0) 0.82(0.03) 0.70(0.11) 0.76(0.07) 0.53(0.16) 0.69(0.16) 1(0) 0.78(0.05) 0.75(0.11) 0.77(0.10) 1(0) 0.88(0.04) 1(0) 0.66(0.08) 1.0(0.01) 0.38(0.11) 1.0(0.00) 0.16(0.06) 1(0) 0.77 (0.08) 1(0) 0.38(0.09) 1(0) 0.78(0.07) 1(0) 0.93(0.01) 0.72(0.07) 0.86(0.03) 0.51(0.11) 0.79(0.07) 1(0) 0.91(0.02) 0.90(0.06) 0.80(0.04) 1(0) 0.92(0.02) 1(0) 0.83(0.02) 1.00(0.00) 0.60(0.04) 1(0) 0.36(0.03) 1(0) 0.78(0.03) 1.00(0.00) 0.49(0.05) 1(0) 0.90(0.03) SCAD TPR TNR 1(0) 0.95(0.06) 0.70(0.10) 0.80(0.11) 0.43(0.16) 0.79(0.16) 1.0(0.01) 0.90(0.07) 0.72(0.11) 0.84(0.14) 1.0(0.01) 0.92(0.05) 1(0) 0.99(0.02) 1.0(0.01) 0.74(0.07) 1.0(0.01) 0.46(0.11) 1(0) 0.99(0.01) 1(0) 0.70(0.07) 1(0) 0.95(0.01) 1(0) 0.99(0.01) 0.67(0.11) 0.91(0.05) 0.47(0.11) 0.84(0.06) 1(0) 0.97(0.02) 0.87(0.06) 0.90(0.03) 1(0) 0.96(0.02) 1(0) 1.00(0.00) 1(0) 0.89(0.02) 1(0) 0.71 (0.03) 1(0) 1.00(0.00) 1.00(0.00) 0.84(0.02) 1(0) 1.00(0.00) gLASSO TPR TNR 0.50(0.05) 0.92(0.01) 0.18(0.02) 0.93(0.01) 0.05(0.02) 0.98(0.02) 0.44(0.04) 0.93(0.01) 0.14(0.02) 0.97(0.01) 0.52(0.04) 0.96(0.01) 0.81(0.05) 0.66(0.03) 0.35(0.01) 0.83(0.01) 0.23 (0.01) 0.87(0.01) 0.75(0.02) 0.81(0.02) 0.37(0.01) 0.87(0.01) 0.93(0.01) 0.91(0.01) 0.38(0.03) 0.97(0.00) 0.10(0.01) 0.98(0.00) 0.02(0.00) 1.0(0.00) 0.31(0.02) 0.98(0.00) 0.07(0.01) 0.99(0.00) 0.26 (0.02) 0.99(0.00) 0.58(0.00) 0.93(0.00) 0.25(0.01) 0.94(0.00) 0.12(0.01) 0.95(0.00) 0.59(0.00) 0.93(0.00) 0.26(0.01) 0.96(0.00) 0.63(0.01) 0.95(0.00) Table 2: Simulation results on model selection. The sample standard errors are in parentheses. matrix normal distribution. This is true also for model selection (results not shown). Even for a relatively small sample size (n = 20), the SMGM method with the SCAD penalty gives the true positive rate and the true negative rate very close to one. Comparing to the results in Table 2 with a comparable sample size, it seems that we are blessed with high dimensionality. For example, when (p, q) = (100, 50) for n = 100, the model selection results are consistently better than those in Table 2 when (p, q) = (20, 20). n 20 p 100 q 20 Ω A1 A2 A2 A3 A3 A3 A1 A2 A2 A3 A3 A3 A1 A2 A2 A3 A3 A3 A1 A2 A2 A3 A3 A3 Γ A1 A1 A2 A1 A2 A3 A1 A1 A2 A1 A2 A3 A1 A1 A2 A1 A2 A3 A1 A1 A2 A1 A2 A3 MLE 504(15) 504(14) 502(16) 504(13) 506(15) 507(16) 59.6(1.4) 59.1(1.4) 59.5(1.4) 59.0(1.1) 59.4(1.3) 59.7(1.3) 471(11) 471(11) 472(12) 473(11) 473(10) 472(10) 68.5(1.3) 68.4(1.3) 68.4(1.1) 68.3(1.3) 68.1(1.3) 68.4(1.4) Ridge 273(7) 268(6) 245(4) 258(6) 239(5) 222(5) 54.3(1.1) 52.1(1.1) 52.3(1.1) 53.0(0.8) 51.4(0.9) 50.7(0.9) 325(6) 323(6) 318(6) 314(6) 310(6) 292(5) 79.4(1.2) 65.9(1.2) 63.7(1.1) 76.9(1.1) 64.4(1.2) LASSO 67.7(6.1) 131(5) 183(11) 133(10) 150(9) 117(6) 12.3(0.7) 28.7(0.8) 32.9(0.9) 24.7(0.8) 30.2(1.1) 19.9(0.8) 68.5(3.5) 160(4) 216(6) 126(6) 176(8) 165(17) 17.3(0.8) 41.8(1.1) 52.5(1.0) 45.6(1.3) 66.2(2.0) 49.2(1.1) SCAD 17.1(2.0) 114.6(3.7) 158(12) 100(6) 118(9) 91(7) 2.60(0.27) 14.2(0.8) 15.3(0.8) 8.06(0.47) 11.7(0.7) 7.2(0.4) 23.9(2.1) 123(5) 148(6) 68.4(4.3) 97(5) 69.2(3.7) 3.41(0.28) 10.9(0.5) 13.3(0.5) 8.21(0.32) 11.3(0.7) 8.21(0.48) Table 3: Simulation results on the Kullback-Leibler loss. The sample standard errors are in parentheses. Based on these simulation results, we see clearly that, by exploiting the covariance matrix of vec(X) as a Kronecker product of two matrices, the proposed method outperforms the graphical LASSO in general, regardless of whether any penalty is used. When the two precision matrices are sparse, the simulation results demonstrate clearly that by imposing appropriate penalties, we can estimate the covariance matrices more accurately with excellent model selection results. In addition, as expected from the asymptotic results, the SCAD penalty gives better performance in terms of Kullback-Leibler loss and model selection than the LASSO. In order to investigate the sensitivity of the algorithm to the initial values, we conduct further numerical simulations. We take Ω = Γ = A3 with n = 10 and p = q = 20. We then take three possible initial values. The ﬁrst is the maximum likelihood estimate computed using our algorithm with identity matrices as initial values when there is no penalty. The second uses identity matrices as initial values. The third uses the true matrices as the initial values. We implement the warm start approach and compare the ﬁnal estimated precision matrices with the tuning parameters chosen as outlined before. The three estimates are denoted as Bj , j = 1, 2, 3 for estimating Ω ⊗ Γ with these three initial values. We then compute the Frobenius norm of D1 = B1 −B3 and D2 = B2 −B3 . We ﬁnd that these norms are eﬀectively all zero across 1000 simulations, suggesting that this algorithm is robust with respect to the initial values. If we change the sparsity parameter a for generating A3 to 0.5, we ﬁnd that the Frobenius norms of 999 D1 ’s are eﬀectively zero, while those of 1000 D2 ’s are zero. The ﬁnal estimates seem to depend on the initial values when the less sparse matrices Ω and Γ are used. However, the eﬀect seems to be minimal, although in this case, convergence cannot be guaranteed due to the need to estimate a large number of parameters. 5 5.1 Data Analysis US agricultural export data The US agricultural export is an important part of US export. In March 2011 alone, the US agricultural export exceeds 13.3 billion US dollars and gives an agricultural trade surplus of about 4.5 billion US dollars. Understanding the export pattern can be useful to understand the current and predict future export trends. We extract the annual US agricultural export data between 1970 and 2009 from the United States Department of Agriculture website. We look at the annual US export data to thirteen regions including North America, Caribbean, Central America, South America, European Union-27, Other Europe, East Asia, Middle East, North Africa, Sub-Saharan Africa, South Asia, Southeast Asia and Oceania. The 36 export items are broadly categorized as bulk items, including for example wheat, rice and so on, intermediate items, consisting of for example wheat ﬂour, soybean oil, and etc., and consumer oriented items, including snack food, breakfast cereals and so on. These product groups are adopted from Foreign Agricultural Service BICO HS-10 codes. Thus, the data set comprises of 40 matrices with dimensionality 14 × 36. A selected 40 years data plot for North America, East Asia and Southeast Asia for these 36 items can be found in Figure 2. 1990 Year 16 14 Log Thousand Log Thousand Log Thousand Southeast Asia East Asia North America 1990 Year Figure 2: USDA export data for three regions over 40 years for 36 items: Each connected line represents one item. The original data are denoted in thousands US dollars. After imputing the one missing value with zero, we take logarithm of the original data plus one. To reduce the potential 20 serial correlations in this multivariate time series data set, we ﬁrst take the lag one diﬀerence for each region item combination, and apply our method to the diﬀerences as if they were independent for further analysis. Box-Pierce tests of lag ten for these 36 × 13 time series after the diﬀerencing operation suggest that most of them can pass the test for serial correlations. We note that this simple operation can be best seen as a rough preprocessing step, and may not achieve independence among data. Therefore more elaborate models may be needed if that is desirable. We choose the tuning parameter using leave-out-one cross validation which minimizes the average Kullback-Leibler loss on the testing data. Since the SCAD penalty gives sparser models, here we only report the results for this penalty. The ﬁnal ﬁtted graphical models for the regions and the items are presented in Figure 3, where there are 43 edges for the regions and 254 edges for the items. It is found that all of the nonzero edges for the regions are negative, indicating the US export to one region is negatively aﬀected by the export to other regions. We then obtain the standard errors using 1000 bootstrap samples for the nonzero edges and plot in Figure 4 the asymptotic 95% pointwise conﬁdence intervals. Among these edges, the magnitude between Europe Union and Other Europe, and that between East Asia and Southeast Asia are the strongest. Interestingly, none of the eleven largest edges corresponds to either North Africa or SubSaharan Africa. To compare the performance of the proposed decomposition to that of the usual graphical LASSO estimate, we use cross validation. Speciﬁcally, each time, one matrix is left out for comparison in terms of its log likelihood. The rest matrices are used to build a model, using either our method or the graphical LASSO method with the tuning parameters chosen by leave-out-one cross validation. We then compute the log likelihood of the matrix that is left out and take average. A simple two-sample t-test shows that our method with SCAD and the LASSO penalty perform similarly (p-value=0.9), while our method with either penalty outperforms the usual graphical LASSO method with both p-values close to 0.01. This shows that our decomposition of the variance matrix is preferred over the decomposition that ignores the matrix structure of the data. 21 South Asia Dairy Products Red Meats, Prep/Pres Tree Nuts Other Europe Pet Foods Other Bulk Commodities Other Consumer Oriented Snack Foods Fruit & Vegetable Juices Sugar, Sweeteners, Bev Bases Southeast Asia Peanuts Soybean Oil Poultry Meat Feeds & Fodders Nursery Products Breakfast Cereals East Asia Red Meats, FR/CH/FR Vegetable Oils (Ex Soybean) Wine and Beer Wheat Caribbean North Africa European Union−27 Eggs & Products Fresh Fruit Live Animals Animal Fats Soybean Meal Processed Fruit & Other Vegetables Intermediate Products Middle East South America Wheat Flour Rice Pulses Central America Fresh Vegetables North America Planting Seeds Soybeans Coarse Grains Tobacco Sub−Saharan Africa Hides & Skins Figure 3: The ﬁtted graphical models of the regions (left) and of the items (right) for the USDA export data from 1970 to 2009. Implied volatilities of equity options The contemporary pricing theory (Merton, 1990) for contingent claims is the most prominent development of Finance in the past decades. However, understanding the pricing mechanism in actual market trading remains not satisfactorily resolved and attractive for many recent investigations; see for example Duan and Wei (2009) and reference therein. How market and economics factors aﬀect the pricing mechanism is a fundamentally important question for both theoretical development and industrial practice. For demonstrating the application of the proposed approach in this area with abundant trading data, we consider the weekly option pricing data of 89 equities in the Standard and Poor (S&P) 100 index, which includes the leading U.S. stocks with exchange-listed options. Constituents of the S&P 100 are selected for sector balance and represent about 57% of the market capitalization of the S&P 500 and almost 45% of the market capitalization of the U.S. equity markets. Since the price of an option is equivalent to the implied volatility from the BlackScholes’s model (Black and Scholes, 1973), the implied volatility data are as informative as option pricing data. We consider a dataset of the implied volatilities for standardized call 22 0.1 0.0 −0.2 −0.4 B C Figure 4: The estimates and the corresponding 95% conﬁdence intervals for the 43 edges between the regions, where the largest eleven edges are marked. ”A” is for CAR and SAM, ”B” for CAM and SAM, ”C” for NAM and EU, ”D” for SAM and EU, ”E” for EU and OE, ”F” for NAM and EA, ”G” for SAM and EU, ”H” for SAM and SEA, ”I” for EA and SEA, ”J” for ME and SEA, ”K” for EU and OC, where EU denotes Europe Union, OE denotes Other Europe, EA denotes East Asia, SEA denotes Southeast Asia, CAM denotes Central America, NAM denotes North America, SAM denotes South America, CAR denotes Caribbean and OC denotes Oceania. options of 89 equities in S&P 100 from January 2008 to November 2010. The options are standardized in the sense that they are all at-the-money (striking price being equal to equity price) and expiring on uniﬁed dates, say 30, 60, 91, 122, 152, 192, 273 and 365 days in the future. High level correlations are expected for the implied volatilities of diﬀerent equities, and so for those expiring on diﬀerent dates. To eliminate the systematic impact among the data, we ﬁrst followed the treatment as in Duan and Wei (2009) to regress the implied volatilities using linear regression against the implied volatilities of the standardized options on the S&P 100 index. We then applied the proposed approach to the residuals to explore the unexplained correlation structures. We plot the connected components with more than two companies of the graphical models in Figure 5. The only other connected component consists of Bank of New York (BK) and JP Morgan Chase (JPM), which are independent of all the other companies. Other than these two, few ﬁnancial ﬁrms are found connected in the estimated graphical model. This may imply the fact that the correlations of the implied volatilities among the ﬁnancial clusters, as well as their impact on other clusters, can be well explained by those induced from the market index. As for the correlations 23 among other ﬁrms on the estimated graph, in total ﬁfty-nine companies are present in the left panel of Figure 5, where 219 out of 3916 possible edges are present. It is remarkable that very clear industrial groups can be identiﬁed. For instance, the tech companies such as Amazon (AMZN), Microsoft (MSFT), Cisco (CSCO), IBM (IBM), QaulComm (QCOM), AT&T (T), Intel (INTC), EMC Corporation (EMC) are tightly connected and form a community with MasterCard (MA). Home Depot and Lowe’s are connected, and the four oil companies ConocoPhilips (COP), Chevron (CVX), Occidental Petroleum Corporation (OXY) and Exxon Mobile (XOM) are closely connected. As for the graph associated with expiration dates on the right panel of Figure 5 , 22 out of 28 possible edges are presented where intuitive interpretation is quite clear. In particular, it seems that the call options to expire in 30, 60 and 365 days are most loosely connected with four edges each, and the call option to expire in 273 days has ﬁve edges, and the options to expire in 91 and 183 days have six edges each, and options to expire 122 days and 152 days are connected to all other options. In summary, we observe clearly industrial and expiration dates pattern from the data analysis even after controlling the level of the implied volatility index of the S&P 100. Such ﬁnding echoes those in the constrained factor analysis on excess returns of equity stocks as in Tsai and Tsay (2010), and can be informative in studying the pricing mechanism of options contingent on equities. Again, caution is needed because we assume the data are independent. To compare the performance of our method to the graphical lasso method, we use the same strategy outlined for analyzing the US agricultural export data. The cross validation procedure shows that our approach with either LASSO or SCAD penalty outperforms the graphical LASSO method signiﬁcantly. CMCSA SO SLB ALL QCOM LMT CL WMB CSCO PEP MMM ABT AVP UTX CVS HAL PGTXN AMZN BHI DIS GD GILD JNJ NSC HON DD BA MSFT 182d 152d 122d BAX NYX S HD 60d HNZ Figure 5: S&P 100. Firms are labeled by their tickers with more detail on http://www.standardandpoors.com/. The colors indicate the community structure extracted via short random walks by Pons and Latapy (2005). We have proposed a novel framework to model high-dimensional matrix-variate data. We demonstrate via simulation and real data analysis that the structural information of this type of data deserves special treatment. Theoretical analysis shows that it is advantageous in modelling such datasets via matrix-variate normal distribution, not only for rate of convergence consideration, but also for illuminating the relationships of the row and the column variables. In this paper, we only study sparsity in the precision matrices. There are obvious ways to extend our work. For example, our framework can be modiﬁed to accommodate a sparse row precision matrix and a sparse column modiﬁed Cholesky matrix (Pourahmadi, 1999; Huang et al., 2006), or two sparse covariance matrices (Bickel and Levina, 2008a, 2008b). The former would be reasonable for the volatility data because the column variables expiration dates are ordered according to time. The proposed framework can be also applied to array type data, where independent arrays with more than two indices are observed 25 (Hoﬀ, 2011). With the many choices to model multiple matrices and a framework to study multidimensional data, these issues are of great interest and will be studied separately. In illustrating our method through data analysis, we have simply treated the time series data as independent after some elementary operations. In practice, this is a rough approximation, and may not by fully satisfactory. It is desirable to develop more elaborate time series models such as vector ARMA models to investigate the mean structure. Given the high dimensionality of the matrix observations, it is also of great interest to develop models that can handle the sparsity in the mean structure as well. A promising related approach for the usual Gaussian graphical model has been done by Rothman et al.(2010), where sparse regression models are employed in addition to sparse covariance estimation. Appendix Proof of Theorem 1: To decompose (3), we deﬁne ∆1 = Ω − Ω0 , ∆2 = Γ − Γ0 and note that tr(Xi ΓXTi Ω) − tr(Xi Γ0 XTi Ω0 ) = tr(Xi Γ0 XTi ∆1 ) + tr(Xi ∆2 XTi Ω0 ) + tr(Xi ∆2 XTi ∆1 ). Further, by Taylor’s expansion {∫ log \|Ω\| − log \|Ω0 \| = tr(Σ0 ∆1 ) − vec (∆1 ) 0 h(v, Ω(1) v )(1 − v)dv vec(∆1 ) −1 −1 where Ω(1) v = Ω0 + v∆1 and h(v, Ω) = Ωv ⊗ Ωv . We deﬁne { n } ∑ 1 tr(Σ0 ∆1 ) T tr(Xi Γ0 Xi ∆1 ) − T1 = , npq i=1 p { n } n ∑ 1 tr(Ψ0 ∆2 ) 1 ∑ T T2 = tr(Xi Ω0 Xi ∆2 ) − , T3 = tr(Xi ∆2 XTi ∆1 ), npq i=1 q npq i=1 {∫ 1 } T4 = p−1 vecT (∆1 ) h(v, Ω(1) v )(1 − v)dv vec(∆1 ), 0 {∫ 1 } (2) −1 T T5 = q vec (∆2 ) h(v, Ωv )(1 − v)dv vec(∆2 ), T6 = {pλ1 (\|ωij \|) − pλ1 (\|ω0ij \|)} + T7 = {pλ2 (\|γij \|) − pλ1 (\|γ0ij \|)} and {pλ1 (\|ωij \|) − pλ1 (\|ω0ij \|)} + (i,j)∈S1 ,i̸=j {pλ2 (\|γij \|) − pλ1 (\|γ0ij \|)} . (i,j)∈S2 ,i̸=j We have the the following decomposition from the deﬁnition (3): g(Ω, Γ) − g(Ω0 , Γ0 ) = T1 + T2 + T3 + T4 + T5 + T6 + T7 . Let α1 = {s1 log p/(nq)}1/2 , β1 = {p log p/(nq)}1/2 , α2 = {s2 log q/(np)}1/2 and β2 = {q log q/(np)}1/2 . For all m dimensional diagonal matrix Dm and symmetric matrix Rm whose diagonal components are zero such that ∥Rm ∥F = C1 and ∥Dm ∥F = C2 for constants C1 and C2 , we deﬁne A = {M : M = α1 Rp + β1 Dp } and B = {M : M = α2 Rq + β2 Dq }, we need to show that [ inf {g(Ω0 + ∆1 , Γ0 + ∆2 ) − g(Ω0 , Γ0 )} > 0 → 1. ∆1 ∈A,∆2 ∈B First, following Rothman et al (2008) and Lam and Fan (2009), we have ∫ T4 ≥ p ∥vec(∆1 )∥ −1 (1 − v) min λ1 (Ω−1 v ⊗ Ωv )dv 0 ≥ (2p)−1 {τ1−1 + o(1)}−2 (C12 α12 + C22 β12 ) and similarly, T5 ≥ (2q)−1 {τ1−1 + o(1)}−2 (C12 α22 + C22 β22 ). Then, we consider T1 and T2 , −1 n ∑ tr(Xi Γ0 Xi ∆1 ) = (npq) n ∑ i=1 tr(Yi YTi ∆1 ) = p−1 tr(Q1 ∆1 ) where Q1 = (nq)−1 ∑n i=1 Yi YTi , and Yi = Xi Γ0 follows matrix-variate normal distri- bution with parameters Σ0 and I by the properties of matrix-variate normal distribution (Gupta and Nagar, 2000). In other words, the columns of Yi are independent and identically distributed normal random vectors with covariance matrix Σ0 . Therefore ∑ ∑ T1 = p−1 tr{(Q1 − Σ0 )∆1 } = p−1 + (Q − Σ0 )ij (∆1 )ij = T11 + T12 1 c (i,j)∈S1 where T11 and T12 are respectively the two sums over S1 and S1c . Since max \|(Q1 − Σ0 )ij \| = i,j Op [{log p/(nq)}1/2 ] by Bickel and Levina (2008a), \|T11 \| ≤ p−1 (s1 + p)1/2 ∥∆1 ∥F max \|(Q1 − Σ0 )ij \| = p−1 Op (α1 + β1 )∥∆1 ∥F . i,j Hence, by choosing C1 and C2 suﬃciently large, T11 is dominated by the positive term T4 . By Condition A2, pλ1 (\|∆ij \|) ≥ λ1 k1 \|∆ij \| for some constant k1 > 0 when n is suﬃciently large. Therefore, ∑ { ∑ } pλ1 (\|∆ij \|) − p−1 (Q1 − Σ0 )ij (∆1 )ij ≥ {λ1 k1 − p−1 max \|(Q1 − Σ0 )ij \|}\|∆1ij \| = λ1 {k1 − op (1)}\|∆1ij \|, which is greater than 0 with probability tending to 1, and the last equality is due to √ −1 −1 Condition A4 so that λ−1 maxi,j \|(Q1 − Σ0 )ij \|}\| = Op {λ−1 log p/(nq)} = op (1). 1 p 1 p Applying exactly the same arguments on T2 , we have shown that T1 + T2 + T4 + T5 + T6 > 0 with probability tending to 1. If the LASSO penalty is applied, Condition A4 for λ1 and λ2 can be relaxed to λ1 = O √ √ log p/(nq)) and λ2 = O(q −1 log q/(np)). As for T7 , applying Taylor’s expansion, pλ1 (\|ωij \|) = pλ1 (\|ω0ij \|)+p′λ1 (\|ω0ij \|)sgn(ω0ij )(ωij −ω0ij )+2−1 p′′λ1 (\|ω0ij \|)(ωij −ω0ij )2 {1+o(1)}. Noting Condition A3, we have ∑ {pλ1 (\|ωij \|) − pλ1 (\|ω0ij \|)} (i,j)∈S1 ,i̸=j ≤ s1 C1 α1 max p′λ1 (\|ω0ij \|) + C1 max p′′λ1 (\|ω0ij \|)α12 /2{1 + o(1)} (i,j)∈S1 = O{p−1 (α12 + β12 )}. Hence, it is dominated by the positive term T4 . Same arguments apply for the penalty on Γ. Therefore T7 is dominated by T4 + T5 . Finally, note that E{tr(Xi ∆2 XTi ∆1 )} = tr (∆2 Ψ0 )tr(Σ0 ∆1 ) by the properties of matrixvariate normal distribution (Gupta and Nagar, 2000). Hence by law of large numbers T3 = (npq) n ∑ tr(Xi ∆2 XTi ∆1 ) = (pq)−1 tr(∆2 Ψ0 )tr(Σ0 ∆2 ){1 + op (1)}. Let ξ1j and ξ2k , j = 1, . . . , p, k = 1, . . . , q be the eigen values of ∆1 and ∆2 . Then by the von Neumann inequality \|tr(Σ0 ∆1 )\| ≤ p ∑ λj (Σ0 )\|ξ1j \| ≤ (τ1 ) p ∑ \|ξ1j \| ≤ τ1−1 p1/2 ∥∆1 ∥F ≤ τ1−1 p1/2 (C12 α12 + C22 β12 )1/2 . By applying the argument on \|tr(Ψ0 )∆2 \|, we have as n → ∞, √ \|T3 \| ≤ 1/ pqτ1−2 (C12 α12 + C22 β12 )(C12 α22 + C22 β22 ) ≤ T4 + T5 . In summary, we conclude that T1 + T2 + T3 + T4 + T5 + T6 + T7 > 0 with probability tending to 1 and this proves Theorem 1. Proof of Theorem 2: For the minimizers Ω and Γ, we have ∂g(Ω, Γ) = 2(tij − p−1 σij + p′λ1 (\|ωij \|))sgn(ωij ) ∂ωij where tij is the (i, j)th element of (npq)−1 n ∑ (Xi ΓXTi ) = p−1 Q1 + (npq)−1 n ∑ i=1 Xi (Γ − Γ0 )XTi . It is seen that the (i, j)th element in (npq)−1 Xi (Γ − Γ0 )XTi is Op {p−1 η2n } following √ Lam and Fan (2009). In addition, p−1 max \|qij − σ0ij \| = Op {p−1 log p/(nq)}} by Bickel 1/2 and Levina (2008a). Further, following Lam and Fan (2009), p−1 \|σ0ij − σij \| = O(p−1 η1n ). 1/2 Therefore we need log p/(nq) + η1n + η2n = O(λ21 p2 ) to ensure that sgn(ωij ) dominates the ﬁrst derivative of g(Ω, Γ), and by the same arguments for the minimizer Γ. Theorem 2 then follows. Proof of Theorem 3: ˆ and Γ ˆ solves 0 = {∂g(Ω, Γ))/∂ T vec(Ω), ∂g(Ω, Γ))/∂ T vec(Γ)}T , for the local miniSince Ω ˆ and Γ ˆ in Theorem 1, we use Taylor’s expansion at the truth. Let A⊗2 = A ⊗ A. mizer Ω Standard calculations imply ∑ n ⊗2 ⊗2 −1 −1 ˆ Σ0 n q i=1 Xi vec(Ω) − vec(Ω0 ) R1n + Λ + n ∑ ⊗2 ˆ − vec(Γ0 ) n−1 p−1 n XTi ⊗2 vec( Γ) Ψ R 2n 0 i=1 S×S S ∑ q −1 ni=1 vec(Xi Γ0 XTi ) − vec(Σ0 ) −1 =n (7) + (bn )S ∑n T −1 p i=1 vec(Xi Ω0 Xi ) − vec(Ψ0 ) S where the remainder terms satisfy ∥R1n ∥ = Op (α1 + β1 ) and ∥R2n ∥ = Op (α2 + β2 ) following the proof of Theorem 1, the subscript S and S × S indicate the sub-vector and submatrix corresponding to those nonzero components of Ω0 and Γ0 excluding ω11 , Λn = ′′ ′ ′ ˆ p′′ T (Ω)}, ˆ diag{pλT1 (Ω), and bn = {pλT1 (\|Ω0 \|)sgn(Ω0 ), pλT1 (\|Ω0 \|)sgn(Ω0 )}T . The follow facts λ2 by results in Gupta and Nagar (2000) imply the moments of the right hand side in (7). First note that E{vec(Xi Γ0 XTi )} = qΣ0 and E{vec(XTi Ω0 Xi )} = pΨ0 . Let S1 = Xi Γ0 XTi , S2 = XTi Ω0 Xi , then ⊗2 cov{vec(S1 ), vec(S1 )} = qΣ⊗2 0 (I + Kpp ), cov{vec(S2 ), vec(S2 )} = pΨ0 (I + Kqq ) where Kab is a commutation matrix of order ab × ab; see Gupta and Nagar (2000) for its deﬁnition and properties. In addition, for any p × q matrix C, E(S1 CS2 ) = Σ0 CΨ0 (2 + pq) and E(S1 )CE(S2 ) = pq = Σ0 CΨ0 . Then for any q × p matrix D, we have tr[(CT ⊗ D)cov{vec(S1 ), vec(S2 )}] = tr((CT ⊗ D)[E{vec(S1 )vecT (S2 )} − E{vec(S1 )}E{vecT (S2 )}]) = tr[E{vec(DS1 C)vecT (S2 )} − E{vec(DS1 C)}E{vecT (S2 )}] = tr{E(S1 CS2 D) − E(S1 )CE(S2 )D} = 2Σ0 CΨ0 D = vecT (Ψ0 )(CT ⊗ D)vec(Σ0 ) = tr{(CT ⊗ D)vec(Ψ0 )vecT (Σ0 )}. This implies cov{vec(S1 ), vec(S2 )} = vec(Σ0 )vecT (Ψ0 ). Furthermore, E(Xi ⊗ Xi ) = vec(Σ0 )vecT (Ψ0 ), E(XTi ⊗ XTi ) = vec(Ψ0 )vecT (Σ0 ). Since p → ∞, q → ∞, n−1 q −1 ∑n i=1 and n−1 p−1 X⊗2 i ∑n i=1 XiT⊗2 are negligible in the left side of (7), so does the covariance between components on the right side of (7). Then following standard arguments in penalized likelihood (Lam and Fan, 2009), we establish Theorem 3. References Allen, G. I. and Tibshirani R. (2010). Transposable regularized covariance models with an application to missing data imputation. Annals of Applied Statistics, 4, 764–790. Banejee, O., El Ghaoui, L. and d’Aspremont, A. (2008). Model selection through sparse maximum likelihood estimation. Journal of Machine Learning Research, 9, 485–516. Bickel, P. and Levina, E. (2008a). Regularized estimation of large covariance matrices. The Annals of Statistics, 36,199–227. Bickel, P. and Levina, E. (2008b). Covariance regularization by thresholding. The Annals of Statistics, 36, 2577–2604. Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 673–654. Dawid, A. P. (1981). Some matrix-variate distribution theory: notational considerations and a Bayesian application. Biometrika, 68, 265–274. Duan, J. C. and Wei, J. (2009). 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The Annals of Statistics, 36, 1509–1533.	{"url":"https://pdffox.com/sparse-matrix-graphical-models-pdf-free.html","timestamp":"2024-11-14T13:32:38Z","content_type":"text/html","content_length":"92473","record_id":"<urn:uuid:1a475499-3553-4b37-acca-db2bb4c3ded0>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00823.warc.gz"}
it w Differential Evolution The Differential Evolution algorithm can be used instead of simann in the optimization part of potfit. Enabling it with the evo compilation switch will create a binary that first uses Differential Evolution and then subsequently the lsq to optimize the potential. Please note that this algorithm is not tested as well as the simann and may perform very poor with tabulated potentials. It however works well with analytic potentials because of the predefined range of the potential parameters. How does it work? Differential Evolution (DE) is a genetic algorithm, it works with populations and generations. At the beginning, $NP$ D-dimensional vectors $\boldsymbol x_{i,G=0}\;,i=1,2,\ldots,NP$ (which each holds all the parameters for the potential) are created as the initial population. The index $G$ indicates the number of the generation. For potfit this means that we have to generate $NP-1$ potentials in addition to the starting potential given by the user. The initialization of the population is done with normally distributed random numbers being added to the starting potential. A DE step consists of several substeps, called mutation, crossover and selection. For each potential $\boldsymbol x_{i,G}\;i=1,2,\ldots,NP$ a mutant vector $\boldsymbol v$ is generated according to $$\boldsymbol v_{i,G+1}=\boldsymbol x_{r_1,G}+F\left(\boldsymbol x_{r_2,G}-\boldsymbol x_{r_3,G}\right)$$ with random indexes $r_1,r_2,r_3 \in {1,2,\ldots,NP}$, integer, mutually different and $F>0$. $F$ is a real and constant factor $\in[0,2]$ which controls the amplification of the differential variation $\left(\boldsymbol x_{r_2,G}-\boldsymbol x_{r_3,G}\right)$. To achieve a greater diversity of the new vectors, a new trial vector $\boldsymbol u$ is created according to $$u_{i,G+1,j}= \begin{cases} v_{i,G+1,j} \quad \text{if} \quad \text{rand}_j[0,1] \leq CR \;\text{or}\; j=\text{rand}(i)\\ x_{i,G,j} \;\;\quad \text{if} \quad \text{rand}_j[0,1] > CR \;\text{or}\; j\ neq\text{rand}(i) \end{cases},\;j=1,2,\ldots,D. $$ Here $\text{rand}_j$ is the $j$th evaluation of a uniform random number generator and $CR$ is the crossover constant $\in[0,1]$. $\text{rand}(i)$ is a randomly chosen index $\in 1,2,\ldots,D$ which ensures that $\boldsymbol u_{i,G+1}$ gets at least one parameter from $\boldsymbol v_{i,G+1}$. To decide wheter or not the trial vector $\boldsymbol u_{i,G+1}$ should become a member of generation $G+1$, it is compared to the target vector $\boldsymbol x_{i,G}$. If the new vector yields a smaller target function than $\boldsymbol x_{i,G}$, then $\boldsymbol u_{i,G+1}$ replaces $\boldsymbol x_{i,G}$, otherwise $\boldsymbol x_{i,G}$ is retained. For more information on DE, please take a look at the references in the DE section of Quotes. algorithms/diffevo.txt ยท Last modified: 2018/01/10 18:26 by daniel	{"url":"https://www.potfit.net/wiki/doku.php?id=algorithms:diffevo","timestamp":"2024-11-01T22:24:10Z","content_type":"text/html","content_length":"21230","record_id":"<urn:uuid:a16670d0-7f50-4784-bc22-45ffb659c833>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00515.warc.gz"}
There has been a series of posts at WUWT by Andy May on SST averaging, initially comparing HADSST with ERSST. They are , and . Naturally I have been involved in the discussions; so has John Kennedy . There has also been Twitter discussion .My initial comment was "Just another in an endless series of why you should never average absolute temperatures. They are too inhomogeneous, and you are at the mercy of however your sample worked out. Just don’t do it. Take anomalies first. They are much more homogeneous, and all the stuff about masks and missing grids won’t matter. That is what every sensible scientist does. The trend was toward HADSST and a claim that SST had been rather substantially declining this century (based on that flaky averaging of absolute temperatures). It was noted that ERSST does not show the same thing. The reason is that HADSST has missing data, while ERSST interpolates. The problem is mainly due to that interaction of missing cells with the inhomogeneity of T. Here is one of Andy's graphs: In these circumstances I usually repeat the calculation that was done replacing the time varying data with some fixed average for each location to show that you get the same claimed pattern. It seems to me obvious that if unchanging data can produce that trend, then the trend is not due to any climate change (there is none) but to the range of locations included in each average, which is the only thing that varies. However at WUWT one meets an avalanche of irrelevancies - maybe the base period had some special property, or maybe it isn't well enough known, the data is manipulated etc etc. I think this is silly, because they key fact is that some set of unchanging temperatures did produce that pattern. So you certainly can't claim that it must have been due to climate change. I set out that in a comment , with this graph: Here A is Andy's average, An is the anomaly average, and Ae is the average made from the fixed base (1961-90) values. Although no individual location in Ae is changing, it descends even faster than So I tried another tack. Using base values is the simplest way to see it, but one can just do a partition of the original arithmetic, and along the way find a useful way of showing the components of the average that Andy is calculating. I set out a first rendition of that . I'll expand on that here, with a more systematic notation and some tables. For simplicity, I will omit area weighting of cells, as Andy did for the early posts. Breakdown of the anomaly difference between 2001 and 2018 Consider three subsets of the cell/month entries (cemos): • Ra is the set with data in both 2001 and 2018 (Na cemos) • Rb is the set with data in 2001 but not in 2018 (Nb cemos) • Rc is the set with data in 2018 but not in 2001 (Nc cemos) I'll mark sums S of cemo data with a 1 for 2001, 2 for 2018, and an a,b,c if they are sums for a subset. I use a similar notation for averages with A plus suffices. I'll set out the notation and some values in a table: Set data N Weights 2001 2018 2001 or 2018 N=18229 S1, A1=S1/(Na+Nb)=19.029 S2,A2=S2/(Na+Nc)=18.216 Ra 2001 and 2018 Na=15026 Wa=Na/N=0.824 S1a, A1a=S1a/Na=19.61 S2a, A2a=S2a/Na=19.863 Rb 2001 but not 2018 Nb=1023 Wb=Nb/N=0.056 S1b, A1b=S1b/Nb=10.52 S2b=0 Rc 2018 but not 2001 Nc=2010 Wc=Nc/N=0.120 S1c=0 S2b, A2b=S2b/Nb=6.865 I haven't given values for the sums S, but you can work them out from the A and N. The point is that they are additive, and this can be used to form Andy's A2-A1 as a weighted sum of the other averages. From additive S: S1=S1a+S1b and S2=S2a+S2c A1(Na+Nb)=A1aNa+A1b*Nb, or and similarly or, dividing by N That expresses A2-A1 as the weighted sum of three terms relating to Ra, Rb and Rc respectively. Looking at these individually • (A2a-A1a)=0.253 are the differences between the data points known for both years. They are the meaningful change measures, and give a positive result • (A1b-A2)=-7.696. The 2001 readings in Rb have no counterpart in 2018, and so no information about increment. Instead they appear as the difference with the 2018 average A2. This isn't a climate change difference, but just reflects whether the points in R2 were from warm or cool places/seasons. • (A2b-A1)=12.164. Likewise these Rc readings in 2018 have no balance in 2001, and just appear relative to overall A1. Note that the second and third terms are not related to CC increases and are large, although this is ameliorated by their smallish weighting. The overall sum that, with weights, makes up the difference is A2-A1 = 0.210 + 0.431 -1.455 = -0.813 So the first term representing actual changes is overwhelmed by the other two, which are biases caused by the changing cell population. This turns a small increase into a large decrease. So why do anomalies help I'll form anomalies by subtracting from each cemo the 2001-2018 mean for that cemo (chosen to ensure all N cemo's have data there). The resulting table has the same form, but very different numbers: Set data N Weights 2001 2018 2001 or 2018 N=18229 S1, A1=S1/(Na+Nb)=-.116 S2,A2=S2/(Na+Nc)=0.136 Ra 2001 and 2018 Na=15026 Wa=Na/N=0.824 S1a, A1a=S1a/Na=-0.118 S2a, A2a=S2a/Na=0.137 Rb 2001 but not 2018 Nb=1023 Wb=Nb/N=0.056 S1b, A1b=S1b/Nb=-0.084 S2b=0 Rc 2018 but not 2001 Nc=2010 Wc=Nc/N=0.120 S1c=0 S2b, A2b=S2b/Nb=0.130 The main thing to note is that the numbers are all much smaller. That is both because the range of anomalies is much smaller than absolute temperatures, but also, they are more homogeneous, and so more likely to cancel in a sum. The corresponding terms in the weighted sum making up A2-A1 are A2-A1 = 0.210 + 0.012 + 0.029 = 0.251 The first term is exactly the same as without anomalies. Because it is the difference of T at the same cemo, subtracting the same base from each makes no change to the difference. And it is the term we want. The second and third spurious terms are still spurious, but very much smaller. And this would be true for any reasonably choice of anomaly base. So why not just restrict to Ra? where both 2001 and 2018 have values? For a pairwise comparison, you can do this. But to draw a time series, that would restrict to cemos that have no missing values, which would be excessive. Anomalies avoid this with a small error. However, you can do better with infilling. Naive anomalies, as used un HADCRUT 4 say, effectively assign to missing cells the average anomaly of the remainder. It is much better to infill with an estimate from local information. This was in effect the Cowtan and Way improvement to HADCRUT. The uses of infilling are described (with links). The GISS V4 land/ocean temperature anomaly was 1.13°C in November 2020, up from 0.88°C in October. That compares with a 0.188deg;C rise in the TempLS V4 mesh index. It was the warmest November in the Jim Hansen's update, with many more details, is here. He thinks that it is clear that 2020 will pass 2016 as hottest year. As usual here, I will compare the GISS and earlier TempLS plots below the jump.	{"url":"https://moyhu.blogspot.com/2020/","timestamp":"2024-11-03T06:24:57Z","content_type":"application/xhtml+xml","content_length":"366103","record_id":"<urn:uuid:a6c1d966-ab53-4ccd-a937-7cf2dba2a437>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00185.warc.gz"}
Binary Exponentiation Let's say you are given two numbers a, n and you have to compute a^n. Naive Approach: O(n) The easiest approach to do this, if one knows how to use "for" loops or implement it (if it is not implemented already) in any given programming language is just a single O(n) iteration. Below is a sample implementation in C++. int a, n; // a, n can be initialized either by taking input or // by assigning hardcoded values a = 3, n = 4 let's say // expo initialized to 1 as it is the multiplicative identity int expo = 1; for (int i = 0; i < n; ++i) { expo = expo * a; // now expo == a^n : true Here, the data type is "int", assuming the value of expo = a^n comes under the constraint of "int". So, any other data type can be used as per requirements. 1. It can be seen that this approach would time out if n > 10^8. 2. If the value of a^n is very large i.e. it can not be stored in "int" or any other provided data type, it will overflow. Possible Solutions 1. Algorithm of lesser complexity can be utilized in this case -- we will see O(log n) algorithm next. 2. This is the reason why most of the time (a^n)%(some number) is to be calculated. Most of the time that "some number" is 1e9 + 7 (which is a prime) in competitive programming problems. Binary Exponentiation Approach: O(log n) For achieving O(log n) complexity, the mathematical fact that any number (in decimal) can be represented uniquely in binary can be utilized. For example, 12 = 18 + 14 + 02 + 01 = (1100)_2 15 = 18 + 14 + 12 + 11 = (1111)_2 Now, also using the fact that a^(m + n) = (a^m)(a^n), we can calculate the values of a^1, a^2, a^4, and so on ... int expo = a; for (int i = 0; i < n; ++i) { expo = (expoexpo); // it's like // expo: a -> a^2 -> a^4 -> a^8 -> a^16 ... // i.e. with ith step the value will be a^(2i) Now, let's say we need to calculate a^n, then there will exist a unique representation of "n" in binary, whose i_th bit can be checked if it is set or not by using a simple boolean expression involving bitwise operator // (n >> i) & 1 == ith bit of n for the proof for this refer to Link now, it can be seen that a^n = (a^(1b_1))x(a^(2b_2))x(a^(4b_3))x... and we can find if the a^(2i) have to multiply or not by using the fact we saw above. So, by a simple modification to the previous code we can calculate a^n. //init a, n int expo = a; // answer initialized to 1 as it is multiplicative identity int answer = 1; for (int i = 0; i < NUMBER_OF_BITS_IN_N; ++i) { if((n >> i) & 1) { // we check if the ith bit is set or not // if it is set then, we multiply expo = a^(2i) answer = answerexpo; expo = (expoexpo); // answer now have the value a^n See, there is only one "O(NUMBER_OF_BITS_IN_N)" for loop, and it is easy to see that the number of bits in n = log_2(n). Hence, the overall complexity = 0(log n) If you are not sure of the number of bits in n, then just simply take MAXIMUM_POSSIBLE_NUMBER_OF_BITS instead which can be ~32 for the int datatype. Modular Binary Exponentiation Considering the second caveat described above, there can be cases where we need to find (a^n)%(some value) -- note that % is the remainder operator (as used in C++). For this, just an easy modification of the code will work, //init a, n, modvalue long long expo = a; // answer initialized to 1 as it is multiplicative identity long long answer = 1; for (int i = 0; i < NUMBER_OF_BITS_IN_N; ++i) { if((n >> i) & 1) { // we check if the ith bit is set or not // if it is set then, we multiply expo = a^(2i) answer = (answerexpo) % modvalue; expo = (expoexpo) % modvalue; // answer now have the value (a^n)% modevalue So, as we saw the binary exponentiation algorithm, it can be used for exponentiating a matrix too, just by changing "" operator with implemented matrix multiplication and taking remainder with each number in the matrix. Further Directions For reading more about binary exponentiation and solving some related problems to get a grasp refer to CP-Algorithms Binary-Exp Top comments (4) Abhijit Tripathy • Good one Rishabh Singhal • thank you! Nishant Sachdeva • • Edited on • Edited Great Read. Well expounded Rishabh Singhal • • Edited on • Edited Thank You! :) For further actions, you may consider blocking this person and/or reporting abuse	{"url":"https://dev.to/edualgo/binary-exponentiation-57oi","timestamp":"2024-11-09T10:49:04Z","content_type":"text/html","content_length":"119909","record_id":"<urn:uuid:e207b68e-b358-468d-9b4f-06431d228dea>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00095.warc.gz"}
zgeqp3: computes a QR factorization with column pivoting of a matrix A - Linux Manuals (l) zgeqp3 (l) - Linux Manuals zgeqp3: computes a QR factorization with column pivoting of a matrix A ZGEQP3 - computes a QR factorization with column pivoting of a matrix A M, N, A, LDA, JPVT, TAU, WORK, LWORK, RWORK, INFO ) INTEGER INFO, LDA, LWORK, M, N INTEGER JPVT( * ) DOUBLE PRECISION RWORK( * ) COMPLEX16 A( LDA, ), TAU( * ), WORK( * ) ZGEQP3 computes a QR factorization with column pivoting of a matrix A: AP = QR using Level 3 BLAS. M (input) INTEGER The number of rows of the matrix A. M >= 0. N (input) INTEGER The number of columns of the matrix A. N >= 0. A (input/output) COMPLEX16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the upper triangle of the array contains the min(M,N)-by-N upper trapezoidal matrix R; the elements below the diagonal, together with the array TAU, represent the unitary matrix Q as a product of min(M,N) elementary reflectors. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M). JPVT (input/output) INTEGER array, dimension (N) On entry, if JPVT(J).ne.0, the J-th column of A is permuted to the front of AP (a leading column); if JPVT(J)=0, the J-th column of A is a free column. On exit, if JPVT(J)=K, then the J-th column of AP was the the K-th column of A. TAU (output) COMPLEX16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors. WORK (workspace/output) COMPLEX16 array, dimension (MAX(1,LWORK)) On exit, if INFO=0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= N+1. For optimal performance LWORK >= ( N+1 )NB, where NB is the optimal blocksize. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. RWORK (workspace) DOUBLE PRECISION array, dimension (2N) INFO (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. The matrix Q is represented as a product of elementary reflectors = H(1) H(2) . . . H(k), where k = min(m,n). Each H(i) has the form H(i) = I - tau v * vaq where tau is a real/complex scalar, and v is a real/complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i). Based on contributions by G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA	{"url":"https://www.systutorials.com/docs/linux/man/l-zgeqp3/","timestamp":"2024-11-06T09:07:01Z","content_type":"text/html","content_length":"9915","record_id":"<urn:uuid:5f975e04-8908-4f8a-a4f5-dee888342f72>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00127.warc.gz"}
Home \| 137 Ventures Founding partner Justin Fishner-Wolfson’s grandfather, Haney Fishner, had a seat on the New York Stock Exchange (NYSE) until 1991. During his tenure, his annunciator number was 137. The annunciator board and its corresponding numbers were used to page members of the exchange and conduct business. In the early 1900s, the over 24 miles of wiring for its annunciator board was one of the most notable features of the NYSE. In 1980, the board was disassembled and replaced by a computerized display system. 137 is a prime number, and further, it also has some unique prime characteristics. 137 is a twin prime, which is when a prime number differs from another prime number by two. Except for the pair (2, 3), this is the smallest possible difference between two primes. 137 and 139 are a twin prime pair. Further, 137 is also a Chen prime. In 1966, a Chinese mathematician, Chen Jingrun, proved that every sufficiently large even number can be written as the sum of a prime and a semi prime (the product of two primes). A prime number p is a Chen prime if p + 2 is either a prime or a product of two primes. 137 satisfies Chen’s theorem, because 137+2=139, which is also prime. In physics, the inverse of 137 approximates the fine structure constant, which represents the probability that an electron will absorb a photon. Since the early 1900’s, physicists have thought that 137 might be at the heart of a Grand Unified Theory (GUT), which could relate the theories of electromagnetism, quantum mechanics and gravity, but physicists have yet to find any link between the 137 and any other physical law in the universe. It was expected that such an important equation would generate an important number, like one or pi, but this was not the case. 137 is usually denoted by α, which in investment terms has its own significance as the excess return above a benchmark index. When a vinculum is added, 137 becomes 1337, l33t or leet, also known as eleet or leetspeak. 1337 is an alternative alphabet that is used primarily online. It uses various combinations of ASCII characters to replace letters. Leetspeak was first used by hackers in the 1980s as a way to prevent websites and news groups from being found by simple keyword searches. It was also used to show status on a bulletin board system (BBS). Since then, it’s become popular in online games to suggest that its user is a skilled hacker (h4x0r).	{"url":"https://www.137ventures.com/","timestamp":"2024-11-03T21:24:54Z","content_type":"text/html","content_length":"42421","record_id":"<urn:uuid:1bf845cc-1b9d-458f-b6eb-3f86aa303557>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00094.warc.gz"}
The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour? You must login to ask question. NCERT Solutions for Class 10 Maths Chapter 12 Important NCERT Questions Areas Related To Circles Class, NCERT Books for Session 2022-2023 CBSE Board and UP Board Others state Board EXERCISE 12.1 Page No:226 Questions No:4	{"url":"https://discussion.tiwariacademy.com/question/the-wheels-of-a-car-are-of-diameter-80-cm-each-how-many-complete-revolutions-does-each-wheel-make-in-10-minutes-when-the-car-is-travelling-at-a-speed-of-66-km-per-hour/","timestamp":"2024-11-12T10:08:20Z","content_type":"text/html","content_length":"160597","record_id":"<urn:uuid:77532f48-bcd2-4d2e-a0d2-400dc87f74a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00244.warc.gz"}
An Introduction to Diophantine Equations - A Problem-Based Approach Fr. 135.60 Tit Andreescu, Titu Andreescu, Dori Andrica, Dorin Andrica, Ion Cucurezeanu An Introduction to Diophantine Equations - A Problem-Based Approach English · Hardback Shipping usually within 4 to 7 working days This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The material is organized in two parts: Part I introduces the reader to elementary methods necessary in solving Diophantine equations, such as the decomposition method, inequalities, the parametric method, modular arithmetic, mathematical induction, Fermat's method of infinite descent, and the method of quadratic fields; Part II contains complete solutions to all exercises in Part I. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants - including Olympiad and Putnam competitors - as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques. List of contents Diophantine Equations.- Elementary Methods for Solving Diophantine Equations.- Some Classical Diophantine Equations.- Pell-Type Equations.- Some Advanced Methods for Solving Diophantine Equations.- Solutions to Exercises and Problems.- Solutions to Elementary Methods for Solving Diophantine Equations.- Solutions to Some Classical Diophantine Equations.- Solutions to Pell-Type Equations.- Solutions to Some Advanced Methods in Solving Diophantine Equations. This problem-solving book is an introduction to the study of Diophantine equations. It introduces the reader to elementary methods necessary in solving Diophantine equations and contains complete solutions to all exercises Additional text From the reviews: “This book is devoted to problems from mathematical competitions involving diophantine equations. … Each chapter contains a large number of solved examples and presents the reader with problems whose solutions can be found in the book’s second part. This volume will be particularly interesting for participants in mathematical contests and their coaches. It will also give a lot of pleasure to everyone who likes to tackle elementary, yet nontrivial problems concerning diophantine equations.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 167 (3-4), September, 2012) “This book explains methods for solving problems with Diophantine equations that often appear in mathematical competitions at various levels. … The book can be recommended to mathematical contest participants, but also to undergraduate students, advanced high school students and teachers.” (Andrej Dujella, Mathematical Reviews, Issue 2011 j) “Diophantus’ Arithmetica is a collection of problems each followed by a solution...The book at hand is intended for high school students, undergraduates and math teachers. It is written in a language that everyone in these groups will be familiar with. The exposition is very lucid and the proofs are clear and instructive. The book will be an invaluable source for math contest participants and other math fans. It will be an excellent addition to any math library.” (Alex Bogomolny, The Mathematical Association of America, October, 2010) “Diophantine analysis, the business of solving equations with integers, constitutes a subdiscipline within the larger field of number theory. One problem in this subject, Fermat's last theorem, till solved, topped most lists of the world's most celebrated unsolved mathematics problems, so the subject attracted much attention from mathematicians and even the larger public. Nevertheless, sophisticated 20th-century tools invented to attackDiophantine equations (algebraic number fields, automorphic forms, L-functions, adelic groups, etc.) have emerged as proper objects of study in their own right. So for a popular subject, modern lower-level works focused on the individual Diophantine equation (and not on big machines aimed generally at classes of such equations) are relatively rare. The present volume…fills this need...Summing Up: Recommended. Lower- and upper-division undergraduates and general readers.” (D.V. Feldman, Choice, July, 2010) From the reviews: "This book is devoted to problems from mathematical competitions involving diophantine equations. ... Each chapter contains a large number of solved examples and presents the reader with problems whose solutions can be found in the book's second part. This volume will be particularly interesting for participants in mathematical contests and their coaches. It will also give a lot of pleasure to everyone who likes to tackle elementary, yet nontrivial problems concerning diophantine equations." (Ch. Baxa, Monatshefte für Mathematik, Vol. 167 (3-4), September, 2012) "This book explains methods for solving problems with Diophantine equations that often appear in mathematical competitions at various levels. ... The book can be recommended to mathematical contest participants, but also to undergraduate students, advanced high school students and teachers." (Andrej Dujella, Mathematical Reviews, Issue 2011 j) "Diophantus' Arithmetica is a collection of problems each followed by a solution...The book at hand is intended for high school students, undergraduates and math teachers. It is written in a language that everyone in these groups will be familiar with. The exposition is very lucid and the proofs are clear and instructive. The book will be an invaluable source for math contest participants and other math fans. It will be an excellent addition to any math library." (Alex Bogomolny, The Mathematical Association of America, October, 2010) "Diophantine analysis, the business of solving equations with integers, constitutes a subdiscipline within the larger field of number theory. One problem in this subject, Fermat's last theorem, till solved, topped most lists of the world's most celebrated unsolved mathematics problems, so the subject attracted much attention from mathematicians and even the larger public. Nevertheless, sophisticated 20th-century tools invented to attackDiophantine equations (algebraic number fields, automorphic forms, L-functions, adelic groups, etc.) have emerged as proper objects of study in their own right. So for a popular subject, modern lower-level works focused on the individual Diophantine equation (and not on big machines aimed generally at classes of such equations) are relatively rare. The present volume...fills this need...Summing Up: Recommended. Lower- and upper-division undergraduates and general readers." (D.V. Feldman, Choice, July, 2010) Product details Authors Tit Andreescu, Titu Andreescu, Dori Andrica, Dorin Andrica, Ion Cucurezeanu Publisher Springer, Basel Languages English Product format Hardback Released 05.10.2010 EAN 9780817645489 ISBN 978-0-8176-4548-9 No. of pages 345 Dimensions 160 mm x 239 mm x 26 mm Weight 692 g Illustrations XI, 345 p. Subject Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra Customer reviews No reviews have been written for this item yet. Write the first review and be helpful to other users when they decide on a purchase. Write a review Thumbs up or thumbs down? Write your own review.	{"url":"https://www.cede.ch/en/books/?id=8129194","timestamp":"2024-11-13T18:05:03Z","content_type":"text/html","content_length":"33572","record_id":"<urn:uuid:b189cd4a-4bfe-4f19-bbfa-bd371fdd0803>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00036.warc.gz"}
Calculator Builder – Create an Online Calculator Introducing the Calculator Builder WordPress plugin – the ultimate WordPress plugin that allows you to create online calculators for any calculation. You can also add style to your calculator and customize it the way you want. Calculator Builder is a great plugin to have awesome and easy-to-use calculators. It has powerful tools to create an intuitive calculator and to use them for different purposes. Display any type of calculator on your website to make it more engaging and user-friendly, such as health and financial ones. The Calculator Builder plugin provides you with elements such as checkboxes, radio buttons, numbers, and dropdowns. It has an amazing set of features that will help you create the online calculator you want very quickly and effectively. Key Features • Intuitive interface – the Calculator Builder plugin has a very intuitive interface and is very engaging. You can create a calculator that perfectly matches your website design, and is highly • Easy to use – another important feature of the Calculator Builder WordPress plugin is that it is super easy to use. With its well-designed interface and structure, you can easily understand how to create calculators even if you are not a developer and don’t have coding skills. However, complex calculators will still require JavaScript skills. • Unlimited items to use for calculators – Calculator Builder provides you with unlimited items to include in your individual calculator. This is also a great feature as not many calculator builders provide you with lots of items. • User-friendly – Sometimes different types of calculators can look confusing, for instance, Financial Calculators. can look confusing. Many users pay attention to the user-friendly aspect of calculators. The Calculator Builder plugin provides you with a great user experience as it is made to make the calculation process easier and quicker. • Highly customizable – CalcHub extension helps you to easily design the calculator the way you want. This will help you to brand your website by making the calculator an inseparable part of it. • Live builder – Լive builder allows you to see the calculator created. This will save you time during the process of creation. • Usage of Vanilla JS: without using the jQuery library • Export/Import tool – Calculator Builder allows you to export and import calculator data • Field types such as “Title”, “Separator”, “Spacer”, “Textarea”, and “Input” Use Cases 1 Finance calculators Do you write a blog? Own a car dealership? Or are otherwise involved in financial services? Then you should consider adding the Calculator Builder plugin to your website which will easily carry out financial calculations and ease the processes concerning any aspect of your finances. You can do the following types of financial calculations: 2 Health calculators Do you own sports, diet, or health websites? Are you involved in health-related services? If yes, then you should start using the Calculator Builder WordPress plugin as a health-related calculator to make your work easier. Examples online calculators The calculator elements of WordPress Calculator Builder include the following: • Radio Button • Dropdown • Checkbox • Number • Textarea • Date • Time • Range Field Types Number – a control used for numbers. When supported, it shows a spinner and applies to default validation. Some devices with dynamic keypads show a numeric keypad. Select – the select element depicts a control with a menu of choices Radio – set the title and select a single value from multiple choices with the same name value – Checkbox – select single values Number and Select – insert Number and Select fields, set the title, choose the addon and write the preferred value Buttons – write the title, and then set the “Calculate” and “Reset” buttons Result – set the field containing the outcome. It is a read-only field. Title – set the title without fields. You can control size and wight of the font; Separator – add the separator to the calculator form as a line Spacer – add space between fieldset Textarea – add textarea to the form Input – you can use the next type of the field: Text, Email, Date, DateTime, Month, Time, Week Range – a slider or dial control. Comparison and Conditional Formula – if(){}, else , if(){}, Else{}, >=, <=, ==, &&, \|\| Math Static Properties – Math.E, Math.LN2, Math.LN10, Math.LOG2E, Math.LOG10E, Math.PI, Math.SQRT1_2, Math.SQRT2 Math Static Methods – Math.pow(), Math.sqrt(), Math.ceil(), Math.round(), Math.random(), Math.max(), Math.min(), Math.log(), Math.abs(), Math.acos(), Math.asin(), Math.atan(), Math.atan2(), Math.cos (), Math.exp(), Math.sin(), Math.tan(), Math.trunc() How to Get Started with the WordPress Calculator Builder plugin PRO Features • Customize the calculator for each individual calculator. • Add the button likes and calculation counter to your calculators and you’ll be able to track it easily. • Easily add an online calculator print button as well as a copy URL button. • Integration WordPress plugin Contact Form 7 with Calculator Builder. Easily send the calculator result by email. Find your Bundle of the Calculators To calculate the result, you must use the variables in the Formula field • Variable x[] – the variable is used for the field that takes part in the calculation • Variable y[] – variable for displaying the result y[1] = x[1] + x[2]; y[1] = x[1] – x[2]; y[1] = x[1] * x[2]; y[1] = x[1] / x[2]; Formula with additional variables You can use the additional variables in the formula field for to facilitate writing the formula and displaying the result. For Example, Formula Monthly payment for Loan: let r = x[2] / 1200; let A = x[1]; let N = x[3]; let result = ( r * A ) / ( 1 - Math.pow((1+r), -N)); y[1] = roundVal(result, 2); roundVal(val, decimals) – function for rounding a number. The first parameter (val) is the number to be rounded, and the second parameter (decimals) is the number of numbers after the decimal point. Conditional formula You can use complex structures to calculate the results. the ability to use the following comparison operators: For Example: if( x[1] < 100 ) { y[1] = x[2] * 2; } else if ( x[1] < 200 ) { y[1] = x[2] * 3; } else { y[1] = x[2] * 4; If you have any questions concerning the plugin ask us at the WordPress forum. Option 1 * Go to the WordPress dashboard * Click “Add New” in the “Plugins” section * Type the plugin name ‘Calculator Builder’ in the search line * Find the plugin and activate Option 2 * Download the ZIP file of the Calculator Builder * Go to the “Plugins” section of the WordPress dashboard * Upload the ZIP file * Activate the “Calculator Builder” Plugin * Build the calculator * Click save * Copy and Paste the shortcode of the calculator where you want it to be * If you want it to appear everywhere on your site, you can insert it for example in your header.php, like this: <?php echo do_shortcode('[Calculator id=1]');?> I recently had the pleasure of working with one of the developers of this WordPress plugin, and I must say, the experience was outstanding! I needed to implement a pet food calculator that dynamically changes field labels based on user input without triggering an immediate calculation. Initially, I struggled with this functionality, but the support I received from the plugin contributor, DmtLo, was exceptional. The support was not only fast but also incredibly helpful, addressing my concerns and providing a solution that was easy to implement. As a result, I was able to further develop and customize the calculator to enhance its functionality. I highly recommend this plugin for anyone in need of a reliable and flexible solution, backed by a responsive and knowledgeable support team. Well Done! thanks for excellent plugin Awesome, easy and looks great Also great support Just starting to use this plugin and have been very encouraged by how responsive the developer has been to questions I’ve had. For something they are spending their time on for free it’s an example to the community. I hope this plugin continues to develop and is able to find a fair way to reward their effort. Works fine, but I left only 4 starts because support is bad. Thanks a lot for this wonderful plugin ! It offers a lot of wonderful features and it is very easy to use it ! 17 (az összes) vélemény olvasása Közreműködők és fejlesztők “Calculator Builder – Create an Online Calculator” egy nyílt forráskódú szoftver. A bővítményhez a következő személyek járultak hozzá: Változási napló • Fixed: main icon in the admin menu. • Fixed: minor bugs • Update: translate file • Fixed: minor bug with saving the calculator • Added: Add the field ‘Time with seconds’ • Added: Value to the field ‘Range’ • Fixed: problem with create the table in database. • Fixed: don’t uncheck the option ‘Calculate when the parameters are changed’ • Fixed: dynamic property for PHP 8.2 • Added: option ‘Perform the calculation only when the button is clicked.’ • Fixed: small issue with shortcode • Changed: links to calculators • Fixed: minor bug in page-list • Added: submit button in the bottom of the form • Added: type ‘required’ for fields: Text, Textarea • Added: option ‘Hide fields of results when changing calculator parameters’ • Optimized: builder script & style • Fixed: minor bug with create variables • Changed: the data table fields type from text to LONGTEXT • Added: possibility resize the form in admin • Fixed: the radio field was omitted in variable field[] • Fixed: checkbox value was sting. Change on number. • Fixed: checkbox value get when the checkbox checked, other = 0; • Added: includes JS and CSS files • Added: support RTL • Added: minification script and style • Added: option for calculation when form load • Added: variables: fieldset, label, field • Added: custom functions: hide, show, addClass, removeClass • Added: button ‘New’ in page created the calculator • Changed: create calculator without the button ‘Calculate’ • Fixed: selected current tag in filter • Fixed: item count in List table • Fixed: Obfuscation function • Fixed: saving parameters in database • Fixed: show calculator on page, custom post • Added: New Fields type: Textarea and Input • Added: new types to Result: HTML block and textarea • Added: tag for calculator • Added: function for copy shortcode • Improvement: plugin admin style • Fixed: builder options ‘addon’ and ‘required’ • Fixed: minor bug • Added: option ‘obfuscation’; • Added: the ability to add more than one calculator per page • Improvement: the work of scripts on the page • Added: New fields: Title, Separator • Added: function for Export/Import calculators • Added: Documentation page • Added: Changelog page • Fixed: saving calculators with conditional symbols • Updated: file for translate .po • Added: link to the Documantation	{"url":"https://hu.wordpress.org/plugins/calculator-builder/","timestamp":"2024-11-06T02:34:52Z","content_type":"text/html","content_length":"150833","record_id":"<urn:uuid:caa299a8-1d20-4d1b-9cf9-da3fb610d7e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00643.warc.gz"}
Multiplication Of Radicals Worksheet Mathworksheets4kids Math, specifically multiplication, develops the foundation of countless scholastic techniques and real-world applications. Yet, for many students, grasping multiplication can posture an obstacle. To address this difficulty, teachers and parents have actually welcomed an effective device: Multiplication Of Radicals Worksheet Mathworksheets4kids. 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naginterfaces.library.mesh.dim2_gen_boundary(coorch, lined, crus, rate, nlcomp, lcomp, nvmax, nedmx, itrace, fbnd=None, data=None, io_manager=None)[source]¶ dim2_gen_boundary generates a boundary mesh on a closed connected subdomain of . For full information please refer to the NAG Library document for d06ba coorchfloat, array-like, shape contains the coordinate of the th characteristic point, for ; while contains the corresponding coordinate. linedint, array-like, shape The description of the lines that define the boundary domain. The line , for , is defined as follows: The number of points on the line, including two end points. The first end point of the line. If , the coordinates of the first end point are those stored in . The second end point of the line. If , the coordinates of the second end point are those stored in . This defines the type of line segment connecting the end points. Additional information is conveyed by the numerical value of as follows: 1. , the line is described in with as the index. In this case, the line must be described in the trigonometric (anticlockwise) direction; 2. , the line is a straight line; 3. if , say (), then the line is a polygonal arc joining the end points and interior points specified in . In this case the line contains the points whose coordinates are stored in , where , , and . crusfloat, array-like, shape The coordinates of the intermediate points for polygonal arc lines. For a line defined as a polygonal arc (i.e., ), if , , for , must contain the coordinate of the consecutive intermediate points for this line. Similarly , for , must contain the corresponding coordinate. ratefloat, array-like, shape is the geometric progression ratio between the points to be generated on the line , for and . If , is not referenced. nlcompint, array-like, shape is the number of line segments in component of the contour. The line of component runs in the direction to if , and in the opposite direction otherwise; for . lcompint, array-like, shape must contain the list of line numbers for the each component of the boundary. Specifically, the line numbers for the th component of the boundary, for , must be in elements to of , where and . The maximum number of the boundary mesh vertices to be generated. The maximum number of boundary edges in the boundary mesh to be generated. The level of trace information required from dim2_gen_boundary. Output from the boundary mesh generator is printed. This output contains the input information of each line and each connected component of the boundary. An analysis of the output boundary mesh is printed on the file object associated with the advisory I/O unit (see FileObjManager). This analysis includes the orientation (clockwise or anticlockwise) of each connected component of the boundary. This information could be of interest to you, especially if an interior meshing is carried out using the output of this function, calling either dim2_gen_inc(), dim2_gen_delaunay() or dim2_gen_front(). The output is similar to that produced when , but the coordinates of the generated vertices on the boundary are also output. You are advised to set , unless you are experienced with finite element mesh generation. fbndNone or callable retval = fbnd(i, x, y, data=None), optional Note: if this argument is None then a NAG-supplied facility will be used. must be supplied to calculate the value of the function which describes the curve on segments of the boundary for which . , the reference index of the line (portion of the contour) described. The values of and at which is to be evaluated. The values of and at which is to be evaluated. dataarbitrary, optional, modifiable in place User-communication data for callback functions. The value of at the specified point. dataarbitrary, optional User-communication data for callback functions. io_managerFileObjManager, optional Manager for I/O in this routine. The total number of boundary mesh vertices generated. coorfloat, ndarray, shape will contain the coordinate of the th boundary mesh vertex generated, for ; while will contain the corresponding coordinate. The total number of boundary edges in the boundary mesh. edgeint, ndarray, shape The specification of the boundary edges. and will contain the vertex numbers of the two end points of the th boundary edge. is a reference number for the th boundary edge and , where and are such that the th edges is part of the th line of the boundary and ; , where and are such that the th edges is part of the th line of the boundary and . (errno ) On entry, the indices of the extremities of line are both equal to . (errno ) On entry, the sum of absolute values of all numbers of line segments is different from . The sum of all the elements of . . (errno ) On entry, . Constraint: . (errno ) On entry, . Constraint: . (errno ) On entry, the geometric progression ratio between the points to be generated on line is . It should be greater than unless the line segment is defined by user-supplied points. (errno ) On entry, the index of the second end point of line is . It should be greater than or equal to and less than or equal to . (errno ) On entry, the index of the first end point of line is . It should be greater than or equal to and less than or equal to . (errno ) On entry, the number of points on line is . It should be greater than or equal to . (errno ) On entry, and . Constraint: . (errno ) On entry, . Constraint: . (errno ) On entry, the line list for the separate connected component of the boundary is badly set: and . It should be less than or equal to and greater than or equal to . (errno ) On entry, end point , with index , does not lie on the curve representing line with index : , , , . (errno ) On entry, end point , with index , does not lie on the curve representing line with index : , , , . (errno ) On entry, and . Constraint: . (errno ) On entry, there is a correlation problem between the user-supplied coordinates and the specification of the polygonal arc representing line with the index in . (errno ) On entry, the absolute number of line segments in the th component of the contour should be less than or equal to and greater than . , and . (errno ) An error has occurred during the generation of the boundary mesh. It appears that is not large enough: . (errno ) An error has occurred during the generation of the boundary mesh. It appears that is not large enough: . (errno ) An error has occurred during the generation of the boundary mesh. Check the definition of each line (the argument ) and each connected component of the boundary (the arguments , and , as well as the coordinates of the characteristic points. Setting may provide more details. Given a closed connected subdomain of , whose boundary is divided by characteristic points into distinct line segments, dim2_gen_boundary generates a boundary mesh on . Each line segment may be a straight line, a curve defined by the equation , or a polygonal curve defined by a set of given boundary mesh points. This function is primarily designed for use with either dim2_gen_inc() (a simple incremental method) or dim2_gen_delaunay() (Delaunay–Voronoi method) or dim2_gen_front() (Advancing Front method) to triangulate the interior of the domain . For more details about the boundary and interior mesh generation, consult the D06 Introduction as well as George and Borouchaki (1998). This function is derived from material in the MODULEF package from INRIA (Institut National de Recherche en Informatique et Automatique). George, P L and Borouchaki, H, 1998, Delaunay Triangulation and Meshing: Application to Finite Elements, Editions HERMES, Paris	{"url":"https://support.nag.com/numeric/py/nagdoc_latest/naginterfaces.library.mesh.dim2_gen_boundary.html","timestamp":"2024-11-12T09:08:03Z","content_type":"text/html","content_length":"307875","record_id":"<urn:uuid:966f0a1f-0491-4006-af07-1fe4f30ce57e>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00115.warc.gz"}
Ronald Fisher Assessment \| Biopsychology \| Comparative \| Cognitive \| Developmental \| Language \| Individual differences \| Personality \| Philosophy \| Social \| Methods \| Statistics \| Clinical \| Educational \| Industrial \| Professional items \| World psychology \| Professional Psychology: Debating Chamber · Psychology Journals · Psychologists File:Ronald Fisher.jpg Sir Ronald Aylmer Fisher, FRS (17 February 1890 – 29 July 1962) was a British eugenicist, evolutionary biologist, geneticist and statistician. He has been described by Richard Dawkins as "The greatest of Darwin’s successors", and the historian of statistics Anders Hald said "Fisher was a genius who almost single-handedly created the foundations for modern statistical science." Biography[ ] Early life[ ] Fisher was born in East Finchley in London, to George and Katie Fisher. His father was a successful businessman, who dealt in fine arts. His boyhood was a happy one, being doted on by his three older sisters, an older brother, and his mother, who died when Fisher was only 14. His father lost his business in several ill-considered transactions only 18 months later. Although Fisher had very bad eyesight, he was a precocious student, winning the Neeld Medal (a competitive essay in Mathematics) at Harrow School at the age of 16. Because of his poor eyesight, he was tutored in mathematics without the aid of paper and pen, which developed his ability to visualize problems in geometrical terms, as opposed to using algebraic manipulations. He was legendary in being able to produce mathematical results without setting down the intermediate steps. He developed a strong interest in biology, and, especially, evolution. He obtained a needed scholarship to the University of Cambridge in 1909 as scholar at Gonville and Caius College. He had a very happy time there, forming many friendships and became enthralled with the heady intellectual atmospherics. In 1911 he became heavily involved in the formation of the Cambridge University Eugenics Society with such luminaries as John Maynard Keynes, R. C. Punnett and Horace Darwin (Charles Darwin's son). The group was quite active, and held monthly meetings, often featuring addresses by leaders of mainstream eugenics organizations, such as the Eugenics Education Society of London, founded by Francis Galton in 1909. By the time Fisher graduated with a degree in mathematics in 1913, it had over 150 members. From the time of graduation when he left Cambridge for a mundane job in London, it would be six years before he found a post that could use his abilities to advantage. Major Leonard Darwin (another of Charles Darwin's sons) and an unconventional and vivacious friend he called Gudruna were almost his only contacts with his Cambridge circle. They sustained him through this difficult period. When the war came, he tried several times to enlist, but was rejected because of his eyesight. For his war work, he took up teaching physics and mathematics at a series of public schools, including Bradfield College in Berkshire. He was miserable, and did poorly at it. A bright spot in his life was that Gudruna matched him to her sister Eileen Guinness. He fell madly in love, and they married when she was only 17 in 1917. With the sisters' help, he set up a subsistence farming operation on the Bradfield estate, where they had a large garden and raised animals, learning to make do on very little. They lived through the war without ever using their food coupons. During this period, Fisher started writing book reviews for the Eugenic Review and gradually increased his interest in genetical and statistical work. He volunteered to undertake all such reviews for the journal, and was hired to a part-time position by Major Darwin. He published several articles on biometry during this period, including the ground-breaking The Correlation to be Expected Between Relatives on the Supposition of Mendalian Inheritance. This paper laid the foundation for what came to be known as biometrical genetics, and introduced the very important methodology of the analysis of variance, which was a considerable advance over the correlation methods used previously. The paper showed very convincingly that the inheritance of continuous variables were consistent with Mendelian principles. With the end of the war he went looking for a new job, and was offered one at the famed Galton Laboratory by Karl Pearson. Because he saw the developing rivalry with Pearson as a professional obstacle, however, he accepted instead a temporary job as a statistician with a small agricultural station in the country in 1919. Early professional years[ ] In 1919 Fisher started work at Rothamsted Experimental Station located at Harpenden in Hertfordshire, England. Here he started a major study of the extensive collections of data recorded over many years. This resulted in a series of reports under the general title Studies in Crop Variation. He was in his prime, and he began a period of amazing productivity. Over the next seven years, he pioneered the principles of the design of experiments and elaborated his studies of analysis of variance. He furthered his studies of the statistics of small samples. Perhaps even more important, he began his systematic approach of the analysis of real data as the springboard for the development of new statistical methods. He began to pay particular attention to the labor involved in the necessary computations, and developed ingenious methods that were as practical as they were founded in rigor. In 1925, this work culminated in the publication of his first book, Statistical Methods for Research Workers. This went into many editions and translations in later years, and became a standard reference work for scientists in many disciplines. In 1935, this was followed by The Design of Experiments, which also became a standard. Fisher invented the techniques of maximum likelihood and analysis of variance, and originated the concepts of sufficiency, ancillarity, Fisher's linear discriminator and Fisher information. His 1924 article "On a distribution yielding the error functions of several well known statistics" presented Karl Pearson's chi-squared and Student's t in the same framework as the normal distribution and his own analysis of variance distribution z. These contributions easily made him a major figure in 20th century statistics. His work on the theory of population genetics also made him one of the three great figures of that field, together with Sewall Wright and J. B. S. Haldane, and as such was one of the founders of the neo-Darwinian modern evolutionary synthesis. Fisher introduced the concept of Fisher information in 1925, many years before Shannon's notion of entropy. Fisher information has been the subject of renewed interest in the last few years, both due to the growth of Bayesian inference in artificial intelligence, and due to B. Roy Frieden's book Physics from Fisher Information, which attempts to derive the laws of physics from a Fisherian starting point. Eugenics[ ] Fisher was an ardent promoter of eugenics, which also stimulated and guided much of his work in genetics of man. His book The Genetical Theory of Natural Selection was started in 1928 and published in 1930. It contained a summary of what was already known to the literature. He developed ideas on sexual selection, mimicry and the evolution of dominance. He famously showed that chance of a mutation increasing the fitness of an organism decreases with the magnitude of the mutation. He also proved that larger populations carry more variation so that they have a larger chance of survival. He set forth the foundations of what was to become known as population genetics. About a third of the book concerned the applications of these ideas to man, and presented what data there was available at the time. He presented a theory that attributed the decline and fall of civilizations to its arrival of a state where the fertility of the upper classes is forced down. Using the census data of 1911 for Britain, he showed that there was an inverse relationship between fertility and social class. Therefore he proposed the abolishment of the economic advantage of small families by instituting subsidies (he called them allowances) to families with larger numbers of children, with the allowances proportional to the earnings of the father. Between 1929 and 1934 the Eugenics Society also campaigned hard for a law permitting sterilization on eugenic grounds. They believed that it should be entirely voluntary, and a right, not a punishment. They published a draft of a proposed bill, and it was submitted to Parliament. Although it was defeated by a 2:1 ratio, this was viewed as progress, and the campaign continued. Fisher played a major role in this movement, and served in several official committees to promote it. In 1934, Fisher moved to increase the power of scientists within the Eugenics Society, but was ultimately thwarted by members with an environmentalist point of view, and he, along with many other scientists, resigned. Method and personality[ ] As an adult, Fisher was noted for his loyalty to his friends. Once he had formed a favorable opinion of any man, he was loyal to a fault. A similar sense of loyalty bound him to his culture. He was a patriot, a member of the Church of England, politically conservative, and a scientific rationalist. Much sought after as a brilliant conversationalist and dinner companion, he very early on developed a reputation for carelessness in his dress and, sometimes, his manners. In later years he was the archetype of the absent-minded professor. Later years[ ] It was Fisher who referred to the growth rate r (used in equations such as the logistic function) as the Malthusian parameter, as a criticism of the writings of Thomas Robert Malthus. Fisher referred to "...a relic of creationist philosophy..." in observing the fecundity of nature and deducing (as Darwin did) that this therefore drove natural selection. He received the recognition of his peers in 1929 when he was inducted into the Royal Society. His fame grew and he began to travel more and lecture to wider circles. In 1931 he spent six weeks at the Statistical Laboratory at Iowa State College in Ames, Iowa. He gave three lectures a week on his work, and met many of the active American statisticians, including George W. Snedecor. He returned again for another visit in 1936. In 1933 he left Rothamsted to become a Professor of Eugenics at University College London. In 1937 he visited the Indian Statistical Institute (in Calcutta), which at the time consisted of one part-time employee, Professor P. C. Mahalanobis. He revisited there often in later years, encouraging its development. He was the guest of honor at its 25th anniversary in 1957 when it had grown to 2000 employees. In 1939 when war broke out, the University tried to dissolve the eugenics department, and ordered all of the animals destroyed. Fisher fought back, but he was exiled back to Rothamsted with a much reduced staff and resources. He was unable to find any suitable war work, and though he kept very busy with various small projects, he became discouraged of any real progress. His marriage disintegrated. His oldest son, a pilot, was killed in the war. In 1943 he was offered the Balfour Chair of Genetics at Cambridge, his alma mater. During the war, this department was also pretty much destroyed, but the University promised him that he would be charged with rebuilding it after the war. He accepted the offer, but the promises were largely unfilled, and the department grew very slowly. A notable exception was the recruitment in 1948 of the Italian researcher Cavalli-Sforza, who established a one man unit of bacterial genetics. His re-appointment was denied by the University in the summer of 1950, much to Fisher's dismay. He now realized he could expect little in the way of support, but he continued his work on mouse chromosome mapping and other projects. They culminated in the publication in 1949 of the idiosyncratic The Theory of Inbreeding. In 1947 he co-founded with Cyril Darlington the journal Heredity: An International Journal of Genetics After retiring from Cambridge in 1957 he spent some time as a senior research fellow at the CSIRO in Adelaide, Australia. He died of colon cancer in 1962. He received many awards for his work and was created a Knight Bachelor by Queen Elizabeth II in 1952. Fisher's important contributions to both genetics and statistics are emphasized by the remark of L.J. Savage, “I occasionally meet geneticists who ask me whether it is true that the great geneticist R.A. Fisher was also an important statistician” (Annals of Statistics, 1976). References[ ] • Fisher, Box Joan (1978) R. A. Fisher: The Life of a Scientist, New York: Wiley, ISBN 0471093009. See also[ ] Bibliography[ ] A selection from Fisher's 395 articles[ ] These are available on the University of Adelaide website: Books by Fisher[ ] Full publication details are available on the University of Adelaide website: Biographies of Fisher[ ] • Fisher, Box Joan (1978) R. A. Fisher: The Life of a Scientist, New York: Wiley, ISBN 0471093009. • Yates F & Mather K (1963) Ronald Aylmer Fisher. Biographical Memoirs of Fellows of the Royal Society of London 9:91-120 Available on University of Adelaide website External links[ ] has a collection of quotations related to:	{"url":"https://psychology.fandom.com/wiki/Ronald_Fisher","timestamp":"2024-11-11T05:12:56Z","content_type":"text/html","content_length":"193910","record_id":"<urn:uuid:b26f2117-06e2-4f9b-b87b-0250985d6cc3>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00137.warc.gz"}
How to perform regression analysis using VAR in STATA - Datapott Analytics The previous article on time series analysis showed how to perform Autoregressive Integrated Moving Average (ARIMA) on the Gross Domestic Product (GDP) of India for the period 1996 – 2016 using STATA . The underlining feature of ARIMA is that it studies the behavior of univariate time series like GDP over a specified time period. Based on that, it recommends an ARIMA equation. This equation then helps to forecast the Gross Domestic Product (GDP) for further years. However, ARIMA is insufficient in defining an econometrics model with more than one variable. For instance, to find the effect of Gross Fixed Capital Formation (GFC) and Private Final Consumption (PFC) on the GDP, ARIMA is not the correct approach. That is where multivariate time series is useful. Consequently, this article explains the process of performing a regression analysis using vector Auto-Regression (VAR) in STATA. Equation of Vector Auto-Regression (VAR) In multivariate time series, the prominent method of regression analysis is Vector Auto-Regression (VAR). It is important to understand VAR for more clarity. Firstly, the term ‘auto-regression’ is used due to the appearance of the lagged value of dependent variables on the right side. Secondly, the term ‘vector’ refers to dealing with the vector of two or more variables. The resultant equation will be as follows: Figure 1: Equation of VAR In the above VAR equation, all three variables are inter-related and simultaneously achieved. Since GFC and PFC play a role in the calculation of GDP, the simultaneity between these variables are To proceed with VAR in STATA, it is important to recognize all the steps, assumptions, and important tests in the process. Steps in performing VAR in STATA As noted in the above equation, the variables are interrelated with lagged values of other variables. However, it is unclear how many lags the variables show 1. Lag selection of interrelation. Therefore, to begin VAR, first, it is imperative to recognize the exact level of lags at which variables are inter-connected or endogenously obtained. 2. Stationarity In the previous articles, the time series data showed that GDP is non-stationary. Therefore it uses the first differencing. The same case could also happen for GFC and PFC. Therefore, the second step would be to check and assure stationarity in data. In the case of co-integration, suppose there are two or more non-stationary variables for regression. While estimating residuals from the regression, the residuals turn out to be stationary. That means, two or more non-stationary series may result in a stationary series. This is called as co-integration. The implication of co-integration 3. Test for Co-integration is that two variables have a long-term casualty and in the long run, the variables might converge towards an equilibrium value. Equilibrium value is steady, therefore they have equal means and variance, or ‘stationary’. Therefore, before initiating VAR, find out if the present model contains any co-integration or equilibrium state. Co-integration indicates a long-term association between two or more non-stationary variables. 4. If Co-integration is not VAR technique where variables are endogenous and dependent on lagged values of other variables. present = We apply VAR. 5. If co-integration is present = apply Vector Error VECM model takes into account the long term and short term causality dynamics. It also offers a possibility to apply VAR to integrated multivariate time series. Correction Model (VECM). 6. VECM diagnostic, tests Based on the constructed VECM model, review the assumptions of autocorrelation and normality, and then proceed to forecast. and forecasting 7. ARCH (Autoregressive Conditionally Time series models incorporating the effects of volatility. Heteroscedastic Model) 8. Extensions of ARCH GARCH (Generalized Autoregressive Conditional Heteroskedasticity) and T-GARCH (Threshold- Generalized Autoregressive Conditional Heteroskedasticity). Table 1: Tests of VAR Models The next article shows the lag selection in a VAR model involving two variables GDP and PFC.	{"url":"https://datapott.com/how-to-perform-regression-analysis-using-var-in-stata/","timestamp":"2024-11-12T23:55:19Z","content_type":"text/html","content_length":"109708","record_id":"<urn:uuid:b33a6224-5c6b-4450-9a2d-8116b1c34328>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00336.warc.gz"}
NetLogo Models Library: IABM Textbook/chapter 3/El Farol Extensions (back to the library) El Farol If you download the NetLogo application, this model is included. You can also Try running it in NetLogo Web This model is from Chapter Three of the book "Introduction to Agent-Based Modeling: Modeling Natural, Social and Engineered Complex Systems with NetLogo", by Uri Wilensky & William Rand. • Wilensky, U. & Rand, W. (2015). Introduction to Agent-Based Modeling: Modeling Natural, Social and Engineered Complex Systems with NetLogo. Cambridge, MA. MIT Press. This model is in the IABM Textbook folder of the NetLogo Models Library. The model, as well as any updates to the model, can also be found on the textbook website: http://www.intro-to-abm.com/. WHAT IS IT? El Farol is a bar in Santa Fe, New Mexico. The bar is popular --- especially on Thursday nights when they offer Irish music --- but sometimes becomes overcrowded and unpleasant. In fact, if the patrons of the bar think it will be overcrowded they stay home; otherwise they go enjoy themselves at El Farol. This model explores what happens to the overall attendance at the bar on these popular Thursday evenings, as the patrons use different strategies for determining how crowded they think the bar will be. El Farol was originally put forth by Brian Arthur (1994) as an example of how one might model economic systems of boundedly rational agents who use inductive reasoning. This is a version of the El Farol model in the Social Science Section of the NetLogo Models Library. This version is intended for use with the IABM textbook. It is NOT intended for textbook users to understand all the code in this model as the point in this section of the textbook is to extend the model by adding new output measures to the model, and not to alter the fundamental algorithm of the An agent will go to the bar on Thursday night if they think that there will not be more than a certain number of people there --- a number given by the OVERCROWDING-THRESHOLD. To predict the attendance for any given week, each agent has access to a set of prediction strategies and the actual attendance figures of the bar from previous Thursdays. A prediction strategy is represented as a list of weights that determines how the agent believes that each time period of the historical data affects the attendance prediction for the current week. One of these weights (the first one) is a constant term which allows the baseline of the prediction to be modified. This definition of a strategy is based on an implementation of Arthur's model as revised by David Fogel et al. (1999). The agent decides which one of its strategies to use by determining which one would have done the best had they used it in the preceding weeks. Interestingly, the optimal strategy from a perfectly rational point-of-view would be to always go to the bar since you are not punished for going when it is crowded, but in Arthur's model agents are not optimizing attending when not crowded, instead they are optimizing their prediction of the attendance. The number of potential strategies an agent has is given by NUMBER-STRATEGIES, and these potential strategies are distributed randomly to the agents during SETUP. As the model runs, at any one tick each agent will only utilize one strategy, based on its previous ability to predict the attendance at the bar. In this version of the El Farol model, agents are given strategies and do not change them once they have them, however since they can change their strategies at any time based on performance, the ecology of strategies being used by the whole population changes over time. The length of the attendance history the agents can use for a prediction or evaluation of a strategy is given by MEMORY-SIZE. This evaluation of performance is carried out in UPDATE-STRATEGIES, which does not change the strategies, but rather updates the performance of each strategy by testing it, and then selecting the strategy that has the best performance given the current data. In order to test each strategy its performance on MEMORY-SIZE past days is computed. To make this work, the model actually records twice the MEMORY-SIZE historical data so that a strategy can be tested MEMORY-SIZE days into the past still using the full MEMORY-SIZE data to make its prediction. HOW TO USE IT The NUMBER-STRATEGIES slider controls how many strategies each agent keeps in its memory. The OVERCROWDING-THRESHOLD slider controls when the bar is considered overcrowded. The MEMORY slider controls how far back, in the history of attendance, agents remember. To run the model, set the NUMBER-STRATEGIES, OVERCROWDING-THRESHOLD and MEMORY size, press SETUP, and then GO. The BAR ATTENDANCE plot shows the average attendance at the bar over time. The green part of the world represents the homes of the patrons, while the blue part of the world represents the El Farol Bar. Over time the attendance will increase and decrease but its mean value comes close to the OVERCROWDING-THRESHOLD. Try running the model with different settings for MEMORY-SIZE and NUMBER-STRATEGIES. What happens to the variability in attendance as you decrease NUMBER-STRATEGIES? What happens to the variability in the plot if you decrease MEMORY-SIZE? Currently the weights that determine each strategy are randomly generated during the model setup. Try altering the weights that are possible during setup so that they only reflect a mix of the following agent strategies: - always predict the same as last week's attendance - an average of the last several week's attendance - the same as 2 weeks ago Can you think of other simple rules one might follow? At the end of Arthur's original paper, he mentions that though he uses a simple learning technique (the "bag of strategies" method that we use here) almost any other kind of machine learning technique would achieve the same results. This method is particularly limiting in that the strategies an agent is given during the setup are all the strategies they have for the entire run of the model. Most other machine learning methods would enable the agents to change their strategies over time. In fact Fogel et al. (1999) implemented a genetic algorithm and got somewhat different results. Try implementing another machine learning technique and see what the results are. Can you think of a better way to measure the success of strategies and calculate the best-strategy? Lists are used to represent strategies and attendance histories. n-values is useful for generating random strategies. Arthur's original model has been generalized as the Minority Game and is included in the Social Sciences section of the NetLogo models library. Wilensky, U. (2004). NetLogo Minority Game model. http://ccl.northwestern.edu/netlogo/models/MinorityGame. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. There is also a participatory simulation version of the Minority Game model in the models library. Stouffer, D. & Wilensky, U. (2004). NetLogo Minority Game HubNet model. http://ccl.northwestern.edu/netlogo/models/MinorityGameHubNet. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. There is an alternative implementation of this model with more parameters that is part of the NetLogo User Community Models. This model is adapted from: Rand, W. and Wilensky, U. (1997). NetLogo El Farol model. http://ccl.northwestern.edu/netlogo/models/ElFarol. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. This model is inspired by a paper by W. Brian Arthur. "Inductive Reasoning and Bounded Rationality", W. Brian Arthur, The American Economic Review, 1994, v84n2, p406-411. David Fogel et al. also built a version of this model using a genetic algorithm. "Inductive reasoning and bounded rationality reconsidered", Fogel, D.B.; Chellapilla, K.; Angeline, P.J., IEEE Transactions on Evolutionary Computation, 1999, v3n2, p142-146. This model is part of the textbook, “Introduction to Agent-Based Modeling: Modeling Natural, Social and Engineered Complex Systems with NetLogo.” If you mention this model or the NetLogo software in a publication, we ask that you include the citations below. For the model itself: • Rand, W., Wilensky, U. (2007). NetLogo El Farol model. http://ccl.northwestern.edu/netlogo/models/ElFarol. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL. Please cite the NetLogo software as: Please cite the textbook as: • Wilensky, U. & Rand, W. (2015). Introduction to Agent-Based Modeling: Modeling Natural, Social and Engineered Complex Systems with NetLogo. Cambridge, MA. MIT Press. Copyright 2007 Uri Wilensky. This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu. (back to the NetLogo Models Library)	{"url":"http://ccl.northwestern.edu/netlogo/models/ElFarol","timestamp":"2024-11-10T20:54:19Z","content_type":"text/html","content_length":"14860","record_id":"<urn:uuid:459ba15f-be2a-4a5f-a447-8871407bbcb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00798.warc.gz"}
Definition 4.5.5.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets. We will say that $q$ is an isofibration if it satisfies the following condition: $(\ast )$ Let $B$ be a simplicial set and let $A \subseteq B$ be a simplicial subset for which the inclusion $A \hookrightarrow B$ is a categorical equivalence. Then every lifting problem \[ \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \] admits a solution.	{"url":"https://kerodon.net/tag/01FV","timestamp":"2024-11-08T18:24:56Z","content_type":"text/html","content_length":"9858","record_id":"<urn:uuid:854142d9-e117-4320-b7e0-06a78f377216>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00150.warc.gz"}
Regarding transmission and reflection of EM-waves • Thread starter Kontilera • Start date In summary, the conversation is about a person seeking help with a problem in the textbook "Classical Electrodynamics" and they are confused about the boundary conditions at the first interface point. After some discussion, they realize that the incident wave is a plane wave and the boundary condition should be E_1 + E_2 = E_3 instead of E_1 + E_2 = E_3 + E_4. I´m trying to read Jacksons 'Classical Electrodynamics' and solving some problems. At the moment I´m stuck at problem 7.3. I started looking at suggested solutions ( ) but I need some help I guess. Looking at how other people have done the boundary conditions at the first interface point (medium to air) with an electric field polarized perpendicular to the plane, they set up the condition of continuity as: E_1 + E_2 = E_3 + E_4. Where E_1 is the incident wave, E_2 the reflected, E_3 the transmitted, and E_4 the reflected wave of the second interface. However the wave E_4 is never actually present at the first interface point! Why should we include this wave in the equation? It is reflected at the second interface point and then travels toward the medium again but at a distant point. Intuitional it seems to me that the boundary condition should be: E_1 + E_2 = E_3. Where does my logic fail? Thanks in advance! However the wave E_4 is never actually present at the first interface point! Are you sure about this? Probably the assignment says that the incoming light wave if a plane wave, which means it has infinite extent in the directions perpendicular to the direction of propagation. Unless the wave is a beam narrower than the gap. Haha of course! I don't know why I was so obsessed with thinking about the whole wave as a arrow... Now everything makes sense, thanks Jano! :-) Last edited: Hello! It's great that you're working on solving problems in Jackson's 'Classical Electrodynamics.' I can understand how you might be confused about the boundary conditions at the first interface point, so let me explain. The equation E_1 + E_2 = E_3 + E_4 is actually the correct condition to use. This is because when an electromagnetic wave hits the first interface, it splits into two waves: one reflected wave (E_2) and one transmitted wave (E_3). The reflected wave (E_4) that is reflected at the second interface also contributes to the overall electric field at the first interface point. Think of it this way: when the electromagnetic wave hits the first interface, some of it is reflected back and some of it passes through and continues into the second medium. The reflected wave (E_4) that is reflected at the second interface then travels back towards the first interface and adds to the electric field there. So, the equation E_1 + E_2 = E_3 + E_4 takes into account all the waves present at the first interface point. I hope this helps clarify things for you. Keep up the good work with your problem solving! FAQ: Regarding transmission and reflection of EM-waves 1. How does the transmission of EM-waves occur? The transmission of EM-waves occurs when the electric and magnetic fields of the wave propagate through a medium or empty space. This propagation is possible due to the alternating nature of the fields, which cause them to continuously induce each other. 2. What factors affect the transmission of EM-waves? The transmission of EM-waves can be affected by the properties of the medium they are passing through, such as its density, composition, and temperature. The frequency and intensity of the wave also play a role in its transmission, as well as any obstacles or obstructions in its path. 3. How is the reflection of EM-waves different from transmission? The reflection of EM-waves occurs when the wave encounters a boundary between two different media and bounces back. This can happen at various angles depending on the angle of incidence and the properties of the media. In contrast, transmission involves the wave passing through the medium without any reflection. 4. Can EM-waves be reflected and transmitted simultaneously? Yes, EM-waves can be both reflected and transmitted when they encounter a boundary between two different media. A portion of the wave may be reflected, while the rest continues to propagate through the medium. This is known as partial reflection and transmission. 5. How does the reflection and transmission of EM-waves impact communication technology? The reflection and transmission of EM-waves play a crucial role in communication technology. For example, radio waves are transmitted and reflected off the Earth's atmosphere to facilitate long-distance communication. Understanding the properties of reflection and transmission allows for the efficient use of EM-waves in various communication technologies, such as satellites and cell	{"url":"https://www.physicsforums.com/threads/regarding-transmission-and-reflection-of-em-waves.663651/","timestamp":"2024-11-08T22:00:18Z","content_type":"text/html","content_length":"88487","record_id":"<urn:uuid:e59f37fc-b109-4627-9443-07f9bb5b4726>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00412.warc.gz"}
Geobound takes a transition class model and a polynomial Lyapunov function as input and symbolically computes the drift, i.e. a multivariate polynomial expressing the expected change in the Lyapunov function for each state. Continuous-time Markov chains (CTMC) are a widely used modelling formalism for systems when probabilistic and timed behavior is of interest. Often, the equilibrium distribution of CTMC can be vital for showing properties about the underlying systems. Many examples can be found in queuing theory and systems biology. So far the models that could be analzed were limited, i.e. the underlying state-space either was finite or the transitions of states had to follow a very restricted pattern. Our tool, Geobound, allows for a more general class of infinite state CTMC, i.e. systems induced by sets of transition classes. The tool first utilizes Lyapunov functions and geometric bounding techniques to retrieve a window inside the state space that encloses a major part of the steady state probability mass and then does a more detailed analysis inside that window revealing even state-wise bounds on the equilibrium distribution. In a nutshell, Geobound takes a transition class model and a polynomial Lyapunov function as input and symbolically computes the drift, i.e. a multivariate polynomial expressing the expected change in the Lyapunov function for each state. Geometric bounds In order to retrieve a window, i.e. a finite subset of the typically infinite state space, that contains at least 1-epsilon of the steady-state probability mass, Geobound scales the drift by that epsilon and the global maximum drift and exploits results by Tweedie [1] that allow the retrieval of the desired window by taking exactly those states that exceed a certain scaled drift value. State-wise bounds In a second analysis step, Geobound computes state-wise bounds on the equilibrium distribution. For that, results by Courtois and Semal [2] are used to retrieve bounds on the steady-state distriubution inside that window conditioned on that window. A more detailed description on the methodology used to retrieve the geometric bounds as well as the state-wise bounds on the equilibrium distribution can be found in the technical report [3]. Geobound uses HOM4PS-2.0 [4] to solve the systems of non-linear equations that emerge when computing the global maximum drift and the SuperLU package in order to compute the state wise bounds. The current version of Geobound is 0.3.3 and is distributed as a pre-built package for Linux 32-bit and comes with HOM4PS2 included. Please follow the installation steps as described in the readme file and make sure that the Java 6 runtime and the Fortran to C (libf2c) library is installed on your system. Geoinf (Geobound + Infamy) The established continuous stochastic logic (CSL) allows to express properties of MPMs, such as probabilistic reachability, survivability, oscillations, switching times between attractor regions, and various others. GeoInf combines the tools Geobound and Infamy to handle complete CSL for MPMs. For a complete description of the used algorithms we refer to our technical report “A CSL Model Checker for Markov Population Models” on arXiv.org. The GeoInf tool chain is based on two tools called Geobound and Infamy that were slightly extended in order to communicate with each other, i.e. Infamy is used to do the model checking and transient analysis while Geobound provides the steady state analysis. Notice: The tools were built on and should be run on recent versions of (for instance, Ubuntu) Linux. More information about Infamy can be found on the Infamy tool homepage. Please download both tools and modify the parameters in the shell scripts shipped along the case studies accordingly to the place where the tools reside. Then, run the shell scripts. Authors: Geobound was written by David Spieler. Infamy was written by Ernst Moritz Hahn. 1. R. L. Tweedie: Sufficient conditions for regularity, recurrence and ergodicity of markov processes. Mathematical Proceedings of the Cambridge Philosophical Society, 78(01):125–136, 1975. 2. P.-J. Courtois and P. Semal: Bounds for the positive eigenvectors of nonnegative matrices and for their approximations by decomposition. J. ACM, 31(4):804–825, 1984. 3. T. Dayar, H. Hermanns, D. Spieler, V.Wolf: Bounding the Equilibrium Distribution of Markov Population Models, 2010, pre-print 4. T. L. Lee, T. Y. Li, C. H. Tsai: Hom4ps-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing, 83(2-3):109–133, 2008.	{"url":"https://mosi.uni-saarland.de/tools/geobound/","timestamp":"2024-11-08T18:26:35Z","content_type":"text/html","content_length":"11196","record_id":"<urn:uuid:7698bf29-4ec1-48ef-bfde-ae77ca9b52f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00345.warc.gz"}
What is Calculation of Equivalent Dose - Problem - Definition What is Calculation of Equivalent Dose – Problem – Definition Calculate the primary photon dose rate, in gray per hour (Gy.h-1), at the outer surface of a 5 cm thick lead shield. Then calculate the equivalent dose rate. Assume that this external radiation field penetrate uniformly through the whole body. Radiation Dosimetry Equivalent dose (symbol H[T]) is a dose quantity calculated for individual organs (index T – tissue). Equivalent dose is based on the absorbed dose to an organ, adjusted to account for the effectiveness of the type of radiation. Equivalent dose is given the symbol H[T]. The SI unit of H[T] is the sievert (Sv) or but rem (roentgen equivalent man) is still commonly used (1 Sv = 100 rem). Unit of sievert was named after the Swedish scientist Rolf Sievert, who did a lot of the early work on dosimetry in radiation therapy. Calculation of Equivalent Dose Rate Assume the point isotropic source which contains 1.0 Ci of ^137Cs, which has a half-life of 30.2 years. Note that the relationship between half-life and the amount of a radionuclide required to give an activity of one curie is shown below. This amount of material can be calculated using λ, which is the decay constant of certain nuclide: About 94.6 percent decays by beta emission to a metastable nuclear isomer of barium: barium-137m. The main photon peak of Ba-137m is 662 keV. For this calculation, assume that all decays go through this channel. Calculate the primary photon dose rate, in gray per hour (Gy.h^-1), at the outer surface of a 5 cm thick lead shield. Then calculate the equivalent dose rate. Assume that this external radiation field penetrate uniformly through the whole body. Primary photon dose rate neglects all secondary particles. Assume that the effective distance of the source from the dose point is 10 cm. We shall also assume that the dose point is soft tissue and it can reasonably be simulated by water and we use the mass energy absorption coefficient for water. See also: Gamma Ray Attenuation See also: Shielding of Gamma Rays The primary photon dose rate is attenuated exponentially, and the dose rate from primary photons, taking account of the shield, is given by: As can be seen, we do not account for the buildup of secondary radiation. If secondary particles are produced or if the primary radiation changes its energy or direction, then the effective attenuation will be much less. This assumption generally underestimates the true dose rate, especially for thick shields and when the dose point is close to the shield surface, but this assumption simplifies all calculations. For this case the true dose rate (with the buildup of secondary radiation) will be more than two times higher. To calculate the absorbed dose rate, we have to use in the formula: • k = 5.76 x 10^-7 • S = 3.7 x 10^10 s^-1 • E = 0.662 MeV • μ[t]/ρ = ^ 0.0326 cm^2/g (values are available at NIST) • μ = 1.289 cm^-1 (values are available at NIST) • D = 5 cm • r = 10 cm The resulting absorbed dose rate in grays per hour is then: Since the radiation weighting factor for gamma rays is equal to one and we have assumed the uniform radiation field, we can directly calculate the equivalent dose rate from the absorbed dose rate as: If we want to account for the buildup of secondary radiation, then we have to include the buildup factor. The extended formula for the dose rate is then: We hope, this article, Calculation of Equivalent Dose – Problem, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.	{"url":"http://www.radiation-dosimetry.org/what-is-calculation-of-equivalent-dose-problem-definition/","timestamp":"2024-11-02T06:05:58Z","content_type":"text/html","content_length":"464613","record_id":"<urn:uuid:d63d952e-5419-44d1-a8c0-05a032bb1734>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00848.warc.gz"}
Calculating the sum and product of the digits of a number. Solving the problem in Python- Devinstructor.com A common task for beginner programmers is to calculate the sum and product of the digits of a number. The number can be input from the keyboard or randomly generated. Here’s how the task is Given a number, find the sum and product of its digits. For example, the sum of the digits of the number 253 is 10 because 2 + 5 + 3 = 10. The product of the digits of the number 253 is 30 because 2 * 5 * 3 = 30. The challenge here is that the number of digits is not known in advance. It could be a three-digit number, as in the example above, or an eight-digit number, or a single-digit number. Generally, it is assumed that this task should be solved arithmetically and using a loop. That is, some arithmetic operations should be performed sequentially on the given number to extract all its digits, then sum and multiply them. This involves using operations of integer division and finding the remainder. If a number is divided by 10, the last digit of the number will be “lost”. For example, 253 ÷ 10 = 25 (remainder 3). On the other hand, the lost digit is the remainder. After extracting this digit, we can add it to the sum of the digits and multiply it by the product of the digits. Suppose n is the number itself, sum is the sum of its digits, and product is the product of the digits. The algorithm to find the sum and product of the digits can be described in words as follows: 1. Assign zero to the variable sum. 2. Assign one to the variable product (since multiplying by zero would result in zero). 3. While the value of the variable n is greater than zero, repeat the following: • Find the remainder when n is divided by 10, i.e., extract the last digit of the number. • Add the extracted digit to the sum and multiply the digit by the product. • Perform integer division of n by 10 to discard the last digit. In Python, the remainder is found using the percent sign (%), and integer division is done using double slashes (//). Here is the Python code for this program: n = int(input("Enter a number: ")) sum = 0 product = 1 while n > 0: digit = n % 10 sum += digit product = digit n //= 10 print("Sum:", sum) print("Product:", product) Example Execution: Enter a number: 253 Sum: 10 Product: 30 The values of the variables can be shortened as follows: while n > 0: digit = n % 10 sum += digit product = digit n //= 10 The above program is only suitable for finding the sum and product of the digits of natural numbers, i.e., integers greater than zero. If the original number can be any integer, negative numbers and zero should be considered. If the number is negative, it does not affect the sum of its digits. In such a case, using the abs function in Python, which returns the absolute value of the argument, will suffice. It will convert the negative number to a positive one, and the loop while n > 0 will work as before. If the number is equal to zero, then according to logic, the sum of its digits and their product should be zero. The loop will not work. Since the initial value of product is 1, you should add a condition to check if the given number is zero. A program that processes all integers can start as follows: n = abs(int(input("Enter a number: "))) sum = 0 product = 1 if n == 0: product = 0 Note that the conditional operator if digit != 0: in Python can be shortened to if digit: because 0 is false. All other numbers are considered true. The mathematical algorithm described above for finding the sum and product of the digits of a number can be called classical or universal. Similarly, you can solve the problem in all imperative languages, regardless of the richness of their toolkit. However, the tools of the programming language may allow you to solve a problem in another way, which is often simpler. For example, in Python, you can avoid converting the entered string to a number but instead extract individual characters from it, which are converted to integers of type int: a = input("Enter a number: ") sum = 0 product = 1 for digit in a: sum += int(digit) product = int(digit) print("Sum:", sum) print("Product:", product) If you add a check to the code to verify that the extracted string character is indeed a digit, the program will become more versatile. With it, you can calculate the sum and product of the digits of not only integers but also real numbers and any string from which digits are extracted. n = input("Enter a string: ") sum = 0 product = 1 for digit in n: if digit.isdigit(): sum += int(digit) product = int(digit) print("Sum:", sum) print("Product:", product) Example Execution: Enter a string: This3 number3is9low! Sum: 15 Product: 81 The string method isdigit checks if the string contains only digits. In our case, the role of the string is played by a single character extracted in the current iteration of the loop. With in-depth knowledge of Python, you can solve the problem in even more sophisticated ways: import functools n = input("Enter a number: ") n = [int(digit) for digit in n if digit.isdigit()] sum = sum(n) product = functools.reduce(lambda x, y: x * y, n) print("Sum:", sum) print("Product:", product) The expression [int(digit) for digit in n if digit.isdigit()] is a list comprehension. If the string “234” is entered, a list of numbers will be obtained: [2, 3, 4]. The built-in sum function returns the sum of the elements given to it as an argument. The function functools.reduce takes two arguments – a lambda expression and, in this case, a list. Here, the variable x accumulates the product, and y takes the next value of the list.	{"url":"https://devinstructor.com/en/python-en/find-the-sum-of-numbers-in-python/","timestamp":"2024-11-08T22:11:48Z","content_type":"text/html","content_length":"88590","record_id":"<urn:uuid:9ca7e912-0733-427d-b260-6b1346c1d216>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00695.warc.gz"}
Fast Code AI What are PINNs? Physics-Informed Neural Networks (PINNs) combine the fundamental laws of physics with the predictive capacity of machine learning, thereby transforming the area of computational science. By integrating differential equations that represent physical rules in standard loss functions, they enable them to predict outcomes for complicated systems when standard models would not be able to. Embedded physical principles steer the network towards physically feasible solutions, minimizing the reliance on big datasets. This makes PINNs particularly useful in situations when data is expensive or limited. In fields where it is essential to comprehend system behavior under varied conditions, such as fluid dynamics, material science, and geophysics, they are very helpful. PINNs application using 2D Heat equation Heat transfer equation in 2D A fundamental concept in thermodynamics and heat transfer, the 2D heat equation shows how heat diffuses through a certain area over time. Traditionally, numerical techniques such as finite element or finite difference analysis have been used to solve problems. Although these techniques are efficient, they can be computationally demanding and time-consuming, particularly when dealing with complicated geometries and boundary conditions. By presenting the 2D heat equation as an optimization problem, we may take use of neural networks' flexibility and effectiveness. The basic goal is to train a neural network without the necessity for a mesh-based discretization of the domain to approximate the temperature distribution in a domain given initial and boundary conditions. Imagine we're working with two outputs, a and b, for which we lack specific target labels. However, we understand these outputs must adhere to the principles of physics. To ensure this adherence, we introduce a penalty in the form of a regularizer whenever a and b deviate from these physical laws. This penalty serves as a guiding signal, allowing the network to learn and internalize the fundamentals of physics. The problem of solving 2D heat equation can be divided into two types: • Stable: Take only spatial coordinates into account to predict the temperature. • Unstable: Take time as an additional input parameter to predict the temperature. Overall Process: • Representation: The neural network takes spatial coordinates (and possibly time) as input and predicts the temperature at those points. • Loss Function: The partial derivative of 2D heat equation is calculated with input and output values and then added to the standard regression loss functions. • Training: Through back-propagation and optimization algorithms, the neural network adjusts its parameters to minimize the loss function, effectively learning the temperature distribution that satisfies the 2D heat equation under the given conditions. Fun Fact: You don’t need very big models or LLMs to train such a network. A simple fully connected network with just 100-150 learnable parameters is sufficient to solve this problem. In our next blog in this series, we will be diving deep in the heat equations for the simple case of a 2D plate and implement a neural network that can learn the physics it in PyTorch. Want to know more about AI ML Technology Incorporate AI ML into your workflows to boost efficiency, accuracy, and productivity. Discover our artificial intelligence services.	{"url":"https://www.fastcode.ai/blogs/pinns","timestamp":"2024-11-02T06:10:27Z","content_type":"text/html","content_length":"34458","record_id":"<urn:uuid:3b6309b8-bbd2-4b07-ac63-3c7622b589b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00011.warc.gz"}
Chapter 2: Exploring Two-Variable Data Notes \| Knowt Two Categorical Variables ➥ Example 2.1 The Cuteness Factor: A Japanese study had 250 volunteers look at pictures of cute baby animals, adult animals, or tasty-looking foods, before testing their level of focus in solving puzzles. The grand total of all cell values, 250 in this example, is called the table total. Pictures viewed is the row variable, whereas level of focus is the column variable. The standard method of analyzing the table data involves first calculating the totals for each row and each column. These totals are placed in the right and bottom margins of the table and thus are called marginal frequencies (or marginal totals). These marginal frequencies can then be put in the form of proportions or percentages. The marginal distribution of the level of focus is: This distribution can be displayed in a bar graph as follows: Similarly, we can determine the marginal distribution for the pictures viewed: The representative bar graph is: Two Quantitative Variables • Many important applications of statistics involve examining whether two or more quantitative (numerical) variables are related to one another. • These are also called bivariate quantitative data sets. • Scatterplot - gives an immediate visual impression of a possible relationship between two variables, while a numerical measurement, called the correlation coefficient - often used as a quantitative value of the strength of a linear relationship. In either case, evidence of a relationship is not evidence of causation. ➥ Example 2.2 Comic books heroes and villains can be compared with regard to various attributes. The scatterplot below looks at speed (measured on a 20-point scale) versus strength (measured on a 50-point scale) for 17 such characters. Does there appear to be a linear association? • Positively associated- When larger values of one variable are associated with larger values of a second variable. • Negatively associated - When larger values of one are associated with smaller values of the other, the variables are called negatively associated. The strength of the association is gauged by how close the plotted points are to a straight line. To describe a scatterplot you must consider form (linear or nonlinear), direction (positive or negative), strength (weak, moderate, or strong), and unusual features (such as outliers and clusters). As usual, all answers must also mention context. Correlation, designated by r, has the formula in terms of the means and standard deviations of the two variables. The correlation formula gives the following: • The formula is based on standardized scores (z-scores), and so changing units does not change the correlation r. • Since the formula does not distinguish between which variable is called x and which is called y, interchanging the variables (on the scatterplot) does not change the value of the correlation r. • The division in the formula gives that the correlation r is unit-free. The value of r always falls between −1 and +1, with −1 indicating perfect negative correlation and +1 indicating perfect positive correlation. It should be stressed that a correlation at or near zero does not mean there is not a relationship between the variables; there may still be a strong nonlinear relationship. Additionally, a correlation close to −1 or +1 does not necessarily mean that a linear model is the most appropriate model. r2 (called the coefficient of determination) - is the ratio of the variance of the predicted values ŷ to the variance of the observed values y. • That is, there is a partition of the y-variance, and r2 is the proportion of this variance that is predictable from a knowledge of x. • We can say that r2 gives the percentage of variation in the response variable, y, that is explained by the variation in the explanatory variable, x. Or we can say that r2 gives the percentage of variation in y that is explained by the linear regression model between x and y. In any case, always interpret r2 in context of the problem. Remember when calculating r from r2 that r may be positive or negative, and that r will always take the same sign as the slope. Alternatively, r2 is 1 minus the proportion of unexplained variance: ➥ Example 2.3 The correlation between Total Points Scored and Total Yards Gained for the 2021 season among a set of college football teams is r = 0.84. What information is given by the coefficient of Solution: r2 = (0.84)2 = 0.7056. Thus, 70.56% of the variation in Total Points Scored can be accounted for by (or predicted by or explained by) the linear relationship between Total Points Scored and Total Yards Gained. The other 29.44% of the variation in Total Points Scored remains unexplained. Least Squares Regression What is the best-fitting straight line that can be drawn through a set of points? By best-fitting straight line we mean the straight line that minimizes the sum of the squares of the vertical differences between the observed values and the values predicted by the line. That is, in the above figure, we wish to minimize It is reasonable, intuitive, and correct that the best-fitting line will pass through ( x̄,ȳ), where x̄ and ȳare the means of the variables X and Y. Then, from the basic expression for a line with a given slope through a given point, we have: The slope b can be determined from the formula where r is the correlation and sx and sy are the standard deviations of the two sets. That is, each standard deviation change in x results in a change of r standard deviations in ŷ. If you graph z -scores for the y-variable against z-scores for the x-variable, the slope of the regression line is precisely r, and in fact, the linear equation becomes, Example 2.4 A sociologist conducts a survey of 15 teens. The number of "close friends" and the number of times Facebook is checked every evening are noted for each student. Letting X and Y represent the number of close friends and the number of Facebook checks, respectively, gives: X: 25 23 30 25 20 33 18 21 22 30 26 26 27 29 20 Y: 10 11 14 12 8 18 9 10 10 15 11 15 12 14 11 1. Identify the variables. 2. Draw a scatterplot. 3. Describe the scatterplot. 4. What is the equation of the regression line? Interpret the slope in context. 5. Interpret the coefficient of determination in context. 6. Predict the number of evening Facebook checks for a student with 24 close friends. 1. The explanatory variable, X, is the number of close friends and the response variable, Y, is the number of evening Facebook checks. 2. Plotting the 15 points (25, 10), (23, 11), . . . , (20, 11) gives an intuitive visual impression of the relationship: 3. The relationship between the number of close friends and the number of evening Facebook checks appears to be linear, positive, and strong. 4. Calculator software gives ŷ= -1.73 + 0.5492x , where x is the number of close friends and y is the number of evening Facebook checks. We can instead write the following: 5. The slope is 0.5492. Each additional close friend leads to an average of 0.5492 more evening Facebook checks. 6. Calculator software gives r = 0.8836, so r2 = 0.78. Thus, 78% of the variation in the number of evening Facebook checks is accounted for by the variation in the number of close friends. 7. –1.73 + 0.5492(24) = 11.45 evening Facebook checks. Students with 24 close friends will average 11.45 evening Facebook checks. • Residual - difference between an observed and a predicted value. • When the regression line is graphed on the scatterplot, the residual of a point is the vertical distance the point is from the regression line. • A positive residual means the linear model underestimated the actual response value. • Negative residual means the linear model overestimated the actual response value. ➥ Example 2.4 We calculate the predicted values from the regression line in Example 2.13 and subtract from the observed values to obtain the residuals: x 30 90 90 75 60 50 y 185 630 585 500 430 400 ŷ 220.3 613.3 613.3 515.0 416.8 351.3 y-ŷ –35.3 16.7 –28.3 –15.0 13.2 48.7 Note that the sum of the residuals is –35.3 + 16.7 – 28.3 – 15.0 + 13.2 + 48.7 = 0 The above equation is true in general; that is, the sum and thus the mean of the residuals is always zero. Outliers, Influential Points, and Leverage • In a scatterplot, regression outliers are indicated by points falling far away from the overall pattern. That is, outliers are points with relatively large discrepancies between the value of the response variable and a predicted value for the response variable. • In terms of residuals, a point is an outlier if its residual is an outlier in the set of residuals. ➥ Example 2.5 A scatterplot of grade point average (GPA) versus weekly television time for a group of high school seniors is as follows: By direct observation of the scatterplot, we note that there are two outliers: one person who watches 5 hours of television weekly yet has only a 1.5 GPA, and another person who watches 25 hours weekly yet has a 3.0 GPA. Note also that while the value of 30 weekly hours of television may be considered an outlier for the television hours variable and the 0.5 GPA may be considered an outlier for the GPA variable, the point (30, 0.5) is not an outlier in the regression context because it does not fall off the straight-line pattern. • Influential Scores - Scores whose removal would sharply change the regression line. Sometimes this description is restricted to points with extreme x-values. An influential score may have a small residual but still have a greater effect on the regression line than scores with possibly larger residuals but average x-values. ➥ Example 2. Consider the following scatterplot of six points and the regression line: The heavy line in the scatterplot on the left below shows what happens when point A is removed, and the heavy line in the scatterplot on the right below shows what happens when point B is Note that the regression line is greatly affected by the removal of point A but not by the removal of point B. Thus, point A is an influential score, while point B is not. This is true in spite of the fact that point A is closer to the original regression line than point B. • A point is said to have high leverage if its x-value is far from the mean of the x-values. Such a point has the strong potential to change the regression line. If it happens to line up with the pattern of the other points, its inclusion might not influence the equation of the regression line, but it could well strengthen the correlation and r2, the coefficient of determination. ➥ Example 2.7 Consider the four scatterplots below, each with a cluster of points and one additional point separated from the cluster. • In A, the additional point has high leverage (its x-value is much greater than the mean x-value), has a small residual (it fits the overall pattern), and does not appear to be influential (its removal would have little effect on the regression equation). • In B, the additional point has high leverage (its x-value is much greater than the mean x-value), probably has a small residual (the regression line would pass close to it), and is very influential (removing it would dramatically change the slope of the regression line to close to 0). • In C, the additional point has some leverage (its x-value is greater than the mean x-value but not very much greater), has a large residual compared to other residuals (so it's a regression outlier), and is somewhat influential (its removal would change the slope to more negative). • In D, the additional point has no leverage (its x-value appears to be close to the mean x-value), has a large residual compared to other residuals (so it's a regression outlier), and is not influential (its removal would increase the y-intercept very slightly and would have very little if any effect on the slope). More on Regression The regression equation 1. If the correlation r = +1, then , and for each standard deviation sx increase in x, the predicted y-value increases by Sy. 2. If, for example, r = +0.4, then , and for each standard deviation sx increase in x, the predicted y-value increases by 0.4 Sy. ➥ Example 2.8 Suppose x = attendance at a movie theater, y = number of boxes of popcorn sold, and we are given that there is a roughly linear relationship between x and y. Suppose further we are given the summary statistics: 1. When attendance is 250, what is the predicted number of boxes of popcorn sold? 2. When attendance is 295, what is the predicted number of boxes of popcorn sold? 3. The regression equation for predicting x from y has the slope: ➥ Example 2.9 Use the same attendance and popcorn summary statistics from Example 2.24 above. 1. When 160 boxes of popcorn are sold, what is the predicted attendance? 2. When 184 boxes of popcorn are sold, what is the predicted attendance? Transformations to Achieve Linearity ➥ Example 2.10 Consider the following years and corresponding populations: Year, x: 1980 1990 2000 2010 2020 Population (1000s), y: 44 65 101 150 230 So, 94.3% of the variability in population is accounted for by the linear model. However, the scatterplot and residual plot indicate that a nonlinear relationship would be an even stronger model.	{"url":"https://knowt.com/note/4c1c8610-55aa-4b07-8675-939e2885a299/Chapter-2-Exploring-Two-Variable-Data","timestamp":"2024-11-09T04:37:44Z","content_type":"text/html","content_length":"236110","record_id":"<urn:uuid:1f84d8b8-c1f1-4b9c-bc65-90c2a1e746e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00258.warc.gz"}
Re: array-fold, array-fold-right, and array-reduce - Simplelists Re: array-fold, array-fold-right, and array-reduce Bradley Lucier 23 Aug 2021 00:51 UTC I began this discussion after reading Joe Marshall's blog post on After looking over fold-{left\|right} in Haskell, OCaml, and SRFI 1, I believe that the definition of fold (fold-left) in SRFI 1 does not conform to the general understanding of this function. SRFI 179 followed SRFI 1's definition of fold to define array-fold. And SRFI 179 seems to have screwed up the definition of array-reduce. In any follow-up SRFI, I think I'd do something like (array-foldl (lambda (accumulate next-element) ...) initial array) (f (f (f ... (f (f init value_0) value_1)...) value_n-1) value_n) (array-foldr (lambda (next-element accumulate) ...) array final) (f value_0 (f value_1 (... (f value_n-1 (f value_n final))))) (array-reduce (lambda (current-value next-element) ...) array) Assumes that current-value and next-element have the same "type", whatever that means. The definition of array-reduce is made a bit easier because the array argument always has elements.	{"url":"https://srfi-email.schemers.org/srfi-179/msg/17428602/","timestamp":"2024-11-04T05:01:17Z","content_type":"text/html","content_length":"13758","record_id":"<urn:uuid:50a15210-8913-4d43-99d1-b1ddbf808cdd>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00752.warc.gz"}
Multiple Charts On One Sheet Excel 2024 - Multiplication Chart Printable Multiple Charts On One Sheet Excel Multiple Charts On One Sheet Excel – You can create a multiplication graph or chart in Stand out using a design. You will discover numerous types of layouts and learn how to format your multiplication graph using them. Here are several tricks and tips to produce a multiplication chart. Once you have a template, all you need to do is backup the method and paste it inside a new mobile. Then you can take advantage of this solution to flourish a series of amounts by another set. Multiple Charts On One Sheet Excel. Multiplication desk template You may want to learn how to write a simple formula if you are in the need to create a multiplication table. First, you have to lock row one of the header line, then flourish the amount on row A by mobile phone B. An alternate way to build a multiplication table is by using blended referrals. In this case, you would key in $A2 into line A and B$1 into row B. The result is really a multiplication kitchen table using a method that really works for both rows and columns. If you are using an Excel program, you can use the multiplication table template to create your table. Just available the spreadsheet along with your multiplication dinner table template and change the brand towards the student’s brand. You can also adjust the page to fit your individual requires. There is an option to modify the shade of the tissues to modify the appearance of the multiplication table, as well. Then, you are able to transform the plethora of multiples to suit your needs. Making a multiplication chart in Shine When you’re making use of multiplication dinner table software, it is simple to build a simple multiplication dinner table in Stand out. Simply create a page with columns and rows numbered from a single to 35. Where columns and rows intersect may be the respond to. If a row has a digit of three, and a column has a digit of five, then the answer is three times five, for example. The same goes for the opposite. Initially, you can enter the numbers that you have to increase. For example, if you need to multiply two digits by three, you can type a formula for each number in cell A1. To help make the figures larger, select the cellular material at A1 and A8, and after that go through the appropriate arrow to decide on a selection of cells. You can then type the multiplication formulation within the cells from the other rows and columns. Gallery of Multiple Charts On One Sheet Excel Multiple Bar Charts On One Axis In Excel Super User How To Quickly Make Multiple Charts In Excel YouTube Adding Multiple Tables On One Excel Sheet Onto Tableau Stack Overflow Leave a Comment	{"url":"https://www.multiplicationchartprintable.com/multiple-charts-on-one-sheet-excel/","timestamp":"2024-11-06T07:46:19Z","content_type":"text/html","content_length":"52724","record_id":"<urn:uuid:12bc3410-4df4-413f-a295-46dec1e6a86f>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00762.warc.gz"}
Nakul Aggarwal -- April 25 Discovering Factorization Surface of Quantum Spin Chains with Machine Learning Entanglement in quantum many-body systems is required for a variety of quantum information tasks, making it crucial to identify the parameter space in which the ground state is fully separable, known as the factorization surface (FS). We employ symbolic regression (SR) to determine a closed-form expression in the parameter regime corresponding to FS of quantum many-body Hamiltonians. We verify the effectiveness of this method by examining the nearest-neighbor (NN) quantum transverse XY model with additional Kaplan-Shekhtman-Entin-Aharony interactions, for which the FS is well-known. We construct an accurate expression for the FS of the XYZ model and estimate the FS for the long-range XY model. We finally discover FS for the NN XY model with Dzyaloshinskii–Moriya type asymmetric interaction for which FS remains unknown till date.	{"url":"https://www.physics.rutgers.edu/gso/SSPAR/","timestamp":"2024-11-05T11:03:21Z","content_type":"text/html","content_length":"172716","record_id":"<urn:uuid:5259d087-b5ac-42b7-8fee-140863f8ad10>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00419.warc.gz"}
uicksort in python QuickSort is a sorting algorithm that is widely used in the industry. It is fast, efficient, and powerful. It uses the divide-and-conquer technique to divide the array into sub-arrays, sort them, and then recursively sort the larger arrays. In this article, we will read quicksort python in detail. We will code it in both Python2 and Python3. Let's get started! Must Recommended Topic, Floor Division in Python, Swapcase in Python Also See, Divmod in Python What is QuickSort in Python? Quicksort is a widely used sorting algorithm that follows the divide and conquer principle. This sorting algorithm works with large data sets but varies from scenario to scenario. The algorithm takes an array as input, and in the quick sort algorithm, you select a pivot element which is nothing but a random number in an array. This number plays a vital role in the performance of the algorithm. Once you select the pivot elements, the elements smaller than pivot are moved toward the left, and numbers greater than the pivot are moved to the right. This process continues by dividing the array into subarrays. In the last, the array is sorted. Also see, Python Operator Precedence Quick Sort Algorithm Let's take a look at the quick sort algorithm. 1. Choose a pivot element - it can be any element. It is preferable that the element is chosen so that roughly half of the elements are smaller than it, and roughly half are bigger than it. 2. Rearrange the elements of the array such that all the elements smaller than the pivot are on its left, and all the elements larger than the pivot are on its right. The pivot is now in its correct position in the sorted array. This is called the partition step. 3. Recursively call quick sort on the elements on the left side of the pivot, and do the same for the elements on the right side of the pivot element. You can also read about Multilevel Inheritance in Python, Fibonacci Series in Python Coding QuickSort in Python We will code quicksort in Python 3 as well as in Python 2. Coding quicksort in Python, or any other language involves applying the concepts we studied above in the language. QuickSort in Python 3 # The partition function, which returns the position of the pivot element. def partition(nums:list[int],start:int,end:int)->int: #the pivot element. We are taking the end element as pivot. pivot:int = nums[end]; # i, which will help us find the final position of the pivot element. i:int = start-1; for j in range(start,end): #if the current element is smaller than the pivot if (nums[j]<pivot): # increment i #swap elements at i and j # finally move the pivot element to its correct position #return the final position of the pivot element return i; #the recursive function that performs quicksort in python 3 #start is the start index of the subarray to sort #end is the end index of the subarray to sort def quicksort(nums:list[int],start:int,end:int): # if start is not less than end, it means the subarray has 0 or 1 elements, and doesn't need # to be sorted. if (start<end): # find the position of the pivot element pivot:int = partition(nums,start,end); #recursively sort left and right subarrays for pivot #utility function to call quick sort def sortArray(nums:list[int])->list[int]: arr = [5,3,2,6,1,4] You can also try this code with Online Python Compiler Run Code QuickSort in Python 2 # The partition function, which returns the position of the pivot element. def partition(nums,start,end): #the pivot element. We are taking the end element as pivot. pivot = nums[end]; # i, which will help us find the final position of the pivot element. i = start-1; for j in range(start,end): #if the current element is smaller than the pivot if (nums[j]<pivot): # increment i #swap elements at i and j # finally move the pivot element to its correct position #return the final position of the pivot element return i; #the recursive function that performs quicksort in python 2 #start is the start index of the subarray to sort #end is the end index of the subarray to sort def quicksort(nums,start,end): # if start is not less than end, it means the subarray has 0 or 1 elements, and doesn't need # to be sorted. if (start<end): # find the position of the pivot element pivot = partition(nums,start,end); #recursively sort left and right subarrays for pivot #utility function to call quick sort def sortArray(nums): arr = [5,3,2,6,1,4] You can also try this code with Online Python Compiler Run Code You can practice by yourself with the help of online python compiler. Time and Space complexity of Quicksort Time Complexities Let's discuss the worst, best, and average time complexities of the quicksort algorithm in Python. Worst Case Complexity is O(n2) Base Case Complexity is O(nlogn) Average Case Complexity O(nlogn) Space Complexity The space complexity for quicksort is O(1) because it is an in-place algorithm. Recursion stack space is not counted here. If we count the stack space, the time complexity will be O(logn) in the average case and O(n) in the worst case. Also see, Convert String to List Python Comparison of Quicksort to Other Sorting Algorithms Quick sort is a sorting algorithm with its own pros and cons. Let's try to better understand this algorithm by comparing it with several other sorting algorithms. When we compare quick sort with insertion sort: • The average time complexity of quicksort is O(nlogn). • The average time complexity of insertion sort is O(n2). • Quick sort works well with large data sets and unsorted data. • Insertion sort works well with small data sets or partially sorted data. When we compare quicksort with Merge sort: • Quicksort works on the principle of divide and conquer but does not require additional space. • Merge sort also works on the same principle but requires additional space. When we compare quicksort with Bubble sort: • Quicksort is best suited to large data sets and is faster than bubble sort. • Bubble sort is an easy-to-understand sorting algorithm but is impractical for large data sets. Quicksort Optimization Techniques There are several techniques to optimize the quicksort algorithm. These techniques help to optimize the performance of the algorithm. The following optimization ways can be used to implement the quicksort technique. 1. A tail recursive optimization technique can be chosen for the recursive calls as it helps control the stack overflow error. A tail recursive technique can be used to implement a quick sort 2. Choosing correctly the pivot element. It is important to select the pivot element correctly as it impacts the performance. 3. A hybrid choice of sorting algorithm can also be implemented. When implemented in small data sets, insertion sort or any other algorithm can be used. Advantages and Disadvantages of Quicksort in python 1. Quiksort algorithm is best suited for large data sets as it works on the principle of divide and conquer. This helps divide the array into smaller subarrays and achieves better performance. 2. The Quicksort algorithm does not use any additional memory consumption to sort the array. 3. The average time complexity of the quick sort algorithm is O(nlogn), which is better than that of other sorting techniques. 1. The worst-case time complexity of quick sort is O(n2), which happens when pivot selection does not divide the array into the correct partition. 2. The algorithm is not stable which means it can change the order of the elements. 3. The algorithm does not work well with partially sorted datasets. Frequently Asked Questions Can we make quicksort stable? One way to do so is by using extra space. We can partition by making lists for elements smaller than, equal to, and greater than the pivot. Then we can concatenate them. However, we should just use merge sort if we are fine with using extra space. What is the memory complexity of quicksort? Quick sort does not require any additional memory space to implement the algorithm. The memory complexity of quicksort is O(1). It uses a constant memory space regardless of the size of an array. What is the best case running time of quicksort? The best-case running time of quicksort happens when the correct selection of pivot happens. the best time complexity of quicksort algorithm is O(nlogn), which is their average time complexity. In this blog, we have gone through the algorithm for quicksort Python. We have also seen a code for quicksort in Python 3 and in Python2. At the end, we discussed the time complexities. We hope you leave this article with a broader knowledge of Data Structures, algorithms, and sorting.Recommended Readings: Recommended Problem - If you liked our article, do upvote our article and help other ninjas grow. You can refer to our Guided Path on Coding Ninjas Studio to upskill yourself in Data Structures and Algorithms, Competitive Programming, System Design, and many more! Keep coding, keep reading Ninjas.	{"url":"https://www.naukri.com/code360/library/how-to-implement-quick-sort-in-python","timestamp":"2024-11-08T02:37:25Z","content_type":"text/html","content_length":"413410","record_id":"<urn:uuid:3de27a52-b6c1-4223-a127-19df6046905b>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00008.warc.gz"}
John F. Nash's Work in Pure Mathematics It’s fair to say that John F. Nash Jr. (1928-2015) didn’t earn much respect from pure mathematicians for his work in game theory. To them, his use of L.E.J. Brouwer (1881-1966) and Shizuo Kakutani (1911-2004)’s fixed-point theorems to show the existence of Nash equilibria in finite games with mixed strategies was and perhaps still is merely considered “not surprising applications of well-known methods” (Milnor, 1998). Indeed, even John von Neumann (1903-1957)—when confronted with Nash’s result—is to have replied “That’s trivial, you know. That’s just a fixed point theorem”. Although—by far—recognized most widely for the work he did in game theory, Nash in the 1950s however also wrote groundbreaking papers on real algebraic geometry, topology and partial differential equations. Indeed, he won the Abel Prize in 2015 for his work on the latter, alongside Louis Nirenberg (1926-2020), considered one of the mos… This post is for paid subscribers	{"url":"https://www.privatdozent.co/p/john-f-nashs-work-in-pure-mathematics","timestamp":"2024-11-13T04:54:20Z","content_type":"text/html","content_length":"144461","record_id":"<urn:uuid:c2f78510-6527-4ce5-9f89-1fd7bc2f71ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00607.warc.gz"}
Easy Refresher: Asset Depreciation • Stephen L. Nelson Asset depreciation answers the question, “How much does the asset cost per period?” To answer that question, you need to: know the cost of the asset, estimate the number of periods in the asset’s useful life, project any salvage value the asset will have at the end of its useful life, and choose a depreciation method. For example, suppose your business purchases a delivery truck for $10,000, uses it for five years, and then sells it for $2,000. Using the simplest depreciation method, straight-line depreciation, you calculate the cost of the truck over the five years as the $10,000 original cost less the $2,000 salvage value for a result of $8,000. Now divide the $8,000 by the five years of useful life. The result—$1,600—is the depreciation expense. Straight-line depreciation is the most popular method because it’s easy to apply and intuitive. The other methods—declining balance, sum-of-the-years’-digits, annuity/sinking fund, and activity—simply allocate the asset cost over the asset’s useful life in different ways. The declining balance depreciation method expenses more of the cost of the asset in the early periods of an asset’s estimated life than in the later periods. It does so using the following formula: (Declining Balance Percentage)(Net Book Value)/(Estimated Life) For example, suppose you want to recalculate the first year’s depreciation expense for the delivery truck using 200% declining balance depreciation. The declining balance percentage is 200%. The net book value, because no depreciation has occurred, is $10,000. The estimated life is five years. Accordingly, you calculate the first year’s depreciation expense as: 200%($10,000/5 years) The declining balance percentage is always greater than 100%. Accordingly, the formula accelerates the depreciation of an asset. Often, federal and state income tax laws determine usage of the declining balance depreciation method. Tax laws allow declining balance depreciation for many types of assets and specify a variety of declining balance percentages to be used, including 125, 150, 175, and 200%, depending on which year you acquire and begin using an asset. Generally, the tax law in effect when you buy and begin using an asset determines the types of assets for which you can use the declining balance method, as well as the declining balance percentage. The sum-of-the-years’-digits depreciation method, like the declining balance method, also expenses more of the cost of an asset in the early periods of an asset’s estimated life than in the later periods. It does so using the following formula: (Periods Left in Estimated Life)/(Sum of the Periods? Digits)(Original Cost-Salvage Value) For example, suppose you want to recalculate the first year’s depreciation expense for the delivery truck using the sum-of-the-years’-digits method. The periods left in the estimated life, because the asset is still new, is five years. The sum of the periods’ (or years’) digits is 1+2+3+4+5, or 15. The original cost less the salvage value is $10,000–$2,000, or $8,000. Accordingly, you calculate the first year’s depreciation expense as: Because the fraction becomes smaller in each succeeding period, the amount of depreciation expensed each year becomes smaller. The annuity and sinking fund depreciation methods are mechanically identical, so this book supplies the same starter workbook for both. Both of these methods expense less of the cost of an asset in the early periods of an asset’s life than in the later periods, so they are roughly the opposite of the declining balance and sum-of-the-years’-digits methods in this regard. The annuity and sinking fund methods also include in their depreciation expenses a specified return on the investment. Generally, the annuity and sinking fund methods violate the Generally Accepted Accounting Principles (GAAP). (Generally Accepted Accounting Principles are the rules and methods that certified public accountants, with help from business and the government, develop and use for financial accounting. Usually, when people refer to Generally Accepted Accounting Principles, they mean the pronouncements of the Financial Accounting Standards Board, an independent professional group.) Because they are contrary to GAAP and because they are complex, these methods are rarely used in practice except in heavily regulated industries such as public utilities in which governmental rate-setting agencies often specify returns on investment. The annuity and sinking fund depreciation methods use the following formula to calculate depreciation expenses: (Original Cost?(Present Value of the Salvage Value))/(Present Value Factor of an Ordinary Annuity for n Periods at i%) where n equals the estimated life, and i equals the specified return on investment. For example, suppose you want to recalculate the first year’s depreciation expense for the delivery truck using the annuity or sinking fund method. Also suppose that you are assured a 10% return on assets by a state regulatory agency. The 10% is the specified return on investment. The original cost is $10,000. The estimated life is five years. The present value of the salvage value is calculated as follows: For each year in the asset’s estimated life, the $2,000 salvage value is divided by the sum of 1 plus the specified return on investment, or (1+10%)^5, or $1,241.84. You can then calculate the present value of an ordinary ($1) annuity for 5 periods using a 10% discount rate using the PV function as follows: for a result of 3.7908. Accordingly, you calculate the depreciation expense as: This depreciation amount also includes assumed investment revenue of 10% on the asset cost of $10,000, or $1,000, meaning the actual amount of the asset being expensed in this period is $2,310.37 minus $1,000 or $1,310.37. In other words, the depreciated value of the truck after one year under this method is $8,689.63. As the net book value of the asset becomes smaller over its useful life, the assumed investment revenue becomes smaller. Consequently, the $2,310.37 of depreciation represents less assumed investment revenue and more actual asset being expensed. The assumed investment revenue amounts to the assumed return on assets allowed by the regulatory agency. The activity method depreciates an asset as it’s used, instead of as time passes, by calibrating the estimated life of an asset in units of use. It does so by using the following formula: (Period Units of Use/Estimated Life in Units of Use)(Original Cost-Salvage Value) For example, suppose you want to recalculate the first year’s depreciation expense for the delivery truck using the activity depreciation method. If a delivery truck lasts for 100,000 miles and you anticipate driving the truck 30,000 miles the first year, you calculate the first year’s depreciation expense as: In general, financial accounting standards and the tax laws guide you in determining asset cost, useful life, and salvage value and in selecting a depreciation method. Accordingly, if you’re building a depreciation schedule to use for tax accounting, your best resources are the publications of the Internal Revenue Service and your tax adviser. Alternatively, if you’re building a depreciation schedule to use for financial accounting, your best resources are the publications of the Financial Accounting Standards Board and your certified public accountant. Note: Be consistent in the financial measurement periods you use in depreciating assets. If you’re building a monthly forecast, calculate depreciation expenses on a monthly basis and enter the useful life in months. Alternatively, if you’re building a quarterly or yearly forecast, calculate your depreciation expenses on a quarterly or yearly basis and enter the estimated life of an asset in quarters or years.	{"url":"https://stephenlnelson.com/articles/asset-depreciation/","timestamp":"2024-11-11T17:17:55Z","content_type":"text/html","content_length":"58268","record_id":"<urn:uuid:9c904283-77fa-41b4-8006-d244105aa391>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00847.warc.gz"}
How to Calculate the Expected Profit From Stock Option Spreads? To calculate the expected profit from stock option spreads, you first need to determine the potential profit and loss at various price levels of the underlying stock. This involves analyzing the strike prices and expiration dates of the options involved in the spread. Next, calculate the net debit or credit of the spread by subtracting the cost of purchasing the options from the premium received from selling the options. This will give you the initial investment required for the spread. Then, determine the breakeven point of the spread by adding or subtracting the net debit/credit from the strike price of the options. After that, calculate the maximum profit potential of the spread by subtracting the net debit from the difference in strike prices of the options. To calculate the expected profit, you can use statistical methods based on the probability of the underlying stock reaching certain price levels. This may involve using option pricing models and historical data to estimate the likelihood of different outcomes. Keep in mind that the expected profit is just an estimate based on certain assumptions and projections, so it's important to consider factors such as market conditions, volatility, and timing when making investment decisions involving option spreads. What is the impact of dividends on stock option spread profitability? Dividends can have a significant impact on stock option spread profitability. When a company pays out a dividend, the stock price typically drops by the amount of the dividend on the ex-dividend date. This can affect the profitability of option spreads in different ways: 1. Long Call Spreads: If you have a long call spread, the drop in stock price due to the dividend payment can reduce the profitability of the trade if the stock price falls below the lower strike price of the spread. This can result in a loss if the stock price is below the breakeven point. 2. Short Call Spreads: On the other hand, if you have a short call spread, the drop in stock price due to the dividend can actually increase the profitability of the trade, as long as the stock price remains above the upper strike price of the spread. This is because the short call option loses value as the stock price declines. 3. Long Put Spreads: For long put spreads, the impact of dividends can be similar to long call spreads. The drop in stock price can reduce the profitability of the trade if the stock price falls below the higher strike price of the spread. 4. Short Put Spreads: With short put spreads, the drop in stock price due to the dividend can increase the profitability of the trade, as long as the stock price remains above the lower strike price of the spread. This is because the short put option gains value as the stock price declines. Overall, dividends can have a complex impact on the profitability of stock option spreads, depending on the direction of the trade and the specific strike prices involved. Traders should carefully consider the timing of dividend payments and their potential impact on their option spread positions. What is the relationship between volatility and expected profit from option spreads? The relationship between volatility and expected profit from option spreads is generally positive. Higher levels of volatility typically result in wider price swings, which can increase the potential profit from option spreads. This is because higher volatility leads to higher option premiums, making it more lucrative for traders to profit from spreads. On the other hand, lower levels of volatility can reduce the profit potential from option spreads as prices are less likely to make large movements. In this case, traders may need to adjust their strategies or consider different types of spreads to maximize profitability. Overall, volatility plays a significant role in determining the expected profit from option spreads, making it an important factor for traders to consider when designing their trading strategies. How to use technical analysis to enhance stock option spread profitability? Technical analysis can be used to enhance the profitability of stock option spreads by providing traders with actionable insights on potential price movements in the underlying security. Here are some ways to incorporate technical analysis into your options trading strategy: 1. Confirming entry and exit points: Technical analysis can help traders identify optimal entry and exit points for their spreads by analyzing key support and resistance levels, trend lines, and other technical indicators. By using technical analysis to time your trades, you can increase the likelihood of profiting from your spread positions. 2. Identifying trends: Technical analysis can help traders identify trends in the underlying stock or index, which can be valuable information when selecting the appropriate spread strategy. For example, if a stock is in a strong uptrend, you may want to consider using bullish spread strategies such as call spreads or bull put spreads to capitalize on the upward momentum. 3. Monitoring volatility: Technical indicators such as Bollinger Bands, Average True Range (ATR), and the Relative Strength Index (RSI) can help traders gauge the level of volatility in the market. By using these indicators to assess market volatility, traders can adjust their options strategies accordingly to account for potential price fluctuations. 4. Confirming market sentiment: Technical analysis can also help traders gauge market sentiment by analyzing volume patterns, price action, and other technical indicators. By monitoring market sentiment, traders can make more informed decisions about the direction of the underlying security and adjust their spread positions accordingly. Overall, incorporating technical analysis into your options trading strategy can help improve the profitability of your spread positions by providing valuable insights on potential price movements and market trends. It is important to use technical analysis in conjunction with other forms of analysis, such as fundamental analysis, to make well-informed trading decisions. What is the role of time decay in stock option spread profitability? Time decay, or theta decay, is the phenomenon where the value of an option decreases as it approaches its expiration date. This is because options have a limited lifespan, and as time goes on, there is less time for the option to potentially move in the desired direction. In stock option spread strategies, time decay can play a significant role in profitability. When establishing an option spread, the trader is essentially betting on the underlying stock's price movement within a specific time frame. As time passes, the option's value will decrease due to time decay, and if the underlying stock does not move as anticipated, the option spread may start to lose value. However, time decay can also work in favor of option spread traders. If the underlying stock stays relatively stagnant or moves in the expected direction, the option spread can benefit from time decay as the options lose value. This can result in the trader being able to close the spread for a profit before expiration. Overall, understanding and managing time decay is crucial in determining the profitability of stock option spread strategies. Traders need to consider the impact of time decay when selecting their option strike prices, expiration dates, and overall strategy to maximize their chances of success.	{"url":"https://coding.ignorelist.com/blog/how-to-calculate-the-expected-profit-from-stock","timestamp":"2024-11-02T05:25:00Z","content_type":"text/html","content_length":"150612","record_id":"<urn:uuid:e7158736-a021-4ec0-b35a-93ecf7700050>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00318.warc.gz"}
A potential difference varying at the rate of 6 times class 12 physics JEE_Main Hint: To solve this question, we need to find out the displacement current because of the changing potential difference between the plates of the capacitor. Then using the formula for the magnetic field due to an infinitely long current carrying wire, we can get the final answer. Formula used: The formula used to solve this question is given by $C = \dfrac{{{\varepsilon _0}A}}{d}$, here $C$ is the capacitance of a parallel plate capacitor having plates of area $A$ separated by a distance of $d$. Complete step-by-step solution: We know that the capacitance of a parallel plate capacitor is given by $C = \dfrac{{{\varepsilon _0}A}}{d}$............(1) We know that the area of a circular plate is given by $A = \pi {r^2}$ Putting this in (1) we get $C = \dfrac{{{\varepsilon _0}\pi {r^2}}}{d}$ $ \Rightarrow d = \dfrac{{{\varepsilon _0}\pi {r^2}}}{C}$ ………...(2) Now, we know that $E = \dfrac{V}{d}$ Differentiating both sides with respect to time t, we have $\dfrac{{dE}}{{dt}} = \dfrac{1}{d} \times \dfrac{{dV}}{{dt}}$ Substituting (2) in the above equation we get $\dfrac{{dE}}{{dt}} = \dfrac{C}{{{\varepsilon _0}\pi {r^2}}}\dfrac{{dV}}{{dt}}$...............(3) Now, we know that the displacement current is given by ${i_d} = {\varepsilon _0}\dfrac{{d{\varphi _E}}}{{dt}}$...............(4) The electric flux is given by ${\varphi _E} = EA$ $ \Rightarrow {\varphi _E} = \pi {r^2}E$ Putting this in (4) we have ${i_d} = {\varepsilon _0}\dfrac{{d\left( {\pi {r^2}E} \right)}}{{dt}}$ $ \Rightarrow {i_d} = {\varepsilon _0}\pi {r^2}\dfrac{{dE}}{{dt}}$........(5) Now, the magnetic field at the edge of the plate can be given by $B = \dfrac{{{\mu _0}{i_d}}}{{2\pi r}}$ Putting (5) in the above expression we get $B = \dfrac{{{\mu _0}{\varepsilon _0}\pi {r^2}}}{{2\pi r}}\dfrac{{dE}}{{dt}}$ $ \Rightarrow B = \dfrac{{{\mu _0}{\varepsilon _0}r}}{2}\dfrac{{dE}}{{dt}}$ Putting (3) in the above expression we get $B = \dfrac{{{\mu _0}{\varepsilon _0}r}}{2}\dfrac{C}{{{\varepsilon _0}\pi {r^2}}}\dfrac{{dV}}{{dt}}$ $ \Rightarrow B = \dfrac{{{\mu _0}C}}{{2r}}\dfrac{{dV}}{{dt}}$.............(6) Now, according to the question, we have the radius of the plate equal to $10cm$. So we have $r = 10cm$ $ \Rightarrow r = 0.1m$ …………..(7) Also, the capacity of the capacitor is given to be equal to \[2\mu F\]. So we have $C = 2\mu F$ $ \Rightarrow C = 2 \times {10^{ - 6}}F$..........(8) Also the variation of voltage with time is given to be equal to $6 \times {10^2}{\text{V}}{{\text{s}}^{ - 1}}$. So we have $\dfrac{{dV}}{{dt}} = 6 \times {10^2}{\text{V}}{{\text{s}}^{ - 1}}$............(9) Substituting (7), (8), and (9) in (7), we get the final value of the magnetic field as $B = 3.33 \times {10^{ - 9}}T$ Hence, the correct answer is option C. Note: The magnetic field obtained from the displacement current is similar to that of the magnetic field produced by an infinitely long current carrying wire. The displacement current passes through the centre of the plates of the capacitor, and hence we obtained the magnetic field at the edge of the plate.	{"url":"https://www.vedantu.com/jee-main/a-potential-difference-varying-at-the-rate-of-6-physics-question-answer","timestamp":"2024-11-14T17:48:30Z","content_type":"text/html","content_length":"160471","record_id":"<urn:uuid:5b8bf0fd-4f64-4b8f-9d94-eab40effd5ea>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00432.warc.gz"}
Tackling Heavy-Tailed Rewards in Reinforcement Learning with Function Approximation: Minimax Optimal and Instance-Dependent Regret Bounds Part of Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track Jiayi Huang, Han Zhong, Liwei Wang, Lin Yang While numerous works have focused on devising efficient algorithms for reinforcement learning (RL) with uniformly bounded rewards, it remains an open question whether sample or time-efficient algorithms for RL with large state-action space exist when the rewards are \emph{heavy-tailed}, i.e., with only finite $(1+\epsilon)$-th moments for some $\epsilon\in(0,1]$. In this work, we address the challenge of such rewards in RL with linear function approximation. We first design an algorithm, \textsc{Heavy-OFUL}, for heavy-tailed linear bandits, achieving an \emph{instance-dependent} $T$-round regret of $\tilde{O}\big(d T^{\frac{1-\epsilon}{2(1+\epsilon)}} \sqrt{\sum_{t=1}^T \nu_t^2} + d T^{\frac{1-\epsilon}{2(1+\epsilon)}}\big)$, the \emph{first} of this kind. Here, $d$ is the feature dimension, and $\nu_t^{1+\epsilon}$ is the $(1+\epsilon)$-th central moment of the reward at the $t$-th round. We further show the above bound is minimax optimal when applied to the worst-case instances in stochastic and deterministic linear bandits. We then extend this algorithm to the RL settings with linear function approximation. Our algorithm, termed as \textsc {Heavy-LSVI-UCB}, achieves the \emph{first} computationally efficient \emph{instance-dependent} $K$-episode regret of $\tilde{O}(d \sqrt{H \mathcal{U}^} K^\frac{1}{1+\epsilon} + d \sqrt{H \mathcal {V}^ K})$. Here, $H$ is length of the episode, and $\mathcal{U}^, \mathcal{V}^$ are instance-dependent quantities scaling with the central moment of reward and value functions, respectively. We also provide a matching minimax lower bound $\Omega(d H K^{\frac{1}{1+\epsilon}} + d \sqrt{H^3 K})$ to demonstrate the optimality of our algorithm in the worst case. Our result is achieved via a novel robust self-normalized concentration inequality that may be of independent interest in handling heavy-tailed noise in general online regression problems.	{"url":"https://proceedings.neurips.cc/paper/2023/hash/b11393733b1ea5890100302ab8a0f74c-Abstract.html","timestamp":"2024-11-14T20:02:12Z","content_type":"text/html","content_length":"10082","record_id":"<urn:uuid:0913a3c3-cafc-4a72-9993-eb4f44ff930d>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00109.warc.gz"}
Normalizing a relativistic model of X-ray reflection. Definition of the reflection fraction and its implementation in relxill Dauser T, Garcia J, Walton DJ, Eikmann W, Kallman T, Mcclintock J, Wilms J (2016) Publication Type: Journal article, Original article Publication year: 2016 Book Volume: 590 Article Number: A76 DOI: 10.1051/0004-6361/201628135 The only relativistic reflection model that implements a parameter relating the intensity incident on an accretion disk to the observed intensity is relxill. The parameter used in earlier versions of this model, referred to as the reflection strength, is unsatisfactory; it has been superseded by a parameter that provides insight into the accretion geometry, namely the reflection fraction. The reflection fraction is defined as the ratio of the coronal intensity illuminating the disk to the coronal intensity that reaches the observer. Methods. The relxill model combines a general relativistic ray-tracing code and a photoionization code to compute the component of radiation reflected from an accretion that is illuminated by an external source. The reflection fraction is a particularly important parameter for relativistic models with well-defined geometry, such as the lamp post model, which is a focus of this paper. Results. Relativistic spectra are compared for three inclinations and for four values of the key parameter of the lamp post model, namely the height above the black hole of the illuminating, on-axis point source. In all cases, the strongest reflection is produced for low source heights and high spin. A low-spin black hole is shown to be incapable of producing enhanced relativistic reflection. Results for the relxill model are compared to those obtained with other models and a Monte Carlo simulation. Conclusions. Fitting data by using the relxill model and the recently implemented reflection fraction, the geometry of a system can be constrained. The reflection fraction is independent of system parameters such as inclination and black hole spin. The reflection-fraction parameter was implemented with the name refl-frac in all flavours of the relxill model, and the non-relativistic reflection model xillver, in v0.4a (18 January 2016). Authors with CRIS profile Involved external institutions How to cite Dauser, T., Garcia, J., Walton, D.J., Eikmann, W., Kallman, T., Mcclintock, J., & Wilms, J. (2016). Normalizing a relativistic model of X-ray reflection. Definition of the reflection fraction and its implementation in relxill. Astronomy & Astrophysics, 590. https://doi.org/10.1051/0004-6361/201628135 Dauser, Thomas, et al. "Normalizing a relativistic model of X-ray reflection. Definition of the reflection fraction and its implementation in relxill." Astronomy & Astrophysics 590 (2016). BibTeX: Download	{"url":"https://cris.fau.de/publications/107159844/","timestamp":"2024-11-05T03:08:22Z","content_type":"text/html","content_length":"15553","record_id":"<urn:uuid:f1ddb4d9-99ac-404c-9ae3-ee92ce267c8a>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00235.warc.gz"}
Spectroscopic determination of the charge radius difference between the alpha and helion particle: a new puzzle? Thursday, June 27, 2024 16:00 - 17:00 Precision spectroscopy with lasers of atoms and molecules is widely used for tests of elements of the Standard Model of physics (such as quantum electrodynamics), and for determinations of fundamental constants and nuclear charge radii. This requires ‘simple’ systems that can be calculated accurately. In the talk I will focus on our precision measurements in helium, were we first (laser) cool 3He (a fermion) and 4He (a boson) to quantum degeneracy and ultimately trap them in a focused laser beam at the ‘magic’ wavelength of 320 nm. Of both trapped and ultra cold isotopes we measure the doubly-forbidden 2 3S1 – 2 1S0 transition frequency at 1557 nm with twelve digits. For the difference in transition frequency (the isotope shift), the biggest theoretical uncertainty is due to the uncertainty of the nuclear charge radius of both isotopes. Therefore we can use our measurement to deduce a charge radius (squared) difference between the helion and alpha particle with unprecedented accuracy [1]. Interestingly, an evaluation recently by the CREMA collaboration of the same charge radius difference from muonic helium ion spectroscopy [2] leads to a value that deviates by 3.6 combined sigma from our measurement. In the talk he will give special attention to the remarkable difference in quantum behaviour between 3He and 4He, and the consequences for cooling to ultra-low temperatures and the spectroscopy we [1] Y. van der Werf et al., arXiv:2306.02333 [2] K. Schuhmann et al., arXiv:2305.11679 Science Park 904 quantum computing, quantum gases and quantum information, quantum matter Prof. Dr. Kjeld Eikema	{"url":"https://physicstalks.amsterdam/event/spectroscopic-determination-of-the-charge-radius-difference-between-the-alpha-and-helion-particle-a-new-puzzle","timestamp":"2024-11-15T04:55:04Z","content_type":"text/html","content_length":"30051","record_id":"<urn:uuid:e5e531c6-e201-42ef-82c2-ebbd5aca661b>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00482.warc.gz"}
The Formula for Calculating the Variance of a Discrete Random Variable Question Video: The Formula for Calculating the Variance of a Discrete Random Variable Mathematics • Third Year of Secondary School Which of the following is the formula we use to calculate the variance of a discrete random variable π ? [A] Var (π ) = π Έ(π )Β² β π Έ(π Β²) [B] Var (π ) = π Έ(π Β²) + π Έ(π )Β² [C] Var (π ) = π Έ(π Β²) β π Έ(π )Β² [D] Var (π ) = π Έ(π ) β π Έ(π )Β² [E] Var (π ) = π Έ(π ) + π Έ(π )Β² Video Transcript Which of the following is the formula we use to calculate the variance of a discrete random variable π ? (a) The variance of π equals the expected value of π squared minus the expected value of π squared. (b) The variance of π equals the expected value of π squared plus the expected value of π squared. (c) The variance of π equals the expected value of π squared minus the expected value of π squared. (d) The variance of π equals the expected value of π minus the expected value of π squared. Or (e) the variance of π equals the expected value of π plus the expected value of π squared. Before we discuss the formula for finding the variance, letβ s just be clear on the different notation used in the five options given. π Έ of π squared means we find the expected value of π first, and then we square this value, whereas π Έ of π squared means we find the squared values of π first and then calculate their expectation. Itβ s sometimes helpful to think of these as the square of the expectation and the expectation of the squares. Now weβ re asked to determine the formula we use to calculate the variance of a discrete random variable. Now the variance is a measure of the extent to which values of that variable differ from their expected value, which we denote as π . We can denote this either as Var of π or as π squared or sometimes π sub π squared if there are multiple variables in the same problem. The formula for calculating the variance from first principles is the variance of π is equal to the expected value of π minus π squared, where π is the expected value of π , which we calculate using the formula the sum of each π ₯-value in the range of the discrete random variable multiplied by the probability that π is equal to that value. Another way of writing this is as the sum of π ₯ minus π squared multiplied by the probability that π is equal to π ₯. This means that we subtract the expected value π from each value that the discrete random variable can take, square these values, multiply by the probability that π is equal to that value, and then add them all up. Returning to the formula we first wrote down though, we can manipulate this. Weβ ll begin by distributing the inner parentheses, so we square π minus π . This gives the expected value of π squared minus two π π plus π squared. As the expectation is linear, this can be distributed over the brackets. So we have the expected value of π squared minus the expected value of two π π plus the expected value of π Now π is just a constant. So the expected value of π squared is just π squared. And in the second term, we can bring two π outside the front of the expectation. So we have the expected value of π squared minus two π multiplied by the expected value of π plus π squared. But π , remember, is the expected value of π , so we can replace π with π Έ of π . And we have π Έ of π squared minus two π Έ of π π Έ of π plus π Έ of π squared. The term in the center becomes negative two multiplied by the expected value of π squared, and we can then group the like terms. Negative two multiplied by π Έ of π squared plus π Έ of π squared is negative π Έ of π squared. So the formula simplifies to the expected value of π squared minus the expected value of π squared. Using the descriptions we wrote down earlier, we can think of this as the expectation of the squares minus the square of the expectation. Looking carefully at the five options we were given, itβ s this one here: the variance of π is equal to the expected value of π squared minus the expected value of π squared. The other options we were given do highlight various common mistakes. For example, in the first option, the terms have been subtracted in the wrong order. In the second option, the terms have been added instead of subtracted. In fact, the most common error isnβ t actually given, which is to forget to square the second term. So the most common incorrect formula used in practice is the expected value of π squared minus the expected value of π . The correct answer is that the formula we use to calculate the variance of a discrete random variable π is the variance of π is equal to the expected value of π squared minus the expected value of π squared.	{"url":"https://www.nagwa.com/en/videos/785162613429/","timestamp":"2024-11-04T09:26:01Z","content_type":"text/html","content_length":"256631","record_id":"<urn:uuid:18536dae-e5fa-4d5b-95b7-3df5fbcf2631>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00609.warc.gz"}
An Introduction to Cosmic Rays and Gamma-Ray Bursts by A. De Rujula Publisher: arXiv 2007 Number of pages: 37 The author reviews the subjects of non-solar cosmic rays (CRs) and long-duration gamma-ray bursts (GRBs). This is a version of an introductory talk to high-energy physicists. Nothing besides the standard model is required to understand CRs of any energy. Download or read it online for free here: Download link (2.3MB, PDF) Similar books Black-Hole Phenomenology Neven Bilic arXivThis set of lectures is an introduction to black-hole astrophysics. The emphasis is made on the phenomenology of X-ray binaries and of supermassive compact objects at galactic centers. Notes to the lectures given at the School on Particle Physics... Lectures on Black Holes, Topological Strings and Quantum Attractors Boris Pioline arXivIn these lecture notes, the author reviews some recent developments on the relation between the macroscopic entropy of four-dimensional BPS black holes and the microscopic counting of states, beyond the thermodynamical, large charge limit. High Energy Astrophysics Jonathan Katz The Benjamin/Cummings PublishingThis book describes the methods and results of modern astrophysical phenomenology and modelling for advanced undergraduates or beginning graduate students. It is meant to be explanatory and expository, rather than complete or definitive. Introduction to Cosmology David H Lyth arXivThese notes form an introduction to cosmology with special emphasis on large scale structure, the cmb anisotropy and inflation. In some places a basic familiarity with particle physics is assumed, but otherwise no special knowledge is needed.	{"url":"http://www.e-booksdirectory.com/details.php?ebook=4524","timestamp":"2024-11-10T08:14:38Z","content_type":"text/html","content_length":"11143","record_id":"<urn:uuid:b3b16aaf-0b66-48b7-aaa4-682b7d86d36e>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00246.warc.gz"}
Connexant cx31993 and USB3 stack As far as I remember the SIQ problem was fixed years ago, so that should not be the problem. But there are enough deficiencies in the WPS that lead to the effect you describe. And I have the suspicion that it relates to the introduction of SMP. What you can do as a test is to reduce to one core by using the /MAXCPU=1 switch of ACPI.PSD and see if that improves things. If it improves things, then my suspicion would prove to be correct.	{"url":"https://www.os2world.com/forum/index.php/topic,3455.15.html?PHPSESSID=j36ahvo8ot32sos4ju9evgve4u","timestamp":"2024-11-13T12:34:02Z","content_type":"application/xhtml+xml","content_length":"42984","record_id":"<urn:uuid:55d50f5c-7ab5-4c5c-80ff-b74012d56354>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00232.warc.gz"}
Google Sheets VLOOKUP from Another Sheet (Easy 2024 Guide) I’m here to discuss how to use VLOOKUP from another sheet in Google Sheets. My goal: Help you increase your productivity with some simple automation. After all, that’s why I love the VLOOKUP Below, I’ll break down the syntax and include screenshots for step-by-step instructions on how to use it. Here’s my guide on how to use VLOOKUP to get data from another sheet. What is VLOOKUP in Google Sheets? VLOOKUP is short for Vertical Lookup. It’s a popular function in Google Sheets because you can use it to call data across different spreadsheets and workbooks. Just as importantly, it’s a time saver. You can use the VLOOKUP function for data analysis on a singular sheet or by calling data across different spreadsheets. If you’re wondering how to use Google Sheets VLOOKUP from another sheet, that’s what I discuss here. In short, you can look up and fetch specific values from large amounts of data quickly and efficiently. Being able to use it across multiple spreadsheets makes it easier to keep your data clean and automatically updates when there are changes between sheets. In this article, we will show you how to Google Sheet VLOOKUP another sheet. How to VLOOKUP from Another Sheet in Google Sheet I want to give a simple step-by-step guide to get started. So here’s how to do VLOOKUP in Google Sheets from a different sheet: 1. Type =VLOOKUP( in a cell in the active sheet 2. Select the cell reference of the value you want to look for. 3. For the range, we can now go to the source sheet where we are performing the VLOOKUP. All you need to do is select the range of cells that you want to look up. 4. Add the index that represents the column you want to return the results from. 5. Add 0 or false for an exact match. 6. Copy the formula to the rest of the cells If that seems a little confusing, don’t worry. I’ll break it down further with real-world examples below. Before You Start We’d recommend having a basic understanding of how VLOOKUP works. An even better idea would be to start out semi-fluent in Google Sheets altogether. To get to this level, you may want to consider taking a Google Sheets course, or even a full productivity course on Udemy to brush up your skills. We even have our own comprehensive course that covers Google Sheets and Google Forms. The Google Sheets VLOOKUP From Another Sheet Formula You know how much I love to break down a Google Sheets function into its most fundamental parts. That’s the case here. And there’s good news. The VLOOKUP syntax is pretty straightforward. As the “vertical” in its name implies, VLOOKUP searches the leftmost column in the established range for a row match and returns the value in the cell identified by the index. It searches a column and returns a value from the matching row, usually from a different column. One of the biggest weaknesses in VLOOKUP is it can’t search for a match in a column on the right and bring back a result from a column to its left. In other words, VLOOKUP can only look right. This is what the formula looks like when broken down by part: Let’s look at each component individually: • =VLOOKUP(): This is the function itself without any parameters. It’s what tells Google Sheets to act. • search_key: This parameter defines what information we’re looking to match. It might be a name, a number, a boolean value, or something different. It might be a static value we’ve defined or it might be a relative value stored in a cell. If we set this as “A2” it will search for the value in cell “A2”. If we set it to “true” it’ll match the text string “true.” • range: This tells the VLOOKUP function where to look for a match and the range in which the value it is supposed to return is located. If we’re searching for a value match in column B and want to return a value in columns C or D, we’d set the range as B:D. • index: This parameter tells VLOOKUP which column value to return. The index is relative to the range, not the sheet. Hence it is a number instead of a letter like columns is usually defined. If our search is matching in column B and returns a value from column C, the index value is 2. If we’re returning column D from a column B match, the index value is 3. • is_sorted: This parameter’s name isn’t as clear as it returns exact matches when it’s set to “false” and the closest match when set to “true.” True is set by default, but false is recommended for most uses. Now let’s have a look at it all put together: How you define the range is extremely important and can be confusing if you’re new to how VLOOKUP works. The defined range needs to include both the data value you’re searching for and the data value you look to return. If you’re trying to search column A and bring back the value in column B, the range needs to include both columns A and B. If you limit the range to A, the VLOOKUP call will fail. Performing VLOOKUP from Another Google Sheet in the Same Workbook Ready for those real world examples I mentioned earlier? This is where you’ll find them. While most of the time, we use the VLOOKUP formula in Google Sheets on the same sheet, many times, you may have to use it to VLOOKUP between two sheets in the same workbook or even across the different workbooks. For example, you may want to fetch the data from specific items in a worksheet while the lookup data is in a different sheet or different workbook. How you use the VLOOKUP function to bring in data from another sheet is somewhat different when you’re working in the same workbook or a different workbook. The formula for the same workbook VLOOKUP looks like this: =VLOOKUP(search_key,{sheet name}!{cell range},index,is_sorted) Notice there’s a “!” between the sheet name and cell range. Also, there are no quotes around the range on a same-workbook VLOOKUP. Let’s look at our example below: This sheet is our Sales sheet. We can get the prices for each product from a different sheet called the Price sheet. Below is the formula that will do this: Here’s how to perform a VLOOKUP Google Sheets from another sheet: Step 1: Type =VLOOKUP( in the cell in the active sheet. Step 2: Select the cell reference of the value you want to look for. In our example, the cell is B2. Step 3: Go to the Price sheet and select the entire range. Step 4: Add the index that represents the column you want to return the results from. In our case, it is column B, so the index is 2. Step 5: Add 0 or false for an exact match. Step 6: To copy the formula to the rest of the cells, we can click and drag the square at the bottom right corner of the cell. The above formula fetches the value from the second column from the sheet named “Price” in the current sheet. Performing VLOOKUP from Another Worksheet in a Different Workbook The process of referencing data using VLOOKUP from another workbook, Google Sheets, is a little more complicated. You need to combine the VLOOKUP function with the IMPORTRANGE function. Now, let’s look at an example where we need to fetch the value from a different Google Sheets workbook =VLOOKUP(search_key,IMPORTRANGE(“{sheetsURL}”,“{sheet name}!{cell range}”),index,is_sorted) This version uses a new command called “IMPORTRANGE()”. This command’s syntax asks you to define the URL of the workbook you’re importing data from, define the specific sheet, and set the range. The formula breaks down like this: IMPORTRANGE(“{sheetsURL}”,“{sheet name}!{cel range}”) • {sheetsURL}: In quotes, add the URL of the Google Sheets file you want to access. Example: “https://docs.Google.com/spreadsheets/d/1AJcuVkYvdiW0NAlfuI” • {sheet name}!{cel range}: Set this the same way you configured it in the same-workbook method. However, notice that this time there are quotes around the sheet name and cell range. Now look at it with a value-added, and all values added : To reiterate, there are three important values you need to set in a cross-sheet VLOOKUP reference: • Workbook URL {sheetsURL} • Sheet Page {sheet name} • Cell Range {cell range}: We’ll use the same product lineup of Gadgets, Gizmos, Thingamabobs, and Widgets from the previous example. This time around, we want to bring in the same information into a sheet called “Outside” in a different workbook. We want to bring in the information from the “Called” sheet in the other workbook: Now let’s say we want to have the “Outside” tab show how many products are in stock, but we only want to worry about updating the “Called” spreadsheet page in the original workbook. Step 1: Click on the cell you want to return the results Step 2: Type in the VLOOKUP formula. Step 3: Select the cell reference for the value you want to look up. Step 4: Type the IMPORTRANGE formula. Step 5: Go to the other work book and on the sheet and copy the URL. Step 6: Paste the link into the formula. Step 7: Get the range you want to lookup together with the sheet name. Step 8: Add the index that represents the column you want to return the results from. Step 9: Add 0 or false for an exact match. Step 10: Click Allow Access. Enter our VLOOKUP formula in the topmost cell — in our case, we’re using =VLOOKUP(A2,IMPORTRANGE("https://docs.Google.com/spreadsheets/d/18nsDPJ-","Called!A2:B5"),2,false)in cell C2. Now our second workbook is referencing the first workbook’s inventory count: This can be extremely useful for someone analyzing the information in the first worksheet without any risk of error in the original data. It’s also very useful for bringing in a fraction of the information in a spreadsheet for easier analysis in another. The VLOOKUP command makes it easier to bring it only the information you want. Performing VLOOKUP in Google Sheets From Multiple Tabs Aside from using it with multiple columns, you can also perform a VLOOKUP from multiple sheets in Google Sheets. To do this, we will need to create an ARRAYFORMULA that will do VLOOKUP in Google Sheets from another tab. In our example sheet, we have 3 different sheet tabs, two of which have the prices of the products. This is the formula we will use: Step 1: Type the ARRAYFORMULA in the cell in the active sheet. Step 2: Type =VLOOKUP( Step 3: Select the cell range of the value you want to look for. Step 3: Add curly brackets and go to the Price sheet and select the entire range. Add a semi-colon after. Step 4: Go to the Price 1 sheet and select the entire range. Close the curly brackets. Step 5: Add the index that represents the column you want to return the results from. In our case, it is column B, so the index is 2. Step 6: Add 0 or false for an exact match. This formula will look up in the two sheets (Price and Price1) and return matching values for the price of the products in each sheet. If it does Into find a match in the first sheet, it will then look in the second sheet. Using VLOOKUP in Google Sheets from Multiple Different Sheets Using the IMPORTRANGE function, you can also perform a Google Sheets VLOOKUP in more than one worksheet. Just like with multiple sheets, you will need to create an array either with curly brackets or with the ARRAYFORMULA. In our example sheets, if we wanted to perform a lookup from multiple sheets in different workbooks, we would use the formula: =VLOOKUP (A2, {IMPORTRANGE ("https://docs.Google.com/spreadsheets/d/1tQfzx2_n97OHJiOUoij0rnDV6sZk0R3tP_Ryrcw/edit#gid=0", "Sheet1!A2:B10") ; IMPORTRANGE("https://docs.Google.com/spreadsheets/d/1aHwZyavTciiGRV31nGL43hlmiWl7JZgytBiftqE/edit#gid=0","Sheet1!A2:B10")}, 2, false) Step 1: Click on the cell you want to return the results Step 2: Type in the =VLOOKUP formula. Step 3: Select the cell reference for the value you want to look up. Step 4: Add curly brackets and type the IMPORTRANGE formula. Step 5: Go to the other work book and on the sheet and copy the URL. Step 6: Paste the link into the formula. Step 7: Get the range you want to lookup together with the sheet name and close the brackets. Step 8: Add a semi-colon and type the IMPORTRANGE formula again for the second workbook. Step 9: Copy and Paste the URL into the formula. Step 10: Get the range and sheet name and add it to the formula. Step 11: Add the index that represents the column you want to return the results from. Step 12: Add 0 or false for an exact match. Step 13: Click Allow Access. This formula will perform a Google Sheets lookup in another sheet in a different workbook and if it fails to find a match there, it will perform the lookup in the second workbook. The formula for each workbook is separated by a semi-colon, and they are enclosed in curly brackets to indicate an array. Some Tips when Using VLOOKUP to Reference Another Sheet/Workbook Here are some tips to keep in mind when referencing another sheet or workbook in the formula: Be specific about the range The Google Sheets VLOOKUP feature can be very performance-hungry and cause a workbook’s performance to come to a crawl. You can avoid slow performance by being specific with the ranges you reference. • Instead of calling entire columns like “A:B” reference the specific starting and ending cells like “A1:B1000”. This cuts down on how much work Google Sheets needs to do to bring in the same amount of information. • If you’re searching column A for information and bringing back the result in column D, use a reference like “A1:D1000” instead of “A1:F1000”. There’s no need to reference columns E and F if in the range if they’re not being used. This is especially important when you’re calling information between different workbooks. When you’re doing a cross-workbook VLOOKUP, it requires Internet bandwidth to transfer data between the two. Use conditional statements to prevent unnecessary calls Another way you can prevent slow-down with cross-sheet VLOOKUP is to use a conditional statement to determine if Google Sheets should run the VLOOKUP at all. For example, if there’s information on the sheet you’re using the VLOOKUP call that tells you there’s no need to run it, use that to your advantage. In our product example, the “Active” sheet lists whether or not a product is in stock. Since we know a product is out of stock, we don’t need to use VLOOKUP to define the stock count. For this, we’ll use the “=if()” function. This function asks if a given condition is true or false, then does something different for each case. The syntax looks like this: =if(logical_expression, value_if_true, value_if_false) In a simple use-case, we can use it to determine if the value in cell A1 is greater than the value in cell B1. So the expression A1>B1 would look like this: =if(A1>B1, “A1 is greater”, “B1 is greater”) The formula will return the text “A1 is greater” if A1 is the larger number and “B1 is greater” if B1 is the larger number. In the case of our product worksheet, if the “In Stock” value is “No” we don’t want to run VLOOKUP. So we set up our if statement like this: • logical_expression: B2=”YES” — This will run the “value if true” if the data in cell B2 is “YES”. • Value_if_true: VLOOKUP(A2, Called!A2:B5,2, false) –This will run the VLOOKUP if the logical expression returns true. • Value_if_false: “out of stock” — This will return the text “out of stock” if the value in cell B2 is anything other than “YES.” If we put it all together, it comes out looking like this: =IF(B2="YES",VLOOKUP(A2,Called!A2:B5,2,false),"out of stock") Notice how cell C3 now says “out of stock” instead of returning the value “0”. In this case, we avoided running a VLOOKUP because we didn’t need to bring back data. While you’re inputting more information into Google Sheets to calculate results, it is creating less work for the program. It is much less work to run many “if checks” than a single VLOOKUP. Using this technique will help you speed up Google Sheets and improve performance. Make sure you have permission For obvious security reasons, Google Sheets won’t let you pull in data from another workbook unless you have the authorization to do so. To reference one workbook from another with VLOOKUP, you need to either be the creator of both or have permission to use both. You can be added as an authorized user either by account or through a sharing URL. Using VLOOKUP to reference information across different sheets and workbooks is an incredibly powerful tool to have at your disposal. The command is particularly helpful with cross-sheet use because it will reflect any changes made to the original sheet across all referenced sheets. Frequently Asked Questions How Do I Do a VLOOKUP From Another Google Sheet? Here’s how to perform a VLOOKUP Google Sheets from another sheet: 1. Type =VLOOKUP( in the cell in the active sheet 2. Select the cell reference of the value you want to look for. 3. For the range, we can now go to the source sheet where we are performing the VLOOKUP. All you need to do is select the range of cells that you want to look up. 4. Add the index that represents the column you want to return the results from. 5. Add 0 or false for an exact match. 6. Copy the formula to the rest of the cells. How Do I Do a VLOOKUP From Multiple Sheets in Google Sheets? To perform a VLOOKUP from multiple sheets: 1. Select the cell reference of the value you want to look for. In our example, the cell is B2. 2. Go to the first sheet and select the entire range. 3. Go to the second sheet and select the range. You can keep selecting from the other sheets if you have more sheets. 4. Add the index that represents the column you want to return the results from. 5. Add 0 or false for an exact match. 6. Copy the formula to the rest of the cells Here’s an example of the formula How Do I Pull Matching Data from Another Sheet in Google Sheets? You can use the VLOOKUP function to pull up data from another sheet in google sheets. All you need is to type in the formula and select the range you want to look up in the other sheet. We’ve shown you just how to do this in this guide. What is the Difference Between VLOOKUP and MATCH in Google Sheets? The VLOOKUP can only search for values in the left column of the range and retrieve values from columns to the right of the search column. On the other hand, the MATCH function is used to search for a value in a single row or column of a specified range, and return the relative position of that value within the row or column. [adthrive-in-post-video-player video-id=”lQPVEINc” upload-date=”2022-08-18T04:54:44.000Z” name=”Screen Recording VLOOKUP.mp4″ description=”” player-type=”default” override-embed=”default”] The wealth of advanced VLOOKUP features and use-cases also work in cross-sheet references. Using VLOOKUP to reference another sheet opens up a world of new possibilities to work with your data. It also helps you save time. In this guide, I showed how to use Google Sheets VLOOKUP from another sheet in the same workbook and in different workbooks. I hope you found this tutorial useful. Looking for more advice? I also wrote about how to use index-match in Google Sheets.	{"url":"https://productivityspot.com/vlookup-another-sheet-google-sheets/","timestamp":"2024-11-08T09:00:35Z","content_type":"text/html","content_length":"458985","record_id":"<urn:uuid:0970ea1b-9d8f-419b-8e29-a172e0a2a93f>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00692.warc.gz"}
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Triple Exponential Moving Average (TEMA) Definition. The Triple Exponential Moving Average (TEMA) is a technical indicator that is used to smooth out the volatility in price data by combining a single exponential moving average (EMA) with a double exponential moving average (DEMA). The TEMA was developed by Jack Hutson in the early 1990s and is considered to be a more sophisticated version of the EMA. The TEMA calculation is as follows: TEMA = (3EMA) - (3DEMA) + DEMA EMA = Exponential Moving Average DEMA = Double Exponential Moving Average The TEMA is a lagging indicator, which means that it is based on past data and is not necessarily predictive of future price action. The TEMA can be used to identify trends and trend reversals, as well as to generate buy and sell signals. The TEMA is not without its critics, who argue that its lagging nature means that it is not as useful as other technical indicators. How do you trade with Tema? First, identify the trend. Tema is a lagging indicator, so it will follow the trend. If the trend is up, you will want to buy; if the trend is down, you will want to sell. Next, identify support and resistance levels. These are levels where the price has bounced in the past, and they can give you an idea of where the price might reverse. Finally, enter your trade. If you are buying, you will want to enter when the price is near support; if you are selling, you will want to enter when the price is near resistance. What is moving average used for? The moving average is a simple technical analysis tool that is used to smooth out price data and better identify trends. There are different types of moving averages, but the most commonly used are the simple moving average (SMA) and the exponential moving average (EMA). The SMA is calculated by taking the average of a given set of prices over a certain period of time. For example, if you wanted to calculate the 10-day SMA, you would take the average of the past 10 days' worth of prices. The EMA is similar to the SMA, but it gives more weight to the most recent prices. This makes it more responsive to changes in the market, but it can also make it more volatile. Moving averages can be used on any time frame, but they are most commonly used on daily or weekly charts. They can be used to identify the overall trend, as well as support and resistance levels. How do you calculate triple exponential moving average in Excel? There is no built-in function to calculate the triple exponential moving average (TEMA) in Excel, but you can easily create a TEMA indicator using a combination of the EMA(), DEMA(), and TEMA() functions. The TEMA indicator is defined as follows: TEMA = 3 * EMA(data) - 3 * DEMA(data) + TEMA(data) EMA(data) = the exponential moving average of the data DEMA(data) = the double exponential moving average of the data TEMA(data) = the triple exponential moving average of the data To calculate the TEMA indicator in Excel, you first need to calculate the EMA, DEMA, and TEMA of the data using the following formulas: EMA(data) = EMA(data, period) DEMA(data) = 2 * EMA(data, period) - EMA(EMA(data, period), period) TEMA(data) = 3 * EMA(data, period) - 3 * DEMA(data, period) + EMA(TEMA(data, period), period) data = the data series period = the number of periods over which to calculate the moving average For example, to calculate the TEMA over a 20-day period, you would use the following formulas: EMA(data) = EMA(data, 20) DEMA(data) = 2 * EMA(data, 20) - EMA(EMA(data, 20), 20) TEMA(data) = 3 * EMA(data, 20) - 3 * DEMA(data, 20) + EMA(TEMA(data, 20), 20) How do you calculate Triangular moving average? A triangular moving average (TMA) is a type of moving average that is similar to other moving averages, such as a simple moving average (SMA) or an exponential moving average (EMA). However, unlike these other moving averages, a TMA places more weight on the middle of the data set, and less weight on the beginning and the end. This makes the TMA more responsive to changes in the data set, and less susceptible to price spikes. To calculate a triangular moving average, you first need to calculate a simple moving average. Then, you need to calculate the weighted average of the data points in the middle of the data set. The weighting is equal to the number of data points in the data set minus the data point's position in the data set. For example, if there are 10 data points in the data set, the first data point would be weighted at 9 (10-1), the second data point would be weighted at 8 (10-2), and so on. Once you have the weighted average of the data points in the middle of the data set, you can add this to the simple moving average to get the triangular moving average. What is moving average technical analysis? Moving average technical analysis is a method of analyzing securities prices using a series of moving averages. The most common moving averages are the Simple Moving Average (SMA) and the Exponential Moving Average (EMA). Moving average technical analysis can be used to identify trends, generate trading signals, and measure the strength of price movements.	{"url":"https://www.infocomm.ky/triple-exponential-moving-average-tema-definition/","timestamp":"2024-11-14T07:47:25Z","content_type":"text/html","content_length":"42634","record_id":"<urn:uuid:590c1219-c9dc-49c7-8688-da166d3289bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00028.warc.gz"}
critical speed of ball mill 76 6 Critical speed is defined as the point at which the centrifugal force applied to the grinding mill charge is equal to the force of gravity. At critical speed, the grinding mill charge clings to the mill inner surface and does not tumble. Most ball mills operate at approximately 75% critical speed, as this is determined to be the optimum speed ... by mill operating parameters. It is hoped that this work can be used as a precursor to the development of a model that can predict liberation given the various ranges of operating parameters. Mill speed, mill charge, ball size, and wet grinding are the parameters which have been selected for the present study. It is hoped that the analysis of the Figure 2-6 Ball Trajectories for 50 mm balls in a 5 m diameter mill at 75% of critical speed .... 19 Figure 2-7 Forces on a ball in contact with a lifter bar (Powell, 1991) ..... 24 Figure 2-8 Parabolic trajectory of a ball at its point of departure from lifter bar (Powell, 1991) 25 76.6 by the square root of the mill diameter in :feet. In the case of a 3t 6" 'diametermill, as proposed in this report, this would be 76.6 divided by the square root of 3.5 (1.8'r) or about 41 RPM. It has been determined that. the optimum speed of a mill with lifter bars is approximately flylJ to 70'/0 of the critical speed, or in this case ... -Critical speed R –radii of the mill r- radii of the ball g –acceleration due to gravity 10. Critical speed of ball mill(in rps)= R=.45/2=.225m r=25/2=12.5m g=9.81m/s2 Nc=1.08rps=64.8 rpm 11. r=60/2= 30mm R=800/2=400mm Nc=.82 rpm Critical speed=1.4*operating speed Operating speed =.82/1.4=.586 rps=35 rpm ... explanation of critical speed for ball mill 76 6. to calculate critical speed of ball mill practical. The critical speed of the ball mill was calculated to be 76rpm Read more Technical Notes 8 Mineral Technologies International Inc complicated in practice and it is not possible to calculate The critical speed of the mill amp c is defined as the speed at which a … 6 zone from ball charge in the mill. - Also they allow to determine the single energy at impact, ... % Critical Speed : 76%. Critical Trajectory Simulations ... – Changing the mill's % critical speed – Mill charge level, optimal ball size, etc. 10. % of critical speed as the industrial ball mill. The ball load. of 50, 40, and 30 mm which corresponds to ball loads of an. ... [76]. However, the high content of sulfur (4-8 wt%), ... The theoretical suitable speed of the mill is 76% of the critical speed. The actual mills fluctuated around 76%. In the current production, when the rotation speed of the ball mill is less than 76% of the critical rotation speed, it is called low-speed grinding; when the rotation speed is higher than the critical rotation speed of 88%, it is ... Take advantage of this Critical Speed Calculator along with the many other various calculators Roton has available at your disposal. Critical Speed - Roton Products, Inc. Shop Products e planation of critical speed for ball mill 76. Critical rotation speed for ball-milling ScienceDirect. Aug 03, 1999 To examine the dependence of critical rotation speed on ball-containing fraction, we measured critical speeds at various ball-containing fractions from 0.3 to 0.95 stepped by 0.05. More The "Critical Speed" for a grinding mill is defined as the rotational speed where centrifugal forces equal gravitational forces at the mill shell's inside surface. This is the rotational speed where balls will not fall away from the mill's shell. ... N c =76.6(D-0.5) where, N c is the critical speed,in revolutions per minute, The Formula derivation ends up as follow: Critical Speed is: N c =76.6 (D 0.5) where: N c is the critical speed,in. turunan rumus kecepatan kritis sentrifugal ball mill . rumus critical speed pada proses grinding. SCM series super thin mill Straight … (24.8 lb) of ball mill feed sample from plant survey. Sample is reconstituted with water to be the same percent solids as the plant ball mill discharge. Mill speed: 35.2 rpm (65 percent of critical speed) Length of time of test: 363 seconds Average torque reading: 1,056 in-lbs Mill HP = (1,056 in-lbs / 12 in./ft) x (35.2 rpm Critical speed (in rpm) = 42.3/sqrt(D - d) with D the diameter of the mill in meters and d the diameter of the largest grinding ball you will … Ball Mill manufacturers in india.Ball Mill critical speed design in India market.The India ball mill manufacturers provides you the ball mill and critical speed for ... >>GET MORE. ... critical speed of ball mill 76 6 - mtmcrusher. A NEW SIMULATING ALGORITHM FOR BALL MILLS. There are two methods for ball mill calcu (5), (6). If the ... † The appendix of the paper lists the mill speed as 12.5 RPM. The mill is fixed speed, so the %critical speed is only a function of mill effective diameter (as liners wear). Doing the math (neglecting the balls) gives a 68.4% critical speed. The ball mill belly length can be achieved with a 18 degree head angle and 1.9 m trunnion diameter. The critical speed is the rotational speed at which balls just start to centrifuge on the mill case and not tumble; it can be calculated based on the equations below. Critical speed (rpm) = 76.6/ √ D − d; D, d in feet (1) = 42.2/ √ D − d; D, d in meters (2) Where D = Internal Mill Diameter d = Maximum Ball Diameter The ratio of the ... the mill. The common range of mill speeds is 65% to 80% of critical, depending on mill type, size and the application. The critical speed of a ball mill is calculated as 54.19 divided by the square root of the radius in feet. The rotational speed is defined as a percentage of the critical speed. Smaller diameter mills https:// Learn about Ball Mill Critical Speed and its effect on inner charge movements. The effect of Ball Mill RPM s... N = speed of mill in rev. per min. N1 = critical speed of mill in rev. per min. Vb = relative velocity of particle at od, ft. per sec. w = weight of portion of charge, lb. W = weight of entire charge, lb. P = fraction of mill volume occupied by charge. g = a constant = 32.2 ft. per sec. per sec. k = a constant = 4π²n²/g = 1.226n². At critical speed, the steel balls inside the tube cease to mill and are pressed against the tube's walls as shown below in Illustration 1. As a result, many mills are set to rotate at approximately 70-80% of critical speed.1. A mill's critical speed is calculated using the formula: N c = 76.6(D0.5), where D is the inside diameter in feet ... Critical speed is C.S. = 76.63 / 11^0.5 = 23.1 rpm Ball and SAG Mills are driven in practice at a speed corresponding to 60-81% of the critical speed, the choice of speed being influenced by economical considerations. Within that range the power is nearly proportional to the speed. Mill rotating speed impacts grinding rates. Ball Mill Critical Speed . · 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. Ball Mill Critical Speed - Mineral Processing & Metallurgy. critical speed of ball mill 76 6 - critical speed of ball mill 76 6 Request a Quotation. Ball mill - Wikipedia. High-quality ball mills are potentially expensive and can grind mixture particles to as small as 5 nm, enormously increasing surface area and reaction rates. The speed of the mill was kept at 63% of the critical speed. The face angle was varied from 90 to 111 degrees for the three types of configuration 1, 2 and 4, as shown in the figure. Also, the height of the lifter bar in configuration 3 was changed to observe the trajectory. It was observed that the ball trajectories could be controlled by the ... Ball Mill Critical Speed Mineral Processing Metallurgy A Ball Mill Critical Speed (actually ball, rod, AG or SAG) is the speed at which the centrifugal for +86 21 … The diameter of the balls used in ball mills play a significant role in the improvement and optimization of the efficiency of the mill [4]. The optimum rotation speed of a mill, which is the speed at which optimum size reduction takes place, is determined in terms of the percentage of the critical speed of the mill [8]. The critical speed is Critical Speed of the mill is the speed of . ... N c = 76.6 √ (1/D) (1) ... The ball milling treatment led to the emergence of cracks on the surface of the 48 h treated specimen due to strain ... The speed of the mill was kept at 63% of the critical speed. The face angle was varied from 90 to 111 degrees for the three types of configuration 1, 2 and 4, as shown in the figure. Also, the height of the lifter bar in configuration 3 was changed to observe the trajectory. improved further by reducing the mill speed from 83.5% of critical speed to a typical ball-milling speed of 69%. The results of these batch grinding tests show that simple changes to equipment may lead to a more efficient use of power, a finer grind and better extraction of gold. 6 (76 percent of critical speed average) Ore specific gravity (sg) 2.77 Ore bulk density 1.64 tonnes/m3 Table 6 – ball mill operating requirements Specification Data Throughput rate (new feed) 795 dry mtph (average per mill) Ore feed size F80 2,200 microns	{"url":"https://www.ilpomodorosnc.it/29163_critical_speed_of_ball_mill_76_6.html","timestamp":"2024-11-05T22:25:19Z","content_type":"text/html","content_length":"37298","record_id":"<urn:uuid:61771c19-b987-4c14-a8e0-6f7df55d3e4e>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00433.warc.gz"}
What Is Limits And Continuity In Calculus? \| Hire Someone To Do Calculus Exam For Me What Is Limits And Continuity In Calculus? In this article If you would like to know more about the limits of Calculus, take a look at the definition of continuity. Continuity, or the use of numbers in calculus is defined as the sum of any two geometric quantities such that (1) a member of a n-element set is a pair of functions from n-element sets satisfying the following three properties: (2) The member function gives a value at some point in the continuum. (3) This gives a new quantity being compared to a particular value. One solution to the second part of the definition is to translate the definitions into the physical domain and to give a definition to show how one can prove the existence of continuations, not even with an infinite number of member functions. How does one prove (or prove) generalizable properties, such as convergence to the continuum? John Maynard discusses in this article how to do that. The definition ofContinuity Consider the countable discrete set as it possesses no compact set. The set of members in it is denoted by ε, and in the extended counterexample, a pair of function f(x) and a set of continuous domain x. The aim is to compute the sets of members in a given n-element set x. Set it as v: The isomorphism is applied to the function f(x) = x/2 like it a real function on the extended subset, f, and one can show again that (v → \0) → \0. Hence you do not need to prove that the set (v → \0) → \0 is full (it is sufficient to give a list of all elements of a n-element set and write their complement), so you can already get the result by combining the f and x results. Furthermore, your proof shows that the family is itself closed and continuous. The topological consequences One of the things that should be emphasized here is the topological consequences of the construction of a proof by taking any n element set. Given f, a family of continuous segments is the set of members of the continuous domain of f which are a discrete subset of the domain of f. To a certain extent this is equivalent to comparing the set of members in a given n-element set f(x) whose value in a n-element set x is f(x) ([1], [2]). That is, compared to the set of members in another n-element set f(x), we shall show that, the element f is continuous iff (x/2) → \0, the family is again continuous iff (f(x/2)) → \0. In this subsection we will describe two further examples. One example that should also stay in this paper comes from the following concept from mathematics: the sets of members of a given domain of a n-element set f form the subfield of the subset of members that belong to the wide function field of f(2n+1). For example, each ball has a set of members defined by f(2k2.5) = ⊙⁡⊙⊙ for all k = 1, 2 ranging from 4 to 66. In the context of the paper and example, it shows that every nonempty subset withWhat Is Limits And Continuity In Calculus? Causation and differentiation are defined as relationships between two things: objects and relationships. You Do My Work The idea then is to think of the relationships on the surface of the two things: the object, which is its name after its name, and its relationship, the thing, which is its name after its name — when we restrict us to each for the rest of the vocabulary we focus on. In calculus — where for many years we relied on binary relationships, like letters and integers (Mulliken & Melzer, 1999) — however we try to get a vocabulary for the full range of what we do to separate the two things, we actually ask ourselves the following questions: What (or even where) is object? How can I determine what is object? What are these two things? What is this relationship? This seems like easy to think of, but what the philosophers who try these questions have said is that you only get to decide directly on what relationships you really (or what the philosopher-clarifier who decides) are. The math’s the philosopher and the philosopher are merely thinking experiment; our intuitions and experiences help us evaluate how we came to form relationships, and then give us clues as to which relationships to trust and which to distrust. The philosopher-clarifier takes your whole life—in your head and in your mind—and then takes your relationship (or what doesn’t you in yourself) and tries to assess whether it matches the relationship. If the relationship is strong, it is so that it cannot be trusted. If the relationship is strong, it is stable, even if you would prefer to expect for certain situations to be governed by it. If it can still be trusted, then it is a strong relationship and your choices are only partly good. If it’s a bad relationship — it is not necessarily a bad one — then you can’t do anything else. If it has strength, it is clear that the relationship is stable, and the philosophers are saying that the poor relationship is in fact a good one; they know when their friend’s not going to agree with you and when they my review here If really that’d be what these questions were about, then this is a question you’d try to answer without being so bogged down by the fact of relationships. The philosopher-clarifier would ask you: What is the relationship between what is object and what is object? Look, this isn’t just about the relationships, it’s about the relationships. If relations are in fact to link — it’s not really about it at all — then this looks something like this: I’m working in three-letter words. The letters don’t refer to cars, and if there’s a clear sense of the relationship that it’s possible to make out over the things that the word points, you’re dealing with a complex problem, which is what sorts of difficulties arose in this domain in the 20th century. If you look at some of the letters that the philosophers have in common with each other, they seem to be pretty well understood and can teach you around a bit about things like this problem, and things like that, as well, in any context where you want to try and solve it. But the problem of the relationship is always very concrete. We may identify which letters are to be found at the top of the pyramid, and the number of them in their sequence.What Is Limits And Continuity In Calculus? “There’s a critical gap between rigorous language and mathematical logic, but then why measure up? If you can answer: You want to know the definition of boundaries between a source and a target that are really grounded in mathematics, then you don’t have to be computational and rigorous. You can do something like The mathematical definition of a boundary also draws from higher order logic itself, and we can do this in two ways: We can test the boundaries of a mathematics domain to try to make sure they operate as hard as they can and don’t hold up as they could. The mathematics definition is built in such a way that when you look at the definition of boundaries in terms of logical functions of a mathematician, it just breaks down and goes “But if you also know for the first part of the definition that limits are not independent properties of any physical fact, then every physical fact but the space-time real function of the universe doesn’t really depend on such limits.” What this gives you is a nice, free way to test boundaries of abstract mathematical objects but also good ones in the object. Do My Math Homework Online Sure, we have the general metaphor, but that isn’t to question the relationship with mathematics in the abstract. But you should also check that you can’t make the boundaries right there. Roughly speaking, the abstract notion of boundaries should be defined in a scientific way, not at a higher level in mathematics. So it’s useful to check what all this is telling you: One example is to say that the boundaries between a system and a component depend on a function which itself is a logical concept of a given class of rules and structure, and where nothing is really built-in and no property could be built-in except for a field of mathematical formalism (i.e. a set of field members). One does that when one uses a function for which the domain is a standard one-time domain. Since the mathematics definition has to be read from a higher level, there are other ways that more formal could be used as well. For example, one could start with a geometry perspective: you can now look at the domain, and the domain you’re trying to test may only depend on that property, but outside of simple logic you need to build a lower level abstract concept as a way to test boundaries: But there are various arguments to the right of using an abstract concept for bounds, and you use this type of argument for limits or not in the extreme: If you’re really serious about doing something, you need to be more serious a lot in order to use this argument so that it uses the abstract concept of boundaries more directly. If you have going on like you’d have with a single algorithm, you might need to use abstract concepts for boundary there. If you don’t, you’ve got a problem to fix. Another example is to add limits as a formalization of the problem in the mathematics definition (the one I’ve used, and was very close at the end): If you look at this in more detail, you can see that constraints come out in a way that makes see this weaker enough that it is easy to say “for a given domain” about boundaries: Con	{"url":"https://hirecalculusexam.com/what-is-limits-and-continuity-in-calculus","timestamp":"2024-11-15T04:09:13Z","content_type":"text/html","content_length":"105442","record_id":"<urn:uuid:6caf7919-c9d1-44ff-85d9-8943829b583b>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00623.warc.gz"}
Current Electricity \| Physics Notes for IITJEE/NEET It is the rate of flow of charge through any cross section. Conventionally, the direction of flow of positive charge is taken as the direction of electric current. It is a scalar quantity and its S.I. unit is ampere (A). • Current carriers in conductor are electrons, (valence e–s) ions in electrolytes, electrons & holes in semiconductor and positive ions /electrons in gases. • Charge of electron = 1.6 × 10–19c • 1 ampere = 6.25 × 1018 electrons/sec • Though direction is associated with current (opposite to the motion of electrons), but it is not a vector quantity as it does not follow rules of vector addition. • For a current to flow through a cross-section, there must be a net flow of charge through that cross-section. In a metal like copper there are around 1028 free electrons per m3 moving randomly in all direction with speeds of the order of 106 m/s even in the absence of electric field. But since the number of electrons passing through a cross-section from left to right is equal to the number of electrons passing from right to left in a given time, therefore the net charge flow is zero and hence the electric current is zero. • A conductor remains uncharged when current flows in it. i.e. Net charge in a current carrying conductor is zero. Current density at a point inside a conductor is defined as the amount of current flowing per unit cross sectional area around that point of the conductor, provided the area is held in a direction normal to the direction of current. i.e. Current density, If area is not normal to current, then area normal to current is A' = A cos θ (see the figure) or I = J A cos θ or Its SI unit is Am–2 Current density can also be related to electric field as where σ is conductivity of the substance & ρ is specific resistance of the substance. J is a vector quantity and its direction is same as that of . Dimensions of J are [M°L–2T°A] • Electric current is a macroscopic physical quantity where as current density is a microscopic physical quantity. • For a given conductor current does not change with change in cross-sectional area. DRIFT VELOCITY [] When the ends of a conductor are connected to the two terminals of a battery, an electric field is set up in the conductor from the positive terminal to the negative terminal. The free electrons in the conductor experiences a force opposite to the direction of the electric field and hence get accelerated. However this process of acceleration is soon interrupted by collision with ions of solid. The average time for which each electron is accelerated before suffering a collision is called the mean free time or mean relaxation time. Thus, the free electrons within the metal, in addition to its random motion acquire a small velocity towards the positive end of conductor. This velocity is called drift velocity. It is given by where e is the charge and m is the mass of electron. is the electric field established in conductor and is the average relaxation time. Negative sign is because the directions of and (for electron) are opposite. where V is the potential difference across ends of the conductor of length l. The uniform current I, flowing through the conductor is given by I = n e A vd where n = number of free electrons per unit volume, A = area of cross-section, vd = drift velocity In vector form, The negative sign is because the direction of drift velocity of electron is opposite to . Mobility - Drift velocity per unit electric field is called mobility. It is denoted by µ. Its S.I. unit is m2/volt-sec. 1. Drift velocity is very small, it is of the order of 10–4m/s which is negligible as compared to thermal speed of e–s at room temperature (105 m/s) 2. The drift velocity is given by 1. where, J = current density 1. e = electronic charge = 1.6 × 10–19 C 2. n = the number of free electrons per unit volume 1. The number of free electrons per unit volume (n) can be determined by the following relation : where N0 = Avogadro number d = density of the metal M = molecular weight and x = number of free electrons per atom 4. For steady current : ; ; This means that for a given material and steady current in case of non-uniform cross-section of material ; ; 5. Variation of drift velocity : ; Vd ∝ E when length is doubled, vd becomes half and when V is doubled, vd becomes twice. When a potential difference is applied across the ends of a conductor, a current I is set up in the conductor. According to Ohm’s law “Keeping the given physical conditions such as temperature, mechanical strain etc. constant, the current (I) produced in the conductor is directly proportional to the potential difference (V) applied across the conductor”. i.e., or ... (1) where K is a constant of proportionality called the conductance of the given conductor. Alternatively, or V = RI ... (2) where the constant R is called the electrical resistance or simply resistance of the given conductor. From above two eqs. it is clear that R = 1/K. If a substance follows Ohm’s law, then a linear relationship exists between V & I as shown by figure 1. These substance are called Ohmic substance. Some substances do not follow Ohm’s law, these are called non-ohmic substance (shown by figure 2) Diode valve, triode valve and electrolytes, thermistors are some examples of non-ohmic conductors. Slope of V-I Curve of a conductor provides the resistance of the conductor slope = tan θ = The SI unit of resistance R is volt/ampere = ohm (Ω) On application of potential difference across the ends of a conductor, the free e–s of the conductor starts drifting towards the positive end of the conductor. While drifting they make collisions with the ions/atoms of the conductor & hence their motion is obstructed. The net hindrance offered by a conductor to the flow of free e–s or simply current is called electrical resistance. It depends upon the size, geometry, temperature and nature of the conductor. For a given conductor of uniform cross-section A and length l, the electrical resistance R is directly proportional to length l and inversely proportional to cross-sectional area A i.e., or or ρ is called specific resistance or electrical resistivity. The SI unit of resistivity is ohm - m. It is the reciprocal of resistivity i.e. . The SI unit of conductivity is Ohm–1m–1 or mho/m. Ohm’s law may also be expressed as, J = σE where J = current density and E = electric field strength Conductivity, where n is free electron density, τ is relaxation time and m is mass of electron. • The value of ρ is very low for conductor, very high for insulators & alloys, and in between those of conductors & insulators for semiconductors. • Resistance is the property of object while resistivity is the property of material. It is incorrect to think that if the length of a resistor is doubled its resistance will become twice. If you look by an eye of physicist you will find that when l change, A will also change. This is discussed in the following article. Case of Reshaping a Resistor On reshaping, volume of a material is constant. i.e., Initial volume = final volume or, Ai li = Af lf ... (i) where li, Ai are initial length and area of cross-section of resistor and lf, Af are final length and area of cross-section of resistor. If initial resistance before reshaping is Ri and final resistance after reshaping is Rf then ... (ii) From eqs. (i) and (ii) , This means that resistance is proportional to the square of the length during reshaping of a resistor wire. Also from eqs. (i) and (ii) , This means that resistance is inversely proportional to the square of the area of cross-section during reshaping of resistor. Since A = π r2 (for circular cross-section) where r is radius of cross section. Resistance of a conductor is given by Rt = R0 (1 + αΔt) Where α = temperature coefficient of resistance and Δt = change in temperature If ρ1 and ρ2 be resistivity of a conductor at temperature t1 and t2, then ρ2 = ρ1 (1 + α Δ T) where α = temperature coefficient of resistivity and where ΔT = t2 – t1 = change in temperature The value of α is positive for all metallic conductors. ∴ ρ2 > ρ1 In other words, with rise in temperature, the positive ions of the metal vibrate with higher amplitude and these obstruct the path of electrons more frequently. Due to this the mean path decreases and the relaxation time also decreases. This leads to increase in resistivity. Please note that the value of α for most of the metals is In case of alloys, the rate at which the resistance changes with temperature is less as compared to pure metals. For example, an alloy manganin has a resistance which is 30-40 times that of copper for the same dimensions. Also the value of α for manganin is very small ≈ 0.00001°C–1. Due to the above properties manganin is used in preparing wires for standard resistance (heaters), resistance boxes etc. Please note that eureka and constantan are other alloys for which ρ is high. These are used to detect small temperature, protect picture tube/ windings of generators, transformers etc. For semiconductors : The resistivity of semiconductors decreases with rise in temperature. For semiconductor the value of α is negative. With rise in temperature, the value of n increases. Please note that decreases with rise in temperature. But the value of increase in n is dominating for the value of ρ in this case. For electrolytes : The resistivity decreases with rise in temperature. This is because the viscosity of electrolyte decreases with increase in temperature so that ions get more freedom to move. For insulators : The resistivity increases nearly exponentially with decrease in temperature. Conductivity of insulators is almost zero at 0 K. Superconductors : There are certain materials for which the resistance becomes zero below a certain temperature. This temperature is called the critical temperature. Below critical temperature the material offers no resistance to the flow of e–s. The material in this case is called a superconductor. The reason for superconductivity is that the electrons in superconductors are not mutually independent but are mutually coherent. This coherent cloud of e–s makes no collision with the ions of super-conductor and hence no resistance is offered to the flow of e–s For example, R = 0 for Hg at 4.2 K and R = 0 for Pb at 7.2 K. These substances are called superconductors at that critical temperature. Superconductors are used • in making very strong electromagnets • to produce very high speed computers • in transmission of electric power • in the study of high energy particle physics and material science When a number of resistances are joined end to end so that same current flows through each, resistor when some potential difference is applied across the combination, the conductor are said to be connected in series. The equivalent resistance in series is given by (Req)s = R1 + R2 + ...+ Rn Equivalent resistance of same resistances connected in series is always greater than the greatest of individual resistance. Potential division rule in series combination Two or more resistors are said to be connected in parallel if the same potential difference exists across all resistors. The equivalent resistance is given by The equivalent resistance in a parallel combination is always less than the value of the least individual resistance in the circuits. Current division rule in parallel combination In a given combination of resistors, when you want to detect whether the resistances are in series or in parallel then see that if the same current flows through two resistors then these are in series and if same potential difference is there across two resistors then these are in parallel potential diff across each resistor is the same & is equal to the applied potential difference. This method is applicable only when the resistors can be clearly identified as in series or parallel. Let us take some example to find resistance between ends A and B. Ex.(v) Infinite series : We observe that there is a repetitive unit extending to infinity on left hand side. We assume that the equivalent resistance of all the unit except one (shown dotted) is equal to X ohm. The equivalent circuit will be as shown below. The equivalent resistance across A and B is Please note that RAB can be taken as X because if you add one unit to the sum of infinite units, then it will be approximately the same. Solve the equation as a normal algebraic equation to find X. Ex.(i) The circuit shown in figure is symmetrical about XAEBY axis. This is because the upper part of the axis is the mirror image of lower part (resistors and current direction both) ∴ IAC = IAD; ICB = IDB ; IAE = IEB ( wheatstone bridge principle) ⇒ ICE = IED = 0 Therefore the circuit can be redrawn. It is now easier to find resistance between X and Y. Ex.(ii) The circuit shown is symmetrical about axis XY. Therefore VB = VH ; VC = VI = VG; VD = VF Therefore the circuit can be redrawn as Ex.(iii) The circuit is asymmetric about the dotted line ∴ IBG = IGC; IFG = IGE and IAG = IGB Therefore the equivalent circuit is The diagram given above is symmetrical but the positions of the resistances are shifted. Let I be the current in the circuit from A. The same leaves the circuit at C. Let current in AB, AD and AE be I1 , I2 and I3 respectively. Since the same current flows in AE and EC, the detached equivalent circuit can be drawn as Equivalent resistance between A and B of the resistors connected as shown in the figure Wheatstone bridge All paths from one point to another which have the same setting of resistances have the same amount of currents. Example : Twelve wires each having resistance r are joined to form a cube. We have to find the equivalent resistance across A and B. By path symmetry, IAB = IBC = IAD = IDC = I ∴ IAE = I– 2I1 ⇒ IGC = I– 2I1, Since current in AB = current in BC ⇒ IBF = 0 Also IAD = IDC ⇒ IDH = 0 The equivalent circuit will be as shown. The resistance now clearly visible as in series and in parallel. ; ; ; Using delta to star conversion If none of the above method works then we may use Kirchhoff’s method which will be discussed later • Resistors are not just in series or in parallel if they look so geometrically, e.g. the resistors in the diagram are not in parallel but in series. These resistors across A and B are in series, as same current passes through them. • This is a common thinking that current which comes out from the positive terminal of a battery is used up till it reaches the negative terminal. But infact the current remains the same in a branch. In fact a potential drop takes place across a resistor. IA = IB = IC = ID = 1 amp VA = VB = +5V VC = VD = 0V This means that a potential drop of 5V takes place across the resistor • 🗴 Incorrect : If two resistances are not in series then it is in parallel and vice-versa. ✓ Correct : The above thinking is incorrect. We may have resistances which are neither in series nor in parallel. It is a system of colour coding used to indicate the values of resistors. For the fixed, moulded composition resistor, four colour bands are printed on one end of the outer casing as shown below. The colour bands are always read left to right from the end that has the bands closest to it. • The first and second colour bands, represent the first and second significant digits respectively, of the resistance value. • The third colour band is for the number of zeros that follow the second digit. • In case the third band is gold or silver, it represents a multiplying factor of 0.1 or 0.01. • The fourth band represents the manufacturer's tolerance. It is a measure of the precision with which the resistor was made. • If the fourth band is not present, the tolerance is assumed to be ± 20%. Standard value of colour codes for carbon resistors To learn the above table of colour codes of resistors let us learn this interesting sentence : BB ROY of Great Britain has a Very Good Wife. In the above sentence the capital letters have the following meaning : B-Black, B-Brown, R-Red, O-Orange, Y-Yellow, G-Green, B-Blue, V-Violet, G-Gold, W-White Remember the colour in the above order and the corresponding digits from 0 to 9 and also the multiplier with the power to 10 from 0 to 9. Commercial resistors are of two types • Wire round resistor made by winding of wires of an alloy manganins, constantan and nichrome. • Carbon resistors have low cost and are compact. A thermistor is a heat sensitive resistor usually made up of semiconductor. The oxides of various metals such as mickel, iron, copper etc. temperature coefficient of thermistor is –ve but is usually large, of the order of 0.04/ºC. The V–I curve of thermistor is as shown. Thermistors are used for resistance thermometer in very low temperature measurement of the order of 10K and to safeguard electronic circuits against current jumps because initially thermistors has high resistance when cold and its resistance drops appreciably when it heats up. It states that the amount of heat produced in a conductor is directly proportional to the • square of the current flowing through the conductor, (q, T – constt) i.e. H ∝ i2 • resistance of the conductor (i, T – constt.) i.e. H ∝ R • time for which the current is passed (i, R, – constt) i.e., H ∝ t Thus, H = i2 RT joule = i2 RT/4.2 cal It is defined as the rate at which work is done in maintaining the current in electric circuit. Electric power, P = VI = I2R = V2/R watt or joule/second. Electric energy : The electric energy consumed in a circuit is defined as the total work done in maintaining the current in an electric circuit for a given time. Electric energy = VIt = Pt = I2 Rt = V2 t / R The S.I. unit of electric energy is joule (denoted by J) where 1 joule = 1 watt × 1 second = 1 volt × 1 ampere × 1 sec. In household circuits the electrical appliances are connected in parallel and the electrical energy consumed is measured in kWh (kilowatt hour). 1 kWh (1 B.O.T. unit) = 1000 Wh = 3.6 × 106 J An emf (electromotive force) device has a positive terminal (at high potential) and a negative terminal (at low potential). This device is responsible for moving positive charge within itself from negative terminal to positive terminal. For this to happen, work is done by some agency in the emf device. The energy required to do this work is chemical energy (as in a battery), mechanical energy (as in electric generator), temperature difference (as in a thermopile). The emf is thus given by the formula The S.I unit of emf is (V) • Electromotive force is not a force but a potential difference. • E.m.f. can be defined as the work done in moving a charge once around a closed circuit. INTERNAL RESISTANCE (r) The potential difference across a real source of emf is not equal to its emf. The reason is that the charge which is moving inside the emf device also suffers resistance. This resistance is called internal resistance of the emf device. E = IR + Ir = V + Ir ⇒ V = E – Ir • Emf is the property of a cell but terminal potential difference depends on the current drawn from the cell. When the terminals of an emf device are connected with a conducting path without any external resistance then E = Ir ⇒ Since internal resistance has a very small value, therefore a very high current flows in the circuit producing a large amount of heat. This condition is called short circuiting. During short circuiting, the terminal potential difference is zero. Equivalent Emf EAB = E1 + E2 + ... + En Equivalent internal resistance, RAB = r1 + r2 + ....... + rn Equivalent emf Equivalent internal resistance If the cells are connected as shown below then they are said to be in mixed grouping. Equivalent emf EAB = nE Equivalent resistance = Where n = no. of cells in a row. and M = no. of rows If this equivalent cell is attached to an external resistance R then 1. The condition for maximum current through external resistance R ⇒ R = nr/m In other words, when external resistance is equal to total internal resistances of all the cells. The maximum current 2. Maximum power dissipation for the circuit shown in fig. For maximum power across the resistor, On solving, we get R = r This is the condition for maximum power dissipation. 3. If identical cells are connected in a loop in order, then emf between any two points in the loop is zero. 4. If n identical cells are connected in series and m are wrongly connected then Enet = nE – 2mE 1. 1st law : The mass of the substance liberated or deposited at an electrode during electrolysis is directly proportional to the quantity of charge passed through the electrolyte. i.e., mass m ∝ q = Zq = Z It, where Z = electrochemical equivalent (E.C.E.) of substance. 2. 2nd law : When the same amount of charge is passed through different electrolytes, the masses of the substance liberated or deposited at the various electrodes are proportional to their chemical where m1 and m2 are the masses of the substances liberated or deposited on electrodes during electrolysis and E1 and E2 are their chemical equivalents. Faraday constant is equal to the amount of charge required to liberate the mass of a substance at an electrode during electrolysis, equal to its chemical equivalent in gram (i.e. one gram equivalent) One faraday (1F) = 96500 C/gram equivalent. 1. If ρ is the density of the material deposited and A is the area of deposition, then the thickness (d) of the layer deposited in electroplating process is . 2. The back e.m.f. for water voltameter is 1.67 V and it is 1.34 V for CuCl2 electrolytes voltameter with platinum electrodes. 3. 96500 C are required to liberate 1.008 g of hydrogen. 4. 2.016 g of hydrogen occupies 22.4 litres at N.T.P. 5. E.C.E. of a substance = E.C.E. of hydrogen × chemical equivalent of the substance. When an electric circuit is composed of two dissimilar metals and the junctions are maintained at different temperature, then an emf is set up in the circuit. This effect is known as thermoelectric or seebeck effect. It is a device in which heat energy is converted into electrical energy. Its working is based on seebeck effect. It has two junctions of two dissimilar metals. Some of the elements forming thermo-electric series Sb, Fe, Zn, Cu, Au, Ag, Pb, Al, Hg, Pt, Ni, Bi Lead (Pb) is thermo-electrically neutral At the cold junction, current flows from the element occuring earlier into the element occurring later in the series. For example: In Cu–Fe thermocouple, current flows from Cu to Fe at hot junction. NEUTRAL TEMPERATURE (Tn) It is that temperature of hot junction for which the thermo emf produced in a thermocouple is maximum. It depends upon the nature of the material of thermocouple but is independent of temperature of cold junction. TEMPERATURE OF INVERSION (Ti) It is that temperature of hot junction for which the thermo emf becomes zero and beyond this temperature, the thermo emf in a thermocouple reverses its direction. It depends upon the nature of the material of thermocouple and temperature of cold junction Let To, Tn, Ti be the temperature of cold junction, neutral temperature and temperature of inversion then With temperature difference T between hot and cold junctions, the thermo-e.m.f. is given by E = αT + βT2 where α and β are Seebeck coefficients At Tn, (dE/dT) = 0 ∴ Tn = – α/2β and Ti = – α/β, when To = 0 S = dE/dT is called thermo-electric power. It states that if current is passed through a junction of two different metals the heat is either evolved or absorbed at that junction. It is the reverse of seebeck effect. The quantity of heat evolved or absorbed at a junction due to Peltier effect is proportional to the quantity of charge crossing that junction. PELTIER COEFFICIENT (π) It is defined as the amount of heat energy evolved or absorbed per second at a junction of two different metals when a unit current is passed through it. The Peltier heat evolved or absorbed at a junction of a thermocouple = πI t where I = current passing through the junction for time t. where T and (T + dT) are the temperature of cold and hot junctions of a thermocouple and dE is the thermo emf produced. (Seebeck coefficient) If a metallic wire has a non uniform temperature and an electric current is passed through it, heat may be absorbed or produced in different sections of the wire. This heat is over and above the joule heat I2Rt and is called Thomson heat. The effect is called Thomson effect. If a charge ΔQ is passed through a small section of given wire having temperature difference ΔT between the ends, Thomson heat, ΔH = σ ΔQ ΔT where σ is constant for a given metal at a given temperature. Thomson emf, σ ΔT, is defined as σ ΔT = ΔH/ΔQ. Note:- σ is positive if heat is absorbed when a current is passed from low temp. to high temperature. σ is numerically equal to P.D. developed between two points of the conductor differing in temp. by 1ºC. • The actual emf developed in a thermocouple loop is the algebraic sum of the net Peltier emf and the net Thomson emf developed in the loop. • If S, π and σ are the Seebeck coefficient, Peltier coefficient, and Thomson coefficient respectively then it is found that • For Peltier effect or Thomson effect, the heat evolved or absorbed is directly proportional to current. But for Joule's law of heating, the heat produced is directly proportional to the square of the current flowing through it. • Thermo-emf set up in a thermocouple when its junctions are maintained at temperature T1 and T3 (i.e. ) is equal to the sum of the emfs set up in a thermocouple when its junctions are maintained first at temperature T1 and T2 (i.e. ) and then at T2 & T3 (i.e. ) i.e. It is called law of intermediate temperature. Many practical combination of resistors cannot be reduced to simple series, parallel combinations. For example the resistors in the figure are neither in series nor in parallel. The use of Ohm’s law is not sufficient to solve such problems. Kirchoff’s laws are used in such cases. We will often use the term junction and loop, so let us first understand the meaning of these words. A junction in a circuit is a point where three or more conductors meet. A loop is a closed conducting path. In the above figure e, f, d, c are junctions. a, b, are not junctions. The various loops are efde, cdfc, eabcf and eabcde. (Based on conservation of charge) At any junction, the sum of currents entering the junction must be equal to the sum of currents leaving it. If this is not so, charges will accumulate at the junction. This cannot happen as this would mean high/low potential maintained at a point in a wire without external influence. When we apply this rule at junction c, we get I = I1 + I2 (Based on energy conservation) The algebraic sum of changes in potential around any closed loop of a circuit must be zero. Sign convention for using loop law If we move a loop element (resistor, emf device, capacitor, inductor etc.) in the direction of increasing potential, we take the potential difference positive and vice-versa. 1. Draw a circuit diagram large enough to show all resistors, emf device, capacitors, currents clearly. 2. Take into account the resistance of voltmeter/ammeter/internal resistance of a cell (if given). 3. Assume the direction of current in all branches. It may be noted here that one branch has only one direction of current. It is best to use junction law simultaneously while drawing currents. This helps to reduce the number of unknown quantities. In the above circuits we arbitrarily assumed the direction of current I1 in branch abcd as anti-clockwise and the direction of current I2 in branch afed as clockwise. In figure 1 we have two unknown currents (I1, I2) whereas figure 2 we have three unknown currents (I1, I2 and I3). The first figure is a better option for solving problems. In figure 1 we used junction rule at d simultaneously while labelling currents. 4. In a branch containing a capacitor, the current is zero when d.c is applied and steady state conditions are achieved. 5. Now we need as many independent equations as there are conditions unknowns. If we have to find a particular unknown, we should ensure that, the unknown appears in one of the equations made by us. 6. For making equations choose the loop and travel the loop completely. We may travel the loop in clockwise or anti-clockwise direction. While using second law use sign conventions properly. 7. Solve the equations formed to find the unknown quantities. If any value of current comes out to be negative then that particular current is in the opposite direction to that assumed. Let us use second law in the loop abcda of figure 1 taking the loop in anti-clockwise direction starting from a. + E2 – I1R4 – (I1 + I2) R3 = 0 For loop afeda, moving the loop in clockwise direction we get – E1 – I2 R1 – I2R2 – (I1 + I2)R3 = 0 NODE METHOD TO APPLY KIRCHOFF’S LAW (OPEN LOOP METHOD) Step 1 : We select a reference node and assume its potential to be (zero/x)V Step 2 : We calculate the voltage of other selected points w.r.t. the reference node Step 3: We find some independent node (whose voltage is not known). We apply Kirchoff’s law to find the relevant values. The condition for balanced wheatstone bridge Note that when battery and galvanometer of a Wheatstone bridge is interchanged, the balance position remains undisturbed, while sensitivity of the bridge changes. In the balanced condition, the resistance in the branch BD may be neglected Example : Resistance connected to BC may be neglected. Note:- In a Wheatstone bridge, the deflection in a galvanometer does not change, if the battery and the galvanometer are interchanged Measuring temperature with the help of Wheatstone bridge At balancing When P = Q then ΔR = S α ΔT 🗴 Incorrect. If the current flows in a wire, there has to be a potential difference. The potential drop takes place only when current passes through a resistor. ✓ Correct. In the diagram, the three resistors are in parallel. The potential at A is equal to the potential at C. Current flows in wire 1 but there is no potential drop across A and C. 🗴 Incorrect. If potential difference between the points is zero, there is zero current between the two points. ✓ Correct. There is no p.d. between A and C still current flows in segment 1. Based on balanced Wheatstone bridge principle. To find unknown resistance Let P be the unknown resistance. At balance point Q is known and l can be calculated. The p.d. across a resistance wire is directly proportional to its length provided I, r and A are constant. V = IR = I ρ ⇒ V α l [ I, ρ and A are constant] PQ is the resistance wire of potentiometer generally made up of constantan or nichrome. One end P is connected to the positive terminal of the battery B while negative terminal is connected to Q through a Rheostat (Rh) and key (K). This is the main circuit. A cell whose emf has to be measured is also connected to the potential wire in such a way that the positive terminal is connected with P and negative terminal is connected to a galvanometer and then to a jockey (J) which is free to slide along the wire There is a potential drop along PQ. The potential drop per unit length along PQ is called potential gradient. When the jockey is pressed on some point, current flows from E to P (⇒). Also current that comes from B after reaching P divides into two parts. One part moves towards A and the other towards E (→). Three cases may arise. • IB > IE. This happens when VPC > E. One side deflection in galvanometer • IB = IE. This happens when VPC = E, Zero deflection in galvanometer • IB < IE . This happens when VPA < E. Other side deflection in galvanometer • At null point since no current flows through E therefore it is said to be in the condition of open circuit. • More is the length of potentiometer, higher is the sensitivity of potentiometer and smaller is the potential gradient. • Potentiometer will work only when B > E. Also the positive terminal of the batteries is connected at P. If any of the above conditions is not followed, we do not get a null point. • Comparison of emfs of cells • To find internal resistance of a cell • Emf can be measured by potentiometer and not voltmeter. It is an instrument used to detect small currents in a circuit. The current required for full scale deflection in the galvanometer is called full scale deflection current and is denoted by Ig. It is defined as the deflection produced in the galvanometer, when unit current flows through it. Current sensitivity, and its unit is rad A–1 Current sensitivity can be increased either by decreasing C i.e. restoring torque per unit twist or increasing B. It is defined as the deflection produced in the galvanometer when a unit voltage is applied across the two terminals of the galvanometer. Voltage sensitivity , its unit is rad V–1 Ammeter is used to measure current in a circuit. Ammeter is always connected in series in the circuit as shown. Conversion of a galvanometer into an ammeter For this, we connect a small resistance S (called shunt) in parallel with the galvanometer. Mathematically, Ig × G = (I – Ig) S where I is the maximum current which ammeter can measure. G is the resistance of galvanometer and Ig is the current of full scale deflection in galvanometer. S is shunt. Since shunt is a small resistance. Therefore the resistance of ammeter is very small. The above arrangement is made so that when we connect ammeter in series to measure current, it does not change the original current to a large extent. The change is in fact very small. Also since galvanometer is a sensitive device and cannot take large currents, this arrangement serves the purpose. Most of the current entering the ammeter passes through the shunt as current always prefer low resistance path. An ideal ammeter is one which has zero resistance. The range of ammeter can be increased but cannot be decreased below Ig. Voltmeter is used to measure potential difference across a resistor. Voltmeter is always connected in parallel across a resistor. Conversion of a galvanometer into a voltmeter For this, we connect a large resistance R in series with the galvanometer. The potential difference which has to be measured is across the external resistance i.e. across points a and b. Let it be V. Then V = Ig (G + R) where V is the maximum potential difference that the voltmeter can measure and R is the large resistance connected in series with the galvanometer The resistance of the voltmeter will be RV = G + R Since R is a large resistance. Therefore resistance of voltmeter is very large. An ideal voltmeter is one which has infinite resistance. The range of voltmeter can be increased and decreased. Note:- When ammeter/voltmeter is connected in the circuit, the current or voltage indicated by these is less than the actual values in their absence.	{"url":"https://www.cleariitmedical.com/2019/05/physics-notes-current-electricity.html","timestamp":"2024-11-06T17:29:38Z","content_type":"application/xhtml+xml","content_length":"961235","record_id":"<urn:uuid:859c3371-ba08-4f43-83a0-0e9192b299ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00884.warc.gz"}
Darren Strash I am a technophile who is committed to open science and equality in education. My goal is to produce innovative high-quality research, and use my extensive research and practical experience to effectively teach and mentor the next generation of computer scientists. In 2011, I received a PhD in Computer Science from University of California, Irvine under the advisement of David Eppstein and Mike Goodrich. From there, I worked in research and development in Intel's Computational Lithography Group until 2014. I then worked as a postdoctoral researcher with Peter Sanders at Karlsruhe Institute of Technology from 2014 to 2016, and as a Visiting Assistant Professor at Colgate University from 2016 to 2018. I am now an Assistant Professor at Hamilton College. If you are looking for more information, here is my CV, my research statement, and my teaching statement. My passion is to reveal and resolve the mismatch between the theory and practice of algorithms, with applications in large scale network analysis and computational geometry. My work often involves first understanding real-world properties of data sets, then designing algorithms that exploit these properties to gain efficiency that is not possible otherwise. This includes both theoretical efficiency and efficiency of algorithms in practice (algorithm engineering). Some specific areas that interest me are combinatorial optimization, subgraph counting/listing, network visualization, shortest paths, range searching, and dynamic data structures. Papers in Refereed Journals [5] Finding Near-Optimal Independent Sets at Scale S. Lamm, P. Sanders, C. Schulz, D. Strash, and R.F. Werneck Journal of Heuristics 23(4), 2017, pp. 207-229. The code is freely available under GPL v2.0. [4] Listing All Maximal Cliques in Large Sparse Real-World Graphs in Near-Optimal Time D. Eppstein, M. Löffler, and D. Strash ACM Journal of Experimental Algorithmics 18(3): 3.1, ACM, 2013. Special issue for SEA 2011. The code and data sets are freely available under GPL v3.0. [3] Category-Based Routing in Social Networks: Membership Dimension and the Small-World Phenomenon D. Eppstein, M.T. Goodrich, M. Löffler, D. Strash, and L. Trott Theoretical Computer Science 514, 2013, pp. 96-104. Special issue for GA 2011. [2] Extended Dynamic Subgraph Statistics using h-index Parametrized Data Structures D. Eppstein, M.T. Goodrich, D. Strash, and L. Trott Theoretical Computer Science 447, 2012, pp. 44-52. Special issue for COCOA 2010. [1] Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings D. Eppstein, M.T. Goodrich, and D. Strash SIAM Journal on Computing 39 (8), 2010, pp. 3814-3829. Papers in Peer-Reviewed Conference Proceedings [17] Reconstructing Generalized Staircase Polygons with Uniform Step Length N. Sitchinava and D. Strash Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017), LNCS, Springer, 2017 (to appear). [16] Distributed Evolutionary k-way Node Separators P. Sanders, C. Schulz, D. Strash, and R. Williger Proceedings of the Conference on Genetic and Evolutionary Computing and Combinatorial Optimization (GECCO 2017), pp. 345-352, ACM, 2017. [15] Shared Memory Parallel Subgraph Enumeration R. Kimmig, H. Meyerhenke, and D. Strash Proceedings of the 31st IEEE International Parallel and Distributed Processing Symposium Workshops (IEEE IPDPSW 2017), pp. 519-529, IEEE, 2017. [14] Temporal Map Labeling: A New Unified Framework with Experiments L. Barth, B. Niedermann, M. Nöllenburg, and D. Strash Proceedings of the 24th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM SIGSPATIAL 2016), pp. 23:1-23:10, ACM, 2016. [13] On the Power of Simple Reductions for the Maximum Independent Set Problem D. Strash Proceedings of the 22nd International Computing and Combinatorics Conference (COCOON 2016), LNCS vol. 9797, pp. 345-356. [12] Accelerating Local Search for the Maximum Independent Set Problem D. Dahlum, S. Lamm, P. Sanders, C. Schulz, D. Strash, and R.F. Werneck Proceedings of the 15th International Symposium on Experimental Algorithms (SEA 2016), LNCS vol. 9685, pp. 118-133. [11] Finding Near-Optimal Independent Sets at Scale S. Lamm, P. Sanders, C. Schulz, D. Strash, and R.F. Werneck Proceedings of the 18th Meeting on Algorithm Engineering and Experiments (ALENEX 2016), pp. 138-150. The code is freely available under GPL v2.0. [10] On Minimizing Crossings in Storyline Visualizations I. Kostitsyna, M. Nöllenburg, V. Polishchuk, A. Schulz, and D. Strash Proceedings of the 23rd International Symposium on Graph Drawing and Network Visualization (GD 2015), LNCS vol. 9411, pp. 192-198. [9] On the Complexity of Barrier Resilience for Fat Regions M. Korman, M. Löffler, R.I. Silveira, and D. Strash Proceedings of the 9th International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics (ALGOSENSORS 2013), LNCS vol. 8243, pp. 201-216. [8] Dynamic Planar Point Location with Sub-logarithmic Local Updates M. Loffler, J.A. Simons, and D. Strash Proceedings of the 13th International Symposium on Algorithms and Data Structures (WADS 2013), LNCS vol. 8037, pp. 499-511. [7] Category-Based Routing in Social Networks: Membership Dimension and the Small-World Phenomenon D. Eppstein, M.T. Goodrich, M. Löffler, D. Strash, and L. Trott Proceedings of the 3rd International Conference on Computational Aspects of Social Networks (CASoN 2011), pp. 102--107. [6] Listing All Maximal Cliques in Large Sparse Real-World Graphs D. Eppstein and D. Strash Proceedings of the 10th International Conference on Experimental Algorithms (SEA 2011), LNCS vol. 6630, pp. 403-414. The code and data sets are freely available under GPL v3.0. [5] Extended Dynamic Subgraph Statistics using h-index Parametrized Data Structures D. Eppstein, M.T. Goodrich, D. Strash, and L. Trott Proceedings of the 4th International Conference on Combinatorial Optimization and Applications (COCOA 2010), LNCS vol. 6508, pp. 128-141. [4] Priority Range Trees M.T. Goodrich and D. Strash Proceedings of the 21st International Symposium on Algorithms and Computation (ISAAC 2010), LNCS vol. 6506, pp. 97-108. [3] Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time D. Eppstein, M. Löffler, and D. Strash Proceedings of the 21st International Symposium on Algorithms and Computation (ISAAC 2010), LNCS vol. 6506, pp. 403-413. [2] Succinct Greedy Geometric Routing in the Euclidean Plane M.T. Goodrich and D. Strash Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009), LNCS vol. 5878, pp. 781-791. [1] Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Crossings D. Eppstein, M.T. Goodrich, and D. Strash Proceedings of the 20th ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pp 150-159. Other Publications [4] Reconstructing a Unit-Length Orthogonally Convex Polygon from its Visibility Graph N. Sitchinava and D. Strash 32nd European Workshop on Computational Geometry (EuroCG 2016), July 2011. [3] Category-Based Routing in Social Networks: Membership Dimension and the Small-World Phenomenon D. Eppstein, M.T. Goodrich, M. Löffler, D. Strash, and L. Trott Workshop on Graph Algorithms and Applications (GA 2011), July 2011. [2] Extending Garbage Collection to Complex Data Structures L. Effinger-Dean, C. Erickson, M. O'Neill, and D. Strash Proceedings of the 3rd Workshop on Semantics, Program Analysis and Computing Environments for Memory Management (SPACE 2006), pp 91-97. [1] Garbage Collection for Trailer Arrays L. Effinger-Dean, C. Erickson, M. O'Neill, and D. Strash Proceedings of the 3rd Workshop on Semantics, Program Analysis and Computing Environments for Memory Management (SPACE 2006), pp 83-90. Software and Data Sets Quick Cliques Software to quickly list all maximal cliques in sparse graphs Code released under GPLv3.0 and data sets Karlsruhe Maximum Independent Sets Software to find near-optimal independent sets in huge complex networks Code released under GPLv2.0 Contact Me Hamilton College Department of Computer Science 198 College Hill Road Clinton, NY 13323 Office: Taylor Science Center 2015A Phone: (315) 859-4188 E-mail: first initial last name AT hamilton DOT edu	{"url":"https://darrenstrash.github.io/","timestamp":"2024-11-13T17:34:00Z","content_type":"text/html","content_length":"45269","record_id":"<urn:uuid:5c073199-ba97-4e7d-84d5-ec405c24ed80>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00297.warc.gz"}
Evolution of packets of surface gravity waves over strong smooth topography Wave packets in a smoothly inhomogeneous medium are governed by a nonlinear Schrödinger (NLS) equation with variable coefficients. There are two spatial scales in the problem: the spatial scale of the inhomogeneities and the distance over which nonlinearity and dispersion affect the packet. Accordingly, there are two limits where the problem can be approached asymptotically: when the former scale is much larger than the latter, and vice versa. In this paper, we examine the limit where the spatial scale of (periodic or random) inhomogeneities is much smaller than that of nonlinearity-dispersion (i.e., the latter effects are much weaker than the former). In this case, the packet undergoes rapid oscillations of the geometric-optical type, and also evolves slowly due to nonlinearity and dispersion. We demonstrate that the latter evolution is governed by an NLS equation with constant (averaged) coefficients. The general theory is illustrated by the example of surface gravity waves in a channel of variable depth. In particular, it is shown that topography increases the critical frequency, for which the nonlinearity coefficient of the NLS equation changes sign (in such cases, no steady solutions exist, i.e., waves with frequencies lower than the critical one disperse and cannot form packets). Dive into the research topics of 'Evolution of packets of surface gravity waves over strong smooth topography'. Together they form a unique fingerprint.	{"url":"https://pure.ul.ie/en/publications/evolution-of-packets-of-surface-gravity-waves-over-strong-smooth-","timestamp":"2024-11-12T13:15:31Z","content_type":"text/html","content_length":"53840","record_id":"<urn:uuid:49514bcb-ba06-4f38-9cd4-eb2bafaf1f9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00688.warc.gz"}
How Do You Find the Lateral and Surface Areas of a Rectangular Prism? The lateral area of a three-dimensional solid is the area of all the lateral faces. In this tutorial, you'll see how to use the dimensions of a rectangular prism to find the lateral area. Take a A rectangle is one of the many fundamental shapes you'll see in math. Rectangles have special properties that can be very useful in helping you solve a problem. This tutorial introduces you to rectangles and explains their interesting qualities! If two figures have the same size and shape, then they are congruent. The term congruent is often used to describe figures like this. In this tutorial, take a look at the term congruent! The term prism is a cool name for a special kind of three-dimensional solid. This tutorial defines the term prism and shows you how to name a prism using the shape of its bases. Check it out! Want to know how the find the lateral and surface areas of a triangular prism? Then check out this tutorial! You'll see how to apply each formula to the given information to find the lateral area and surface area. Take a look!	{"url":"https://virtualnerd.com/pre-algebra/perimeter-area-volume/surface-area/lateral-surface-areas-examples/rectangular-prism-lateral-surface-areas","timestamp":"2024-11-13T13:06:08Z","content_type":"text/html","content_length":"28169","record_id":"<urn:uuid:706ba5e8-ce7c-4cc3-bbc6-b39174da95db>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00276.warc.gz"}
Comments on Physics with an edge: MiHsC with horizons, no waves./* There is no relativistic light speed limit for...Zephir: There is no relativistic light speed limit...The explanation of inertia with radiation pressure...Someone was very concerned by this very point and ...Czeko: The example does need non-locality, but thi...@mike Does the principle of locality is broken in ...Interesting preprint from Matt Walker & Abraha...Czeko: The standard explanation of Unruh radiation...Odd question: what generates Unruh radiation?Maybe spacetime is 5-D and we're on a 3-sphere...Analytic D: Thanks for sharing this. I could do wi...Mike: If you lose a dimension and spacetime degene...ZeroIsEverything: :) Your joke didn't fall fla...Ryan: Indeed, you're right and I'm hoping ...I went on a 16 mile hike in September and somethin...Mike, reading your comment, I imagined this situat...Losing a dimension makes sense in light of the hol...Alain: The Higgs mechanism is only responsible for...Czeko: Thanks. I'm working on a sheltering mod...interesting. I think about what is the way matter ...Best explanation so far. Maybe mass shadows Unru... tag:blogger.com,1999:blog-4637778157419388168.post7933357166285280246..comments2024-10-26T11:40:33.076-07:00Mike McCullochhttp://www.blogger.com/profile/ 00985573443686082382noreply@blogger.comBlogger21125tag:blogger.com,1999:blog-4637778157419388168.post-68270330259672581432017-01-17T13:26:08.614-08:002017-01-17T13:26:08.614-08:00/* There is no relativistic light speed limit for monochromatic waves (as Unruh waves are) since such waves carry no information. /<br /><br />In relativity the Unruh radiation is essentially black body radiation: it's neither monochromatic, neither superluminal. The filtering white light to monochromatic doesn't make it superluminal.Zephirhttps://www.blogger.com/profile/ 06010623752049244967noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-42525323755524565372017-01-17T13:01:16.299-08:002017-01-17T13:01:16.299-08:00Zephir: There is no relativistic light speed limit for monochromatic waves (as Unruh waves are) since such waves carry no information.Mike McCullochhttps://www.blogger.com/profile/ 00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-42541846343245910082017-01-17T07:21:18.474-08:002017-01-17T07:21:18.474-08:00The explanation of inertia with radiation pressure of Unruh waves is sorta recursive by itself, as it depends on inertia of Unruh waves. And how large/distant the Rindler horizon is in comparison to cosmic horizon? The Unruh radiation propagates with speed of light, so that no inertia induced by its shielding can be a momentary effect.<br /><br />http://physicsfromtheedge.blogspot.cz/2015/10/ explaining-mihsc-with-schematics.html<br /><br />https://i.imgur.com/OKTzJma.jpgZephirhttps://www.blogger.com/profile/ 06010623752049244967noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-17632663180919838572015-11-09T23:34:12.989-08:002015-11-09T23:34:12.989-08:00Someone was very concerned by this very point and throwed your theory to the bin because of that. Is there a formal formulation explaining why the locality principle is broken and in which way it doesn't matter at all?<br /> 04231020181834141834noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-77823321668241943492015-11-09T13:30:44.198-08:002015-11-09T13:30:44.198-08:00Czeko: The example does need non-locality, but this doesn't bother relativity because the selection or deselection of Unruh waves by the horizon can be achieved at the phase velocity. Unruh waves, infinite in extent and each of constant frequency cannot carry information, so their phase velocities can be greater than c.Mike McCullochhttps://www.blogger.com/profile/ 00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-69232023016067171932015-11-09T09:32:54.477-08:002015-11-09T09:32:54.477-08:00@mike Does the principle of locality is broken in your example?<br />Czekohttps://www.blogger.com/profile/ 04231020181834141834noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-23440390441590220002015-11-06T17:58:57.306-08:002015-11-06T17:58:57.306-08:00Interesting preprint from Matt Walker & Abraham Loeb, which might add clues to the nature of "dark mass/energy" and support the Inertial Horizon view:<br /><br />http://arxiv.org/abs/1401.1146<br /><br />Is the Universe Simpler than LCDM?<br /><br />Matthew G. Walker, Abraham Loeb<br /><br />(Submitted on 6 Jan 2014 (v1), last revised 28 May 2014 (this version, v2))<br /><br />In the standard cosmological model, the Universe consists mainly of two invisible substances: vacuum energy with constant mass-density rho_v=\Lambda/(8pi G) (where Lambda is a `cosmological constant' originally proposed by Einstein and G is Newton's gravitational constant) and cold dark matter (CDM) with mass density that is currently rho_{DM,0}\sim 0.3 rho_v. This `LCDM' model has the virtue of simplicity, enabling straightforward calculation of the formation and evolution of cosmic structure against the backdrop of cosmic expansion. Here we review apparent discrepancies with observations on small galactic scales, which LCDM must attribute to complexity in the baryon physics of galaxy formation. Yet galaxies exhibit structural scaling relations that evoke simplicity, presenting a clear challenge for formation models. In particular, tracers of gravitational potentials dominated by dark matter show a correlation between orbital size, R, and velocity, V, that can be expressed most simply as a characteristic acceleration, a_{DM}\sim 1 km^2 s^{-2} pc^{-1} \approx 3 x 10^{-9} cm s^{-2} \approx 0.2c\sqrt{G rho_v}, perhaps motivating efforts to find a link between localized and global manifestations of the Universe's dark components.qraalhttps://www.blogger.com/profile/ 13436948899560519608noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-83261106196042344322015-11-02T14:34:36.073-08:002015-11-02T14:34:36.073-08:00Czeko: The standard explanation of Unruh radiation is that a Rindler horizon splits up ever-forming zpf virtual particle pairs so the unpaired particles become real, but I'm moving towards an informational understanding and I can derive the usual formula like this: when a Rindler horizon forms it reduces the uncertainty in position Delta_x, so by Heisenberg Delta_p must increase. Since E=pc, energy is produced. Energy from information loss. See Appendix C of my book (the appendices are available for free as a pdf on the world scientific webpage for my book).Mike McCullochhttps://www.blogger.com/profile/ 00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-13722777104900590712015-11-02T13:24:01.495-08:002015-11-02T13:24:01.495-08:00Odd question: what generates Unruh radiation?Czekohttps://www.blogger.com/profile/ 04231020181834141834noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-77250228314211525142015-11-01T23:45:15.585-08:002015-11-01T23:45:15.585-08:00Maybe spacetime is 5-D and we&# 39;re on a 3-sphere 'surface' of a 4-space?qraalhttps://www.blogger.com/profile/ 13436948899560519608noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-25611163147405077132015-10-31T03:19:04.413-07:002015-10-31T03:19:04.413-07:00Analytic D: Thanks for sharing this. I could do with going on a long hike as well. Fresh air, open moors, lots of time to think: 'Zen and the Art of Motorcycle Maintenance', 'Not all who wander are lost'..etc.<br /><br />I've been trying to get gravity into the MiHsC framework three ways: 1) using sheltering, 2) from the uncertainty principle involving interactions between Planck masses (published) and 3) from changes in horizon areas. You're suggesting a mix of 1 and the interaction part of option 2, which is the way to go: IMO the answer will use all three, but the maths seems to be suggesting something further than my intuition can go at the moment. I can get G in F=GMm/r^2 to within half a percent, but as I said it falls flat because I lose a dimension. I need that long walk..Mike 00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-49080687191576448592015-10-30T00:38:25.762-07:002015-10-30T00:38:25.762-07:00Mike:<br />If you lose a dimension and spacetime degenerates into a e.g. 2D holographic surface, I guess it's flatlander time for everyone then? ;)ZeroIsEverythinghttps://www.blogger.com/profile/ 13236152077605874591noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-31663833368182569002015-10-29T07:41:19.032-07:002015-10-29T07:41:19.032-07:00ZeroIsEverything: :) Your joke didn't fall flat.Mike McCullochhttps://www.blogger.com/profile/ 00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-53528936686415866652015-10-29T05:13:16.121-07:002015-10-29T05:13:16.121-07:00Ryan: Indeed, you're right and I'm hoping to explain the lost dimension w/ the holographic principle. The alternative is that gravity just can't be captured this way.. We'll see!Mike McCullochhttps:// www.blogger.com/profile/00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-63107839542278265692015-10-28T17:18:10.928-07:002015-10-28T17:18:10.928-07:00I went on a 16 mile hike in September and something about the solitude and the motion led me to thinking about an MiHsC related sheltering model for hours as I wore out my legs.<br /><br />I'm convinced that gravity is due to particle pairwise sheltering. Two massive particles at distance <i>r</i> will block respective waves from the other's horizon that have a node at it's partner, causing an equal pressure pushing them together. Obviously, <i>r/2</i> will have more nodes and thus more pressure. But this is only a linear relationship. The inverse square law comes into play with gravity because QM implies that the number of long waves increases with the surface area at a given radial distance, however linearly fewer wavelengths will have nodes as that wavelength increases, generating an inverse square relationship for sheltering.<br /><br />And the empirical <i>mm</i> term is due to the round robin of sheltering, since particle A1 shelters B1 and B2, as does A2 shelters B1 and B2, for four pairs of sheltering to produce the force, which matches the Newtonian description. B1 sheltering B2 produces no net force towards A and vice versa.<br /><br />This has some implications with high acceleration away from large bodies as well as providing an explicit mechanism for mutual acceleration (I didn't have a mental image of that until now). For example, like you've said, a ring with mass rotating fast enough would experience no gravity from bodies above/below the axis of rotation.<br /><br />I have a feeling though that Mike's derivation is a bit off from my incredibly rough picture here, though.Analytic Dhttps://www.blogger.com/profile/ 14307179997233629815noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-85325572387590044752015-10-28T14:45:26.637-07:002015-10-28T14:45:26.637-07:00Mike, reading your comment, I imagined this situation:<br /><br /><i>On the street</i><br /><br />Police officer:<br />- Sir, is this your dimension?!<br /><br />Mike:<br />- Why, thank you very much. I didn't really miss it. You can keep it, and have a nice day.ZeroIsEverythinghttps://www.blogger.com/profile/ 13236152077605874591noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-62094929456910786672015-10-28T10:14:19.580-07:002015-10-28T10:14:19.580-07:00Losing a dimension makes sense in light of the holographic principle:<br /><br />https://en.wikipedia.org/wiki/Holographic_principle#Energy.2C_matter.2C_and_information_equivalenceRyan Pavlickhttps://www.blogger.com/profile/ 00889118190372082068noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-29608587689559465002015-10-28T08:30:16.270-07:002015-10-28T08:30:16.270-07:00Alain: The Higgs mechanism is only responsible for electron & quark mass, only 0.1% of known mass, so it is negligible.Mike McCullochhttps://www.blogger.com/profile/ 00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-90099101916610293042015-10-28T08:27:23.641-07:002015-10-28T08:27:23.641-07:00Czeko: Thanks. I'm working on a sheltering model & I can get to Newton, but oddly I seem to lose a dimension on the way!.. Can't say more yet..Mike McCullochhttps://www.blogger.com/profile/ 00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-3836952368806793692015-10-28T06:15:30.940-07:002015-10-28T06:15:30.940-07:00interesting.<br />I think about what is the way matter interact with unruh radiation, to behave like mass...<br /><br />If I understand well, if a radiation interact with matter, it is because of Higgs boson ? and the sensibility to higgs is what creat various particles mass?<br /><br />unruhradiation which interact with higgs boson and particle maybe change the pattern, creating casimir effect ?<br /><br />is Higgs mechanism compatible with MiHsC ,inertia? or do they compete?Alain_Cohttps://www.blogger.com/profile/ 08352476615242858677noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-64188301323844469112015-10-28T01:10:06.553-07:002015-10-28T01:10:06.553-07:00Best explanation so far. <br /> <br />Maybe mass shadows Unruh radiation?Czekohttps://www.blogger.com/profile/04231020181834141834noreply@blogger.com	{"url":"https://physicsfromtheedge.blogspot.com/feeds/7933357166285280246/comments/default","timestamp":"2024-11-02T11:33:37Z","content_type":"application/atom+xml","content_length":"46204","record_id":"<urn:uuid:01930c46-45a2-4eaf-9c24-88cc845b0bdb>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00823.warc.gz"}
Round Up/Down to Nearest with ROUND/ROUNDUP/ROUNDDOWN - Knowledge Hub: Your learning resources Round Up/Down to Nearest with ROUND/ROUNDUP/ROUNDDOWN August 24, 2024 Hack and Tricks The ROUND function is a mathematical operation that rounds a number to a predetermined number of decimal places in order to adjust it to a specified degree of precision. In computations involving money and science, where precision is essential, this function is frequently used. The value to be rounded is denoted by “number,” & the number of decimal places to be used is indicated by “num_digits,” in the syntax for the ROUND function, which is =ROUND(number, num_digits). A positive integer “num_digits” causes the number to be rounded to that many decimal places. Key Takeaways • The ROUND function in Excel is used to round a number to a specified number of digits. • The ROUNDUP function in Excel is used to round a number up to a specified number of digits. • The ROUNDDOWN function in Excel is used to round a number down to a specified number of digits. • When using ROUND/ROUNDUP/ROUNDDOWN in Excel, it’s important to understand the syntax and the different arguments that can be used. • Practical examples of rounding up/down to the nearest whole number or decimal can help illustrate the use of ROUND/ROUNDUP/ROUNDDOWN in Excel. If “num_digits” is negative, on the other hand, the number is rounded to the left of the decimal point. To illustrate, the result of applying =ROUND(3.14159, 2) is 3.14, whereas the result of applying =ROUND(3.14159, 0) is 3. Numbers ending in .5 or more are rounded up, and those ending in less than .5 are rounded down. These are the standard rounding rules that the ROUND function Knowing how to use this function correctly is crucial for accurate calculations in a variety of scenarios where rounding is needed. It is a useful tool. Arguments and Syntax. The ROUNDUP function has the following syntax: =ROUNDUP(number, num_digits). You can specify how many decimal places you want to round up a value to by using the “num_digits” argument in addition to the “number” argument, which contains the value you want to round up. Usage & Examples. Using the formula =ROUNDUP(3.14159, 2), for instance, you could round a number like 3.14159 to two decimal places and get 3.15. To round a given number to the closest whole number, use the formula =ROUNDUP(3.14159, 0), which yields a result of 4. Applications in Practice. In circumstances where you wish to round a number up, regardless of its decimal value, the ROUNDUP function comes in handy. It can be applied to pricing, finance calculations, and any other situation where rounding up is required. The ROUNDDOWN function always rounds a number down to the closest designated decimal place, which is the opposite of the ROUNDUP function. The ROUNDDOWN function can be written with the syntax = ROUNDDOWN(number, num_digits). You specify the value to be rounded down to the number of decimal places by using the “num_digits” argument in addition to the “number” argument. For instance, the formula =ROUNDDOWN(3.14159, 2) would yield 3.14 if you had the number 3.14159 & you wanted to round it down to two decimal places. Using the formula =ROUNDDOWN(3.14159, 0) to round the same number down to the nearest whole number would yield a result of 3. When a number needs to be rounded down consistently, regardless of its decimal value, the ROUNDDOWN function comes in handy. When rounding down is required, it can be applied to floor plans, inventory management, and other situations. The ROUND, ROUNDUP, and ROUNDDOWN functions in Excel are frequently utilized for data analysis, financial modeling, and other mathematical computations. The function name and the arguments enclosed in parenthesis are all that are needed to use these functions in Excel. When utilizing the ROUND function, for instance, to round a number to two decimal places, you would type =ROUND(A1, 2), where A1 is the cell that holds the number you wish to round. To carry out more intricate calculations, these functions can also be paired with other Excel formulas and functions. For instance, you can round the result of a sum to a given number of decimal places by using the ROUND function in conjunction with the SUM function. For financial reporting and budgeting, this can be especially helpful. In order to help users better understand how to use these functions, Excel also offers a helpful feature known as “Function Arguments.”. Excel displays a list of all available functions, their arguments, and a description for each argument as soon as you begin typing a function in a cell. Those who are unfamiliar with these features or just want a fast refresher on how they operate may find this useful. There are several real-world situations where rounding to the nearest whole number is useful. For example, it is standard procedure to round up to the nearest cent when computing sales tax on a purchase. Rounding up would result in a total tax of $0.63 rather than $0.62 if a purchase totals $10.49 and the sales tax rate is 6%. In another instance, it might be essential to round down when measuring ingredients for a recipe to make sure that no extra is added. Rounding down would ensure that no extra flour is added if a recipe calls for 1 point 75 cups of flour but only 1 point 5 cups are available. These real-world examples show how rounding up or down can be used in various contexts to guarantee consistency and accuracy in computations. Thinking About Negative Numbers. Negative values need special consideration when using rounding functions, as you should always keep in mind. For example, using the ROUND function, -0.5 will be rounded to -1 & -0.4 to 0. If this isn’t taken into consideration, computations may come out incorrect. Recognizing the Objective of Every Task. Using these functions without comprehending their function or how they operate is another common error. It’s important to keep in mind that each rounding function has a distinct purpose: ROUND always rounds down, ROUNDUP always rounds up, and ROUNDDOWN always rounds to the nearest value. Inaccurate results may arise if these functions are used interchangeably without taking into account their unique purpose. Aware of Accuracy. When utilizing these functions in computations, precision must also be taken into consideration. When calculating in multiple steps, rounding too soon can cause cumulative errors that impact the outcome. It is essential to comprehend the function and operation of these rounding functions in order to use them efficiently. Spend some time learning about their syntax and how they deal with various numerical values, both positive & negative. Make use of Excel’s “Function Arguments” feature when using these functions to make sure you are entering the right arguments for each function. This can guarantee that you are utilizing these features correctly and help prevent mistakes. When utilizing these functions in computations, precision is another crucial factor to take into account. To prevent cumulative errors, use caution when & where to round numbers in multi-step computations. Finally, when utilizing these functions in computations, always double-check your results. When rounding numbers, it’s simple to make mistakes. By taking the extra time to check your work, you can find errors before they affect your final results. In conclusion, proficient application of the ROUND, ROUNDUP, and ROUNDDOWN functions is necessary for precise mathematical computations in a variety of domains, including science, finance, & daily life. Your rounding calculations will be accurate & dependable if you are familiar with their syntax and intended use, stay away from common pitfalls, and adhere to recommended usage guidelines.	{"url":"https://free-info-site.com/round-up-down-to-nearest-with-round-roundup-rounddown/","timestamp":"2024-11-13T19:16:53Z","content_type":"text/html","content_length":"74763","record_id":"<urn:uuid:5e276641-81d1-47eb-9fc5-a15792ccf3ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00823.warc.gz"}
Building My Own A/B Testing Platform Over the course of my career, I've worked across a number of projects that use external providers for A/B testing. These tools are great, they're established, trusted and ready to go, but they can be expensive and lack a lot of flexibility that some projects end up needing further down the line. That got me thinking a couple of weeks back, I have a bit of time, I'd like to learn a bit more about how A/B testing works in the background, why not build my own platform? And so, that's exactly what I started doing and will hopefully walk you through in this post. Initial Requirements As a starting point, I set out an initial list of requirements that I wanted to meet: • Bayesian and Frequentist A/B Testing: Supports both Bayesian and Frequentist approaches to A/B testing, providing flexibility depending on your analysis requirements. • Sequential Testing: Allows for running frquentist and bayesian A/B tests sequentially, stopping early if a significant result is found. • Bucketing: Provides functionality for bucketing users into test groups using various strategies. • Multiple Testing Corrections: Includes multiple testing correction algorithms such as Bonferroni, Benjamini-Hochberg, and Holm corrections. • Plotting: Offers built-in plotting tools for visualizing A/B test results. Choosing the Tech Stack To keep dependencies manageable and ensure ease of setup, I chose Poetry for dependency management and Tox for testing automation. The CLI is built using a basic command-line interface setup, allowing users to execute tests directly from their terminals, which I found ideal for scripting automated test runs. Additionally, I built a lightweight FastAPI framework to run the web interface, providing a way to visualize data and interact with experiments through a basic dashboard: Web Interface The A/B testing platform interface, showing the initial setup and a running test. Through the interface, users can upload a list of users and the count of conversions for the event being tracked (example). They then select the type of test (Bayesian or Frequentist), if they want to use sequential testing and the stopping threshold for the test. They can then submit the test and will receive the results, with plots showing the test progress and outcomes shortly after. Alongside the web interface, I also built a command line interface that allows users to take advantage of similar functionality to the web app, but from the terminal. This is useful when running tests over scripts or alongside automated systems, which I would like to investigate more than just building a web interface (although, I may look to make this much fancier in the future). Bayesian and Frequentist For this project, I wanted to start an initial implementation of both Bayesian and frequentist testing options, which allows users to choose the best method for their needs. To implement this, I used the PyMC3 library, which has already done a lot of the heavy lifting for Bayesian testing, which I like considering I'm not big on maths. It looks like this: # Defining the Bayesian model using PyMC with pm.Model() as model: # Priors for variant A and B (Beta distributions with prior data) prior_a = pm.Beta( "prior_a", alpha=self.prior_successes + 1, beta=self.prior_failures + 1 prior_b = pm.Beta( "prior_b", alpha=self.prior_successes + 1, beta=self.prior_failures + 1 # Likelihoods (Binomial distributions) based on observed data for each variant likelihood_a = pm.Binomial( likelihood_b = pm.Binomial( print(f"Running {'sequential' if sequential else 'non-sequential'} Bayesian A/B test") if sequential: # Burn-in and thinning parameters for MCMC burn_in = 100 # Ignore the first 100 samples for model stabilization thinning = 5 # Only keep every 5th sample to reduce autocorrelation for i in range(burn_in, num_samples + 1, thinning): trace = pm.sample(1, return_inferencedata=True, tune=0, target_accept=0.95) posterior_prob = ( (trace.posterior["prior_b"] > trace.posterior["prior_a"]) if posterior_prob > stopping_threshold: f"Stopping early at sample {i} with posterior probability {posterior_prob:.2f}" # Sample from the posterior distribution trace = pm.sample( tune=1000 # Increase the number of tuning steps # Calculate the uplift based on the chosen method self.uplift_method = uplift_method self.uplift_dist = calculate_uplift(trace, uplift_method) # Display the results self.plots = display_results(trace, self.uplift_dist, uplift_method) Frequentist is the more traditional approach of the two, instead of using prior knowledge to make decisions, you start your test with a blank slate, knowing nothing and assuming nothing about either option. To make a decision, you calculate a p-value as data comes in, telling you the likelihood of seeing results if the null hypothesis is true. For example, given a p-value 0.05, you would have 5% chance that you would see a difference as big as the one you did when there is no difference between the two options. This can be implemented with NumPy and SciPy, so is a fair bit simpler: import numpy as np import scipy.stats as st def calculate_pvalue(stat, alt_hypothesis, alpha): """Calculate the p-value based on the test statistic and the hypothesis type.""" if alt_hypothesis == "one_tailed": return 1 - st.norm.cdf(np.abs(stat)) if stat > 0 else st.norm.cdf(stat) elif alt_hypothesis == "two_tailed": return 2 * (1 - st.norm.cdf(np.abs(stat))) def calculate_power( prop_null, trials_null, trials_alt, effect_size, alpha, alt_hypothesis """Calculate the power of the test given a specific effect size.""" se_pooled = np.sqrt( prop_null * (1 - prop_null) * (1 / trials_null + 1 / trials_alt) z_alpha = ( st.norm.ppf(1 - alpha / 2) if alt_hypothesis == "two_tailed" else st.norm.ppf(1 - alpha) z_effect = effect_size / se_pooled return 1 - st.norm.cdf(z_alpha - z_effect) def calculate_stat_pvalue(self, sequential, stopping_threshold): pooled_prop, se_pooled = calculate_pooled_prop_se(self) stat = (self.prop_alt - self.prop_null) / se_pooled if sequential: return self.perform_sequential_testing(stopping_threshold) pvalue = calculate_pvalue(stat, self.alt_hypothesis, self.alpha) return stat, pvalue def calculate_pooled_prop_se(self): pooled_prop = (self.success_null + self.success_alt) / (self.trials_null + self.trials_alt) se_pooled = np.sqrt( pooled_prop * (1 - pooled_prop) * (1 / self.trials_null + 1 / self.trials_alt) return pooled_prop, se_pooled User Bucketing and Sequential Testing For user bucketing, I wanted to keep things simple but also supporrt both the frequentist and bayesian approaches. To do this, I implemented a simple function that would be given existing data from a data warehouse and then bucket users between a number of groups based on fixed percentages. You can find the code for this here. As you can probably see in the code examples above, I also implemented sequential testing, which is a method of A/B testing that allows you to stop a test early if you have enough data to make a decision. This can be useful if you have a lot of data and want to make a decision as soon as possible. Multiple Testing Corrections One key functionality for this platform is handling multiple testing scenarios. I incorporated corrections like Bonferroni and Benjamini-Hochberg to reduce the chance of false positives. This feature allows the platform to maintain accuracy when running numerous tests simultaneously—a common challenge in data-driven environments. All the code for this can be found here. Next Steps At the moment, this project is completely exploratory. I'm not a statistician, but I hope to use this as a way to understand more about the methods used behind A/B testing and how they can be used for a scalable platform. I'd also like to see how far I could take this project, potentially a fancier interface and some more advanced features. I'll be looking to post some more updates on this project as I go, so keep an eye out for those! Check Out the Code If you're interested in exploring the codebase or contributing to the project, you can find the repository here. Feel free to reach out with any questions or feedback — I'm always open to collaboration and new ideas!	{"url":"https://nicholasgriffin.dev/blog/building-my-own-ab-testing-platform","timestamp":"2024-11-02T01:04:06Z","content_type":"text/html","content_length":"242879","record_id":"<urn:uuid:54fda2a9-98c6-4870-bea0-31609ed36821>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00756.warc.gz"}
Problem 37 at the end of Chapter 8. There are those who insist that the initial working title for Episode XXVII of the Star Wars series was P = NP --- but this is surely apocryphal. In any case, if you're so inclined, it's easy to find NP-complete problems lurking just below the surface of the original Star Wars movies. Consider the problem faced by Luke, Leia, and friends as they tried to make their way from the Death Star back to the hidden Rebel base. We can view the galaxy as an undirected graph G = (V,E), where each node is a star system and an edge (u,v) indicates that one can travel directly from u to v. The Death Star is represented by a node s, the hidden Rebel base by a node t. Certain edges in this graph represent longer distances than others; thus, each edge e has an integer length l[e] > 0. Also, certain edges represent routes that are more heavily patrolled by evil Imperial spacecraft; so each edge e also has an integer risk r[e] > 0, indicating the expected amount of damage incurred from special-effects-intensive space battles if one traverses this edge. It would be safest to travel through the outer rim of the galaxy, from one quiet upstate star system to another; but then one's ship would run out of fuel long before getting to its destination. Alternately, it would be quickest to plunge through the cosmopolitan core of the galaxy; but then there would be far too many Imperial spacecraft to deal with. In general, for any path P from s to t, we can define its total length to be the sum of the lengths of all its edges; and we can define its total risk to be the sum of the risks of all its edges. So Luke, Leia, and company are looking at a complex type of shortest-path problem in this graph: they need to get from s to t along a path whose total length and total risk are both reasonably small. In concrete terms, we can phrase the Galactic Shortest Path problem as follows: given a set-up as above, and integer bounds L and R, is there a path from s to t whose total length is at most L, and whose total risk is at most R? Prove that Galactic Shortest Path is NP-complete.	{"url":"https://www.cs.cornell.edu/courses/cs482/2005sp/Homework/hw07.htm","timestamp":"2024-11-14T04:22:14Z","content_type":"text/html","content_length":"6769","record_id":"<urn:uuid:9e0d2787-f96a-45f3-976c-ff450cecc448>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00218.warc.gz"}
JPS 2008 Spring Meeting@Abstracts for the Meetings Held by the Physical Society@Definition@Forum@www.GrammaticalPhysics.ac The new quantum grammar version of Ehrenfest condition Live lesson culture school / Yuuichi Uda The new quantum grammar here is the new grammar proposed by me at JPS 2006 Spring Meeting 27pXA-6. At JPS 2007 Spring Meeting 28pSL-11, I published a candidate for new grammar version of Shrodinger equation representing the physical law in this new grammar. However, I am still not quite confident that the equation proposed then is plausible. Accordingly, this time, I propose a condition which I think to be pretty plausible as a candidate for the condition obeyed by the new grammar version of Schrodinger equation. I should like to check afterwards if the equation showed by me at JPS 2007 Spring Meeting 28pSL-11 obeys this condition. The new grammar proposed by me at JPS 2006 Spring Meeting 27pXA-6 represents a quantum history whose quantum state at time t is represented by a wave function ψ(□,t) in the ordinary quantum mechanics by such a functional Φ as Φ[χ]＝exp[α∫dtφ(χ(t),t)]; ψ(x,t)＝exp φ(x,t), concerning a system with one degree of freedom. Accordingly, this time, I propose the new grammar version of Ehrenfest condition as the following equation. m(d/dt)(d/dt)∫Dχ Φ[χ]χ(t)Φ[χ] ＝－∫Dχ Φ[χ]V'(χ(t))Φ[χ] ・・・・・・※ Here, Φ[χ]* is the complex conjugate of Φ[χ], and V' is a function defined as V'(x)≡(d/dx)V(x) using potential energy V. The measure of the functional integral ∫Dχ is selected to be ∞ ∞ lim Π ∫dx(nε). ε→+0 n＝－∞ －∞ Condition ※ is pretty plausible because it is very similar to both the expression of Ehrenfest theorem in the existing quantum mechanics and the equation of motion in the classical mechanics. The new grammar version of the equation probably must be made as the condition ※ is derived from it. This reason can be used as a pretty powerful guiding principle for finding out the new grammar version of the equation. Ref. Joint Meeting Of Pacific Region Particle Physics Communities Monday 30 October 2006 International Outreach. This article is a rewrite of the following article. JPS 2008 Spring Meeting@Abstracts for the Meetings Held by the Physical Society@Definition of Grammatical Physics@Grammatical Physics@Forum@Vintage(2008-2014)	{"url":"http://grammaticalphysics.ac/forum/define/abst/200803.html","timestamp":"2024-11-11T13:58:38Z","content_type":"text/html","content_length":"8760","record_id":"<urn:uuid:1a95f697-7b0e-4eea-9fae-e230c35ee0db>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00872.warc.gz"}
Sliding Window Maximum Difficulty: Hard Asked in: Amazon, Microsoft Understanding the Problem You are given an integer array arr of size n . Assume a sliding window of size k starting from index 0 . In each iteration, the sliding window moves to the right by one position till n-k . Write a program to return an array representing the maximum number in all sliding windows. Problem Note • The first element of the resultant array is max(arr[0...k]) , then the second element is max(arr[1...k+1]) and so on. • The size of the resultant array will be n-k+1 . • You are expected to solve this question in O(n) time complexity Example 1 Input: arr[] = [4, 3, 8, 9, 0, 1], k = 3 Output: [8, 9, 9, 9] Explanation: The window size is 3 and the maximum at different iterations are as follows: max(4, 3, 8) = 8 max(3, 8, 9) = 9 max(8, 9, 0) = 9 max(9, 0, 1) = 9 Hence, we get arr = [8, 9, 9, 9] as output. Example 2 Input: arr[] = [9, 8, 6, 4, 3, 1], k = 4 Output: [9, 8, 6] Explanation: The window size is 4 and the maximum at different iterations are as follows: max(9, 8, 6, 4) = 9 max(8, 6, 4, 3) = 8 max(6, 4, 3, 1) = 6 Hence, we get arr = [9, 8, 6] as output. Example 3 Input: arr[] = [1, 2, 3, 4, 10, 6, 9, 8, 7, 5], k = 3 Output: [3, 4, 10, 10, 10, 9, 9, 8] Explanation: The window size is 3 and the maximum at different iterations are as follows: max(1, 2, 3) = 3 max(2, 3, 4) = 4 max(3, 4, 10) = 10 max(4, 10, 6) = 10 max(10, 6, 9) = 10 max(6, 9, 8) = 9 max(9, 8, 7) = 9 max(8, 7, 5) = 8 Hence, we get arr = [3, 4, 10, 10, 10, 9, 9, 8] as output. We will be discussing 4 solutions • Naive Approach: Iterating through the elements for all windows separately. • Using a Self-balancing tree : Use the tree to store the elements of the window and do all operations at log(n) time. • Using Heap Data Structure : Use a max-heap data structure to find the maximum element in the window at constant time. • Using Deque : Use a double-ended queue to store the elements of the window. Before moving on to the solutions, you can try this question here. Naive Approach The naive approach to this problem will be to “examine” window of size k starting at each element of the array. We iterate through the whole array and calculate maximum among the window of next k elements from each element of the array till the last element of the window and array coincide. Solution steps 1. Traverse all the N elements of the array (Outer loop) 2. At each step, calculate maximum element among next k elements (Inner loop) 3. Store the maximum element of each window in the answer array 4. Return the answer array Pseudo Code int[] slidingWindowMaximum(int arr[], int n, int k) ans = [] // there are N-k+1 windows for (i = 0 to N-k) max = arr[i] for(j = i to i+k-1) if(arr[j] > max) max = arr[j] return ans Time Complexity: O(NK) Space Complexity: O(1) Using self-balancing BST The self-balancing BST like Red-Black Tree, AVL tree, etc. adjust their height automatically for every insertion and deletion, taking O(log N) time on an average. The idea is to make a BST of size K and at each sliding, we will insert the next element and delete the least-recent added element in O(log N) . Solution steps 1. Create a self-balancing BST for the first K elements. 2. Get the maximum element from the BST and store it in the solution array. 3. Move the window (having K elements) forward. This step needs “dropping” the first element (arr[i]) in the window and “adding” the element after the last element (arr[i+k]) to the window. 4. Iterate through the whole array from 0 to N-K and repeat the above steps. 5. Return the solution array. Pseudo Code int[] slidingWindowMaximum(int arr[], int n, int k) ans = [] root = Tree() // self-balancing Tree // insert first K elements for(i = 0 to K-1) // Now the tree contains the first k elements d = 0 // to store the elements to be deleted for(i = k to n-1) // find the maximum element and append it // to the answer array d += 1 // increase the d for next window return ans // The maximum element will be at the rightest in the tree int findMax(Tree root) if(root.right == NULL) return root return findMax(root.right) Time Complexity: O(N log K) Space Complexity: O(K) (To store the elements in BST). Using a heap data structure The intuition of using a max heap for a better solution to the problem can come from finding a maximum of K elements in less than O(K) time. Within every window, we can use a heap to store the K elements of the current and get the maximum element (top of the max heap) in O(log K) time. For sliding the window forward, we need to “drop” the leftmost element of the window from the heap and add the new element to it. We have done this for all the windows in the array. The heap can be implemented using a priority queue. Note: In various languages like C++, there is no predefined function to remove a particular element from the priority queue. We can overcome this problem by using another priority queue which keeps the record of the leftmost elements only. Solution steps 1. Take two priority queues - one for keeping the heap and other for keeping the elements to be “dropped” from the heap when the window slides forward. 2. Push the first K elements of the array into the heap. 3. Store the top of the heap in result array as it is the maximum element in the first window. 4. Now, Iterate through the array and at each step we have two cases • The leftmost element of the window was the maximum element (top of the heap):- For this, remove the top of the heap to slide the window forward. • The leftmost element of the window was not the maximum element of the window:- For this, keep the leftmost element ( arr[i-k] ) in the other heap and keep popping the elements from both the heaps until the top of heaps match. 5. Add that element of array to the heap and store the result. 6. After the iteration, return the array. Pseudo Code int[] slidingWindowMaximum(int arr[], int n, int k) heap = PriorityQueue() toDrop = PriorityQueue() // Push First K elements in the heap for(i = 0 to k-1) ans = [] // store the maximum in ans //iterate for rest elements for(i = k to n) // pop the heap if leftmost element is previous element was maximum if(arr[i-k] == heap.top()) // push the leftmost element to toDrop heap // pop elements from both heap till their top matches while(!toDrop.isEmpty() and toDrop.top() == heap.top()) // push the element to the heap return ans Time Complexity: O(N * log K) Space Complexity: O(N) (In the worst case, "dropped" priority queue can have size N if the first element in the array is maximum). Using Deque We can use a double-ended queue to keep only the indices of those elements which are useful. The use of deque is to add and drop elements from both the ends of a queue. We will slide the window of K elements by “dropping” the first element and “adding” the next element after the window to move it forward. The deque will keep the index of the maximum element at the front and also at a time, it will delete all the unnecessary elements from the window. You can look at the solution steps for more 1. Create a dequeue to store elements. 2. Iterate through the array, insert the first K elements in the array. While insertion we will take care of the window such that there are no unnecessary indices. To remove these indices, we will remove all the elements from the back of the queue that is smaller than the current array element. 3. After the iteration for the first K element, the maximum element's index is at the front of the queue. 4. Now, Iterate through the remaining part of the array and remove the element from the front if they are out of the current window. 5. Again, insert the element in the dequeue and before inserting delete those unnecessary indices which are smaller than the current array element. 6. Now, after each iteration, you will get the maximum element of the current window. Pseudo Code int[] slidingWindowMaximum(int arr[], int n, int k) q = Dequeue() ans = [] // for First K elements for( i = 0 to k-1) while(!q.empty() and arr[i] >= arr[q.back()]) // Remove the indices of elements that are smaller than the current elements // the element at the front has index of the highest element in the window // for rest elements for(i = k to n-1) // drop the elements that are out of window while(!q.empty() and q.front() <= i-k) // remove those elements smaller than the current element from back while(!q.empty() and arr[i] >= arr[q.back()]) return ans Time Complexity: O(2*N) (Every element is added and removed at most once) Space Complexity: O(K) Comparison of different solutions Suggested problems to solve • Find Median of sliding window in an array. • Smallest Window that contains all character of the string itself. • Count Distinct Elements in every window of size K. • Maximum and Minimum in the window of a given size in an array Please comment down below if you find an improvement in the above explanation. Happy Coding! Enjoy Algorithms!	{"url":"https://afteracademy.com/blog/sliding-window-maximum/","timestamp":"2024-11-07T10:33:21Z","content_type":"application/xhtml+xml","content_length":"86800","record_id":"<urn:uuid:f6db369b-bcbb-472a-90a2-c7efb31469b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00640.warc.gz"}